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Primitively 2-universal senary integral quadratic forms
Byeong-Kweon Oh and Jongheun Yoon
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
This work of the second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2019R1A2C1086347 and NRF-2020R1A5A1016126).
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
[2020]Primary 11E12, 11E20
For a positive integer $m$, a (positive definite integral) quadratic form is called primitively $m$-universal if it primitively represents all quadratic forms of rank $m$. It was proved in [7] that there are exactly $107$ equivalence classes of primitively $1$-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively $2$-universal quadratic forms is six, and there are exactly $201$ equivalence classes of primitively $2$-universal senary quadratic forms.
§ INTRODUCTION
A quadratic form of rank $n$ is a quadratic homogeneous polynomial
\[
f(x_1, \dots, x_n) = \sum_{i, j = 1}^n f_{ij}x_i x_j \qquad (f_{ij} = f_{ji}\in\q)\text{,}
\]
where the corresponding symmetric matrix $M_f = (f_{ij})$ is nondegenerate. We say $f$ is integral if $M_f$ is an integral matrix, and we say $f$ is positive definite if $M_f$ is positive definite. Throughout this article, we always assume that any quadratic form is integral and positive definite.
For two (positive definite integral) quadratic forms $f$ and $g$ of rank $n$ and $m$, respectively, we say $g$ is represented by $f$ if there is an integral matrix $T\in M_{n, m}(\z)$ such that $T^t M_f T = M_g$. We say $g$ is isometric to $f$ if the above integral matrix $T$ is invertible. The equivalence class of $f$ is the set of all quadratic forms which are isometric to $f$. We say $g$ is primitively represented by $f$ if the above integral matrix $T$ can be extended to an invertible matrix in $GL_n(\z)$ by adding suitable $(n-m)$ columns. In particular, a positive integer $a$ is primitively represented by $f$ if and only if there are integers $x_1$, …, $x_n$ such that
\[
f(x_1, \dots, x_n) = a \quad\text{and}\quad \gcd(x_1, \dots, x_n) = 1\text{.}
\]
For a positive integer $m$, a quadratic form is called (primitively) $m$-universal if it (primitively, respectively) represents all quadratic forms of rank $m$. Lagrange's four-square theorem states that the quaternary quadratic form corresponding to the identity matrix $I_4$ is $1$-universal. The complete classification of $1$-universal quadratic forms up to isometry has been done by Ramanujan, Dickson, Conway–Schneeberger, and Bhargava (see [16], [5], and [1]). In 1998, Kim, Kim, and the first author proved in [9] that there are exactly eleven equivalence classes of $2$-universal quinary quadratic forms. For more information on $n$-universal quadratic forms, one may see [8] and [12].
Let $f$ and $g$ be quadratic forms of rank $n$ and $m$, respectively. We say $g$ is represented by $f$ over the ring of $p$-adic integers $\z_p$ for some prime $p$, if there is a matrix $T\in M_{n, m}(\z_p)$ such that $T^t M_f T = M_g$. We further say $g$ is primitively represented by $f$ over $\z_p$ if the above matrix $T$ can be extended to an invertible matrix in $GL_n(\z_p)$ by adding suitable $(n-m)$ columns in $\z_p^n$. Clearly, if $g$ is (primitively) represented by $f$ over $\z$, then $g$ is also (primitively) represented by $f$ over $\z_p$ for any prime $p$. However, the converse is not true in general. In fact, there is an effective criterion whether or not $g$ is represented by $f$ over $\z_p$ for any prime $p$ (for this, see [14]). However, as far as the authors know, there is no known effective criterion whether or not $g$ is primitively represented by $f$ over $\z_p$.
Finding primitively $1$-universal quadratic forms was first considered by Budarina in [2]. She classified all primitively $1$-universal quaternary quadratic forms satisfying some special local properties. Later, she also classified in [3] all primitively $2$-universal quadratic forms which has squarefree odd discriminant and of class number one. Recently, Earnest and Gunawardana classified in [6] all quadratic forms that primitively represent all integers corresponding to unary quadratic forms over $\z_p$ for any prime $p$ including $p=2$. Furthermore, they provided a complete list of all $1$-universal quaternary quadratic forms that are almost primitively $1$-universal. Here, a quadratic form is called almost primitively $1$-universal if it represents almost all positive integers primitively. Ju, Kim, Kim, Kim, and the first author finally proved in [7] that there are exactly $107$ equivalence classes of primitively $1$-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively $2$-universal quadratic forms is six, and that there are exactly $201$ equivalence classes of primitively $2$-universal senary quadratic forms (see Table <ref>).
The subsequent discussion will be conducted in the more suitable language of quadratic spaces and lattices. Let $F$ be either $\q$ or $\q_p$ for some prime $p$. An (quadratic) $F$-space is a finite dimensional vector space $V$ equipped with a nondegenerate symmetric bilinear form
\[
B: V\times V\to F
\]
and the associated quadratic form $Q: V\to F$ is defined by $Q(v) = B(v, v)$ for any $v\in V$. Let $R$ be either $\z$ or $\z_p$ for some prime $p$. An (quadratic) $R$-lattice is a finitely generated free $R$-module $L$ equipped with a nondegenerate symmetric bilinear form
\[
B: L\times L\to R
\]
and an associated quadratic form $Q: L\to R$ is defined by $Q(v) = B(v, v)$ for any $v\in L$. We will denote by $Q(L)$ the set of $Q(v)$ for any vector $v\in L$. The scale $\fs L$ of $L$ is the ideal $B(L, L)$ in $R$, and the norm $\fn L$ of $L$ is the ideal in $R$ generated by the set $Q(L)$.
Given an $R$-basis $e_1, \dotsc, e_n$ for an $R$-lattice $L$, we call the corresponding symmetric $n\times n$ matrix $M_L = (B(e_i, e_j))$ the Gram matrix of $L$, and in this case, we write $L\cong (B(e_i, e_j))$. We call a $\z$-lattice $L$ positive definite if $M_L$ is positive definite. For a symmetric $n\times n$ matrix $N$ over $R$, we let $\langle N\rangle$ (or simply $N$) stand for any $\z$-lattice whose Gram matrix is $N$. For instance, the $\z$-lattice $I_n$ stands for the $n$-ary lattice whose Gram matrix is the identity matrix of rank $n$. Moreover, if the Gram matrix of $L$ is a diagonal matrix $\diag(a_1, \dotsc, a_n)$ ($a_i\in R$), then we simply write $L\cong \langle a_1, \dotsc, a_n\rangle$. For a $\z$-lattice $L$ and a positive integer $a$, $L^a$ denotes the $\z$-lattice obtained from scaling $L$ by $a$ so that
\[
M_{L^a}=a\cdot M_L\text{.}
\]
For a $\z$-lattice $L$, we define $L_p = \z_p\otimes L$, which is a $\z_p$-lattice for any prime $p$. From now on, we always assume that any $\z$-lattice $L$ is integral, that is, $\fs(L)\subseteq\z$, and positive definite, that is, $M_L$ is positive definite.
Let $m$ and $n$ be positive integers such that $m\le n$. An $m\times n$ ($n\times m$) matrix $A$ over $R$ is called primitive if it can be extended to an invertible matrix in $GL_n(R)$ by adding suitable $n-m$ rows (columns, respectively), or equivalently, if the greatest common divisor of the determinants of all $m\times m$ submatrices of $A$ is a unit in $R$. A finite sequence $v_1, \dots, v_m$ of vectors in $R^n$ is called primitive if it can be extended to a basis for $R^n$, or equivalently, if the coefficient matrix of $v_1, \dots, v_m$ is primitive. Finally, a submodule $M$ of $R^n$ is primitive if and only if there exists a basis for $M$ which is primitive, or equivalently, $M$ is a direct summand of $R^n$. Note that a vector $v = (a_1, \dots, a_n)\in R^n$ is primitive if and only if
\[
\gcd(a_1, \dots, a_n) \in R^\times\text{.}
\]
For an $R$-lattice $L$, we will denote by $Q^\ast(L)$ the set of $Q(v)$ for any primitive vector $v\in L$.
For two $R$-lattices $M$ and $L$, a representation from $M$ to $L$ is a linear map $\sigma : M \to L$ such that $B(\sigma(x), \sigma(y)) = B(x, y)$ for any $x$, $y\in L$. If in addition $\sigma(M)$ is a primitive sublattice of $L$, then such a representation is called primitive. If there exists a (primitive) representation $\sigma : M \to L$ then we say that $M$ is (primitively, respectively) represented by $L$. If such a representation $\sigma$ is bijective then we say that $M$ is isometric to $L$, and in that case, we denote it by $\sigma: M\cong L$. An $R$-lattice $L$ is called (primitively) $n$-universal if $L$ (primitively, respectively) represents all $R$-lattices of rank $n$.
For an $R$-lattice $L$, the class of $L$ consists of all $R$-lattices that are isometric to $L$. Let $L$ and $M$ be $\z$-lattices. If $M_p$ is isometric to $L_p$ for any prime $p$, then we say that $M$ is in the genus of $L$. It is well-known that genus is partitioned into finitely many classes. It is well-known that if $M_p$ is (primitively) represented by $L_p$ for any prime $p$, then $M$ is (primitively, respectively) represented by some $\z$-lattice $L'$ in the genus of $L$ (see <cit.>).
We always assume that $\mathbb{H}$ and $\mathbb{A}$ are binary $R$-lattices whose Gram matrices are
\[
\mathbb{H}\cong \left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right) \quad\text{and}\quad \mathbb{A}\cong \left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\text{.}
\]
If $R = \z_2$, then $\mathbb{H}$ is an isotropic even unimodular lattice, whereas $\mathbb{A}$ is an anisotropic even unimodular lattice. For any odd prime $p$, let $\Delta_p$ be any nonsquare unit in $\z_p$. Any unexplained notation and terminology can be found in [10] or [15].
§ THE MINIMAL RANK OF PRIMITIVELY $2$-UNIVERSAL $\Z$-LATTICES
It is well known in [9] that the minimal rank of $2$-universal $\z$-lattices is five, which implies that the minimal rank of primitively $2$-universal $\z$-lattices is at least five. The aim of this section is to show that the minimal rank of primitively $2$-universal $\z$-lattices is, in fact, six.
For a prime $p$, let $L$ be an anisotropic $\z_p$-lattice. Let $L = L_0 \mathbin{\perp} \cdots \mathbin{\perp} L_t$ be a Jordan decomposition of $L$ such that $L_i = 0$ or $\fs(L_i) = p^i\z_p$. If $L_t\ne 0$, then any integer in $4p^{t+1}\z_p$ is not primitively represented by $L$.
It is well-known that for any anisotropic unimodular $\z_p$-lattice $M$, if $Q(x)\in 4p\z_p$ for some $x\in M$, then $x\in pM$. For an $\alpha\in\z_p$, suppose that $4p^{t+1}\alpha$ is primitively represented by $L$. Then there is a primitive vector $x\in L$ such that $Q(x) = 4p^{t+1}\alpha$. Let $x = x_0 + \cdots + x_t$ for some $x_i\in L_i$ ($0\le i\le t$). Then there is a $k$ such that $x_k$ is primitive in $L_k$. If $p$ is odd, then $\ord_p (Q(x_k)) = k$, and thus
\[
Q(x - x_k) \equiv -Q(x_k) \pmod{4p^{k+1}}\text{.}
\]
Hence, $Q(x - x_k) = - \epsilon^2 Q(x_k)$ for some unit $\epsilon\in\z_p$. Since $x-x_k\notin L_k$, the vector $(x-x_k) + \epsilon x_k$ is a nonzero isotropic vector. This is a contradiction.
Now, assume that $p = 2$. First, suppose that $L_k$ admits an orthogonal basis $e_1, \dots, e_m$. If $x_k = \alpha_1 e_1 + \cdots + \alpha_m e_m$, then $\ord_2 (Q(\alpha_j e_j)) = k$ for some $1\le j\le m$. Thus $Q(x - \alpha_j e_j) \equiv -Q(\alpha_j e_j) \Mod{4p^{k+1}}$, which implies $Q(x - \alpha_j e_j) = - \epsilon^2 Q(\alpha_j e_j)$ for some unit $\epsilon\in\z_2$. Since $x-\alpha_j e_j\notin \z_2 e_j$, the vector $(x - \alpha_j e_j) + \epsilon \alpha_j e_j$ is a nonzero isotropic vector. This is a contradiction. If $L_k$ does not admit an orthogonal basis, then $L_k\cong \mathbb{A}^{2^k}$ and $\ord_2 (Q(x_k)) = k+1$. Since $L_k$ represents every integer in $2^{k+1}\z_2^\times$, there is a vector $z\in L_k$ such that $Q(z) = -Q(x - x_k)$. Since $x-x_k\notin L_k$, $z + (x - x_k)$ is a nonzero isotropic vector. This is a contradiction.
For a positive integer $n$, let $U$ be an $n$-dimensional quadratic space over $\q_p$ and let $W$ be an anisotropic quadratic space over $\q_p$. Then any $\z_p$-lattice on $U\mathbin{\perp} W$ is not primitively $(n+1)$-universal.
Let $L$ be any $\z_p$-lattice on $U\mathbin{\perp} W$ and $\ell$ be any $\z_p$-lattice on $U$. It is enough to show that there is an $a\in\z_p$ such that $L$ cannot primitively represent $\ell\mathbin{\perp}\langle a\rangle$. To do this, we show that there is an $a\in\z_p$ that is not primitively represented by $\sigma(\ell)^\perp$ for any primitive representation $\sigma : \ell \to L$.
We know that there are only finitely many possibilities of $\sigma(\ell)$ up to isometry of $L$ by <cit.>. Therefore, for any possible primitive representation $\sigma : \ell \to L$, there are only finitely many $\sigma(\ell)^\perp$ up to isometry. For any such $\sigma(\ell)^\perp$, we have $\q_p \sigma(\ell)^\perp\cong W$, for $\q_p \sigma(\ell)\cong U$. Hence, $\sigma(\ell)^\perp$ is an anisotropic $\z_p$-lattice. Therefore by Lemma <ref>, there exists an integer $r$ such that any integer in $p^r\z_p$ is not primitively represented by $\sigma(\ell)^\perp$. Now, if we choose any integer $a$ in the intersection of all possible $p^r\z_p$'s, then $\ell\mathbin{\perp}\langle a\rangle$ is not primitively represented by $L$. This completes the proof.
For any quinary $\z$-lattice $L$, there are infinitely many binary $\z$-lattices that are not primitively represented by $L$.
Since $\q L$ represents any positive rational number, we may write $\q L = U \mathbin{\perp} \langle dL\rangle$ for some quaternary $\q$-space $U$. Since $U$ is a quaternary space of discriminant $1$, it follows that for each prime $p<\infty$, $U_p$ is isotropic if and only if $S_p U = (-1, -1)$. However, since $S_\infty U = 1 \ne (-1, -1)$, there is a prime $q$ such that $S_q U \ne (-1, -1)$ by Hilbert reciprocity law. This implies that $U_q$ is anisotropic. Hence, $L_q$ is not primitively $2$-universal from the above lemma. Therefore, there are infinitely many binary $\z$-lattices that are not primitively represented by $L$.
The minimal rank of primitively $2$-universal $\z$-lattices is $6$.
By Lemma <ref>, the minimal rank of primitively $2$-universal $\z$-lattices is at least six. Furthermore, since $I_6$ is primitively $2$-universal over $\z_p$ for any prime $p$ and is of class number one, $I_6$ is primitively $2$-universal (see Theorem <ref>). The theorem follows from this.
The prime $q$ in the proof of the Lemma <ref> such that $S_q U \ne (-1, -1)$ or equivalently $U_q$ is anisotropic is called the core prime of $L$ see also <cit.>.
The existence of a core prime of a quinary $\z$-lattice will play a significant role when we determine binary $\z$-lattices that are primitively represented by a quinary $\z$-lattice in Sections <ref> and <ref>.
§ CANDIDATES OF PRIMITIVELY $2$-UNIVERSAL SENARY $\Z$-LATTICES
In this section, we find all candidates of primitively $2$-universal senary $\z$-lattices.
A $\z$-sublattice $\z e_1 + \cdots + \z e_k$ of a $\z$-lattice $L$ is called a $k$-section of $L$ if there are vectors $e_{k+1}, \dots, e_n$ in $L$ such that $\{e_1, \dots, e_n\}$ is a Minkowski reduced basis for $L$. Recall that a $k$-section of $L$ is not unique in general.
Let $L$ be a primitively $2$-universal senary $\z$-lattice. We find all possible $k$-sections of $L$ for each $k = 1, \dots, 6$, inductively. To find all possible candidates of $(k+1)$-sections containing a specific $k$-section, we apply Lemma <ref> repeatedly, which is quite well known (see <cit.>).
Let $M$ and $N$ be $\z$-lattices of rank $m$ and $n$ respectively. Suppose that the ordered set $\{e_1, \dots, e_n\}$ is a Minkowski reduced basis for $N$. Suppose further that $M$ is represented by $N$, but not by the $k$-section $\z e_1 + \cdots + \z e_k$.
* We have $Q(e_{k+1}) \le C_4(k+1) \max\{\mu_m(M), Q(e_k)\}$, where the constant $C_4(j)$ depending only on $j$ is defined in [4] and $\mu_m(M)$ is the $m$-th successive minimum of $M$ see <cit.>.
* Suppose further that $n\le m+4$ and $N$ is $m$-universal. Then for any $x = x_1 e_1 + \cdots + x_n e_n \in L$, we have $Q(x)\ge Q(e_j)$ for any $j$ such that $x_j\ne 0$. Also, we have $Q(e_{k+1})\le \mu_m(M)$.
(1) Suppose on the contrary that
\[
Q(e_{k+1}) > C_4(k+1) \max\{\mu_m(M), Q(e_k)\}\text{.}
\]
Since $Q(e_{k+1}) \le C_4(k+1) \mu_{k+1}(N)$, we have
\[
\mu_{k+1}(N) > \max\{\mu_m(M), Q(e_k)\}\text{.}
\]
Since we are assuming that $M$ is represented by $N$, the above inequality implies that
\[
M \rightarrow \operatorname{span}_\q\{x\in L : Q(x)<\mu_{k+1}(N)\}\cap N = \z e_1 \oplus \cdots \oplus \z e_k\text{,}
\]
which is a contradiction.
(2) Since $I_m$ is represented by $N$, we have $N = I_m\mathbin{\perp} N'$ for some $\z$-sublattice $N'$ of $N$. Assume that the ordered set $\{e_{m+1}, \dots, e_n\}$ is a Minkowski reduced basis for $N'$. Since $\operatorname{rank} N'\le 4$, for any vector
\[
x = x_{m+1} e_{m+1} + \cdots + x_n e_n\in N'\text{,}
\]
we have $Q(x)\ge Q(e_j)$ for any $j$ such that $x_j \ne 0$ (see <cit.>). Hence, the former assertion holds trivially. Furthermore, it is well known that $Q(e_j) = \mu_j(N)$ for any $j$ with $1\le j\le n$. Now, on the contrary to the latter assertion, suppose that $Q(e_{k+1}) > \mu_m(M)$, then $\mu_{k+1}(N) > \mu_m(M)$. Since we are assuming that $M$ is represented by $N$, this inequality implies that
\[
M \rightarrow \operatorname{span}_\q\{x\in L : Q(x)<\mu_{k+1}(N)\}\cap N \subseteq \z e_1 \oplus \cdots \oplus \z e_k\text{,}
\]
which is a contradiction.
For a $\z$-lattice $M$, a binary $\z$-lattice $\ell$ is called a primitive binary exception of $M$ if $\ell$ is not primitively represented by $M$. If a $\z$-lattice $M$ is not primitively $2$-universal, we define the truant of $M$ to be, among all the primitive binary exceptions of $M$, the least isometry class with respect to the following total order in terms of Gram matrices:
\[
\begin{pmatrix}a_1 & b_1 \\ b_1 & c_1\end{pmatrix} \prec \begin{pmatrix}a_2 & b_2 \\ b_2 & c_2\end{pmatrix} \quad \Leftrightarrow \quad \begin{cases}
c_1 < c_2\text{,} & \text{ or}\\
c_1 = c_2\text{ and }a_1 < a_2\text{,} & \text{ or}\\
c_1 = c_2\text{, }a_1 = a_2\text{, and }b_1 < b_2\text{,}
\end{cases}
\]
where we assume that all binary lattices are Minkowski reduced, that is, $0 \le 2b_i \le a_i \le c_i$ for any $i = 1, 2$.
Now, we find all candidates of a $k$-section of a primitively $2$-universal senary $\z$-lattice for each $k = 1, \dots, 6$, inductively. Since $I_2$ is the truant of any lattice of rank less than two, clearly the $2$-section of any primitively $2$-universal $\z$-lattice must be $I_2$. Since $I_2$ cannot primitively represent $\langle 1, 2\rangle$, we must have $1\le Q(e_3)\le 2$ by the last lemma. Thus, the $3$-section must be either $I_3$ or $I_2\mathbin{\perp}\langle 2\rangle$. It is easily seen that the truant of each $3$-section is $\ang{2, 2}$ and $\mathbb{A}$, respectively. From this, the $4$-section must be one of
\[
I_4\text{,}\quad I_3\mathbin{\perp}\langle 2\rangle\text{,}\quad\text{or}\quad I_2\mathbin{\perp}\mathbb{A}\text{.}
\]
By repeating this and by removing candidates that have the truant, we finally obtain the following list of $201$ candidates of isometry classes of primitively $2$-universal (P$2$U for short) senary $\z$-lattices:
The $201$ candidates of P$2$U senary $\z$-lattices
Type $5$-section Candidates Possible $k$'s A $I_5$ $I_5\mathbin{\perp} \langle k\rangle$ $k=1$, $2$
2*B 2*$I_4\mathbin{\perp}\langle2\rangle$ $I_4\mathbin{\perp}\langle2,k\rangle$ $2\le k\le 5$
$I_4\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&k\end{smallmatrix}\right)$ $k=2$, $3$, $5$, $6$
C $I_4\mathbin{\perp}\langle3\rangle$ $I_4\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&k\end{smallmatrix}\right)$ $k=3$
3*[-4pt]D 3*[-4pt]$I_3\mathbin{\perp}\langle2,2\rangle$ $I_3\mathbin{\perp}\langle2,2,k\rangle$ $2\le k\le 6$
$I_3\mathbin{\perp}\langle2\rangle\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&k\end{smallmatrix}\right)$ $3\le k\le 8$
$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&0&1\\0&2&1\\1&1&k\end{smallmatrix}\right)$[-9pt]0pt24pt $k=3$, $5\le k\le 7$
2*[-4pt]E 2*[-4pt]$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)$ $I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\langle k\rangle$ $2\le k\le 5$, $k=7$, $8$
$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&k\end{smallmatrix}\right)$[-9pt]0pt24pt $2\le k\le 9$
F $I_3\mathbin{\perp}\langle2,3\rangle$ $I_3\mathbin{\perp}\langle2\rangle\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&k\end{smallmatrix}\right)$ $k=3$
3*[-6pt]G 3*[-6pt]$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)$ $I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)\mathbin{\perp}\langle k\rangle$ $k=3$, $4$, $6\le k\le 18$
$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1&0\\1&3&1\\0&1&k\end{smallmatrix}\right)$[-9pt]0pt24pt $3\le k\le 19$
$I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&3&1\\1&1&k\end{smallmatrix}\right)$[-9pt]0pt24pt $3\le k\le 19$
4*[-12pt]H 4*[-12pt]$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\langle2\rangle$ $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\langle2,k\rangle$ $3\le k\le 5$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&k\end{smallmatrix}\right)$ $k=2$, $4$, $5$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&0&1\\1&2&0&1\\0&0&2&0\\1&1&0&k\end{smallmatrix}\right)$[-12pt]0pt30pt $3\le k\le 6$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&0&1\\1&2&0&1\\0&0&2&1\\1&1&1&k\end{smallmatrix}\right)$[-12pt]0pt30pt $k=3$, $4$, $6$
3*[-13pt]I 3*[-13pt]$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&2\end{smallmatrix}\right)$ $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&2\end{smallmatrix}\right)\mathbin{\perp}\langle k\rangle$[-9pt]0pt24pt $k=2$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&2&0\\1&1&0&k\end{smallmatrix}\right)$[-12pt]0pt30pt $2\le k\le 5$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&2&1\\1&1&1&k\end{smallmatrix}\right)$[-12pt]0pt30pt $2\le k\le 4$
J $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\langle3\rangle$ $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&k\end{smallmatrix}\right)$ $k=3$
4*[-18pt]K 4*[-18pt]$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)$ $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)\mathbin{\perp}\langle k\rangle$[-9pt]0pt24pt $3\le k\le 20$, $22\le k\le 24$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&0\\1&2&1&0\\1&1&3&1\\0&0&1&k\end{smallmatrix}\right)$[-12pt]0pt30pt $3\le k\le 24$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&3&0\\1&1&0&k\end{smallmatrix}\right)$[-12pt]0pt30pt $3\le k\le 25$
$I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&3&1\\1&1&1&k\end{smallmatrix}\right)$[-12pt]0pt30pt $3\le k\le 25$
We refer to any of the above candidates by the expression such as (type) or (type)$_k$. For instance, $89$ candidates in the last four rows are called type K. Among them, $21$, $22$, $23$, and $23$ candidates in each of four rows are of types Ki, Kii, Kiii, and Kiv, respectively. For instance, a lattice of type Kiv is any $\z$-lattice of the form $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&3&1\\1&1&1&k\end{smallmatrix}\right)$. Finally, K$^{\mbox{\scriptsize iv}}_3$ stands for the lattice $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&3&1\\1&1&1&3\end{smallmatrix}\right)$.
Moreover, when we refer to each of the candidates in Table <ref>, we always assume that the basis $e_1, \dots, e_6$ corresponds exactly to the Gram matrix given in the table. For instance, when we consider the $\z$-lattice K$^{\mbox{\scriptsize iv}}_3 = \z e_1 + \dotsb + \z e_6$, we always assume that
\[
(B(e_i, e_j)) = \left(\begin{smallmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&2&1&1&1\\0&0&1&2&1&1\\0&0&1&1&3&1\\0&0&1&1&1&3\end{smallmatrix}\right)\text{.}
\]
From now on, by abusing of terminology, the $s$-section of $\z e_1 + \cdots + \z e_6$ always implies the $\z$-sublattice $\z e_1 + \cdots + \z e_s$ for any $s = 1$, $2$, …, $6$.
§ THE PROOF OF PRIMITIVE $2$-UNIVERSALITY (ORDINARY CASES)
In this section, we prove that some candidates given in Table <ref> are, in fact, primitively $2$-universal. Let $L = \z e_1 + \dotsb \z e_6$ be one of candidates of primitively $2$-universal $\z$-lattices, where the corresponding symmetric matrix $(B(e_i, e_j))$ is given in Table <ref>. We prove the primitive $2$-universality of $L$ for the cases when $L$ itself or the $5$-section of $L$ is of class number one. Since the key ingredients of the proofs, which we will explain more precisely later, are quite similar to each other, we only provide the proofs of some representative cases. For the complete proofs, one may see the second author's dissertation [17].
§.§ Case 1: $L$ has class number one
As explained in the introduction, if a $\z$-lattice is of class number one, then it primitively represents any $\z$-lattice that is primitively represented over $\z_p$ for any prime $p$. Hence, if $L$ is of class number one, then $L$ is primitively $2$-universal if and only if $L_p$ primitively represents all binary $\z_p$-lattices for any prime $p$.
If $L$ has class number one, then $L$ is primitively $2$-universal. In fact, there are exactly $10$ such $\z$-lattices in Table <ref>.
One may easily show that
\[
L \cong \text{A$_1$, A$_2$, B$^{\mbox{\scriptsize i}}_2$, B$^{\mbox{\scriptsize ii}}_2$, B$^{\mbox{\scriptsize ii}}_3$, D$^{\mbox{\scriptsize iii}}_3$, E$^{\mbox{\scriptsize ii}}_2$, E$^{\mbox{\scriptsize ii}}_3$, or I$^{\mbox{\scriptsize ii}}_3$.}
\]
Note that $dL = 1$, $2$, $4$, $3$, $5$, $8$, $4$, $7$, $4$, and $8$, respectively. Since $\mathbb{H}\mathbin{\perp}\mathbb{H}$ is primitively represented by $L_p$ for any odd prime $p$, $L_p$ is primitively $2$-universal. One may directly show that $L_2$ is also primitively $2$-universal.
In fact, for senary lattices, Lemma <ref> generalizes Budarina's result [3], where $L$ is required to be of class number one and to be of squarefree odd discriminant. One may verify from the proof that there are exactly four such cases out of our $201$ candidates.
§.§ Case 2: the $5$-section of $L$ has class number one and splits $L$ orthogonally
In the remaining of this section, we consider the case when the $5$-section of $L$, say $M$, has class number one. Since $M$ is a primitive sublattice of $L$, $L$ primitively represents any $\z$-lattice that is primitively represented by $M$. Hence, if $M$ is of class number one, then $L$ primitively represents any $\z$-lattice that is locally primitively represented by $M$. Note that by Lemma <ref>, $M_q$ is not primitively $2$-universal for some prime $q$, and hence there are infinitely many $\z$-lattices that are not primitively represented by $M$.
Recall that any core prime $q$ of $M$ satisfies $S_q U \ne (-1, -1)$, where $\q M \cong U\mathbin{\perp} \langle dM \rangle$ (see Definition <ref>). One may easily check that the $5$-section of $L$ whose type is not of H has class number one, and the $5$-section $I_2\mathbin{\perp}\mathbb{A}\mathbin{\perp} \langle 2 \rangle$ of $L$ with type H has class number two. Note that the genus mate of this lattice is $I_4\mathbin{\perp} \langle 6 \rangle$.
Hereafter, $\alpha$, $\beta$ denote integers in $\z_p$, and $\epsilon$, $\delta$ denote units in $\z_p$, unless stated otherwise, where the prime $p$ could be easily verified from the context.
For the $5$-section $M$ of each type given in Table <ref>, the core prime $q$ of $M$ and local structures over $\z_q$ of any binary $\z$-lattice $\ell$ that is not primitively represented by $M$ are given as in the table.
The core prime and the local structures
Type $M$ $q$ Local structures
A $I_5$ $2$ $\ell_2\cong \langle 1, 8\alpha \rangle$ or $\fn (\ell_2)\subseteq 4\z_2$
B $I_4\mathbin{\perp} \langle 2 \rangle$ $2$ $\ell_2\cong \langle 2, 16\alpha \rangle$ or $\fn (\ell_2)\subseteq 8\z_2$
D $I_3\mathbin{\perp} \langle 2,2 \rangle$ $2$
$\ell_2\cong \langle 1, 16\alpha \rangle$, $\langle 4, 16\alpha \rangle$, $\langle 20, 16\alpha \rangle$, or
$\fn (\ell_2)\subseteq 16\z_2$
E $I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)$ $3$ $\ell_3\cong \langle 3, 9\alpha \rangle$ or $\fs (\ell_3)\subseteq 9\z_3$
G $I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)$ $5$ $\ell_5\cong \langle 5, 25\alpha \rangle$ or $\fs (\ell_5)\subseteq 25\z_5$
I $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&2\end{smallmatrix}\right)$ $2$
$\ell_2\cong\langle 1, 32\alpha \rangle$, $\langle 5, 16\epsilon \rangle$, $\langle 4, 32\alpha \rangle$, or
$\fn (\ell_2)\subseteq 16\z_2$
K $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)$[-9pt]0pt24pt $7$ $\ell_7\cong \langle 7, 49\alpha \rangle$ or $\fs (\ell_7)\subseteq 49\z_7$
One may easily verify that the prime $q$ given in Table <ref> is the only core prime for each $5$-section $M$, and any binary $\z_p$-lattice is primitively represented by $M_p$ for any prime $p\ne q$. Hence, a binary $\z$-lattice $\ell$ is primitively represented by $M$ if and only if $\ell_q$ is primitively represented by $M_q$.
Since all the other cases can be proved in similar manners, we only provide the proofs of the cases when $L$ is of types I and K.
First, assume that $L$ is of type I. Note that $M_2\cong \mathbb{H}\mathbin{\perp} N$, where $N\cong \ang{3,7,12}$ and
\[
Q^\ast(N) = \{3,7\}(\z_2^\times)^2\cup 2\z_2^\times\cup \{12, 20, 28\}(\z_2^\times)^2\cup 8\z_2^\times\text{.}
\]
Hence, $M_2$ primitively represents all binary lattices of the form $\ang{\alpha, \theta}$, $\mathbb{H}^{2^a}$, and $\mathbb{A}^{2^a}$, where $\theta\in Q^\ast(N)$ and $0\le a\le 2$. Moreover, $M_2$ primitively represents $\ang{\theta, \theta'}$, where $\theta$, $\theta'\in \{1, 5, 4\}$, $\ang{1, 16\epsilon}$, $\ang{5, 32\alpha}$, and $\ang{4, 16\epsilon}$.
Now, assume that $L$ is of type K. Note that $M_7\cong \mathbb{H}\mathbin{\perp} N$, where $N\cong \ang{1,1,7\cdot\Delta_7}$ and $Q^\ast(N) = \{1,\Delta_7,7\cdot\Delta_7\}(\z_7^\times)^2$. In this case, the lemma follows directly from $\ang{7,7}\cong \ang{7\cdot\Delta_7,7\cdot\Delta_7}$. This completes the proof.
In Table <ref>, if a binary $\z$-lattice $\ell$ is not primitively represented by $M$, then the local structures of $\ell$ necessarily satisfies one of the conditions given in the fourth column in the same row. However, we do not know whether or not any binary $\z$-lattice satisfying one of the local structures is necessarily not primitively represented by $M$.
We first complete the proof of the case when the $5$-section splits $L$ orthogonally.
Suppose that $L$ has class number at least two. If $M$ is of class number one and splits $L$ orthogonally, then $L$ is primitively $2$-universal. In fact, there are exactly $51$ such lattices in Table <ref>.
By the above lemma, $L$ is of type
\[
\text{B\textsuperscript{i}(except B$^{\mbox{\scriptsize i}}_2$), D\textsuperscript{i}, E\textsuperscript{i}, G\textsuperscript{i}, I\textsuperscript{i}, or K\textsuperscript{i}.}
\]
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a $\z$-lattice which is not primitively represented by $M$. We assume that $\ell$ is Minkowski reduced, that is, $0\le 2b\le a\le c$. To show that $\ell$ is primitively represented by $L$, we may consider two $\z$-lattices
\[
\ell' \cong \begin{pmatrix}a-k&b\\b&c\end{pmatrix}\text{,}\qquad \ell'' \cong \begin{pmatrix}a&b\\b&c-k\end{pmatrix}\text{.}
\]
If either $\ell'$ or $\ell''$ is primitively represented by $M$, then clearly $\ell$ is primitively represented by $L\cong M\mathbin{\perp}\ang{k}$. Moreover, $\ell'$ ($\ell''$) is primitively represented by $M$ if and only if $\ell'_q$ ($\ell''_q$, repsectively) is primitively represent by $M_q$ for the core prime $q$ of $M$.
Since all the other cases can be proved in similar manners, we only provide the proofs of the cases when $L$ is of types Di or Gi.
First, assume that $L$ is of type Di. By Lemma <ref>, we may assume that
\[
\ell_2\cong\ang{1, 16\alpha}\text{,}\quad \ang{4, 16\alpha}\text{,}\quad \ang{20, 16\alpha}\text{,} \quad \text{or} \quad \fn (\ell_2)\subseteq 16\z_2\text{.}
\]
Suppose that $k = 3$ or $5$. Assume that $a\not\equiv 0\Mod{16}$. Since $d\ell''_2 \not\equiv 0\Mod{16}$, $\ell''_2$ is primitively represented by $M_2$. Hence, $\ell''$ is primitively represented by $M$ if $c\ge 7$, in which case $\ell''$ is positive definite. Now, assume that $a\equiv 0\Mod{16}$. Since $a-k$ is odd, $\ell'_2$ is split by $\ang{a-k}$. Furthermore, since $a-k \not\equiv 1\Mod8$, $\ell'_2$ is primitively represented by $M_2$. Hence, $\ell'$ is primitively represented by $M$. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 6$.
Now, suppose that $k = 2$ or $6$. Assume that $a\not\equiv 0\Mod8$. Since $d\ell''_2 \not\equiv 0\Mod{16}$, $\ell''_2$ is primitively represented by $M_2$. Hence, $\ell''$ is primitively represented by $M$ if $c\ge 9$, in which case $\ell''$ is positive definite. Now, assume that $c$ is odd. Since $c \equiv 1\Mod8$, $\ell''_2$ is split by $\ang{c-k}$. Furthermore, since $c-k \not\equiv 1\Mod8$, $\ell''$ is primitively represented by $M$ if $c\ge 9$. Finally, assume that $a\equiv 0\Mod8$ and $c$ is even. Since $\ell_2$ is not unimodular, we must have $\fs (\ell_2) \subseteq 4\z_2$, which implies
\[
c-k \equiv 2\Mod4 \quad\text{and}\quad \fs (\ell''_2) = 2\z_2\text{.}
\]
Hence, $\ell''$ is primitively represented by $M$ if $c\ge 9$. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 8$.
Finally, suppose that $k = 4$. Assume that $a\not\equiv 0\Mod4$. Since $d\ell''_2 \not\equiv 0\Mod{16}$, $\ell''_2$ is primitively represented by $M_2$. Hence, $\ell''$ is primitively represented by $M$ if $c\ge 6$, in which case $\ell''$ is positive definite. Now, assume that $c$ is odd. Since $c \equiv 1\Mod8$, $\ell''_2$ is split by $\ang{c-4}$. Furthermore, since $c-4 \not\equiv 1\Mod8$, $\ell''$ is primitively represented by $M$ if $c\ge 6$. Hence, we may assume that $a\equiv 0\Mod4$ and $c$ is even. Since $\ell_2$ is not unimodular, we have
\[
\fs (\ell_2) \subseteq 4\z_2 \quad\text{and}\quad d\ell_2 \equiv 0\Mod{64}\text{.}
\]
If $a\not\equiv 0\Mod{16}$, then $d\ell''_2 \not\equiv 0\Mod{64}$; hence $\ell''$ is primitively represented by $M$ if $c\ge 6$. Now, assume that $a\equiv 0\Mod{16}$. Since $\fs (\ell'_2) = 4\z_2$, $\ell'_2$ is split by $\ang{a-4}$. Furthermore, since $a-4 \equiv -4\Mod{16}$, $\ell'_2$ is primitively represented by $M_2$. Hence, $\ell'$ is primitively represented by $M$ for $a\ge 16$. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 5$.
Next, assume that $L$ is of type Gi. By Lemma <ref>, we may assume that
\[
\ell_5\cong \ang{5, 25\alpha} \quad \text{or} \quad \fs (\ell_5)\subseteq 25\z_5\text{.}
\]
Suppose that $k\ne 10$ or $15$. Since $\fs (\ell''_5) = \z_5$, $\ell''_5$ is primitively represented by $M_5$. Hence, $\ell''$ is primitively represented by $M$ if $c\ge 25$, in which case $\ell''$ is positive definite. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 24$.
Now, suppose that $k = 10$ or $15$. Assume that $a\not\equiv 0\Mod{25}$. Since $d\ell''_5 \not\equiv 0\Mod{125}$, $\ell''_5$ is primitively represented by $M_5$. Hence, $\ell''$ is primitively represented by $M$ if $c\ge 21$, in which case $\ell''$ is positive definite. Now, assume that $a\equiv 0\Mod{25}$. Since $\fs (\ell'_5) = 5\z_5$, $\ell'_5$ is split by $\ang{a-k}$. Furthermore, since $a-k\equiv 15$ or $10\Mod{25}$, $\ell'_5$ is primitively represented by $M_5$. Hence, $\ell'$ is primitively represented by $M$ since $a\ge 25$. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 20$.
If $L\cong$ $_3$ or $_3$, then we may take $M \cong I_3\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$, which is a primitive sublattice of $L$. If $L\cong$ , then we may take $M \cong I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)$, which is a primitive sublattice of $L$. Since the class number of $M$ is one and $M$ splits $L$ orthogonally, the proofs of these three candidates are quite similar to that of the above theorem.
Let $R = \z$ or $\z_p$ for some prime $p$. Let $O$ be an $R$-lattice and let $\mathfrak B=\{e_1, \dots, e_n\}$ be the fixed (ordered) basis for the $R$-lattice $O$.
We define
\[
O'= R\begin{bmatrix}
a_{11} & \dots & a_{1n}\\
\vdots & & \vdots\\
a_{m1} & \dots & a_{mn}
\end{bmatrix}
\]
by the $R$-sublattice of $O$ generated by $m$ vectors $a_{11}e_1 + \cdots + a_{1n}e_n, \dots,\allowbreak a_{m1}e_1 + \cdots + a_{mn}e_n$.
Note that if the rank of $O'$ is $m$, then the symmetric matrix corresponding to $O'$ is that
\[
M_{O'}= \begin{pmatrix}
a_{11} & \dots & a_{1n}\\
\vdots & & \vdots\\
a_{m1} & \dots & a_{mn}
\end{pmatrix} M_O \begin{pmatrix}
a_{11} & \dots & a_{m1}\\
\vdots & & \vdots\\
a_{1n} & \dots & a_{mn}
\end{pmatrix}\text{.}
\]
When only the corresponding symmetric matrix $M_O$ is given instead of the basis for $O$, we assume that $\mathfrak B$ is the basis for $O$ such that $(B(e_i,e_j))=M_O$.
§.§ Case 3: the $5$-section of $L$ has class number one and does not split $L$ orthogonally
Recall that we are assuming that $L$ is one of $201$ candidates of primitively $2$-universal senary $\z$-lattices given in Table <ref>, and $\mathfrak B=\{e_1,\dots,e_6\}$ is the basis for $L$ such that the corresponding symmetric matrix $(B(e_i,e_j))$ is given in Table <ref>. Furthermore, $M=\z e_1+\dots+\z e_5$ is the $5$-section of $L$.
Let $L$ and $N$ be $\z$-lattices given as follows:
* The $\z$-lattice $L$ is of type , and $N \cong qI_3\mathbin{\perp}\mathbb{A}$.
* The $\z$-lattice $L$ is of type or , and $N \cong qI_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)$.
* The $\z$-lattice $L$ is of type , or , and $N \cong qI_2\mathbin{\perp}\left(\begin{smallmatrix}3&1&1\\1&5&-2\\1&-2&5\end{smallmatrix}\right)$.
Here, $q$ is the core prime of the $5$-section $M$ of $L$.
Let $\ell \cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice.
If $\ell' \cong \left(\begin{smallmatrix}qa-ds^2 & qb-dst\\qb-dst & qc-dt^2\end{smallmatrix}\right)$ is positive definite and is primitively represented by $N$ for some integers $s$ and $t$, where $d = dL$ is the discriminant of $L$, then $\ell$ is primitively represented by $L$.
One may easily show that there is a representation $\psi: L^q \to N\mathbin{\perp} \langle d\rangle$ such that $\psi(L^q)\cap N=M^q$.
Since all the other cases can be proved in similar manners, we only provide the proof of the case when $L$ is of type Kiv. Note that $M \cong I_2 \mathbin{\perp} \left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)$ and $q=7$. Since $N$ primitively represents $\ell'$, there are integers $c_i$'s and $d_i$'s for $i=1,\dots,5$ such that the primitive sublattice of $N$
\[
\z\begin{bmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{bmatrix} \cong \ell' \cong \begin{pmatrix}7a-(7k-5)s^2 & 7b-(7k-5)st\\7b-(7k-5)st & 7c-(7k-5)t^2\end{pmatrix}\text{,}
\]
where $\left(\begin{smallmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{smallmatrix}\right)$ is a primitive matrix. Note that
\[
\left(\begin{smallmatrix}3&1&1\\1&5&-2\\1&-2&5\end{smallmatrix}\right) \equiv 3\left(\begin{smallmatrix}1\\-2\\-2\end{smallmatrix}\right)\begin{pmatrix}1&-2&-2\end{pmatrix}\Mod7\text{.}
\]
Thus, we have
\[
\left.\begin{aligned}
3(c_3 - 2c_4 - 2c_5)^2 & \equiv 5s^2\\
3(c_3 - 2c_4 - 2c_5)(d_3 - 2d_4 - 2d_5) & \equiv 5st\\
3(d_3 - 2d_4 - 2d_5)^2 & \equiv 5t^2
\end{aligned}\right\} \pmod7\text{.}
\]
Hence, after replacing $(s, t)$ by $(-s, -t)$, if necessary, we may assume that
\[
c_3 - 2c_4 - 2c_5 + 2s \equiv d_3 - 2d_4 - 2d_5 + 2t \equiv 0\pmod7\text{.}
\]
Therefore, there are integers $a_3$, $a_4$, $a_5$, $b_3$, $b_4$, and $b_5$ satisfying
\[
\begin{pmatrix}c_3 & d_3\\c_4 & d_4\\c_5 & d_5\\s & t\end{pmatrix} = \begin{pmatrix}-1&0&2&0\\2&1&1&1\\1&-1&0&0\\0&0&0&1\end{pmatrix} \begin{pmatrix}a_3 & b_3\\a_4 & b_4\\a_5 & b_5\\s & t\end{pmatrix}\text{.}
\]
Now, consider the sublattice
\[
O = \z\begin{bmatrix}c_1 & c_2 & a_3 & a_4 & a_5 & s\\d_1 & d_2 & b_3 & b_4 & b_5 & t\end{bmatrix}
\]
of $L$. Since
\begin{multline*}
\begin{pmatrix}a_3 & a_4 & a_5 & s\\b_3 & b_4 & b_5 & t\end{pmatrix} \begin{pmatrix}14&7&7&7\\7&14&7&7\\7&7&21&7\\7&7&7&7q\end{pmatrix} \begin{pmatrix}a_3 & b_3\\a_4 & b_4\\a_5 & b_5\\s & t\end{pmatrix}\\
= \begin{pmatrix}c_3 & c_4 & c_5 & s\\d_3 & d_4 & d_5 & t\end{pmatrix} \begin{pmatrix}3&1&1&0\\1&5&-2&0\\1&-2&5&0\\0&0&0&7q-5\end{pmatrix} \begin{pmatrix}c_3 & d_3\\c_4 & d_4\\c_5 & d_5\\s & t\end{pmatrix}\text{,}
\end{multline*}
the $\z$-lattice $O$ is isometric to $\ell$. Now, since
\[
\begin{pmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{pmatrix} = \begin{pmatrix}c_1 & c_2 & -a_3 + 2a_5 & 2a_3 + a_4 + a_5 + s & a_3 - a_4\\d_1 & d_2 & -b_3 + 2b_5 & 2b_3 + b_4 + b_5 + t & b_3 - b_4\end{pmatrix}
\]
is a primitive matrix, so is
\[
\begin{pmatrix}c_1 & c_2 & -a_3 + 2a_5 & 2a_3 + a_4 + a_5 + s & a_3 - a_4 & a_5\\d_1 & d_2 & -b_3 + 2b_5 & 2b_3 + b_4 + b_5 + t& b_3 - b_4 & b_5\end{pmatrix}\text{.}
\]
Therefore, the matrix
\[
\begin{pmatrix}c_1 & c_2 & a_3 & a_4 & a_5 & s\\d_1 & d_2 & b_3 & b_4 & b_5 & t\end{pmatrix}
\]
is primitive, which implies that $O$ is a primitive sublattice of $L$. This completes the proof.
If $L$ is of type
\[
\text{\textup{E\textsuperscript{ii}, G\textsuperscript{ii}, G\textsuperscript{iii}, K\textsuperscript{ii}, K\textsuperscript{iii}, or K\textsuperscript{iv}},}
\]
then $L$ is primitively $2$-universal. There are exactly $110$ such $\z$-lattices in Table <ref>.
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0 \le 2b\le a\le c$) be a $\z$-lattice which is not primitively represented by $M$. Since all the other cases can be proved in similar manners, we only provide the proofs of the cases when $L$ is of type K. Note that
\[
\det(\text{K\textsuperscript{ii}})=7k-3\text{,}\quad \det(\text{K\textsuperscript{iii}})=7k-6\text{,}\quad \text{and}\quad \det(\text{K\textsuperscript{iv}})=7k-5\text{.}
\]
Assume that $L$ is of type K. By Lemma <ref>, we may assume that
\[
\ell_7\cong\ang{7, 49\alpha} \quad \text{or} \quad \fs (\ell_7)\subseteq 49\z_7\text{.}
\]
Observe that $N = 7I_2\mathbin{\perp}\left(\begin{smallmatrix}3&1&1\\1&5&-2\\1&-2&5\end{smallmatrix}\right)$ is of class number one, and that $7$ is the only core prime of $N$. Furthermore, $N_7\cong \mathbb{H}^7\mathbin{\perp} O$, where $O = \ang{\Delta_7, 7, 7}$. Thus $N_7$ primitively represents all binary $\z_7$-lattices of the form $\ang{7\alpha, \theta}$, where $\theta\in Q^\ast(O) = \{\Delta_7, 7, 7\cdot\Delta_7\}(\z_7^\times)^2$. Hence, $N_7$ primitively represents
\[
\ell' \cong \left(\begin{smallmatrix}7a & 7b\\7b & 7c - (7k-3)\end{smallmatrix}\right)\text{,}\quad \left(\begin{smallmatrix}7a & 7b\\7b & 7c - (7k-6)\end{smallmatrix}\right)\text{,}\quad \text{and}\quad \left(\begin{smallmatrix}7a & 7b\\7b & 7c - (7k-5)\end{smallmatrix}\right)\text{.}
\]
Therefore $N$ primitively represents $\ell'$ if it is positive definite. Hence, by Lemma <ref>, $\ell$ is primitively represented by $L$ if $c \ge 33$. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 32$.
In fact, we do not use the fact that $M$ is the $5$-section of $L$ in the above theorem.
If $L\cong$ , , or , then we may take a primitive sublattice $M$ of $L$ as in the following table. Then $q$ is the only core prime of $M$, and one may apply Lemma <ref> for each pair of $L$ and $N$ in the table. Therefore, the proofs of primitive $2$-universalities of these three candidates are quite similar to Theorem <ref>.
The core prime of $M$ and the choice of $N$
$L$ $M$ $q$ $N$
D$^{\mbox{\scriptsize ii}}_3$ $I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)$ $5$ $5I_3\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&3\end{smallmatrix}\right)$
H$^{\mbox{\scriptsize iv}}_3$ or I$^{\mbox{\scriptsize iii}}_3$ $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&2&1\\1&1&3\end{smallmatrix}\right)$[-9pt]0pt24pt $7$ $7I_2\mathbin{\perp}\left(\begin{smallmatrix}3&1&1\\1&5&-2\\1&-2&5\end{smallmatrix}\right)$
Summing up all, we have proved the primitive $2$-universalities of $175$ $\z$-lattices among $201$ candidates, and the primitive $2$-universalities of $26$ candidates remain unproven.
§ THE PROOF OF PRIMITIVE $2$-UNIVERSALITY (EXCEPTIONAL CASES)
Let $L$ be one of the remaining $26$ candidates of primitively $2$-universal $\z$-lattices which we do not consider in Section <ref>. We try to find quinary or quaternary primitive sublattices of $L$ which have class number one or two to prove primitive $2$-universality of $L$, in each exceptional case. Since the key ingredients of the proofs, which we will explain more precisely later, are quite similar to each other, we only provide the proofs of some representative cases. For the complete proofs, one may see the second author's dissertation [17]. At the end of this section, we provide some essential data needed for computations.
§.§ Case 4: Type H lattices
In this subsection, we prove the primitive $2$-universalities of the remaining type H lattices. The main obstacle for type H lattices is that the $5$-section $I_2\mathbin{\perp}\mathbb{A}\mathbin{\perp}\ang{2}$ of $L$ is of class number two, and the genus mate $I_4\mathbin{\perp}\ang{6}$ is not represented by $L$. The following lemma gives some information on binary $\z$-lattices that are primitively represented by the $5$-section of $L$, though it has class number two.
Let $\ell \cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice and let $m$ and $k\ge 2$ be positive integers. Suppose that
\[
\begin{pmatrix}2a-(2k-1)s^2 & 2b-(2k-1)st\\2b-(2k-1)st & 2c-(2k-1)t^2\end{pmatrix}
\]
is positive definite and is primitively represented by $2I_m\mathbin{\perp}\ang{4,1}$ for some integers $s$ and $t$. Then the binary $\z$-lattice $\ell$ is primitively represented by $I_m\mathbin{\perp}\ang{2}\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&k\end{smallmatrix}\right)$.
Since the proof is quite similar to that of Lemma <ref>, it is left to the readers.
If a binary $\z$-lattice $\ell$ is not primitively represented by the $\z$-lattice $M = I_2\mathbin{\perp}\ang{2}\mathbin{\perp}\mathbb{A}$, then either $\ell_2\cong \ang{6, 16\alpha}$ or $\fn (\ell_2)\subseteq 8\z_2$.
Fix a basis for $M$ corresponding to the Gram matrix in the statement of the lemma. Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0\le 2b\le a\le c$) be a binary $\z$-lattice which does not satisfy the conclusion. Note that $M$ primitively represents $\ang{1, 1, 2, 2}$, which has class number one, and $\ang{1, 1, 2, 2}\cong \mathbb{H}\mathbin{\perp}\mathbb{H}$ is primitively $2$-universal over $\z_p$ for any odd prime $p$.
Hence, we may assume that
\[
\ell_2\cong\begin{cases}
\ang{1, -1}\text{, }\ang{\epsilon, 4\delta}\text{, }\ang{\epsilon, 16\alpha}\text{, }\\
\ang{2, -2}\text{, }\ang{2\epsilon, 8\alpha}\text{, }\\
\ang{4\epsilon, 4\delta}\text{ with }\epsilon\delta\equiv -1\Mod4\text{, }\ang{4\epsilon, 16\alpha}\text{,}\\
\mathbb{H}^2\text{,}\\
\mathbb{H}\text{ or }\mathbb{A}\text{.}
\end{cases}
\]
We define the binary $\z$-lattices
\[
\ell'(u,t) \cong \left(\begin{smallmatrix}2a-3t^2 & 2ua + 2b\\2ua + 2b & 2u^2a + 4ub + 2c\end{smallmatrix}\right) \quad\text{and}\quad \ell''(u,t) \cong \left(\begin{smallmatrix}2a + 4ub + 2u^2c & 2b + 2uc\\2b + 2uc & 2c-3t^2\end{smallmatrix}\right)\text{.}
\]
Note that
\[
\ell \cong \left(\begin{smallmatrix}a & ua + b\\ua + b & u^2a + 2ub + c\end{smallmatrix}\right) \cong \left(\begin{smallmatrix}a + 2ub + u^2c & b + uc\\b + uc & c\end{smallmatrix}\right)
\]
for any integer $u$. Hence, if $\ell'(u,t)$ or $\ell''(u,t)$ is primitively represented by $N\cong 2I_2\mathbin{\perp}\ang{4, 1}$ for some integers $u$ and $t$, then $\ell$ is primitively represented by $M$ by Lemma <ref>. Since $N_p\cong \mathbb{H}\mathbin{\perp}\mathbb{H}$ for any odd prime $p$, $N_p$ is primitively $2$-universal over $\z_p$ for any odd prime $p$. Note that
\[
\fs (\ell'(u,1)_2) = \fs (\ell''(u,1)_2) = \z_2\text{.}
\]
First, assume that $\fs(\ell_2) = \z_2$. Assume that $a$ is odd. Since $d\ell''(0,1)\equiv 2\Mod4$, $\ell''(0,1)_2$ is primitively represented by $N_2$. Note that $\ell''(0,1)$ is positive definite. Hence, $\ell''(0,1)$ is primitively represented by $N$. Now, assume that $a$ is even. Then $a\equiv 0\Mod4$ and $c$ is odd. In this case, $d\ell'(0,1)\equiv 2\Mod4$ and $\ell'(0,1)$ is positive definite. Hence, $\ell'(0,1)$ is primitively represented by $N$.
Now, assume that $\ell_2\cong\langle 2, -2\rangle$. Note that $a\equiv b\equiv c\equiv 0\Mod2$. First, suppose that $a\equiv 2\Mod4$. If $a \not\equiv -2\Mod{16}$, then
\[
d\ell''(0,1)\equiv 4\Mod8 \quad \text{and} \quad d\ell''(0,1)\not\equiv -4\Mod{32}\text{.}
\]
Hence, $\ell''(0,1)$ is primitively represented by $N$. If $a\equiv -2\Mod{16}$, then
\[
\fs (\ell''(0,2)_2) = 4\z_2 \quad \text{and} \quad d\ell''(0,2)\equiv 32\Mod{64}\text{.}
\]
Hence, $\ell''(0,2)$ is primitively represented by $N$. Now, suppose that $a\equiv 0\Mod4$. Then $a\equiv 0\Mod{16}$ and $c\equiv 2\Mod4$. If $c \not\equiv -2\Mod{16}$, then $\ell'(0,1)$ is primitively represented by $N$, and if $c\equiv -2\Mod{16}$, then $\ell'(0,2)$ is primitively represented by $N$.
Next, assume that $\ell_2\cong\langle 2\epsilon, 8\alpha\rangle$. Note that $a\equiv b\equiv c\equiv 0\Mod2$. If $a\equiv 2\Mod4$ and $a \not\equiv 6\Mod{16}$, then
\[
d\ell''(0,1)\equiv 4\Mod8 \quad \text{and} \quad d\ell''(0,1)\not\equiv -4\Mod{32}\text{.}
\]
Hence, $\ell''(0,1)$ is primitively represented by $N$. Similarly, if $c\equiv 2\Mod4$ and $c \not\equiv 6\Mod{16}$, then $\ell'(0,1)$ is primitively represented by $N$. Now, assume that $a\equiv 6\Mod{16}$ or $c\equiv 6\Mod{16}$. Since we are assuming that
\[
\ell_2\not\cong\langle 6, 16\alpha\rangle\text{,}
\]
we have $d\ell\equiv 16\Mod{32}$. Assume that $a\equiv c\equiv 6\Mod{16}$. Then, $b\equiv 2\Mod4$. Hence, there is an $\eta\in\{1, -1\}$ such that
\[
a + 2\eta b + c \equiv 6\Mod8\text{.}
\]
Since $d\ell''(\eta, 1)\equiv 8\Mod{16}$, we have
\[
\ell''(\eta, 1)_2\cong\langle\epsilon, 8\delta\rangle\text{,}
\]
which is primitively represented by $N_2 \cong \langle 1, 2, 2, 4 \rangle$. Furthermore, since $\ell''(\eta, 1)$ is positive definite, it is primitively represented by $N$. Next, assume that $a\equiv 6\Mod{16}$ and $c\not\equiv 6\Mod{16}$. Then either $c\equiv 8\Mod{16}$ and $b\equiv 0\Mod8$, or $c\equiv 0\Mod{16}$ and $b\equiv 4\Mod8$. In any case,
\[
a - 2b + c\equiv -2\Mod{16}\text{.}
\]
Since $d\ell''(-1,1)\equiv 12\Mod{32}$, we have
\[
\ell''(-1, 1)_2\cong\langle-3, -4\rangle\text{,}
\]
which is primitively represented by $N_2 \cong \langle 1, 2, 2, 4 \rangle$. Furthermore, since $\ell''(-1, 1)$ is positive definite, it is primitively represented by $N$. Finally, assume that $a\not\equiv 6\Mod{16}$ and $c\equiv 6\Mod{16}$. Then, similarly to the above, $\ell'(-1, 1)$ is primitively represented by $N$ in this case.
Now, assume that $\fs(\ell_2) = 4\z_2$. Note that $a\equiv b\equiv c\equiv 0\Mod4$. If $a\equiv 4\Mod8$, then $d\ell''(0,1)\equiv 8\Mod{16}$. Since $\ell''(0,1)$ is positive definite, $\ell''(0,1)$ is primitively represented by $N$. Similarly, if $c\equiv 4\Mod8$, then $\ell'(0,1)$ is primitively represented by $N$ in this case.
Next, assume that $\ell_2 \cong \mathbb{H}^2$. Note that $a\equiv b-2\equiv c\equiv 0\Mod4$. Assume that $a\equiv 4\Mod8$. Since $d\ell''(0,1)\equiv 8\Mod{16}$, we have
\[
\ell''(0, 1)_2\cong\langle\epsilon, 8\delta\rangle\text{,}
\]
which is primitively represented by $N_2 \cong \langle 1, 2, 2, 4 \rangle$. Since $\ell''(0,1)$ is positive definite, $\ell''(0,1)$ is primitively represented by $N$. Next, assume that $c\equiv 4\Mod8$. Then, similarly to the above, $\ell'(0,1)$ is primitively represented by $N$ in this case. Finally, assume that $a\equiv c\equiv 0\Mod8$. Since
\[
a - 2b + c\equiv 4\Mod8\text{,}
\]
we have $d\ell''(-1, 1)\equiv 8\Mod{16}$. Since $\ell''(-1,1)$ is positive definite, $\ell''(-1,1)$ is primitively represented by $N$.
Finally, assume that $\ell_2 \cong \mathbb{H}$ or $\mathbb{A}$. Note that $a\equiv b-1\equiv c\equiv 0\Mod2$. Assume that $a\equiv 6\Mod8$. Since $d\ell''(0,1)\equiv 8\Mod{16}$, we have
\[
\ell''(0, 1)_2\cong\langle\epsilon, 8\delta\rangle\text{,}
\]
which is primitively represented by $N_2 \cong \langle 1, 2, 2, 4 \rangle$. Hence, $\ell''(0,1)$ is primitively represented by $N$. Assume that $c\equiv 6\Mod8$. Then, similarly to the above, $\ell'(0,1)$ is primitively represented by $N$ in this case. Now, suppose that neither $a$ nor $c$ is congruent to $6$ modulo $8$. Then we have
\[
a \equiv 2\Mod8 \quad \text{or} \quad a \equiv 0\Mod 4\text{,}
\]
and the same with $c$. First, assume that $a\equiv c\equiv 2\Mod8$. Then, there is an $\eta\in\{1, -1\}$ such that
\[
a + 2\eta b + c \equiv 6\Mod8\text{.}
\]
Since $d\ell''(\eta, 1)\equiv 8\Mod{16}$, $\ell''(\eta, 1)$ is primitively represented by $N_2 \cong \langle 1, 2, 2, 4 \rangle$. Hence, $\ell''(\eta, 1)$ is primitively represented by $N$ if it is positive definite, that is, if $a\ge 7$, or if $a = 2$ and $c\ge 15$. Clearly, $\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}2&1\\1&10\end{smallmatrix}\right)$ are primitively represented by $M$. Next, assume that $a\equiv 0\Mod4$ and $c\equiv 2\Mod8$. Then
\[
4a - 4b + c\equiv 6\Mod8\text{.}
\]
Since $d\ell'(-2, 1)\equiv 8\Mod{16}$, we have $\ell'(-2, 1)_2 \cong\langle\epsilon, 8\delta\rangle$, which is primitively represented by $N_2$. Since
\[
\ell'(-2, 1) \cong \left(\begin{smallmatrix}2a-3 & -4a + 2b\\-4a + 2b & 8a - 8b + 2c\end{smallmatrix}\right)
\]
is positive definite, it is primitively represented by $N$. Now, assume that $a\equiv 2\Mod8$ and $c\equiv 0\Mod4$. Then, similarly to the above, $\ell''(-2, 1)$ is primitively represented by $N$ if it is positive definite, that is, if $a\ge 11$, or if $a = 10$ and $c\ge 4$. The case when $a = 2$ will be postponed to the end of this proof. Finally, assume that $a\equiv c\equiv 0\Mod4$. Then, there is an $\eta\in\{1, -1\}$ such that
\[
a + 2\eta b + c \equiv 6\Mod8\text{.}
\]
Hence, $\ell''(\eta, 1)$ is primitively represented by $N$ if it is positive definite, that is, if $a\ge 7$, or if $a = 4$ and $c\ge 5$. Clearly $\left(\begin{smallmatrix}4&1\\1&4\end{smallmatrix}\right)$ is primitively represented by $M$.
Now, suppose that $a=2$, $b=1$, and $c\equiv 0\Mod4$. If $c\not\equiv 0\Mod{16}$, then $\ang{1,1,2}$ represents $c-2$. If we choose a vector $v$ in the $3$-section of $M$ such that $Q(v) = c-2$, then clearly, $\z[e_4, v + e_5]$ is a primitive sublattice of $M$ isometric to $\left(\begin{smallmatrix}2&1\\1&c\end{smallmatrix}\right)$. If $c\equiv 0\Mod{16}$, then $\ang{1,1,2}$ primitively represents $c-14$. If we choose a vector $w$ in the $3$-section of $M$ such that $Q(v) = c-14$, then clearly,
\[
\z[e_4, w - 2e_4 + 3e_5]
\]
is a primitive sublattice of $M$ isometric to $\left(\begin{smallmatrix}2&1\\1&c\end{smallmatrix}\right)$.
For type Hi lattices, the $5$-section splits $L$. Hence, one may prove the primitive $2$-universalities of type Hi lattices in the similar way to Theorem <ref> by using the above lemma. Note that for the proofs of primitive $2$-universalities of type Hii, Hiii, and Hiv lattices, we need Lemma <ref> given below as well as Lemma <ref>. In particular, the proof of the primitive $2$-universality of the lattice H$^{\mbox{\scriptsize iii}}_6$ will be provided in Subsection <ref> since we need Lemma <ref> additionally. The proof of the primitive $2$-universality of the lattice H$^{\mbox{\scriptsize ii}}_5$ will be provided in Theorem <ref>.
Let $\ell \cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice and let $q$ and $r$ be positive integers.
* Suppose that $\ang{1,1,2,2}$ primitively represents $\left(\begin{smallmatrix}2a-A & 2b-B\\2b-B & 2c-C\end{smallmatrix}\right)$, where
\[
\begin{pmatrix}A&B\\B&C\end{pmatrix} = \begin{pmatrix}s_1 & s_2\\t_1 & t_2\end{pmatrix} \begin{pmatrix}2q-1&0\\0&2r-1\end{pmatrix} \begin{pmatrix}s_1 & t_1\\s_2 & t_2\end{pmatrix}
\]
for some integers $s_1$, $s_2$, $t_1$, and $t_2$ such that at least one of $A$ and $C$ is odd. Then $\ell$ is primitively represented by $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&q\end{smallmatrix}\right)\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&r\end{smallmatrix}\right)$.
* Suppose that $\ang{1,2,2,4}$ primitively represents $\left(\begin{smallmatrix}2a-A & 2b-B\\2b-B & 2c-C\end{smallmatrix}\right)$, where
\[
\begin{pmatrix}A&B\\B&C\end{pmatrix} = \begin{pmatrix}s_1 & s_2\\t_1 & t_2\end{pmatrix} \begin{pmatrix}2q-1&1\\1&2r-1\end{pmatrix} \begin{pmatrix}s_1 & t_1\\s_2 & t_2\end{pmatrix}
\]
for some integers $s_1$, $s_2$, $t_1$, and $t_2$. Then $\ell$ is primitively represented by $I_2\mathbin{\perp}\ang{2}\mathbin{\perp}\left(\begin{smallmatrix}2&1&1\\1&q&1\\1&1&r\end{smallmatrix}\right)$.
* Suppose that $\ang{1,1,2,2}$ primitively represents $\left(\begin{smallmatrix}2a-A & 2b-B\\2b-B & 2c-C\end{smallmatrix}\right)$, where
\[
\begin{pmatrix}A&B\\B&C\end{pmatrix} = \begin{pmatrix}s_1 & s_2\\t_1 & t_2\end{pmatrix} \begin{pmatrix}2q-1&1\\1&2r-2\end{pmatrix} \begin{pmatrix}s_1 & t_1\\s_2 & t_2\end{pmatrix}
\]
for some integers $s_1$, $s_2$, $t_1$, and $t_2$ such that at least one of $A$ and $C$ is odd. Then $\ell$ is primitively represented by $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&0&0&1\\0&2&1&1\\0&1&q&1\\1&1&1&r\end{smallmatrix}\right)$.
Since the proof is quite similar to that of Lemma <ref>, it is left to the readers.
The $\z$-lattice $\cong I_2\mathbin{\perp}\mathbb{A}\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&5\end{smallmatrix}\right)$ is primitively $2$-universal.
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0\le 2b\le a\le c$) be a $\z$-lattice which is not primitively represented by the $5$-section of $L$. Then, by Lemma <ref>, we may assume that $\ell$ satisfies either
(i) $a$ or $c\equiv 6\Mod{16}$ and $d\equiv 0\Mod{32}$, or
(ii) $a\equiv c\equiv 0\Mod8$ and $b\equiv 0\Mod4$.
According to Lemma <ref>, if a $\z$-lattice
\[
\ell' \cong \begin{pmatrix}2a-3&2b\\2b&2c-(2q-1)\end{pmatrix}\text{,}
\]
is primitively represented by $N\cong \ang{1,1,2,2}$, then $\ell$ is primitively represented by $L$. Furthermore, according to Lemma <ref>, if
\[
\ell'' \cong \begin{pmatrix}2a-(2q-1)&2b\\2b&2c\end{pmatrix}\quad\text{or}\quad \ell''' \cong \begin{pmatrix}2a&2b\\2b&2c-(2q-1)\end{pmatrix}
\]
is primitively represented by $N'\cong \ang{1,2,2,4}$, then $\ell$ is primitively represented by $I_2\mathbin{\perp}\ang{2}\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&5\end{smallmatrix}\right)$, and hence by $L$.
Assume that case (i) holds. Note that $a\equiv b\equiv c\equiv 0\Mod2$ in this case. If $a\equiv c\equiv 2\Mod4$, then $d\ell'\equiv 3\Mod8$. Thus $\ell'_2$ is primitively represented by $N_2$ in this case. Since $\ell'$ is positive definite, it is primitively represented by $N$. If $a-2\equiv c\equiv 0\Mod4$, then $\fs (\ell'''_2) = \z_2$ and $d\ell'''\equiv 20\Mod{32}$. Thus, one may easily show that $\ell'''_2$ is primitively represented by $N'_2$ in this case. Since $\ell'''$ is positive definite, it is primitively represented by $N'$. Similarly, if $a\equiv c-2\equiv 0\Mod4$, then $\ell''$ is primitively represented by $N'$.
Now, assume that case (ii) holds. Clearly, we have $d\ell'\equiv 3\Mod8$. Since $\ell'$ is positive definite, it is primitively represented by $N$.
§.§ Case 5: $L$ has a quaternary primitive sublattice of class number one or two
In this subsection, we prove the primitive $2$-universalities of the lattices
\[
\text{B$^{\mbox{\scriptsize ii}}_5$, B$^{\mbox{\scriptsize ii}}_6$, D$^{\mbox{\scriptsize iii}}_5$, I$^{\mbox{\scriptsize ii}}_4$, I$^{\mbox{\scriptsize ii}}_5$, I$^{\mbox{\scriptsize iii}}_4$, and J$_3$.}
\]
Note that the $5$-section of $L$ has class number one and does not split $L$ orthogonally. In these cases, we make use of some information on the primitive binary exceptions of quaternary primitive sublattices. The method of the proof depends heavily on whether the discriminant of such a quaternary sublattice which we are considering is a perfect square or not.
First, assume $L\cong$ B$^{\mbox{\scriptsize ii}}_5$ or B$^{\mbox{\scriptsize ii}}_6$. The $4$-section $I_4$ is a quaternary orthogonal summand of class number one whose discriminant is a perfect square. Hence, the primitive $2$-universalities could be proved in a similar manner to that of Theorem <ref>.
Recall that a finite sequence of vectors $v_1, \dots, v_m$ in $\z^n$ ($m\le n$) is primitive if and only if the greatest common divisor $g$ of the determinants of all $m\times m$ submatrices of the coefficient matrix of $v_1, \dots, v_m$, which is defined by the $m\times n$ matrix whose rows are $v_1, \dots, v_m$, is $\pm 1$. Also, we say that $v_1, \dots, v_m$ is $p$-primitive for a prime $p$ if $g$ is prime to $p$. Then it is clear that $v_1, \dots, v_m$ is primitive if and only if it is $p$-primitive for any prime $p$.
Let $L = \z e_1 + \cdots + \z e_{n+1}$ be a free $\z$-module of rank $n+1$, and let $M = \z e_1 + \cdots + \z e_n$. Suppose that $v_1, \dots, v_m$ are vectors in $M$ for some $1 \le m \le n$.
* Suppose that $v_1, \dots, v_m$ is $p$-primitive for some prime $p$. Then $v_1, \dots,\allowbreak v_{m-1}, v_m+pw$ is also $p$-primitive for any $w\in M$.
* Suppose that $v_1, \dots, v_m$ is $p$-primitive for some odd prime $p$. Then for any $w\in M$, either $v_1, \dots, v_{m-1}, v_m+w$ or $v_1, \dots, v_{m-1}, v_m-w$ is also $p$-primitive.
* Suppose that $v_1, \dots, v_m$ is primitive. For a vector $y = y_1 e_1 + \cdots + y_n e_n + y_{n+1} e_{n+1}\in L$, put
\begin{align*}
\mathcal{P}(y) & = \{p : p\text{ is a prime that divides }\gcd(y_1, \dots, y_n, y_{n+1})\}\text{,}\\
\mathcal{P}(y_{n+1}) & = \{p : p\text{ is a prime that divides }y_{n+1}\}\text{.}
\end{align*}
If $\mathcal{P}(y_{n+1})\setminus\mathcal{P}(y) = \varnothing$, then $v_1, \dots, v_{m-1}, v_m+y$ is primitive. If $\mathcal{P}(y_{n+1})\setminus\mathcal{P}(y) = \{p\}$ for some odd prime $p$, then either $v_1, \dots, v_{m-1},\allowbreak v_m+y$ or $v_1, \dots, v_{m-1}, v_m-y$ is primitive.
(1) The lemma follows from the fact that the determinant of any $m\times m$ submatrix of the $m\times n$ coefficient matrix of $v_1, \dots, v_m$ is congruent to the determinant of the corresponding $m\times m$ submatrix of the $m\times n$ coefficient matrix of $v_1, \dots, v_{m-1}, v_m+pw$ modulo $p$.
(2) Suppose on the contrary that both
\[
v_1, \dots, v_{m-1}, v_m+w \quad\text{and}\quad v_1, \dots, v_{m-1}, v_m-w
\]
are not $p$-primitive. This implies that the determinant of any $m\times m$ submatrix of the $m\times n$ coefficient matrices $C(\eta)$ of $v_1, \dots, v_{m-1}, v_m + \eta w$ is a multiple of $p$ for any $\eta\in\{1, -1\}$. Observe that, by multilinearity of the determinant, the determinant of any $m\times m$ submatrix of $C(\eta)$ is equal to
\begin{multline*}
\det(\text{the corresponding $m\times m$ submatrix of $C$})\\
+ \eta\cdot\det(\text{the corresponding $m\times m$ submatrix of $C'$})\text{,}
\end{multline*}
where $C$ is the $m\times n$ coefficient matrix of $v_1, \dots, v_m$, and $C'$ is that of $v_1, \dots, v_{m-1}, w$. Since $p$ is odd, if the determinants of any $m\times m$ submatrix of $C(1)$ and the corresponding $m\times m$ submatrix of $C(-1)$ are multiples of $p$ simultaneously, then so are the determinants of the corresponding $m\times m$ submatrices of $C$ and $C'$. This implies that $v_1, \dots, v_m$ is not $p$-primitive, which is a contradiction.
(3) We have to show that the greatest common divisor of the determinants of $m\times m$ submatrices of $m\times (n+1)$ coefficient matrix of $v_1, \dots, v_{m-1}, v_m + \eta y$ is $1$ for some $\eta\in\{1, -1\}$. Let $g_1$ be the greatest common divisor of the determinants of all $m\times m$ submatrices containing the $(n+1)$-th column, and $g_2$ be the greatest common divisor of the determinants of those not containing the column. Then it suffices to show that $(g_1, g_2) = 1$. Let
\[
w = y_1 e_1 + \cdots + y_n e_n\in M\text{.}
\]
Then $g_2$ is equal to the greatest common divisor of the determinants of all $m\times m$ submatrices of $m\times n$ coefficient matrix of $v_1, \dots, v_{m-1}, v_m + \eta w$. Note that $g_1 = |y_{n+1}|$. Hence it suffices to show that $v_1, \dots, v_{m-1}, v_m + \eta w$ is $q$-primitive for any $q\in \mathcal{P}(y_{n+1})$. If $\mathcal{P}(y_{n+1})\setminus\mathcal{P}(y) = \varnothing$, it follows directly from (1). If $\mathcal{P}(y_{n+1})\setminus\mathcal{P}(y) = \{p\}$, it follows from (1) that both $v_1, \dots, v_{k-1}, v_k + w$ and $v_1, \dots, v_{k-1}, v_k - w$ are $q$-primitive for any $q\in\mathcal{P}(y_{n+1})$ such that $q\ne p$ (equivalently, for any $q\in \mathcal{P}(y)$), and it follows from (2) that at least one of the two is $p$-primitive.
Now, assume that $L\cong$ J$_3$. Note that the quaternary sublattice $\langle 1, 2\rangle\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ of $L$ with a perfect square discriminant has class number one. Below we provide a complete proof of the primitive $2$-universality of J$_3$, which is one of the most complicated cases among all candidates.
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice such that $a\equiv 1\Mod2$ and $c\equiv 0\Mod{16}$. If $\left(\begin{smallmatrix}a&b\\b&c-6\end{smallmatrix}\right)$ is primitively represented by the $\z$-lattice $\ang{1, 2}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$, then $\ell$ is primitively represented by the $\z$-lattice $\ang{1}\mathbin{\perp}\mathbb{A}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$.
The $\z$-lattice $L\cong$ $\cong I_2\mathbin{\perp}\mathbb{A}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ is primitively $2$-universal.
(1) We fix bases for $\z$-lattices $\ang{1, 2}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ and $\ang{1}\mathbin{\perp}\mathbb{A}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ corresponding to the given Gram matrices. With respect to such bases, if there exists a primitive sublattice $\z\left[\begin{smallmatrix}c_1 & c_2 & c_3 & c_4\\d_1 & d_2 & d_3 & d_4\end{smallmatrix}\right]\cong \left(\begin{smallmatrix}a&b\\b&c-6\end{smallmatrix}\right)$ of $\ang{1, 2}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ for $c_i$, $d_i\in\z$, then clearly, the sublattice
\[
\z\begin{bmatrix}c_1 & c_2 & 0 & c_3 & c_4\\d_1 & d_2-1 & 2 & d_3 & d_4\end{bmatrix}
\]
of $\ang{1}\mathbin{\perp}\mathbb{A}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ is isometric to $\ell$. Since the greatest common divisor of the determinants of all $2\times 2$ submatrices containing the third column is $2$, it suffices to show that $\left(\begin{smallmatrix}c_1 & c_3 & c_4\\d_1 & d_3 & d_4\end{smallmatrix}\right)$ is $2$-primitive. Since $a$ is odd, exactly one of $c_1$, $c_3$, $c_4$ is odd or all of them are odd. Furthermore, since
\[
d_1^2 + 2d_2^2 + 3d_3^2 + 2d_3 d_4 + 3d_4^2 \equiv 10\Mod{16}\text{,}
\]
$d_1$ is even and $d_3$, $d_4$ are odd. Therefore, $\left(\begin{smallmatrix}c_1 & c_3 & c_4\\d_1 & d_3 & d_4\end{smallmatrix}\right)$ is $2$-primitive.
(2) Denote by $M$ the $5$-section of $L$. Let $M' = \z[e_1, e_3, e_4, e_5, e_6]$ and $N = \z[e_1, e_3, e_5, e_6]$ be primitive sublattices of $L$. Note that
\[
M' \cong \ang{1}\mathbin{\perp}\begin{pmatrix}2&1\\1&2\end{pmatrix}\mathbin{\perp}\begin{pmatrix}3&1\\1&3\end{pmatrix}\quad \text{and} \quad N \cong \ang{1,2}\mathbin{\perp}\begin{pmatrix}3&1\\1&3\end{pmatrix}\text{.}
\]
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0\le 2b\le a\le c$) be a binary $\z$-lattice which is primitively represented by neither $M$ nor $N$. Then $\ell$ satisfies either
(i) $a$ or $c\equiv 1\Mod8$ and $d\ell\equiv 0\Mod{16}$, or
(ii) $a\equiv c\equiv 0\Mod4$ and $b$ is even.
If $c\le 32$, then one may directly check that $\ell$ is primitively represented by $L$. Hence, we may assume that $c\ge 33$. Also, we assume that $a\ne 4$ for the moment. Consider the $\z$-lattices
\begin{align*}
\ell' = \ell'_u(s,t;s') & \cong \begin{pmatrix}a + 2ub + u^2c - s^2 - 6s'^2 & b + uc - st\\b + uc - st & c - t^2\end{pmatrix}\text{,}\\
\ell'' = \ell''_u(s,t;t') & \cong \begin{pmatrix}a - s^2 & ua + b - st\\ua + b - st & u^2a + 2ub + c - t^2 - 6t'^2\end{pmatrix}\text{,}
\end{align*}
where $u$, $s$, $t$, $s'$ and $t'$ are integers. Observe that the orthogonal complement of $N$ in $M'$ is $N^\perp = \z[- e_3 + 2e_4]\cong\ang{6}$. Hence, by Lemma <ref>, if $\ell'_u(s, t; s')$ ($\ell''_u(s, t; t')$) is primitively represented by $N$ for some $s'$ ($t'$, respectively) even, then $\ell$ is primitively represented by $L$. Moreover, by (1) given above, if all of the following three conditions hold, then $\ell$ is primitively represented by $L$:
(a) $\ell'_u(s, t; 1)$ ($\ell''_u(s, t; 1)$) is primitively represented by $N$;
(b) $a + 2ub + u^2c - s^2$ ($u^2a + 2ub + c - t^2$) $\equiv 0\Mod{16}$;
(c) $c - t^2$ ($a - s^2$, respectively) is odd.
Assume that case (i) holds. First, suppose that $a$ is odd. We consider the $\z$-lattice
\[
\ell''_u(0,0;1)\cong\bigl(\begin{smallmatrix}a & ua + b\\ua + b & u^2a + 2ub + c - 6\end{smallmatrix}\bigr)\text{.}
\]
Since $d\ell''_u(0,0;1)\equiv 2\Mod4$, $\ell''_u(0,0;1)_2$ is primitively represented by $N_2\cong \ang{1,2}\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$. Hence, for any integer $u$, $\ell''_u(0,0;1)$ is primitively represented by $N$. Since $a$ is odd, there is a $u\in\{-2, -1, 0, 1\}$ such that $ua + b\equiv 0\Mod4$, which implies that $u^2a + 2ub + c\equiv 0\Mod{16}$. Hence, we are done by (1). Now, suppose that $a$ is even. Then $a\equiv 0$, $4\Mod{16}$ and $c$ is odd. Hence, similarly to the above, if we take an integer $u\in\{-2, -1, 0, 1\}$ such that $u^2a + 2ub + c\equiv 0\Mod{16}$, then $\ell'_u(0,0;1)$ is primitively represented by $N$.
Now, assume that case (ii) holds. First, suppose that either $a\equiv 0$, $4\Mod{16}$ or $c\equiv 0$, $4\Mod{16}$. If we define a $\z$-lattice $\ell'''$ by
\[
\ell''' = \begin{cases}
\ell''_0(1,0;1) & \text{if }a\equiv 0\Mod{16}\text{,}\\
\ell''_0(1,2;1) & \text{if }a\equiv 4\Mod{16}\text{,}\\
\ell'_0(0,1;1) & \text{if }c\equiv 0\Mod{16}\text{,}\\
\ell'_0(2,1;1) & \text{if }c\equiv 4\Mod{16}\text{,}
\end{cases}
\]
then $d\ell'''\equiv 2\Mod4$. Hence, $\ell'''$ is primitively represented by $N$. Therefore, by (1), $\ell$ is primitively represented by $L$ in each case. Now, suppose that $a$, $c\equiv 8$ or $12\Mod{16}$. First, assume that $b\equiv 2\Mod4$. One may easily show that there is an $\eta\in\{1, -1\}$ such that
\[
a + 2\eta b + c \equiv 0\text{ or }4\Mod{16}\text{.}
\]
Hence, one of $\ell''_\eta(1,0;1)$ or $\ell''_\eta(1,2;1)$ is primitively represented by $N$. Therefore, by (1), $\ell$ is primitively represented by $L$. Now, assume that $b\equiv 0\Mod4$. If we define a $\z$-lattice $\ell^{(4)}$ by
\[
\ell^{(4)} = \begin{cases}
\ell''_0(0,1;0) & \text{if }a\equiv 8\Mod{16}\text{,}\\
\ell'_0(1,0;0) & \text{if }c\equiv 8\Mod{16}\text{,}\\
\ell''_0(0,4;0) & \text{if }a\equiv c\equiv 12\Mod{16}\text{ and }b\equiv 0\Mod8\text{,}\\
\ell''_0(0,0;2) & \text{if }a\equiv c\equiv 12\Mod{16}\text{ and }b\equiv 4\Mod8\text{,}
\end{cases}
\]
then one may easily show that
\[
d\ell^{(4)}\equiv 8\Mod{16}\text{,}\quad \ell^{(4)}_2\cong\langle 12, -4\rangle\text{,}\quad \text{or}\quad d\ell^{(4)}\cong 32\Mod{64}\text{.}
\]
Hence, $\ell^{(4)}$ is primitively represented by $N$, which implies that $\ell$ is primitively represented by $L$ in each case.
Finally, we consider the remaining case when $a = 4$. Note that $b = 0$ or $b = 2$. It is well known that the $4$-section $N'\cong I_2\mathbin{\perp}\mathbb{A}$ of $L$ is primitively $1$-universal (see [2]). If we choose a primitive vector $v$ in $N'$ such that $Q(v) = c$, then clearly, $\z[e_5 - e_6, v]$ is a primitive sublattice of $L$ which is isometric to $\ang{4, c}$. If we choose a vector $w$ in $N'$ such that $Q(w) = c - 3$, then clearly,
\[
\z[e_5 - e_6, w + e_5]
\]
is a primitive sublattice of $L$ which is isometric to $\left(\begin{smallmatrix}4&2\\2&c\end{smallmatrix}\right)$.
Next, we consider the case when $L\cong$ D$^{\mbox{\scriptsize iii}}_5$ or I$^{\mbox{\scriptsize ii}}_5$. The quaternary primitive $\z$-sublattice $N$ of $L$ given in Table <ref> has class number two and its genus mate $N'$ is also primitively represented by $L$. Hence, any binary $\z$-lattice that is represented by the genus of $N$ is primitively represented by $L$. Note that $dN = 16$ and the sublattice $\z[e_3, e_4, e_5, e_6]$ of $L$ is isometric to $N$ for both cases.
The $\z$-lattice $N$ and its genus mate $N'$
$L$ $N$ $N'$
D$^{\mbox{\scriptsize iii}}_5$ $\langle 1\rangle \mathbin{\perp}\left(\begin{smallmatrix}2&0&1\\0&2&1\\1&1&5\end{smallmatrix}\right)$[-9pt]0pt24pt $I_3\mathbin{\perp}\langle 16\rangle$
I$^{\mbox{\scriptsize ii}}_5$ $\left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&2&0\\1&1&0&5\end{smallmatrix}\right)$[-12pt]0pt30pt $I_2\mathbin{\perp}\left(\begin{smallmatrix}4&2\\2&5\end{smallmatrix}\right)$
Below we provide a complete proof of the primitive $2$-universality of D$^{\mbox{\scriptsize iii}}_5$. The proof of the primitive $2$-universality of I$^{\mbox{\scriptsize ii}}_5$ is quite similar to this. Recall that $\alpha$, $\beta$ denote integers in $\z_p$, and $\epsilon$, $\delta$ denote units in $\z_p$, unless stated otherwise, where the prime $p$ could be easily verified from the context.
If a quaternary $\z_2$-lattice $I_3\mathbin{\perp}\ang{16}$ does not primitively represent a binary $\z_2$-lattice $\ell_2$, then $\ell_2$ satisfies one of the following conditions:
$\ell_2\cong\ang{-1, \alpha}$, $\ang{1, 12}$, $\ang{1, 8}$, $\ang{1, 40}$, $\ang{5, 24}$, $\ang{5, -8}$, $\ang{1, 64\alpha}$, $\ang{3, 32\alpha}$, $\ang{5, 64\alpha}$, $\ang{2, 10}$, $\ang{6, 6}$, $\ang{2\epsilon, 12}$, $\ang{2, 8}$, $\ang{-2, 24}$, $\ang{2\epsilon, 32\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{2\epsilon, 128\alpha}$, $\fn (\ell_2)\subseteq 4\z_2$, or $\q_2 \ell_2\cong \q_2 \mathbb{H}$.
Let $\ell \cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice. Suppose that, for some integers $s$, $t$, and $t'$,
\[
\ell'\cong \begin{pmatrix}a-2s^2 & b-2st\\b-2st & c-2t^2 - 8t'^2\end{pmatrix}
\]
is positive definite and $\ell'_2$ is primitively represented by the quaternary $\z_2$-lattice $I_3\mathbin{\perp}\ang{16}$. Then $\ell$ is primitively represented by the $\z$-lattice $L\cong$ .
The $\z$-lattice $L\cong$ $\cong I_3\mathbin{\perp}\left(\begin{smallmatrix}2&0&1\\0&2&1\\1&1&5\end{smallmatrix}\right)$ is primitively $2$-universal.
(1) One may easily verify the assertion by a direct computation.
(2) Consider two primitive sublattices of $L$,
\[
N = \z[e_3, e_4, e_5, e_6]\quad \text{and}\quad N' = \z[e_1, e_2, e_3, e_4 + e_5 - 2e_6]\cong I_3\mathbin{\perp}\ang{16}\text{,}
\]
where the ordered basis $\{e_i\}_{i=1}^6$ for $L$ corresponds to the Gram matrix given in the statement. Moreover, we fix an ordered basis for $N'$ corresponding to the Gram matrix given above. Note that the class number of $N$ is two and $N'$ is the other class is the genus of $N$. Hence, if $\ell'$ satisfies all conditions given above, then $\ell'$ is primitively represented by either $N$ or $N'$.
If $\ell'$ is primitively represented by $N$, then one may directly check that $\ell$ is primitively represented by $L$. Now, suppose that the primitive sublattice
\[
\z\begin{bmatrix}c_1 & c_2 & c_3 & c_4\\d_1 & d_2 & d_3 & d_4\end{bmatrix}
\]
of $N'$ is isometric to $\ell'$. Then clearly, the sublattice
\[
\z\begin{bmatrix}c_1 & c_2 & c_3 & c_4 + s & c_4 & -2c_4\\d_1 & d_2 & d_3 & d_4 + t & d_4 + 2t' & -2d_4\end{bmatrix}
\]
of $L$ is isometric to $\ell$. To see that such a sublattice is primitive, note that the greatest common divisor of all $2\times 2$ submatrices of the above coefficient matrix divides $(g_1, g_2)$, where $g_1$ and $g_2$ are the greatest common divisor of all $2\times 2$ submatrices of
\[
\begin{pmatrix}c_1 & c_2 & c_3 & -2c_4\\d_1 & d_2 & d_3 & -2d_4\end{pmatrix}\quad \text{and}\quad \begin{pmatrix}c_1 & c_2 & c_3 & c_4\\d_1 & d_2 & d_3 & d_4 + 2t'\end{pmatrix}\text{,}
\]
respectively. Since $\left(\begin{smallmatrix}c_1 & c_2 & c_3 & c_4\\d_1 & d_2 & d_3 & d_4\end{smallmatrix}\right)$ is primitive, $g_1$ divides $2$ and $g_2$ is odd. Hence, $(g_1, g_2) = 1$.
(3) Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0\le 2b\le a\le c$) be a $\z$-lattice which is primitively represented by neither the genus of $N$ nor the $5$-section of $L$. Then by Lemma <ref> and by (1) given above, we may assume that $\ell$ satisfies one of the following conditions:
(i) $\ell_2\cong \langle 1, -16\rangle$;
(ii) $\ell_2\cong \langle 1, 64\alpha\rangle$ for some $\alpha\in\z_2$;
(iii) $\ell_2\cong\langle 4, 16\alpha\rangle$ or $\langle 20, 16\alpha\rangle$ for some $\alpha\in\z_2$;
(iv) $\fn(\ell_2)\subseteq 16\z_2$.
If $c\le 21$, then one may directly check that $\ell$ is primitively represented by $L$. Hence, we may assume that $c\ge 22$. If either
\[
\ell'(s, t; s') \cong \left(\begin{smallmatrix}a-2s^2 - 8s'^2 & b-2st\\b-2st & c-2t^2\end{smallmatrix}\right)\quad\text{or}\quad \ell''(s, t; t') \cong \left(\begin{smallmatrix}a-2s^2 & b-2st\\b-2st & c-2t^2 - 8t'^2\end{smallmatrix}\right)
\]
satisfies all conditions given in (2), then $\ell$ is primitively represented by $L$.
Assume that case (i) holds. First, suppose that $a$ is odd. Note that $a\equiv 1\Mod8$. Since $d\ell''(0,2;1)\equiv 32\Mod{64}$, $\ell''(0,2;1)_2 \cong \langle 1, 32\epsilon \rangle$ is primitively represented by $N_2$. Since $\ell''(0,2;1)$ is positive definite, $\ell$ is primitively represented by $L$. Now, suppose that $a$ is even. Note that $c\equiv 1\Mod8$, and we have
\[
a\equiv 20\Mod{32}\text{,} \quad a\equiv -16\Mod{64}\text{,} \quad \text{or} \quad a\equiv 0\Mod{128}\text{.}
\]
Since $\ell'(2,0;1)$ is positive definite, $\ell$ is primitively represented by $L$, by the similar reasoning.
Next, assume that case (ii) holds. Suppose that $a$ is odd. Note that $a\equiv 1\Mod8$. Since $d\ell''(0,0;1)\equiv -8\Mod{64}$, $\ell''(0,0;1)_2 \cong \langle 1, -8 \rangle$ is primitively represented by $N_2$. Furthermore, since $\ell''(0,0;1)$ is positive definite, $\ell$ is primitively represented by $L$. Now, suppose that $a$ is even. Note that $c\equiv 1\Mod8$, and we have
\[
a\equiv 4\Mod{32}\text{,} \quad a\equiv 16\Mod{64}\text{,} \quad \text{or} \quad a\equiv 0\Mod{64}\text{.}
\]
If $a\ge 16$, then $\ell'(0,0;1)$ is positive definite. Hence, $\ell$ is primitively represented by $L$ in this case. If $a = 4$, then $\ell''(0,0;1)_2 \cong \langle 1, 32\epsilon \rangle$ and $\ell''(0,0;1)$ is positive definite. Hence, $\ell$ is primitively represented by $L$.
Now, assume that case (iii) holds. First, suppose that $a\equiv c\equiv 4\Mod{16}$. Note that $b\equiv 4\Mod8$. One may easily show that there is an $\eta\in\{1, -1\}$ such that $d\ell''(1, \eta; 0)\equiv 32\Mod{64}$. Then, $\ell''(1, \eta; 0)_2 \cong \langle 2, 16\epsilon \rangle$ is primitively represented by $N_2$. Since $\ell''(1, \eta; 0)$ is positive definite, $\ell$ is primitively represented by $L$. Next, suppose that $a\equiv 4\Mod{16}$, $b\equiv 0\Mod8$, and $c\equiv 0\Mod{16}$. In this case, either $\ell''(1,0;0)$ or $\ell''(1,2;0)$ is isometric to $\langle 2, 16\epsilon \rangle$ over $\z_2$. Furthermore, since it is positive definite, $\ell$ is primitively represented by $L$. Similarly to the above, if $a\equiv 0\Mod{16}$, $b\equiv 0\Mod8$, and $c\equiv 4\Mod{16}$, then $\ell$ is primitively represented by $L$.
Finally, assume that case (iv) holds. If we define a $\z$-lattice $\ell'''$ by
\[
\ell''' = \begin{cases}
\ell''(0,1;0) & \text{if }a\equiv 16\Mod{32}\text{,}\\
\ell'(0,1;0) & \text{if }c\equiv 16\Mod{32}\text{,}\\
\ell'(1,1;0) & \text{if }a\equiv c\equiv 0\Mod{32}\text{ and }b\equiv 8\Mod{16}\text{,}\\
\ell''(1,0;1) & \text{if }a\equiv 0\Mod{32}\text{, }b\equiv 0\Mod{16}\text{, and }c\equiv 0\Mod{64}\text{,}\\
\ell'(0,1;1) & \text{if }a\equiv 0\Mod{64}\text{, }b\equiv 0\Mod{16}\text{, and }c\equiv 0\Mod{32}\text{,}\\
\ell'(1,1;1) & \text{if }a\equiv c\equiv 16\Mod{32}\text{ and }b\equiv 0\Mod{32}\text{,}\\
\ell'(1,0;0) & \text{if }a\equiv 32\Mod{64}\text{, }b\equiv 16\Mod{32}\text{, }c\equiv -32\Mod{128}\text{,}\\
\ell'(0,1;0) & \text{if }a\equiv -32\Mod{128}\text{, }b\equiv 16\Mod{32}\text{, }c\equiv 32\Mod{64}\text{,}\\
\ell'(1,1;0) & \text{if }a\equiv c\equiv 32\Mod{128}\text{ and }b\equiv -16\Mod{64}\text{,}\\
\ell'(1,3;0) & \text{if }a\equiv c\equiv 32\Mod{128}\text{ and }b\equiv 16\Mod{64}\text{,}
\end{cases}
\]
then one may easily show that
\[
d\ell'''\equiv 16\Mod{128}\text{,}\quad d\ell'''\equiv 32\Mod{64}\text{,}\quad \text{or}\quad d\ell'''\equiv 64\Mod{256}\text{.}
\]
Hence, $\ell$ is primitively represented by $L$ in each case.
Finally, assume that $L\cong$ I$^{\mbox{\scriptsize ii}}_4$ or I$^{\mbox{\scriptsize iii}}_4$. Note that the quaternary orthogonal summand $\z[e_3, e_4, e_5, e_6]$ of $L$ has a nonsquare discriminant. Hence, such a quaternary $\z$-lattice is not primitively $2$-universal over $\z_p$ for infinitely many primes $p$.
The $\z$-lattice $N\cong \left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&2&0\\1&1&0&4\end{smallmatrix}\right)$ primitively represents any binary $\z$-lattice $\ell'$ satisfying all of the following three conditions:
(a) $\fn (\ell'_2)\subseteq 2\z_2$ and $\ell'_2$ represents some element in $\{6, -2, 4, 20\}$;
(b) $\ell'_3$ represents $\Delta_3$ or $3$;
(c) $\ell'_p$ represents a unit in $\z_p$ for any odd prime $p$ with $\bigl(\frac3p\bigr) = -1$, where $\bigl(\frac{\cdot}{\cdot}\bigr)$ is the Legendre symbol.
If the quinary $\z$-lattice $M\cong \ang{2}\mathbin{\perp} N$ does not primitively represent a positive definite binary $\z$-lattice $\ell$, then $\ell$ satisfies $\fn (\ell_2) = \z_2$, $\ell_3\cong \ang{3\cdot\Delta_3, 9\alpha}$ for some $\alpha\in\z_3$, or $\fs (\ell_3)\subseteq 9\z_3$.
The $\z$-lattice $\cong I_2\mathbin{\perp} N$ is primitively $2$-universal.
(1) Note that $N$ has class number one and $dN = 12$. Hence, any binary $\z$-lattice $\ell'$ is primitively represented by $N$ if and only if $\ell'_p$ is primitively represented by $N_p$ for any prime $p$. Since $N_2\cong \mathbb{A}\mathbin{\perp}\ang{-2, -2}\cong \mathbb{H}\mathbin{\perp}\ang{6, -2}$, $\ell'_2$ is primitively represented by $N_2$ if $\fn (\ell'_2)\subseteq(2)$ and $\ell'_2$ represents some element in $\{6, -2, 4, 20\}\subset\z_2$. Since $N_3\cong \mathbb{H}\mathbin{\perp}\ang{\Delta_3, 3}$, $\ell'_3$ is primitively represented by $N_3$ if $\ell'_3$ represents $\Delta_3$ or $3$. Now, suppose $p\ne 2$, $3$. If $\bigl(\frac3p\bigr) = 1$ then $N_p\cong \mathbb{H}\mathbin{\perp} \mathbb{H}$, and hence $N_p$ is primitively $2$-universal. If $\bigl(\frac3p\bigr) = -1$, that is, $dN_p = \Delta_p$, then $N_p\cong \mathbb{H}\mathbin{\perp}\ang{1, -\Delta_p}$. Hence, $\ell'_p$ is primitively represented by $N_p$ if it represents a unit in $\z_p$. Note that for any odd prime $p$, $\bigl(\frac3p\bigr) = -1$ if and only if $p\equiv 5$, $7\Mod{12}$.
(2) Note that $M$ has class number one, $dM = 24$, and $3$ is the only core prime of $M$. Hence, any binary lattice $\ell$ is primitively represented by $M$ if and only if $\ell_p$ is primitively represented by $M_p$ for $p=2$, $3$. Note that $M_2\cong \mathbb{H}\mathbin{\perp} \mathbb{H}^2\mathbin{\perp} \ang{6}$ primitively represents any binary lattice $\ell_2$ satisfying $\fn (\ell_2)\subseteq 2\z_2$, and $M_3\cong \mathbb{H}\mathbin{\perp} \ang{1,1,3}$ primitively represents all binary lattices representing $1$, $\Delta_3$, or $3$.
(3) Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ ($0\le 2b\le a\le c$) be a binary $\z$-lattice which is primitively represented by neither the $5$-section of $L$ nor the primitive sublattice
\[
M = \z[e_1+e_2, e_3, e_4, e_5, e_6]\cong \ang{2}\mathbin{\perp} N
\]
of $L$. Then by Lemma <ref> and by (1) given above, we may assume that $\ell$ satisfies one of the following conditions:
(i) $\ell_2\cong \langle 1, 32\alpha\rangle$ for some $\alpha\in\z_2$;
(ii) $\ell_2\cong \langle 5, 16\epsilon\rangle$ for some $\epsilon\in\z_2^\times$;
(iii) $a\equiv b\equiv c\equiv 0\Mod{12}$.
First, assume that case (iii) holds. Observe that $M^\perp = \z[e_1 - e_2] \cong \ang{2}$ and $e_1 - e_2 = - (e_1 + e_2) + 2e_1$. Hence, if $\ell'\cong \left(\begin{smallmatrix}a&b\\b&c-2\cdot 2^2\end{smallmatrix}\right)$ is primitively represented by $M$, then $\ell$ is primitively represented by $L$ by Lemma <ref>. In fact, $\ell'$ is primitively represented by $M$ for $\fs (\ell'_2)\subseteq 4\z_2$ and $\fs (\ell'_3) = \z_3$.
Denote by $O$ the $5$-section of $L$. Then $O^\perp = \z(- e_3 - e_4 + e_5 + 2e_6)\cong \ang{12}$. Hence, if
\[
\ell''\cong \begin{pmatrix}a-12\cdot 2^2&b\\b&c\end{pmatrix}\quad \text{or}\quad \ell'''\cong \begin{pmatrix}a&b\\b&c-12\cdot 2^2\end{pmatrix}
\]
is primitively represented by $O$, then $\ell$ is primitively represented by $L$ by Lemma <ref>. Moreover, $\ell''$ ($\ell'''$) is primitively represented by $O$ if and only if $\ell''$ ($\ell'''$) is positive definite and $\ell''_2$ ($\ell'''_2$, respectively) is primitively represented by $O_2$.
Now, assume that case (i) holds. If $c\le 64$, then one may directly check that $\ell$ is primitively represented by $L$. Now, we assume that $c\ge 65$. First, suppose that $a$ is odd. Since $d\ell'''\equiv 16\Mod{32}$, $\ell'''_2$ is primitively represented by $O_2$. Furthermore, since $\ell'''$ is positive definite, it is primitively represented by $O$. Now, suppose that $a$ is even. Since $c$ is odd, similarly to the above, $\ell''$ is primitively represented by $O$ if $a\ge 64$ so that $\ell''$ is positive definite. Hence, we may assume that $a < 64$. Note that $c\equiv 1\Mod8$ and one of the following conditions holds:
(I) $a\equiv 4\Mod{32}$ and $b\equiv 2\Mod4$;
(II) $a\equiv 16\Mod{32}$ and $b\equiv 4\Mod8$;
(III) $a\equiv 0\Mod{32}$ and $b\equiv 0\Mod8$.
We define
\[
\ell^{(4)} \cong \begin{cases}
\left(\begin{smallmatrix}a&b\\b&c-1-4\end{smallmatrix}\right) & \text{if }a=48\text{,}\\
\left(\begin{smallmatrix}a-4&b\\b&c-1\end{smallmatrix}\right) & \text{if }a=36\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-9-4\end{smallmatrix}\right) & \text{if }a=32\text{ and }c\equiv 1\Mod{16}\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-1-4\end{smallmatrix}\right) & \text{if }a=32\text{ and }c\equiv 9\Mod{16}\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-9-4\end{smallmatrix}\right) & \text{if }a=16\text{ and }c\equiv 0\Mod3\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-1-4\end{smallmatrix}\right) & \text{if }a=16\text{ and }c\not\equiv 0\Mod3\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-1\end{smallmatrix}\right) & \text{if }a=4\text{ and }c\equiv 0,1\Mod3\text{,}\\
\left(\begin{smallmatrix}a&b\\b&c-9\end{smallmatrix}\right) & \text{if }a=4\text{ and }c\equiv 2\Mod3\text{.}
\end{cases}
\]
Then by (1), $\ell^{(4)}$ is primitively represented by $N$. Hence, $\ell$ is primitively represented by $L$ in each case. The proof of case (ii) is quite similar to this.
The $\z$-lattice $N\cong \left(\begin{smallmatrix}2&1&1&1\\1&2&1&1\\1&1&2&1\\1&1&1&4\end{smallmatrix}\right)$ primitively represents a positive definite binary $\z$-lattice $\ell'$ if all of the following three conditions hold:
(a) $\fn\ell'_2\subseteq 2\z_2$ and $\ell'_2$ represents twice an odd integer;
(b) $\ell'_{13}$ represents $1$ or $13$;
(c) $\ell'_p$ represents a unit in $\z_p$ for any odd prime $p$ with $\bigl(\frac{13}{p}\bigr) = \bigl(\frac{p}{13}\bigr) = -1$, where $\bigl(\frac{\cdot}{\cdot}\bigr)$ is the Legendre symbol.
The lattice $\cong I_2\mathbin{\perp} N$ is primitively $2$-universal.
Since the proof is quite similar to that of the above theorem, it is left to the readers.
§.§ Case 6: The remaining cases
In this subsection, we prove the primitive $2$-universalities of the remaining lattices
\[
\text{D$^{\mbox{\scriptsize ii}}_k$ ($4\le k\le 8$), D$^{\mbox{\scriptsize iii}}_6$, D$^{\mbox{\scriptsize iii}}_7$, H$^{\mbox{\scriptsize iii}}_6$, and I$^{\mbox{\scriptsize iii}}_2$.}
\]
Note that the $5$-section of $L$ does not split $L$ orthogonally in all cases.
Assume that $L\cong$ D$^{\mbox{\scriptsize ii}}_k$. Note that one may prove the primitive $2$-universality of $L$ in the similar manner to Theorem <ref> by using Lemma <ref> instead of Lemma <ref>.
Next, assume that $L\cong$ D$^{\mbox{\scriptsize iii}}_k$ for $k = 6$ or $7$.
Let $\ell \cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a $\z$-lattice and let $m$, $k\ge 2$ be positive integers. Suppose that there exists a primitive sublattice
\[
\z\begin{bmatrix}c_1 & \cdots & c_m & c_{m+1} & c_{m+2}\\d_1 & \cdots & d_m & d_{m+1} & d_{m+2}\end{bmatrix} \cong \begin{pmatrix}a-(k-1)s^2 & b-(k-1)st\\b-(k-1)st & c-(k-1)t^2\end{pmatrix}
\]
of the $\z$-lattice $I_{m+2}$ for some integers $c_i$, $d_i$, $s$, and $t$. If
\[
c_{m+1} + c_{m+2} + s \equiv d_{m+1} + d_{m+2} + t \equiv 0\Mod2\text{,}
\]
then $\ell$ is primitively represented by $I_m\mathbin{\perp}\left(\begin{smallmatrix}2&0&1\\0&2&1\\1&1&k\end{smallmatrix}\right)$.
Since the proof is quite similar to that of Lemma <ref>, it is left to the readers.
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice. Suppose that there exists a primitive sublattice $\z\left[\begin{smallmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{smallmatrix}\right]$ of the $\z$-lattice $I_5$, which is isometric to $\ell$, for some integers $c_i$ and $d_i$ such that the set
\[
\{(\overline{\vphantom{d}c_i + c_j}, \overline{d_i + d_j})\mid 1\le i < j\le 5\}
\]
is a proper subset of $(\z/2\z)^2$, where $\overline{n}$ is the residue class of $n$ modulo $2$ for any integer $n$. Then $\ell$ satisfies one of the followings:
$a$ or $c\equiv 1\Mod4$ and $d\ell\equiv 0\Mod4$;
$a\not\equiv 1\Mod8$, $c\not\equiv 1\Mod8$, and $d\ell\equiv 2\Mod4$.
Consider the set $C = \{(\overline{\vphantom{d}c_i}, \overline{d_i})\mid 1\le i \le 5\}$. Since $\z\left[\begin{smallmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{smallmatrix}\right]$ is a primitive sublattice of $I_5$, the set $C$ contains at least two nonzero vectors in $(\z/2\z)^2$. Furthermore, one may easily show from the assumption that $C$ is one of
\[
\{(\overline{1}, \overline{0}), (\overline{0}, \overline{1})\}\text{,}\quad \{(\overline{1}, \overline{0}), (\overline{1}, \overline{1})\}\text{,}\quad \{(\overline{0}, \overline{1}), (\overline{1}, \overline{1})\}\text{,}
\]
which corresponds to each of the followings, respectively:
$a+c\equiv 1\Mod4$ and $b$ is even;
$a\equiv 5\Mod8$ and $b\equiv c\Mod2$;
$a\equiv b\Mod2$ and $c\equiv 5\Mod8$.
The lemma follows directly from this.
The $\z$-lattice $\cong I_3\mathbin{\perp}\left(\begin{smallmatrix}2&0&1\\0&2&1\\1&1&k\end{smallmatrix}\right)$ for $k = 6$ or $7$ is primitively $2$-universal.
Let $\ell\cong \left(\begin{smallmatrix}a&b\\b&c\end{smallmatrix}\right)$ be a binary $\z$-lattice such that $0\le 2b\le a\le c$. Note that the $5$-section $M \cong I_3 \mathbin{\perp} \langle 2, 2 \rangle$ in this case has class number one. Hence, we may assume that $\ell$ is not primitively represented by $M$ locally, that is, one of the following conditions holds:
(i) $\ell_2\cong\langle 1, 16\alpha\rangle$ for some $\alpha\in\z_2$;
(ii) $\ell_2\cong\langle 4, 16\alpha\rangle$ or $\langle 20, 16\alpha\rangle$ for some $\alpha\in\z_2$;
(iii) $\fn(\ell_2)\subseteq 16\z_2$.
Note that we have $a\equiv 1\Mod8$, $a\equiv 4\Mod{16}$, or $a\equiv 0\Mod{16}$. Assume that both of the $\z$-lattices
\[
\ell(1) \cong \begin{pmatrix}a-(k-1)&b\\b&c\end{pmatrix}\text{,}\qquad \ell(2) \cong \begin{pmatrix}a&b\\b&c-(k-1)\end{pmatrix}
\]
are positive definite. Then one may easily show that $\ell(s)$ is primitively represented by $I_5$ for some $s = 1$, $2$. Let $N = \z\left[\begin{smallmatrix}c_1 & c_2 & c_3 & c_4 & c_5\\d_1 & d_2 & d_3 & d_4 & d_5\end{smallmatrix}\right]$ be a primitive binary $\z$-sublattice of $I_5$ which is isometric to $\ell(s)$. If there is an $(i, j)$ with $1\le i<j\le 5$ such that
\[
c_i + c_j + (2-s) \equiv d_i + d_j + (s-1)\equiv 0\Mod2\text{,}
\]
then $\ell$ is primitively represented by by Lemma <ref>. If there does not exist such an $(i, j)$, then by Lemma <ref>, one of the followings must hold:
(a) $a-(2-s)(k-1)\equiv 1\Mod4$ or $c-(s-1)(k-1)\equiv 1\Mod4$, and $d\ell(s)\equiv 0\Mod4$;
(b) $a-(2-s)(k-1)\not\equiv 1\Mod8$, $c-(s-1)(k-1)\not\equiv 1\Mod8$, and $d\ell(s)\equiv 2\Mod4$.
However, one may easily verify that none of (a) and (b) holds in each case. For instance, consider case (i) when $k = 6$. If $a$ is odd, then $a\equiv 1\Mod8$. Since $d\ell(2)\equiv 3\Mod8$, $\ell(2)$ is primitively represented by $I_5$ so that we may take $s = 2$. Since $d\ell(2)$ is odd, $\ell(2)$ satisfies neither (a) nor (b). Now, suppose that $a$ is even. Then $a\equiv 0\Mod4$ and $c\equiv 1\Mod8$. Therefore, similarly to the above, $\ell(1)$ is primitively represented by $I_5$ and hence $\ell(1)$ does not satisfy any of (a) and (b).
Now, we have to consider the case when neither $\ell(1)$ nor $\ell(2)$ is positive definite. Note that if $a\ge 9$, then both $\ell(1)$ and $\ell(2)$ are positive definite. Hence, we may assume that $a = 1$ or $a = 4$. If $a = 1$, then $b = 0$ and $c\equiv 0\Mod8$ by (i). Since $\ell(2)$ is positive definite if $c\ge 9$, one may apply the same method as the above to prove the theorem. One may directly check that $\ell$ is primitively represented by $L$ if $c\le 8$. Now, assume that $a = 4$. Then we have either $b = 0$ or $b = 2$. If $b = 0$, then $c\equiv 0\Mod{16}$ by (ii). Hence, $\ell(2)$ is positive definite, and we may apply the same method to the above to prove the theorem. If $b = 2$, then $c\equiv 1\Mod8$ by (i). Note that $I_3$ represents $c-k\equiv 2, 3\Mod8$ by Legendre's three-square theorem. If we choose a vector $v$ in the $3$-section of $L$ such that $Q(v) = c-k$, then clearly, $\z[e_4 + e_5, v + e_6]$ is a primitive sublattice of $L$ isometric to $\left(\begin{smallmatrix}4&2\\2&c\end{smallmatrix}\right)$.
Now, assume $L\cong$ H$^{\mbox{\scriptsize iii}}_6$ or I$^{\mbox{\scriptsize iii}}_2$. We may apply Lemma <ref> by putting $M = \z[e_1 + e_2, e_3, e_4, e_5, e_6]$ and $n=5$. One may prove the primitive $2$-universality of $L$ for these cases in a similar manner to the proofs of Theorems <ref> or <ref> by using Lemmas <ref> and <ref>.
Finally, we provide some essential data needed for computations in this section.
For each given quaternary $\z_2$-lattice $N$, the binary $\z_2$-lattice $\ell$ that is not primitively represented by $N$ satisfies one of the conditions given in Table <ref>.
Since one may prove the lemma by direct computations, the proof is left to the readers.
The local structures
$N$ Binary $\z_2$-lattices that are not primitively represented by $N$
$I_4$ $\ell_2\cong\ang{\epsilon, 4\alpha}$, $\ang{2, 6}$, $\ang{2\epsilon, 8\alpha}$, $\fn (\ell_2)\subseteq 4\z_2$, or $\q_2 \ell\cong \q_2 \mathbb{H}$
$\ell_2\cong\ang{\epsilon, 4\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{\epsilon, 16\alpha}$, $\ang{2\epsilon, 8\alpha}$,
$\ang{4\epsilon, 4\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{4\epsilon, 16\alpha}$, $\mathbb{A}$, $\fn (\ell_2)\subseteq 8\z_2$,
or $\q_2 \ell\cong \q_2 \mathbb{H}$
$\ell_2$ is unimodular, $\ell_2\cong\ang{\epsilon, 16\alpha}$, $\ang{2\epsilon, 8\delta}$ with $\epsilon\delta\equiv 3\Mod8$,
$\ang{2\epsilon, 32\alpha}$, $\ang{4\epsilon, 16\alpha}$, $\ang{8\epsilon, 8\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{8\epsilon, 32\alpha}$,
$\mathbb{A}^2$, $\fn (\ell_2)\subseteq 16\z_2$, or $\q_2 \ell\cong \q_2 \mathbb{H}$
$\ell_2\cong\ang{1, 1}$, $\ang{3, -1}$, $\ang{1, 20}$, $\ang{-1, -4}$,
$\ang{\epsilon, 16\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{\epsilon, 64\alpha}$, $\ang{2, 2}$, $\ang{2, 6}$, $\ang{2, 10}$,
$\ang{2\epsilon, 16\alpha}$ with $\epsilon\equiv 1\Mod4$, $\ang{2\epsilon, 32\alpha}$ with $\epsilon\equiv -1\Mod4$,
$\ang{12, 12}$, $\ang{4, 20}$, $\ang{4\epsilon, 16\delta}$ with $\epsilon\delta\equiv 3\Mod8$, $\ang{4\epsilon, 64\alpha}$,
$\fn (\ell_2)\subseteq 8\z_2$, or $\q_2 \ell \cong \q_2 \mathbb{H}$
For the $5$-section $M$ and its core prime $q$ of each type given in Table <ref>, if a binary $\z$-lattice $\ell$ is not primitively represented by $M$, then $\ell$ satisfies one of the conditions given in the table.
Since one may prove the lemma by direct computations, the proof is left to the readers.
The core prime and the local structures
Type $M$ $q$ Local structures
C $I_4\mathbin{\perp}\ang{3}$ $2$ $\ell_2\cong\ang{3, 8\alpha}$ or $\fn (\ell_2)\subseteq 4\z_2$
F $I_3\mathbin{\perp}\ang{2, 3}$ $3$
$\ell_2\cong\ang{4\epsilon, 4\delta}$ or $\mathbb{A}$, or
$\ell_3\cong\ang{6, 9\alpha}$ or $\fs (\ell_3)\subseteq 9\z_3$
J $I_2\mathbin{\perp}\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)\mathbin{\perp}\ang{3}$ $2$ $\ell_2\cong\ang{1, 8\alpha}$ or $\fn (\ell_2)\subseteq 4\z_2$
- $I_3\mathbin{\perp}\left(\begin{smallmatrix}3&1\\1&3\end{smallmatrix}\right)$ $2$
$\ell_2\cong\ang{3, 7}$, $\ang{-1, 4}$, $\ang{2, 2}$, $\ang{2, 64\alpha}$,
$\ang{10, 32\epsilon}$, $\ang{8, 64\alpha}$, or $\fs (\ell_2)\subseteq 16\z_2$
[00]
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# Close encounters of black hole - star binaries with stellar-mass black holes
Taeho Ryu1,2, Ruggero Valli1, Rüdiger Pakmor1, Rosalba Perna3,4, Selma E. de
Mink1,5, Volker Springel1
1 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85748
Garching, Germany
2 Physics and Astronomy Department, Johns Hopkins University, Baltimore, MD
21218, USA
3 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY
11794-3800, USA
4 Center for Computational Astrophysics, Flatiron Institute, New York, NY
10010, USA
5 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science
Park 904, 1098XH Amsterdam, The Netherlands E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Dynamical interactions involving binaries play a crucial role in the evolution
of star clusters and galaxies. We continue our investigation of the
hydrodynamics of three-body encounters, focusing on binary black hole (BBH)
formation, stellar disruption, and electromagnetic (EM) emission in dynamical
interactions between a BH-star binary and a stellar-mass BH, using the moving-
mesh hydrodynamics code AREPO. This type of encounters can be divided into two
classes depending on whether the final outcome includes BBHs. This outcome is
primarily determined by which two objects meet at the first closest approach.
BBHs are more likely to form when the star and the incoming BH encounter first
with an impact parameter smaller than the binary’s semimajor axis. In this
case, the star is frequently disrupted. On the other hand, when the two BHs
encounter first, frequent consequences are an orbit perturbation of the
original binary or a binary member exchange. For the parameters chosen in this
study, BBH formation, accompanied by stellar disruption, happens in roughly 1
out of 4 encounters. The close correlation between BBH formation and stellar
disruption has possible implications for EM counterparts at the binary’s
merger. The BH that disrupts the star is promptly surrounded by an optically
and geometrically thick disk with accretion rates exceeding the Eddington
limit. If the debris disk cools fast enough to become long-lived, EM
counterparts can be produced at the time of the BBH merger.
###### keywords:
black hole physics – gravitation – stellar dynamics
††pubyear: 2022††pagerange: Close encounters of black hole - star binaries
with stellar-mass black holes–LABEL:lastpage
## 1 Introduction
Dynamical interactions between stars and the compact objects they leave behind
play an important role in dense environments, such as globular and nuclear
star clusters and disks of Active Galactic Nuclei (AGNs). On global, large
scales they can influence cluster thermodynamics (Hut et al., 1992), while on
local, small-scales close interactions can alter the original birth
composition of isolated stars and binaries.
Dynamical formation of binary black holes (BBHs) is one of the leading
pathways (e.g., Downing et al., 2010; Portegies Zwart & McMillan, 2000;
Samsing et al., 2014; Rodriguez et al., 2015; Antonini et al., 2016; Askar et
al., 2017; Banerjee, 2018; Perna et al., 2019; Fragione et al., 2019; Di Carlo
et al., 2019; Rodriguez et al., 2019; Arca Sedda et al., 2020; Mapelli et al.,
2021) to forming the binaries which have been observed via gravitational wave
(GW) emission at their merger by the LIGO and Virgo observatories (The LIGO
Scientific Collaboration et al., 2021). Subsequent dynamical encounters
between BBHs and tertiary BHs can further influence the orbital parameters of
the binaries, and hence their timescale to merger by GW radiation (e.g., Trani
et al., 2019; Samsing et al., 2020; Wang et al., 2021; Arca Sedda et al.,
2021).
Despite the relatively high fractions of stars and compact objects in
binaries, hydrodynamic simulations of close encounters involving binaries have
begun only recently. Lopez et al. (2019) and Ryu et al. (2022, Paper 1 in the
following) studied close encounters between BBHs and single stars. They found
that, in addition to altering the spin of the accreting BHs, tidal disruption
events (TDEs) can have a significant impact on the binary BH’s orbit, in ways
which can be quantitatively different than the case of pure scattering. The EM
signatures produced by these close encounters can also differ significantly
from those of TDEs by isolated BHs: depending on the geometry of the
encounter, the accretion rate can display periodic modulations with the
orbital period. Detections of such events can provide constraints on the
formation of BBH mergers (Samsing et al., 2019).
More recently, Ryu et al. (2023, Paper 2 in the following) performed the first
investigation of close encounters between binary stars and single BHs. Their
hydrodynamic simulations showed a variety of possible outcomes, from full
disruptions of both stars, to a full disruption of one star and a partial
disruption of the other, to dissociation into bound and unbound single stars.
Among these cases of dissociation, interesting outcomes include the formation
of a runaway star, and of a fast-moving BH that accretes the tidally disrupted
debris of the other star. In other outcomes, the binary stars are dissociated,
and one of the stars is exchanged with the intruding BH, resulting in the
formation of an X-ray binary.
Here we extend the line of investigation begun in Paper 1 and Paper 2 by
performing a suite of hydrodynamic simulations of nearly parabolic close
encounters between BH-star binaries and single BHs. Similarly to Paper 2, we
use the moving-mesh code AREPO (Springel, 2010; Pakmor et al., 2016;
Weinberger et al., 2020), whose quasi-Lagrangian approach to hydrodynamics is
well adapted to the problem. Our study aims to elucidate how such encounters
can lead to a variety of outcomes, including EM transients due to the
disruption of the star and the formation (via member exchange) of tight BBHs
surrounded by debris material, potentially leading to a situation in which the
BBH merger could be accompanied by an EM counterpart.
Our paper is organized as follows: § 2 presents the estimate for the rate of
this type of encounters in globular clusters. § 3 describes the details of the
numerical simulations and the initial conditions. We present our simulation
results in § 4, with particular emphasis on a classification of the outcomes
and its dependence on encounter parameters. We discuss these results in the
context of their possible EM counterparts in § 5, and we finally summarize our
work and conclude in § 6.
## 2 Encounter rate in globular clusters
We begin by estimating the encounter rate of three-body interactions between
BH-star binaries and single stars in globular clusters. Following Paper 2, we
first calculate the differential rate of a single BH encountering a BH-star
binary as ${\rm d}\mathcal{R}/{\rm d}N_{\bullet}\simeq n\Sigma v_{\rm rel}$.
Here, $n$ is the binary number density in the vicinity of the BH, $n\simeq
f_{\rm b}n_{\rm s}$, where $f_{\rm b}$ is the non-interacting star - BH binary
fraction, $f_{\rm b}\simeq 10^{-4}-10^{-5}$ (Morscher et al., 2015; Kremer et
al., 2018), and $n_{\rm s}$ is the number density of stellar-mass objects near
the center of the clusters. The variable $v_{\rm rel}$ represents the relative
velocity between the binary and the BH while $\Sigma$ is the encounter cross-
section. For $v_{\rm p}=\sqrt{{2G(M_{\bullet}+\;M_{\star})}/{r_{\rm
p}}}\gg\sigma$, we can write $\Sigma\simeq\uppi
G(M_{\bullet}+\;M_{\star})r_{\rm p}/\sigma^{2}$ where $\sigma$ is the velocity
dispersion. Adopting our results that strong encounters occur when $r_{\rm
p}<a$, and assuming that this relation applies to binaries of any size and
mass ratio, we can approximate $\Sigma\simeq\uppi
G(M_{\bullet}+\;M_{\star})a\sigma^{-2}$. Then, we find that ${\rm
d}\mathcal{R}/{\rm d}N_{\bullet}$ can be expressed as
$\displaystyle\frac{{\rm d}\mathcal{R}}{{\rm d}N_{\bullet}}$
$\displaystyle\simeq\frac{n\uppi G(M_{\bullet}+\;M_{\star})a}{\sigma},$
$\displaystyle\simeq 4\times 10^{-13}\;\mathrm{yr}^{-1}\left(\frac{f_{\rm
b}}{10^{-4}}\right)\left(\frac{n_{\rm s}}{10^{5}{\rm
pc}^{-3}}\right)\left(\frac{M_{\bullet}+\;M_{\star}}{20\,{\rm
M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{a}{100\,{\rm
R}_{\odot}}\right)\left(\frac{\sigma}{15\;\mathrm{km}\sec}\right)^{-1}.$ (1)
Assuming more than a few tens of single stellar-mass black holes exist in
dense clusters at the present day (Morscher et al., 2015; Askar et al., 2018;
Kremer et al., 2018), and $\simeq$150 globular clusters in the Milky Way
(Harris, 2010), the rate of strong three-body encounters per Milky Way-like
galaxy is,
$\displaystyle\mathcal{R}$ $\displaystyle\simeq 6\times
10^{-9}\;\mathrm{yr}^{-1}\left(\frac{N_{\bullet}}{15000}\right)\left(\frac{f_{\rm
b}}{10^{-4}}\right)\left(\frac{n_{\rm s}}{10^{5}{\rm
pc}^{-3}}\right)\left(\frac{M_{\bullet}+\;M_{\star}}{20\,{\rm
M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{a}{100\,{\rm
R}_{\odot}}\right)\left(\frac{\sigma}{15\;\mathrm{km}\sec}\right)^{-1}.$ (2)
Two outcomes that can be produced in this type of encounters are EM transients
due to disruption of the star and formation of BBHs. In particular, because
BBHs would likely form in $\simeq 25\%$ of all encounters (see § 4.4), the
rate $\mathcal{R}$ of BBH-forming events would be
$O(10^{-9})\;\mathrm{yr}^{-1}$. The rate for encounters involving massive
stars would be relatively high, compared to that for low-mass stars before all
the massive stars turn into compact objects. However, as discussed in §5.3,
the total number of this type of encounters over the full cluster lifetime
would be higher for less massive stars because of their longer lifetime and
higher abundance. Note that $f_{\rm b}$ depends on cluster parameters such as
the initial binary fraction and the cluster age (Morscher et al., 2015), and
calculating $\mathcal{R}$ requires a detailed modeling of cluster evolution,
as well as the star formation history. Thus, for a more precise estimate of
$\mathcal{R}$ a more careful consideration of cluster evolution history is
required.
## 3 Simulation details
### 3.1 Numerical methods
Our numerical methods and setup are essentially the same as in Paper 2. We
perform a suite of 3D hydrodynamic simulations of close encounters using the
massively parallel gravity and magnetohydrodynamics moving-mesh code AREPO
(Springel, 2010; Pakmor et al., 2016; Weinberger et al., 2020), which combines
advantages of the two conventional hydrodynamical schemes, the Eulerian
finite-volume method and the Lagrangian smoothed particle method, such as
shock capturing without introducing an artificial viscosity, low advection
errors, an efficient tracking of supersonic flow, and an automatically
adaptive adjustment of spatial resolution. We use the HELMHOLTZ equation of
state (Timmes & Swesty, 2000) which accounts for radiation pressure, assuming
local thermodynamic equilibrium. We include $8$ isotopes ($\mathrm{n}$,
$\mathrm{p}$, ${}^{4}\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{14}\mathrm{N}$,
${}^{16}\mathrm{O}$, ${}^{20}\mathrm{Ne}$, ${}^{24}\mathrm{Mg}$, Pakmor et al.
2012). We follow the advection of the elements which are then used for the
update of the thermodynamics quantities (e.g., pressure). We do not follow the
nuclear reactions, which should be fine given the short duration of the
simulations and the reaction rates expected for the temperatures and densities
that occur in our simulations.
Figure 1: Top panel: The radial density profile of the main-sequence stars
with $M_{\star}=10\,{\rm M}_{\odot}$ (red) relaxed for five stellar dynamical
times. Bottom panel: The relative error with respect to the MESA model, as a
function of mass. The dashed grey line in the top panel indicates the profiles
for the MESA model, which are just sitting below the solid line.
### 3.2 Stellar model
The initial state of the star was taken as an evolved main-sequence (MS) star
computed using the stellar evolution code MESA (version r22.05.1) (Paxton et
al., 2011; Paxton et al., 2013, 2015, 2018, 2019; Jermyn et al., 2023). The
star has an initial mass of $10\,{\rm M}_{\odot}$ and a metallicity $Z=0.006$,
which is lower than solar and consistent with what is found for globular
clusters (e.g., VandenBerg et al., 2013), whose high stellar density
facilitates dynamical interactions.111Note that the exact value of the
metallicity does not affect our main results because the two most often cases
in our simulations are fly-bys of stars around the BHs, or near-collisional
disruptions of stars, and the outcomes of these two cases are not
significantly affected by a slight change in the stellar internal structure
due to a different metallicity.. Convection is modeled according to the mixing
length theory with a mixing length parameter of $1.5$. We use the Ledoux
(1947) criterion to determine the boundary of the convective regions and
include an exponential overshoot prescription (Herwig, 2000) with parameters
$f=0.014$ and $f_{0}=0.004$. We treat semiconvection as in Langer et al.
(1983, 1985) assuming it is fully efficient. Wind mass loss is modeled with
the prescription from Vink et al. (2001). We evolve the star until about
halfway through the main sequence, which we choose as the time when the
central hydrogen mass fraction drops to 0.3, at which point the star has
developed a $2.6\,{\rm M}_{\odot}$ convective helium core with a radius of
$0.8\,{\rm R}_{\odot}$. The stellar radius is $R_{\star}=5.4\,{\rm
R}_{\odot}$, and the central density is $\simeq 11\,{\rm g\,cm^{-3}}$. Since
we evolve the stellar model as a single star, we neglect possible past
interactions that could have affected the structure. For example, if the black
hole binary system formed through binary evolution, the star may have accreted
mass (e.g., Renzo & Götberg, 2021). If the system formed through dynamical
capture, the star may have lost mass. We expect such effects to be small and
not affect our results significantly.
We use the MESA stellar model as initial state for the AREPO simulation. After
mapping the 1D MESA model into a 3D AREPO grid with $N\simeq 5\times 10^{5}$
cells, the 3D single star is first relaxed. It usually takes up to five
stellar dynamical time until it is fully relaxed. The stellar dynamical time
is defined as $\sqrt{R_{\star}^{3}/GM_{\star}}$ where $R_{\star}$ and
$M_{\star}$ are the radius and mass of the star, respectively. Note that we
increase the resolution for each single star by almost a factor of 2 compared
to that in Paper 2, showing the results were converged with $N\gtrsim
2.5\times 10^{5}$. This is to more conservatively guarantee the convergence of
our results, and to better resolve stars that may be partially disrupted
during encounters. In addition to that, the resolution becomes finer over time
by adopting refinement near the BHs (§ 3.4): some simulations with violent
interactions have $10^{7}$ cells at the end of the simulations. The density
profiles of the relaxed stars considered in our simulations are depicted in
Figure 1. As shown in the figure, the relative difference of our 3D star with
respect to the MESA model is $1\%$ for the inner region up to $2\,{\rm
M}_{\odot}$. The match is better than $10\%$ throughout most of the star,
except for the surface. We expect that this is sufficient for the aims of this
study.
Figure 2: Schematic diagrams for the initial configuration of the BH-star
binary (blue solid circle and red solid star) and single BH (black circle) for
a prograde case with an inclination angle $i<90^{\circ}$ and phase angle
$\phi=0^{\circ}$, projected onto the $x-y$ plane (left) and the $x-z$ plane
(right). The arrows indicate the instantaneous direction of motion. The open
symbols, on the same circle with the solid symbols, indicate the case with
$\phi>0^{\circ}$.
### 3.3 Black holes
As in Paper 2, we model the BH using a sink particle assuming it is not
rotating initially. It only interacts gravitationally with gas and grows in
mass via accretion of gas. We set the gravitational softening length of the BH
($\simeq 0.01\,{\rm R}_{\odot}$) to be ten times the minimum softening length
of the cells of the stars.
We follow the same procedure for accretion described in Paper 2. However, we
significantly improve the resolution near the BH using refinement (see § 3.4),
which leads to more accurate estimates of the accretion rate with stricter
conditions for accretion than in Paper 2. We search for cells bound to the BH
(i.e. negative orbital energy relative to the BH) within $10^{3\,}r_{\rm g}$
(c.f., $1.5\times 10^{4}\,r_{\rm g}$ in Paper 2) where $r_{\rm
g}=GM_{\bullet}/c^{2}$ is the gravitational radius of the BH and $M_{\bullet}$
denotes the mass of the BH. We still apply the same inverse-distance kernel
(Monaghan & Lattanzio, 1985) to put more weight onto closer cells. Although
the change in the momentum and the mass of the BH due to accretion is taken
into account, our simulations do not include potential radiative feedback
produced by accretion.
### 3.4 Mesh refinement
The simulation code can adjust the local mesh resolution by adaptively
splitting or merging cells if certain prescribed refinement criteria are
satisfied (for more details, see § 6 in Springel 2010). We apply the
refinement technique to cells in the vicinity of each BH to better resolve the
stream structure there. At every time step, the code refines cells near a BH
if all of the following conditions are met:
1. 1.
the distance from the BH fulfils $r<5000~{}r_{\rm g}$,
2. 2.
the cell density is $\rho>2\times 10^{-4}\;\mathrm{g}\;\mathrm{cm}^{-3}$,
3. 3.
the cell mass exceeds $>6\times 10^{22}\;\mathrm{g}$,
4. 4.
and $\Delta d/r>0.26$ for $500~{}r_{\rm g}<r<5000~{}r_{\rm g}$ and $\Delta
d>500\,r_{\rm g}$ for $r<500\,r_{\rm g}$, where $\Delta d$ is the cell size.
The refinement radius in condition (i) is chosen to be larger than the
accretion radius ($1000~{}r_{\rm g}$ in this work) to ensure that gas streams
inside the accretion radius are well resolved. Condition (ii) is designed to
apply the refinement only to the cells that represent “real” gas, not vacuum
regions. Criterion (iii) avoids a runaway creation of low-mass cells. Finally,
the resolution limit imposed through condition (iv) guarantees that there are
at least $O(10^{2})$ cells within the accretion radius. On the other hand, at
every time step, the code can also derefine cells within $r<5000\,r_{\rm g}$
around each BH if the cell mass is $<1.5\times 10^{22}\;\mathrm{g}$, meaning
the mass of cells within $r<5000\,r_{\rm g}$ never becomes smaller than this
mass resolution limit.
We ran a few simulations with five different resolution limits within the
range $0.05\leq\Delta d\leq 0.4$. This confirmed that the global evolution of
the systems, such as their final interaction outcomes, is not affected by the
refinement. The accretion rate has converged when the cell size fulfils
$\Delta d/r<0.3$. Note that the number of cells within a volume at distance
$r$ increases approximately by a factor of 8 when $\Delta d$ decreases by a
factor of 2.
### 3.5 Star – black hole binary
Before we carry out our encounter experiments, we relax binaries consisting of
a fully relaxed star and a BH for $10$ stellar dynamical times. We
parameterize the binary’s semimajor axis $a$ using an approximate analytic
estimate of the Roche lobe radius (Eggleton, 1983),
$\displaystyle\frac{r_{\rm
RL}}{a}=\frac{0.49\,q^{2/3}}{0.6\,q^{2/3}+\ln(1+q^{1/3})},$ (3)
where $r_{\rm RL}$ is the volume averaged Roche lobe radius of the star,
$q=M_{\star}/M_{\bullet}$ is the mass ratio, and $a$ is the orbital
separation. We define $a_{\rm RL}\equiv a(R_{\rm RL}=R_{\star})$ as the
separation at which the star fills its Roche lobe. For $q=0.5$ and $r_{\rm
RL}=R_{\star}$, $a_{\rm RL}\simeq 3.12\,R_{\star}\simeq 16.9\,{\rm
R}_{\odot}$.
We have performed this binary relaxation process for every binary with
different orbital parameters (3 different binaries in total). The semimajor
axis and the eccentricity of the relaxed binaries differ by less than 1% from
their initial values.
Model number | Model name | $a$ | $b$ | $\phi~{}[^{\circ}]$ | $i$ | $t_{\rm p}$ | $P$ | $v_{\rm orb}$
---|---|---|---|---|---|---|---|---
Unit | $a_{\rm RL}$ | $\,{\rm R}_{\odot}$ | - | $\,{\rm R}_{\odot}$ | ∘ | ∘ | hours | days | ${\rm km\,s^{-1}}$
1 | $a4b2\phi 0i30$ | 4 | 67.5 | 2 | 67.5 | 0 | 30 | 50 | 12 | 291
2 | $a4b1\phi 0i30$ | 4 | 67.5 | 1 | 33.8 | 0 | 30 | 18 | 12 | 291
3 | $a4b1/2\phi 0i30$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 30 | 6.3 | 12 | 291
4 | $a4b1/4\phi 0i30$ | 4 | 67.5 | 1/4 | 8.45 | 0 | 30 | 2.3 | 12 | 291
5 | $a4b2\phi 180i30$ | 4 | 67.5 | 2 | 67.5 | 180 | 30 | 50 | 12 | 291
6 | $a4b1\phi 180i30$ | 4 | 67.5 | 1 | 33.8 | 180 | 30 | 18 | 12 | 291
7 | $a4b1/2\phi 180i30$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 30 | 6.3 | 12 | 291
8 | $a4b1/4\phi 180i30$ | 4 | 67.5 | 1/4 | 8.45 | 180 | 30 | 2.3 | 12 | 291
9 | $a4b2\phi 0i150$ | 4 | 67.5 | 2 | 67.5 | 0 | 150 | 50 | 12 | 291
10 | $a4b1\phi 0i150$ | 4 | 67.5 | 1 | 33.8 | 0 | 150 | 18 | 12 | 291
11 | $a4b1/2\phi 0i150$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 150 | 6.3 | 12 | 291
12 | $a4b1/4\phi 0i150$ | 4 | 67.5 | 1/4 | 8.45 | 0 | 150 | 2.3 | 12 | 291
13 | $a4b2\phi 180i150$ | 4 | 67.5 | 2 | 67.5 | 180 | 150 | 50 | 12 | 291
14 | $a4b1\phi 180i150$ | 4 | 67.5 | 1 | 33.8 | 180 | 150 | 18 | 12 | 291
15 | $a4b1/2\phi 180i150$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 150 | 6.3 | 12 | 291
16 | $a4b1/4\phi 180i150$ | 4 | 67.5 | 1/4 | 8.45 | 180 | 150 | 2.3 | 12 | 291
17 | $a2b1/2\phi 0i30$ | 2 | 33.7 | 1/2 | 8.44 | 0 | 30 | 1.2 | 4.1 | 412
18 | $a2b1/2\phi 180i30$ | 2 | 33.7 | 1/2 | 8.44 | 180 | 30 | 1.2 | 4.1 | 412
19 | $a2b1/2\phi 0i150$ | 2 | 33.7 | 1/2 | 8.44 | 0 | 150 | 1.2 | 4.1 | 412
20 | $a2b1/2\phi 180i150$ | 2 | 33.7 | 1/2 | 8.44 | 180 | 150 | 1.2 | 4.1 | 412
21 | $a6b1/2\phi 0i30$ | 6 | 101 | 1/2 | 25.3 | 0 | 30 | 12 | 22 | 238
22 | $a6b1/2\phi 180i30$ | 6 | 101 | 1/2 | 25.3 | 180 | 30 | 12 | 22 | 238
23 | $a6b1/2\phi 0i150$ | 6 | 101 | 1/2 | 25.3 | 0 | 150 | 12 | 22 | 238
24 | $a6b1/2\phi 180i150$ | 6 | 101 | 1/2 | 25.3 | 180 | 150 | 12 | 22 | 238
25 | $a4b1/2\phi 0i0$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 0 | 6.3 | 12 | 291
26 | $a4b1/2\phi 0i60$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 60 | 6.3 | 12 | 291
27 | $a4b1/2\phi 0i120$ | 4 | 67.5 | 1/2 | 16.9 | 0 | 120 | 6.3 | 12 | 291
28 | $a4b1/2\phi 180i0$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 0 | 6.3 | 12 | 291
29 | $a4b1/2\phi 180i60$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 60 | 6.3 | 12 | 291
30 | $a4b1/2\phi 180i120$ | 4 | 67.5 | 1/2 | 16.9 | 180 | 120 | 6.3 | 12 | 291
31 | $a4b1/2\phi 45i30$ | 4 | 67.5 | 1/2 | 16.9 | 45 | 30 | 6.3 | 12 | 291
32 | $a4b1/2\phi 90i30$ | 4 | 67.5 | 1/2 | 16.9 | 90 | 30 | 6.3 | 12 | 291
33 | $a4b1/2\phi 135i30$ | 4 | 67.5 | 1/2 | 16.9 | 135 | 30 | 6.3 | 12 | 291
34 | $a4b1/2\phi 225i30$ | 4 | 67.5 | 1/2 | 16.9 | 225 | 30 | 6.3 | 12 | 291
35 | $a4b1/2\phi 270i30$ | 4 | 67.5 | 1/2 | 16.9 | 270 | 30 | 6.3 | 12 | 291
36 | $a4b1/2\phi 315i30$ | 4 | 67.5 | 1/2 | 16.9 | 315 | 30 | 6.3 | 12 | 291
Table 1: The initial model parameters for encounters between a circular binary
($q=0.5$) with total mass of $30\,{\rm M}_{\odot}$ and a $10\,{\rm M}_{\odot}$
BH. The model name (second column) contains the information of key initial
parameters: for the model names with the format $a(1)b(2)\phi(3)i(4)$, the
numerical values encode (1) the initial semimajor axis of the binary $a/a_{\rm
RL}$, (2) the impact parameter $b$, (3) the initial phase angle $\phi$ in
degrees, and (4) the initial inclination angle $i$ in degrees. Here, $a_{\rm
RL}\simeq 3.12\,R_{\star}\simeq 16.9\,{\rm R}_{\odot}$ is the separation when
the star in the binary fills its Roche lobe. The last three columns show the
dynamical time $t_{\rm p}$ at pericenter (see its definition in the text), in
units of hours, the orbital period $P$ of the binary, and the relative
velocity $v_{\rm orb}$ of the binary members.
### 3.6 Initial conditions
Following the same terminology as in Paper 2, we refer to quantities with a
subscript containing $b-\bullet$ as those relating to the orbit between a
binary and a single BH. We assume a parabolic encounter with eccentricity
$1-e_{b-\bullet}=10^{-5}$ between a single $10\,{\rm M}_{\odot}$ BH with a
binary consisting of a $20\,{\rm M}_{\odot}$ BH and a $10\,{\rm M}_{\odot}$
star. The exact choices of the system parameters are somewhat arbitrary, but
BHs with such masses have been found in X-ray binaries (Binder et al., 2021,
e.g.). Encounters between objects of comparable masses are expected in the
dense centers of young mass-segregated star clusters. We later discuss
potential effects of different masses and orbits of encountering objects in §
5.3, based on our simulation results. We consider three semi-major axes for
the initial binary systems: $a/a_{\rm RL}=2$, $4$ and $6$, corresponding to an
orbital period of 4, 12, and 22 days, respectively. We assume the binaries are
circular at the start of our simulations. This is primarily to simplify the
initial conditions, but this may not be unreasonable given that close binaries
are often found to be circular (Almeida et al., 2017).
The distance between the binary’s center of mass and the BH at the first
closest approach $r_{\rm p,b-\bullet}$ is parameterized using the impact
parameter $b$, i.e., $r_{\rm p,b-\bullet}=0.5\,ba$ where $a$ is the binary
semimajor axis. We consider $b=1/4$, $1/2$, $1$, and $2$ for $a/a_{\rm RL}=4$,
and $1/2$ for $a/a_{\rm RL}=2$ and $6$. The binary’s angular momentum
direction is always along the $z$-axis in our simulations. We illustrate the
initial configuration of the stellar binary and the BH in Figure 2.
We investigate the dependence of encounter outcomes on key encounter
parameters, that is inclination angle $i=0$, $30^{\circ}$, $60^{\circ}$,
$120^{\circ}$ and $180^{\circ}$, $b=1/4$, 1/2, 1 and 2, and the phase angle
$\phi=0^{\circ}-180^{\circ}$ with $\Delta\phi=45^{\circ}$. We define $\phi$ as
the initial angle between the line connecting the two members in the binary
and the coordinate $x-$axis (see Figure 2). We start by studying the
dependence on the two phase angles of the binary ($\phi=0^{\circ}$ and
$180^{\circ}$) by fixing all the other parameters. To achieve this, we
initially rotate the binary while the initial separation between the center of
mass of the binary and the BH is fixed at $r=5\,a$. This allows us to examine
the outcomes from the first contact of the single BH with a different member
of the binary. However, given the relatively high computational costs, instead
of simulating encounters with every combination of $i$ and $b$, we perform
simulations for the encounters of the intermediate-size binaries ($a/a_{\rm
RL}=4$) with different combinations of $b=1/4$, 1/2, 1 and 2, and
$i=30^{\circ}$, $150^{\circ}$, and $\phi=0^{\circ}$ and $180^{\circ}$. For the
smallest and largest binaries ($a/a_{\rm RL}=2$ and 6), we only consider
$i=30^{\circ}$ and $150^{\circ}$ while $b=1/2$. In addition, we further
examine the dependence of $i$ on the outcome properties by considering $i=0$,
$60^{\circ}$, $120^{\circ}$ and $180^{\circ}$ (for $b=1/2$). Last, we also
study the impact of the phase angle $\phi$ on the encounter outcomes by
simulating encounters with six additional phase angles ($\phi=45^{\circ}$,
$90^{\circ}$, $135^{\circ}$, $225^{\circ}$, $270^{\circ}$, and $315^{\circ}$).
In Table 1, we summarize the initial parameters considered in our simulations.
Each of the models is integrated in time up to a few $100\,t_{\mathrm{p}}$ as
needed to identify the final outcomes. Here, $t_{\rm p}=\sqrt{r_{\rm
p}^{3}/GM}$ is the dynamical time at $r=r_{\rm p}$, where $M$ is the total
mass of the three objects ($40\,{\rm M}_{\odot}$). The value of $t_{\rm p}$
for each model is given in Table 1.
The total computational cost for each run varies, mainly depending on how long
the interactions last until the final outcomes are produced. Using 200 - 300
CPU-cores of the Intel Xeon CascadeLake-AP processor (Xeon Platinum 9242), the
total compute time per run has been around 70000 - 100000 core hours.
Figure 3: An example of a BBH-forming encounter, Model 20.$a2b1/2\phi
180i150$, showing the density distribution in the binary orbital plane at a
few different times in units of $t_{\rm p}$. The color bar gives the
logarithmic density in ${\rm g}\,{\rm cm}^{-3}$. The time is measured since
the expected pericenter passage between the binary’s center of mass and the
single BH. At $t/t_{\rm p}\simeq-16$ (top-$1^{\rm st}$), the binary (star -
green dot) and the single BH (yellow dot) approach each other. At $t/t_{\rm
p}=-3.21$ (top-$3^{\rm rd}$), the incoming BH strongly encounters with the
star in the binary, followed by the collision of the star (top-$4^{\rm th}$).
The BH that disrupted the star is gravitationally captured by the other BH,
forming a merging binary with $a\simeq 6.70\,{\rm R}_{\odot}$ and $e\simeq
0.943$, corresponding to $t_{\rm GW}\simeq 10^{4}\;\mathrm{yr}$ (bottom
panels).
Model number | Model name | BBH? | STAR? | Binary type | $a$ | $e$ | $\log_{10}t_{\rm GW}$ | $v$ | Single type | $v$
---|---|---|---|---|---|---|---|---|---|---
- | - | - | - | - | $\,{\rm R}_{\odot}$ | - | yr | km/s | - | km/s
1 | $a4b2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 76.2 | 0.796 | - | - | | -
2 | $a4b1\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 55.3 | 0.347 | - | - | |
3 | $a4b1/2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 110 | 0.352 | - | - | | -
4 | $a4b1/4\phi 0i30^{\star\star}$ | - | - | - | - | - | - | - | - | -
5 | $a4b2\phi 180i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 63.3 | 0.1 | 12.2 | 56.0 | $\star$ | $167$
6 | $a4b1\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 257 | 0.685 | 13.3 | 61.0 | - | -
7 | $a4b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 114 | 0.518 | 12.5 | 31.3 | - | -
8 | $a4b1/4\phi 180i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 41.9 | 0.727 | 9.91 | 60.5 | $\star$ | $183$
9 | $a4b2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 62.0 | 0.156 | - | 57 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 172
10 | $a4b1\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 42.2 | 0.724 | - | 67.8 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 203
11 | $a4b1/2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 50.3 | 0.722 | - | 57.1 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 170
12 | $a4b1/4\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 50 | 0.731 | - | 57.8 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 172
13 | $a4b2\phi 180i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 63.3 | 0.161 | - | 52.4 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 157
14 | $a4b1\phi 180i150^{\star}$ | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 66.1 | 0.857 | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | -
15 | $a4b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 12 | 0.661 | 8.031 | 38.6 | - | -
16 | $a4b1/4\phi 180i150^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 109 | 0.310 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
17 | $a2b1/2\phi 0i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 18.0 | 0.396 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
18 | $a2b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 61.0 | 0.406 | 11.7 | 64.0 | - | -
19 | $a2b1/2\phi 0i150$ | No | Destroyed | - | - | - | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20),\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 60.8, 234
20 | $a2b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 6.70 | 0.943 | 4.36 | 47.6 | - | -
21 | $a6b1/2\phi 0i30^{\star\star}$ | - | - | - | - | - | - | - | - |
22 | $a6b1/2\phi 180i30$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 354 | 0.787 | 13.2 | 69.1 | - | -
23 | $a6b1/2\phi 0i150$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 76.0 | 0.728 | - | 65.7 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 199
24 | $a6b1/2\phi 180i150$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)\longleftrightarrow\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 149 | 0.972 | 8.64 | 79.2 | - | -
25 | $a4b1/2\phi 0i0^{\star\star}$ | - | - | - | - | - | - | - | |
26 | $a4b1/2\phi 0i60^{\star}$ | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 101 | 0.725 | 11.5 | - | $\star$ | -
27 | $a4b1/2\phi 0i120$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 68 | 0.488 | - | 42.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 126
28 | $a4b1/2\phi 180i0$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 370 | 0.831 | 13 | 38.6 | - | -
29 | $a4b1/2\phi 180i60$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 155 | 0.713 | 12.3 | 38.0 | - | -
30 | $a4b1/2\phi 180i120$ | Yes | Destroyed | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 94.6 | 0.774 | 11.0 | 94.5 | - | -
31 | $a4b1/2\phi 45i30^{\star\star}$ | - | - | - | - | - | - | - | |
32 | $a4b1/2\phi 90i30$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 37 | 0.301 | - | 75.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 227
33 | $a4b1/2\phi 135i30$ | Yes | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 48.0 | 0.816 | 9.556 | 60.6 | $\star$ | 181
34 | $a4b1/2\phi 225i30^{\star}$ | - | - | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)-\star(10)$ | 58.4 | 0.555 | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ | -
35 | $a4b1/2\phi 270i30$ | No | Survived | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)-\star(10)$ | 64.5 | 0.460 | - | 69.6 | $\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 232
36 | $a4b1/2\phi 315i30$ | No | Destroyed | - | - | - | - | - | $\leavevmode\hbox to8.61pt{\vbox to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20),\leavevmode\hbox to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip 3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ | 93.1, 209
Table 2: The outcomes of each model: the model number (first column), the
model name (second column), whether a BBH forms (third column), and whether
the star survives or is destroyed (fourth column). The next five columns show
the type of final binary product and its properties (semimajor axis,
eccentricity, GW-driven merger time scale only for BBHs, and ejection
velocity). The last two columns indicate the type of singles as a final
product and their ejection velocity. The star symbol (⋆) at the end of some
model names indicates the case where the three objects form a quasi-stable
triple (see its definition in the main text) and the final outcomes are not
determined until the end of simulations. For this case, the properties of the
inner binary are provided. The double star symbols (⋆⋆) indicate cases where
the three objects do not form any quasi-stable object and the final outcomes
are not determined. The symbol $\leavevmode\hbox to8.61pt{\vbox
to8.61pt{\pgfpicture\makeatletter\hbox{\hskip 4.30554pt\lower-4.30554pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.30554pt}{0.0pt}\pgfsys@curveto{4.30554pt}{2.37791pt}{2.37791pt}{4.30554pt}{0.0pt}{4.30554pt}\pgfsys@curveto{-2.37791pt}{4.30554pt}{-4.30554pt}{2.37791pt}{-4.30554pt}{0.0pt}\pgfsys@curveto{-4.30554pt}{-2.37791pt}{-2.37791pt}{-4.30554pt}{0.0pt}{-4.30554pt}\pgfsys@curveto{2.37791pt}{-4.30554pt}{4.30554pt}{-2.37791pt}{4.30554pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(20)$ indicates the
$20\,{\rm M}_{\odot}$ BH initially in the binary, $\leavevmode\hbox
to6.43pt{\vbox to6.43pt{\pgfpicture\makeatletter\hbox{\hskip
3.21388pt\lower-3.21388pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@curveto{3.01389pt}{1.66454pt}{1.66454pt}{3.01389pt}{0.0pt}{3.01389pt}\pgfsys@curveto{-1.66454pt}{3.01389pt}{-3.01389pt}{1.66454pt}{-3.01389pt}{0.0pt}\pgfsys@curveto{-3.01389pt}{-1.66454pt}{-1.66454pt}{-3.01389pt}{0.0pt}{-3.01389pt}\pgfsys@curveto{1.66454pt}{-3.01389pt}{3.01389pt}{-1.66454pt}{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(10)$ marks the
$10\,{\rm M}_{\odot}$ single incoming BH, and $\star(10)$ stands for the star
initially in the binary. Arrows indicate the active BH in the BBHs; double-
headed arrows signify that both BHs are active at the end of the corresponding
simulation.
Figure 4: The orbital properties of the final binaries, for binary black holes
(top panels) and black hole – star binaries (bottom panels). The left panels
compare the initial semimajor axis with the final one, and the right panels
show the semimajor axis and the eccentricity of the final binaries. The black
diagonal lines in the left panels depict the cases where the size of the
initial binaries and final binaries are identical. The grey curves in the top-
right panels indicate the gravitational wave-driven merger time scales of 14
Gyr, 1 Gyr and 1 Myr, respectively, for a binary consisting of $10\,{\rm
M}_{\odot}$ and $20\,{\rm M}_{\odot}$ black holes. In the top panels, the
solid (hollow) markers indicate BBHs that would (not) merge in a Hubble time.
On the other hand, the hollow markers in the bottom panels indicate the models
where the final outcome is an unstable triple and, for this case, the orbital
properties of the inner binary are presented.
Figure 5: An example of a non-BBH-forming encounter, Model 17.$a2b1/2\phi
0i30$. We depict the density distribution in the binary orbital plane at a few
different times in units of $t_{\rm p}$. At $t/t_{\rm p}\simeq-16$
(top-$1^{\rm st}$), the binary (star - green dot) and the single BH (yellow
dot) approach each other. At $t/t_{\rm p}=-3.21$ (top-$3^{\rm rd}$), the two
BHs encounter, resulting in the ejection of the initially single BH, while the
initial binary orbit is significantly perturbed to become an interacting
binary (bottom panels). Because of periodic interactions at pericenter, the
binary orbit continues to evolve until the end of the simulation. The
semimajor axis and eccentricity measured at the end of the simulation are
$\simeq 18\,{\rm R}_{\odot}$ and $\simeq 0.4$, respectively.
## 4 Results
### 4.1 Classification of outcomes
The outcomes of three-body encounters between BH-star binaries and single BHs
can be divided into three classes, depending on the final products.
1. 1.
BBH-forming encounters: This class refers to encounters in which a BBH
emerges. In this case, the impact parameter is mostly $\lesssim 1/2-1$. The
incoming single BH frequently interacts first with the star by the time the
binary’s center of mass and the single BH arrive at pericenter (models with
“Yes” in the fifth column in Table 2). In this situation, the star in the
binary nearly collides with the incoming single BH. We show one example for
this type of encounter in Figure 3. The incoming single BH loses a significant
amount of its kinetic energy and is gravitationally captured by the other BH
initially in the binary. Because of the member exchange due to a violent star-
removing encounter, the size of the final binary is not necessarily correlated
with the size of the initial binary. To illustrate this, we compare in the
top-left panel of Figure 4 the final $a$ of the BBHs with the semimajor axis
of the initial BH-star binaries. The final $a$ covers over a wide range of
values and is not necessarily comparable to $a$ of the initial binary. These
violent interactions can lead to the formation of merging BBHs, as illustrated
in the top-right panels: the GW-driven merger time scale of 5 out of 14 final
BBHs is less than a Hubble time. Note that the absolute magnitude of the
binding energy of the merging BBHs is much larger than the typical kinetic
energy of stars in both globular and nuclear clusters ($\simeq\sigma^{2}$
where $\sigma$ is the velocity dispersion). This suggests that subsequent
interactions with other stars would not dissociate these “hard” binaries, but
rather make them more compact (Heggie, 1975) and more eccentric (Valtonen &
Karttunen, 2006), which would facilitate their mergers. In addition, the
disruption of the star prior to the BBH formation means that at least one
member of the BBH is frequently surrounded by gas upon binary formation. When
the BBH is compact, both BHs accrete gas.
2. 2.
Non-BBH-forming encounters: In this class, the outcomes are member exchanges
between the two BHs or perturbations of the initial binary’s orbit (models
with “No” in the fifth column of Table 2). This mostly occurs when the two BHs
interact at the first contact between the binary and the single BH. We show
one example for this type of encounter in Figure 5, resulting in an orbit
perturbation. The impact of the encounters is relatively weak compared to the
BBH-forming encounters. As shown in the bottom-left panel of Figure 4, the
final value of $a$ scatters within less than a factor of 2 around the initial
$a$. The eccentricity of the final BH-star binary is widely distributed
between 0.1 and 0.9 (bottom-right panel), similarly to those of final BBHs
(top-right panel). In this type of encounters, EM transient phenomena, such as
tidal disruption events, collisions, or interacting binaries, can be created
(e.g., Model 17. $a2b1/2\phi 0i30$). In addition, the single BHs are ejected
at $\gtrsim 60\,{\rm km\,s^{-1}}$, comparable to or higher than the escape
velocity of globular clusters (i.e., tens of ${\rm km\,s^{-1}}$; Gnedin et al.
2002; Antonini & Rasio 2016).
3. 3.
Undetermined: This class refers to cases where final outcomes are not
determined (12 models in total, Models with “-” in the fifth column in Table 2
and with superscript $\star$ or $\star\star$). Among these 12 models, there
are eight encounters (models designated with superscript $\star$) in which the
three objects form an unstable hierarchical triple, which we define as a
triple where the outer binary is on a very large eccentric orbit so that the
pericenter distance of the outer binary is smaller than the semimajor axis of
the inner binary. In the table, we provide the orbital parameters of the inner
binary. In the rest (models with superscript $\star\star$), interactions
become extremely prolonged so that a final outcome has not (yet) emerged.
From now on, we will focus on the first two classes, i.e., BBH-forming and
Non-BBH-forming encounters. These types of final outcomes and their properties
are summarized in Table 2.
Figure 6: Schematic diagram showing three dominant configurations resulting in
the perturbation of the original binary’s orbit in our simulations. The black
arrows indicate the direction of motion of objects, and long grey arrows the
trajectory of the incoming BH (black circle). In the first configuration (top
panels), for the prograde encounter with $\phi<90^{\circ}$ and $b\lesssim 1$,
the incoming BH undergoes a strong encounter with the BH (blue circle)
initially in the binary (small closest approach distance compared to the
binary semimajor axis), quickly turns around and advances to the left (top-
right panel). This quick turn-around motion gives a momentum kick (green
arrow) to the blue circle to the right with respect to the star (red circle).
The orbit of the initial binary is perturbed. The second configuration
(bottom-left panel) is a distant fly-by where the incoming BH does not
significantly interact with any of the binary members, and this happens when
$b\gtrsim 1$. The last configuration (bottom-right panel) shows the case where
the incoming BH passes through the binary without strong interactions with any
of the binary members (e.g., $b\simeq 1/4$ and $a/a_{\rm RL}=4$).
Figure 7: Schematic diagram showing the dominant configuration resulting in an
exchange of binary members. The black arrows indicate the direction of motion
of objects, and long grey arrows (right panel) the trajectory of the incoming
BH (black circle). In retrograde encounters with $\phi<90^{\circ}$, like
configuration 1 for the orbit perturbation (Figure 6), the incoming BH
strongly interacts with the BH (blue circle) initially in the binary, quickly
turns around and advances to the right. This motion results in a momentum kick
(green arrow) to the blue circle to the left with respect to the star (red
circle). The initially single BH gravitationally captures the star and forms a
binary.
### 4.2 Dynamical processes
The two most crucial factors to determine the outcomes for the parameter space
considered in our simulations are 1) the types of objects that meet at the
closest encounter (BH - BH or BH - star), and 2) the net direction of the
momentum kick relative to the bystander object (i.e., the object in the binary
that does not interact with the incoming BH at the first closest approach)
imparted by the interaction between the two meeting objects. As explained in
the previous section, the first aspect substantially affects the chances of
the survival of the star. The second aspect determines which objects end up in
the final binary (i.e., member exchange, binary perturbation) and how the
final binary’s orbit looks like.
For the non-BBH-forming encounters with $b\simeq 1/2-1$, the most frequent
outcomes are either a member exchange or a perturbation of the original
binary. The latter happens in retrograde encounters. This case can be
categorized into three configurations, which are illustrated in Figure 6. In
the first configuration (top panels), the incoming BH strongly interacts with
the other BH and turns around at a small pericenter distance compared to the
binary semimajor axis. The initially single BH is rapidly ejected from the
system in the direction roughly opposite to the incoming direction. This
interaction imparts a momentum kick to the BH that perturbs the binary orbit.
In the other two configurations, the incoming BH either moves around or passes
through the binary, without significant interactions with any of the binary
members (bottom panels).
The dominant channel for member exchange is depicted in Figure 7. For the non-
BBH-forming encounters in a prograde orbit, the two BHs meet first and pass
through their points of closest approach. Like the first configuration for
orbit perturbation, their relative motion gives a momentum kick to the motion
of the BH originally in the binary, relative to the star. The momentum kick
gives an additional acceleration in the BH’s receding motion from the star.
The initially single BH, after turning around the other BH, moves in a similar
direction with the star and gravitationally captures it.
For the BBH-forming encounters, the star and the initially single BH undergo
close encounters, naturally resulting in a tidal disruption event or stellar
collision. Both events can also impart a momentum kick to the disrupting BH.
In our simulations, the momentum kick is not large enough to prevent the two
BHs from forming a bound pair. For example, if the star and the incoming BH
undergo a head-on collision, the incoming BH dramatically slows down and forms
a merging BBH with the other BH (e.g., Model 20. $a2b1/2\phi 180i150$). We
caution that the head-on collision between the two equal-mass objects in our
simulations is an extreme case yielding a dramatic drop in the BH’s kinetic
energy. The net effect of such star-removing events on the motion of the
disrupting BH and the subsequent formation of a BBH depends on the mass ratio,
relative velocity, and the direction of the momentum kick.
### 4.3 Binary black hole formation
Typical semimajor axes of BBHs formed in the BBH-forming encounters range
within $10-400\,{\rm R}_{\odot}$ while eccentricities vary within $0.1-0.97$.
Correspondingly, the GW-driven merger time scales of those merging binaries
are in the range $10^{4}-10^{13}\;\mathrm{yr}$. Five of our models among these
encounters are merging BBHs with $a\simeq 7-150\,{\rm R}_{\odot}$ and $e\simeq
0.6-0.97$. As explained in § 4.2, the dominant formation channel is the
gravitational capture of the incoming BH by the BH originally in the binary
after strong interactions between the incoming BH and the star. Naturally, a
disruption event or a collision precedes the BBH formation. As a result,
either the disrupting BH or both BHs accrete matter by the time they form a
stable binary.
### 4.4 Dependence of outcomes on parameters
We examine the dependence of outcomes on a few key encounter parameters, phase
angle $\phi$, impact parameter $b$, inclination angle $i$, and semimajor axis
$a$, by varying one parameter at a time, keeping the rest of them fixed. Our
simulations suggest that the two most important parameters that affect the
formation of BBHs in this scenario of three-body encounters are the impact
parameter and the phase angle.
1. 1.
Phase angle $\phi$: this is found to be one of the key parameters that
separates BBH-forming encounters from non-BBH-forming encounters. For the
former, very likely outcomes are BBHs, frequently accompanied by a disruption
of the star. On the other hand, for the latter, frequent outcomes are
eccentric BH-star binaries produced via member exchange or weak tidal
perturbations of the initial stellar orbit. In addition, even for the BBH-
forming encounters, the direction of the encounter between the initially
isolated BH and the star at the first closest approach relative to the other
BH determines the size of the semimajor axis of the BBH: if the momentum kick
imparted on the encountering BH is given such that it adds to the encountering
BH’s momentum, a large binary forms (e.g., Model 6. $a4b1\phi 180i30$ and
Model 22. $a6b1/2\phi 180i30$). Although our study is not appropriate for rate
estimates, the dependence of outcomes on the phase angle may indicate that
roughly $\simeq 25\%$ of these three-body encounters between objects of
similar mass with $b\lesssim 1$ may possibly lead to BBH formation with a high
chance of creating EM transients.
2. 2.
Impact parameter $b$: in general, the initial binary and the single BH can
interact significantly (member exchange or stellar collisions) at the first
closest approach when $r_{\rm p}\lesssim a$, which is also found in Ryu et al.
(2022); Ryu et al. (2023). Fly-by only occurs at $r_{\rm p}>a$ (Models 1.
$a4b2\phi 0i30$, and Model 9. $a4b2\phi 0i150$). For this case, the initial
binary orbit is weakly perturbed, resulting in a 10 - 20% change in the
semimajor axis. Relatively weak interactions also take place when the impact
parameter is too small compared to the size of the binary, i.e., $r_{\rm
p}<a/8$ (e.g., Model 4. $a4b1/4\phi 0i30$, and Model 16. $a4b1/4\phi
180i150$), as the single BH penetrates through the binary without interacting
strongly with any of the binary members (see the bottom-right panel of Figure
6).
3. 3.
Inclination angle $i$: prograde encounters tend to result in strong
interactions between the first two encounter objects, frequently leading to
outcomes that involve member exchange (e.g., models with $\phi=0^{\circ}$ and
$i=30^{\circ}$) or stellar collisions (e.g., models with $\phi=180^{\circ}$
and $i=150^{\circ}$). This is because the relative velocity between the two
encountering objects is smaller, implying a larger gravitational focusing
cross section ($\propto v^{2}/\sigma^{2}$ where $\sigma$ is a typical relative
velocity at infinity). A typical configuration for member exchange in prograde
encounters is drawn in Figure 7. On the other hand, the first interactions in
retrograde orbits are relatively weak due to the large relative speed between
the two encountering objects. As a result, frequent outcomes are perturbations
of the initial binary orbit, as depicted in Figure 6.
4. 4.
Semimajor axis $a$: given the same pericenter distance relative to $a$
($r_{\rm p}\simeq 0.25a$) for the simulations with varying $a$, the type of
the final outcomes does not show a strong dependence on $a$. However, the size
of the final binary is closely correlated with that of the initial binary,
e.g., $a\gtrsim 76\,{\rm R}_{\odot}$ of final binaries in models with
$a/a_{\rm RL}=6$ (or $a=101\,{\rm R}_{\odot}$) and $a\lesssim 61\,{\rm
R}_{\odot}$ in models with $a/a_{\rm RL}=2$ (or $a=32\,{\rm R}_{\odot}$).
Figure 8: Face-on (left) and edge-on (right) density distribution of disks
around the BH that disrupts the star at the first closest approach in four
selected models with $i=30^{\circ}$ or $150^{\circ}$, for Model 6. $a4b1\phi
180i30$ ($1^{\rm st}$ row), Model 15. $a4b1/2\phi 180i150$ ($2^{\rm nd}$ row),
Model 18. $a2b1/2\phi 180i30$ ($3^{\rm rd}$ row), and Model 22. $a6b1/2\phi
180i30$ ($4^{\rm th}$ row), at the end of the simulations. The white
horizontal bar at the bottom-left corner of each panel shows the spatial
scale, $4\,{\rm R}_{\odot}$, except for the second row of panels where it is
$2\,{\rm R}_{\odot}$.
Figure 9: Profiles of the structure of the disks in simulations with
$i=30^{\circ}$ or $150^{\circ}$ where a BBH forms, including the four models
shown in Figure 8: The aspect ratio, defined as the ratio of the density scale
height to the cylindrical radius $r$ (top-left), the ratio of the mass-
weighted average of the azimuthal velocity along the midplane within the scale
height to the Keplerian velocity $v_{\rm kep}$ (top-right), the average
density along the midplane within the scale height (bottom-left), and the
mass-weighted average of the temperature along the midplane within the scale
height (bottom-right). All the reported quantities are measured at the end of
the simulations. Figure 10: The accretion rates of the initially single BHs
that fully destroy the star in BBH-forming simulations with
$\phi=180^{\circ}$, and $i=30^{\circ}$ or $150^{\circ}$.
### 4.5 Accretion
Our simulations show that stars can be disrupted in three-body interactions
between BH-star binaries and single BHs via strong interactions with very
small impact parameters, i.e., collisions. In such events, a merging BBH can
subsequently form, and at least one of the BHs is surrounded by an accretion
disk which can create EM transient phenomena. To zero-th order, the disk
structure and features of the accretion rate can be imprinted onto light
curves of such events.
The refinement scheme adopted for the simulations allows us to resolve the gas
structure down to $0.01\,{\rm R}_{\odot}\simeq 10^{3}\,r_{\rm g}$ from the BH.
Although the regions that we can resolve are still too far from the BH to be
directly related to the accretion process, we can provide an accurately
resolved large scale structure of the disks formed in star-destroying events,
which can be used as initial conditions for detailed disk simulations. Here,
we define a disk as a group of gas cells tightly bound to the BH and
coherently orbiting in the azimuthal direction. The outer edge of the disk is
defined as the radius containing 99% of the total bound mass orbiting at a
velocity exceeding 1% of the local Keplerian speed $v_{\rm
kep}(r)=\sqrt{G[M(<r)+M_{\bullet}]/r}$, where $M(<r)$ is the mass enclosed
within $r$.
We show in Figure 8 both the face-on (left panels) and edge-on (right panels)
density distributions of the disks around the BH that destroys the star at the
first encounter in four example models, and in Figure 9 the radial profiles of
the aspect ratio, the density, the temperature, and the rotational velocity
for all models where an accretion disk forms. The aspect ratio $h/r$ is
defined as the ratio of the first-moment density scale height, averaged over a
given cylindrical radius, to the cylindrical radius. Here, we excluded Model
20. $a2b1/2\phi 180i150$ in this analysis because the BH in that model is
surrounded by a nearly spherical gas cloud, not by a disk. But we provide the
accretion rate for that model also, shown in Figure 10.
We find that the disks are thick and pressure-supported, and mostly confined
within $r\simeq 30\,{\rm R}_{\odot}$. In general, the aspect ratio $h/r$ (top-
left panel of Figure 9) is comparable to or greater than order unity up to the
outer edge of the disks. $h/r$ declines from $h/r\simeq 3-5$ to $h/r\simeq 1$
outwards. The rotational velocity $v^{\phi}$ near the mid-plane is sub-
Keplerian ($v^{\phi}/v_{\rm kep}\simeq 0.1-0.6$), indicating the disk is not
rotationally supported. The velocity ratio remains the same out to the outer
disk edge. The density of the inner region stays flat at
$\rho\simeq(0.1-5)\;\mathrm{g}\;\mathrm{cm}^{-3}$ up to 0.1 - 0.2 of the disk
size, then declines steeply following a $r^{-4}$ power-law. On the other hand,
the temperature does not show such flatness at $r\lesssim\,{\rm R}_{\odot}$,
but continuously decreases following a $r^{-1}$ power-law.
Finally, we present in Figure 10 the accretion rate of the initially single
BHs that fully destroy the star at the first closest encounter. The general
trend is that, upon disruption or collision, the accretion rate dramatically
increases up to $\dot{M}\simeq(10^{-6}-10^{-5})\,{\rm M}_{\odot}\,{\rm
s}^{-1}$ and it takes around 80-100 hours until $\dot{M}$ declines by a factor
of 100 from its peak. When the binary is eccentric and the pericenter distance
is sufficiently close, a periodic perturbation from the other BH at periastron
results in periodic bursts on a time scale $\simeq$ the orbital period (e.g.,
Model 15. $a4b1/2\phi 180i150$, and Model 20. $a2b1/2\phi 180i150$). Although
the accretion rate is super-Eddington, the total accreted mass is at most
$0.1\,{\rm M}_{\odot}$ ($\lesssim 0.4\%$) until the end of the simulation, and
the magnitude of the BH spin driven by accretion can be as large as $0.01$.
We have to caution that such extremely high accretion rates for stellar-mass
BHs would result in strong outflows (e.g., Sądowski et al., 2014), which would
regulate the accretion rate. Although we have realized a significant
improvement in resolving gas motions near the BHs compared to Paper 2 thanks
to using refinement, since feedback from the BHs is not included in our
simulations it is likely that our accretion rates are overestimated.
Nonetheless, if the luminosity is mostly driven by accretion, the features
revealed in the accretion rate (e.g., periodic bursts) could possibly be
imprinted in the light curves.
Figure 11: The evolution of $a$ and $e$ of the five merging BBHs formed in
three-body interactions due to GW emission. The markers depict $a$ and $e$ of
the final merging binary black holes. The four grey horizontal lines indicate
the semimajor axes at which the rest-frame GW frequency (twice the orbital
frequency) is $f_{\rm GW}=10^{-4}$, $10^{-2}$, $1$, and $10^{2}$ Hz,
respectively.
## 5 Discussion
### 5.1 Formation of merging binary black holes
Our simulations show that close three-body encounters between a BH-star binary
and a single BH can create a merging BBH (see the top-left panel of Figure 4).
One possibly dominant formation process we identified is the close interaction
between the star and the incoming BH at the first closest approach, resulting
in a stellar disruption, followed by the formation of a BBH. 5 out of 11 BBHs
formed in our simulations would merge in a Hubble time via GW emission. The
semimajor axes of the merging BBHs are $\lesssim 114\,{\rm R}_{\odot}$ and
their eccentricities are quite high, $0.66\lesssim e\lesssim 0.97$. If the
required conditions are met ($r_{\rm p}\simeq 0.5\,a$, encounters between the
star and the incoming BH at the first closest approach), this type of
encounters can form, albeit likely rarely, a very compact eccentric BBH:
$t_{\rm GW}\simeq 10^{4}$ yr in Model 20. $a2b1/2\phi 180i150$.
To see whether the merging BBHs can have residual eccentricities when they
enter the frequency band of LIGO (10 Hz to 10kHz), we evolve the five binaries
assuming their orbits evolve purely via GW emission until $t_{\rm GW}=P$,
where $P$ is the binary orbital period. We solve Equations 5.6 and 5.7 in
Peters (1964) simultaneously using a 4th-order Runge-Kutta method with an
adoptive step size of $10^{-3}t_{\rm GW}$. As a sanity check, we confirmed
that our numerical solutions are consistent with the analytic solution
(Equation 5.11 in Peters 1964) within fractional errors of $\lesssim 10^{-8}$.
Figure 11 shows the evolution of $a$ and $e$ of the five merging BBHs,
starting from those found in our simulations (marked as scatters near the top-
left corner). As shown in the figure, by the time the BBHs enter the LIGO
frequency band, their residual eccentricities would be very small
($e<10^{-5}$).
Nonetheless, the circumbinary gas produced by the disruption of the star may
affect the (at least early) orbital evolution, which may hence deviate from
the purely GW-driven evolution considered above. The gas-binary interaction
and resulting binary evolution remains an active topic of study. A growing
number of numerical works have suggested that a binary surrounded by a
circumbinary disk can expand (e.g., Miranda et al., 2017; Muñoz et al., 2019;
Duffell et al., 2020) and can be driven into an eccentric orbit (e.g., Zrake
et al., 2021; D’Orazio & Duffell, 2021), depending on the disk and binary
parameters, as opposed to the predictions from the commonly held picture of
surrounding gas driving binaries into shrinking circular orbits (e.g.,
Armitage & Natarajan, 2002). However, given the limited parameter space
explored in previous work, it is not straightforward to predict the evolution
of our unequal-mass, very eccentric BBHs surrounded by a possibly misaligned
disk, based on the results from the previous work.
The remaining 6 BBHs with GW-driven merger timescales longer than a Hubble
time are hard binaries in typical stellar cluster environments. This means
that those binaries could become potential GW event candidates via weak
interactions with other objects and a few strong interactions like the ones
considered in this study.
### 5.2 Electromagnetic counterparts of binary black hole merger
The close association of BBHs and stellar disruptions can have important
implications for EM counterparts of BBH mergers. At the time the BBH forms,
there would be a prompt EM transient phenomenon due to the stellar disruption.
The very high accretion rate (Figure 10), along with the accretion-driven BH
spin and magnetic field of debris inherited from the star, suggests that a jet
can be launched. For such a case, the luminosity powered by the jet would
track the accretion rate as $\propto\dot{M}c^{2}$ with an uncertain efficiency
factor. We also found that both BHs can be surrounded by the stellar debris
and accrete, possibly suggesting that both BHs may be able to launch jets
simultaneously, potentially leading to a unique observational signature.
In addition to the prompt EM emission, the existence of the surrounding gas
when the BBH forms may result in a possible EM counterpart at the time of
merger. This is a quite similar situation as found in Ryu et al. (2022) where
an initially hard BBH encounters with a single star and becomes surrounded by
gas debris after disrupting the star. Perna et al. (2016) studied the
evolution of an initially hyper-Eddington accretion disk which cools and shuts
down the magnetorotational instability before the disk material is fully
accreted. Under these conditions, the “dead disk” is expected to survive until
the BBHs merge, and to heat up and re-ignite during the merger process, hence
yielding a possible EM counterpart to the GW event.
### 5.3 Varieties of encounters
Although we consider three-body encounters between a circular binary and a
single object with similar masses (the largest mass ratio is 0.5), there could
be a variety of these types of events involving, e.g., initially eccentric
binaries and a wide range of masses of encountering objects.
Encounters involving massive stars (i.e., $10\,{\rm M}_{\odot}$) are likely to
occur during the early evolutionary stages of star clusters unless there is
another episode of star formation, since stars with mass $>10\,{\rm
M}_{\odot}$ would collapse to compact objects in tens of Myrs. Therefore, over
the full cluster lifetime, the overall rate would indeed be higher for
encounters involving less massive MS stars because such binaries would survive
longer. Using Monte Carlo simulations of globular clusters, Kremer et al.
(2018) showed that up to 10 detached BH-MS binaries can exist in clusters at
|
2021
[1]Wataru Iwashita
[1]Department of Mechanical Science and Bioengineering, Osaka University, 1-3
Machikaneyama, Toyonaka, 560-8531, Osaka, Japan
2]Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, 252-5258, Kanagawa, Japan
# Control of Static Friction by Designing Grooves on Friction Surface
<EMAIL_ADDRESS>Hiroshi Matsukawa Michio Otsuki * [
###### Abstract
This study numerically investigated the friction of viscoelastic objects with
grooves. A 3D viscoelastic block with grooves on a rigid substrate is slowly
pushed from the lateral side under uniform pressure on the top surface. The
local friction force at the interface between the block and the substrate
obeys Amontons’ law. Numerical results obtained using the finite element
method reveal that the static friction coefficient decreases with increasing
groove width and depth. The propagation of the precursor slip is observed
before bulk sliding. Furthermore, bulk sliding occurs when the area of slow
precursor slip reaches a critical value, which decreases with increasing
groove size. A theoretical analysis based on a simplified model reveals that
the static friction coefficient is related to the critical area of the
precursor, which is determined by the instability of the precursor. A scaling
law for the critical area is theoretically predicted, and it indicates that
the decrease in the effective viscosity due to the formation of the grooves
leads to a decrease in the static friction coefficient. The validity of the
theoretical prediction is numerically confirmed.
###### keywords:
Static friction coefficient, Groove design, Precursor slip, Amontons’ law,
Viscoelastic object
### Graphical Abstract
## 1 Introduction
Friction forces occur in different situations, such as sliding parts of
machines and contact surfaces between tires and the ground, and prevent the
relative motion between two objects in contact. Friction forces are desirable
in applications requiring low slippage, whereas they are undesirable in
sliding parts of machines due to energy loss. Therefore, the control of
friction forces is important in engineering Bowden1950 ; Persson2000 ;
Popov2017_text ; Rabinowicz1995 ; Dowson1998 ; Bhushan2013 ; Baumberger2006 .
One of the known methods to control friction forces is designing friction
surfaces by forming grooves. Generally, grooves on the surfaces of tires and
sliding parts of machines are formed to reduce undesirable lubrication in wet
conditions, which leads to a decrease in the friction coefficient and results
in accidental slippage Li2004 ; Li2005 ; Li2006 ; Yamaguchi2012 . However, the
dependence of friction force on grooves in dry conditions has not been
established clearly.
Generally, for friction between solids in dry conditions, Amontons’ law is
expected to hold Bowden1950 ; Persson2000 ; Popov2017_text ; Rabinowicz1995 ;
Dowson1998 ; Bhushan2013 ; Baumberger2006 . According to Amontons’ law, the
friction coefficient does not depend on the external pressure or size and
shape of the object. However, the phenomenological explanation of Amontons’
law is based on the adhesion of microscopic asperities at the friction
interface Bowden1950 ; Persson2000 ; Popov2017_text ; Rabinowicz1995 ;
Dowson1998 ; Bhushan2013 ; Baumberger2006 ; Archard1957 ; Dieterich1996 , and
implicitly assumes the uniformity of the stress field. For macroscopic objects
associated with the non-uniform stress field, Amontons’ law may not hold.
Therefore, the friction coefficient may depend on the shape of the macroscopic
objects in dry conditions.
In fact, recent studies on the friction of objects with flat friction surfaces
have shown that Amontons’ law is not satisfied when the non-uniformity of the
stress field is significant Bouissou1998 ; Ben-David2011 ; Otsuki2013 ;
Katano2014 ; IwashitaSciRep2023 . In Refs. 16; 18, numerical simulations and
analysis of a simplified model have revealed the mechanism of breakdown of
Amontons’ law in viscoelastic materials. The analysis clarified that the local
precursor slip before bulk sliding due to the non-uniform stress field leads
to the breakdown of Amontons’ law, and that the static friction coefficient
exhibits characteristic load dependence. The relationship between the
precursor slip and breakdown of Amontons’ law and the load dependence of the
static friction coefficient have been verified in experiments on acrylic glass
blocks Katano2014 . Precursor slip relates to earthquake Yamaguchi2011 ;
Obara2016 ; Kato2021 ; Petrillo2020 ; Xu2023 and fracture Svetlizky2014 ;
Bayart2016 ; Svetlizky2017 ; Berman2020 ; Gvirtzman2021 ; Kammer2015 ;
Castellano2023 , and has been extensively studied in experiments Katano2014 ;
Yamaguchi2011 ; Xu2023 ; Rubinstein2007 ; Ben-David2011 ; Svetlizky2014 ;
Bayart2016 ; Svetlizky2017 ; Berman2020 ; Gvirtzman2021 ; Maegawa2010 and
numerical simulations Otsuki2013 ; IwashitaSciRep2023 ; Petrillo2020 ;
Maegawa2010 ; Burridge1967 ; deSousa1992 ; Braun2009 ; Capozza2012 ; Ozaki2014
; Scheibert2010 ; Amundsen2012 ; Tromborg2014 ; Radiguet2013 ; Kammer2015 ;
Castellano2023 ; Albertini2021 ; Taloni2015 ; deGeus2019 . However, many of
these studies have considered only flat friction surfaces, and the effects of
grooves in the friction surface on frictional properties and precursor slip
are yet to be discovered.
Recently, several studies have been conducted to reveal the dependence of the
friction coefficient on the macroscopic shape of the friction surface. In
Refs. 45; 46; 47; 48; 49; 50, shapes of friction surfaces were represented by
a spatial dependence of the local friction coefficient in 1D or 2D spring-
block models. The frictional properties of the models vary with the spatial
pattern of the local friction coefficient Capozza2015 ; Costagliola2016 ;
Costagliola2017 ; Costagliola2018 ; Costagliola2022IJSS ; Maegawa2017 , and
these results have been applied to experiments of macroscopic objects
Berardo2019 ; Balestra2022 . However, it is unclear to what extent the results
of the spring-block models with a spatial pattern of local friction
coefficient reflect the effect of the actual surface shape. Experiments with
rubber and gel blocks have also revealed the dependence of the friction
coefficient on the macroscopic shape of the friction surface Maegawa2016 ;
Gao2023 . However, it is unclear whether the results for relatively soft
objects such as rubber and gel can be applied to harder materials where
Amontons’ law is locally satisfied.
In this study, using the finite element method (FEM), we numerically
investigate the friction of a 3D viscoelastic material with grooves in a dry
condition, where the friction force locally obeys Amontons’ law. The
dependence of the static friction coefficient on the groove shape is
investigated. We find that the static friction coefficient is a decreasing
function of the groove width and depth. We also observe that local precursor
slip occurs before bulk sliding of the viscoelastic material. The bulk sliding
occurs when the area of the precursor slip reaches a critical value. The
static friction coefficient is scaled by the normalized critical area of the
precursor slip. The propagation of the precursor slip is analytically studied
based on a simplified model. We derive the conditions for the onset of bulk
sliding and the dependence of the static friction coefficient on the groove
shape. The results show that the static friction coefficient decreases due to
the decrease in effective viscosity as the groove width and depth increase.
Both pillars in the friction surface and main body supporting them play an
important role. Our results aid in the improvement of sliding interface design
by making grooves for both wet and dry conditions.
## 2 Model and Methods
Figure 1: Schematic of the system. (a) Grooved viscoelastic block moving on a
rigid substrate. (b) Cross-section perpendicular to the $y$ direction
indicated as I in (a). (c) The bottom of the block indicated as II in (a). The
blue region represents the contact area between the rigid substrate and the
block.
We consider grooved viscoelastic blocks on a rigid substrate under a uniform
external pressure $P_{\mathrm{ext}}$ with width $W$, length $L$, and height
$H$ along the $x$, $y$, and $z$ axes, respectively, as shown in Fig. 1. The
rigid substrate is at $z=0$. A rigid plate with a width of $W$ and a height of
$0.5H$ pushes the side of the block at $y=0$ and $0.5H\leq z\leq H$ with a
slow constant velocity $V$ along the $y$ direction. This study considers a
longitudinal groove parallel to the $y$ direction, as shown in Fig. 1. The
number of pillars in the friction surface of the block is denoted by $n_{x}$.
The pillars are equally spaced with width $l_{\mathrm{g}}$. The height and
width of the pillar are denoted by $d$ and $W/n_{x}-l_{\mathrm{g}}$,
respectively. The cross-section perpendicular to the $y$ direction is
symmetrical, as shown in Fig. 1b. The pillar is in contact with the rigid
substrate, as shown in Fig. 1c. The ratio $\phi$ of the area of the non-
contact surface to the area of the bottom surface is given by
$\phi=l_{\mathrm{g}}n_{x}/W$, and the contact area of the friction surface is
given by $LW(1-\phi)$. Here, $\phi=0$ corresponds to a rectangular block
without grooves.
The equation of motion for the viscoelastic object is given by
$\rho\ddot{u}_{i}=\sum_{j}\partial_{j}\sigma_{ij}$ (1)
with density $\rho$, displacement vector $\bm{u}$, and stress tensor
$\bm{\sigma}$; where $\sigma_{ij}$ is the $ij$ component of $\bm{\sigma}$,
${u}_{i}$ is the $i$ component of $\bm{u}$, and $\ddot{u}_{i}$ is its second-
order time derivative. The stress $\bm{\sigma}$ is given by
$\bm{\sigma}=\bm{\sigma}^{\mathrm{(E)}}+\bm{\sigma}^{\mathrm{(V)}}$ with the
elastic stress $\bm{\sigma}^{\mathrm{(E)}}$ obeying Hooke’s law and the
viscous stress $\bm{\sigma}^{\mathrm{(V)}}$ proportional to strain rate. We
assume that the viscoelastic material of the block is isotropic. The $ij$
component of the elastic stress tensor $\sigma^{(\mathrm{E})}_{ij}$ is given
by
$\sigma^{\mathrm{(E)}}_{ij}=\frac{E}{1+\nu}\epsilon_{ij}+\frac{\nu
E}{(1+\nu)(1-2\nu)}\sum_{k}\epsilon_{kk}\delta_{ij}$ (2)
with Young’s modulus $E$, Poisson’s ratio $\nu$, Kronecker’s delta
$\delta_{ij}$, and strain tensor $\epsilon_{ij}$. The $ij$ component of the
viscous stress tensor $\sigma^{(\mathrm{V})}_{ij}$ is given by
$\sigma^{\mathrm{(V)}}_{ij}=\eta_{1}\dot{\epsilon}_{ij}+\eta_{2}\sum_{k}\dot{\epsilon}_{kk}\delta_{ij}$
(3)
with the two viscosity coefficients $\eta_{1}$ and $\eta_{2}$ and the strain
rate tensor $\dot{\epsilon}_{ij}$ Landau1986 .
The boundary conditions on the top surface of the block at $z=H$ are given by
$\sigma_{zz}=-P_{\mathrm{ext}}$ and $\sigma_{xz}=\sigma_{yz}=0$. At surfaces
except the top and bottom of the block, free boundary conditions
($\sum_{j}\sigma_{ij}n_{j}=0$) are applied, where $n_{j}$ is the $j$ component
of the normal vector $\bm{n}$ to the surface. The boundary conditions at the
contact surface with the rigid plate at $y=0$ are given by
$\sigma_{xy}=\sigma_{zy}=0$ and $\dot{u}_{y}=V$, where $\dot{u}_{y}$ is the
velocity along the $y$ direction. We set $V$ sufficiently small to push the
block quasi-statically.
The friction between the block bottom and substrate obeys Amontons’ law
locally. Since the substrate is rigid, the $z$-direction displacement $u_{z}$
satisfies $u_{z}\geq 0$. At the bottom, the tangential stress vector
$\bm{t}(x,y)=(\sigma_{xz},\sigma_{yz})$ at the position $(x,y)$ is given by
$\bm{t}=-\frac{\bm{v}}{v}\,\sigma^{\mathrm{(fric)}}\ ,$ (4)
$\sigma^{\mathrm{(fric)}}(x,y)=\mu(v(x,y))\,p(x,y)\ ,$ (5)
where $\sigma^{\mathrm{(fric)}}$ is the frictional stress,
$\bm{v}(x,y)=(\dot{u}_{x},\dot{u}_{y})$ is the slip velocity vector with
velocities along the $i$ direction $\dot{u}_{i}$, and
$v(x,y)=\lvert\bm{v}\rvert$ is the slip velocity ccm2006 . The bottom pressure
$p(x,y)=-\sigma_{zz}(x,y,z=0)$ is set to satisfy $u_{z}\geq 0$, where $p=0$
for $u_{z}>0$. Here, $\mu(v)$ is the local friction coefficient depending on
$v$. In the static region with $v(x,y)=0$, $\mu(v)$ is lower than
$\mu_{\mathrm{S}}$, and set to balance the local internal shear stress with
the frictional stress. In the slip region with $v(x,y)>0$, $\mu(v)$ is given
by
$\mu(v)=\left\\{\begin{array}[]{ll}\mu_{\mathrm{S}}-\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)v/v_{\mathrm{c}},{}&0<v<v_{\mathrm{c}}\\\
\mu_{\mathrm{K}},{}&v\geq v_{\mathrm{c}}\end{array}\right.\ ,$ (6)
where $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$ are the local static and
dynamic friction coefficients, respectively. Here, $v_{\mathrm{c}}$ is the
characteristic velocity. The local Amontons’ law is expected to hold when a
local region considered in the interface contains a sufficiently large number
of real contact points, and has a negligibly small spatial variation in
internal stress Archard1957 ; Dieterich1996 ; Dieterich1994 .
We numerically solve Eq. (1) using FEM. The viscoelastic block is divided into
cubes with length $\Delta x$, comprising six tetrahedrons. The displacements
of its nodes are evolved based on Eq. (1), and the displacement and velocity
within each element are approximated using linear interpolation. The local
friction coefficient $\mu(v)$ is approximately given as
$\mu(v)=\left\\{\begin{array}[]{lll}\mu_{\mathrm{S}}\,v/v_{\mathrm{e}},{}&0\leq
v\leq v_{\mathrm{e}}\\\
\mu_{\mathrm{S}}-\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)v/v_{\mathrm{c}},{}&v_{\mathrm{e}}<v<v_{\mathrm{c}}\\\
\mu_{\mathrm{K}},{}&v\geq v_{\mathrm{c}}\end{array}\right.$ (7)
with a sufficiently small velocity scale $v_{\mathrm{e}}$. The state with
$0\leq v\leq v_{\mathrm{e}}$ corresponds to the static region, and the state
with $v>v_{\mathrm{e}}$ corresponds to the slip region. We set
$v_{\mathrm{e}}/V=2.5\times 10^{-2}$ to satisfy $v_{\mathrm{e}}\ll
V,v_{\mathrm{c}}$, and use $\Delta x/H=1/48$, $\Delta
t/(H\sqrt{\rho/E})\thickapprox 10^{-6}$, and $V\sqrt{\rho/E}=2.83\times
10^{-5}$. In our simulation, we first apply a uniform pressure
$P_{\mathrm{ext}}$ to the top surface and relax the system to an equilibrium
state. From the time $t=0$ after the relaxation, the rigid plate pushes the
side of the block with a constant velocity $V$, and the calculation continues
until a periodic stick-slip is observed.
We set the length and width of the block to $L/H=4$ and $W/H=1$, respectively.
Qualitatively similar results are obtained for $L/H=2$, as shown in Appendix
A. We adopt $\nu=0.34$, $\eta_{1}/(H\sqrt{\rho E})=2.83$,
$\eta_{2}/\eta_{1}=1$, $\mu_{\mathrm{S}}=0.38$, $\mu_{\mathrm{K}}=0.1$, and
$v_{\mathrm{c}}\sqrt{\rho/E}=4.81\times 10^{-4}$ following previous
simulations Otsuki2013 ; IwashitaSciRep2023 . We select the number of pillars
in the friction surface as $n_{x}=3$, and confirm that the dependence of the
numerical results on $n_{x}$ is small, as shown in Appendix B. In this study,
we investigate the dependence on the external pressure $P_{\mathrm{ext}}$,
fraction of non-contact area $\phi$, and groove depth $d$.
## 3 Results
### 3.1 Numerical Simulation
Figure 2: Ratio of friction force $F_{\mathrm{T}}$ to applied normal force
$F_{\mathrm{N}}$ against displacement of the rigid plate $U$ for
$P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. The thin and thick solid lines
represent the results for $\phi=0$ and $\phi=0.5$, respectively. The thin and
thick horizontal solid lines represent macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ for $\phi=0$ and $\phi=0.5$, respectively. The dotted and
dashed lines represent $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$,
respectively.
Figure 2 shows the friction force $F_{\mathrm{T}}$ against the displacement of
the rigid plate $U=Vt$ at time $t$ for $P_{\mathrm{ext}}/E=0.003$ and
$d/H=0.5$. Here, $F_{\mathrm{T}}$ is given by the force on the rigid plate in
the $y$ direction. In Fig. 2, $F_{\mathrm{T}}$ is normalized by the normal
load $F_{\mathrm{N}}=P_{\mathrm{ext}}LW$ applied to the top surface of the
block. The thin and thick solid lines represent the results for $\phi=0$ and
$\phi=0.5$, respectively. For each $\phi$, $F_{\mathrm{T}}/F_{\mathrm{N}}$
increases approximately linearly with $U$, and rapidly decreases after
reaching a peak value. When the rapid decrease occurs, the entire system
slides, and the block returns to a static state after reaching a minimum value
close to the local dynamic friction coefficient $\mu_{\mathrm{K}}$. The
increase and decrease in $F_{\mathrm{T}}/F_{\mathrm{N}}$ repeat periodically,
which corresponds to stick-slip motion. We define the maximum value of
$F_{\mathrm{T}}/F_{\mathrm{N}}$ in the periodic stick-slip region as the
macroscopic static friction coefficient $\mu_{\mathrm{M}}$, which is lower
than the local static friction coefficient $\mu_{\mathrm{S}}$. Figure 2 shows
that $\mu_{\mathrm{M}}$ for the block with grooves is lower than that for the
flat block.
In Fig. 3, we plot the macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against the groove depth $d$ for different values of $\phi$
with $P_{\mathrm{ext}}/E=0.003$ and $0.006$. Note that the results for
$\phi=0$ are independent of $d$. For each $P_{\mathrm{ext}}$,
$\mu_{\mathrm{M}}$ is a decreasing function of $d$. As $d$ approaches 0,
$\mu_{\mathrm{M}}$ converges to that for $\phi=0$. The macroscopic static
friction coefficient $\mu_{\mathrm{M}}$ is a decreasing function of $\phi$.
These results indicate that the static friction force decreases as the size of
the groove increases. Comparing Fig. 3a and b, we find that $\mu_{\mathrm{M}}$
is a decreasing function of $P_{\mathrm{ext}}$, which is consistent with the
results of previous studies on rectangular blocks without grooves Otsuki2013 ;
Katano2014 ; IwashitaSciRep2023 .
Figure 3: Macroscopic static friction coefficient $\mu_{\mathrm{M}}$ against
$d$ for different values of $\phi$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b)
$P_{\mathrm{ext}}/E=0.006$. The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively. Figure 4: Spatial
distributions of the slip velocity $v$ in the friction surface at $z=0$ for
$U=U_{1},U_{2},U_{3}$, and $U_{4}$ shown in Fig. 2 for
$P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and $d/H=0.5$. The blue area
represents the static region. The yellow-green and yellow areas represent the
slip regions with $v\leq V$ and $v>V$, respectively. The rigid plate pushes
the block at $y=0$. The white area represents the groove region.
In Fig. 4, we present the spatial distributions of the slip velocity $v$ in
the friction surface at $z=0$ for the displacements $U=U_{1},U_{2},U_{3}$, and
$U_{4}$ shown in Fig. 2 with $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and
$d/H=0.5$. Here, we select $U_{1}/L=4.6\times 10^{-3}$, $U_{2}/L=5\times
10^{-3}$, $U_{3}/L=5.4\times 10^{-3}$, and $U_{4}/L=5.57\times 10^{-3}$ in the
periodic stick–slip region. In Fig. 4, the blue area represents the static
region, and the yellow-green and yellow areas represent the sliding regions
with $v\leq V$ and $v>V$, respectively. The quasi-static precursor slip with
$v\leq V$ begins to propagate from the region near the rigid plate at $y=0$
for $U=U_{1}$, and the area of precursor slip expands quasi-statically as $U$
increases to $U_{2}$ and $U_{3}$. After $U_{3}$, the area of precursor slip
develops rapidly, and the entire system begins to slide, leading to bulk
sliding at $U_{4}$ (see Supplementary Videos). During bulk sliding, the slip
velocity $v$ exceeds $V$. We confirm that the displacement due to these slips
is approximately along the $y$ direction for all parameters.
Figure 5: Normalized precursor slip area $\tilde{S}$ against $U/L$ for
$P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. The thin and thick lines represent
the results for $\phi=0$ and $\phi=0.5$, respectively. The thin and thick
horizontal lines represent the normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ for $\phi=0$ and $\phi=0.5$, respectively.
Figure 5 shows the normalized slip area $\tilde{S}=S/[LW(1-\phi)]$ against
$U/L$ for $P_{\mathrm{ext}}/E=0.003$ and $d/H=0.5$. Here, the precursor slip
area $S$, defined by the sum of the yellow-green and yellow areas in Fig. 4,
is normalized by the contact area in friction surface $LW(1-\phi)$. When
$\tilde{S}=0$, the entire friction surface is static, while $\tilde{S}=1$
indicates bulk sliding where the entire friction surface is sliding. The thin
and thick solid lines represent the results for $\phi=0$ and $\phi=0.5$,
respectively. The normalized precursor slip area $\tilde{S}$ increases
gradually with $U$. When $\tilde{S}$ reaches a threshold value
$\tilde{S}_{\mathrm{c}}$, the propagation speed of $\tilde{S}$ suddenly
increases, and $\tilde{S}$ reaches unity, which corresponds to the bulk
sliding. The oscillation of $\tilde{S}$ in Fig. 5 and small drops of
$F_{\mathrm{T}}/F_{\mathrm{N}}$ in Fig. 2 before bulk sliding is caused by the
sequence of the bounded rapid precursors (BRPs) Otsuki2013 . Each BRP reduces
the stress and $\tilde{S}$, but they both recover quickly due to a slight
increase in the driving force. The BRP becomes significant depending on the
values of the parameters. We evaluate the critical area
$\tilde{S}_{\mathrm{c}}$ as the maximum value of $\tilde{S}$ in the sequence
of the BRP. Figure 5 shows that $\tilde{S}_{\mathrm{c}}$ decreases with
increasing $\phi$.
Figure 6: Normalized critical area of precursor slip $\tilde{S}_{\mathrm{c}}$
against $d$ for different values of $\phi$ with (a) $P_{\mathrm{ext}}/E=0.003$
and (b) $P_{\mathrm{ext}}/E=0.006$.
In Fig. 6, we show the normalized critical area of the precursor slip
$\tilde{S}_{\mathrm{c}}$ against the groove depth $d$ for different values of
$\phi$ with $P_{\mathrm{ext}}/E=0.003$ and $0.006$. We find that
$\tilde{S}_{\mathrm{c}}$ is a decreasing function of $d$. As $d$ approaches 0,
$\tilde{S}_{\mathrm{c}}$ approaches that for $\phi=0$. We also find that
$\tilde{S}_{\mathrm{c}}$ decreases with increasing $\phi$. Comparing Fig. 6a
and b, we see that $\tilde{S}_{\mathrm{c}}$ is a decreasing function of
$P_{\mathrm{ext}}$, which is consistent with the results of previous studies
on blocks without grooves Otsuki2013 ; Katano2014 ; IwashitaSciRep2023 .
Figure 7: Macroscopic static friction coefficient $\mu_{\mathrm{M}}$ against
$\tilde{S}_{\mathrm{c}}$ for different values of $\phi$ and $d$. The filled
and open symbols represent the results for $P_{\mathrm{ext}}/E=0.003$ and
$P_{\mathrm{ext}}/E=0.006$, respectively. The solid line represents the
analytical results given by Eq. (20). The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.
The dependence of the macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ on $\phi$ and $d$ shown in Fig. 3 is similar to that of
$\tilde{S}_{\mathrm{c}}$ shown in Fig. 6. This similarity indicates a close
relation between $\mu_{\mathrm{M}}$ and $\tilde{S}_{\mathrm{c}}$. In fact, as
shown in Fig. 7, $\mu_{\mathrm{M}}$ is an almost linear function of
$\tilde{S}_{\mathrm{c}}$ for different values of $\phi$ and $d$ with
$P_{\mathrm{ext}}/E=0.003$ and $0.006$. Figure 7 also shows that
$\mu_{\mathrm{M}}$ lies between $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$.
This scaling of $\mu_{\mathrm{M}}$ using $\tilde{S}_{\mathrm{c}}$ is
consistent with the results of previous studies on blocks without grooves
Otsuki2013 ; Katano2014 ; IwashitaSciRep2023 .
Figure 8: Spatial distribution of stress in the friction surface at $U=U_{3}$
for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$, and $d/H=0.5$. (a) Spatial
distribution of ratio of frictional stress $\sigma^{\mathrm{(fric)}}$ to
bottom pressure $p$. (b) Spatial distribution of $p$. The rigid plate pushes
the block at $y=0$. The white area in (a) represents the region without
contact. The white area in (b) represents the groove area.
Figure 8a shows the spatial distribution of the ratio of the frictional stress
$\sigma^{\mathrm{(fric)}}$ to the bottom pressure $p$ in the friction surface
at $U=U_{3}$ for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$ and $d/H=0.5$. Here,
$U=U_{3}$ indicates the state just before bulk sliding, as shown in Figs. 2
and 5. Note that the local static friction in the static state with $v=0$
takes any values for $0<\sigma^{\mathrm{(fric)}}/p<\mu_{\mathrm{S}}$. As shown
in Supplementary Videos and previous studies Otsuki2013 ; IwashitaSciRep2023 ,
the ratio $\sigma^{\mathrm{(fric)}}/p$ returns to the value near
$\mu_{\mathrm{K}}$ in the entire area just after the bulk sliding. As the
block is pushed, $\sigma^{\mathrm{(fric)}}/p$ reaches the local static
friction coefficient $\mu_{\mathrm{S}}$ near the region pushed by the rigid
plate. The area with $\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{S}}$
gradually increases as $U$ increases. The region with
$\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{S}}$ corresponds to the
slip region at $U=U_{3}$ in Fig. 4, while $\sigma^{\mathrm{(fric)}}/p$ remains
near $\mu_{\mathrm{K}}$ in the static region.
Figure 8b shows the spatial distribution of the bottom pressure $p$ at
$U=U_{3}$ for $P_{\mathrm{ext}}/E=0.003$, $\phi=0.5$ and $d/H=0.5$. Although a
uniform pressure $P_{\mathrm{ext}}$ is applied at the top surface, the spatial
average of $p$ becomes $P_{\mathrm{ext}}/(1-\phi)$, because the contact area
in the friction surface, $LW(1-\phi)$, is smaller than the area of the top
surface, $LW$, due to the grooves. We confirm that the bottom pressure is
$p\approx P_{\mathrm{ext}}/(1-\phi)$ in most areas except for the regions near
$y=0$ and $L$. The spatial distribution of $p$ is almost independent of the
time $t$, as shown in Supplementary Videos.
### 3.2 Theoretical Analysis
We theoretically analyze the effect of the longitudinal grooves shown in Sect.
3.1 based on a simplified model Otsuki2013 ; IwashitaSciRep2023 . The
precursor slip is approximately uniform in the $x$ direction and propagates
toward the $y$ direction, as shown in Fig. 4. Therefore, we neglect
displacements in the $z$ and $x$ directions and consider only the
$y$-dependent displacement along the $y$ direction. Since the bottom pressure
$p$ at $z=0$ is approximately uniform, as shown in Fig. 8b, we assume
$p=P_{\mathrm{ext}}/(1-\phi)$. Additionally, since the deformation is
significant in the region near the bottom before bulk sliding in our 3D
simulations, as shown in Appendix C, we focus on the slip and deformation in
the region $0\leq z/H\leq\alpha$ with a constant $\alpha$, as shown in the red
shaded area in Fig. 9.
Figure 9: (a) Schematic of the derivation of the simplified model for grooved
viscoelastic block. (b) Cross-section perpendicular to the $y$ direction
indicated as I in (a). The red shaded areas represent the region from the
bottom to the height $z=\alpha H$. The dotted rectangle represents the element
with infinitesimal width $\mathrm{d}y$.
We consider the equation of motion for a thin element at $y$ with small width
$\mathrm{d}y$ indicated by the dotted rectangle in Fig. 9a. The mass of the
element is given by $\rho A(\phi,d)\mathrm{d}y$, where $A(\phi,d)$ is the
cross-sectional area of the red region in Fig. 9b excluding the groove. In
Fig. 9a, the forces acting on the left and right surfaces of that element are
given by $A(\phi,d)\sigma_{yy}(y,t)$ and
$A(\phi,d)\sigma_{yy}(y+\mathrm{d}y,t)$, respectively. Here, the normal stress
in the $y$ direction is denoted by $\sigma_{yy}$. The friction force acting on
the bottom is given by $\mu P_{\mathrm{ext}}W\mathrm{d}y$. The equation of
motion for the displacement $q_{y}(y,t)$ of the thin element along the $y$
direction is given by
$\rho
A(\phi,d)\mathrm{d}y\,\ddot{q}_{y}(y,t)=A(\phi,d)\left[\sigma_{yy}(y+\mathrm{d}y,t)-\sigma_{yy}(y,t)\right]-\mu(\dot{q}_{y}(y,t))P_{\mathrm{ext}}W\mathrm{d}y\
,$ (8)
where $\dot{q}_{y}(y,t)$ and $\ddot{q}_{y}(y,t)$ are the first and second-
order time derivatives of $q_{y}(y,t)$, respectively. We assume a plane stress
state, where the normal stress $\sigma_{yy}(y,t)$ is given by
$\sigma_{yy}(y,t)=E_{1}\frac{\partial q_{y}(y,t)}{\partial
y}+\eta_{\mathrm{t}}\frac{\partial\dot{q}_{y}(y,t)}{\partial y}$ (9)
with the elastic modulus $E_{1}=E/[(1+\nu)(1-\nu)]$ and viscous modulus
$\eta_{\mathrm{t}}=\eta_{1}(\eta_{1}+2\eta_{2})/(\eta_{1}+\eta_{2})$.
The cross-sectional area $A(\phi,d)$ is given by
$A(\phi,d)=A_{0}[1-\kappa(\phi,d)]$ (10)
with the cross-sectional area $A_{0}=\alpha HW$ for $\phi=0$. Here,
$\kappa(\phi,d)$ is the decreasing rate of the cross-sectional area by the
groove,
$\kappa(\phi,d)=\left\\{\begin{array}[]{lll}\phi\,d/(\alpha H),{}&0\leq
d\leq\alpha H\\\ \phi,{}&\alpha H<d\leq H\end{array}\right.\ .$ (11)
Substituting Eqs. (9) and (10) into Eq. (8) and taking the limit of
$\mathrm{d}y\rightarrow 0$, we obtain
$\rho(1-\kappa)\ddot{q}_{y}(y,t)=(1-\kappa)\left[E_{1}\frac{\partial^{2}q_{y}(y,t)}{\partial
y^{2}}+\eta_{\mathrm{t}}\frac{\partial^{2}\dot{q}_{y}(y,t)}{\partial
y^{2}}\right]-\frac{\mu(\dot{q}_{y}(y,t))P_{\mathrm{ext}}}{\alpha H}\ .$ (12)
The boundary conditions are given by $\partial q_{y}(L,t)/\partial y=0$ for
the free boundary at $y=L$ and $q_{y}(0,t)=U$ for the fixed boundary at $y=0$.
We set $t=0$ just after the bulk sliding, where the friction coefficient is
given by $\mu=\mu_{\mathrm{K}}$. When a precursor slip occurs with the
normalized slip area $\tilde{S}$ for $U>0$, the friction coefficient is given
by $\mu=\mu_{\mathrm{S}}$ in the region $0\leq y/L\leq\tilde{S}$, because the
slip distances of the precursors are significantly smaller than that in bulk
sliding. In the other regions, $\mu$ remains $\mu_{\mathrm{K}}$ due to the
frictional stress drop after the bulk sliding. This is confirmed by direct
numerical calculations of Eq. (12) and qualitatively consistent with the
results in Sect. 3.1. For sufficiently slow driving with
$\ddot{q}_{y}\thickapprox 0$ and $\dot{q}_{y}\thickapprox 0$, the quasi-static
solution of $q_{y}$ in Eq. (12) is analytically derived as described in
Appendix D. In this quasi-static solution $q_{\mathrm{a}}(y)$, $\tilde{S}$ is
given as an increasing function of $U$.
We conduct a stability analysis based on Eq. (12) following the procedure in
the previous studies Otsuki2013 ; IwashitaSciRep2023 . Substituting
$q_{y}(y,t)=q_{\mathrm{a}}(y)+\delta q(y,t)$ into Eq. (12) with the
perturbation $\delta q(y,t)$, we obtain the equation for $\delta q(y,t)$ as
$\rho(1-\kappa)\delta\ddot{q}(y,t)=(1-\kappa)\left[E_{1}\frac{\partial^{2}\delta
q(y,t)}{\partial
y^{2}}+\eta_{\mathrm{t}}\frac{\partial^{2}\delta\dot{q}(y,t)}{\partial
y^{2}}\right]-\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}}{v_{\mathrm{c}}\alpha
H}\delta\dot{q}(y,t)\ .$ (13)
Note that $\delta q(y,t)$ has a non-zero value in the region
$0<y/L<\tilde{S}$, and $\delta q(y,t)$ remains zero in the other region due to
static friction. Since the perturbation $\delta q(y,t)$ is zero for $y=0$ and
$\tilde{S}<y/L<1$, $\delta q(y,t)$ is expressed as
$\delta q(y,t)=\sum_{m}q_{m}e^{\lambda_{m}t}\sin{k_{m}\xi}\ ,$ (14)
where $m$ is a positive integer, $q_{m}$ is a constant, $\lambda_{m}$ is the
eigenvalue of the time evolution operator with $k_{m}=m\pi$ and
$\xi=y/(\tilde{S}L)$. Substituting Eq. (14) into Eq. (13), multiplying by
$2\sin{k_{n}\xi}$ with positive integer $n$, and integrating in
$0<y<\tilde{S}L$, we obtain
$(1-\kappa)\rho
L^{2}\lambda_{m}^{2}+(1-\kappa)E_{1}\frac{k_{m}^{2}}{\tilde{S}^{2}}+(1-\kappa)\eta_{\mathrm{t}}\frac{k_{m}^{2}}{\tilde{S}^{2}}\lambda_{m}-\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}L^{2}}{v_{\mathrm{c}}\alpha
H}\lambda_{m}=0\ .$ (15)
The perturbation $\delta q(y,t)$ is unstable in the case of
$\real\lambda_{m}>0$ and $\imaginary\lambda_{m}\neq 0$. The latter condition,
$\imaginary\lambda_{m}\neq 0$, indicates the oscillatory motion. However, the
backward motion associated with the oscillation is inhibited by the static
friction. Then, the instability induces only forward motion, that is, the BRP,
and does not increase further. Therefore, the perturbation develops and causes
bulk sliding only in the case of $\real\lambda_{m}>0$ and
$\imaginary\lambda_{m}=0$. In Eq. (15), we find that the stability conditions
of the system are generally determined by the competition between the
viscosity represented by the third term and velocity-weakening friction
represented by the fourth term on the left-hand side. These terms are
considered as the stabilizing and destabilizing factors, respectively. The
stabilizing factor decreases due to $\tilde{S}^{-2}$ in the third term as the
precursor slip area $\tilde{S}$ increases. When $\tilde{S}$ reaches the
critical area $\tilde{S}_{\mathrm{c}}$, the destabilizing factor overwhelms
the stabilizing factor, and the perturbation $\delta q(y,t)$ becomes unstable.
Therefore, $\tilde{S}$ increases rapidly just after reaching
$\tilde{S}_{\mathrm{c}}$, as shown in Fig. 5, and bulk sliding occurs. The
viscous term is proportional to $1-\kappa$. The velocity-weakening friction
term is proportional to the load on the top of the block but independent of
$\kappa$. Thus, if $\kappa(\phi,d)$ increases by increasing $\phi$ and $d$,
the viscosity becomes effectively smaller, which leads to the decrease of
$\tilde{S}_{\mathrm{c}}$.
As $\tilde{S}$ increases, the mode with $m=1$ in Eq. (14) becomes unstable
first, which determines $\tilde{S}_{\mathrm{c}}$. By definition, the maximum
value of $\tilde{S}_{\mathrm{c}}$ does not exceed $1$, and for
$\tilde{S}_{\mathrm{c}}<1$, $\tilde{S}_{\mathrm{c}}$ satisfies
$\pi^{2}(1-\kappa)\eta_{\mathrm{t}}\tilde{S}_{\mathrm{c}}^{-2}+2\pi(1-\kappa)L\sqrt{\rho
E_{1}}\tilde{S}_{\mathrm{c}}^{-1}=\frac{\left(\mu_{\mathrm{S}}-\mu_{\mathrm{K}}\right)P_{\mathrm{ext}}L^{2}}{v_{\mathrm{c}}\alpha
H}\ ,$ (16)
which is derived from Eq. (15). Therefore, $\tilde{S}_{\mathrm{c}}$ is given
by
$\tilde{S}_{\mathrm{c}}=\min(\tilde{S}_{\mathrm{c}}^{*},1)\ ,$ (17)
where $\min(a,b)$ is a function that takes the smaller value between $a$ and
$b$, and $\tilde{S}_{\mathrm{c}}^{*}$ is the solution of Eq. (16) given by
$\tilde{S}_{\mathrm{c}}^{*}=\dfrac{\pi\eta_{\mathrm{t}}}{L\left(-\sqrt{\rho
E_{1}}+\sqrt{\rho
E_{1}+\frac{\mu_{\mathrm{S}}-\mu_{\mathrm{K}}}{1-\kappa}\,\frac{P_{\mathrm{ext}}\eta_{\mathrm{t}}}{v_{\mathrm{c}}\alpha
H}}\right)}\ .$ (18)
For $\tilde{S}_{\mathrm{c}}\ll 1$, $\tilde{S}_{\mathrm{c}}$ is approximately
given by
$\tilde{S}_{\mathrm{c}}\simeq\pi\left\\{\frac{[1-\kappa(\phi,d)]\alpha}{\mu_{\mathrm{S}}-\mu_{\mathrm{K}}}\,\frac{\eta_{\mathrm{t}}v_{\mathrm{c}}}{P_{\mathrm{ext}}H}\right\\}^{\frac{1}{2}}\frac{H}{L}\
.$ (19)
This result indicates that the normalized critical area of the precursor slip
$\tilde{S}_{\mathrm{c}}$ is a decreasing function of $P_{\mathrm{ext}}$ and
the size of grooves, because $\kappa(\phi,d)$ in Eq. (19) increases with
$\phi$ and $d$, as described in Eq. (11). These analytical results are
qualitatively consistent with those of the FEM simulations shown in Fig. 6.
The macroscopic static friction coefficient $\mu_{\mathrm{M}}$ can be
analytically derived in our simplified model Otsuki2013 ; IwashitaSciRep2023 .
Since the ratio of the local frictional stress to bottom pressure is
$\mu_{\mathrm{S}}$ in the slip region and $\mu_{\mathrm{K}}$ in the static
region, $\mu_{\mathrm{M}}$ just before the bulk sliding is given by
$\mu_{\mathrm{M}}=\mu_{\mathrm{K}}+(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})\tilde{S}_{\mathrm{c}}\
.$ (20)
This result is qualitatively consistent with the FEM simulations shown in Fig.
7, where Eq. (20) is represented by the solid line.
Substituting Eq. (20) into Eq. (19), we obtain
$\mu_{\mathrm{M}}-\mu_{\mathrm{K}}\simeq\pi\left\\{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})[1-\kappa(\phi,d)]\alpha\,\frac{\eta_{\mathrm{t}}v_{\mathrm{c}}}{P_{\mathrm{ext}}H}\right\\}^{\frac{1}{2}}\frac{H}{L}\
.$ (21)
This equation, together with Eq. (11), implies that the macroscopic static
friction coefficient $\mu_{\mathrm{M}}$ is a decreasing function of
$P_{\mathrm{ext}}$, $\phi$ and $d$. The analytical results are qualitatively
consistent with the FEM simulations shown in Fig. 3.
Figure 10: Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ against $\kappa(\phi,d)$ for different values of
$\phi$ and $d$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b)
$P_{\mathrm{ext}}/E=0.006$. The symbols represent the results of the FEM
simulations. The solid lines represent the analytical results given by Eq.
(17) with $\alpha=0.2$. Figure 11: Macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ for different values of $\phi$ and
$d$ with (a) $P_{\mathrm{ext}}/E=0.003$ and (b) $P_{\mathrm{ext}}/E=0.006$.
The symbols represent the results of the FEM simulations. The solid lines
represent the analytical results given by Eqs. (17) and (20) with
$\alpha=0.2$. The dotted and dashed lines indicate $\mu_{\mathrm{S}}$ and
$\mu_{\mathrm{K}}$, respectively.
Equations (19) and (21) indicate that $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ for different $\phi$ and $d$ are scaled by the decreasing
rate of cross-sectional area $\kappa(\phi,d)$. Figures 10 and 11 respectively
show $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ obtained from the FEM
simulations against $\kappa(\phi,d)$. Both $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ are scaled by $\kappa(\phi,d)$ and decrease with increasing
$\kappa(\phi,d)$. The solid line in Fig. 10 represents the analytical result
given by Eq. (17). The solid line in Fig. 11 represents the result given by
Eqs. (17) and (20). Here, we set $\alpha=0.2$ to semi-quantitatively reproduce
the results of the FEM simulations in Sect. 3.1. The deformation before bulk
sliding is significant only in the region $z/H<0.5$, as shown in Appendix C.
This result is consistent with the estimate of $\alpha=0.2$. The numerical
results of FEM are semi-quantitatively consistent with the theoretical
analysis.
These results explain the decreases of $\mu_{\mathrm{M}}$ and
$\tilde{S}_{\mathrm{c}}$ with the increases of $\phi$ and $d$ in Figs. 3 and
6. Here, $\phi$ represents the decrease in the contact area of the pillars,
and $d$ represents the decrease in the size of the main body. These values
determine the decreasing rate of the cross-sectional area $\kappa(\phi,d)$.
Thus, it can be concluded that $\mu_{\mathrm{M}}$ and $\tilde{S}_{\mathrm{c}}$
decrease with increasing $\phi$ or $d$ because of the decrease in effective
viscoelasticity due to the reduction of the cross-sectional area, leading to
the decline of the stability and bulk sliding with a smaller size of the
precursor slip.
## 4 Discussion
Generally, grooves on friction surfaces are designed to control lubrication
properties in wet conditions. It is considered that the friction coefficient
at the wet interface increases with the width and depth of the grooves,
because they can eject more lubricant from the friction interface Li2004 ;
Li2005 ; Li2006 ; Yamaguchi2012 . However, this study reveals that the groove
size also affects friction in dry conditions. The static friction coefficient
in dry conditions decreases with increases in groove width and depth. This is
opposite to the usual consideration for the friction in the wet case. Even in
wet conditions, the friction force at the solid-solid interface determines the
total friction force after the ejection of the lubricant. These results should
aid in improving the design of sliding interfaces with grooves for both wet
and dry conditions.
The influence of the groove shape on friction in dry conditions has recently
been investigated based on different models, where the effect of grooves is
represented by the spatial distribution of the local friction coefficient
Capozza2015 ; Costagliola2016 ; Costagliola2017 ; Maegawa2017 ;
Costagliola2018 ; Costagliola2022IJSS ; Berardo2019 ; Balestra2022 . These
previous works report a decrease in the static friction coefficient by forming
longitudinal grooves, which is consistent with our results. However, the
effect of the depth of the longitudinal grooves is ignored in their models,
while our study is based on a realistic 3D system and reveals its importance.
Moreover, previous studies have investigated different patterns of grooves
including transversal grooves. Their results have suggested that complex
shapes in the friction surface used in various industrial products such as
shoe soles and tires and adopted on the surface of living things such as
snakes Baum2014BJN ; Baum2014TL ; Costagliola2017 affect the frictional
properties. Therefore, the extension of our study to these complex shapes will
lead to more efficient guiding principles for groove design.
In this study, the parameter values for virtual materials are adopted to
reduce the computational load, which does not correspond to those for real
materials. These are selected to compare the results with those in previous
simulations Otsuki2013 ; IwashitaSciRep2023 . However, the mechanism of
changes in the friction coefficient revealed by our theory in Sect. 3.2 is
universal and independent of specific parameter values. The numerical results
for flat friction surfaces in Ref. 16 have been reproduced in experiments of
acrylic glass Katano2014 . Therefore, we expect our results will be
experimentally verified in future work.
## 5 Conclusion
Friction surfaces of products such as shoe soles, tires, and sliding parts of
machines have grooves. Several studies on grooves have focused on controlling
lubrication properties via their design. However, it has been empirically
known that grooves affect friction even in dry conditions, although no
theoretical explanation exists. In this study, we have performed numerical
simulations of viscoelastic objects using 3D FEM to clarify the effect of
longitudinal grooves on static friction in a dry condition. We have revealed
that the static friction coefficient is a decreasing function of the groove
size, and that precursor slip occurs before bulk sliding. The static friction
coefficient is scaled by the normalized critical area of the precursor slip.
Based on the simplified model, we have theoretically derived the equation for
the static friction coefficient depending on the groove size. The theoretical
result indicates that the static friction coefficient decreases with the
decreasing rate of the cross-sectional area in the viscoelastic object. The
decrease in the cross-sectional area reduces the effective viscosity, which
enhances the instability of the precursor slip and decreases the static
friction coefficient. Our results provide new guiding principles for groove
design for static friction control beyond the empirical laws for both wet and
dry conditions. Investigation of different types of grooves and their effect
on dynamic friction will be the subject of future studies.
## Data Availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
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*Acknowledgments This research used computational resources of Fugaku at the RIKEN Center for Computational Science (Project: hp220372), Yukawa-21 at YITP, Kyoto University, SQUID at the Cybermedia Center, Osaka University, ohtaka and kugui at ISSP, the University of Tokyo, the supercomputers at RCCS, Okazaki, Japan (Project: 23-IMS-C126), and JSS3 at JAXA. We would like to thank Editage (www.editage.com) for English language editing.
*Funding This work was supported by the Japan Society for the Promotion of Science KAKENHI (Grant Numbers JP21H01006, JP22KJ2190, JP23K03248, and JP23K03252).
## Author information
*Contributions All authors contributed to the study conception and design. The FEM simulation and analysis based on a simplified model are performed by WI. The first draft of the manuscript was written by WI, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
## Ethics declarations
*Competing interests The authors declare no competing interests.
## Supplementary information
The online version contains supplementary material.
## Appendix A Results for $L/H=2$
Figure 12: (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $d$ for different values of $\phi$ with $L/H=2$ and
$P_{\mathrm{ext}}/E=0.003$. The dotted and dashed lines indicate
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.
In this appendix, we show the results for $L/H=2$. Figure 12a and b
respectively show $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ against $d$
for different values of $\phi$ with $P_{\mathrm{ext}}/E=0.003$. We find that
$\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ are decreasing functions of
$\phi$ and $d$, which is consistent with the results for $L/H=4$, as shown in
Figs. 3 and 6. Compared to Figs. 3a and 6a for the identical pressure, we find
that $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ increase with decreasing
$L$.
Figure 13a and b respectively show $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ with $P_{\mathrm{ext}}/E=0.003$.
We find that $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$ are decreasing
functions of $\kappa(\phi,d)$. The solid lines show the analytical results for
$\alpha=0.2$ based on Eqs. (17) and (20). It is shown that $\kappa(\phi,d)$
for $\alpha=0.2$ can scale $\tilde{S}_{\mathrm{c}}$ and $\mu_{\mathrm{M}}$
even for the system with $L/H=2$. The decreases in $\tilde{S}_{\mathrm{c}}$
and $\mu_{\mathrm{M}}$ by increasing $L$ is consistent with Eqs. (19) and (21)
for the theoretical analysis in Sect. 3.2.
Figure 13: (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $\kappa(\phi,d)$ for different values of $\phi$ and
$d$ with $L/H=2$ and $P_{\mathrm{ext}}/E=0.003$. The solid lines represent the
analytical results of Eqs. (17) and (20) with $\alpha=0.2$. The dotted and
dashed lines indicate $\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.
## Appendix B Dependence on Number of Pillars $n_{x}$ in Friction Surface
Figure 14: (a) Normalized critical area of precursor slip
$\tilde{S}_{\mathrm{c}}$ and (b) macroscopic static friction coefficient
$\mu_{\mathrm{M}}$ against $n_{x}$ for $P_{\mathrm{ext}}/E=0.003$ and $0.006$
with $L/H=4$, $d/H=0.5$, and $\phi=0.5$. The dotted and dashed lines represent
$\mu_{\mathrm{S}}$ and $\mu_{\mathrm{K}}$, respectively.
In this appendix, we verify the dependence of the $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ on the number of pillars of the block $n_{x}$ for $L/H=4$,
$d/H=0.5$, and $\phi=0.5$. Figure 14a and b show $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ against $n_{x}$ for $P_{\mathrm{ext}}/E=0.003$ and $0.006$,
respectively. In Fig. 14, we find that $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ are almost independent of $n_{x}$. According to the
theoretical results given by Eqs. (19) and (21), $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ depend on the decreasing rate of the cross-sectional area
$\kappa(\phi,d)$, which is independent of $n_{x}$, as described in Eq. (11).
This analytical result explains the behaviors of $\tilde{S}_{\mathrm{c}}$ and
$\mu_{\mathrm{M}}$ in Fig. 14.
## Appendix C Acceleration Distribution
Figure 15 demonstrates the spatial distribution of the $y$-direction
acceleration $\ddot{u}_{y}$ for $P_{\mathrm{ext}}/E=0.003$, $d/H=0.5$, and
$\phi=0.5$ in the cross-section at $x=0.5W$ with $U/L=5.43\times 10^{-3}$,
just before bulk sliding. The critical length in the $y$ direction of the
precursor slip is estimated to be $\tilde{S}_{\mathrm{c}}L$, because the
precursor slip propagates uniformly in the $x$-direction, as shown in Fig. 4.
The acceleration around the region of $y/H=0$ and $0.5\leq z/H\leq 1$ is
small, because the displacement is fixed at the rigid plate position at the
region. Moreover, the acceleration is negligible in the static region at the
bottom ($z=0$) with $\sigma^{\mathrm{(fric)}}/p\thickapprox\mu_{\mathrm{K}}$
for $\tilde{S}_{\mathrm{c}}L\leq y\leq L$, as shown in Fig. 8a. Therefore, the
region with significant acceleration before bulk sliding is concentrated in
the region for $0\leq y\leq\tilde{S}_{\mathrm{c}}L$ near the bottom, as shown
in Fig. 15.
Figure 15: Spatial distribution of acceleration along the $y$ direction,
$\ddot{u}_{y}$, on cross-section at $x=0.5W$ for $P_{\mathrm{ext}}/E=0.003$,
$d/H=0.5$, and $\phi=0.5$ at $U/L=5.43\times 10^{-3}$, just before bulk
sliding. The gray thick line for $y/H=0$ and $0.5\leq z/H\leq 1$ indicates the
position of the rigid plate.
## Appendix D Quasi-static Solution
Using the equation of motion (12), boundary conditions, and assumption of
friction coefficient $\mu$ in Sect. 3.2, we obtain the quasi-static solution
$q_{0}(y)$ at $U=0$ as
$q_{0}(y)=\frac{\mu_{\mathrm{K}}P_{\mathrm{ext}}}{2(1-\kappa)E_{1}\alpha
H}\left(y^{2}-2Ly\right)\ .$ (22)
The quasi-static solution $q_{\mathrm{a}}(y)$ for $U>0$ is obtained as
$q_{\mathrm{a}}(y)=\left\\{\begin{array}[]{cc}q_{1}(y),&~{}~{}~{}0\leq
y\leq\tilde{S}L\\\ q_{0}(y),&~{}~{}~{}\tilde{S}L\leq y\leq
L\end{array}\right.\ .$ (23)
With the assumption of $\mu$ and connectivity condition at $y=\tilde{S}L$,
$q_{1}(\tilde{S}L)=q_{0}(\tilde{S}L)$ and
$\mathrm{d}q_{1}(\tilde{S}L)/\mathrm{d}y=\mathrm{d}q_{0}(\tilde{S}L)/\mathrm{d}y$,
we obtain $q_{1}$ as
$q_{1}(y)=q_{0}(y)+\frac{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}}{2(1-\kappa)E_{1}\alpha
H}\left(y^{2}-2\tilde{S}Ly+\tilde{S}^{2}L^{2}\right)\ .$ (24)
To satisfy the boundary condition at $y/L=0$, $\tilde{S}$ is derived as
$\tilde{S}=\left[\frac{2(1-\kappa)E_{1}\alpha
HU}{(\mu_{\mathrm{S}}-\mu_{\mathrm{K}})P_{\mathrm{ext}}L^{2}}\right]^{\frac{1}{2}}\
.$ (25)
|
Accepted in IEEE International Conference on Robotics and Automation (ICRA)
2022
# CaTGrasp: Learning Category-Level Task-Relevant
Grasping in Clutter from Simulation
Bowen Wen1,2, Wenzhao Lian1, Kostas Bekris2 and Stefan Schaal1 1Intrinsic
Innovation LLC in CA, USA. {wenzhaol<EMAIL_ADDRESS>This research was
conducted during Bowen’s internship at Intrinsic. 2Rutgers University in NJ,
USA. {bw344<EMAIL_ADDRESS>Bowen Wen and Kostas Bekris were
partially supported by the US NSF Grant IIS-1734492. The opinions expressed
here are of the authors and do not reflect the views of the sponsor.
###### Abstract
Task-relevant grasping is critical for industrial assembly, where downstream
manipulation tasks constrain the set of valid grasps. Learning how to perform
this task, however, is challenging, since task-relevant grasp labels are hard
to define and annotate. There is also yet no consensus on proper
representations for modeling or off-the-shelf tools for performing task-
relevant grasps. This work proposes a framework to learn task-relevant
grasping for industrial objects without the need of time-consuming real-world
data collection or manual annotation. To achieve this, the entire framework is
trained solely in simulation, including supervised training with synthetic
label generation and self-supervised, hand-object interaction. In the context
of this framework, this paper proposes a novel, object-centric canonical
representation at the category level, which allows establishing dense
correspondence across object instances and transferring task-relevant grasps
to novel instances. Extensive experiments on task-relevant grasping of
densely-cluttered industrial objects are conducted in both simulation and
real-world setups, demonstrating the effectiveness of the proposed framework.
Code and data are available at https://sites.google.com/view/catgrasp.
## I Introduction
Robot manipulation often requires identifying a suitable grasp that is aligned
with a downstream task. An important application domain is industrial
assembly, where the robot needs to perform constrained placement after
grasping an object [1, 2]. In such cases, a suitable grasp requires stability
during object grasping and transporting while avoiding obstructing the
placement process. For instance, a grasp where the gripper fingers cover the
thread portion of a screw can impede its placement through a hole. Grasping a
screw in this manner is not a task-relevant grasp.
There are many challenges, however, in solving task-relevant grasps. (a)
Grasping success and task outcome are mutually dependent [3]. (b) Task-
relevant grasping involves high-level semantic information, which cannot be
easily modeled or represented. (c) 6D grasp pose annotation in 3D is more
complicated than 2D image alternatives. Achieving task-relevant grasps
requires additional semantic priors in the label generation process than
geometric grasp generation often studied in stable grasp learning [4, 5]. (d)
Generalization to highly-variant, novel instances within the category requires
effective learning on category-level priors. (e) In densely cluttered
industrial object picking scenarios, as considered in this work, the growing
number of objects, the size of the grasp solution space, as well as
challenging object properties (e.g., textureless, reflective, tiny objects,
etc.), introduce additional combinatorial challenges and demand increased
robustness.
With recent advances in deep learning, many efforts resort to semantic part
segmentation [6, 7] or keypoint detection [8] via supervised learning on
manually labeled real-world data. A line of research is circumventing the
complicated data collection process by training in simulation [9, 10]. It
still remains an open question, however, whether category-level priors can be
effectively captured through end-to-end training. Moreover, most of the
existing work considers picking and manipulation of isolated objects [11, 12,
13, 14, 15]. The complexity of densely cluttered industrial object picking
scenarios considered in this work, make the task-relevant grasping task even
more challenging.
Figure 1: Given a database of 3D models of same category, the proposed method
learns: (a) an object-centric NUNOCS representation that is canonical for the
object category, (b) a heatmap that indicates the task achievement success
likelihood dependent on the hand-object contact region during the grasp, and
(c) a codebook of stable 6D grasp poses. The heatmap and the grasp poses are
transferred to real-world, novel unseen object instances during testing for
solving task-relevant grasping.
To tackle the above challenges, this work aims to learn category-level, task-
relevant grasping solely in simulation, circumventing the requirement of
manual data collection or annotation efforts. In addition, during the test
stage, the trained model can be directly applied to novel object instances
with previously unseen dimensions and shape variations, saving the effort of
acquiring 3D models or re-training for each individual instance. This is
achieved by encoding 3D properties - including shape, pose and gripper-object
contact experience that is relevant to task performance - shared across
diverse instances within the category. Therefore, once trained, this category-
level, task-relevant grasping knowledge not only transfers across novel
instances, but also effectively generalizes to real-world densely cluttered
scenarios without the need for fine-tuning. In summary, the contributions of
this work are the following:
* $\bullet$
A novel framework for learning category-level, task-relevant grasping of
densely cluttered industrial objects and targeted placement. To the best of
the authors’ knowledge, this is the first work that effectively tackles task-
relevant grasping of industrial objects in densely cluttered scenarios in a
scalable manner without human annotation.
* $\bullet$
Instead of learning sparse keypoints relevant to task-relevant manipulation,
as in [10, 8], this work models dense, point-wise task relevance on 3D shapes.
To do so, it leverages hand-object contact heatmaps generated in a self-
supervised manner in simulation. This dense 3D representation eliminates the
requirement of manually specifying keypoints.
* $\bullet$
It introduces the "Non-Uniform Normalized Object Coordinate Space" (NUNOCS)
representation for learning category-level object 6D poses and 3D scaling,
which allows non-uniform scaling across three dimensions. Compared to the
previously proposed Normalized Object Coordinate Space (NOCS) representation
[16] developed in the computer vision community for category-level 6D pose and
1D uniform scale estimation, the proposed representation allows to establish
more reliable dense correspondence and thus enables fine-grained knowledge
transfer across object instances with large shape variations.
* $\bullet$
The proposed framework is solely trained in simulation and generalizes to the
real-world without any re-training, by leveraging domain randomization [17],
bi-directional alignment [18], and domain-invariant, hand-object contact
heatmaps modeled in a category-level canonical space. To this end, a synthetic
training data generation pipeline, together with its produced novel dataset in
industrial dense clutter scenarios, is presented.
## II Related Work
Stable Grasping \- Stable grasping methods focus on robust grasps. The methods
can be generally classified into two categories: model-based and model-free.
Model-based grasping methods require object CAD models to be available
beforehand for computing and storing grasp poses offline w.r.t. specific
object instances. During test stage, the object’s 6D pose is estimated to
transform the offline trained grasp poses to the object in the scene[19, 20,
21, 22]. More recently, model-free methods relax this assumption by directly
operating over observations such as raw point cloud [4, 23] or images [24, 25,
26, 27, 28], or transferring the category-level offline trained grasps via
Coherent Point Drift [29]. Representative works [4, 5, 30] train a grasping
evaluation network to score and rank the grasp candidates sampled over the
observed point cloud. For the sake of efficiency, more recent works [31, 32,
33, 34, 24, 35] develop grasp pose prediction networks, which directly output
6D grasp proposals along with their scores, given the scene observation.
Differently, the proposed CaTGrasp aims to compute grasps that are not only
stable but also task-relevant.
Task-Relevant Grasping \- Task-relevant grasping requires the grasps to be
compatible with downstream manipulation tasks. Prior works have developed
frameworks to predict affordance segmentation [36, 37, 7, 38, 39, 40] or
keypoints [41] over the observed image or point cloud. This, however, often
assumes manually annotated real world data is available to perform supervised
training [42, 43, 12], which is costly and time-consuming to obtain. While [6,
44] alleviates the problem via sim-to-real transfer, it still requires manual
specification of semantic parts on 3D models for generating synthetic
affordance labels. Instead, another line of research [9, 10, 45] proposed to
learn semantic tool manipulation via self-interaction in simulation. While the
above research commonly tackles the scenarios of tool manipulation or
household objects, [46] shares the closest setting to ours in terms of
industrial objects. In contrast to the above, our work considers more
challenging densely cluttered scenarios. It also generalizes to novel unseen
object instances, without requiring objects’ CAD models for pose estimation or
synthetic rendering during testing as in [46].
Category-Level Manipulation \- In order to generalize to novel objects without
CAD models, category-level manipulation is often achieved by learning
correspondence shared among similar object instances, via dense pixel-wise
representation [47, 48, 49] or semantic keypoints [8, 10]. In particular,
sparse keypoint representations are often assigned priors about their semantic
functionality and require human annotation. While promising results on
household objects and tools have been shown[8, 10], it becomes non-trivial to
manually specify semantic keypoints for many industrial objects (Fig. 4),
where task-relevant grasp poses can be widely distributed. Along the line of
work on manipulation with dense correspondence, [47] developed a framework for
learning dense correspondence over 2D image pairs by training on real world
data, which is circumvented in our case. [48, 49] extended this idea to multi-
object manipulation given a goal configuration image. Instead of reasoning on
2D image pairs which is constrained to specific view points, this work
proposes NUNOCS representation to establish dense correspondence in 3D space.
This direct operation in 3D allows to transfer object-centric contact
experience, along with a 6D grasp pose codebook. Additionally, the task-
relevant grasping has not been achieved in [47, 48, 49].
Figure 2: Overview of the proposed framework. Right: (a) Given a collection of
CAD models for objects of the same category, the NUNOCS representation is
aggregated to generate a canonical model for the category. The CAD models are
further utilized in simulation to generate synthetic point cloud data for
training all the networks (3D U-Net, NUNOCS Net and Grasping Q Net).
Meanwhile, the category-level grasp codebook and hand-object contact heatmap
are identified via self-interaction in simulation. Top-left: (b) A 3D U-Net is
leveraged to predict point-wise centers of objects in dense clutter, based on
which the instance segmentation is computed by clustering. Center: (c) The
NUNOCS Net operates over an object’s segmented point cloud and predicts its
NUNOCS representation to establish dense correspondence with the canonical
model and compute its 9D pose $\xi_{o}\in\\{SE(3)\times R^{3}\\}$ (6D pose and
3D scaling). This allows to transfer the precomputed category-level knowledge
to the observed scene. Bottom-left: (d) Grasp proposals are generated both by
transferring them from a canonical grasp codebook and directly by sampling
over the observed point cloud. IK-infeasible or in-collision (using FCL [50])
grasps are rejected. Then, the Grasping Q Net evaluates the stability of the
accepted grasp proposals. This information is combined with a task-relevance
score computed from the grasp’s contact region. The entire process can be
repeated for multiple object segments to find the currently best grasp to
execute according to $P(T,G)=P(T|G)P(G)$. Red dashed arrows occur in the
offline training stage only.
## III Problem Statement
We assume novel unseen objects of the same type have been collected into a
bin, forming a densely cluttered pile as in common industrial settings. The
objective is to compute task-relevant 6D grasp poses $\xi_{G}\in SE(3)$ that
allow a downstream constrained placement task. The grasping process is
repeated for each object instance until the bin is cleared. The inputs to the
framework are listed below.
* $\bullet$
A collection of 3D models $\mathcal{M}_{C}$ belonging to category $C$ for
training (e.g., Fig 4 left). This does not include any testing instance in the
same category, i.e., $M^{\text{test}}_{C}\notin\mathcal{M}_{C}$.
* $\bullet$
A downstream placement task $T_{C}$ corresponding to the category (e.g., Fig
5), including a matching receptacle and the criteria of placement success.
* $\bullet$
A depth image $I_{D}$ of the scene for grasp planning at test stage.
## IV Approach
Fig. 2 summarizes the proposed framework. Offline, given a collection of
models $\mathcal{M}_{C}$ of the same category, synthetic data are generated in
simulation (Sec. IV-E) for training the NUNOCS Net (Sec. IV-A), Grasping Q Net
(Sec. IV-B) and 3D U-Net (Sec. IV-D). Then, self-interaction in simulation
provides hand-object contact experience, which is summarized in task-relevant
heatmaps for grasping (Sec. IV-C). The canonical NUNOCS representation allows
the aggregation of category-level, task-relevant knowledge across instances.
Online, the category-level knowledge is transferred from the canonical NUNOCS
model to the segmented target object via dense correspondence and 9D pose
estimation, guiding the grasp candidate generation and selection.
### IV-A Category-level Canonical NUNOCS representation
Previous work [47] learned dense correspondence between object instances using
contrastive loss. It requires training on real-world data and operates over 2D
images from specific viewpoints. Instead, this work establishes dense
correspondence in 3D space to transfer knowledge from a trained model database
$\mathcal{M}_{C}$ to a novel instance $M_{C}^{\text{test}}$.
Inspired by [16], this work presents the Non-Uniform Normalized Object
Coordinate Space (NUNOCS) representation. Given an instance model $M$, all the
points are normalized along each dimension, to reside within a unit cube:
$\displaystyle
p^{d}_{\mathbb{C}}=(p^{d}-p^{d}_{min})/(p^{d}_{max}-p^{d}_{min})\in[0,1];d\in\\{x,y,z\\}.$
The transformed points exist in the canonical NUNOCS $\mathbb{C}$ (Fig. 1
bottom-right). In addition to being used for synthetic training data
generation (Sec. IV-E), the models $\mathcal{M}_{C}$ are also used to create a
category-level canonical template model, to generate a hand-object contact
heatmap (Sec. IV-C) and a stable grasp codebook (Sec. IV-B). To do so, each
model in $\mathcal{M}_{C}$ is converted to the space $\mathbb{C}$, and the
canonical template model is represented by the one with the minimum sum of
Chamfer distances to all other models in $\mathcal{M}_{C}$. The transformation
from each model to this template is then utilized for aggregating the stable
grasp codebook and the task-relevant hand-object contact heatmap.
For the NUNOCS Net, we aim to learn
$\Phi:\mathcal{P}_{o}\rightarrow\mathcal{P}_{\mathbb{C}}$, where
$\mathcal{P}_{o}$ and $\mathcal{P}_{\mathbb{C}}$ are the observed object cloud
and the canonical space cloud, respectively. $\Phi(\cdot)$ is built with a
PointNet-like architecture [51] given it is light-weight and efficient. The
learning task is formulated as a classification problem by discretizing
$p^{d}_{\mathbb{C}}$ into 100 bins. Softmax cross entropy loss is used as we
found it more effective than regression by reducing the solution space [16].
Along with the predicted dense correspondence, the 9D object pose
$\xi_{o}\in\\{SE(3)\times R^{3}\\}$ is also recovered. It is computed via
RANSAC[52] to provide an affine transformation from the predicted canonical
space cloud $\mathcal{P}_{\mathbb{C}}$ to the observed object segment cloud
$\mathcal{P}_{o}$, while ensuring the rotation component to be orthonormal.
Compared to the original NOCS representation [16], which recovers a 7D pose of
novel object instances, the proposed NUNOCS allows to scale independently in
each dimension when converting to the canonical space. Therefore, more fine-
grained dense correspondence across object instances can be established via
measuring their similarity ($L_{2}$ distance in our case) in $\mathbb{C}$.
This is especially the case for instances with dramatically different 3D
scales, as shown in the wrapped figure, where colors indicate dense
correspondence similarity in $\mathbb{C}$ and one example correspondence for
NUNOCS and NOCS respectively. A key difference from another related work on
VD-NOC [53], which directly normalizes the scanned point cloud in the camera
frame, is that the proposed NUNOCS representation is object-centric and thus
agnostic to specific camera parameters or viewpoints.
### IV-B Stable Grasp Learning
During offline training, grasp poses are uniformly sampled from the point
cloud of each object instance, covering the feasible grasp space around the
object. For each grasp $G$, the grasp quality is evaluated in simulation. To
compute a continuous score $s_{G}\in[0,1]$ as training labels, 50 neighboring
grasp poses are randomly sampled in the proximity of $\xi_{G}\in SE(3)$ and
executed to compute the empirical grasp success rate. The intuition is that
grasp stability should be continuous over its 6D neighborhood. Once the grasps
are generated, they are then exploited in two ways.
First, given the relative 9D transformation from the current instance to the
canonical model, the grasp poses are converted into the NUNOCS space and
stored in a stable grasp codebook $\mathcal{G}$. During test time, given the
estimated 9D object pose $\xi_{o}\in\\{SE(3)\times R^{3}\\}$ of the observed
object’s segment relative to the canonical space $\mathbb{C}$, grasp proposals
can be generated by applying the same transformation to the grasps in
$\mathcal{G}$. Compared with traditional online grasp sampling over the raw
point cloud [5, 4], this grasp knowledge transfer is also able to generate
grasps from occluded object regions. In practice, the two strategies can be
combined to form a robust hybrid mode for grasp proposal generation.
Second, the generated grasps are utilized for training the Grasping Q Net,
which is built based on PointNet [51]. Specifically, in each dense clutter
generated (as in Sec. IV-E), the object segment in the 3D point cloud is
transformed to the grasp’s local frame given the object and grasp pose. The
Grasping Q Net takes the point cloud as input and predicts the grasp’s quality
$P(G)$, which is then compared against the discretized grasp score $s_{G}$ to
compute softmax cross entropy loss. This one-hot score representation has been
observed to be effective for training [30].
### IV-C Affordance Self-Discovery
In contrast to prior work [7], which manually annotates parts of segments, or
uses predefined sparse keypoints [8], this work discovers grasp affordance via
self-interaction. In particular, the objective is to compute
$P(T|G)=P(T,G)/P(G)$ automatically for all graspable regions on the object. To
achieve this, a dense 3D point-wise hand-object contact heatmap is modeled.
For each grasp in the codebook $G\in\mathcal{G}$ (generated as in Sec. IV-B),
a grasping process is first simulated. The hand-object contact points are
identified by computing their signed distance w.r.t the gripper mesh. If it’s
a stable grasp, i.e., the object is lifted successfully against gravity, the
count $n(G)$ for all contacted points on the object are increased by $1$.
Otherwise, the grasp is skipped. For these stable grasps, a placement process
is simulated, i.e., placing the grasped object on a receptacle, to verify the
task relevance (Fig. 2 top-right). Collision is checked between the gripper
and the receptacle during this process. If the gripper does not obstruct the
placement and if the object can steadily rest in the receptacle, the count of
joint grasp and task success $n(G,T)$ on the contact points is increased by
$1$. After all grasps are verified, for each point on the object point cloud,
its task relevance can be computed as $P(T|G)=n(G,T)/n(G)$. Examples of self-
discovered hand-object contact heatmaps are shown in Fig. 3. Interestingly,
these heatmaps achieve similar performance to human annotated part-
segmentation [36] but can be interpreted as a “soft” version.
Eventually, for each of the training objects within the category, the hand-
object contact heatmap $P(T|G)$ is transformed to the canonical model. The
task-relevant heatmaps over all training instances are aggregated and averaged
to be the final canonical model’s task-relevance heatmap. During testing, due
to the partial view of the object’s segment, the antipodal contact points
$p_{c}$ are identified between the gripper mesh and the transformed canonical
model (Fig. 2 bottom-left). For each grasp candidate, the score
$P_{G}(T|G)=\frac{1}{|p_{c}|}\sum_{p_{c}}P_{p_{c}}(T|G)$ is computed. It is
then combined with the predicted $P_{G}(G)$ from Grasping Q Net (Sec. IV-B) to
compute the grasp’s task-relevance score: $P_{G}(T,G)=P_{G}(T|G)P_{G}(G)$.
Figure 3: Examples of task-relevant hand-object contact heatmaps $P(T|G)$.
Warmer color indicates higher values of $P(T|G)$. The white areas are the
small concave regions for which the rigid parallel-jaw gripper can’t touch.
They remain unexplored and set to the default $P(T|G)=0.5$ though they are
also unlikely to be touched during testing. The collected contact heatmap is
object-centric and domain-invariant. Once identified in simulation, it is
directly applied in the real world.
### IV-D Instance Segmentation in Dense Clutter
This work employs the Sparse 3D U-Net [54, 55] due to its memory efficiency.
The network takes as input the entire scene point cloud $\mathcal{P}\in
R^{N\times 3}$ voxelized into sparse volumes and predicts per point offset
$\mathcal{P}_{\text{offset}}\in R^{N\times 3}$ w.r.t. to predicted object
centers. The training loss is designed as the $L_{2}$ loss between the
predicted and the ground-truth offsets [56]. The network is trained
independently, since joint end-to-end training with the following networks has
been observed to cause instability during training.
During testing, the predicted offset is applied to the original points,
leading the shifted point cloud to condensed point groups
$\mathcal{P}+\mathcal{P}_{\text{offset}}$, as shown in Fig. 2. Next, DBSCAN
[57] is employed to cluster the shifted points into instance segments.
Additionally, the segmented point cloud is back-projected onto the depth image
$I_{D}$ to form 2D segments. This provides an approximation of the per-object
visibility by counting the number of pixels in each segment. Guided by this,
the remaining modules of the framework prioritize the top layer of objects
given their highest visibility in the pile during grasp candidate generation.
### IV-E Training Data Generation in Simulation
The entire framework is trained solely on synthetic data. To do so, synthetic
data are generated in PyBullet simulation [58], aiming for physical
plausibility [59, 18], while leveraging domain randomization [17] and bi-
directional domain alignment on the depth modality [18]. At the start of each
scene generation, an object instance type and its scale is randomly chosen
from the associated category’s 3D model database $\mathcal{M}_{C}$. The number
of object instances in the bin, the camera pose relative to the bin, and
physical parameters (such as bounciness and friction) are randomized. To
generate dense clutter, object poses are randomly initialized above the bin.
The simulation is executed until the in-bin objects are stabilized. The
ground-truth labels for NUNOCS, grasping quality and instance segmentation are
then retrieved from the simulator.
Figure 4: Left: The 3 object categories: Nuts, HMN and Screws. For each
category, the first row is a collection of 3D models used for learning in
simulation. The second row is the novel unseen instances with challenging
properties (tiny, texture-less, glossy, etc), used during testing both in
simulation and in the real-world. Right: Instance-level grasp performance
evaluated in simulation (top) and real-world (bottom). For each object
instance, the group of 4 stacked bars from left to right are PointNetGPD [30],
Ours-NA, Ours-NOCS and Ours, where the column corresponding to Ours is marked
with a black boundary. Stable grasps include both task-irrelevant and task-
relevant grasps, while the missing blanks are grasp failures.
## V Experiments
This section aims to experimentally evaluate $3$ questions: i) Is the proposed
dense correspondence expressive and reliable enough to represent various
object instances within a category? ii) How well does the proposed model, only
trained in simulation, generalize to real-world settings for task-relevant
grasping? iii) Does the proposed category-level knowledge learning also
benefit grasp stability? Our proposed method is compared against:
* $\bullet$
PointNetGPD [30]: A state-of-the-art method on robust grasping. Its open-
source code111https://github.com/lianghongzhuo/PointNetGPD is adopted. For
fair comparison, the network is retrained using the same synthetic training
data of industrial objects as our method. At test time, it directly samples
grasp proposals over the raw point cloud without performing instance
segmentation [30].
* $\bullet$
Ours-NA: A variant of our method that does not consider task-relevant
affordance but still transfers category-level grasp knowledge. Only $P(G)$ is
used for ranking grasp candidates.
* $\bullet$
Ours-NOCS: A variant of our method by replacing the NUNOCS representation with
NOCS [16] for solving the category-level pose, while the remainings are the
same as our framework. This serves as an ablation to study the effectiveness
of capturing cross-instance large variations by using the proposed NUNOCS
representation.
Some other alternatives are not easy to compare against directly. For
instance, KETO [10] and kPAM [8] focused on singulated household object
picking from table-top, which is not easy to adapt to our setting. In
addition, kPAM requires human annotated real-world data for training.
### V-A Experimental Setup
In this work, 3 different industrial object categories are included: Nuts,
Screws and HMN series connectors. Their training and testing splits are
depicted in Fig. 4(left). For each category, the first row is a collection of
3D models crawled online. This inexpensive source for 3D model training is
used to learn category-level priors. The second row is 4 novel unseen object
instances used for testing. Different from the training set, the testing
object instances are real industrial objects purchased from
retailers222https://www.digikey.com; https://www.mcmaster.com for realistic
evaluation. They are examined and ensured to be novel unseen object instances
separated from the training set. The testing object instances are chosen so as
to involve sufficient variance to evaluate the cross-instance generalization
of the proposed method, while being graspable by the gripper in our
configuration. The CAD models of the testing object instances are solely used
for setting up the simulation environment.
Evaluations are performed in similar setups in simulation and the real-world.
The hardware is composed of a Kuka IIWA14 arm, a Robotiq Hand-E gripper, and a
Photoneo 3D camera, as in the wrapped figure. Simulation experiments are
conducted in PyBullet, with the corresponding hardware components modeled and
gravity applied to manipulated objects. At the start of the bin-picking
process, a random number (between 4 to 6) of object instances of the same type
are randomly placed inside the bin to form a cluttered pile. Experiments for
each of the 12 object instances have been repeated 10 times in simulation and
3 times in real-world, with different arbitrarily formed initial pile
configurations. This results in approximately 600 and 180 grasp evaluations in
simulation and real-world respectively for each evaluated approach. For each
bin-clearing scenario, its initial pile configuration is recorded and set
similarly across all evaluated methods for fair comparison.
After each grasp, its stability is evaluated by a lifting action. If the
object drops, the grasp is marked as failure. For stable grasps, additional
downstream category-specific placement tasks are performed to further assess
the task-relevance. A stable grasp is further examined and marked as a task-
relevant grasp, if the placement also succeeds. Otherwise, it is marked as a
task-irrelevant grasp, though being stable. The placement receptacles are CAD
designed and 3D printed for each object instance with tight placement
tolerances ($<3mm$). For evaluation purposes, the placement planning is
performed based on manually annotated 6D in-hand object pose post-grasping.
This effort is beyond the scope of this work.
| Nuts | HMN | Screws | Total
---|---|---|---|---
PointnetGPD | 53.3% | 49.2% | 45.0% | 49.2% [t]
Ours-NA | 51.1% | 58.3% | 50.0% | 53.1%
Ours-NOCS | 75.6% | 71.7% | 80.0% | 75.7%
Ours | 97.8% | 88.3% | 93.3% | 93.1% [b]
TABLE I: Results of task-relevant grasp percentage out of the total grasp
attempts in simulation. For each method, approximately 600 grasps are
conducted.
| Nuts | HMN | Screws | Total
---|---|---|---|---
PointnetGPD | 33.3% | 35.0% | 38.0% | 35.4% [t]
Ours-NA | 40.0% | 43.3% | 42.9% | 42.1%
Ours-NOCS | 70.0% | 58.3% | 52.5% | 60.3%
Ours | 93.3% | 83.3% | 86.7% | 87.8% [b]
TABLE II: Results of task-relevant grasp percentage out of the total grasp
attempts in real-world. For each method, approximately 180 grasps are
conducted. Figure 5: Qualitative comparison of the grasping and placement
evaluation in the real-world. The snapshots are taken for one of the test
objects per category during the bin-clearing process, where the initial pile
configuration is similar across different methods. For the placement
verification images (second row), red boxes indicate either failure grasps, or
stable but task-irrelevant grasps. Blue boxes indicate task-relevant grasps
resulting in successful placement. Note that given the similar pile
configuration, the methods do not necessarily choose the same object instance
as the target due to different grasp ranking strategies. See the supplementary
media for the complete video.
### V-B Results and Analysis
The quantitative results in simulation and real-world are shown in Table I and
II respectively. The success rate excludes the task-irrelevant or failed
grasps. As demonstrated in the two tables, Ours significantly surpasses all
baselines measured by the success rate on task-relevant grasping in both
simulation and real-world. Example real-world qualitative results are shown in
Fig. 5.
Fig. 4 (right) decomposes the semantic grasping attempts of all the methods at
the object instance level. Thanks to the task-relevant hand-object contact
heatmap modeling and knowledge transfer via 3D dense correspondence, most of
the grasps generated by Ours and Ours-NOCS are task-relevant semantic grasps.
In comparison, a significant percentage of grasps planned by PointNetGPD and
Ours-NA are not semantically meaningful, i.e., the objects, though stably
grasped, cannot be directly manipulated for the downstream task without
regrasping.
The fact that Ours reaches comparable or better performance than Ours-NOCS
indicates that the proposed NUNOCS is a more expressive representation for 3D
dense correspondence modeling and task-relevant grasp knowledge transfer. In
particular, for the object “HMN2” (Fig. 4 left), the performance gap is more
noticeable as its 3D scales vary significantly from the canonical model along
each of the 3 dimensions. Additionally, the number of 3D training models
available for the HMN category is more limited compared to the Screws or Nuts
category. This requires high data-efficiency to capture the large variance.
Despite these adverse factors, Ours is able to learn category-level task-
relevant knowledge effectively, by virtue of the more representative NUNOCS
space.
Although our proposed method targets at task-relevance, it also achieves a
high stable grasp success rate, as shown in Fig. 4 (right). This demonstrates
the efficacy of the proposed hybrid grasp proposal generation, where
additional grasps transferred from the category-level grasp codebook span a
more complete space around the object including occluded parts. This is also
partially reflected by comparing Ours-NA and PointNetGPD, where PointNetGPD
generates grasp candidates by solely sampling over the observation point
cloud.
Comparing the overall performance across simulation and real-world experiments
indicates a success rate gap of a few percent. This gap is noticeably larger
for Screws, of which the instances are thin and tiny. Therefore, when they
rest in a cluttered bin, it is challenging to find collision-free grasps and
thus requires high precision grasp pose reasoning. In particular, as shown in
Fig. 4, the instance “Screw3” challenges all evaluated methods in terms of
grasp stability. Improving the gripper design [60] for manipulating such tiny
objects is expected to further elevate the performance. In addition, during
gripper closing, the physical dynamics of Screws is challenging to model when
they roll inside the bin in simulation. With more advanced online domain
adaptation techniques [61], the performance is expected to be boosted.
## VI Conclusion
This work proposed CaTGrasp to learn task-relevant grasping for industrial
objects in simulation, avoiding the need for time-consuming real-world data
collection or manual annotation. With a novel object-centric canonical
representation at the category level, dense correspondence across object
instances is established and used for transferring task-relevant grasp
knowledge to novel object instances. Extensive experiments on task-relevant
grasping of densely cluttered industrial objects are conducted in both
simulation and real-world setups, demonstrating the method’s effectiveness. In
future work, developing a complete task-relevant framework with visual
tracking [62] in the feedback loop for manipulating novel unseen objects is of
interest.
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|
# Gravitational Contact Interactions —
— Does the Jordan Frame Exist?
In Honor of Graham G. Ross
Christopher T. Hill<EMAIL_ADDRESS>Fermi National Accelerator Laboratory
P.O. Box 500, Batavia, Illinois 60510, USA
###### Abstract
Scalar–tensor theories, with Einstein Hilbert and non-minimal interactions,
$\sim M^{2}R/2-\alpha\phi^{2}R/12$, have graviton exchange induced contact
interactions. These always pull the theory back into a formal Einstein frame
in which $\alpha$ does not exist. In the Einstein frame, contact terms are
induced, that are then absorbed back into other couplings to define the
effective potential (or equivalently, the renormalization group). We show how
these effects manifest themselves in a simple model by computing the effective
potential in both frames displaying the discrepancy.
††preprint: FERMILAB-CONF-22-699
## I Introduction
Today we are here to honor the life and career of our colleague, Graham
Ross.111 Invited Talk, ”Workshop on the Standard Model and Beyond,
Graham G. Ross Memorial Session,” Corfu, Sept. 1, 2022. I’ve known Graham for
nearly 50 years, beginning at Caltech when I was a graduate student in the
mid-1970’s and he was a Research Scientist. He became a mentor and good
friend.
Back then, we worked on the new theory, QCD, exploring the photon structure
functions, photon , and nonleptonic weak interactions, nonleptonic , with
renormalization group (RG) methods. In the latter we followed the pioneering
work of Shifman , where I was first introduced to “contact interactions” via
the so-called “penguin diagrams” (so named in ref.penguins ). Contact
interactions will play a central role in my talk today.
In 1981, I visited Oxford where Graham had joined the faculty. There we
discussed the possibility that the top quark would be very heavy. He had some
ideas based upon the RG equation for the top quark Yukawa coupling, $g_{top}$.
He proposed that $g_{top}/g_{QCD}$ would flow into the infra-red to a fixed
ratio (a tracker solution), which predicted a top quark mass of about 110 GeV
(with Brian Pendleton PR ). Note that at this time the experimental limit on
the top quark mass was of order $\sim 10$ GeV, and most people believed it
would appear at about $\sim 3\times m_{b}\sim 15$ GeV.
Inspired by the Pendleton-Ross idea, I studied the RG equations and found the
predicted top quark mass would most probably be near an “infrared quasi-fixed
point,” or “focus point,” $\sim 220$ GeV fixed assuming standard model
running to the high scales. People were quite skeptical, however, about a
heavy top quark (some people promised to “eat their hats” if the top mass
exceeded $\sim 40-90$ GeV but I don’t recall beholding any such feasts). In
the late 1980’s Graham and I overlapped at CERN for a year and we worked on
various generalizations of axions schiz , which, particularly in the neutrino
sector, proved to have potentially interesting cosmological implications late
.
In the 1990’s it was becoming clear that the top quark was surprisingly heavy.
Bardeen, Lindner and I considered a composite Higgs boson as a boundstate of
top and anti-top quarks BHL . We found that the composite solution implied
that $g_{top}$ should be near the RG quasi–fixed point value, inspired from
the work of Pendleton and Ross a decade earlier. The top quark finally weighed
in at the Tevatron in 1995 at $175$ GeV, shy of fixed point prediction by
about 20%. I remain of the opinion that this is suggesting an intimate
dynamical relationship between the top quark and Higgs boson, though we may
not yet have the full theory (see CTH3 for a sketch of some new ideas).
Graham Ross was a universal physicist, able to move from phenomenology to
theory comfortably. Above all, he paid close attention to experimental
results. He was a key player in developing many ideas surrounding the MSSM in
the ‘80’s and ‘90’s. However, circa 2014 he became increasingly interested in
alternatives, such as the Weyl invariant gravity theories which had become an
active arena of research Weyl ; authors ; HillRoss ; staro . We then renewed
our collaboration, and that carried us to his untimely passing.
I feel a great loss, of a dear friend and colleague, an amiable person of the
highest integrity, an energetic researcher with a deep passion for real world
physics and its essential interface with experiment, and someone I’ve known
for much of my life. We leave a half dozen unfinished papers on our laptops.
I would like to talk today about one of my more recent works with Graham Ross,
because I feel it is foundational to quantum field theory and model
independent. This work came out of the Covid pandemic, and we had daily
multiple email exhanges, sometime skype calls, during this time.
The subject of my talk is the field theoretic structure of non-minimal
gravitationally coupled scalars in the regime in which the Planck mass is
present (as opposed to any putative pre-Planckian era). The question that
emerges from this is: “Does the Jordan frame really exist?” You’ll see what I
think as the story unfolds. All of this is described in the paper HillRoss ,
and some of the implications are tackled in staro . A follow-on paper with one
of Graham’s former students, Dumitru Ghilencea, is nearing completion Ghil .
## II Non-Minimal Couplings of Scalars to Curvature
Over the years there has been considerable interest in Brans-Dicke, scalar-
tensor, and scale or Weyl invariant theories. These have in common fundamental
scalar fields, $\phi_{i}$, that couple to gravity through non-minimal
interactions, $\sim M^{2}R+F(\phi_{i})R$. The Planck mass, $M$ can co-exist
with these non-minimal couplings, or the scalars may acquire VEV’s that
dynamically generate it authors . When the theory is proscribed with
non–minimal interactions we say it is given in a “Jordan frame.”
A key tool in the analysis of these models is the Weyl transformation, Weyl .
This involves a redefinition of the metric, $g^{\prime}=\Omega(\phi_{i})g$, in
which $g$ comingles with the scalars. $\Omega$ can be chosen to lead to a new
effective theory, typically one that has a pure Einstein-Hilbert action, $\sim
M^{2}R$, in which the non-mimimal interactions have been removed. This is
called the “Einstein frame.” Alternatively, one might use a Weyl
transformation to partially remove a subset of scalars from the non-minimal
interactions $\sim M^{2}R+F^{\prime}(\phi_{i})R$, where $F^{\prime}$ is
optimized for some particular application.
It is a priori unclear, however, how or whether the original Jordan frame
theory can be physically equivalent to the Einstein frame form and how the
Weyl transformation is compatible with a full quantum theory Duff Ruf .
Nonetheless, many authors consider this to be a valid transformation and a
symmetry of Weyl invariant theories, and many loop calculations permeate the
literature which attempt to exploit apparent simplifications offered by the
Jordan frame.
Graham Ross and I showed that any theory with non-minimal couplings contains
contact terms HillRoss . These are generated by the graviton exchange
amplitudes in tree approximation and they are therefore ${\cal{O}}(\hbar^{0})$
and therefore classical. The contact terms occur because emission vertices
from the non-minimal interaction are proportional to $q^{2}$ of the graviton,
while the Feynman propagator is proportional to $1/q^{2}$. The cancellation of
$q^{2}\times 1/q^{2}$ therefore leads to point-like interactions that must be
included into the effective action of the theory at any given order of
perturbation theory. The result is that the non-minimal interactions disappear
from the theory and Planck suppressed higher dimension operators appear with
modified couplings.
The structural form of the theory when the contact term interactions are
included corresponds formally to a Weyl transformation of the metric that
takes the theory to the pure Einstein frame with higher dimension, Planck
suppressed, interactions. In the pure Einstein-Hilbert action there are no
classical contact terms, but at loop level they can be generated, and must be
removed as part of the renormalization group. However, by virtue of the
contact terms, nowhere is a metric redefinition performed (hence the issue of
a Jacobian in the measure of the gravitational path integral in going to the
Einstein frame becomes moot).
This means that, provided we are interested in the theory on mass scales below
$M$, the Jordan frame is an illusion and doesn’t really exist physically. In
the Jordan frame the contact terms are hidden, but they are always present,
ergo, even though the action superficially appears to be Jordan, it isn’t, and
remains always in an Einstein frame.
Efforts to compute quantities, such as effective potentials (or equivalently,
$\beta$-functions), in the Jordan frame, while ignoring the contact terms,
will yield incorrect results. Nonetheless, though the non-minimal interactions
are not present in the classical Einstein frame, they are regenerated by loops
and the potentials and RG equations are modified by this effect.
In a simple theory in the Jordan frame, where the nonminimal coupling is
$-\alpha\phi^{2}R/12$, this raises the question of how to understand the fate
of $\alpha$? With a single scalar with quartic and other interactions,
$\lambda_{i}$, the usual “naive” calculation of a $\beta$-function in the
Jordan frame (“naive” means ignoring contact terms) yields the form:
$\displaystyle\frac{\partial\alpha(\mu)}{\partial\ln(\mu)}=\beta_{\alpha}(\lambda_{i})\equiv(1-\alpha)\gamma_{\alpha}(\lambda_{i})$
(1)
where the factor $(1-\alpha)$ reflects the fact that when $M^{2}=0$ and
$\alpha=1$ (conformal limit) the kinetic term of $\phi$ disappers and $\phi$
becomes static parameter).
However, the contact terms (or a Weyl tranformation) remove $\alpha$ and leave
an Einstein frame with only the Einstein-Hilbert term, $M^{2}R$ and the $\phi$
couplings $\lambda_{i}^{\prime}$ (in what follows primed couplings refer the
Einstein frame and un-primed to Jordan frame). This means that $N$ couplings,
$(\alpha,\lambda_{i})$, in the Jordan frame have become $N-1$ couplings,
$(\lambda_{i}^{\prime})$, in the Einstein frame. Therefore any physical
meaning ascribed to $\alpha$ or the $\beta_{\alpha}$ function is apparantly
lost.
Three Feynman diagrams (shown below as D1, D2, D3, in Figs.(2,3,4)) contribute
to $\beta_{\alpha}$ in the Jordan frame. One of them multiplies $\alpha$ in
the Jordan frame (D3) and yields the $(1-\alpha)$ factor in eq.(1). But even
with $\alpha=0$ in the Einstein frame, two diagrams (D1 and D2) exist and
reintroduce a perturbative $\delta\alpha$, for a small step in scale
$\delta\mu/\mu$. This is then removed by the contact terms, but leads to
correction terms in the renormalization of the $\lambda_{i}^{\prime}$. We are
therefore sensitive to the same scale breaking information in the Einstein
frame that one has in the Jordan frame, which is encoded into
$\gamma_{\alpha}$. This does not, however, imply that the resulting
calculations in the Jordan and Einstein frames are then consistent! I will
explicitly demonstrate the inconsistency through calculation pf effective
potentials (the RG equations of the couplings can always be read off from the
effective potentials).
If we stayed in the Jordan frame, with nonzero $\alpha$, and naively computed
the same effective potential (“naively” means ignoring contact terms), into an
Einstein frame, we would obtain a different result. The difference is a term
proportional to $\alpha$ in the Jordan frame. Equivalently, going initially to
the Einstein frame and running with the RG, does not commute with running
initially in the Jordan frame and subsequently going to the Einstein frame!
We turn presently to a brief discussion of contact terms in general and review
a simple toy model from HillRoss that is structurally similar to the
gravitational case. We then summarize gravitational contact terms (and refer
the reader to HillRoss for details). We then exemplify the Einstein frame
renormalization group compared to the naive Jordan frame result, which ignores
contact terms, and illustrate the discrepancy.
## III Contact Interactions
Generally speaking “contact interactions” are point-like operators that are
generated in the effective action of the theory in perturbation theory. They
may arise in the UV from ultra-heavy fields that are integrated out, such as
the Fermi weak interaction that arises from integrating out the heavy
$W$-boson. They may also arise in the IR when a vertex in the theory is
proportional to $q^{2}$ and cancels against a $1/q^{2}$ propagator. The
gravitational contact term we discuss presently is of the IR form.
### III.1 Contact Terms in Non-gravitational Physics
Contact terms arise in a number of phenomena. Diagrammatically they can arise
in the IR when a vertex for the emission of, e.g., a massless quantum, of
momentum $q_{\mu}$, is proportional to $q^{2}$. This vertex then cancels the
$1/q^{2}$ from a massless propagator when the quantum is exchanged. This
$q^{2}/q^{2}$ cancellation leads to an effective pointlike operator from an
otherwise single-particle reducible diagram.
For example, in electroweak physics a vertex correction by a $W$-boson to a
massless gluon emission induces a quark flavor changing operator, e.g.,
describing $s\rightarrow d$+gluon, where $s(d)$ is a strange (down) quark.
This has the form of a local operator:
$\displaystyle g\kappa\overline{s}\gamma_{\mu}T^{A}d_{L}D_{\nu}G^{A\mu\nu}$
(2)
where $G^{A\mu\nu}$ is the color octet gluon field strength and $\kappa\propto
G_{Fermi}$.
This implies a vertex for an emitted gluon of 4-momentum $q$ and polarization
and color, $\epsilon^{A\mu}$, of the form
$g\kappa\overline{s}\gamma_{\mu}T^{A}d_{L}\epsilon^{A\mu}\times q^{2}+...$.
However, the gluon propagates, $\sim 1/q^{2}$, and couples to a quark current
$\sim g\epsilon^{A\mu}\overline{q}\gamma_{\mu}T^{A}q$. This results in a
contact term:
$\displaystyle
g^{2}\kappa\left(\frac{q^{2}}{q^{2}}\right)\overline{s}\gamma^{\mu}T^{A}d_{L}\overline{q}\gamma_{\mu}T^{A}q\;=\;g^{2}\kappa\overline{s}\gamma^{\mu}T^{A}d_{L}\overline{q}\gamma_{\mu}T^{A}q$
The result is a 4-body local operator which mediates electroweak transitions
between, e.g., kaons and pions Shifman , also known as “penguin diagrams”
penguins . Note the we can rigorously obtain the contact term result by use of
the gluon field equation within the operator of eq.(2),
$\displaystyle D_{\nu}G^{A\mu\nu}=g\overline{q}\gamma^{\mu}T^{A}q.$ (4)
This is justified as operators that vanish by equations of motion, known as
“null operators,” will generally have gauge noninvariant anomalous dimensions
and are unphysical Deans .
Another example occurs in the case of a cosmic axion, described by an
oscillating classical field, $\theta(t)=\theta_{0}\cos(m_{a}t)$, interacting
with a magnetic moment, $\vec{\mu}(x)\cdot\vec{B}$, through the
electromagnetic anomaly $\kappa\theta(t)\vec{E}\cdot\vec{B}$. A static
magnetic moment emits a virtual spacelike photon of momentum $(0,\vec{q})$.
The anomaly absorbs the virtual photon and emits an on-shell photon of
polarization $\vec{\epsilon}$, inheriting energy $\sim m_{a}$ from the cosmic
axion. The Feynman diagram, with the exchanged virtual photon, yields an
amplitude,
$\propto(\theta_{0}\mu^{i}\epsilon_{ijk}q^{j})(1/\vec{q}^{\;2})(\kappa\epsilon^{k\ell
h}q_{\ell}m_{a}\epsilon_{h})\sim(\kappa\theta_{0}m_{a}\vec{q}^{\;2}/\vec{q}^{\;2})\vec{\mu}\cdot\vec{\epsilon}$.
The $\vec{q}^{\;2}$ factor then cancels the $1/\vec{q}^{\;2}$ in the photon
propagator, resulting in a contact term which is an induced, parity violating,
oscillating electric dipole interaction:
$\sim\kappa\theta(t)\vec{\mu}\cdot\vec{E}$. This results in cosmic axion
induced electric dipole radiation from any magnet, including an electron CTHa
.
### III.2 Illustrative Toy Model of Contact Terms
To illustrate the general IR contact term phenomenon, consider a single
massless real scalar field $\phi$ and operators $A$ and $B,$ which can be
functions of other fields, with the action given by:
$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi-A\partial^{2}\phi-B\phi$ (5)
Here $\phi$ has a propagator ${i}/{q^{2}}$, but the vertex of a diagram
involving $A$ has a factor of $\partial^{2}\sim-q^{2}$. This yields a
pointlike interaction, $\sim q^{2}\times({i}/q^{2})$, in a single particle
exchange of $\phi$, and therefore implies contact terms.
At lowest order in perturbation theory consider the diagram with $\phi$
exchange in Fig.(1). This involves two time-ordered products of interaction
operators:
$\displaystyle{T}\;\;i\\!\\!\int\\!A\partial^{2}\phi\times
i\\!\\!\int\\!B\phi\rightarrow\frac{iq^{2}}{q^{2}}\ AB\;\;\;=\;i\int\\!AB$
$\displaystyle\frac{1}{2}{T}\;\;i\\!\\!\int\\!A\partial^{2}\phi\times
i\\!\\!\int\\!A\partial^{2}\phi\rightarrow\frac{i(q^{2})^{2}}{2q^{2}}A^{2}\\!=\\!\frac{i}{2}\int\\!A\partial^{2}A.$
where $d^{4}x$ is understood in the integrals amd the $\frac{1}{2}$ factor in
the $A\partial^{2}A$ term comes from the second order in the expansion of the
path integral $\exp(i\int A\partial^{2}\phi)$. Note that we also produce a
nonlocal interaction $-{i}B^{2}/{2q^{2}}.$
Figure 1: Contact terms in the toy model are generated by diagrams with
exchange of $\phi$ (dashed). In gravity, with non-minimal term
$\sim\int\\!\sqrt{-g}F\left(\phi\right)R$ and matter field Lagrangian
$\sim\int\\!\sqrt{-g}L\left(\phi\right)$ then $A$ is replaced by $F(\phi)$ and
$B$ is replaced by $L(\phi)$, and the dashed line is a graviton propagator.
We thus see that we have diagrammatically obtained a local effective action:
$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi+\frac{1}{2}A\partial^{2}A+AB+\makebox{
{long distance}}$ (7)
Of course, we can see this straightforwardly by “solving the theory,” by
defining a shifted field:
$\displaystyle\phi=\phi^{\prime}-\frac{1}{\partial^{2}}\left(\partial^{2}A+B\right)$
(8)
Substituting this into the action $S$ and integrating by parts yields:
$\displaystyle
S=\int\frac{1}{2}\partial\phi^{\prime}\partial\phi^{\prime}+\frac{1}{2}A\partial^{2}A+AB+\frac{1}{2}B\frac{1}{\partial^{2}}B$
(9)
An equivalent effective local action that describes both short and large
distance is then,
$\displaystyle
S=\int\frac{1}{2}\partial\phi\partial\phi+\frac{1}{2}A\partial^{2}A+AB-{B\phi}$
(10)
The contact terms have become pointlike components of the effective action,
while the remaining long distance effects are produced by the usual massless
$\phi$ exchange. Note that the derivatively coupled operator $A$ has no long
distance interactions due to $\phi$ exchange. Moreover, in the effective
action of eq.(10) we have implicitly “integrated out” the $A\partial^{2}\phi$,
which is no longer part of the action and is replaced by new operators
$\frac{1}{2}A\partial^{2}A+AB$. We will see that this is exactly what happens
with gravity, where the $A\partial^{2}\phi$ term is schematically the
nonminimal $F(\phi)R(g)\sim F(\phi)\partial^{2}h$ term in a weak field
expansion of gravity $g=\eta+h$.
One can also adapt the use of equations of motion to obtain eq.(10) from the
action eq.(5) but this requires care. For example, the insertion of the $\phi$
equation of motion, into $A\partial^{2}\phi$ correctly gives the $AB$ term but
misses the factor of $1/2$ in the $A\partial^{2}A$ term. We can therefore do a
trick of defining a “modified equation of motion,” where we supply a factor of
$1/2$ on the term $A\partial^{2}A$, e.g. substitute:
$\displaystyle\partial^{2}\phi=-\partial^{2}A-B\rightarrow-\frac{1}{2}\partial^{2}A-B$
(11)
in place of the $\partial^{2}\phi$ in the second term of eq.((5)) to obtain
eq.(10).
## IV Gravitational Contact Terms
Consider a general theory involving scalar fields $\phi_{i},$ an Einstein-
Hilbert term and a non-minimal interaction:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;S=\int\sqrt{-g}\left(\frac{1}{2}M^{2}R\left(g_{\mu\nu}\right)+\frac{1}{2}F\left(\phi_{i}\right)R\left(g_{\mu\nu}\right)+L\left(\phi_{i}\right)\right)$
$\displaystyle=S_{1}+S_{2}+S_{3}$ (12)
where we use the metric signature and curvature tensor conventions of CCJ .
$S_{1}$ is the kinetic term of gravitons
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}M^{2}\int\sqrt{-g}R$ (13)
and becomes the Fierz-Pauli action in a weak field expansion.
$S_{2}$ is the non-minimal interaction, and takes the form:
$\displaystyle
S_{2}=\frac{1}{2}\int\sqrt{-g}F\left(\phi_{i}\right)R\left(g_{\mu\nu}\right)$
(14)
$S_{3}$ is the matter action with couplings to the gravitational weak field:
$\displaystyle S_{3}=\int\sqrt{-g}\;L\left(\phi_{i}\right)$ (15)
The Lagrangian takes the form
$\displaystyle
L\left(\phi_{i}\right)=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi_{i}\partial_{\nu}\phi_{i}-W(\phi_{i})$
(16)
with potential $W(\phi_{i})$. The matter lagrangian has stress tensor and
stress tensor trace:
$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\phi_{i}\partial_{\nu}\phi_{i}-g_{\mu\nu}\left(\frac{1}{2}g^{\rho\sigma}\partial_{\rho}\phi_{i}\partial_{\sigma}\phi_{i}-W(\phi_{i})\right).$
$\displaystyle T$ $\displaystyle=$
$\displaystyle-\partial^{\sigma}\phi_{i}\partial_{\sigma}\phi_{i}+4W(\phi_{i})$
(17)
There are then three ways to obtain the contact term:
(1) GRAVITON EXCHANGE CONTACT TERM
In reference HillRoss the corresponding Feynman diagrams of Figure 1 with are
evaluated arising from single graviton exchange between the interaction terms.
We treat the theory perturbatively, expanding around flat space. Hence we
linearize gravity with a weak field $h_{\mu\upsilon}$:
$\displaystyle
g_{\mu\upsilon}\approx\eta_{\mu\upsilon}+\frac{h_{\mu\upsilon}}{M}.$ (18)
The scalar curvature is then:
$\displaystyle R$ $\displaystyle=$ $\displaystyle R_{1}+R_{2}$ $\displaystyle
MR_{1}$ $\displaystyle=$ $\displaystyle\biggl{(}\partial^{2}h\
-\partial^{\mu}\partial^{\nu}h_{\mu\nu}\biggr{)}$ $\displaystyle M^{2}R_{2}$
$\displaystyle=$
$\displaystyle-\frac{3}{4}\partial^{\rho}h^{\mu\nu}\partial_{\rho}h_{\mu\nu}-\frac{1}{2}h^{\mu\nu}\partial^{2}h_{\mu\nu}+...$
(19)
(see HillRoss for the complete expression for $R_{2}$). $S_{1}$ then becomes:
$\displaystyle
S_{1}=\frac{1}{2}M^{2}\int\sqrt{-g}R=\frac{1}{2}M^{2}\int\\!\\!\left(R_{1}+R_{2}+\frac{1}{2}\frac{h}{M}R_{1}\right)$
$\displaystyle=\frac{1}{2}\int
h^{\mu\nu}\biggl{(}\frac{1}{4}\partial^{2}\eta_{\mu\nu}\eta_{\rho\sigma}-\frac{1}{4}\partial^{2}\eta_{\mu\rho}\eta_{\nu\sigma}$
$\displaystyle\qquad\qquad-\frac{1}{2}\partial_{\rho}\partial_{\sigma}\eta_{\mu\nu}+\frac{1}{2}\partial_{\mu}\partial_{\rho}\eta_{\nu\sigma}\biggr{)}h^{\rho\sigma}.$
(20)
Note that the leading term, $\int R_{1}$, is a total divergence and is
therefore zero in the Einstein-Hilbert action, and what remains of eq.(IV) is
the Fierz-Pauli action. This is key to the origin of the contact terms. The
non-minimal interaction, $S_{2}$, then takes the leading form:
$\displaystyle
S_{2}=\frac{1}{2}\int\\!\\!\sqrt{-g}\;F\left(\phi\right)R\left(g\right)\rightarrow\frac{1}{2}\int\\!\\!F\left(\phi\right)R_{1}\left(g\right)$
$\displaystyle\qquad=\int\frac{1}{2M}F\left(\phi\right)\Pi^{\mu\nu}h_{\mu\nu}$
(21)
where it is useful to introduce the transverse derivative,
$\displaystyle\Pi^{\mu\nu}=\partial^{2}\eta^{\mu\nu}\
-\partial^{\mu}\partial^{\nu}.$ (22)
$S_{2}$ involves derivatives, and is the analogue of the $A\partial^{2}\phi$
term in eq.(5). It will therefore generate contact terms in the gravitational
potential due to single graviton exchange. $S_{3}$ is the analogue of the
$B\phi$ term in eq.(5), and this situation will closely parallel the toy
model.
In HillRoss we developed the graviton propagator, following the nice lecture
notes of Donoghue et. al., Donoghue . We remark that we found a particularly
useful gauge choice,
$\displaystyle\partial_{\mu}h^{\mu\nu}=w\partial^{\nu}h$ (23)
where $w$ defines a single parameter family of gauges. The familiar De Donder
gauge corresponds to $w=\frac{1}{2}$, while the choice $w=\frac{1}{4}$ is
particularly natural in this application, and the gauge invariance of the
result is verified by the $w$-independence (we verify the Newtonian potential
from graviton exchange between static masses in $w$ gauge; see HillRoss ).
We can the compute single graviton exchange between the interaction terms of
the theory. A diagram with a single $S_{2}$ vertex and single $S_{3}$ vertex
is the analogue of $AB$ in the toy model and yields:
$\displaystyle-i\left\langle T\;S_{2}S_{3}\right\rangle=\int
d^{4}x\;\frac{F\left(\phi_{i}\right)}{2M^{2}}T(\phi_{i})$ (24)
Also we have the pair $\left\langle S_{2}S_{2}\right\rangle$ which corresponds
to $\frac{1}{2}A\partial^{2}A$ in the toy model and yields:
$\displaystyle-i{\left\langle T\;S_{2}S_{2}\right\rangle}$ $\displaystyle=$
$\displaystyle-\int
d^{4}x\;\frac{3}{4M^{2}}\;F\left(\phi_{i}\right)\\!\partial^{2}F\left(\phi_{i}\right)$
Hence, the action becomes
$\displaystyle S=S_{1}+S_{3}+S_{CT}$ (26)
where
$\displaystyle S_{CT}=\int
d^{4}x\biggl{(}-\frac{3}{4M^{2}}F\partial^{2}F+\frac{1}{2M^{2}}FT\biggr{)}$
(27)
Note the sign of the $F\partial^{2}F$ is opposite (repulsive) to that of the
toy model $A\partial^{2}A$. Since this is a tree diagram effect, it is
classical.
(2) WEYL TRANSFORMATION
In eq.(IV) we define:
$\displaystyle\Omega^{2}=\left(1+\frac{F\left(\phi_{i}\right)}{M^{2}}\right)$
(28)
and perform a Weyl transformation on the metric:
$\displaystyle g_{\mu\nu}(x)\rightarrow\Omega^{-2}g_{\mu\nu}(x)\qquad
g^{\mu\nu}(x)\rightarrow\Omega^{2}g^{\mu\nu}(x)$
$\displaystyle\sqrt{-g}\rightarrow\sqrt{-g}\Omega^{-4}$ $\displaystyle
R(g)\rightarrow\Omega^{2}R(g^{\prime})+6\Omega^{3}D\partial\Omega^{-1}$ (29)
and the action of eq.(V) becomes:
$\displaystyle
S\rightarrow\int\sqrt{-g}\biggl{(}\frac{1}{2}M^{2}R\left(g\right)$
$\displaystyle-3M^{2}\partial_{\mu}\left(1+\frac{F}{M^{2}}\right)^{1/2}\\!\\!\partial^{\mu}\left(1+\frac{F}{M^{2}}\right)^{-1/2}$
$\displaystyle+\frac{1}{2}\left(1+\frac{F}{M^{2}}\right)^{-1}\\!\\!\\!\partial_{\mu}\phi_{i}\partial^{\nu}\phi^{i}-\left(1+\frac{F}{M^{2}}\right)^{-2}\\!\\!\\!W(\phi_{i})\biggr{)}$
Keeping terms to $O({1\over M^{2}})$ and integrating by parts we have:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=S_{1}+S_{3}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int
d^{4}x\bigg{(}-\frac{3F\left(\phi_{i}\right)\partial^{2}F\left(\phi_{i}\right)}{4M^{2}}+\frac{F\left(\phi_{i}\right)T\left(\phi_{i}\right)}{2M^{2}}\bigg{)}$
(31)
The Weyl transformed action is identically consistent with the contact terms
of eq.(27) above, to first order in $1/M^{2}$.
Hence, contact terms arise in gravity with non-minimal couplings to scalar
fields due to graviton exchange and their form is equivalent to a Weyl
redefinition of the theory to the Einstein frame. However we emphasie that the
contact terms are not a Weyl tranformation since they do not involve a
rescaling of the metric, and there would therefore be no Jacobian ghost terms
introduced into the path integral. Hence any theory with a non-minimal
interaction $\sim F(\phi)R$ will lead to contact terms at order $1/M^{2}$.
What we say presently only applies on scales below $M$, hence the Jordan frame
would, at best, apply only in a UV completion of the theory to pre-Planckian
scales.
(3) USE OF $R$ (MODIFIED) EQUATION OF MOTION
The Einstein equation with the nonminimal term is:
$\displaystyle
M^{2}G_{\alpha\beta}=-T_{\alpha\beta}-D_{\mu}(D_{\nu}F(\phi_{i}))+g_{\mu\nu}D^{2}F(\phi_{i})$
$\displaystyle M^{2}R=T-3D^{2}F(\phi_{i})$ (32)
We can use a simple trick to obtain the contact term via a modified the
equation of motion for $R$. We supply a factor of $1/2$ in the last term which
is the analogue of the $\partial^{2}A$ term as in eq.(11):
$\displaystyle
R^{\prime}=\frac{1}{M^{2}}\biggl{(}T-\frac{3}{2}D^{2}F(\phi_{i})\biggr{)}$
(33)
Then substitute $R^{\prime}$ for $R$ in the non-minimal term $FR$ of the
action of eq.(V)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;S\rightarrow\int\sqrt{-g}\left(\frac{1}{2}M^{2}R\left(g_{\mu\nu}\right)+\frac{1}{2}F\left(\phi_{i}\right)R^{\prime}\left(g_{\mu\nu}\right)+L\left(\phi_{i}\right)\right)$
$\displaystyle\qquad=S_{1}+S_{3}$ $\displaystyle\qquad+\int
d^{4}x\bigg{(}-\frac{3F\left(\phi_{i}\right)\partial^{2}F\left(\phi_{i}\right)}{4M^{2}}+\frac{F\left(\phi_{i}\right)T\left(\phi_{i}\right)}{2M^{2}}\bigg{)}$
In the RG calculation we will only need the exact equation of motion for $R$
in the Einstein frame (without the pseudo $-3D^{2}F/2$ term), so this
ambiguity does not arise.
## V Effective Potetial in a Simple Model
Consider the following action:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Jordan}=\int\sqrt{-g}\biggl{(}\frac{1}{2}\partial\phi\partial\phi-\frac{\lambda_{1}}{4}\phi^{4}-\frac{\lambda_{2}\phi^{6}}{12M^{2}}$
$\displaystyle-\frac{\gamma}{12}\frac{\phi^{2}}{M^{2}}\partial\phi\partial\phi-\frac{\alpha}{12}\phi^{2}R+\frac{1}{2}M^{2}R\biggr{)}$
(35)
This is the most general action for a real scalar field with a $Z_{2}$
symmetry $\phi\rightarrow-\phi$ valid to O($M^{-2}$) with Einstein gravity and
assuming $\phi$ is massless, $m^{2}=0$. Here we do not include a term
$\phi^{4}R/M^{2}$ since, after use of equations of motion, $R\sim M^{-2}$ such
a term would enter the physics at O($M^{-4}$).
We can go to the Einstein frame by implementing the CT. We find that the
effect of single graviton exchange to eq.(V) yields a new action to order
$M^{-2}$:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Einstein}=\\!\int\\!\sqrt{-g}\biggl{(}\frac{1}{2}\partial\phi\partial\phi-\frac{\lambda^{\prime}_{1}}{4}\phi^{4}-\frac{\lambda^{\prime}_{2}\phi^{6}}{12M^{2}}$
$\displaystyle-\frac{\gamma^{\prime}\phi^{2}}{12M^{2}}\partial\phi\partial\phi+\frac{1}{2}M^{2}R\biggr{)}$
(36)
where:
$\displaystyle\gamma^{\prime}=\gamma-\alpha-\alpha^{2}$
$\displaystyle\lambda_{1}^{\prime}=\lambda_{1}$
$\displaystyle\lambda_{2}^{\prime}=\lambda_{2}+\alpha\lambda_{1}$ (37)
We see that to first order in $M^{-2}$ in $S_{Einstein}$ we have three
interaction terms, though the original action $S_{Jordan}$ displayed four
interaction terms. In the Einstein frame action we see that $\alpha$ has
disappeared having been absorbed into redefining the other coupling constants.
This indicates that the nonminimal term in $S_{Jordan}$ with coupling $\alpha$
is unphysical. Moreover, if we are careful to incorporate the effects of the
contact term, then $S_{Einstein}$ will “close” under renormalization.
We define,
$\displaystyle Z=1-\frac{\gamma^{\prime}\phi^{2}}{6M^{2}}$ (38)
and the Einstein action becomes:
$\displaystyle
S_{Einstein}=\\!\int\\!\sqrt{-g}\biggl{(}\frac{1}{2}Z\partial\phi\partial\phi-\frac{\lambda^{\prime}_{1}}{4}\phi^{4}-\frac{\lambda^{\prime}_{2}\phi^{6}}{12M^{2}}$
$\displaystyle+\frac{1}{2}M^{2}R\biggr{)}$ (39)
We do a background field calculation where we shift
$\phi\rightarrow\phi_{0}+\sqrt{\hbar}Z_{0}^{-1/2}\hat{\phi}$, where we define
$\displaystyle Z_{0}=1-\frac{\gamma^{\prime}\phi_{0}^{2}}{6M^{2}}$ (40)
We will treat $\phi\rightarrow\phi_{0}$ as a constant and compute the one
loop, $\cal{O(\hbar)}$ effective potential in $\phi_{0}$ integrating out the
quantum fluctuations $\hat{\phi}$ .222 In a forthcoming paper with Ghilenca
Ghil we will give the general effective action where the backgound field is
treated as non-constant and dynamical. The present analysis parallels a
Coleman-Weinberg potential CW . Expand the action with the shifted field to
O$(\hat{\phi}^{2})$, dropping terms odd in $\hat{\phi}$ for constant
$\phi_{0}$.
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Einstein}=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\sqrt{-g}\biggl{(}\frac{1}{2}\partial\hat{\phi}\partial\hat{\phi}-\frac{1}{2}B\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\biggr{)}$
where,
$\displaystyle Z_{0}=1-\frac{\gamma^{\prime}\phi_{0}^{2}}{6M^{2}}$
$\displaystyle
B=Z_{0}^{-1}\biggl{(}{3\lambda^{\prime}_{1}}\phi_{0}^{2}+\frac{5\lambda^{\prime}_{2}}{2M^{2}}\phi_{0}^{4}\biggr{)}$
$\displaystyle={3\lambda^{\prime}_{1}}\phi_{0}^{2}+\frac{5\lambda^{\prime}_{2}+\gamma^{\prime}\lambda^{\prime}_{1}}{2M^{2}}\phi_{0}^{4}$
$\displaystyle
V_{0}(\phi_{0})=\frac{\lambda^{\prime}_{1}}{4}\phi_{0}^{4}+\frac{\lambda^{\prime}_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}$
First we consider the non-curvature terms, with flat Minkowsky space metric
$g_{\mu\nu}=\eta_{\mu\nu}$. We obtain the effective potential from the log of
the path integral described briefly in the Appendix, eq.(71), obtained by
integrating out $\hat{\phi}^{2}$. Presently we plug $m^{2}\rightarrow B$ into
the path integral expression of eq.(71),
$\displaystyle\Gamma_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}B^{2}L+O\biggl{(}\frac{B^{3}}{\Lambda^{2}}\biggr{)}$
(43)
and we define the log as:
$\displaystyle L=\frac{1}{16\pi^{2}}\ln\frac{\Lambda}{\mu}$ (44)
with a generic infrared cut-off mass scale $\mu$. Hence, squaring $B$ to
O($\hbar$), the resulting potential is to ${\cal{O}}(\phi_{0}^{6})$:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{0}=-\biggl{(}\frac{9\lambda^{\prime
2}_{1}}{2}\phi_{0}^{4}+\frac{(15\lambda^{\prime}_{2}\lambda^{\prime}_{1}+3\gamma^{\prime}\lambda^{\prime}_{1}{}^{2})}{2}\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}L$
(45)
Note that $\mu\rightarrow\phi_{0}$ in the logarithm this is essentialy a
Coleman-Weinberg potential CW .
### V.1 Inclusion of Gravitational Effects
Consider the weak field approximation to gravity. We choose $w$-gauge of
eq.(23) and obtain
$\displaystyle\partial_{\alpha}h^{\alpha\beta}=w\partial^{\beta}h,\qquad
R=(1-w)\partial^{2}h/M$ (46)
Up to linear terms in $h_{\mu\nu}\hat{\phi}^{2}/M$ we have,
$\displaystyle
S_{Einstein}\rightarrow\int\Biggl{(}\frac{1}{2}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{1}{4}\frac{h}{M}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{1}{2}M^{2}R$
$\displaystyle-\frac{h^{\mu\upsilon}}{2M}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}-\frac{1}{2}\biggl{(}1+\frac{1}{2}\frac{h}{M}\biggr{)}B\hat{\phi}^{2}-V_{0}(\phi_{0})\Biggr{)}$
(47)
The contributions to the potential from the Feynman diagrams of Figs.(1,2) are
then,
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D1}=-\frac{\bigl{(}1+2w\bigr{)}}{12M}B\partial^{2}hL$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D2}=\frac{B}{4M}\partial^{2}hL$
(48)
Full details will be given in Ghil . Noting that,
$\displaystyle\frac{1}{4M}\partial^{2}h-\frac{(1+2w)}{12M}\partial^{2}h=\frac{(1-w)}{6M}\partial^{2}h=\frac{1}{6M}R$
Hence we have the net contribution to the potential,
$\displaystyle\Gamma_{\alpha}\equiv\Gamma_{D1}+\Gamma_{D2}=\frac{1}{6}BR\;L=\frac{\lambda_{1}\phi_{0}^{2}}{2}RL$
(50)
We thus see that there is a $\delta\alpha\phi_{0}^{2}R/12$ term generated from
the $3\lambda^{\prime}_{1}\phi_{0}^{2}$ term in B:
$\displaystyle\delta\alpha=6\lambda^{\prime}_{1}L+O(1/M^{2})$ (51)
Note that there are other terms proportional to $R$, but we only keep leading
terms in $1/M^{2}$ in $\delta\alpha$, since $R\sim 1/M^{2}$.
To implement the contact term we use, in eq.(50), the leading order $R$
equation of motion in the Einstein frame,
$\displaystyle
R=\frac{1}{M^{2}}T=\frac{1}{M^{2}}4\times\biggl{(}\frac{\lambda_{1}}{4}\phi_{0}^{4}-\frac{9\lambda^{2}_{1}}{2}\phi_{0}^{4}L\biggr{)}+{\cal{O}}\biggl{(}\frac{1}{M^{4}}\biggr{)}$
$\displaystyle\rightarrow\lambda_{1}\frac{\phi_{0}^{4}}{M^{2}}+{\cal{O}}(\hbar)$
(52)
We then have
$\displaystyle\Gamma_{\alpha}=\frac{1}{2}\lambda^{\prime}_{1}\phi^{2}_{0}R\;L\rightarrow\frac{\lambda^{\prime
2}_{1}\phi_{0}^{6}}{2M^{2}}L+{\cal{O}}\frac{1}{M^{4}}$ (53)
Therefore, combining all effects
$\displaystyle\Gamma_{Einstein}\equiv V_{0}+\Gamma_{0}+\Gamma_{\alpha}=$
$\displaystyle\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}$
$\displaystyle-\frac{1}{2}\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma-[\lambda_{1}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}L$
(54)
where the term in $[..]$ comes from the gravitational effects of D1 and D2.
Figure 2: Diagram D1. Figure 3: Diagram D2.
From eq.(V.1) we can extract the $\beta$-functions for $\lambda_{1}$ and
$\lambda_{2}$. Define $D=16\pi^{2}\partial/\partial\ln\mu$, and we have:
$\displaystyle D\lambda_{1}=18\lambda_{1}^{2}$ $\displaystyle
D\lambda_{2}=90\lambda_{1}\lambda_{2}-6\lambda_{1}^{2}+18\lambda^{2}\gamma^{\prime}$
(55)
In Ghil we obtain the RG equation for $\gamma^{\prime}$, but this requires an
effective action for non-constant $\phi_{0}$.
### V.2 Comparison to a Conventional Calculation of $\beta_{\alpha}$
in Jordan Frame Neglecting Contact Terms
We now compute the potential in the Jordan frame where we neglect the contact
term, as is conventionally done. Here we again use the background field
method.
Return to $S_{Jordan}$, and begin by shifting
$\phi\rightarrow\phi_{0}+\sqrt{\hbar}\hat{\phi}$ and expanding to
O$\hat{\phi}^{2}$ where $\phi_{0}$ is a constant classical background field
and $\sqrt{\hbar}\hat{\phi}$ is a quantum fluctuation. Henceforth we will
suppress the factor of $\sqrt{\hbar}$ but we will compute loops to order
$\hbar$. The shifted action where we drop the linear terms in $\hat{\phi}$
becomes to order $\hbar$:
$\displaystyle
S_{Jordan}=\int\sqrt{-g}\biggl{(}\frac{1}{2}g^{\mu\upsilon}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}$
$\displaystyle-\frac{1}{2}Z_{0}^{-1}A\hat{\phi}^{2}R-\frac{1}{2}B\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\biggr{)}$
(56)
where, we have the relations of eq.(V), but with unprimed couplings replacing
the primed ones, and
$\displaystyle A=\frac{\alpha}{6}$ (57)
Expanding to linear terms in $h_{\mu\nu}/M$ in weak field gravity the action
eq.(V.2) becomes
$\displaystyle
S\rightarrow\int\Biggl{(}\frac{1}{2}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+\frac{h}{4M}\eta^{\mu\nu}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}-\frac{h^{\mu\upsilon}}{2M}\partial_{\mu}\hat{\phi}\partial_{\nu}\hat{\phi}$
$\displaystyle-\frac{1}{2}Z_{0}^{-1}AR{\hat{\phi}^{2}}-\frac{1}{2}B\hat{\phi}^{2}-B\frac{h}{4M}\hat{\phi}^{2}-V_{0}(\phi_{0})+\frac{1}{2}M^{2}R\Biggr{)}$
Figure 4: Diagram D2.
Neglecting the contact terms we see, in addition to the non-curvature
potential obtained previously in eq.(45), the action eq.(V.2) now (naively)
generates three diagrams linear in the curvature, D1, D2, and D3 of
Figs(2,3,4). The additional D3 diagram (which is absent in the Einsteain
frame) is:
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{D3}=-Z_{0}^{-1}ABRL=-\frac{\alpha}{6}BRL+O\frac{1}{M^{2}}R$
(59)
Hence we have from eqs.(50,59) for (D1+D2+D3):
$\displaystyle\Gamma_{D3}+\Gamma_{\alpha}=-\frac{1}{6}(\alpha-1)BR\;L$
$\displaystyle=-\frac{1}{6}(\alpha-1)\biggl{(}{3\lambda_{1}}\phi_{0}^{2}+\frac{5\lambda_{2}+\gamma\lambda_{1}}{2M^{2}}\phi_{0}^{4}\biggr{)}RL$
(60)
Therefore, combining all effects:
$\displaystyle\Gamma_{Jordan}=$
$\displaystyle\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}+\frac{\alpha}{12}\phi_{0}^{2}R+\frac{\gamma}{12M^{2}}\phi_{0}^{2}R$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma)\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$
$\displaystyle-\frac{1}{6}(\alpha-1)\biggl{(}{3\lambda_{1}}\phi_{0}^{2}+\frac{5\lambda_{2}+\gamma\lambda_{1}}{2M^{2}}\phi_{0}^{4}\biggr{)}RL$
(61)
This contains the conventional radiative correction and renormaliztion group
running of $\alpha$
$\displaystyle\alpha\rightarrow\alpha+6\lambda_{1}(1-\alpha)L$ (62)
We again use the leading order $R$ equation of motion in the Einstein frame as
in eq.(V.1) to implement the contact term, but must now keep the
$\lambda^{2}_{1}L$ term,
$\displaystyle
R\rightarrow(\lambda_{1}-18\lambda^{2}_{1}L)\frac{\phi_{0}^{4}}{M^{2}}+{\cal{O}}(\hbar)$
(63)
hence
$\displaystyle\Gamma_{Jordan}\equiv
V_{0}+\Gamma_{0}+\Gamma_{\alpha}+\Gamma_{D3}$
$\displaystyle=\frac{\lambda_{1}}{4}\phi_{0}^{4}+\frac{\lambda_{2}+\alpha\lambda_{1}}{12}\frac{\phi_{0}^{6}}{M^{2}}-\frac{2\lambda^{2}_{1}\alpha\phi_{0}^{6}}{M^{2}}L$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{2}_{1}\phi_{0}^{4}+(15{\lambda_{1}\lambda_{2}}+3\lambda^{2}_{1}\gamma-[\lambda_{1}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{\prime}_{1}}{4}\phi_{0}^{4}+\frac{\lambda^{\prime}_{2}}{12}\frac{\phi_{0}^{6}}{M^{2}}+\frac{\lambda^{\prime
2}_{1}\alpha\phi_{0}^{6}}{2M^{2}}L\biggl{(}8-{3}\alpha\biggr{)}$
$\displaystyle-\frac{1}{2}L\biggl{(}9\lambda^{\prime
2}_{1}\phi_{0}^{4}+(15{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}+3\lambda^{\prime
2}_{1}\gamma^{\prime}-[\lambda^{\prime}_{1}{}^{2}])\frac{\phi_{0}^{6}}{M^{2}}\biggr{)}$
where we’ve converted to the Einstein frame primed variables via eqs.(V).
Comparing to eq.(V.1) we therefore see an inconsistency between the potentials
$\Gamma_{Einstein}$ and $\Gamma_{Jordan}$ at O($\hbar$):
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{Einstein}-\Gamma_{Jordan}=\frac{(8-3\alpha)\alpha\lambda^{\prime
2}_{1}\phi_{0}^{6}}{2M^{2}}\hbar L$ (65)
The fault lies in the Jordan frame calculation which does not implement the
contact term.
One might attempt to interpret this as a quantum “frame anomaly.” We see some
vague parallels with the axial anomaly here, but we think the issue is more of
an error that one is making in computing naively. Such an anomaly
interpretation would require identifying an operator that plays a special roll
in the Weyl tranformation and that has matrix elements given by the rhs of
eq,(65). Perhaps this can be related to the Weyl current, but we have not yet
explored this possibility.
## VI CONCLUSIONS
The Weyl transformation acting on the Jordan frame, to remove non-minimal
interactions leading to the minimal Einstein frame, is identical to
implementing the contact terms HillRoss . If one didn’t know about the Weyl
transformation one would discover it in the induced contact terms in the
single graviton exchange potential involving non-minimal couplings. The Weyl
transformation is powerful as it is fully non-perturbative. Technically it can
provide a useful check on the normalization and implementation of the graviton
propagators in various gauges. But the contact term stipulates that the
mapping to the Einstein frame is dynamical and inevitable, and does not
involve field redefinitions.
In a model with non-minimal coupling $-\alpha\phi^{2}R/12$ this implies that
the parameter $\alpha$ doesn’t really exist physically. Computing
$\beta$-functions in a Jordan frame without implementing the contact term will
yield incorrect results. Implementing the contact term yields the Einstein
frame and results computed there will have no contact term ambiguities.
However, the Einstein frame has a loop induced contact term which is then
absorbed back into the potential terms by the contact terms (equivalently, a
mini-Weyl transformation, or use of the $R$ equation of motion). The use of
the $R$ equation of motion on the non-minimal term is analogous to the use of
the gluon field equation for the electroweak penguin. It is likely the Deans
and Dixon Deans constraints on null operators applies to gravity as well.
This implies generally that there are pitfalls in directly interpreting any
physics in the Jordan frame.
We emphasize that our analysis applies strictly to a theory with a Planck mass
term. A Weyl invariant theory, where $M=0$, is nonperturbative and our
analysis is then inapplicable, and the Jordan frame may then be physically
relevant. Indeed, there is no conventional gravity in this limit since the
usual $M^{2}R$ (Fierz-Pauli) graviton kinetic term does not then exist. Hence
in this limit one would have to appeal to a UV completion, e.g., string theory
or $R^{2}$ gravity, etc.
In the case of an $R^{2}$ UV completion theory we view the formation of the
Planck mass by, e.g., inertial symmetry breaking, i.e., as a dynamical phase
transition, similar to a disorder-order phase transition in a material medium
FHR . An intriguing point to note is that if we have an $R^{2}$ UV completion,
then the graviton propagator is $\propto 1/q^{4}$. But then the $AR\sim
Ahq^{2}$ vertex implies a $1/q^{2}$ graviton exchange amplitude, which means
an inverse square law “pseudo-gravitational force” exists even above the
Planck scale to fields that couple non-minimally. This is remarkable to us and
one of many issues to develop further in this context.
Acknowledgements
Part of this work was done at Fermilab, operated by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the United States Department of
Energy.
## Appendix A Projection-Regulated Feynman Loops
The loop induced effective potential for $\phi_{0}$ provides a useful way to
extract all of the $\beta$-functions of the various coupling constants. The
potential $\Gamma(\phi_{0})$ is the log of the path integral: $\Gamma=i\ln P$.
In the case of a real scalar field with mass term we consider the free action,
$\displaystyle\frac{1}{2}\int
d^{4}x\biggl{(}\partial\phi\partial\phi-m^{2}\phi^{2}\biggr{)}$ (66)
we have for the path integral:
$\displaystyle
P=\underset{k}{\prod}\biggl{(}k^{2}-m^{2}\biggr{)}^{-1/2}=\det\biggl{(}k^{2}-m^{2}\biggr{)}^{-1/2}$
(67)
whre $k=(k_{0},\vec{k})$ is the $4$-momentum hence, we have:
$\displaystyle\Gamma=i\ln
P=-\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\ln\biggl{(}k^{2}-m^{2}+i\epsilon\biggr{)}$
(68)
This can be evaluated with a Wick rotation to a Euclidean momentum,
$k\rightarrow k_{E}=(ik_{0},\vec{k})$, and a Euclidean momentum space cut off
$\Lambda$:
$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\Lambda}\frac{d^{4}k_{E}}{(2\pi)^{4}}\ln\biggl{(}\frac{k_{E}^{2}+m^{2}}{\Lambda^{2}}\biggr{)}$
(69)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{1}{64\pi^{2}}\biggl{(}\Lambda^{4}\ln\frac{\Lambda^{2}+m^{2}}{\Lambda^{2}}-m^{4}\ln\frac{\Lambda^{2}+m^{2}}{m^{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\qquad-\frac{1}{2}\Lambda^{4}+\Lambda^{2}m^{2}\biggr{)}+(\makebox{irrelevant
constants})$
The cutoff can be viewed is a spurious parameter, introduced to make the
integral finite and arguments of logs dimensionless, but and not part of the
defining action. The only physically meaningful dependence upon $\Lambda$ is
contained in the logarithm, where it reflects scale symmetry breaking by the
quantum trace anomaly. Powers of $\Lambda$, e.g.,
$\Lambda^{4},\Lambda^{2}m^{2}$, spuriously break scale symmetry and are not
part of the classical action Bardeen .
It is therefore conceptually useful to have a definition of the loops in which
the spurious powers of $\Lambda$ do not arise. This can be done by defining
the loops applying projection operators on the integrals. The projection
operator
$\displaystyle
P_{n}=\biggl{(}1-\frac{\Lambda}{n}\frac{\partial}{\partial\Lambda}\biggr{)}$
(70)
removes any terms proportional to $\Lambda^{n}.$ Since the defining classical
Lagrangian has mass dimension 4 and involves no terms with $\Lambda^{2}m^{2}$
or $\Lambda^{4},$ we define the regularized loop integrals as:
$\displaystyle\Gamma$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{2}P_{2}P_{4}\int_{0}^{\Lambda}\frac{d^{4}k_{E}}{(2\pi)^{4}}\ln\biggl{(}\frac{k_{E}^{2}+m^{2}}{\Lambda^{2}}\biggr{)}$
(71) $\displaystyle=$
$\displaystyle-\frac{1}{64\pi^{2}}m^{4}\biggl{(}\ln\frac{\Lambda^{2}}{m^{2}}\biggr{)}+O\biggl{(}\frac{m^{6}}{\Lambda^{2}}\biggr{)}$
where we take the limit $\Lambda>>m$ to suppress $O(m^{6}/{\Lambda^{2}})$
terms and we are interested only in the log term (not additive constants) This
means that the additive, non-log terms, e.g. $c^{\prime}m^{2}$, are
undetermined, and the only physically meaningful result is the
$\ln(\Lambda^{2}/m^{2})$ term. $\Lambda$ can be swapped for a renormalization
scale $\mu$. Interestingly, if we define the integral as $\int\rightarrow
P_{1}P_{2}P_{3}...P_{\infty}\int$, the action on the logs will lead to the
Euler constant that arises in dimensional regularization, hinting at a mapping
to the dimensionally regularized result.
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|
# Sensitivity of YAC to measure the light-component spectrum of primary cosmic
rays at the “knee” energies
L M Zhai1, J Huang1, D Chen2, M Shibata3, Y Katayose3, Ying Zhang1, J S Liu1,
Xu Chen1, X B Hu1,4 and Y H Lin1 1 Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China 2 National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China 3 Faculty of Engineering, Yokohama National University,
Yokohama 240-8501, Japan 4 Department of Physics, Shandong University, Jinan
250100, China<EMAIL_ADDRESS>
###### Abstract
A new air-shower core-detector array (YAC : Yangbajing Air-shower Core-
detector array) has been developed to measure the primary cosmic-ray
composition at the “knee” energies in Tibet, China, focusing mainly on the
light components. The prototype experiment (YAC-I) consisting of 16 detectors
has been constructed and operated at Yangbajing (4300 m a.s.l.) in Tibet since
May 2009. YAC-I is installed in the Tibet-III AS array and operates together.
In this paper, we performed a Monte Carlo simulation to check the sensitivity
of YAC-I+Tibet-III array to the cosmic-ray light component of cosmic rays
around the knee energies, taking account of the observation conditions of
actual YAC-I+Tibet-III array. The selection of light component from others was
made by use of an artificial neural network (ANN). The simulation shows that
the light-component spectrum estimated by our methods can well reproduce the
input ones within 10% error, and there will be about 30% systematic errors
mostly induced by the primary and interaction models used. It is found that
the full-scale YAC and the Tibet-III array is powerful to study the cosmic-ray
composition, in particular, to obtain the energy spectra of protons and helium
nuclei around the knee energies.
###### pacs:
98.70.Sa, 96.50.sb, 96.50.sd
††: J. Phys. G: Nucl. Part. Phys.
Keywords: cosmic rays, hadronic interaction, knee, composition, energy
spectrum
## 1 Introduction
It is well known that the all-particle spectrum of primary cosmic rays follows
a power law of dJ/dE $\propto$ E-γ, but steepens at energies around
4$\times$1015 eV where the power index $\gamma$ changes sharply from $\sim$2.7
to $\sim$3.1 [1, 2]. Such structure of the all-particle energy spectrum is
called the “knee”, which is considered to be closely related to the origin,
acceleration and propagation mechanism of cosmic rays. In order to explain the
existence of the knee, many hypotheses and mechanisms [3, 4] have been
proposed. Although all these approaches can well describe the knee structure,
there are much discrepancies in the prediction of the individual components at
the knee region. Thus, it is critical to measure the primary chemical
composition or mass group at energies 50-10 000 TeV, especially, to measure
the primary energy spectra of individual component and determine a break
energy of the spectral index for individual component. Direct cosmic-ray
measurements on board balloons or satellites are the best way to study the
chemical composition, while the maximum energy they can cover is up to 1014
eV/nucleon at most due to limited detection area or exposure time. We may have
no choice but to rely on ground-based air-shower (AS) measurements to study
the primary chemical composition around the knee.
The study of cosmic-ray composition around the knee was done by a hybrid
experiment of the emulsion chambers (ECs), the burst detectors (BDs) and the
AS array (Tibet-II), where ECs and BDs of total area 80 m2 were set up near
the center of the AS array and operated for three years [5, 6, 7]. The
threshold energy of ECs capable of analyzing the fine structure of AS cores is
about 1 TeV, so that it is not difficult to separate the AS events induced by
light-component of protons and helium nuclei, while the energy range of
primary particles is limited to be above $\sim$200 TeV for protons and
$\sim$400 TeV for helium nuclei [6]. This experiment suggests that the flux of
light component is less than $\sim$30% of the total, resulting in that the
knee is dominated by nuclei heavier than helium [7]. A demerit of this
experiment is that there are few statistics of the high-energy core events due
to the high detection threshold energy of the experiment as mentioned above.
To improve this problem, a new air-shower core detector named YAC (Yangbajing
Air shower Core detector) has been developed and improved so as to meet our
requirements.
One important improvement is to lower the detection threshold energy of
primary particles to several times 10 TeV, about one order of magnitude
smaller than the previous experiment. With this improvement, the energy
spectra of individual components measured by YAC will overlap with those of
direct measurements, which may help us to examine the knee of light component,
such as “proton knee” or “helium knee”. Another important improvement of YAC
is its ability to count the number of shower particles passing through each
detector in a wide dynamic ranging from 1 to 106 particles, making it possible
to observe the primary cosmic rays in the energy range from $\sim$10 TeV to
$\sim$10 PeV.
Until now, we have constructed and operated YAC-I as a prototype of full-scale
YAC comprising 400 core detectors. YAC-I is a small array consisting of 16
core detectors which were placed near the center of the Tibet-III AS array as
shown in Fig. 1, while being able to observe a lot of AS-core events in the
energy around $10^{14}$ eV by the operation of a few months. As the primary
composition around this energy region is fairly well known by the direct
observations [8, 9, 10], the data from YAC-I may be used to test the
interaction models such as SIBYLL2.1, QGSJETII-04 and EPOS-LHC being currently
used in the Monte Carlo (MC) simulations. In this paper, we discuss the
performance and sensitivity of YAC for observing light-component spectrum of
primary particles through MC simulations based on the YAC-I experiment.
## 2 YAC-I Experiment
Figure 1: Schematic view of YAC-I and Tibet-III AS array. Open squares :
scintillation detectors of the Tibet-III array. For details of the Tibet-III,
see the paper [1]. Filled red squares : core detectors of the YAC-I array.
YAC-I consists of 16 detector units of 0.2 m2 each, which cover the area of
about 10 m2. Each detector of YAC-I consists of a lead plate with the
thickness of 3.5 cm (6.3 radiation lengths), a supporting iron plate with the
thickness of 0.9 cm (0.5 radiation lengths) and a plastic scintillator with
the thickness of 1 cm.
The YAC-I array consists of 16 core detectors which are placed on a 4$\times
4$ square grid covering the area of about 10 m2 as shown in Fig. 1 and has
been operating since May 2009, together with the Tibet-III AS array. Each core
detector of YAC-I comprises a plastic scintillator with the size of 40 cm
$\times$ 50 cm and a lead plate with the thickness of 3.5 cm (6.3 radiation
lengths) being put on the scintillator. The lead plate is used to select AS
particles and cores capable of having sufficient energy to create cascade
showers in the lead plate and pass through the scintillator. The plastic
scintillator in the core detector is divided into 10 pieces of the width of 4
cm and the scintillation lights are collected through wavelength shifting
(WLS) fibers as shown in Fig. 2. Such design ensures the geometrical
uniformity of detector response within 5 %. The details about the hardware of
YAC detector is described in [11, 12]. Two photomultiplier tubes (PMTs) of
high-gain (HAMAMATSU: R4125) and low-gain (HAMAMATSU: R5325) are equipped to
cover a wide dynamic range from 1 MIP(Minimum Ionization Particle) to 106 MIPs
as seen in Fig. 2. The corresponding linearity and saturation effect of PMT
and scintillator were examined by use of cosmic-ray muons and electron beams
provided by the beam facility of BEPCII (Beijing Electron Positron Collider,
IHEP, China) [13]. The stability of the PMT gain was checked and corrected
using cosmic-ray muons.
Figure 2: Schematic view of the scintillation detector, the top view (a) and
the back view (b). The scintillator is divided into 10 pieces of 4 cm width
and scintillation lights from each piece are collected through the wavelength
shifting (WLS) fibers.
In this experiment, the YAC-I array works to observe air-shower cores and
their accompanying ASs are observed simultaneously with the Tibet-III AS
array. The Tibet-III provides information on the arrival time and direction of
each air shower and its AS size corresponding to the energy of primary
particle [1]. When a YAC-I event is triggered, its accompanying AS is
simultaneously recorded by the Tibet-III array and the matching between YAC-I
and Tibet-III events is made by their arrival time stamps recorded by a GPS
clock.
## 3 Monte Carlo Simulation
We have carried out a full Monte Carlo (MC) simulation on the development of
air showers in the atmosphere using the simulation code Corsika [14] (version
7.3500). Three hadronic interaction models, including SIBYLL2.1 [15], EPOS-LHC
(v3400) [16] and QGSJETII-04 [17], are used to generate the air-shower events
in the atmosphere. For the primary cosmic rays, we examined three composition
models, namely, “He-poor”, “He-rich” and “Gaisser-fit” models, in order to
evaluate the systematic errors attributable to primary composition models. The
“He-poor” model is based on the HD (Heavy Dominant) model mentioned in the
paper [1], and it is slightly revised to match with the new all-particle
energy spectrum [1], to be the new “He-poor” model. The “He-rich” model is the
“Model B (a lightly harder spectrum than the previous on by taking account of
the nonlinear effects)” mentioned in the paper [4]. The “Gaisser-fit” model is
the “three-population” model mentioned in the paper [18]. The proton spectra
of the former two models are fitted to the direct measurements at the low
energy and consistent with the spectrum obtained from the Tibet-EC experiment
at the high energy. The He spectrum of He-poor model coincides with the
results from RUNJOB, but the He spectrum of He-rich model coincides with the
results from JACEE, ATIC2 and CREAM. The Gaisser-fit model fits to a higher He
model (almost same as our He-rich model) at the low energy range and to the
KASCADE-QGSJET data at high energy range in which light components (P and He)
dominate in the chemical composition. In all models mentioned above, each
component is summed up so as to match with the all-particle spectrum with a
sharp knee, which was obtained with the Tibet-III AS array [1].
Table 1 is a summary of the fractions of the components (P, He, Medium and Fe)
of the three composition models in given energy regions for three primary
models. The energy spectra of individual components (or mass groups) for three
primary models are shown in Fig. 3. It is seen that all the individual
components of the three models in the low energy range (less than 100 TeV) are
in good agreement with direct measurements while differ significantly at
higher energy. The all-particle spectra of three models, however, coincide
with each other and reproduce the sharp knee structure as well.
Table 1: The fractions of individual components in the assumed primary cosmic-
ray spectra of He-poor, He-rich and Gaisser-fit models.
* Composition | Components | $10^{13}-10^{14}$ eV | $10^{14}-10^{15}$ eV | $10^{15}-10^{16}$ eV
---|---|---|---|---
Models | (%) | (%) | (%)
He-poor | P | 31.5 | 22.7 | 9.6
He | 22.4 | 18.8 | 9.7
Medium | 26.6 | 26.6 | 26.1
Fe | 19.5 | 31.9 | 54.6
He-rich | P | 31.1 | 26.3 | 10.0
He | 25.1 | 28.7 | 17.5
Medium | 32.4 | 34.4 | 50.3
Fe | 11.3 | 10.6 | 22.2
Gaisser-fit | P | 32.8 | 29.0 | 19.6
He | 34.4 | 37.4 | 37.4
Medium | 20.1 | 20.4 | 25.3
Fe | 12.7 | 13.2 | 17.7
In this simulation, primary cosmic rays at the top of the atmosphere within
the zenith angles smaller than 60 degrees are thrown into the atmosphere
isotropically and the minimum energy of primary cosmic rays is set to 40 TeV.
All shower particles in the atmosphere are traced down to the minimum energy
of 1 MeV. The AS events generated are randomly dropped onto the area of 32.84
m $\times$ 32.14 m, which is a 15 m wider in each side of the YAC-I array.
This dropping area is large enough to collect the AS events more than 99.5%
under our core-event selection conditions (see below in the text). Observation
of the MC events is made with the same method as that of the experiment.
The detector responses to shower particles falling on the detectors of
(YAC-I+Tibet-III) array are calculated using the Geant4 [21] (version 9.5),
where the detector performance, trigger efficiency and effective area are
adequately taken into account based on the experimental conditions. The number
of charged particles passing through the scintillator is defined as the PMT
output (charge) divided by that of the single-particle peak. The single-
particle peak is determined by a probe calibration [1, 11] using cosmic rays,
typically muons. The value of single-particle peak is measured as 1.98 MeV for
YAC-I detectors and 6.28 MeV for the Tibet-III detectors. These values are
used in this MC simulation.
Figure 3: Primary cosmic-ray composition for He-poor, He-rich and Gaisser-fit
models compared with those of direct measurements (ATIC2 [8], JACEE [19],
RUNJOB [20], CREAM [9, 10]) and the sum of all components (all-particle
spectrum) compared with the results obtained by the Tibet-III experiment [1] .
The main purpose of this work is to check the sensitivity of YAC to observe
the light component of primary cosmic rays as well as to evaluate the
systematic errors by adopting different primary composition models and
interaction models mentioned above. For this, we selected five combinations of
interaction models and primary composition models. Four combinations of
SIBYLL2.1+He-rich, SIBYLL2.1+He-poor, EPOS-LHC+He-poor and EPOS-LHC+Gaisser-
fit are to check the sensitivity of YAC to the light component and
uncertainties due to the adoption of the different composition models. Other
three combinations of SIBYLL2.1+He-poor, QGSJETII-04+He-poor and EPOS-LHC+He-
poor are to check the interaction models and also uncertainties under the same
primary composition model. It has value to point out here that there is no
serious difference among the current interaction models on the particle
production in the forward region and proton-air inelastic cross sections in
our concerned energy region from 10 TeV to $10^{4}$ TeV since all the models
are well tuned using recent accelerator data including LHC, while there are
big differences among primary composition models because of a lack of direct
observation data at energies above $\sim$200 TeV.
The number of air-shower events generated for each model is 7.40$\times$107,
6.57$\times$107, 4.67$\times$107, 6.25$\times$107 and 5.18$\times$107,
respectively, as shown in Table 2. The analysis of these MC events was made by
the same method used in the experiment.
## 4 Analysis
Information on the size $N_{e}$ and arrival direction of each air shower event
hitting both YAC-I and Tibet-III arrays can be easily obtained from the MC
events observed with the Tibet-III AS array simultaneously. Details of its
analysis are found in the paper [1].
From the YAC-I array, we can obtain the following five quantities reflecting
the characteristic of AS cores : (1) ${N_{hit}}$, the number of “fired”
detectors with ${N_{b}}$$\geq$ 200, where $N_{b}$ is the number of particles
(burst size) observed by each core-detector ; (2) $\sum$${N_{b}}$, the total
sum of $N_{b}$ of fired detector ; (3) ${N_{b}}$top, the maximum ${N_{b}}$
among the fired detectors ; (4) $\langle R\rangle$, the mean lateral spread
defined as $\langle R\rangle$ = $\sum$${r_{i}}$/($N_{hit}$-1) ; (5) $\langle
N_{b}R\rangle$, the mean energy-flow spread defined as $\langle
N_{b}R\rangle=\sum({N_{b}^{i}}\times{r_{i}})/N_{hit}$, where $N_{b}^{i}$
denotes the number of particles observed in $i$-th fired detector and $r_{i}$
represents the lateral distance from the burst center ($X_{c},Y_{c}$), where
($X_{c},Y_{c}$) = $\left(\frac{\sum N_{b}^{i}x_{i}}{\sum N_{b}^{i}},\frac{\sum
N_{b}^{i}y_{i}}{\sum N_{b}^{i}}\right)$. It is confirmed that the five
quantities mentioned above are basic and enough to separate the light
component (P+He) from others. A use of ANN-method [22] may further improve the
quality of separation.
Table 2: Statistics of the data sets selected in MC simulation.
* Models | Primaries | Core events
---|---|---
(E $\geq$ 40 TeV) | (Mode energy: $\sim$200 TeV)
SIBYLL2.1+He-rich | 7.40$\times$107 | 64 331
SIBYLL2.1+He-poor | 6.57$\times$107 | 47 580
QGSJETII-04+He-poor | 4.67$\times$107 | 31 928
EPOS-LHC+He-poor | 6.25$\times$107 | 42 137
EPOS-LHC+Gaisser-fit | 5.18$\times$107 | 49 390
In order to obtain the light-component spectrum using the data from both
arrays of YAC-I and Tibet-III, we select the high-energy core events by
imposing the conditions of ${N_{b}}\geq 200$, $N_{hit}\geq 4$,
${N_{b}}^{top}\geq 1500$ and ${N_{e}}\geq 80\,000$. The mode energy of primary
particles producing such high energy core events is then estimated to be about
200 TeV. The statistics of the data-sets selected based on the five models are
listed in Table 2. Shown in Fig. 4 is the effective $S\Omega$ of YAC-I array
to observe the AS-core events satisfying the event select conditions, where
$S$ denotes the detection area and $\Omega$ the solid angle. The effective
$S\Omega$ depends weakly on the model used, but its difference is found to be
smaller than 25% in our concerned energy range.
Figure 4: The effective $S\Omega$ of YAC-I array to observe the light-
component for various models used in MC.
## 5 Results and Discussion
We check the sensitivity of YAC array to the interaction models and primary
cosmic-ray models using the high-energy core events selected under the
conditions discussed in the previous section.
### 5.1 Total burst-size spectrum and mean lateral spread of AS-cores
It is well known that the absolute intensity of the total burst sizes depends
sensitively on the increase of cross sections, inelasticity, and also on the
primary cosmic-ray composition. Shown in Fig. 5-(a) is the integral total
burst-size spectrum ($\sum N_{b}$ (SumNb)) obtained by the respective MC model
for comparison. The $\sum N_{b}$ spectra obtained by five MC models are
compared each other by taking the flux ratio to that by the SIBYLL2.1+He-rich
model in Fig. 5-(b). It is seen that the EPOS-LHC+Gaisser-fit model gives the
highest flux in all $\sum N_{b}$ region. According to our MC simulation, the
observed AS cores in the size region of $\sum N_{b}$ = 2$\times 10^{3}$ \-
4$\times 10^{5}$ are produced mostly by the light component (P+He) with its
primary energies of several times 1014 \- 1015 eV. The fraction of light
component in the primary of this energy region is about 66% for Gaisser-fit
model while about 55% for the He-rich model and 42% for the He-poor model as
seen in Table 1. It should be, however, noted that about 70% of the observed
high-energy core events are originated by the light component, that is, the
contribution from other nuclei is fairly small.
Figure 5: (a) Integral $\sum N_{b}$ (SumNb) spectra obtained from five MC
models, (b) The intensity ratios of $\sum N_{b}$ to that obtained by the
SIBYLL2.1+He-rich model, (c) The mean energy-flow lateral spreads $\langle
N_{b}R\rangle$ in the respective energy interval for five MC models.
Here, we discuss the uncertainties due to different interaction models in
reference to the spectrum of high-energy core events. Before entering in
discussion, we should first remind that the production rate of high-energy
core events is most sensitive to the energy per nucleon of primary particles.
Thus, the mean energy per nucleon of helium nuclei is 1/4 of protons when
compared at the same primary energy and also the interaction mean free path of
helium nuclei in the air is about a half of that of protons. For the He-poor
model, it is seen that the flux of protons is slightly higher than that of
helium nuclei or almost same in the energy region over about 300 TeV and also
their power indices are almost same as learned from Fig. 3 and Table 1. If we
combine this with above discussion, it may be allowed to ignore the
contribution from helium nuclei to the core events observed in this primary
model, that is, it is regarded as those produced by protons. Under this
assumption, it may be noticed from Fig. 5-(b) that the flux values by
SIBYLL2.1, QGSJETII-04 and EPOS-LHC using the same primary model of He-poor
match well with an error of smaller than 10%, which may be attributed to
uncertainties due to the interaction models used. A deviation of QGSJETII-04
in the core-size region above about 2$\times 10^{5}$ may be due to a low
statistics of the events, that is, within a statistical fluctuation (at most
2$\sigma$ level).
On the other hand, when we arrange the light component flux of protons and
helium nuclei in descending order in the $10^{2}$ -$10^{4}$ TeV region, it
becomes as Gaisser-fit $>$ He-rich $>$ He-poor. It is then confirmed that the
flux of core events is in the same order as the primary light-component flux
and also the shape of each primary spectrum is well reflected in the
corresponding core-event spectrum, as seen in Fig. 5-(b). A typical example is
seen in the case of Gaisser-fit primary model. This means that high energy
core-event observation with YAC is very sensitive to the primary light-
component spectrum.
A correlation between $\langle N_{b}R\rangle$ and $\sum N_{b}$ is shown in
Fig. 5-(c). This figure tells us that the Gaisser-fit primary model gives
smaller lateral spread than others, while the QGSJETII-04+He-poor model gives
larger spread than others. As protons with the long interaction mean free path
can penetrate deep in the atmosphere and produce AS cores near the observation
level, resulting in giving smaller lateral spread. The Gaisser-fit primary is
light-component dominant as seen in Fig. 3, so that the core spread by this
model should be smaller than others. When the primary model is fixed as He-
poor, the mean spread of QGSJETII-04 is slightly larger than that of
SIBYLL2.1. This may slightly depend on the interaction model since the energy
spectrum of secondary particles in the very forward region (Feynman $x\sim$
0.1 - 0.3) produced at collision in the SIBYLL2.1 model is harder than that of
QGSJETII-04, that is, the former contains slightly larger number of very high-
energy secondaries than the latter. The difference of the intensity and
lateral spread between both models could be attributed to the number of very
high energy particles as those penetrate deep in the atmosphere. It should be
noted that the lateral spread of AS-cores is mostly caused by Coulomb
scattering of shower electrons and positrons in the atmosphere, not by
transverse momentum of secondaries produced at collisions except within the
depth of 1-2 radiation lengths from the interaction point in the atmosphere.
In connection with this, the energy loss of AS cores generated by QGSJETII-04
may be faster than those by SIBYLL2.1, resulting in giving lower AS-core
intensity. Hence, a precise AS experiment like the (YAC + Tibet-III AS) array
will be able to examine the interaction models to some extent.
### 5.2 Sensitivity of YAC-I to observe the light-component spectrum around
the knee
In this analysis, we use the ANN technique to separate the light-component
from others. This method is shown to be quite effective for such purpose as
confirmed by our previous works [6, 7]. In this ANN analysis, we use the
following seven quantities : (1) $N_{hit}$, (2) $\sum$$N_{b}$, (3)
$N_{b}^{top}$, (4) $\langle R\rangle$, (5) $\langle N_{b}R\rangle$, (6)
$N_{e}$ and (7) $\theta$ (zenith angle). These are input to the ANN with 35
hidden nodes and 1 output unit. To train the ANN in separating light-component
(P+He) from other nuclei, the input patterns for light-component and others
are set to 0 and 1, respectively. We then define a critical value of $T_{c}$
to calculate the corresponding purity and selection efficiency of the selected
(P+He)-like events.
Figure 6: ANN output pattern value (T) distribution of training (P+He) events
based on the EPOS-LHC+He-poor model. The average selection purity and
efficiency over whole energy range of (P+He)-like events are 95%, 76% at
${T_{c}}$ = 0.4. Table 3: The ratios of (P+He)/All at three phases of
analysis, before and after ANN training based on the MC models. In this table
the second column represents the ratio of primary (P+He) flux to the all-
particle flux in the energy region above 40 TeV. The third column represents
the ratio of true (P+He) events contained in the observed high-energy core
events selected by the condition mentioned in the text. The fourth column
represents the ratio of true (P+He) events contained in the ANN trained core
events with $T\leq T_{c}=0.4$. The fifth column represents the ratio of the
number of ANN trained core events with the output $T\leq 0.4$ to that of all
ANN trained core events ($0\leq T\leq 1$).
* Models | Primaries | Core events | After ANN training
---|---|---|---
(P+He)/all(%) | (P+He)/all(%) | Purity (%) | Efficiency (%)
SIBYLL2.1+He-rich | 56.2 | $71.1\pm 0.2$ | $92.0\pm 0.2$ | $70.1\pm 0.3$
SIBYLL2.1+He-poor | 46.8 | $69.7\pm 0.2$ | $94.0\pm 0.2$ | $71.1\pm 0.3$
QGSJETII-04+He-poor | 46.8 | $69.5\pm 0.3$ | $94.6\pm 0.2$ | $77.1\pm 0.4$
EPOS-LHC+He-poor | 46.8 | $69.3\pm 0.2$ | $94.6\pm 0.2$ | $75.8\pm 0.3$
EPOS-LHC+Gaisser-fit | 67.0 | $87.4\pm 0.1$ | $96.1\pm 0.2$ | $67.8\pm 0.3$
Figure 6 shows the ANN output distribution trained using the EPOS-LHC+He-poor
model. As seen in this figure, the events with ${T_{c}}\leq 0.4$ could be
regarded as the (P+He)-like events, and the average selection purity and
efficiency over whole energy range of (P+He)-like events are 95% and 76%,
respectively. Table 3 is a summary of the ratios of the (P+He)/All before and
after ANN training analysis based on the five MC models over the whole energy
range. Thanks to the performance of the YAC-I array, the ratio of the
(P+He)/All before ANN training (core events) has already reached $\sim$70% by
the core-event selection conditions ($\sim$87% for Gaisser-fit model, as seen
in Table 3). With the ANN training, the purity of selected (P+He)-like events
is further increased up to $\sim$95% as learned from Table 3. This high
quality data set is used for reconstructing the light-component primary
spectrum. Table 3 teaches us that after ANN training the difference of the
selection purity and efficiency is within $\sim$6% among three hadronic
interaction models, and $\sim$8% among three primary composition models.
Overall uncertainties due to ANN training for the MC models are then estimated
to be about 10%.
### 5.3 Expected light-component spectrum
Using the ANN trained by the EPOS-LHC+He-poor events, we select the (P+He)
like events from all the observed events and also obtain the selection purity
and efficiency. The primary energy $E_{0}$ of each selected events is then
estimated using the AS size $N_{e}$ obtained by the Tibet-III. The relation
between air shower size $N_{e}$ and primary energy $E_{0}$ is expressed as
$E_{0}=\alpha\times N_{e}^{\beta},$
where the parameters of $\alpha$ and $\beta$ are estimated from AS events
generated by the MC for the Tibet-III AS array, while $\alpha$ and $\beta$
depending on the zenith angle of air showers. Details of this procedure is
described in the paper [1]. Shown in Fig. 7 is the correlation between $E_{0}$
and $N_{e}$ of the (P+He)-like events selected by the ANN method. The solid
line denotes the best fit curve for this correlation, and the parameter values
of $\alpha=0.794$ and $\beta=1.005$ are then obtained for air showers with
$\sec\theta\leq 1.1$. The energy resolution is also estimated as about 25% at
energies around 200 TeV.
Figure 7: Scatter plots of the primary energy ($E_{0}$) and the estimated
shower size ($N_{e}$) of (Proton+Helium)-like events based on EPOS-LHC+He-poor
model with $\sec\theta\leq 1.1$. Solid line shows the fitting result of
$E_{0}$ = 0.794$\times$ ${N_{e}}^{1.005}$ GeV.
We also checked a dependence of the correlation of $N_{e}$ and $E_{0}$ on the
interaction and primary composition model, and obtained the similar
relationship in the other models. We then found that there is less than 10%
difference for the determination of the primary energy by use of different
interaction and primary composition models.
Figure 8 shows the estimated primary energy spectrum of the light-component
(P+He) in comparison with the assumed primary spectrum. It is seen that the
estimated energy spectrum well reproduces the assumed one within about 10%
errors. Almost same results are obtained for other MC models.
Figure 8: The estimated energy spectrum of (Proton+Helium) compared with the
assumed (input) one based on SIBYLL2.1+He-rich model.
In this work, we discuss the systematic errors coming from the models used in
the MC simulation for deriving the primary (P+He) spectrum using the
(YAC-I+Tibet-III) array. The systematic errors caused by each step of the
analysis procedures are investigated including the dependence of the MC data
on the interaction models, the primary composition models and the algorithms
for the primary mass identification. We summarize these systematic errors as
follows: 1) errors due to the observation efficiency ($S\Omega$ of (P+He))
depending on the interaction and primary composition models are found to be
smaller than 25%; 2) errors due to the selection of (P+He)-like events with
ANN training are about 10%, in which the model dependence on the purity and
efficiency is totally included; 3) errors due to the estimation of primary
energy with use of the conversion from $N_{e}$ to $E_{0}$ are 10%, in which
the dependence on both interaction and primary composition models are taken
into account; 4) errors due to the reconstruction procedures of the primary
light-component spectrum from the observed core events are estimated to be
smaller than 10%. The total systematic errors are then estimated to be about
30% as the square root of quadratic sum of those four systematic errors, which
may be somewhat overestimated because of a little correlation among four error
estimation parts.
## 6 Summary
In this paper, we have carried out a full MC simulation to examine the
capability of measuring the energy spectrum of primary light-component (P+He)
of cosmic rays at the knee energies using the (YAC-I+Tibet-III) array. The
models used in this MC simulation are SIBYLL2.1, EPOS-LHC (v3400) and
QGSJETII-04 for the interaction and He-poor, He-rich and Gaisser-fit for the
primary cosmic-ray composition. The Corsika code was used to generate AS
events in the atmosphere and the Geant4 code was used to treat the shower
particles entering in the detectors. The air-shower core events observed with
the YAC-I array were analyzed to select those induced by the light component
using the ANN technique. In this paper, we focused on the sensitivity of the
YAC-I array to observe the light component of cosmic rays around the knee and
discussed the systematic errors coming from the the models used in the MC
indispensable for obtaining the result which should be independent on the
model as possible. It is shown that the YAC+Tibet AS array is powerful to
study the primary cosmic-ray chemical composition, in particular, to obtain
the energy spectrum of light-component (P+He) of cosmic rays at the knee
energies. A full-scale YAC consisting of 400 core detectors covering more than
5000 m2 could be operated together with the Tibet-III array in the very near
future.
The authors would like to express their thanks to the members of the Tibet
AS$\gamma$ collaboration for the fruitful discussion. This work is supported
by the Grants from the National Natural Science Foundation of China
(Y11122005B, Y31136005C and Y0293900TF) and the Chinese Academy of Sciences
(H9291450S3) and the Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, CAS. The Knowledge Innovation Fund (H95451D0U2 and
H8515530U1) of IHEP, China also provide support to this study.
## References
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|
11institutetext: 1 University of New South Wales, Australia
<EMAIL_ADDRESS>22institutetext: 2 Division of Fire Safety
Engineering, Lund University, Sweden<EMAIL_ADDRESS>
# Revisiting the paper ”Simulating dynamical features of escape panic”: What
have we learnt since then?
Milad Haghani1 Enrico Ronchi2
###### Abstract
The paper ”Simulating dynamical features of escape panic” by Helbing, Farkas,
and Vicsek, published over two decades ago in Nature, has left an indelible
mark on the field of crowd dynamics. With nearly 3,000 citations to date,
according to the Web of Science records, and significant influence, it has
shaped the crowd dynamics field. This analysis investigates the overall
influence of this paper through a variety of indicators, mapping its reach
across research areas. The intellectual foundation of the paper is traced,
examining the references cited. The terminological impact is also explored,
showing how the paper made use of terms like ”panic” and ”herding”. Moreover,
the alignment of the assumptions of the paper with empirical evidence is
discussed, finding discrepancies in key assertions about panic behaviour. The
numerical simulations of the paper and observations have significantly
influenced the field, such as for the case of the ”faster-is-slower”
phenomenon. The paper remains a key pillar in crowd dynamics, nevertheless, we
advocate for a new course of the field shifting away from the terminology
adopted in the paper and focusing more on empirical evidence.
###### keywords:
Social force model pedestrian dynamics crowd dynamics evacuation simulation
## 1 Introduction
More than two decades have passed since the publication of one of the most
influential papers in the field of crowd dynamics authored by Helbing, Farkas,
and Vicsek, which was featured in Nature (Helbing et al. [1]). Remarkably,
this paper has obtained nearly 3,000 citations according to Web of Science and
6,000 citations according to Google Scholar up to now. Its influence extends
beyond mere numerical counts, shaping the trajectory of crowd dynamics
research in profound ways since its inception. The paper is not just the most
cited paper in the field; it has also fundamentally shaped the discourse and
development of crowd dynamics as a standalone discipline. This prompts us to
consider how differently our understanding of crowd dynamics would have
evolved had this groundbreaking paper not appeared on the pages of Nature. Has
this influence predominantly steered the field in a positive direction, or
have there been negative consequences? If the impact has been mixed, what
aspects have been beneficial and what aspects have not? Our investigation
operates on two distinct levels: a bibliometric analysis and a content-focused
examination. We undertake several key inquiries:
(1) Overall influence: Through analysis of bibliometric indicators, we
quantify the influence of the paper based on objective metrics, while also
mapping the extent of influence that this paper has had and the areas of
research to which its influence has permeated.
(2) Intellectual Foundation: We trace the intellectual lineage that led to the
genesis of this paper by dissecting its references, and in some cases, the
references within those references. By doing so, we aim to unveil the origins
and foundations of concepts like escape panic as referred by the authors.
(3) Terminological Influence: We explore how the language and terminology
employed in this paper have shaped the broader lexicon of crowd dynamics. To
gauge this influence, we analyse the language employed in the field prior to
and subsequent to the publication of this paper.
(4) Assumptions vs. Empirical Evidence: Helbing et al. [1] outlined a set of
assumptions characterising crowd (escape) panic. Since then, numerous
empirical studies have emerged, testing these assumptions through various
experimental methods. We critically assess whether this accumulated body of
evidence aligns with the conceptual characteristics laid out in the original
paper.
(5) Numerical Simulations and Experimental Replication: Following the
introduction of the social force modelling concept, Helbing et al. [1]
conducted a series of numerical simulations that yielded noteworthy
conclusions. These findings have permeated the crowd dynamics literature. We
evaluate the extent to which subsequent experimental studies have investigated
these numerical observations.
Overall, this study endeavours to provide an evidence-based revisiting of the
influence, intellectual underpinnings, terminological evolution, and
validation of assumptions and numerical findings within the referenced work.
Such an assessment can help guide the trajectory of the field, offering
insights into whether a conscious paradigm shift is necessary to address the
issues associated with its influence. From here on, we refer to the paper as
the HFV paper (after the initials of the authors’ surnames).
## 2 Overall Influence
At the time of this analysis, we identified nearly 3,000 papers that have
referenced the HFV paper, according to the Web of Science, making it the
second most cited paper ever published in the crowd dynamics literature. It
follows an earlier work by Helbing and Molnar [2] where they laid the
foundation of the Social Force (SF) model of pedestrians and introduced the
model for the first time. The rate of annual citations for the HFV paper
exhibited a consistent linear increase since its publication and that trend
continued until 2015, when there was a noticeable change in this growing trend
(See fig:1). This may coincide with a paradigm shift in crowd dynamics
research around that year. Researchers began to shift their focus
significantly towards experimental work, diverging from the predominant trend
of numerical work up until that point.
Figure 1: The number of citing articles of HFV paper every year since its
publication (part (a)), the countries of origin of the citing articles and the
clusters of citing articles and their relative prominence (part (c)).
One noteworthy aspect is the substantial portion of these citations
originating from papers authored by researchers based in China, indicating the
exceptional popularity and influence of this paper within this community. The
majority of citing articles belong to the fields of Physics and Computer
Science, with a notable concentration of these articles in the journal Physica
A. Within the field of crowd dynamics, the HFV paper has also left its mark on
influential papers such as [3] and [4] which introduced the concept of
cellular automata modelling to the field in 2001. To gain a deeper
understanding of the areas where the HFV paper has exerted its influence, we
conducted a document co-citation analysis among the reference lists of the
nearly 3,000 citing papers. This analysis identifies clusters of references
that are co-cited by these papers. In other words, it reveals groups of
references that tend to appear together in the citations to the HFV paper by
its citing articles. We identified ten such clusters, and by analysing the key
phrases in the titles of these citing articles, we identified and categorised
the themes of studies that have referenced the HFV paper. These themes are
listed in app:1 in order of significance, showing how the influence of this
paper has extended beyond the field of crowd dynamics. Additionally, app:1
provides a list of references that have been most frequently co-cited with the
HFV paper.
## 3 Intellectual Foundation
The HFV paper cites a total of twenty references. In our analysis, we do not
intend to scrutinise every single reference, as not all appear to be equally
fundamental in shaping its intellectual foundation. Notably, the initial
references cited in the HFV paper have played a pivotal role in setting the
stage and justifying the study and modelling of escape panic. An intriguing
observation concerning these initial references is that not all of them appear
to support the argument; in fact, some seem to directly contradict the
statements they accompany. For instance, the paper by Keating [5] is cited
following a statement asserting ”Sometimes this behaviour [panic] is triggered
in life-threatening situations such as fires”. The reference title [5], The
myth of panic, and its content strongly suggests that it does not align with
the statement, but rather points in the opposite direction. Similarly, the
subsequent statement reads, ”At other times, stampedes can arise during the
rush for seats”. One of the two references cited following this statement is
the 1987 study by Johnson [6]. However, the content and conclusions of
Johnson’s paper appear to contradict the statement. This study [6] analyses
empirical evidence related to a tragic incident before a rock concert and
reports “evidence showing that panic did not cause the death and injury of
numerous young people”. Furthermore, it highlights that post-disaster
interviews and event reconstructions revealed no signs of stampede; instead,
most competition stemmed from people’s attempts to escape a crowd crush. This
contradicts the notion that panic can arise from people’s rush for seats, as
stated by HFV. Interestingly, in an editorial written by David Low about the
HFV paper in the same issue of Nature, Johnson’s paper is inaccurately cited
following this statement, ”The consequences of crushing, trampling, and panic
in crowds are well known”. References [5] and [6] are repeatedly cited by HFV
throughout the paper without them offering support in favour of the arguments
that they accompany.
Another example arises in the statement, ”Panicking individuals tend to show
maladaptive and relentless mass behaviour like jamming and life-threatening
overcrowding, which has often been attributed to social contagion”. The 1957
work of Quarantelli [7] is frequently cited following many of these
statements. Published in 1957, it has been hard to trace this reference in
contemporary databases. However, we managed to locate a more recent working
paper by Quarantelli from 2001 [8], which we believe would naturally reflect
the author’s observations and reports from earlier stages of their career. The
working paper is titled The Sociology of Panic and it clearly indicates that
the Quarantelli’s findings could not have possibly been consistent with the
portrayal presented in the HFV paper. Taken together, these observations cast
doubt on some of the foundational ideas upon which the HFV paper appears to
rely. In fact, certain aspects of the work cited seem to fundamentally
contradict key premises of the paper or at least reflect a possible inaccurate
use of terminology.
## 4 Terminological Influence
An examination of the text within the article reveals the prominent usage of
two terms: panic (mentioned 37 times) and herding (mentioned 10 times). The
notable prevalence of these terminologies within the field of pedestrian
dynamics can be directly attributed to the influential impact of this article.
As shown in [9], prior to the publication of this article, there was minimal
mention of these two terms in the context of studies on crowds and
evacuations. Similarly, the terminology of crowd dynamics (mentioned 3 times),
which has been detected in the titles, abstracts, or keywords of more than 700
articles since 2000, was scarcely identifiable in any publications within the
aforementioned domains before the publication of this article. It is
reasonable to conclude that the nomenclature by which this field is recognised
owes its existence to the terminological influence exerted by this article.
## 5 Assumptions vs. Empirical Evidence
The foundation of the HFV paper hinges on the concept of panic, which it
delineates through a set of defining characteristics. In this analysis, we
prospectively reevaluate some of these key assumptions to determine whether
they align with the empirical evidence that has emerged in the field since
that publication. People move or try to move considerably faster than normal.
This behaviour does not necessarily indicate panic per se. Individuals start
pushing, and interactions among people become physical in nature. Empirical
evidence does not support the notion that people engage in competitive
behaviour when responding to a life-threatening situation [10, 11]. At exits,
arching and clogging are observed. Arching is not a particular characteristic
of panic; rather, it is observable to some extent in laboratory experiments
when a crowd of pedestrians is instructed to pass through a narrow bottleneck
[12]. Jams build up. Similar to the previous point, the formation of a
pedestrian traffic jam is linked to a situation where the inflow of
pedestrians exceeds the capacity of a passage. People show a tendency toward
mass behaviour, that is, to do what other people do. This assumption is
particularly important because it is readily operationalisable in evacuation
models. In many cases, in fact, crowd models have been assessed on their
ability to replicate such phenomenon. However, empirical evidence accumulated
since the HFV paper suggests that the matter of imitation is complex. Social
influence has been observed in several experimental studies, but there are
several factors and variables affecting its extent, e.g., crowd size,
familiarity with the environment, and perceived urgency [13]. While there is
evidence of some level of imitation regarding decision adaptation, it does not
lead to blind herd-like behaviour where all individuals follow a single action
(e.g., as visually depicted by the Figure 1 of the Editorial piece by David
Low [14] on the HFV paper, marking one of the most influential illustrations
produced in this field). In addition, empirical evidence has overwhelmingly
suggested that humans take into consideration a range of factors in flights
situations and not purely the decisions of others [15].
## 6 Numerical Simulations and Experimental Replication
The HFV article subjected its proposed crowd escape panic model to numerical
testing, leading to recommendations that have significantly influenced both
the field and public perception. Among these observations, the most
consequential is the so-called faster-is-slower phenomenon. The study
considered the evacuation of a crowd of 200 agents through a 1m-wide exit
while gradually increasing a parameter known as desired velocity. The outcome
revealed that as the desired velocity increased, system efficiency (in this
context, the inverse of total evacuation time) initially improved. However,
further increases in this parameter led to reduced system efficiency, implying
longer evacuation times. This finding was translated into the recommendation
that the most efficient way to evacuate and survive a crowded space with
limited exit capacity, relative to occupancy, is for individuals to remain
patient and keep lower speeds when passing through bottlenecks. Another
suggestion was that asymmetrical placement of columns in front of exits can
improve outflow. Several dimensions warrant consideration: (1) Empirical
testing: Empirical studies have in some instances confirmed and in other
contradicted the faster-is-slow and column phenomena [16, 17, 18]. In a
controlled experiment, participants were instructed to exit forcefully
(without resorting to aggression), resulting in significantly higher flow
efficiency compared to a calm and patient manner [16]. (2) Model results need
context: Numerical findings could be very parameter specific. According to
Figure 1 (c) of the original paper, the minimum time required for 200 people
to evacuate a room with a single 1m-wide exit is approximately 120s. Empirical
observations indicate that this number of people may complete this evacuation
in different (even faster) times depending on several factors. (3) Desired
velocity in non-free flow conditions: The concept of desired velocity is hard
to grasp physically, except in free-flow conditions. In scenarios with
bottlenecks or obstacles, it remains unclear how different levels of desired
flow correspond to various levels of escape motivation. It is possible that in
reality, assuming that the model components can manifest physically in the
real world (at least in terms of desired velocity), we are always within the
monotonically decreasing limb of the graph in Figure 1(c) of the HFV article.
This would imply that an increase in evacuation time does not manifest for
typical (non-aggressive) forces applied by people at a bottleneck. This
observation has had profound implications and has influenced both research
studies that employed ants and mice as models for human crowds [18] as well as
public perceptions regarding efficient evacuations. The main argument is that
the panic concept assumed by the model may be misinterpreted as an
encouragement to stay on the beginning of the left-hand side of the
flow/density relationship, whereas we know that flow/density reaches a maximum
with a speed which is at an optimal density value, and not at slow speed
level.
## 7 Conclusions
Our analysis suggests that the HFV paper stands as one of the most fundamental
and influential contributions to the field of crowd dynamics. It has not only
profoundly shaped the terminologies employed within this field but has also
significantly influenced the trajectory of research undertaken by scholars. It
is evident that the introduction of the social force model of pedestrian
dynamics marks a major paradigm shift in crowd research. This model introduced
a paradigm that has spurred numerous developments in agent-based modelling of
pedestrian traffic and beyond, establishing itself as a standard benchmark.
Patterns of referencing to the article demonstrate its broad-reaching impact,
reflecting its use as an intellectual foundation and source of inspiration in
a multitude of domains. Our work highlights aspects that facilitate the
interpretation of the HFV paper: (1) Model limitations: There are a set of
misconceptions regarding the model applicability. The model excels mostly when
integrated with other layers of modelling, addressing various aspects of
pedestrian behaviour and decision-making beyond the operational level and
step-taking behaviour [19]. (2) Calibration procedure: Presenting the model
along with a detailed sensitivity analysis to identify critical model
parameters would have helped understanding their impact on simulation output.
A standardised calibration procedure could have facilitated future model
applications. In the social force model paradigm, the forces lack physical
manifestations, making them metaphorical or imaginary concepts. This renders
them unmeasurable (though some have attempted to measure these forces in
experimental settings [20]), making it exceedingly difficult to link
experimental observations to the model components and calibrate parameters
using established procedures. This led to partial calibration attempts [19].
(3) Model validation: The numerical case studies presented in the paper,
including the faster-is-slower phenomenon and the obstacle effect, have been
later scrutinised showing they may not hold for all scenarios. These findings
have directed research efforts toward replicating them, in some instances
relying on non-valid methods, such as animal experiments [18]. Relying on
assumptions about panic, unsupported by references, has significantly
influenced the direction of research in this field.
In summary, the HFV paper has played a pivotal role in the establishment of
crowd dynamics as a research field. While we recognise the methodological and
modelling benefits derived from this paper, we advocate for a paradigm shift
in crowd dynamics research. It is essential to recognise that the mere
publication of this paper in Nature should not serve as an unequivocal
endorsement of the quest to delineate and explore the concept of crowd panic.
Thus, we call for a reimagining of the field, focused on empirical evidence to
chart a new course for research in crowd dynamics.
Milad Haghani: Conceptualization, Methodology, Software, Validation, Formal
Analysis, Investigation, Writing – Original Draft, Visualization Enrico
Ronchi: Conceptualization, Methodology, Validation, Formal Analysis,
Investigation, Writing – Review and Editing
## References
* [1] Helbing, D., Farkas, I., Vicsek, T.: Simulating dynamical features of escape panic. Nature 407(6803), 487–490 (2000)
* [2] Helbing, D., Molnar, P.: Social force model for pedestrian dynamics. Physical review E 51(5), 4282 (1995)
* [3] Helbing, D.: Traffic and related self-driven many-particle systems. Reviews of modern physics 73(4), 1067 (2001)
* [4] Burstedde, C., Klauck, K., Schadschneider, A., Zittartz, J.: Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Physica A: Statistical Mechanics and its Applications 295(3-4), 507–525 (2001)
* [5] Keating, J.P.: The myth of panic. Fire Journal 76(3), 57–61 (1982)
* [6] Johnson, N.R.: Panic at “the who concert stampede”: an empirical assessment. Social Problems 34(4), 362–373 (1987)
* [7] Quarantelli, E.L.: The behavior of panic participants. Sociology and Social Research 41, 187–194 (1957)
* [8] Quarantelli, E.L.: Panic, sociology of (2001)
* [9] Feliciani, C., Corbetta, A., Haghani, M., Nishinari, K.: How crowd accidents are reported in the media: Lexical and sentiment analyses. arXiv preprint arXiv:2309.14633 (2023)
* [10] Drury, J., Cocking, C., Reicher, S., Burton, A., Schofield, D., Hardwick, A., Graham, D., Langston, P.: Cooperation versus competition in a mass emergency evacuation: A new laboratory simulation and a new theoretical model. Behavior research methods 41(3), 957–970 (2009)
* [11] Bartolucci, A., Casareale, C., Drury, J.: Cooperative and competitive behaviour among passengers during the costa concordia disaster. Safety science 134, 105055 (2021)
* [12] Uesten, E., Schumann, J., Sieben, A.: Exploring the dynamic relationship between pushing behavior and crowd dynamics. Collective Dynamics 8, 1–29 (2023)
* [13] Haghani, M., Cristiani, E., Bode, N.W., Boltes, M., Corbetta, A.: Panic, irrationality, and herding: three ambiguous terms in crowd dynamics research. Journal of advanced transportation 2019 (2019)
* [14] Low, D.J.: Following the crowd. Nature 407(6803), 465–466 (2000)
* [15] Kinateder, M., Comunale, B., Warren, W.H.: Exit choice in an emergency evacuation scenario is influenced by exit familiarity and neighbor behavior. Safety science 106, 170–175 (2018)
* [16] Haghani, M., Sarvi, M., Shahhoseini, Z.: When ‘push’does not come to ‘shove’: Revisiting ‘faster is slower’in collective egress of human crowds. Transportation research part A: policy and practice 122, 51–69 (2019)
* [17] Zuriguel, I., Echeverría, I., Maza, D., Hidalgo, R.C., Martín-Gómez, C., Garcimartín, A.: Contact forces and dynamics of pedestrians evacuating a room: the column effect. Safety science 121, 394–402 (2020)
* [18] Haghani, M.: Empirical methods in pedestrian, crowd and evacuation dynamics: Part i. experimental methods and emerging topics. Safety science 129, 104743 (2020)
* [19] Haghani, M., Sarvi, M.: Crowd model calibration at strategic, tactical, and operational levels: Full-spectrum sensitivity analyses show bottleneck parameters are most critical, followed by exit-choice-changing parameters. Transportation Letters pp. 1–28 (2023)
* [20] Zhao, Y., Lu, T., Su, W., Wu, P., Fu, L., Li, M.: Quantitative measurement of social repulsive force in pedestrian movements based on physiological responses. Transportation research part B: methodological 130, 1–20 (2019)
## Appendix A Appendix 1
Clusters of topics and research areas where the HFV paper has been cited,
based on a document co-citation analysis of the citing articles of the HFV
paper:
Cluster 1: pedestrian evacuation; social force model; bidirectional pedestrian
flow; new lattice; visual field.
Cluster 2: crowd space; predictive crowd analysis technique; dense crowd;
emotional contagion; personality trait.
Cluster 3: jamming transition; pedestrian evacuation; fire evacuation model;
pedestrian dynamics; self-driven particle.
Cluster 4: collective escape; pedestrian counter flow; emergency escape;
pedestrian contact force; pedestrian dynamics.
Cluster 5: collective motion; self-propelled particle; collective decision;
nonlinear dynamics.
Cluster 6: pedestrian trajectory prediction; evacuation crowd dynamics; human
behaviour; social interaction.
Cluster 7: dynamic decision behaviour; information service; discrete opinion
dynamics; critical market crash.
Cluster 8: modelling pedestrian crowd; panic condition; crowd guidance;
disaster evacuation; aircraft disembarking.
Cluster 9: switching phenomena; financial market; correlated randomness; trend
switching processes; trend switching.
Cluster 10: granular material; clogging transition; horizontal hopper; hopper
angle; large reflective obstacle.
References most frequently co-cited by the HFV paper:
1104 Co-citations: Helbing D, 1995, PHYS REV E, V51, P4282,
DOI 10.1103/PhysRevE.51.4282
565 Co-citations: Burstedde C, 2001, PHYSICA A, V295, P507,
DOI 10.1016/S0378-4371(01)00141-8
392 Co-citations: Kirchner A, 2002, PHYSICA A, V312, P260,
DOI 10.1016/S0378-4371(02)00857-9
380 Co-citations: Helbing D, 2005, TRANSPORT SCI, V39, P1,
DOI 10.1287/trsc.1040.0108
306 Co-citations: Helbing D, 2007, PHYS REV E, V75, P0,
DOI 10.1103/PhysRevE.75.046109
291 Co-citations: Helbing D, 2001, REV MOD PHYS, V73, P1067,
DOI 10.1103/RevModPhys.73.1067
285 Co-citations: Hughes RL, 2002, TRANSPORT RES B-METH, V36, P507,
DOI 10.1016/S0191-2615(01)00015-7
270 Co-citations: Reynolds C.W., 1987, ACM SIGGRAPH COMPUTE, V21, P25,
DOI 10.1145/37402.37406
268 Co-citations: Moussaid M, 2011, P NATL ACAD SCI USA, V108, P6884,
DOI 10.1073/pnas.1016507108
261 Co-citations: Helbing D, 2002, PEDESTRIAN AND EVACUATION DYNAMICS, V0, P21
229 Co-citations: Muramatsu M, 1999, PHYSICA A, V267, P487,
DOI 10.1016/S0378-4371(99)00018-7
207 Co-citations: Vicsek T, 1995, PHYS REV LETT, V75, P1226,
DOI 10.1103/PhysRevLett.75.1226
201 Co-citations: Helbing D, 2000, PHYS REV LETT, V84, P1240,
DOI 10.1103/PhysRevLett.84.1240
193 Co-citations: Kirchner A, 2003, PHYS REV E, V67, P0,
DOI 10.1103/PhysRevE.67.056122
181 Co-citations: Henderson LF, 1971, NATURE, V229, P381,
DOI 10.1038/229381a0
169 Co-citations: Treuille A, 2006, ACM T GRAPHIC, V25, P1160,
DOI 10.1145/1141911.1142008
167 Co-citations: Helbing D, 2003, PHYS REV E, V67, P0,
DOI 10.1103/PhysRevE.67.067101
166 Co-citations: Hughes RL, 2003, ANNU REV FLUID MECH, V35, P169,
DOI 10.1146/annurev.fluid.35.101101.161136
163 Co-citations: Zheng XP, 2009, BUILD ENVIRON, V44, P437,
DOI 10.1016/j.buildenv.2008.04.002
|
# Compact localised states in magnonic Lieb lattices
Grzegorz Centała Jarosław W. Kłos<EMAIL_ADDRESS>Institute of Spintronics
and Quantum Information, Faculty of Physics, Adam Mickiewicz University,
Poznań, Uniwersytetu Poznańskiego 2, Poznań 61-614, Poland
(March 26, 2023)
###### Abstract
Lieb lattice is one of the simplest bipartite lattices where compact localized
states (CLS) are observed. This type of localisation is induced by the
peculiar topology of the unit cell, where the modes are localized only on one
sublattice due to the destructive interference of partial waves. The CLS exist
in the absence of defects and are associated with the flat bands in the
dispersion relation. The Lieb lattices were successfully implemented as
optical lattices or photonic crystals. This work demonstrates the possibility
of magnonic Lieb lattice realization where the flat bands and CLS can be
observed in the planar structure of sub-micron in-plane sizes. Using forward
volume configuration, we investigated numerically (using the finite element
method) the Ga-dopped YIG layer with cylindrical inclusions (without Ga
content) arranged in a Lieb lattice of the period 250 nm. We tailored the
structure to observe, for the few lowest magnonic bands, the oscillatory and
evanescent spin waves in inclusions and matrix, respectively. Such a design
reproduces the Lieb lattice of nodes (inclusions) coupled to each other by the
matrix with the CLS in flat bands.
Keywords
flat bands, compact localized states, Lieb lattice, spin waves, finite element
method
††preprint: APS/123-QED
## I Introduction
There are many mechanisms leading to wave localization in systems with long-
range order, i.e. in crystals or quasicrystals. The most typical of these
require (i) the local introduction of defects, including the defects in the
form of surfaces or interfaces [1] (ii) the presence of global disorder [2],
(iii) the presence of external fields [3] or (iv) the existence of many-body
phenomena [4]. However, since at least the late 1980s, it has been known that
localization can occur in unperturbed periodic systems in the absence of
fields and many-body effects, and is manifested by the presence of flat, i.e.,
dispersion-free bands in the dispersion relation. The pioneering works are
often considered to be the publications of B. Sutherland [5] and E. H. Lieb
[6], who found the flat bands of zero energy [7] for bipartite lattices with
use of the tight-binding model Hamiltonians, where the hoppings occur only
between sites of different sublattices. The simplest realization of this type
of system is regarded as the Lieb lattice [6, 8], where the nodes of one
square sublattice, of coordination number $z=4$, connect to each other only
via nodes with a coordination number $z=2$ from other two square sublattices
(Fig. 1). In the case of extended Lieb lattices [9, 10], the nodes of $z=2$
form chains: dimmers, trimmers, etc.(Fig. 2). An intuitive explanation for the
presence of the flat bands is the internal isolation of excitations located in
one of the sublattices. The cancelling of excitations at one sublattice is the
result of forming destructive interference and local symmetry within the
complex unit cell [11]. When only one of the sublattices is excited, the other
sublattice does not mediate the coupling between neighbouring elementary
cells, and the phase difference between the cells is irrelevant to the energy
(or the frequency) of the eigenmode on the whole lattice - i.e. the Bloch
function. Modes of this type are therefore degenerated for different wave
vector values in infinite lattices. We are dealing here with the localization
on specific arrangements of structure elements, which are isolated from each
other. Such kinds of modes are called compact localized states (CLS) [12, 13,
14, 15, 16] and show a certain resistance to the introduction of defects [17,
18]. The flat band systems with CLS are the platform for the studies of
Anderson localization [19], and unusual properties of electric conductivity
[20]. A similar localization is observed in the quasicrystals, where the
arrangements of the elements composing the structure are replicated
aperiodically and self-similarly throughout the system [21, 22] and the
excitation can be localized on such patterns. The CLS in finite Lieb lattices
have a form of loops (plaquettes) occupying the majority nodes ($z=2$). These
states are linearly dependent and do not form a complete basis for the flat
band. Therefore occupancy gaps need to be filled (for infinite lattice) by
states occupying only one sublattice of majority nodes, localizes at lines,
called noncontractible loop states (NLS) [14, 15, 23].
The topic of Lieb lattices and other periodic structures with compact
localization and flat bands was renewed [8] about 10 years ago when physical
realizations of synthetic Lieb lattices began to be considered for electronic
systems [24, 25], optical lattices [26, 27], superconducting systems [28, 29],
in phononics [30] and photonics [31, 14]. In a real system, where the
interaction cannot be strictly limited to the nearest elements of the
structure, the bands are not perfectly flat. Therefore, some authors use the
extended definition of the flat band to consider the bands that are flat only
along particular directions or in the proximity of high-symmetry Brillouin
zone points [32]. In tight-binding models, this effect can be included by
considering the hopping to at least next-nearest-neighbours [33, 34].
Similarly, the crossing of the flat band by Dirac cones can be transformed
into anti-crossing and lead to opening gaps, separating the flat band from
dispersive bands. This effect can in induced by the introduction of spin-orbit
term to tight-binding Hamiltonian (manifested by the introduction of Peierls
phase factor to the hopping) or by dimerization of the lattice (by alternative
changes of hoppings or site energies) [33, 34, 35, 36, 37]. The later scenario
can be easily observed in real systems where the position of rods/wells
(mimicking the sites of Lieb lattice) and contrast between them can be easily
altered [38]. Opening the narrow gap between flat band and dispersive bands
for Lieb lattice is also fundamentally interesting because it leads to the
appearance of so-called Landau-Zener Bloch oscillations [39].
The isolated and perfectly flat bands for Lieb lattices are topologically
trivial – their Chern number is equal to zero [40]. For weakly dispersive
(i.e. almost flat) bands the Chern numbers can be non-zero [41]. However, when
the flat band is intersected by dispersive bands then it can exhibit the
discontinuity of Hilbert–Schmidt distance between eigenmodes corresponding to
the wave vectors just before and just after the crossing. Such an effect is
called singular band touching [16]. This limiting value of Hilbert–Schmidt
distance is bulk invariant, different from the Chern number.
One of the motivations for the photonic implementation of systems with flat,
or actually nearly flat bands [42], was the desire to reduce the group
velocity of light in order to compress light in space, which leads to the
concentration of the optical signal and an increase in the light-matter
interaction, or the enhancement of non-linear effects. Another, more obvious
application is the possibility of realizing delay lines that can buffer the
signal to adjust the timing of optical signals [43]. A promising alternative
to photonic circuits are magnonic systems, which allow signals of much shorter
wavelengths to be processed in devices several orders of magnitude smaller
[44, 45]. For this reason, it seems natural to seek a magnonic realization of
Lieb lattices.
In this paper, we propose the realization of such lattices based on a magnonic
structure in the form of a perpendicularly magnetized magnetic layer with
spatially modulated material parameters or spatially varying static internal
field. Lieb lattices have been studied also in the context of magnetic
properties, mainly due to the possibility of enhancing ferromagnetism in
systems of correlated electrons [46], where the occurrence of flat bands with
zero kinetic energy was used to expose the interactions. There are also known
single works where the spin waves have been studied in the Heisenberg model in
an atomic Lieb lattice, such as the work on the magnon Hall effect [47]. But
the comprehensive studies of spin waves in nanostructures that realize
magnonic Lieb lattices and focus on wave effects in a continuous model have
not been carried out so far. In this work, we demonstrate the possibility of
realization of magnonic lattices in planar structure based on low spin wave
damping material: yttrium iron garnet (YIG) where the iron is partially
substituted by gallium (Ga). We present the dispersion relation with a weakly
depressive (flat) band exhibiting the compact localized spin waves. The flat
is almost intersected at the $M$ point of the 1st Brillouin zone by highly
dispersive bands, similar to Dirac cones. We discuss the spin wave spectra and
compact localized modes both for simple and extended Lieb lattices.
The introduction is followed by the section describing the model and numerical
method we used, which precedes the main section where the results are
presented and discussed. The paper is summarized by conclusion and
supplemented with additional materials where we showed: (A) the results for
extended Lieb-7 lattice, (B) an alternative magnonic Lieb lattice design via
shaping the demagnetizing field, and (C) an attempt of formation magnonic Lieb
lattice by dipolarly coupled magnetic nanoelements, (D) discussion of small
differences in the demagnetizing field of majority and minority nodes
responsible for opening a small gap in the Lieb lattice spectrum.
## II Structure
Magnonic crystals (MCs) are regarded as promising structures for magnonic-
based device applications [48, 45]. In our studies, we consider planar MCs to
design the magnonic Lieb lattice, owing to the relative ease of fabrication of
such structures and their experimental characterization [49, 50, 51]. We
proposed realistic systems that mimic the main features of the tight-binding
model of Lieb lattice [33, 16].
Investigated MCs consist of yttrium iron garnet doped with gallium (Ga:YIG)
matrix and yttrium iron garnet (YIG) cylindrical inclusions arranged in Lieb
lattice Fig. 1. Doping YIG with Gallium is a procedure where magnetic ${\rm
Fe}^{3+}$ ions are replaced by non-magnetic ${\rm Ga}^{3+}$ ions. This method
not only decreases saturation magnetization $M_{S}$ but, simultaneously,
arises uni-axial out-of-plane anisotropy, that ensures the out-of-plane
orientation of static magnetization in Ga:YIG layer at a relatively low
external field applied perpendicularly to the layer. Discussed geometry, i.e.
forward volume magnetostatic spin wave configuration, does not introduce an
additional anisotropy in the propagation of spin waves, related to the
orientation of static magnetization.
Figure 1: Basic magnonic Lieb lattice. The planar magnonic structure consists
of YIG cylindrical nanoelements embedded within Ga:YIG. Dimensions of the
ferromagnetic unit cell are equal to 250x250x59 nm and the unit cell contains
three inclusions of 50 nm diameter. (a) The structure of basic Lieb lattice,
and (b) top view of the Lieb lattice unit cell where the node (inclusion) from
minority sublattice $A$ and two nodes (inclusions) from two majority
sublattices $B$ are marked.
The design of the Lieb lattice requires the partial localization of spin wave
in inclusions, which can be treated as an approximation of the nodes from the
tight-binding model. Furthermore, the neighbouring inclusions in the lattice
have to be coupled strongly enough to sustain the collective spin wave
dynamics, and weakly enough to minimize the coupling between further
neighbours. Therefore, the geometrical and material parameters were selected
to ensure the occurrence of oscillatory excitations in the (YIG) inclusions
and exponentially evanescent spin waves in the (Ga:YIG) matrix. The size of
inclusions was chosen small enough to separate three lowest magnonic bands
with almost uniform magnetization precession inside the inclusion from the
bands of higher frequency, where the spin waves are quantised inside the
inclusions. Also, the thickness of the matrix and inclusion was chosen in a
way that there are no nodal lines inside the inclusion. The condition which
guarantee the focussing magnetization dynamics inside the inclusions is
fulfilled in the frequency range below the ferromagnetic resonance (FMR)
frequency of the out-of-plane magnetized layer made of Ga:YIG (matrix
material): $f_{\rm FMR,Ga:YIG}=$4.95 GHz and above the FMR frequency of out-
of-plane magnetized layer made of YIG (inclusions material): $f_{\rm
FMR,YIG}=2.42$ GHz. These limiting values were obtained using the Kittel
formula for out-of-plane magnetised film: $f_{\rm
FMR}=\frac{\gamma}{2\pi}\text{\textbar}{\mu_{0}H_{\rm 0}+\mu_{0}H_{\rm
ani}-\mu_{0}M_{S}\text{\textbar}}$, where we used the following values of
material parameters [52] for YIG: gyromagnetic ratio $\gamma=177~{}\frac{\rm
GHz}{\rm T}$, magnetization saturation $\mu_{0}M_{S}=182.4$ mT, exchange
stiffness constant $A=3.68~{}\frac{\rm pJ}{\rm m}$, (first order) uni-axial
anisotropy field $\mu_{0}H_{\rm ani}=-3.5~{}$mT, and for Ga:YiG:
$\gamma=179~{}\frac{\rm GHz}{\rm T}$, $\mu_{0}M_{S}=20.2$ mT,
$A=1.37~{}\frac{\rm pJ}{\rm m}$, $\mu_{0}H_{\rm ani}=94.1$ mT. Since the
greatest impact of the first order uniaxial anisotropy field ($\mu_{0}H_{\rm
ani}$), we decided to neglect higher order terms of uni-axial anisotropy and
cubic anisotropy of (Ga:)YIG. Due to the presence of out-of-plane anisotropy
and relatively low saturation magnetization, we could consider a small
external magnetic field $\mu_{0}H_{0}=100$ mT to reach saturation state.
Figure 2: Extended magnonic Lieb lattice – Lieb - 5. Dimensions of the unit
cell are 375x375x59 nm and contain 5 inclusions of size 50 nm in diameter.
Also, we maintain the same separation (distance between centres of
neighbouring sites is 125 nm) as for considered basic Lieb lattice – Fig. 1.
(a) The structure of Lieb-5 lattice, and (b) top view on Lieb-5 lattice unit
cell where the node (inclusion) from minority sublattice $A$ and four nodes
(inclusions) from two majority sublattices $B$ are marked.
It is worth noticing that without the evanescent spin waves in the
ferromagnetic matrix, the appropriate coupling between inclusions would not be
possible. Therefore the realization of the Lieb lattice in form of the array
of ferromagnetic nanoelemets embedded in air/vacuum seems to be very
challenging – see the exemplary results in Supplementary Information C.
We also tested the possibility of other realizations of magnonic Lieb
lattices. One solution seemed to be the design of a structure in which the
concentration of the spin wave amplitude in the Lieb lattice nodes would be
achieved through an appropriately shaped profile of the static demagnetizing
field – Supplementary Information B. However, the obtained results were not as
promising as for YIG/Ga:YIG system.
In the main part of the manuscript, we present the results for the basic Lieb
lattice (showed in Fig. 1) and extended Lieb-5 lattice (showed in Fig. 2),
based on YIG/Ga:YIG structures. The further extension of the Lieb lattice may
be realized by increasing the number of $B$ nodes between neighbouring $A$
nodes. Supplementary Information A presents the results for Lieb-7, where for
each site (inclusion) from minority sublattice $A$, we have six nodes
(inclusions), grouped in three-element chains, from majority sublattices $B$.
## III Methods
The spin waves spectra and the spatial profiles of their eigenmodes were
obtained numerically in a semi-classical model, where the dynamics of
magnetization vector $\textbf{M}(\textbf{r},t)$ is described by the Landau-
Lifshitz equation [53]:
$\frac{d\textbf{M}}{dt}=-\gamma\mu_{0}[\textbf{M}\times\textbf{H}_{\rm
eff}+\frac{\alpha}{M_{S}}\textbf{M}\times(\textbf{M}\times\textbf{H}_{\rm
eff})].$ (1)
The symbol $\textbf{H}_{\rm eff}(\textbf{r},t)$ denotes effective magnetic
field.
In numerical calculations, we neglected the damping term since $\alpha$ is
small both for YIG and for YIG with Fe substituted partially by Ga (for
$\alpha_{\rm Ga:YIG}=$6.1\text{\times}{10}^{-4}$$ and $\alpha_{\rm
YIG}=$1.3\text{\times}{10}^{-4}$$ [52]). The effective magnetic field $H_{\rm
eff}$ contains the following components: the external field $H_{0}$, exchange
field $H_{\rm ex}$, bulk uniaxial anisotropy field $H_{\rm ani}$ and dipolar
field $H_{\rm d}$:
$\textbf{H}_{\rm
eff}(\textbf{r},t)=H_{0}\hat{\mathbf{z}}+\frac{2A}{\mu_{0}M^{2}_{S}}\laplacian\textbf{M}(\textbf{r},t)+H_{\rm
ani}(\textbf{r})\hat{\mathbf{z}}-\gradient\varphi(\textbf{r},t),$ (2)
where the $z-$direction is normal to the plane of the magnonic crystal. We
assume that the sample is saturated in $z-$direction and magnetization vector
precesses around this direction. The material parameters ($M_{S}$, $A$,
$\alpha$ and $\gamma$) are constant within matrix and inclusions.
Using the magnetostatic approximation the dipolar term of the effective
magnetic field may be expressed as a gradient of magnetic scalar potential:
$\textbf{H}_{\rm d}(\textbf{r},t)=-\gradient\varphi(\textbf{r},t)$ (3)
By using the Gauss equation magnetic scalar potential may be associated with
magnetisation as follows:
$\laplacian\varphi(\textbf{r},t)=\divergence\textbf{M}(\textbf{r},t)$ (4)
Spin-wave dynamics is calculated numerically using the finite-element method
(FEM). We used the COMSOL Multiphysics [54] to implement the Landau-Lifshitz
equation (Eq. 1) and performed FEM computation for the defined geometry of
magnonic Lieb lattices. The COMSOL Multiphysics is the software used for
solving a number of physical problems, since many implemented modules it
becomes more and more convenient. Nevertheless, all the equations were
implemented in the Mathematics module which contains different forms of
partial differential equations. Eq. 1 was solved by using eigenfrequency
study, on the other hand, to solve Eq. 4 we used stationary study. To obtain
free decay of scalar magnetic potential in the model we applied $5~{}\mu$m of
a vacuum above and underneath the structure. At the bottom and top surface of
the model with vacuum, we applied the Dirichlet boundary condition. We use the
Bloch theorem for each variable (magnetostatic potential and components of
magnetization vector) at the lateral surfaces of a unit cell. We selected the
symmetric unit cell with minority node $A$ in the centre to generate a
symmetric mesh which does not perturb the four-fold symmetry of the system –
this approach is of particular importance for the reproduction of the
eigenmodes profiles in high-symmetry points. In our numerical studies, we used
2D wave vector $\textbf{k}=k_{x}\hat{\textbf{x}}+k_{y}\hat{\textbf{y}}$ as a
parameter for eigenvalue problem which was selected along the high symmetry
path $\Gamma-X-M-\Gamma$ to plot the dispersion relation. We considered the
lowest 3, 5 and 7 bands for basic Lieb lattice, Lieb-5 lattice and Lieb-7
lattice, respectively.
## IV Results
The tight-biding model of the basic Lieb lattice, with hopping restricted to
next-neighbours gives three bands in the dispersion relation. The top and
bottom bands are symmetric with respect to the second, perfectly flat band,
and intersect with this dispersionless band at $M$ point of 1st Brillouin
zone, with constant slope forming two Dirac cones[27, 8]. In a realistic
magnonic system, the spin wave spectrum showing the particle-hole symmetry
with a zero energy flat band is difficult to reproduce because (i) the
dipolarly dominated spin waves, propagating in magnetic film, experience a
significant reduction of the group velocity with an increase of the wave
vector (this tendency is reversed for much larger wave vectors were the
exchange interaction starts to dominate) [53], (ii) the dipolar interaction is
long-range. The first effect makes the lowest band wider than the third band,
and the latter one – induces the finite width of the second band [33]. We are
going to show, that this weakly dispersive band supports the existence of CLS.
Therefore, we will still refer to it as flat band, which is a common practice
for different kinds of realization of Lieb lattices in photonics or optical
lattices.
The results obtained for the basic magnonic Lieb lattice, (Fig. 1), are shown
in Fig. 3. As we predicted, three lowest bands form a band structure which is
similar to the dispersion relation known from the tight-binding model [10].
However, in a considered realistic system there is an infinite number of
higher bands, not shown in Fig. 3(a). For higher bands, spin waves can
propagate in an oscillatory manner in the matrix hence the system does not
mimic the Lieb lattice where the excitations should be associated with the
nodes (inclusions) of the lattice.
Figure 3: Dispersion relation for the basic magnonic Lieb lattice, containing
three inclusions in the unit cell: one inclusion $A$ from minority sublattice
and two inclusions $B$ from majority sublattices (see Fig. 1). (a) The
dispersion relation is plotted along the high symmetry path
$\Gamma$-X-M-$\Gamma$ (see the inset). The lowest band (blue) and the highest
band (red) create Dirac cones almost touching (b) in the M point. The middle
band (green) is relatively flat in the vicinity of the M point.
Due to the fourfold symmetry of the system, the dispersion relation could be
inspected along the high symmetry path $\Gamma-X-M-\Gamma$. Frequencies of the
first three bands are in the range $f_{\rm FMR,YIG}~{}-~{}f_{\rm FMR,Ga:YIG}$.
Their total width is about $\approx 0.78$ GHz. The first and third band form
Dirac cones at $M$ point, separated by a tiny gap $\approx 15$ MHz. The
possible mechanism responsible for opening the gap is a small difference in
the demagnetizing field in the areas of inclusions $A$ (from the minority
lattice) and inclusions $B$ (from two majority sublattices) – see
Supplementary Information D. Inclusions $A$ ($B$) have four (two) neighbours
of type $B$ ($A$). Although inclusions $A$ and $B$ have the same size and are
made of the same material, the static field of demagnetization inside them
differs slightly due to the different neighbourhoods. This effect is
equivalent to the dimerization of the Lieb lattice by varying the energy of
the nodes in the tight-binding model, which leads to the opening of a gap
between Dirac cones and parabolic flattening of them in very close proximity
to the $M$-point. It is worth noting that in the investigated system, the gap
opens between the first and second bands, while the second and third bands
remain degenerated at point $M$, with numerical accuracy.
The middle band can be described as weakly dispersive. The band is more flat
on the $X-M$ path and, in particular, in the vicinity of $M$ point – see Fig.
3(b). The small width of the second band can be attributed to long-range
dipolar interactions which govern the magnetization dynamics in a considered
range of sizes and wave vectors. It is known that even the extension of the
range of interactions to next-nearest-neighbours in the tight-binding model
induces the finite width of the flat band for the Lieb lattice.
Figure 4: The spin wave profiles obtained for the basic magnonic Lieb lattice,
composed of three inclusions in the unit cell (see Fig. 1). The modes are
presented for each band exactly at $M$ (left column) and in its proximity
($M^{\leftarrow}$) on the path $M-\Gamma$ (right column). In the presented
profiles, the saturation and the colour denote the amplitude and phase of the
dynamic, in-plane component of magnetization. The compact localized states
(CLS) are presented at the point $M^{\leftarrow}$ for the second band – right
column. The CLS do not occupy minority sublattice $A$. The inclusions $B$, in
which the magnetization dynamics is focused, are quite well isolated from each
other. One can easily notice that the lattice is decorated by loops (marked by
grey patches) where the phase of the precessing magnetization flips between
inclusions ($+$ and $-$ signs). Exactly at point $M$ – left column, we observe
the degeneracy of the second and third bands. The spin waves occupy $B$
inclusions only in one majority sublattice, i.e. along vertical or horizontal
lines, filliping the phase from inclusion to inclusion which gives the pattern
characteristic to noncontractible loop states (NLS) - marked by grey stripes.
To prove that the second band supports the CLS regardless of its finite width,
we plotted the profiles of spin wave eigenmodes at $M$ point and in its close
vicinity. The results are presented in Fig. 4. The profiles were shown for
infinite lattice and are presented in the form of square arrays containing 3x3
unit cells, where the dashed lines mark their edges. It is visible that the
spin waves are concentrated in the cylindrical inclusions, where the amplitude
and phase of precession is quite homogeneous. In calculations, we used the
Bloch boundary conditions applied for a single unit cell, which means that at
$M$ point the Bloch function is flipped after translation by lattice period,
in both principal directions of the lattice and we will not see the single
closed loops of CLS or lines of NLS. Exactly at $M$ point, all three bands
have zero group velocity. Therefore, the corresponding modes (left column) are
not propagating. The lowest band ($M_{1}$) occupy only inclusions $A$ from the
minority sublattice where the static demagnetizing field is slightly lower
than inside inclusions $B$ (see Supplementary Information D), which justifies
its lower frequency and lifting the degeneracy with two higher modes $M_{2}$
and $M_{3}$ of the same frequency. Each of the modes $M_{2}$ and $M_{3}$
occupy only one of two sublattices $B$, therefore they can be interpreted as
NLS. To observe the pattern typical for CLS, we need to move slightly away
from $M$ point. The first and third modes have then the linear dispersion with
high group velocity and the second band remains flat. We selected the point
$M^{\leftarrow}$ shifted from $M$ point toward $\Gamma$ point by 5% of
$M-\Gamma$ distance (right column). We can see that the first and third modes
$M^{\leftarrow}_{1}$, $M^{\leftarrow}_{3}$ occupy now all inclusions and the
mode $M^{\leftarrow}_{2}$ from the flat band has a profile typical for CLS,
predicted by tight-binding models [55, 8, 56, 9, 10]:
$\text{\textbar}m_{\textbf{k}}>=[\underbrace{-e^{i\frac{k_{y}}{2}}}_{B},\underbrace{0}_{A},\underbrace{e^{i\frac{k_{x}}{2}}}_{B}]$
(5)
where $m_{\textbf{k}}$ is the complex amplitude of the Bloch function in the
base of unit cell (i.e. on two inclusion $B$ from majority sublattices and one
inclusion $A$ from minority sublattice), $\textbf{k}=[k_{x},k_{y}]$ is
dimensionless wave vector. From (Eq. 5), we can see that (i) CLS do not occupy
the minority nodes $A$ and (ii) close to $M$ point the phases at two nodes
$B$, from different majority sublattices, are opposite. These two features are
reproduced for $M^{\leftarrow}_{2}$ mode in investigated magnonic Lieb
lattice. In the profile of this mode, we marked (by a grey patch) the
elementary loop of CLS which is easily identified in finite systems. Here, in
an infinite lattice with Bloch boundary conditions, the loops are infinitely
replicated with $\pi$ phase shift after each translation $x-$ and
$y-$direction. The localization at the inclusions $B$ and the absence of the
spin wave dynamics in inclusions $A$ is observed regardless of the wave
vector. Therefore, the coupling can take place only between the next
neighbours (inclusions $B$), i.e. on larger distances and mostly due to
dipolar interactions, that makes the second band not perfectly flat.
Figure 5: Dispersion relation for the extended magnonic Lieb lattice Lieb-5,
containing five inclusions in the unit cell: one inclusion $A$ from minority
sublattice and four inclusions $B$ from majority sublattices (see Fig. 2). (a)
The dispersion relation is plotted along the high symmetry path
$\Gamma$-X-M-$\Gamma$ (see the inset). The first, third and fifth bands (dark
blue, red and cyan) are strongly dispersive bands, while the second and fourth
bands (green and magenta) are less dispersive and related to the presence of
CLS. The system does not support the appearance of Dirac cones, even in case
when the interaction is fictitiously limited only to inclusions, according to
tight-binding model. (b) The zoomed regions in the vicinity of $\Gamma$ (in
dark green frame) and $M$ points show the essential gaps with relatively low,
parabolic-like curvatures for top and bottom bands.
Let’s discuss now the presence of flat bands and CLS in an extended magnonic
Lieb lattice (Lieb-5), containing five inclusions in the unit cell: one
inclusion $A$ form minority sublattice and four inclusions $B$ from majority
sublattices, as it is presented in Fig. 2. In the considered structure, we add
two additional inclusions $B$ into the unit cell in such a way that
neighbouring inclusions $A$ are linked by the doublets of inclusions $B$. The
sizes of inclusions, distances between them, the thickness of the layer and
the material composition of the structure remained the same as for the basic
Lieb lattice, discussed earlier (Fig. 1).
The dispersion relation obtained for the magnonic Lieb-5 lattice can be found
in Fig. 5(a). The properties of the extended Lieb lattices are well described
in the literature [10, 57, 58, 59]. The tight-binding model description of
Lieb-5 lattices, with information about their dispersion relation and the
profiles of the eigenmodes, are presented in numerous papers[10, 55, 16].
Therefore, it is possible to compare the obtained results with the theoretical
predictions of the tight-binding model.
Figure 6: The profiles obtained for the extended Lieb lattice consisted of 5
inclusions in the unit cell. The modes are presented for bands No. 3-5 in
$\Gamma$ point and its proximity $\Gamma^{\leftarrow}$ (the first and second
column). In the third and fourth columns, we presented the profiles for bands
No. 1-3 at $M$ point and its vicinity $M^{\leftarrow}$. Each profile of
eigenmode is presented on a grid composed of 3x3 unit cells - dashed lines
mark the edges of unit cells. The scheme of the unit cell is presented in top-
left corner. Exactly at $\Gamma$ (and $M$) point the bands No. 3 and 4 (No. 2
and 3) are degenerated and profiles: $\Gamma_{3}$ and $\Gamma_{4}$ ($M_{2}$
and $M_{3}$) have non-standard (for CLS) complementary form – i.e. their
combinations $\Gamma_{3}\pm i\Gamma_{4}$ ($M_{2}\pm iM_{3}$) gives NLS. To
obtain proper profiles of CLS, where the phase of procession flips around CLS
loop, we need to explore the vicinity of $\Gamma$ ($M$) point – see the grey
patches for the mode $\Gamma^{\leftarrow}_{4}$ ($M^{\leftarrow}_{2}$) with $+$
and $-$ signs.
The tight-binding model of Lieb-5 lattice predicts two flat bands with CLS:
the second (green) and fourth (magenta) band in the spectrum. The flat bands
in the tight-binding model are not intersected by Dirac cones but they are
degenerated at $\Gamma$ and $M$ point with the third band (red). These
features are reproduced in investigated magnonic Lieb-5 lattice (Fig. 2). The
dispersion relation for this system is presented in the Fig. 5(a). Also, we
have marked, with two rectangles (dark green and violet), the vicinities of
$\Gamma$ and $M$ points, where the flat bands (the fourth and second bands)
become degenerated with the third, dispersive band – Fig. 5(b). It is easy to
notice the essential frequency gaps ($\approx 33$ MHz and $\approx 84~{}$MHz
at $\Gamma$ and $M$ points, respectively), which qualitatively corresponds to
the prediction of the tight-binding model. It is worth noting that although
the low dispersion bands (the second and fourth band) are in general not
perfectly flat. Nevertheless, around the point $\Gamma$ and $M$ points the
bands are flattened and the $\Gamma-X$ and $X-M$ sections are very flat for
the fourth and second band, respectively.
The spin wave profiles of CLS at the high symmetry points: $\Gamma$ and $M$
are presented in Fig. 6. Exactly at $\Gamma$ and $M$ (the first and third
column), we can see the pairs of degenerated mods $\Gamma_{3}$, $\Gamma_{4}$
and $M_{2}$, $M_{3}$ which exhibit features of CLS predicated by the tight-
binding model (see the loops of sites on grey patches): (i) modes occupy only
the inclusions $B$ from majority sublattices, (ii) doublets of inclusions $B$
in the loops of CLS have opposite (the same) phases at $\Gamma$ ($M$) point.
The significant difference is that; once we switch one to another $B$-$B$
doublet, circulating the CLS loop the phase of precession charges by
$\pm\pi/2$ not by 0 or $\pi$. However, when we make combinations of
degenerated modes: $\Gamma_{3}\pm i\Gamma_{4}$ or $M_{2}\pm iM_{3}$, then we
obtain the NLS occupying the horizontal or vertical lines, where the
precession at exited $B$ inclusion will be in- or out-of-phase. The CLS modes
are clearly visible when we move slightly away from the high symmetry point
where the degeneracy occurs. In the proximity of $\Gamma$ and $M$ point, one
can see the CLS modes $\Gamma^{\leftarrow}_{4}$ and $M^{\leftarrow}_{2}$ for
which the phase of precession takes the relative values close to 0 or $\pi$.
The small discrepancies, are visible as a slight change in the colours
representing the phase, resulting from the fact that we are not exactly in
high symmetry points but shifted by 5% on the path $\Gamma-M$.
The extension of the presented analysis to magnonic Lieb - 7 lattice, where
the inclusions $A$ are liked by the chains composed of three inclusions $B$,
is presented in Supplementary Information A.
## V Conclusions
We proposed a possible realisation of the magnonic Lieb lattices where the
compact localized spin wave modes can be observed in flat bands. The presented
system qualitatively reproduces the spectral properties and the localization
features of the modes, predicted by the tight-binding model and observed for
photonic and electronic counterparts. The magnonic platform for the
experimental studies of Lieb lattices seems to be attractive due to the larger
flexibility in designing magnonic systems and the steering of its magnetic
configuration by external biases. The idea of the magnonic Lieb latices allows
considering many problems related to dynamics, localization and interactions
in flat band systems taking the advantage of the magnonic systems: presence
and possibility of tailoring of long-range interactions, intrinsic non-
linearity, etc.
## VI Acknowledgements
This work has received funding from National Science Centre Poland grants
UMO-2020/39/O/ST5/02110, UMO-2021/43/I/ST3/00550 and support from the Polish
National Agency for Academic Exchange grant BPN/PRE/2022/1/00014/U/00001.
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## Supplementary Information
### VI.1 Double extended Lieb lattice
We can generate further extensions of the magonic Lieb lattice by adding more
inclusions $B$, i.e. by introducing additional majority sublattices. We
considered here a doubly extended Lieb lattice (Lieb-7) to check to what
extent the magnonic system corresponds to the tight-binding model. The
mentioned lattice consists of seven nodes; six belong to majority sublattices
$B$ and one belongs to minority sublattice $A$ (Fig. 7). The magnetic
parameters were kept as for basic and Lieb-5 lattices, considered in the
manuscript. The geometrical parameters have changed only as a result of the
introduction of additional inclusions $B$. Therefore, the unit cell has
increased to the size 500x500 nm.
Figure 7: Doubly extended magnonic Lieb lattice: Lieb-7. Dimensions of the
ferromagnetic unit cell are equal to 500x500x59 nm. The unit cell contains
seven inclusions of 50 nm diameter. (a) The structure of extended Lieb
lattice, and (b) top view on Lieb-7 lattice unit cell where the node
(inclusion) from minority sublattice $A$ and two nodes (inclusions) from two
majority sublattices $B$ are marked.
In the case of a doubly extended Lieb lattice (Lieb-7), we expect (according
to the works [10, 59]) to obtain seven bands in the dispersion relation. The
tight-binding model predicts that the bands will be symmetric with respect to
the fourth band, exhibiting particle-hole symmetry. However, due to the
dipolar interaction, we did not expect such symmetry. Another feature which
one may deduce from the tight-binding model is that bands No. 2, 4 and 6
should be flat while bands No. 1, 3, 5 and 7 are considered dispersive.
Moreover, bands No. 3 and 5 suppose to form a Dirac cone intersecting flat
band No. 4 at the $\Gamma$ point.
We calculated the dispersion relation for magnonic Lieb-7 lattice (Fig. 8(a)),
which share many properties with those characteristic for the tight-binding
model [59]: (i) third and fifth bands form the Dirac cones which almost
intersect the flatter forth band at $\Gamma$ point; (ii) the third (and fifth)
band has a parabolic shape at $M$ point where it is degenerated with the
second (and six) band which is weakly dispersive. The mentioned regions of
dispersion are presented as 3D plots in Fig. 8(b). Also, we are going to
discuss shortly the profiles of spin wave eigenmodes (including CLS) in these
two regions of the dispersion relation, which are presented in Fig. 9.
Figure 8: Dispersion relation for the double extended magnonic Lieb lattice
(Lieb-7) containing seven inclusions in the unit cell: one inclusion $A$ from
minority sublattice and six inclusions $B$ from majority sublattices (see Fig.
2). (a) The dispersion relation is plotted along the high symmetry path
$\Gamma$-X-M-$\Gamma$ (see the inset). The first, third, fifth and seven bands
(dark blue, red, cyan and orange) are dispersive, while the second, fourth and
sixth bands (green, magenta and dark green bands) are the flatter bands,
supporting the magnonic CLS. Dirac cones occur at the $\Gamma$ point and
almost interact with the flatter fourth band, while at $M$ point, we observe
the degeneracy of the dispersive parabolic third (fifth) band with a flatter
second (six) band. (b) The zoomed vicinity of $\Gamma$ point (dark green
frame) and $M$ point (violet frame) regions are presented in 3D. Figure 9:
The profiles of eigenmodes were obtained for magnonic Lieb-7. The modes are
presented for bands No. 3-5 at $\Gamma$ point and 5-7 at $M$ point. The modes
denoted as $\Gamma_{3}$ and $\Gamma_{4}$ are degenerated whereas the
$\Gamma_{5}$ is separated from them by extremely small gap $\approx~{}2$ MHz.
At $M$ point, we showed the profiles for bands No. 5, 6 and 7. The modes
$M_{5}$ and $M_{6}$ are degenerated and separated from $M_{7}$ by essential
gap – predicted by the tight-binding model.
Dirac cones appear at the $\Gamma$ point for bands No. 3 and 5. At this point,
as for the basic magnonic Lieb lattice (Fig. 3), there is a very narrow gap of
the width $\approx~{}2$ MHz. The profiles $\Gamma_{4}$ and $\Gamma_{5}$ (left
column in Fig. 9) represent the degenerated states originating from flat and
dispersive bands. Both of them do not occupy the inclusions $A$ and are more
focused on two inclusions $B$ arranged in horizontal ($\Gamma_{4}$) and
vertical lines ($\Gamma_{5}$) – see grey stripes. Therefore, their profiles
are similar to NLS, where the first and third inclusion $B$ in each three-
element chain, linking inclusions $A$, precesses out-of-phase and the second
(central) inclusion $B$ remains unoccupied.
At the $M$ point, the $M_{5}$ and $M_{6}$ bands are degenerated. For these
bands, the spin waves are localized in all inclusions $B$ and do not occupy
inclusions $A$ (see right column of Fig. 6) The first and third inclusion $B$
in each three-element chain, linking inclusions $A$, precess in-phase, whereas
the second (central) inclusion $B$ precesses out-of-phase with respect to the
first and third one. This pattern of occupation of inclusions and the phase
relations between them is similar to one observed for CLS (see grey patches
marking the loops of inclusions in the left column of Fig. 6), but has one
significant difference. The phase difference between successive three-element
chains of inclusions $B$, in the loop, is equal to $\pm\pi/2$. However, the
linear combination of the modes $M_{5}\pm iM_{6}$ produces, similarly to the
case of the Lieb-5 lattice, the NLS. To observe the proper profiles of CLS or
NLS, we need to shift slightly from the high symmetry points $\Gamma$ and $M$
to cancel the degeneracy.
### VI.2 Realization of Lieb lattice
by shaping demagnetizing field
We have considered also an alternative realisation method for a magnonic Lieb
lattice in a ferromagnetic layer. This approach is based on shaping the
internal demagnetizing field. The structure under consideration is presented
in Fig. 10. It consists of a thin (28.5 nm) and infinite CoFeB layer on which
a Py antidot lattice (ADL), of 28.5 nm thickness, is deposited. The
cylindrical holes in ADL are arranged in shape of the basic Lieb lattice. The
size of the unit cell and diameter of holes remains the same as for the basic
Lieb lattice proposed in the main part of the manuscript (see Fig. 1). Due to
the absence of perpendicular magnetic anisotropy (PMA), we decided to apply a
much larger external magnetic field ($H_{0}=1500$ mT) to saturate the
ferromagnetic material in an out-of-plane direction.
Figure 10: Basic magnonic Lieb lattice where spin wave excitations in the
CoFeB layer are shaped by demagnetizing field from Py antidot lattice.
Dimensions of the ferromagnetic unit cell are equal to 250x250x59 nm and
contain 3 inclusions of 50 nm diameter. (a) structure of basic Lieb lattice,
(b) top view on basic Lieb lattice unit cell and differentiation to nodes of
sublattice $A$ and $B$.
We assumed the same gyromagentic ratio for both materials
$\gamma=187~{}{\rm\frac{GHz}{T}}$, the following values of material parameters
for CoFeB [60]: saturation magnetization - $M_{S}=1150~{}{\rm\frac{kA}{m}}$,
exchange stiffness constant - $A=15~{}{\rm\frac{pJ}{m}}$. For Py, we used
material parameters [61]: saturation magnetization -
$M_{S}=796~{}{\rm\frac{kA}{m}}$, exchange stiffiness constant -
$A=13~{}{\rm\frac{pJ}{m}}$.
The deposition of the ADL made of Py (material of lower $M_{S}$) above the
CoFeB layer (material of higher $M_{S}$) is critical for spin wave
localization in CoFeB below the exposed parts (holes) of the ADL. The
demagnetization field produced on CoFeB/Air interface creates wells partially
confining the spin waves. However, this pattern of internal demagnetizing
field becomes smoother with increasing distance from the ADL.
Figure 11: The dispersion relation obtained for basic Lieb lattice formed by
demagnetizing field of antidot lattice (see Fig. 10). (a) The dispersion
relation, (b) the 3D plot of dispersion relation in the region marked with the
green frame in (a). Results were obtained for $H_{0}=1500$ mT applied out-of-
plane.
The obtained dispersion relation is shown in Fig. 11. It is worth noting that
the lowest band is very dispersive, while the highest band is flattened more
than in the case of the structure presented in the main part of the manuscript
(see Fig. 3). The middle band, which suppose to support CLS, varies in extent
similar to the third band. For this structure, Dirac cones in the $M$ point
cannot be clearly unidentified.
### VI.3 Lieb lattice formed by YIG inclusions
in non-magnetic matrix
The periodic arrangement of ferromagnetic cylinders surrounded by nonmagnetic
material (e.g. air) seems to be the simplest realization of the Lieb lattice.
To refer this structure to the bi-component system investigated in the main
part of the manuscript, we assumed the same material and geometrical
parameters for inclusions as for the structure presented in Fig. 1.
The advantage of this system is that the confinement of spin waves within the
areas of inclusions is ensured for arbitrarily high frequency. We are not
limited here by the FMR frequency of the matrix, as it was for bi-component
Lieb lattices (Figs. 1, 2). However, the coupling of magnetization dynamics
between the inclusions is here provided solely by the dynamical demagnetizing
field, i.e. the evanescent spin waves do not participate in the coupling.
Therefore, the interaction between inclusions is much smaller in general,
which leads to a significant narrowing of all magnonic bands (Fig. 11). The
widths of the second and third band can be even smaller than the gap
separating from the first bands – Fig. 11(b). Such strong modification of the
spectrum makes the applicability of the considered system for the studies of
magnonic CLS questionable.
Figure 12: Dispersion relations for basic Lieb lattice. (a) The results
obtained for YIG inclusions in Ga:YIG matrix (dashed lines) and YIG inclusions
without matrix (solid lines). (b) The zoomed dispersion relation obtained for
YIG inclusions without matrix, marked in (a) by the frame.
### VI.4 Demagnetizing field
in YIG—Ga:YIG Lieb lattice
The difficulty in designing the magnonic system is not only due to the
adjustment of geometrical parameters of the system but also due to the shaping
of the internal magnetic field $\textbf{H}_{\rm eff}$.
The components of the effective magnetic field can be divided into long-range
and short-range. The realization of our model is inseparably linked to the
long-range dipole interactions through which the coupling between inclusions
is possible. This kind of interaction is sensitive to the geometry of the
ferromagnetic elements forming the magnonic system.
In Lieb lattice, the nodes of minority sublattice $A$ have four neighbours and
the nodes of majority sublattice $B$ have two. As a result, identical
inclusions (in terms of their shapes and material parameters) become
distinguishable, because of slightly different values of the internal
demagnetising field. This has consequences for the formation of a frequency
gap between Dirac cones at point $M$ in the dispersion relation obtained for
the basic Lieb lattice. In the literature, this phenomenon has been described
for the tight-binding model and is called node dimerisation of the lattice
[37].
In Fig. 13 we have shown the profile of the $z$-component of the demagnetising
field. For each inclusion through which the cut line passes, we have marked
the minimum value of the demagnetising field. The slightly lower value of
internal filed for inclusions $A$ is responsible for a tiny lowering of the
frequency for the mode $M_{1}$ (concentrated in inclusions $A$) respect the
degenerated modes $M_{2}$ and $M_{3}$ (confined in inclusions $B$).
Figure 13: Profile of static demagnetizing field plotted at cut through (a)
Lieb lattice unit cell. (b) The $z$-component of the demagnetizing field along
the cut line is shown in (a). In the plot, we have marked peaks for the areas
of inclusions $A$ and $B$. Please note the slightly different values of
demagnetizing in the centre of $A$ and $B$ inclusion due to different the
number neighboring of nodes: four for inclusion $A$, two for inclusion $B$.
|
Figure 4: The graph adjacency matrix ${\bf V}$, characterizing the connections
between different nodes of DEGM after lifelong learning.
(a) The edges of DEGM after the MSFIRC lifelong learning. (b) The edges of
DEGM after the CCCOSCZC lifelong learning.
Figure 5: Edge information of DEGM after lifelong learning. The red and blue represent the basic and specific node, respectively. Criteria | Dataset | BE | LGM | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---|---
SL | MNIST | 26.3 | 685.3 | 22.3 | 22.3 | 21.9
SVHN | 47.0 | 941.7 | 30.1 | 29.0 | 39.3
Fashion | 43.8 | 663.4 | 37.7 | 27.4 | 36.6
IFashion | 45.9 | 1148.4 | 35.6 | 27.4 | 38.4
RMNIST | 27.9 | 704.2 | 20.2 | 22.1 | 25.3
Cifar10 | 994.4 | 1241.1 | 615.3 | 608.1 | 892.1
Average | 197.5 | 897.4 | 126.9 | 122.7 | 175.6
Table 4: The results under MSFIRC lifelong learning. Criteria | Dataset | BE | LGM | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---|---
SL | CelebA | 213.9 | 535.6 | 229.2 | 217.0 | 215.4
CACD | 414.9 | 814.3 | 368.3 | 281.95 | 347.3
3D-Chair | 649.1 | 2705.9 | 324.0 | 291.46 | 513.8
Omniglot | 875.1 | 5958.9 | 225.6 | 195.7 | 343.2
ImageNet* | 758.4 | 683.1 | 689.6 | 652.8 | 769.1
Car | 745.1 | 583.7 | 588.8 | 565.9 | 709.8
Zappos | 451.1 | 431.2 | 263.4 | 275.8 | 280.7
| CUB | 492.0 | 330.2 | 461.3 | 569.6 | 638.6
| Average | 575.0 | 1505.4 | 393.8 | 381.3 | 477.2
Table 5: The results under CCCOSCZC lifelong learning.
We show the adjacency matrix of the nodes from the graph ${\bf V}$ of DEGM
after MSFIRC lifelong learning in Fig. 4a where ”C1” represents the first
node. We can observe that DEGM creates three basic nodes and three specific
nodes, respectively. Three specific nodes have edges from the first and second
basic nodes. We also show ${\bf V}$ of DEGM after CCCOSCZC lifelong learning
in Fig. 4b. DEGM creates basic nodes when learning the first, third, fourth
and fifth task. We also show the edge information between members of
$\mathcal{S}$ and members of $\mathcal{G}$ in Fig. 5 where red colour
represents the basic nodes and blue colour represents the specific nodes.
### L.2 Results for generalization bounds
To estimate the discrepancy, we train an auxiliary model on the distribution
$\mathbb{P}^{i}\otimes{\tilde{\mathcal{P}}_{(i+1)}}$ for each $(i+1)$-th task
learning. Then we calculate the discrepancy by using Definition 3 of the paper
for each training epoch.
In the following, we train a single model with GR under MNIST, Fashion and
IFashion (MFI) lifelong learning. The images from all databases have pixel
values within the range $[0,255]$. We implement the decoder by a neural
network that outputs the mean vector of a Gaussian distribution with the
diagonal covariance matrix (diagonal element is $1.0$). We estimate the
reconstruction error term by using :
$\displaystyle{\log{p_{\theta}}\left({{\bf x}\,|\,{\bf
z}}\right)}=-\frac{1}{{2\sigma^{2}}}{\left\|{{\bf{x}}-{\mu_{\theta}}\left({\bf{z}}\right)}\right\|^{2}}-\frac{1}{2}\log
2\pi\sigma^{2}$ (131)
It can be noted that we also normalize this reconstruction error by dividing
$28\times 28$, as done in Chen et al. (2016b). We evaluate the average risk
and the discrepancy distance for each training epoch (See details in Lemma 1
from the paper). To calculate $|KL_{1}-KL_{2}|$ and discrepancy distance, we
randomly choose 10000 number of samples from ${\mathcal{P}}_{(1:t)}$ and
${\mathbb{P}}^{t-1}\otimes{\tilde{\mathcal{P}}}_{t}$ at the $t$-th task
learning, respectively. The results are presented in Fig. 6a where the source
risk is continuously decreased while the discrepancy is increased as learning
more tasks. We also evaluate the risk, $|KL_{1}-KL_{2}|$ and discrepance
distance on MNIST, shown in Fig. 6b. In order to select the generated images
that are belonging to MNIST, we train a task-specific classifier that predicts
the task label for giving samples.
(a) Evaluation on all tasks. (b) Evaluation on MNIST
Figure 6: The risk, $|KL_{1}-KL_{2}|$ and discrepancy distance estimated by a
single VAE model with GR under MFI lifelong learning.
In the following, we investigate the results for Lemma 2 of the paper. We
consider a sequence of MNIST, Fashion, IFashion (MFI) learning tasks. We
consider a mixture model ${\bf M}=\\{{\mathcal{M}}_{1},{\mathcal{M}}_{2}\\}$
consisting of two components after LLL in which ${\mathcal{M}}_{1}$ is fixed
after the first task learning while ${\mathcal{M}}_{2}$ is used to learn
Fashion and IFashion, respectively. We also consider to learn a single model
${\mathcal{M}}$ with GR under MFI lifelong learning. We evaluate the average
target risk (NLL estimated by ELBO) for each training epoch and the results
are reported in Fig. (7) where ”single” and ”mixture” represent
${\mathcal{M}}$ and ${\bf M}$, respectively. As shown from the results, the
expansion mechanism can achieve a tight GB, as discussed in Lemma 2 of the
paper.
Figure 7: Target risk on the single model and the mixture model.
In the following, we provide additional empirical results for the theoretical
analysis. It notes that in the following experiments, we resize all databases
as $32\times 32\times 3$ and the pixel values of all images are normalized as
$[0,1]$.
We evaluate the target risk (square loss) of a single model on four datasets
(MNIST, SVHN, Fashion, IFashion) under MSFI (See Theorem-2). We plot the
results in Fig. 8-a where ”All” represents the accumulated target risks for
all tasks (left hand side of Equation-7 in the paper). In the next, we train a
single model under MSFI and SFMI setting, respectively. We plot the target
risk on MNIST in Fig. 8-b where ”Early” and ”Recent” denote that the MNIST is
used as the first and third task, respectively. It observes that the model
tends to forget early tasks than recent tasks, as demonstrated in Equation-14
of the paper and discussed in Theorem 3. We also evaluate the accumulated
target risk estimated by DEGM under MSFI lifelong learning, which is shown in
Fig. 8-c. It observes that there have no accumulated errors for each task
during lifelong learning since the number of components match the number of
task, as discussed in Theorem 3.
Figure 8: ”a” shows the target risk (LHS of Eq.(7) in Theorem-2 of the
paper)) across four tasks under MSFI lifelong learning. ”b” shows the target
risk (LHS of Eq.(7) in Theorem-2 of the paper) on MNIST under MSFI lifelong
learning. ”c” shows the target risk (LHS of Eq.(7) in Theorem-2 of the paper)
estimated by the proposed mixture model under (MSFI) lifelong learning. It
notes that all results are the accumulated target error and we do not
calculate the average result.
In the following, we investigate the performance of DEGM and a single model
when changing the order of tasks. We randomly generate three different orders
: Fashion, SVHN, MNIST, Cifar10, IMNIST, IFashion (FSMCII); Cifar10, IMNIST,
Fashion, MNIST, SVHN, IFashion (CIFMSI); IFashion, IMNIST, Cifar10, Fashion,
SVHN, MNIST (IICFSM). We evaluate the accumulated target risk for all tasks
and the results are presented in Fig. 9 where ”order1”, ”order2” and ”order3”
denote FSMCII, CIFMSI and IICFSM, respectively. It observes that a single
model is sensitive to the choice of orders of tasks since the final
accumulated target risk for all tasks are different when training the model
under different orders of tasks. Although CN-DPM can avoid the forgetting
during the training, the performance is still changed when changing the order
of tasks. This is mainly because newly created components only reuse the
transferable information from the first task and would lead to negative
transfer when learning an entire different task. However, the proposed DEGM is
robust to the change of orders of tasks.
Figure 9: The accumulated target risks of DEGM, a single model and CN-DPM*
with different orders of tasks. “a”, ”b” and ”c” represent the results
achieved by DEGM, a single model and CN-DPM*, respectively.
### L.3 Evaluation by using other criterion
In addition to SL and NLL, we introduce to use the structural similarity index
measure (SSIM) Hore and Ziou (2010) and the Peak-Signal-to-Noise Ratio (PSNR)
Hore and Ziou (2010) as criteria. We report the results evaluated by the above
criterion in Table 6 and Table 7.
Criteria | SL | SSMI | PSNR
---|---|---|---
BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM*
MNIST | 26.3 | 685.3 | 22.3 | 22.3 | 21.9 | 0.88 | 0.19 | 0.90 | 0.90 | 0.90 | 21.0 | 7.0 | 21.8 | 21.8 | 21.8
SVHN | 47.0 | 941.7 | 30.1 | 29.0 | 39.3 | 0.58 | 0.20 | 0.66 | 0.67 | 0.61 | 13.7 | 5.0 | 15.5 | 15.7 | 14.3
Fashion | 43.8 | 663.4 | 37.7 | 27.4 | 36.6 | 0.68 | 0.15 | 0.72 | 0.79 | 0.73 | 18.4 | 3.7 | 19.0 | 20.6 | 19.2
IFashion | 45.9 | 1148.4 | 35.6 | 27.4 | 38.4 | 0.72 | 0.11 | 0.76 | 0.81 | 0.76 | 18.2 | 5.0 | 19.4 | 20.6 | 19.1
RMNIST | 27.9 | 704.2 | 20.2 | 22.1 | 25.3 | 0.87 | 0.20 | 0.91 | 0.90 | 0.89 | 16.5 | 7.0 | 22.2 | 21.8 | 21.2
Cifar10 | 994.4 | 1241.1 | 615.3 | 608.1 | 892.1 | 0.29 | 0.23 | 0.49 | 0.50 | 0.34 | 16.5 | 15.4 | 18.9 | 18.9 | 17.0
Average | 197.5 | 897.4 | 126.9 | 122.7 | 175.6 | 0.67 | 0.18 | 0.74 | 0.76 | 0.70 | 18.1 | 7.2 | 19.5 | 19.9 | 18.8
Table 6: The performance of various models under the MSFIRC learning setting. Criteria | SL | SSMI | PSNR
---|---|---|---
BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM* | BE | LGM | DEGM | DEGM-2 | CN-DPM*
CelebA | 213.9 | 535.6 | 229.2 | 217.0 | 215.4 | 0.69 | 0.48 | 0.66 | 0.69 | 0.69 | 23.5 | 19.3 | 23.2 | 23.4 | 23.5
CACD | 414.9 | 814.3 | 368.3 | 281.95 | 347.3 | 0.57 | 0.47 | 0.62 | 0.68 | 0.63 | 20.6 | 17.33 | 21.2 | 22.4 | 21.4
3D-Chair | 649.1 | 2705.9 | 324.0 | 291.46 | 513.8 | 0.73 | 0.42 | 0.84 | 0.86 | 0.79 | 19.0 | 13.54 | 22.4 | 23.1 | 20.5
Omniglot | 875.1 | 5958.9 | 225.6 | 195.7 | 343.2 | 0.73 | 0.22 | 0.92 | 0.93 | 0.89 | 17.9 | 9.2 | 24.0 | 24.6 | 22.1
Sub-ImageNet | 758.4 | 683.1 | 689.6 | 652.8 | 769.1 | 0.37 | 0.42 | 0.41 | 0.43 | 0.37 | 18.5 | 18.9 | 19.0 | 19.2 | 18.5
Car | 745.1 | 583.7 | 588.8 | 565.9 | 709.8 | 0.39 | 0.48 | 0.47 | 0.49 | 0.42 | 18.0 | 19.0 | 19.0 | 19.2 | 18.2
Zappos | 451.1 | 431.2 | 263.4 | 275.8 | 280.7 | 0.68 | 0.60 | 0.75 | 0.74 | 0.73 | 20.0 | 20.2 | 22.4 | 22.3 | 22.1
CUB | 492.0 | 330.2 | 461.3 | 569.6 | 638.6 | 0.35 | 0.48 | 0.45 | 0.43 | 0.35 | 19.0 | 20.9 | 19.3 | 18.6 | 18.0
Average | 575.0 | 1505.4 | 393.8 | 381.3 | 477.2 | 0.60 | 0.45 | 0.64 | 0.66 | 0.61 | 19.6 | 17.3 | 21.3 | 21.6 | 20.5
Table 7: The performance of various models under the CCCOSCZC learning
setting.
### L.4 Ablation study
First, we evaluate the effectiveness of the proposed dynamic expansion
mechanism used in our model. We train all models under MSFIRC lifelong
learning and present the results in Fig 10a where we compare the average
score, the training times and the memory use. It observes that even if the
DEGM uses few training epochs but does not degenerate the performance. In
addition, DEGM outperforms CN-DPM* that only transfers features from a single
shared model. Furthermore, the proposed DEGM can achieve a similar performance
as the baseline that trains individual VAEs for each task.
Figure 10: The results for various baselines under MSFIRC. In the left chart,
“A” represent the average square loss and PSNR for all tasks. “B” represent
the overall training times (seconds). “C” represents the number of parameters
($10^{8}$) for various baselines. In the right chart, “A” represent the
average square loss for all tasks. “B” represent number of parameters
($10^{8}$). “C” represents the number of basic components. X-axis represents
different threshold values $thr$
The effects of thresholds in ORVAE
In the following, we investigate the performance of DEGM with different
thresholds $\tau$ under MSFIRC lifelong learning and report results in Fig.
10b. As reduce $\tau$, DEGM tends to increase the number of basic components
and gradually improve performance. It observes that the choice of $\tau=400$
can trade-off between the performance and the model’s complexity since it does
not significantly improve the performance when decreasing $thr$ from 400. In
the next, in order to evaluate the effects of the proposed expansion mechanism
and the adaptive weight, we introduce several new baselines in the following.
DEGM-4:
This baseline generates information flows from all learned components to a new
component. For instance, if a new component receives the information flow from
members of $\mathcal{S}$, we will sum up the latent codes and intermediate
representations from the sub-inference and sub-decoder of these components.
DEGM-4 does not use the adaptive weight.
DEGM-5:
We implement this baseline by considering to create the edges without using
the adaptive weight. The training process for this baseline is described as
follows: Once the $t$-th task learning was finished, As similar done to DEGM,
we have a set of $K$ measures, denoted by $KS=\\{k{s_{1}},\dots,k{s_{K}}\\}$,
which can be used to build the edges from a new node to members of
$\mathcal{G}$. We set a threshold $\tau$ which is used to update $\bf{V}$ such
that if each $ks_{i}<\tau$, then ${\bf{V}}(t+1,{\mathcal{GI}(i)})=1$,
otherwise ${\bf{V}}(t+1,{\mathcal{GI}(i)})=0$. This means that if
$ks_{i}>\tau$, then the construction of a new component does not reuse the
information and parameters from the $i$-th component in DEGM.
DEGM-6:
For this baseline, we consider the adaptive weight for each edge is equal.
This means that the importance of each basic component is treated as the same
for a new task.
DEGM-7:
We implement this baseline by considering to create only a single edge for a
new component to a certain basic component that has the maximum sample log-
likelihood for data of the new task.
CN-DPM*-1;
This baseline builds a new components and creates connections with previously
learned components, as similar to DEGM-4.
CN-DPM*-2:
We implement this baseline by using the large model which contains $1.3\times
10^{9}$ number of parameters.
We report the results in Table 8. It observes that the adaptive weights in the
expansion mechanism can further improve the performance. We also find that
although CN-DPM*-2 uses the more parameters, our DEGM still outperforms CN-
DPM* by a large margin.
Criteria | Dataset | DEGM | DEGM-4 | CN-DPM*-1 | CN-DPM*-2 | DEGM-5 | DEGM-6 | DEGM-7
---|---|---|---|---|---|---|---|---
SL | MNIST | 22.30 | 22.18 | 22.12 | 22.67 | 22.88 | 21.48 | 22.35
SVHN | 30.18 | 30.73 | 40.53 | 38.74 | 30.56 | 29.44 | 29.20
Fashion | 37.73 | 41.22 | 45.03 | 38.51 | 37.65 | 37.26 | 41.29
IFashion | 35.62 | 49.26 | 36.19 | 36.90 | 41.27 | 37.15 | 43.95
RMNIST | 20.23 | 60.72 | 24.79 | 24.39 | 27.86 | 25.73 | 25.78
Cifar10 | 615.34 | 610.38 | 929.55 | 877.35 | 617.34 | 617.36 | 614.48
Average | 126.90 | 135.58 | 183.03 | 173.09 | 129.59 | 128.07 | 129.51
Table 8: The results of various models under MSFIRC lifelong learning.
### L.5 The number of parameters used in various methods
We list the number of parameters used in various methods in Table 9 and Table
10, respectively. We can observe that the proposed DEGM architecture requires
less parameters than other parameters.
Model | LGM | BE | DEGM | DEGM-2 | CN-DPM*
---|---|---|---|---|---
N | ${1.5\times 10^{8}}$ | ${9.4\times 10^{8}}$ | ${1.6\times 10^{8}}$ | ${8.7\times 10^{8}}$ | ${4.2\times 10^{8}}$
Table 9: The number of parameters of various models under MSFIR learning setting. Model | LGM | BE | DEGM | DEGM-2 | CN-DPM* | LIMix
---|---|---|---|---|---|---
N | ${1.9\times 10^{9}}$ | ${3.9\times 10^{9}}$ | ${3.2\times 10^{8}}$ | ${1.3\times 10^{9}}$ | ${9.4\times 10^{9}}$ | ${9.4\times 10^{9}}$
Table 10: The number of parameters of various models under CCCOSCZC learning
setting.
### L.6 Visual results
We show the reconstructions from DEGM under MSFIRC and CCCOSCZC lifelong
learning in Fig. 11 and Fig. 12, respectively.
(a) MNIST.
(b) SVHN.
(c) Fashion.
(d) IFasion.
(e) RMNIST.
(f) Cifar10.
(g) Task 1.
(h) Task 2.
(i) Task 3.
(j) Task 4.
(k) Task 5.
(l) Task 6.
Figure 11: Image reconstructions when using DEGM under the MSFIRC lifelong
learning. The first row represents testing data samples and the second row are
their reconstructions using DEGM.
(a) Task1.
(b) Task2.
(c) Task3.
(d) Task4.
(e) Task5.
(f) Task6.
(g) Task7.
(h) Reconstruction of Task1.
(i) Reconstruction of Task2.
(j) Reconstruction of Task3.
(k) Reconstruction of Task4.
(l) Reconstruction of Task5.
(m) Reconstruction of Task6.
(n) Reconstruction of Task7.
(o) Task8.
(p) Reconstruction of Task8.
Figure 12: Image reconstructions when using DEGM under the CCCOSCZC lifelong
learning. The first row represents testing data samples and the second row are
their reconstructions using DEGM.
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# Spatial and temporal characteristics of spontaneous parametric down-
conversion with varying focal planes of interacting beams
Richard Bernecker<EMAIL_ADDRESS>Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, D-07743 Jena, Germany.
Baghdasar Baghdasaryan Theoretisch-Physikalisches Institut, Friedrich-
Schiller-Universität Jena, D-07743 Jena, Germany. Helmholtz-Institut Jena,
D-07743 Jena, Germany. Fraunhofer Institute for Applied Optics and Precision
Engineering IOF, Albert-Einstein-Strasse 7, 07745 Jena, Germany Stephan
Fritzsche Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität
Jena, D-07743 Jena, Germany. Helmholtz-Institut Jena, D-07743 Jena, Germany.
Abbe Center of Photonics, Friedrich-Schiller-University Jena, Albert-Einstein-
Str. 6, 07745 Jena, Germany
###### Abstract
Spontaneous parametric down-conversion (SPDC) is a widely used process to
prepare entangled photon pairs. In SPDC, a second-order nonlinear crystal is
pumped by a coherent laser beam to generate photon pairs. The photon pairs are
usually detected by single-mode fibers (SMF), where only photons in a Gaussian
mode can be collected. The collection modes possess typical Gaussian
parameters, namely a beam waist and a focal plane position. The collection
efficiency of photons highly depends on the choice of both parameters. The
exact focal plane position of the pump beam relative to those of the detection
modes is difficult to determine in a real experiment. Usually, theoretical and
experimental studies assume that the focal plane positions of the pump and the
generated beams are positioned in the center of the crystal. The displacement
of beam focal planes can lead to deviations from expected results and the
coupling efficiency into SMF can decrease. In this work, we theoretically
consider variable positions of focal planes and investigate how they influence
the spatial and temporal properties or the purity of photon pairs. We present
SPDC arrangements, in which the knowledge of the exact position of the focal
planes is essential, as well as scenarios, where focal plane displacements do
not contribute significantly to experimental outcomes. These findings are of
particular interest for achieving higher efficiency in SPDC experiments.
###### pacs:
Valid PACS appear here
Quantum-based technologies are increasingly explored and integrated into
today’s applications. In this context, quantum entanglement is an inseparable
part of quantum communication protocols [1, 2]. The process of spontaneous
parametric down-conversion (SPDC) is the most reputable source of photonic
entanglement [3, 4] and provides an experimental platform for fundamental
quantum science [5].
In SPDC, a nonlinear crystal is illuminated with a strong, high-energetic
laser field called pump beam. Photon pairs with lower energies, also known as
signal and idler photons, are subsequently down-converted. The generated
signal and idler photons fulfill the energy $\omega_{p}=\omega_{s}+\omega_{i}$
and the momentum $\bm{k}_{p}=\bm{k}_{s}+\bm{k}_{i}$ conservations. The
momentum conservation, also known as the phase-matching (PM) condition,
ensures constructive interference between the three interacting beams and
inherently determines the spectral and spatial properties of signal and idler
photons. Spectrally and spatially engineered photons are important ingredients
in current research on quantum information [6, 7, 8], quantum computing [9],
and quantum communication [10, 11]. Additionally to the PM, the pump beam
properties also have a large impact on the spectral and spatial properties of
signal and idler photons [12, 13, 14]. The pump characteristics include the
beam width, the focal plane position relative to the crystal, and its spatial
and temporal degrees of freedom (DOFs) [15, 16].
Figure 1: Schematic paths of pump and signal beam in a non-linear crystal. For
the sake of simplicity, only the signal beam is shown, the idler beam can be
similarly imagined. The beams are described as Gaussian beams with beam widths
$w_{p}$ and $w_{s}$. Most calculations assume that the focal planes of the
pump, signal, and idler lie in the center of the crystal $z=0$ as shown in the
left picture. On the contrary, we allow in the right picture that the pump,
signal, and idler focal planes are not fixed. The parameters for focal plane
shifts along the propagation axis are $z_{p}$ for the pump beam and
$z_{s},z_{i}$ for the generated beams.
Besides the generation of photon pairs, optimal experimental verification is
an essential part of quantum fundamental research too. Usually, the spatial
shape of signal and idler photons are detected by multi-plane light conversion
(MPLC) [17, 18], where an arbitrary spatial mode is projected to a Gaussian
mode [19], in order to couple it into a single-mode fiber (SMF) [20, 21]. The
coupling efficiency into SMF depends on the beforehand chosen focal planes and
beams widths of the pump, signal, and idler modes. Optimizing the coupling
efficiency of collecting a photon pair in fundamental Gaussian modes (FGMs) is
a particularly interesting aspect from an experimental perspective [22, 23,
24, 25]. We distinguish between the single-mode coupling efficiency for a
certain frequency (within a narrowband filter bandwidth) [26], and the
spectral brightness, which pertains to the maximal collection probability on a
broadband of frequencies [27, 28].
In the past, there were several theoretical and experimental approaches to
modify the pump or the detection scheme of the generated modes with linear
optical systems [29, 30, 31, 32, 33], in order to improve the pair collection
efficiency. The variation of the pump intensity or the pump beam width has
also been explored to optimize the photon pair collection efficiency or
transverse spatial correlations [34, 35, 36]. In this regard, the manipulation
of the purity between the signal and idler photons via pump focusing was shown
[37, 38]. Moreover, it has been proposed in Refs. [39, 40] how to consider the
change of the angular spectrum of the pump at different positions beyond the
crystal center. In analogy to the change of the pump focal plane position, the
position of the nonlinear crystal has been varied to control the time delay
between signal and idler photons and the coincidence rate [41].
All these considerations were primarily concerned with optimal pump focusing
or the right choice of optical elements in beam paths, in order to achieve
enhanced photon collection efficiency in SPDC. In this work, we theoretically
consider variable focal planes for pump, signal, and idler explicitly and
describe their impact on experimentally measurable quantities such as the
coupling efficiency into SMF, spatial and temporal correlations of generated
photons and the spectral purity between down-converted photons. In this
regard, we will compare scenarios of focal planes fixed at different positions
and investigate if the measurement probability of signal and idler photons
remain unaffected. We discuss setup conditions with noteworthy influence on
the spectral brightness and the coupling efficiency. The alignment of focal
plane positions in these scenarios will require more careful effort in order
to increase the coupling efficiency. We also contemplate scenarios where the
precise position of focal planes becomes insignificant.
## I Theory
We start our investigation with the theoretical description of the SPDC
process. Our group published a paper on the characterization of spatio-
temporal DOFs in SPDC, where a general expression for the SPDC-state, also
known as biphoton state, has been derived [42] and later verified
experimentally [19]. First, we briefly recap the derivation of the expression
from [42], where the focal planes of the pump, signal, and idler beams are
assumed to be at $z=0$, i.e in the crystal center. This assumption, as shown
in Fig. 1 on the left, is common in theoretical as well as experimental
studies. Next, we extend the expression to consider variable focal planes for
the pump, signal, and idler beams, which is illustrated on the right side of
Fig. 1.
### I.1 Biphoton state of SPDC decomposed in Laguerre-Gaussian Basis
The common expression of the general biphoton state in the wave vector
representation of the interacting beams is [40]
$\displaystyle\ket{\Psi}_{\text{SPDC}}=\iint$ $\displaystyle
d\bm{q}_{s}\>d\bm{q}_{i}\>d\omega_{s}\>d\omega_{i}\>\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$
$\displaystyle\times\hat{a}_{s}^{\dagger}(\bm{q}_{s},\omega_{s})\>\hat{a}_{i}^{\dagger}(\bm{q}_{i},\omega_{i})\ket{\text{vac}},$
(1)
where we consider the paraxial approximation by the separation into
longitudinal and transversal components of the wave vector
$\bm{k}=\bm{q}+k_{z}(\omega)\>\bm{z}$, where $z$ is the propagation direction
of the pump beam. The paraxial approximation is valid in most experimental
SPDC setups since typical optical apparatuses support only paraxial rays about
the central axis. The biphoton state (1) is an integral over all possible
transverse wave vectors $\bm{q}_{s,i}$ and frequencies $\omega_{s,i}$ of a
signal and idler pair that is created from the vacuum state $\ket{\text{vac}}$
by creation operators $\hat{a}_{s,i}^{\dagger}(\bm{q}_{s,i},\omega_{s,i})$ of
signal and idler photons, respectively.
The so-called biphoton mode function
$\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$ encodes the coupling
between the wave vectors of the pump, signal, and idler beams [39]:
$\displaystyle\Phi(\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})$
$\displaystyle=N\>V_{p}(\bm{q}_{s}+\bm{q}_{i})\>S_{p}(\omega_{s}+\omega_{i})$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\operatorname{\mathrm{e}}^{i(k_{p,z}-k_{s,z}-k_{i,z})z},$
(2)
where $N$ is the normalization constant, $V_{p}$ is the spatial distribution
of the pump beam, whereas $S_{p}$ characterizes the spectral DOF of the pump.
The integral over the propagation direction $z$ describes the phase mismatch
${\Delta k_{z}=k_{p,z}-k_{s,z}-k_{i,z}}$ in the $z$ direction. The exact
expression of $\Delta k_{z}$ depends on the features of the crystal and the
geometry between the interacting beams and the crystal.
Since discrete modes are easier to detect and manipulate in experiments [43,
44] than continuous modes, the continuous transverse spatial variables in (1)
are often discretized by a set of optical modes. Furthermore, an appropriate
choice of the set can reduce the dimensionality of a state. In Ref. [42],
Laguerre-Gaussian (LG) modes have been used as a basis for the description of
the spatial distribution of the down-converted photons. This choice is
reasonable since LG modes carry well-defined projection of orbital angular
momentum (OAM) [45], which is conserved in collinear SPDC [46, 47, 48]. The
biphoton state decomposed in the LG basis $\ket{p,\ell,\omega}=\int
d\bm{q}\,\mathrm{LG}_{p}^{\ell}(\bm{q})\,\hat{a}^{\dagger}(\bm{q},\omega)\ket{\text{vac}}$
reads then
$\displaystyle\ket{\Psi}_{\text{SPDC}}=\iint d\omega_{s}\>d\omega_{i}$
$\displaystyle\sum_{p_{s},\ell_{s}}\sum_{p_{i},\ell_{i}}C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$
$\displaystyle\times\ket{p_{s},\ell_{s},\omega_{s}}\ket{p_{i},\ell_{i},\omega_{i}},$
(3)
where the overlap amplitudes $C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ of the LG
decomposition are frequency-dependent. The probability to find signal and
idler photons in spatial modes $(p_{s}|\ell_{s})$ and $(p_{i}|\ell_{i})$ at
frequencies $\omega_{s}$ and $\omega_{i}$ is given by
$P_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})=|C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})|^{2}$.
We can also call this quantity the single-mode coupling efficiency. On the
other hand, the maximal value of
$P_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$ over all possible
energies $\omega_{s}$ and $\omega_{i}$ is called the spectral brightness.
The following assumptions and approximations have been applied in Ref. [42],
in order to derive a comprehensive expression for
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}(\omega_{s},\omega_{i})$:
* •
Pump, signal, and idler beams are focused in the middle of the crystal.
* •
A small deviation $\Omega$ of generated frequencies from the central frequency
$\omega_{0}$ has been assumed, i.e. $\Omega\ll\omega_{0}$, so that we can
write $\omega=\omega_{0}+\Omega$. The central frequencies fulfill the energy
conservation $\omega_{p,0}=\omega_{s,0}+\omega_{i,0}$.
* •
In the paraxial regime, where $|\bm{q}|\ll k$, the so-called Fresnel
approximation can be applied on
$k_{z}=\sqrt{k^{2}-\mathinner{\\!\left\lvert\bm{q}\right\rvert}^{2}}$:
$k_{z}=k(\Omega)\sqrt{1-\frac{|\bm{q}|^{2}}{k(\Omega)^{2}}}\approx
k+\frac{\Omega}{u}+\frac{G\Omega^{2}}{2}-\frac{|\bm{q}|^{2}}{2k},$ (4)
with the group velocity $u=1/(\partial k/\partial\Omega)$ and the group
velocity dispersion $G=\partial/\partial\Omega\,(1/u)$ at the corresponding
central frequency.
* •
Momentum conservation for the central frequencies $\Delta
k=k_{p}-k_{s}-k_{i}=0$ is assumed. When a periodically poled crystal with
poling period $\Lambda$ along the crystal axis is used, this is achieved by
$\Delta k=k_{p}-k_{s}-k_{i}-\frac{2\pi}{\Lambda}=0$.
We shall expand now the formalism from Ref. [42], to support variable
positions of the focal planes for the pump, signal, and idler beams.
### I.2 Shift of focal plane positions
We briefly recap the angular spectrum propagation of beams. Mathematically,
electromagnetic field distributions can be described by a propagator factor
obtained via the angular spectrum representation in momentum space. This is a
well-investigated formalism, with the following main key ideas. We can choose
the $z$-direction as the propagation axis and write the Fourier transformation
of an arbitrary field $\bm{E}$ in the transverse $x$-$y$-plane of a fixed
point $z=\text{const.}$ as
$\bm{\tilde{E}}(\bm{q};z)\propto\iint_{-\infty}^{\infty}dx\>dy\>\bm{E}(x,y,z)\operatorname{\mathrm{e}}^{-ik_{x}x}\>\operatorname{\mathrm{e}}^{-ik_{y}y}.$
(5)
Here are $\bm{q}=(k_{x},k_{y})$ the transverse wave vector components. The
amplitude in momentum-space $\bm{\tilde{E}}(\bm{q};z)$ can also be used for
the inverse Fourier transform for the field in real space
$\bm{E}(x,y,z)\propto\iint_{-\infty}^{\infty}dk_{x}\>dk_{y}\>\bm{\tilde{E}}(\bm{q};z)\operatorname{\mathrm{e}}^{ik_{x}x}\>\operatorname{\mathrm{e}}^{ik_{y}y}.$
(6)
When we split the field in a spatial and time-dependent part
$\bm{E}=\bm{E}(x,y,z)\operatorname{\mathrm{e}}^{-i\omega t}+\text{c.c}$ and
also write $|\bm{k}|=k=\frac{\omega^{2}n^{2}}{c^{2}}$, the Helmholtz equation
reads as
$\left(\bm{\nabla}^{2}+k^{2}\right)\bm{E}(x,y,z)=0.$ (7)
We insert expression (6) into the Helmholtz equation and arrive at a
differential equation for the spatial evolution:
$(\partial_{z}^{2}+k^{2}-k_{x}^{2}-k_{y}^{2})\>\bm{\tilde{E}}(\bm{q};z)=0.$
(8)
When setting $k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}$, the solution of the
angular spectrum of an electric field evolving along the z-axis can be written
as
$\bm{\tilde{E}}(\bm{q};z)=\bm{\tilde{E}}(\bm{q};0)\>e^{\pm ik_{z}z}$ (9)
(see also Refs. [40, 39]). The positive signs in the exponential indicate a
wave propagation into $z>0$, while the negative sign describes a wave
propagating into the region $z<0$.
We apply Eq. (9) to the pump, signal, and idler beams. The PM function renewed
with the pump, signal, and idler focused at positions $z_{p}$, $z_{s}$, and
$z_{i}$ respectively (see Fig. 1), is now
$\displaystyle\Phi($
$\displaystyle\bm{q}_{s},\bm{q}_{i},\omega_{s},\omega_{i})=V_{p}(\bm{q}_{s}+\bm{q}_{i})\>S_{p}(\omega_{s}+\omega_{i})$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\operatorname{\mathrm{e}}^{i\left[k_{p,z}(z+z_{p})-k_{s,z}(z+z_{s})-k_{i,z}(z+z_{i})\right]}.$
(10)
Note that we continue to consider paraxial beams for pump, signal, and idler
and use the Fresnel approximation from Eq. (4) for $k_{z}$.
To specify our setup, we assume a Gaussian envelope of pulse duration $T_{0}$
for the spectral part of the pump. Due to
$\omega_{p}-\omega_{p,0}=\Omega_{p}=\Omega_{s}+\Omega_{i}$ we can write
$S_{p}(\Omega_{p})=\frac{T_{0}}{\sqrt{\pi}}\exp{\biggl{(}-\frac{T_{0}^{2}}{4}\>(\Omega_{s}+\Omega_{i})^{2}\biggl{)}}.$
(11)
The spatial distribution of the pump beam is also described as a Gaussian with
beam width $w_{p}$,
$\mathrm{V}(\bm{q}_{\mathrm{s}}+\bm{q}_{\mathrm{i}})\>=\>\frac{w_{p}}{\sqrt{2\pi}}\>\exp{\biggl{(}-\frac{w_{p}^{2}}{4}|\bm{q}_{\mathrm{s}}+\bm{q}_{\mathrm{i}}|^{2}\biggr{)}.}$
(12)
This reduces and simplifies the expression from Ref.[42] enormously. The final
formula reads then
$\displaystyle
C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}\propto\delta_{\ell_{s},-\ell_{i}}\>\exp{\biggl{(}-\frac{T_{0}^{2}}{4}\>(\Omega_{s}+\Omega_{i})^{2}\biggl{)}}\sum_{s=0}^{p_{s}}\sum_{i=0}^{p_{i}}\>(T_{s}^{p_{s},\ell_{s}})^{*}\>(T_{i}^{p_{i},\ell_{i}})^{*}\Gamma[h]\>\Gamma[b]$
$\displaystyle\times\int_{-L/2}^{L/2}dz\>\exp{\biggl{[}iz\biggl{(}\frac{\Omega_{s}+\Omega_{i}}{u_{p}}-\frac{\Omega_{s}}{u_{s}}-\frac{\Omega_{i}}{u_{i}}+\frac{G_{p}\>(\Omega_{s}+\Omega_{i})^{2}}{2}-\frac{G_{s}\>\Omega_{s}^{2}}{2}-\frac{G_{i}\>\Omega_{i}^{2}}{2}\biggl{)}\biggl{]}}\>\frac{D^{\ell_{i}}}{H^{h}\>B^{b}}\>{{}_{2}}{\tilde{F}}_{1}\biggl{[}h,b,1+\ell_{i},\frac{D^{2}}{H\,B}\biggl{]}$
(13)
with the abbreviations
$\displaystyle h$ $\displaystyle=$ $\displaystyle
1+s+\frac{1}{2}\>(\ell_{i}+\mathinner{\\!\left\lvert\ell_{s}\right\rvert}),$
$\displaystyle b$ $\displaystyle=$ $\displaystyle
1+i+\frac{1}{2}\>(\ell_{i}+\mathinner{\\!\left\lvert\ell_{i}\right\rvert}),$
$\displaystyle D$ $\displaystyle=$
$\displaystyle-\frac{w_{p}^{2}}{4}-\frac{i}{2k_{p}}(z+z_{p}),$ $\displaystyle
H$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{s}^{2}}{4}-\frac{i}{2}\Bigl{[}\frac{(z+z_{p})}{k_{p}}-\frac{(z+z_{s})}{k_{s}}\Bigl{]},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{w_{p}^{2}}{4}+\frac{w_{i}^{2}}{4}-\frac{i}{2}\Bigl{[}\frac{(z+z_{p})}{k_{p}}-\frac{(z+z_{i})}{k_{i}}\Bigl{]},$
$\displaystyle T^{p,\ell}_{k}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{p+k}(i)^{\ell}}{(p-k)!\>(|\ell|+k)!k!}\>\sqrt{\frac{p!\,(p+|\ell|)!}{\pi}}\,\biggr{(}\frac{w}{\sqrt{2}}\biggl{)}^{2k+|\ell|+1},$
and the regularized hypergeometric function ${{}_{2}}{\tilde{F}}_{1}$ [49].
The collecting widths of the generated signal and idler modes are denoted by
$w_{s}$ and $w_{i}$. The expression (13) gives full insight into the spatial
distribution of the biphoton state decomposed in LG modes and also into the
spatio-temporal coupling in SPDC [50, 51]. Note that the overlap amplitudes
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ from Eq. (13) depend only on $|\ell|$,
where $\ell=\ell_{s}=-\ell_{i}$ (see Ref. [19]), but we keep the notation of
Eq. (13) for a proper illustration of our results.
## II Results and discussion
### II.1 Justifying the choice of the same focal plane positions for signal
and idler, $z_{s}=z_{i}$
In this section, we study the impact of $z_{p}$, $z_{s}$, and $z_{i}$ on the
probability to detect the signal and idler photons in FGMs, or in other words,
the efficiency of direct coupling of generated photons into SMF. In the
following sections, unless otherwise stated, we assume spectrally a
continuous-wave pump such that
$S_{p}(\omega_{s}+\omega_{i})=\delta(\Omega_{s}+\Omega_{i})$ which leads to
the same amount of deviation from the center frequency for signal and idler
photons, $\Omega_{s}=-\Omega_{i}$.
Figure 2: Normalized amplitude of the single-mode coupling for
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$ depending on the pump focal position
$z_{p}$ for a crystal with $L=$30\text{\,}\mathrm{mm}$$ centered at
$z=$0\text{\,}\mathrm{mm}$$. The focal plane positions of signal and idler are
set at $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$. Two scenarios are examined: (a)
$\gamma=\frac{$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ and (b)
$\gamma=\frac{$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$. The full width at half
maximum is independent of the crystal length but determined by the beam width
ratio $\gamma$. As $\gamma$ increases, the influence of the pump focal plane
shifts on the normalized amplitude decrease.
In general, we identify that the coupling efficiency into SMF decreases, when
the pump focal plane is displaced from the crystal center (see Fig. 2). The
amplitude becomes more robust to the change of $z_{p}$ if the beam width ratio
$\gamma=\frac{w_{p}}{w_{s}}$ increases ($w_{s}=w_{i}$ is assumed).
Figure 3: The single-mode coupling efficiency for FGMs in dependence of signal
and idler focal plane shits $z_{s},z_{i}$ for different pump focal plane
shifts $z_{p}$. The setup parameters are
$L=$1\text{\,}\mathrm{mm}$,w_{p}=w_{s}=$10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,w_{i}=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The black star refers to the pair
($z_{s}|z_{i}$) that maximizes the coupling efficiency. (a) The pump has its
focal plane in the center of the crystal, so optimal coupling would imply that
the focal planes of the signal and idler move in opposite directions. (b) and
(c) If $z_{p}\neq$0\text{\,}\mathrm{mm}$$, the optimum amplitudes are reached
for signal and idler focal planes shifted from the center. These findings are
in line with the advanced-wave picture.
In order to increase the amplitude for a given pump focal plane shift $z_{p}$,
we should adjust the signal and idler focal plane positions. The optimal
choice of a pair $z_{s}$ and $z_{i}$ strongly depends on the beam width of
collection modes, $w_{s}$ and $w_{i}$. We distinguish two scenarios of equal
$w_{s}=w_{i}$ or unequal beam widths.
Fig. 3 represents the coupling efficiency for signal photons filtered at
$\lambda_{s}=$810\text{\,}\mathrm{nm}$$ for unequal beam widths $w_{s}\neq
w_{i}$ as a function of $z_{s}$ and $z_{i}$. The calculations have been
carried out for three different pump focal plane positions $z_{p}$. The star
corresponds to the ($z_{s}|z_{i}$) combination that optimizes the single-mode
coupling efficiency. If $z_{p}$ and for instance, $z_{s}$ are fixed in the
experimental setup, the efficiency strongly depends on the focal plane
position of the corresponding partner $z_{i}$. An edge case is visualized in
Fig. 3 (a), where the pump is fixed in the crystal center. The maximum is
attained when the signal focus plane and the idler focus plane displacements
are equal in magnitude but point in opposite directions, i.e. $z_{i}=-z_{s}$.
The initial dependence of the amplitude on $z_{s}$ and $z_{i}$ from (a) is
more distorted and moves away from the plot center the larger $z_{p}$ is.
Moreover, the maximum amplitudes lay not on the diagonal $z_{s}=z_{i}$ and
move further away for increasing $z_{p}$.
However, in experiments $w_{s}=w_{i}$ is more common and we observe more
symmetric dependencies of the single mode coupling efficiency on $z_{s}$ and
$z_{i}$. Fig. 4 shows the same as Fig. 3 but for a constant focal plane
position $z_{p}=$10\text{\,}\mathrm{mm}$$ and the same beam width for signal
and idler $w_{s}=w_{i}$. We distinguish between two different crystal lengths
$L$ and beam width ratios $\gamma$. Since we choose
$z_{p}\neq$0\text{\,}\mathrm{mm}$$, the area of high efficiency lies not
around $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$. This area resembles a circle
laying on the diagonal $z_{s}=z_{i}$. The higher the length or the beam width
ratio, the bigger the red circle which means a larger range of $z_{s},z_{i}$
where the amplitude is constant. This enables fixing the focal planes of
signal and idler at the same location.
Figure 4: Like in Fig. 3 The single-mode coupling efficiency for FGMs is shown
in dependence of signal and idler focal plane shits for a fixed pump focal
plane position $z_{p}=$0\text{\,}\mathrm{mm}$$. We considered
$w_{p}=$20\text{\,}\mathrm{\SIUnitSymbolMicro
m}$/$40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,w_{s}=w_{i}=$20\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The black star refers to the pair
($z_{s}|z_{i}$) that maximizes the coupling efficiency. The high amplitude
areas widen and the exact focal plane position of signal and idler is less
relevant for thicker crystals or higher beam width ratios. Figure 5: Mode
distribution of the biphoton state in LG basis filtered at
$\lambda_{s}=$810\text{\,}\mathrm{mm}$$ for four different arrangements of
focal plane shifts $z_{p}$ and $z_{s}=z_{i}$: (i) pump, signal, and idler
focal planes are assumed to be at $z=$0\text{\,}\mathrm{mm}$$, (ii) only the
pump beam is shifted (by $5\text{\,}\mathrm{mm}$), but signal and idler remain
in the center of the crystal, (iii) pump, signal, and idler are shifted by the
same amount of $5\text{\,}\mathrm{mm}$, (iv) value $z_{s}=z_{i}$ that
maximizes the coincidence amplitude for the given pump shift $z_{p}$. The
upper show a crystal with length $L=$10\text{\,}\mathrm{mm}$$, the lower row
represents $L=$25\text{\,}\mathrm{mm}$$. The mode numbers $(p_{s}|\ell_{s})$
for signal and $(p_{i}|\ell_{i})$ for idler run over $p=0,1,2$ and
$\ell=-2,\dots,2$. The thick bars mark p=0 and the two following thin bars
p=1,2. Each row is normalized by its maximum (corresponding to crystal
length). If the pump is not focused in the center, the signal and idler beams
should be shifted as well, to maximize the coupling efficiency.
Our results show that for a fixed frequency when considering the single-mode
coupling efficiency, $z_{p}=$0\text{\,}\mathrm{mm}$$ not always implies
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ for the highest amplitude. This is shown
in more detail in chapter II.2.2. If the total frequency spectrum on contrary
is considered, $z_{p}=$0\text{\,}\mathrm{mm}$$ always implies
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ for maximal ”spectral brightness”. The
focal planes of all beams should lie in the center of the crystal.
Our findings that focal plane shifts of the pump, signal, or idler beam from
the crystal center affect the efficiency and have to compensate each other is
consistent with the advanced-wave picture (AWP) [29, 40, 52, 53, 54], which
provides a classical analog to understand biphoton coincidence experiments.
To summarize, the choice of pump, signal, and idler focal plane position can
greatly influence the coupling efficiency. When choosing equal collecting
widths for signal and idler $w_{s}=w_{i}$, it is sufficient to assume signal
and idler focal plane positions at the same spot $z_{s}=z_{i}$ for maximum
efficiency.
### II.2 Spatial and temporal characteristics for $z_{s}=z_{i}$
Figure 6: Same as in Fig. 5, but for a crystal with length
$L=$1\text{\,}\mathrm{mm}$$ and different beam width ratios,
$\gamma=\frac{w_{p}}{w_{s,i}}=1$ (upper row) and $\gamma=\sqrt{2}$ (lower
row). When all three beam focal planes are positioned in the middle of the
crystal, the coincidence amplitude for $(p_{s}|\ell_{s})=(0|0)$ and
$(p_{i}|\ell_{i})=(0|0)$ is the highest. The amplitude decreases if the focal
plane of the pump beam is shifted. For this particular pump shift exists a
certain shift for signal and idler $z_{s}=z_{i}\neq$0\text{\,}\mathrm{mm}$$
(d), that maximizes the amplitude and improves the efficiency. The focal plane
positions of signal and idler in c) are distant from the maximizing focal
plane shift. The probability of measuring photon pairs in these modes is very
low.
The subsequent sections address how the pump, signal, and idler focal plane
positions influence the spatio-temporal biphoton state. As we discussed in the
previous section, we can set $z_{s}=z_{i}$ for equal signal and idler beam
widths. This assumption provides for our results 99.99 % agreement compared to
the actual maximizing focal plane shits $z_{s}$ and $z_{i}$. We distinguish
four different combinations of $z_{p}$ relative to $z_{s}=z_{i}$:
1. (i)
The focal plane of the pump, signal, and idler are laying all in the middle of
the crystal, i.e. $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.
2. (ii)
The focal plane of the pump is shifted by a certain amount $z_{p}$
(experimentally perhaps unintentionally and therefore not noticed). However,
signal and idler beams are still positioned at the crystal center,
$z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.
3. (iii)
The pump, signal, and idler focal planes are shifted by the same amount as the
pump in (ii) so that they are focused on the same spot, i.e.
$z_{p}=z_{s}=z_{i}\neq$0\text{\,}\mathrm{mm}$$.
4. (iv)
The pump beam is positioned on the same spot as in (ii) and (iii), but the
focal plane positions of signal and idler are chosen in such a configuration,
that maximizes the amplitude for the Fundamental Gaussian Mode
$|C_{0,0}^{0,0}|^{2}$ for the given $z_{p}$.
In the following, we will probe a ppKTP crystal pumped with a coherent laser
operating at $\lambda_{p}=$405\text{\,}\mathrm{nm}$$. The procedure (i)-(iv)
will be accomplished for different crystal lengths $L$ and beam width ratios
$\gamma$.
#### II.2.1 Influence of $z_{p},z_{s}$, and $z_{i}$ on spatial biphotons
state
Figure 7: Influence of focal plane shifts on the wavelength spectrum for
signal photons. We show crystal configurations with
$L=$1\text{\,}\mathrm{mm}$,$20\text{\,}\mathrm{mm}$$ and
$\gamma=\frac{2}{3},1$. The different focal plane positions of pump, signal,
and idler are illustrated in each plot with different colors. Again, a pump
shift of $z_{p}=$5\text{\,}\mathrm{mm}$$ was chosen. The red curves represent
$z_{s}=z_{i}=z_{s}^{\text{max}}$. The highest spectral brightness is achieved
when the focal planes of pump, signal, and idler are laying in the center of
the crystal. Note that for thinner crystals the spectrum is much wider. It is
also easy to notice that for longer crystals, focal plane shifts have a more
vivid influence on the brightness. The larger the focal plane shift, the
further the maximum from the initial position is shifted.
Fistly we analyze if the spatial DOF of generated photons is affected by the
change of the focal plane positions. We apply the narrowband regime and assume
signal photons filtered at $\lambda_{s}=$810\text{\,}\mathrm{nm}$$. The
normalized coupling efficiency of finding a pair of photons in the Laguerre-
Gaussian modes with radial number $p$ and OAM number $\ell$ are shown in FIGs.
5 and 6. We truncate the infinite space of mode numbers $p$ and $\ell$ to the
subspace of $p_{s,i}\in\\{0,1,2\\}$ and $\ell_{s,i}\in\\{-2,-1,0,1,2\\}$,
where the highest contributions of the overlap amplitudes
$C_{p_{s},p_{i}}^{\ell_{s},\ell_{i}}$ occur. The OAM conservation is easy to
see for both figures since only the modes fulfilling the condition
$\ell_{p}=\ell_{s}+\ell_{i}=0$ are non-zero. The scenarios (i)-(iv) are
depicted in four columns, shown from left to right.
Fig. 5 corresponds to the mode distribution for the crystal lengths
$L=$10\text{\,}\mathrm{mm}$$ and $L=$25\text{\,}\mathrm{mm}$$, where the beam
width ratio is fixed to the value $\gamma=\sqrt{2}$. The pump focal plane
shift according to (ii) is $z_{p}=$5\text{\,}\mathrm{mm}$$. The FGMs
$(p_{s}|\ell_{s})=(p_{i}|\ell_{i})=(0|0)$ have always the largest amplitude in
each frame. Moreover, this amplitude is the highest for the crystal with
$L=$10\text{\,}\mathrm{mm}$$ when all beams are center-focused, see case (i).
The probability decreases when $z_{p}\neq$0\text{\,}\mathrm{mm}$$ (ii).
However, this can be compensated by focal plane shifts of signal and idler. If
the signal and idler modes are shifted to the same position as the pump
according to scenario (iii), the amplitude decreases further. The focal plane
position $z_{s}=z_{i}=$2.17\text{\,}\mathrm{mm}$$ in (iv) maximizes the FGMs
for $z_{p}=$5\text{\,}\mathrm{mm}$$. The ratios of the amplitude in comparison
to (i) are in (ii) 0.89, in (iii) 0.76, and in (iv) 0.94. The more distant the
focal plane positions are from $z_{s}^{\text{max}}$, the lower the efficiency.
The situation is different for the crystal length $L=$20\text{\,}\mathrm{mm}$$
in the second row. The scenario (i)
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ is no longer the best choice for
achieving the highest efficiency. The ratios relative to scenario (i) now are
the following: (ii) 0.89, (iii) 1.03, and (iv) 1.04. The coupling efficiency
is not maximized at $z_{p}=z_{i}=$0\text{\,}\mathrm{mm}$$, since we filtered
the signal at $810\text{\,}\mathrm{nm}$. We will show in the next section that
the coupling into SMF strongly depends on the considered wavelength.
Particularly, we find the optimal value at
$z_{s}=z_{i}=$5.67\text{\,}\mathrm{mm}$$ that maximize the coupling efficiency
for given $z_{p}=$5\text{\,}\mathrm{mm}$$.
Fig. 6 analyzes the same as Fig. 5, but for for different beam width ratio
values $\gamma=1,\sqrt{2}$. The crystal length remains constant at
$L=$1\text{\,}\mathrm{mm}$$. Each column indicates the scenarios (i)-(iv) of
focal plane positions exactly like those described before. We make similar
observations from left to right as in the upper row of Fig. 5: when all focal
planes are in the center, the highest amplitude is achieved. These
probabilities decrease again for a shifted pump. The optimal shift $z_{s,i}$
for a displaced pump focal plane ($z_{p}=$5\text{\,}\mathrm{mm}$$) is close to
the middle of the crystal with $z_{s}=z_{i}=$0.17\text{\,}\mathrm{mm}$$. The
efficiency is significantly reduced for the shifts
$z_{p}=z_{s}=z_{i}=$5\text{\,}\mathrm{mm}$$, so these focal plane positions
are not recommended. Higher beam width ratios $\gamma$ even allow more mode
combinations, which corresponds to an increasing spiral bandwidth [47].
We can conclude, that the single-mode coupling efficiency of the spatial
spectrum is affected by focal plane position. There is always a certain
combination of $z_{p},z_{s}$, and $z_{i}$ that maximizes the efficiency for a
given setup. However, the positioning of all beam focal planes in the center
of the crystal may not be the most effective choice to achieve the peak
amplitude for all frequencies. The optimal choice of $z_{p},z_{s}$, and
$z_{i}$ depends strongly on the filtered frequency.
#### II.2.2 Influence of $z_{p},z_{s}$, and $z_{i}$ on spectral biphotons
state
Figure 8: (a) The maximizing signal/idler focal shifts $z_{s}^{\text{max}}$
for pump focal shifts $z_{p}$ from $-10\text{\,}\mathrm{mm}$ to
$10\text{\,}\mathrm{mm}$. We compare six different beam width ratios $\gamma$
from $\frac{1}{2}$ to $2$. The larger $\gamma$, the more horizontal the
curves. A horizontal curve indicates that signal and idler focal plane should
be positioned in the crystal center for all shifts of the pump focal plane.
(b) For every point $z_{s}^{\text{max}}(z_{p})$ from above in (a) the
corresponding amplitude is shown. The values are normalized to the maximum
amplitude. The amplitudes are comparatively small for larger beam width ratios
$\gamma$, but a shift of the pump focal plane has only a small influence on
the amplitude. (c) Like in a) the dependence $z_{s}^{\text{max}}(z_{p})$ is
shown. Four different crystal lengths from $1\text{\,}\mathrm{mm}$ to
$30\text{\,}\mathrm{mm}$ are compared. The smaller $L$, the more horizontal
the curves. (d) The corresponding amplitude for every point
$z_{s}^{\text{max}}(z_{p})$ from above in (c) is shown. The amplitudes for
larger $L$ are higher, which is only possible when the demanding PM conditions
in longer crystals are properly fulfilled.
Apart from the spatial DOF, we should also analyze the influence of the focal
plane shifts on the spectral DOF of the biphoton state. We consider here the
spectral response of the Fundamental Gaussian Mode
$|C_{0,0}^{0,0}(\Omega)|^{2}$. It is enough to look only at the spectral
response of the signal mode since the spectrum of signal and idler modes are
symmetric with respect to the central frequency due to the continuous-wave
pump, $\Omega_{s}=-\Omega_{i}$.
Fig. 7 shows $|C_{0,0}^{0,0}(\Omega)|^{2}$ for different focal plane positions
and for different combinations of parameters
$L=$1\text{\,}\mathrm{mm}$,$20\text{\,}\mathrm{mm}$$ and
$\gamma=\frac{2}{3},1$. The four colors in each plot represent four different
setups of combinations of $z_{p}=$5\text{\,}\mathrm{mm}$$ and $z_{s}=z_{i}$
according to scenarios (i)-(iv). The spectrum of signal photons is much
broader for the thin crystal regime on the left compared to a thick crystal
shown in the right column of Fig. 7. In terms of focal plane shifts, the blue
curves corresponding to $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ show
always the highest brightness. When the pump focal plane is shifted (green,
red, yellow curves), the magnitude of the corresponding amplitudes changes.
Furthermore, in (d), we readily discernible that shifts of signal and idler
directly shape the frequency spectrum and position of the maximum. The larger
the signal and idler shifts are, the more the maximum is moved away from the
value of the blue curve. We observe in Figs. 7 (b) and (d), that for small
wavelengths the blue curve lies under the green and red curves. This implies
that the focusing of all modes in the middle of the crystal is not preferable
anymore at this frequency. However, when considering the possible highest
brightness, the condition $z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$ always
provides the highest brightness.
### II.3 Generalizing the influence of $z_{p},z_{s}$, and $z_{i}$ on the
spectral brighness
We observed from FIGs. 5-7 that for a given shift
$z_{p}=$5\text{\,}\mathrm{mm}$$, a $z_{s}^{\text{max}}$ exists, which
maximizes the coupling efficiency. In this section, we generalize our results
and consider the dependence $z_{s}^{\text{max}}(z_{p})$ for different beam
width ratios $\gamma$ and crystal lengths $L$. The upper plots of Fig. 8 show
the shift $z_{s}=z_{i}$ for a given $z_{p}$ , that maximizes the spectral
brightness. The normalized amplitude is shown at the bottom of the figures.
This means points that overlap vertically belong to the same $z_{p}$ value,
see the example in Fig. 8 (a) and (b). Note that the maximum spectral
brightness is not always achieved at the same frequency for different focal
plane considerations.
In Fig. 8 (a) we plot the maximizing signal and idler focal plane shifts
$z_{s}^{\text{max}}$ for given pump focal shifts in the range from
$-10\text{\,}\mathrm{mm}$ to $10\text{\,}\mathrm{mm}$ for a crystal of length
${L=$20\text{\,}\mathrm{mm}$}$. Six different values for the beam width ratio
are displayed. The maximizing shift $z_{s}^{\text{max}}$ changes linearly with
$z_{p}$, whereby the slope of the line depends on the beam width ratio. The
higher $\gamma$, the smaller the slope of the lines. Hence signal and idler
focal plane can be positioned in the middle at
$z_{s}=$0\text{\,}\mathrm{mm}$$, regardless of $z_{p}$. The shift of the pump
focal plane has no significant impact.
The spectral brightness is highest for $z_{p}=z_{s}=$0\text{\,}\mathrm{mm}$$
for all $\gamma$ in Fig. 8 (b). Whenever the pump focal plane is shifted away
from the crystal center, the amplitude drops. At high values for $\gamma$, the
$z_{s}^{\text{max}}(z_{p})$ dependence becomes almost constant. The
explanation is that a high beam width ratio results in a less divergent pump
with big width and a more constant beam radius over the length of the crystal.
This means no major noticeable change in the system and therefore less impact
on the yield. The optimum beam width ratio for maximum spectral brightness in
Fig. 8 is $\gamma=\frac{3}{2}$.
It is important to mention, that the photons for every
$z_{s}^{\text{max}}(z_{p})$ value are spectrally filtered at the corresponding
maximum. The frequency that maximizes the spectral brightness lies in a very
small range between $809.88\text{\,}\mathrm{nm}$ and
$809.90\text{\,}\mathrm{nm}$.
Similarly, Fig. 8 (c) and (d) show the maximizing beam shifts for given pump
shifts between $-10\text{\,}\mathrm{mm}$ to $10\text{\,}\mathrm{mm}$ at a
constant beam width ratio
$\gamma=\frac{3}{2}=\frac{$30\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}{$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ displaying four different
crystal lengths. Again, linear lines can be seen in the top chart. The thinner
$L$, the more horizontal these lines are. This corresponds to the expectations
of a thin crystal since the pump beam radius does not change significantly
over the length of the crystal [55, 56]. As a consequence, pump shifts have
almost no effect on the spectral brightness in thin crystals. Pump focal plane
shifts should be taken into account by a proper signal and idler focal plane
positions $z_{s}^{\text{max}}$ in thicker crystals.
### II.4 Spatio-temporal correlations between signal and idler photons
Figure 9: Spectral purity $\text{Tr}(\rho^{2}_{s\text{,SMF}})$ as a function
of the focal plane position of the pump $z_{p}$ and signal beam $z_{s}$ with
the assumption $z_{s}=z_{i}$ and a setup with crystal length
$L=$30\text{\,}\mathrm{mm}$$, beam width ratio $\gamma=\frac{1}{\sqrt{2}}$ and
pulse duration $T_{0}=$0.5\text{\,}\mathrm{ps}$$. The purity reaches its
maximum when all beams are focused in the crystal center,
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$.
In general, we can distinguish two kinds of correlations in the SPDC process:
the correlation between signal and idler photons (see FIGs. 5 and 6) or the
correlation between spatial and spectral DOF of generated photons. The spatio-
spectral correlation implies that the spatial characteristics of signal
(idler) photons can not be considered independently of the spectral DOF, they
are coupled. Mathematically, it means that the biphoton mode function can not
be written as the product state of spatial and spectral DOFs
$\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\neq\Phi_{\bm{q}}(\bm{q}_{s},\bm{q}_{i})\Phi_{\Omega}(\Omega_{s},\Omega_{i})$.
Correspondingly, the correlation between signal and idler photons implies that
the biphoton mode function can not be written as
$\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\neq\Phi_{s}(\bm{q}_{s},\Omega_{s})\Phi_{i}(\bm{q}_{i},\Omega_{i})$.
Here, we quantify explicitly how the focal plane shifts affect both types of
correlations. We consider for this section the Gaussian envelope Eq. (11) for
the spectral DOF of the pump. We look at the purity of the spatial (spectral)
biphoton state [50]
$\displaystyle\text{Tr}(\rho^{2}_{\bm{q}})$ $\displaystyle=\int
d\bm{q}_{s}\>d\Omega_{s}\>d\bm{q}_{i}\>d\Omega_{i}\>d\bm{q}_{s}^{\prime}\>d\Omega_{s}^{\prime}\>d\bm{q}_{i}^{\prime}\>d\Omega_{s}^{\prime}$
$\displaystyle\times\Phi(\bm{q}_{s},\bm{q}_{i},\Omega_{s},\Omega_{i})\>\Phi^{*}(\bm{q}_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{s},\Omega_{i})$
$\displaystyle\times\Phi(\bm{q}_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{s}^{\prime},\Omega_{i}^{\prime})\>\Phi^{*}(\bm{q}_{s},\bm{q}_{i},\Omega_{s}^{\prime},\Omega_{i}^{\prime})$
(14)
as a measure for the correlations between space and frequency DOF. It turns
our that all dependencies of the spatial purity $\text{Tr}(\rho^{2}_{\bm{q}})$
on $z_{p},z_{s}$, and $z_{i}$ cancel out. Since $z_{s}$ and $z_{i}$ are
parameters associated with detection mechanisms, this seems logical.
Therefore, the spatio-spectral correlation can not be manipulated by the beam
shifts $z_{p},z_{s}$, and $z_{i}$.
The situation is different for the purity of the signal (idler) photon [50]
$\displaystyle\text{Tr}(\rho^{2}_{\text{s}})$ $\displaystyle=\int
d\bm{q}_{s}\>d\Omega_{s}\>d\bm{q}_{i}\>d\Omega_{i}\>d\bm{q}_{s}^{\prime}\>d\Omega_{s}^{\prime}\>d\bm{q}_{i}^{\prime}\>d\Omega_{s}^{\prime}$
$\displaystyle\times\Phi(\bm{q}_{s},\Omega_{s},\bm{q}_{i},\Omega_{i})\>\Phi^{*}(\bm{q}_{s}^{\prime},\Omega_{s}^{\prime},\bm{q}_{i},\Omega_{i})$
$\displaystyle\times\Phi(\bm{q}_{s}^{\prime},\Omega_{s}^{\prime},\bm{q}_{i}^{\prime},\Omega_{i}^{\prime})\>\Phi^{*}(\bm{q}_{s},\Omega_{s},\bm{q}_{i}^{\prime},\Omega_{i}^{\prime}),$
(15)
where its dependence on $z_{p},z_{s}$, and $z_{i}$ does not drop off. The
purity (15) gives the strength of the correlation between signal and idler
photons, i.e., how entangled the two photons are. In the last years, one of
the most important goals of photonic quantum technologies has been the
reduction of the correlation between signal and idler photons. The heralded
pure single photons from SPDC are believed to be a good candidate for an
indistinguishable single-photon source [57, 58, 59], which is required for a
successful photonic boson sampling [60]. Usually, the spatial DOF of the
biphoton state is doped off by just collecting the photons into SMF, which
accepts only the Gaussian mode. We can then talk about the spectral purity of
the biphoton state which can be estimated by
$\displaystyle\text{Tr}(\rho^{2}_{\text{s,SMF}})$
$\displaystyle=\int\>d\Omega_{s}\>d\Omega_{i}\>d\Omega_{s}^{\prime}\>d\Omega_{i}^{\prime}\>$
$\displaystyle\times
C^{0,0}_{0,0}(\Omega_{s},\Omega_{i})\>C^{0,0}_{0,0}(\Omega_{s}^{\prime},\Omega_{i}^{\prime})\>$
$\displaystyle\times[C^{0,0}_{0,0}(\Omega_{s}^{\prime},\Omega_{i})]^{*}\>[C^{0,0}_{0,0}(\Omega_{s},\Omega_{i}^{\prime})]^{*}$
(16)
Fig. 9 shows the dependence of the spectral purity
$\text{Tr}(\rho^{2}_{s\text{,SMF}})$ on $z_{p}$ and $z_{s}$ for a crystal
length $L=$30\text{\,}\mathrm{mm}$$ and pulse duration of
$T_{0}=$0.5\text{\,}\mathrm{ps}$$. The combination
$z_{p}=z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$, which corresponds to the crystal
center, yields the maximum purity. Additionally, high levels of purity can
also be observed along the diagonal $z_{p}=z_{s}=z_{i}$.
## III Conclusions
In this work, we have assumed paraxial pump and collecting signal and idler
beams with defined beam widths and focal plane positions. We theoretically
investigated the dependence of spatial and temporal DOFs of the biphoton state
on these focal plane positions. In addition, the single-mode coupling
efficiency and the spectral brightness of Fundamental Gaussian Modes were
studied. Generally, the spectral brightness reaches the maximum if all
involved beams are positioned in the center of the crystal. The single-mode
coupling efficiency strongly depends on the frequency: for certain narrow-band
filtered frequencies, positioning all focal planes in the crystal center would
not attain the highest efficiency.
Depending on the setup, small deviations of the focal plane positions from the
crystal center can have a large impact on the coupling efficiency. In sense of
the advanced-wave picture, pump, signal, and idler focal plane shifts must
compensate each other for higher efficiency. However, equal positioning of
signal and idler focal planes is sufficient in most setups. Thus we advise
choosing a suitable signal and idler focal plane position
$z_{s}=z_{i}=z_{s}^{\text{max}}$ for higher efficiency if the pump beam is not
center-focused. Regardless of $z_{p}$, $z_{s}=z_{i}=$0\text{\,}\mathrm{mm}$$
can be assumed for high beam width ratios $\gamma$ or short lengths $L$, see
results in section II.3.
We also find that correlations between space and frequency degrees of freedom
are not affected by focal plane shifts. In contrast, the entanglement between
signal and idler photons depends on the focal plane positions of involved
beams. We recommend placing all focal planes of SPDC beams in the center of
the crystal to achieve the highest purity.
## IV Acknowledgment
The authors thank Carlos Sevilla-Gutiérrez for very helpful discussions and
suggestions.
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|
# Some remarks on semantics and expressiveness of the Sentential Calculus with
Identity
Steffen Lewitzka Universidade Federal da Bahia – UFBA, Departamento de Ciência
da Computação, Instituto de Computação, 40170-110 Salvador BA, Brazil,
<EMAIL_ADDRESS>
###### Abstract
Suszko’s Sentential Calculus with Identity $\mathit{SCI}$ results from
classical propositional calculus $\mathit{CPC}$ by adding a new connective
$\equiv$ and axioms for identity $\varphi\equiv\psi$ (which we interpret here
as ‘propositional identity’). We reformulate the original semantics of
$\mathit{SCI}$ in terms of Boolean prealgebras establishing a connection to
‘hyperintensional semantics’. Furthermore, we define a general framework of
dualities between certain $\mathit{SCI}$-theories and Lewis-style modal
systems in the vicinity of $\mathit{S3}$. Suszko’s original approach to two
$\mathit{SCI}$-theories corresponding to $\mathit{S4}$ and $\mathit{S5}$ can
be formulated as a special case. All these dualities rely particularly on the
fact that Lewis’ ‘strict equivalence’ is axiomatized by the
$\mathit{SCI}$-principles of ‘propositional identity’.
Keywords: non-Fregean logic, Boolean prealgebra, hyperintensional semantics,
modal logic
## 1 Introduction
The set $Fm_{\equiv}$ of formulas of the Sentential Calculus with Identity
$\mathit{SCI}$ is inductively defined in the usual way over an infinite set
$V$ of propositional variables $x_{0},x_{1},...$, logical connectives $\bot$,
$\top$, $\neg$, $\vee$, $\wedge$, $\rightarrow$ and an identity connective
$\equiv$ for building formulas of the form $(\varphi\equiv\psi)$. As a
deductive system, $\mathit{SCI}$ extends classical propositional logic
$\mathit{CPC}$ by the identity axioms (id1)–(id7) below. That is,
$\mathit{SCI}$ can be axiomatized by all formulas having the form of a
classical tautology together with the following identity axioms:
(id1) $\varphi\equiv\varphi$
(id2) $(\varphi\equiv\psi)\rightarrow(\varphi\rightarrow\psi)$
(id3) $(\varphi\equiv\psi)\rightarrow(\neg\varphi\equiv\neg\psi)$
(id4)–(id7)
$((\varphi_{1}\equiv\psi_{1})\wedge(\varphi_{2}\equiv\psi_{2}))\rightarrow((\varphi_{1}*\varphi_{2})\equiv(\psi_{1}*\psi_{2}))$,
where $*\in\\{\vee,\wedge,\rightarrow,\equiv\\}$, respectively.
With Modus Ponens MP as inference rule, the notion of derivation is defined in
the usual way. We write $\varPhi\vdash_{\mathit{SCI}}\varphi$ if there is a
derivation of $\varphi\in Fm_{\equiv}$ from the set $\varPhi\subseteq
Fm_{\equiv}$. The introduction of $\mathit{SCI}$ is a consequence of R.
Suszko’s work on non-Fregean logics which, in turn, was motivated by his
attempts to formalize ontological aspects of Wittgenstein’s Tractatus logico-
philosophicus (see, e.g. [14]). Recall that, according to G. Frege, the
denotation (referent, Bedeutung) of a formula is nothing but a truth-value.
This principle, called by Suszko the Fregean Axiom, can be formalized as
$(\varphi\leftrightarrow\psi)\rightarrow(\varphi\equiv\psi)$ if we assume the
classical interpretation of connectives and read $\varphi\equiv\psi$ as
‘$\varphi$ and $\psi$ have the same denotation’. The essential feature of a
non-Fregean logic is the failure of Fregean Axiom. $\mathit{SCI}$ can be seen
as a basic non-Fregean logic extending $\mathit{CPC}$. The identity axioms
express our basic intuition on propositional identity: it should be a
congruence relation on formulas that refines equivalence
$\leftrightarrow$.111Indeed,
$(\varphi\equiv\psi)\rightarrow(\varphi\leftrightarrow\psi)$ as well as
$(\varphi\equiv\psi)\rightarrow(\psi\equiv\varphi)$ and
$((\varphi\equiv\psi)\wedge(\psi\equiv\chi))\rightarrow(\varphi\equiv\chi)$
are derivable. The ‘compatibility’ with connectives of the language is
expressed by axioms (id3) and (id4)–(id7). As already pointed out in [2],
replacing (id3)–(id7) by the single scheme
(1) $(\varphi\equiv\psi)\rightarrow(\chi[x:=\varphi]\equiv\chi[x:=\psi]),$
which we call the Substitution Principle SP, results in a deductively
equivalent system.333This fact can be shown by induction on $\chi$. SP
essentially says that formulas with the same denotation can be replaced by
each other in any context. This principle can be seen as a particular instance
of a general ontological law known in the literature as the indiscernibility
of identicals or Leibniz’s law. In a formal context, SP represents a necessary
condition for the existence of a natural propositional semantics. In fact, if
we interpret logical connectives and further operators of the object language
semantically as functions on propositions, then SP says that all these
functions are well-defined: identical arguments yield identical function
values. For instance, SP holds in classical and intuitionistic propositional
logic with propositional identity $\varphi\equiv\psi$ given as equivalence
$\varphi\leftrightarrow\psi$. If we assume the propositional modal language
and define propositional identity as strict equivalence:
$(\varphi\equiv\psi):=\square(\varphi\leftrightarrow\psi)$, then SP is a
derivable principle in Lewis modal systems $\mathit{S3}$–$\mathit{S5}$ but not
in the weaker systems $\mathit{S1}$ and $\mathit{S2}$, cf. [9, 11]. However,
it is enough to add SP to system $\mathit{S1}$ in order to get a logic with a
natural algebraic semantics. This logic was introduced in [9] under the name
$\mathit{S1}$+$\mathit{SP}$. In the present paper, we shall refer to it by the
simpler label $\mathit{S1SP}$. We then get the hierarchy
$\mathit{S1SP}\subsetneq\mathit{S3}\subsetneq\mathit{S4}\subsetneq\mathit{S5}$
of Lewis (-style) modal logics for which we can use the same framework of
algebraic semantics based on Boolean algebras (we shall explore this kind of
semantics in section 4).
The interpretability of $\mathit{SCI}$ in Lewis system $\mathit{S3}$ indicates
a strong connection between $\mathit{SCI}$-theories and Lewis-style modal
systems. Essential aspects of that connection were already revealed by Suszko,
Bloom [13, 2] showing that specific extensions of $\mathit{SCI}$ correspond,
in some sense, to modal logics $\mathit{S4}$ and $\mathit{S5}$, respectively.
Instead of interpreting $\mathit{SCI}$-theories in Lewis modal systems,
Suszko’s approach restores modal logic within $\mathit{SCI}$-extensions via
the definition $\square\varphi:=(\varphi\equiv\top)$.
In the present paper, we study dualities between $\mathit{SCI}$-theories and
Lewis-style modal systems (not restricted to $\mathit{S4}$ and $\mathit{S5}$)
in a systematical way and establish precise criteria for the existence of such
dualities. We consider here both object languages separately – the language of
$\mathit{SCI}$ versus the language of propositional modal logic – and define
appropriate translations between them. In contrast to the original model-
theoretic approach (cf. [1, 2]), we introduce $\mathit{SCI}$-models explicitly
as Boolean prealgebras (or Boolean prelattices). In this way, we find a bridge
to an approach known in the literature as ‘hyperintensional semantics’ (see,
e.g., [4, 12]) and present $\mathit{SCI}$ as a basic classical logic for
(hyper-) intensional modeling and reasoning.
## 2 Intensionality as a measure for the discernibility of propositions
Originally introduced by M. J. Cresswell [3], the notion of
‘hyperintensionality’ has been interpreted in different ways in the literature
and there seems to be no formal standard definition. Usually, an operator (of
a given logic) is regarded as extensional if its application to formulas with
the same truth-value results again in formulas having the same truth-value,
otherwise the operator may be seen as intensional.444We consider here only
classical logics. In the context of possible worlds semantics, an operator is
often regarded as hyperintensional if its application to formulas having the
same truth-values at all possible (accessible) worlds does not necessarily
result in formulas with the same truth-value at the actual world. For
instance, the modal operator of normal modal logics is intensional (but not
hyperintensional). Therefore, modal logics are often regarded as intensional
logics. Possible worlds semantics, however, is not an appropriate framework
for dealing with hyperintensional operators. There are proposals in the
literature conceiving hyperintensional semantics in terms of Boolean
prelattices (see, e.g., [4, 12, 11]), and we will follow a similar approach.
For this purpose, let us regard a proposition as the denotation of a formula
at a given model. A proposition can be, e.g., a truth-value (in classical
propositional logic), a set of possible worlds (in normal modal logics), an
element of some algebraic structure, etc. Under these assumptions, we propose
to explain intensionality as a measure of discernibility of propositions. The
more propositions can be distinguished in models of the underlying classical
logic the higher the degree of intensionality. In $\mathit{CPC}$, only two
propositions, the True and the False, can be distinguished. Current modal
logics provide much more (infinitely many) propositions: even if two formulas
$\varphi$ and $\psi$ have the same truth-value at the actual world, they may
have different truth-values at some accessible world and thus denote different
propositions: the Fregean Axiom does not hold – the denotation of a formula is
more than a classical truth-value. Nevertheless, many propositions remain
indiscernible: logically equivalent formulas such as $\neg\neg\varphi$ and
$\varphi$ will always denote the same proposition in classical modal logics.
The aim of hyperintensional semantics is to overcome such limitations of the
possible worlds framework (motivations come, e.g., from the study of natural
language semantics) and to provide a more fine-grained approach that allows to
discern even more propositions. This goal can be perfectly achieved working
with $\mathit{SCI}$ and appropriate axiomatic extensions. By a proposition we
will mean more specifically the element of a given $\mathit{SCI}$-model. The
degree of intensionality of a model is the largest number of propositions that
can be distinguished. We shall see that all expressible intensions can be
discerned in logic $\mathit{SCI}$. In fact, there is an $\mathit{SCI}$-model
where any two different formulas denote different propositions, see Theorem
3.12 below. We call such a model intensional since the denotation of a formula
can be identified with its intension, i.e. its syntactical form. In this
sense, $\mathit{SCI}$ is a logic of highest degree of intensionality and, of
course, is able to model hyperintensional operators. Imposing appropriate
axioms, we get specific $\mathit{SCI}$-theories where specific propositions
become indiscernible. In particular, $\mathit{CPC}$ as well as some Lewis-
style modal logics can be represented as specific $\mathit{SCI}$-theories.
While models of $\mathit{CPC}$ are extensional, models of modal logics lie
somewhere between the extremes of extensional and intensional model. In the
following, we will present $\mathit{SCI}$ as an (hiper-) intensional
logic.555In contrast to our view, Bloom and Suszko explicitly deny the
intensional character of $\mathit{SCI}$. “Some people, upon discovering that
the identity connective was not truth-functional, have thought that
$\mathit{SCI}$ is an intensional logic. We emphatically deny this. The essence
of intensionality is that the rule ”equals may be replaced by equals” fails.
However, this rule does hold in the SCI … ” (cf. p. 1 of [2]). Actually, that
rule is formalized by SP which is valid in $\mathit{SCI}$.
## 3 Boolean prealgebras as models of $\mathit{SCI}$
Recall that a preorder on a set $M$ is a binary relation on $M$ satisfying
reflexivity and transitivity. If a preorder is also antisymmetic, then it is a
partial order. We also expect the reader to be familiar with the concepts of
Boolean algebra, filters and ultrafilters (on Boolean algebras) and quotient
Boolean algebras. We apply a somewhat unusual notation for Boolean algebras
(with operators) which has the advantage that for any new connective or symbol
of the underlying object language a corresponding operator for the algebraic
semantics can easily be identified. In particular, for the connectives
$\vee,\wedge,\neg,\bot,\top,\rightarrow$ of our classical logic, we denote the
corresponding operations of a given Boolean (pre-) algebra $\mathcal{B}$ by
$f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow}$ (or, more
precisely, by $f_{\vee}^{\mathcal{B}}$, etc., if we wish to emphasize the
given context of (pre-) algebra $\mathcal{B}$).
###### Definition 3.1.
A structure
$\mathcal{B}=(B,f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},\preceq)$
of type $(2,2,1,0,0,2)$ with a preorder $\preceq$ on universe $B$ is a Boolean
prealgebra if the relation $\approx$ defined by $a\approx b$
$:\Leftrightarrow$ ($a\preceq b$ and $b\preceq a$) is a congruence relation on
$B$ such that the quotient $\mathcal{B}/{\approx}$ is a Boolean algebra, and
for all $a,b\in B$ we have: $a\preceq b\Leftrightarrow f_{\wedge}(a,b)\approx
a$. In this case, we call $\approx$ the associated congruence, and we call
quotient $\mathcal{B}/{\approx}$ the associated Boolean algebra. If the given
structure $\mathcal{B}$ itself is a Boolean algebra, then we denote the
underlying lattice order by $\leq$.666Even if $\mathcal{B}$ is a Boolean
algebra, the lattice order $\leq$ may differ from the given preorder
$\preceq$.
Of course, every Boolean algebra together with its lattice order (regarded as
a preorder) is trivially a Boolean prealgebra. Recall that every Boolean
algebra is a Heyting algebra. Those Heyting algebras which are not Boolean
algebras are non-trivial though natural examples of Boolean prealgebras. In
order to see this, consider any designated ultrafilter $U$ of a given Heyting
algebra (which exists by Zorn’s Lemma) and the preorder $a\preceq b$
$:\Leftrightarrow f_{\rightarrow}(a,b)\in U$, where $f_{\rightarrow}(a,b)$ is
the relative pseudo-complement of $a$ w.r.t. $b$. Then the resulting quotient
algebra modulo $\approx$ is the two-element Boolean algebra.777Of course,
there may exist further congruence relations on a given Heyting algebra that
result in a Boolean quotient algebra. Considering the intuitionistic tautology
$(x\rightarrow y)\leftrightarrow(x\rightarrow(x\wedge y))$, one easily checks
that also the condition $a\preceq b\Leftrightarrow f_{\wedge}(a,b)\approx a$
holds for all elements $a,b$ of the Heyting algebra.888Recall that all
intuitionistic tautologies are interpreted by the top element $f_{\top}$ of
any Heyting algebra under any assignment, and also recall that the following
condition is valid in every Heyting algebra: $f_{\rightarrow}(a,b)=f_{\top}$
iff $a\leq b$.
Note that we cannot do without that second condition in Definition 3.1. Even
if the resulting quotient $\mathcal{B}/{\approx}$ of structure $\mathcal{B}$
is a Boolean algebra, condition $a\preceq b\Leftrightarrow
f_{\wedge}(a,b)\approx a$ is not necessarily true. Consider, for instance, the
$4$-element Boolean algebra $Pow(2)$ with the preorder $\preceq$ given by set-
theoretic inclusion on $Pow(2)$ extended by the tuple $(\\{1\\},\\{2\\})$, so
we have in particular $\\{1\\}\preceq\\{2\\}$. Relation $\approx$ is the
identity on $Pow(2)$ and the resulting quotient algebra is, of course, again
the Boolean algebra $Pow(2)$. However, the second condition of Definition 3.1
fails since we have $\\{1\\}\preceq\\{2\\}$, but $\\{1\\}\subsetneq\\{2\\}$,
i.e. $\\{1\\}\cap\\{2\\}\neq\\{1\\}$.
If one deals with Boolean algebras, then one usually considers only the
operations of supremum (join) $f_{\vee}$, infimum (meet) $f_{\wedge}$,
complement $f_{\neg}$, least element $f_{\bot}$ and greatest element
$f_{\top}$. Further relevant operations, such as implication
$f_{\rightarrow}(a,b):=f_{\vee}(f_{\neg}(a),b)$, are definable. This, however,
does not hold in general for Boolean prealgebras. For instance, although we
have $f_{\rightarrow}(a,b)\approx f_{\vee}(f_{\neg}(a),b)$, the propositions
(i.e. elements) $f_{\rightarrow}(a,b)$ and $f_{\vee}(f_{\neg}(a),b)$ may be
distinct.
###### Lemma 3.2.
Let $\mathcal{B}$ be a Boolean prealgebra with preorder $\preceq$, and let
$\mathcal{B}/{\approx}$ be the associated Boolean algebra with lattice order
$\leq^{\mathcal{B}/\approx}$. Then for all $a,b\in B$: $a\preceq
b\Leftrightarrow\overline{a}\leq^{\mathcal{B}/\approx}\overline{b}$.
###### Proof.
Let $\mathcal{B}$ be a Boolean prealgebra with preorder $\preceq$. Then for
all $a,b\in B$, $a\preceq b\Leftrightarrow
f^{\mathcal{B}}_{\wedge}(a,b)\approx a\Leftrightarrow
f_{\wedge}^{\mathcal{B}/\approx}(\overline{a},\overline{b})=\overline{a}\Leftrightarrow\overline{a}\leq^{\mathcal{B}/\approx}\overline{b}$.
∎
###### Definition 3.3.
Let $\mathcal{B}$ be a Boolean prealgebra with associated Boolean algebra
$\mathcal{B}/{\approx}$, and let $F\subseteq B$ be closed under $\approx$,
i.e. $a\in F\Leftrightarrow b\in F$ whenever $a\approx b$, for any $a,b\in B$.
Then we say that $F$ is a filter of $\mathit{B}$ if the set
$F^{\mathcal{B}/{\approx}}=\\{\overline{a}\mid a\in F\\}$ is a filter (in the
usual sense) of Boolean algebra $\mathcal{B}/{\approx}$. The notions of proper
filter and ultrafilter of a Boolean prealgebra are defined analogously.
###### Corollary 3.4.
Let $\mathcal{B}$ be a Boolean prealgebra. A subset $F$ is a filter of
$\mathcal{B}$ if and only if the following conditions are satisfied for all
$a,b\in B$:
* •
If $a,b\in F$, then $f_{\wedge}(a,b)\in F$.
* •
If $a\in F$ and $a\preceq b$, then $b\in F$.
A filter $F$ is a proper filter iff $F\neq B$ iff $f_{\bot}\notin F$. A filter
$F$ is an ultrafilter iff $F$ is maximal among all proper filters.
###### Corollary 3.5.
Let $\mathcal{B}$ be a Boolean prealgebra. If $\mathcal{B}$ is itself a
Boolean algebra, then its lattice order $\leq$ refines the given preorder
$\preceq$, i.e., for all $a,b\in B$: $a\leq b$ implies $a\preceq b$.
###### Proof.
Suppose $\mathcal{B}$ is a Boolean algebra. Then for any $a,b\in B$: $a\leq b$
$\Leftrightarrow$ $a=f^{\mathcal{B}}_{\wedge}(a,b)$ $\Rightarrow$
$\overline{a}=f^{\mathcal{B}/{\approx}}_{\wedge}(\overline{a},\overline{b})$
$\Leftrightarrow$ $\overline{a}\leq^{\mathcal{B}/{\approx}}\overline{b}$
$\Leftrightarrow$ $a\preceq b$, where the last step follows from Lemma 3.2. ∎
###### Definition 3.6.
An $\mathit{SCI}$-model $\mathcal{M}$ is a structure
$\mathcal{M}=(M,\mathit{TRUE},f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
where
$(M,f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},\preceq)$
is a Boolean prealgebra, $\mathit{TRUE}\subseteq M$ is a designated
ultrafilter and $f_{\equiv}$ is an additional binary function satisfying for
all $m,m^{\prime}\in M$:
$f_{\equiv}(m,m^{\prime})\in\mathit{TRUE}\Leftrightarrow m=m^{\prime}$. The
elements of the universe $M$ are called propositions, and $\mathit{TRUE}$ is
the designated set of true propositions.
An assignment (or valuation) of an $\mathit{SCI}$-model $\mathcal{M}$ is a
function $\gamma\colon V\rightarrow M$. Any assignment $\gamma$ extends in the
canonical way to a function from $Fm_{\equiv}$ to $M$ which we again denote by
$\gamma$. More precisely, we have $\gamma(\bot)=f_{\bot}$,
$\gamma(\top)=f_{\top}$, and
$\gamma(\varphi*\psi)=f_{*}(\gamma(\varphi),\gamma(\psi))$ for
$*\in\\{\vee,\wedge,\rightarrow,\equiv\\}$.
###### Definition 3.7.
If $\mathcal{M}$ is an $\mathit{SCI}$-model and $\gamma$ is an assignment of
$\mathcal{M}$, then we call the tuple $(\mathcal{M},\gamma)$ an
$\mathit{SCI}$-interpretation. The satisfaction relation between
interpretations and formulas is defined as follows:
$(\mathcal{M},\gamma)\vDash\varphi:\Leftrightarrow\gamma(\varphi)\in\mathit{TRUE}$
If $(\mathcal{M},\gamma)\vDash\varphi$ for all assignments $\gamma\in M^{V}$,
then we write $\mathcal{M}\vDash\varphi$ and say that $\mathcal{M}$ validates
$\varphi$ (or $\varphi$ is valid in $\mathcal{M}$). For $\varPhi\subseteq
Fm_{\equiv}$, we define as usual
$(\mathcal{M},\gamma)\vDash\varPhi:\Leftrightarrow(\mathcal{M},\gamma)\vDash\varphi\text{
for all }\varphi\in\varPhi$. The relation of logical consequence is defined in
the standard way for any set $\varPhi\cup\\{\varphi\\}\subseteq Fm_{\equiv}$:
$\varPhi\Vdash_{\mathit{SCI}}\varphi:\Leftrightarrow Mod(\varPhi)\subseteq
Mod(\\{\varphi\\})$, where for any $\varPsi\subseteq Fm_{\equiv}$,
$Mod(\varPsi)$ is the class of all $\mathit{SCI}$-interpretations satisfying
$\varPsi$.
###### Corollary 3.8.
The connective of propositional identity has the intended meaning, i.e. for
any interpretation $(\mathcal{M},\gamma)$ and any $\varphi,\psi\in
Fm_{\equiv}$: $(\mathcal{M},\gamma)\vDash\varphi\equiv\psi$ iff
$\gamma(\varphi)=\gamma(\psi)$ iff $\varphi$ and $\psi$ denote the same
proposition in $(\mathcal{M},\gamma)$.
###### Proof.
$(\mathcal{M},\gamma)\vDash\varphi\equiv\psi$ iff
$\gamma(\varphi\equiv\psi)\in\mathit{TRUE}$ iff
$f_{\equiv}(\gamma(\varphi),\gamma(\psi))\in\mathit{TRUE}$ iff
$\gamma(\varphi)=\gamma(\psi)$. ∎
In [2], the authors consider only the logical connectives $\neg$ and
$\rightarrow$, and consequently define an $\mathit{SCI}$-model as a structure
$\mathcal{A}=(A,f_{\neg},f_{\rightarrow},f_{\equiv})$ that satisfies certain
conditions according to [Definition 1.6 [2]] (we use here our specific
notation for the semantic operations $f_{\neg},f_{\rightarrow},f_{\equiv}$ in
order to keep the presentation consistent). By the following result, that
original definition is essentially equivalent to our Definition 3.6 of
$\mathit{SCI}$-model presented above. This is not obvious since both
definitions are formulated in very different ways. In particular, the original
definition given in [2] hides the prelattice structure which is an explicit
part of our concept of $\mathit{SCI}$-model.
###### Theorem 3.9 (Equivalence of the two semantics).
Our semantics based on Boolean prealgebras is equivalent to original semantics
of $\mathit{SCI}$ in the following sense. Let
$\mathcal{M}=(M,\mathit{TRUE},f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
be an $\mathit{SCI}$-model according to Definition 3.6. Then the pair
$<\mathcal{A},\mathit{TRUE}>$, where
$\mathcal{A}=(M,f_{\neg},f_{\rightarrow},f_{\equiv})$, is a model of
$\mathit{SCI}$ according to [Definition 1.6 [2]]. On the other hand, if
$<\mathcal{A},B>$, with $\mathcal{A}=(A,f_{\neg},f_{\rightarrow},f_{\equiv})$,
is a model according to [Definition 1.6 [2]], then
$\mathcal{M}=(M,B,f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
is an $\mathit{SCI}$-model in our sense, where $a\preceq b\Leftrightarrow
f_{\rightarrow}(a,b)\in B$, and the additional operations
$f_{\vee},f_{\wedge},f_{\bot},f_{\top}$ can be defined by the usual Boolean
equations (e.g. $f_{\top}:=f_{\rightarrow}(a,a)$ for some fixed $a\in A$,
etc.).
###### Proof.
If
$\mathcal{M}=(M,\mathit{TRUE},f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
is an $\mathit{SCI}$-model in our sense, then, using the terminology of
[Definition 1.6 [2]], the set $\mathit{TRUE}$ is clearly closed, proper, prime
and normal. Since $\mathcal{M}$ is based on a Boolean prealgebra, we have
$f_{\top}\preceq h(\varphi)\in\mathit{TRUE}$ for any classical propositional
tautology $\varphi$ and any valuation (assignment) $h$ of $\mathcal{M}$. Since
$f_{\equiv}(a,b)\in\mathit{TRUE}\Leftrightarrow a=b$, also the identity axioms
of $\mathit{SCI}$ are all interpreted by elements of $\mathit{TRUE}$ under any
assignment $h$. Thus, $\mathit{TRUE}$ is also admissible and therefore a
prime, normal filter according to [Definition 1.6 [2]]. Thus, the reduct
$\mathcal{A}=(M,f_{\neg},f_{\rightarrow},f_{\equiv})$ along with prime, normal
filter $\mathit{TRUE}\subseteq M$ yields an $\mathit{SCI}$-model in the
original sense. Now let us suppose $<\mathcal{A},B>$, with
$\mathcal{A}=(A,f_{\neg},f_{\rightarrow},f_{\equiv})$, is a model in the sense
of [2]. We have to extract from that concept a preorder $\preceq$ that yields
a prelattice and the desired $\mathit{SCI}$-model in the sense of Definition
3.6 above. For elements $a,b\in A$, we define $a\preceq b:\Leftrightarrow$
$f_{\rightarrow}(a,b)\in B$. Since $B$ is a prime, normal filter (in the
terminology of [Definition 1.6 [2]], $B$ is, in a sense, deductively closed
(i.e. if $h(\varPhi)\subseteq B$ and $\varPhi\vdash_{\mathit{SCI}}\varphi$,
then $h(\varphi)\in B$, for any set $\varPhi\cup\\{\varphi\\}$ of formulas and
any valuation $h$ of $\mathcal{A}$). It follows that $\preceq$ is a preorder
on $A$, and $a\approx b:\Leftrightarrow$ ($a\preceq b$ and $b\preceq a$)
defines a congruence relation of the structure $(A,f_{\neg},f_{\rightarrow})$.
In particular, for any propositional formulas $\varphi,\psi$ (without identity
connective), if $\varphi\leftrightarrow\psi$ is a theorem of $\mathit{CPC}$,
then $h(\varphi)\approx h(\psi)$ under any valuation $h$. Thus, all Boolean
equations are valid in the quotient structure of
$(A,f_{\neg},f_{\rightarrow})$ modulo $\approx$, and that quotient structure
must be a Boolean algebra (actually, it is the two-element Boolean algebra).
We may define additional Boolean operations, such as $f_{\wedge}$ … , in the
obvious way. Furthermore, one easily verifies that the equivalence $a\preceq
b\Leftrightarrow f_{\wedge}(a,b)\approx a$ is valid. Thus,
$(A,f_{\neg},f_{\rightarrow},\preceq)$ is a Boolean prealgebra. Finally, the
equivalence $f_{\equiv}(a,b)\in B\Leftrightarrow a=b$ is warranted by the fact
that $B$ is normal (in the sense of [Definition 1.6 [2]]). Thus,
$\mathcal{M}=(A,B,f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
is an $\mathit{SCI}$-model according to Definition 3.6 above. ∎
One easily verifies that any $\mathit{SCI}$-interpretation (in our sense)
satisfies the axioms of $\mathit{SCI}$. Completeness of $\mathit{SCI}$ w.r.t.
our semantics follows from the original completeness theorem of $\mathit{SCI}$
(see, e.g. [2]) together with Theorem 3.9. Nevertheless, we will sketch out in
the following an independent proof. Suppose $\varPhi$ is a set of formulas
which is consistent in $\mathit{SCI}$. By Zorn’s Lemma, there is an extension
$\varPsi\supseteq\varPhi$ which is maximal consistent in logic $\mathit{SCI}$.
By the axioms of propositional identity, the relation $\cong$ defined by
$\varphi\cong\psi:\Leftrightarrow(\varphi\equiv\psi)\in\varPsi$
is a congruence relation on $Fm_{\equiv}$ (symmetry, transitivity and
compatibility with operations follow from applications of (1), i.e. the
Substitution Property SP). Moreover, by (id2), $\varphi\cong\psi$ implies:
$\varphi\in\varPsi\Leftrightarrow\psi\in\varPsi$. For $\varphi\in
Fm_{\equiv}$, let $[\varphi]$ be the congruence class of $\varphi$ modulo
$\cong$. Then we put $M:=\\{[\varphi]\mid\varphi\in Fm_{\equiv}\\}$,
$\mathit{TRUE}:=\\{[\varphi]\mid\varphi\in\varPsi\\}$ and define operations
$f_{\neg}([\varphi]):=[\neg\varphi]$,
$f_{*}([\varphi],[\psi]):=[\varphi*\psi]$, for
$*\in\\{\vee,\wedge,\rightarrow,\equiv\\}$, and $f_{\bot}:=[\bot]$,
$f_{\top}:=[\top]$. The relation $\preceq$ on $M$ defined by
$[\varphi]\preceq[\psi]:\Leftrightarrow\varphi\rightarrow\psi\in\varPsi$
is a preorder on $M$. By SP, $\preceq$ is well-defined. Next we show that the
structure
$\mathcal{M^{\prime}}=(M,f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},\preceq)$
is a Boolean prealgebra. The relation $\approx$ given by
$[\varphi]\approx[\psi]:\Leftrightarrow\varphi\leftrightarrow\psi\in\varPsi\Leftrightarrow([\varphi]\preceq[\psi]\text{
and }[\psi]\preceq[\varphi])$
is obviously a congruence relation of $\mathcal{M^{\prime}}$. Since $\varPsi$
is maximal consistent, it contains in particular all equivalences
$\varphi\leftrightarrow\psi$ which are valid in $\mathit{CPC}$. These
equivalences axiomatize as equations ‘$\varphi=\psi$’ the class of Boolean
algebras. It follows that the quotient of $\mathcal{M^{\prime}}$ modulo
$\approx$ is a Boolean algebra whose elements are the congruence classes of
the elements $[\varphi]\in M$ modulo $\approx$. Moreover, for any elements
$[\varphi],[\psi]$ we have: $[\varphi]\preceq[\psi]$ iff
$\varphi\rightarrow\psi\in\varPsi$ iff
$(\varphi\wedge\psi)\leftrightarrow\varphi\in\varPsi$ iff
$f_{\wedge}([\varphi],[\psi])\approx[\varphi]$. Hence, $\mathcal{M}^{\prime}$
is a Boolean prealgebra in accordance with Definition
3.1.999$\mathcal{M^{\prime}}$ is not necessarily a Boolean algebra. For
example,
$f_{\vee}([\varphi],[\psi])=[\varphi\vee\psi]\neq[\psi\vee\varphi]=f_{\vee}([\psi],[\varphi])$
is possible. Even if $\mathcal{M^{\prime}}$ is a Boolean algebra, the preorder
$\preceq$ may be strictly coarser than the underlying lattice order (cf. Lemma
3.5). In fact, $\preceq$ is the lattice order iff $\varPsi$ contains all
instances of the Fregean Axiom
$(\varphi\equiv\psi)\leftrightarrow(\varphi\leftrightarrow\psi)$. By
construction, we have for any elements $[\varphi]$, $[\psi]$:
$[\varphi]=[\psi]$ iff $\varphi\cong\psi$ iff $\varphi\equiv\psi\in\varPsi$
iff $[\varphi\equiv\psi]=f_{\equiv}([\varphi],[\psi])\in\mathit{TRUE}$. Thus,
$\mathcal{M}:=(M,\mathit{TRUE},f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\equiv},\preceq)$
is an $\mathit{SCI}$-model. We consider the assignment $\gamma\in M^{V}$
defined by $x\mapsto[x]$. By induction on formulas, it follows that
$\gamma(\varphi)=[\varphi]$. Then we have
$(\mathcal{M},\gamma)\vDash\varphi\Leftrightarrow\gamma(\varphi)=[\varphi]\in\mathit{TRUE}\Leftrightarrow\varphi\in\varPsi.$
In particular, $(\mathcal{M},\gamma)\vDash\varPhi$ and whence $\varPhi$ is
satisfiable. We have proved soundness and completeness of $\mathit{SCI}$
w.r.t. the semantics given by the class of $\mathit{SCI}$-models.
###### Theorem 3.10 (Soundness and Completeness).
For any set $\varPhi\cup\\{\varphi\\}\subseteq Fm_{\equiv}$, the following
holds:
$\varPhi\Vdash_{\mathit{SCI}}\varphi\Leftrightarrow\varPhi\vdash_{\mathit{SCI}}\varphi$.
Classical propositional logic $\mathit{CPC}$ is extensional in the sense that
the denotation (reference, Bedeutung) of any formula is given by its truth-
value relative to the underlying assignment: either true or false.
Consequently, the Fregean Axiom holds:
$(\varphi\leftrightarrow\psi)\leftrightarrow(\varphi\equiv\psi)$. It is known
that this situation can be modeled in $\mathit{SCI}$ by presenting a two-
element model where all true formulas denote one element (the true
proposition) and all false formulas denote the other one (the false
proposition).
###### Example 3.11.
There exists an extensional $\mathit{SCI}$-model, i.e. a two-element model
$\mathcal{M}$ where the denotation of a formula is nothing but a classical
truth value: for every assignment $\gamma\colon V\rightarrow\\{0,1\\}$ and all
$\varphi,\psi\in Fm$, $(\mathcal{M},\gamma)\vDash\varphi\equiv\psi$ iff
$(\mathcal{M},\gamma)\vDash\varphi\leftrightarrow\psi$ iff $\varphi$ and
$\psi$ have the same classical truth-value.
Of course, the desired extensional model will be based (up to isomorphism) on
the two-element Boolean algebra $\mathcal{B}$ with universe $\\{0,1\\}$. Let
$\preceq$ be the natural total order on $\\{0,1\\}$. The resulting relation
$\approx$ is the identity and the associated quotient algebra is $\mathcal{B}$
itself. We define an additional Boolean operation
$f_{\equiv}\colon\\{0,1\\}\times\\{0,1\\}\rightarrow\\{0,1\\}$ by
$f_{\equiv}(x,y)=1$ $:\Leftrightarrow$ $x=y$. Then $\mathcal{B}$ together with
$f_{\equiv}$ and the unique ultrafilter $\mathit{TRUE}:=\\{1\\}$ yields an
$\mathit{SCI}$-model $\mathcal{M}$. Obviously, for any assignment
$\gamma\colon V\rightarrow\\{0,1\\}$ and for any formulas $\varphi,\psi\in
F_{\equiv}$, we have $(\mathcal{M},\gamma)\vDash\varphi\equiv\psi$ iff
$\gamma(\varphi)=\gamma(\psi)$ iff $\varphi$ and $\psi$ have the same
classical truth-value.
It is clear that the above two-valued $\mathit{SCI}$-model along with all
possible assignments yields essentially the standard two-valued semantics of
classical propositional logic $\mathit{CPC}$. In fact, $\mathit{CPC}$ is
represented by the $\mathit{SCI}$-theory $\mathit{SCI}^{ext}$ that results
from $\mathit{SCI}$ by adding Fregean Axiom
$(\varphi\leftrightarrow\psi)\rightarrow(\varphi\equiv\psi)$. Theory
$\mathit{SCI}^{ext}$ contains
$(\varphi\leftrightarrow\psi)\leftrightarrow(\varphi\equiv\psi)$ and thus
$(\varphi\leftrightarrow\psi)\equiv(\varphi\equiv\psi)$ as theorems. By SP,
$(\varphi\leftrightarrow\psi)$ and $(\varphi\equiv\psi)$ then can be replaced
by each other in every context. One easily shows that $\mathit{SCI}^{ext}$ is
sound and complete w.r.t. the class of all extensional (i.e., two-element)
$\mathit{SCI}$-models. We have for any $\varphi\in Fm_{\equiv}$:
$\vdash_{\mathit{SCI}^{ext}}\varphi\text{ }\Leftrightarrow\text{
}\vdash_{\mathit{CPC}}\varphi^{*},$
where $\varphi^{*}$ is the result of replacing every subformula of the form
$\psi\equiv\chi$ in $\varphi$ by $\psi\leftrightarrow\chi$.
Another important example of $\mathit{SCI}$-model, as opposed to an
extensional model, is an intensional model where the denotation of a formula
is determined by its intension, i.e. its syntactical form. In such a model,
any two (syntactically) different formulas have different denotations. The
denotation of a formula can be identified with its intension. In the
following, we present a construction of such a model. Intensional models have
also been constructed for a logic that extends $\mathit{SCI}$ by propositional
quantifiers and a truth predicate (see, e.g. the discussion and a construction
presented in [8]).101010The construction of an intensional model for such a
first-order logic is not trivial because of the impredicativity of
propositional quantifiers. Note that bound variable $x$ in formula $\forall
x\varphi$ ranges over the universe of all propositions which contains in
particular the proposition denoted by $\forall x\varphi$ itself.
###### Example 3.12.
There exists an intensional $\mathit{SCI}$-model, i.a. a model $\mathcal{M}$
along with an assignment $\gamma$ such that for all $\varphi,\psi\in
Fm_{\equiv}$,
$(\mathcal{M},\gamma)\vDash\varphi\equiv\psi\Leftrightarrow\varphi=\psi.$
Let us construct model $\mathcal{M}$. We define a rank $R\colon
Fm_{\equiv}\rightarrow\mathbb{N}$ on formulas as follows:
* •
$R(x)=R(\bot)=R(\top)=R(\varphi\equiv\psi)=0$, for any $x\in V$ and
$\varphi,\psi\in Fm_{\equiv}$.
* •
If $\varphi,\psi\in Fm$ such that $R(\varphi)$ and $R(\psi)$ are already
defined, then $R(\neg\varphi)=R(\varphi)+1$ and
$R(\varphi*\psi)=max\\{R(\varphi),R(\psi)\\}+1$, where
$*\in\\{\vee,\wedge,\rightarrow\\}$.
We consider the given enumeration of the set of variables
$V=\\{x_{0},x_{1},x_{2},...\\}$ and define the set $\mathit{TRUE}$ by
induction on rank $R$ as the smallest set such that the following conditions
are satisfied:
* •
For formulas of rank $0$, we have: $\bot\not\in\mathit{TRUE}$,
$\top\in\mathit{TRUE}$, $x_{i}\in\mathit{TRUE}$ iff $i$ is an even index,
$\varphi\equiv\psi\in\mathit{TRUE}$ iff $\varphi=\psi$.
* •
Suppose membership of all formulas of rank $\leq n\in\mathbb{N}$ w.r.t.
$\mathit{TRUE}$ is already determined. Let $\varphi$, $\psi$ be formulas such
that $max\\{R(\varphi),R(\psi)\\}=n$. Then:
* –
$\varphi\wedge\psi\in\mathit{TRUE}$ if $\varphi\in\mathit{TRUE}$ and
$\psi\in\mathit{TRUE}$
* –
$\varphi\vee\psi\in\mathit{TRUE}$ if $\varphi\in\mathit{TRUE}$ or
$\psi\in\mathit{TRUE}$
* –
$\neg\varphi\in\mathit{TRUE}$ if $\varphi\notin\mathit{TRUE}$
* –
$\varphi\rightarrow\psi\in\mathit{TRUE}$ if $\varphi\notin\mathit{TRUE}$ or
$\psi\in\mathit{TRUE}$
Membership w.r.t. $\mathit{TRUE}$ determines a classical truth-value for every
formula. The relation $\preceq$ on $M:=Fm_{\equiv}$ defined by
$\varphi\preceq\psi:\Leftrightarrow$ $\varphi\rightarrow\psi\in\mathit{TRUE}$
is a preorder. Moreover, the relation $\approx$ defined by
$\varphi\approx\psi$ $:\Leftrightarrow$ ($\varphi\preceq\psi$ and
$\psi\preceq\varphi$) $\Leftrightarrow$ ‘both $\varphi$ and $\psi$ belong to
$\mathit{TRUE}$ or both $\varphi$ and $\psi$ belong to
$M\smallsetminus\mathit{TRUE}$’ is a congruence relation on the structure
$(M,\vee,\wedge,\neg,\bot,\top,\rightarrow)$. The associated quotient algebra
is the two-element Boolean algebra $\\{0,1\\}$ where $1$ is the image of
$\mathit{TRUE}$ under the canonical homomorphism. Moreover,
$\varphi\preceq\psi$ $\Leftrightarrow$
$\varphi\rightarrow\psi\in\mathit{TRUE}$ $\Leftrightarrow$
[$(\varphi\wedge\psi)\rightarrow\varphi\in\mathit{TRUE}$ and
$\varphi\rightarrow(\varphi\wedge\psi)\in\mathit{TRUE}$] $\Leftrightarrow$
$(\varphi\wedge\psi)\approx\varphi$. Hence,
$\mathcal{M^{\prime}}=(M,\vee,\wedge,\neg,\bot,\top,\rightarrow,\preceq)$
is a Boolean prealgebra. Together with ultrafilter $\mathit{TRUE}$ and the
operation $f_{\equiv}$ on $M=Fm_{\equiv}$ defined by
$f_{\equiv}(\varphi,\psi):=(\varphi\equiv\psi)$ we then obtain the
$\mathit{SCI}$-model
$\mathcal{M}=(M,\mathit{TRUE},\vee,\wedge,\neg,\bot,\top,\rightarrow,f_{\equiv},\preceq).$
Consider the assignment $\gamma\colon V\rightarrow Fm_{\equiv}$, $x\mapsto x$.
Then, by induction on formulas, $\gamma(\varphi)=\varphi$ for any $\varphi\in
Fm_{\equiv}$. Furthermore, for all $\varphi,\psi\in Fm_{\equiv}$:
$\begin{split}(\mathcal{M},\gamma)\vDash\varphi\equiv\psi&\Leftrightarrow\gamma(\varphi\equiv\psi)=f_{\equiv}(\gamma(\varphi),\gamma(\psi))\in\mathit{TRUE}\\\
&\Leftrightarrow\gamma(\varphi)=\gamma(\psi)\Leftrightarrow\varphi=\psi.\end{split}$
As a consequence, already observed by Suszko, only trivial identities are
theorems of $\mathit{SCI}$.
###### Corollary 3.13.
For all $\varphi,\psi\in Fm_{\equiv}$,
$\vdash_{\mathit{SCI}}\varphi\equiv\psi\Leftrightarrow\varphi=\psi$.
###### Proof.
If $\varphi=\psi$, then by identity axiom (id1):
$\vdash_{\mathit{SCI}}\varphi\equiv\psi$. On the other hand, if
$\varphi\neq\psi$, then we have $(\mathcal{M},\gamma)\nvDash\varphi\equiv\psi$
for the intensional model constructed above and thus $\varphi\equiv\psi$ is
not logically valid. Soundness yields
$\nvdash_{\mathit{SCI}}\varphi\equiv\psi$. ∎
In the remainder of this section, we show that some relevant modal principles
can be restored in the pure $\mathit{SCI}$, i.e. in $\mathit{SCI}$ with no
additional axioms. The representation of certain Lewis-style modal systems by
means of appropriate $\mathit{SCI}$-extensions will be the topic of the next
section.
For $\varphi\in Fm_{\equiv}$, we define
(2) $\square\varphi:=(\varphi\equiv\top).$
###### Theorem 3.14.
Let $\mathcal{M}$ be an $\mathit{SCI}$-model. Then the following are
equivalent:
1. (i)
$\mathcal{M}$ is based on a Boolean algebra, i.e. its
$\\{f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top}\\}$-reduct is a Boolean
algebra.
2. (ii)
For all formulas $\chi$ having the form of a classical tautology, and for all
formulas $\varphi$ and $\psi$, model $\mathcal{M}$ validates $\square\chi$ and
$(\varphi\equiv\psi)\leftrightarrow\square(\varphi\leftrightarrow\psi)$.
###### Proof.
If $\mathcal{M}$ is a Boolean algebra, then all theorems of $\mathit{CPC}$, as
well as their substitution instances, are evaluated by the top element
$f_{\top}$ under any assignment. It is also known that the equivalence
$f_{\rightarrow}(m,m^{\prime})=f_{\top}$ $\Leftrightarrow$ $m\leq m^{\prime}$
holds in every Boolean algebra (actually, in every Heyting algebra). Then it
is clear that (i) implies (ii). Now, suppose (ii) holds true. Then
$\mathcal{M}$ validates in particular $\square(\varphi\leftrightarrow\psi)$
whenever $\varphi\leftrightarrow\psi$ is a classical tautology. Since
$(\varphi\equiv\psi)\leftrightarrow\square(\varphi\leftrightarrow\psi)$ is
valid in $\mathcal{M}$, we have $\mathcal{M}\vDash\varphi\equiv\psi$ for all
Boolean equations $\varphi\equiv\psi$ that axiomatize the class of Boolean
algebras. Hence, $\mathcal{M}$ itself is based on a Boolean algebra. ∎
###### Definition 3.15.
$\mathit{SCI}^{+}$ is the logic that results from $\mathit{SCI}$ by adding the
following axioms:
* •
$\square\chi$ whenever $\chi$ has the form of a classical tautology,
* •
$(\varphi\equiv\psi)\leftrightarrow\square(\varphi\leftrightarrow\psi)$.
The next result then follows from Theorem 3.14.
###### Corollary 3.16.
The $\mathit{SCI}$-extension $\mathit{SCI}^{+}$ is sound and complete w.r.t.
the class of those $\mathit{SCI}$-models which are based on Boolean algebras.
As a consequence, $\mathit{SCI}^{+}$ coincides with the known
$\mathit{SCI}$-theory $\mathit{WB}$.111111Theory $\mathit{WB}$ is discussed in
some works on non-Fregean logic (see, e.g. [15] for a detailed presentation).
The question arises whether theory $\mathit{SCI}^{+}$ contains further
interesting modal laws. Using Corollary 3.16, we may argue semantically
showing that the following formulas are theorems of $\mathit{SCI^{+}}$:
* •
$\square\varphi\rightarrow\varphi$
* •
$\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi)$.
In fact, given a Boolean algebra, the top element $f_{\top}$ is contained in
every ultrafilter; and for any elements $m,m^{\prime}$: if $m\leq m^{\prime}$,
then $m=f_{\top}$ implies $m^{\prime}=f_{\top}$. Thus, the validity of the
above formulas is justified. However, some principles of normal Lewis systems
are not valid. For instance, the full necessitation rule does not hold. As a
contra-example, we consider the Boolean algebra $2^{2}$ with elements
$\varnothing$, $\\{0\\}$, $\\{1\\}$, $\\{0,1\\}$ and set-theoretic inclusion
as lattice order, along with the ultrafilter
$\mathit{TRUE}=\\{\\{0\\},\\{0,1\\}\\}$ and operation $f_{\equiv}$ defined by
$f_{\equiv}(m,m^{\prime}):=\\{0\\}\in\mathit{TRUE}$ if $m=m^{\prime}$, and
$f_{\equiv}(m,m^{\prime})=\\{1\\}\notin\mathit{TRUE}$ otherwise. Then
$f_{\square}(\\{0\\})=f_{\equiv}(\\{0\\},f_{\top})=\\{1\\}\not\leq\\{0\\}$.
Thus, $\square(\square\varphi\rightarrow\varphi)$ is not valid.
###### Definition 3.17.
In some analogy to Lewis modal system $\mathit{S3}$, we define the following
extension of $\mathit{SCI}^{+}$: $\mathit{SCI}_{3}$ is the logic that results
from $\mathit{SCI}^{+}$ by adding all formulas of the form
$\square(\varphi\rightarrow\psi)\rightarrow\square(\square\varphi\rightarrow\square\psi)$
as theorems.
Note, however, that the alleged analogy to Lewis system $\mathit{S3}$ is
rather weak. For instance,
$\square(\square(\varphi\rightarrow\psi)\rightarrow\square(\square\varphi\rightarrow\square\psi))$
is a theorem of $\mathit{S3}$ but not of $\mathit{SCI}_{3}$.
###### Definition 3.18.
An $\mathit{SCI}$-model $\mathcal{M}$ is an $\mathit{SCI}_{3}$-model if
$\mathcal{M}$ is based on a Boolean algebra and satisfies the following
condition for all $m,m^{\prime}\in M$:
$m\leq m^{\prime}\Rightarrow f_{\square}(m)\leq f_{\square}(m^{\prime}),$
where $f_{\square}(m):=f_{\equiv}(m,f_{\top})$. That is, $f_{\square}$ is
monotonic on $M$.
###### Corollary 3.19.
Logic $\mathit{SCI}_{3}$ is sound and complete w.r.t. the class of
$\mathit{SCI}_{3}$-models.
###### Proof.
One easily checks that every $\mathit{SCI}_{3}$-model validates formulas of
the form
$\square(\varphi\rightarrow\psi)\rightarrow\square(\square\varphi\rightarrow\square\psi)$.
In order to prove completeness, it is enough to show that the constructed
model in the proof of Theorem 3.10 above satisfies the condition of
monotonicity of $f_{\square}$. Since $\mathit{SCI}_{3}$ contains
$\mathit{SCI}^{+}$, we already know that that model is a Boolean algebra. So
for two elements $[\varphi]$ and $[\psi]$, suppose $[\varphi]\leq[\psi]$
(where $\leq$ is the lattice order). Then
$(\varphi\rightarrow\psi)\equiv\top\in\varPsi$. That is,
$\square(\varphi\rightarrow\psi)\in\varPsi$ and thus
$\square(\square\varphi\rightarrow\square\psi)\in\varPsi$. But then
$(\square\varphi\rightarrow\square\psi)\equiv\top\in\varPsi$ and thus
$[\square\varphi\rightarrow\square\psi]=[\top]$, i.e.
$f_{\square}([\varphi])=[\square\varphi]\leq[\square\psi]=f_{\square}([\psi])$.
∎
We are interested in conditions that ensure, in some precise sense, complete
restorations of some Lewis-style modal systems, in particular of
$\mathit{S3}$–$\mathit{S5}$. It turns out that principle
$(\varphi\equiv\psi)\leftrightarrow\square(\varphi\leftrightarrow\psi)$, valid
in $\mathit{SCI}^{+}$, is too weak for this purpose. In fact, we must
postulate the equation
$(\varphi\equiv\psi)\equiv\square(\varphi\leftrightarrow\psi)$, i.e. we must
identify propositional identity with strict equivalence. These topics will be
studied in section 5.
## 4 Some Lewis-style modal systems and their algebraic semantics
The goal of this section is to revise some Lewis-style modal systems in the
vicinity of $\mathit{S3}$ (more precisely, systems based on a logic called
$\mathit{S1SP}$) which in the subsequent section then will be shown to be
dual, in some precise sense, to certain $\mathit{SCI}$-theories. Our object
language is now the language of propositional modal logic $Fm_{\square}$, i.e.
the set of formulas inductively defined over the set of variables
$V=\\{x_{0},x_{1},...\\}$, logical connectives
$\bot,\top,\vee,\wedge,\neg,\rightarrow$ and the modal operator $\square$.
Thus, the languages $Fm_{\equiv}$ and $Fm_{\square}$ share the ‘pure’
propositional part based on the logical connectives. We introduce an ‘identity
connective’ defined by strict equivalence:
(3)
$(\varphi\equiv\psi):=(\square(\varphi\rightarrow\psi)\wedge\square(\psi\rightarrow\varphi)).$
It is evident that under this interpretation, all Lewis modal systems
$\mathit{S1}$–$\mathit{S5}$ satisfy Suszko’s identity axioms (id1)
$\varphi\equiv\varphi$ and (id2)
$(\varphi\equiv\psi)\rightarrow(\varphi\rightarrow\psi)$. Moreover,
$\mathit{S3}$ also satisfies the remaining identity axioms, i.e. SP ( where,
of course, identity is given as strict equivalence according to (3) above).
$\mathit{S3}$ is the weakest Lewis modal system containing SP (cf. [9, 11]).
In the following, we recall definitions of some relevant Lewis-style modal
systems and consider an algebraic semantics which can be immediately
translated into $\mathit{SCI}$-semantics, and vice-versa. We adopt that
particular approach to algebraic semantics from [9].
Lewis system $\mathit{S1}$ can be defined in the following way (cf., e.g.,
[5]). All formulas of the following form are axioms:
* •
tautologies (and their substitution-instances) of $\mathit{CPC}$
* •
$\square\varphi\rightarrow\varphi$
* •
$(\square(\varphi\rightarrow\psi)\wedge\square(\psi\rightarrow\chi))\rightarrow\square(\varphi\rightarrow\chi)$
(transitivity of strict implication)
The inference rules are Modus Ponens MP, Axiom Necessitation AN “If $\varphi$
is an axiom, then $\square\varphi$ is a theorem”, and Substitution of Proved
Strict Equivalents SPSE “If $\varphi\equiv\psi$ is a theorem, then so is
$\chi[x:=\varphi]\equiv\chi[x:=\psi]$”.
Lewis system $\mathit{S3}$ results from $\mathit{S1}$ by adding
(S3)
$\square(\varphi\rightarrow\psi)\rightarrow\square(\square\varphi\rightarrow\square\psi)$
as an axiom scheme to $\mathit{S1}$. Of course, rule (AN) now applies also to
(S3). Rule SPSE can be ignored since it is derivable from the rest.
Lewis system $\mathit{S4}$ results from $\mathit{S3}$ by adding
(S4) $\square\varphi\rightarrow\square\square\varphi$
as an axiom scheme (rule (AN) now applies also to (S4)). Finally,
$\mathit{S5}$ results from $\mathit{S4}$ by adding
(S5) $\neg\square\varphi\rightarrow\square\neg\square\varphi$
as an axiom scheme.
We do not consider Lewis system $\mathit{S2}$ since it is apparently not
susceptible to our algebraic semantics. Recall, however, that $\mathit{S2}$
can be captured by a non-normal Kripke-style semantics. There is no known
natural semantics for Lewis system $\mathit{S1}$ (cf. [5]). If we strengthen
the $\mathit{S1}$-rule SPSE to our stronger Substitution Principle SP,
$(\varphi\equiv\psi)\rightarrow(\chi[x:=\varphi]\equiv\chi[x:=\psi])$, and add
it as a theorem scheme to $\mathit{S1}$ (i.e., SP is regarded a scheme of
theorems; recall that AN is not applicable to theorems), then we obtain modal
system $\mathit{S1+SP}$ which was introduced and studied in [9]. Simplifying
notation, we will refer to that system as $\mathit{S1SP}$ instead of
$\mathit{S1+SP}$. In contrast to $\mathit{S1}$, the stronger system
$\mathit{S1SP}$ has a natural model-theoretic semantics which we will recall
below.
In system $\mathit{S1}$, derivations from the empty set, i.e. derivations of
theorems, are defined as usual. For
$\mathcal{L}\in\\{\mathit{S1SP},\mathit{S3},\mathit{S4},\mathit{S5}\\}$ and
$\varPhi\cup\\{\varphi\\}\subseteq Fm_{\square}$, we write
$\varPhi\vdash_{\mathcal{L}}\varphi$ if there is a derivation of $\varphi$
from $\varPhi$, i.e. a finite sequence $\varphi_{1},...,\varphi_{n}=\varphi$
such that for each $\varphi_{i}$, $i\leq i\leq n$, the following holds:
$\varphi_{i}\in\varPhi$ or $\varphi_{i}$ is an axiom of $\mathcal{L}$ or
$\varphi_{i}$ is obtained by AN (i.e. $\varphi_{i}=\square\psi$ for some axiom
$\psi$ of $\mathcal{L}$) or $\varphi_{i}$ is obtained by MP applied to
preceding formulas of the sequence. Note that we can do without the full
Necessitation Rule “If $\varphi$ is a theorem, then so is $\square\varphi$”.
In fact, by induction on derivations one shows that the full Necessitation
Rule is derivable in $\mathit{S4}$.
The following result is proven in [[9], Lemma 2.3] where it is originally
formulated for logic $\mathit{S1SP}$. The proof given there makes use of SP.
However, one recognizes that SP can be replaced by the $\mathit{S1}$-rule SPSE
in the proof. Hence, the result also holds in the weaker system $\mathit{S1}$.
###### Lemma 4.1 ([9]).
Every instance of the following principle N is a theorem of $\mathit{S1}$:
$\square\varphi\leftrightarrow(\varphi\equiv\top).$
N expresses the fact that there exists exactly one necessary proposition,
namely the proposition denoted by $\top$.
N would easily follow from distribution principle K,
$\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi)$.121212Consider
classical tautology $\varphi\leftrightarrow(\varphi\leftrightarrow\top)$, rule
AN, principle K and MP. However, K is not available in $\mathit{S1}$.
Nevertheless, using N and SP we are able to show the following (cf. [9], Lemma
2.4):
###### Lemma 4.2 ([9]).
Distribution principle K holds in $\mathit{S1SP}$, i.e. formulas of the form
$\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi)$
are theorems of $\mathit{S1SP}$.
###### Lemma 4.3.
Equivalences
$\square(\varphi\wedge\psi)\leftrightarrow(\square\varphi\wedge\square\psi)$
are theorems of $\mathit{S1SP}$.
###### Proof.
We show that
$\square(\varphi\wedge\psi)\rightarrow(\square\varphi\wedge\square\psi)$ is a
theorem. By Lemma 4.1,
$\square(\varphi\wedge\psi)\leftrightarrow((\varphi\wedge\psi)\equiv\top)$. In
particular, we have the following valid implication:
$\square(\varphi\wedge\psi)\rightarrow\square(\top\rightarrow(\varphi\wedge\psi))$.
By the transitivity axiom of strict implication of $\mathit{S1}$,
$(\square(\top\rightarrow(\varphi\wedge\psi))\wedge\square((\varphi\wedge\psi)\rightarrow\varphi))\rightarrow\square(\top\rightarrow\varphi)$.
Note that $\square((\varphi\wedge\psi)\rightarrow\varphi))$ results from an
application of rule AN. Then transitivity of implication yields
$\square(\varphi\wedge\psi)\rightarrow\square(\top\rightarrow\varphi)$, i.e.
$\square(\varphi\wedge\psi)\rightarrow(\varphi\equiv\top)$. By principle N, we
get $\square(\varphi\wedge\psi)\rightarrow\square\varphi$. Similarly, we get
$\square(\varphi\wedge\psi)\rightarrow\square\psi$ and thus
$\square(\varphi\wedge\psi)\rightarrow(\square\varphi\wedge\square\psi)$.
Now, we show the converse
$(\square\varphi\wedge\square\psi)\rightarrow\square(\varphi\wedge\psi)$
making use of SP. Note that $\square\psi\leftrightarrow(\psi\equiv\top)$ and
$(\psi\equiv\top)\rightarrow\square(\varphi\wedge
y)[y:=\psi]\equiv\square(\varphi\wedge y)[y:=\top]$ are instances of N and SP,
respectively. By transitivity of implication,
$\square\psi\rightarrow(\square(\varphi\wedge\psi)\equiv\square(\varphi\wedge\top))$
is a theorem. Thus,
$\square\psi\rightarrow(\square(\varphi\wedge\top)\rightarrow\square(\varphi\wedge\psi))$
is a theorem. By rule AN, $\varphi\equiv(\varphi\wedge\top)$ is a theorem.
Then we may apply $\mathit{S1}$-rule SPSE (or the stronger SP) and derive
$\square\psi\rightarrow(\square\varphi\rightarrow\square(\varphi\wedge\psi))$
which modulo $\mathit{CPC}$ is equivalent to
$(\square\varphi\wedge\square\psi)\rightarrow\square(\varphi\wedge\psi)$. ∎
By Lemma 4.3, we may write strict equivalence
$\square(\varphi\rightarrow\psi)\wedge\square(\psi\rightarrow\varphi)$
equivalently and shorter as $\square(\varphi\leftrightarrow\psi)$ in systems
containing $\mathit{S1SP}$. In $\mathit{S1SP}$, we may also strengthen the
result of Lemma 4.1 as follows.
###### Lemma 4.4.
The following scheme $\square N$ is derivable in $\mathit{S1SP}$:
$\square\varphi\equiv(\varphi\equiv\top).$
###### Proof.
Note that $\varphi\leftrightarrow(\varphi\leftrightarrow\top)$ is a
propositional tautology. Rule AN yields
$\Box(\varphi\leftrightarrow(\varphi\leftrightarrow\top))$, i.e.
$\varphi\equiv(\varphi\leftrightarrow\top)$. Consider the instance
$(\varphi\equiv(\varphi\leftrightarrow\top))\rightarrow(\square
x[x:=\varphi]\equiv\square x[x:=(\varphi\leftrightarrow\top)]$ of SP and apply
MP. This yields theorem
$\square\varphi\equiv\square(\varphi\leftrightarrow\top)$ ∎
In the following definitions, by a Boolean algebra expansion we always mean a
structure
$\mathcal{M}=(M,\mathit{TRUE},f_{\vee},f_{\wedge},f_{\neg},f_{\bot},f_{\top},f_{\rightarrow},f_{\square})$
which is based on a Boolean algebra with the usual operations along with a
designated ultrafilter $\mathit{TRUE}$ and an additional unary function
$f_{\square}$. The induced lattice order is always denoted by $\leq$.
###### Definition 4.5.
Let $\mathcal{M}$ be a Boolean algebra expansion satisfying the following
conditions for all $a,b,c\in M$:
(1) $f_{\square}(a)\in\mathit{TRUE}\Leftrightarrow a=f_{\top}$
(2) $f_{\square}(a)\leq a$
(3)
$f_{\wedge}(f_{\square}(f_{\rightarrow}(a,b)),f_{\square}(f_{\rightarrow}(b,c)))\leq
f_{\square}(f_{\rightarrow}(a,c))$
Then we call $\mathcal{M}$ an $\mathit{S1SP}$-algebra.
Note that conditions (2) and (3) reflect corresponding axioms of
$\mathit{S1}$.
###### Lemma 4.6.
In every $\mathit{S1SP}$-algebra it holds that
$f_{\square}(f_{\wedge}(a,b))\in\mathit{TRUE}\Leftrightarrow
f_{\wedge}(f_{\square}(a),f_{\square}(b))\in\mathit{TRUE},$
for all elements $a,b$, i.e. formulas of the form
$\square(\varphi\wedge\psi)\leftrightarrow(\square\varphi\wedge\square\psi)$
are valid in the class of $\mathit{S1SP}$-algebras. Moreover, modal principle
$K$,
$\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi),$
is valid in the class of $\mathit{S1SP}$-algebras.
###### Proof.
By (1), $f_{\square}(f_{\wedge}(a,b))\in\mathit{TRUE}$ $\Leftrightarrow$
$a=f_{\top}$ and $b=f_{\top}$ $\Leftrightarrow$
$f_{\square}(a)\in\mathit{TRUE}$ and $f_{\square}(b)\in\mathit{TRUE}$
$\Leftrightarrow$ $f_{\wedge}(f_{\square}(a),f_{\square}(b))\in\mathit{TRUE}$.
The second assertion can be shown as follows: For a given
$\mathit{S1SP}$-algebra, suppose
$f_{\square}(f_{\rightarrow}(a,b))\in\mathit{TRUE}$ and
$f_{\square}(a)\in\mathit{TRUE}$. The former implies
$f_{\rightarrow}(a,b)=f_{\top}$, i.e. $a\leq b$. The latter implies
$a=f_{\top}$. It follows $b=f_{\top}$ and thus
$f_{\square}(b)\in\mathit{TRUE}$. ∎
Notice that validity of modal principle $K$ in the class of
$\mathit{S1SP}$-algebras does not mean that all instances of $K$ are
interpreted by the top element of the given Boolean algebra (as it is the case
in normal modal logics). It only means that such instances are interpreted by
some element of the ultrafilter $\mathit{TRUE}$, a designated ultrafilter that
contains in particular the element $f_{\square}(f_{\top})$. In fact, we cannot
choose an arbitrary ultrafilter $\mathit{TRUE}$ of the Boolean algebra:
condition (1) of Definition 4.5 must be fulfilled. In this aspect, our
semantic approach differs from the usual one where the involved class of modal
algebras usually forms an equational class, i.e. a variety of algebras. Recall
that a modal algebra in the usual sense is a Boolean algebra with an operator
$f_{\square}$ satisfying the following sronger conditions for all elements
$a,b$:
$\begin{split}&f_{\square}(f_{\wedge}(a,b))=f_{\wedge}(f_{\square}(a),f_{\square}(b))\text{
and }\\\ &f_{\square}(f_{\top})=f_{\top}.\end{split}$
It is known that the class of all modal algebras in this sense constitutes
algebraic semantics for normal modal system $\mathit{K}$.
Given the modal language $Fm_{\square}$ and an $\mathit{S1SP}$-algebra
$\mathcal{M}$, the notion of an assignment (valuation) $\gamma\colon
V\rightarrow M$ is defined as before as a ‘homomorphism’ from $Fm_{\square}$
to $\mathcal{M}$, in particular:
$\gamma(\square\varphi)=f_{\square}(\gamma(\varphi))$. Also the notion of
satisfaction is given in the same way:
$(\mathcal{M},\gamma)\vDash\varphi\Leftrightarrow\gamma(\varphi)\in\mathit{TRUE}$.
$\mathit{S1SP}$-algebras were introduced in [9] (not under this name) to
provide a kind of algebraic semantics for Lewis-style modal logic
$\mathit{S1SP}$:
###### Theorem 4.7 ([9]).
$\mathit{S1SP}$ is (strongly) sound and complete with respect to the class of
all $\mathit{S1SP}$-algebras.
###### Definition 4.8.
A Boolean algebra expansion $\mathcal{M}$ is an $\mathit{S3}$-algebra if the
following hold for all $a,b\in M$:
(1) $f_{\square}(a)\in\mathit{TRUE}\Leftrightarrow a=f_{\top}$
(2) $f_{\square}(a)\leq a$
(S3) $f_{\square}(f_{\rightarrow}(a,b))\leq
f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(b)))$
###### Lemma 4.9.
Every $\mathit{S3}$-algebra is an $\mathit{S1SP}$-algebra, i.e. particularly
condition (3) of Definition 4.5 is satisfied. Moreover, in every
$\mathit{S3}$-algebra, the modal operator $f_{\square}$ is a monotone function
and it holds that
$f_{\square}(f_{\wedge}(a,b))=f_{\wedge}(f_{\square}(a),f_{\square}(b)),$
for all elements $a,b$.
###### Proof.
Condition (S3) ensures that $f_{\square}$ is a monotone function: $a\leq b$
iff $f_{\rightarrow}(a,b)=f_{\top}$ iff
$f_{\square}(f_{\rightarrow}(a,b))\in\mathit{TRUE}$
$\overset{(S3)}{\Rightarrow}$
$f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(b)))\in\mathit{TRUE}$
iff
$f_{\rightarrow}(f_{\square}(a),f_{\square}(b))=f_{\top}$ iff
$f_{\square}(a)\leq f_{\square}(b)$. Note that $f_{\wedge}(a,b)\leq a$ and
$f_{\wedge}(a,b)\leq b$. Monotonicity implies
$f_{\square}(f_{\wedge}(a,b))\leq f_{\wedge}(f_{\square}(a),f_{\square}(b)).$
On the other hand, $\varphi\rightarrow(\psi\rightarrow(\varphi\wedge\psi))$ is
a propositional tautology and therefore denotes the top element, under any
assignment. Thus,
$f_{\square}(f_{\rightarrow}(a,f_{\rightarrow}(b,f_{\wedge}(a,b))))\in\mathit{TRUE}$,
for any elements $a,b$. Condition (S3) along with ‘Modus Ponens’ yields
$f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(f_{\rightarrow}(b,f_{\wedge}(a,b)))))\in\mathit{TRUE}$,
i.e. $f_{\square}(a)\leq f_{\square}(f_{\rightarrow}(b,f_{\wedge}(a,b)))$.
Again by (S3), we get $f_{\square}(f_{\rightarrow}(b,f_{\wedge}(a,b)))\leq
f_{\square}(f_{\rightarrow}(f_{\square}(b),f_{\square}(f_{\wedge}(a,b))))$.
Thus,
$\begin{split}&f_{\square}(a)\leq
f_{\square}(f_{\rightarrow}(f_{\square}(b),f_{\square}(f_{\wedge}(a,b)))\leq
f_{\rightarrow}(f_{\square}(b),f_{\square}(f_{\wedge}(a,b))\text{ and
hence}\\\
&f_{\rightarrow}(f_{\square}(a),f_{\rightarrow}(f_{\square}(b),f_{\square}(f_{\wedge}(a,b)))=f_{\top}.\end{split}$
The term on the left hand side of the last equation is an interpretation of
the formula $x\rightarrow(y\rightarrow z)$ which is logically equivalent to
$(x\wedge y)\rightarrow z$. Hence,
$f_{\rightarrow}(f_{\wedge}(f_{\square}(a),f_{\square}(b)),f_{\square}(f_{\wedge}(a,b)))=f_{\top}$,
i.e.
$f_{\wedge}(f_{\square}(a),f_{\square}(b))\leq f_{\square}(f_{\wedge}(a,b)).$
Finally,
$f_{\wedge}(f_{\square}(a),f_{\square}(b))=f_{\square}(f_{\wedge}(a,b))$.
In order to see that every $\mathit{S3}$-algebra is an
$\mathit{S1SP}$-algebra, it is enough to show that condition (3) of Definition
4.5 follows from the conditions of Definition 4.8:
$((\varphi\rightarrow\psi)\wedge(\psi\rightarrow\chi))\rightarrow(\varphi\rightarrow\chi))$
is a propositional tautology and is therefore interpreted by the top element
$f_{\top}$ of any model. By (1),
$f_{\square}(f_{\rightarrow}(f_{\wedge}(f_{\rightarrow}(a,b),f_{\rightarrow}(b,c)),f_{\rightarrow}(a,c)))\in\mathit{TRUE}.$
Applying (S3) and ‘Modus Ponens’, we get
$f_{\square}(f_{\rightarrow}(f_{\square}(f_{\wedge}(f_{\rightarrow}(a,b),f_{\rightarrow}(b,c))),f_{\square}(f_{\rightarrow}(a,c))))\in\mathit{TRUE}.$
Since
$f_{\wedge}(f_{\square}(a),f_{\square}(b))=f_{\square}(f_{\wedge}(a,b))$, as
shown above, we obtain the following:
$f_{\square}(f_{\rightarrow}(f_{\wedge}(f_{\square}(f_{\rightarrow}(a,b)),f_{\square}(f_{\rightarrow}(b,c))),f_{\square}(f_{\rightarrow}(a,c))))\in\mathit{TRUE}$.
Applying condition (1) yields
$f_{\rightarrow}(f_{\wedge}(f_{\square}(f_{\rightarrow}(a,b)),f_{\square}(f_{\rightarrow}(b,c))),f_{\square}(f_{\rightarrow}(a,c)))=f_{\top},$
i.e.,
$f_{\wedge}(f_{\square}(f_{\rightarrow}(a,b)),f_{\square}(f_{\rightarrow}(b,c)))\leq
f_{\square}(f_{\rightarrow}(a,c))$, which is precisely condition (3) of
Definition 4.5. ∎
###### Definition 4.10.
We call a Boolean algebra expansion $\mathcal{M}$ a strong
$\mathit{S4}$-algebra if the following conditions hold for all elements $a,b$:
(1) $f_{\square}(a)\in\mathit{TRUE}\Leftrightarrow a=f_{\top}$
(2) $f_{\square}(a)\leq a$
(K) $f_{\square}(f_{\rightarrow}(a,b))\leq
f_{\rightarrow}(f_{\square}(a),f_{\square}(b))$
(S4) $f_{\square}(a)\leq f_{\square}(f_{\square}(a))$
###### Lemma 4.11.
Every strong $\mathit{S4}$-algebra is an $\mathit{S3}$-algebra.
###### Proof.
It is enough to show that condition (S3) holds in every strong
$\mathit{S4}$-algebra. First, we observe that conditions (1) and (S4) imply
that $f_{\square}(f_{\top})=f_{\top}$. Then by (K),
$f_{\square}(f_{\rightarrow}(f_{\square}(f_{\rightarrow}(a,b)),f_{\rightarrow}(f_{\square}(a),f_{\square}(b))))=f_{\top}$.
Again by (K),
$f_{\rightarrow}(f_{\square}(f_{\square}(f_{\rightarrow}(a,b))),f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(b))))=f_{\top}$.
That is,
$f_{\square}(f_{\square}(f_{\rightarrow}(a,b)))\leq
f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(b)))$. Applying
condition (S4), we obtain condition (S3):
$f_{\square}(f_{\rightarrow}(a,b)))\leq
f_{\square}(f_{\rightarrow}(f_{\square}(a),f_{\square}(b)))$. ∎
If there is a notion of strong $\mathit{S4}$-algebra, one may expect that
there is a notion of $\mathit{S4}$-algebra, too. Indeed,
$\mathit{S4}$-algebras have been studied in the literature under different
labels such as topological Boolean algebras or interior algebras. An
$\mathit{S4}$-algebra (alias interior algebra alias topological Boolean
algebra) is usually defined as a Boolean algebra with an operator
$f_{\square}$ (which can be viewed as an interior operator) such that the
following conditions (IA1)–(IA4) are satisfied for all elements $a,b$:
(IA1) $f_{\square}(a)\leq a$
(IA2) $f_{\square}(f_{\square}(a))=f_{\square}(a)$
(IA3) $f_{\square}(f_{\wedge}(a,b))=f_{\wedge}(f_{\square}(a),f_{\square}(b))$
(IA4) $f_{\square}(f_{\top})=f_{\top}$.
###### Theorem 4.12.
Every strong $\mathit{S4}$-algebra is an $\mathit{S4}$-algebra.
###### Proof.
Suppose $\mathcal{M}$ is a strong $\mathit{S4}$-algebra in the sense of
Definition 4.10. Then (IA1) above holds trivially. (IA2) follows from (IA1)
along with condition (S4). (IA3) is condition (S3) which holds by Lemma 4.11.
By condition (1), $f_{\square}(f_{\top})\in\mathit{TRUE}$. Then, by condition
(S4), $f_{\square}(f_{\square}(f_{\top}))\in\mathit{TRUE}$. Again by (1),
$f_{\square}(f_{\top})=f_{\top}$, i.e. (IA4) is satisfied. ∎
The converse of Theorem 4.12 is not true. As a contra-example we consider any
interior algebra with more than two elements where the interior operator
$f_{\square}$ is the identity: $a\mapsto f_{\square}(a)=a$. For every
ultrafilter $U$, there exists an element $a\in U$ such that $a<f_{\top}$. Then
condition (1) of Definition 4.10 of a strong $\mathit{S4}$-algebra cannot be
satisfied by all elements. An interior algebra gives rise to a strong
$\mathit{S4}$-algebra if there is an ultrafilter $\mathit{TRUE}$ such that for
any element $a$, $a<f_{\top}$ implies $f_{\square}(a)\notin\mathit{TRUE}$.
Thus, the class of strong $\mathit{S4}$-algebras is properly contained in the
class of all $\mathit{S4}$-algebras. Nevertheless, for a completeness result
concerning Lewis modal system $\mathit{S4}$, it is enough to consider only
strong $\mathit{S4}$-algebras.
###### Definition 4.13.
A Boolean algebra expansion $\mathcal{M}$ is called an $\mathit{S5}$-algebra
if all elements $a$ satisfy the following:
$\begin{split}f_{\square}(a)=\begin{cases}&f_{\top},\text{ if }a=f_{\top}\\\
&f_{\bot},\text{ else}\end{cases}\end{split}$
Note that Definition 4.13 does not impose any condition on the designated
ultrafilter $\mathit{TRUE}$ of the given Boolean algebra expansion. Actually,
if we only consider the algebraic properties of an $\mathit{S5}$-algebra, then
the designated ultrafilter can be disregarded. The resulting notion of an
$\mathit{S5}$-algebra then is equivalent to the usual definitions of
$\mathit{S5}$-algebras found in the literature. For example, an
$\mathit{S5}$-algebra can be characterized as an interior algebra in which
every open element is closed, i.e. where
$f_{\Diamond}(f_{\square}(a))=f_{\square}(a)$ holds for every element $a$,
with closure operator $f_{\Diamond}(a):=f_{\neg}(f_{\square}(f_{\neg}(a)))$.
In fact, one easily verifies:
###### Corollary 4.14.
Let $\mathcal{M}$ be a Boolean algebra expansion. The following are
equivalent:
* •
$\mathcal{M}$ is an $\mathit{S5}$-algebra.
* •
$\mathcal{M}$ is an interior algebra satisfying for all $a\in M$:
$f_{\Diamond}(f_{\square}(a))=f_{\square}(a)$.
In particular, every $\mathit{S5}$-algebra is an $\mathit{S4}$-algebra (i.e.
an interior algebra). Given any $\mathit{S5}$-algebra, condition (1) of
Definition 4.10 is (trivially) satisfied, independently of the choice of the
designated ultrafilter $\mathit{TRUE}$. Thus, every $\mathit{S5}$-algebra is
also a strong $\mathit{S4}$-algebra.
Recall that the relation of satisfaction between
$\mathit{S1SP}$-interpretations $(\mathcal{M},\gamma)$ and formulas
$\varphi\in Fm_{\square}$ is given similarly as for $\mathit{SCI}$ models:
$(\mathcal{M},\gamma)\vDash\varphi:\Leftrightarrow\gamma(\varphi)\in\mathit{TRUE}$.
Also the concept of logical consequence is defined in the usual way. Extending
the proof of Theorem 4.7 in a straightforward way, we get
###### Theorem 4.15.
$\mathit{S3}$ ($\mathit{S4}$, $\mathit{S5}$) is strongly sound and complete
w.r.t. the class of all $\mathit{S3}$-algebras ((strong)
$\mathit{S4}$-algebras, $\mathit{S5}$-algebras), respectively.
## 5 Dualities between $\mathit{SCI}$-theories and Lewis-style modal logics
The goal of this section is to show that under certain assumptions, some
Lewis-style modal logics are, in a precise sense, in duality with certain
theories formalized in the language of $\mathit{SCI}$, more precisely, with
certain axiomatic extensions of $\mathit{SCI}$. The crucial conditions for
these dualities are the following:
1. (I)
‘The $\mathit{SCI}$ principles of propositional identity are valid. In
particular, SP is valid.’
2. (II)
‘Propositional identity = strict equivalence’, i.e.,
$(\varphi\equiv\psi)\equiv\square(\varphi\leftrightarrow\psi)$ holds.
3. (III)
‘Necessity = identity with proposition $\top$. In particular, there is exactly
one necessary proposition: the proposition denoted by $\top$’, i.e.,
$\square\varphi\equiv(\varphi\equiv\top)$ holds.
4. (IV)
‘All classical tautologies are necessary: If $\varphi$ is a classical
tautology (i.e. an instance of a theorem of $\mathit{CPC}$), then
$\square\varphi$ is valid.’
From a semantic point of view, (III) and (IV) will ensure that the envolved
$\mathit{SCI}$-models are Boolean algebras (cf. Theorem 3.14 and the remark in
the last paragraph of section 3.)
We remark here that a similar type of dualities between propositional logics
with an identity connective and normal modal systems is established by T.
Ishii [6]. His propositional calculus $\mathit{PCI}$ is also defined in the
language of $\mathit{SCI}$ though the axioms (and rules) for the identity
connective differ in some aspects from $\mathit{SCI}$. Ishii shows duality
between $\mathit{PCI}$ and normal system $\mathit{K}$, as well as a series of
further dualities between extensions of $\mathit{PCI}$ and corresponding
normal modal systems.131313Ishii does not use the term ‘duality’.
We now establish translations between the propositional languages of
$\mathit{SCI}$ and of modal logic, i.e. between $Fm_{\equiv}$ and $Fm_{\Box}$.
###### Definition 5.1.
The translation $\mathit{box}\colon Fm_{\equiv}\rightarrow Fm_{\square}$ is
inductively defined as follows: $\mathit{box}(x):=x$,
$\mathit{box}(\bot):=\bot$, $\mathit{box}(\top):=\top$,
$\mathit{box}(\neg\varphi):=\neg\mathit{box}(\varphi)$,
$\mathit{box}(\varphi*\psi):=(\mathit{box}(\varphi)*\mathit{box}(\psi))$, for
$*\in\\{\wedge,\vee,\rightarrow\\}$, and
$\mathit{box}(\varphi\equiv\psi):=(\square(\mathit{box}(\varphi)\rightarrow\mathit{box}(\psi))\wedge\square(\mathit{box}(\psi)\rightarrow\mathit{box}(\varphi))_{.}$
On the other hand, the translation $\mathit{id}\colon Fm_{\square}\rightarrow
Fm_{\equiv}$ is inductively defined as follows: $\mathit{id}(x):=x$,
$\mathit{id}(\bot):=\bot$, $\mathit{id}(\top):=\top$,
$\mathit{id}(\neg\varphi):=\neg\mathit{id}(\varphi)$,
$\mathit{id}(\varphi*\psi):=(\mathit{id}(\varphi)*\mathit{id}(\psi))$, for
$*\in\\{\wedge,\vee,\rightarrow\\}$, and
$\mathit{id}(\square\varphi):=(id(\varphi)\equiv\top).$
For $\varPhi\subseteq Fm_{\equiv}$, we let
$\mathit{box}(\varPhi):=\\{\mathit{box}(\psi)\mid\psi\in\varPhi\\}$; and for
$\varPhi\subseteq Fm_{\Box}$, the set $\mathit{id}(\varPhi)$ is defined
analogously.
Induction on formulas ensures that $\mathit{box}(\varphi)\in Fm_{\square}$ for
any $\varphi\in Fm_{\equiv}$; and $\mathit{id}(\varphi)\in Fm_{\equiv}$ for
any $\varphi\in Fm_{\square}$. If the underlying logics are strong enough,
then the translations $\mathit{box}$ and $\mathit{id}$ are inverse to each
other in the sense of the next result.
Recall that in the language of modal logic $Fm_{\square}$, we use the
following abbreviation:
$(\varphi\equiv\psi):=(\square(\varphi\rightarrow\psi)\wedge\square(\psi\rightarrow\varphi))$,
cf. (3) above. Since we are working with modal systems containing
$\mathit{S1SP}$, we may define equivalently
$(\varphi\equiv\psi):=\square(\varphi\leftrightarrow\psi)$, cf. Lemma 4.6.
Also recall that in the language $\mathcal{L}_{\equiv}$ of $\mathit{SCI}$, we
use the abbreviation $\square\varphi:=(\varphi\equiv\top)$, cf. (2).
###### Theorem 5.2.
* •
Let $\mathcal{L}$ be a modal logic in the language $Fm_{\square}$ containing
$\mathit{S1SP}$. Then for any $\varphi\in Fm_{\square}$:
$\vdash_{\mathcal{L}}\varphi\equiv\mathit{box}(\mathit{id}(\varphi)).$
* •
Let $\mathcal{L}_{\equiv}$ be an axiomatic extension of $\mathit{SCI}$ in the
language $Fm_{\equiv}$ containing theorems of the form
$(\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi)$.161616That is,
formulas of the form
$(\chi\equiv\psi)\equiv((\chi\leftrightarrow\psi)\equiv\top)$ are theorems.
Then for any $\varphi\in Fm_{\equiv}$:
$\vdash_{\mathcal{L}_{\equiv}}\varphi\equiv\mathit{id}(\mathit{box}(\varphi)).$
###### Proof.
Under the assumptions of the first item, we show the assertion by induction on
$\varphi\in Fm_{\square}$. If $\varphi$ is an atomic formula, we get
$\mathit{box}(\mathit{id}(\varphi))=\varphi$. Then the assertion holds because
$\square(\varphi\leftrightarrow\varphi)$ is a theorem of $\mathcal{L}$ (apply
the rule of Axiom Necessitation (AN) to $\varphi\leftrightarrow\varphi$). Now
suppose $\varphi=\square\psi$ for some $\psi\in Fm_{\square}$.
$\begin{split}\mathit{box}(\mathit{id}(\varphi))&=\mathit{box}(\mathit{id}(\square\psi))\\\
&=\mathit{box}(\mathit{id}(\psi)\equiv\top),\text{ by definition of
}\mathit{id}\\\
&=\square(\mathit{box}(\mathit{id}(\psi)\leftrightarrow\top)),\text{ by
definition of }\mathit{box}\\\
&\equiv_{\mathcal{L}}\square(\psi\leftrightarrow\top),\text{ by induction
hypothesis and SP}\\\ &=(\psi\equiv\top)\\\
&\equiv_{\mathcal{L}}\square\psi,\text{ recall that
}\square\psi\equiv(\psi\equiv\top)\text{ is a theorem of }\mathit{S1SP}\\\
&=\varphi\end{split}$
Hence, $\varphi\equiv_{\mathcal{L}}\mathit{box}(\mathit{id}(\varphi))$, i.e.
$\vdash_{\mathcal{L}}\varphi\equiv\mathit{box}(\mathit{id}(\varphi))$. The
remaining cases of the induction step follow straightforwardly. Now, we assume
the hypotheses of the second item and show its assertion by induction on
$\varphi\in Fm_{\equiv}$. The induction base is clear; and in the induction
step, only the case $\varphi=(\psi\equiv\chi)$ requires some attention:
$\begin{split}\mathit{id}(\mathit{box}(\varphi))&=\mathit{id}(\mathit{box}(\psi\equiv\chi))\\\
&=\mathit{id}(\square(\mathit{box}(\psi)\leftrightarrow\mathit{box}(\chi))),\text{
by definition of }\mathit{box}\\\
&=(\mathit{id}(\mathit{box}(\psi))\leftrightarrow\mathit{id}(\mathit{box}(\chi)))\equiv\top,\text{
by definition of }\mathit{id}\\\
&\equiv_{\mathcal{L}_{\equiv}}(\mathit{id}(\mathit{box}(\psi))\equiv\mathit{id}(\mathit{box}(\chi))),\text{
by assumptions on }\mathcal{L}_{\equiv}\\\
&\equiv_{\mathcal{L}_{\equiv}}(\psi\equiv\chi),\text{ by induction hypothesis
and SP}\\\ &=\varphi\end{split}$
∎
If $\mathcal{L}$ is a modal logic and $\mathcal{L}_{\equiv}$ is an
$\mathit{SCI}$-extension satisfying the hypotheses required in Theorem 5.2,
then we are able to establish a condition (actually, two equivalent
conditions) under which both logics have, in a precise sense, the same
expressive power, i.e. are dual to each other:
###### Definition 5.3.
Let $\mathcal{L}$ be a modal logic in the language $Fm_{\square}$ containing
$\mathit{S1SP}$. Let $\mathcal{L}_{\equiv}$ be an extension of $\mathit{SCI}$
in the language $Fm_{\equiv}$ containing theorems of the form
$(\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi)$. Furthermore, suppose
one of the following two conditions is true:
1. (i)
For any $\varPhi\cup\\{\varphi\\}\subseteq Fm_{\equiv}$,
$\varPhi\vdash_{\mathcal{L_{\equiv}}}\varphi\Longleftrightarrow\mathit{box}(\varPhi)\vdash_{\mathcal{L}}\mathit{box}(\varphi)$.
2. (ii)
For any $\varPhi\cup\\{\varphi\\}\subseteq Fm_{\square}$,
$\varPhi\vdash_{\mathcal{L}}\varphi\Longleftrightarrow\mathit{id}(\varPhi)\vdash_{\mathcal{L_{\equiv}}}\mathit{id}(\varphi)$.
Then we say that $\mathcal{L}_{\equiv}$ and $\mathcal{L}$ are dual to each
other, and we call $\mathcal{L}_{\equiv}$ the (dual) $\mathit{SCI}$-theory of
modal logic $\mathcal{L}$; and we call $\mathcal{L}$ the (dual) modal theory
of $\mathcal{L}_{\equiv}$.
Actually, it would be enough to consider only one of the conditions (i), (ii)
in Definition 5.3, as the next result shows.
###### Lemma 5.4.
Let $\mathcal{L}$ be a modal logic and let $\mathcal{L}_{\equiv}$ be its dual
$\mathit{SCI}$-theory according to Definition 5.3. Then both conditions (i)
and (ii) of Definition 5.3 are satisfied.
###### Proof.
Let $\mathcal{L}_{\equiv}$ be the $\mathit{SCI}$-theory of modal system
$\mathcal{L}$ and suppose that fact is witnessed by condition (i) of
Definition 5.3. We show that condition (ii) follows. Let
$\varPhi\cup\\{\varphi\\}\subseteq Fm_{\square}$ and suppose
$\varPhi\vdash_{\mathcal{L}}\varphi$. There are
$\varphi_{1},...,\varphi_{n}\in\varPhi$ such that
$\vdash_{\mathcal{L}}(\varphi_{1}\wedge...\wedge\varphi_{n})\rightarrow\varphi$.
By Theorem 5.2,
$\vdash_{\mathcal{L}}\mathit{box}(\mathit{id}((\varphi_{1}\wedge...\wedge\varphi_{n})\rightarrow\varphi))$.
Then condition (i) yields
$\vdash_{\mathcal{L}_{\equiv}}\mathit{id}((\varphi_{1}\wedge...\wedge\varphi_{n})\rightarrow\varphi)$.
Taking into account the definition of $\mathit{id}$, that implies
$\mathit{id}(\varPhi)\vdash_{\mathcal{L}_{\equiv}}\mathit{id}(\varphi)$. The
implication from right-to-left of (ii) follows similarly. Analogously, one
establishes condition (i) under the assumption that condition (ii) holds true.
∎
###### Lemma 5.5.
Let $\mathcal{L}$ be a modal logic and let $\mathcal{L}_{\equiv}$ be its dual
$\mathit{SCI}$-theory. Then the following hold:
(a) For any $\varphi\in Fm_{\equiv}$,
$\vdash_{\mathcal{L}}\mathit{box}(\square\varphi)\equiv\square\mathit{box}(\varphi)$,
i.e.
$\mathit{box}(\square\varphi)\equiv_{\mathcal{L}}\square\mathit{box}(\varphi)$.
(b) For any $\varphi,\psi\in Fm_{\square}$,
$\vdash_{\mathcal{L}_{\equiv}}\mathit{id}(\varphi\equiv\psi)\equiv(\mathit{id}(\varphi)\equiv\mathit{id}(\psi))$,
which we also write as
$\mathit{id}(\varphi\equiv\psi)\equiv_{\mathcal{L}_{\equiv}}(\mathit{id}(\varphi)\equiv\mathit{id}(\psi))$.
###### Proof.
Under the given assumptions, we have:
$\mathit{box}(\square\varphi)=\mathit{box}(\varphi\equiv\top)=\square(\mathit{box}(\varphi)\leftrightarrow\top)=(\mathit{box}(\varphi)\equiv\top)\equiv_{\mathcal{L}}\square\mathit{box}(\varphi)$.
The last equation holds because $\square\psi\equiv(\psi\equiv\top)$ is a
theorem of $\mathit{S1SP}$ and thus of $\mathcal{L}$, for any $\psi\in
Fm_{\square}$.
On the other hand:
$\mathit{id}(\varphi\equiv\psi)=\mathit{id}(\square(\varphi\leftrightarrow\psi))=(\mathit{id}(\varphi)\leftrightarrow\mathit{id}(\psi))\equiv\top)\equiv_{\mathcal{L}_{\equiv}}(\mathit{id}(\varphi)\equiv\mathit{id}(\psi))$.
The last equation holds because formulas of the form
$(\chi\equiv\xi)\equiv((\chi\leftrightarrow\xi)\equiv\top)$ are theorems of
$\mathcal{L}_{\equiv}$. ∎
As expected, particular examples of Definition 5.3 are the
$\mathit{SCI}$-theories of modal systems $\mathit{S1SP}$, $\mathit{S3}$,
$\mathit{S4}$ and $\mathit{S5}$ which we are going to define in the following
as deductive systems in the language of $\mathit{SCI}$. Recall that we have
$\square\varphi:=(\varphi\equiv\top)$.
###### Definition 5.6.
We consider the language $Fm_{\equiv}$ of $\mathit{SCI}$ and define deductive
systems on the base of the following axiom schemes (CPC) + (1)–(5):
(CPC) any formula $\varphi$ having the form of a classical tautology, i.e.
$\varphi$ is the substitution instance of a theorem of $\mathit{CPC}$
(1) $(\chi\equiv\psi)\leftrightarrow\square(\chi\leftrightarrow\psi)$
(2) $\square\varphi\rightarrow\varphi$
(3’)$(\square(\varphi\rightarrow\psi)\wedge\square(\psi\rightarrow\chi))\rightarrow\square(\varphi\rightarrow\chi)$
(3)
$\square(\varphi\rightarrow\psi)\rightarrow\square(\square\varphi\rightarrow\square\psi)$
(4) $\square\varphi\rightarrow\square\square\varphi$
(5) $\neg\square\varphi\rightarrow\square\neg\square\varphi$.
Then logic $\mathit{S1SP}_{\equiv}$ is axiomatized by the axiom schemes (CPC),
(1), (2), (3’) together with the scheme of theorems SP
$(\varphi\equiv\psi)\rightarrow(\chi[x:=\varphi]\equiv\chi[x:=\psi])$. That
is, $\mathit{S1SP}_{\equiv}$ is given by the following deductive system. For
$\varPhi\cup\\{\varphi\\}\subseteq Fm_{\equiv}$, we write
$\varPhi\vdash_{\mathit{S1SP}_{\equiv}}\varphi$ if there is a derivation, i.e.
a sequence $\varphi_{1},...,\varphi_{n}=\varphi$, such that for every
$\varphi_{i}$, $1\leq i\leq n$: $\varphi_{i}\in\varPhi$ or $\varphi_{i}$ is an
instance of (CPC), (1)–(3’) or SP or $\varphi_{i}$ is obtained by rule MP or
$\varphi_{i}$ is obtained by rule AN (i.e. there is some $1\leq j<i$ such that
$\varphi_{j}$ is an axiom, i.e. an instance of (CPC) + (1)–(3’) and
$\varphi_{i}=\square\varphi_{j}$).
The deductive system $\mathit{S3}_{\equiv}$ is defined analogously but with
axiom schemes (CPC), (1), (2), (3) (and without theorem scheme SP). Similarly,
logic $\mathit{S4}_{\equiv}$ is given by the axioms (CPC) and (1)–(4). If
additionally we consider axiom scheme (5), then we obtain system
$\mathit{S5}_{\equiv}$.181818Of course, rule AN only applies to the given
axioms of the respective underlying system.
###### Lemma 5.7.
$(\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi)$ is a theorem of
$\mathit{S1SP}_{\equiv}$.
###### Proof.
Applying rule AN to (1) results in
$\square((\chi\equiv\psi)\leftrightarrow\square(\chi\leftrightarrow\psi))$.
Formula
$((\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi))\leftrightarrow\square((\chi\equiv\psi)\leftrightarrow\square(\chi\leftrightarrow\psi))$
is an instance of (1). Modus Ponens yields
$(\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi)$. ∎
###### Theorem 5.8.
$\mathit{SCI}\subseteq\mathit{SCI^{+}}\subseteq\mathit{S1SP}_{\equiv}\subseteq\mathit{S3}_{\equiv}\subseteq\mathit{S4}_{\equiv}\subseteq\mathit{S5}_{\equiv}$.
###### Proof.
The first inclusion is trivial by the definitions (cf. Definition 3.15).
Claim 1: $\mathit{SCI}^{+}\subseteq\mathit{S1SP}_{\equiv}$.
It is enough to show $\mathit{SCI}\subseteq\mathit{S1SP}_{\equiv}$. Recall
that SP is euivalent to the identity axioms (id3)–(id7) (modulo the rest of
$\mathit{SCI}$). So we only need to show that (id1) $\varphi\equiv\varphi$ and
(id2) $(\varphi\equiv\psi)\rightarrow(\varphi\rightarrow\psi)$ are theorems of
$\mathit{S1SP}_{\equiv}$. (id1) derives considering axiom
$\varphi\leftrightarrow\varphi$, rule AN and scheme (1). (id2) derives from
(1)+(2). Thus Claim 1 is true.
Claim 2:
$\square(\varphi\wedge\psi)\rightarrow(\square\varphi\wedge\square\psi)$ is a
theorem of $\mathit{S3}_{\equiv}$.
Apply AN to the tautologies $(\varphi\wedge\psi)\rightarrow\varphi$ and
$(\varphi\wedge\psi)\rightarrow\psi$ and consider axiom schemes (3) and (2).
Using propositional calculus, Claim 3 follows.
Claim 3: $\mathit{S1SP}_{\equiv}\subseteq\mathit{S3}_{\equiv}$.
It is enough to show that scheme (3) is stronger than (3’), and that scheme SP
is derivable in $\mathit{S3}_{\equiv}$. Of course,
$(\varphi\rightarrow\psi)\rightarrow((\psi\rightarrow\chi)\rightarrow(\varphi\rightarrow\chi))$
is a propositional tautology and thus an axiom. Applying rule AN, (3), (2) and
modus ponens then yields
$\square(\varphi\rightarrow\psi)\rightarrow(\square(\psi\rightarrow\chi)\rightarrow\square(\varphi\rightarrow\chi))$.
Modulo $\mathit{CPC}$, this is equivalent to (3’). (Note that we argued as in
original modal logic.) Thus, (3) is stronger than (3’) (modulo the rest).
Finally, in order to show that principle SP is derivable, we derive the
identity axioms (id3)–(id7) of $\mathit{SCI}$ which are equivalent to SP
modulo the rest. Consider the tautology
$(\varphi\leftrightarrow\psi)\rightarrow(\neg\varphi\leftrightarrow\neg\psi)$
and apply AN, (3), (2) and MP. We derive
$\square(\varphi\leftrightarrow\psi)\rightarrow\square(\neg\varphi\leftrightarrow\neg\psi)$.
By scheme (1) and transitivity of implication, we get
$(\varphi\equiv\psi)\rightarrow(\neg\varphi\equiv\neg\psi)$, i.e. (id3). Now
we consider the tautology
$(\varphi\leftrightarrow\psi)\rightarrow((\varphi^{\prime}\leftrightarrow\psi^{\prime})\rightarrow((\varphi\vee\varphi^{\prime})\leftrightarrow(\psi\vee\psi^{\prime})))$.
By AN and axioms,
$\square(\varphi\leftrightarrow\psi)\rightarrow(\square(\varphi^{\prime}\leftrightarrow\psi^{\prime})\rightarrow\square((\varphi\vee\varphi^{\prime})\leftrightarrow(\psi\vee\psi^{\prime})))$.
In this formula, we may replace formulas of the form
$\square(\chi_{1}\leftrightarrow\chi_{2})$ by $\chi_{1}\equiv\chi_{2}$,
according to (1). This results in
$(\varphi\equiv\psi)\rightarrow((\varphi^{\prime}\equiv\psi^{\prime})\rightarrow((\varphi\vee\varphi^{\prime})\equiv(\psi\vee\psi^{\prime})))$
which is equivalent to
$((\varphi\equiv\psi)\wedge(\varphi^{\prime}\equiv\psi^{\prime}))\rightarrow((\varphi\vee\varphi^{\prime})\equiv(\psi\vee\psi^{\prime}))$,
i.e. (id4). Similarly, we derive (id5) and (id6). Towards (id7), we consider
the propositional tautology
$(\varphi\leftrightarrow\psi)\rightarrow((\varphi^{\prime}\leftrightarrow\psi^{\prime})\rightarrow((\varphi\leftrightarrow\varphi^{\prime})\leftrightarrow(\psi\leftrightarrow\psi^{\prime})))$
and derive
(*)
$\square(\varphi\leftrightarrow\psi)\rightarrow(\square(\varphi^{\prime}\leftrightarrow\psi^{\prime})\rightarrow\square((\varphi\leftrightarrow\varphi^{\prime})\leftrightarrow(\psi\leftrightarrow\psi^{\prime})))$
in a similar way as before. Using Claim 2 and axiom scheme (3), we get
$\square((\varphi\leftrightarrow\varphi^{\prime})\leftrightarrow(\psi\leftrightarrow\psi^{\prime}))\rightarrow\square(\square(\varphi\leftrightarrow\varphi^{\prime})\leftrightarrow\square(\psi\leftrightarrow\psi^{\prime}))$.
Considering (*) and transitivity of implication, we derive
$\square(\varphi\leftrightarrow\psi)\rightarrow(\square(\varphi^{\prime}\leftrightarrow\psi^{\prime})\rightarrow\square(\square(\varphi\leftrightarrow\varphi^{\prime})\leftrightarrow\square(\psi\leftrightarrow\psi^{\prime})))$.
Now, in the same way as before, we apply (1) and corresponding replacements to
derive
(**)
$(\varphi\equiv\psi)\rightarrow((\varphi^{\prime}\equiv\psi^{\prime})\rightarrow(\square(\varphi\leftrightarrow\varphi^{\prime})\equiv\square(\psi\leftrightarrow\psi^{\prime})))$.
Note that the proof of Lemma 5.7 also works in $\mathit{S3}_{\equiv}$. By
schemes (3’) and (1), the connective $\equiv$ is transitive in
$\mathit{S3}_{\equiv}$. Putting these observations together and considering
the equations
‘$(\varphi\equiv\varphi^{\prime})\equiv\square(\varphi\leftrightarrow\varphi^{\prime})\equiv\square(\psi\leftrightarrow\psi^{\prime})\equiv(\psi\equiv\psi^{\prime})$’,
we are able to derive
$(\square(\varphi\leftrightarrow\varphi^{\prime})\equiv\square(\psi\leftrightarrow\psi^{\prime}))\rightarrow((\varphi\equiv\varphi^{\prime})\equiv(\psi\equiv\psi^{\prime}))$.
This together with (**) and transitivity of implication yields
$(\varphi\equiv\psi)\rightarrow((\varphi^{\prime}\equiv\psi^{\prime})\rightarrow((\varphi\equiv\varphi^{\prime})\equiv(\psi\equiv\psi^{\prime})))$
which is equivalent to (id7). Thus, Claim 3 is true. Finally, the inclusions
$\mathit{S3}_{\equiv}\subseteq\mathit{S4}_{\equiv}\subseteq\mathit{S5}_{\equiv}$
are clear by Definition 5.6. ∎
We are now able to establish the intended dualities between some of our
$\mathit{SCI}$-theories and corresponding modal systems.
###### Theorem 5.9.
The logics $\mathit{S1SP}_{\equiv}$, $\mathit{S3}_{\equiv}$,
$\mathit{S4}_{\equiv}$ and $\mathit{S5}_{\equiv}$ introduced in Definition 5.6
are the dual $\mathit{SCI}$-theories of the modal logics $\mathit{S1SP}$,
$\mathit{S3}$, $\mathit{S4}$ and $\mathit{S5}$, respectively.
###### Proof.
We prove the duality between $\mathit{S3}$ and $\mathit{S3}_{\equiv}$. The
remaining dualities follow in the same way. First, let us check that the
logics $\mathcal{L}:=\mathit{S3}$ and
$\mathcal{L}_{\equiv}:=\mathit{S3}_{\equiv}$ satisfy the conditions of
Definition 5.3. On the one hand, we know that $\mathit{S3}$ is the weakest
Lewis modal system containing principle SP (c.f. [9, 11]) and thus contains
$\mathit{S1SP}$. On the other hand, by Theorem 5.8 and Lemma 5.7, we know that
$\mathcal{L}_{\equiv}=\mathit{S3}_{\equiv}$ contains $\mathit{SCI}$ and
theorems $(\chi\equiv\psi)\equiv\square(\chi\leftrightarrow\psi)$. It remains
to check one of the equivalent conditions (i) or (ii) of Definition 5.3. We
show that (i) holds. So let $\varPhi\cup\\{\varphi\\}\subseteq Fm_{\equiv}$
and suppose $\varPhi\vdash_{\mathit{S3}_{\equiv}}\varphi$. We show
$\mathit{box}(\varPhi)\vdash_{\mathit{S3}}\mathit{box}(\varphi)$ by induction
on the length $n\geq 1$ of derivations of $\varphi$ from $\varPhi$ in
$\mathit{S3}_{\equiv}$. If $n=1$, then we distinguish the following cases
(a)–(d).
(a) $\varphi\in\varPhi$. Then trivially
$\mathit{box}(\varphi)\in\mathit{box}(\varPhi)$ and thus
$\mathit{box}(\varPhi)\vdash_{\mathit{S3}}\mathit{box}(\varphi)$.
(b) $\varphi$ has the form of a classical tautology. Since translation
$\mathit{box}$ preserves logical connectives, it follows that
$\mathit{box}(\varphi)$ is of the same form, i.e., has the form of a classical
tautology, too, and as such is an axiom of $\mathit{S3}$.
(c) $\varphi$ is an instance of scheme (1), say
$\varphi=(\chi\equiv\psi)\leftrightarrow((\chi\leftrightarrow\psi)\equiv\top)$.
By definition of $\mathit{box}$:
$\mathit{box}(\varphi)=\square(\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi))\leftrightarrow\square((\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi))\leftrightarrow\top)$.
Considering the definition of the identity connective
$(\varphi_{1}\equiv\varphi_{2}):=\square(\varphi_{1}\leftrightarrow\varphi_{2})$
in $\mathit{S3}$, this yields
$\mathit{box}(\varphi)=(\mathit{box}(\chi)\equiv\mathit{box}(\psi))\leftrightarrow((\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi))\equiv\top)$.
By Lemma 4.4,
$\square(\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi))\equiv((\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi))\equiv\top)$
is a theorem of $\mathit{S3}$. Applying SP, we get
$\begin{split}\mathit{box}(\varphi)&\equiv_{\mathit{S3}}((\mathit{box}(\chi)\equiv\mathit{box}(\psi))\leftrightarrow\square(\mathit{box}(\chi)\leftrightarrow\mathit{box}(\psi)))\\\
&=(\mathit{box}(\chi)\equiv\mathit{box}(\psi))\leftrightarrow(\mathit{box}(\chi)\equiv\mathit{box}(\psi)).\end{split}$
Of course, any such trivial biconditional is a theorem of $\mathit{S3}$ and so
is $\mathit{box}(\varphi)$.
(d) $\varphi$ is an instance of scheme (2), say
$\varphi=(\square\psi\rightarrow\psi)$. By Lemma 5.5(a),
$\mathit{box}(\varphi)\equiv_{\mathit{S3}}\square\mathit{box}(\psi)\rightarrow\mathit{box}(\psi)$.
The latter is an axiom of $\mathit{S3}$.
(e) $\varphi$ is an instance of scheme (3), say
$\varphi=\square(\psi\rightarrow\chi)\rightarrow\square(\square\psi\rightarrow\square\chi)$.
As in (d), we apply Lemma 5.5(a) and get
$\mathit{box}(\varphi)\equiv_{\mathit{S3}}\square(\mathit{box}(\psi)\rightarrow\mathit{box}(\chi))\rightarrow\square(\square\mathit{box}(\psi)\rightarrow\square\mathit{box}(\chi))$.
The latter is an axiom of $\mathit{S3}$.191919Note that the same argument is
applicable if we consider the axioms (3’), (4), (5). If $\varphi$ is such an
axiom, then $\mathit{box}(\varphi)$ is the corresponding axiom of modal system
$\mathit{S1SP}$, $\mathit{S4}$, $\mathit{S5}$, respectively.
Examining the cases (b)–(e) above, we conclude in particular the following
Fact: For any axiom $\chi$ of $\mathit{S3}_{\equiv}$, we have
$\mathit{box}(\chi)\equiv_{\mathit{S3}}\chi^{\prime}$, where $\chi^{\prime}$
is an axiom of modal system $\mathit{S3}$.
Now, suppose $\varphi$ is derived in $n+1$ steps and the assertion is true for
all derivations of length $\leq n$. We may assume that $\varphi$ is obtained
by an application of the rules MP or AN. In the former case, there are $\psi$
and $\psi\rightarrow\varphi$ derived in $\leq n$ steps, and the induction
hypothesis yields
$\mathit{box}(\varPhi)\vdash_{\mathit{S3}}\mathit{box}(\varphi)$. In the
latter case, $\varphi=\square\chi$ for some axiom $\chi$ of
$\mathit{S3}_{\equiv}$ that occurs in the given derivation. By Lemma 5.5(a)
and the Fact above,
$\mathit{box}(\varphi)\equiv_{\mathit{S3}}\square\mathit{box}(\chi)$ and
$\mathit{box}(\chi)\equiv_{\mathit{S3}}\chi^{\prime}$, where $\chi^{\prime}$
is an axiom of modal system $\mathit{S3}$. Since SP holds in $\mathit{S3}$, we
may replace $\mathit{box}(\chi)$ by $\chi^{\prime}$ in every context. Applying
SP in $\mathit{S3}$, we get
$\mathit{box}(\varphi)\equiv_{\mathit{S3}}\square\chi^{\prime}$. Since
$\chi^{\prime}$ is an axiom of $\mathit{S3}$, formula $\square\chi^{\prime}$
is a theorem of $\mathit{S3}$ by the rule of Axiom Necessitation. Hence,
$\mathit{box}(\varphi)$ is a theorem of $\mathit{S3}$. We have finished the
induction and thus the proof of the Theorem. ∎
We have established dualities between some particular $\mathit{SCI}$-theories
and corresponding Lewis-style modal logics by means of the respective
deductive systems (cf. Definition 5.3). How can these dualities be described
semantically? One easily recognizes that a given $\mathit{S1SP}$-algebra can
be transformed into an $\mathit{SCI}$-model defining
$f_{\equiv}(a,b):=f_{\square}(f_{\leftrightarrow}(a,b))$, where
$f_{\leftrightarrow}(a,b)$ is defined in the obvious way. This corresponds to
the theorem $(\varphi\equiv\psi)\equiv\square(\varphi\leftrightarrow\psi)$ of
$\mathit{S1SP}$. The resulting $\mathit{SCI}$-model then will be a model of
$\mathit{S1SP}_{\equiv}$. The other way round, any given $\mathit{SCI}$-model
which is a model of $\mathit{S1SP}_{\equiv}$ can be transformed into an
$\mathit{S1SP}$-algebra defining $f_{\square}(a):=f_{\equiv}(a,f_{\top})$.
This corresponds to the theorem $\square\varphi\equiv(\varphi\equiv\top)$ of
modal system $\mathit{S1SP}$. We conclude that the $\mathit{SCI}$-theory
$\mathit{S1SP}_{\equiv}$ is sound and complete w.r.t. the class of exactly
those $\mathit{SCI}$-models which can be obtained from
$\mathit{S1SP}$-algebras by the above presented transformation. So from a
semantic point of view, the duality between $\mathit{SCI}$-theory
$\mathit{S1SP}_{\equiv}$ and modal system $\mathit{S1SP}$ is given by those
respective classes of models (and the transformations in both directions).
Analogously, we can describe the remaining dualities semantically. Detailed
proofs derive straightforwardly from the above results.
Our view on intensionality as a measure for the discernibility of propositions
(‘the more propositions can be distinguished in models of the underlying logic
the higher degree of intensionality’) is presented here in a rather informal
and intuitive way. An interesting task for future work could be a precise
formalization of that concept – in classical as well as in non-classical
settings. The dualities established in this paper generalize and extend
earlier results (e.g. [2, 9]) or are in analogy with similar results that hold
in propositional logics distinct from $\mathit{SCI}$ (cf. [6]). The question
arises which further (hyper-) intensional logics can be represented in a
framework based on $\mathit{SCI}$ or based on a logic with different axioms
for propositional identity. Can all (hyper-) intensional logics be captured by
an appropriate axiomatization of propositional identity? These and similar
questions remain to be further investigated.
## References
* [1] S. L. Bloom and R. Suszko, Semantics for the sentential calculus with identity, Studia Logica 28, 77–81, 1971.
* [2] S. L. Bloom and R. Suszko, Investigation into the sentential calculus with identity, Notre Dame Journal of Formal Logic 13(3), 289–308, 1972.
* [3] M. J. Cresswell, Hyperintensional logic, Studia Logica 34(1), 25–38, 1975.
* [4] C. Fox, S. Lappin, Foundations of Intensional Semantics, Blackwell Publishing, 2005.
* [5] G. E. Hughes and M. J. Cresswell, A new introduction to modal logic, Routledge, 1996.
* [6] T. Ishii, Propositional calculus with identity, Bulletin of the Section of Logic 27(3), University of Łódź, 1998.
* [7] S. Lewitzka, $\in_{K}$: A non-Fregean Logic of Explicit Knowledge, Studia Logica 97(2), 233–264, 2011.
* [8] S. Lewitzka, Construction of a canonical model for a first-order non-Fregean logic with a connective for reference and a total truth predicate, Logic Journal of the IGPL 20(6), 1083–1109, 2012.
* [9] S. Lewitzka, Algebraic semantics for a modal logic close to S1, Journal of Logic and Computation 26(5), 1769–1783, 2016, first published online: November 27, 2014.
* [10] S. Lewitzka, A modal logic amalgam of classical and intuitionistic propositional logic, Journal of Logic and Computation 27(1), 201–212, 2017, first published online: July 20, 2015.
* [11] S. Lewitzka, Denotational semantics for modal systems S3–S5 extended by axioms for Propositional quantifiers and identity, Studia Logica 103(3), 507–544, 2015.
* [12] C. Pollard, Hyperintensions, Journal of Logic and Computation 18(2), 257–282, 2008.
* [13] R. Suszko, Identity connective and modality, Studia Logica 27, 7–39, 1971.
* [14] R. Suszko, Abolition of the fregean axiom, Lecture Notes in Mathematics, 453:169–239 (1975), in: R. Parikh (ed.), Logic Colloquium, Springer Verlag, 2006.
* [15] R. Wawrzynczak, Some Boolean theories in SCI, Bulletin of the Section of Logic 2(3), 197–204, 1973.
|
# A Two-Speed Actuator for Robotics with Fast Seamless Gear Shifting
Alexandre Girard1 and H. Harry Asada2 This work was supported by The Boeing
Company. 1 A. Girard is with the Department of Mechanical Engineering,
Universite de Sherbrooke, Qc, Canada.2 H. H. Asada is with Department of
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA,
USA.3 ©IEEE. Personal use of this material is permitted. Permission from IEEE
must be obtained for all other uses, in any current or future media, including
reprinting/republishing this material for advertising or promotional purposes,
creating new collective works, for resale or redistribution to servers or
lists, or reuse of any copyrighted component of this work in other works.
DOI:10.1109/IROS.2015.7354047
###### Abstract
This paper present a novel dual-speed actuator adapted to robotics. In many
applications, robots have to bear large loads while moving slowly and also
have to move quickly through the air with almost no load. This lead to
conflicting requirements for their actuators. Multiple gear ratios address
this issue by allowing an effective use of power over a wide range of torque-
speed load conditions. Furthermore, very different gear ratios also lead to
drastic changes of the intrinsic impedance, enabling a non-back-drivable mode
for stiff position control and a back-drivable mode for force control. The
proposed actuator consists of two electric motors coupled to a differential;
one has a large gear ratio while the other is almost direct-drive and equipped
with a brake. During the high-force mode the brake is locked, only one motor
is used, and the actuator behaves like a regular highly-geared servo-motor.
During the high-speed mode the brake is open, both motor are used at the same
time, and the actuator behaves like a direct drive motor. A dynamic model is
developed and novel controllers are proposed for synergic use of both motors.
The redundancy of motors is exploited for maintaining full control of the
output during mode transitions, allowing for fast and seamless switching even
when interacting with unknown environments. Results are demonstrated with a
proof-of-concept linear actuator.
## I INTRODUCTION
In many robotic systems, including legged robots and wearable robots,
actuators are often required to operate in distinctively different torque-
speed load conditions. A legged robot, for example, has to move its leg
forward quickly through the air and, once touching the ground, it has to bear
a large load [1]. These two operating conditions, high speed at low torque vs.
high torque at low speed, are often an order of magnitude different, while the
required output power is similarly low. When the torque-speed load conditions
do not vary significantly, a single gear ratio can be picked so that the
actuator always operate under nearly optimal conditions. For distinctively
different torque-speed conditions, the actuator will be far for its optimal
operating conditions with a gear ratio picked from the middle ground. As
illustrated on Fig. 1 for a typical electromagnetic (EM) actuator, extremum
torque-speed conditions are not optimal in term of efficiency and power
output. This often leads to the use of oversized and inefficient actuators,
which is inhibitory particularly for mobile robots.
Figure 1: Limitations of EM motors for extremum torque-speed operations
Automobiles with internal combustion (IC) engines use transmissions with
multiple gear ratios to match torque-speed conditions. IC engines have a very
narrow speed range in which they can effectively deliver power; a transmission
with multiple gear ratios is a necessity for the engine to work effectively
for a wide range of output speed. EM motors are more flexible than IC engines,
but still far from ideal sources. EM motors cannot output high power at low
speed because of thermal dissipation and magnetic flux limits related to
material properties; are limited in speed by the supply tension and others;
and are very inefficient when producing large forces at low speed [2]. In
robotic, since it is often the extremums, i.e. maximum torque and speed, that
determine the actuator design instead of the power requirement, much can be
gained with multiple gear ratios.
It will be a significant breakthrough if a type of multiple speed transmission
can be used effectively in robotics. The use of multiple reduction ratios
would enable actuators to meet diverse speed-torque load conditions. Even a
small, lightweight actuator can generate large torques and move at high speed
if equipped with both large and small gear ratio. Moreover, multiple speed
transmission can allow an actuator to work closer to its optimal operating
conditions, improving overall efficiency significantly. Furthermore, gear
shifting significantly changes the intrinsic impedance of an actuator. Since
the impedance is proportional to the square of gear ratio, the impedance may
vary in a range that is several order-of-magnitude different. The actuator may
be made back-drivable while using its small reduction ratio, an important
property in many applications where the robot physically interacts with the
environment [3]. Also the same actuator may be made non-back-drivable while
using its large reduction ratio, allowing the actuator to support loads
without consuming energy and enabling high-stiffness position control.
Despite the desirable features, gear shifting is more technically challenging
in robotics applications than in vehicle applications. For powertrains, the
load is mostly a large inertia, while for robots, the loads may exhibit a rich
range of dynamics including spring-like and damper-like loads. Hence, unlike
vehicle applications, leaving the load free momentarily during transitions
(from one gear ratio to another) is not acceptable in the context of robotics;
the actuator must be engaged with the load at all time to maintain control of
the output. Hence, an effective gear shifting methodology is necessary for
seamless transitions under diverse load conditions.
This paper presents a novel two-speed actuator that meets the requirements of
the robotics context. The system consist of a highly geared EM motor and a
direct drive EM motor equipped with a locking brake, both engaged with a
3-port gearbox. In high-force mode the actuator is non-back-drivable and can
produce large torque at low speed, while in high-speed mode the actuator is
back-drivable and capable of force control and high-speed motion. Moreover,
the two motors are coordinated to fully control the output load, while
performing fast and seamless mode transitions. In the following, related works
are briefly reviewed (section II), then the principle of the proposed two-
speed actuator is presented (section III). Next, a dynamic model is derived
(section IV) and control algorithms for fast seamless transition and
synergistic coordination of the dual motors will be discussed (section V). The
results are demonstrated experimentally with a proof-of-concept linear
actuator (section VI).
## II Related Works
Traditional robots generally use actuators that behave as displacement-sources
because of their high intrinsic impedance. These include geared EM motors and
hydraulics cylinders. Using a force sensor, it is possible to control the
output force using those actuators, but the bandwidth is rather limited. To
guarantee the stability of the force-feedback scheme only half the intrinsic
inertia can be canceled [3]. Since 70’s, roboticists have been attempting to
build actuators that can behave naturally as a force-source such as series-
elastic actuators, pneumatic cylinder and air-muscles [4][5]. However, because
of the physical limitation of compliant transmission materials, the achievable
bandwidth is limited and precise position control is hardly achievable.
Another interesting approach is using force-controllable clutches to decouple
a geared EM motor with high-impedance [6]. However, those actuators using
friction-drive or electro/magnetorehologic fluids clutches have poor
efficiency and limited bandwidth and fidelity due to physical limitation and
difficulties for accurate control. Direct drive EM actuators are the best
force-source actuators with high fidelity, high bandwidth. However, the very
low force density and low efficiency at low speeds make them impractical for
mobile robot applications [2]. On the other hand, actuators with non-back-
drivable mechanisms have the advantage for pure position control tasks and
they can bear very large load without any power consumption.
Since both small and large intrinsic impedances are advantageous in different
scenario, several group have developed variable intrinsic impedance actuators.
Novel design concepts, such as those using a variable stiffness spring [7] or
antagonist non-linear devices [8], can alter the stiffness continually, yet
the variable range is rather limited. A promising approach using a series-
compliance that can be locked with a brake vary the impedance in a broader
range [9]. Another concept is a dual actuator systems using EM motors in a
serial configuration, one controlling the position and the other the impedance
[10]. Furthermore, so-called macro-micro actuators, can improve the bandwidth
of force-source type of actuators by exploiting the high-bandwidth of a small
actuator in parallel, allowing for wider-range impedance control and improved
position control[11].
While the actuator work in robotics have been focused on impedance and
bandwidth issues, in the powertrain field the torque-speed matching issue is
predominant, since power density and efficiency are critical for mobile
systems. The idea of using multiple gear ratios with electric motors has been
explored occasionally, to improve efficiency and power density [12]. A twin
motor configuration has been proposed for smooth gear shifting, where each
motor shifts at a different timing [13]. Also, a dual motor configuration
using a planetary coupling and non-back-drivable worm-gears was proposed for a
mobile robot powertrain [14]. Multiple speed powertrains provide effective
solutions for torque-speed matching, however robots require more distinct gear
ratios and a gear shifting methodology adapted to the wide dynamic behaviors
of the load.
Actuator | Force | Stiff position | Ideal op. | Holding
---|---|---|---|---
technology | control | control | speed11footnotemark: 1 | efficiency
Geared EM | Poor | Good | Low | Good
Direct drive EM | Good | Fair | High | Poor
Series elastic | Fair | Poor | Low22footnotemark: 2 | Good
Micro-Macro | Good | Fair | Low22footnotemark: 2 | Good
Serial Dual Motors | Good | Fair | High | Poor
Clutch actuators | Fair | Poor | Low22footnotemark: 2 | Poor
Pneumatic | Fair | Poor | High | Good
Hydraulic | Poor | Good | Low | Good
1 Operating speed at which the actuator is powerful and efficient.
2 Assuming the primary actuator is a geared EM motor.
TABLE I: Actuator technologies for robotics
Table I summarizes the comparison of the actuators discussed above. Note that
the actuators developed for good force control performance have lost many
advantages of traditional high impedance actuators. The proposed dual-speed
actuator with two distinctively different gear ratios is an alternative
approach that combines the advantages of both types of actuators. Compared to
the existing variable impedance actuators, the new design can achieve order-
of-magnitude different impedance instead of continuous small variations.
Moreover, the proposed dual-speed design aims to improve available power and
efficiency over a wide range of operating speed, which has rarely been
addressed in the robotics literature. All actuators listed in Table I are
powerful and efficient for either low or high output speed alone. Furthermore,
unlike the powertrains of automobiles, the proposed concept has distinctively
different reduction ratio and the output load is always fully under control
even during the transitions.
## III Dual-speed dual-motor actuator principle
The new design concept, referred to as a Dual-Speed Dual-Motor (DSDM)
actuator, consists of a direct drive motor (M1) equipped with a locking brake
and an geared EM motor (M2) with a large reduction ratio coupled to the same
output through a differential, see Fig. 2. The differential can be viewed as a
series-type junction where the speeds add up and the force is shared.
Figure 2: DSDM actuator concept
The envisioned implementation of the DSDM concept is to embed all the
components into a single compact unit, as illustrated by Fig. 3. A lot of
weight and space could be saved by combining the reduction and the
differential gearing and having all the components inside a single housing.
Figure 3: Possible architecture of an integrated DSDM concept
The DSDM can be used in two modes, high-force mode when the brake is closed
and high-speed mode when the brake is open. The result is like having two very
different reduction ratio you can choose from during operation. Fig. 4
conceptually illustrates the principle with a leverage analogy, M1 acts like a
force source connected almost directly to the output and M2 acts like a
displacement source with a large lever arm relative to the output.
Figure 4: Dual input system
During the high-force mode, see Fig. 5a, the brake is closed and M2 drives the
output with a large mechanical advantage. The result is a low-speed
displacement-source type of actuation like a geared EM motor. During the high-
speed mode, see Fig. 5b, M1 drive the output almost directly, creating a high-
speed force-source actuator like a direct drive EM motor. Additionally, both
motors can be used simultaneously to drive the output even faster.
(a) High force mode
(b) High speed mode
Figure 5: Two modes of operation
Fig. 6 illustrates the operating range of the DSDM actuator plotted on the
standard torque-speed plane. The high-force mode region is determined by the
performance of M2 alone, since M1 is locked. The high-speed mode region can
exceed the performance of M1 alone, as M2 can be used simultaneously to
increase the output speed. The fail safe zone indicate the guaranteed
performance of the DSDM actuator in case of failure in either motor.
Figure 6: DSDM actuator operation region, with a difference between M1 and M2
gearing ratio of only 4 for illustration purposes
### III-A Weight advantage
A DSDM actuator will be lighter than a single motor for applications with a
wide range of operating speed. Suppose that an actuator must generate 10 W
output power at two operating points: 0.5 Nm of torque at a speed of 20
rad/sec and 0.1 Nm at 100 rad/sec. A single EM motor that satisfies these
requirements at both operating points tends to be oversized in terms of power,
to reach both operating points, see Fig. 7a. A DSDM actuator can reach the
same operating points using two smaller motors with appropriate gear ratios,
see Fig. 7b. On the other hand the DSDM actuator uses more components: two
motors instead of one, more gearing and an additional brake. The DSDM concept
pays-off when the difference in speed between two required operating points
becomes larger. Fig. 8 shows the estimated weight of actuators in relation to
the ratio of operating speeds ($\lambda=\frac{w_{1}}{w_{2}}$), while the
required power output is kept at 10 W. The actuator weight is computed
assuming that the mass of each component is proportional to its maximum output
torque, with values taken from commercially available components in the 10 -
100 watts range: 2 kg/Nm for motors, 0.1 kg/Nm for gearboxes and differentials
and 0.2 kg/Nm for brakes [15]. As shown in Fig. 8, the DSDM concept becomes
advantageous when there is a large speed difference between the operating
points. This is because only the gearbox and brake need to be scaled up for
the DSDM actuator to meet the high torque requirement of the low-speed
operating point, while the motor size must be increased for the single motor
solution.
(a) One motor solution
(b) DSDM solution
Figure 7: Case study of two actuator solution for two 10 W operating points
Figure 8: Weight of a single motor compared to the DSDM concept for two 10 W
operating points at different speeds $w_{1}=100$ rad/sec,
$w_{2}=w_{1}/\lambda$
### III-B Efficiency advantage
Since EM motor are not efficient when producing large torque at low speeds,
the DSDM concept can be very advantageous in term of efficiency. The power
loss $P_{loss}$ in an EM motor is proportional to the square of the motor
output torque $\tau_{m}$. Hence, for a given output torque $\tau_{out}$, the
motor with the larger gear ratio $R$ will be more efficient:
$\displaystyle
P_{loss}=rI^{2}=\frac{r}{k_{m}^{2}}\tau_{m}^{2}=\frac{r\tau_{out}^{2}}{k_{m}^{2}R^{2}}$
(1)
where $r$ is the winding resistance and $k_{m}$ is the motor torque constant.
Taking the same torque-speed requirements used for analyzing the weight
advantage (see Fig. 8), the single motor would have power loss $\lambda^{2}$
times greater than the DSDM at the low speed operating point.
## IV Modeling
### IV-A 3-ports planetary gear junction
If a planetary gear box is used to implement the serial junction with the
planet carrier connected to the output, M1 to the sun gear and M2 to the ring
gear, the kinematic relation of the system is given by
$\displaystyle
w_{o}=\underbrace{\left[\frac{1}{N+1}\right]}_{1/R_{1}}w_{1}+\underbrace{\left[\frac{N}{r_{2}(N+1)}\right]}_{1/R_{2}}w_{2}$
(2)
where $r_{2}$ is the additional reduction of M2, $N$ is the ratio of gear
teeth of the ring gear over the sun gear, and $w_{o}$, $w_{1}$ and $w_{2}$ are
angular velocities of the output shaft (port $o$), M1 input shaft (port $1$)
and M2 input shaft (port $2$). Assuming the planet gears are massless, the
torque relation of the system is given by:
$\displaystyle-\tau_{o}=\underbrace{\left[N+1\right]}_{R_{1}}\tau_{1}=\underbrace{\left[\frac{r_{2}(N+1)}{N}\right]}_{R_{2}}\tau_{2}$
(3)
Hence, the 3-ports planetary coupling can be interpreted as a 0-junction, in
the bond graph terminology, with different mechanical advantages ($R_{1}$ and
$R_{2}$) on each input ports.
### IV-B Dynamics
Fig. 9 shows a lumped-parameter dynamic model of a DSDM when the brake is open
(high-speed mode). $J_{i}$ and $b_{i}$ are the inertia and damping of the
respective i-th ports, $I_{1}$, $I_{2}$, $k_{1}$, $k_{2}$ are the currents and
the torque constants of M1 and M2. Applying Newton’s law on each ports yields
the following equations of motions:
$\displaystyle\tau_{ext}-\tau_{o}$ $\displaystyle=Z_{o}(s)w_{o}$ (4)
$\displaystyle k_{1}I_{1}-\tau_{1}$ $\displaystyle=Z_{1}(s)w_{1}$ (5)
$\displaystyle k_{2}I_{2}-\tau_{2}$ $\displaystyle=Z_{2}(s)w_{2}$ (6)
where $Z_{i}(s)=J_{i}s+b_{i}$ represents the mechanical impedance of the i-th
ports. Note that the system is coupled due to the constraint given by eq. (2)
and (3), and that there is only two degrees of freedom among the three ports.
Fig. 10, illustrate the coupled equations motion in block diagram form.
Figure 9: Lumped-parameter dynamic model of a DSDM Figure 10: Dynamics of a
DSDM
It is then possible to eliminate one variable and express the dynamic as the
following system of two equations:
$\displaystyle\left[\begin{array}[]{l}k_{1}I_{1}\scriptstyle+\textstyle\frac{\tau_{ext}}{R_{1}}\\\
k_{2}I_{2}\scriptstyle+\textstyle\frac{\tau_{ext}}{R_{2}}\\\
\end{array}\right]=\left[\begin{array}[]{c
c}\frac{Z_{o}(s)}{R_{1}}&\scriptstyle Z_{1}(s)\\\
\frac{Z_{o}(s)}{R_{2}}\scriptstyle+R_{2}Z_{2}(s)&\scriptstyle-\textstyle\frac{R_{2}}{R_{1}}\scriptstyle
Z_{2}(s)\\\ \end{array}\right]\left[\begin{array}[]{l}w_{o}\\\ w_{1}\\\
\end{array}\right]$ (13)
The equations can be converted to state space form:
$\displaystyle\left[\begin{array}[]{c}\dot{w_{o}}\\\
\dot{w_{1}}\end{array}\right]$
$\displaystyle=\frac{-1}{J_{T}}\left[\begin{array}[]{c c}b_{T}&0\\\
R_{1}b_{o}&0\\\ \end{array}\right]\left[\begin{array}[]{c}w_{o}\\\
w_{1}\end{array}\right]+\underline{B}\left[\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right]$ (22) $\displaystyle\text{with}\quad\underline{B}$
$\displaystyle=\frac{1}{J_{T}}\left[\begin{array}[]{c
c}k_{1}R_{1}&k_{2}R_{1}\frac{R_{1}J_{1}}{R_{2}J_{2}}\\\
k_{1}(R_{1}^{2}+\frac{R_{1}^{2}J_{o}}{R_{2}^{2}J_{2}})&-k_{2}\frac{R_{1}J_{o}}{R_{2}J_{2}}\\\
\end{array}\right]$ (25) $\displaystyle J_{T}$
$\displaystyle=\scriptstyle\left[J_{o}+R_{1}^{2}J_{1}+\left(\frac{R_{1}}{R_{2}}\right)^{2}\frac{J_{1}}{J_{2}}J_{o}\right]$
(26) $\displaystyle b_{T}$
$\displaystyle=\scriptstyle\left[b_{o}+\left(\frac{R_{1}}{R_{2}}\right)^{2}\frac{J_{1}}{J_{2}}b_{o}\right]$
(27)
where the external torque and the damping at each input port are eliminated
for brevity. From eq.(22) it can be seen that if $R_{1}<<R_{2}$ then M1
current $I_{1}$ dominates the output responses during high-speed mode.
## V Control
### V-A Output nullspace
From the kinematic input-output view point, the DSDM actuator has one
redundant degree of freedom. In other words, there is an infinite number of
combinations of $w_{1}$ and $w_{2}$ producing the same output speed $w_{0}$,
from eq.(2):
$\displaystyle\left[w_{o}\right]=\left[\begin{array}[]{c
c}\frac{1}{R_{1}}&\frac{1}{R_{2}}\end{array}\right]\left[\begin{array}[]{c}w_{1}\\\
w_{2}\\\ \end{array}\right]$ (31)
A vector perpendicular to the above coefficient vector forms the null space of
the DSDM actuator system. Any input combination in this direction produces
zero output speed:
$\displaystyle\left[\begin{array}[]{c}w_{1}\\\ w_{2}\\\
\end{array}\right]=\underbrace{\left[\begin{array}[]{c}1\\\ -R_{2}/R_{1}\\\
\end{array}\right]}_{\text{Nullspace
Projection}}u\;\rightarrow\;w_{0}=0\quad\forall u\in\Re$ (36)
Interestingly, a similar expression can be obtained for the dynamics of the
output in response to current inputs at the two motors, from eq.(22):
$\displaystyle J_{T}\dot{w}_{o}+b_{T}w_{o}=$
$\displaystyle\left[\begin{array}[]{c
c}k_{1}R_{1}&k_{2}R_{1}\frac{R_{1}J_{1}}{R_{2}J_{2}}\end{array}\right]\left[\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right]$ (40)
Hence, there is a one degree of freedom space of inputs $I_{1}$ and $I_{2}$
that do not affect the output:
$\displaystyle\left[\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right]=\underbrace{\left[\begin{array}[]{c}\frac{J_{1}}{k_{1}}\\\
-\frac{R_{2}J_{2}}{R_{1}k_{2}}\end{array}\right]}_{\text{Nullspace
Projection}}u\;\rightarrow\;J_{T}\dot{w}_{o}+b_{T}w_{o}=0\quad\forall u\in\Re$
(45)
### V-B High-force mode
During high-force mode, with M1 locked by the brake, the actuator behaves like
a regular geared EM motor and standard speed or position control scheme can be
used.
### V-C High-speed mode
Figure 11: High-speed mode control loop
With a small reduction ratio $R_{1}$ the DSDM actuator can be controlled like
a direct drive motor: the output torque can be controlled directly by
controlling M1 current, see inner feedback loop in Fig. 11. The output loop
can be used for various objectives. With the measurement of output position
and velocity, impedance control, friction compensation and feedback
linearization can be implemented. Moreover, exploiting the nullspace, a
secondary objective can be brought into the system without influencing the
output. In case the first motor is overloaded, for example, the second motor
can reduce the load by projecting inputs through the null space, producing no
effect upon the output, but changing the proportion of the two input commands.
Note that the nullspace projection vector, see eq.(45), depends only on
parameters associated with the motors. Therefore, it is possible to project
the secondary controller inputs on the output nullspace even if the output
dynamic parameters that include the load inertia and damping are unknowns. The
secondary controller can be used not only for load balancing but also for
minimizing power consumption, avoiding speed saturation of M1, etc.
### V-D Seamless mode transitions
Unlike powertrains, where the load is mostly a large inertia and can be
disengaged momentarily during gear shiftings, robotic actuators face a wide
range of dynamic load, which must be controlled continuously even during a
transition. For instance, if it drives a spring-like load, even a very short
disengagement from the load might lead to large undesirable output motion.
This section addresses transition control between the two modes.
#### V-D1 From high-force mode to high-speed mode
This transition control is simple. The locking brake can be released anytime,
M1 is then instantaneously freed and the controller can immediately switch to
the high-speed control mode.
#### V-D2 From high-speed mode to high-force mode
For this transition, M1 speed $w_{1}$ must be brought to zero so that the
locking brake can be engaged. Two algorithms, a kinematic and a dynamic
approach, can be considered. The former assumes local high gain velocity
feedback controls for the individual motors. Hence velocities $w_{1}$ and
$w_{2}$ can be treated as control inputs. Using the nullspace projection
vector from eq.(36), the kinematic control law can be written as
$\displaystyle\left[\begin{array}[]{c}w_{1}\\\ w_{2}\\\
\end{array}\right]=\underbrace{\left[\begin{array}[]{c}1\\\ -R_{2}/R_{1}\\\
\end{array}\right]}_{\text{Nullspace
Projection}}u_{1}+\left[\begin{array}[]{c}0\\\ R_{2}\\\
\end{array}\right]u_{2}$ (52)
leading to
$\displaystyle w_{o}=u_{2}\quad\text{with}\quad w_{1}=u_{1}$ (53)
Therefore during transitions, using $u_{1}$ velocity $w_{1}$ can be driven to
zero, while fully controlling the output velocity using $u_{2}$. The kinematic
control law is valid only when high fidelity velocity controls are available.
Alternatively the dynamic control algorithm does not require this. While
running the high-speed control scheme of Fig. 11, a braking law for $w_{1}$
can be used in parallel as the secondary controller projected on the output
nullspace:
$\displaystyle\left[\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right]=\underbrace{\left[\begin{array}[]{c}\frac{J_{1}}{k_{1}}\\\
-\frac{R_{2}J_{2}}{R_{1}k_{2}}\end{array}\right]}_{\text{Nullspace
Projection}}\underbrace{-Cw_{1}}_{\text{Braking
Law}}+\left[\begin{array}[]{c}\frac{1}{k_{1}R_{1}}\\\
0\end{array}\right]\tau_{d}$ (60)
leads to:
$\displaystyle J_{T}\dot{w}_{o}+b_{T}w_{o}=$
$\displaystyle\,\tau_{d}\quad\text{and}\quad\dot{w}_{1}=-Cw_{1}+f(w_{o},\tau_{d})$
(61)
Hence, the output is not influenced by the braking law due to orthogonality,
and is still controlled using the desired torque $\tau_{d}$ determined by the
high-speed mode controller. On the other hand, $w_{1}$ is directly influenced
by the braking law but also by the output speed and the desired output torque
$\tau_{d}$. Mathematically, it would be possible to also fully uncouple
$\dot{w}_{1}$ equation, but the control law would not be practical in the
scenario of $R_{1}<<R_{2}$ when considering torque and speed saturations.
Increasing the gain $C$ will lead to faster braking of $w_{1}$, however
$I_{2}$ will saturate if the gain is too large. A large $C$ can still be used
for faster braking at a cost of deviation from the desired torque $\tau_{d}$.
There is a trade-off, passed the $I_{2}$ saturation point, between fast
braking of $w_{1}$ for fast transition and high-fidelity output torque
control.
### V-E Automatic mode selection
To simplify the programming of a DSDM actuator by an end-user, it is
interesting to integrate an automatic mode selection in the control scheme. As
shown in Fig. 12, the mode selection could be made based the on the load
conditions and a user-provided desired impedance for each task. When a large
impedance is desired, the actuator automatically select the mode that allow
the fastest convergence to the target given the load conditions. Hence, the
actuator will use high-speed mode, except when encountering resistance it will
automatically ”down shift” to high-force mode. If a low impedance or torque
limitation is desired, then the DSDM actuator must stay in high-speed mode to
enforce those specifications.
Figure 12: State machine for automatic mode selection
## VI EXPERIMENTAL VALIDATIONS
### VI-A Test bench design
A test bench was developed to demonstrate the functionality of the DSDM
concept, see Fig. 13. Discrete components (motors, brake, gearhead and
differential) are used for the ease of implementation and future
modifications. Two 20 Watts DC motor (Maxon RE 25) are used for both M1 and M2
positions. M1 is equipped with an electromechanical brake (Maxon AB 28) and M2
with a gearhead reductor of 18:1 (Maxon GP 32). The differential is a custom
made 3-ports planetary gear, see Fig. 14, with an additional reduction of 3:1
on the ring gear input. The planetary gearing is designed with a ratio $N$ of
3 (see eq.(2)), hence $R_{1}$= 4 and $R_{2}$= 72. The output is connected to a
20 mm lead ballscrew, leading to specifications given by Table II.
Figure 13: Proof-of-concept DSDM linear actuator Mode | Max Force | Max Speed | Apparent Output Mass
---|---|---|---
High-force | 600 N | 40 mm/sec | 750 kg
High-speed | 30 N | 0.7 m/sec | 2 kg
TABLE II: Prototype specifications using two 20 W motors
(a) Prototype
(b) CAD section view
Figure 14: Custom 3-ports gearbox
### VI-B Controller Implementations
The controller was implemented using a NI Compact Rio platform and two NI 9505
DC motor drives. All inner current control loops are implemented on the FPGA
at a 20 kHz sampling rate and the outer control loops are implemented in the
real time OS at a 500 Hz sampling rate. Two encoders provide the position at
both motor shafts and the output position is computed from those values using
the kinematic relation of eq.(2). The current is sensed with a Hall effect
sensor (ACS712) and the speed data is computed using filtered discrete
differentiation of the position data.
### VI-C High-force mode
For the high-force mode experiment, M2 is controlled using an experimentally
tuned PI with an anti-windup scheme. Fig. 15 shows the response to a 200 mm
step of the reference position, with a perturbation at $t\,$= 5 sec (human
applying about 50 lbs), see experiment #1 in the video. The results show that
the actuator is speed limited (maximum supply voltage reached) for most of the
course, but very strong as illustrated by a small speed loss during the
perturbation. The actuator precision is also very good as the final
positioning error is less than a micron (neglecting the backlash as the
position sensor is upstream of the gearbox). Hence, the actuator acts like a
strong and precise displacement source as desired in this mode.
Figure 15: Output motion using high-force mode for a 200 mm target step
### VI-D High-speed mode
For the high-speed mode experiment, the inner current controllers are pure
integrator leading to closed-loop bandwidth well over 500 Hz. Part of the
prototype coulomb friction (mainly in the ballscrew) is canceled using the
feedback linearization loop, see Fig. 11, and the prescribed impedance is a
pure stiffness, relying on the natural friction of the system to provide the
damping. Fig. 16 shows the results of, as before, the response to a 200 mm
reference position step, see experiment #2 in the video. The result with the
high-speed mode is much more dynamic than the high-force mode response, and a
maximum speed of 0.3 m/sec was reach during the motion. The actuator overshoot
a little bit before stabilizing on the target. Also, when the desired
impedance is set to zero, the actuator is easy to back-drive by hand, as
illustrated by experiment #3 in the video. Moreover, the actuator can limit
its output force and handle collision without stability issues as demonstrated
in experiment #5 in the video. An inherent advantage of not relying on force
feedback to control the impedance.
Figure 16: Output motion using high-speed mode for a 200 mm target step
### VI-E Mode transitions
Fig. 17 shows the results from an experiment were the actuator first control
the output speed using high-force mode, then switch to high-speed mode (M2
speed is then brought to zero for demonstration purposes) and then switch back
to high-force mode, see experiment #4 in the video. The results show that the
transitions are seamless and that the output is always under control, here
maintaining a constant speed. Since the actuator is fighting a dissipative
load (the friction in the ballscrew), a constant force must be applied at all
time, if not the output would suddenly stop. Note, the noisy behavior when
using high-speed mode at low speed is due to a limited angular velocity
resolution on $w_{1}$.
Figure 17: Mode transitions with a constant output speed target
Fig. 18 shows the results from an experiment were the actuator is controlled
in position (target at 300 mm) and automatically select the mode as proposed
in section V-E. Here the actuator first accelerate quickly using high-speed
mode since there is no resistance and switch back to high-force mode after
impacting a ball to continue its course (experiment #6 in the video). Hence
the transition is done while in contact with a spring-like load. The results
show that the DSDM is still able to switch mode quickly even during this
dynamic scenario. This experiment also illustrate how a single task can take
advantage of the two operating modes of the actuator; fast reaching and large
force as soon as the contact is made.
Figure 18: Automatic down shift during contact with a spring-like load
## VII CONCLUSION AND OUTREACH
In this paper, the use of distinctively different gear ratios with EM motors
in robotics is proposed. This allow for an effective use of power over a wide
range of torque-speed conditions, enabling smaller and more efficient actuator
systems. Furthermore, very different gear ratios lead to drastic intrinsic
impedance change, enabling a displacement-source type of behavior with the
large non-back-drivable reduction ratio and good force control capability with
the small reduction ratio. Additionally, a novel dual-motor implementation of
dual-speed is proposed to address the problematics of gear shifting in the
context of robotics. The proposed DSDM actuator can keep full control of the
output during gear shifting, with a controller exploiting the redundancy of
inputs. The DSDM actuator is advantageous for many actual robotic
applications, but also enabled new applications that are impossible with
regular actuators because the requirements are too conflicting to meet with a
single motor of usable size.
Also, many aspect of using multiple gear ratios in a robotic context have yet
to be explored. Gear shifting is a very non-linear process and it would be
interesting to explore with greater depth how it should be integrated in a
robot routine. Regarding the DSDM concept, in this first iteration two
identical motors were used. However, it could be more effective to use
asymmetric motors optimized for their particular roles. For instance M1 could
be optimized for force bandwidth and M2 for power density and precision.
Generalizing, it would even be possible to use different technologies for the
force source and the displacement source, recall Fig. 4.
## ACKNOWLEDGMENTS
Special thanks to Lluis Penalver-Aguila and James Torres for the help with the
detailed drawing and the manufacturing of the gearbox.
## References
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# Universality in High Energy Collisions of small and large systems
P. Castorina1,2, A. Iorio2, D. Lanteri1,3, H. Satz4 and M. Spousta2
###### Abstract
Strangeness enhancement and collective flow are considered signatures of the
quark gluon plasma formation. These phenomena have been detected not only in
relativistic heavy ion collisions but also in high energy, high multiplicity
events of proton-proton and proton-nucleus (“small systems”) scatterings. A
universal behavior emerges by considering the parton density in the transverse
plane as the dynamical quantity to specify the initial condition of the
collisions, which in $e^{+}e^{-}$ annihilation at the available energies is
too low to expect collective effects.
## 1 Introduction
Recent experimental results in proton-proton ($pp$) and proton-nucleus ($pA$)
collisions [1, 2, 3, 4, 5, 6, 7, 8, 9] support the conclusion that the system
created in high energy, high multiplicity collisions with these “small”
initial settings is essentially the same as that one produced with “large”
initial nucleus-nucleus ($AA$) configurations.
The ALICE collaboration reported [1] an enhanced production of multi-strange
hadrons, previously observed in $PbPb$ collisions [10], in high energy, high
multiplicity $pp$ events, shown in fig. 1.(left).
Moreover, the energy loss in $AA$ collisions was shown to scale in small and
large systems [11] and another important similarity among $pp$, $pA$, and $AA$
collisions was identified in several measurements of long-range di-hadron
azimuthal correlations [12, 4, 6, 3], indicating universality in flow-like
patterns.
In this contribution, based on ref. [13], the universal behavior will be
discussed by considering a dynamical variable, previously introduced to
predict the strangeness enhancement in $pp$ [14, 15, 16], which corresponds to
the initial entropy density of the collisions and takes into account the
transverse size (and its fluctuations) of the initial configuration in high
multiplicity events.
The next sections are devoted to the strangeness enhancement, mean transverse
momentum and elliptic flow. The final section contains our comments,
clarifying why collective effects cannot arise in $e^{+}e^{-}$ annihilation at
LEP (or lower) energies.
## 2 Universality in strangeness production and in mean transverse momentum
In fig.1.(left) the multi-strange hadron production data are plotted versus
the charge multiplicity at mid-rapidity. A stronger evidence of the universal
behavior in $pp,pA,AA$ can be discoverd by considering the initial entropy
density $s_{0}$, given in the one-dimensional hydrodynamic formulation [17] by
the form
$s_{0}\tau_{0}\simeq\frac{1.5}{A_{T}}\;\frac{dN^{x}_{ch}}{dy}=\frac{1.5}{A_{T}}\;\frac{N_{part}^{x}}{2}\left.\frac{dN^{x}_{ch}}{dy}\right|_{y=0}\;,$
(1)
with $x\simeq pp$, $pA$, $AA$. Here $A_{T}$ is the transverse area,
$(dN^{x}_{ch}/dy)_{y=0}$ denotes the number of produced charged secondaries,
normalized to half the number of participants $N_{part}^{x}$, in reaction $x$,
and $\tau_{0}$ is the formation time. The quantity $s_{0}\tau_{0}$ is directly
related to the number of partons per unit of transverse area and therefore,
due to the large fluctuations in high multiplicity events, a reliable
evaluation of the transverse area for different collisions as a function of
the multiplicity is required.
In studying the strangeness enhancement and the average $p_{t}$, we use
results from Glauber Monte Carlo (MC) [18] to obtain $A_{T}$ and the
multiplicity for $AA$ and for $pPb$ collisions. For $pp$ scaterings the
effective transverse area is sensitive to the fluctuations of the gluon field
configurations and therefore we apply the CGC parameterization of the
transverse size as a function of $N_{ch}$ [19]. The universal trend in
elliptic flow needs a more refined analysis (see sec. 3).
Alice data are plotted versus $s_{0}\tau_{0}$ in fig. 1.(right), showing a
complete smooth trend of the strangeness production among $pp,pA,AA$ data.
Figure 1: The strangeness production quantified in terms of the ratio of
yields of K, $\Lambda$, $\Xi$, and $\Omega$ hadrons to pions evaluated as a
function of the multiplicity (left panel) and of the initial entropy density
(right panel). Data from Ref. [1] (and refs. therein).
This universal behavior is confirmed by a similar analysis based on the
Statistical Hadronization model (SHM) [20], where strangeness production is
reduced with respect to the predicted rates by one further parameter,
$0<\gamma_{s}\leq 1$. The predicted rate for a hadron species containing
$\nu=1$, $2$, $3$ strange quarks is suppressed by the factor
$\gamma_{s}^{\nu}$.
The energy dependence of $\gamma_{s}$ is reported in fig. 2.(left), suggesting
a different behavior in strangeness production in $pp$ versus $AA$ collisions.
On the other hand, by plotting $\gamma_{s}$ versus the parton density in the
transverse plane ( see fig. 2.(right)), a smooth behavior for small and large
setting emerges, with a enhancement for $pp$ in high energy, high multiplicity
events. The universal trend shows that $\gamma_{s}$ increases with the parton
density in the transverse plane, up to the fixed point $\gamma_{s}=1$, where
any suppression disappears.
Figure 2: The strangeness suppression factor $\gamma_{s}$ as a function of the
energy (left) and of the initial entropy density (right) evaluated for data
from Refs. [21, 22, 23]. The Phobos parameterization [24] for the relation
between charge multiplicity, energy and the number of participants is applied
for RHIC data.
An analogous study can be done for the scaling of the average $p_{t}$, as
discussed in [19]. We analyze the average $p_{t}$ in the low transverse
momentum region where the soft, non-perturbative, effects in the particle
production are more important than in the higher $p_{t}$ range. The behavior
of the average $p_{t}$ is evaluated in the region $0.15<p_{t}<1.15$ GeV (fig.
3.(left)) and $0.15<p_{t}<2$ GeV (fig. 3-(right)) for different colliding
systems as a function of the previous dynamical variable. One can see that the
average $p_{t}$ for soft particle production follows the same slowly
increasing trend for all the collisional systems.
Figure 3: Average $p_{t}$ as a function of initial entropy density evaluated
in the interval of $0.15<p_{t}<1.15$ GeV (left) and $0.15<p_{t}<2$ GeV ([13]
for refs. to experimental data).
## 3 Elliptic flow and partecipants eccentricity
In non-central collisions, the beam direction and the impact parameter vector
define a reaction plane for each event. If the nucleon density within the
nuclei is continuous, the initial nuclear overlap region has an “almond-like”
shape and the impact parameter determines uniquely the initial geometry of the
collision. In a more realistic description, where the position of the
individual nucleons that participate in inelastic interactions is considered,
the overlap region has a more irregular shape and the event-by-event
orientation of the almond fluctuates around the reaction plane. Therefore, in
the analysis of the elliptic flow where the fluctuations are important, the
geometrical eccentricity is replaced by the participant eccentricity,
$\epsilon_{part}$, defined using the actual distribution of participants. The
size of the fluctuation in $\epsilon_{part}$ and its correlated transverse
area $S$ (different from the geometrical one, $A_{T}$) are evaluated by
Glauber MC.
The scaling of $v_{2}/\epsilon_{part}$ versus the initial entropy density is
depicted in fig. 4.(left) for $AA$ and $pp$. One can see that the $pp$ trend,
at lower values, is smoothly followed by the data-points from $AA$ collisions.
To clarify the different role of $A_{T}$ versus $S$, fig. 4.(right) shows that
the scaling in $v_{2}/\epsilon_{part}$ is not observed if one considers
$A_{T}$ rather than $S$ in evaluating the initial entropy density.
Figure 4: The $v_{2}/\epsilon_{part}$ values for $pp$, $PbPb$, $AuAu$, and
$CuCu$ evaluated as a function of entropy density (left) (for data references
see [13]). The $v_{2}/\epsilon_{part}$ values for $pp$ and $PbPb$ evaluated as
a function of entropy density when the geometrical transverse area $A_{T}$,
rather tha $S$, is used in the evaluation of the initial energy density for
data (right).
## 4 Comments and Conclusions
By the scaling variable $s_{0}\tau_{0}$, one can evaluate at which
multiplicity the same behavior in high-multiplicity $pp$ and $PbPb$ collisions
is expected, by solving the equation
$(dN/d\eta)_{AA}/A_{T}^{AA}=x/A_{T}^{pp}(x)$ for $x$ being the multiplicity in
$pp$. The result is shown in Tab. 2 of Ref. [13].
A final comment concerns the possible detection of collective effects in
$e^{+}e^{-}$ annihilation. Indeed, more recently, LEP data have been
reconsidered [25] to check if a flow-like behavior is generated with this
initial, small, non hadronic, setting. The answer is negative and confirmed at
lower energy by the BELLE collaboration [26]. Moreover, there is no
strangeness enhancement in $e^{+}e^{-}$ annihilation [27] as a function of the
available energy.
According to the universality point of view, strangeness enhancement and
collective flow are both indications of the formation of an initial system
with high parton number density in the transverse plane. Therefore, as
confirmed in Ref. [28], in $e^{+}e^{-}$ annihilation at LEP or lower energies
there is no chance of observing the enhancement of the strangeness production
or flow-like effects because the parton density in the transverse plane is too
small. Acknowledgements
P.C. is partially supported by UNCE/SCI/013.
## References
* [1] Jaroslav Adam et al., Nature Phys., 13:535–539, 2017.
* [2] Betty Bezverkhny Abelev et al.,Phys. Lett., B728:25–38, 2014.
* [3] Vardan Khachatryan et al., Phys. Lett., B765:193–220, 2017.
* [4] Georges Aad et al,Phys. Rev. Lett., 116(17):172301, 2016.
* [5] Vardan Khachatryan et al., JHEP, 09:091, 2010.
* [6] Vardan Chachatryan et al.Phys. Rev. Lett., 116(17):172302, 2016.
* [7] C. Aidala et al. Nature Phys., 15(3):214–220, 2019.
* [8] Edward K. G. Sarkisyan, Aditya Nath Mishra, Raghunath Sahoo, and Alexander S. Sakharov, Phys. Rev., D94(1):011501, 2016.
* [9] Constantin Loizides,Nucl. Phys., A956:200–207, 2016.
* [10] Betty Bezverkhny Abelev et al, Phys. Lett., B728:216–227, 2014.[Erratum: Phys. Lett.B734,409(2014)].
* [11] A. Adare et al. Phys. Rev., C93(2):024911, 2016.
* [12] Georges Aad et al., Phys. Rev. Lett., 110(18):182302, 2013.
* [13] P.Castorina, A.Iorio, D.Lanteri, M.Spousta and H.Satz, Phys. Rev. C 101, 054902 (2020).
* [14] P. Castorina and Helmut Satz. Eur. Phys. J., A52(7):200, 2016.
* [15] P. Castorina, S. Plumari, and H. Satz.Int. J. Mod. Phys., E25(08):1650058, 2016.
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* Loizides et al. [2018] Constantin Loizides, Jason Kamin, and David d’Enterria.
* [19] L. McLerran and P. Tribedy.Nucl. Phys., A945:216–225, 2016 and references therein.
* [20] F. Becattini, review paper International School on Quark-Gluon Plasma and Heavy Ion Collisions: past, present, future, 1 2009.
* [21] F. Becattini, 33rd Workshop of the INFN Eloisatron Project, Erice, Italy, October 19-25, 1996, pages 74–104, 1996.
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* [26] A.Abdesselam et al. , Measurement of two-particle correlation in hadronic $e^{+}e^{-}$ collisions at Belle, arXiv:2008.04187
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* [29]
|
# Training With Data Dependent Dynamic
Learning Rates
Shreyas Saxena
Apple
<EMAIL_ADDRESS>Nidhi Vyas
Apple
<EMAIL_ADDRESS>Dennis DeCoste
Apple
<EMAIL_ADDRESS>
###### Abstract
Recently many first and second order variants of SGD have been proposed to
facilitate training of Deep Neural Networks (DNNs). A common limitation of
these works stem from the fact that they use the same learning rate across all
instances present in the dataset. This setting is widely adopted under the
assumption that loss-functions for each instance are similar in nature, and
hence, a common learning rate can be used. In this work, we relax this
assumption and propose an optimization framework which accounts for difference
in loss function characteristics across instances. More specifically, our
optimizer learns a dynamic learning rate for each instance present in the
dataset. Learning a dynamic learning rate for each instance allows our
optimization framework to focus on different modes of training data during
optimization. When applied to an image classification task, across different
CNN architectures, learning dynamic learning rates leads to consistent gains
over standard optimizers. When applied to a dataset containing corrupt
instances, our framework reduces the learning rates on noisy instances, and
improves over the state-of-the-art. Finally, we show that our optimization
framework can be used for personalization of a machine learning model towards
a known targeted data distribution.
## 1 Introduction
Figure 1: Eigen-value spectrum of Hessian for ResNet-18 trained on CIFAR100.
For three classes, we plot top-10 eigen-values over the course of training.
The loss function characteristics are different for each class.
Owing to the non-convex nature of most loss functions, coupled with poorly
understood learning dynamics, optimization of Deep Neural Networks (DNNs) is a
challenging task. Recent work can be broadly divided in two categories: (1)
adaptive first order variants of SGD, such as ADAMw [24], RMSprop [12] and
Adam [20], and (2) second order methods such as K-FAC [26]. These methods
adapt the optimization process per parameter and obtain improved convergence
and generalization. The objective function for these frameworks is to minimize
finite-sum problems:
$\mathcal{L}(\theta)=\frac{1}{N}\sum_{i}^{N}L^{i}(\theta)$, where
$L^{i}(\theta)$ denotes the loss on a single training data point. Standard
optimization frameworks assume that the $N$ loss functions come from the same
distribution, and share characteristics such as the curvature, Lipshitz
constant, etc. However, in practice, this is not true. Figure 1 highlights the
difference in loss function characteristics over the course of training for
classes present in CIFAR100.
In contrast to prior work, which models the learning dynamics of sum of finite
problems, in this work, we account for differences in characteristics of
underlying loss functions in the sum. More specifically, for each instance in
the dataset, we associate a dynamic learning rate, and learn it along with
model parameters. The SGD update rule for our framework takes the following
form:
$\theta^{t+1}=\theta^{t}-\frac{\lambda}{N}\sum_{i}^{N}(\mathbf{w}^{i,t}_{inst}\cdot\frac{\partial
L^{i}}{\partial\theta^{t}})$
where $L^{i}$, $\lambda$, $\mathbf{w}^{i,t}_{inst}$ denotes loss, model
parameter learning rate, and multiplicative learning rate correction for
instance $i$ at time step $t$. Having a dynamic learning rate per instance
allows our optimization framework to focus on different modes of data during
the course of optimization. Instead of using a heuristic, we learn the
learning rate for each instance via meta-learning using a held out meta set.
We also extend our framework to learn a dynamic learning rate per class.
The main contributions of our work are:
1. 1.
We present an optimization framework which learns an adaptive learning rate
per instance in the dataset to account for differences in the loss function
characteristics across instances. We extend this framework to learn adaptive
learning rates per class. We also show that learning dynamic weight-decay
facilitates learning dynamic learning rates on data.
2. 2.
We show that learning adaptive learning rate on data leads to variance
reduction, and explains faster convergence and improved accuracy.
3. 3.
We show that in presence of noisy data, our framework reduces the learning
rates on noisy instances, and prioritizes learning from clean instances. Doing
so, we our method outperforms state-of-the-art by a significant margin.
4. 4.
We show that our framework can be used for personalization of machine learning
models towards a targeted distribution.
## 2 Learning the learning rate for data
As mentioned earlier, the main goal of our work is to learn a dynamic learning
rate for instances present in the train dataset. In this section, we first
formalize this intution and present the framework for learning instance level
learning rates. Next, we will show how our framework can be extended to learn
class level learning rates. We derive our method using stochastic gradient
descent (SGD) as the optimizer for model paramters, however, extension to
other class of optimizers can be done in a similar manner.
### 2.1 Learning instance level learning rate
Let
$\left\\{\left(\mathbf{x}^{i}_{train},y^{i}_{train}\right)\right\\}^{N}_{i=1}$
denote train set, where $\mathbf{x}^{i}\in\mathbb{R}^{d}$ denotes a single
data point in the train set and $y^{i}\in\\{1,...,k\\}$ denotes the
corresponding target. Let $f(\mathbf{x},\theta^{t})$ and $\theta^{t}$ denote
the model and model’s parameters at step $t$ respectively. Let
$L(\mathbf{x}^{i}_{train},y^{i}_{train};\theta^{t})$ denote an arbitrary
differentiable loss function on data point $i$ at time step $t$. In what
follows, we denote $L(\mathbf{x}^{i}_{train},y^{i}_{train};\theta^{t})$ as
$L_{train}^{i}(\theta^{t})$. Let $\mathbf{w}_{inst}^{t}\in\mathbb{R}^{N}$
denote the instance level parameters at time step $t$. Instance level
parameters weigh contribution of instances in the gradient update (see
Equation 1), and can be interpreted as a multiplicative learning rate
correction over the learning rate of model parameter’s optimizer. Note,
learning the learning rate for each instance is equivalent to learning
weighting for each instance. For the ease of explanation, we choose the latter
formulation. Our goal in optimization is to solve for optimal $\hat{\theta}$
by minimizing the weighted loss on the train set:
$\displaystyle\mathcal{L}_{train}(\theta^{t},\mathbf{w}_{inst}^{t})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}w^{i,t}_{inst}\cdot
L_{train}^{i}(\theta^{t})$ (1)
$\displaystyle\hat{\theta}(\mathbf{w}_{inst}^{t})$
$\displaystyle=\arg\min_{\theta}\mathcal{L}_{train}(\theta,\mathbf{w}_{inst}^{t})$
(2)
The solution to above equation is a function of $\mathbf{w}_{inst}^{t}$, which
is not known a priori. Setting $\mathbf{w}_{inst}^{t}$ as $\mathbf{1}$ at all
time steps recovers the standard gradient descent optimization framework, but
that might not be optimal. This brings us to the question: What is the optimal
value of $\mathbf{w}^{t}_{inst}$ i.e. the learning rate for data points at
time step $t$?
#### Learning dynamic instance level learning rate via meta-learning
In contrast to model parameters, whose optimal value is approximated by
minimizing the loss on train set, we can not approximate the optimal value of
$\mathbf{w}^{t}_{inst}$ by minimizing the loss on train set. Doing so, leads
to a degenerate solution, where $\mathbf{w}_{inst}^{t}=\mathbf{0}$. In
principle, the optimal value of $\mathbf{w}^{t}_{inst}$ is the one, which when
used to compute the gradient update at time step $t$, minimizes the error on a
held-out set (referred as meta set) at convergence, i.e
$\hat{\mathbf{w}}_{inst}^{t}=\arg\min_{\mathbf{w}_{inst}^{t}}\mathcal{L}_{meta}(\hat{\theta}(\mathbf{w}_{inst}^{t}))$.
Here $\hat{\theta}$ denotes model parameters at convergence. The sequence of
model updates from time step $t$ till convergence
($\theta^{t}\rightarrow\hat{\theta}$) can be written as a feed forward
computational graph, allowing us to backpropagate meta-gradient (gradient on
meta set) to $\mathbf{w}^{t}_{inst}$. However, this is not feasible in
practice due to: (1) heavy compute for backpropagating through time steps, (2)
heavy memory foot-print from saving all intermediate representations and (3)
vanishing gradient due to backpropagation through time steps.
To alleviate this issue, we approximate the meta-gradient at convergence with
the meta-gradient at time step $t+1$. More formally, we sample a mini-batch
from train set, and write one step SGD update on model parameters
(${\theta^{t+1}}$) as a function of instance parameters at time step $t$ (see
equation 3). The one step update is used to compute loss on meta set
$\mathcal{L}_{meta}({\theta^{t+1}})$, which is then used to compute the meta-
gradient on instance parameters:
$\displaystyle{\theta^{t+1}}(\mathbf{w}_{inst}^{t})$
$\displaystyle=\theta^{t}-\frac{\lambda}{N}\sum_{i=1}^{N}\mathbf{w}^{i,t}_{inst}\frac{\partial
L_{train}^{i}(\theta^{t})}{\partial\theta^{t}}$ (3)
$\displaystyle\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}})}{\partial
w^{i,t}_{inst}}$
$\displaystyle=\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}})}{\partial{\theta^{t+1}}}\cdot\frac{\partial{\theta^{t+1}}}{\partial
w^{i,t}_{inst}}$ (4)
$\displaystyle\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}})}{\partial
w^{i,t}_{inst}}$
$\displaystyle=\frac{-\lambda}{N}\cdot\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}})}{\partial{\theta^{t+1}}}\cdot\big{[}{\frac{\partial
L_{train}^{i}}{\partial\theta^{t}}}\big{]}^{T}$ (5)
Here, $\lambda$ and $N$ corresponds to the learning rate of the model
optimizer and number of samples in train mini-batch respectively. Using the
meta-gradient on instance-parameters we update the instance-parameters using
first order gradient update rule (see equation 6).
$\displaystyle w^{i,t+1}_{inst}$
$\displaystyle=w^{i,t}_{inst}-\lambda_{w_{inst}}\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}}^{*})}{\partial
w^{i,t}_{inst}}$ (6)
The pseduo code for our method is outlined in Algorithm 1.
Algorithm 1 Learning algorithm for learning the learning rate per instance
0: Train set $\mathcal{D}_{train}$, meta set $\mathcal{D}_{meta}$, model
learning rate $\lambda$, instance parameter learning rate $\lambda_{inst}$,
max iterations $T$.
0: Model parameters at convergence $\theta^{T}$.
1: Initialize model parameters $\theta^{0}$ and instance level parameters
$\mathbf{w}_{inst}^{0}$.
2: for $t=0$ to $T-1$ do
3: $\\{x_{train}^{i},y_{train}^{i}\\}\leftarrow$
SampleMiniBatch($\mathcal{D}_{train}$).
4: $\\{x_{meta}^{j},y_{meta}^{j}\\}\leftarrow$
SampleMiniBatch($\mathcal{D}_{meta}$).
5: $\\{\mathbf{w}_{inst}^{t}\\}\leftarrow$ SampleInstanceParameters.
6: Update model parameters, and express $\theta^{t+1}$ as a function of
$\mathbf{w}_{inst}^{t}$ by Eq. (3).
7: Update $\mathbf{w}_{inst}^{t+1}$ by Eq. (6).
8: end for
#### Analysis of meta-gradient on instance level parameters
From equation 5, we can observe that the meta-gradient on instance parameter
$\mathbf{w}_{inst}^{i,t}$ is proportional to the dot product of $i^{th}$
training sample’s gradient on model parameters at time step $t$ with meta-
gradient on model parameters of samples in the meta set at time step $t+1$.
Therefore, training samples whose gradient aligns with the gradients on meta-
set will obtain a higher weight, leading to an increased learning rate. The
converse holds true as well. For example, if an instance in train set has
wrong label, its gradient will not align with gradient of clean instances in
the meta set. Over the course of learning, the corrupt instance will end up
obtaining a lower value of instance parameter.
### 2.2 Learning class level learning rate
While instance parameters have the flexibility to adapt to each instance
present in the dataset, number of parameters grow with the size of dataset. To
alleviate this issue, another way we can partition a dataset is by leveraging
the class membership of data points. More specifically, we can learn a
learning rate for each class, shared by all the instances present within the
class. Let $\mathbf{w}_{class}^{t}\in\mathbb{R}^{N}$ denote the class
parameters at time step $t$. Similar to equation (3), we can write the one
step look ahead update as a function of class parameters, and use it compute
the meta-gradient on the class parameters:
$\displaystyle\frac{\partial\mathcal{L}_{meta}({\theta_{t+1}}^{*})}{\partial\mathbf{w}^{c,t}_{class}}$
$\displaystyle=-\frac{\lambda.n_{c}}{N}\cdot\frac{\partial\mathcal{L}_{meta}({\theta^{t+1}}^{*})}{\partial{\theta_{t+1}}^{*}}\cdot\big{[}\frac{1}{n_{c}}{\sum\limits_{\begin{subarray}{c}i\\\
y_{i}=c\end{subarray}}^{n_{c}}\frac{\partial
L_{train}^{i}}{\partial\theta_{t}}}\big{]}^{T}$ (7)
Here, $w^{c,t}_{class}$ denotes the weight for $c$ (target class for $i^{th}$
train data point), and $n_{c}$ denotes number of samples in class $c$. As seen
in equation 7, the meta-gradient on class parameters is proportional to the
dot-product of meta-gradient on model parameters with gradient on model
parameters from train set, averaged over instances belonging to class $c$. We
provide detailed derivation in supplementary material.
### 2.3 Meta-learning weight decay regularization
Use of weight-decay as a regularizer is a de-facto standard in training Deep
Neural Networks (DNNs). However, the exact role weight-decay plays in
optimization of modern DNNs is not well understood [8, 13, 24, 39]. In this
section, we highlight the importance of learning the weight-decay coefficient
along with the learning rate for dataset, class or instances. For ease of
explanation, let us consider the case where we are interested in learning
instance level learning rate, and we have the standard weight-decay term added
as a regularizer.
$\displaystyle\mathcal{L}_{train}(\theta^{t},w_{dataset}^{t})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\mathbf{w}^{i,t}_{inst}\cdot
L_{train}^{i}(\theta^{t})+\lambda_{wd}\|\theta^{t}\|$ (8)
Here $\lambda_{wd}$ denotes the weight-decay coefficient. During the course of
optimization, regardless of the magnitude of the first term, the contribution
of the second term in gradient update is fixed. In our experiments, we found
this to be problematic. When meta-gradient reduces the magnitude of instance
parameter $\mathbf{w}^{i,t}_{inst}$, it leads to a relative increase of
weight-decay component in the gradient update. This leads to destablization of
the training. We solved this problem by treating the weight-decay coefficient
as a learnable parameter, which is learnt along with the learning rate of
class and instances using meta learning setup.
## 3 Experiments
### 3.1 Implementation details
Unless stated otherwise, the following implementation details hold true for
all experiments in the paper. We use SGD optimizer (without momentum and
weight-decay) to learn instance and class level learning rates. We use same
batch-size to sample batches from train-set and meta-set. Apart from clamping
negative learning-rates to 0, we do not employ any form of regularization, and
rely on the meta-gradient to regularize the learning process. We perform
$k$-fold cross validation, where the held-out set is used as both: meta-set
and validation-set (for picking best configuration). For reporting final
numbers, we average out dynamic learning rate trajectory for each class and
instance, and use it to train on the full train-set. We ensure all methods use
the same amount of training data. We report mean and standard-deviation
computed over 3 runs.
### 3.2 Image Classification on CIFAR100
In this section, we show efficacy of our optimization framework when applied
to the task of image classification on CIFAR100 [21] dataset. CIFAR100 dataset
contains 100 classes, 50,000 images in the train set and 10,000 images in the
test set. Therefore, in our framework, along with the model parameters, we
learn 100 and 50,000 dynamic learning rates for class and instances
respectively. We evaluate our framework with ResNet18 [11], VGG16 [35]. We use
standard setup for training both architectures (details in supplementary).
In Figure 2, we compare our optimization framework to other optimizers
commonly used in the deep learning community. Similar to results in [40], when
tuned appropriately, apart from ADAM, all other optimizers obtain performance
comparable to SGD. In both settings, learning class or instance level learning
rate, we outperform these standard optimizers by a significant margin. These
gains over standard optimizers can be attributed to the fact that our
framework adapts the optimization process over samples in dataset instead of
model-parameters.
Across both architectures, learning instance level learning rates performs
more favorably compared to learning class level learning rates. This validates
our hypothesis: loss functions for samples within a class might have different
characteristics, and might benefit from learning sample specific learning
rates. However, class level parameters get more frequent updates compared to
instance level parameters, and hence can achieve faster convergence (see
Figure 2, middle).
| ResNet18 | VGG16
---|---|---
SGD | 77.5 $\pm$ 0.0 | 75.4 $\pm$ 0.2
Momentum [30] | 77.1 $\pm$ 0.3 | 74.1 $\pm$ 0.2
Adam [20] | 74.4 $\pm$ 0.2 | 72.2 $\pm$ 0.5
AdamW [24] | 77.4 $\pm$ 0.1 | 74.5 $\pm$ 0.2
Polyak [31] | 78.0 $\pm$ 0.3 | 74.5 $\pm$ 0.3
LookAhead [40] | 77.2 $\pm$ 0.4 | 75.6 $\pm$ 0.2
Instance-level | 78.6 $\pm$ 0.2 | 76.5 $\pm$ 0.1
Class-level | 78.3 $\pm$ 0.2 | 76.2 $\pm$ 0.2
Figure 2: Table: Comparison of class and instance level optimizer with
different optimizers on CIFAR100. We do a grid search on learning rate and
weight-decay for other optimizers (see supplementary). Lookahead and Polyak
are wrapped around SGD. Figures: Test and train accuracy for different
optimizers for VGG16. Using our optimizer leads to reduced variance, and
better generalization.
### 3.3 Analysis of optimization framework
In this section, we will analyze different components of our optimization
framework.
#### Variance reduction via dynamic learning rates on data
One key hypothesis of our work is: loss functions for different data points
can have different characteristics, and hence might benefit from different
learning rates. In Figure 3 we empirically verify this property on CIFAR100,
using class-level learning rates for training VGG16. In Figure 3 (A, B) we
plot the train loss and test accuracy for the best (green) and the worst (red)
performing class at convergence. Shared learning rate used by SGD works well
for the green class, but is not able to optimize the red class (until learning
rate decay at epoch 150). In contrast, our method accounts for class
performance on the meta-set, and reduces the learning rate for each class in
proportion to that (see Figure 3, C and D). This result provides an
interesting view to our method from the perspective of variance reduction. In
contrast to standard setting where methods have been proposed to reduce
variance in the gradient estimator for the entire mini-batch, our work
performs selective variance reduction. Lowering the learning rate for classes
with worse performance will lower the overall variance in gradient estimator,
leading to faster and stable convergence. In light of recent work [6], which
shows ineffectiveness of standard variance reduction framework in deep
learning, our results indicate that performing selective variance reduction
could be an interesting direction to explore.
Figure 3: To speed up convergence, our framework reduces learning rate on
classes with worse performance on meta set (see text for details). Learning
dynamics for class with best (green) and worst (red) performance at
convergence. A:Train loss. B: Test accuracy. C: Dynamic learning rate for the
classes, along with mean learning rate for all classes (black). D: At each
epoch, we plot the correlation of learning rate of a class with their
performance on heldout meta-set and train-set.
#### Importance of history
| History | ResNet18 | VGG16
---|---|---|---
Instance | ✓ | 78.6 $\pm$ 0.2 | 76.5 $\pm$ 0.1
✗ | 77.8 $\pm$ 0.2 | 75.4 $\pm$ 0.2
Class | ✓ | 78.3 $\pm$ 0.2 | 76.2 $\pm$ 0.2
✗ | 77.8 $\pm$ 0.1 | 75.7 $\pm$ 0.1
Table 1: Impact of retaining history on CIFAR100 for image-classification.
As mentioned earlier, learning learning rates on data points can be
interpreted as learning a weighting on them. Some recent works [16, 32, 34,
38] have used meta-learning based approaches to dynamically assign weights to
instances. In general, these works [16, 34, 38] train a secondary neural
network to assign weights to data-points throughout the course of training, or
[32] perform an online approximation of weights for each sample in the mini-
batch. These frameworks are Markovian in nature, since weights estimated at
each time step are independent of past predictions. In contrast, our framework
treats learning rates on data as learnable parameters, and benefits from past
history of optimization. In Table 1, we establish the importance of retaining
optimization history, by evaluating our framework without the use of history.
Specifically, after each time step, we update the model parameters using the
updated value of instance and class learning rates. Post model parameter
update, we reset the instance and class learning rates to their initial value
of 1. As shown in Table 1, not reusing the history of learnt learning rates
leads to a significant drop in performance across all settings and
architectures on CIFAR100. Another place of comparison with this prior work is
in noisy setting (see Section 3.4), where we outperform these methods by a
significant margin.
#### Importance of learning a dynamic weight-decay
Recently, the importance of weight-decay in optimization of DNNs has gained
much interest [8, 13, 24, 39]. [8] empirically demonstrate that weight-decay
plays an important role in the first few epochs, and does not play as much a
role in the later stages of training. Our results indicates otherwise. In this
work, we show that learning a dynamic weight-decay leads to significant change
in SGD dynamics and also facilitates the learning of learning rates on data.
Figure 4 highlights the change in dynamics of SGD optimizer when weight-decay
is learnt along with the model parameters. Compared to baseline (fixed weight-
decay), the learnt weight-decay adapts to different stages of optimization
(see Figure 4, right). More specifically, post learning rate drop, when model
is most prone to overfitting, dynamic weight decay coefficient increases. This
leads to a temporary drop in performance, but results in better generalization
at convergence. As see in the table (Figure 4, left), learning a dynamic
weight-decay improves performance of all three optimizers.
Dynamic weight decay | ✗ | ✓
---|---|---
SGD | 77.5 $\pm$ 0.0 | 78.0 $\pm$ 0.1
Class-level | 77.7 $\pm$ 0.1 | 78.3 $\pm$ 0.2
Instance-level | 78.2 $\pm$ 0.1 | 78.6 $\pm$ 0.2
Figure 4: Evaluation of dynamic weight-decay on CIFAR100 with ResNet18. Left:
Dynamic weight-decay improves all three optimizers. Center: Train (dashed) and
validation (solid) accuracy for SGD optimizer, with fixed and dynamic weight-
decay. Right: Plot showing the value of dynamic weight-decay when learnt along
with SGD. When the learning rate drops (epoch 80, 120), weight-decay
coefficient increases so as to counteract overfitting, and obtains better
performance at convergence.
### 3.4 Results on Robust Learning
Learning instance-level learning rates can be useful when some of the labels
in the dataset are noisy, where the framework should decrease learning rates
on corrupt instances. In this section, we validate our framework in a
controlled corrupted label setting.
To compare with the relevant state-of-the-art, we follow the common setting in
([16, 32]) to train deep CNNs, where the label of each image is independently
changed to a uniform random class with probability $p$, where $p$ is noise
fraction. The labels of validation data remain clean for evaluation. We
compare our approach with recent state-of-the-art approaches in this setting
[16, 32, 33, 34]. MentorNet [16] and Meta-Weight-Net [34] train an auxillary
neural network to assign weights to samples in the mini-batch. L2RW [32] uses
a held-out set to perform an online approximation of weights for samples in
the mini-batch. Data parameters [33] introduced learnable temperature
parameters per data-point, which scale the gradient contribution of each data
point. Unfortunately, all of these works report results in two distinct
settings: setting A [32] and setting B [34]. While both settings use
WRN-28-10, they differ in learning rate schedules (see details in
supplementary). To make a fair comparion, we report results in both settings.
Similar to setup in [32, 34], we keep 1000 clean images as meta-set. As seen
in Figure 5, our method outperforms other state-of-the-art methods in both
settings under different levels of noise.
| 40% | 60%
---|---|---
Baseline [34] | $51.1\pm 0.4$ | $30.9\pm 0.3$
Focal Loss [23] | $51.2\pm 0.5$ | $27.7\pm 3.7$
Co-teaching [10] | $46.20\pm 0.15$ | $35.7\pm 1.2$
Using 1000 clean images
MentorNet [16] | $61.4\pm 4.0$ | $36.9\pm 1.5$
L2RW [32] | $60.8\pm 0.9$ | $48.2\pm 0.3$
MWNet [34] | $67.7\pm 0.3$ | $58.8\pm 0.1$
Ours (instance-level) | $\bf{69.0\pm 0.2}$ | $\bf{59.6\pm 0.3}$
Setting A
| 40% | 80%
---|---|---
Baseline [32] | $50.66\pm 0.24$ | $8.0$
MentorNet PD [16] | $56.9$ | $14.0$
Data Parameters [33] | $\bf{70.93\pm 0.15}$ | $35.8\pm 1.0$
Using 1000 clean images
MentorNet DD [16] | $67.5$ | $35.0$
L2RW [32] | $61.34\pm 2.06$ | -
Ours (instance-level) | $70.8\pm 0.1$ | $\bf{42.5\pm 0.3}$
Setting B
Figure 5: Tables: On CIFAR100, our methods outperforms state-of-the-art under
varying levels of uniform noise in two settings (see text for details).
Figure: Plot of mean and standard-deviation over instance-level learning rates
for clean and corrupts instances. Our framework learns to reduce the learning
rate on corrupt instances.
### 3.5 Personalizing DNN models
In a traditional setting, machine learning models are trained under empirical
risk minimization (ERM) framework, where train and test set are assumed to be
sampled from the same distribution. These models are optimized to work well
across the entire data distribution. However, this training setup would not be
ideal when at test time only a subset of train distribution is of interest.
This situation can come up in various practical problems: (1) targeting a
certain demographic for recommender systems [28], (2) personalizing models in
health for a certain anomaly or demographic [14, 27], etc. In this setting, an
important question one needs to answer is: What training data should I train
the model on?
We simulate this scenario using the CIFAR100 dataset, which contains 100 fine
grained classes, and 20 super classes (mutually disjoint, contains 5 classes).
In this scenario, despite having the entire dataset annotated, we are
interested in one super class at test time. Below we detail different methods
which can be used in this scenario along with our proposed solution.
Biased training: The problem can be reduced to ERM framework by training the
model on instances belonging to classes present in the super class. This
approach makes an assumption that the other classes present in the dataset are
completely disjoint from the super class. However, some of the discarded
classes might share common low-level features which might be useful to train
the early layers of deep neural network.
Full training: To address the aforementioned limitation, and take advantage of
all the annotated data, one can train the model on all classes of the CIFAR100
dataset. However, this approach makes an assumption that training on all
classes would be beneficial for the classes present in super class.
Transfer Learning: Train model on all 100 classes (full training), followed by
training on the classes present in super class (biased training). A limitation
of this approach is that pretraining model on all 100 classes might bias the
model.
Our solution: The limitation of approaches mentioned above lies in the fact
that it involves making a hard choice regarding the classes present in the
training set. We relax this constraint by using dynamic class-level learning
rates, which can guide the optimization process towards a biased subset
dynamically. The meta set is comprised of instances belonging to the super-
class.
We benchmark our method and the baselines in table below on the CIFAR100
dataset. As seen in table in Figure 6, using our proposed solution we
outperform the other baselines by a significant margin.
Super Class | People | Aquatic | Vehicle 1 | Electrical | Reptiles
---|---|---|---|---|---
Mammals | Devices
Biased Training | $54.1\pm 1.3$ | $69.7\pm 4.5$ | $91.1\pm 0.5$ | $87.0\pm 0.5$ | $77.0\pm 0.5$
Full Training | $55.5\pm 1.6$ | $61.1\pm 1.4$ | $84.7\pm 0.5$ | $77.5\pm 1.1$ | $65.1\pm 1.4$
Transfer Learning | $54.6\pm 1.4$ | $66.4\pm 4.1$ | $91.0\pm 0.2$ | $87.7\pm 0.7$ | $76.2\pm 0.3$
Ours (class-level) | $\bf{61.0\pm 1.2}$ | $\bf{70.7\pm 0.3}$ | $\bf{93.0\pm 0.5}$ | $\bf{88.1\pm 0.8}$ | $\bf{79.5\pm 0.7}$
Figure 6: Table: For the task of personalization towards a super-class, using
dynamic class level learning rates outperforms baselines which involve a
static choice of train data. Figure: Repetable trajectories of learning rate
for classes not present in the super class.
## 4 Related Work
Optimization of DNNs has gained a lot of interest recently [1, 13, 17, 19, 24,
26, 40] . While a full detailed review is beyond the scope of this paper,
here, we give a brief overview of related work most relevant to the material
present in the paper.
Learning adaptive learning rates on data can be interpreted as learning a
weighting on each data point. [16, 34, 38] train an auxillary neural network
to assign weights to data points. [32] performs an online approximation, where
it uses one step look ahead on meta-set to estimate the weights for samples in
the mini-batch. These approaches are Markovian in nature, since weights
estimated at each time step are independent of past-predictions. In contrast,
our method leverages the past history of optimization, and outperforms these
state-of-the-art methods for robust learning (see Section 3.4).
Our work also has connections to importance sampling. [15, 17, 18] propose
approximations for the gradient norm of instances which are used for sampling
data points. Theoretically, sampling data points with high gradient norm
should lead to faster convergence. However, this would not work well in real
world dataset which contain noisy data. In the same spirit, our work also has
connections to the field of curriculum learning [4, 9, 36] and self-paced
learning [7, 22, 29, 37]. These approaches either design hand-crafted
heuristics, or use loss value of data points as a proxy to decide the ordering
of data. All these approaches require coming up with a heuristics which might
not work from one problem domain to another. In contrast, our work can be
interpreted as a soft differentiable form of importance sampling, where the
importance of a sample (learning rate) or curriculum is learnt through meta
gradient.
Data parameters [33] introduced learnable temperature parameters for each
instance and class in the dataset. These parameters controlled the gradient
contribution for each data point, and were learnt using gradient descent. In
similar spirit, [2] introduced learnable robustness parameter per data point,
which generalized different regression losses. Both of these works learn
parameters per data point for robust estimation in classification or
regression. In comparison, our formulation is more generic and can admit any
differentiable loss function. More importantly, due to its meta-learning
framework, our approach allows for robust estimation of noise as well as
personalization.
Other work has proposed optimizing a few hyperparameters, such as kernel
parameters [5], weight decay [3] or others [25], using gradients during
training. However, to the best of our knowledge, none have done so for
learning dynamic learning rates across training per se, nor done so at scale
(e.g. one rate per instance) to achieve state-of-the-art performance.
## 5 Conclusion
In this paper, we have proposed an optimization framework which accounts for
differences in loss function characteristics across instances and classes
present in the dataset. More specifically, our framework learns a dynamic
learning rate for each instance and class present in the dataset. Learning a
dynamic learning rate allows our framework to focus on different modes of
training data. For instance, when presented with noisy dataset, our framework
reduces the learning rate on noisy instances, and focuses on optimizing model
parameters using clean instances. When applied for the task of image-
classification, across different CNN architectures, our framework outperforms
standard optimizers. Finally, for the task of personalization of machine
learning models towards a known data distribution, our framework outperforms
strong baselines.
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|
commenta
# Piecewise deterministic generative models
Andrea Bertazzi
Ecole Polytechnique
<EMAIL_ADDRESS>
&Alain Oliviero-Durmus
Ecole Polytechnique
<EMAIL_ADDRESS>
Dario Shariatian
Inria
<EMAIL_ADDRESS>
&Umut Simsekli
Inria
<EMAIL_ADDRESS>
&Eric Moulines
Ecole Polytechnique
<EMAIL_ADDRESS>
###### Abstract
We introduce a novel class of generative models based on piecewise
deterministic Markov processes (PDMPs), a family of non-diffusive stochastic
processes consisting of deterministic motion and random jumps at random times.
Similarly to diffusions, such Markov processes admit time reversals that turn
out to be PDMPs as well. We apply this observation to three PDMPs considered
in the literature: the Zig-Zag process, Bouncy Particle Sampler, and
Randomised Hamiltonian Monte Carlo. For these three particular instances, we
show that the jump rates and kernels of the corresponding time reversals admit
explicit expressions depending on some conditional densities of the PDMP under
consideration before and after a jump. Based on these results, we propose
efficient training procedures to learn these characteristics and consider
methods to approximately simulate the reverse process. Finally, we provide
bounds in the total variation distance between the data distribution and the
resulting distribution of our model in the case where the base distribution is
the standard $d$-dimensional Gaussian distribution. Promising numerical
simulations support further investigations into this class of models.
## 1 Introduction
Diffusion-based generative models (Ho et al., 2020; Song et al., 2021) have
recently achieved state-of-the-art performance in various fields of
application (Dhariwal and Nichol, 2021; Croitoru et al., 2023; Jeong et al.,
2021; Kong et al., 2021). In their continuous time interpretation (Song et
al., 2021), these models leverage the idea that a diffusion process can bridge
the data distribution $\mu_{\star}$ to a base distribution $\pi$, and its time
reversal can transform samples from $\pi$ into synthetic data from
$\mu_{\star}$. As shown in the 1980s by Anderson (1982), the time reversal of
a diffusion process, i.e., the backward process, is itself a diffusion with
explicit drift and covariance functions that are related to the score
functions of the time-marginal densities of the original, forward diffusion.
Consequently, the key element of these generative models is learning these
score functions using techniques such as (denoising) score-matching
(Hyvärinen, 2005; Vincent, 2011).
In this work, we explore the potential of using piecewise deterministic Markov
processes (PDMPs) as noising processes instead of diffusions. PDMPs were
introduced around forty years ago (Davis, 1984, 1993) and since then have been
successfully applied in various fields, including communication networks
(Dumas et al., 2002), biology (Berg and Brown, 1972; Cloez, Bertrand et al.,
2017), risk theory (Embrechts and Schmidli, 1994), and the reliability of
complex systems (Zhang et al., 2008). More recently, PDMPs have been
intensively studied in the context of Monte Carlo algorithms (Fearnhead et
al., 2018) as alternatives to Langevin diffusion-based methods and Metropolis-
Hastings mechanisms. This renewed interest in PDMPs has led to the development
of novel processes, such as the Zig-Zag process (ZZP) (Bierkens et al.,
2019a), the Bouncy Particle Sampler (BPS) (Bouchard-Côté et al., 2018), and
the Randomised Hamiltonian Monte Carlo (RHMC) (Bou-Rabee and Sanz-Serna,
2017). Compared to Langevin-based methods, PDMPs offer several advantages,
such as better scalability and reduced computational complexity in high-
dimensional settings (Bierkens et al., 2019a).
In this paper, we propose a new family of generative models based on PDMPs.
Our contributions are the following:
1. 1)
Leveraging the existing literature on time reversals of Markov jump processes
(Conforti and Léonard, 2022), we characterise the time reversal of any PDMP
under appropriate conditions. It turns out that this time reversal is itself a
PDMP with characteristics related to the original PDMP; see Proposition 1.
2. 2)
We further specify the characteristics of the time-reversal processes
associated with the three aforementioned PDMPs: ZZP, BPS, and RHMC. For these
processes, Proposition 2 shows the corresponding time-reversals are PDMPs with
simple reversed deterministic motion and with jump rates and kernels that
depend on (ratios of) conditional densities of the velocity of the forward
process before and after a jump. In contrast to common diffusion models, the
emphasis is on distributions of the velocity, similar to the case of the
underdamped Langevin diffusion (Dockhorn et al., 2022), which includes an
additional velocity vector akin to the PDMPs we consider. Moreover, the
structure of the backward jump rates and kernels closely connects to the case
of continuous time jump processes on discrete state spaces (Sun et al., 2023;
Lou et al., 2024).
3. 3)
We define our _piecewise deterministic generative models_ employing either
ZZP, BPS, or RHMC as forward process, transforming data points to a noise
distribution of choice, and develop methodologies to estimate the backward
rates and kernels. Then, we define the corresponding backward process based on
approximations of the time reversed ZZP, BPS, and RHMC obtained with the
estimated rates and kernels. In Section 4 we test our models on simple toy
distributions.
4. 4)
We obtain a bound for the total variation distance between the data
distribution and the distribution of our generative models taking into account
two sources of error: first, the approximation of the characteristics of the
backward PDMP, and second, its initialisation from the limiting distribution
of the forward process; see Theorem 1.
## 2 PDMP based generative models
### 2.1 Piecewise deterministic Markov processes
Informally, a PDMP (Davis, 1984, 1993) on the measurable space
$(\mathbb{R}^{D},\mathcal{B}(\mathbb{R}^{D}))$ is a stochastic process that
follows deterministic dynamics between random times, while at these times, the
process can evolve stochastically on the basis of a Markov kernel. In order to
define a PDMP precisely, we need three components, which we call
_characteristics_ of the PDMP: a _vector field_
$\Phi:\mathbb{R}_{+}\times\mathbb{R}^{D}\to\mathbb{R}^{D}$, which governs the
deterministic motion, a _jump rate_
$\lambda:\mathbb{R}_{+}\times\mathbb{R}^{D}\to\mathbb{R}_{+}$, which defines
the law of random event times, and finally a _jump kernel_
$Q:\mathbb{R}_{+}\times\mathbb{R}^{D}\times\mathcal{B}(\mathbb{R}^{D})\to[0,1]$,
which is applied at event times and defines the new location of the process.
Now we can describe the formal construction of a PDMP with the characteristics
$(\Phi,\lambda,Q)$. To this end, consider the differential flow
$\varphi:(t,s,z)\mapsto\varphi_{t,t+s}(z)$, which solves the ODE,
$\mathrm{d}z_{t+s}=\Phi(t+s,z_{t+s})\mathrm{d}s$ for $s\geqslant 0$, i.e.
$z_{t+s}=\varphi_{t,t+s}(z_{t})$. We define by recursion on $n\in\mathbb{N}$
the process on $(Z_{t})_{t\in\left[0,\mathrm{T}_{n}\right]}$ on
$\left[0,\mathrm{T}_{n}\right]$ and the increasing sequence of jump times
$(T_{n})_{n\in\mathbb{N}}$ starting from an initial state $Z_{0}$ and setting
$\mathrm{T}_{0}=0$. Assume that $(\mathrm{T}_{i})_{i\in\\{0,\ldots,n\\}}$ and
$(Z_{t})_{t\in\left[0,\mathrm{T}_{n}\right]}$ are defined for some
$n\in\mathbb{N}$. We now define
$(Z_{t})_{t\in\left[\mathrm{T}_{n},\mathrm{T}_{n+1}\right]}$. First, we define
$\tau_{n+1}=\inf\left\\{t>0:\int_{0}^{t}\lambda(T_{n}+u,\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+u}(Z_{\mathrm{T}_{n}}))\mathrm{d}u\geqslant
E_{n+1}\right\\}$ (1)
where $E_{n+1}\sim\text{Exp}(1)$, and set the $n+1$-th jump time
$\mathrm{T}_{n+1}=\mathrm{T}_{n}+\tau_{n+1}$. The process is then defined on
$\left[\mathrm{T}_{n},\mathrm{T}_{n+1}\right)$ by
$Z_{\mathrm{T}_{n}+t}=\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+t}(Z_{\mathrm{T}_{n}})$
for $t\in[0,\tau_{n+1})$. Finally, we set $Z_{\mathrm{T}_{n+1}}\sim
Q(T_{n+1},\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+\tau_{n+1}}(Z_{\mathrm{T}_{n}}),\cdot)$.
The process $(Z_{t})_{t\geqslant 0}$ is a Markov process by (Jacobsen, 2005,
Theorem 7.3.1). We note that a PDMP typically has several types of jumps
belonging to a family of jump rates and kernels
$(\lambda_{i},Q_{i})_{i\in\\{1,\ldots,\ell\\}}$. A PDMP of such type can be
obtained with the construction we have described by setting
$\lambda(t,z)=\sum_{i=1}^{\ell}\lambda_{i}(t,z)\;,\quad
Q(t,z,\mathrm{d}z^{\prime})=\sum_{i=1}^{\ell}\frac{\lambda_{i}(t,z)}{\lambda(t,z)}Q_{i}(t,z,\mathrm{d}z^{\prime})\;.$
(2)
An alternative, equivalent construction of a PDMP with $\lambda,Q$ satisfying
(2) is given in Section A.1. Finally, we say a PDMP is homogeneous (as opposed
to the non-homogeneous case we have described) when the characteristics do not
depend on time, that is $\Phi:\mathbb{R}^{D}\to\mathbb{R}^{D}$,
$\lambda:\mathbb{R}^{D}\to\mathbb{R}_{+}$, and
$Q:\mathbb{R}^{D}\times\mathcal{B}(\mathbb{R}^{D})\to[0,1]$. In all this work,
we suppose that the PDMPs that we consider are non-explosive in the sense of
Davis (1993), that is it is such that $\mathrm{T}_{n}\to+\infty$ as
$n\to+\infty$, almost surely (see Durmus et al. (2021) for conditions ensuring
this).
We now introduce the three PDMPs we consider throughout the paper. All these
PDMPs are time-homogeneous and live on a state space of the form
$\mathsf{E}=\mathbb{R}^{d}\times\mathsf{V}$, for
$\mathsf{V}\subset\mathbb{R}^{d}$, assuming $V_{0}\in\mathsf{V}$. Then,
$Z_{t}$ can be decomposed as $Z_{t}=(X_{t},V_{t}),$ where
$X_{t}\in\mathbb{R}^{d}$ is the component of interest and has the
interpretation of the position of a particle, whereas $V_{t}\in\mathsf{V}$ is
an auxiliary vector playing the role of the particle’s velocity. In the
sequel, if there is no risk of confusion, we take the convention that any
$z\in\mathbb{R}^{d}\times\mathsf{V}$, and we write $z=(x,v)$ for
$x\in\mathbb{R}^{d}$ and $v\in\mathsf{V}$. All the PDMPs below have a
stationary distribution of the form $\pi(\mathrm{d}x)\otimes\nu(\mathrm{d}v)$,
where $\pi$ has density proportional to $x\mapsto\mathrm{e}^{-\psi(x)}$, for
$\psi:\mathbb{R}^{d}\to\mathbb{R}$ a continuously differential potential, and
$\nu$ is a simple distribution on $\mathsf{V}$ for the velocity vector (e.g.
standard normal if $\mathsf{V}=\mathbb{R}^{d}$ or uniform distribution if
$\mathsf{V}$ is a compact set).
##### The Zig-Zag process
The Zig-Zag process (ZZP) (Bierkens et al., 2019a) is a PDMP with the state
space $\mathsf{E}^{\mathrm{Z}}=\mathbb{R}^{d}\times\\{-1,1\\}^{d}$. The
deterministic motion is determined by the homogeneous vector field
$\Phi^{\mathrm{Z}}(x,v)=(v,0)^{\operatorname{T}}$, i.e. the particle moves
with constant velocity $v$. For $i\in\\{1,\ldots,d\\}$ we define the jump
rates
$\lambda^{\mathrm{Z}}_{i}(x,v):=(v_{i}\partial_{i}\psi(x))_{+}+\lambda_{r}$,
where $(a)_{+}=\max(0,a)$, $\partial_{i}$ denotes the $i$-th partial
derivative, and $\lambda_{r}\geqslant 0$ is a user chosen refreshment rate.
The corresponding (deterministic) jump kernels are given by
$Q^{\mathrm{Z}}_{i}((x,v),(\mathrm{d}y,\mathrm{d}w))=\updelta_{(x,\mathscr{R}_{i}^{\mathrm{Z}}v)}(\mathrm{d}y,\mathrm{d}w)$,
where $\updelta_{z}$ denotes the Dirac measure at $z\in\mathsf{E}$. Here,
$\mathscr{R}^{\mathrm{Z}}_{i}$ is the operator that reverses the sign of the
$i$-th component of the vector to which it is applied, i.e.
$\mathscr{R}^{\mathrm{Z}}_{i}v=(v_{1}\ldots,v_{i-1},-v_{i},v_{i+1},\ldots,v_{d})$.
The ZZP falls within our definition of PDMP taking $\lambda,Q$ as in (2). As
shown in Bierkens et al. (2019a), the ZZP has invariant distribution
$\pi\otimes\nu$, where $\nu$ is the uniform distribution over $\\{\pm
1\\}^{d}.$ Moreover, Bierkens et al. (2019b) shows that for any
$\lambda_{r}\geqslant 0$ the law of the ZZP converges exponentially fast to
its invariant distribution e.g. when $\pi$ is a standard normal distribution.
##### The Bouncy Particle sampler
The Bouncy Particle sampler (BPS) (Bouchard-Côté et al., 2018) is a PDMP with
state space is
$\mathsf{E}^{\mathrm{B}}=\mathbb{R}^{d}\times\mathsf{V}^{\mathrm{B}}$, where
$\mathsf{V}^{\mathrm{B}}=\mathbb{R}^{d}$ or
$\mathsf{V}^{\mathrm{B}}=\mathsf{S}^{d-1}:=\\{v\in\mathbb{R}^{d}\,:\,\left\|v\right\|=1\\}$.
The deterministic motion is governed as ZZP by the homogeneous vector field
defined for $z=(x,v)\in\mathsf{E}$ by
$\Phi^{\mathrm{B}}(x,v)=(v,0)^{\operatorname{T}}$. Now we introduce two jump
rates which correspond to two types of random events: _reflections_ and
_refreshments_. Reflections enforce that $\mu(x,v)=\pi(x)\nu(v)$ is the
invariant density of the process, where
$\pi(\mathrm{d}x)\propto\exp(-\psi(x))\mathrm{Leb}(\mathrm{d}x)$ is a given
distribution and $\nu$ is either a standard normal distribution when
$\mathsf{V}^{\mathrm{B}}=\mathbb{R}^{d}$ or the uniform distribution on
$\mathsf{S}^{d-1}$ when $\mathsf{V}^{\mathrm{B}}=\mathsf{S}^{d-1}$.
Reflections are associated to the homogeneous jump rate
$(x,v)\mapsto\lambda^{\mathrm{B}}_{1}(x,v)=\langle
v,\nabla\psi(x)\rangle_{+}$, while refreshments are associated to
$(x,v)\mapsto\lambda^{\mathrm{B}}_{2}(x,v)=\lambda_{r}$ for $\lambda_{r}>0$.
The corresponding jump kernels are
$Q^{\mathrm{B}}_{1}((x,v),(\mathrm{d}y,\mathrm{d}w))=\updelta_{(x,\mathscr{R}^{\mathrm{B}}_{x}v)}(\mathrm{d}y,\mathrm{d}w)\;,\quad
Q^{\mathrm{B}}_{2}((x,v),(\mathrm{d}y,\mathrm{d}w))=\updelta_{x}(\mathrm{d}y)\nu(\mathrm{d}w),$
where $\mathscr{R}^{\mathrm{B}}_{x}v=v-2(\nicefrac{{\langle
v,\nabla\psi(x)\rangle}}{{\lvert\nabla\psi(x)\rvert^{2}}})\nabla\psi(x)\;.$
The operator $\mathscr{R}^{\mathrm{B}}_{x}$ _reflects_ the velocity $v$ off
the hyperplane that is tangent to the contour line of $\psi$ passing though
point $x$. The norm of the velocity is unchanged by the application of
$\mathscr{R}^{\mathrm{B}}$, and this gives the interpretation that
$\mathscr{R}^{\mathrm{B}}$ is an elastic collision of the particle off such
hyperplane. As observed in Bouchard-Côté et al. (2018), BPS requires a
strictly positive $\lambda_{r}$ to avoid being reducible, that is to make sure
the process can reach any area of the state space. Exponential convergence of
the BPS to its invariant distribution was shown in Deligiannidis et al.
(2019); Durmus et al. (2020).
##### Randomised Hamiltonian Monte Carlo
Randomised Hamiltonian Monte Carlo (RHMC) (Bou-Rabee and Sanz-Serna, 2017)
refers to the PDMP with state space
$\mathsf{E}^{\mathrm{H}}=\mathbb{R}^{d}\times\mathbb{R}^{d}$ which is
characterised by Hamiltonian deterministic flow and refreshments of the
velocity vector from the standard normal distribution. The flow is governed by
the homogeneous vector field defined by
$(x,v)\mapsto\Phi^{\mathrm{H}}(x,v)=(v,-\nabla\psi(x))^{\operatorname{T}}$,
where $\psi$ is the potential of $\pi$. The jump rate coincides with the
refreshment part of BPS, i.e., it is the constant function
$\lambda^{\mathrm{H}}:(x,v)\mapsto\lambda_{r}>0$ and jump kernel
$Q^{\mathrm{H}}((x,v),(\mathrm{d}y,\mathrm{d}w))=\updelta_{x}(\mathrm{d}y)\nu(\mathrm{d}w).$
When the stationary distribution $\pi$ is a standard Gaussian, the
deterministic dynamics $(x_{t},v_{t})_{t\geqslant 0}$ satisfy
$\mathrm{d}x_{t}=v_{t}\mathrm{d}t$, $\mathrm{d}v_{t}=-x_{t}\mathrm{d}t$, which
for $t\geqslant 0$ has solution $x_{t}=x_{0}\cos(t)+v_{0}\sin(t)$ and
$v_{t}=-x_{0}\sin(t)+v_{0}\cos(t)$, where $(x_{0},v_{0})$ is the initial
condition. It is well known that Hamiltonian dynamics preserve the density
$\mu(x,v)=\pi(x)\nu(v)$ (Neal, 2010), where $\nu$ is the standard normal
distribution, while velocity refreshments are necessary to ensure the process
is irreducible. Exponential convergence of the law of this PDMP to $\mu$ was
shown in Bou-Rabee and Sanz-Serna (2017).
###### Remark 1 (Noise schedule)
For a given time-homogeneous PDMP with characteristics $(\Phi,\lambda,Q)$ and
a given positive function $t\mapsto\beta(t)$ on $\mathbb{R}_{+}$, we can
define the corresponding time transformed, non-homogeneous PDMP with
characteristics $(\Phi_{\beta},\lambda_{\beta},Q)$ where $Q$ is unchanged,
while the deterministic flow and jump rates are non-homogeneous and given by
$\Phi_{\beta}(t,z)=\beta(t)\Phi(z)$ and
$\lambda_{\beta}(t,z)=\beta(t)\lambda(z)$. The PDMP
$(\Phi_{\beta},\lambda_{\beta},Q)$ has the same stationary distribution of the
PDMP $(\Phi,\lambda,Q)$, where $\beta(t)$ plays the role of the noise
schedule.
### 2.2 Time reversal of PDMPs
Similarly to the case of diffusion processes, we need to define appropriate
time reversals of PDMPs to be able to map noise to samples from the data
distribution. For a given PDMP $(Z_{t})_{t\in[0,\mathrm{T}_{f}]}$ with initial
distribution $\mu_{0}$, its _time reversal_ is the process that at time
$t\in[0,\mathrm{T}_{f}]$ has distribution $\mu_{0}P_{\mathrm{T}_{f}-t},$ where
$\mu_{0}P_{t}$ denotes the law of $Z_{t}.$ It follows that the law of the time
reversal at time $\mathrm{T}_{f}$ is $\mu_{0}$, which is the key observation
in the context of generative modelling. Characterisations of the law of time
reversed Markov processes with jumps were obtained in Conforti and Léonard
(2022) and in the following statement we adapt their Theorem 5.7 to our
setting, showing that the time reversal of a PDMP with characteristics
$(\Phi,\lambda,Q)$ is a PDMP with reversed deterministic motion and jump rates
and kernels satisfying (3).
###### Proposition 1
Consider a non-explosive PDMP $(Z_{t})_{t\geqslant 0}$ with characteristics
$(\Phi,\lambda,Q)$ and initial distribution $\mu_{0}$ on $\mathbb{R}^{D}$. In
addition, let $\mathrm{T}_{f}$ be a time horizon. Suppose that $\Phi$ is
locally bounded, $(t,z)\mapsto\lambda(t,z)$ is continuous in both its
variables, and
$\int_{0}^{\mathrm{T}_{f}}\mathbb{E}[\lambda(t,Z_{t})]\mathrm{d}t<\infty$.
Assume the technical conditions H 3, H 4, postponed to the supplement. Then,
the corresponding time reversal process is a PDMP with characteristics
$(\overleftarrow{\Phi},\overleftarrow{\lambda},\overleftarrow{Q})$, where
$\overleftarrow{\Phi}(t,z)=-\Phi(\mathrm{T}_{f}-t,z)$ and
$\overleftarrow{\lambda},\overleftarrow{Q}$ are the unique solutions to the
following balance equation: for almost all
$t\in\left[0,\mathrm{T}_{f}\right]$,
$\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}y)\overleftarrow{\lambda}(t,y)\overleftarrow{Q}(t,y,\mathrm{d}z)=\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}z)\lambda(\mathrm{T}_{f}-t,z)Q(\mathrm{T}_{f}-t,z,\mathrm{d}y)\;,$
(3)
where $\mu_{0}P_{t}$ stands for the distribution of $Z_{t}$ starting from
$\mu_{0}$.
The proof is postponed to Section A.3. In the next proposition we derive
expressions for the backward jump rate and kernel satisfying (3) corresponding
to a forward PDMP with characteristics with the same structure as those of
ZZP, BPS, and RHMC. We state the result assuming the PDMP has only one jump
type, but the generalisation to the case of $\ell>1$ jump mechanisms of the
form (2) can be immediately obtained applying Proposition 2 to each pair
$(\lambda_{i},Q_{i})$ for $i\in\\{1,\ldots,\ell\\}$. We refer to Section A.5
for the details.
###### Proposition 2
Consider a non-explosive PDMP $(X_{t},V_{t})_{t\geqslant 0}$ with
characteristics $(\Phi,\lambda,Q)$ and initial distribution
$\mu_{0}^{X}\otimes\mu_{0}^{V}$ on $\mathbb{R}^{2d}$. In addition, let
$\mathrm{T}_{f}$ be a time horizon. Suppose that $\Phi$ and $\lambda$ satisfy
the same conditions as Proposition 1, in particular the technical conditions H
3, H 4 postponed to the supplement. Suppose in addition that for any
$t\in\left(0,\mathrm{T}_{f}\right]$, the conditional distribution of $V_{t}$
given $X_{t}$ has a transition density $(x,v)\mapsto p_{t}(v|x)$ with respect
to some reference measure $\mu_{\mathrm{ref}}^{V}$ on $\mathbb{R}^{d}$.
1. (1)
_(Deterministic jumps)_. Suppose
$Q((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v)$
where for any $y\in\mathbb{R}^{d}$,
$\mathscr{R}_{y}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is an involution which
preserves $\mu_{\mathrm{ref}}^{V}$, i.e.,
$\mathscr{R}_{y}^{-1}=\mathscr{R}_{y}$ and
$\mu_{\mathrm{ref}}^{V}(\mathrm{d}\mathscr{R}_{y}w)=\mu_{\mathrm{ref}}^{V}(\mathrm{d}w)$.
Then for almost all $t\in\left[0,\mathrm{T}_{f}\right]$ and any
$(y,w)\in\mathbb{R}^{2d}$ such that $p_{\mathrm{T}_{f}-t}(w\rvert y)>0$ it
holds that
$\overleftarrow{\lambda}(t,(y,w))=\frac{p_{\mathrm{T}_{f}-t}(\mathscr{R}_{y}w\rvert
y)}{p_{\mathrm{T}_{f}-t}(w\rvert
y)}\lambda(\mathrm{T}_{f}-t,(y,\mathscr{R}_{y}w))\;,\,\overleftarrow{Q}((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v)\;.$
2. (2)
_(Refreshments)._ Suppose
$Q((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\nu(\mathrm{d}v\rvert
y)$, where $\nu$ is a transition kernel on
$\mathbb{R}^{d}\times\mathcal{B}(\mathbb{R}^{d})$, and
$\lambda(t,(y,w))=\lambda(t,y)$. Suppose also for any $y\in\mathbb{R}^{d}$,
$\nu(\cdot\rvert y)$ is absolutely continuous with respect to
$\mu_{\mathrm{ref}}^{V}$. Then for almost all
$t\in\left[0,\mathrm{T}_{f}\right]$ and any $(y,w)\in\mathbb{R}^{2d}$ such
that $p_{\mathrm{T}_{f}-t}(w\rvert y)>0$ it holds that
$\overleftarrow{\lambda}(t,(y,w))=\frac{(\nicefrac{{\mathrm{d}\nu}}{{\mathrm{d}\mu_{\mathrm{ref}}^{V}}})(w\rvert
y)}{p_{\mathrm{T}_{f}-t}(w\rvert
y)}\lambda(\mathrm{T}_{f}-t,y),\,\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)p_{\mathrm{T}_{f}-t}(v\rvert
x)\mu_{\mathrm{ref}}^{V}(\mathrm{d}v).$
The proof is postponed to Section A.4. We remark that, when
$\mu_{0}^{V}(\mathsf{V})=1$ for $\mathsf{V}\subset\mathbb{R}^{d}$, the
reference measure can simply be chosen such that
$\mu_{\mathrm{ref}}^{V}(\mathsf{V})=1$. Applying Proposition 2 we are able to
derive explicit expressions for the characteristics of the time reversals of
ZZP, RHMC, and BPS. The rigorous statements and their proofs can be found in
Section A.6. For ZZP and BPS we assume the following condition on the
potential of $\pi$, the stationary distribution for the position vector of the
forward process. This is satisfied e.g. by any multivariate normal
distribution.
###### H 1
$\psi\in\mathcal{C}^{2}(\mathbb{R}^{d})$ and
$\sup_{x\in\mathbb{R}^{d}}\|\nabla^{2}\psi(x)\|<+\infty$.
For BPS and RHMC we suppose that for any $t\in\left(0,\mathrm{T}_{f}\right]$,
the conditional distribution of $V_{t}$ given $X_{t}$ has a transition density
$(x,v)\mapsto p_{t}(v|x)$ with respect to the Lebesgue measure. Moreover, for
all samplers we assume H 4.
##### Time reversal of ZZP
In order to apply Proposition 2 we additionally assume that
$\int\lvert\partial_{i}\psi(x)\rvert\mathrm{d}\mu_{\star}(x)<\infty$ for all
$i=1,\dots,d$. We find that the deterministic motion is defined by
$\overleftarrow{\Phi}^{\mathrm{Z}}(y,w)=(-w,0)^{\operatorname{T}}$ for any
$(y,w)\in\mathbb{R}^{2d}$, while the backward rates and kernels are for
$i=1,\dots,d$ and for all $(y,w)\in\mathbb{R}^{2d}$ such that
$p_{\mathrm{T}_{f}-t}(w\rvert y)>0$,
$\displaystyle\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(y,w))=\frac{p_{\mathrm{T}_{f}-t}(\mathscr{R}_{i}^{\mathrm{Z}}w\rvert
y)}{p_{\mathrm{T}_{f}-t}(w\rvert
y)}\lambda^{\mathrm{Z}}_{i}(y,\mathscr{R}_{i}^{\mathrm{Z}}w)\;,\quad\overleftarrow{Q}_{i}^{\mathrm{Z}}((y,w),(\mathrm{d}x,v))=\updelta_{(y,\mathscr{R}^{\mathrm{Z}}_{i}w)}(\mathrm{d}x,v)\;.$
(4)
##### Time reversal of BPS
Whereas in Section A.6 we consider the case where the velocity of BPS is
initialised on $\mathsf{S}^{d-1},$ we can formally apply Proposition 2 to the
case of $\nu$ is the standard $d$-dimensional Gaussian distribution assuming
that $\int|\nabla\psi(x)\rvert\mathrm{d}\mu_{\star}(x)<\infty$. The drift of
the backward BPS is clearly the same as for the backward ZZP, while jump rates
and kernels are for all $t\in[0,\mathrm{T}_{f}]$ and $(y,w)\in\mathbb{R}^{2d}$
such that $p_{\mathrm{T}_{f}-t}(w\rvert y)>0$
$\displaystyle\overleftarrow{\lambda}_{1}^{\mathrm{B}}(t,(y,w))=\frac{p_{\mathrm{T}_{f}-t}(\mathscr{R}^{\mathrm{B}}_{y}w\rvert
y)}{p_{\mathrm{T}_{f}-t}(w\rvert
y)}\lambda_{1}^{\mathrm{B}}(y,\mathscr{R}^{\mathrm{B}}_{y}w),\quad\overleftarrow{Q}_{1}^{\mathrm{B}}((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{(y,\mathscr{R}^{\mathrm{B}}_{y}w)}(\mathrm{d}x,\mathrm{d}v)\;,$
$\displaystyle\overleftarrow{\lambda}_{2}^{\mathrm{B}}(t,(y,w))=\lambda_{r}\frac{\nu(w)}{p_{\mathrm{T}_{f}-t}(w|y)}\;,\qquad\quad\,\,\overleftarrow{Q}_{2}^{\mathrm{B}}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=p_{\mathrm{T}_{f}-t}(v\rvert
y)\updelta_{y}(\mathrm{d}x)\mathrm{d}v\;.$ (5)
Time reversal of RHMC. The deterministic motion of the backward RHMC follows
the system of ODEs
$\overleftarrow{\Phi}^{\mathrm{H}}(x,v)=(-v,\nabla\psi(x))^{\operatorname{T}}$,
which, when the limiting distribution $\pi$ is Gaussian, has solution
$x_{t}=x_{0}\cos(t)-v_{0}\sin(t)$ and $v_{t}=x_{0}\sin(t)+v_{0}\cos(t)$. The
backward refreshment rate and kernel coincide with those of BPS as given in
(5).
###### Remark 2 (Variance exploding PDMPs)
Similarly to the case of diffusion models (Song et al., 2021), we can define
_variance exploding PDMPs_ choosing $\psi(x)=0$ for all $x\in\mathbb{R}^{d}$,
that is when $\pi(\mathrm{d}x)$ is the Lebesgue measure. In this case, the
deterministic motion of RHMC coincides with ZZP and BPS, and all three
processes have only velocity refreshment events.
### 2.3 Approximating the characteristics of time reversals of PDMPs
For our three main examples, ZZP, BPS, and RHMC, the results of Section 2.2
show that the jump rates and the jump kernels of the corresponding backward
PDMPs involve the conditional densities of the velocity of the forward process
given its position at times $t\in\left[0,\mathrm{T}_{f}\right]$. Since such
conditional densities are unavailable in analytic form, in this section we
provide methods to learn the jump rates and kernels of each of these time
reversed PDMPs.
##### Approximating the jump rates of the backward ZZP via ratio matching
In the case of ZZP, we need to approximate for any $i\in\\{1,\ldots,d\\}$, the
rates in (4). Since the terms
$\lambda_{i}^{\mathrm{Z}}(x,\mathscr{R}^{\mathrm{Z}}_{i}v)$ are known, it is
sufficient to estimate the density ratios
$r_{i}^{\mathrm{Z}}(x,v,t):=\nicefrac{{p_{t}(\mathscr{R}^{\mathrm{Z}}_{i}v\rvert
x)}}{{p_{t}(v\rvert x)}}$ for all states $(x,v)$ such that $p_{t}(v\rvert
x)>0$. To this end, we introduce a class of functions
$\\{s^{\theta}:\mathbb{R}^{d}\times\\{-1,1\\}^{d}\times[0,\mathrm{T}_{f}]\to\mathbb{R}_{+}^{d}\,:\,\theta\in\Theta\\}$
for some parameter set $\Theta\subset\mathbb{R}^{d_{\theta}}$ and aim to find
a parameter $\theta_{\star}\in\Theta$ such that for any
$i\in\\{1,\ldots,d\\}$, the $i$-th component of $s^{\theta_{\star}}$, denoted
by $s_{i}^{\theta_{\star}}(\cdot)$, is an approximation of
$r_{i}^{\mathrm{Z}}$. We then approximate the backward ZZP by using the rates
$\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))=s^{\theta_{\star}}_{i}(x,v,\mathrm{T}_{f}-t)\,\lambda^{\mathrm{Z}}_{i}(x,\mathscr{R}^{\mathrm{Z}}_{i}v)$.
To address the problem of fitting $\theta$, we consider different loss
functions inspired by the ratio matching (RM) problem considered in Hyvärinen
(2007).
From a discrete probability density $p_{\pm}$ on $\\{-1,1\\}^{d}$, RM consists
in learning the $d$ ratios
$v\mapsto\nicefrac{{p_{\pm}(\mathscr{R}_{i}v)}}{{p_{\pm}(v)}}$ for
$i\in\\{1,\ldots,d\\}$. This problem was motivated in Hyvärinen (2007) as a
means to estimate $p_{\pm}$ without requiring its normalising constant,
similarly to score matching applied to estimate continuous probability
densities (Hyvärinen, 2005). In our context we are interested only in the
ratios, hence as opposed to Hyvärinen (2007) we do not model the conditional
distributions $(x,v)\mapsto p_{t}(v\rvert x)$, but directly the ratios
$r_{i}^{\mathrm{Z}}$. Adapting the ideas of Hyvärinen (2007) to our context,
we introduce the function $\mathbf{G}:r\mapsto(1+r)^{-1}$ and define the
_Explicit Ratio Matching_ objective function
$\displaystyle{\ell}_{\mathrm{E}}(\theta)$
$\displaystyle=\int_{0}^{\mathrm{T}_{f}}\mathrm{d}t\,\omega(t)\sum_{i=1}^{d}\mathbb{E}\Big{[}\\{\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))-\mathbf{G}(r_{i}(X_{t},V_{t},t))\\}^{2}$
(6)
$\displaystyle\qquad\qquad+\\{\mathbf{G}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))-\mathbf{G}(r_{i}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))\\}^{2}\Big{]}\;.$
where $\omega:[0,\mathrm{T}_{f}]\to\mathbb{R}_{+}^{*}$ is a probability
density, and $(X_{t},V_{t})_{t\geqslant 0}$ is a ZZP initialised from
$\mu_{\star}\otimes\mathrm{Unif}(\\{-1,1\\}^{d})$. This objective function
considers simultaneously the square error in the estimation of both
$(x,v,t)\mapsto r_{i}(x,v,t)$ and $(x,v,t)\mapsto
r_{i}(x,\mathscr{R}_{i}^{\mathrm{Z}}v,t)$, where the function $\mathbf{G}$
improves numerical stability, particularly when one of the two ratios is very
small. Clearly ${\ell}_{\mathrm{E}}(\theta)=0$ if and only if
$s_{i}^{\theta}(x,v,t)=r_{i}(x,v,t)$ for almost all $x,v,t$ and all $i$.
Moreover, the choice of $\mathbf{G}$ allows us to optimise without knowledge
of the true ratios, as shown in the following result.
###### Proposition 3
It holds that
$\operatorname*{arg\,min}_{\theta}{\ell}_{\mathrm{E}}(\theta)=\operatorname*{arg\,min}_{\theta}{\ell}_{\mathrm{I}}(\theta)$
for
${\ell}_{\mathrm{I}}(\theta)=\\!\\!\int_{0}^{\mathrm{T}_{f}}\\!\\!\\!\\!\mathrm{d}t\,\omega(t)\sum_{i=1}^{d}\mathbb{E}\Big{[}\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},V_{t},t))+\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))-2\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))\Big{]}\;,$
(7)
where $(X_{t},V_{t})_{t\in\mathbb{R}_{+}}$ is a ZZP starting from
$\mu_{\star}\otimes\mathrm{Unif}(\\{-1,1\\}^{d})$.
Therefore we aim to solve the minimisation problem associated with
${\ell}_{\mathrm{I}}$, which has for empirical counterpart
$\theta\mapsto\sum_{n=1}^{N}\sum_{i=1}^{d}\left(\mathbf{G}^{2}(s_{i}^{\theta}(X^{n}_{\tau^{n}},V^{n}_{\tau^{n}},\tau^{n}))+\mathbf{G}^{2}(s_{i}^{\theta}(X^{n}_{\tau^{n}},\mathscr{R}_{i}^{\mathrm{Z}}V^{n}_{\tau^{n}},\tau^{n}))-2\mathbf{G}(s_{i}^{\theta}(X^{n}_{\tau^{n}},V^{n}_{\tau^{n}},\tau^{n}))\right)$
(8)
where $\\{\tau^{n}\\}_{n=1}^{N}$ are i.i.d. samples from $\omega$, independent
of $\\{(X^{n}_{t},V^{n}_{t})_{t\geqslant 0}\\}_{n=1}^{N}$, which are $N$
i.i.d. realisations of the ZZP respectively starting at the $n$-th training
data point with velocity $V^{n}_{0}$, where $\\{V^{n}_{0}\\}_{n=1}^{N}$ are
i.i.d. observations of $\mathrm{Unif}(\\{-1,1\\}^{d})$. We remark that when
$d$ is large, this loss can be computed efficiently by subsampling over the
dimensions.
##### Approximating the characteristics of BPS and RHMC
For BPS and RHMC, Proposition 2 shows that if we aim to sample from the
backward process, we have to estimate both ratios of the conditional density
of the velocity of the forward PDMP given its position at any time
$t\in[0,\mathrm{T}_{f}]$, and also to be able to sample from such densities as
prescribed by the backward jump kernel (5). In order to address both
requirements, we introduce a parametric family of conditional probability
distributions $\\{p_{\theta}:\theta\in\Theta\\}$ of the form $(x,v,t)\mapsto
p_{\theta}(v\rvert x,t)$, where $\Theta\subset\mathbb{R}^{d_{\theta}}$, which
we model with the framework of normalising flows (NFs) (Papamakarios et al.,
2021). The advantage of NFs lays in their feature that, once the network is
learned, it is possible both to obtain an estimate of the density at a given
state and time, and also to generate samples which are approximately from
$(x,v,t)\mapsto p_{t}(v|x)$. Focusing on BPS, we now illustrate how we can use
NFs to learn the backward jump rates and kernels. We aim to find a parameter
$\theta_{\star}^{\mathrm{B}}$ such that
$p_{\theta_{\star}^{\mathrm{B}}}(v\rvert x,t)$ approximates $p_{t}(v\rvert
x)$, that is the conditional density of the forward BPS with respect to the
Lebesgue measure. The optimal parameter $\theta_{\star}^{\mathrm{B}}$ can be
estimated by maximum likelihood, which is equivalent to minimising the loss
$\ell_{\mathrm{ML}}(\theta)=-\int_{0}^{\mathrm{T}_{f}}\mathrm{d}t\;\omega(t)\mathbb{E}\left[\log
p_{\theta}(V_{t}\rvert X_{t},t)\right],$ (9)
where $\omega:[0,\mathrm{T}_{f}]\to\mathbb{R}_{+}^{*}$ is a probability
density, and $(X_{t},V_{t})_{t\geqslant 0}$ is a a BPS initialised from
$\mu_{\star}\otimes\nu$, with $\nu$ denoting the density of the
$d$-dimensional standard normal distribution. Once we have obtained the
optimal parameter
$\theta_{\star}^{\mathrm{B}}=\operatorname*{arg\,min}_{\theta}\ell_{\mathrm{ML}}(\theta)$,
we can define our approximation of the backward refreshment mechanism of BPS
taking the rate
$\bar{\lambda}_{2}^{\mathrm{B}}(t,(x,v))=\lambda_{r}\times\nicefrac{{\nu(v)}}{{p_{\theta_{\star}^{\mathrm{B}}}(v\rvert
x,\mathrm{T}_{f}-t)}}$ and the kernel
$\bar{Q}_{2}^{\mathrm{B}}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=p_{\theta_{\star}^{\mathrm{B}}}(v\rvert
y,\mathrm{T}_{f}-t)\updelta_{y}(\mathrm{d}x)\mathrm{d}v$. Similarly, we
estimate the backward reflection ratio of BPS as
$\bar{\lambda}^{\mathrm{B}}_{1}(t,(x,v))=\lambda^{\mathrm{B}}_{1}(x,\mathscr{R}^{\mathrm{B}}_{x}v)\times\nicefrac{{p_{\theta_{\star}^{\mathrm{B}}}(\mathscr{R}^{\mathrm{B}}_{x}v\rvert
x,\mathrm{T}_{f}-t)}}{{p_{\theta_{\star}^{\mathrm{B}}}(v\rvert
x,\mathrm{T}_{f}-t)}}$.
### 2.4 Simulating the backward process
We now discuss how we can simulate the backward PDMP with exact backward flow
map $(t,x,v)\mapsto\varphi_{-t}(x,v)$ and jump characteristics
$\overline{\lambda}$ and $\overline{Q}$ that are approximations of the jump
rates and kernels of the time reversed PDMPs obtained as discussed in Section
2.3. We recall that the backward rates have the general form
$\overline{\lambda}(t,(x,v))=s_{\theta}(x,v,\mathrm{T}_{f}-t)\lambda(x,\mathscr{R}v)$,
where $s_{\theta}$ is an estimate of a density ratio and $\mathscr{R}$ is a
suitable involution. Even though such a PDMP can in principle be simulated
following the construction of Section 2.1, the generation of the random jump
times via (1) requires the integration of
$\overline{\lambda}(t,\varphi_{-t}(x,v))$ with respect to $t$. Analytic
expressions for such integral are unavailable since $\overline{\lambda}$ is
defined through a neural network. A standard approach in the literature (see
e.g. Bertazzi et al. (2022, 2023)) is to discretise time and effectively
approximate the integral in (1) with a finite sum. Here we focus on
approximations based on splitting schemes discussed in Bertazzi et al. (2023),
adapting their ideas to the non-homogeneous case. Such splitting schemes
approximate a PDMP with a Markov chain defined on the time grid
$\\{t_{n}\\}_{n\in\\{0,\ldots,N\\}}$, with $t_{0}=0$ and
$t_{N}=\mathrm{T}_{f}$. The key idea is that the deterministic motion and the
jump part of the PDMP are simulated separately in a suitable order, obtaining
second order accuracy under suitable conditions (see Theorem 2.6 in Bertazzi
et al. (2023)). Now, we give an informal description of the splitting scheme
that we use for RHMC, that is based on splitting DJD in Bertazzi et al.
(2023), where D stands for deterministic motion and J for jumps. We define our
Markov chain based on the step sizes
$\\{\delta_{j}\\}_{j\in\\{1,\ldots,N\\}}$, where $\delta_{j}=t_{j}-t_{j-1}$.
Suppose we have defined the Markov chain on
$\\{t_{k}\\}_{k\in\\{0,\ldots,n\\}}$ for $n<N$ and that the state at time
$t_{n}$ is $(x_{t_{n}},v_{t_{n}})$. The next state is obtained following three
steps. _First_ , the particle moves according to its deterministic motion for
a half-step, that is we define an intermediate state
$(x_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}},v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}})=\varphi_{-\nicefrac{{\delta_{n+1}}}{{2}}}(x_{t_{n}},v_{t_{n}})$.
_Second_ , we turn our attention to the jump part of the process. In this
phase, the particle is only allowed to move through jumps and there is no
deterministic motion. This means that the rate is frozen to the value
$\overline{\lambda}(t_{n}+\nicefrac{{\delta_{n+1}}}{{2}},(x_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}},v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}}))$
and thus the integral in (1) can be computed trivially. The proposal for the
next event time is then given by
$\tau_{n+1}\sim\mathrm{Exp}(\overline{\lambda}(t_{n}+\nicefrac{{\delta_{n+1}}}{{2}},(x_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}},v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}})))$.
If $\tau_{n+1}\leqslant\delta_{n+1}$, we draw
$w\sim\overline{Q}(t_{n}+\nicefrac{{\delta_{n+1}}}{{2}},(x_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}},v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}}),\cdot)$
and set $v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}}=w$, else we do not alter the
velocity vector. _Finally_ we conclude with an additional half-step of
deterministic motion, letting
$(x_{t_{n+1}},v_{t_{n+1}})=\varphi_{-\nicefrac{{\delta_{n+1}}}{{2}}}(x_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}},v_{t_{n}+\nicefrac{{\delta_{n+1}}}{{2}}}).$
We refer to Appendix C for a detailed description of the schemes used for each
process.
## 3 Error bound in total variation distance
In this section, we give a bound on the total variation distance between the
data distribution $\mu_{\star}$ and the law of the synthetic data generated by
a PDMP with initial distribution $\pi\otimes\nu$ and approximate
characteristics obtained e.g. with the methods described in Section 2.3. We
obtain our result comparing the law of such PDMP to the law of the exact time
reversal obtained in Section 2.2, that is the PDMP with the analytic
characteristics of Proposition 2 and with initial distribution
$\mathcal{L}(X_{\mathrm{T}_{f}},V_{\mathrm{T}_{f}})$, i.e. the law of the
forward PDMP at time $\mathrm{T}_{f}$ when initialised from
$\mu_{\star}\otimes\nu$. In our theorem, we then take into account two of the
three sources of error of our models, neglecting the discretisation error of
the methods discussed in Section 2.4.
First, we shall assume the following condition, which deals with the error
introduced by initialising the backward PDMP from $\pi\otimes\nu.$
###### H 2
The forward PDMP with semigroup $(P_{t})_{t\geqslant 0}$ is such that there
exist $\gamma,C>0$ for which
$\lVert\pi\otimes\nu-\mu_{\star}\otimes\nu P_{t}\rVert_{\mathrm{TV}}\leqslant
Ce^{-\gamma t}.$ (10)
In Section D.1 we give a brief discussion on the conditions on $\mu_{\star}$
and $\pi$ required to ensure H 2. Informally, for ZZP, BPS, and RHMC H 2 is
verified with $C<\infty$ when e.g. $\pi$ is a multivariate standard Gaussian
distribution and the tails of $\mu_{\star}$ are sufficiently light. We are now
ready to state our result.
###### Theorem 1
Consider a non-explosive PDMP $(X_{t},V_{t})_{t\geqslant 0}$ with initial
distribution $\mu_{\star}\otimes\nu$, stationary distribution $\pi\otimes\nu$,
and characteristics $(\Phi,\lambda,Q)$. Let $\mathrm{T}_{f}$ be a time
horizon. Suppose the assumptions of Proposition 1 as well as H 2 hold. Let
$(\overline{X}_{t},\overline{V}_{t})_{t\in[0,\mathrm{T}_{f}]}$ be a non-
explosive PDMP initial distribution $\pi\otimes\nu$ and characteristics
$(\overline{\Phi},\overline{\lambda},\overline{Q})$, where
$\overline{\Phi}(t,(x,v))=\Phi(\mathrm{T}_{f}-t,(x,v))$ for all
$t\in[0,\mathrm{T}_{f}]$ and $(x,v)\in\mathbb{R}^{2d}$. Then it holds that
$\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\leqslant
Ce^{-\gamma\mathrm{T}_{f}}+2\mathbb{E}\left[1-\exp\left(-\int_{0}^{\mathrm{T}_{f}}g_{\mathrm{T}_{f}-t}(X_{t},V_{t})\mathrm{d}t\right)\right]$
(11)
where
$\displaystyle
g_{t}(x,v)=\frac{(\overleftarrow{\lambda}\wedge\overline{\lambda}\;)(t,(x,v))}{2}\lVert\overleftarrow{Q}(t,(x,v),\cdot)-\overline{Q}(t,(x,v),\cdot)\rVert_{\mathrm{TV}}+\big{\rvert}\overleftarrow{\lambda}(t,(x,v))-\overline{\lambda}(t,(x,v))\big{\rvert}$
(12)
and $\overleftarrow{\lambda},\overleftarrow{Q}$ are as given by Proposition 1.
The proof is postponed to Section D.2. For the sake of illustration, we obtain
a simple upper bound to (11) in the case of ZZP (for the details, see Section
D.3). Assuming the conditions of Theorem 1 are satisfied and also
$\mathbb{E}[\lvert
r_{i}^{\mathrm{Z}}(X_{t},V_{t},\mathrm{T}_{f}-t)-s^{\theta}_{i}(X_{t},V_{t},\mathrm{T}_{f}-t)\rvert\lambda_{i}^{\mathrm{Z}}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})]\leqslant
M$ for all $t\in[0,\mathrm{T}_{f}]$ and $i\in\\{1,\ldots,d\\}$, for the ZZP we
find
$\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\leqslant
Ce^{-\gamma\mathrm{T}_{f}}+4M\mathrm{T}_{f}d\;.$ (13)
Compared to the bounds obtained for diffusion based generative models (see
e.g. Chen et al. (2023)), we observe that the expected error in the estimation
of the score is substituted in the expected error for the estimation of
backward rates.
## 4 Numerical simulations
In this section, we test our piecewise deterministic generative models on
simple synthetic datasets.
##### Design
We compare the generative models based on ZZP, BPS, and RHMC with the improved
denoising diffusion probabilistic model (i-DDPM) given in Nichol and Dhariwal
(2021). For all of our models, we choose the standard normal distribution as
target distribution for the position vector, as well as for the velocity
vector in the cases of BPS and RHMC. The accuracy of trained generative models
is evaluated by the kernel maximum mean discrepancy (MMD). We refer to
Appendix E for a detailed description of the parameters and networks choices.
##### Sample quality
Dataset | i-DDPM | BPS | RHMC | ZZP
---|---|---|---|---
Checkerboard | 2.49 ± 0.98 | 1.96 ± 1.51 | 4.27 ± 3.36 | 0.81 ± 0.19
Fractal tree | 8.04 ± 5.58 | 2.25 ± 1.70 | 4.41 ± 4.35 | 1.12 ± 0.58
Gaussian grid | 23.19 ± 9.72 | 4.59 ± 4.03 | 4.01 ± 3.32 | 4.43 ± 4.05
Olympic rings | 2.03 ± 1.60 | 2.07 ± 1.19 | 2.41 ± 2.24 | 1.43 ± 0.86
Rose | 6.77 ± 5.81 | 1.92 ± 1.57 | 2.16 ± 1.59 | 0.90 ± 0.35
Table 1: MMD $\downarrow$, in units of $1e-3$, averaged over 6 runs, with the
corresponding standard deviations.
Fractal tree
ZZP
i-DDPM
Olympic rings
BPS
i-DDPM
Figure 1: Comparative results on two-dimensional generation of synthetic
datasets.
In Table 1 we report the MMD score for five, $2$-dimensional toy
distributions. We observe that the PDMP based generative models perform well
compared to i-DDPM in all of these five datasets. In particular, ZZP and
i-DDPM are implemented with the same neural network architecture, hence ZZP
appears to compare favourably to i-DDPM with the same model expressivity. The
results of Table 1 are supported by the plots of generated data shown in
Figure 1, illustrating how ZZP and BPS are able to generate more detailed
edges compared to i-DDPM. In Figure 2, we compare the output of RHMC and
i-DDPM for a very small number of reverse steps. We observe how in this
setting the data generated by RHMC are noticeably closer to the true data
distribution compared to i-DDPM. This phenomenon is observed also for BPS as
shown in Table 2, and is intuitively caused by the refreshment kernel, which
is able to generate velocities that correct wrong positions. Respecting this
intuition, ZZP does not perform as well as BPS and RHMC for a small number of
reverse steps since its velocities are constrained to $\\{-1,1\\}$.
Nonetheless, ZZP generates the most accurate results in our experiments given
a large enough number of reverse steps. Additional results can be found in
Appendix E, including some promising results applying the ZZP to the MNIST
dataset.
steps | 2 | 5 | 10 | 25 | 50 | 100 | 250
---|---|---|---|---|---|---|---
i-DDPM | 696.28 | 192.17 | 45.08 | 12.34 | 11.78 | 8.72 | 11.71
BPS | 165.09 | 22.18 | 5.48 | 1.58 | 3.01 | 3.66 | 2.07
RHMC | 26.48 | 3.00 | 1.75 | 0.60 | 0.99 | 1.72 | 1.03
ZZP | 358.25 | 89.49 | 11.31 | 1.20 | 0.71 | 1.04 | 0.42
Table 2: MMD $\downarrow$ on the 2D rose dataset, for the different methods at
various number of backward steps, based on one run.
Gaussian grid
i-DDPM, 2 steps
i-DDPM, 10 steps
RHMC, 2 steps
RHMC, 10 steps
Rose dataset
i-DDPM, 2 steps
i-DDPM, 10 steps
RHMC, 2 steps
RHMC, 10 steps
Figure 2: Comparing RHMC and i-DDPM for small number of reverse steps.
## 5 Discussion and conclusions
We have introduced new generative models based on piecewise deterministic
Markov processes, developing a theoretically sound framework with specific
focus on three PDMPs from the sampling literature. While this work lays the
foundations of this class of methods, it also opens several directions worth
investigating in the future.
Similarly to other generative models, our PDMP based algorithms are sensitive
to the choice of the network architecture that is used to approximate the
backward characteristics. Therefore, it is crucial to investigate which
architectures are most suited for our algorithms in order to achieve state of
the art performance in real world scenarios. For instance, in the case of BPS
and RHMC it could be beneficial to separate the estimation of the density
ratios and the generation of draws of the velocity conditioned on the position
and time. For the case of ZZP, efficient techniques to learn the network in a
high dimensional setting need to be investigated, while network architectures
that resemble those used to approximate the score function appear to adapt
well to the case of density ratios. Moreover, there are several alternative
PDMPs that could be used as generative models and that we did not consider in
detail in this paper, as for instance variance exploding alternatives.
## Acknowledgments and Disclosure of Funding
AB, AOD, and EM are funded by the European Union (ERC, Ocean, 101071601). US
and DS are funded by the European Union (ERC, Dynasty, 101039676). Views and
opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. US is additionally funded by the
French government under management of Agence Nationale de la Recherche as part
of the "Investissements d’avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE
3IA Institute). The authors are grateful to the CLEPS infrastructure from the
Inria of Paris for providing resources and support.
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## Appendix
The Appendix is organised as follows. Appendix A includes the proofs and
details regarding Section 2.1 and Section 2.2. Appendix B contains details and
proofs regarding the framework of density ratio matching. Appendix C contains
the pseudo-codes of the splitting schemes that are used to simulate the
backward PDMPs. Appendix D contains the proof for Theorem 1 and some related
details Finally, Appendix E contains the details on the numerical simulations,
as well as additional results.
## Appendix A PDMPs and their time reversals
### A.1 Construction of a PDMP with multiple jump types
In this section we describe the formal construction of a non-homogeneous PDMP
with the characteristics $(\Phi,\lambda,Q)$ where $\lambda,Q$ are of the form
$\lambda(t,z)=\sum_{i=1}^{\ell}\lambda_{i}(t,z)\;,\quad
Q(t,z,\mathrm{d}z^{\prime})=\sum_{i=1}^{\ell}\frac{\lambda_{i}(t,z)}{\lambda(t,z)}Q_{i}(t,z,\mathrm{d}z^{\prime})\;.$
(14)
Recall the differential flow $\varphi:(t,s,z)\mapsto\varphi_{t,t+s}(z)$, which
solves the ODE, $\mathrm{d}z_{t+s}=\Phi(t+s,z_{t+s})\mathrm{d}s$ for
$s\geqslant 0$, i.e. $z_{t+s}=\varphi_{t,t+s}(z_{t})$. Similarly to the case
of one type of jump only, we start the PDMP from an initial state $Z_{0}$,
assume it is defined as $(Z_{t})_{t\in\left[0,\mathrm{T}_{n}\right]}$ on
$\left[0,\mathrm{T}_{n}\right]$ for some $n\in\mathbb{N}$, and we now define
define $(Z_{t})_{t\in\left[\mathrm{T}_{n},\mathrm{T}_{n+1}\right]}$. First, we
define the proposals $(\tau^{i}_{n+1})_{i\in\\{1,\ldots,\ell\\}}$ for next
event time as
$\tau^{i}_{n+1}=\inf\left\\{t>0:\int_{0}^{t}\lambda_{i}(T_{n}+u,\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+u}(Z_{\mathrm{T}_{n}}))\mathrm{d}u\geqslant
E_{n+1}^{i}\right\\}$ (15)
where $E_{n+1}^{i}\sim\text{Exp}(1)$ for $i\in\\{1,\ldots,\ell\\}.$ Then
define
$i^{*}=\operatorname*{arg\,min}_{i\in\\{1,\ldots,\ell\\}}\tau^{i}_{n+1}$ and
set the next jump time to
$T_{n+1}=T_{n}+\tau^{i^{*}}_{n+1}.$
The process is then defined on $\left[\mathrm{T}_{n},\mathrm{T}_{n+1}\right)$
by
$Z_{\mathrm{T}_{n}+t}=\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+t}(Z_{\mathrm{T}_{n}})$
for $t\in[0,\tau_{n+1})$. Finally, we set $Z_{\mathrm{T}_{n+1}}\sim
Q_{i^{*}}(T_{n+1},\varphi_{\mathrm{T}_{n},\mathrm{T}_{n}+\tau_{n+1}}(Z_{\mathrm{T}_{n}}),\cdot)$.
### A.2 Extended generator
In order to define the generator of a PDMP, Davis [1993, Theorem 26.14]
requires the set of conditions Davis [1993, (24.8)]. Notably, the PDMP is
required to be non-explosive in the sense that the expected number of random
events after any time $t$ starting the PDMP from any state should be finite.
These conditions are verified for the forward PDMPs we consider.
###### H 3
The characteristics $(\Phi,\lambda,Q)$ satisfy the standard conditions [Davis,
1993, (24.8)].
Assuming H 3, Davis [1993, Theorem 26.14] gives that the extended generator of
a PDMP with characteristics $(\Phi,\lambda,Q)$ is given by
$\mathscr{L}_{t}f(z)=\langle\Phi(t,z),\nabla_{z}f(z)\rangle+\lambda(t,z)\int_{\mathbb{R}^{d}}(f(y)-f(z))Q(t,z,\mathrm{d}y)\;,$
(16)
for all functions $f\in\mathrm{dom}(\mathscr{L}_{t})$, that is the space of
measurable functions such that
$M_{t}^{f}=f(Z_{t})-f(Z_{0})-\int_{0}^{t}\mathscr{L}_{s}f(Z_{s})\mathrm{d}s$
is a local martingale. We also introduce the Carré du champ
$\Gamma_{t}(f,g):=\mathscr{L}_{t}(fg)-f\mathscr{L}_{t}g-g\mathscr{L}_{t}f,$
with domain
$\mathrm{dom}(\Gamma_{t}):=\\{f,g:f,g,fg\in\mathrm{dom}(\mathscr{L}_{t})\\}$
which in the case of a PDMP with generator (16) takes the form
$\Gamma_{t}(f,g)(z)=\lambda(t,z)\int_{\mathbb{R}^{d}}(f(y)-f(z))(g(y)-g(z))Q(t,z,\mathrm{d}y)\;.$
(17)
### A.3 Proof of Proposition 1
In order to prove Proposition 1 we apply Conforti and Léonard [2022, Theorem
5.7] and hence in this section we verify the required assumptions. Before
starting, we state the following technical condition which we omitted in
Proposition 1 and is assumed in Conforti and Léonard [2022, Theorem 5.7].
###### H 4
It holds
$\mathrm{C}_{c}^{2}(\mathbb{R}^{d})\subset\mathrm{dom}(\mathscr{L}_{t})$ for
any $t\in\mathbb{R}_{+}$.
We now turn to verifying the remaining assumptions in Conforti and Léonard
[2022, Theorem 5.7]. The “General Hypotheses” of Conforti and Léonard [2022]
are satisfied since we assume the vector field $\Phi$ is locally bounded, the
switching rate $(t,z)\mapsto\lambda(t,z)$ is a continuous function, and the
jump kernel $Q$ is such that $Q(t,x,\cdot)$ is a probability distribution. In
particular these assumptions imply that
$\sup_{t\in[0,T],\lvert
z\rvert\leqslant\rho}\int_{\mathbb{R}^{d}}(1\wedge\lvert
z-y\rvert^{2})\lambda(t,z)Q(t,z,\mathrm{d}y)\leqslant\sup_{t\in[0,T],\lvert
z\rvert\leqslant\rho}\lambda(t,z)<\infty\quad\text{for all }\rho\geqslant 0.$
(18)
Then, Conforti and Léonard [2022, Theorem 5.7] requires a further
integrability condition, which is satisfied when
$\int_{[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}}(1\wedge\lvert
z-y\rvert^{2})\mu_{0}P_{t}(\mathrm{d}z)\lambda(t,z)Q(t,z,\mathrm{d}y)<\infty.$
(19)
It is then sufficient to have that
$\int_{0}^{\mathrm{T}_{f}}\mathbb{E}[\lambda(t,Z_{t})]\mathrm{d}t<\infty$ (20)
Finally, Conforti and Léonard [2022, Theorem 5.7] requires some technical
assumptions which we now discuss. Introduce the class of functions that are
twice continuously differentiable and compactly supported, denoted by
$\mathcal{C}^{2}_{c}(\mathbb{R}^{d})$, and for
$f\in\mathcal{C}^{2}_{c}(\mathbb{R}^{d})$ consider the two following
conditions:
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}\mu_{0}P_{t}(\mathrm{d}z)\lvert\mathscr{L}_{t}f(z)\rvert\mathrm{d}t<\infty,$
(21)
$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}\mu_{0}P_{t}(\mathrm{d}z)\lvert\Gamma_{t}(f,g)(z)\rvert\mathrm{d}t<\infty\qquad\text{
for all }g\in\mathcal{C}^{2}_{c}(\mathsf{\mathbb{R}}^{d}).$ (22)
We define
$\mathcal{F}:=\\{f\in\mathcal{C}^{2}_{c}(\mathsf{E}):\eqref{eq:integrab_generator},\eqref{eq:integrab_carreduchamp}\text{
hold }\\}$. We need to verify that
$\mathcal{F}\equiv\mathcal{C}^{2}_{c}(\mathsf{E}).$ Let us start by
considering (21): we find
$\displaystyle\int_{0}^{T}\mu_{0}P_{t}(\mathrm{d}z)\lvert\mathscr{L}_{t}f(z)\rvert\mathrm{d}t$
(23)
$\displaystyle\leqslant\int_{0}^{T}\mu_{0}P_{t}(\mathrm{d}z)\left(\lvert\langle\Phi(t,z),\nabla
f(z)\rangle\rvert+\lambda(t,z)\int\lvert u(y)-u(z)\rvert
Q(t,z,\mathrm{d}y)\right)\mathrm{d}t.$ (24)
Since $f\in\mathcal{C}^{2}_{c}(\mathsf{E})$ we have that
$\lvert\langle\Phi(t,z),\nabla f(z)\rangle\rvert$ is compactly supported and
hence integrable, while the second term is finite assuming
$\int_{0}^{T}\mathbb{E}[\lambda(t,Z_{t})]\mathrm{d}t<\infty.$ Under the latter
assumption, (22) can be easily verified.
### A.4 Proof of Proposition 2
Let us denote the initial condition of the forward PDMP by
$\mu_{0}=\mu_{0}^{X}\otimes\mu_{0}^{V}$. First of all, notice that, for a PDMP
with position-velocity decomposition and homogeneous jump kernel, the flux
equation (3) becomes
$\mu_{0}P_{\tilde{t}}(\mathrm{d}y,\mathrm{d}w)\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\tilde{t}}(\mathrm{d}x,\mathrm{d}v)\lambda(\tilde{t},(x,v))Q((x,v),(\mathrm{d}y,\mathrm{d}w))$
(25)
where $\tilde{t}=\mathrm{T}_{f}-t.$ Moreover, since the jump kernel leaves the
position vector unchanged we obtain that this is equivalent to
$\mu_{0}P_{\tilde{t}}(\mathrm{d}w\rvert
y)\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\tilde{t}}(\mathrm{d}v\rvert
y)\lambda(\tilde{t},(y,v))Q((x,v),(\mathrm{d}y,\mathrm{d}w)),$
where $\mu_{0}P_{t}(\mathrm{d}w\rvert y)$ is the conditional law of the
velocity vector given the position vector at time $t$ with initial
distribution $\mu_{0}.$
Suppose first that
$Q((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v)$
for an involution $\mathscr{R}_{y}$. Then we find
$\mu_{0}P_{\tilde{t}}(\mathrm{d}w\rvert
y)\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\tilde{t}}(\mathrm{d}\mathscr{R}_{y}w\rvert
y)\lambda(\tilde{t},(y,\mathscr{R}_{y}w))\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v)$
where we used that
$\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v)=\updelta_{\mathscr{R}_{y}v}(\mathrm{d}w)$
since $\mathscr{R}_{y}$ is an involution. Under our assumptions we have
$\mu_{0}P_{\tilde{t}}(\mathrm{d}\mathscr{R}_{y}w\rvert
y)=p_{\tilde{t}}(\mathscr{R}_{y}w\rvert
y)\mu_{\mathrm{ref}}^{V}(\mathrm{d}w),\quad\mu_{0}P_{\tilde{t}}(\mathrm{d}w\rvert
y)=p_{\tilde{t}}(w\rvert y)\mu_{\mathrm{ref}}^{V}(\mathrm{d}w),\;$
since we assumed
$\mu_{\mathrm{ref}}^{V}(\mathrm{d}w)=\mu_{\mathrm{ref}}^{V}(\mathrm{d}\mathscr{R}_{y}w).$
Hence we find for any $(y,w)\in\mathbb{R}^{2d}$ such that
$p_{\tilde{t}}(w\rvert y)>0$
$\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\frac{p_{\tilde{t}}(\mathscr{R}_{y}w\rvert
y)}{p_{\tilde{t}}(w\rvert
y)}\lambda(\tilde{t},(y,\mathscr{R}_{y}w))\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v).$
This can only be satisfied if
$\overleftarrow{\lambda}(t,(y,w))=\frac{p_{\tilde{t}}(\mathscr{R}_{y}w\rvert
y)}{p_{\tilde{t}}(w\rvert y)}\lambda(\tilde{t},(y,\mathscr{R}_{y}w)),\quad
Q(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\updelta_{\mathscr{R}_{y}w}(\mathrm{d}v).$
Consider now the second case, that is
$Q((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{y}(\mathrm{d}x)\nu(\mathrm{d}v\rvert
y)$ and $\lambda(t,(y,w))=\lambda(t,y)$. The flux equation (3) can be
rewritten as
$\mu_{0}P_{\tilde{t}}(\mathrm{d}w\rvert
y)\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\tilde{t}}(\mathrm{d}v\rvert
y)\lambda(\tilde{t},y)\updelta_{y}(\mathrm{d}x)\nu(\mathrm{d}w\rvert y)$
Under our assumptions we have $\nu(\mathrm{d}w\rvert
y)=(\mathrm{d}\nu/\mathrm{d}\mu_{\mathrm{ref}}^{V})(w\rvert
y)\mu_{\mathrm{ref}}^{V}(\mathrm{d}w)$ and
$\mu_{0}P_{\tilde{t}}(\mathrm{d}w\rvert y)=p_{\tilde{t}}(w\rvert
y)\mu_{\mathrm{ref}}^{V}(\mathrm{d}w)$ for some measure
$\mu_{\mathrm{ref}}^{V}$. Hence for any $(y,w)\in\mathbb{R}^{2d}$ such that
$p_{\tilde{t}}(w\rvert y)>0$ we obtain
$\overleftarrow{\lambda}(t,(y,w))\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\frac{(\mathrm{d}\nu/\mathrm{d}\mu_{\mathrm{ref}}^{V})(w\rvert
y)}{p_{\tilde{t}}(w\rvert
y)}\lambda(\tilde{t},y)p_{\tilde{t}}(\mathrm{d}v\rvert
y)\updelta_{y}(\mathrm{d}x).\;$ (26)
This is satisfied when
$\overleftarrow{\lambda}(t,(y,w))=\frac{(\mathrm{d}\nu/\mathrm{d}\mu_{\mathrm{ref}}^{V})(w\rvert
y)}{p_{\tilde{t}}(w\rvert
y)}\lambda(\tilde{t},y),\quad\overleftarrow{Q}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\tilde{t}}(\mathrm{d}v\rvert
y)\updelta_{y}(\mathrm{d}x).$
### A.5 Extension of Proposition 2 to multiple jump types
Proposition 2 considers PDMPs with one type of jump, while here we discuss the
case of characteristics of the form (14), which is e.g. the case of ZZP and
BPS. In this setting we can assume the backward jump rate and kernel have a
similar structure, that is
$\overleftarrow{\lambda}(t,z)=\sum_{i=1}^{\ell}\overleftarrow{\lambda}_{i}(t,z)\;,\quad\overleftarrow{Q}(t,z,\mathrm{d}z^{\prime})=\sum_{i=1}^{\ell}\frac{\overleftarrow{\lambda}_{i}(t,z)}{\overleftarrow{\lambda}(t,z)}\overleftarrow{Q}_{i}(t,z,\mathrm{d}z^{\prime})\;,$
(27)
in which case the balance condition (3) can be rewritten as
$\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}y)\sum_{i=1}^{\ell}\overleftarrow{\lambda}_{i}(t,y)\overleftarrow{Q}_{i}(t,y,\mathrm{d}z)=\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}z)\sum_{i=1}^{\ell}\lambda_{i}(\mathrm{T}_{f}-t,z)Q_{i}(\mathrm{T}_{f}-t,z,\mathrm{d}y)\;.$
(28)
It is then enough that
$\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}y)\overleftarrow{\lambda}_{i}(t,y)\overleftarrow{Q}_{i}(t,y,\mathrm{d}z)=\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}z)\lambda_{i}(\mathrm{T}_{f}-t,z)Q_{i}(\mathrm{T}_{f}-t,z,\mathrm{d}y)\;$
(29)
holds for all $i\in\\{1,\dots,\ell\\}.$ It follows that it is sufficient to
apply Proposition 2 to each pair $(\lambda_{i},Q_{i})$ to obtain
$(\overleftarrow{\lambda}_{i},\overleftarrow{Q}_{i})$ such that (3) holds.
### A.6 Time reversals of ZZP, BPS, and RHMC
In this section we give rigorous statements regarding time reversals of ZZP,
BPS, and RHMC. For all samplers we rely on Proposition 2 and hence we focus on
verifying its assumptions. In the cases of ZZP and RHMC we assume the
technical condition H 4 since proving it rigorously is out of the scope of the
present paper. We remark that this can be proved with techniques as in Durmus
et al. [2021], which show H 4 in the case of BPS.
###### Proposition 4 (Time reversal of ZZP)
Consider a ZZP $(X_{t},V_{t})_{t\in[0,\mathrm{T}_{f}]}$ with initial
distribution $\mu_{\star}\otimes\nu$, where $\nu=\mathrm{Unif}(\\{\pm
1\\}^{d})$ and invariant distribution $\pi\otimes\nu$, where $\pi$ has
potential $\psi$ satisfying H 1. Assume that H 4 holds and that
$\int\mu_{\star}(\mathrm{d}x)\lvert\partial_{i}\psi(x)\rvert<\infty$ for all
$i=1,\dots,d.$ Then the time reversal of the ZZP has vector field
$\overleftarrow{\Phi}^{\mathrm{Z}}(x,v)=(-v,0)^{T}$
and jump rates and kernels are given for all $(y,w)\in\mathbb{R}^{2d}$ such
that $P_{\mathrm{T}_{f}-t}(w\rvert y)>0$ by
$\displaystyle\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(y,w))=\frac{p_{\mathrm{T}_{f}-t}(\mathscr{R}_{i}^{\mathrm{Z}}w\rvert
y)}{p_{\mathrm{T}_{f}-t}(w\rvert
y)}\lambda^{\mathrm{Z}}_{i}(y,\mathscr{R}_{i}^{\mathrm{Z}}w),\quad\overleftarrow{Q}_{i}^{\mathrm{Z}}((y,w),(\mathrm{d}x,v))=\updelta_{(y,\mathscr{R}^{\mathrm{Z}}_{i}w)}(\mathrm{d}x,v)$
(30)
for $i=1,\dots,d$.
##### Proof
We verify the conditions of Proposition 2 corresponding to deterministic
transitions and rely on Section A.5 to apply the proposition to each pair
$(\lambda^{\mathrm{Z}}_{i},Q^{\mathrm{Z}}_{i})$. First notice the vector field
$\Phi(x,v)=(v,0)^{T}$ is clearly locally bounded and
$(t,x)\mapsto\lambda(x,v)$ is continuous since $\psi$ is continuously
differentiable. Moreover, the ZZP can be shown to be non-explosive applying
Durmus et al. [2021, Proposition 9]. Then, we need to verify (20). First,
observe that
$\mathbb{E}[\lambda_{i}(X_{t},V_{t})]\leqslant\mathbb{E}[\lvert\partial_{i}\psi(X_{t})\rvert].$
Then
$\displaystyle\mathbb{E}[\lvert\partial_{i}\psi(X_{t})\rvert]$
$\displaystyle=\mathbb{E}\left[\left\lvert\partial_{i}\psi(X_{0})+\int_{0}^{1}\langle
X_{t}-X_{0},\nabla\partial_{i}\psi(X_{0}+s(X_{t}-X_{0}))\rangle\mathrm{d}s\right\rvert\right]$
(31)
$\displaystyle\leqslant\mathbb{E}[\lvert\partial_{i}\psi(X_{0})\rvert]+\mathbb{E}\left[\int_{0}^{1}\lvert\langle
X_{t}-X_{0},\nabla^{2}\psi(X_{0}+s(X_{t}-X_{0}))\mathsf{e}_{i}\rangle\rvert\mathrm{d}s\right]$
(32)
where $\mathsf{e}_{i}$ is the $i$-th vector of the canonical basis. Notice
that $\lvert X_{t}-X_{0}\rvert\leqslant t\sqrt{d}$. Thus we find
$\displaystyle\mathbb{E}[\lvert\partial_{i}\psi(X_{t})\rvert]$
$\displaystyle\leqslant\mathbb{E}[\lvert\partial_{i}\psi(X_{0})\rvert]+t\sqrt{d}\sup_{x\in\mathbb{R}^{d}}\lVert\nabla^{2}\psi(x)\rVert$
(33)
and therefore
$\displaystyle\int_{0}^{\mathrm{T}_{f}}\mathbb{E}[\lambda(X_{t},V_{t})]\mathrm{d}t\leqslant\mathrm{T}_{f}\sum_{i=1}^{d}\left(\mathbb{E}\lvert\partial_{i}\psi(X_{0})\rvert+\frac{\mathrm{T}_{f}}{2}\sqrt{d}\sup_{x\in\mathbb{R}^{d}}\lVert\nabla^{2}\psi(x)\rVert\right).$
(34)
Since $\mathbb{E}_{\mu_{\star}}\lvert\partial_{i}\psi(X)\rvert<\infty$ and
because we are assuming H 1, we obtain (20). Finally, notice that
$P_{t}(\mathrm{d}v\rvert x)$ is absolutely continuous with respect to the
counting measure on $\\{1,-1\\}^{d}$, which is clearly invariant with respect
to $\mathscr{R}^{\mathrm{Z}}_{i}$. $\square$
###### Proposition 5 (Time reversal of BPS)
Consider a BPS $(X_{t},V_{t})_{t\in[0,\mathrm{T}_{f}]}$ with initial
distribution $\mu_{\star}\otimes\nu$, where
$\nu=\mathrm{Unif}(\mathsf{S}^{d-1})$, and invariant distribution
$\pi\otimes\nu$, where $\pi$ has potential $\psi$ satisfying H 1. Assume that
$\mathbb{E}_{\mu_{\star}}[|\nabla\psi(X)\rvert]<\infty.$ Then there exists a
density $p_{t}(w\rvert
y):=\nicefrac{{\mathrm{d}(\mu_{0}P_{t})(\mathrm{d}w\rvert
y)}}{{\nu(\mathrm{d}w)}}.$ Moreover, the time reversal of the BPS has vector
field
$\overleftarrow{\Phi}^{\mathrm{B}}(x,v)=(-v,0)^{T},$
while the jump rates and kernels are given for all
$t,y,w\in[0,\mathrm{T}_{f}]\times\mathbb{R}^{2d}$ such that
$p_{\mathrm{T}_{f}-t}(w\rvert y)>0$ by
$\displaystyle\overleftarrow{\lambda}_{1}^{\mathrm{B}}(t,(y,w))=\frac{p_{\tilde{t}}(\mathscr{R}^{\mathrm{B}}_{y}w\rvert
y)}{p_{\tilde{t}}(w\rvert
y)}\lambda_{1}^{\mathrm{B}}(y,\mathscr{R}^{\mathrm{B}}_{y}w),\,\,\overleftarrow{Q}_{1}^{\mathrm{B}}((y,w),(\mathrm{d}x,\mathrm{d}v))=\updelta_{(y,\mathscr{R}^{\mathrm{B}}_{y}w)}(\mathrm{d}x,\mathrm{d}v),$
(35)
$\displaystyle\overleftarrow{\lambda}_{2}^{\mathrm{B}}(t,(y,w))=\lambda_{r}\frac{1}{p_{\mathrm{T}_{f}-t}(w|y)},\quad\overleftarrow{Q}_{2}^{\mathrm{B}}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=\mu_{0}P_{\mathrm{T}_{f}-t}(\mathrm{d}v\rvert
y)\updelta_{y}(\mathrm{d}x)\mathrm{d}v,$ (36)
where $\tilde{t}=\mathrm{T}_{f}-t$.
###### Remark 3
Under the assumption that $\nu$ is the uniform distribution on the sphere, it
is natural to take $\mu_{\mathrm{ref}}^{\mathrm{V}}=\nu$, which gives that
$\nicefrac{{\mathrm{d}\nu}}{{\mathrm{d}\mu_{\mathrm{ref}}^{V}}}=1$ and hence
the backward refreshment rate is as in (36). When $\nu$ is the $d$-dimensional
Gaussian distribution, the natural choice is to let
$\mu_{\mathrm{ref}}^{\mathrm{V}}$ be the Lebesgue measure and hence we obtain
a rate as given in (5).
##### Proof
We verify the general conditions of Proposition 2, then focusing on the
deterministic jumps and the refreshments relying on Section A.5. The BPS was
shown to be non-explosive for any initial distribution in Durmus et al. [2021,
Proposition 10]. Since $\lambda(t,(x,v))=\lambda(x,v)=\langle
v,\nabla\psi(x)\rangle_{+}$, with a similar reasoning of the proof of
Proposition 4 we have
$\displaystyle\mathbb{E}[\lambda(X_{t},V_{t})]=\mathbb{E}\left[\left(\langle
V_{t},\nabla\psi(X_{0})\rangle+\int_{0}^{1}\langle
V_{t},\nabla^{2}\psi(X_{0}+s(X_{t}-X_{0}))(X_{t}-X_{0})\rangle\mathrm{d}s\right)_{+}\right].$
(37)
Taking advantage of $\lvert V_{t}\rvert=1$ we have $\lvert
X_{t}-X_{0}\rvert\leqslant t$ and thus we find
$\displaystyle\mathbb{E}[\lambda(X_{t},V_{t})]$
$\displaystyle\leqslant\mathbb{E}\left[|\nabla\psi(X_{0})\rvert+\int_{0}^{1}\lvert\nabla^{2}\psi(X_{0}+s(X_{t}-X_{0}))(X_{t}-X_{0})\rvert\mathrm{d}s\right]$
(38)
$\displaystyle\leqslant\mathbb{E}\left[|\nabla\psi(X_{0})\rvert\right]+t\sup_{x\in\mathbb{R}^{d}}\lVert\nabla^{2}\psi(x)\rVert.$
(39)
This is sufficient to obtain (20) since
$\mathbb{E}[|\nabla\psi(X_{0})\rvert]<\infty$ and we assume H 1. Moreover, H 4
holds by Durmus et al. [2021, Proposition 23]. Finally notice that
$P_{t}(\mathrm{d}v\rvert x)$ is absolutely continuous with respect to
$\mu_{\mathrm{ref}}^{\mathrm{V}}=\mathrm{Unif}(\mathsf{S}^{d-1})$, which
satisfies
$\mu_{\mathrm{ref}}^{\mathrm{B}}(\mathscr{R}^{\mathrm{B}}(x)v)=\mu_{\mathrm{ref}}^{\mathrm{B}}(v)$
for all $x,v\in\mathbb{R}^{d}\times\mathsf{S}^{d-1}$. All the required
assumptions in Proposition 2 are thus satisfied. $\square$
###### Proposition 6 (Time reversal of RHMC)
Consider a RHMC $(X_{t},V_{t})_{t\in[0,\mathrm{T}_{f}]}$ with initial
distribution $\mu_{\star}\otimes\nu$, where $\nu$ is the $d$-dimensional
standard normal distribution, and invariant distribution $\pi\otimes\nu$,
where $\pi$ has potential $\psi\in\mathcal{C}^{1}(\mathbb{R}^{d})$. Suppose
that H 4 holds and that for any $y\in\mathbb{R}^{d}$, $P_{t}(\mathrm{d}w|y)$
is absolutely continuous with respect to Lebesgue measure, with density
$p_{t}(w\rvert y)$. Then the time reversal of the RHMC has vector field
$\overleftarrow{\Phi}^{\mathrm{H}}(x,v)=(-v,\nabla\psi(x))^{T},$
while the jump rates and kernels are given for all $(y,w)\in\mathbb{R}^{2d}$
such that $p_{\mathrm{T}_{f}-t}(w\rvert y)>0$ by
$\displaystyle\overleftarrow{\lambda}_{2}^{\mathrm{H}}(t,(y,w))=\lambda_{r}\frac{\nu(w)}{p_{\mathrm{T}_{f}-t}(w|y)},\quad\overleftarrow{Q}_{2}^{\mathrm{H}}(t,(y,w),(\mathrm{d}x,\mathrm{d}v))=p_{\mathrm{T}_{f}-t}(v\rvert
y)\updelta_{y}(\mathrm{d}x)\mathrm{d}v.$ (40)
##### Proof
First of all, RHMC is non-explosive by Durmus et al. [2021, Proposition 8].
Then $\Phi$ is locally bounded and (20) is trivially satisfied. Finally, we
can take $\mu_{\mathrm{ref}}^{\mathrm{V}}$ to be the Lebesgue measure.
$\square$
## Appendix B Density ratio matching
### B.1 Ratio matching with Bregman divergences
We now describe a general approach to approximate ratios of densities based on
the minimisation of Bregman divergences [Sugiyama et al., 2011], which as we
discuss is closely connected to the loss of Hyvärinen [2007].
For a differentiable, strictly convex function $f$ we define the Bregman
divergence $\mathrm{B}_{f}(r,s):=f(r)-f(s)-f^{\prime}(s)(r-s)$. Given two
time-dependent probability density functions on $\mathbb{R}^{2d}$, $p,q$, we
wish to approximate their ratio
$r(x,v,t)=\nicefrac{{p_{t}(x,v)}}{{q_{t}(x,v)}}$ for $t\in[0,\mathrm{T}_{f}]$
with a parametric function
$s_{\theta}:\mathbb{R}^{d}\times\mathbb{R}^{d}\times[0,\mathrm{T}_{f}]\to\mathbb{R}_{+}$
by solving the minimisation problem
$\min_{\theta}\int_{0}^{\mathrm{T}_{f}}\omega(t)\,\mathbb{E}\Big{[}\mathrm{B}_{f}(r(X_{t},V_{t},t),s_{\theta}(X_{t},V_{t},t))\Big{]}\mathrm{d}t,$
(41)
where the expectation is with respect to the joint density $q_{t}(x,v)$, that
is $(X_{t},V_{t})\sim q_{t}$, while $\omega$ is a probability density function
for the time variable. Well studied choices of the function $f$ include e.g.
$f(r)=r\log r-r$, that is related to a KL divergence, or $f(r)=(r-1)^{2}$,
related to the square loss, or $f(r)=r\log r-(1+r)\log(1+r),$ which
corresponding to solving a logistic regression task. Ignoring terms that do
not depend on $\theta$ we can rewrite the minimisation as
$\min_{\theta}\int_{0}^{\mathrm{T}_{f}}\\!\\!\\!\omega(t)\left(\mathbb{E}_{p_{t}}\big{[}f^{\prime}(s_{\theta}(X_{t},V_{t},t))s_{\theta}(X_{t},V_{t},t)-f(s_{\theta}(X_{t},V_{t},t))\big{]}-\mathbb{E}_{q_{t}}\big{[}f^{\prime}(s_{\theta}(X_{t},V_{t},t))\big{]}\right)\mathrm{d}t.$
(42)
Notably this is independent of the true density ratio and thus it is a
formulation with similar spirit to _implicit score matching._ Naturally, in
practice the loss can be approximated empirically with a Monte Carlo average.
### B.2 Details and proofs regarding Hyvärinen’s ratio matching
#### B.2.1 Connection to Bregman divergences
In the next statement, we show that the loss ${\ell}_{\mathrm{I}}$ defined in
(6), or equivalently its explicit counterpart ${\ell}_{\mathrm{E}}$ (see
Proposition 3), can be put in the framework of Bregman divergences.
###### Corollary 1
Recall $\mathbf{G}(r)=(1+r)^{-1}$ and let $f(r)=\nicefrac{{(r-1)^{2}}}{{2}}.$
The task $\min_{\theta}{\ell}_{\mathrm{E}}(\theta)$ is equivalent to
$\displaystyle\min_{\theta}$
$\displaystyle\sum_{i=1}^{d}\;\mathbb{E}_{p_{t}}\Big{[}\mathrm{B}_{f}(\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t)),\mathbf{G}(r_{i}(X_{t},V_{t},t)))$
(43)
$\displaystyle\quad+\mathrm{B}_{f}(\mathbf{G}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t)),\mathbf{G}(r_{i}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t)))\Big{]}$
(44)
##### Proof
The result follows by straightforward computations. $\square$
#### B.2.2 Proof of Proposition 3
The proof follows the same lines as Hyvärinen [2007, Theorem 1]. We find
$\displaystyle{\ell}_{\mathrm{E}}(\theta)$
$\displaystyle=C+\int_{0}^{\mathrm{T}_{f}}\omega(t)\sum_{i=1}^{d}\mathbb{E}_{p_{t}}\Big{[}\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},V_{t},t))+\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))$
(45)
$\displaystyle\quad-2\mathbf{G}(r_{i}(X_{t},V_{t},t))\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))-2\mathbf{G}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))\mathbf{G}(r_{i}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))\Big{]}\mathrm{d}t,$
(46)
where $C$ is a constant independent of $\theta.$ Then plugging in the
expression of $\mathbf{G}$ we can rewrite the last term as
$\displaystyle\mathbb{E}_{p_{t}}\Big{[}\mathbf{G}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))\mathbf{G}(r_{i}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))\Big{]}$
(47) $\displaystyle=\int\sum_{v\in\\{\pm
1\\}^{d}}p_{t}(x,v)\mathbf{G}(s_{i}^{\theta}(x,\mathscr{R}_{i}^{\mathrm{Z}}v,t))\frac{p_{t}(x,\mathscr{R}_{i}^{\mathrm{Z}}v)}{p_{t}(x,v)+p_{t}(x,\mathscr{R}_{i}^{\mathrm{Z}}v)}\mathrm{d}x$
(48)
$\displaystyle=\mathbb{E}_{p_{t}}\Big{[}\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))\frac{p_{t}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})}{p_{t}(X_{t},V_{t})+p_{t}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})}\Big{]}.$
(49)
Therefore we find
$\displaystyle{\ell}_{\mathrm{E}}(\theta)=C+\int_{0}^{\mathrm{T}_{f}}\omega(t)\sum_{i=1}^{d}\mathbb{E}_{p_{t}}\Bigg{[}\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},V_{t},t))+\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))$
(50)
$\displaystyle\quad-\frac{2\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))\,p_{t}(X_{t},V_{t})}{p_{t}(X_{t},V_{t})+p_{t}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})}-\frac{2\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))\,p_{t}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})}{p_{t}(X_{t},V_{t})+p_{t}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t})}\Bigg{]}\mathrm{d}t$
(51)
$\displaystyle=C+\\!\int_{0}^{\mathrm{T}_{f}}\\!\\!\\!\omega(t)\sum_{i=1}^{d}\mathbb{E}_{p_{t}}\Big{[}\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},V_{t},t))+\mathbf{G}^{2}(s_{i}^{\theta}(X_{t},\mathscr{R}_{i}^{\mathrm{Z}}V_{t},t))-2\mathbf{G}(s_{i}^{\theta}(X_{t},V_{t},t))\Big{]}\mathrm{d}t.$
(52)
## Appendix C Discretisations of time reversed PDMPs with splitting schemes
Here we discuss the splitting schemes we use to discretise the backward PDMPs.
We give the pseudo-code for RHMC in Algorithm 1, and discuss the cases of ZZP
and BPS below. For further details on this class of approximations we refer
the reader to Bertazzi et al. [2023].
Initialise either from $(\overline{X}_{0},\overline{V}_{0})\sim\pi\otimes\nu$
or
$(\overline{X}_{0},\overline{V}_{0})\sim\pi(\mathrm{d}x)p_{\theta^{*}}(\mathrm{d}v\rvert
x,\mathrm{T}_{f})$;
for _$n=0,\dots,N-1$_ do
$(\widetilde{X},\widetilde{V})=\varphi^{\mathrm{H}}_{-\nicefrac{{\delta_{n+1}}}{{2}}}(\overline{X}_{t_{n}},\overline{V}_{t_{n}})$
;
$\widetilde{t}=\mathrm{T}_{f}-t_{n}-\frac{\delta_{n+1}}{2}$ ;
Estimate ratio:
$\overline{s}=\nicefrac{{\nu(\widetilde{V})}}{{p_{\theta^{*}}(\;\widetilde{V}\rvert\;\widetilde{X},\;\widetilde{t}\;)}}$
;
Draw proposal $\tau_{n+1}\sim\mathrm{Exp}(\overline{s}\lambda_{r})$ ;
if _$\tau_{n+1}\leqslant\delta_{n+1}$_ then
Draw $\widetilde{V}\sim
p_{\theta^{*}}(\,\cdot\,\rvert\widetilde{X},\;\widetilde{t}\;)$;
end if
$(\overline{X}_{t_{n+1}},\overline{V}_{t_{n+1}})=\varphi^{\mathrm{H}}_{-\nicefrac{{\delta_{n+1}}}{{2}}}(\widetilde{X},\widetilde{V})$;
end for
Algorithm 1 Splitting scheme DJD for the time reversed RHMC
Initialise $(\overline{X}_{0},\overline{V}_{0})\sim\pi\otimes\nu$;
for _$n=0,\dots,N-1$_ do
$\widetilde{X}=\overline{X}_{t_{n}}-\frac{\delta_{n+1}}{2}\;\overline{V}_{t_{n}}$
;
$\widetilde{V}=\overline{V}_{t_{n}}$ ;
$\widetilde{t}=\mathrm{T}_{f}-t_{n}-\frac{\delta_{n+1}}{2}$ ;
Estimate density ratios:
$s^{\theta^{*}}(\widetilde{X},\widetilde{V},\widetilde{t}\;)$ ;
for _$i=1\dots,d$_ do
With probability
$(1-\exp(-\delta_{n+1}\;s^{\theta^{*}}_{i}(\widetilde{X},\widetilde{V},\widetilde{t}\;)\;\lambda_{i}(\widetilde{X},\mathscr{R}^{\mathrm{Z}}_{i}\widetilde{V})))$
set $\widetilde{V}=\mathscr{R}_{i}^{\mathrm{Z}}\widetilde{V}$ ;
end for
$\overline{X}_{t_{n+1}}=\widetilde{X}-\frac{\delta_{n+1}}{2}\;\widetilde{V}$ ;
$\overline{V}_{t_{n+1}}=\widetilde{V}$ ;
end for
Algorithm 2 Splitting scheme DJD for the time reversed ZZP
Initialise either from $(\overline{X}_{0},\overline{V}_{0})\sim\pi\otimes\nu$
or
$(\overline{X}_{0},\overline{V}_{0})\sim\pi(\mathrm{d}x)p_{\theta^{*}}(\;\cdot\;\rvert
x,\mathrm{T}_{f})$ ;
for _$n=0,\dots,N-1$_ do
$\widetilde{V}=\overline{V}_{t_{n}}$ ;
Estimate density ratio:
$\overline{s}_{2}=\nicefrac{{\nu(\widetilde{V})}}{{p_{\theta^{*}}(\;\widetilde{V}\rvert\;\overline{X}_{t_{n}},\;\mathrm{T}_{f}-t_{n})}}$
;
With probability
$(1-\exp(-\lambda_{r}\overline{s}_{2}\;\frac{\delta_{n+1}}{2}))$ draw
$\widetilde{V}\sim
p_{\theta^{*}}(\;\cdot\;\rvert\overline{X}_{t_{n}},\mathrm{T}_{f}-t_{n})$ ;
$\widetilde{X}=\overline{X}_{t_{n}}-\frac{\delta_{n+1}}{2}\;\overline{V}_{t_{n}}$
;
$\widetilde{t}=\mathrm{T}_{f}-t_{n}-\frac{\delta_{n+1}}{2}$ ;
Estimate density ratio:
$\overline{s}_{1}=\nicefrac{{p_{\theta^{*}}(\mathscr{R}^{\mathrm{B}}_{\widetilde{X}}\widetilde{V}\rvert\;\widetilde{X},\;\widetilde{t}\;)}}{{p_{\theta^{*}}(\;\widetilde{V}\rvert\;\widetilde{X},\;\widetilde{t}\;)}}$
;
With probability
$(1-\exp(-\delta_{n+1}\overline{s}_{1}\lambda_{1}(\widetilde{X},\mathscr{R}^{\mathrm{B}}_{\widetilde{X}}\widetilde{V})))$
set $\widetilde{V}=\mathscr{R}^{\mathrm{B}}_{\widetilde{X}}\widetilde{V}$ ;
$\overline{X}_{t_{n+1}}=\widetilde{X}-\frac{\delta_{n+1}}{2}\;\widetilde{V}$ ;
Estimate density ratio:
$\overline{s}_{2}=\nicefrac{{\nu(\widetilde{V})}}{{p_{\theta^{*}}(\;\widetilde{V}\rvert\;\overline{X}_{t_{n+1}},\;\mathrm{T}_{f}-t_{n+1})}}$
;
With probability
$(1-\exp(-\lambda_{r}\overline{s}_{2}\;\frac{\delta_{n+1}}{2}))$ draw
$\widetilde{V}\sim
p_{\theta^{*}}(\;\cdot\;\rvert\overline{X}_{t_{n}},\mathrm{T}_{f}-t_{n+1})$ ;
$\overline{V}_{t_{n+1}}=\widetilde{V}$ ;
end for
Algorithm 3 Splitting scheme RDBDR for the time reversed BPS
### C.1 Simulating the backward ZZP
For ZZP we apply the splitting scheme DJD discussed in Section 2.4, with the
only difference that we allow multiple velocity flips during the jump step
similarly to Bertazzi et al. [2023]. Algorithm 2 gives a pseudo-code.
### C.2 Simulating the backward BPS
In the case of BPS, we follow the recommendations of Bertazzi et al. [2023]
and adapt their splitting scheme RDBDR, where R stands for refreshments, D for
deterministic motion, and B for bounces. We give a pseudo-code in Algorithm 3.
We remark that an alternative is to use the scheme DJD for BPS, simulating
reflections and refreshments in the J part of the splitting. This choice has
the advantage of reducing the number of model evaluations.
## Appendix D Discussion and proof for Theorem 1
### D.1 Discussion on H 2
In this section we discuss H 2 in the case of ZZP, BPS, and RHMC. For all
three of these samplers, existing theory shows convergence of the form
$\lVert\updelta_{(x,v)}P_{t}-\pi\otimes\nu\rVert_{V}\leqslant
C^{\prime}e^{-\gamma t}V(x,v),$ (53)
where $V:\mathbb{R}^{2d}\to[1,\infty)$ is a positive function and
$\lVert\mu\rVert_{V}:=\sup_{\lvert g\rvert\leqslant V}\lvert\mu(g)\rvert$ is
the $V$-norm. When the initial condition of the process is
$\mu_{\star}\otimes\nu$, we obtain the bound
$\displaystyle\lVert\mu_{\star}\otimes\nu
P_{t}-\pi\otimes\nu\rVert_{V}\leqslant C^{\prime}e^{-\gamma
t}\mu_{\star}\otimes\nu(V),$ (54)
which translates to a bound in TV distance, since we assume $V\geqslant 1$.
Conditions on $\pi$ ensuring (53) can be found for ZZP in Bierkens et al.
[2019b], for BPS in Deligiannidis et al. [2019], Durmus et al. [2020], and for
RHMC in Bou-Rabee and Sanz-Serna [2017]. Observe that we can set the constant
$C$ in H 2 to $C=C^{\prime}\mu_{\star}\otimes\nu(V)$. Clearly, $C$ is finite
whenever $\mu_{\star}\otimes\nu(V)<\infty.$ Since $V$ is such that
$\lim_{\lvert z\rvert\to\infty}V(z)=+\infty$, showing $C$ is finite requires
suitable tail conditions on the initial distribution $\mu_{\star}\otimes\nu.$
### D.2 Proof of Theorem 1
First notice that
$\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\leqslant\lVert\mu_{\star}\otimes\nu-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}},\overline{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\;,$
(55)
hence we focus on bounding the right hand side. Under our assumptions, the
forward PDMP $(X_{t},V_{t})_{t\in[0,\mathrm{T}_{f}]}$ admits a time reversal
that is a PDMP
$(\overleftarrow{X}_{t},\overleftarrow{V}_{t})_{t\in[0,\mathrm{T}_{f}]}$ with
characteristics
$(\overleftarrow{\Phi},\overleftarrow{\lambda},\overleftarrow{Q})$ satisfying
the conditions in Proposition 1. Therefore, it holds
$\mu_{\star}\otimes\nu=\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})$
and so (55) can be written as
$\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\leqslant\lVert\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}},\overline{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\;,$
We introduce the intermediate PDMP
$(\widetilde{X}_{t},\widetilde{V}_{t})_{t\in[0,\mathrm{T}_{f}]}$ with initial
distribution $\mathcal{L}(X_{\mathrm{T}_{f}},V_{\mathrm{T}_{f}})$ and
characteristics $(\overleftarrow{\Phi},\overline{\lambda},\overline{Q})$. In
particular, $(\widetilde{X}_{t},\widetilde{V}_{t})_{t\in[0,\mathrm{T}_{f}]}$
has the same characteristics as
$(\overline{X}_{t},\overline{V}_{t})_{t\in[0,\mathrm{T}_{f}]}$, but different
initial condition By the triangle inequality for the TV distance we find
$\displaystyle\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}$
$\displaystyle\leqslant\lVert\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})-\mathcal{L}(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}+\lVert\mathcal{L}(\widetilde{X}_{T},\widetilde{V}_{\mathrm{T}_{f}})-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}},\overline{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}.$
(56)
Applying the data processing inequality to the second term, we find the bound
$\lVert\mu_{\star}-\mathcal{L}(\overline{X}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}\leqslant\lVert\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})-\mathcal{L}(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}+\lVert\mathcal{L}(X_{\mathrm{T}_{f}},V_{\mathrm{T}_{f}})-\pi\otimes\nu\rVert_{\mathrm{TV}}.$
(57)
The second term in (57) can be bounded applying H 2, hence it is left to bound
the first term. We introduce the Markov semigroups
$P_{t},\overleftarrow{P}_{t},\overline{P}_{t}:\mathbb{R}_{+}\times\mathbb{R}^{2d}\times\mathcal{B}(\mathbb{R}^{2d})\to[0,1]$
defined respectively as
$P_{t}((x,v),\cdot):=\mathbb{P}_{(x,v)}((X_{t},V_{t})\in\cdot)$,
$\overleftarrow{P}_{t}((x,v),\cdot):=\mathbb{P}_{(x,v)}((\overleftarrow{X}_{t},\overleftarrow{V}_{t})\in\cdot)$,
and
$\widetilde{P}_{t}((x,v),\cdot):=\mathbb{P}_{(x,v)}((\widetilde{X}_{t},\widetilde{V}_{t})\in\cdot)$.
Recall that for any probability distribution $\eta$ on
$(\mathbb{R}^{2d},\mathcal{B}(\mathbb{R}^{2d}))$, $\eta
P_{t}(\cdot)=\int_{\mathbb{R}^{2d}}\eta(\mathrm{d}x,\mathrm{d}v)P_{t}((x,v),\cdot)$,
and similarly for $\eta\widetilde{P}_{t}(\cdot)$ and
$\eta\overleftarrow{P}_{t}(\cdot)$. Finally, to ease the notation we denote
$Q_{\mathrm{T}_{f}}:=\mathcal{L}(X_{\mathrm{T}_{f}},V_{\mathrm{T}_{f}})=(\mu_{\star}\otimes\nu)P_{\mathrm{T}_{f}}$.
Then we can rewrite the first term in (57) as
$\displaystyle\lVert\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})-\mathcal{L}(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}$
$\displaystyle=\lVert
Q_{\mathrm{T}_{f}}\overleftarrow{P}_{\mathrm{T}_{f}}-Q_{\mathrm{T}_{f}}\widetilde{P}_{\mathrm{T}_{f}}\rVert_{\mathrm{TV}}$
(58) $\displaystyle\leqslant\int
Q_{\mathrm{T}_{f}}(\mathrm{d}x,\mathrm{d}v)\lVert\updelta_{(x,v)}\overleftarrow{P}_{\mathrm{T}_{f}}-\updelta_{(x,v)}\widetilde{P}_{\mathrm{T}_{f}}\rVert_{\mathrm{TV}}.$
(59)
Therefore we wish to bound
$\lVert\updelta_{(x,v)}\overleftarrow{P}_{\mathrm{T}_{f}}-\updelta_{(x,v)}\widetilde{P}_{\mathrm{T}_{f}}\rVert_{\mathrm{TV}}.$
A bound for the TV distance between two PDMPs with same initial condition and
deterministic motion, but different jump rate and kernel was obtained in
Durmus et al. [2021, Theorem 11] using the coupling inequality
$\displaystyle\lVert\updelta_{(x,v)}\overleftarrow{P}_{\mathrm{T}_{f}}-\updelta_{(x,v)}\widetilde{P}_{\mathrm{T}_{f}}\rVert_{\mathrm{TV}}\leqslant
2\mathbb{P}_{(x,v)}\left((\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})\neq(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\right),$
(60)
and then bounding the right hand side. Following the proof of Durmus et al.
[2021, Theorem 11] we have that a synchronous coupling of the two PDMPs
satisfies
$\mathbb{P}_{(x,v)}\left((\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})\neq(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\right)\leqslant
2\mathbb{E}_{(x,v)}\left[1-\exp\left(-\int_{0}^{\mathrm{T}_{f}}g_{t}(\overleftarrow{X}_{t},\overleftarrow{V}_{t})\mathrm{d}t\right)\right],$
where
$\displaystyle g_{t}(x,v)$
$\displaystyle=\frac{1}{2}\left(\overleftarrow{\lambda}(t,(x,v))\wedge\overline{\lambda}(t,(x,v))\right)\left\lVert\overleftarrow{Q}(t,(x,v),\cdot)-\overline{Q}(t,(x,v),\cdot)\right\rVert_{\mathrm{TV}}$
$\displaystyle\quad+\left\rvert\overleftarrow{\lambda}(t,(x,v))-\overline{\lambda}(t,(x,v))\right\rvert.$
Since
$\mathcal{L}(\overleftarrow{X}_{t},\overleftarrow{V}_{t})=\mathcal{L}(X_{\mathrm{T}_{f}-t},V_{\mathrm{T}_{f}-t})$
for $t\in[0,\mathrm{T}_{f}]$, we can rewrite this bound as
$\displaystyle\mathbb{P}_{(x,v)}\left((\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})\neq(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\right)$
$\displaystyle\leqslant
2\mathbb{E}_{(x,v)}\left[1-\exp\left(-\int_{0}^{\mathrm{T}_{f}}g_{\mathrm{T}_{f}-t}(X_{t},V_{t})\mathrm{d}t\right)\right].$
(61)
Plugging this bound in (59) we obtain
$\displaystyle\lVert\mathcal{L}(\overleftarrow{X}_{\mathrm{T}_{f}},\overleftarrow{V}_{\mathrm{T}_{f}})-\mathcal{L}(\widetilde{X}_{\mathrm{T}_{f}},\widetilde{V}_{\mathrm{T}_{f}})\rVert_{\mathrm{TV}}$
$\displaystyle\leqslant
2\mathbb{E}\left[1-\exp\left(-\int_{0}^{\mathrm{T}_{f}}g_{\mathrm{T}_{f}-t}(X_{t},V_{t})\mathrm{d}t\right)\right].$
(62)
This concludes the proof.
### D.3 Application to the ZZP
Here we give the details on the bound (13), which considers the case of ZZP.
First, we upper bound the function $g_{t}$ defined in (12). We focus on the
first term in (12), that is
$g^{1}_{t}(x,v)=\frac{(\overleftarrow{\lambda}^{\mathrm{Z}}\wedge\bar{\lambda}^{\mathrm{Z}}\;)(t,(x,v))}{2}\lVert\overleftarrow{Q}^{\mathrm{Z}}(t,(x,v),\cdot)-\bar{Q}^{\mathrm{Z}}(t,(x,v),\cdot)\rVert_{\mathrm{TV}}.$
(63)
We find
$\displaystyle\lVert\overleftarrow{Q}^{\mathrm{Z}}(t,(x,v),\cdot)-\bar{Q}^{\mathrm{Z}}(t,(x,v),\cdot)\rVert_{\mathrm{TV}}=\sup_{A}\left\lvert\sum_{i=1}^{d}\mathbbm{1}_{(x,\mathscr{R}_{i}^{\mathrm{Z}}v)\in
A}\left(\frac{\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))}{\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))}-\frac{\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))}{\bar{\lambda}^{\mathrm{Z}}(t,(x,v))}\right)\right\rvert$
(64)
$\displaystyle\leqslant\sup_{A}\left\lvert\sum_{i=1}^{d}\mathbbm{1}_{(x,\mathscr{R}_{i}^{\mathrm{Z}}v)\in
A}\left(\frac{\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))-\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))}{\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))}+\frac{\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))}{\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))}-\frac{\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))}{\bar{\lambda}^{\mathrm{Z}}(t,(x,v))}\right)\right\rvert$
(65)
$\displaystyle\leqslant\left(\sum_{i=1}^{d}\frac{\lvert\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))-\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))\rvert}{\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))}\right)+\frac{\lvert\bar{\lambda}^{\mathrm{Z}}(t,(x,v))-\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))\rvert}{\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))}$
(66)
In the last inequality we used that $\bar{\lambda}^{\mathrm{Z}}$ is non-
negative. Therefore we find
$\displaystyle g^{1}_{t}(x,v)$
$\displaystyle\leqslant\frac{1}{2}\left(\sum_{i=1}^{d}\lvert\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))-\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))\rvert\right)+\frac{1}{2}\lvert\bar{\lambda}^{\mathrm{Z}}(t,(x,v))-\overleftarrow{\lambda}^{\mathrm{Z}}(t,(x,v))\rvert$
(67)
$\displaystyle\leqslant\sum_{i=1}^{d}\lvert\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))-\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))\rvert.$
(68)
Noticing that
$\lvert\overleftarrow{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))-\bar{\lambda}_{i}^{\mathrm{Z}}(t,(x,v))\rvert=\lvert
r_{i}^{\mathrm{Z}}(x,v,t)-s^{\theta}_{i}(x,v,t)\rvert\;\lambda_{i}^{\mathrm{Z}}((x,\mathscr{R}^{\mathrm{Z}}_{i}v).$
we find
$g_{t}(x,v)\leqslant 2\sum_{i=1}^{d}\lvert
r_{i}^{\mathrm{Z}}(x,v,t)-s^{\theta}_{i}(x,v,t)\rvert\;\lambda_{i}^{\mathrm{Z}}((x,\mathscr{R}^{\mathrm{Z}}_{i}v)$
Finally, we use the inequality $1-e^{-z}\leqslant z$, which holds for
$z\geqslant 0$, to conclude that
$\displaystyle\mathbb{E}\left[1-\exp\left(-\int_{0}^{\mathrm{T}_{f}}g_{\mathrm{T}_{f}-t}(X_{t},V_{t})\mathrm{d}t\right)\right]$
(69) $\displaystyle\quad\leqslant
2\sum_{i=1}^{d}\mathbb{E}\left[\int_{0}^{\mathrm{T}_{f}}\lvert
r_{i}^{\mathrm{Z}}(X_{t},V_{t},\mathrm{T}_{f}-t)-s^{\theta}_{i}(X_{t},V_{t},\mathrm{T}_{f}-t)\rvert\;\lambda_{i}^{\mathrm{Z}}(X_{t},\mathscr{R}^{\mathrm{Z}}_{i}V_{t})\mathrm{d}t\right].$
(70)
## Appendix E Experimental details
We run our experiments on 50 Cascade Lake Intel Xeon 5218 16 cores, 2.4GHz.
Each experiment is ran on a single CPU and takes between 1 and 5 hours to
complete, depending on the dataset and the sampler at hand.
### E.1 Continuation of Section 4
##### 2D datasets
In our experiments we consider the five datasets displayed in Figure 3. The
Gaussian grid consists of a mixture of nine Gaussian distribution with
imbalanced mixture weights $\\{.01,.02,.02,.05,.05,.1,.1,.15,.2,.3\\}$. We
load $100000$ training samples for each dataset, and use $10000$ test samples
to compute the evaluation metrics. We use a batch size of $4096$ and train our
model for $25000$ steps.
##### Detailed setup
For ZZP and i-DDPM we use a neural network consisting of eight time-
conditioned multi-layer perceptron (MLP) blocks with skip connections, each of
which consisting of two fully connected layers of width $256$. The time
variable $t$ passes through two fully connected layers of size $1\times 32$
and $32\times 32$, and is fed to each time conditioned block, where it passes
through an additional $32\times 64$ fully connected layer before being added
element-wise to the middle layer. The model size is $6.5$ million parameters.
For ZZP, we apply the softplus activation function
$x\mapsto\nicefrac{{1}}{{\beta}}\log(1+\exp(\beta x))$ to the output of the
network, with $\beta=1$, to constrain it to be positive and stabilise
behaviour for outputs close to $1$.
In the case of RHMC and BPS, we use neural spline flows [Durkan et al., 2019]
to model the conditional densities of the forward processes, as it shows good
performance among available architectures. We leverage the implementation from
the zuko package [Rozet et al., 2022]. We set the number of transforms to $8$,
the hidden depth of the network to $8$ and the hidden width to $256$. To
condition on $x,t$, we feed them to three fully connected layers of size
$d\times 8$, $8\times 8$ and $8\times 8$, where $d$ is either the dimension of
$X_{t}$, or $d=1$ in the case of the time variable. The resulting vectors are
then concatenated and fed to the conditioning mechanism of zuko. The resulting
model has $3.8$ million parameters.
For the simulation of backward PDMPs with splitting schemes we use a quadratic
schedule for the time steps, that is $(\delta_{n})_{n\in\\{1,\ldots,N\\}}$
given by
$\delta_{n}=\mathrm{T}_{f}\times((\nicefrac{{n}}{{N}})^{2}-(\nicefrac{{n-1}}{{N}})^{2})$.
For i-DDPM, we follow the design choices introduced in Nichol and Dhariwal
[2021] and in particular we use the variance preserving process, the cosine
noise schedule, and linear time steps. We set the refreshment rate of each
forward PDMP to $1$, the time horizon to $5$, and take advantage of the
approaches described in Section 2.3 to learn the characteristics of the
backward processes.
All experiments are conducted using PyTorch [Paszke et al., 2019]. The
optimiser is Adam [Kingma and Ba, 2015] with learning rate 5e-4 for all neural
networks.
##### Additional results
In Table 3 we show the accuracy in terms of the refreshment rate, while in
Table 4 we show different choices of the time horizon. In both cases, we
consider the Gaussian mixture data and we use the 2-Wasserstein metric to
characterise the quality of the generated data. Figure 3 shows the generated
data by the best model for each process.
refresh rate | 0.0 | 0.1 | 0.5 | 1.0 | 2.0 | 5.0 | 10.0
---|---|---|---|---|---|---|---
process | | | | | | |
BPS | 0.286 | 0.069 | 0.048 | 0.052 | 0.045 | 0.048 | 0.072
RHMC | 0.324 | 0.040 | 0.041 | 0.033 | 0.040 | 0.047 | 0.072
ZZP | 0.040 | 0.045 | 0.040 | 0.036 | 0.038 | 0.035 | 0.057
Table 3: Mean of 2-Wasserstein $W_{2}\downarrow$, on Gaussian grid dataset, averaged over $10$ runs. time horizon | 2 | 5 | 10 | 15
---|---|---|---|---
process | | | |
BPS | 0.031 | 0.040 | 0.040 | 0.041
HMC | 0.025 | 0.036 | 0.036 | 0.033
ZZP | 0.031 | 0.033 | 0.030 | 0.046
Table 4: Mean 2-Wasserstein $W_{2}\downarrow$ for different time horizon,
averaged over $10$ runs.
Checkerboard
ZZP
RHMC
BPS
i-DDPM
Fractal tree
ZZP
RHMC
BPS
i-DDPM
Olympic rings
ZZP
RHMC
BPS
i-DDPM
Rose
ZZP
RHMC
BPS
i-DDPM
Gaussian mixture
ZZP
RHMC
BPS
i-DDPM
Figure 3: Generation for the various datasets.
### E.2 MNIST digits
Figure 4: Generation for the ZZP trained on MNIST.
Finally, we consider the task of generating handwritten digits training the
ZZP on the MNIST dataset. In Figure 4 we show promising results obtained with
the same design choices described in Section E.1, apart from the following
differences. The optimiser is Adam [Kingma and Ba, 2015] with learning rate
2e-4. We use a U-Net following the implementation of Nichol and Dhariwal
[2021], choosing the parameters of the network as follows: we set the hidden
layers to $[128,256,256,256]$, fix the number of residual blocks to $2$ at
each level, and add self-attention block at resolution $16\times 16$, using
$4$ heads. We use an exponential moving average with a rate of $0.99$. At
every layer, we use the silu activation function, while we apply the softplus
to the output of the network, with $\beta=0.2$. We train the model for 40000
steps with batch size $128$.
|
Quantum circuit representation of Bayesian networks
address1]Sima E. Borujeni
address1]Saideep Nannapanenimycorrespondingauthor
[mycorrespondingauthor]Corresponding author. Phone: +1 316-978-6240
address2]Nam H. Nguyen
[fn1]Present address: Boeing Research & Technology, Huntington Beach, CA 92647, USA
address2]Elizabeth C. Behrman
address3]James E. Steck
[address1]Department of Industrial, Systems, and Manufacturing Engineering, Wichita State University
1845 Fairmount St, Box 35, Wichita, KS, 67260 USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
[address2]Department of Mathematics, Statistics, and Physics, Wichita State University
1845 Fairmount St, Box 33, Wichita, KS, 67260 USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
[address3]Department of Aerospace Engineering, Wichita State University
1845 Fairmount St, Box 44, Wichita, KS, 67260 USA
Probabilistic graphical models such as Bayesian networks are widely used to model stochastic systems to perform various types of analysis such as probabilistic prediction, risk analysis, and system health monitoring, which can become computationally expensive in large-scale systems. While demonstrations of true quantum supremacy remain rare, quantum computing applications managing to exploit the advantages of amplitude amplification have shown significant computational benefits when compared against their classical counterparts. We develop a systematic method for designing a quantum circuit to represent a generic discrete Bayesian network with nodes that may have two or more states, where nodes with more than two states are mapped to multiple qubits. The marginal probabilities associated with root nodes (nodes without any parent nodes) are represented using rotation gates, and the conditional probability tables associated with non-root nodes are represented using controlled rotation gates. The controlled rotation gates with more than one control qubit are represented using ancilla qubits. The proposed approach is demonstrated for three examples: a 4-node oil company stock prediction, a 10-node network for liquidity risk assessment, and a 9-node naive Bayes classifier for bankruptcy prediction. The circuits were designed and simulated using Qiskit, a quantum computing platform that enable simulations and also has the capability to run on real quantum hardware. The results were validated against those obtained from classical Bayesian network implementations.
Bayesian network Quantum Circuit Qiskit QubitFinance
§ INTRODUCTION
Bayesian Networks, also known as Bayesian belief networks, are probabilistic graphical models used to represent knowledge about an uncertain domain. A Bayesian network is represented as a directed acyclic graph with nodes and edges, where nodes represent the random variables and edges represent the probabilistic dependence between nodes [Murphy & Russell, 2002].
Bayesian networks are used to perform two types of analysis: forward analysis, which provides a probabilistic prediction of the lower-level nodes in the Bayesian networks using probability distributions of the higher-level nodes, and inverse analysis, which infers the values of higher-level nodes using data on the lower-level nodes. The inverse analysis is commonly referred to as Bayesian inference since the inference analysis is carried out using the Bayes theorem. The forward analysis is typically performed through Monte Carlo analysis and has been used to perform uncertainty propagation [Nannapaneni et al., 2016], performance evaluation [Zhu & Deshmukh, 2003], reliability and risk analysis [Garvey et al., 2015], and prognostics [Ferreiro et al., 2012], whereas the inverse analysis has been used for system identification [Lee & Song, 2016], health monitoring [Kothamasu et al., 2006], and system diagnostics [Li et al., 2017]. Bayesian networks have been used to carry out a variety of analyses in several domains of science and engineering such as mechanical [Xu, 2012], aerospace [Li et al., 2017, Nannapaneni et al., 2018], and manufacturing systems [Büyüközkan et al., 2015, Nannapaneni et al., 2018], industrial systems [Cai et al., 2016], healthcare [Kahn Jr et al., 1997, Kalet et al., 2015], infrastructure systems [Hosseini & Barker, 2016, Nannapaneni et al., 2017], biomedical systems [Miyauchi & Nishimura, 2018], and transportation [Sun et al., 2006, Pettet et al., 2017].
Some of the issues with the current implementations of Bayesian networks is the high computational expense in the presence of large number of nodes (random variables) for both forward and inverse analyses. One possible way to obviate this difficulty might be to use the principles of quantum computing (sometimes referred to as quantum-assisted computing). This is because quantum computers make use of “superposition" which is
the ability of quantum systems to be simultaneously in multiple different states.
Several algorithms have been developed using the principles of quantum mechanics that have demonstrated superior computational performance over the corresponding classical algorithms, and the most notable of these are Shor's algorithm [Shor, 1994] and Grover's algorithm [Grover, 1996]. Shor's algorithm is used for the factorization of integers; this algorithm has exponential speedup when compared to the best known classical algorithms. Grover's algorithm is used for search in an unstructured search space (such as an unstructured database), and has quadratic speedup.
Due to its computational benefits, the Grover's algorithm has been used as a sub-routine in the development of many quantum algorithms for classification such as quantum support vector machines [Rebentrost et al., 2014], for clustering such as quantum k-means clustering [Aïmeur et al., 2007], and for combinatorial optimization [Baritompa et al., 2005]. With regard to Bayesian networks, Ozols et al [Ozols et al., 2013] used the principles of amplitude amplification [Brassard et al., 2002] to develop an algorithm for Bayesian inference (inverse analysis) known as quantum rejection sampling, which is a quantum version of the rejection sampling algorithm [Gilks & Wild, 1992] used for inference in classical Bayesian networks. Woerner and Egger [Woerner & Egger, 2019] used the principles of amplitude amplification and estimation [Brassard et al., 2002] to perform risk analysis (forward analysis), and demonstrated it with two toy problems from the financial industry. In this paper, we consider the representation of Bayesian networks in a quantum computing paradigm to facilitate the use of those quantum algorithms for forward and inverse analyses.
There are primarily two types of architectures that have widely been used to realize quantum computing: quantum gate models [Dallaire-Demers & Wilhelm, 2016] and quantum annealing [Bunyk et al., 2014]. The quantum gate architecture uses a series of quantum gates that act on individual qubits to achieve the desired computation.
More details regarding qubits and gates are available in Sections <ref> and <ref>. A quantum circuit is a graphical representation of the sequence of gates implemented on various qubits to do the desired computation. Quantum annealing architecture uses the principles of quantum annealing [Boixo et al., 2013], which is a quantum equivalent to classical simulated annealing algorithm, to make the desired computation. According to Ajagekar et al [Ajagekar et al., 2020], quantum annealing architecture is better suited for optimization problems whereas quantum gate architecture facilitates universal quantum computation. As the goal of this paper is the representation of Bayesian networks, we use the quantum gate architecture as opposed to the quantum annealing architecture.
Low et al [Low et al., 2014] discussed the principles of quantum circuit design to represent a Bayesian network with discrete nodes that have two states, and also discussed the circuit design for implementing quantum rejection sampling for inference. In this paper, we consider the representation of generic discrete Bayesian networks with nodes that may have two or more states, and also discuss the decomposition of complex gates using elementary gates (discussed in Section <ref>) such that they can be implemented on available quantum computing platforms.
Paper Contributions: The overall contributions made through this paper are: (1) Decomposition of a multi-qubit gate into elementary gates to represent a discrete variable with more than two states; (2) A systematic procedure to design a quantum circuit to represent a generic discrete Bayesian network with nodes that may have two or more states; and (3) Illustration of the proposed quantum circuit representation to three Bayesian networks used for oil price stock prediction, liquidity risk assessment, and bankruptcy prediction, and validating the results against classical Bayesian network implementations.
Paper Organization: Section <ref> provides a brief background to qubits, quantum gates, and Bayesian networks. Section <ref> details the proposed methods for designing a quantum circuit to represent a Bayesian network. Section <ref> details the application of the proposed methods to three Bayesian networks from the financial industry followed by concluding remarks in Section <ref>.
§ BACKGROUND
In this section, we provide a brief background to qubits, different quantum gates that are used to perform qubit transformations, and Bayesian Networks.
§.§ Qubit
A qubit (or a quantum bit) is an elementary unit of information in quantum computing, similar to a classical bit (or simply a bit) in classical computing. A bit is always in one of the either two basis states - 0 and 1, whereas a qubit can be in both the basis states simultaneously. This property of a qubit is known as quantum superposition. In quantum computing, the Dirac notation is used to represent the two basis states as $\Ket{0}$ and $\Ket{1}$. In general, any two orthonormal states can be used as the basis states but the commonly used basis states or computational basis are $\Ket{0}$ and $\Ket{1}$. Eq. (<ref>) provides their vector representations as
\begin{equation}
\label{eqn:basis}
\Ket{0} = \begin{bmatrix}
{1} \\
{0} \\
\end{bmatrix} \hspace{2cm}
\Ket{1}= \begin{bmatrix}
{0} \\
{1} \\
\end{bmatrix}
\end{equation}
A general pure state of a qubit is a superposition, which is linear combination of the two basis states written as $ \Ket{\Psi} = c_1 \Ket{0} + c_2 \Ket{1}$ or
\begin{equation}
\begin{bmatrix}
\end{bmatrix} = c_1\begin{bmatrix}
{1} \\
{0} \\
\end{bmatrix}
{0} \\
{1} \\
\end{bmatrix}
\end{equation}
where $c_1$ and $c_2$ are complex numbers, which represent the probability amplitudes corresponding to $\Ket{0}$ and $\Ket{1}$ respectively. When a measurement is made, the qubit collapses to one of the two basis states. The probabilities of observing the qubit in $\Ket{0}$ and $\Ket{1}$ states are computed as the inner product of their probability amplitudes and their complex conjugates, represented as $c_1^{\dagger} c_1 = |c_1|^2$ and $c_2^{\dagger} c_2 = |c_2|^2$ respectively, where $c_1^{\dagger}$ and $c_2^{\dagger}$ are the complex conjugates of $c_1$ and $c_2$ respectively. Since a qubit can be either in $\Ket{0}$ or in $\Ket{1}$, the sum of their probabilities is equal to unity ($|c_1|^2 + |c_2|^2=1$).
When multiple qubits are used in computing, their joint state can be obtained through a tensor product of individual qubits. If $\Ket{\Psi_1}=a_1\Ket{0}+a_2\Ket{1}$ and $\ket{\Psi_2} = b_1\Ket{0}+b_2\Ket{1}$ represent two qubits with real-valued probability amplitudes, then their joint state is represented as $\Ket{\Psi_1} \otimes \Ket{\Psi_2} = a_1b_1\Ket{00}+a_1b_2\Ket{01}+a_2b_1\Ket{10}+a_2b_2\Ket{11}$. But the joint state need not always be a product state (tensor product of individual states). There are states of the joint system that can not be written as a product of individual states; these are called entangled states. Entanglement is quantum correlation stronger than any possible classical correlation.
An example of an entangled state of two qubits is the Bell state, defined as $\dfrac{1}{\sqrt{2}} \Ket{00} + \dfrac{1}{\sqrt{2}} \Ket{11}$ [Nielsen & Chuang, 2002]. Clearly, it can not be written as a tensor product of two qubits, $\Ket{\Psi_1}=a_1\Ket{0}+a_2\Ket{1}$ and $\ket{\Psi_2} = b_1\Ket{0}+b_2\Ket{1}$, because then $a_1b_1 = a_2b_2 = \dfrac{1}{\sqrt{2}}$ but also $a_1b_2 = a_2b_1 = 0$. Thus, neither qubit has an individual state, but the two are perfectly correlated. If one of the two qubits is measured to be in $\Ket{0}$ state, then the other qubit is also in $\Ket{0}$ due to the presence of entanglement between them.
[line cap=round, line join=round, >=Triangle]
(-2.5,-2.49) rectangle (2.70,2.70);
[shift=(0,0), fill, fill, fill opacity=0.1] (0,0) – (56.7:0.4) arc (56.7:90.:0.4) – cycle;
[shift=(0,0), fill, fill, fill opacity=0.1] (0,0) – (-135.7:0.4) arc (-135.7:-33.2:0.4) – cycle;
(0,0) circle (2cm);
[rotate around=0.:(0.,0.),dash pattern=on 3pt off 3pt] (0,0) ellipse (2cm and 0.9cm);
(0,0)– (0.70,1.07);
[->] (0,0) – (0,2);
[->] (0,0) – (-0.81,-0.79);
[->] (0,0) – (2,0);
[dotted] (0.7,1)– (0.7,-0.46);
(0,0)– (0.7,-0.46);
(-0.08,-0.3) node[anchor=north west] $\phi$;
(0.01,0.9) node[anchor=north west] $\theta$;
(-1.01,-0.72) node[anchor=north west] $\mathbf {X}$;
(2.07,0.3) node[anchor=north west] $\mathbf {Y}$;
(-0.5,2.6) node[anchor=north west] $\mathbf {Z=|0\rangle}$;
(-0.4,-2) node[anchor=north west] $-\mathbf {Z=|1\rangle}$;
(0.4,1.65) node[anchor=north west] $|\Psi\rangle$;
[fill] (0,0) circle (1.5pt);
[fill] (0.7,1.1) circle (0.5pt);
Bloch sphere representation of a qubit
A Bloch sphere (shown in Figure <ref>) is often used for geometrical representation of a qubit. In a Bloch sphere, the positive Z-axis corresponds to the $\Ket{0}$ state while the negative Z-axis corresponds to $\Ket{1}$ state. A pure state of a qubit is represented by a point on the Bloch sphere and can be represented as $\cos(\frac{\theta}{2}) \Ket{0} + e^{i\phi}\sin(\frac{\theta}{2}) \Ket{1}$ for given values of $(\theta,\phi)$.
§.§ Quantum gates
Quantum gates are mathematical operations performed on the qubits to change their probability amplitudes to gain the desired computations. The quantum gates are similar to classical gates (such as the AND gate) acting on classical bits. Geometrically, one-qubit gates represent unitary rotations about various axes in the Bloch sphere.
There are two elementary gates in quantum computing - $U_3$ and CNOT, which act on a single qubit and two qubits respectively. Any other multi-qubit gates can be decomposed into these elementary gates. We discuss in detail the one-qubit and two-qubit elementary gates. A more comprehensive review of gates is available in Nielsen and Chuang's textbook [Nielsen & Chuang, 2002].
§.§.§ One-qubit gates
We review the generic $U_3$ gate [McKay et al., 2018], and some of its special cases - $X$ (sometimes referred to as Pauli-$X$), $R_Y$ and $R_Z$ gates.
$U_3$ gate has three parameters $\theta$, $\phi$ and $\lambda$, and it can be used to construct any arbitrary single qubit gate. The matrix representation of this gate is given as
\begin{equation}
\label{eqn:U_3matrix}
U_3(\theta, \phi, \lambda) = \begin{bmatrix}
\cos\Big(\dfrac{\theta}{2}\Big) & -e^{i\lambda} \sin\Big(\dfrac{\theta}{2}\Big) \\[2mm]
e^{i\phi} \sin\Big(\dfrac{\theta}{2}\Big) & e^{i(\phi + \lambda)} \cos\Big(\dfrac{\theta}{2}\Big) \\
\end{bmatrix}
\end{equation}
where $\theta$ represents the angle of rotation about the Y-axis, and $\phi$ and $\lambda$ represent the angles of rotation around the Z-axis. The generic $U_3$ gate is often represented as the $U$ gate.
$R_Y$ gate: The $R_Y$ gate is a single-qubit gate, which corresponds to a rotation of angle $\theta$ (radians) about the y-axis on the Bloch sphere. $R_Y$ gate can be represented as a special case of $U_3$ gate as
\begin{equation}
R_Y(\theta)=U_3(\theta, 0, 0)=
\begin{bmatrix}
\cos\Big(\dfrac{\theta}{2}\Big) & -\sin\Big(\dfrac{\theta}{2}\Big) \\[2mm]
\sin\Big(\dfrac{\theta}{2}\Big) & \cos\Big(\dfrac{\theta}{2}\Big)\\
\end{bmatrix}
\end{equation}
$R_Z$ gate or Phase-shift gate:
$R_Z(\lambda)$ is another single qubit gate, which corresponds to rotation about the Z-axis by an angle $\lambda$ on the Bloch sphere. $R_Z$ gate can be represented as a special case of $U_3$ as
\begin{equation}
\begin{bmatrix}
1 & \hspace{2mm} 0 \\
0 & \hspace{2mm} e^{i\lambda}\\
\end{bmatrix}
\end{equation}
Using the matrix representations of $R_Y$ and $R_Z$ gates, the $U_3$ gate given in Eq. (<ref>) can be decomposed into two phase-shift rotations and one rotation about the Y-axis as
\begin{equation}
\label{eq:U_3_3matricies}
\begin{aligned}
U_3(\theta, \phi, \lambda) &= \begin{bmatrix}
1 & 0 \\[2mm]
0 & e^{i\phi} \\
\end{bmatrix}
\begin{bmatrix}
\cos\Big(\dfrac{\theta}{2}\Big) & -\sin\Big(\dfrac{\theta}{2}\Big) \\[2mm]
\sin\Big(\dfrac{\theta}{2}\Big) & \cos\Big(\dfrac{\theta}{2}\Big) \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\[2mm]
0 & e^{i\lambda} \\
\end{bmatrix} \\
& = R_Z(\phi)R_Y(\theta)R_Z(\lambda)
\end{aligned}
\end{equation}
For more detail about the rotation gates and decomposition of an arbitrary single qubit gate, refer to [Nielsen & Chuang, 2002] and [Cross et al., 2017].
X gate: The $X$ gate is the quantum-equivalent to the classical NOT gate or sometimes referred to as flip gate, as it flips $\Ket{0}$ to $\Ket{1}$ and $\Ket{1}$ to $\Ket{0}$. The matrix notation of the $X$ gate is equal to
\begin{equation}
X = U_3\Big(\pi, -\frac{\pi}{2}, \frac{\pi}{2}\Big) = \begin{bmatrix}
0 & \hspace{2mm} 1 \\
1 & \hspace{2mm} 0\\
\end{bmatrix}
\end{equation}
§.§.§ Two-qubit gates
An elementary two-qubit gate is the controlled-NOT (CNOT or $CX$) gate. The two qubits on which the CNOT gate is implemented are referred to as the control qubit and target qubit. When the control qubit is $\Ket{0}$, the target qubit remains unchanged whereas when the control qubit is $\Ket{1}$, the $X$ gate is implemented on the target qubit. The CNOT gate does not have any effect on the control qubit. In the usual computational basis ($\Ket{00}, \Ket{01}, \Ket{10}, and \Ket{11}$ states),the matrix representation of a CNOT gate is given as
\begin{equation}
\centering
\begin{bmatrix}
1 & 0 & 0 & 0 \\
{0} & {1} & {0} & {0} \\
{0} & {0} & {0} & {1} \\
{0} & {0} & {1} & {0} \\
\end{bmatrix}
\end{equation}
For example, consider two qubits given as $\Ket{\Psi_1}=a_1\Ket{0}+a_2\Ket{1}$ and $\ket{\Psi_2} = b_1\Ket{0}+b_2\Ket{1}$ on which a CNOT gate is implemented with $\Ket{\Psi_1}$ as the control qubit. The combined quantum state before the application of CNOT gate is given by their tensor product, $\Ket{\Psi_1} \otimes \Ket{\Psi_2} = a_1b_1 \Ket{00} + a_1b_2 \Ket{01} + a_2b_1 \Ket{10} + a_2b_2 \Ket{11}$. The quantum state after the application of the CNOT gate is equal to $a_1b_1 \Ket{00} + a_1b_2 \Ket{01} + a_2b_1 \Ket{11} + a_2b_2 \Ket{10}$. The $\Ket{00}$ and $\Ket{01}$ remain unchanged since the control qubit is $\Ket{0}$ whereas $\Ket{10}$ and $\Ket{11}$ become $\Ket{11}$ and $\Ket{10}$ respectively as the $X$ gate is applied when the control qubit is $\Ket{1}$. Similar to the CNOT gate, we have the $CU$ gate, which implements the $U$ (or the $U_3$) gate when the control qubit is $\Ket{1}$. Given the matrix of the $U$ gate in Eq. (<ref>), the matrix representation of $CU$ can be written as
\begin{equation}
\begin{bmatrix}
{1} & \hspace{4mm}{0} & {0} & {0} \\
{0} & \hspace{4mm}{1} & {0} & {0} \\
{0} & \hspace{4mm}{0} & \cos\Big(\dfrac{\theta}{2}\Big)& -e^{i\lambda} \sin\Big(\dfrac{\theta}{2}\Big) \\[2mm]
{0} & \hspace{4mm} {0} & e^{i\phi} \sin\Big(\dfrac{\theta}{2}\Big)& e^{i(\phi + \lambda)} \cos\Big(\dfrac{\theta}{2}\Big) \\
\end{bmatrix}
\end{equation}
A variant of the $CU$ gate is the $CR_Y(\theta)$ gate, which implements the rotation gate $R_Y(\theta)$ on the target qubit when the control qubit is equal to $\Ket{1}$. The matrix representation of the $CR_Y(\theta)$ can be written as
\begin{equation}
\begin{bmatrix}
{1} & \hspace{4mm}{0} & {0} & {0} \\
{0} & \hspace{4mm}{1} & {0} & {0} \\
{0} & \hspace{4mm}{0} & \cos\Big(\dfrac{\theta}{2}\Big)& - \sin\Big(\dfrac{\theta}{2}\Big) \\[2mm]
{0} & \hspace{4mm} {0} & \sin\Big(\dfrac{\theta}{2}\Big)& \cos\Big(\dfrac{\theta}{2}\Big) \\
\end{bmatrix}
\end{equation}
§.§.§ Three-qubit gates
We will discuss two three-qubit gates that are later used in the proposed methodology: CCNOT (or $CCX$ or Toffoli) and $CCR_Y(\theta)$. Out of the three qubits on which the CCNOT and $CCR_Y(\theta)$ are implemented, two qubits act as control qubits and the other is the target qubit. In the case of CCNOT gate, when both control qubits are $\Ket{1}$, we implement the $X$ gate on the target qubit. In the case of the $CCR_Y(\theta)$ gate, we implement the $R_Y(\theta)$ gate when the two control qubits are in the $\Ket{1}$ state. The three-qubits are not elementary gates but can be decomposed into a combination of single-qubit and CNOT gates. For example, the CCNOT can be represented using a combination of nine single qubit gates and six CNOT gates [Shende & Markov, 2008]. Similar to $CCX$ and $CCR_Y(\theta)$, we can define an $C^nX$ and $C^nR_Y(\theta)$ gates with $n$ control qubits and one target qubits. Figure <ref> provides the representation of various gates discussed above in a quantum circuit. A quantum circuit represents a graphical representation of a sequence of gates that are implemented on various qubits to obtain a desired computation. Measurement gate in Figure <ref> performs the measurement operation on a qubit.
$X$ $R_Y$ $R_Z$ $U_3$ $CU$
@C=1em @R=.7em X @C=1em @R=.7em R_Y @C=1em @R=.7em R_Z @C=1em @R=.7em U_3 @C=1em @R=.7em 1
CNOT CCNOT $CR_Y$ $CCR_Y$ Measurement
@C=1em @R=.7em 1
@C=1em @R=.7em 1
@C=1em @R=.7em 1
@C=1em @R=.7em 1
@C=1em @R=.7em
Representation of commonly used one, two, and three qubit gates
@C=1em @R=0.9em
U = @C=1em @R=0.9em
1 1
A B C
Decomposition of a CU gate into a combination of single qubit and CNOT gates
According to [Nielsen & Chuang, 2002], the $CU$ can be decomposed into a combination of single-qubit and CNOT gates as given in Figure <ref>; this decomposition can mathematically be represented as
\begin{equation}
\label{eqn:cu}
CU = (I\otimes A)CX(I\otimes B)CX(I\otimes C)
\end{equation}
where $A = R_Z(\phi)R_Y(\theta/2)$, $B = R_Y(-\theta/2)R_Z(-(\lambda+\phi)/2)$, $C = R_Z((\lambda-\phi)/2)$, and $I$ represents the identity matrix. $I\otimes A$ represents the tensor product of two matrices, $I$ and $A$, where $I$ and $A$ are single-qubit gates implemented on the first and second qubits respectively. $I\otimes A$ is the simplified representation of the two gates acting on the two qubits.
§.§ Bayesian Networks
As mentioned in Section <ref>, Bayesian networks (BNs) are probabilistic graphical models that represent a probabilistic framework to model stochastic/uncertain systems. In a probabilistic framework, a stochastic system can be represented as a joint probability distribution defined over the set of random variables. A BN consists of nodes and edges, where nodes represent the random variables, and edges represent the dependence between nodes, which is quantified using conditional probability tables (CPT, for discrete variables) and conditional probability distributions (CPD, for continuous variables). In this paper, we consider the design of quantum circuits to represent discrete Bayesian networks.
Let us consider a Bayesian network with $s$ nodes, where $\mathbb{V}=\{V_1,V_2,...,V_s\}$ represents the set of all nodes or random variables. An edge from node $V_i$ to node $V_j$ represents the dependence between the variables $V_i$ and $V_j$, and that the values of $V_j$ are dependent on the values of $V_i$. Here, $V_i$ is called the parent node and node $V_j$ is referred to as the child node. The nodes without any parent nodes are typically referred to as root nodes. Using the graphical representation of a Bayesian network, the joint probability over the nodes (random variables) can be decomposed into a product of marginal and conditional probabilities as
\begin{equation}
P(V_1,V_2,...,V_s)= \prod_{i=1}^{s} P(V_i| \Pi_{V_i})
\end{equation}
where $\Pi_{V_i}$ denotes the set of parents nodes associated with $V_i$. For root nodes, the $P(V_i| \Pi_{V_i})$ becomes equal to the marginal distribution, $P(V_i)$.
Consider a simple Bayesian network with 3 nodes as shown in Figure <ref> where A, B and C are discrete random variables with two states True (or “1") and False (or “0"). In this BN, $A$ and $B$ are root nodes whereas $C$ is a child node with $A$ and $B$ as parent nodes. We have the marginal distributions for root nodes (shown in Figure <ref> and a CPT for the child node, which represents a probability distribution of the child node conditioned on the values taken by the parent nodes. Given a CPT, we can calculate its marginal distribution by integrating over the distributions of the parent nodes, i.e., $P(C) = \sum_{A, B} P(C|A, B)\times P(A, B)$. Due to the independence between nodes A and B (from Figure <ref>), the joint probabilities of A and B can be written as the product of individual probabilities, i.e., $P(A, B) = P(A)\times P(B)$, and therefore, $P(C) = P(C|A, B)\times P(A)\times P(B)$.
We use the marginal probabilities of various nodes in a Bayesian network as a measure to check the accuracy of the proposed quantum circuit representation approach in Section <ref>. After we design the quantum circuit, we estimate the marginal probabilities by simulating the quantum circuits, and compare them against the values from classical Bayesian network implementations. After providing a brief background to quantum computing and Bayesian networks, we will now discuss the representation of a Bayesian network through a quantum circuit.
An example of a 3-node Bayesian network
§ QUANTUM CIRCUIT OF A BAYESIAN NETWORK
We use the following principles for the design of a quantum circuit to represent a Bayesian network.
* Map each node in a Bayesian network to one or more qubits (depending on the number of discrete states of the node)
* Map the marginal/conditional probabilities of each node to the probability amplitudes (or probabilities) associated with various states of the qubit(s).
* Realize the required probability amplitudes of quantum states using (controlled) rotation gates.
@C=0.5em @R=0.9em
q_0:0 R_Y(θ_A)[0em]0 X 1 X [0em]0 X 1 X [0em]0 1 [0em]0 1
q_1:0 R_Y(θ_B)[0em]0 X 1 X [0em]0 1 [0em]0 X 1 X [0em]0 1
q_2:0 [0em]0 R_Y(θ_C,00) [0em]0 R_Y(θ_C,01) [0em]0 R_Y(θ_C,10) [0em]0 R_Y(θ_C,11)
Conceptual quantum circuit for the 3-node Bayesian network
Let us discuss how these ideas can be used to design a quantum circuit for the 3-node Bayesian network in Figure <ref>. All the nodes in Figure <ref> have two states (0 and 1), and since a qubit can represent two states ($\Ket{0}$ and $\Ket{1}$), we can map each node to a different qubit. Also, let us map state 0 of each node to $\Ket{0}$ and state 1 to $\Ket{1}$. Let the three nodes A, B, and C be mapped to three qubits $q_0, q_1,$ and $q_2$ respectively. Let the initial state of the three qubits be $\Ket{0}$. In the cases of $q_0$ and $q_1$, we will need to apply rotation gates ($R_Y$) with angles ($\theta_A$ and $\theta_B$) that result in superposed quantum states whose probabilities correspond the probabilities of nodes A and B respectively. The calculation of those angles are later discussed in Section <ref>. In the case of qubit $q_2$ (that corresponds to C), we will have a different rotation angle conditioned on the states of $q_0$ and $q_1$ since we have a different set of probabilities for node C conditioned on the values of parent nodes (A, B). Since there are four combinations of parent node values, we will have four rotation angles, one for each parent node combination. These conditional rotations are implemented using controlled rotation gates, whose angles depend on the conditional probabilities of C; these rotations are represented as $\theta_{C,ij}$, where $i,j=0,1$ and represent the states of $q_0$ and $q_1$ respectively.
Figure <ref> provides a conceptual quantum circuit for the 3-node Bayesian network. First, we implement the single qubit rotations to obtain the probabilities associated with A and B. Depending on the values of A and B, we implement controlled rotations to realize the conditional probabilities associated with C. In a controlled rotation, since the rotations are applied only when the control qubit is $\Ket{1}$, we use the $X$ gate to flip the $\Ket{0}$ to $\Ket{1}$ state to obtain the conditional probabilities when the parent node value(s) are 0. As there are two parent nodes, we implement a $CCR_Y$ gate to realize the conditional probabilities of C for every combination of the parent nodes. If there are $n$ parent nodes, then we would implement a $C^nR_Y$ gate. The overall quantum circuit can be obtained by composing all the single-qubit and controlled rotations in a sequential manner (as shown in Figure <ref>). In this paper, we refer to this approach as the C-QBN approach, which stands for Compositional approach for Quantum Bayesian networks.
We discuss below the computation of rotation angles to realize nodal probabilities in Section <ref>, representation of two-state child nodes with one or more parent nodes in Section <ref>, and representation of nodes with more than two states in Section <ref>.
§.§ Rotation angle computation
As mentioned above, we can represent a two-state root node using a single qubit. By applying an $R_Y$ gate with an appropriate angle, the probabilities of the root node can be mapped to the probabilities (and thus probability amplitudes) of the basis states, $\Ket{0}$ and $\Ket{1}$. Let $\theta_{V_i}$ represent the rotation angle associated with a two-state root node, $V_i$. Given the initial state of a qubit as $\Ket{0}$, the application of $R_Y(\theta)$ will transform $\Ket{0}$ to $\cos{\bigg(\dfrac{\theta}{2}\bigg)}\Ket{0} + \sin{\bigg(\dfrac{\theta}{2}\bigg)}\Ket{1}$. Therefore, the probabilities associated with the $\Ket{0}$ and $\Ket{1}$ states are equal to $\cos^2{\bigg(\dfrac{\theta}{2}\bigg)}$ and $\sin^2{\bigg(\dfrac{\theta}{2}\bigg)}$ respectively. If $P(V_i=0)$ and $P(V_i=0)$ represent the probabilities of states 0 and 1 of $V_i$, then the rotation angle can be computed as
\begin{equation}
\label{eqn:theta}
\theta_{V_i} = 2\times\tan^{-1}\sqrt{\dfrac{P(\Ket{1})}{P(\Ket{0})}} = 2\times\tan^{-1}\sqrt{\dfrac{P(V_i=1)}{P(V_i=0)}}
\end{equation}
In Eq. (<ref>), $P(\Ket{0})$ and $P(\Ket{1})$ represent the probabilities of a qubit to be in $\Ket{0}$ and $\Ket{1}$ respectively. Since we map the nodal probabilities to the probabilities of quantum states, $P(\Ket{0})$ and $P(\Ket{1})$ are replaced with $P(V_i=0)$ and $P(V_i=1)$ respectively. Therefore, two-state root nodes can be represented using an $R_Y$ gate with a rotation angle of $2\times\tan^{-1}\sqrt{\dfrac{P(V_i=1)}{P(V_i=0)}}$. In Figure <ref>, the rotation angle to find the probabilities of A and B can be calculated as $\theta_A = 2\times\tan^{-1}\sqrt{\dfrac{0.8}{0.2}} = 2.214$ and $\theta_B = 2\times\tan^{-1}\sqrt{\dfrac{0.7}{0.3}} = 1.982$.
Eq. (<ref>) can also be used to compute the rotation angles associated with conditional probabilities of two-state child nodes. Let $V_i$ and $\Pi_{V_i}$ represent a child node and set of its parent nodes respectively. For each combination of parent node values, $\Pi_{V_i} = \Pi^*_{V_i}$, we have probabilities for $V_i=0$ and $V_i=1$ denoted as $P(V_i=0|\Pi_{V_i} = \Pi^*_{V_i})$ and $P(V_i=1|\Pi_{V_i} = \Pi^*_{V_i})$ respectively. The rotation angle associated with $V_i$ when $\Pi_{V_i} = \Pi^*_{V_i}$, which is denoted by $\theta_{V_i, \Pi^*_{V_i}}$ can be calculated as
\begin{equation}
\label{eqn:theta2}
\theta_{V_i, \Pi^*_{V_i}} = 2\times\tan^{-1}\sqrt{\dfrac{P(V_i=1|\Pi_{V_i} = \Pi^*_{V_i})}{P(V_i=0|\Pi_{V_i} = \Pi^*_{V_i})}}
\end{equation}
The rotation angles associated with node C (qubit $q_2$ in Figure <ref>) can be calculated using Eq. (<ref>) as $\theta_{C,00} = 2\times\tan^{-1}\sqrt{\dfrac{0.85}{0.15}} = 2.346$, $\theta_{C,01} = 2\times\tan^{-1}\sqrt{\dfrac{0.7}{0.3}} = 1.982 $, $\theta_{C,10} = 2\times\tan^{-1}\sqrt{\dfrac{0.6}{0.4}} = 1.772 $, and $\theta_{C,11} = 2\times\tan^{-1}\sqrt{\dfrac{0.9}{0.1}} = 2.498$. The conditional probabilities associated with child nodes are realized through controlled rotations. As controlled rotation gates are not elementary gates, they need to be decomposed into single-qubit and two-qubit elementary gates; the decomposition is discussed below in Section <ref>.
§.§ Representing two-state child nodes with two-state parent nodes
First, we consider the representation of child nodes with one parent node, and then we consider the case with multiple parent nodes. A two-state child node with one parent node can be represented using two $CR_Y$ gates (assuming the parent node has two states) with rotation angles computed using Eq. (<ref>) conditioned on the parent node value. We consider parent nodes with more than two states in Section <ref>. The $CR_Y$ gate is a special case of a $CU$ gate discussed in Section <ref>, where $U=R_Y$. In Section <ref>, we discussed the decomposition of a $CU$ gate in terms of elementary single-qubit and CNOT gates shown in Figure <ref> and given in Eq. (<ref>). Since $R_Y(\theta) = U_3\Big(\dfrac{\theta}{2},0,0\Big)$, the single qubit gates A, B, and C in Figure <ref> can be calculated as $A = R_Z(0)R_Y\Big(\dfrac{\theta}{2}\Big) = R_Y\Big(\dfrac{\theta}{2}\Big), B = R_Y\Big(-\dfrac{\theta}{2}\Big)R_Z\Big(-\dfrac{(0+0)}{2}\Big) = R_Y\Big(-\dfrac{\theta}{2}\Big),$ and $C = R_Z\Big(\dfrac{(0-0)}{2}\Big) = I$. Figure <ref> illustrates the decomposition of the $CR_Y$ gate into single-qubit and CNOT gates. After considering child nodes with one parent node, we now consider child nodes with more than one parent nodes.
@C=1em @R=0.9em
R_Y (θ) = @C=1em @R=0.9em
1 1
R_Y(θ/2) R_Y(-θ/2)
Decomposition of a $CR_Y$ gate into single-qubit and CNOT gates
Let $n$ represent the number of parent nodes for a child node, $V_i$. The conditional probabilities of $V_i$ can be stated using $C^nR_Y$ gate where the $n$ control qubits are the $n$ qubits corresponding to the $n$ parent nodes and the target qubit represents the child node. For the child node C in Figure <ref>, n=2, and therefore, we used a $CCR_Y$ or $C^2R_Y$ gate to show the conditional probabilities of C in Figure <ref>. $C^nR_Y$ is not an elementary gate and will need to be decomposed into elementary gates. One of the techniques to build the $C^nR_Y$ is the use of additional “dummy" qubits known as ancilla qubits [Nielsen & Chuang, 2002]. Following [Nielsen & Chuang, 2002], implementation $C^nR_Y$ requires $n-1$ ancilla qubits. Using ancilla qubits, the $C^nR_Y$ gate is decomposed into a combination of $2(n-1)$ CCNOT gates, and one $CR_Y$ gate. For illustration, Figure <ref> details the representation of $C^5R_Y$ gate using four ancilla qubits. The $CR_Y$ gate can again be decomposed into a combination of single qubit and CNOT gates as detailed in Figure <ref>.
In Figure <ref>, $q_i, i=0\dots 4$ represent the control qubits (parent nodes), and $q_5$ is the target qubit (child node). As there are five control qubits, we use four ancilla qubits ($a_j, j=0\dots 3$). In total, we have 8 CCNOT gates ($2\times (5-1)$), two CNOTs and two single-qubit rotation gates. In Figure <ref>, there are two control qubits ($n=2$) for $q_2$ (node C), we will use one ancilla qubit to make the $CCR_Y$ gates. Note that we do not need a different set of ancilla qubits for implementing various controlled rotations (conditional probabilities), and we can the same set of ancilla qubits for all the $C^nR_Y$ rotations. Moreover, we can use the same set of ancilla qubits to build the $C^nR_Y$ rotations associated with various child nodes. Consider a Bayesian network with $s$ two-state nodes given by ${V_1, V_2,\dots V_s}$ and let $|.|$ denote the cardinality operator. For a node $V_i$, $|\Pi_{V_i}|$ provides the number of parent nodes of a $V_i$. For a root node, $|\Pi_{V_i}| = 0$ and for a child node, $|\Pi_{V_i}| > 0$. Therefore, the total number of qubits required to represent a Bayesian network with two-state nodes (denoted as $m_{BN,2}$) can be calculated as
\begin{equation}
\label{eqn:total_2qubit}
m_{BN,2} = s + \max \Big(|\Pi_{V_1}|, |\Pi_{V_2}|, \dots |\Pi_{V_s}|\Big) - 1
\end{equation}
where $s$ qubits are used to represent $s$ nodes in the Bayesian network, and an additional $\max \Big(|\Pi_{V_1}|, |\Pi_{V_2}|, \dots |\Pi_{V_s}|\Big)-1$ ancilla qubits are used to represent the multi-qubit conditional rotations.
@C=0.5em @R=0.9em
q_0:0 1
q_1:0 1
q_2:0 1
q_3:0 1 =
q_4:0 1
q_5:0 R_Y (θ) @C=0.6em @R=0.9em
q_0:0 1 1
q_1:0 4 4
q_2:0 3 3
q_3:0 3 3
q_4:0 3 3
a_0:0 1 1
a_1:0 1 1
a_2:0 1 1
a_3:0 1 1
q_5:0 R_Y(θ/2) R_Y(-θ/2)
Representation of $C^5R_Y$ gate using ancilla qubits
§.§ Representing discrete variables with more than two states
When a node has more than two states, then we need to use more than one qubit to represent it, as one qubit can represent only two states. Consider a random variable $V_i$ with $n_i$ states, denoted as $V_{i,j}, j=0,1,\dots n_i-1$, and $P(V_{i,j})$ represents the probability of state $V_{i,j}$; therefore, $\sum_{j=0}^{n_i-1} P(V_{i,j})=1$.
Let $m_i$ represent the number of qubits to represent $V_i$. The number of states that are represented by $m_i$ qubits is $2^{m_i}$, which needs to be greater than or equal to $n_i$.
Thus, value of $m_i$ can be calculated as the smallest integer that is greater than or equal to ${\log_2 n_i}$, which can be represented using the ceiling function as $m_i = \ceil[\big]{\log_2 n_i}$, where $\ceil[\big]{.}$ is the ceiling function.
Let $\Ket{q_j} j=l\dots l+m_i-1$ represent $m_i$ qubits in the quantum circuit used to represent $V_i$. In addition to qubits representing node $V_i$, there could be other qubits in the quantum circuit, which are used to represent other nodes and/or ancilla qubits. Here, $l$ is used to represent the indices of the qubits used to represent node $V_i$. The superposition state, $\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}$ can be written in terms of the basis states as
\begin{equation}
\label{eqn:mstate}
\Ket{q_l q_{l+1} \dots q_{l+m_i-1}} = \sum_{j=0}^{2^{m_i}-1} \alpha_t \Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j
\end{equation}
@C=1em @R=0em
0 4U_0,1 …m_i-1
0 U_0,1 …m_i-1
0 U_0,1 …m_i-1 =
⋯ U_0,1 …m_i-1 ⋯
0 U_0,1 …m_i-1 @C=1em @R=0em
0 R_Y(θ_l) 1 X 1 X
0 3U_1 …m_i-1, q_l = 1 3U_1 …m_i-1, q_l = 0
0 U_1 …m_i-1, q_l = 1 U_1 …m_i-1, q_l = 0
⋯ U_1 …m_i-1, q_l = 1 ⋯ U_1 …m_i-1, q_l = 0 ⋯
0 U_1 …m_i-1, q_l = 1 U_1 …m_i-1, q_l = 0
Decomposing a $m_i$ qubit rotation into single qubit and Controlled $m_i$ - 1 qubit rotations
In Eq. (<ref>), $\Ket{q_l q_{l+1} \dots q_{l+k-1}}_j$ represents a basis state ($\Ket{00\dots 0}$ for instance) and $\alpha_j$ represents its probability amplitude. Let us map the $n_i$ states of the variable to $n_i$ basis states represented by these $m_i$ qubits. Hence, state $V_{i,j}$ can be mapped to the quantum state $\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j$. The probability amplitudes of these $n_i$ quantum states can be calculated using the available state probabilities and the probability amplitudes of the remaining quantum states are set to zero. Therefore, $\alpha_j = \sqrt{P(V_{i,j})}$ when $j<n_i$ and $\alpha_j = 0$ when $n_i\leq j < 2^{m_i}$.
Our goal is to identify the gate $U$ that acts on $m_i$ qubits and produces the desired state probabilities, i.e., $U\Ket{0}^{\otimes m_i} = \Ket{q_l q_{l+1} \dots q_{l+m_i-1}}$ which is equlal to $\sum_{j=0}^{n_i-1} \sqrt{P(V_{i,j})} \Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j$. Since $U$ is a multi-qubit gate, it needs to be decomposed into a set of elementary gates discussed in Section <ref>. Our approach is to decompose the probability distribution defined over $m_i$ qubits into a combinations of marginal and conditional distributions that can be implemented using single-qubit and controlled-rotations. First, we rotate the first qubit ($q_l$) to obtain its associated probability values corresponding to its $\Ket{0}$ and $\Ket{1}$ states respectively using a single-qubit $R_Y$ rotation, and we implement a different multi-qubit rotation on the remaining $k-1$ when $q_l=\Ket{0}$ and $q_l=\Ket{1}$.
Figure <ref> details the decomposition of a $m_i$ qubit rotation into a combination of single-qubit and controlled $m_i$-1 qubit rotations, where $U_{0,1 \dots m_i-1}$ is the $m_i$ qubit rotation, and $U_{1 \dots m_i-1, q_l=\Ket{1}}$ and $U_{1 \dots m_i-1, q_1=\Ket{0}}$ are the rotations implemented on $m_i-1$ qubits ($q_{l+1},q_{l+2} \dots q_{l+m_i-1}$) when $q_l=\Ket{1}$ and $q_l=\Ket{0}$ respectively. $R_Y(\theta_l)$ represents the rotation to obtain the probabilities associated with qubit $q_l$, and can be calculated using Eq. (<ref>). The probabilities of $\Ket{0}$ and $\Ket{1}$ states of $q_l$ can be calculated using an indicator function, $\mathbb{I}_{q_l}$, defined as
\begin{equation}
\label{eqn:ql}
\mathbb{I}_{q_l}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}})=
\left\{
\begin{array}{ll}
1 & \mbox{if } \Ket{q_l}=\Ket{1} \\
0 & \mbox{if } \Ket{q_l}=\Ket{0}
\end{array}
\right.
\end{equation}
Using Eq. (<ref>), the probability that $q_l=\Ket{1}$ can be calculated as $P(q_l=\Ket{1}) = \sum_{j=0}^{2^{m_i}-1} \alpha_j^2 \mathbb{I}_{q_l}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j)$, and $P(q_l=\Ket{0}) = 1-P(q_l=\Ket{1})$. The rotation angle, $\theta_l$ in Figure <ref> can be calculated using Eq. (<ref>) as
\begin{equation}
\label{eqn:rot_ql}
\begin{aligned}
\theta_l & = 2\times \tan^{-1} \sqrt{\dfrac{P(q_l=\Ket{1})}{P(q_l=\Ket{0})}} \\
& = 2\times \tan^{-1}\sqrt{\dfrac{\sum_{j=0}^{2^{m_i}-1} \alpha_j^2 \mathbb{I}_{q_l}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_i)}{1-\sum_{j=0}^{2^{m_i}-1} \alpha_j^2 \mathbb{I}_{q_l}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j)}}
\end{aligned}
\end{equation}
The above procedure for decomposing $m_i$ qubit rotation into a single qubit and controlled $m_i$-1 qubit rotations can again be used to decompose the controlled $m_i$-1 qubit rotation resulting in controlled single-qubit and controlled-controlled $m_i$-2 qubit rotations. We will illustrate the decomposition of $CU_{1 \dots m_i-1, q_l=\Ket{1}}$ in Figure <ref>.
@C=1em @R=0.5em
3U_1 …m_i-1, q_l=1
U_1 …m_i-1, q_l=1 =
⋯ U_1 …m_i-1, q_l=1 ⋯
U_1 …m_i-1, q_l=1
@C=1em @R=0.8em
1 1 1 1 1
R_Y(θ_l+1,q_l=1) 1 X 1 X
2U_2 …m_i-1, q_l=1, q_l+1=11 2U_2 …m_i-1, q_l=1, q_l+1=0
⋯ ⋯ U_2 …m_i-1, q_l=1, q_l+1=11 ⋯ U_2 …m_i-1, q_l=1, q_l+1=0 ⋯
U_2 …m_i-1, q_l=1, q_l+1=11 U_2 …m_i-1, q_l=1, q_l+1=0 22571.5em–
Decomposing a controlled $m_i$-1 qubit rotation into controlled single qubit and controlled-controlled $m_i$-2 qubit rotations
First, we implement a $CR_Y$ gate on qubit $q_{l+1}$ to obtain the probabilities of $q_{l+1}$ when $q_l=\Ket{1}$. The probabilities of $q_{l+1}=\Ket{0}$ and $q_{l+1}=\Ket{1}$ when $q_l=\Ket{1}$ are calculated using another indicator function defined over $q_l$ and $q_{l+1}$ as
\begin{equation}
\label{eqn:ql1}
\begin{split}
&\mathbb{I}_{q_l=1, q_{l+1}}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}})\\
&= \left\{
\begin{array}{ll}
1 & \mbox{if } \Ket{q_l}=\Ket{1} \text{and} \Ket{q_{l+1}}=\Ket{1} \\
0 & \mbox{if } \Ket{q_l}=\Ket{1} \text{and} \Ket{q_{l+1}}=\Ket{0}
\end{array}
\right.
\end{split}
\end{equation}
\begin{equation}
\label{eqn:rot_ql1}
\begin{split}
&\theta_{l+1, q_l=\Ket{1}} = 2\times \tan^{-1} \sqrt{\dfrac{P(q_{l+1}=\Ket{1}|q_l=\Ket{1})}{P(q_{l+1}=\Ket{0}|q_l=\Ket{1})}}\\
& = 2\times \tan^{-1}\sqrt{\dfrac{P(q_{l+1}=\Ket{1},q_l=\Ket{1})}{P(q_{l+1}=\Ket{0},q_l=\Ket{1})}} = 2\times \\
& \tan^{-1}\sqrt{\dfrac{\sum_{j=0}^{2^{m_i}-1} \alpha_j^2 \mathbb{I}_{q_l=\Ket{1}, q_{l+1}}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j)}{1-\sum_{i=0}^{2^{m_i}-1} \alpha_j^2 \mathbb{I}_{q_l=1, q_{l+1}}(\Ket{q_l q_{l+1} \dots q_{l+m_i-1}}_j)}}
\end{split}
\end{equation}
$CCU_{2 \dots m_i-1, q_l=\Ket{1}, q_{l+1}=\Ket{1}}$ and $CCU_{2 \dots m_i-1, q_l=\Ket{1}, q_{l+1}=\Ket{0}}$ are again decomposed into a set of controlled-controlled single qubit rotations and triply controlled $m_i$-3 qubit rotations following the procedure described above. This decomposition is carried out until we reach $m_i-1$ controlled qubit rotations implemented on the qubit $q_{l+m_i-1}$. In this way, a $m_i$-qubit rotation required to realize the probabilities associated with a discrete variable with more than two states is achieved through uncontrolled/controlled/multi-controlled qubit rotations. Multi-controlled qubit rotations can be implemented using ancilla qubits as detailed in Figure <ref>. When the multi-state variable is a child node, then we will have a different $m_i$ qubit rotation for each combination of the parent nodes. Depending on the number of parent nodes, Each controlled/multi-qubit controlled $m_i$-qubit rotation can be represented following the above sequential decomposition process.
In Section <ref>, we discussed the implementation of controlled rotations to realize conditional probabilities of a child node when both the parent and child nodes have two states. Here, let us consider the cases when a combination of multi-state variables and two-state variables are parent nodes for a multi-state child node. Consider a variable $V_i$ with $n_i$ states with $\Pi_{V_i}$ as the set of parent nodes. Let $\Pi_{V_{i,j}}$ represent the $j^{th}$ parent node and $n_{\Pi_{V_{i,j}}}$ represent the number of discrete states in the $j^{th}$ parent node. Number of qubits required to represent $\Pi_{V_i}$ can be calculated as
\begin{equation}
\label{eqn:nq_parent}
m_{q, \Pi_{V_i}} = \sum_{i=1}^{|\Pi_{V_i}|} \ceil{\log_2 n_{\Pi_{V_{i,j}}}}
\end{equation}
where $m_{q, \Pi_{V_i}}$ is the number of qubits required to represent $\Pi_{V_i}$ and $|\Pi_{V_i}|$ represents the cardinality of the set of parent nodes. If $n_i$ represents the number of states of child node $V_i$, then the highest order of $C^nR_Y$ gate required to realize the conditional probabilities of $V_i$ can be calculated as $n = m_{q, \Pi_{V_i}} + \ceil{\log_2 n_i}$. In order to implement this $C^nR_Y$ gate, we will need $n-1 = m_{q, \Pi_{V_i}} + \ceil{\log_2 n_i}-1$ ancilla qubits. Therefore, the total number of qubits required to obtain a Bayesian network with a combination of two-state and multi-state variables is given as
\begin{equation}
\label{eqn:nBN}
m_{BN} = \Bigg(\sum_{i=1}^m \ceil{\log_2 n_i}\Bigg) + \max_i \Big(m_{q, \Pi_{V_i}} + \ceil{\log_2 n_i} - 1\Big)
\end{equation}
where $m_{BN}$ denoted the number of qubits required to represent a given BN, $m_{q, \Pi_{V_i}} + \ceil{\log_2 n_i} - 1$ is the number of ancilla qubits required to realize the conditional probabilities of node $V_i$. As mentioned in Section <ref>, the same set of ancilla qubits can be used for various child nodes. Since qubits can represent only discrete states, any continuous variables need to be discretized to be represented using qubits. If a continuous variable is discretized into more than two states, then the above procedure to handle discrete variables with more than two levels can be used. If the discretization involves only two states, a single qubit can be used to represent it.
§ ILLUSTRATION EXAMPLES
For illustration of the proposed methodology, we consider three examples with varying properties from the financial industry: (1) a 4-node Bayesian network for an oil company stock price prediction; (2) a 10-node Bayesian network used for liquidity risk assessment; and (3) a Naive Bayes classifier with 8 features (a total of 9 nodes) used for bankruptcy prediction. The first two BNs consider random variables with only two states whereas the Naive Bayes classifier considers nodes that have two and three states. In particular, there are six features with three states, two features with two states, and a class variable with two classes. We considered both 4-node and 10-node BNs to demonstrate the scalability of the proposed methods.
For each of the three examples, we will design quantum circuits using the proposed C-QBN approach, and then execute them using Qiskit, which is a Python-based simulated quantum computing platform developed by IBM [Wille et al., 2019]. We demonstrate the proposed methods on a simulated platform instead of using real quantum computers as hardware implementation of circuits of large depths (large number of gates) are affected by noise, which leads to incorrect results [Mandviwalla et al., 2018, Martin et al., 2019]. Since simulations are not affected by hardware noise, we use them to demonstrate the proposed methods. The probabilities of various states of the BNs are computed, and these results are compared with the probabilities obtained from simulating the examples on a classical Bayesian network platform such as Netica [Netica, 2019].
§.§ 4-node BN: Oil Company Stock Price
This 4-node Bayesian network example to assess an oil company stock price is obtained from [Shenoy & Shenoy, 2000]. The four variables in this network are the interest rate (IR), stock market (SM), oil industry (OI), and oil company stock price (SP). IR has two states - high and low; SM has two states - good and bad; OI has two states - good and bad; and SP has two states - high and low. Here, we represent low/bad with state 0 and high/good with state 1. The dependence between these four variables and associated conditional probability tables are given in Figure <ref>.
The BN in Figure <ref> has two root nodes, i.e. nodes without parent nodes (IR, OI), one node with only one parent node (SM), and one node with two parent nodes (SP). Since SP has two parent nodes, we use one ancilla qubit to represent its conditional probability values as discussed in Section <ref>. This results in a five-qubit system (four qubits to represent four variables in Figure <ref> and an ancilla qubit). We discuss below the construction of quantum circuits corresponding to this BN using the C-QBN approach described in Section <ref>.
Bayesian network for an oil company stock price [Shenoy & Shenoy, 2000]
Quantum circuit: Figure <ref> provides the quantum circuit corresponding to the BN in Figure <ref> constructed using the C-QBN approach. The five qubits are denoted as $q_i, i=0\dots 4$ and the measurement bit is denoted as $c$.
The variables - IR, OI, SM, and SP are denoted using the qubits $q_4, q_3, q_2$ and $q_0$ respectively, and the ancilla qubit is $q_1$. We chose $q_1$ as the ancilla qubit for the purpose of illustration. In reality, any qubit can be chosen as an ancilla qubit in Qiskit. We used this mapping as the representation of an n+1-qubit state is given as $\Ket{q_nq_{n-1}\dots q_0}$, i.e., the state of the $n+1^{th}$ qubit ($q_n$) is written first while the first qubit ($q_0$) is written at the end. By following this mapping, the parent nodes appear ahead of their associated child nodes.
After mapping the variables to various qubits, we now identify the appropriate gates to be implemented on those qubits to obtain the required marginal or conditional probability values. Let us begin with the root nodes (IR and OI). The rotation angles required to represent those root nodes were calculated using Eq. (<ref>) as $\theta_{IR} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.25}{0.75}}\bigg) = \dfrac{\pi}{3}$ and $\theta_{OI} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.4}{0.6}}\bigg) = 1.37$.
For realizing child nodes, we compute the associated rotation angles for various combinations of the parent node(s).
Quantum circuit of the 4-node oil company stock price BN. Variables IR, OI, SM and SP are mapped to $q_4$, $q_3$, $q_2$, and $q_0$ respectively, and $q_1$ is the ancilla qubit
To represent SM node, we compute its rotation angles when IR=0 and IR=1 as $\theta_{SM,0} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.7}{0.3}}\bigg) = 1.982$ and $\theta_{SM,1} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.2}{0.8}}\bigg)=0.928$ respectively. Here, $\theta_{SM,j}$ corresponds to the rotation angle of SM for a given value $j$ of its parent node, IR. As discussed in Section <ref>, the controlled rotations are decomposed into a combination of uncontrolled rotations and CNOT gates. For example, the conditional probability values of SM when IR=1 are realized by implementing a controlled-rotation gate $CR_Y(\theta_{SM,1})$. Following Section <ref>, this gate is implemented as $\bigg(I\otimes R_Y\bigg(\dfrac{\theta_{SM,1}}{2}\bigg)\bigg)CX\bigg(I\otimes R_Y\bigg(\dfrac{-\theta_{SM,1}}{2}\bigg)\bigg)CX$. The controlled-rotation when IR=0 is implemented by first flipping the states of the $q_4$ (IR) qubit using an $X$ gate, and then applying the $CR_Y(\theta_{SM,0})$ gate. Thus, the conditional probability values of SM are realized for various values of its parent node (IR). We now consider the SP node.
Since SP has two parent nodes, we have four rotation angles ($\theta_{SP,00}$, $\theta_{SP,01}$, $\theta_{SP,10}$, $\theta_{SP,11}$) for various values of the two parent nodes; these values were calculated using Eq. (<ref>) as $0.644, 1.772, \frac{\pi}{2}, 2.22$ respectively. Here, $\theta_{SP,jk}$ corresponds to the rotation angle of SP for given values $j$ and $k$ of parent nodes OI and SM respectively. Following Section <ref>, the controlled-controlled rotations are implemented using an ancilla qubit. At the end of the circuit, we add the measurement gates and stores the measured qubit values in a classical bit register ($c$ in Figure <ref>). We consider only four measurements across the qubits associated with various nodes in the BN, and do not consider a measurement gate for the ancilla qubit as it does not represent any variable in the BN.
Circuit simulation: The BN is simulated using the circuit constructed with the C-QBN approach, and the accuracy of the results is compared with those obtained using Netica, which is a classical Bayesian network software.
After a simulation is made, the system is measured, which returns a single quantum state (such as $\Ket{1010}$). A total of 8192 shots were carried out in each simulation, and the measured states after each shot are used to estimate the the probabilities of all the states. We used 8192 shots as that was the maximum number of shots possible on the real IBM quantum computers such as the 5-qubit IBM QX5 [Mandviwalla et al., 2018].
When a measurement is made, one of the 16 states is observed as we were measuring only four qubits corresponding to four nodes in the BN. The probability associated with each state ($P(\Ket{q_4q_3q_2q_0})$) can be computed using the Monte Carlo approach as
\begin{equation}
P(\ket{q_4q_3q_2q_0}) = \dfrac{n_{\ket{q_4q_3q_2q_0}}}{N}
\end{equation}
where $P(\Ket{q_4q_3q_2q_0})$ is the probability of state $\Ket{q_4q_3q_2q_0}$; $n_{\Ket{q_4q_3q_2q_0}}$ and $N$ represent the number of times $\Ket{q_4q_3q_2q_0}$ is observed and the total number of shots (8192) respectively. The marginal probabilities of each of the nodes can be estimated using Equation by marginalizing over the state probabilities calculated using Eq. (<ref>).
\begin{equation}
\label{eqn:marginalize}
P(\Ket{q_i}) = \sum_{q_j, j=4,3,2,0, j\neq i} P(\Ket{q_4q_3q_2q_0})
\end{equation}
Comparison of simulation results: As discussed above, we ran 8192 shots of the quantum circuit to estimate the marginal probabilities. Since the marginal probabilities are estimated from data, there could be variation across multiple runs of the quantum circuit. In order to quantify the variation across runs, we ran the circuit $r$ times and obtained the marginal probability values from each run. Given the simulation results from $r$ runs, we compute the $(1-\alpha)$ confidence intervals of the estimated marginal probabilities and checked if the marginal probabilities from Netica fall within the estimated intervals.
Let $p_i^m, i=1\dots r, m={IR, OI, SM, SP}$ represent the marginal probability value of the $m^{th}$ variable in the $i^{th}$ run, then the sample mean and standard deviation can be calculated as $\bar{p}^m = \dfrac{\sum_{i=1}^{r} p_i^m}{r}$ and $s^m = \sqrt{\dfrac{\sum_{i=1}^{r} (p_i^m-\bar{p}^m)^2}{r-1}} $ respectively. Given the sample mean and standard deviation, the $(1-\alpha)$ confidence interval were calculated as $\bar{p}^m \pm t_{\frac{\alpha}{2}} \dfrac{s^m}{\sqrt{r}}$. Here, $t_{\frac{\alpha}{2}}$ is the t-statistic corresponding to the $(1-\alpha)$ confidence interval. In this study, we chose $r=10$ and $\alpha=0.05$. The sample mean and standard deviation, and the 95% CIs of the marginal probabilities using both the methods are provided in Table <ref>, along with the marginal probabilities obtained using Netica. The variation in the probability values obtained using the C-QBN approach, can be attributed to the variability in the measurement process.
From Table <ref>, it can be observed that the marginal probabilities from Netica fall within their estimated confidence intervals obtained from the quantum circuit simulations. Since each node has two states, we provided the probabilities of only one of the states as the probabilities of the other states can be computed from the given states. For example, $P(IR=1)=1-P(IR=0)$. Thus, the C-QBN approach was used to represent the 4-node Bayesian network.
Simulation results using Qiskit with 8192 shots of the oil company stock price BN
Comparison of marginal probabilities in the 4-node Bayesian network
Netica 3c|C-QBN
Value Probability Mean SD 95% CI
IR=0 0.75 0.7504 0.0042 [0.7471, 0.7536]
SM=0 0.425 0.4252 0.0064 [0.4219, 0.4284]
OI=0 0.6 0.5999 0.0050 [0.5962, 0.6037]
SP=0 0.499 0.4994 0.0045 [0.4950, 0.5022]
§.§ 10-node BN: Liquidity Risk Assessment
Bayesian network for liquidity risk assessment [Tavana et al., 2018]
Here, we consider designing a quantum circuit to represent a 10-node Bayesian network obtained from [Tavana et al., 2018] used for liquidity risk assessment in banking. The 10 variables in the Bayesian network are described in Table <ref>, and the dependence between various variables are shown in Figure <ref>, and the conditional probability tables are given in Figure <ref>.
In Table <ref>, B refers to the bank under consideration, and O refers to other banks. This BN has one root node ($X_6$), six nodes with one parent node ($X_7, X_8, X_9, X_1, X_2, X_3$), two nodes with two parent nodes ($X_4, X_5$), and one node with three parent nodes ($X_{10}$). Since the maximum number of parent nodes is three, we need two ancilla qubits to represent the conditional probability values in addition to the ten qubits used to represent the ten nodes in the BN totaling to 12 qubits. The representation of the root node ($X_6$), child nodes with either one or two parent nodes follows the same procedure as detailed in Section <ref>. Therefore, we discuss below the representation of $X_{10}$, which is the child node with three parent nodes ($X_1, X_2, X_4$) using the C-QBN approach.
Variables in the 10-node liquidity risk assessment Bayesian network [Tavana et al., 2018]
Variable Description
$X_1$ $\text{Liquidity ratio} = \dfrac{\text{Liquid assets of \textbf{B}}}{\text{Current liabilities of \textbf{B}}}$
$X_2$ $\dfrac{\text{Credits of \textbf{B} in \textbf{O}}}{\text{Credits of \textbf{O} in \textbf{B}}}$
$X_3$ $\dfrac{\text{Long term deposits of \textbf{B}}}{\text{Short term deposits of \textbf{B}}}$
$X_4$ $\dfrac{\text{Credits of \textbf{B} in \textbf{O}}}{\text{Credits of \textbf{O} in \textbf{B}}}$
$X_5$ $\dfrac{\text{Total loan of \textbf{B}}}{\text{Total deposits of \textbf{B}}}$
$X_6$ $\dfrac{\text{bonds of \textbf{B}}}{\text{Total assets of \textbf{B}}}$
$X_7$ $\dfrac{\text{Volatile deposits of \textbf{B}}}{\text{Total liabilities of \textbf{B}}}$
$X_8$ $\dfrac{\text{Short investments of \textbf{B}}}{\text{Total assets of \textbf{B}}}$
$X_9$ $\dfrac{\text{Credits of \textbf{B} in central bank}}{\text{Total deposits of \textbf{B}}}$
$X_{10}$ $\text{Bank liquidity risk} = \dfrac{\text{Long term deposits of \textbf{B}}}{\text{Short term deposits of \textbf{B}}}$
CPT for liquidity risk assessment Bayesian network [Tavana et al., 2018]
Quantum circuit: Figure <ref> provides the quantum circuit of the 10-node BN constructed the C-QBN approach. The 12 qubits are denoted as $q_i, i=0\dots 11$ and the measurements of various qubits are stored in classical bits denoted as $c$. In this circuit $X_1$ is mapped to qubit $q_7$, $X_2$ represented by $q_3$, $X_3$ to $q_4$, $X_4$ to $q_6$, $X_5$ to $q_5$, $X_6$ to $q_{11}$, $X_7$ to $q_{10}$, $X_8$ to $q_9$, $X_9$ to $q_8$. Finally, $X_{10}$ is represented using $q_0$, and $q_1, q_2$ are the ancilla qubits. Since $X_{10}$ are three parent nodes, we need to implement $C^3R_Y$ gate for each of the 8(=$2^3$) combinations of the parent nodes. Following Section <ref>, the $C^3R_Y$ gate requires the use of two ancilla qubits, and is build through a combination of one, two, and three-qubit gates($R_Y$, CNOT, CCNOT).
For example, the conditional probability values of $X_5$ when $X_8=1$ and $X_4=0$ would be obtained by implementing a controlled-controlled rotation gate $CCR_Y$. Since $X_5$ has two parent nodes, we have four rotation angles($\theta_{X_5,00}$, $\theta_{X_5,01}$, $\theta_{X_5,10}$, $\theta_{X_5,11}$) for various values of the two parent nodes. Following Section <ref>, the controlled-controlled rotations for $X_5$ are implemented using one of the two available ancilla qubits ($q_2$ is used here).
Circuit simulation and results: Following the 4-node BN, we ran the quantum circuits obtained using the C-QBN ten times each with 8192 shots. The marginal probabilities of nodes from Netica, along with the sample mean, sample standard deviation and the 95% confidence intervals of the sample mean are provided in Table <ref>.
Simulation result using Qiskit with 8192 shots of the liquidity Risk Assessment BN
Comparison of marginal probabilities in the 10-node Bayesian network
Netica 3c|C-QBN
Value Probability Mean SD 95% CI
$X_1=0$ 0.431 0.4306 0.0025 [0.4287, 0.4326]
$X_2=0$ 0.863 0.8638 0.0019 [0.8624, 0.8653]
$X_3=0$ 0.976 0.9757 0.0009 [0.9750, 0.9764]
$X_4=0$ 0.57 0.5694 0.0024 [0.5676, 0.5712]
$X_5=0$ 0.527 0.526 0.0034 [0.5235, 0.5286]
$X_6=0$ 0.98 0.9805 0.0007 [0.9800, 0.9810]
$X_7=0$ 0.977 0.9767 0.0009 [0.9760, 0.9773]
$X_8=0$ 0.0261 0.0267 0.0009 [0.0260, 0.0274]
$X_9=0$ 0.956 0.9559 0.0014 [0.9548, 0.9570]
$X_{10}=0$ 0.24 0.2397 0.002 [0.2382, 0.2412]
From the results in Table <ref>, it can be observed that the true marginal probabilities (obtained from Netica) fall within the confidence intervals obtained using the C-QBN approach. These results help conclude that the C-QBN approach is able to simulate the 10-node Bayesian network with two and three parent nodes, each with two states.
§.§ 9-node Naive Bayes Classifier: Bankruptcy Prediction
Here, we consider designing a quantum circuit to represent a Naive Bayes classifier used for bankruptcy prediction; this model is obtained from [Sun & Shenoy, 2007]. There are eight features in this classifier that correspond to several financial-accounting, market-based, and other extraneous factors. The financial-accounting factors are Cash/Total Assets (CH), a variable related to the variation in cash and short-term marketable securities (LM), a binary variable to check if the net income was negative in the last two years (IT), and a variable that represents the ratio of the change in net income and the sum of the absolute net income in the last two years (CHN). The market-based factors are a variable that represents the natural logarithm of firm's size relative to the CRSP NYSE/AMEX/NASDAQ market capitalization index (M) and a variable that represents the difference of the firm's stock return and the value-weighted CRSP NYSE/AMEX/NASDAQ index return in the previous year (R).
Naive Bayes classifier for bankruptcy prediction [Sun & Shenoy, 2007]
CPT for bankruptcy prediction naive Bayes classifier [Sun & Shenoy, 2007]
The extraneous factors are variables that relate to the Compustat codings (AU) and Industry Failure Rate (IFR). The bankruptcy classification status is denoted with the variable B. Figure <ref> shows the Naive Bayes model, and the associated conditional probability tables are provided in Figure <ref>. The variables B, AU and IT have two states $\{0,1\}$ while all the remaining variables have three states $\{0,1,2\}$. We are using this example for the sake of illustration, and the readers are referred to [Sun & Shenoy, 2007] for more details about the variables and the model. Since each the variables B, AU and IT has two states, it can be represented using a single qubit. Each of the remaining variables has three states; therefore, each node is represented using $\ceil{\log_2 3} = 2$ qubits.
The total number of qubits used to represent the 9-node Naive Bayes classifier is 16 ($3\times1 + 6\times2 + 1$). The representation of the root node (B), and child nodes with one parent node (AU, IT) follows the procedure described in Section <ref>. Here, we discuss the representation of child nodes with more than two levels (CH, LM, M, R, CHN, IFR) using the C-QBN approach.
Quantum circuit: The 16 qubits are denoted as $q_i, i=0 \dots 15$ and the measurements of various qubits are stored in the classical bit register $c$. The nine variables B, AU, IT, CH, LM, M, R, CHN, and IFR are mapped to $q_{15}, q_{14},q_{13}, (q_{12}, q_{11}),(q_{10}, q_{9}),(q_{8}, q_{7}),(q_{6}, q_{5}), (q_{4}, q_{3}),$ and $(q_{2}, q_{1})$ respectively. Any qubit can be chosen as the ancilla qubit, and in this example, we chose $q_0$ for the sake of illustration. We discuss the representation of CH, and the same procedure can be applied to other nodes as well. We map the three states of CH {0,1,2} to the states $\ket{00}$, $\ket{01}$, $\ket{10}$ of qubits $q_{12}$ and $q_{11}$. Figure <ref> shows the associated quantum circuit with gates associated with B and CH nodes only. We apply the appropriate $CU$ transformations to realize the conditional probability values for various values of B {0,1}. Here, the control qubit is $q_{15}$ and the target is a two-qubit system $(q_{12}, q_{11})$.
First, let us consider the case when $q_{15}=\Ket{1}$, i.e., B=1. When $q_{15}=\Ket{1}$, the transformation $U$ should result in the probability values of 0.19, 0.63 and 0.18 for $(q_{12}, q_{11})$ states of $\ket{00}$, $\ket{01}$ and $\ket{10}$ respectively. The probability of state $\Ket{11}$ is fixed at 0. The multi-qubit rotation $U$ is not an elementary transformation, we will decompose it into a combination of one and two-qubit elementary transformations. The probability of $\Ket{0}$ and $\Ket{1}$ states of $q_{12}$ can be calculated as $P(\Ket{00}) + P(\Ket{01}) = 0.19+0.63 = 0.82$ and $P(\Ket{10}) + P(\Ket{11}) = 0.18 + 0 = 0.18$ respectively. We realize the marginal probabilities of $q_{12}$ using a single-qubit $R_Y$ gate with the rotation angle $\theta_{q_{12}, q_{15} = \Ket{1}} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.18}{0.82}}\bigg) = 0.5725$.
After realizing the marginal probabilities of $q_{12}$, we consider the conditional probabilities of $q_{11}$ given $q_{12}$. When $q_{12}=\Ket{1}$, the probability of $q_{11}=\Ket{0}$ is computed using Eq. (<ref>) as
\begin{equation}
\label{eqn:cond_prob}
\begin{split}
P(q_{11}=\Ket{0}|q_{12}=\Ket{1}) &= \dfrac{P(q_{12}=\Ket{1},q_{11}=\Ket{0})}{P(q_{12}=\Ket{1})}\\
&= \dfrac{0.18}{0.18} = 1
\end{split}
\end{equation}
Therefore, $P(q_{11}=\Ket{1}|q_{12}=\Ket{1}) = 1-P(q_{11}=\Ket{0}|q_{12}=\Ket{1}) = 0$. Thus, the probability values associated with the $\ket{10}$ and $\ket{11}$ states are applied through a $CR_Y$ where $q_{12}$ and $q_{11}$ are the control and target qubits respectively, and with a rotation angle, $\theta_{q_{11}, q_{12}=\Ket{1} , q_{15}=\Ket{1}} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{1}{0}}\bigg) = \pi$. Similarly, when $q_{12}=\Ket{0}$, the probabilities of $q_{11}=\Ket{0}$ and $q_{11}=\Ket{1}$ is computed as $P(q_{11}=\Ket{0}|q_{12}=\Ket{0}) = \dfrac{P(q_{12}=\Ket{0},q_{11}=\Ket{0})}{P(q_{12}=\Ket{0})} = \dfrac{0.19}{0.82} = 0.232$ and $P(q_{11}=\Ket{1}|q_{12}=\Ket{0}) = 1-\dfrac{P(q_{12}=\Ket{0},q_{11}=\Ket{0})}{P(q_{12}=\Ket{0})} = 0.768$. Therefore, the rotation angle to show the conditional probability values when $q_{12}=\Ket{0}$ is $\theta_{q_{11}, q_{12}=\Ket{0}, q_{15}=\Ket{1}} = 2\times \tan^{-1}\bigg(\sqrt{\dfrac{0.768}{0.232}}\bigg) = 2.1365$. In this way, the two-qubit rotation gate $U$ is decomposed into a combination of single and two-qubit ($CR_Y$) gates. Since $U$ is implemented when $q_{15}=\Ket{1}$, the controlled-rotations in $U$ gate decomposition become controlled-controlled rotations with B as an additional control qubit. Following Section <ref>, the controlled-controlled rotations are implemented using an ancilla qubit. Similar decomposition procedure is followed to implement the two-qubit $U$ gate when $q_{15}=\Ket{0}$. In this way, the conditional probabilities associated with CH are realized. This procedure is then repeated to show the conditional probability values of three-level nodes LM, M, R, CHN, and IFR. Thus, the quantum circuit of the 9-node naive Bayes classifier is constructed using the C-QBN approach.
Comparison of marginal probabilities in the 9-node Naive Bayes classifier
Netica 3c|C-QBN
Value Probability Mean SD 95% CI
B=0 0.5 0.5015 0.0051 [0.4977, 0.5054]
AU=0 0.595 0.5992 0.0050 [0.5954, 0.6030]
IT=0 0.565 0.5637 0.0047 [0.5602, 0.5672]
CH=0 0.24 0.2429 0.0076 [0.2372, 0.2486]
CH=1 0.63 0.6269 0.0075 [0.6213, 0.6325]
LM=0 0.23 0.2306 0.0060 [0.2261, 0.2352]
LM=1 0.635 0.6368 0.0040 [0.6337, 0.6398]
M=0 0.295 0.2960 0.0042 [0.2928, 0.2991]
M=1 0.595 0.5957 0.0053 [0.5917, 0.5998]
R=0 0.395 0.3940 0.0058 [0.3897, 0.3984]
R=1 0.475 0.4760 0.0050 [0.4723, 0.4798]
CHN=0 0.255 0.2576 0.0045 [0.2542, 0.2610]
CHN=1 0.585 0.5836 0.0061 [0.5790 , 0.5882]
IFR=0 0.14 0.1384 0.0039 [0.1355, 0.1413]
IFR=1 0.72 0.7212 0.0043 [0.7179, 0.7244]
Circuit simulation and results: Similar to the previous examples, we ran the circuit 10 times, each with 8192 shots.The mean, standard deviation, and the 95% CI of the marginal probabilities are given in Table <ref>, from which it can be observed that the probability values from Netica fall within the 95% CI obtained from the C-QBN approach.
For variables with three states (CH, LM, M, R, CHN, IFR), we provided the probabilities of two states as the probability of the third state can be computed using the probabilities of the given states. For example, $P(M=2)=1-P(M=0)-P(M=1)$. These results help conclude that the proposed circuit construction approach can represent the 9-node Naive Bayes classifier. The histogram in Figure <ref> shows different quantum state probabilities in the partial circuit using nodes B and CH.
Each state represents a joint probability of the corresponding variables. The first value in each of the states corresponds to B and the other two are used to represent the three states of CH. Since CH has only three states irrespective of the value of $B$, the probability of state $\Ket{11}$ is fixed at zero. Therefore, the probabilities associated with $\Ket{011}$ and $\Ket{111}$ states are equal to zero and do not appear in the histogram.
Simulation results of nodes B and CH in the 9-node naive Bayes classifier using Qiskit with 8192 shots
§ CONCLUSION
This paper detailed the design of a quantum circuit to represent a generic discrete Bayesian network with nodes that may have two or more states. The quantum circuit design follows three steps. The first step is to map a Bayesian network node to one or more qubits depending on the number of states. The second step is mapping the marginal or conditional probabilities of nodes to probability amplitudes/probabilities associated with the qubits to be in $\Ket{0}$ and $\Ket{1}$ states. The third step is to realize the required probability amplitudes using single-qubit and (multi-qubit) controlled rotation gates. We used ancilla qubits for the implementation of multi-qubit rotation gates. When a node is mapped to more than one qubit, the multi-qubit rotations required to realize the required probabilities are decomposed into a combination of single-qubit and multi-qubit controlled rotations.
The proposed approach was demonstrated with three Bayesian networks: a Bayesian network with four nodes and each with two states used for an oil company stock prediction, a Bayesian network with ten nodes and each with two states used for liquidity risk assessment, and a Naive Bayes classifier with nine nodes (eight features). Of the nine nodes, three nodes had two states and six nodes had three states. The quantum circuits are designed and simulated on Qiskit [McKay et al., 2018], which is a Python-based simulator for quantum computing. We simulated each circuit with 8192 shots, and calculated the marginal probabilities associated with each node. Since the results from quantum circuit are stochastic, we repeated the simulations 10 times, each time with 8192 shots. Using the results from 10 simulations, we estimated the 95% confidence intervals. To validate the simulation results, we simulated the Bayesian networks using a classical Bayesian network software (Netica) and tested if the Qiskit results match the results from the classical software. We found that the marginal probabilities of all the nodes obtained from the classical implementations were within the 95% confidence intervals obtained from Qiskit.
All the quantum circuits in this work were designed manually. Future work should consider automating the design of quantum circuit for a given Bayesian network. We used a simulation platform in this work to validate the accuracy of the methods. In future, we will implement the proposed methods on real quantum computers, and study the effect of hardware noise on the circuit implementation.
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Quantum circuit of the 10-node liquidity risk assessment BN. Variables $X_6$, $X_7$, $X_8$, $X_9$, $X_1$, $X_4$, $X_5$, $X_3$, $X_2$ and $X_{10}$ are mapped to $q_{11}$, $q_{10}$, $q_{9}$, $q_{8}$, $q_{7}$, $q_{6}$, $q_{5}$, $q_{4}$, $q_{3}$, and $q_{0}$ respectively, and $q_1$ and $q_0$ are the ancilla qubits.
Quantum circuit of the 10-node liquidity risk assessment BN. Variables $X_6$, $X_7$, $X_8$, $X_9$, $X_1$, $X_4$, $X_5$, $X_3$, $X_2$ and $X_{10}$ are mapped to $q_{11}$, $q_{10}$, $q_{9}$, $q_{8}$, $q_{7}$, $q_{6}$, $q_{5}$, $q_{4}$, $q_{3}$, and $q_{0}$ respectively, and $q_1$ and $q_0$ are the ancilla qubits. (contd)
Quantum circuit of the 10-node liquidity risk assessment BN. Variables $X_6$, $X_7$, $X_8$, $X_9$, $X_1$, $X_4$, $X_5$, $X_3$, $X_2$ and $X_{10}$ are mapped to $q_{11}$, $q_{10}$, $q_{9}$, $q_{8}$, $q_{7}$, $q_{6}$, $q_{5}$, $q_{4}$, $q_{3}$, and $q_{0}$ respectively, and $q_1$ and $q_0$ are the ancilla qubits. (contd)
Quantum circuit of the 10-node liquidity risk assessment BN. Variables $X_6$, $X_7$, $X_8$, $X_9$, $X_1$, $X_4$, $X_5$, $X_3$, $X_2$ and $X_{10}$ are mapped to $q_{11}$, $q_{10}$, $q_{9}$, $q_{8}$, $q_{7}$, $q_{6}$, $q_{5}$, $q_{4}$, $q_{3}$, and $q_{0}$ respectively, and $q_1$ and $q_0$ are the ancilla qubits. (contd)
Quantum circuit representing nodes B and CH in the 9-node naive Bayes classifier. Variables B, AU, IT, CH, LM, M, R, CHN, and IFR are mapped to $q_{15}, q_{14},q_{13}, (q_{12}, q_{11}),(q_{10}, q_{9}),(q_{8}, q_{7}),(q_{6}, q_{5}), (q_{4}, q_{3}),$ and $(q_{2}, q_{1})$ respectively, and $q_0$ is the ancilla qubit.
|
# Early Period of Training Impacts Out-of-Distribution Generalization
Chen Cecilia Liu, Iryna Gurevych
Ubiquitous Knowledge Processing Lab, Department of Computer Science,
Hessian Center for AI (hessian.AI),
Technical University of Darmstadt
###### Abstract
Prior research has found that differences in the early period of neural
network training significantly impact the performance of in-distribution (ID)
tasks. However, neural networks are often sensitive to out-of-distribution
(OOD) data, making them less reliable in downstream applications. Yet, the
impact of the early training period on OOD generalization remains understudied
due to its complexity and lack of effective analytical methodologies. In this
work, we investigate the relationship between learning dynamics and OOD
generalization during the early period of neural network training. We utilize
the trace of Fisher Information and sharpness, with a focus on gradual
unfreezing (i.e. progressively unfreezing parameters during training) as the
methodology for investigation. Through a series of empirical experiments, we
show that 1) selecting the number of trainable parameters at different times
during training, i.e. realized by gradual unfreezing – has a minuscule impact
on ID results, but greatly affects the generalization to OOD data; 2) the
absolute values of sharpness and trace of Fisher Information at the initial
period of training are not indicative for OOD generalization, but the relative
values could be; 3) the trace of Fisher Information and sharpness may be used
as indicators for the removal of interventions during early period of training
for better OOD generalization.
## 1 Introduction
While deep neural networks have achieved impressive results in their training
tasks, they are often sensitive to distribution shifts (i.e. out-of-
distribution, OOD) during inference. As in many applications of deep neural
networks, the training and testing data may come from different distributions,
such as training with perfect images or texts but inference with noise-
corrupted data [21, 46], data obtained from different time periods [37, 58],
or across different languages and domains [55, 54, 40, 34, 19], to name a few.
Failure to generalize to the OOD setting degrades the models’ robustness and
reliability.
A plethora of research has found that differences in the early period of
training have a significant impact on the in-distribution (ID) performance
[17, 1, 47, 15, inter alia] for a wide range of settings. These settings
include classification, multimodal learning, training from scratch, parameter-
efficient fine-tuning or even federated learning. The wide observation of such
a period in machine learning applications suggests that the early period of
learning is generally important for neural network training [33], which is
sometimes analogous to biological processes (such as the critical learning
period in animals [1, 32]).
In particular, prior literature identifies that modifications (or
interventions) to the optimization process are critical ways to shape the
early period of training. Training techniques such as weight decay [17],
learning rates [27, 47], data augmentations [17, 42] (such as with MixUp [61])
or adding noise to weights [16] impact learning dynamics early on, and can
significantly improve or hamper the final task results depending on the time
of application or removal. Yet, there is very limited work exploring how the
early period of training impacts generalization to OOD data during testing, as
well as “when” the early period of training significantly affects the outcome
of final (OOD) results.
[41] found that using gradual unfreezing [24] (i.e., progressively releasing
trainable parameters during training at fixed time intervals) can impact the
trace of Fisher Information in the early period of training; however, the work
was focused on which parameters to select for better cross-lingual transfer
(where cross-lingual transfer is a form of natural OOD generalization).
Besides the trace of Fisher Information, other sharpness metrics have also
been used to study the generalization of network training, especially after
the success of methods such as Sharpness-Aware Minimization (SAM) [14, 36,
62], however, sharpness is less used in prior work to study the early period
of training.
In this work, we delve deeper and advance our understanding of how the early
period of training impacts OOD generalization through a series of empirical
investigations. We first employ gradual unfreezing [24] as a method to
intervene in the dynamics of the early period of training from scratch for OOD
generalization and investigate how the early period of training impacts OOD
generalization. This is done in a simple and controlled setting. Next, we
investigate which metrics are effective for studying the early period of
training for OOD generalization. Here, we utilize Fisher Information and
sharpness and show how these metrics change in the early period of training
with gradual unfreezing, and examine their impact on the final solutions.
After this, we focus on whether there is an optimal time (i.e. “when”) to
remove intervention to achieve a good trade-off between ID and OOD
generalization. Along the way, we also try to answer the questions: how early
does the “early period” begin and can our observations extend to other
settings without gradual unfreezing or with different model architectures?
To summarize, in this paper, we show that 1) when considering the number of
trainable parameters at a time, i.e. realized by gradual unfreezing – has a
minuscule impact on ID results, but greatly affects the generalization on OOD
data; 2) we also show that the value of sharpness and trace of Fisher
Information during the initial period of training are not indicative for OOD
generalization, but their relative values could be; 3) the trace of Fisher
Information and sharpness may signify the removal of interventions during the
early period of training.
## 2 Related Work
Early period of neural network training. Under the standard usage of the term
generalization (in-distribution, where training and testing data are assumed
to be from the same distribution), prior work [17] shows that the early period
of training of neural networks exhibits a “critical learning period” when
trained from scratch. Regularization and interventions applied in this
critical period affect final task results.
[27] indicates that when learning with a lower learning rate, Fisher
Information exhibits an “explosion” which impedes generalization. Applying
regularization to the trace of Fisher Information alleviates the negative
impact of the high Fisher Information. [42] shows the termination of MixUp
early in training and switching to standard empirical risk minimization helps
with better ID generalization. [60, 16] shows that even winning “lottery
tickets” emerge in the early period of training with large learning rates. The
critical learning period is also found in many other settings, such as in
multimodal models [32], in linear models [33], in transformers [47] and
Federated learning [57]. However, prior work focuses on in-distribution
generalization, ignoring the setting of OOD generalization.
[41] shows the early period of training relates to cross-lingual transfer
performance in the parameter-efficient fine-tuning (PEFT) setting with
transformer models, connecting the early period of training with progressive
parameter unfreezing and OOD generalization (as cross-lingual data are
naturally-occurring OOD data). We utilize findings in [41] and base our work
on gradual unfreezing [24]. In this paper, we focus on 1) a general setup
(i.e. training from scratch), and 2) the characterization of the early period
of training and its relationship to OOD generalization.
Fisher Information, sharpness and generalization. Fisher Information has been
studied in many prior works such as [5, 45] to investigate and improve
optimization behaviour. Similarly, sharpness is another popular metric used to
study optimization behaviour and its relationship to generalization.
[28] found a correlation between sharpness and the ratio of learning rate to
batch size. [29, 11, 49] give theoretical guarantees on the generalization
error using sharpness-related measures and conduct large empirical experiments
showing that sharpness-based measures correlate with generalization. However,
there have been debates on whether sharp minima (such as a high largest
eigenvalue of the training Hessian, $\lambda_{max}$) imply poor generalization
[9] and demonstrate the limits of $\lambda_{max}$ in explaining in-
distribution generalization [30]. Most of the current research efforts are
towards analyzing the loss landscape at convergence, for in-distribution
generalization.
For OOD, [2] shows adaptive sharpness is not a stable measurement for OOD
generalization of the final solution. However, the relationship between
metrics such as Fisher Information, sharpness and OOD generalization in the
early period of training is unclear.
## 3 Preliminaries
### 3.1 Fisher Information Matrix (FIM)
To investigate the training process, we first look at the Fisher Information
[13]. Fisher Information reflects the local curvature and measures the amount
of information with respect to network parameters, i.e. how sensitive the
network predictions are to the changes in parameters. A larger Fisher
Information indicates that a small change in network parameters can change the
output significantly, which can be interpreted as a “sharper” loss landscape.
Let $x$ be the inputs and $y$ be the labels of a dataset $D$. Given a neural
network that is parameterized by $w$. The Fisher Information is defined as:
$\displaystyle F(w)=\operatorname{\mathbb{E}}_{P_{x,y}}\left[\nabla_{w}\log
p_{w}(\hat{y}|x)\nabla_{w}\log p_{w}(\hat{y}|x)^{T}\right].$ (1)
Estimating the full Fisher Information is usually expensive, prior work shows
that the trace of the Fisher Information ($\operatorname{\texttt{tr}(F)}$)
correlates well with the full Fisher Information and can be used for real
applications and capture signals during the learning process [1, 27, 53, inter
alia]. Using the empirical data distribution $\hat{Q}(x)$:
$\displaystyle\operatorname{\texttt{tr}(F)}=\operatorname{\mathbb{E}}_{x\sim\hat{Q}(x)}\operatorname{\mathbb{E}}_{\hat{y}\sim
p_{w}(\hat{y}|x)}||\nabla_{w}\log p_{w}(\hat{y}|x)||^{2}.$ (2)
### 3.2 Sharpness
Let $\mathcal{L_{D}}(w)=\frac{1}{|D|}\sum_{(x,y)\in D}\log p_{w}(y|x)$ be the
loss over training datasets of a neural network parameterized by $w$, and
$\delta$ be a small perturbation drawn from a noise distribution, such as a
Gaussian distribution $\mathcal{N}(0,\rho^{2}diag(c^{2}))$. The definition of
worst-case and average-case sharpness are [14, 36, 2, 22]:
$\displaystyle
S^{\rho}_{avg}=\operatorname{\mathbb{E}}_{\delta\sim\mathcal{N}(0,\rho^{2}diag(c^{2}))}\mathcal{L_{D}}(w-\delta)-\mathcal{L_{D}}(w),$
(3)
$\displaystyle S^{\rho}_{worst}=\max_{\|\delta\odot
c^{-1}\|_{p}\;\leq\;\rho}\mathcal{L_{D}}(w-\delta)-\mathcal{L_{D}}(w),$ (4)
where $\odot c^{-1}$ is element-wise multiplication.
The sharpness here refers to how rapidly the loss changes with respect to the
changes in the model parameters.111The sharpness can be negative. While both
the Fisher Information and sharpness are used for investigating loss
landscapes and generalization, they offer different views of the training
process.
Both $S^{\rho}_{avg}$ and $S^{\rho}_{worst}$ are studied in the prior
literature for generalization. For instance, [29, 10, 36] show that the worst-
case sharpness correlates better with generalization, at convergence. While
prior work believe that flatter (less sharp) minima in the loss landscape lead
to better generalization in neural networks [22, 31, 26, 4], these metrics’
attribution to the early period of training and how their early period trends
are related to OOD generalization is understudied.
### 3.3 Gradual Unfreezing
Gradual unfreezing [24] is a simple tuning method that progressively increases
the number of trainable parameters (i.e. unfreeze, layer-by-layer) of the
neural network from the top to the bottom of the network with a fixed interval
of training steps, $k$ (i.e. the unfreezing time). In this paper, we used a
modified formulation of gradual unfreezing [41], where we progressively
unfreeze the parameters during the early period in the training top-down and
for “blocks” of parameters (a block of parameters can be a single layer or
several consecutive layers, in our case, we use the namespace of the
parameters used in standard implementations to determine blocks). See Appendix
A for details and the algorithm.
## 4 Experimental Setup
To investigate our research questions, we perform two sets of experiments.
First in a controlled setup, with datasets such as CIFAR10 [35] and models
such as ResNet [20] to observe patterns and validate our hypothesis. Then, we
experiment with the complex setup with transformer models. We use $\rho=0.01$
to calculate the sharpness (both average-case and worst-case) with 15 samples,
and $L2$ norm for the worst-case sharpness. We normalize the
$\operatorname{\texttt{tr}(F)}$ by the number of trainable parameters. We use
the Auto-PGD algorithm [8] as implemented in [2] (we refer the readers to the
original papers for details) for computing worst-case sharpness, as it is a
hyperparameter-free estimation method.
#### Controlled Setup.
In this setup, following [21], we use ResNet-18 and VGG-11 as the neural
networks for this study (training from scratch). We perform experiments on
classic image classification datasets and use MNIST [38], CIFAR10 [35] and
CIFAR100 [35] for training. For out-of-distribution evaluation, we use the
corrupted corresponding evaluation datasets, named MNIST-C [48], CIFAR10-C
[21] and CIFAR-100-C [21] (results averaged across corruptions and
severities). The ID evaluation sets are the original test sets respectively.
We use the SGD optimizer and the CIFAR datasets are applied with standard
augmentations (i.e. random crops and flips). We report results over 6 random
seeds for MNIST (due to the high variance in OOD results), and we use 4 random
seeds for other datasets. Other hyperparameters such as learning rate or
weight decay are specified in Appendix B. In our experiments, the default
learning rate specified in Appendix B is denoted as $lr_{d}$ and we also
experimented with reduced learning rates which are 1/10th of the default,
specified as 0.1*$lr_{d}$.
#### Complex Setup.
We also conduct experiments using transformers. As pre-train then fine-tuning
becomes a popular way to adapt general foundational models to downstream
tasks, we examine the cross-lingual transfer (train with English data, test
with other languages) task using parameter-efficient fine-tuning (PEFT) with
the LoRA [25] adapters (HuggingFace PEFT [44] implementation). This setting is
parallel to our controlled setting because: 1) only English data (i.e. ID
data) is used for training and validation (other language data are for
testing), this is a natural setting of OOD generalization with parallel
evaluation protocol to the image tasks; 2) using LoRA adapters allows us to
inject randomly initialized parameters for learning, which is analogous to our
controlled setting.
We train with SQuAD [51] (English, question and answering task) and MNLI [56]
(English, natural language inference task), and evaluate on XQuAD [3], MLQA
[39] and XNLI [7]. We use XLM-RoBERTa [6] as the pre-trained multilingual
transformer backbones and AdamW [43] as the optimizer. We report results
across 4 seeds for all experiments in this setting. Please see Appendix B for
hyperparameters.
## 5 Gradual Unfreezing in the Early Period of Training Can Improve Out-of-
Distribution Generalization
Recall from § 2, where gradual unfreezing improved OOD performance for PEFT
with transformers. Here, we first validate that gradual unfreezing applied to
the early period of training in our controlled setting (training from scratch)
could also help OOD generalization. By examining three different datasets and
two model architectures in Table LABEL:tab:ood, progressive parameter
unfreezing (i.e. gradual unfreezing) does not influence ID results by a large
margin (mostly minor degradation, but can also positively impact the ID
results). However, gradual unfreezing has a non-negligible positive impact on
the OOD results. This observation is also applicable to different learning
rates, although the default (larger) learning rate empirically is better for
both ID and OOD for CIFAR datasets. Here, we provide evidence that gradual
unfreezing can improve OOD performance when training from scratch even if it
was proposed for transfer learning [24], and validate the usability of gradual
unfreezing as an intervention for our study.
Table 1: Classification results on various datasets and model architectures. RN18 is ResNet-18. The default learning rate ($lr_{d}$) cases are the same as described in §4, 0.1*$lr_{d}$ indicates the learning rate is 1/10th of the default learning rate. GU: indicates that gradual unfreezing is applied to the early period of training, here we observe the results that are close to the ID results, but with better OOD results. | MNIST RN18 | CIFAR10 RN18 | CIFAR100 RN18 | CIFAR10 VGG11
---|---|---|---|---
Method | ID / OOD | ID / OOD | ID / OOD | ID / OOD
$lr_{d}$ | 99.06/33.36 | 93.32/72.36 | 71.07/45.10 | 88.62/71.63
$lr_{d}$ \+ GU | 98.98/63.99 | 93.26/72.95 | 71.03/46.34 | 88.53/72.16
0.1*$lr_{d}$ | 99.26/58.46 | 91.66/71.14 | 69.95/44.59 | 86.93/68.41
0.1*$lr_{d}$ \+ GU | 99.18/62.51 | 91.51/71.26 | 70.67/46.03 | 87.01/69.45
## 6 Early Period of Training Impacts Out-of-Distribution Generalization
### 6.1 Evidence of Impact on Out-of-Distribution Generalization
[17, 1] among others show that for ID generalization, there is a “critical
learning period” of the neural network. ID generalization degrades with the
delaying application of regularization, as well as the removal of
regularization.
Using gradual unfreezing, we experimented with different unfreezing steps $k$
(ranging from 1 to equally dividing the total training steps among the number
of trainable parameter blocks) to measure its impact on both ID and OOD test
results. Indeed (as in Figure 1), it is possible that withholding trainable
parameters can influence the OOD generalization as early as after training on
a single batch of data. The effect is especially prominent for simpler
datasets like MNIST.
Prolonging the unfreezing interval of parameters during training initially
results in minimal change in the ID test performance with a larger learning
rate, subsequently leading to quick deterioration of the ID results. The
deterioration of ID results over unfreezing intervals aligns with trends
observed in the early stages of training using other interventions, as
reported in prior work [17, 1], while the effects on OOD results serve as
evidence that the early period of training can impact OOD generalization.
Gradual unfreezing casts interesting trends on the OOD generalization and
shows the trade-off between ID and OOD generalizations. Before the quick
deterioration of ID results, there is a short window where OOD results could
be improved. As soon as the ID results start decreasing rapidly, the OOD
results first increase, then rapidly decrease for CIFAR10/100 with ResNet, but
not in MNIST. There seems to be a range of $k$ in the early period of
training, such that the ID results decayed minimally, but with better OOD
results, and we will examine this in more detail in § 6.4.
Figure 1: Change in ID and OOD evaluation results when unfreezing parameters
at different times (compared to standard training). Sub-figures (a)-(d) are
with the $lr_{d}$, and Sub-figures (e)-(h) are with 0.1*$lr_{d}$. The shortest
unfreezing time $k$ is 1 (training with 1 batch of data). The x-axis is in the
log scale.
### 6.2 Learning Dynamics in the Early Period of Training
Observing Figure 2, by freezing the number of trainable parameters at a time
(and gradually unfreezing them), we can induce higher Fisher Information and
larger $S^{\rho}_{avg}$, $S^{\rho}_{worst}$ at the beginning of training
compared to the standard training procedure, although there is an anomaly in
the $\operatorname{\texttt{tr}(F)}$ of CIFAR100 with ResNet. In general, the
longer we withhold parameters, the higher the level of sharpness and
$\operatorname{\texttt{tr}(F)}$ we can sustain, unfreezing parameters reduce
these metrics.
While there are variations between $S^{\rho}_{avg}$, $S^{\rho}_{worst}$ and
$\operatorname{\texttt{tr}(F)}$ they are all sensitive to the early period of
training and interventions. $S^{\rho}_{avg}$ shows more consistent trends
across datasets and architectures compared to the other two metrics. Due to
the randomness in estimating $S^{\rho}_{avg}$, $S^{\rho}_{worst}$, and
$\operatorname{\texttt{tr}(F)}$, it is also evident that a single, absolute
largest value of these metrics during the early period of training may not be
a consistent indication of OOD generalization (or ID generalization, in fact).
This indicates that the discussion for a high or low value of
$S^{\rho}_{avg}$, $S^{\rho}_{worst}$, or $\operatorname{\texttt{tr}(F)}$
during the early period of training should be relative rather than absolute.
Empirically, our findings differed from prior work on ID generalization (such
as [27]) that demonstrates the ‘explosion’ of $\operatorname{\texttt{tr}(F)}$
during the early period of training (due to using a small learning rate) is
harmful. Here, a higher $\operatorname{\texttt{tr}(F)}$ induced by parameter
freezing does not hurt generalization, in both ID and OOD. When considering
the trainable parameters as a variable, having initial higher sharpness or
$\operatorname{\texttt{tr}(F)}$ can be advantageous up to a certain time frame
during training. More interestingly, some sub-figures (such as Figure 2 (f)
and (j)) show a rapid increase of sharpness after about 100 training steps
(see the $k$=750 curve). The $\operatorname{\texttt{tr}(F)}$ and sharpness
transition from a transient phase to a relatively stabilized value when
withholding parameters for long. Introducing new trainable parameters induces
a quick drop in the corresponding metrics. As a result, the optimization
trajectory and learning mechanism change when manipulating the trainable
parameters through unfreezing during the early period of training (such as the
network resorting to higher rank features at the initial learning period, more
details in Appendix E.1).
Overall, these results suggest that while a lower sharpness or
$\operatorname{\texttt{tr}(F)}$ during the early period of training may be
good for ID generalization, when factoring in the trainable parameters (as
also shown empirically in Table LABEL:tab:ood), a lower initial sharpness or
$\operatorname{\texttt{tr}(F)}$ could lead to worse OOD generalization. This
observation applies strictly to the very early period of training, and the
eventual reduction of sharpness or $\operatorname{\texttt{tr}(F)}$ after the
initial period is still desirable (evident in Figure 1, Figure 2, and work
like SAM [14, 36, 52]). Importantly, while sharpness and
$\operatorname{\texttt{tr}(F)}$ are effective metrics to study the early
period of training, we need to look at their relative trends for OOD
generalization (in fact, also in ID).
Figure 2: Unfreezing parameters at different times affects the learning
dynamics in the early period of training (with $lr_{d}$). Sub-figures (a)-(d):
The $\operatorname{\texttt{tr}(F)}$ when unfreezing parameters at k= [250,750]
steps versus standard training. Sub-figures (e)-(h): The average-case $L2$
sharpness (Eqn. 3) when unfreezing parameters at k=[250,750] steps versus
standard training. Sub-figures (i)-(l): The worst-case $L2$ sharpness (Eqn. 4)
when unfreezing parameters at k=[250,750] steps for ResNet and k=[250,5000]
for VGG, versus standard training. The y-axis is in the log scale and is
normalized between 0 and 1000 for visualization. All figures are with the
default learning rates to the first 2000 training steps, we also observe
similar trends with 0.1*$lr_{d}$ (see Appendix E.4).
### 6.3 Final Solutions and Out-of-Distribution Results
Changing the learning dynamics in the early period of training inevitably
results in different final solutions. For example with CIFAR10 on ResNet, the
largest eigenvalue of the Hessian ($\lambda_{max}$, calculated on a subset of
training data) of final solutions shows a negative correlation with OOD
results (see Appendix C). However, such a negative correlation is not
consistent nor always statistically significant across different setups. Our
results complement the findings in [2], which serve as additional evidence of
the need for developing new robust metrics and further thorough investigation
for OOD generalization.
### 6.4 Learning Dynamics Could Signify the Time Period to Remove
Interventions
While the value of $\operatorname{\texttt{tr}(F)}$ or sharpness during the
initial period of learning (or at the end of learning) may not be indicative
of good OOD generalization in general, they could signify the time to release
parameters for minimal ID decay and better OOD results. When unfreezing
parameters (Figure 2), we observed that in many cases, those metrics
experience a “transient” period (the first 50-100 steps, characterized by
rapid growth or drop of the sharpness or $\operatorname{\texttt{tr}(F)}$),
followed by a ‘stabilization’ phase where the rate of change in metric values
slows down (unless parameters are released).
Notice that the best range of $k$ for overall best ID and OOD results is after
the stabilization of sharpness and $\operatorname{\texttt{tr}(F)}$ (in Figure
1 and Figure 2), but not for too long. For instance in Figure 1 (b), although
the OOD results improved significantly, the ID results deteriorated
drastically after 800-1000 training steps (at least 1 point drop for
CIFAR10/100). This observation leads to the conjecture that the best time to
remove the intervention (i.e. unfreezing parameters in our case) while keeping
reasonable ID results (less than 0.5 points decrease in accuracy) and
achieving better OOD results must satisfy two constraints: 1) after the
initial rapid change of sharpness or $\operatorname{\texttt{tr}(F)}$ (the
transient phase), and 2) not too far into the stabilization phase.
The second constraint is self-evident in Figure 1, as a larger $k$ hurts both
ID and OOD results. To quickly validate the first constraint, we use MNIST to
pick the earliest ending step of the transient period among three metrics
($S^{\rho}_{worst}$, $S^{\rho}_{avg}$ and $\operatorname{\texttt{tr}(F)}$).
Then we experiment with 10 different $k$ values each consecutively (10 steps
apart) that are smaller or greater than that $k$ value. For the smaller $k$s,
we get 98.93/52.72 as the median ID/OOD results, for the larger $k$s, we get
98.91/53.54 as the median ID/OOD results, which validates the constraints. The
stabilization of examined metrics indicates when to introduce new trainable
parameters.
Next, we use a heuristic algorithm that satisfies the above-mentioned
constraints to determine the stabilization time of the three metrics (we first
detect a significant change in metrics, then detect the stabilization point of
the metrics, the algorithm is in Appendix D). The OOD results are then
compared with with ten random $k$ values per dataset ($k\leq 800$) to
determine the winning rate (i.e., the percentage of times where the value
picked by the algorithm is better than a sampled value), and the results are
in Table LABEL:tab:k_by_metric.
Empirically, using such an algorithm is better than doing a random
hyperparameter search the majority of the time. In most cases, the degradation
of ID accuracy is within 0.5 points, except when using the VGG network.
Nonetheless, this further validates that the stabilization of
$S^{\rho}_{worst}$, $S^{\rho}_{avg}$ and $\operatorname{\texttt{tr}(F)}$ could
signify the removal of interventions (in our case gradual unfreezing) to trade
some ID performance for OOD. While $\operatorname{\texttt{tr}(F)}$ shows
better results, there isn’t a clear winning metric for intervention removal
due to: 1) metrics exhibiting high noise during training; and 2) the
determined stabilization points from different metrics collide or are very
close to each other. We will defer the exploration of more sophisticated
algorithms for future work. However, it’s worth noting the existence of an
optimal time that effectively balances good ID and ODD results.
Table 2: Results using the heuristic algorithm (Appendix D) to determine the
$k$ for gradual unfreezing (GU), best OOD results are bolded. The algorithm
can determine the same value of $k$ in different metrics in multiple cases
(hence the same results). WR stands for winning rate (OOD). See Appendix D for
visualization of $k$ overlay on Figure 2.
| MNIST RN18 | CIFAR10 RN18 | CIFAR100 RN18 | WR | CIFAR10 VGG11 | WR
---|---|---|---|---|---|---
Method | ID / OOD | ID / OOD | ID / OOD | - | ID / OOD |
Standard | 99.06/33.36 | 93.32/72.36 | 71.07/45.10 | - | 88.62/71.63 | -
GU${}_{S^{\rho}_{worst}}$ | 98.78/52.48 | 93.06/72.75 | 70.68/45.19 | 60% | 87.69/71.47 | 40%
GU${}_{S^{\rho}_{avg}}$ | 98.78/52.48 | 93.02/72.58 | 70.67/45.35 | 60% | 87.71/72.37 | 100%
GU${}_{\operatorname{\texttt{tr}(F)}}$ | 98.91/54.12 | 93.02/73.56 | 70.78/45.82 | 83% | 88.40/71.86 | 60%
## 7 Generality of Findings on the Early Period of Training
### 7.1 Higher Initial Sharpness via Learning Rate
Figure 3: The $S^{\rho}_{avg}$ profile during the early period of training,
the learning rate switch happens at 200 steps (all parameters trainable).
Using gradual unfreezing is a specific case where high sharpness at the
initial learning period could benefit OOD generalizations. A critical question
is: is there another way to intervene in the early period of training with
higher sharpness, that also positively impacts OOD generalization?
Recall that a higher learning rate typically results in lower sharpness (and a
lower learning rate results in higher sharpness, as indicated in [27]). Based
on our findings in the previous sections, we hypothesize that using a lower
learning rate at the initial period of learning, then switching to a higher
learning rate later (high sharpness to low sharpness), may help OOD
generalization (surprisingly, this is exactly a simple form of learning rate
warm-up!).
To validate, we use the CIFAR10 dataset on ResNet18 with two learning rates:
we initially use 1/10th of the default learning rate, then increase the
learning rate to the default value after $k$ steps (i.e. low-to-high, denoted
as lrl2h, and the reverse is lrh2l. $k$ is determined using random
hyperparameter search). The results are in Table 3 (an example sharpness
profile is in Figure 3, with more details in Appendix E.2); indeed, lrl2h can
provide better OOD results (and without degrading ID results). Further, with
the learning rate switching step $k$ determined using
$\operatorname{\texttt{tr}(F)}$, $S^{\rho}_{avg}$ and $S^{\rho}_{worst}$, the
OOD results are 72.57 and 72.94 (same $k$ for both sharpness metrics)
respectively.
Table 3: Effects of switch learning rates (increasing or decreasing) in the early period of training. Method | ID/OOD | Method | ID/OOD
---|---|---|---
$lr_{d}$ | 93.32/72.36 | $lr_{l2h}$ | 93.35/72.89
0.1*$lr_{d}$ | 91.66/71.14 | $lr_{h2l}$ | 91.58/71.18
### 7.2 Empirical Validations in Transformers
For our complex experimental setup (described in §4), Figure 4 shows the
learning dynamics of XLM-R with MNLI in the early period, withholding
trainable parameters increases the sharpness and
$\operatorname{\texttt{tr}(F)}$, note that in Figure 4 (c) the
$S^{\rho}_{worst}$ value is negative, withholding trainable parameters still
increase the $S^{\rho}_{worst}$ during training based on Eqn. 4 (the learning
dynamics for the SQuAD dataset is in Appendix E.3).
Figure 4: Learning Dynamics of XLM-R with LoRA training with MNLI, y-axis for
figures are in the log scale, the original value sharpness value in (c) is
negative where we take the absolute value before visualization. All values are
normalized between 0 and 1000 for visualization.
Similarly, the results using $\operatorname{\texttt{tr}(F)}$ to determine the
$k$ for unfreezing is in Table 4 ($k$ values are in Appendix D, results with
$S^{\rho}_{worst}$ / $S^{\rho}_{avg}$ are in Table 7 in the Appendix) and the
winning rate is 80%. The ID results are not sacrificed in this experimental
setting, hence further pointing towards that the stabilization of sharpness
and $\operatorname{\texttt{tr}(F)}$ could signify ‘when’ to remove
intervention in the early period of training for better OOD generalization.
Table 4: Cross-lingual transfer results of standard training and using $\operatorname{\texttt{tr}(F)}$ to determine unfreezing interval $k$ for gradual unfreezing (GU). WR stands for winning rate, averaged over 10 randomly sampled $k$ per training dataset. EM is the exact match score. | XQuAD | | MLQA | | XNLI | WR
---|---|---|---|---|---|---
Method | F1- En/X-ling | EM- En/X-ling | F1- X-ling | EM- X-ling | Acc- En/X-ling |
Standard | 82.96/68.72 | 71.39/52.64 | 56.27 | 40.93 | 83.17/71.84 | -
GU${}_{\operatorname{\texttt{tr}(F)}}$ | 83.77/70.70 | 72.33/54.40 | 58.47 | 42.31 | 83.36/72.49 | 80%
## 8 Conclusions
In this work, we investigate the early period of training and its impact on
out-of-distribution generalization. We demonstrate that using gradual
unfreezing to modulate the number of trainable parameters during the early
period of training can affect out-of-distribution generalization in various
settings, including transformers, and reveal that the number of trainable
parameters at a time is an important factor that was missing in the previous
literature. We observe different patterns to previous work in in-distribution
generalization, where higher sharpness and $\operatorname{\texttt{tr}(F)}$
during the early period of training may be beneficial for out-of-distribution
generalization. We reveal that metric values during the early period of
training may not be indicative of the out-of-distribution generalization,
however, they could signify “when” to remove interventions such as gradual
unfreezing (or to increase the learning rate, §7.1) for better results.
The significance of effective training and fine-tuning, along with the growing
emphasis on research into techniques that involve modifying only partial
parameters of the final model, such as freezing parameters (e.g. Adapters [23,
50, 25] or pruning) or network expansions [59, 12, 18] cannot be overstated.
Our empirical investigations highlight the need to develop a more theoretical
understanding of the early period of training and OOD generalization, as well
as the creation of new theoretical metrics for better indication of OOD
generalization.
## 9 Acknowledgement
This work was funded by the German Federal Ministry of Education and Research
(BMBF) under the promotional reference 13N15897 (MISRIK).
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## Appendix A Gradual Unfreezing
Following the notations and algorithm in [41], let $\textrm{FORWARD}(*)$ be
the standard forward pass, and $\textrm{BACKWARD}(*)$ calculates gradients and
performs updates for trainable parameters. The modified gradual unfreezing
algorithm is in Algorithm 1.
In our experiments, we partition the blocks by their natural namespaces, as
the following:
ResNet18: The definition block follows the standard implementation of ResNet,
with an input convolution layer and a batch norm group together as the
additional block. The model parameters are partitioned into 5 blocks, and a
classification head.
VGG11: The definition block follows the standard implementation of VGG, with 8
blocks in total. The classification head consists of 3 linear layers with a
ReLU function in between.
Algorithm 1 Gradual Unfreezing
1:A model’s eventual trainable parameters are partitioned into blocks
$j\in\\{0,\dots,L-1\\}$ parameterized by $\theta_{j}$, with a task-specific
classification head $C$, and an unfreezing interval $k$. A set $\mathcal{S}$
of the indices of parameter blocks to unfreeze.
2:
3:Initialize $C$, $\theta_{j}$ for all $j$
4:$\mathcal{S}\leftarrow\\{C\\}$
5:$j\leftarrow L-1$
6:for $i=1\dots\text{N}$ do
7: Sample a data batch $b\sim D$
8: if $i\mod k==0$ and $i\leq kL$ then
9: $\mathcal{S}\leftarrow\mathcal{S}\cup\\{\theta_{j}\\}$
10: $j\leftarrow j-1$
11: end if
12: $\textrm{FORWARD}(*)$
13: $\textrm{BACKWARD}(\mathcal{S})$
14:end for
XLM-RoBERTa + LoRA: The experiment follows [41]. Each parameter block consists
of 2 sets of LoRA adapters added to the query and value of the backbone
transformer from the same layer. The LoRA parameters are partitioned into 12
blocks, and a classification head, where the classification head and the last
layer of LoRA adapters are trainable initially.
## Appendix B Hyperparameters
For all our experiments, the hyperparameters are listed in Table 5. We use the
default hyperparameters for the AdamW optimizer, except for the learning rate.
All other hyperparameters for the transformer experiments follow [41], and we
use the HuggingFace PEFT [44] implementations of LoRA.
For calculating $S^{\rho}_{worst}$ and $S^{\rho}_{avg}$, we use $L2$ norm and
$\rho=0.01$ with 15 examples. We follow the setup in [2] and use the
implementation with 2048 data points from the training data (un-augmented when
calculating sharpness metrics) for all experiments. We use a batch size of
256, except for SQuAD (the batch size is 32) for calculating all the metrics.
The sharpness and $\operatorname{\texttt{tr}(F)}$ are recorded every 10
batches (steps) for all datasets.
Table 5: Hyperparameters used in all experiments.
| MNIST | CIFAR10 | CIFAR10 | CIFAR100 | SQuAD | MNLI
---|---|---|---|---|---|---
| RN18 | RN18 | VGG11 | RN18 | XLM-R | XLM-R
optimizer | SGD | SGD | SGD | SGD | AdamW | AdamW
lr scheduler | const. | const. | const. | const. | linear | linear
$lr_{d}$ | 0.01 | 0.1 | 0.15 | 0.01 | 0.0005 | 0.0005
batch size | 128 | 128 | 128 | 128 | 32 | 128
training epochs | 10 | 200 | 200 | 200 | 15 | 15
weight decay | 0.01 | 0 | 0 | 0.0005 | 0.01 | 0.01
momentum | 0.9 | 0 | 0 | 0.9 | - | -
LoRA r | - | - | - | - | 8 | 8
LoRA alpha | - | - | - | - | 8 | 8
LoRA dropout | - | - | - | - | 0.2 | 0.2
## Appendix C Properties of the Final Solutions and OOD Results
We plot the final solution’s $\lambda_{max}$ (largest eigenvalue of training
data feature), $S^{\rho}_{worst}$ and $S^{\rho}_{avg}$ against the OOD test
results in Figure 5 respectively.
While in general the sharpness measures and OOD have negative correlations
(i.e. the smaller sharpness values the better, especially $S^{\rho}_{worst}$
has a consistent negative correlation), they are not always statistically
significant (e.g., for MNIST). The learning rate has a big impact on the final
solutions’ sharpness. Furthermore, such as in Figure 5 (c), we can even attain
slightly positive correlations. Our results complement the findings in [2],
which serve as evidence pointing towards the need for developing robust new
metrics and thorough investigation for OOD generalization.
Figure 5: Final feature $\lambda_{max}$, $S^{\rho}_{worst}$, and
$S^{\rho}_{avg}$ versus the OOD test results (coloured by learning rate),
labelled with Kendall’s $\tau$ and p-value.
## Appendix D Algorithm to Determine the Unfreeze Time
To verify our hypothesis, we use a simple algorithm with heuristic to
determine the unfreezing time $k$, Algorithm 2 presents the flow, $\tau$ is 3
or 8 and $\epsilon$ is 0.02 (i.e. the percentage of change in the signal is
within 2%). The algorithm takes $t_{\Delta_{\hat{S}}}$ as the input, which is
the index marking the end of the rapid increase of the signal using a similar
logic.
Algorithm 2 Find Stabilization
1:procedure
find_stabilization_by_mean($\hat{S},t_{\Delta_{\hat{S}}},\tau,\epsilon$)
$\triangleright$ $\hat{S}$ is an array of normalized signal when only the head
is trainable, $t_{\Delta_{\hat{S}}}$ is the index marking the end of the rapid
increasing of the signal, $\tau$ is the window for smoothing the signals,
$\epsilon$ is the threshold in changes of the signal for stabilization.
2: if $t_{\Delta_{\hat{S}}}$ > 0 then
3: $\hat{S}$ = $\hat{S}$[$t_{\Delta_{\hat{S}}}$:]
4: end if
5: $\mu_{\hat{S}}$ = moving_average($\hat{S},\tau$)
6: $\Delta_{\mu_{\hat{S}}}$ = np.abs(np.diff($\mu_{\hat{S}}$))
7: for i, $\delta$ in enumerate($\mu_{\hat{S}}$) do
8: if $\delta\leq\epsilon$ then
9: index = i $\triangleright$ The first time where the change is smaller than
$\tau$.
10: break
11: end if
12: end for
13: if $t_{\Delta_{\hat{S}}}$ > 0 then
14: index = index + $t_{\Delta_{\hat{S}}}$
15: end if
16: return index
17:end procedure
Using the heuristic algorithm, we determine the value $k$ for experiments in
Table 6, where we observe the determined $k$ are very close to each other
except for VGG with CIFAR10 and XLM-R with SQuAD. All the $k$ value are shown
visually in Figure 6, overlaying on top of the learning dynamics.
Table 6: Different $k$ determined by Algorithm 2.
Metric | MNIST RN18 | CIFAR10 RN18 | CIFAR100 RN18 | CIFAR10 VGG11 | SQuAD XLM-R | MNLI XLM-R
---|---|---|---|---|---|---
$S^{\rho}_{worst}$ | 270 | 230 | 260 | 960 | 810 | 780
$S^{\rho}_{avg}$ | 270 | 270 | 250 | 1010 | 1090 | 720
$\operatorname{\texttt{tr}(F)}$ | 210 | 260 | 230 | 250 | 1310 | 790
Table 7 shows the complete results for our complex experimental setup
(described in §4). While all results are better than the standard training,
empirically, $\operatorname{\texttt{tr}(F)}$ is a metric that gives a better
winning rate compared to a random hyperparameter search.
Table 7: Cross-lingual transfer results of standard training and using all 3 metrics to determine the unfreezing interval $k$ for gradual unfreezing (GU). WR stands for winning rate, averaged over 10 randomly sampled $k$ per training dataset. | XQuAD | | MLQA | | XNLI | WR
---|---|---|---|---|---|---
Method | F1- En/X-ling | EM- En/X-ling | F1- X-ling | EM- X-ling | Avg- En/X-ling |
Standard | 82.96/68.72 | 71.39/52.64 | 56.27 | 40.93 | 83.17/71.84 | -
GU${}_{S^{\rho}_{worst}}$ | 83.78/70.09 | 72.10/54.17 | 57.86 | 42.02 | 82.83/72.13 | 45%
GU${}_{S^{\rho}_{avg}}$ | 83.84/70.00 | 72.12/53.69 | 58.10 | 42.03 | 83.03/72.27 | 40%
GU${}_{\operatorname{\texttt{tr}(F)}}$ | 83.77/70.70 | 72.33/54.40 | 58.47 | 42.31 | 83.36/72.49 | 80%
Figure 6: Learning dynamics with determined value of $k$ (vertical line) on
the figure.
## Appendix E Additional Learning Dynamics
### E.1 Ranks
Figure 7 shows the evolution of feature ranks before the classification head
for the first 2000 training steps. We observe standard training typically
starts with a lower feature rank, as the training progresses the feature rank
gradually increases. When withholding parameters from training, feature ranks
are high at the initial period of learning, and as we gradually release
parameters, the feature ranks reduce compared to the initial value.
Figure 7: Change of feature ranks before the classification head. The sudden
decrease in feature ranks is due to unfreezing the trainable parameters.
### E.2 Change Learning Rate from Low to High
We provide an example of $S^{\rho}_{avg}$ during training when switching from
a low learning rate to a high learning rate at $k=200$ in Figure 8. From
Figure 8, we can see the $S^{\rho}_{avg}$ is high initially, then quickly
drops when the learning rate increases at step k. More interestingly,
switching the learning rate induces similar effects as release parameters from
frozen.
Figure 8: Learning Dynamics of CIFAR10 on ResNet18, when training with 0.1*lrd
at the initial 200 steps, then switched to lrd after 200 steps using standard
training.
### E.3 SQuAD
In Figure 9, we present the learning dynamics for XLM-R with SQuAD in the
early period of learning. The learning dynamics show a similar trend as the
SQuAD dataset, the $S^{\rho}_{worst}$ value is also negative, and withholding
trainable parameters increases the $S^{\rho}_{worst}$ during training based on
Eqn. 4 in our main paper.
Figure 9: Learning Dynamics of XLM-R with LoRA training with SQuAD, y-axis for
figures are in the log scale, the original value sharpness value in sub-figure
(c) is negative where we take the absolute value before visualization. All
values are normalized between 0 and 1000 for visualization.
### E.4 1/10th of the Default Learning Rate
Figure 10 shows the learning dynamics in the early period of training using
1/10th of the default learning rate (i.e. 0.1*lrd). The trends are similar to
using the default learning rate.
Figure 10: Unfreezing parameters at different times affect the learning
dynamics in the early period of training. Sub-figures (a)-(c): The
$\operatorname{\texttt{tr}(F)}$ when unfreezing parameters at k= [250,750]
steps versus standard training. Sub-figures (d)-(f): The average-case $L2$
sharpness (Eqn. 3) when unfreezing parameters at k= [250,750] steps versus
standard training. Sub-figures (g)-(i): The worst-case $L2$ sharpness (Eqn. 4)
when unfreezing parameters at k= [250,750] steps versus standard training. The
y-axis is in the log scale.
|
# Attention is all you need for boosting graph convolutional neural network
Yinwei Wu The final year project work was carried out under the 3+1+1
Educational Framework at the National University of Singapore (Chongqing)
Research Institute. College of Software Engineering, Sichuan University,
Chengdu, China
###### Abstract
Graph Convolutional Neural Networks (GCNs) possess strong capabilities for
processing graph data in non-grid domains. They can capture the topological
logical structure and node features in graphs and integrate them into nodes’
final representations. GCNs have been extensively studied in various fields,
such as recommendation systems, social networks, and protein molecular
structures. With the increasing application of graph neural networks, research
has focused on improving their performance while compressing their size. In
this work, a plug-in module named Graph Knowledge Enhancement and Distillation
Module (GKEDM) is proposed. GKEDM can enhance node representations and improve
the performance of GCNs by extracting and aggregating graph information via
multi-head attention mechanism. Furthermore, GKEDM can serve as an auxiliary
transferor for knowledge distillation. With a specially designed attention
distillation method, GKEDM can distill the knowledge of large teacher models
into high-performance and compact student models. Experiments on multiple
datasets demonstrate that GKEDM can significantly improve the performance of
various GCNs with minimal overhead. Furthermore, it can efficiently transfer
distilled knowledge from large teacher networks to small student networks via
attention distillation.
## Acknowledgement
This work was carried out under the 3+1+1 Educational Framework at the
National University of Singapore(Chongqing) Research Insitute. The author is
honored to have the support of Prof. Wang Xinchao from the National University
of Singapore, Mr. Wang Xu from Sichuan University and Mr. Jing Yongcheng from
University of Sydney.
###### Contents
1. 1 Introduction
2. 2 Related work
1. 2.1 Graph convolutional neural network
2. 2.2 Graph Neural network knowledge distillation
3. 2.3 Attention mechanism
3. 3 Background
1. 3.1 Notations
2. 3.2 Graph nerual network and Graph representation learning
3. 3.3 Multi-headed self-Attention mechanism
4. 3.4 Knowledge distillation
4. 4 Methods
1. 4.1 Graph knowledge enhancement module
1. 4.1.1 Reason
2. 4.1.2 Method
2. 4.2 Graph knowledge distillation module
1. 4.2.1 Reason
2. 4.2.2 Method
5. 5 Experiments
1. 5.1 Experimental setup
1. 5.1.1 Dataset
2. 5.1.2 Types of GCNs
3. 5.1.3 Experiment environment
2. 5.2 The enhancement effect of GKEDM for GCN (RQ1)
3. 5.3 Analysis of additional parameters GKEDM introduced (RQ2)
4. 5.4 GKEDM Attention Map Distillation (RQ3)
5. 5.5 Comparison of GKEDM distillation and related work (RQ4)
6. 6 Shortcomings and future work
7. 7 Conclusion
###### List of Figures
1. 1 The existence of over-smoothing in graph neural networks
2. 2 GKEDM knowledge enhancement module:GKEDM knowledge enhancement module consists of two phases. In the first phase, GKEDM trains a GCN. In the second stage, GKEDM extracts the trained GCN’s backbone and concats it with the GKEDM knowledge enhancement module for fine-tuning.
3. 3 Knowledge distillation module of GKEDM:The GKEDM knowledge distillation module will drive the student network to mimic the topology of the teacher network.
4. 4 Relationship between $\alpha$ setting and distillation effect
###### List of Tables
1. 1 Notations Table
2. 2 Dataset statistics
3. 3 :The effect of GCNII after GKEDM enhancement
4. 4 :The effect of MoNet after GKEDM enhancement
5. 5 :The effect of GRAPHSAGE after GKEDM enhancement
6. 6 :Comparison of number of Param. and Acc.
7. 7 :Knowledge distillation effect
8. 8 :Relationship between $\alpha$ setting and distillation effect
9. 9 :Comparison of GKEDM distillation and related work
## 1 Introduction
Graph Convolutional Neural Networks (GCNs) are powerful representation
learning tools for graph data in non-grid domains. GCNs can learn graph data
structures through generalization of convolution, which is widely used in
computer vision domains. Graph neural networks have made influential
achievements in social networks, recommendation systems, biomedical research
and other domains. Message-passing based GCNs are the prevailing architecture
currently. This type of GCN leverages a message passing algorithm on the graph
to capture node interaction and aggregates node features hierarchically to
extract global features. Through message-passing, nodes obtain information
about themselves and neighbors to update their representations. Message-
passing algorithms typically involve neighbor aggregation, non-linear
activations, and techniques like convolution, pooling, and residual for
complex feature transformations and model optimization. GCNs based on message
propagation mechanism stack multiple graph convolutional layers to achieve a
larger receptive field, enabling better topology capturing of the graph.
However, One major challenge hindering the advancement of GCNs is the issue of
over-smoothing. This refers to a phenomenon in which node representation in a
multi-layer GCN gradually becomes indistinct or overly smoothed as they pass
through layers, making it challenging to distinguish between them. The root
cause of this problem is that in each layer, representations are updated by
averaging or weighted averaging with neighboring nodes’ representations, which
can result in a loss or muddling of information and cause node features to
become increasingly similar. This limitation hinders GCNs from adding extra
layers and therefore restricts their potential to become more expressive
models.
with the deepening of the research, the problem of over-smoothing has
gradually been alleviated [1][2]. As the over-smoothing problem was
alleviated, researchers began gradually increasing the number of parameters of
the GCNs in order to boost their expressive capabilities, just as in the image
and text domains. This trend makes it a challenge to deploy graph neural
networks under resource and time constraints.
Knowledge distillation was proposed by [3] in 2015, and has now become the
mainstream method of model compression. Knowledge distillation provides
additional supervision signals for the training of the student network to
improve its performance. The additional supervision signals come from a
trained teacher model with a large number of parameters, which extracted the
hidden distribution of the complex training data. Initially, knowledge
distillation relied solely on soft target probabilities imparted by teachers
to facilitate the learning of student networks. Recently, to further
investigate the potential of knowledge distillation, many studies are no
longer confined to the soft target probabilities. Instead, they also utilize
the intermediate layers of the model for distillation [4][5].
Knowledge distillation in traditional neural networks for image and text data
has been extensively studied. However, research on distilling knowledge in
graph neural networks is still in its infancy [6][7][8]. Graph data contains
unique information, such as its topological structure, and is distinct from
grid data. To improve distillation efficiency, careful selection and design of
the information to be distilled is necessary.
In this work, a novel plug-in module for GCNs named GKEDM (Graph Knowledge
Enchance and Distillation Module), is proposed. GKEDM is capable of replacing
the output layer of GCN. Its knowledge enhancement module employs an attention
mechanism for local topology information aggregation to boost node
representation, thus improving network performance.
Simultaneously, to alleviate the computational burden resulting from
parameters, a custom attention graph knowledge distillation method that
suitable for GKEDM modules is introduced. This method enables the teacher
network to transfer local topology to the student model, achieving the goal of
distillation.
To demonstrate the universality of GKEDM, I performed enhancement and
knowledge distillation on different kinds of GCN. Experimental results across
multiple datasets demonstrate that GKEDM significantly enhances the
capabilities of graph neural networks and optimizes the efficiency of
knowledge distillation.
The contributions of this paper are:
* •
A plug-in GCN enhancement module based on graph local topology attention is
proposed, named GKEDM. It can be directly applied to GCNs and improve its
performance.
* •
A novel knowledge distillation method suitable for GKEDM is introduced, which
can effectively transfer the topology knowledge of GCN from teacher network to
student network.
* •
The experimental results show that the GKEDM is versatile and effective in
various GCNs and datasets.
## 2 Related work
This work is related to graph convolutional neural networks, knowledge
distillation on graph convolutional neural networks, and attention mechanisms.
### 2.1 Graph convolutional neural network
In recent years, the domain of graph neural networks(GNN) has gained
significant research attention owing to its powerful processing capabilities
on non-grid data. After the convolution operation on the graph was defined
[9], an increasing number of convolution-based approaches are being used to
enhance the representational capacity of GNNs. That specific type of GNN is
referred to as GCN. Graph Attention Network(GAT) [10] is an attention
mechanism-based GCN designed for processing graph data. Compared to other
GCNs, GAT utilizes a trainable inter-node attention mechanism to evaluate the
relevance of each node to the target node. It then performs weighted
aggregation of representations of neighboring nodes to better capture the
local topology. his attention mechanism adapts to the representations of
neighboring nodes, making GAT more flexible and comprehensible while
processing graph data. GraphSAGE [11] leverages self-supervised learning to
update a node’s embedding based on its neighbors, providing flexibility in
handling diverse graph data types.With neighbor sampling, it can efficiently
process large-scale graph data. GCNII [12] adopts residule block to prevent
over-smoothing while training, thereby enabling the stacking of more GCN
layers. MONET [13] incorporates a learnable distance-based kernel function
that transforms similarity between nodes into Gaussian Kernel Functions at
multiple scales. This approach retains information at various levels,
resulting in improved node representations. Rather than designing a new
structure of GCN, this work focuses on improving the performance of the
existing GCNs, and performing knowledge distillation and model compression on
them.
### 2.2 Graph Neural network knowledge distillation
The data to be processed by convolutional neural networks(CNN) performing on
grid domain data is often a fixed-size matrix, such as image data. Different
from traditional CNN, the GCNs need to deal with the non-regular graph data
composed of nodes and edges. With additional features that grid domain data
does not have, such as node features, graph topology, communities, etc., GCNs
usually need to be carefully designed. Taking into account the aforementioned
dissimilarities, the knowledge distillation approach for GCNs must also
address the distillation of supplementary graph data structure captured. GCN
knowledge distillation was first proposed in [6]. Yang et al. captured the
realationship between nodes by introducing an LSP(Local Structure preserving)
module to achieve local topology distillation in graphs. More specifically,
LSP uses the distance of the node representation in the kernel space to
measure the similarity between nodes, and regards the similarity of the first-
order neighborhood of nodes as a distribution. By letting the student network
mimic the node neighborhood distribution of the teacher network, the purpose
of distillation topology knowledge can be achieved. Subsequently, other
methods of graph neural network knowledge distillation were proposed. In [8],
He et al. used an adversarial way to perform model compression, named
GraphAKD. In GraphAKD, the student network as a generator needs to produce
output similar to the teacher network to fool the discriminator, and the
discriminator needs to distinguish between the output of the student and the
output of the teacher. Contrastive learning distillation has been proposed in
the image domain [14], and contrastive learning in the graph neural network
domain has also been used as a way of representation learning [15] to prove
effective. CKJ et al. [7] combined graph neural network distillation and
contrastive learning to propose G-CRD. Through contrastive learning, G-CRD
makes the representation of the student network node approximate the
representation of the corresponding node of the teacher network, and at the
same time, it is far away from the representation of other nodes of the
teacher network, implicitly preserving the relationship between nodes, and
achieving the purpose of topology distillation. The knowledge distillation
approach introduced in this research differs from previous approaches, as it
focuses on a specific architecture for graph neural networks with GKEDM
enhancement.
### 2.3 Attention mechanism
Attention mechanism (AM) is an important technique for enhancing the
expressive ability of neutral network models. Attention allows for varying
weights to be assigned to different segments of input data. This enhances
neural networks’ ability to manage complex data structures and multiple
sources of information. The AM is actually reflected in the very early gate
mechanism, but it received extensive attention after the Transformer [16]
framework came out. Compared to traditional convolution, AM has stronger
relationship capturing abilities and significant potential applications in the
era of big data. Transformer was first applied to the natural language
processing domain, and then migrated to the image domain [17], which has also
proved to be very effective. All the current giant models with amazing results
are basically evolved based on Transformer architecture.
In the graph neural network domain, the attention mechanism was first used in
GAT [10]. Unlike Transformer, the GAT does not use the framework of Query,
Key, and Value, but the idea based on attention still achieves good results.
Since the GAT is implemented based on the message passing mechanism, there can
be problems with over-smoothing when training large deep models. In order to
solve the oversmoothing problem and design a Transformer suitable for graph
data, Ying et al. abandoned the architecture based on the message passing
mechanism, and carefully designed a very important feature in the Transformer
in the proposed Graphormer [2], which is positional encoding. Positional
encoding has been proposed in the Transformer of the original version for
natural language processing. Since Transformer does not have explicit position
information, and order information is crucial in text sequences, a mechanism
is needed to represent the position of each input. To solve this problem,
Transformer added positional encoding to explicitly identify the position
information of the input for the model. Positional encoding has been proved to
have a significant influence on the results in the design of Transformer, so
subsequent Transformer have also adopted a variety of positional encodings
suitable for different data structures. Graphormer can excel in graph
structures by creating appropriate positional encodings to denote the roles,
locations, and other characteristics of nodes within the graph. However,
Graphormer, as a derivative architecture of Transformer, also has the most
important problem, that is computational complexity. And because of the
particularity of the graph structure, this problem becomes more serious.Each
node must query and compute with all nodes in the graph, resulting in a
computational complexity of $O(n^{2})$. This challenge presents a major
obstacle for the application of Graphormer to graph datasets containing tens
of thousands of nodes.
This work employs an attention mechanism that leverages the strengths of GAT
and Graphormer and incorporates a self-attention mechanism based on local
structure, thereby reducing computational load. This work incorporates a self-
attention mechanism utilizing Query, Key, and Value architectures, while also
implementing positional encoding based on the graph Laplacian matrix and
random walk to enhance the model’s expressive capability.
## 3 Background
The purpose of this work is to enhance and distill GCNs based on attention
mechanism. This section first introduces some basic principles and algorithms
of graph representation learning based on message passing mechanism, and is
followed by an introduction related to the self-attention mechanism and
knowledge distillation.
### 3.1 Notations
Due to the large number of symbols involved, this subsection provides a
comprehensive listing of all mathematical symbolic representations employed in
this work, as well as their corresponding descriptions, for the sake of
clarity and convenience. The notations are shown in Table 1:
Notation | Description
---|---
$G=(\mathcal{V},E)$ | A graph $G$ containing the set of edges $E$ and the set of nodes $\mathcal{V}$
$v_{n}\in\mathcal{V}$ | Denotes the node $n$
$e_{ij}\in E$ | Denotes the edge connecting nodes $i$ to $j$
$H_{i}^{T},H_{i}^{S}$ | Denotes all the node representations at the output of layer $i$ of the teacher GCN and
all the node representations at the output of layer $i$ of the student,
respectively
$h_{n}^{i}$ | Denotes the representation of node $n$ at layer $i$
$z_{n}$ | Denotes the logits of node $n$ predicted by the GCN
$y_{n}$ | Denotes the label of node $n$
${y_{n}^{pred}}$ | Denotes the prediction result of the network for node $n$
$x_{n}$ | Denotes the initial features of node $n$
$N(v_{n})$ | Denotes 1-hop neighbors of node $n$
$Q,K,V$ | Denotes $Query,Key,Value$ in attention mechanism
$d$ | Denotes the length of the node embedding vector
Table 1: Notations Table
### 3.2 Graph nerual network and Graph representation learning
Graph neural networks (GNNs) are neural networks that can generate node
embeddings based on node and edge features as well as graph topology. These
embeddings can be further used for various downstream tasks. Types of tasks
that GNN can perform include node prediction, graph prediction, edge
prediction, and relationship prediction. This work primarily focuses on node
classification of graph convolutional neural networks (GCN). Therefore, this
section delves into the GCN and its message passing mechanism, which is a
fundamental algorithm for performing classification predictions.
GCN receives graph that composed of edge set $E$ and node set $\mathcal{V}$ as
input. For each node $v_{n}\in\mathcal{V}$ there exists an initial
$d-dimensional$ input feature vector $x_{n}$, and in general, this initial
feature is the input to layer 0 of the GCN. Also, each node has a
corresponding label $y_{n}$, and 1-hop neighbor node set $N(v_{n})$. For the
node classification task, the GCN needs to predict a label ${y_{n}^{pred}}$
for each node based on its feature information and the topology of the graph,
and make ${y_{n}^{pred}}$ and $y_{n}$ as similar as possible in terms of
metrics. A message passing GCN consists of a stack of convolutional layers,
each of which can be viewed as an encoder. In each layer of the GCN, the nodes
embed the topology in their own information by collecting information from
their neighbors. This stage is known as $AGGREGATE$. Then the node enters the
second $UPDATE$ phase to generate and update its own node representation in
the next layer. The above aggregation-update process can be expressed in
mathematical language as the equation (1):
$\displaystyle message_{N(v)}^{i}=AGGREGATE^{(i)}({h_{u}^{i}|\forall u\in
N(v)})$ (1) $\displaystyle
h_{v}^{i+1}=UPDATE^{(i)}(h_{v}{{}^{i}},message_{N(v)}^{i})$
In the above equation, $AGGREGATE$ is an aggregation function with
permutation-invariant, $message$ is the information of the local neighborhood
that the node aggregated, and $h_{v}^{i+1}$ is the input representation of the
node in the next layer, specifically, $h_{v}^{0}=x_{v}$. After a total of $i$
layers of message passing convolutional layers are stacked, the node can
aggregate the information of the nodes in the $i$-order neighborhood and thus
has a more powerful representation. For the node classification task, the GCN
ultimately passes the node’s final representation to a linear classifier, such
as a multilayer perceptron (MLP). This step calculates the probability
distribution $z_{v}$ for each class and determines the loss $L_{CE}$ using the
cross-entropy loss function as stated in Eq. (2):
$\displaystyle z_{v}$ $\displaystyle=MLP(h_{v}^{i})$ (2) $\displaystyle
L_{CE}$ $\displaystyle=H(y_{v},z_{v})$
$H$ denotes the cross-entropy loss function. After the training is completed
by gradient descent, the prediction results of the nodes can be obtained by
$softmax(z_{v})$.
### 3.3 Multi-headed self-Attention mechanism
Self-Attention mechanism based on $Key,Query,Value$ has received extensive
attention after being proposed in Transformer[16]. Its main principle is to
map the set of input feature vectors $H\in\mathbb{R}^{n\times d}$ to three new
sets of features located in different expression spaces ,
$K\in\mathbb{R}^{n\times d},Q\in\mathbb{R}^{n\times d},V\in\mathbb{R}^{n\times
d}$, by a function as shown in Eq. (3):
$\displaystyle K=f_{k}(H),$ (3) $\displaystyle Q=f_{q}(H),$ $\displaystyle
V=f_{v}(H),$
For any two input nodes, the similarity is computed by point multiplication
using the $Query$ features of the source node and the $Key$ features of the
target node. Once the similarity between the target node and all the source
nodes is obtained, the target node weights and sums the value features of
these source nodes based on the calculated similarity to produce its own new
representation. The mathematical expression is shown in the Eq. (4):
$\displaystyle Attention(Q,K,V)=Softmax(QK^{T}/\sqrt{d})V,$ (4)
$QK^{T}$ is a $n\times n$ matrix that represents the similarity between two
two nodes, and $\sqrt{d}$ is a normalization factor. The final resulting
$Attention(Q,K,V)$ matrix has the same shape as $V$, so that the Transformer
can improve the expressiveness of the model by stacking multiple identical
attention layers.
In order to enable the model to observe more representation space, Multi-head
Self Attention (MSA) is one of the main extensions to enhance the
representational power of Transformer models. The MSA is based on the idea of
splitting, where the input vector is partitioned into multiple subvectors and
each subvector is assigned a separate attention head. Each attention header
contains an independent set of attention weight matrices for computing the
self-attention. Finally, the self-attention results of all heads are
aggregated to form the final output vector. For a MSA using $k$ heads, the
model will use $K$ sets of different functions to obtain $K$ sets $Q,K,V$ and
perform the self-attentive mechanism separately. Each head will provide a
matrix in the shape of $V$. Finally, for all the results of self-attention, a
linear mapping or averaging can be used to obtain the output. This approach
can enhance the performance of the Transformer to some extent.
### 3.4 Knowledge distillation
Knowledge distillation is a technique for enhancing the performance of student
models through the transfer of advanced and complex knowledge from large and
high-performance teacher models. By feeding the output of the teacher model as
a supervised signal to the student model, the student model can leverage the
experience and knowledge of the teacher model to enhance its ability to learn
the distinguishing features and patterns of the task, ultimately improving
overall performance. The knowledge distillation first proposed in [3] by
having the student output label probabilities to match the teacher output soft
label probabilities, with a specific loss function formula of Eq. (5):
$\displaystyle L_{KD}=H(softmax(z^{T}/T_{1}),softmax(z^{S}/T_{1}))$ (5)
where $H$ denotes the cross-entropy loss or KL divergence, and $T_{1}$ denotes
the distillation temperature, which allows for smoother labels for
distillation. $z^{T}$ and $z^{S}$ are the labeling probabilities of teachers
and students, respectively.
## 4 Methods
In this section, we will briefly introduce the motivation for proposing GKEDM
and present the principles and specific algorithms of GKEDM.
### 4.1 Graph knowledge enhancement module
#### 4.1.1 Reason
Message passing based GCN embeds topology into the representation vector of
nodes by aggregation of multiple convolutional layers. However, overstacking
the number of convolutional layers can cause over-smoothing, i.e., the
representation of all nodes tends to be the same and cannot be differentiated.
One of the reasons for this phenomenon is that GCNs indiscriminately aggregate
information from node neighbors, and some information from important neighbors
may be underestimated, while some noisy information from completely unrelated
neighbors may be amplified. Over-smoothing phenomenon becomes more and more
significant as the number of layers of the GCN increases, which in turn leads
to degradation of the model performance.
From the figure (1), it can be seen that in a variety of GCNs implemented
based on the message passing mechanism, the prediction performance reaches
saturation and declines as the number of layers increases.Therefore, it is
more common to build shallow graph GCNs with only 2-3 layers. While specially
designed graph GCNs, such as GCNII, address the over-smoothing problem by
utilizing a residual connection structure similar to that in image domains,
the indiscriminate aggregation of neighboring node information still adversely
impacts model performance to a certain extent. The additional computational
effort required by this approach cannot be ignored. It is crucial to find ways
to improve the performance of GCNs while controlling the amount of parameters.
Figure 1: The existence of over-smoothing in graph neural networks
The MSA, which is widely used in Transformer, can calculate the similarity of
two nodes’ representations, and then selectively choose the information passed
by neighbors for weighted reception. By selectively receiving different
information through the attention mechanism, nodes are able to filter out
useful information and discard noisy information, which is more conducive to
the functional differentiation of nodes, i.e., making different types of nodes
more distinguishable. It is beneficial for downstream node classification
tasks.
However, given the unique characteristics of graph data’s non-grid structure,
applying self-attention mechanisms to graph neural networks requires special
consideration. If applies the attention mechanism to all node pairs and
aggregates the global topology, although it can increase the expressiveness of
the model, the $O(n^{2})$ time complexity will be a huge burden. Therefore,
GKEDM chooses to trade-off the expressiveness and computational overhead by
choosing to aggregate node information in the first-order neighborhood of
nodes with MSA. Unlike GAT, GKEDM adopts the paradigm of Transformer multi-
headed attention mechanism and adds location encoding to enhance the
expressiveness and differentiation of the nodes.
#### 4.1.2 Method
The core goal of GKEDM is to enhance and coalesce node attribute information
and graph topology information on the backbone of various types of GCN
networks that have been trained to obtain a better node representation. the
algorithm flow of the knowledge enhancement part of GKEDM is shown in Figure
3.
Figure 2: GKEDM knowledge enhancement module:GKEDM knowledge enhancement
module consists of two phases. In the first phase, GKEDM trains a GCN. In the
second stage, GKEDM extracts the trained GCN’s backbone and concats it with
the GKEDM knowledge enhancement module for fine-tuning.
GKEDM’s knowledge enhancement module is a two-stage algorithm. In the first
stage, GKEDM first pre-trains a GCN, as shown in the upper part of Figure 3.
We assume that the lower layers of GCN are mainly responsible for learning the
representation of nodes, while at higher layers it is more concerned with
aggregating the topology into the graph node representation. Thus the main
task at this stage is to make the lower convolutional layers of the GCN learn
the basic representation of a node through optimization.
After learning the basic representation of nodes, GKEDM replaces the last
convolutional layer of the GCN with a MSA based layer in order to allow the
network to better aggregate the information of neighboring nodes in the higher
convolutional layers and thus enhance the capture of topology. The MSA based
graph convolutional layer can perform more effective message aggregation.
Instead of choosing to add an additional MSA based layer after the last layer
of the convolutional layer of the trained GCN, GKEDM chose to perform a
replacement, an approach with two considerations:
* •
The excessive stacking of graph convolutional layers may cause information
loss and smoothing of node features. Therefore, reducing the number of layers
of graph convolutional neural network can avoid the over-smoothing phenomenon.
* •
Reducing the number of layers of graph convolution layers can reduce the
computational burden to some extent.
To apply the GKEDM module on the GCN, for a pre-trained GCN with $k+1$ layers,
a node feature matrix $H_{k}\in\mathbb{R}^{n\times d}$ is generated at the
$k$th layer after feeding graph $G$. At this point $H_{k}$ can be considered
to have learned the basic representation of each node. To further
differentiate the nodes and enhance their distinguishability, thus
facilitating the downstream node classification task, GKEDM computes a
positional encoding (PE) for each node. PE has proven to be very effective for
applications in Transformer in the image and text domains. The principle of
its action is to allow the network to distinguish the positional information
of the input. In graph, nodes may have different responsibilities,
communities, etc., and PE can provide the network with effective priori by
using this information. GKEDM uses the Laplace position code [18]. The
Laplacian matrix is a very important mathematical tool to measure the graph
properties and it can represent the location characteristics of the nodes in
the graph. Laplacian PE is done by extracting the top $m$ smallest non-
repeating eigenvalues in the graph Laplacian matrix as the eigenvectors for
location coding. After combining the above position encoding, the new input
$\hat{H_{k}}$ of GKEDM is computed as Eq. (6):
$\displaystyle\hat{H_{k}}=H_{k}+f(PE(G))$ (6)
$PE$ is the Laplace PE, and $f$ is a linear mapping layer that maps the PE to
the $d$-dimension node hidden vector. After obtaining $\hat{H_{k}}$, the MSA
layer is computed using Eq. (3) and Eq. (4) to get $Attention$, and residual
connection is used to stabilize the performance.
$\displaystyle K$ $\displaystyle=f_{k}(\hat{H_{k}}),$ (7) $\displaystyle Q$
$\displaystyle=f_{q}(\hat{H_{k}}),$ $\displaystyle V$
$\displaystyle=f_{v}(\hat{H_{k}}),$ $\displaystyle A$
$\displaystyle=Softmax(QK^{T}/\sqrt{d})$ $\displaystyle H_{k+1}$
$\displaystyle=AV+\hat{H_{k}},$
Finally, the GKEDM can be optimized by the loss function of Eq. (2) and the
stochastic gradient descent algorithm.
### 4.2 Graph knowledge distillation module
#### 4.2.1 Reason
Although the knowledge enhancement module of GKEDM is able to improve the
performance of the GCN through MSA, it introduces additional parameters
proportional to the length $d$ of the hidden vector of the node
representations due to the computation of $Q,K,V$. When $d$ is large, adding
GKEDM for knowledge enhancement introduces a large computational burden.
Knowledge distillation is a common model compression technique applied in
neural networks. Since the classification performance of GCN has been greatly
improved by GKEDM, knowledge distillation can be performed at the cost of a
small performance loss, and a trade-off between computational speed and
accuracy can be made to achieve both improved accuracy and reduced model size.
#### 4.2.2 Method
There’s some research on how to apply knowledge distillation to graph neural
networks [6][8][7]. In this work, a customized distillation method for GKEDM,
named _Attention Map Distillation (AMD_), is used, which is different from the
above method. AMD is closer in principle to [6]’s Local Structure Perserving
(LSP) structure, so we first give a brief introduction to the LSP structure.
The LSP structure treats the 1-hop neighborhood of a node as distribution, and
the purpose of its distillation is to train the student network to mimic the
1-hop neighborhood distribution learned by the teacher network. To be more
precise, as GCN goes deeper and incorporates topological information into node
representations, the distribution of node neighborhoods can be illustrated as
the similarity between neighboring node representations and central node
representations in the hidden space. In the LSP, the similarity between nodes
is calculated with three kernel functions:
$\displaystyle K(h_{n}^{i},h_{n}^{j})=\left\\{\begin{aligned}
&((h_{n}^{i})^{\top}h_{n}^{j})^{d},&Poly\\\
&e^{-\frac{1}{2\sigma^{2}}}||h_{n}^{i}-h_{n}^{j}||^{2}&RBF\\\
&(h_{n}^{i})^{\top}h_{n}^{j},&Linear\end{aligned}\right.$ (8)
The kernel function can map the low-dimensional features into the high-
dimensional feature space, thus making the samples that can not be separated
in the low-dimensional space linearly separable in the high-dimensional space.
Among the above three kernel functions, the RBF kernel function is the most
widely used, which can map the samples into an infinite dimensional feature
space with good nonlinear fitting ability and adaptability. LSP uses the KL
divergence to measure the difference in the 1-hop neighborhood distribution
around each node for teachers and students, and uses this as a loss function
for optimization:
$\displaystyle L_{LSP}=\sum_{i\in\mathcal{V}}D_{KL}(\mathop{softmax}_{i,j\in
E}(K((h_{n}^{i})^{S},(h_{n}^{j})^{S}))||\mathop{softmax}_{i,j\in
E}(K((h_{n}^{i})^{T},(h_{n}^{j})^{T})))$ (9)
It can be seen that the LSP uses a predefined kernel function to measure the
similarity of two nodes. It may have some limitations:
* •
The assumption of using kernel functions is that the representations of two
similar nodes in the kernel function space are also similar, which is not
guaranteed. Similar node representations do not necessarily remain similar
after the mapping of kernel functions. The kernel function approach lacks
flexibility.
* •
The teacher network and the student network may have difference in parametric
size, so the student network may limit its own exploration by completely
imitating the structure of the teacher network in the kernel function space.
To this end, in comparison to the LSP structure that uses a predefined
unlearnable kernel function to characterize the similarity between
representations, the AMD proposed in this paper uses learnable parameters to
measure the distance between nodes. AMD improves the effectiveness of
knowledge distillation by training to fit a more suitable measure of node
similarity. A similar approach to attention graph distillation was proposed in
[19], but its role is in the field of natural language processing, while this
paper is the first work to apply attention distillation to graph neural
network distillation.
The knowledge distillation module of GKEDM is shown in the figure:
Figure 3: Knowledge distillation module of GKEDM:The GKEDM knowledge
distillation module will drive the student network to mimic the topology of
the teacher network.
The GCN for knowledge enhancement using GKEDM contains a MSA layers introduced
by GKEDM. and the attention map generated by its MSA layer can be used as
additional supervised information for knowledge distillation on the student
network after the teacher network has completed training. The attention maps
$A^{T}$ and $A^{S}$ of teacher and student respectively can be obtained by the
equation (6), which are representations of the similarity of nodes to their
neighbors, and thus a description of the node neighborhood structure. By
driving the student network to mimic the neighborhood structure of the teacher
network, GKEDM is able to achieve the purpose of attention distillation.
Different from LSP, the attention map of this method is obtained by $Query$,
$key$ computation containing learnable parameters, and gains more flexibility
through the MSA. This approach can overcome the distillation instability
problem of teacher and student networks due to size mismatch. In particular,
we train the teacher and student attention graphs by minimizing the KL
divergence of:
$\displaystyle L_{A}=L_{KL}(A^{T}||A^{S})$ (10)
With attention graph distillation, $Key$ and $Query$ pair information between
teacher network nodes is transferred to students. However, in addition to Key
and Query information, $Value$ are also very important representations of node
descriptions in the MSA. In order to reach a more efficient distillation,
Value-Value relations between teacher node pairs are also delivered as an
additional attentional information for students to learn:
$\displaystyle L_{VR}=L_{KL}((V^{T})^{\top}V^{T}||(V^{S})^{\top}V^{S})$ (11)
$V^{T}$ and $V^{S}$ are derived from the teacher and student, respectively.
The above equation measures the dissimilarity between node pairs by the
$Value-Value$ pairs. It is worth noting that this approach can be applied to
Key-Key pairs and Query-Query pairs, which will be compared in the
experimental section of this paper. The total loss function of the attention
distillation is finally expressed as:
$\displaystyle L_{attention}=L_{A}+L_{VR}$ (12)
And the final loss function of the student network training is the cross-
entropy loss plus the attention loss, which is expressed by the Eq. 13
$\displaystyle L=L_{CE}+\alpha L_{attention}$ (13)
$\alpha$ is the weight of the attention loss function, which indicates the
intensity of attention distillation.
## 5 Experiments
GKEDM is proposed to enhance the GCN for node knowledge extraction and to
compress GCN’s knowledge to a smaller network by distillation. In order to
verify the feasibility and generality of GKEDM, we set up the experiments to
answer the following questions.
RQ1:Can GKEDM work for different types of GCN on datasets of different sizes
and tasks?
RQ2:Does GKEDM’s improvement in performance come at the cost of additional
parameters?
RQ3:Does GKEDM’s attention map distillation really work?
RQ4:How does GKEDM’s attention map distillation compare to other distillation
methods?
In this section, we first introduce the dataset, the GCN and the experimental
environment used, followed by experiments to answer each of the above four
questions.
### 5.1 Experimental setup
#### 5.1.1 Dataset
We conducted experiments on a series of datasets containing different numbers
of graphs and nodes for the node classification task. These datasets include
both multi-class tasks and multi-label tasks. The statistical information of
the datasets is summarized in the table (2). More detailed information is as
follows .
* •
PPI(Protein-Protein Interaction) [20] is a protein interaction dataset. Each
protein is represented as a node and the interactions are represented as
edges. Its contains a total of 24 graphs and is a multi-label classification
task.
* •
The task of the FLICKR [21] dataset is to classify images on the web by their
description and properties.
* •
CORA FULL [22] is an expanded CORA dataset. It contains a graph representing
paper citation relationships, with each node representing a paper. The task of
this dataset is to predict the kind of papers.
Dataset | Number of nodes | Number of node’s feature | Task
---|---|---|---
PPI | 2,372 | 50 | Multi-label(121 labels)
FLICKR | 89,250 | 500 | Multi-class(7 classes)
CORA FULL | 19,793 | 8710 | Multi-class(70 classes)
Table 2: Dataset statistics
#### 5.1.2 Types of GCNs
In addition to experiments on different datasets, to demonstrate that GKEDM
does not work only for specific GCN, we also conducted experiments on
different GCNs:
* •
GCN [9] is the most classical semi-supervised graph neural network model. It
learns node representations by defining convolutional operations on graph data
structures.
* •
GCNII [12] avoids the over-smoothing problem existing in GCNs by residual
connections, and thus can be stacked to higher layers.
* •
GRAPHSAGE [11] is a GCN that learns node representations by sampling and
aggregating the neighbors of nodes.
* •
MONET [13] is a network that can be used on non-Eulerian domains, and is
therefore suitable for the graph. It enhances graph networks by using kernel
functions.
#### 5.1.3 Experiment environment
All experiments were performed on a LINUX system using pytorch framework
version 1.12.1+cu116. The GNN framework is DGL, version 1.0.2+cu116, and the
hardware used is a RTX3090.
### 5.2 The enhancement effect of GKEDM for GCN (RQ1)
To improve the performance of GCNs on node classification tasks, this work
introduces an enhancement module, GKEDM. GKEDM aims at weighting the
aggregated node neighborhood information and updating the basic representation
of nodes by introducing an attention mechanism. However, datasets of different
sizes and tasks may differ in the structure of the graph, and the semantics of
the node features. Also, since the principles of different kinds of GCNs are
different, the node representations they learn may also differ. To verify
whether the attention mechanism can ignore these differences and achieve its
original purpose of aggregating valid information, this experiment uses
different GCNs trained on different datasets and compares the classification
accuracy of the network without GKEDM with that of the network with GKEDM.
Since the PPI dataset is a multi-label dataset, the metric used is the
F1-score. The experimental results are shown in the table (3)(4)(5):
Model | Dataset | Layers | Params(M) | Original Acc. | Acc. with GKEDM | Impv.
---|---|---|---|---|---|---
GCNII | PPI | 16 | 2.14 | 0.82 | 0.97 | 0.15
GCNII | PPI | 16 | 0.54 | 0.55 | 0.85 | 0.30
GCNII | CORA FULL | 1 | 0.28 | 0.64 | 0.85 | 0.21
GCNII | CORA FULL | 8 | 3.3 | 0.70 | 0.89 | 0.19
Table 3: :The effect of GCNII after GKEDM enhancement Model | Dataset | Layers | Params(M) | Original Acc. | Acc. with GKEDM | Impv.
---|---|---|---|---|---|---
MoNet | PPI | 3 | 0.03 | 0.80 | 0.93 | 0.13
MoNet | PPI | 3 | 0.54 | 0.82 | 0.97 | 0.15
MoNet | CORA FULL | 3 | 5.7 | 0.65 | 0.86 | 0.21
MoNet | CORA FULL | 2 | 0.28 | 0.66 | 0.81 | 0.15
Table 4: :The effect of MoNet after GKEDM enhancement Model | Dataset | Layers | Params(M) | Original Acc. | Acc. with GKEDM | Impv.
---|---|---|---|---|---|---
GRAPHSAGE | PPI | 5 | 0.24 | 0.76 | 0.92 | 0.16
GRAPHSAGE | PPI | 3 | 0.03 | 0.66 | 0.85 | 0.19
GRAPHSAGE | CORA FULL | 2 | 0.14 | 0.69 | 0.84 | 0.15
GRAPHSAGE | CORA FULL | 2 | 0.28 | 0.65 | 0.86 | 0.21
Table 5: :The effect of GRAPHSAGE after GKEDM enhancement
From the experimental results, it can be seen that the performance of the GCNs
on the node classification tasks is significantly enhanced after applied
GKEDM, and the accuracy improvement of up to 30% can be achieved. The
enhancement effect is demonstrated across various GCNs and datasets,
highlighting the generality of GKEDM for GCN enhancement. It is worth noting
that, as shown in Table 3 for experiments on GCNII, different degrees of
performance enhancement are observed even for same models with large
differences in the number of layers and parametric quantities. It indicating
that the GKEDM is a general module that works effectively for features of
different scales and semantics.
### 5.3 Analysis of additional parameters GKEDM introduced (RQ2)
For neural networks, increasing the number of parameters can improve the
fitting ability of the model, which is one of the most direct ways to improve
the performance. The additional trainable parameters allow the model to
improve the learning and to capture the information that could not be captured
originally. The knowledge enhancement module of GKEDM uses an MSA layer to
replace the last layer of the original GCN. The attention layer maps node
representations to $Query$, $Key$, $Value$, thus introducing additional
parameters. Although the GKEDM-enhanced model has better performance, the
introduction of additional parameters makes the number of parameters more
significant compared to the original model. Therefore, it is not convincing to
compare only the accuracy of the original model with that of the GKEDM
enhanced model. The purpose of this experiment is to demonstrate that the
improvement of GKEDM for model performance does not rely on the introduction
of additional parameters, but on more efficient parameter utilization. This
experiment uses the number of parameters and accuracy of the model as metrics
for comparison, and the experimental results are shown in Table 6:
Model | Dataset | Total Param.(M) | With GKEDM | Acc.
---|---|---|---|---
GCNII | PPI | 2.14 | No | 0.82
GCNII | PPI | 0.66 | Yes | 0.85
GCNII | CORA FULL | 3.3 | No | 0.64
GCNII | CORA FULL | 0.28 | Yes | 0.85
GRAPHSAGE | PPI | 0.24 | No | 0.76
GRAPHSAGE | PPI | 0.14 | Yes | 0.85
GRAPHSAGE | CORA FULL | 0.29 | No | 0.65
GRAPHSAGE | CORA FULL | 0.14 | Yes | 0.84
MoNet | PPI | 0.54 | No | 0.82
MoNet | PPI | 0.22 | Yes | 0.93
MoNet | CORA FULL | 5.7 | No | 0.65
MoNet | CORA FULL | 0.28 | Yes | 0.84
Table 6: :Comparison of number of Param. and Acc.
As illustrated in Table 6, we performed a comparison of performance and number
of parameters for the same GCN with or without the addition of GKEDM. From the
results, it can be seen that the GCN with GKEDM is able to surpass the
performance of the original GCN with a small number of parameters, which
proves that GKEDM does not rely on number of parameters to improve the
expressive power of the model. From the comparison of MoNet on the dataset
CORA FULL below Table 6, even a small model with only one twentieth of the
number of parameters can outperform a large model in the final performance
through the enhancement of GKEDM. This experiment illustrates the ability of
GKEDM to improve the parameter utilization of GCNs and the effectiveness of
the GKEDM for GCNs enhancement.
### 5.4 GKEDM Attention Map Distillation (RQ3)
The main purpose of this experiment is to verify the effectiveness of GKEDM
attention map distillation. The task of knowledge distillation is mainly to
improve the performance of the small student model by passing the dark
knowledge of the large teacher model as an additional supervisory signal.
Therefore, the number of teacher network participants should be more than the
student network and the performance should be appropriately separated from the
student network. Therefore, this experiment was chosen to be conducted on
FLICKER and PPI where the distillation effect could be seen. Meanwhile, in
order to try the effect of different combinations of attentional collation, we
also tried Key-Relation and Query-Relation in addition to distillation of
Value-Relation. The experimental results are shown in the following figure:
Model | Dataset | Teacher Param.(M) | Teacher Acc. | Student Param.(M) | Student original Acc. | Distill Acc. | Type
---|---|---|---|---|---|---|---
GraphSAGE | PPI | 0.74 | 0.9452 | 0.15 | 0.8624 | 0.8755 | a+v
GraphSAGE | PPI | 0.74 | 0.9452 | 0.15 | 0.8624 | 0.8750 | a+v+q+k
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.7196 | a+v
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.7147 | a+v+q+k
MoNet | PPI | 1.04 | 0.9753 | 0.33 | 0.9264 | 0.9399 | a+v
MoNet | PPI | 1.04 | 0.9753 | 0.33 | 0.9264 | 0.9399 | a+v+q+k
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9281 | a+v
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9302 | a+v+q+k
Table 7: :Knowledge distillation effect
The column $Type$ in table 7 indicates the type of attention distillation
used. $a,v,q,k$ represent the attention map, Value-Value relationship, Query-
Query relationship and Key-Key relationship respectively. From the
experimental results, it can be seen that the student network performance is
improved to some extent by adding attention map distillation. There was no
significant difference between distillation using only attention map and value
relations and distillation using attention graphs and all relations
simultaneously. The above results illustrate the effectiveness of attention
distillation.
To further investigate the effect of the parameter setting of $\alpha$ in Eq.
13 on the distillation effect, we set up a control experiment of attentional
distillation with different $\alpha$. The experiments use GRAPHSAGE, and the
distillation method uses the attention map and Value relations as additional
supervised signals. The experimental results are shown in Table 8 and Figure
4.
$\alpha$ weight | Distillation Imprv.
---|---
0.01 | +0.132
0.05 | +0.172
0.1 | +0.213
0.15 | +0.108
0.2 | +0.046
0.3 | +0.048
0.5 | +0.035
1 | -0.057
Table 8: :Relationship between $\alpha$ setting and distillation effect Figure
4: Relationship between $\alpha$ setting and distillation effect
It can be observed that the attention distillation effect increases with
increasing weight at the beginning and reaches its highest value at
$\alpha=0.1$. Subsequently, as the $\alpha=0.1$ increase, the distillation
effect begins to decline, even impairs the performance of student network.
This demonstrates that attention map distillation is better to be served as an
additional supervisory signal. By constraining the student network to emulate
the local topology of the teacher network during training, it allows the
student network to learn to better aggregate information from topology. When
this constraint is small, the student network can learn the appropriate node
representation according to the scale of its own features. When this
constraint is large, it has a negative impact on the learning of the student
network, where the student nodes are not free to explore the parameter space
and are forced to align to the distribution of the teacher network. However,
there is a scale inconsistency problem between the student model and the
teacher model. Forcing the student model to align to teacher model can hurt
the performance of the student model.
### 5.5 Comparison of GKEDM distillation and related work (RQ4)
In order to further demonstrate the effectiveness of GKEDM distillation, this
experiment compares the effectiveness of distillation using GKEDM with that of
related distillation methods. Other distillation methods specifically compared
are:
* •
KD:It is the first method used to improve the performance of small models by
distilling the dark knowledge of a large teacher model. The core idea is to
use the output probability distribution of the teacher model as a soft label
to guide the learning process of the student model, thus improving the
generalization performance of the student model.
* •
FITNET:FITNET uses an intermediate layer distillation approach. In the
distillation process, the intermediate layer output of the teacher model is
used as the target of the corresponding intermediate layer of the student
model, and the student model learns the knowledge of the teacher model by
minimizing the distance between its intermediate layer output and the
intermediate layer output of the teacher model. When the middle layer feature
size of the student network is different from that of the teacher network, the
student network output size and the teacher network output size can be made to
align by a linear mapping method.
* •
LSP:It is the first method to apply knowledge distillation to graph neural
networks. Through the Local Structure Preservation (LSP) module, the teacher
network is able to guide the student network to learn the topology of graph
data, thus improving its performance.
The experimental results are shown in Table 9. In the KD method, the
hyperparameters of the student network training are set as follows: soft label
distillation weight is 0.8 and hard label weight is 0.2. The parameter
settings of the LSP are referred to the original paper. The weight of LSP loss
is 100, and the kernel method is set as RBF.
Model | Dataset | Teacher Param.(M) | Teacher Acc. | Student Param. (M) | Student original Acc. | Distill Acc. | Type
---|---|---|---|---|---|---|---
GraphSAGE | PPI | 0.74 | 0.9213 | 0.15 | 0.8865 | 0.8527 | KD
GraphSAGE | PPI | 0.74 | 0.9213 | 0.15 | 0.8865 | 0.8823 | LSP
GraphSAGE | PPI | 0.74 | 0.9213 | 0.15 | 0.8865 | 0.8500 | FITNET
GraphSAGE | PPI | 0.74 | 0.9213 | 0.15 | 0.8865 | 0.9019 | a+v+q+k(ours)
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.6977 | KD
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.7049 | LSP
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.6771 | FITNET
GraphSAGE | FLICKR | 0.4 | 0.8183 | 0.2 | 0.6983 | 0.7155 | a+v(ours)
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9166 | KD
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9297 | LSP
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9282 | FITNET
MoNet | PPI | 1.04 | 0.9753 | 0.21 | 0.9023 | 0.9302 | a+v+q+k(ours)
Table 9: :Comparison of GKEDM distillation and related work
From the experimental results, it can be seen that attention distillation
using GKEDM achieved better results than the previous method on different GCNs
and different datasets, which illustrates the importance of attention for
improving the performance of GCNs. Compared with the unlearnable distance
measurement used in LSP, the distance measurement based on learnable
parameters of GKEDM is more flexible and reduces the constraints on the
student model, thus achieving better results.
## 6 Shortcomings and future work
Although GKEDM has achieved good results on various GCNs and datasets, the
following problems remain:
* •
Since GKEDM is still a GCN implemented based on the message passing mechanism,
it also suffers from the problem of over-smoothing when the number of layers
is too large. It has been verified that the performance degradation still
occurs after stacking the attention layers of GKEDM. This problem reduces the
scope of GKEDM applications. The limitation of the number of parameters limits
the effectiveness of GKEDM on large datasets. The performance of GKEDM on
large-scale models and datasets has not been fully tested in this work.
* •
Although GKEDM distillation is effective, the GCN has to be added with GKEDM
module and retrained. For large datasets, this process is very time and
resource consuming. In addition, the distillation process of GKEDM is
relatively complex and is not an end-to-end process.
Therefore, the future work could be:
* •
Investigate the efficacy of GKEDM on large GCNs and large datasets, then
conceive a more universal framework.
* •
Simplifying the distillation process, enables GKEDM to save computational time
and computational resources while maintaining the distillation effect.
## 7 Conclusion
In this work, we propose a knowledge enhancement and distillation module for
GCN, called GKEDM, which aims to improve the performance of GCNs on node
classification tasks and to lightweight them. The enhancement module of GKEDM
enhances the ability of GCNs to learn graph node representations by
introducing a convolutional layer based on an attention mechanism, thus
improving their performance on node classification tasks. Experiments show
that GKEDM can provide performance enhancement for different kinds of GCNs on
different datasets.
To make the model more lightweight through knowledge distillation, we also
propose a distillation module suitable for GKEDM. This module allows the
student network to mimic the neighborhood structure of the teacher network
through attention map distillation. The experimental results show that the
distillation module of GKEDM achieves the best results on several commonly
used node classification task datasets, proving the effectiveness of the
module. We believe that the method can provide useful guidance for broader
research on GCNs and can also serve as an effective solution to improve the
performance of node classification tasks. We believe that the method can
provide useful guidance for broader research on GCNs and can also serve as an
effective solution to improve the performance of GCNs on node classification
tasks. we hope this work can stimulate more research on GCNs and provide
support for broader applications in the future.
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|
e1e-mail<EMAIL_ADDRESS>11institutetext: Department of
Mechanical Science and Bioengineering, Graduate School of Engineering Science,
Osaka University, 1–3 Machikaneyama, Toyonaka, Osaka 560–8531, Japan
We numerically investigate the mechanical and geometrical properties of dense
wet granular particles with irreversible attractive interaction. The shear
modulus exhibits two inflection points as the packing fraction increases, and
the bulk modulus shows a non-monotonic behavior. The coordination number also
exhibits two inflection points. The peak position in the pair correlation
function shifts to a lower value due to attractive interaction. The Voronoi
tessellation of the particle configuration reveals that the probability
density function for the volume of the Voronoi cell broadens as the packing
fraction approaches the jamming point.
# Mechanical and geometrical properties of jammed wet granular materials
Kiwamu Yoshiie1,addr1 and Michio Otsukiaddr1
(Received: date / Accepted: date)
## 1 Introduction
Amorphous materials, such as granular materials, emulsions, foams, and
colloidal suspensions, behave like solids with rigidity when the packing
fraction $\phi$ exceeds a critical volume fraction $\phi_{c}$, while they
exhibit liquid-like behaviors for $\phi<\phi_{c}$. This transition, known as
the jamming transition, has been extensively studied for years liu1998jamming
; liu2010jamming ; van2009jamming ; behringer2018physics .
The mechanical properties of the materials exhibit critical behaviors near the
critical fraction $\phi_{c}$. The pressure $P$, the shear modulus $G$, and the
bulk modulus $B$ for repulsive frictionless particles exhibit power law
scalings as a function of $\phi-\phi_{c}$ ohern2002random ; ohern2003jamming .
Critical scaling laws for the rheological properties have been observed for
systems under steady shear olsson2007critical ; hatano2007criticality ;
hatano2008scaling ; tighe2010model ; otsuki2009critical . Recently, the
nonlinear elasticity of jammed amorphous materials coulais2014shear ;
otsuki2014avalanche ; boschan2016beyond ; otsuki2022softening , the effect of
friction somfai2007critical ; silbert2010jamming ; otsuki2017discontinuous ;
otsuki2021shear , and the frequency dependence of the complex shear modulus
tighe2011relaxations ; dagois2017softening have been studied.
The geometrical properties also change drastically in the vicinity of
$\phi_{c}$. When the packing fraction exceeds $\phi_{c}$, the coordination
number $Z$ of frictionless particles changes from zero to the isostatic value
$Z_{\mathrm{iso}}$. The excess coordination number $Z-Z_{\mathrm{iso}}$
exhibits a power law scaling as a function of $\phi-\phi_{c}$ ohern2002random
; ohern2003jamming . Moreover, in the three-dimensional systems consisting of
frictionless mono-dispersed spheres with diameter $d$, the pair correlation
function $g(r)$ diverges as $r\to d$ at $\phi_{c}$, which indicates that a lot
of particles are on the verge of making contact silbert2006structural .
Earlier studies on the jamming transition have focused on purely repulsive
particles. However, cohesion, such as the capillary force in wet granular
materials, is nonnegligible in various realistic situations. It is well known
that the irreversible capillary force affects the dynamics of granular
materials herminghaus2005dynamics ; strauch2012wet ; herminghaus2013wet ;
mitarai2006wet , which might result from the change in their mechanical and
geometrical properties. Some researchers have studied the rheological
properties of cohesive particles in systems at constant volume
chaudhuri2012inhomogeneous ; gu2014rheology ; irani2014impact ;
irani2016athermal ; irani2019discontinuous or under constant pressure
rognon2008dense ; khamseh2015flow ; yamaguchi2018rheology ; badetti2018shear ;
mandal2021rheology ; vo2020additive ; vo2020evolution ; vo2020role ;
macaulay2021viscosity . The yield stress or the apparent friction coefficient
increases due to the attractive interaction between particles. The attractive
force also causes a gel-like contact network for low $\phi$ head2007well ;
zheng2016shear , shear bands irani2014impact ; irani2016athermal ;
irani2019discontinuous ; singh2014effect , and clusters of particles
yamaguchi2018rheology ; vo2020evolution ; macaulay2019shear ; lois2008jamming
. For two-dimensional particles with a simple reversible attractive force, it
is reported that the transition point $\phi_{c}$ for rigidity decreases as the
strength of cohesion increases koeze2018sticky , and the critical scaling laws
in $G$ and $B$ for repulsive particles are broken Koeze2020Elasticity .
However, the behavior of three-dimensional wet granular materials with the
irreversible capillary force near $\phi_{c}$ remains unclear.
In this study, we numerically investigate the mechanical and geometrical
properties of three-dimensional frictionless wet granular materials. In Sect.
2, we explain our model and setup. Section 3 presents the numerical results on
the mechanical properties. In Sect. 4, we deal with the geometrical
properties. Section 4 consists of three parts. The $\phi$-dependence of the
coordination number is shown in Sect. 4.1. We demonstrate the change in the
pair correlation function for $\Gamma_{s}>0$ in Sect. 4.2. In Sect. 4.3, we
analyze the geometry of the particle configuration using the Voronoi
tessellation. Section 5 presents discussion and conclusions of our results.
The dependence of the critical fraction $\phi_{c}$ on the strength of cohesion
is shown in A. In B, we investigate the effect of the capillary force on
scaling laws for $G$.
## 2 Setup
We consider three-dimensional wet granular materials consisting of $N$
frictionless identical particles in a cubic box of linear size $L$. We apply
oscillatory shear under Lees-Edwards boundary conditions using the SLLOD
method to measure the mechanical properties evans2008non . The equation of
motion is given by
$\displaystyle\dfrac{\differential\bm{r}_{i}}{\differential t}$
$\displaystyle=\dfrac{\bm{p}_{i}}{m}+\dot{\gamma}(t)y_{i}\bm{e}_{x},$ (1)
$\displaystyle\dfrac{\differential\bm{p}_{i}}{\differential t}$
$\displaystyle=\sum_{j\neq
i}F_{ij}\bm{n}_{ij}-\dot{\gamma}(t)p_{i,y}\bm{e}_{x}$ (2)
with the mass $m$, the unit vector along the x-direction $\bm{e}_{x}$, and the
shear rate $\dot{\gamma}(t)$. Here, $\bm{r}_{i}=(x_{i},y_{i},z_{i})$ and
$\bm{p}_{i}=m_{i}(\frac{d}{dt}{\bm{r}}_{i}-\dot{\gamma}(t)y_{i})\bm{e}_{x}$
are the position and peculiar momentum of particle $i$, respectively. The
force between particles $i$ and $j$ is denoted by $F_{ij}$. The force $F_{ij}$
consists of the dissipative force $F_{ij}^{(\mathrm{d})}$ and the static force
$F_{ij}^{(\mathrm{s})}$ as
$\displaystyle F_{ij}=F_{ij}^{(\mathrm{d})}+F_{ij}^{(\mathrm{s})}.$ (3)
The normal unit vector $\bm{n}_{ij}$ is given by
$\bm{n}_{ij}=\bm{r}_{ij}/r_{ij}$ with $\bm{r}_{ij}=\bm{r}_{i}-\bm{r}_{j}$ and
$r_{ij}=|\bm{r}_{ij}|$.
The dissipative force is given by
$\displaystyle F_{ij}^{(\mathrm{d})}$
$\displaystyle=-\xi_{\mathrm{n}}v^{\rm(n)}_{ij}\Theta(\Delta_{ij})$ (4)
with the normal velocity
$v_{ij}^{\rm(n)}=(\frac{d}{dt}\bm{r}_{i}-\frac{d}{dt}\bm{r}_{j})\cdot\bm{n}_{ij}$
and the viscous constant $\xi_{\mathrm{n}}$, the diameter $d$, and the overlap
length $\Delta_{ij}=d-r_{ij}$. The contact between particles is formed at
$\Delta_{ij}=0$ ($r_{ij}=d$). Here, $\Theta(x)$ is the Heaviside step function
satisfying $\Theta(x)=1$ for $x\geq 0$ and $\Theta(x)=0$ otherwise. The static
force $F_{ij}^{(\mathrm{s})}$ consists of the elastic contact force
$F_{ij}^{(\mathrm{e})}$ and the attractive capillary force
$F_{ij}^{(\mathrm{cap})}$ as
$\displaystyle F_{ij}^{(\mathrm{s})}$
$\displaystyle=F_{ij}^{(\mathrm{e})}+F_{ij}^{(\mathrm{cap})}.$ (5)
The elastic contact force is given by
$\displaystyle F_{ij}^{(\mathrm{e})}$
$\displaystyle=k_{\mathrm{n}}\Delta_{ij}\Theta(\Delta_{ij})$ (6)
with the spring constant $k_{\mathrm{n}}$.
Figure 1: Static force $F_{ij}^{(\mathrm{s})}$ against $\Delta_{ij}$. The red
line represents the behavior when the particles are approaching. The blue line
represents the behavior when the particles are separating.
The capillary force $F_{ij}^{(\mathrm{cap})}$ is modeled as roy2016micro ;
roy2017general ; roy2018liquid ; shi2020steady
$\displaystyle
F_{ij}^{(\mathrm{cap})}=\begin{cases}-2\pi\Gamma_{s}d\cos\theta&\mbox{if
}\Delta_{ij}\geq 0,\\\\[4.0pt]
f^{\rm(a)}(\hat{s}_{ij})\Theta\left(\Delta_{ij}+d_{\mathrm{c}}\right)&\mbox{if
}\Delta_{ij}<0,\mbox{separation},\\\\[4.0pt] 0&\mbox{if
}\Delta_{ij}<0,\mbox{approaching}\\\\[4.0pt] \end{cases}$ (7)
with
$\displaystyle
f^{\rm(a)}(\hat{s}_{ij})=\dfrac{-2\pi\Gamma_{s}d\cos\theta}{1+1.05\hat{s}_{ij}+2.5\hat{s}_{ij}^{2}}.$
(8)
Here, $\theta$ is the contact angle, $d_{c}$ is the rapture length of the
capillary bridge, $\hat{s}_{ij}=s_{ij}\sqrt{d/V_{b}}$ is the normalized
separation distance, with the separation distance $s_{ij}=-\Delta_{ij}$. The
surface tension is denoted by $\Gamma_{s}$, which characterizes the strength
of cohesion. The volume of the liquid bridge is denoted by $V_{b}$, which
characterizes the rapture length $d_{c}$ as $d_{c}=(1+\theta/2)V_{b}^{1/3}$
lian1993theoretical ; willett2000capillary . When the particles approach
before the contact, $F_{ij}^{(\mathrm{cap})}$ is zero. After the particles
contact at $\Delta_{ij}=0$, the capillary bridge is formed, i.e., the
attractive force becomes active. When the particles are separating, the force
does not follow the same path; the attractive force is active until the
capillary bridge disappears at $\Delta_{ij}=-d_{c}$.
The static force $F^{(\mathrm{s})}$ given by Eqs. (5)-(8) is irreversible due
to the capillary force, as shown in Fig. 1. When the particles are approaching
before contact, $F^{(\mathrm{s})}$ is zero and follows the red line. After the
capillary bridge is formed at $\Delta_{ij}=0$, the capillary force
$F^{(\mathrm{cap})}$ is active, and $F^{(\mathrm{s})}$ follows the blue line
until the bridge is broken at $\Delta_{ij}=-d_{c}$. As shown in Fig. 1, the
static force becomes zero at $\Delta_{ij}=\delta_{0}$ ($r_{ij}=d-\delta_{0}$)
with $\delta_{0}=2\pi\Gamma_{s}d\cos\theta/k_{\mathrm{n}}>0$, where positive
$F_{ij}^{(\mathrm{e})}$ and negative $F_{ij}^{(\mathrm{cap})}$ cancel each
other out.
The particles are randomly placed with an initial packing fraction
$\phi_{\mathrm{ini}}=0.45$ without any overlap. The system is gradually
compressed until the packing fraction reaches a given value $\phi$. In each
compression step, we increase the packing fraction by $\Delta\phi=0.000025$
with the affine transformation of the particle configuration and the system
size. The particles are relaxed to a mechanical equilibrium state with the
kinetic temperature $T<T_{\mathrm{th}}$ following Eqs. (1) and (2) with
$\dot{\gamma}(t)=0$. Here, the kinetic temperature is given by $T=\sum
m|\bm{v}_{i}|^{2}/(2N)$.
After the compression, we apply the oscillatory shear strain as
$\displaystyle\gamma(t)=\gamma_{0}\sin\omega t$ (9)
for $N_{\mathrm{cyc}}$ cycles. Here, $\gamma_{0}$ and $\omega$ are the strain
amplitude and the angular frequency, respectively, which are set small enough.
In the last cycle, we measure the shear (storage) modulus $G$ as doi1988theory
$\displaystyle G(\phi)=\frac{\omega}{\pi}\int_{0}^{2\pi/\omega}\differential
t\ \sigma_{xy}(\phi,t)\sin\omega t/\gamma_{0}$ (10)
with the shear stress for a given $\phi$
$\displaystyle\sigma_{xy}(\phi,t)=-\dfrac{1}{2L^{3}}\sum_{i}\sum_{i<j}(r_{ij,x}F_{ij,y}+r_{ij,y}F_{ij,x}).$
(11)
We also measure the pressure as
$\displaystyle
P(\phi)=\dfrac{1}{3L^{3}}\sum_{i}\sum_{i<j}\bm{r}_{ij}\cdot\bm{F}_{ij}$ (12)
after the last cycle, and calculate the bulk modulus as
$\displaystyle B(\phi)=\phi\frac{\differential P}{\differential\phi}.$ (13)
Here, we have ignored the kinetic parts of the shear stress and the pressure
because the contact stress is dominant in our dense system da2005rheophysics .
We use $N=3000$, $\gamma_{0}=1.0\times 10^{-5}$, $\omega=1.0\times
10^{-4}\sqrt{k_{\mathrm{n}}/m}$, $N_{\mathrm{cyc}}=100$, and
$T_{\mathrm{th}}=10^{-8}k_{\mathrm{n}}d_{0}^{2}$. We choose $d_{c}=5.0\times
10^{-4}d_{0}$, $V_{b}=7.5\times 10^{-11}d_{0}^{3}$, $\theta=\pi/9$, and
$\Gamma_{s}/k_{\mathrm{n}}=0,~{}3.0\times 10^{-3},~{}1.5\times
10^{-2},~{}3.0\times 10^{-2}$ for the parameters of the capillary force
following ref. roy2017general ; roy2018liquid ; shi2020steady . Here,
$\Gamma_{s}=0$ corresponds to dry particles. We adopt the Adams-Morton and
Adams-Bashforth methods with a time step $\Delta
t=0.005\sqrt{m/k_{\mathrm{n}}}$ for the time evolution of $\bm{r}_{i}$ and
$\bm{p}_{i}$, respectively. We have numerically confirmed that $N$ and
$N_{\mathrm{cyc}}$ are large enough, and $\gamma_{0}$, $\omega$,
$T_{\mathrm{th}}$, and $\Delta t$ are small enough not to influence our
results.
## 3 Mechanical properties
In Fig. 2, we plot the shear modulus $G$ against the volume fraction $\phi$
for various $\Gamma_{s}$. For each $\Gamma_{s}$, $G$ becomes non-zero as the
packing fraction $\phi$ exceeds a critical fraction $\phi_{c}$. As
$\Gamma_{s}$ increases, the critical fraction $\phi_{c}$ decreases, as shown
in A. The shear modulus $G$ increases with $\phi$. For wet particles with
$\Gamma_{s}>0$, there are two inflection points (open symbols) where the
curvature of $G(\phi)$ changes sign with
$\frac{\differential[2]}{\differential\phi^{2}}G(\phi)=0$. The position of the
inflection point with higher $\phi$ is almost independent of $\Gamma_{s}$
($\phi\simeq 0.63$). The inflection point does not exist for dry particles
with $\Gamma_{s}=0$. The inflection points indicate that the simple power law
scaling for repulsive particles ohern2002random ; ohern2003jamming is not
satisfied for wet granular materials. Similar behaviors are reported for two-
dimensional particles with a simple reversible attractive interaction
Koeze2020Elasticity , but only one inflection point exists in the system.
Figure 2: Shear modulus $G$ against $\phi$ for various $\Gamma_{s}$ with
$\phi>\phi_{c}$. Open symbols represent inflection points. Figure 3: Pressure
$P$ against $\phi$ for various $\Gamma_{s}$ with $\phi>\phi_{c}$. The inset
shows $P$ in the vicinity of $\phi_{c}$. The dashed line represents $P=0$.
Figure 3 displays the pressure $P$ against the volume fraction $\phi$ for
various $\Gamma_{s}$ with $\phi>\phi_{c}$. The inset of Fig. 3 shows $P$ near
$\phi_{c}$. For each $\Gamma_{s}$, $P$ increases with $\phi$. The pressure is
positive and almost $0$ even in the vicinity of $\phi_{c}$, as shown in the
inset of Fig. 3. For high $\phi>0.65$, $P$ seems independent of $\Gamma_{s}$.
See B for the relation between $G$ and $P$.
Figure 4: Bulk modulus $B$ against $\phi$ for various $\Gamma_{s}$ with
$\phi>\phi_{c}$.
In Fig. 4, we demonstrate the bulk modulus $B$ against the packing fraction
$\phi$ for various $\Gamma_{s}$ with $\phi>\phi_{c}$. The bulk modulus $B$ is
not a monotonic function of $\phi$ for $\Gamma_{s}>0$, while $B$ for
$\Gamma_{s}=0$ does not exhibit such non-monotonic behavior. As the packing
fraction $\phi$ decreases, $B$ rapidly increases near $\phi_{c}$, which is not
shown in the previous study Koeze2020Elasticity . The non-monotonic behavior
in $B$ of wet granular materials indicates that $B$ does not obey the power
law scaling for dry repulsive particles ohern2002random ; ohern2003jamming .
## 4 Geometrical properties
In this section, we analyze the geometrical properties of wet granular
materials. In Sect. 4.1, we show the $\phi$-dependence of the coordination
number. Section 4.2 demonstrates the change in the pair correlation function
for $\Gamma_{s}>0$. In Sect. 4.3, we discuss the probability density function
for the volume of the Voronoi cell obtained by the Voronoi tessellation.
### 4.1 Coordination number
We plot the excess coordination number $Z-Z_{\mathrm{iso}}$ against the volume
fraction $\phi$ for various $\Gamma_{s}$ with $\phi>\phi_{c}$ in Fig. 5. Here,
we calculate the coordination number $Z$ after the final oscillatory shear as
$\displaystyle Z=2N_{\mathrm{con}}/N,$ (14)
where $N_{\mathrm{con}}$ is the total number of contacts with $F_{ij}\neq 0$.
For three-dimensional frictionless particles, the isostatic value
$Z_{\mathrm{iso}}$ equals $6$ van2009jamming . As $\phi$ increases, so does
$Z-Z_{\mathrm{iso}}$, increasing from $0$. There are two inflection points
(open symbols) in $Z-Z_{\mathrm{iso}}$ for each $\Gamma_{s}$. Their positions
are almost the same as those for $G$ in Fig. 2. The existence of two
inflection points is natural if the relation $G\propto Z-Z_{\mathrm{iso}}$ for
repulsive particles ohern2002random holds for wet granular particles, which
is discussed in B.
Figure 5: Excess coordination number $Z-Z_{\mathrm{iso}}$ against $\phi$ for
various $\Gamma_{s}$ with $\phi>\phi_{c}$. Open symbols represent the
inflection points.
### 4.2 Pair correlation function
Figure 6: Pair correlation function $g(r)$ against $r$ for various $\phi$ with
$\Gamma_{s}/k_{\mathrm{n}}=0$ (a) and $\Gamma_{s}/k_{\mathrm{n}}=3.0\times
10^{-3}$ (b). Dashed and dash-dotted lines represent $r=d$ and
$r=d-\delta_{0}$, respectively. The blue shaded region corresponds to
$r<d-\delta_{0}$. The red shaded region corresponds to $d-\delta_{0}\leq r\leq
d+d_{c}$. Figure 7: Plot of $\Psi_{\mathrm{+}}$ and $\Psi_{\mathrm{-}}$
against $\phi$ for various $\Gamma_{s}$ with $\phi>\phi_{c}$. Circles ,
squares, and triangles represent $\Gamma_{s}=3.0\times 10^{-3}$,
$\Gamma_{s}=1.5\times 10^{-3}$, and $\Gamma_{s}=3.0\times 10^{-4}$
respectively. Open (filled) symbol corresponds to $\Psi_{\mathrm{+}}$
($\Psi_{\mathrm{-}}$).
In Fig. 6, we demonstrate the pair correlation function $g(r)$ against $r$ in
the static state after the final oscillatory shear for various
$\phi>\phi_{c}$. The pair correlation function $g(r)$ is given by
$\displaystyle g(r)=\dfrac{L^{3}}{N^{2}}\left\langle\sum_{i}\sum_{j\neq
i}\delta^{3}(r-r_{ij})\right\rangle.$ (15)
The blue area in Fig. 6 ($r<d-\delta_{0}$) corresponds to the region with the
static force $F_{ij}^{(\mathrm{s})}>0$, while the red area
($d-\delta_{0}<r<d+d_{c}$) represents the region with
$F_{ij}^{(\mathrm{s})}<0$. The first peak of $g(r)$ rapidly increases as
$\phi\to\phi_{c}$. For $\Gamma_{s}/k_{\mathrm{n}}=0$ (Fig. 6 (a)), the peak
position approaches $r=d$ as $\phi\to\phi_{c}$ silbert2006structural . For
$\Gamma_{s}/k_{\mathrm{n}}=3.0\times 10^{-3}$ (Fig. 6 (b)), the peak position
decreases to $r=d-\delta_{0}$, at which $F_{ij}^{(\rm s)}=0$. We should note
that the gel-like structure characterized by power law decay in the structure
factor zheng2016shear is not observed in our simulation with $\phi>\phi_{c}$.
In Fig. 6 (b), $g(r)$ has a large value even in the red region with
$d-\delta_{0}<r<d+d_{c}$ near $\phi_{c}$, which indicates that many contacts
have a negative static force $F_{ij}^{(\mathrm{s})}$. Here, we introduce
$N_{+}$ and $N_{-}$ as the number of contacts with $F_{ij}^{(\mathrm{s})}>0$
and $F_{ij}^{(\mathrm{s})}<0$, respectively, and plot the ratios
$\Psi_{+}=N_{+}/N_{\mathrm{con}}$ and $\Psi_{-}=N_{-}/N_{\mathrm{con}}$ with
the total number of contacts $N_{\mathrm{con}}=N_{+}+N_{-}$ against $\phi$ for
various $\Gamma_{s}>0$ in Fig. 7. Note that
$N_{+}=\int_{0}^{d-\delta_{0}}g(r)\differential r$ and
$N_{-}=\int^{d+d_{c}}_{d-\delta_{0}}g(r)\differential r$ if all pairs with
$r_{ij}<d-\delta_{0}$ are contacting. For high $\phi$, $F_{ij}^{(\mathrm{s})}$
is positive for almost all contacts ($\Psi_{+}=1$ and $\Psi_{-}=0$), which is
consistent with high $P$ for large $\phi$ in Fig. 3. As $\phi$ decreases to
$\phi_{c}$, $\Psi_{+}$ and $\Psi_{-}$ approach $0.5$. This indicates that the
pressure $P$ of wet granular materials decreases as $\phi\to\phi_{c}$ because
the number of contacts with $F_{ij}^{(\mathrm{s})}<0$ increases. This behavior
is different from that of dry particles with purely repulsive interaction,
where the positive contact force decreases to $0$, keeping
$N_{+}=N_{\mathrm{con}}\simeq Z_{\rm iso}N/2$ ($\Psi_{+}=1$).
### 4.3 Voronoi tessellation
The structure of disordered particles has been studied using the Voronoi
tessellation bernal1964bakerian ; finney1970random ; finney1970random2 ;
oger1996voronoi ; jullien1996computer ; yang2002voronoi ; aste2007invariant ;
xu2007analysis . A Voronoi cell associated with each particle contains an
ensemble of points closer to a given sphere center than any other. In ref.
xu2007analysis , it is reported that the probability density function for the
volume of the Voronoi cell changes due to the capillary force, but the
$\phi$-dependence is not investigated.
Figure 8: Probability density function $p(V^{*})$ against $V^{*}$ with
$\Gamma_{s}/k_{\mathrm{n}}=0$ (a) and $\Gamma_{s}/k_{\mathrm{n}}=3.0\times
10^{-3}$ (b) for various $\phi>\phi_{c}$.
We obtain the Voronoi cell from the particle configuration after the final
oscillatory shear using the VORO++ code library rycroft2009voro++ . We define
the volume of the Voronoi cell as $V$ and plot the probability density
functions $p(V^{*})$ of the normalized volume $V^{*}=V/\bar{V}$ with the
average of the volume $\bar{V}=L^{3}/N$ in Fig. 8 for different
$\phi>\phi_{c}$. Note that $\bar{V}$ is related to the packing fraction $\phi$
as $\bar{V}=\pi d^{3}/(6\phi)$. For dry particles
($\Gamma_{s}/k_{\mathrm{n}}=0$), the probability density function is almost
independent of $\phi$ (Fig. 8(a)). This independence might indicate that the
volume of the Voronoi cell is affinely deformed by changing $\phi$. For wet
particles ($\Gamma_{s}/k_{\mathrm{n}}=3.0\times 10^{-3}$), the width of the
probability density functions increases, and the peak position of $p(V^{*})$
is shifted to lower $V^{*}$ as $\phi$ decreases (Fig. 8(b)), which is
consistent with an experiment of wet particles xu2007analysis .
Figure 9: Variance of $V^{*}$ against $\phi$ for various $\Gamma_{s}$ with
$\phi>\phi_{c}$.
In Fig. 9, the variance of the normalized volume $V^{*}$ is plotted against
$\phi$ for different $\Gamma_{s}$ with $\phi>\phi_{c}$. The variances for
different $\Gamma_{s}$ collapse onto a master curve. However, the range of
$\phi$ for each $\Gamma_{s}$ depends on $\phi_{c}$ and broadens as
$\Gamma_{s}$ increases. For $\Gamma_{s}/k_{\mathrm{n}}=0$, the variance is
almost independent of $\phi$, which is consistent with the probability density
function of $V^{*}$ in Fig. 8 (a). However, the variance for $\Gamma_{s}>0$
increases as $\phi$ approaches $\phi_{c}$. This corresponds to the increase in
the width of $p(V^{*})$ in Fig. 8 (b). The packing fraction where the variance
increases is close to the inflection point for $G$ ($\phi\simeq 0.63$) in Fig.
2.
## 5 Conclusion and discussion
We numerically studied the mechanical and geometrical properties of wet
granular materials with $\phi>\phi_{c}$. For $\Gamma_{s}>0$, the shear modulus
$G(\phi)$ has two inflection points, and the bulk modulus $B(\phi)$ exhibits a
non-monotonic behavior. These mechanical properties are qualitatively
different from those for dry particles with $\Gamma_{s}=0$, where $G(\phi)$
and $B(\phi)$ obey simple power law scalings near $\phi_{c}$ ohern2002random ;
ohern2003jamming . The excess coordination number $Z(\phi)-Z_{\mathrm{iso}}$
also has two inflection points. The peak position in the pair correlation
function $g(r)$ becomes lower than the diameter $d$ due to the attractive
capillary force. The probability density function for the volume of the
Voronoi cell broadens as the packing fraction approaches $\phi_{c}$. These
results indicate that the geometrical properties change with the mechanical
properties due to the capillary force.
The breakdown of simple power-law behaviors in $G(\phi)$ and $B(\phi)$ has
been reported in ref. Koeze2020Elasticity for two-dimensional particles with
a simple reversible attractive force. However, it is also reported that $G$
and $B$ satisfy critical scaling laws, including the strength of attraction.
The attractive force in ref. Koeze2020Elasticity differs from the
irreversible capillary force, and the critical scaling laws do not apply to
the mechanical properties shown in this paper because of the two inflection
points for $G(\phi)$ and the non-monotonic behavior in $B(\phi)$, which are
not observed in ref. Koeze2020Elasticity . The critical scaling laws near
$\phi_{c}$ in wet granular materials will be the subject of future study.
In this study, we have neglected contact friction between particles to focus
on the effect of the attractive capillary force. Recent studies have reported
that the contact friction affects the critical behaviors near $\phi_{c}$ for
dry repulsive particles somfai2007critical ; silbert2010jamming ;
otsuki2017discontinuous ; otsuki2021shear . However, it is unclear whether the
friction force changes the mechanical properties of attractive wet particles
shown in this study. Further work is necessary to resolve this issue.
## Acknowledgment
K. Y. and M.O. thank S. Takada, T. Nakamura, and H. Mizuno for helpful
discussions. Numerical computation in this work was conducted at the Yukawa
Institute Computer Facility. We would like to thank Editage (www.editage.com)
for English language editing. K.Y. is partially supported by Leave a Nest Co.,
Ltd., Hosokawa Powder Technology Foundation (Grant No. HPTF20506), and the
Grant-in-Aid for Japan Society for Promotion of Science JSPS Research Fellow
(Grant No. 21J13720). M.O. is partially supported by Scientific Grant-in-Aid
of Japan Society for the Promotion of Science, KAKENHI (Grants No. 19K03670
and No. 21H01006).
## Author contribution statement
K.Y. carried out the numerical simulations. K.Y. and M.O. interpreted the
results and wrote the manuscript.
## Appendix A Critical fraction
This appendix shows the $\Gamma_{s}$-dependence of the critical fraction
$\phi_{c}$. Here, we define the critical fraction $\phi_{c}$ as the packing
fraction where $G$ exceeds a threshold $G_{\mathrm{th}}$ with
$G_{\mathrm{th}}/k_{\mathrm{n}}=1.0\times 10^{-4}$. We have checked that
$\phi_{c}$ does not change if we use a smaller
$G_{\mathrm{th}}/k_{\mathrm{n}}=5.0\times 10^{-5}$. In Fig. 10, we plot the
critical fraction $\phi_{c}$ against $\Gamma_{s}$. The critical fraction
$\phi_{c}$ decreases as $\Gamma_{s}$ increases, which is consistent with the
results of ref. Koeze2020Elasticity .
Figure 10: Critical fraction $\phi_{c}$ against $\Gamma_{s}$.
## Appendix B Critical scaling of $G$
Figure 11: Shear modulus $G$ against $P$ for various $\Gamma_{s}$. The dashed
line represents $G\propto P^{1/2}$.
In this appendix, we numerically investigate the effect of the capillary force
on the scaling laws for $G$ of dry repulsive particles. For frictionless
particles with the linear elastic interaction $F_{ij}^{\mathrm{(e)}}$ given by
Eq. (6), $G$ satisfies $G\propto P^{\alpha}$ with $\alpha\simeq 1/2$ and
$G\propto(Z-Z_{\mathrm{iso}})$ ohern2002random ; ohern2003jamming .
Figure 11 displays $G$ against $P$ for various $\Gamma_{s}$ obtained from the
data in Figs. 2 and 3. For dry particles with $\Gamma_{s}=0$, $G$ almost
satisfies the scaling law $G\propto P^{\alpha}$ with $\alpha=1/2$. For wet
particles with $\Gamma_{s}>0$, there is a region where $G$ behaves as a power
law function with an exponent lower than $1/2$ for
$P/(k_{\mathrm{n}}d^{-1})>10^{4}$. However, the power-law behavior seems to
break down as $P$ decreases, and $G$ seems to become zero at a finite $P$.
In Fig. 12, we show the shear modulus $G$ against the excess coordination
number $Z-Z_{\mathrm{iso}}$ for different $\Gamma_{s}$ obtained from the data
in Figs. 2 and 5. For dry particles with $\Gamma_{s}=0$,
$G\propto(Z-Z_{\mathrm{iso}})$ is satisfied. For $\Gamma_{s}>0$, there is a
region where $G$ is proportional to $Z-Z_{\mathrm{iso}}$, but the scaling
relation is broken for smaller $Z-Z_{\mathrm{iso}}$ near $\phi_{c}$.
Figure 12: Shear modulus $G$ against $Z-Z_{\mathrm{iso}}$ for various
$\Gamma_{s}$. The dashed line represents $G\propto Z-Z_{\mathrm{iso}}$.
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|
remark ∎ example ∎
# Deformation Decomposition versus Energy Decomposition for Chemo- and Poro-
Mechanics
Janel Chua<EMAIL_ADDRESS>Department of Civil and Environmental
Engineering, Carnegie Mellon University Mina Karimi Department of Mechanical
and Civil Engineering, California Institute of Technology Patrick Kozlowski
Department of Civil and Environmental Engineering, Carnegie Mellon University
Mehrdad Massoudi National Energy Technology Laboratory, 626 Cochran Mill Road
Pittsburgh, PA 15236 Santosh Narasimhachary Siemens Corporation Kai Kadau
Siemens Energy Inc George Gazonas DEVCOM Army Research Laboratory, Aberdeen
Proving Ground, 21005 MD, USA Kaushik Dayal Department of Civil and
Environmental Engineering, Carnegie Mellon University Center for Nonlinear
Analysis, Department of Mathematical Sciences, Carnegie Mellon University
Department of Mechanical Engineering, Carnegie Mellon University
###### Abstract
We briefly compare the structure of two classes of popular models used to
describe poro- and chemo- mechanics wherein a fluid phase is transported
within a solid phase. The multiplicative deformation decomposition has been
successfully used to model permanent inelastic shape change in plasticity,
solid-solid phase transformation, and thermal expansion, which has motivated
its application to poro- and chemo- mechanics. However, the energetic
decomposition provides a more transparent structure and advantages, such as to
couple to phase-field fracture, for models of poro- and chemo- mechanics.
††preprint: Journal of Applied Mechanics, Vol. 91, 014501, 2024.
https://doi.org/10.1115/1.4062967
## 1 Introduction
There is significant current interest in modeling problems of fluid transport
in porous media as well as fluid phase transport in solid materials, i.e.,
poro- and chemo- mechanics. The motivations range from modeling hydrogels [1],
to transport in geological structures [2], to hydrogen embrittlement of metals
[3], among various other applications. An approach that has been proposed in
the literature to model poro- and chemo- mechanics is to decompose the
deformation gradient into an elastic part – that causes stress – and an
inelastic part – that accounts for the shape change due to fluid transport;
this is the “multiplicative decomposition”. The application of the
multiplicative decomposition to poro- and chemo- mechanics is motivated by the
success of this strategy in modeling thermoelasticity, plasticity, twinning,
solid-solid phase transformations, and related phenomena that involve
inelastic deformation, e.g. reviewed in [4, 5].
However, an important distinction between thermoelasticity, plasticity,
twinning on the one hand, and poro- and chemo- mechanics on the other hand, is
that the former class of phenomena do not involve the introduction of material
into the bulk of the solid, whereas the latter class do. The introduced
material has energy and stress that is distinct from the energy and stress of
the solid. This motivates an approach that is based on additively combining
the energies of the solid and the fluid, e.g. [1, 6, 7] and many others.
In this note, we briefly contrast the overall structure of these two
approaches, and argue that the additive decomposition of the energy is the
more appealing alternative. We also highlight [8], which critically examined a
model micromechanism for biological growth, and consequently argued against a
multiplicative decomposition in that context.
### Definitions and Notation.
We use $\displaystyle{\mathbold F}$ for the deformation gradient,
$\displaystyle{\mathbold T}$ for the 1st Piola-Kirchoff (P-K) stress, and
$\displaystyle\mu$ for the chemical potential. For simplicity, we assume a
single fluid phase that is defined by the densities in the deformed and the
reference configurations $\displaystyle\rho$ and $\displaystyle\rho_{0}$;
i.e., $\displaystyle\rho$ and $\displaystyle\rho_{0}$ are the mass of the
fluid phase per unit deformed and reference volumes. For simplicity, we follow
the affine deformation assumption that the volume of the fluid phase in the
deformed and reference configurations are related by $\displaystyle
J=\det{\mathbold F}$, implying that $\displaystyle\rho=J^{-1}\rho_{0}$.
The energy density is written in terms of $\displaystyle{\mathbold F}$ and
$\displaystyle\rho_{0}$, rather than $\displaystyle{\mathbold F}$ and
$\displaystyle\rho$, because the former pair of arguments can be independently
varied in a simple way that decouples deformation and transport.
## 2 Multiplicative Deformation Decomposition into Elastic and Inelastic
Parts
The central idea in the multiplicative deformation decomposition is to write
the deformation gradient $\displaystyle{\mathbold F}$ as the product of an
elastic part $\displaystyle{\mathbold F}_{e}$, that causes stress, and an
inelastic part $\displaystyle{\mathbold F}_{i}\left(\rho_{0}\right)$, that is
driven by the coupled field $\displaystyle\rho_{0}$. That is,
$\displaystyle{\mathbold F}={\mathbold F}_{e}{\mathbold F}_{i}$, and the free
energy density is typically of the general form given by:
$W({\mathbold F},\rho_{0})=W_{e}\left({\mathbold F}{\mathbold
F}_{i}^{-1}\left(\rho_{0}\right)\right)+W_{i}(\rho_{0})$ (1)
The elastic energy $\displaystyle W_{e}$ is minimized when
$\displaystyle{\mathbold F}={\mathbold F}_{i}\left(\rho_{0}\right)$ up to
rotations.
The resulting P-K stress has the form:
${\mathbold T}=\frac{\partial W}{\partial{\mathbold F}}=\frac{\partial
W_{e}}{\partial{\mathbold F}_{e}}{\mathbold F}_{i}^{-T}$ (2)
In general, $\displaystyle{\mathbold F}_{i}$ is invertible. Consequently,
$\displaystyle{\mathbold T}={\bf 0}\iff\frac{\partial
W_{e}}{\partial{\mathbold F}_{e}}={\bf 0}$.
The (referential) chemical potential is the key quantity that governs the
transport of the fluid phase. It is defined as the energy-conjugate to
$\displaystyle\rho_{0}$, e.g. [9], and has the form:
$\mu=\frac{\partial W}{\partial\rho_{0}}=\frac{\partial
W_{e}}{\partial{\mathbold F}_{e}}\frac{\ \mathrm{d}{\mathbold F}_{i}^{-T}}{\
\mathrm{d}\rho_{0}}:{\mathbold F}+\frac{\ \mathrm{d}W_{i}}{\
\mathrm{d}\rho_{0}}$ (3)
where $\displaystyle:$ represents a double contraction over 2-nd order
tensors.
We note the key undesirable features of this class of models. Consider a
homogeneous body described by such a model with zero applied traction on the
entire boundary and uniform $\displaystyle\rho_{0}$. A solution to this
boundary-value problem is $\displaystyle{\mathbold T}\equiv{\bf 0}$, implying
that $\displaystyle{\mathbold F}={\mathbold F}_{i}$ on the entire body, up to
a rigid rotation. Hence, even if the solid material is highly deformed due to
fluid infiltration, e.g. due to hydrogen in a metallic lattice with stretched
atomic bonds or due to fluid in a hydrogel with stretched polymer chains, the
elastic energy $\displaystyle W_{e}$ is minimized. While it is possible to
augment the inelastic energy $\displaystyle W_{i}$ to depend on the
deformation, this would not allow the interpretation of the decomposition of
$\displaystyle{\mathbold F}$ as an elastic and inelastic part.
## 3 Additive Energy Decomposition into Solid Strain Energy and Fluid Energy
The central idea in the additive energy decomposition is to additively combine
the energetic contributions of the solid and fluid phases to find the total
free energy. An example of such a form is:
$W({\mathbold F},\rho_{0})=\alpha W_{s}({\mathbold
F})+(1-\alpha)JW_{f}\left(J^{-1}\rho_{0}\right)$ (4)
where $\displaystyle J=\det{\mathbold F}$. The referential volume fraction of
the solid phase is $\displaystyle\alpha$, and we assume a single fluid phase;
for the case with multiple fluids with the possibility of evolving volume
fractions, we refer to [2] and references therein.
The form of the fluid contribution $\displaystyle
JW_{f}\left(J^{-1}\rho_{0}\right)$ is motivated by the requirement that the
energy density $\displaystyle W_{f}$ of a simple fluid depends only on the
density in the deformed state, i.e., $\displaystyle\rho=J^{-1}\rho_{0}$, when
we consider the isothermal setting. Further, the leading factor of
$\displaystyle J$ accounts for the fact that $\displaystyle W_{f}$ is the
energy per unit deformed volume, whereas the hyperelastic energy density
$\displaystyle W$ is per unit reference volume. An important assumption here
is that of affine deformation; i.e., both the solid skeleton and the fluid
volume deform under $\displaystyle{\mathbold F}$ affinely but this can be
relaxed [2].
The resulting P-K stress has the form:
$\begin{split}{\mathbold T}&=\frac{\partial W}{\partial{\mathbold
F}}=\alpha\frac{\partial W_{s}}{\partial{\mathbold
F}}+(1-\alpha)\frac{\partial J}{\partial{\mathbold
F}}\left(W_{f}\left(J^{-1}\rho_{0}\right)-J^{-1}\rho_{0}\frac{\partial
W_{f}}{\partial\rho}\right)=\alpha\frac{\partial W_{s}}{\partial{\mathbold
F}}+(1-\alpha)J{\mathbold
F}^{-T}\left(W_{f}\left(\rho\right)-\rho\frac{\partial
W_{f}}{\partial\rho}\right)\\\ &={\mathbold T}_{s}+(1-\alpha)J{\mathbold
F}^{-T}p\end{split}$ (5)
where we have defined the solid stress $\displaystyle{\mathbold
T}_{s}:=\alpha\frac{\partial W_{s}}{\partial{\mathbold F}}$; used the relation
$\displaystyle\frac{\partial J}{\partial{\mathbold F}}=J{\mathbold F}^{-T}$;
and used the relation that the fluid pressure111 The pressure $\displaystyle
p$ is the derivative of the Helmholtz free energy with respect to volume,
keeping temperature and mass fixed [10]. In terms of the density
$\displaystyle\rho$ – which is inversely proportional to the volume when the
mass is fixed – and in terms of the Helmholtz free energy _density_ , we have:
$p=\frac{\partial\
}{\partial\left(\frac{1}{\rho}\right)}\left(\frac{1}{\rho}W_{f}(\rho)\right)=W_{f}(\rho)-\rho\frac{\partial
W_{f}}{\partial\rho}$ (6) is given by $\displaystyle
p=\left(W_{f}\left(\rho\right)-\rho\frac{\partial
W_{f}}{\partial\rho}\right)$. We can then define the P-K fluid stress
$\displaystyle{\mathbold T}_{f}:=(1-\alpha)J{\mathbold F}^{-T}p$,
corresponding to a Cauchy stress
$\displaystyle\mathbold{\sigma}_{f}=(1-\alpha)p{\mathbold I}$.
The chemical potential for this model has the form:
$\mu=\frac{\partial W}{\partial\rho_{0}}=(1-\alpha)\frac{\partial
W_{f}}{\partial\rho}$ (7)
which corresponds to the standard thermodynamic expression for fluids.
We consider again a homogeneous body with zero applied traction on the entire
boundary and uniform $\displaystyle\rho_{0}$. In this energetic decomposition
model, a solution to this boundary-value problem is that the total stress
$\displaystyle{\mathbold T}\equiv{\bf 0}$, implying that the fluid and solid
stresses $\displaystyle{\mathbold T}_{s}$ and $\displaystyle{\mathbold T}_{f}$
balance each other but neither is necessarily zero. Given a fluid pressure
$\displaystyle p\neq 0$, there will generally be a fluid stress
$\displaystyle{\mathbold T}_{f}\neq{\bf 0}$ which in turn requires a solid
stress $\displaystyle{\mathbold T}_{s}\neq{\bf 0}$. With this state of fluid
and solid stress, $\displaystyle W_{s}$ will not reach its minimum, and the
body will deform. Hence, the deformation of the solid material due to fluid
infiltration, e.g. the stretching of atomic bonds or polymer chains, will be
reflected in the solid stress $\displaystyle{\mathbold T}_{s}$ and energy
$\displaystyle W_{s}$.
## 4 A Remark on Phase-field Fracture Modeling for Poro- and Chemo- Mechanics
The phase-field approach provides a powerful method for modeling fracture,
e.g. [11]. Briefly, a phase-field $\displaystyle\phi$ tracks the level of
damage, with $\displaystyle\phi=1$ denoting the intact undamaged material and
$\displaystyle\phi=0$ denoting the completely damaged or fractured material.
An energetic framework uses an energy density with contributions that include
$\displaystyle\phi^{2}W({\mathbold F})+G_{c}(1-\phi)^{2}$, where the first
term accounts for the elastic energy and the second for the work to fracture,
with $\displaystyle G_{c}$ being the Griffith parameter. This structure of the
energy density sets up a competition between elastic energy and the work to
fracture: minimizing over $\displaystyle\phi$ drives $\displaystyle\phi\to 0$
when the elastic energy becomes larger due to deformation than the work to
fracture. Given this reasoning, it is natural to develop poro- and chemo-
mechanical models of phase-field fracture wherein only the energy of the solid
phase $\displaystyle W_{s}$ contributes to the fracture energetic balance.
That is, in a simple model, we would replace (4) by the expression
$\displaystyle\phi^{2}W_{s}({\mathbold
F})+JW_{f}\left(J^{-1}\rho_{0}\right)+G_{c}(1-\phi)^{2}$ to model fracture
which releases the stress in the solid but does not directly affect the fluid.
### Competing Interest Statement.
The authors have no competing interests to declare.
### Acknowledgments.
We acknowledge financial support from ARO (MURI W911NF-19-1-0245) and NSF
(DMREF 2118945, DMS 2108784); NSF for XSEDE computing resources provided by
Pittsburgh Supercomputing Center; and Noel Walkington and Tony Rollett for
useful discussions. Kaushik Dayal acknowledges an appointment to the National
Energy Technology Laboratory sponsored by the U.S. Department of Energy.
## References
* [1] Wei Hong, Xuanhe Zhao, Jinxiong Zhou, and Zhigang Suo. A theory of coupled diffusion and large deformation in polymeric gels. Journal of the Mechanics and Physics of Solids, 56(5):1779–1793, 2008.
* [2] Mina Karimi, Mehrdad Massoudi, Noel Walkington, Matteo Pozzi, and Kaushik Dayal. Energetic formulation of large-deformation poroelasticity. International Journal for Numerical and Analytical Methods in Geomechanics, 46(5):910–932, 2022.
* [3] Emilio Martínez-Pañeda, Alireza Golahmar, and Christian F Niordson. A phase field formulation for hydrogen assisted cracking. Computer Methods in Applied Mechanics and Engineering, 342:742–761, 2018.
* [4] Vlado A Lubarda. Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev., 57(2):95–108, 2004.
* [5] Souhayl Sadik and Arash Yavari. On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math. Mech. Solids, 22(4):771–772, 2017.
* [6] Virginia von Streng, Rami Abi-Akl, Bianca Giovanardi, and Tal Cohen. Morphogenesis and proportionate growth: A finite element investigation of surface growth with coupled diffusion. Journal of the Mechanics and Physics of Solids, 146:104211, 2021\.
* [7] Timothy J Truster and Arif Masud. A unified mixture formulation for density and volumetric growth of multi-constituent solids in tissue engineering. Computer Methods in Applied Mechanics and Engineering, 314:222–268, 2017.
* [8] Isaac Vikram Chenchiah and Patrick D Shipman. An energy-deformation decomposition for morphoelasticity. Journal of the Mechanics and Physics of Solids, 67:15–39, 2014\.
* [9] Olivier Coussy. Poromechanics. John Wiley & Sons, 2004.
* [10] Morton E Gurtin. An introduction to continuum mechanics. Academic press, 1982.
* [11] John D Clayton and Jarek Knap. A geometrically nonlinear phase field theory of brittle fracture. International Journal of Fracture, 189(2):139–148, 2014.
|
# Multimodal Speech Enhancement Using Burst Propagation
Leandro A. Passos, Ahmed Khubaib, Mohsin Raza, and Ahsan Adeel
University of Wolverhampton, Wolverhampton, England, UK
<EMAIL_ADDRESS>
###### Abstract
This paper proposes the MBURST, a novel multimodal solution for audio-visual
speech enhancements that consider the most recent neurological discoveries
regarding pyramidal cells of the prefrontal cortex and other brain regions.
The so-called burst propagation implements several criteria to address the
credit assignment problem in a more biologically plausible manner: steering
the sign and magnitude of plasticity through feedback, multiplexing the
feedback and feedforward information across layers through different weight
connections, approximating feedback and feedforward connections, and
linearizing the feedback signals. MBURST benefits from such capabilities to
learn correlations between the noisy signal and the visual stimuli, thus
attributing meaning to the speech by amplifying relevant information and
suppressing noise. Experiments conducted over a Grid Corpus and CHiME3-based
dataset show that MBURST can reproduce similar mask reconstructions to the
multimodal backpropagation-based baseline while demonstrating outstanding
energy efficiency management, reducing the neuron firing rates to values up to
$70\%$ lower. Such a feature implies more sustainable implementations,
suitable and desirable for hearing aids or any other similar embedded systems.
###### Index Terms:
Burstpropagation, Multimodal Learning, Audio-Visual Speech Enhancement
## I Introduction
The World Health Organization states that $430$ million people suffer from
moderate to higher hearing loss nowadays and estimates that nearly $2.5$
billion will present hearing impairment to some degree by 2050 [1]. The
problem impacts the individual social relationships and the perception of
surrounding sounds [2], which may also lead to loneliness and psychological
distresses [3, 4]. Therefore, intelligent computer systems that boost and
clean the audio signal are highly desirable.
In this context, many efforts have been applied toward machine learning-based
approaches for speech enhancement [5, 6]. A particularly interesting approach
comprises multimodal approaches that combine correlated audio and visual (AV)
information to amplify relevant information and suppress noise [7]. The idea
can be illustrated by a dialogue in a pub, where people talk and live music is
played in the background, and the listener focuses on reading the speaker’s
lips and expressions to mind the context and infer meaning from the
conversation. The work of Adeel et al. [8] comprises related applications
regarding IoT and 5G for lip-reading hearing aids, encrypted audio speech
reconstruction [9], and speech enhancement in different conditions using deep
learning [10, 11]. Further, the recent work of Passos et al. [12] investigates
audio and visual information fusion using Graph Neural Networks and canonical
correlation analysis, as well as a cortical cell-inspired model that
approximates the computational model to a more biologically plausible approach
[13].
Despite such approximation, the methods mentioned above and most of the
machine and deep learning solutions rely on backpropagation, an algorithm
inspired by an antiquate modeling of neuronal information flow where inputs
are linearly combined and exposed to an activation function, whose output
feeds consecutive layers. Further, the outcome is compared to an expected
target, and the error is back-propagated, assigning the credit for any
mistakes or successes to neurons that are multiple synapses away from the
output and updating the weights associated with such neurons accordingly. In
this scenario, Payeur [14] proposed a novel approach inspired by more recent
studies on the physiological mechanism of pyramidal neurons, where the
learning is regulated by the frequency of bursts, the so-called
Burstpropagation.
Burstpropagation tackles the credit assignment problem in a more biologically
plausible manner by addressing the primary principles of pyramidal neurons
suggested by Körding and König [15] as follows. First, it employs a burst-
dependent learning rule to steer the sign and magnitude of plasticity through
feedback stimuli. Second, it multiplexes the feedback and feedforward
information across multiple layers using different connections, i.e., distinct
weights are used during the forward and the backward propagation processes.
Third, it performs the alignment between feedback and feedforward connections
by approximating the loss-function gradients through burst-dependent learning.
Finally, the fourth principle regards the linearization of the feedback
signals concerning the credit information, which can be performed using
recurrent connections.
Therefore, this paper proposes the so-called MBURST, a multimodal approach
that combines audio and visual information to enhance speech quality using
burst propagation. The model preserves biological properties of real neurons
like short-term synaptic plasticity [16], dendritic excitability [17],
synaptic transmission, inhibitory microcircuit, and burst-dependent synaptic
plasticity [14]. Experiments conducted over a Grid Corpus and CHiME3-based
dataset for clean audio mask reconstruction show that MBURST can generate
clean audio masks with similar quality to a multimodal backpropagation-based
baseline while providing a dramatic reduction in the energy consumption,
reaching values up to $70\%$ lower. Such an attribute is extremely desirable
for real-world applications like embedded systems on hearing aid devices.
Thus, this paper comprises two main contributions:
* •
it provides the MBURST, a biologically plausible and energy-efficient method
that combines audio and visual information for speech enhancement; and
* •
it introduces burst propagation, a state-of-the-art algorithm inspired by
pyramidal neurons, in the context of multimodal learning.
The remainder of this paper is presented as follows. Section II provides a
theoretical background regarding Burstpropagation and the proposed approach.
Further, Sections III and IV describe the methodology employed to conduct the
experiments and the results, respectively. Finally, Section V states the
conclusions and future work.
## II Theoretical Background
This section briefly introduces the concepts behind burst propagation, as well
as the proposed burst propagation-based multimodal approach, namely MBURST.
### II-A Burst Propagation
The burst-dependent algorithm, or burst propagation, defines the weighted sum
at the neuron’s input as “somatic potentials”. Similarly, the neuron’s output
is termed the event rate. The ’somatic potentials’ of a dense layer are
defined as:
$v_{l}=W_{l}e_{l-1},$ (1)
where $l\in\\{1,2,\cdots,L\\}$ denotes the network’s layers, $W_{l}$ is the
weight connecting layer $l-1$ to layer $l$ and $e_{l}$ is the $l$th layer’s
event rate, defined by:
$e_{l}=f(v_{l}),$ (2)
where $f_{l}$ stands for any activation function, e.g., the Rectifier Linear
Unit (ReLU) or Sigmoid, for layer $l$. Regarding convolution layers, the
multiplications are simply replaced by a convolution operator where $W_{l}$
represents layer $l$’s kernel.
Since the burst rate drives the model’s learning procedure, the first step in
the process comprises computing the output layer ($l=L$) bursting probability:
$p_{L}=\zeta(\bar{p}_{L}-h(e_{L})\odot\Delta_{e_{L}}\mathcal{L})$ (3)
where $\bar{p}_{L}$ is a hyperparameter that defines the reference bursting
probability, $\zeta$ is a squashing function that guarantees
$P_{L,i}\in\left[0,1\right]$, $\Delta_{e_{L}}\mathcal{L}$ stands dor the loss
function derivative, and $h(e_{l})$ is a vector-valued function defined by:
$h(e_{l})\equiv f^{{}^{\prime}}(v_{l})\odot e_{l}^{-1}.$ (4)
The bursting probability is used to calculate the bursting rate with and
without teaching, i.e., $b_{l}$ and $\bar{b}_{l}$, respectively, at any layer
$l$:
$\begin{array}[]{c}b_{l}=p_{l}\odot e_{l},\\\ \bar{b}_{l}=\bar{p}_{l}\odot
e_{l}.\end{array}$ (5)
Further, the bursting rates are propagated to the previous layers through
feedback weights, namely $Y$. The result is employed to compute the current
hidden layer’s “dendritic potential” with ($u_{l}$) and without
($\bar{u}_{l}$) teacher. Notice that, for convolutional layers, $Y$ stands for
a kernel, and a convolution operation replaces the multiplication.
$\begin{array}[]{c}u_{l}=h(e_{l})\odot(Y_{l}b_{l+1}),\\\
\bar{u}_{l}=h(e_{l})\odot(Y_{l}\bar{b}_{l+1}).\end{array}$ (6)
In the sequence, the somatic potentials are used to calculate the hidden
layer’s bursting probability and the reference bursting probability:
$\begin{array}[]{c}p_{l}=\sigma(\beta u_{l}+\alpha),\\\
\bar{p}_{l}=\sigma(\beta\bar{u}_{l}+\alpha),\end{array}$ (7)
where $\beta=1$ and $\alpha=0$ are are constants that control the dendritic
transfer function and $\sigma$ is a sigmoid function.
Finally, the change in the forward and feedback weights, $W$ and $Y$,
respectively, are computed as follows:
$\begin{array}[]{c}\Delta W_{l}=\Delta Y_{l}=-(b_{l}-\bar{b}_{l})\odot
e_{l-1}^{T}.\end{array}$ (8)
Notice that, once again, the multiplication is replaced by a backward
convolution in the context of convolutional layers.
### II-B MBURST
The proposed multimodal burst propagation-based approach to speech enhancement
combines audio and visual information for clean audio estimation. The model
takes advantage of recent neurological research on pyramidal cells to learn
coherent relationships between noisy audio and visual information and extract
clean information by amplifying relevant information and suppressing noise.
The model comprises two burst-dependent convolutional neural networks that
work in parallel to extract features from noisy audio inputs and their
respective frames. The outcomes from both channels are flattened and exposed
to a burst-dependent fully-connected layer for embedding extraction. The
embeddings are concatenated into a single feature vector, which is used to
feed a similar burst-dependent dense layer for classification. Figure 1
depicts the process pipeline.
Figure 1: Proposed method. The left image depicts the multimodal network with
noisy audio and visual inputs. Both channels comprise two layers of burst-
based convolutions, followed by flattening and a burst-based dense layer for
feature embedding. Finally, such embeddings are concatenated and used to feed
another burst-based dense for classification. Right-top and right-bottom
frames illustrate the burst-based convolutional and dense layers,
respectively. Notice such layers comprise a forward weight matrix (kernel)
$\bm{W}$ and a backward weight matrix $\bm{Y}$.
## III Methodology
This section describes the dataset and the environment configuration
considered in the experiments.
### III-A Dataset
The experiments performed in this work were conducted over a dataset based on
roughly $1,000$ sentences composed of a six-word sequence extracted from
Speakers $1$ in Grid Corpus [18] dataset. The clear Grid utterances are
blended with non-stationary random noises, e.g., bus, cafe, street, and
pedestrian, using four distinct Signal-to-noise ratios (SNR), i.e.,
$\\{-12dB,-6dB,0dB,6dB\\}$, extracted from the $3$rd CHiME Challenge
(CHiME3)[19].
#### III-A1 Data pre-processing
The original audio was resampled at $16$ kHz. Each utterance was divided into
approximately $75$ frames with $1,244$ samples per frame and a $25\%$
increment rate. Further, a $622$-bin power spectrum was computed using STFTs
and a hamming window procedure. A Viola-Jones lip detector [20] and an object
tracker [21] extract the speakers’ lip images from the GRID Corpus films,
recorded at $25$ frames per second. A $92\times 50$-inch area around the lip’s
center was picked. The recovered lip sequences were up-sampled three times to
match the $75$ STFT frames of the audio signal.
#### III-A2 Ideal Binary Mask
Hu and Wang [22] proposed the so-called Ideal Binary Mask (IBM), a binary mask
with a positive value to a time-frequency (TF) unity whose goal energy is
greater than the interference energy and zero otherwise. Among a number of the
desired qualities in speech enhancement, the binary masks retain the speech-
dominated units while zeroing out the noise-dominated units of noisy speech
and providing high intelligibility scores on speech reconstruction, even
considering meager signal-to-noise ratio (SNR) settings [23].
IBM employs a premixing of target and interference signals, i.e., the speech
and noise signals, computed through a criterion function:
$\displaystyle IBM(t,f)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&\mbox{if
}10\log_{10}\left(\frac{X(t,f)^{2}}{N(t,f)^{2}}\right)\geq LC,\\\
0&\mbox{otherwise}\end{array}\right.$ (11)
where $X$ stands for the speech with no background noise, $N$ represents the
noise portion, and $LC$ is the local criterion used to differentiate between
speech and background noise, i.e., a threshold. Further, the frame of
reference $t$ and the frequency range $f$ denote the time and frequency
dimensions, respectively.
The method assigns a positive value to the more prevalent TF units if their
SNR is greater than or equal to the LC. Conversely, TF units dominated by
noise are presumed to have SNRs lower than LC. Thus they are given assigned a
value of 0 and muted. The proportion of the TF-positive speech versus noise-
dominant samplings relies on a proper LC selection. Since SNRs greater than
$0$ dB are speech dominating and those less than $0$ dB are noise dominant, an
LC of 0 dB is supposedly optimal. However, studies suggest that such selection
may imply disproportionate removal of TF units on mixed noisy and speech
instances due to the addition of noise known as artifacts, therefore
suggesting employing an LC of $5$ dB [24, 24] for optimum performance.
(a) (b) (c) (d)
Figure 2: Multimodal-based mask reconstructions: (a) Original mask, (b)
reconstruction using Unimodal, (c) Multimodal, and (d) the proposed MBURST.
### III-B Experimental Setup
This work implements MBURST, a burst propagation-based multimodal architecture
for speech enhancement. The architecture comprises two parallel networks,
i.e., an audio and a visual channel, each containing two convolutional layers
with $32$ channels, followed by flattening and a dense layer with $256$ units
for embedding extraction. Finally, such embeddings are concatenated and used
to feed the top layer, which is encharged to estimate the clean audio masks.
It also implements two baselines for comparison purposes, i.e., a similar
network using the same configuration but the backpropagation algorithm for
learning, namely Multimodal and a unimodal version also trained using
backpropagation that comprises the same architecture except by the visual
channel, namely unimodal.
The convolutions consider kernels of size $3\times 3$, paddings equal to $1$,
and a stride of size $2$ for the visual and $1$ for the audio channels. This
difference in the stride concerns the shape of the audio input, i.e., since
the input audio comprises the current signal, composed of $500$ features, plus
the seven prior frames, forming an $8\times 500$ matrix, we opted to use
stride $1$ to maintain this shape due to the reduced size.
The experiments were conducted using the weighted binary cross entropy as the
loss function since the rate of white pixels in the masks (relevant
information) is several times smaller than black ones. The same justificative
is considered for employing the F1-score as the evaluation metric. Further,
the optimization of the parameters is conducted using Adam with a learning
rate of $10^{-3}$ and weight decay of $10^{-6}$. Regarding burst propagation,
the reference bursting probability follows the standard value adopted by the
original work [14], setting $\bar{P}_{L}=0.2$. The dataset was randomly split
into training ($80\%$), testing ($15\%$), and proxy ($5\%$) sets. The training
is conducted for $4$ runs, considering different dataset splits, during $40$
epochs.
Finally, MBURST, as well as the baseline architectures, were implemented in
Python using Pytorch framework [25], and the code is available on
GitHub111Available at:https://github.com/Leandropassosjr/MBURST..
## IV Experimental Results
This section presents the results and discussion under the perspectives of
clean audio signal mask reconstruction and energy efficiency.
### IV-A Clean Audio Signal Mask Reconstruction
Table I presents the average F1-score and accuracy, as well as their
respective standard deviations, over the training and test sets for the
proposed MBURST and the baseline architectures. In this scenario, one can
notice that the Unimodel was not capable of providing competitive results,
which is expected since the visual context is essential to introduce context
to the noisy signal and boost the reconstruction. On the other hand, both the
Multimodal and MBURST obtained similar results, with the Multimodal approach
achieving slightly better values (less than $2\%$ on average), which can be
possibly explained by activation sparsity induced by the burst rate mechanism.
Such results reinforce the relevance of visual context for clean speech
reconstruction.
TABLE I: F1-score and Accuracy concerning the Unimodal, Multimodal, and the
proposed MBURST for clean audio mask reconstruction over the train and test
sets.
Data set | Metric | Unimodal | Multimodal | MBURST
---|---|---|---|---
Train | F1 | $0.677\pm 0.000$ | $0.802\pm 0.005$ | $0.782\pm 0.001$
Acc. | $82.919\pm 0.043$ | $91.898\pm 0.220$ | $90.602\pm 0.067$
Test | F1 | $0.696\pm 0.000$ | $0.790\pm 0.003$ | $0.768\pm 0.002$
Acc. | $81.758\pm 0.047$ | $90.236\pm 0.164$ | $88.798\pm 0.233$
Figure 2 depict some examples of the reconstructions themselves. Notice that
such images were generated by gathering groups of $150$ randomly selected
sequential samples from the proxy data set. In this context, one can observe
that the masks reflect the results provided in Table I, i.e., Unimodal
presents a poor mask reconstruction, while both the Multimodal and the MBURST
provide more accurate and very similar results. Once again, the results
suggest that correlated visual context is essential to extract more relevant
information from noisy audio signals and improve the reconstruction quality.
### IV-B Energy Analysis
This work considers the energy efficiency in terms of neurons’ activation
rate, which shows itself as MBURST’s major triumph. In this aspect, one can
observe in Figure 3 that the MBURST neurons’ firing rate decreases
dramatically to values below $0.2$ in the first iterations and strides towards
values close to $0.1$ in the subsequent iterations for both training and
testing sets, showing an energy gain around $70\%$ and $65\%$ over the
Multimodal and Unimodal approaches, respectively. Regarding the
backpropagation-based approaches, the Unimodel performed better than
Multimodel, suggesting that the backpropagation was not robust enough to
combine the noisy audio and visual information in an energy efficiency
fashion, thus implying the worst performance achieved by the Multimodel
approach in this context.
(a)
(b)
Figure 3: Energy efficiency rate over (a) train and (b) test sets.
Table II analytically reinforces MBURST’s robustness in the context of energy
efficiency by presenting the area under the curve over the average energy rate
measured during the training and testing procedures. In this context, MBURST
shows itself $48\%$ and $58\%$, on average over train and test sets, more
efficient than Unimodal and Multimodal, respectively.
TABLE II: Area under the curve considering the average energy rate during
training and testing for the Unimodal, Multimodal, and the proposed MBURST.
Data set | Unimodal | Multimodal | MBURST
---|---|---|---
Train | $11.12$ | $13.70$ | $5.63$
Test | $11.51$ | $14.14$ | $6.08$
## V Conclusions
This paper proposed the MBURST, a burst propagation-based multimodal approach
that implements a more biologically plausible solution for the task of AV
speech enhancement. One can draw two main conclusions from experiments
conducted over a Grid Corpus and CHiME3-based dataset. First, multimodal AV
architectures are more efficient than a standard unimodal approach for clean
audio mask reconstruction since correlated visual information provides
contextual meaning to noisy audio. Second, MBURST can produce similar results
to a traditional backpropagation multimodal architecture in the context of
accuracy and reconstruction task metrics, with a massive advantage regarding
energy efficiency due to the burst-rate-based activation mechanism.
Regarding future work, we aim to extend the burst propagation concepts to
recent studies comprising context-sensitive neocortical neurons [26] and
canonical cortical networks [13].
## Acknowledgments
The authors are grateful to the Engineering and Physical Sciences Research
Council (EPSRC) grant EP/T021063/1.
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* [25] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, “Pytorch: An imperative style, high-performance deep learning library,” in _Advances in Neural Information Processing Systems 32_. Curran Associates, Inc., 2019, pp. 8024–8035. [Online]. Available: http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf
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|
# LaplaceNet: A Hybrid Energy-Neural Model
for Deep Semi-Supervised Classification
Philip Sellars1, Angelica I. Aviles-Rivero1 and Carola-Bibiane Schönlieb1 P.
Sellars, Angelica I. Aviles-Rivero and Carola-Bibiane Schönlieb are with the
Department of Theoretical Physics and Applied Mathematics, Univeristy of
Cambridge, Cambridge, UK<EMAIL_ADDRESS>.
###### Abstract
Semi-supervised learning has received a lot of recent attention as it
alleviates the need for large amounts of labelled data which can often be
expensive, requires expert knowledge and be time consuming to collect. Recent
developments in deep semi-supervised classification have reached unprecedented
performance and the gap between supervised and semi-supervised learning is
ever-decreasing. This improvement in performance has been based on the
inclusion of numerous technical tricks, strong augmentation techniques and
costly optimisation schemes with multi-term loss functions. We propose a new
framework, LaplaceNet, for deep semi-supervised classification that has a
greatly reduced model complexity. We utilise a hybrid energy-neural network
where graph based pseudo-labels, generated by minimising the graphical
Laplacian, are used to iteratively improve a neural-network backbone. Our
model outperforms state-of-the-art methods for deep semi-supervised
classification, over several benchmark datasets. Furthermore, we consider the
application of strong-augmentations to neural networks theoretically and
justify the use of a multi-sampling approach for semi-supervised learning. We
demonstrate, through rigorous experimentation, that a multi-sampling
augmentation approach improves generalisation and reduces the sensitivity of
the network to augmentation. Code available at
https://github.com/psellcam/LaplaceNet.
## I Introduction
The advent of deep learning has been key in achieving outstanding performance
in several computer vision tasks including image classification [1, 2, 3, 4,
5], object detection e.g. [6, 7, 8] and image segmentation [9, 10, 11].
Training deep learning models often relies upon access to large amounts of
labelled training data. In real-world scenarios we often find that labels are
scarce, expensive to collect, prone to errors (high uncertainty) and might
require expert knowledge. Therefore, relying on a well-representative dataset
to achieve good performance is a major limitation for the practical deployment
of machine learnt methods. These issues have motivated the development of
techniques which are less reliant on labelled data.
Semi-supervised learning aims to extract information from unlabelled data, in
combination with a small amount of label data, and produce results comparable
to fully supervised approaches. In recent years, the developments in deep
learning have motivated new directions in semi-supervised learning (SSL) for
image classification. The major benefit of these new deep approaches being the
ability to learn feature representations rather than rely upon hand-crafted
features. In the last few years, deep SSL papers have reached unprecedented
performance e.g. [12, 13], and the gap between supervised and semi-supervised
models is much smaller now that it was even five years ago, with semi-
supervised methods surpassing certain supervised techniques.
What techniques have been crucial to the improved performance of deep semi-
supervised methods? Although, there is no universal answer, there are several
shared commonalities between the current SOTA. The works of [13, 12, 14]
demonstrated that a key factor for improving performance is the use of strong
augmentations strategies such as AutoAugment [15], RandAugment [16], Cutout
[17] and CTAugment [14]. Additionally, the use of confidence thresholding [12,
18] and temperature sharpening [13, 19] are thought to be vital in improving
performance for pseudo-labeling methods. Other papers [20, 21, 19] have shown
great improvement from using interpolating techniques such as MixUp [22].
Several SOTA have also promoted large batch sizes [12] with a large ratio of
unlabelled to labelled data per batch.
Recent approaches in SSL have proposed costly optimisation schemes involving
multi-term loss functions to improve the generalisation of their models [14,
20]. Some approaches [12] use separate loss terms for unlabelled and labelled
data, whilst consistency regularisation approaches such as [13] use a standard
supervised loss in combination with a specialised consistency loss. Other
approaches go even further [20, 14] and use three or more loss terms which
promote entropy minimisation, class balancing or simultaneously minimise
several consistency losses.
Over-costly computational approaches and unnecessary complexity, make it hard
to directly say what tools or approaches are important for improved
generalisation and make it difficult to use SSL methods in realistic settings.
Furthermore, despite the significant improvements found in using
augmentations, there has been little effort in the field of SSL to investigate
how best to include strong augmentations techniques in the learning framework.
With these points in mind, in this work, we introduce a new deep SSL framework
for image classification which offers state-of-the-art performance with
massively reduced model complexity. Our main contributions are:
* –
We propose a graph based pseudo-label approach for semi-supervised image
classification which we name LaplaceNet. We demonstrate through extensive
testing, that our approach produces state-of-the-art results on benchmark
datasets CIFAR-10, CIFAR-100 and Mini-ImageNet. We do so with vastly reduced
model complexity compared to the current state-of-the-art. We show that a
single loss, the classic supervised loss, is all that is required for
fantastic performance in the SSL domain.
* –
We show that using an energy-based graphical model for pseudo-label generation
produces more accurate pseudo-labels, with a small computational overhead,
than using the network’s predictions directly. Furthermore, we demonstrate
that energy-based pseudo-label approaches can produce state-of-the-art results
without the techniques (temperature sharpening, confidence thresholding, soft
labels) that are currently thought to be essential for pseudo-label methods.
* –
Instead, we offer further evidence that strong augmentation is by far and away
the most important tool for improving the performance of semi-supervised
models in the natural image domain. With this in mind, we propose,
theoretically justify and experimentally demonstrate that a multi-sample
averaging approach to strong augmenation not only improves generalisation but
reduces the sensitivity of the model’s output to data augmentation.
## II Related Work
The problem of improving image classification performance using SSL has been
extensively investigated from the classic perspective e.g. [23, 24, 25, 26,
27, 28], in which one seeks to minimise a given energy functional that
exploits the assumed relationship between labelled and unlabelled data [29].
However, classical approaches tended to rely on hand-crafted features that
limited their performance and generalisation capabilities. With the
popularisation of deep learning and its ability to learn generalisable feature
representations, many techniques have incorporated neural networks to mitigate
problems of generalisation. These modern state-of-the-art methods are
dominated by two approaches, consistency regularisation and pseudo-labelling,
which differ in how they incorporate unlabelled data into the loss function.
### II-A Consistency Regularisation Techniques
One of the fundamental assumptions that allows semi-supervised learning to
help performance is the cluster assumption, which states that points in the
same cluster are likely to be in the same class. This can be seen to be
equivalent to the low-density assumption which states that the decision
boundaries of the model should lie in low-density regions of the data
distribution. Following from the above assumptions, if we have access to some
labelled data $Z_{l}=\\{x_{i},y_{i}\\}_{i=1}^{n_{l}}$ and a large amount of
unlabelled data $Z_{u}=\\{x_{i}\\}_{i=n_{l}+1}^{n_{l}+n_{u}}$, we should seek
to move our decisions boundaries to be in low density regions of the joint
labelled and unlabelled data distributions.
Consistency regularisation seeks to implement the low-density assumption by
encouraging the model $f$ to be invariant to perturbations $\delta$ to the
data $x$. As a result the decision boundaries are pushed to low-density
regions. Mathematically, given some data perturbing function
$u:\mathcal{X}\rightarrow\mathcal{X}$, such that $u(x)=x+\delta$, consistency
based approaches seek to minimise some consistency loss $L_{\text{con}}$ in
the general form of
$L_{\text{con}}=||f(u(x))-f(x)||_{2}^{2}.$ (1)
A large number of papers have applied this idea to SSL including the
$\prod-$Model and temporal ensembling [30], Virtual Adversarial Training (VAT)
[31], Mean Teacher [32], the Interpolation Consistency Training (ICT) [21]
RemixMatch [14] and MixMatch [19]. The downside of consistency regularisation
techniques is the vagueness in choosing an appropriate $\delta$. This
vagueness is reflected in the wide range of perturbations which have been used
in the field. Virtual Adversarial Training uses adversarial training to learn
an effective $\delta$ for each point. Mean Teacher [32] decided to apply a
perturbation to the model itself, and replaces $f(u(x))$ with an exponential
moving average of the model $f_{\text{EMA}}(x)$. Interpolation Consistency
Training [21] seeks to train the model to provide consistent predictions at
interpolations of unlabelled points. The authors of [13] demonstrated that by
replacing simple noising perturbations with stronger augmentation
perturbations (eg, RandAugment [16] or CTAugment [14]) leads to a substantial
performance improvements.
Although these techniques have demonstrated great performance, it is unclear
how best to set the perturbations $\delta$ and how best to incorporated them
in learning frameworks. In our work, we avoid using model based perturbations
and instead focus on the the application of strong data augmentation.
### II-B Pseudo-Labelling Techniques
Another family of methods, termed pseudo-label approaches, focus on estimating
labels for the unlabelled data points and then using them in a modified loss
function. Forcing the network to make predictions on unlabelled points
minimises the entropy of the unlabelled predictions [29] and moves the
decision boundaries to low-density regions. Additional, dependent on the
accuracy of the pseudo-labels, we increase the amount of labelled data the
model has access to and reduce overfitting to the initally small label set.
There are many ways to incorporate unlabelled data / pseudo-label pairs into
the loss function but the most common ways are to either create a specific
loss term for the unlabelled data pseudo-label pairs [12, 18] or by using
composite batches containing both labelled and unlabelled data and keeping the
standard supervised classification loss [33, 20].
The first application of this idea to the deep learning setting was presented
by Lee [34]. Viewing the output of the neural network $f(x)$ as a discrete
probability distribution, Lee assigned a hard pseudo-label $\hat{y}$ for each
unlabelled data point according to its most likely prediction
$\hat{y}_{i}=\operatorname*{arg\,max}f(x_{i})$. These pseudo-labels were then
used in a two termed loss function of the form
labelled loss unlabelled loss (2)
$\displaystyle\hat{L}_{ssl}=\color[rgb]{0,0,0}\frac{1}{n_{l}}\sum_{i=1}^{n_{l}}l_{s}(f(x),y)+\color[rgb]{0,0,0}\eta\color[rgb]{0,0,0}\frac{1}{n_{u}}\sum_{i=1}^{n_{u}}l_{s}(f(x),\hat{y}),$
where $l_{s}$ is some loss function and $\eta$ is a weighting parameter. The
pseudo-labels are recalculated every-time the unlabeled data is passed through
the network. As an alternative to hard labels, [19] used the full output
probability distribution of the network as a soft label for each point.
However, it was found that sharpening this distribution helped ensured the
model’s prediction entropy was minimised.
As pointed out by Arazo et al [20] there is a potential pitfall in this style
of approach. Networks are often wrong and the neural network can overfit to
its own incorrectly guessed pseudo-labels in a process termed confirmation
bias. Arazo et al proposed using MixUp [22], soft labels and a minimum ratio
of labeled to unlabeled data to reduce confirmation bias. An alternative
method to reduce confirmation bias is to use uncertainty quantification for
the produced pseudolabels. These methods calculate a confidence score $r_{i}$
for each pseudo-label $\hat{y}_{i}$. The works of [12, 33] used the entropy of
the probability distribution to give $r_{i}$ whilst [35] used the distance
between unlabelled points and labelled points in feature space. One can then
either weight the loss terms by $r_{i}$ or exclude pseudo-labels whose $r_{i}$
is below some threshold $\tau$ in an attempt to prevent the network learning
from low confidence predictions. This style of approach is based upon the idea
that the neural network is well calibrated, i.e that the model’s softmax score
is a good indicator of the actually likelihood of a correct prediction.
However, recent research has suggested that modern neural networks are not as
well calibrated as may be intuitively thought [36]. In our work we demonstrate
that, whilst a intuitive solution, uncertainty quantification is not needed
for our pseudo-label approach.
In a completely different direction to network predictions, it has been shown
from a classical perspective [25] that energy based models such as graphs are
well suited to the task of label propagation. Therefore, several works [33,
37, 38] have shown good performance by iteratively feeding the feature
representation of a neural network to a graph, performing pseudo-label
generation on the graph and then using those labels to train the network.
However, graphical approaches have yet to show that they can produce state-of-
the-art results compared to model based approaches such as [12, 13]. In our
work, we present a graphical approach which surpasses the performance of model
based approaches, demonstrating that graphical approaches have a lot of
promise for practical applications.
## III Proposed Technique
This section details our proposed semi-supervised method. We cover the
generation of pseudo-labels, the optimisation of the model alongside a full
algorithm and we explore our multi-sample augmentation approach.
### III-A Problem Statement:
From a joint distribution $\mathcal{Z}=(\mathcal{X},\mathcal{Y})$ we have a
dataset $Z$ of size $n=n_{l}+n_{u}$ comprised of a labelled part of joint
samples $Z_{l}=\\{x_{i},y_{i}\\}_{i=1}^{n_{l}}$ and a unlabelled part
$Z_{u}=\\{x_{i}\\}_{i=n_{l}+1}^{n}$ of single samples on $\mathcal{X}$. The
labels come from a discrete set of size C $y\in\\{1,2,..,C\\}$. Our task is to
train a classifier $f$, modelled by a neural network with parameter vector
$\theta$, which can accurately predict the labels of unseen data samples from
the same distribution $\mathcal{X}$. The classifier $f$ can be viewed as the
composition of two functions $z$ and $g$ such that $f(x)=g(z(x))$.
$z:\mathcal{X}\rightarrow\mathbb{R}^{d_{p}}$ is the embedding function mapping
our data input to some $d_{p}$ dimensional feature space and
$g:\mathbb{R}^{d_{p}}\rightarrow\mathbb{R}^{C}$ projects from the feature
space to the classification space.
### III-B Pseudo-labels Generation
As a pseudo-label based approach, we iteratively assign a pseudo-label
$\hat{y}$ to all data points in $Z_{u}$ once per epoch. In this work, we
generate hard pseudo-labels using a graph based approach first proposed by
Zhou et al [26] and first adapted to deep networks by Iscen et al [33] which
has been thoroughly studied in the classical machine learning literature.
We first extract the feature representation of the dataset $V$ by using the
embedding function of the neural network $z$ so that
$V=\\{z(x_{1}),..,z(x_{n})\\}$. Unlike other works we do not apply
augmentation to the data whilst producing the pseudo-labels. Using $V$ and a
similarity metric $d$, we use $d(v_{i},v_{j})=\langle v_{i},v_{j}\rangle$, we
construct a symmetric weighted adjacency matrix $W\in\mathbb{R}^{n\times n}$.
The elements $w_{ij}\in W$ are given by $W_{ij}=d(v_{i},v_{j})$ and represent
the pairwise similarities between data points. We then sparsify $W$ using the
following nearest neighbour approach, which reads:
$W_{ij}=\begin{cases}d(v_{i},v_{j}),&\text{if $i$ is one of the $k$ nearest
neighbor of $j$,}\\\ &\text{or vice versa.}\\\ 0&\text{otherwise}.\end{cases}$
(3)
We then construct the degree matrix $D:=\text{diag}(W\mathbbm{1}_{n})$ and use
this to normalise the affinity matrix $\mathcal{W}=D^{-1/2}WD^{-1/2}$, which
prevent nodes with high degree having a large global impact. Finally, we use
the initial label information to create the labelled matrix
$Y\in\mathbb{R}^{n\times C}$
$Y_{ij}=\begin{cases}1,&\text{if $y_{i}=j$,}\\\
0&\text{otherwise}.\end{cases}$ (4)
We can then propagate the information contained in $Y$ across the graph
structure $\mathcal{W}$ by minimising the graphical Laplacian of the
prediction matrix $F\in\mathbb{R}^{n\times C}$ plus a fidelity term to the
supplied labelled data:
$\mathcal{Q}(F)=\frac{1}{2}\sum_{i,j=1}^{n}\mathcal{W}_{ij}\left|\left|\frac{F_{i}}{\sqrt{D_{ii}}}-\frac{F_{j}}{\sqrt{D_{jj}}}\right|\right|^{2}+\frac{\mu}{2}\sum_{i=1}^{n}||F_{i}-Y_{i}||^{2},$
(5)
where $\mu$ is a scalar weight. The first term enforces points which are close
according to the metric $d$ to share a similar label whilst the second term
encourages initially labelled points to keep their label. To side-step the
computationally infeasible closed form solution, we use the conjugate gradient
approach to solve the linear system $\left(I-\gamma\mathcal{W}\right)F=Y$,
where $\gamma(1+\mu)=1$. Using $F$ the pseudo-labels $\hat{y}_{i}$ are given
by
$\hat{y}_{i}=\operatorname*{arg\,max}_{j}F_{ij}.$ (6)
A common problem in label propogation is that the psuedo-labels produced by
the graph may be unbalanced over the classes and Iscen et al [33] attempted to
weight the optimisation problem to avoid this possibility. We found that the
weighting approach of Iscen et al actually made the performance of the model
worse than leaving the predictions as is. An alternate approach to counter
class in-balances is distribution alignment [14], which enforces the
distribution of the pseudo-label predictions to match some given prior
distribution. The implementation of this idea by ReMixMatch focused on
applying this idea to the network predictions and wasn’t optimal for a graph
based framework.
Instead we propose a novel smoother version of distribution alignment which
can be applied during or just after the conjugate gradient approach. We give a
full algorithm for this in Algorithm 1. The algorithm is an iterative approach
which smoothly deforms the pseudo-label predictions $F$ by the ratio $R$
between the prior distribution $D$ and the pseudo-label distribution of the
unlabelled points $D_{U}$. Thereby promoting the prediction of
underrepresented classes and vice versa. To ensure the deformation is smooth
we clip the range of $R$ values to be close to one. We show in the
experimental section that this approach improves the performance of the model.
Algorithm 1 Smooth Distribution Alignment
1: Input: Pseudo-label Prediction $F\in\mathbb{R}^{n\times C}$, Prior
Distribution $D\in\mathbb{R}^{C}$, labelled and unlabelled indexes
$L=\\{l_{i}\\}_{i=1}^{n_{l}}$ and $U=\\{u_{i}\\}_{i=1}^{n_{u}}$ and max
iteration $T$
2: Output: Adjusted Pseudo-label Prediction $F\in\mathbb{R}^{n\times C}$
3: for $t_{i}=1$, $t_{i}{+}{+}$, while $t_{i}<T$ do
4: $D_{U}\in\mathbb{R}^{C}\leftarrow\text{Initialise with zeros}$ Get the
pseudo-label distribution:
5: for $u_{i}\in U$ do
6: $D_{U}[\operatorname*{arg\,max}_{j}F[u_{i}]]\mathrel{+}=\frac{1}{n_{u}}$
7: end for
8: $R=D/D_{u}$ Clip values for smooth deformation:
9: $R[R>1.01]=1.01$ and $R[R<0.99]=0.99$ # Deform the current predictions:
10: for $c_{i}=1$, $c_{i}{+}{+}$, while $c_{i}<C$ do
11: $F[U,c_{i}]\mathrel{*}=R[c_{i}]$
12: end for
13: Row normalise $F$ to give valid distributions.
14: end for
### III-C Semi-Supervised Loss
In the deep semi-supervising setting, particularly in the current SOTA [12]
[19], several works seek to minimise a semi-supervised loss $\hat{L}_{ssl}$
composed of two or more terms, one each for the labelled and unlabelled data
points and potentially others covering entropy minimisation etc., which has
the following form:
labelled loss unlabelled loss other terms (7)
$\displaystyle{L}_{\text{gen}}=\color[rgb]{0,0,0}\frac{1}{n_{l}}\sum_{i=1}^{n_{l}}l_{s}(f(x),y)+\color[rgb]{0,0,0}\eta\color[rgb]{0,0,0}\frac{1}{n_{u}}\sum_{i=1}^{n_{u}}l_{s}(f(x),\hat{y})+.....,$
where $\eta$ is a balancing parameter. For our approach we wanted to strip
away as much complexity from the loss function as possible in an effort to see
what elements are required for good performance. We move away from using a
multi-term loss and instead only use the standard supervised loss which has
worked so well in supervised image classification. To include our unlabelled
data we use mixed batches of size $b$ which are made up of $b_{l}$ labelled
samples and $b_{u}$ unlabelled samples to which we have assigned a pseudo-
label $\hat{y}$. Our semi-supervised loss, $L_{ssl}$, is given by:
$L_{ssl}=\frac{1}{b}\sum_{i=1}^{b}l_{s}(f(x_{i}),y_{i}).$ (8)
Note that in (8) $y_{i}$ may be a ground truth label or a pseudo-label. What
is remarkable about this loss is its simplicity. There is no confidence
thresholding of the pseudo-labels, additional weighting parameters, no
consistency based terms or other regularisations. Instead we rely upon the
strength of the combination of an a energy based graphical approach to pseudo-
labels estimation and the clever use of strong augmentation to increase
generalisation.
### III-D Training the model
For initialisation purposes, we quickly extract some baseline knowledge from
the dataset by minimising a supervised loss $L_{sup}$, for one hundred passes
through $Z_{l}$. This supervised loss reads:
$L_{\text{sup}}=\frac{1}{b}\sum_{i=1}^{b}l_{s}(f(x_{i}),y_{i}),$ (9)
where $b$ is the batch size and $l_{s}$ is the cross entropy loss. We emphasis
that (9) uses only the tiny labelled set $Z_{l}$, and is performed once before
the main semi-supervised optimisation. We then begin our main learning loop
which alternates between updating the pseudo-label predictions and minimising
the semi-supervised loss $L_{ssl}$ for one epoch, where we define one epoch to
be one pass through the unlabelled data $Z_{u}$. This cycle then runs for a
total of $S$ optimisation steps and the fully trained model is then tested on
the relevant testing set. Note that we do use Mixup [22] on both $L_{sup}$ and
$L_{ssl}$ with a beta distribution parameters $\alpha$. In Algorithm 2, we
give an overview of training our model for $S$ optimisation steps.
Algorithm 2 Training Scheme for LaplaceNet
1: Input: labelled data $Z_{l}=\\{x_{i},y_{i}\\}_{i=1}^{n_{l}}$, unlabelled
data $Z_{u}=\\{x_{i}\\}_{i=n_{l}+1}^{n}$, untrained model $f$ with trainable
parameters $\theta$ and embedding function $z$. Hyper-parameters: Number of
optimisation steps $S$ # Initialisation:
2: for $i=1$ to $100$ do
3: optimise $L_{sup}$ over $Z_{l}$
4: end for
5: Set current step to zero $s_{i}=0$ # Main Optimisation Process:
6: while $s_{i}<S$ do
7: Extract features: $V=\\{z(x_{i})\\}_{i=1}^{n}$
8: Construct Graph Matrix $W$
9: Degree Normalisation $\mathcal{W}=D^{\frac{-1}{2}}WD^{\frac{-1}{2}}$
10: Propagate Information via $\mathcal{Q}(F)$
11: Distributed Alignment on $F$
12: $\hat{y_{i}}=\operatorname*{arg\,max}F_{i}\text{ }\forall\text{
}n_{l}+1\leq i\leq n$
13: for $i=1$ to $\lfloor\frac{n_{u}}{b_{u}}\rfloor$ do
14: $B_{L}=\\{x_{i},y_{i}\\}_{i=1}^{b_{l}}\subset Z_{l}$ ,
$B_{u}=\\{x_{i},\hat{y}_{i}\\}_{i=1}^{b_{u}}\subset Z_{u}$
15: Composite Batch $B=B_{L}\cup B_{u}$
16: Optimise $L_{ssl}$ , $s_{i}++$
17: end for
18: end while
### III-E Multi-Sampling Augmentation
Since the work of [13], several approaches have implemented the use of strong
augmentations [18, 14, 12] to the problem of semi-supervised learning, with
each work having a different way of including augmentation to their framework.
Very recent works [18, 14] have begun using multiple augmented versions of the
same unlabelled image. As yet there is no motivation for why this multiple
sampling idea is preferable to alternatives such as larger batch sizes or
running the code for more steps. In this section we offer a theoretical
motivation for why multi-sampling improves generalization along with a
mathematically bound on its performance gain. With this knowledge in mind we
provide a simple method for including augmentation averaging into our SSL
framework and demonstrate this approach increases accuracy and reduces the
sensitivity of the model to data augmentation.
We view an augmentation strategy $A$ as a set $T$ of transformations
$t:\mathcal{X}\rightarrow\mathcal{X}$ and denote it as
$T=\\{t_{1},t_{2},..,t_{\delta}\\}$. The current standard approach, that the
majority of existing techniques follow, is to simply sample $t\sim T$ once for
each data point and compute some augmented loss $L_{Aug}$:
$L_{\text{Aug}}=\frac{1}{n}\sum_{i=1}^{n}l_{s}(f(\color[rgb]{0,0,0}t(x_{i})\color[rgb]{0,0,0}),y_{i}).$
(10)
However, we argue that such a simple implementation, might not extract the
full information present in the augmentation. If we want to encourage our
model output to be more resistant to data augmentations from $T$, and as a
result produce a more generalisable model, we need to perform a multi-sample
approach. To justify this, we consider the following loss $L_{T}$:
$L_{T}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{t\sim T}[l(f(t(x_{i})),y)],$ (11)
which measures the risk of the model over the entire augmentation set. If we
want to minimise (11) then we must minimise the expected augmentation error
over the entire transformation set for each data point $\mathbb{E}_{t\sim
T}[l(f(t(x_{i})),y)]$. To see how a multi-sample approach helps us do just
that we use Hoeffding’s inequality which provides us with a probability bound
that the sum of bounded independent random variables deviates from its
expected value by more than a certain amount.
Let $Z_{1},..,,Z_{n_{a}}$ be a sequence of i.i.d random variables. Assume that
$\mathbb{E}[Z]=\mu$ and $\mathbb{P}[a\leq Z_{i}\leq b]=1$ for every $i$. Then,
by Hoeffding’s inequality, for any $\epsilon>0$, one has:
$\mathbb{P}\left[\left|\frac{1}{n_{a}}\sum_{i=1}^{n_{a}}Z_{i}-\mu\right|>\epsilon\right]\leq
2\text{exp}(-2m\epsilon^{2}/(b-a)^{2}).$ (12)
We can rewrite (12) in context of the previously defined augmentation loss
where we replace $Z_{1},..,,Z_{n_{a}}$ with $n_{a}$ samples from $T$ :
$f(t_{1}(x)),f(t_{2}(x)),..,f(t_{n_{a}}(x))$
$\displaystyle\mathbb{P}\left[\left|\frac{1}{n_{a}}\sum_{j=1}^{n_{a}}l(f(t(x_{i})),y_{i})-\mathbb{E}_{t\sim
T}[l(f(t(x_{i})),y_{i})]\right|>\epsilon\right]$ (13) $\displaystyle\leq
2\text{exp}(-2n_{a}\epsilon^{2}/b^{2}).$
As we increase $n_{a}$, we converge in probability to the desired loss
$\mathbb{E}_{t\sim T}[l(f(t(x_{i})),y)]$ for each data point. Subsequently, we
should optimise to a lower of $L_{T}$ meaning that the model output will
fluctuate less over the augmentation set $T$ for the used training data, and
in the process make our model more generalisable. Furthermore, we can see that
the probability is bounded by an exponent whose power is $\propto-n_{a}$.
Therefore, as we increase $n_{a}$ the rate of decrease for the bound also
decreases, making the first few samples far more important than later ones.
This result explains prior behaviours reported but not reasoned in past papers
such as [14]. When using a $n_{a}$ sample the computational complexity
increases as $O(n_{a})$ but as there should be dimishing returns for
increasing $n_{a}$ it should only be necessary to use $n_{a}$ values slightly
above one. As we have shown that a multi-sample approach should offer generic
performance increases for suitable $T$ we change (8) and (9) to a multi-sample
version. For (8) this becomes
$L_{ssl}=\frac{1}{b}\frac{1}{n_{a}}\sum_{i=1}^{b}\sum_{j=1}^{n_{a}}l_{s}(f(t_{j}(x_{i})),y_{i}).$
(14)
where the index $j$ represents repeated samples from $T$. In the ablation
section, we perform a thorough experimental evaluation to test the theoretical
predictions we have made in this section.
Augmentation Implementation Similarly to other approaches we use two different
augmentation strategies: one for labelled data and another for unlabelled
data. However, we apply strong augmentations to both labelled and unlabelled
data, unlike past approaches [12] which reported divergences using this
approach. For strong augmentations we make use of RandAugment [16], and CutOut
augmentation [17]. For completeness we list the full data transformations for
labelled and unlabeld data in Table I and the implementation of RandAugment
and CutOut in the supplementary material.
TABLE I: The augmentation transformations used for labelled and unlabelled data. For normalisation we use the official channel labelled Transform | Unlabelled Transform
---|---
Random Horizontal Flip
Random Crop and Pad
RandAugment Sample | RandAugment Sample
- | RandAugment Sample
CutOut
Normalisation
## IV Implementation and Evaluation
In this section we detail the implementation of LaplaceNet, including hyper
parameter values and training schemes, and the evaluation protocol we used to
measure our model’s performance and compare against the current state-of-the-
art.
### IV-A Dataset Description
We use three image classification datasets: CIFAR-10 and CIFAR-100 [39] and
Mini-ImageNet [40]. Following standard protocol, we evaluate our method’s
performance on differing amounts of labelled data for each datasets.
(a) CIFAR-10,CIFAR-100 Containing 50k training images and 10k test images,
these datasets contain 10 and 100 classes respectively. The image size is
small at $32$ by $32$ pixels. We perform experiments using 500,1k,2k and 4k
labels for CIFAR-10 and 4k and 10k labels for CIFAR-100.
(b) Mini-ImageNet A subset of the popular ImageNet dataset, containing 100
classes each with 500 training and 100 test images. The resolution of the
images is $84\times 84$ pixels and represents a much harder challenge than the
CIFAR-10/CIFAR-100 datasets. We use 4k and 10k labels in our experiment.
### IV-B Implementation Details
Architectures For a fair comparison to older works we use the ”13-CNN”
architecture [32] and for comparison to recent state-of-the-art works we use a
WideResNet (WRN) 28-2 and a WRN-28-8 [41] architecture. We additional use a
ResNet-18 [5] for Mini-Imagenet. For all models we set the drop-out rate to
$0$. For the ”13-CNN” we add a $l_{2}$-normalisation layer to the embedding
function. Infrastructure For all experiments, we use 1-2 Nvidia P100 GPUs.
Training Details: We train with stochastic gradient descent (SGD) using
Nesterov momentum $n_{m}$ with value $0.9$ and weight decay $\omega$ with
value $0.0005$. We use an initial learning rate of $l_{r}=0.3$ and use
$S=250000$ optimisation steps in total. We utilise a cosine learning rate
decay such that the learning rate decays to zero after 255000 steps. We do not
make use of any $EMA$ model averaging. Parameters We list the parameter values
used in Table II. Most parameter values are common parameter settings from the
deep learning field and are not fine-tuned to our application. Being able to
work with reasonably generic parameters is well suitable to the task of SSL
where using fine-tuning over validation sets is often impossible in practical
applications.
TABLE II: List of hyperparameters used in the paper across the CIFAR-10/100
and Mini-Imagenet datasets.
Parameter | CIFAR-10 | CIFAR-100 | Mini-ImageNet
---|---|---|---
$\alpha$ | 1.0 | 0.5 | 0.5
$\mu$ | 0.01 | 0.01 | 0.01
$k$ | 50 | 50 | 50
$S$ | $2.5\times 10^{5}$ | $2.5\times 10^{5}$ | $2.5\times 10^{5}$
$b$ | 300 | 100 | 100
$b_{l}$ | 48 | 50 | 50
$l_{r}$ | 0.03 | 0.03 | 0.1
$n_{m}$ | 0.9 | 0.9 | 0.9
$\omega$ | $5\times 10^{-4}$ | $5\times 10^{-4}$ | $5\times 10^{-4}$
$n_{a}$ | 3 | 3 | 3
TABLE III: Top-1 error rate on the CIFAR-10/100 datasets for our method and other methods using the 13-CNN architecture. We denote with $\dagger$ experiments we have ran. Dataset | CIFAR-10 | CIFAR-100
---|---|---
Method | 500 | 1000 | 2000 | 4000 | 4000 | 10000
Supervised Baseline | 37.12 $\pm$ 0.89 | 26.60 $\pm$ 0.22 | 19.53 $\pm$ 0.12 | 14.02 $\pm$ 0.10 | 53.10 $\pm$ 0.34 | 36.59 $\pm$ 0.47
Consistency Based Approaches
$\Pi$-Model | - | - | - | 12.36 $\pm$ 0.31 | - | 39.19 $\pm$ 0.36
MT$\dagger$ | 27.45 $\pm$ 2.64 | 21.55 $\pm$ 1.48 | 15.73 $\pm$ 0.31 | 12.31 $\pm$ 0.20 | 45.36 $\pm$ 0.49 | 36.08 $\pm$ 0.51
VAT | - | - | - | 11.36 $\pm$ 0.34 | - | -
MT-LP | 24.02 $\pm$ 2.44 | 16.93 $\pm$ 0.70 | 13.22 $\pm$ 0.29 | 10.61 $\pm$ 0.28 | 43.73 $\pm$ 0.20 | 35.92 $\pm$ 0.47
SNTG | – | 18.41$\pm$0.52 | 13.64$\pm$0.32 | 9.89$\pm$0.34 | – | 37.97$\pm$0.29
MT-fast-SWA | - | 15.58 $\pm$ 0.12 | 11.02 $\pm$ 0.12 | 9.05 $\pm$ 0.21 | - | 34.10 $\pm$ 0.31
MT-ICT | - | 15.48 $\pm$ 0.78 | 9.26 $\pm$ 0.09 | 7.29 $\pm$ 0.02 | - | -
Dual Student | – | 14.17$\pm$0.38 | 10.72$\pm$0.19 | 8.89$\pm$0.09 | – | 32.77$\pm$0.24
Pseudo-Labelling Approaches
TSSDL$\dagger$ | - | 21.13 $\pm$ 1.17 | 14.65 $\pm$ 0.33 | 10.90 $\pm$ 0.23 | - | -
LP$\dagger$ | 32.40 $\pm$ 1.80 | 22.02 $\pm$ 0.88 | 15.66 $\pm$ 0.35 | 12.69 $\pm$ 0.29 | 46.20 $\pm$ 0.76 | 38.43 $\pm$ 1.88
DAG | 9.30 $\pm$ 0.73 | 7.42 $\pm$ 0.41 | 7.16 $\pm$ 0.38 | 6.13 $\pm$ 0.15 | 37.38 $\pm$ 0.64 | 32.50 $\pm$ 0.21
Pseudo-Label Mixup | 8.80 $\pm$ 0.45 | 6.85 $\pm$ 0.15 | - | 5.97 $\pm$ 0.15 | 37.55 $\pm$ 1.09 | 32.15 $\pm$ 0.50
LaplaceNet $\dagger$ | 5.68 $\pm$ 0.08 | 5.33 $\pm$ 0.02 | 4.99 $\pm$ 0.12 | 4.64 $\pm$ 0.07 | 31.64 $\pm$ 0.02 | 26.60 $\pm$ 0.23
TABLE IV: Top-1 error rate for CIFAR-10/100. All methods, except MixMatch, are tested using the same code-base and use the same model code, the same optimiser (SGD) with the same optimisation parameters, the same number of optimisation steps and the same RandAugment implementation. We denote with $\dagger$ experiments we have ran. Dataset | CIFAR-10 | CIFAR-100
---|---|---
Method | 500 | 2000 | 4000 | 4000 | 10000
Other Methods
MixMatch | 9.65 $\pm$ 0.94 | 7.03 $\pm$ 0.15 | 6.34 $\pm$ 0.06 | — | —
Same Codebase
UDA $\dagger$ | 6.88 $\pm$ 0.74 | 5.61 $\pm$ 0.16 | 5.40 $\pm$ 0.19 | 36.19 $\pm$ 0.39 | 31.49 $\pm$ 0.19
FixMatch(RA) $\dagger$ | 5.92 $\pm$ 0.11 | 5.42 $\pm$ 0.11 | 5.30 $\pm$ 0.08 | 34.87 $\pm$ 0.17 | 30.89 $\pm$ 0.18
LaplaceNet $\dagger$ | 5.57 $\pm$ 0.60 | 4.71 $\pm$ 0.05 | 4.35 $\pm$ 0.10 | 33.16 $\pm$ 0.22 | 27.49 $\pm$ 0.22
TABLE V: Top-1 error rate for Mini-ImageNet. We compare against methods which have used an identical ResNet-18 architecture. Method | 4000 | 10000
---|---|---
Supervised Baseline | 66.04 $\pm$ 0.32 | 52.89 $\pm$ 0.33
Consistency Regularisation Methods
MT | 72.51 $\pm$ 0.22 | 57.55 $\pm$ 1.11
MT-LP | 72.78 $\pm$ 0.15 | 57.35 $\pm$ 1.66
Pseudo-Label Methods
LP | 70.29 $\pm$ 0.81 | 57.58 $\pm$ 1.47
Pseudo-Label Mixup | 56.49 $\pm$ 0.51 | 46.08 $\pm$ 0.11
LaplaceNet | 46.32 $\pm$ 0.27 | 39.43 $\pm$ 0.09
### IV-C Evaluation Protocol
We evaluate the performance of LaplaceNet on the CIFAR-10/CIFAR-100 and Mini-
Imagenet datasets and compare against the current SOTA models for semi-
supervised learning. For ease of comparison, we split the current SOTA into
two groups.
1. 1.
Methods which used the $13$-CNN architecture [32]: $\Pi$-Model [30], Mean
Teacher(MT) [32], Virtual Adversarial Training (VAT) [31], Label Propogation
for Deep Semi-Supervised Learning (LP) [33], Smooth Neighbors on Teacher
Graphs (SNTG) [42], Stochastic Weight Averaging(SWA) [43], Interpolation
Consistency Training (ICT) [21], Dual Student [44], Transductive Semi-
Supervised Deep Learning(TSSDL) [35], Density-Aware Graphs (DAG) [38] and
Pseudo-Label Mixup [20]. Unfortunately, due to the natural progress in the
field, each paper has different implementation choices which are not
standardised. Despite this, comparisons to this group are still useful as a
barometer for model performance.
2. 2.
Recent methods which used the WRN [41] (MixMatch [19], FixMatch (RandAugment
variant) [12] and UDA [13]). To guarantee a fair comparison to these
techniques, and as suggested by [45], we used a shared code-base for UDA and
FixMatch which reimplemented the original baselines. Additionally we then
ensured UDA and FixMatch used the same model code, the same optimiser with the
same parameters, the same number of optimisation steps and the same
RandAugment implementation as our approach.
Evaluation Protocol For each dataset we use the official train/test partition
and use the Top-1 error rate as the evaluation metric. For each result we give
the mean and standard deviation over five label splits.
## V Results and Discussion
In this section, we discuss the experiments we performed to evaluate and
compare our model against the state-of-the-art (SOTA). Additionally, we detail
several ablation experiments which explore the benefits of graph-based pseudo-
labels, the effect of augmentation averaging and evaluating the importance of
individual components.
### V-A Comparison to SOTA
Firstly, we test our model on the less complex CIFAR-10 and CIFAR-100
datasets. In Table III, we compare LaplaceNet against the first group of
methods using the $13$-CNN network. Our approach, by some margin, produces the
best results on CIFAR-10 and CIFAR-100 and represents a new SOTA for pseudo-
labels methods. We obtain a lower error rate using 500 labels than the recent
work of Arazo et al [20] obtain using 4000 labels. For CIFAR-100 LaplaceNet is
a full $6\%$ more accurate than any other approach and the first method to
achieve an error rate below $30\%$ on CIFAR-100 using 10k labels. In Table IV
we compare against the second group of methods using the WRN-28-2 network.
LaplaceNet is again the best performing method, outperforming the recent works
of UDA [13] and FixMatch [12]. In particular we find a significant increase in
performance on the more complex CIFAR-100 dataset and beat the other
considered methods by more than $3\%$ with 10k labels.
To test the performance of LaplaceNet on a more complex dataset, we evaluate
our model on the Mini-ImageNet dataset, which is a subset of the well known
ImageNet dataset and in Table V we compare our results against all others
methods which have used this dataset. Once again, we find our method performs
very well, producing an error rate a $10\%$ and $7\%$ better than any other
method on 4k and 10k labelled images respectively. Demonstrating our approach
can be applied to complex problems in the field. Additionally, we are more
than $20\%$ more accurate that the nearest graphical approach (LP).
To test the effect of increasing network size on our performance we also ran
our model on CIFAR-10/100 using an WRN-28-8(26 million parameters)
architecture and and compared that to the WRN-28-2(1.6 million parameters)
architecture in Table VI. Unsurprisingly, we achieved a large performance
improvement using a WRN-28-8 on both CIFAR-10 and CIFAR-100, with an $2.87$
error rate on CIFAR-10 using 4k labels and an $22.11\%$ error rate on
CIFAR-100 using 10k labels.
TABLE VI: The effect on Top-1 error rate by scaling up the neural network in
size from a WRN-28-2 to a WRN-28-8 on the CIFAR-10/100 datasets.
Dataset | CIFAR-10 | CIFAR-100
---|---|---
Model | 500 | 4000 | 4000 | 10000
WRN-28-2 | 5.57 $\pm$ 0.60 | 4.35 $\pm$ 0.10 | 33.16 $\pm$ 0.22 | 27.49 $\pm$ 0.22
WRN-28-8 | 3.81 $\pm$ 0.37 | 2.87 $\pm$ 0.18 | 26.61 $\pm$ 0.10 | 22.11 $\pm$ 0.23
(a)
(b)
Figure 1: Experimental comparison of the effect of using pseudo-labels
produced in a graphical framework versus pseudo-labels generated by the neural
network on the Top-1 error rate on the CIFAR-100 dataset ((a) 4k and (b) 10k
labelled images) with the 13-CNN network. Using graphically produced pseudo-
labels we achieve a much higher accuracy than using the network predictions.
### V-B Graph Based Pseudo-Labels
Many pseudo-label based techniques [12] [20] have produced state-of-the-art
results using pseudo-labels generated directly by the network rather than
using an energy based approach such as label propogation on a constructed
graph, which is computationally more complex. Therefore, in this section we
examine whether there is any advantage in using a graph based approach? To
test the importance of graph based pseudo-labels, we created two variants of
LaplaceNet, both without distribution alignment and with $n_{a}=1$.
1. 1.
The pseudo-labels are generated directly from the network predictions:
$\hat{y}_{i}=\text{argmax }f(x_{i})\text{ }\forall\text{ }i>l$
2. 2.
The pseudo-labels are generated from the graph, as in Equation 6,
$\hat{y}_{i}=\text{argmax}_{j}\text{ }F_{ij}\text{ }\forall\text{ }i>l$
We then compared the Top-1 error rate of these two variants on the CIFAR-100
dataset, see Fig 1. The graph-variant greatly outperformed the direct
prediction variant, emphasising the clear advantage that graphically produced
pseudo-labels have. What is contributing to this advantage? As an energy-based
approach, propogation on the graph incorporates information on the global
structure of the data, whilst the network is making a purely local decision at
each point. Arazo et al [20] showed that naive network based pseudo-label
approach could not generate an accurate solution for the ”two moons” toy
dataset, despite the fact that this problem has been solved by graphical
methods for some time [25]. Thus demonstrating that purely local decisions are
detrimental to accuracy when the global structure of data isn’t taken into
account.
### V-C Augmentation Averaging
In this paper we justify a multiple augmentation approach to further improve
semi-supervised models. In this section, we present the experimental
verification of our theoretical predictions about augmentation averaging as
well as comparing its effect to potential alternative techniques. To test the
effect of augmentation averaging we ran our approach on the CIFAR-100 dataset
using the 13-CNN network for a range of values $n_{a}=[1,3,5]$. Additionally
we compared the changed caused by augmentation averaging to the more common
approaches of scaling the batch size $b$ and labelled batch size $b_{l}$ by
$[1,3,5]$ and scaling the number of optimisation steps $S$ by $[1,3,5]$
To quantify the effect of a given change we use two measures: the augmentation
invariance of the classifier, which we define in this paper, and Top-1 error.
Augmentation invariance measures the extent to which the classifier’s
performance changes under data augmentation. Given an augmenation function
$u:\mathcal{X}\rightarrow\mathcal{X}$ and a classifier $f_{\theta}$ the
augmentation invariance $V$ with respect to a dataset $Z$ made up of $n$
point-label pairs $Z=\\{x_{i},y_{i}\\}_{i=1}^{n}$ is given by
$V_{Z}=\frac{\frac{1}{n}\sum_{i=1}^{n}\mathbbm{1}_{\operatorname*{arg\,max}f_{\theta}(u(x_{i}))=y_{i}}}{\frac{1}{n}\sum_{i=1}^{n}\mathbbm{1}_{\operatorname*{arg\,max}f_{\theta}(x_{i})=y_{i}}},$
(15)
which can be viewed as the performance ratio with and without data
augmentation. We consider both the augmentation invariance of our model with
respect to the fully labelled training and test data in order to give a full
picture of the model’s invariance, but we still only use a subset of the
labelled data for training.
TABLE VII: The effect of removing individual components from the baseline model on Top-1 error rate for CIFAR-100 on the 13-CNN network. | CIFAR-100
---|---
Model | 4k | 10k
Baseline | 32.41 $\pm$ 0.25 | 27.37 $\pm$ 0.20
Component Removed | |
RandAugment | 44.43 $\pm$ 0.66 | 34.75 $\pm$ 0.23
Distribution Alignment | 33.26 $\pm$ 0.24 | 29.07 $\pm$ 0.07
MixUp | 33.74 $\pm$ 0.26 | 28.02 $\pm$ 0.20
Figure 2: A comparison on the effect of increasing batch size versus
increasing the number of augmentation samples on Top-1 error rate, test data
augmentation invariance and training data augmentation invariance for the
CIFAR-100 dataset. Increasing the amount of augmentation averaging decreased
the error rate whilst also decreasing the sensitivity of the model’s output
predictions to augmented data. Increasing the batch size had a similar effect
on the model’s sensitivity, but it offered no improvement to model accuracy.
In Fig 2 we present our findings. We found that naively scaling the number of
optimisation steps without changing the hyperparameters led to the model
diverging as we spent too many epochs at a high learning rate. Therefore, we
provide results for the other two considered techniques which can be directly
compared. As theorised in Section III we find that increasing the number of
augmentation samples decreased the sensitivity of the model’s predictions to
augmentation on both the training and test data.An almost identical effect was
found by scaling the batch size. However, the major difference between the two
is their effect on Top-1 error rate. We found scaling the batch size offered
no improvement to Top-1 error, in-fact the largest batch size offered the
worst outcome, whilst increasing the number of augmentation samples noticeably
improved the model’s accuracy. Additionally as theorised in Section III, we
see that the gain in performance from $n_{a}=1\rightarrow 3$ is much greater
than $n_{a}=3\rightarrow 5$, supporting our statements regarding the
exponential bound in probability. These results suggests that scaling the
number of augmentation samples could be a great option for semi-supervised
models using suitable strong augmentations.
### V-D Component Evaluation
As LaplaceNet combines several different techniques, we tested the importance
of strong augmentation, distribution alignment and MixUp to the overall
accuracy of the model. We created a baseline model ($n_{a}=1$) and then remove
each component one at a time and tested the performance on the CIFAR-100
dataset, see Table VII. Whilst the removal of each component decreased the
performance of the model, it is clear the most crucial component to model
performance is strong augmentation and removing it drastically reduces model
accuracy. However, unlike other works [20] we find that whilst MixUp [22]
offers a small advantage is it not critical for composite batch pseudo-label
approaches. This may be due to the advantages of graph-based approaches
overcoming the flaws of naive neural network predictions.
## VI Conclusion
We propose a new graph based pseudo-label approach for semi-supervised image
classification, LaplaceNet, that outperforms the current state-of-the-art on
several datasets whilst having a much lower model complexity. Our model
utilises a simple single term loss function without the need for additionally
complexity whilst additionally avoiding the need for confidence thresholding
or temperature sharpening which was thought to be essential for state-of-the-
art performance. We instead generate accurate pseudo-labels through a graph
based technique with distribution alignment. We also explore the role that
augmentation plays in semi-supervised learning and justify a multi-sampling
approach to augmentation which we demonstrate through rigorous experimentation
improves both the generalisation of the network as well as the model’s
sensitivity to augmented data.
## Acknowledgment
PS thanks the UK Engineering and Physical Sciences Research Council (EPSRC)
and the National Physical Laboratory (NPL) for supporting this work. AIAR
gratefully acknowledges the financial support of the CMIH and CCIMI University
of Cambridge. CBS acknowledges support from the Philip Leverhulme Prize, the
Royal Society Wolfson Fellowship, the EPSRC grants EP/S026045/1 and
EP/T003553/1, EP/N014588/1, EP/T017961/1, the Wellcome Innovator Award
RG98755, the Leverhulme Trust project Unveiling the invisible, the European
Union Horizon 2020 research and innovation programme under the Marie
Skodowska-Curie grant agreement No. 777826 NoMADS, the Cantab Capital
Institute for the Mathematics of Information and the Alan Turing Institute.
## Appendix A Augmentation Pool
In this work we use RandAugment [16] rather than a learnt augmentation
strategy such as AutoAugment [15] which has a large computational cost. In
Table IX we detail the augmentation pool used. Additionally, we apply CutOut
[17] augmentation after RandAugment sampling.
We use two different augmentation strategies in our work: one for labelled
data and one for unlabelled data. We use ”strong” augmentations, RandAugment
and CutOut, on both labelled and unlabelled data with the only difference
being that we sample once from RandAugment for labelled data and twice for
unlabelled data. Given a pre-selected list of transformations, RandAugment
randomly samples from the list with each transformation having a magnitude
parameter. Rather than optimising this parameter on a validation set, which
may not exist in typical semi-supervised applications, we sample a random
magnitude from a pre-set range. This is same as is done in FixMatch [12] and
UDA [13]. We list the transformation pool for RandAugment and the
implementation of CutOut in Table IX.
TABLE VIII: Computational time taken for our approach using 4k labelled images on the CIFAR-100 dataset using the 13-CNN architecture. We provide the time taken for a number of different settings used in the results section. All experiments were performed using one NVIDIA P100 GPU. Model | Computational Time (Hours)
---|---
Baseline | 7.52 $\pm$ 0.04
Component Removal
No Distribution Alignment | 6.18 $\pm$ 0.01
No Strong Augmentation | 5.84 $\pm$ 0.03
No Graphical Propogation | 6.32 $\pm$ 0.01
Model Scaling
$3\times$-Batch-size | 12.28 $\pm$ 0.03
$5\times$-Batch-size | 17.23 $\pm$ 0.06
$3\times$-Samples | 12.88 $\pm$ 0.01
$5\times$-Samples | 18.14 $\pm$ 0.11
## Appendix B Computational Time
To give clarity on the how long our code takes to run we provide the
computational run times of LaplaceNet on the CIFAR-100 dataset using the
13-CNN model for a variety of settings, see Table VIII. Each experiment was
run on one P100 NVIDIA GPU. From Table VIII, we see that the time increased
caused by increasing the batch size or increasing the number of samples is
very similar. Component-wise, removing strong augmentation gives the largest
decrease in computational time whilst removing the graphical propogation saved
just over an hour on CIFAR-100. This represent a very small time trade off
given the advantages present in using graphical pseudo-labels.
TABLE IX: List of Transformations used in our application of RandAugment as
well their description and magnitude range. Additionally, we list the CutOut
transformation used at the end of RandAugment sampling.
Transformation | Description | Range
---|---|---
RandAugment Transformations
Autocontrast | Maximises the image contrast by setting the darkest (lightest) pixel to black (white) | —–
Brightness | Adjusts the brightness of the image. where $B=0$ returns a black image. image | $B\in[0.05,0.95]$
Color | Adjusts the colour balance of the image. $C_{l}=0$ returns a black and white image. | $C_{l}\in[0.05,0.95]$
Contrast | Controls the contrast of the image. $C_{o}=0$ returns a gray image. | $C_{o}\in[0.05,0.95]$
Equalise | Equalises the image histogram. | —–
Identity | Returns the original image. | —–
Posterise | Reduces each pixel to $B$ bits. | $B\in[4,8]$
Rotate | Rotates the image by $\theta$ degrees. | $\theta\in[-30,30]$
Sharpness | Adjusts the sharpness of the image, where $S=0$ returns a blurred image | $S\in[0.05,0.95]$
Shear X | Shears the image along the horizontal axis with rate $R$. | $R\in[-0.3,0.3]$
Shear Y | Shears the image along the vertical axis with rate $R$ | $R\in[-0.3,0.3]$
Solarize | Inverts all pixels above a threshold value of $T$ | $T\in[0,1]$
Translate X | Translates the image horizontally by ($\lambda$×image width) pixels. | $\lambda\in[-0.3,0.3]$
Translate Y | Translates the image vertically by ($\lambda$×image height) pixels | $\lambda\in[-0.3,0.3]$
CutOut Augmentation
CutOut | Sets a random square patch of side-length ($L$×image width) pixels to grey | $L\in[0,0.5]$
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|
# Injectivity of sampled Gabor phase retrieval in spaces with general
integrability conditions
Matthias Wellershoff University of Maryland, Department of Mathematics,
William E. Kirwan Hall, 4176 Campus Drive, College Park, MD 20742,
<EMAIL_ADDRESS>
###### Abstract
It was recently shown that functions in $L^{4}([-B,B])$ can be uniquely
recovered up to a global phase factor from the absolute values of their Gabor
transforms sampled on a rectangular lattice. We prove that this remains true
if one replaces $L^{4}([-B,B])$ by $L^{p}([-B,B])$ with $p\in[1,\infty]$. To
do so, we adapt the original proof by Grohs and Liehr and use a classical
sampling result due to Beurling. Furthermore, we present a minor modification
of a result of Müntz–Szász type by Zalik. Finally, we consider the
implications of our results for more general function spaces obtained by
applying the fractional Fourier transform to $L^{p}([-B,B])$ and for more
general nonuniform sampling sets.
Keywords Phase retrieval, Gabor transform, Sampling theory, Time-frequency
analysis
Mathematics Subject Classification (2010) 94A12, 94A20
## 1 Introduction
In this paper, we consider the _Gabor transform_ of functions $f\in
L^{p}(\mathbb{R})$, $p\in[1,\infty]$, given by
$\mathcal{G}f(x,\omega):=2^{1/4}\int_{\mathbb{R}}f(t)\mathrm{e}^{-\pi(t-x)^{2}}\mathrm{e}^{-2\pi\mathrm{i}t\omega}\,\mathrm{d}t,\qquad(x,\omega)\in\mathbb{R}^{2},$
and try to understand if one can recover $f$ from measurements of the absolute
value $\left\lvert\mathcal{G}f\right\rvert$ on discrete sets
$S\subset\mathbb{R}^{2}$. This so-called sampled Gabor phase retrieval problem
has recently been studied extensively [2, 3, 10, 11]. It is an elegant
mathematical problem in the sense that it is rather easy to state while, at
the same time, being less easy to solve. Moreover, it is connected to certain
audio processing applications such as the phase vocoder [8, 18].
A hallmark of all phase retrieval problems is that signals cannot be fully
recovered from phaseless measurements. For the Gabor phase retrieval problem,
we can see that the functions $f$ and $\mathrm{e}^{\mathrm{i}\alpha}f$, where
$\alpha\in\mathbb{R}$, generate the same measurements
$\left\lvert\mathcal{G}(\mathrm{e}^{\mathrm{i}\alpha}f)\right\rvert=\left\lvert\mathrm{e}^{\mathrm{i}\alpha}\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}f\right\rvert.$
Hence, we are not able to distinguish between $f$ and
$\mathrm{e}^{\mathrm{i}\alpha}f$ on the basis of their Gabor transform
magnitude samples. We will therefore consider the equivalence relation $\sim$
on $L^{p}(\mathbb{R})$ defined by
$f\sim
g:\iff\operatorname{\exists}\alpha\in\mathbb{R}:f=\mathrm{e}^{\mathrm{i}\alpha}g.$
(1)
With the help of this relation, we can introduce the phase retrieval operator
$\mathcal{A}:V/{\sim}\to[0,\infty)^{S}$, where $V$ is a subspace of
$L^{p}(\mathbb{R})$, by
$\mathcal{A}(f)(x,\omega):=\left\lvert\mathcal{G}f(x,\omega)\right\rvert,\qquad(x,\omega)\in
S,$
for $f\in V/{\sim}$. The _sampled Gabor phase retrieval problem_ is the
problem of inverting $\mathcal{A}$ when $S\subset\mathbb{R}^{2}$ is discrete.
We note that it has long been known that one can invert $\mathcal{A}$ for
$V=L^{2}(\mathbb{R})$ and $S=\mathbb{R}^{2}$. In applications, one does
typically not have access to measurements of the Gabor transform magnitude on
the entire time-frequency plane, however, and we thus believe that the sampled
Gabor phase retrieval problem is a natural first step towards a better
understanding of settings encountered in practice.
Relatively little was known about the inversion of $\mathcal{A}$ for discrete
sets $S$ until recently when a series of breakthroughs was presented in the
papers [2, 3, 10, 11]. For the genesis of this paper, the work in [11] was
most important. The authors of that paper show that sampled Gabor phase
retrieval is unique when $V=L^{4}([-B,B])$ and
$S=\mathbb{Z}\times(4B)^{-1}\mathbb{Z}$. The assumption that signals lie in
$L^{4}([-B,B])$ does not seem very natural, however, such that we are
interested in extending the result to spaces with more general integrability
conditions, and notably to $L^{2}([-B,B])$ as well as $L^{1}([-B,B])$. We will
do so here and prove the following main result.
###### Theorem 1.1.
Let $p\in[1,\infty]$, $B>0$ and $b\in(0,\tfrac{1}{4B})$. Then, the following
are equivalent for $f,g\in L^{p}([-B,B])$:
1. 1.
$f=\mathrm{e}^{\mathrm{i}\alpha}g$ for some $\alpha\in\mathbb{R}$,
2. 2.
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$\mathbb{N}\times b\mathbb{Z}$.
We observe that this theorem is almost optimal in view of the results
presented in [2]: there, for any lattice $S\subset\mathbb{R}^{2}$ in the time-
frequency plane, explicit examples $f,g\in L^{2}(\mathbb{R})$ were constructed
which do not agree up to global phase but which satisfy that
$\left\lvert\mathcal{G}f(x,\omega)\right\rvert=\left\lvert\mathcal{G}g(x,\omega)\right\rvert,\qquad(x,\omega)\in
S.$
In particular, it is necessary to restrict the Gabor phase retrieval problem
to a proper subspace $V$ of $L^{2}(\mathbb{R})$ in order to obtain a
uniqueness result from samples. It may not surprise the reader that one may
further generalise Theorem 1.1 in multiple ways to include more general
function spaces obtained by taking fractional Fourier transforms of elements
in $L^{p}([-B,B])$ or more general nonuniform sampling sets. Both of these
generalisation have already been suggested in [11] and we adapt them here. We
remark that during the review process for this paper, other manuscripts
addressing aspects of sampled short-time Fourier transform phase retrieval
have emerged, such as [1, 12, 13, 19].
#### Outline
In Section 2, we introduce some basic concepts needed for the further
understanding of this paper. Specifically, we introduce the fractional Fourier
transform and the Paley–Wiener spaces along with some of their most relevant
properties. Additionally, we provide two core insights: the first of those
being that the Gabor transform of a bandlimited function is bandlimited in the
first argument (cf. Lemma 2.4) and the second of those being a result of
Müntz–Szász type by Zalik (cf. Theorem 2.6).
In Section 3, we prove our main result, which states that for sufficiently
dense uniformly discrete sets $X$ and certain countable sets $\Omega$,
functions in $\mathrm{PW}_{B}^{p}$, $p\in[1,\infty]$, are uniquely determined
(up to global phase) from their Gabor magnitudes on $X\times\Omega$. We
finally extend this result by applying the fractional Fourier transform to
rotate the time-frequency plane.
#### Notation
We use the convention
$\mathcal{F}f(\xi)=\int_{\mathbb{R}^{d}}f(t)\mathrm{e}^{-2\pi\mathrm{i}(t,\xi)}\,\mathrm{d}t,\qquad\xi\in\mathbb{R}^{d},$
for the Fourier transform on the Schwartz space $\mathcal{S}(\mathbb{R}^{d})$.
It is well-known that the Fourier transform can be extended to the space of
tempered distributions $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and thereby to
all $L^{p}(\mathbb{R}^{d})$, $p\in[1,\infty]$. The inverse of the Fourier
transform is given by
$\mathcal{F}^{-1}g(t)=\int_{\mathbb{R}^{d}}g(\xi)\mathrm{e}^{2\pi\mathrm{i}(\xi,t)}\,\mathrm{d}\xi,\qquad
t\in\mathbb{R}^{d},$
for $g\in\mathcal{S}(\mathbb{R}^{d})$ and can likewise be extended to
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$. We use the translation operators
$\\{\operatorname{T}_{x}\\}_{x\in\mathbb{R}}$ given by
$\operatorname{T}_{x}f(t):=f(t-x),\qquad t\in\mathbb{R},$
for $x\in\mathbb{R}$, as well as the modulation operators
$\\{\operatorname{M}_{\omega}\\}_{\omega\in\mathbb{R}}$ given by
$\operatorname{M}_{\omega}f(t):=\mathrm{e}^{2\pi\mathrm{i}t\omega}f(t),\qquad
t\in\mathbb{R},$
for $\omega\in\mathbb{R}$. We denote the normalised Gaussian by
$\phi(t)=2^{1/4}\mathrm{e}^{-\pi t^{2}}$ where $t\in\mathbb{R}$. The
cardinality of a finite set $X\subset\mathbb{R}$ is denoted by $\left\lvert
X\right\rvert$. Finally, we denote the reflection and complex conjugation of
functions by $f^{\\#}(t):=\overline{f(-t)}$ for $t\in\mathbb{R}$, and the
duality pairing between the space of tempered distributions
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and the Schwartz space
$\mathcal{S}(\mathbb{R}^{d})$ by $\langle f,\psi\rangle$, where
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and
$\psi\in\mathcal{S}(\mathbb{R}^{d})$. With this, we can define the Gabor
transform of a tempered distribution $f\in\mathcal{S}^{\prime}(\mathbb{R})$
via $\mathcal{G}f(x,\omega):=\langle
f,\operatorname{M}_{\omega}\operatorname{T}_{x}\phi\rangle$.
## 2 Preliminaries
### 2.1 The fractional Fourier transform
The _fractional Fourier transform_ of a function $f\in\mathcal{S}(\mathbb{R})$
is defined by
$\mathcal{F}_{\theta}f(\xi):=c_{\theta}\mathrm{e}^{\pi\mathrm{i}\xi^{2}\cot\theta}\int_{\mathbb{R}}f(t)\mathrm{e}^{\pi\mathrm{i}t^{2}\cot\theta}\mathrm{e}^{-2\pi\mathrm{i}\frac{t\xi}{\sin\theta}}\,\mathrm{d}t,\qquad\xi\in\mathbb{R},$
for $\theta\in\mathbb{R}\setminus\pi\mathbb{Z}$, where
$c_{\theta}\in\mathbb{C}$ is the square root of $1-\mathrm{i}\cot\theta$ with
positive real part, and by $\mathcal{F}_{2k\pi}f:=f$ as well as
$\mathcal{F}_{(2k+1)\pi}f(\xi):=f(-\xi)$, for $\xi\in\mathbb{R}$, where
$k\in\mathbb{Z}$. One can show that the fractional Fourier transform satisfies
[14, Theorem 2.1 on p. 406]
$\int_{\mathbb{R}^{d}}\mathcal{F}_{\theta}f(\xi)g(\xi)\,\mathrm{d}\xi=\int_{\mathbb{R}^{d}}f(t)\mathcal{F}_{\theta}g(t)\,\mathrm{d}t,\qquad
f,g\in\mathcal{S}(\mathbb{R})$
and that it maps a Schwartz function to a Schwartz function [17, Theorem 5.3
on p. 170]. Therefore, we can extend the fractional Fourier transform to the
tempered distributions via
$\langle\mathcal{F}_{\theta}f,\psi\rangle:=\langle
f,\mathcal{F}_{\theta}\psi\rangle,\qquad\psi\in\mathcal{S}(\mathbb{R}),$
for $f\in\mathcal{S}^{\prime}(\mathbb{R})$.
The fractional Fourier transform is a powerful tool in time-frequency
analysis. One of its most crucial properties is that it corresponds to a
rotation of the time-frequency plane [4]. To describe this property, we
introduce the rotation operator
$\operatorname{R}_{\theta}:\mathbb{R}^{2}\to\mathbb{R}^{2}$, defined by
$\operatorname{R}_{\theta}(x,\omega):=(x\cos\theta-\omega\sin\theta,x\sin\theta+\omega\cos\theta)$,
where $\theta\in\mathbb{R}$ and $x,\omega\in\mathbb{R}$.
###### Lemma 2.1.
Let $\theta\in\mathbb{R}$ and $f\in\mathcal{S}^{\prime}(\mathbb{R})$. It holds
that
$\mathcal{G}\mathcal{F}_{\theta}f(x,\omega)=\mathrm{e}^{\pi\mathrm{i}\left(x^{2}-\omega^{2}\right)\sin\theta\cos\theta+2\pi\mathrm{i}x\omega\sin^{2}\theta}\cdot\mathcal{G}f(\operatorname{R}_{-\theta}(x,\omega)),\qquad(x,\omega)\in\mathbb{R}^{2}.$
###### Proof.
Let $(x,\omega)\in\mathbb{R}^{2}$ be arbitrary but fixed. We consider
$\mathcal{G}\mathcal{F}_{\theta}f(x,\omega)=\langle\mathcal{F}_{\theta}f,\operatorname{M}_{\omega}\operatorname{T}_{x}\phi\rangle=\langle
f,\mathcal{F}_{\theta}\operatorname{M}_{\omega}\operatorname{T}_{x}\phi\rangle$
such that the lemma boils down to understanding the commutative properties of
the fractional Fourier transform and the modulation and translation operators.
According to [4, Table I on p. 3086], it holds that
$\displaystyle\mathcal{F}_{\theta}\operatorname{T}_{x}=\mathrm{e}^{\pi\mathrm{i}x^{2}\sin\theta\cos\theta}\cdot\operatorname{M}_{-x\sin\theta}\operatorname{T}_{x\cos\theta}\mathcal{F}_{\theta},$
$\displaystyle\mathcal{F}_{\theta}\operatorname{M}_{\omega}=\mathrm{e}^{-\pi\mathrm{i}\omega^{2}\sin\theta\cos\theta}\cdot\operatorname{M}_{\omega\cos\theta}\operatorname{T}_{\omega\sin\theta}\mathcal{F}_{\theta}.$
Using that the fractional Fourier transform of the Gaussian is the Gaussian,
we therefore find
$\displaystyle\mathcal{F}_{\theta}\operatorname{M}_{\omega}\operatorname{T}_{x}\phi$
$\displaystyle=\mathrm{e}^{\pi\mathrm{i}\left(x^{2}-\omega^{2}\right)\sin\theta\cos\theta}\cdot\operatorname{M}_{\omega\cos\theta}\operatorname{T}_{\omega\sin\theta}\operatorname{M}_{-x\sin\theta}\operatorname{T}_{x\cos\theta}\phi$
$\displaystyle=\mathrm{e}^{\pi\mathrm{i}\left(x^{2}-\omega^{2}\right)\sin\theta\cos\theta+2\pi\mathrm{i}x\omega\sin^{2}\theta}\cdot\operatorname{M}_{\omega\cos\theta-x\sin\theta}\operatorname{T}_{x\cos\theta+\omega\sin\theta}\phi$
which proves the lemma. ∎
### 2.2 The Paley–Wiener spaces
In the following, we will mostly work with bandlimited functions. Precisely,
we consider the _Paley–Wiener spaces_ defined by
$\mathrm{PW}^{p}_{B}:=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R})\,\middle|\,f=\mathcal{F}F\mbox{
for some }F\in L^{p}([-B,B])\right\\},$
for $B>0$ and $p\in[1,\infty]$. One may see that the Paley–Wiener spaces are
nested — $\mathrm{PW}_{B}^{q}\subset\mathrm{PW}_{B}^{p}$ for $1\leq p\leq
q\leq\infty$ — which is due to the nestedness of $L^{p}$-spaces over closed
intervals — $L^{q}([-B,B])\subset L^{p}([-B,B])$ for $1\leq p\leq
q\leq\infty$. It therefore follows that
$\mathrm{PW}_{B}^{p}\subset\mathrm{PW}_{B}^{2}\subset L^{2}(\mathbb{R})$ for
$p\in[2,\infty]$ since the Fourier transform is unitary on
$L^{2}(\mathbb{R})$. Additionally, $\mathrm{PW}_{B}^{p}\subset
L^{q}(\mathbb{R})$ for $p\in[1,2]$, where $q\in[2,\infty]$ is the Hölder
conjugate of $p$, by the Hausdorff–Young inequality. Hence, the elements of
Paley–Wiener spaces are $L^{p}$-functions. In fact, one can show that the
elements of Paley–Wiener spaces extend to entire functions of exponential-
type. This is one direction of the well-known Paley–Wiener the- orem.
An important property of bandlimited functions is that one may recover them
from samples. Let us call a subset $X\subset\mathbb{R}$ _uniformly discrete_
if there exists an $\epsilon>0$ such that, for all $x,y\in X$ with $x\neq y$,
it holds that $\left\lvert x-y\right\rvert>\epsilon$. We say that $X$ is a
_set of uniqueness_ for $\mathrm{PW}_{B}^{p}$ if
$f(x)=0\mbox{ for all }x\in X\implies f=0,$
for $f\in\mathrm{PW}_{B}^{p}$. Similarly, we say that $X$ is a _set of
sampling_ for $\mathrm{PW}_{B}^{1}$ if there exists a constant $K>0$ such that
$\sup_{t\in\mathbb{R}}\left\lvert f(t)\right\rvert\leq K\sup_{x\in
X}\left\lvert f(x)\right\rvert,\qquad f\in\mathrm{PW}_{B}^{1}.$
Clearly, every set of sampling for $\mathrm{PW}_{B}^{1}$ is a set of
uniqueness for $\mathrm{PW}_{B}^{p}$ by the nestedness of the Paley–Wiener
spaces. Hence, $f\in\mathrm{PW}_{B}^{p}$ is uniquely determined by
$(f(x))_{x\in X}$ if $X$ is a set of sampling for $\mathrm{PW}_{B}^{1}$. Sets
of sampling for $\mathrm{PW}_{B}^{1}\subset C_{0}(\mathbb{R})$ were
characterised by Beurling in a series of seminar lectures given at Princeton
[6]. Specifically, Beurling introduces
$\underline{n}(r):=\inf_{t\in\mathbb{R}}\left\lvert
X\cap[t,t+r]\right\rvert,\qquad r>0,$
along with the _lower uniform density_
$\mathrm{l.u.d.}(X):=\lim_{r\to\infty}\frac{\underline{n}(r)}{r}$
and proves the following result.
###### Theorem 2.2 ([6, Theorem 5 on p. 346]).
A uniformly discrete set $X\subset\mathbb{R}$ is a set of sampling for
$\mathrm{PW}_{B}^{1}$ if and only if
$\mathrm{l.u.d.}(X)>2B.$
###### Remark 2.3.
Notably, there are uniformly discrete sets $X$ that form a set of uniqueness
for $\mathrm{PW}_{B}^{2}$ with $\mathrm{l.u.d.}(X)=0$ [15]. This illustrates
that $\mathrm{l.u.d.}(X)>2B$ is sufficient but not necessary for $X$ being a
set of uniqueness for $\mathrm{PW}_{B}^{p}$ in general.
Finally, we note that the Gabor transform of a bandlimited function is
bandlimited itself in the time variable (after modulation) and that therefore
the square of the Gabor transform magnitudes is bandlimited.
###### Lemma 2.4.
Let $p\in[1,\infty]$, $B>0$, $\omega\in\mathbb{R}$ and
$f\in\mathrm{PW}_{B}^{p}$. Then,
$x\mapsto\mathrm{e}^{2\pi\mathrm{i}x\omega}\mathcal{G}f(x,\omega)\in\mathrm{PW}_{B}^{p}$
and therefore
$x\mapsto\lvert\mathcal{G}f(x,\omega)\rvert^{2}\in\mathrm{PW}_{2B}^{q}$, where
$q\in[1,\infty]$ is such that
$1+\frac{1}{q}=\frac{2}{p}.$
###### Proof.
First, we write $\mathcal{G}f(x,\omega)$ in terms of the inverse Fourier
transform of the signal $\mathcal{F}^{-1}f$ and the Gaussian $\phi$ (cf. [9,
Lemma 3.1.1 on p. 39]):
$\mathrm{e}^{2\pi\mathrm{i}x\omega}\mathcal{G}f(x,\omega)=\mathcal{F}\left(\mathcal{F}f\cdot\operatorname{T}_{\omega}\phi\right)(-x)=\mathcal{F}\left(\mathcal{F}^{-1}f\cdot\operatorname{T}_{-\omega}\phi\right)(x).$
Next, we use the facts that $\mathcal{F}^{-1}f\in L^{p}([-B,B])$ and $\phi\in
L^{\infty}(\mathbb{R})$ to see that
$\mathcal{F}^{-1}f\cdot\operatorname{T}_{-\omega}\phi$ is also in
$L^{p}([-B,B])$. This implies that
$x\mapsto\mathrm{e}^{2\pi\mathrm{i}x\omega}\mathcal{G}f(x,\omega)$ is in the
Paley-Wiener space $\mathrm{PW}_{B}^{p}$.
We then compute the squared modulus of $\mathcal{G}f(x,\omega)$ as a
convolution of two functions in Fourier space: specifically, let
$F_{\omega}:=\mathcal{F}^{-1}f\cdot\operatorname{T}_{-\omega}\phi$. Then,
$\lvert\mathcal{G}f(x,\omega)\rvert^{2}=\mathcal{F}F_{\omega}(x)\cdot\overline{\mathcal{F}F_{\omega}(x)}=\mathcal{F}F_{\omega}(x)\cdot\mathcal{F}F_{\omega}^{\\#}(x)=\mathcal{F}\left(F_{\omega}\ast
F_{\omega}^{\\#}\right)(x).$
Using Young’s convolution inequality, we can show that $F\ast
F_{\omega}^{\\#}\in L^{q}(\mathbb{R})$, where $q\in[1,\infty]$ satisfies
$1+\frac{1}{q}=\frac{2}{p}.$
Finally, we can bound the support of $F\ast F_{\omega}^{\\#}$ by $[-2B,2B]$.
Putting everything together, we conclude that
$x\mapsto\lvert\mathcal{G}f(x,\omega)\rvert^{2}$ is in the Paley-Wiener space
$\mathrm{PW}_{2B}^{q}$. ∎
We have shown that $x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert^{2}$
belongs to the Paley–Wiener space
$\mathrm{PW}_{2B}^{q}\subset\mathrm{PW}_{2B}^{1}$. Therefore, we can apply
Theorem 2.2 to conclude that
$(\left\lvert\mathcal{G}f(x,\omega)\right\rvert)_{x\in X}$ completely
determines $x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert$, for
$\omega\in\mathbb{R}$ fixed, provided that $X\subset\mathbb{R}$ is a uniformly
discrete set such that $\mathrm{l.u.d.}(\Lambda)>4B$.
###### Remark 2.5.
A similar result can be proven for the short-time Fourier transform with
window $\psi\in\mathcal{F}L^{q}(\mathbb{R})$, where $q\in[1,\infty]$, by
following the same proof strategy. In this way, one may arrive at a partial
sampling result for short-time Fourier transform phase retrieval in which one
assumes knowledge of measurements on $X\times\mathbb{R}$, where $X$ is a set
of uniqueness for the appropriate Paley–Wiener space.
### 2.3 Zalik’s theorem
For the proof of our main result, we need the following result of Müntz–Szász
type due to Zalik. It asserts that certain translates of Gaussians are
complete in the spaces $L^{p}([a,b])$ and $C([a,b])$.
###### Theorem 2.6 (Zalik’s theorem; [20, Theorem 4 on p. 302]).
Let $p\in[1,\infty)$, $-\infty<a<b<\infty$, $c>0$, and let
$\Omega\subset\mathbb{R}$ be a countable set. Then,
$\left\\{t\mapsto\mathrm{e}^{-c(t-\omega)^{2}}\,\middle|\,\omega\in\Omega\right\\}$
is complete in $L^{p}([a,b])$ and $C([a,b])$ if and only if
$\sum_{\omega\in\Omega\setminus\\{0\\}}\left\lvert\omega\right\rvert^{-1}$ (2)
diverges.
###### Proof.
The result for $L^{p}([a,b])$ follows from a small modification of the
original proof in [20]. For $C([a,b])$ consider the following argument which
is also inspired by the original proof: suppose that the sum in equation (2)
diverges, and let $f\in\mathcal{C}([a,b])$ as well as $\epsilon>0$ be
arbitrary but fixed. Define
$g(x):=f\left(\frac{\log x}{2c}\right)\mathrm{e}^{\frac{\log^{2}x}{4c}},\qquad
x\in\left[\mathrm{e}^{2ca},\mathrm{e}^{2cb}\right],$
and note that $g\in C([\mathrm{e}^{2ca},\mathrm{e}^{2cb}])$. According to [16,
Theorem 6.1 on p. 30], $\\{x\mapsto x^{\omega}\,|\,\omega\in\Omega\\}$ is
complete in $C([\mathrm{e}^{2ca},\mathrm{e}^{2cb}])$. Therefore, there exist
$\Omega_{0}\subset\Omega$ finite and
$(\lambda_{\omega})_{\omega\in\Omega_{0}}\in\mathbb{C}$ such that
$\sup_{x\in[\mathrm{e}^{2ca},\mathrm{e}^{2cb}]}\left\lvert\sum_{\omega\in\Omega_{0}}\lambda_{\omega}x^{\omega}-g(x)\right\rvert<\epsilon.$
By the change of variable $x=\mathrm{e}^{2ct}$, it follows that
$\sup_{t\in[a,b]}\left\lvert\sum_{\omega\in\Omega_{0}}\lambda_{\omega}\mathrm{e}^{2c\omega
t}-f(t)\mathrm{e}^{ct^{2}}\right\rvert<\epsilon$
and therefore
$\sup_{t\in[a,b]}\left\lvert\sum_{\omega\in\Omega_{0}}\lambda_{\omega}\mathrm{e}^{c\omega^{2}}\cdot\mathrm{e}^{-c(t-\omega)^{2}}-f(t)\right\rvert\leq\sup_{t\in[a,b]}\left\lvert\sum_{\omega\in\Omega_{0}}\lambda_{\omega}\mathrm{e}^{2c\omega
t}-f(t)\mathrm{e}^{ct^{2}}\right\rvert<\epsilon$
showing that $\\{t\mapsto\mathrm{e}^{-c(t-\omega)^{2}}\,|\,\omega\in\Omega\\}$
is complete in $C([a,b])$.
Now, suppose that
$\\{t\mapsto\mathrm{e}^{-c(t-\omega)^{2}}\,|\,\omega\in\Omega\\}$ is complete
in $C([a,b])$. Then, a modification of the argument above shows that
$\\{x\mapsto x^{\omega}\,|\,\omega\in\Omega\\}$ is complete in
$C([\mathrm{e}^{2ca},\mathrm{e}^{2cb}])$. Therefore, [16, Theorem 6.1 on p.
30] implies that the sum in equation (2) diverges. ∎
###### Remark 2.7.
By the same proof, we can extend Zalik’s theorem to a countable subset
$\Omega$ of $\mathbb{C}$ under the condition that there exists a $\delta>0$
and a finite subset $\Omega_{0}\subset\Omega$ such that
$\left\lvert\operatorname{Re}\left(\omega-\frac{1}{2}\right)\right\rvert\geq\delta\left\lvert\omega-\frac{1}{2}\right\rvert,\qquad\omega\in\Omega\setminus\Omega_{0}.$
## 3 Main results
After laying out the groundwork, we present our main theorem, which
generalises a result previously established by Grohs and Liehr for
$\mathrm{PW}_{B}^{4}$ [11]. Specifically, we extend their result to hold for
the entire family of Paley–Wiener spaces $\mathrm{PW}_{B}^{p}$ for
$p\in[1,\infty]$. This encompasses the case $p=2$ which is commonly studied in
signal processing, as well as the challenging case $p=1$. Our result shows
that for sufficiently dense uniformly discrete sets $X$ and certain countable
sets $\Omega$, functions in $\mathrm{PW}_{B}^{p}$ are uniquely determined (up
to global phase) from their Gabor magnitudes on $X\times\Omega$.
###### Theorem 3.1 (Main result).
Let $p\in[1,\infty]$ and $B>0$. Let $X\subset\mathbb{R}$ be a uniformly
discrete set with $l.u.d.(X)>4B$ and let $\Omega\subset\mathbb{R}$ be a
countable set such that
$\sum_{\omega\in\Omega\setminus\\{0\\}}\left\lvert\omega\right\rvert^{-1}$
diverges. Then, the following are equivalent for $f,g\in\mathrm{PW}_{B}^{p}$:
1. 1.
$f=\mathrm{e}^{\mathrm{i}\alpha}g$ for some $\alpha\in\mathbb{R}$,
2. 2.
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$X\times\Omega$.
###### Proof of Theorem 3.1.
The following arguments are inspired by [11]. It is obvious that item 1
implies item 2. Therefore, we assume that $f,g\in\mathrm{PW}_{B}^{p}$ are such
that $\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$
on $X\times\Omega$. Let us now fix an arbitrary $\omega\in\Omega$ and note
that Lemma 2.4 implies that
$x\mapsto\lvert\mathcal{G}f(x,\omega)\rvert^{2},x\mapsto\lvert\mathcal{G}g(x,\omega)\rvert^{2}\in\mathrm{PW}_{2B}^{1}$.
It follows by Theorem 2.2 that
$\left\lvert\mathcal{G}f(x,\omega)\right\rvert=\left\lvert\mathcal{G}g(x,\omega)\right\rvert,\qquad
x\in\mathbb{R}.$
Following the argument in the proof of Lemma 2.4, it is readily shown that
$\left(\mathcal{F}^{-1}f\cdot\operatorname{T}_{-\omega}\phi\right)\ast\left(\mathcal{F}^{-1}f\cdot\operatorname{T}_{-\omega}\phi\right)^{\\#}=\left(\mathcal{F}^{-1}g\cdot\operatorname{T}_{-\omega}\phi\right)\ast\left(\mathcal{F}^{-1}g\cdot\operatorname{T}_{-\omega}\phi\right)^{\\#}$
in $L^{1}([-2B,2B])$. We reformulate the above to
$\int_{-B}^{B}\mathcal{F}^{-1}f(\eta)\overline{\mathcal{F}^{-1}f(\eta-\xi)}\phi(\eta+\omega)\phi(\eta-\xi+\omega)\,\mathrm{d}\eta\\\
=\int_{-B}^{B}\mathcal{F}^{-1}g(\eta)\overline{\mathcal{F}^{-1}g(\eta-\xi)}\phi(\eta+\omega)\phi(\eta-\xi+\omega)\,\mathrm{d}\eta$
for almost all $\xi\in\mathbb{R}$. By completing the square, we see that
$\phi(\eta+\omega)\phi(\eta-\xi+\omega)=\sqrt{2}\cdot\mathrm{e}^{-2\pi(\eta+\omega-\xi/2)^{2}}\cdot\mathrm{e}^{-\frac{\pi\xi^{2}}{2}}$
and therefore
$\int_{-B}^{B}\left(\mathcal{F}^{-1}f(\eta)\overline{\mathcal{F}^{-1}f(\eta-\xi)}-\mathcal{F}^{-1}g(\eta)\overline{\mathcal{F}^{-1}g(\eta-\xi)}\right)\mathrm{e}^{-2\pi(\eta+\omega-\xi/2)^{2}}\,\mathrm{d}\eta=0,$
(3)
for almost all $\xi\in\mathbb{R}$. According to Fubini’s theorem,
$\eta\mapsto
H_{\xi}(\eta):=\mathcal{F}^{-1}f(\eta)\overline{\mathcal{F}^{-1}f(\eta-\xi)}-\mathcal{F}^{-1}g(\eta)\overline{\mathcal{F}^{-1}g(\eta-\xi)}\in
L^{1}([-B,B])$
for almost all $\xi\in\mathbb{R}$. Let us now fix an arbitrary
$\xi\in\mathbb{R}$ such that the above two assertions hold and consider the
set $\Omega^{\xi}:=\tfrac{\xi}{2}-\Omega$. Then,
$\Omega^{\xi}\subset\mathbb{R}$ is countable and
$\sum_{\omega\in\Omega^{\xi}\setminus\\{0\\}}\left\lvert\omega\right\rvert^{-1}$
diverges. Therefore, Theorem 2.6 implies that the set
$\left\\{\eta\mapsto\mathrm{e}^{-2\pi(\eta-\omega)^{2}}\,\middle|\,\omega\in\Omega^{\xi}\right\\}$
is complete in $C([-B,B])$. It follows that, for all $t\in\mathbb{R}$ and all
$\epsilon>0$, there exist a finite set $\Omega_{0}^{\xi}\subset\Omega^{\xi}$
and a sequence
$(\lambda_{\omega}^{\xi})_{\omega\in\Omega_{0}^{\xi}}\in\mathbb{C}$ such that
$\sup_{\eta\in[-B,B]}\left\lvert\sum_{\omega\in\Omega_{0}^{\xi}}\lambda_{\omega}^{\xi}\mathrm{e}^{-2\pi(\eta-\omega)^{2}}-\mathrm{e}^{2\pi\mathrm{i}\eta
t}\right\rvert<\epsilon.$
According to equation (3), we have
$\displaystyle\left\lvert\int_{-B}^{B}H_{\xi}(\eta)\mathrm{e}^{2\pi\mathrm{i}\eta
t}\,\mathrm{d}\eta\right\rvert$ $\displaystyle\leq\int_{-B}^{B}\left\lvert
H_{\xi}(\eta)\right\rvert\left\lvert\sum_{\omega\in\Omega_{0}^{\xi}}\lambda_{\omega}^{\xi}\mathrm{e}^{-2\pi(\eta-\omega)^{2}}-\mathrm{e}^{2\pi\mathrm{i}\eta
t}\right\rvert\,\mathrm{d}\eta$ $\displaystyle\leq\epsilon\cdot\left\lVert
H_{\xi}\right\rVert_{1}.$
Since the above holds for all $\epsilon>0$, we conclude
$\mathcal{F}^{-1}H_{\xi}(t)=\int_{-B}^{B}H_{\xi}(\eta)\mathrm{e}^{2\pi\mathrm{i}\eta
t}\,\mathrm{d}\eta=0,\qquad t\in\mathbb{R}.$
Therefore, $H_{\xi}=0$ in $L^{1}(\mathbb{R})$ which implies that
$\mathcal{F}^{-1}f(\eta)\overline{\mathcal{F}^{-1}f(\eta-\xi)}=\mathcal{F}^{-1}g(\eta)\overline{\mathcal{F}^{-1}g(\eta-\xi)}$
(4)
for almost all $\eta\in\mathbb{R}$.
The rest of the proof is classical. We present the following argument inspired
by [5, Theorem 2.5 on p. 588] for the convenience of the reader: another
application of Fubini’s theorem implies that
$\xi\mapsto\mathcal{F}^{-1}f(\eta)\overline{\mathcal{F}^{-1}f(\eta-\xi)}\in
L^{1}(\mathbb{R})$
for almost all $\eta\in\mathbb{R}$ (and the same is true for $g$). If we fix
such an $\eta$, then we can apply the Fourier transform to equation (4) and
obtain
$\mathcal{F}^{-1}f(\eta)\overline{f(t)}=\mathcal{F}^{-1}g(\eta)\overline{g(t)},\qquad
t\in\mathbb{R}.$
If $f=0$, then $\mathcal{F}^{-1}g(\eta)\overline{g(t)}=0$ such that either
$g=0$ or $\mathcal{F}^{-1}g=0$ almost everywhere in which case $g=0$. Hence,
the theorem is proven if $f=0$. So let us assume that $f\neq 0$. Then, there
exists $t_{0}\in\mathbb{R}$ such that $f(t_{0})\neq 0$. Therefore,
$\mathcal{F}^{-1}f=\tau\cdot\mathcal{F}^{-1}g,\qquad\tau:=\frac{\overline{g(t_{0})}}{\overline{f(t_{0})}}$
in $L^{1}([-B,B])$ and thus $f=\tau g$. This proves the theorem since it
implies $\left\lvert f(t_{0})\right\rvert=\left\lvert g(t_{0})\right\rvert$
which shows that $\left\lvert\tau\right\rvert=1$. ∎
We can use the fractional Fourier transform to rotate our result in the time-
frequency plane. To this end, we introduce the spaces
$\mathcal{F}_{\theta}L^{p}([-B,B]):=\left\\{f\in\mathcal{S}^{\prime}(\mathbb{R})\,\middle|\,f=\mathcal{F}_{\theta}F\mbox{
for some }F\in L^{p}([-B,B])\right\\},$
for $B>0$ and $p\in[1,\infty]$. These spaces are nested similarly to the
Paley-Wiener spaces, with
$\mathcal{F}_{\theta}L^{q}([-B,B])\subset\mathcal{F}_{\theta}L^{p}([-B,B])$
for $1\leq p\leq q\leq\infty$. Thus, for $p\geq 2$, we have
$\mathcal{F}_{\theta}L^{p}([-B,B])\subset\mathcal{F}_{\theta}L^{2}([-B,B])\subset
L^{2}(\mathbb{R})$ since the fractional Fourier transform is unitary on
$L^{2}(\mathbb{R})$. Moreover, for $1\leq p\leq 2$, we have
$\mathcal{F}_{\theta}L^{p}([-B,B])\subset L^{q}(\mathbb{R})$, where $q$ is the
Hölder conjugate of $p$, since the Hausdorff-Young inequality extends to the
fractional Fourier transform [7, Theorem 4.2 on p. 88].
With these spaces in hand, we can now state and prove a generalisation of our
main result.
###### Theorem 3.2 (Generalised main result).
Let $p\in[1,\infty]$, $B>0$ and $\theta\in\mathbb{R}$. Let
$X\subset\mathbb{R}$ be a set of uniqueness for $\mathrm{PW}_{2B}^{1}$ and let
$\Omega\subset\mathbb{R}$ be a countable set such that
$\sum_{\omega\in\Omega\setminus\\{0\\}}\left\lvert\omega\right\rvert^{-1}$
diverges. Then, the following are equivalent for
$f,g\in\mathcal{F}_{\theta}L^{p}([-B,B])$:
1. 1.
$f=\mathrm{e}^{\mathrm{i}\alpha}g$ for some $\alpha\in\mathbb{R}$,
2. 2.
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$\operatorname{R}_{\theta}(\Omega\times X)$.
###### Proof.
Clearly, item 1 implies item 2. We will therefore assume that
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$\operatorname{R}_{\theta}(\Omega\times X)$. According to Lemma 2.1, we have
$\left\lvert\mathcal{G}\mathcal{F}_{-\theta}f\right\rvert=\left\lvert\mathcal{G}\mathcal{F}_{-\theta}g\right\rvert$
on $\Omega\times X$. Note that
$\mathcal{F}_{-\theta}f,\mathcal{F}_{-\theta}g\in L^{p}([-B,B])$ such that
$\mathcal{F}\mathcal{F}_{-\theta}f,\mathcal{F}\mathcal{F}_{-\theta}g\in\mathrm{PW}_{B}^{p}$.
Another application of Lemma 2.1 yields
$\left\lvert\mathcal{G}\mathcal{F}\mathcal{F}_{-\theta}f\right\rvert=\left\lvert\mathcal{G}\mathcal{F}\mathcal{F}_{-\theta}g\right\rvert$
on $X\times\Omega$. Therefore, Theorem 3.1 implies that
$\mathcal{F}\mathcal{F}_{-\theta}f=\mathrm{e}^{\mathrm{i}\alpha}\cdot\mathcal{F}\mathcal{F}_{-\theta}g$
for some $\alpha\in\mathbb{R}$. Finally, $f=\mathrm{e}^{\mathrm{i}\alpha}g$
follows from Fourier inversion. ∎
#### Acknowledgements
The author would like to thank Rima Alaifari for her comments which have
improved the presentation of the paper as well as the first reviewer whose
remarks have inspired a generalisation of the main results. Additionally, the
author acknowledges funding through the SNSF Grant 200021_184698.
#### Declaration of generative AI and AI-assisted technologies in the writing
process
During the review process for this work, the author used ChatGPT to correct
punctuation and orthography. After using this tool, the author reviewed and
edited the content as needed and takes full responsibility for the content of
the publication.
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|
# ChiBench: a Benchmark Suite for Testing Electronic Design Automation Tools
Rafael Sumitani 0009-0003-5226-1966 UFMGBrazil<EMAIL_ADDRESS>,
João Victor Amorim 0009-0001-6775-2579 UFMGBrazil<EMAIL_ADDRESS>,
Augusto Mafra 0000-0000-0000-0000 CadenceBrazil<EMAIL_ADDRESS>,
Mirlaine Crepalde 0000-0000-0000-0000 CadenceBrazil<EMAIL_ADDRESS>and
Fernando M Quintão Pereira 0000-0002-0375-1657 UFMGBrazil
<EMAIL_ADDRESS>
###### Abstract.
Electronic Design Automation (EDA) tools are software applications used by
engineers in the design, development, simulation, and verification of
electronic systems and integrated circuits. These tools typically process
specifications written in a Hardware Description Language (HDL), such as
Verilog, SystemVerilog or VHDL. Thus, effective testing of these tools
requires benchmark suites written in these languages. However, while there
exist some open benchmark suites for these languages, they tend to consist of
only a handful of specifications. This paper, in contrast, presents ChiBench,
a comprehensive suite comprising 50 thousand Verilog programs. These programs
were sourced from GitHub repositories and curated using Verible’s syntactic
analyzer and JasperTM’s HDL semantic analyzer. Since its inception, ChiBench
has already revealed bugs in public tools like Verible’s obfuscator and
parser. In addition to explaining some of these case studies, this paper
demonstrates how ChiBench can be used to evaluate the asymptotic complexity
and code coverage of typical electronic design automation tools.
Benchmark, Verilog, Testing
††ccs: Software and its engineering Compilers
## 1\. Introduction
EDA (Electronic Design Automation) tools are software applications used by
engineers in the design, development, simulation, and verification of
electronic systems and integrated circuits (ICs). These tools cover various
stages of the electronic design process, from conceptualization and design
entry to implementation, verification, and testing. Examples of EDA tools
include Cadence Jasper for formal verification111Available at
https://www.cadence.com/en_US/home/tools/system-design-and-
verification/formal-and-static-verification.html and Verible’s
tool222Available at https://github.com/chipsalliance/verible. These tools
operate on similar types of input data: programs in some Hardware Description
Language (HDL), such as Verilog, SystemVerilog, VHDL or SystemC. Thus, the
effective development and testing of such tools require benchmarks in these
languages.
#### Verilog Benchmarks (or their lack thereof)
A Benchmark Suite is a collection of programs used to test computing systems
that process such programs. There exist open source benchmark collections
tailored for EDA tools, such as the ISCAS Benchmark Circuits (Brglez et al.,
1989) (31 circuits), the MCNC Benchmark Circuits (Koźmiński, 1991) (19
circuits in the YAL format), the EPFL Combinational Benchmark Suite (Amaru et
al., 2019) (23 circuits), the RAW Benchmark Suite (Babb et al., 1997) (twelve
programs), the KOIOS collection (19 circuits implementing different neural
networks) and the Titan23 suite of 23 circuits (Murray et al., 2015). These
collections contain a small number of programs: typically less than 50. This
fact is unfortunate because, in the works of Wang and O’Boyle (2018):
“Although there are numerous benchmark sites publicly available, the number of
programs available is relatively sparse compared to the number that a typical
compiler will encounter in its lifetime.” This paper mitigates this problem,
releasing a much larger collection of Verilog circuits.
#### The Contribution of this Work
This paper describes ChiBench, an open collection of 50K Verilog programs,
mined from open-source GitHub repositories. These programs were curated in a
three-step process, which Section 2 explains: first, Verilog codes were
automatically scraped from public code repositories. In a second phase, each
program was parsed using Verible’s syntactic analyzer, to ensure compliance
with the IEEE Standard 1364 (Thomas and Moorby, 1996). Finally, these programs
were sieved via Jasper’s HDL semantic analyzer, to ensure that each circuit
contains all the required dependencies.
This paper illustrates three different usage scenarios where ChiBench has been
employed, using two tools from the Verible project as test subjects. First,
Section 3.1 demonstrates how ChiBench programs can be used to investigate the
asymptotic complexity of EDA tools. Second, Section 3.2 explains how these
programs can test EDA tools by gauging the coverage of ChiBench as a test
dataset. Finally, Section 3.3 briefly describes two bugs discovered in open-
source tools through ChiBench-based tests.
## 2\. The Construction of a Benchmark Suite
In order to build our Benchmark Suite, we have mined programs from open-source
GitHub repositories, using GitHub’s REST API 333Available at
https://docs.github.com/en/rest. We use GitHub’s API to build a list of
candidate Verilog repositories. Said list is sorted by popularity (measured as
the number of stargazers). We remove from the candidate list repositories that
are not available for public usage, due to the lack of a license. Thus, for
each repository $R$ in the sorted list, we have implemented a Python script
that proceeds as follows:
1. (1)
Clone $R$ and locally copy all its .v files;
2. (2)
Assigns a unique name to each .v file, based on its repository and its local
path;
3. (3)
Remove any special characters from the file’s name to avoid encoding issues.
We repeat the above sequence of steps for all the repositories in the base
list, until reaching a predefined number of files. This threshold is set upon
calling the mining script. Currently, ChiBench provides only Verilog programs;
however, the mining script can be easily adapted to fetch files in any other
language that is tagged in GitHub, such as VHDL.
### 2.1. Curating the Data
After we have copied the necessary number of Verilog files from GitHub, we
proceed to select valid programs. To this effect, we only keep files that are
syntactically and semantically valid. Thus, this process involves passing the
files through two sieves. The first sieve, the syntax analysis, happens via
the Verible syntactic analyzer. At this stage, if Verible’s parser cannot
build an abstract syntax tree for a file, we discard it. Example 2.1
illustrates one such situation.
###### Example 2.1.
The program in Figure 1, which specifies an 8-bit counter, will be filtered
out by the syntactic filter. It contains a missing semicolon at Line 7. Such
syntactically invalid files are uncommon in the mining process. Nevertheless,
they occur, as the repositories contain, for instance, files that are still
under development.
Figure 1. Verilog specification filtered out by our syntactic verification.
Verilog specification filtered out by our syntactic verification.
Once we remove any syntactically invalid programs, we use Jasper’s HDL
semantic analyzer to filter out any semantically invalid programs444Jasper’s
HDL semantic analyzer is triggered with the analyze built-in command.. Notice
that Jasper’s HDL analyzer also rejects invalid syntax. However, Jasper’s HDL
analyzer is more computationally expensive than Verible’s because it also
considers semantic analysis, being more restricted to Verilog language
standards. Consequently, in order to reduce the number of programs sent for
semantic analysis, we chose to filter out syntactically invalid programs
before using Jasper. Example 2.2 better explains what is the role of the
semantic analysis.
###### Example 2.2.
Figure 2 shows an example of a program that fails the semantic sieve due to a
type inconsistency. In this case, the IEEE standard forbids the declaration of
data ports with the wire type. Our data generation process considers each file
independently, thus, this error might occur. Nevertheless, our experience is
that most Verilog programs can be successfully validated as a single
compilation unit.
Figure 2. Verilog specification that fails the semantic test.
Verilog specification filtered out by our semantic verification.
### 2.2. Licensing
ChiBench only contains files that provide permissive licenses. In other words,
we remove from the public distribution of this collection any Verilog
specification that comes from either a repository without a license or that
comes from a repository whose license prevents distribution. Figure 3 shows
the number of licenses in each repository used to build ChiBench. Each
ChiBench file contains, as a comment, the URL of the repository from where it
comes, plus its license.
Figure 3. Licenses of repositories used to build ChiBench.
Licenses of repositories used to build ChiBench.
## 3\. Evaluation
The goal of this section is to demonstrate that ChiBench is a useful
collection of benchmarks. To this end, we shall investigate the following
three research questions:
RQ1::
Can ChiBench Programs be used to infer, empirically, the asymptotic complexity
of EDA tools?
RQ2::
How much coverage should we expect to obtain by using ChiBench as a dataset to
test EDA tools?
RQ3::
Are ChiBench-based tests able to uncover zero-day bugs in well-known EDA
tools?
RQ4::
How are ChiBench’s programs characterized in terms of size?
#### Experimental Setup
The evaluation described in this Section uses two tools available in the
Verible Project (Release v0.0-3622-g07b310a3, Mar 13th, 2024) as test
subjects: the parser and the obfuscator. Experiments run on an AMD Ryzen 9
5900X 12-Core 3.7GHz processor featuring Ubuntu Linux 22.04.4 LTS (kernel
6.5.0-27-generic). Coverage is measured via Clang’s source-based code coverage
feature (available in Clang 14.0.0).
### 3.1. RQ1 – Asymptotic Analysis
The asymptotic complexity of a tool is an expression that relates the running
time of that tool as a function of its input size. Determining an analytical
formula for the asymptotic complexity of a tool is a challenging problem, as
this formula is influenced by many details of that tool’s implementation.
Thus, as an alternative, to understand the asymptotic behavior of
implementations of algorithms, researchers resort to the so-called empirical
complexity analysis. In the words of Sumitani et al. (2023): “Empirical
Complexity Analysis is a branch of computer science that tries to infer the
asymptotic complexity of algorithms through the observation of multiple
executions of that algorithm with different inputs”. In this paper, we show
how ChiBench can be effectively used as a means to carry out empirical
complexity analysis.
Figure 4. The impact of the number of tokens on Verible’s parsing time. This
figure contains about 50K points—one point per program in the ChiBench
collection.
The impact of the number of tokens on Verible’s parsing time.
#### Discussion
Figure 4 relates the number of tokens in ChiBench programs with the time that
Verible’s syntactic analyzer takes to parse these programs. To perform this
experiment, we used the programs in the ChiBench collection as input to
Verible’s parser. As Figure 4 shows, the running time behavior of Verible is
linear. In this regard, Pearson’s coefficient relating running time and number
of tokens is 0.99, with a p-value of less than $2^{-16}$. Therefore, Verible’s
parser is expected to run in linear time on the number of tokens that form the
program with very strong probability.
### 3.2. RQ2 – Coverage Analysis
Coverage is a metric that quantifies the effectiveness of a test suite. In
this paper, we report coverage in two ways. Either as the number of lines in
the source code of a program that the test suite exercises. Or else as the
number of branches exercised by the test case in the binary representation of
that same program. Thus, we define the coverage ratio as either the number of
lines covered divided by the total lines of code or as the number of branches
covered divided by the total number of branches. The higher the coverage
ratio, the better the test suite. This section analyzes the coverage ratio of
ChiBench.
Figure 5. Code coverage of different tools of the Verible Framework, using the
128 largest programs from ChiBench.
Code coverage of different tools of the Verible Framework, using the 128
largest programs from ChiBench.
#### Discussion
Figure 5 presents the code coverage for Verible’s obfuscator and parser
assessed with the 128 largest programs from ChiBench. The coverage pattern for
both tools is similar; however, the parser demonstrates a notably higher
coverage. For lines covered it ranges from 10% to 19% against 4.5% to 7% for
the obfuscator. In terms of branches covered, the parser starts at 31% and
reaches 36% whereas the obfuscator begins at 11% and ends at nearly 12%. The
red lines represent the code coverage when using all programs from ChiBench.
For the parser, we achieved 42% coverage for lines and 48% for branches. In
comparison, the obfuscator reached 14% for lines and nearly 16% for branches.
We hypothesize that this disparity arises because the parser is a larger
feature and, consequently, calls a larger variety of functions from Verible’s
modules. In contrast, the obfuscator is a more self-contained tool. Although
these numbers seem low in principle, we remind the reader that Verible is a
large framework, which includes several tools, such as a linter, a code
formatter, and a language server. The code that forms these parts, although
part of the Verible binary library, remained largely unseen during this
experiment.
### 3.3. RQ3 – Actual Bug Reports
One of the primary goals of a benchmark suite is to uncover bugs in the
implementation of tools that read this suite. ChiBench was initially conceived
to stress-test any EDA tool and to demonstrate the effectiveness of this
collection as a bug-finding mechanism, we apply it onto open-source tools
available in the Verible Framework.
#### Discussion
We have used ChiBench to evaluate the correctness of Verible’s parser and code
obfuscator. In this case, we define as a buggy evaluation the analysis of a
program from ChiBench that results in a “crash”; that is, an execution of the
subject tool that leads to either an assertion or to an operating system
exception (such as a segmentation fault). Under these definitions, we have
observed one buggy evaluation on the parser and another on the obfuscator.
Both have been reported to the Verible’s community:
* •
https://github.com/chipsalliance/verible/issues/2159:
Verible’s obfuscator crashes when reading a program that only contains the
pragma directive.
* •
https://github.com/chipsalliance/verible/issues/2181:
Verible’s parser crashes instead of reporting syntax errors related to
instantiation type.
Interestingly, the second issue—which has been acknowledged as a true bug—was
not activated by a program in ChiBench itself, but by a program derived from
the suite: to maximize the diversity of ChiBench programs, we have used this
collection to generate new programs. Generation works as follows: we use the
50K programs in ChiBench to calculate the probability that each production
rule in Verible’s parser is exercised when parsing a program. Then, we use
this probabilistic grammar to generate new programs. In this case, every type
is set as logic.
### 3.4. RQ4 – Size Characterization
ChiBench programs are mined from open-source repositories; hence, we speculate
that they approximate the average Verilog program that represents typical
circuits. This section provides some characterization of such programs. To
this end, we analyze their size distribution.
#### Discussion
Figure 6 shows a density curve representing the size distribution of ChiBench
programs. We measure size as the number of tokens that the Verible lexer
produces for each Verilog file. Notice that this metric does not depend on
user-defined names. Figure 6 makes it clear that most Verilog specifications
are small. In total, ChiBench contains 50,611 Verilog programs. Out of this
lot, 66.87% contain less than 1,000 tokens, and 79.24% contain less than
2,000. The largest program in ChiBench contains 25,690,281 tokens, and the
smallest contains only 4. The average number of tokens is 33,350.9, and the
median number is 489.
Figure 6. Distribution of ChiBench programs per size, measured in number of
tokens.
Distribution of ChiBench programs per size, measured in number of tokens.
## 4\. Related Work
As we have already mentioned in Section 1, there exist already collections of
benchmarks formed by hardware specification languages (Brglez et al., 1989;
Koźmiński, 1991; Amaru et al., 2019; Babb et al., 1997; Murray et al., 2015).
However, these collections are small: never containing more than 50 circuits.
Nevertheless, with the rising popularity of large language models, this
scenario has seen changes in the last year.
As an example of this new trend, at the end of 2023, Thakur et al. (2024)
released VeriGen, a version of the CodeGen (Nijkamp et al., 2023) fine-tuned
for the synthesis of Verilog specifications. In the process of tuning CodeGen,
Thakur et al. have collected 50K Verilog circuits. However, in contrast to
ChiBench, the dataset used by Thakur et al. has not undergone any form of
filtering; hence, we do not know if these programs are semantically valid.
This dataset is publicly available555At
https://huggingface.co/datasets/shailja/Verilog_GitHub; however, each program
is a single line string. Parsing these programs automatically to reconstruct
the original Verilog specification is not possible, due to the presence of
line comments in the codes. This shortcoming is not a problem in the context
of Thakur et al.’s work, given that they are interested in building a large
language model that is based on k-grams of Verilog codes. Yet, the dataset
used to train the model could not be used, for instance, to stress-test EDA
tools. We believe that ChiBench might be useful to support the implementation
of models like VeriGen. To this effect, independent evaluations of VeriGen
have found that “A primary contributing factor to this shortfall [the
inability to uncover bugs in EDA tool] is the insufficiency of HDL code
resources for training” (Nijkamp et al., 2023). This perceived lack of
benchmarks seems to be a common issue among researchers working on large
language models for hardware specifications (Yao et al., 2024; Tsai et al.,
2024).
## 5\. Conclusion
This paper has described ChiBench, a collection of 50K Verilog programs mined
from open-source GitHub repositories. In addition to explaining the
methodology to build this suite, this paper showed how ChiBench can be
effectively used to test and analyze the behavior of electronic design
automation tools. The infrastructure used in the construction of ChiBench can
be adjusted to other languages, such as VHDL. Thus, future work might involve
maximizing the diversity of ChiBench programs to broaden coverage of EDA
tools.
#### Software
ChiBench is available at https://github.com/lac-dcc/chimera. Each benchmark
available in ChiBench provides, as a header comment, its original license, in
addition to the repository from where that specification was obtained.
## Acknowledgement
This project is sponsored by Cadence Design Systems. Additionally, the authors
acknowledge the support of CNPq (grant 406377/2018-9), FAPEMIG (grant
PPM-00333-18), and CAPES (grant “Edital CAPES PrInt”).
## References
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* Sumitani et al. (2023) Rafael Sumitani, Lucas Silva, Frederico Campos, and Fernando Pereira. 2023. A Class of Programs that Admit Exact Complexity Analysis via Newton’s Polynomial Interpolation. In _SBLP_. ACM, New York, NY, USA, 50–55. https://doi.org/10.1145/3624309.3624311
* Thakur et al. (2024) Shailja Thakur, Baleegh Ahmad, Hammond Pearce, Benjamin Tan, Brendan Dolan-Gavitt, Ramesh Karri, and Siddharth Garg. 2024. VeriGen: A Large Language Model for Verilog Code Generation. _ACM Trans. Des. Autom. Electron. Syst._ 29, 3, Article 46 (apr 2024), 31 pages. https://doi.org/10.1145/3643681
* Thomas and Moorby (1996) Donald E. Thomas and Philip R. Moorby. 1996. _The VERILOG Hardware Description Language_ (3rd ed.). Kluwer Academic Publishers, USA.
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|
# Emergence of 6-particle “hexciton” states in WS2 and MoSe2 monolayers
J. Choi1,2, J. Li1,3, D. Van Tuan4, H. Dery4,5, S. A. Crooker1 1National High
Magnetic Field Laboratory, Los Alamos, NM 87545, USA 2Advanced
Instrumentation Institute, Korea Research Institute of Standards and Science,
Daejeon 34113, Korea 3Wuhan National High Magnetic Field Center and School of
Physics, Huazhong University of Science and Technology, Hubei 430074, China
4Department of Electrical and Computer Engineering, University of Rochester,
Rochester, New York 14627, USA 5Department of Physics and Astronomy,
University of Rochester, Rochester, New York 14627, USA
###### Abstract
When doped with a high density of mobile charge carriers, monolayer
transition-metal dichalcogenide (TMD) semiconductors can host new types of
composite many-particle exciton states that do not exist in conventional
semiconductors. Such multi-particle bound states arise when a photoexcited
electron-hole pair couples to not just a single Fermi sea that is quantum-
mechanically distinguishable (as for the case of conventional charged excitons
or trions), but rather couples simultaneously to multiple Fermi seas, each
having distinct spin and valley quantum numbers. Composite six-particle
“hexciton” states were recently identified in electron-doped WSe2 monolayers,
but under suitable conditions they should also form in all other members of
the monolayer TMD family. Here we present spectroscopic evidence demonstrating
the emergence of many-body hexcitons in charge-tunable WS2 monolayers (at the
A-exciton) and MoSe2 monolayers (at the B-exciton). The roles of
distinguishability and carrier screening on the stability of hexcitons are
discussed.
The monolayer transition-metal dichalcogenide (TMD) semiconductors such as
WSe2, MoSe2, WS2, and MoS2 host a multitude of excitonic complexes, due in
part to the spin-orbit-split nature of their conduction and valence bands at
the $K$ and $K^{\prime}$ valleys of the Brillouin zone Urbaszek:2018 ;
Muller:2018 . When they are optically allowed and possess a non-zero
oscillator strength, these bound complexes manifest as discrete resonances in
optical absorption spectra. Early studies of nominally undoped TMD monolayers
revealed pronounced absorption peaks from the fundamental electron-hole
optical transition (i.e., the $X^{0}$ neutral exciton) Splendiani:2010 ;
Mak:2010 ; Li:2014 . Subsequent studies of charge-tunable TMD monolayers
demonstrated the emergence, at lower energy, of additional strong absorption
lines when the monolayer was populated with a background Fermi sea of holes or
electrons (i.e., the $X^{\pm}$ charged excitons) Ross:2013 ; Mak:2013 ;
Wang:2017NL . Whether, and under what conditions, a $X^{\pm}$ complex is most
accurately described as a simple three-particle ‘trion’ (wherein the
photoexcited exciton binds a carrier from the Fermi sea Ross:2013 ; Mak:2013 ;
Wang:2017NL ; Kheng:1993 ; Astakhov:2002 ; BarJoseph:2005 ), or a four-
particle ‘tetron’ (a trion additionally correlated with the resulting hole
that is left behind in the Fermi sea Bronold:2000 ; Suris:2001 ; Suris:2003 ;
Combescot:2018 ; Rana:2020 ), or an ‘exciton-polaron’ (an exciton dressed by
collective excitations of the Fermi sea Efimkin:2017 ; Sidler:2016 ), remains
an active area of study and discussion Reichman:2019 ; Glazov:2020 ; Rana:2021
; Efimkin:2021 ; Fey:2020 ; Liu:2021 .
Regardless of interpretation, all of these descriptions share a common
understanding: $X^{\pm}$ are bound states arising from the interaction of a
photoexcited electron-hole (e-h) pair with the subset of carriers in the Fermi
sea that have distinguishable quantum numbers. In most conventional III-V and
II-IV semiconductors such as GaAs or ZnSe, where band extrema occur at the
single central $\Gamma$-point valley of the Brillouin zone Kheng:1993 ;
Astakhov:2002 ; BarJoseph:2005 , this means that $X^{\pm}$ forms with mobile
carriers having opposite spin to that of the photoexcited electron or hole
because Pauli exclusion prevents strong short-range interactions with same-
spin carriers. However, the multi-valley nature of monolayer TMDs expands the
basis set of available quantum numbers, and band-edge electrons and holes can
be distinguished not only by their spin (up or down) but also by their valley
degree of freedom ($K$ or $K^{\prime}$). TMD monolayers therefore permit,
under suitable conditions, photoexcited e-h pairs to interact with Fermi seas
containing more than one type of quantum-mechanically distinguishable carrier.
As demonstrated recently Li:2022 ; vanTuanPRL:2022 ; vanTuan:2022 , this leads
to qualitatively new types of multi-particle composite exciton ground states
that can be described as bound six-particle “hexcitons” (when photoexcited e-h
pairs interact with two distinguishable Fermi seas) or even eight-particle
“oxcitons” (when interacting with three distinguishable Fermi seas). These
bound many-body excitonic states have large oscillator strengths and manifest
as discrete absorption resonances in linear optical spectroscopy, and emerge
at energies even further below $X^{0}$ and $X^{\pm}$. We emphasize that these
composite hexcitons are optically-allowed ground states of the interacting
exciton-Fermi sea system, and are therefore distinct from the many types of
optically-forbidden dark excitons and trions that appear only in
photoluminescence studies Urbaszek:2018 ; Muller:2018 ; Robert:2017 ;
Malic:2018 ; He:2020 ; Yang:2022 , and moreover should not be confused with
multi-exciton complexes (such as biexcitons) that appear only at higher
photoexcitation intensity Urbaszek:2018 ; Muller:2018 ; Barbone:2018 ;
ZLi:2018 ; Chen:2018 .
To date, such multi-particle hexcitons have been identified and studied only
in electron-doped WSe2 monolayers Li:2022 ; vanTuanPRL:2022 ; vanTuan:2022 .
This is due to i) the excellent optical quality of exfoliated WSe2, ii) the
ability to electrostatically dope WSe2 to high electron densities, and iii)
because composite hexcitons in WSe2 are expected at the low-energy A-exciton
optical transitions where spectral linewidths are typically much sharper than
at the higher-energy B-exciton. This latter fact arises from the positive sign
of the conduction band (CB) spin-orbit splitting in WSe2 ($\Delta_{c}>0$)
Song:2013 ; Kormanyos:2015 , which mandates that A-exciton optical transitions
photoexcite electrons to the upper CBs in $K$ and $K^{\prime}$, where Pauli
exclusion does not prevent them from interacting with both of the
distinguishable Fermi seas of electrons that occupy the two lower CBs (as
depicted in the band diagram in Fig. 1, one Fermi sea resides in the same
valley but has opposite spin, the other has same spin but resides in the
opposite valley).
However, under appropriate conditions, composite hexcitons should also emerge
in all members of the monolayer TMD family. For example, monolayer WS2 also
has a positive $\Delta_{c}$ and CB structure similar to WSe2, and therefore
hexcitons should also appear at its A-exciton under conditions of high
electron doping. In contrast, the negative $\Delta_{c}$ of monolayer MoSe2
Song:2013 ; Kormanyos:2015 precludes the existence of hexcitons at its
A-exciton (instead, only conventional $X^{\pm}$ should appear), but does allow
for the formation of hexcitons at its B-exciton transition. To date, neither
of these predictions have been explicitly tested. Here, using low-temperature
optical absorption measurements of electrostatically-gated WS2 and MoSe2
monolayers, we demonstrate and investigate the emergence of composite
hexcitons in both WS2 monolayers (at the A-exciton) and MoSe2 monolayers (at
the B-exciton). Based on the data, the roles of distinguishability and carrier
screening on the stability of hexcitons are discussed.
Figure 1: a) Schematic of the charge-tunable monolayer TMD structures. b,c)
Gate ($V_{g}$) dependent optical absorption spectra at 4 K and $B$=0 from
monolayer WSe2 and WS2, respectively, at their A-exciton band edges. Similar
to WSe2, WS2 exhibits strong $X^{0}$ absorption at charge neutrality,
conventional $X_{s,t}^{-}$ absorption at small electron density ($n_{e}$), and
the emergence at even lower energy of a strong “hexciton” absorption resonance
at larger $n_{e}$ (dashed oval, labeled as $H^{-}$). d) Band diagram of WSe2
and WS2. Extending the picture of $X^{\pm}$ as 4-particle tetrons, the
hexciton corresponds to a 6-particle correlated state that forms when a
photoexcited e-h pair couples simultaneously to both of the quantum-
mechanically distinguishable Fermi seas residing in the two lower CBs. This
occurs in WSe2 and WS2 at the A-exciton because $\Delta_{c}>0$, such that
optical transitions promote electrons to the upper CBs, where they are
distinguishable from electrons in the Fermi sea. Blue/red colors denote spin-
up/down bands. For clarity, only optical transitions in the $K$ valley are
depicted. The redshift of the hexciton ceases (yellow arrows) when the Fermi
sea begins to fill the upper CBs, at $n_{e}\approx 5(4)\times 10^{12}$ cm-2
for WSe2 (WS2).
Figure 1a depicts the charge-tunable TMD monolayer samples studied in this
work. Single monolayer flakes of WS2, MoSe2, and WSe2 were mechanically
exfoliated and sandwiched between thin slabs of hexagonal boron nitride (hBN).
Few-layer graphite flakes were used to electrically contact the monolayer, and
to serve as top and bottom gates. In this work, the top and bottom gates were
tied together and gate voltage $V_{g}$ was used to electrostatically dope the
monolayers with a background Fermi sea of electrons or holes. Dual gating
allows us to attain high carrier densities approaching 1013/cm2. Each
assembled structure was then positioned and placed directly over the core of a
single-mode optical fiber, to ensure a rigid and robust alignment between the
optical path and the doped TMD monolayer. This experimental approach Li:2020 ;
Stier:2018 mitigates the drift and vibration that can otherwise complicate
optical studies of TMD monolayers at low temperatures and in the pulsed
magnetic fields used in this work.
The sample-on-fiber assembly was then mounted on a purpose-built probe and
loaded into a liquid helium cryostat. Broadband white light from a Xenon lamp
was directed down the single mode fiber. Following transmission through the
sample the light passed through a thin-film circular polarizer and was then
retro-reflected and directed back into a multimode collection fiber. The
transmitted light was dispersed in a 300 mm spectrometer and detected by a
charge-coupled device (CCD). In this way the optical absorption from the doped
TMD monolayer was directly measured as $1-T/T_{0}$, where $T$ is the spectrum
of the transmitted light and $T_{0}$ is a reference spectrum. We note that
absorption spectra typically permit a straightforward evaluation and
visualization of exciton oscillator strengths, in comparison to reflectivity
studies where lineshapes depend sensitively on interference effects from the
surrounding layer structure.
Figure 1b shows a map of the gate-dependent absorption spectra from a WSe2
monolayer at low temperature (4K) and at zero magnetic field, in the spectral
range of its A-exciton. When $V_{g}\approx-2$V, the monolayer is at its charge
neutrality point and only the neutral exciton ($X^{0}$) absorption resonance
is observed. At increasingly negative or positive $V_{g}$, the monolayer
becomes lightly doped with mobile holes or electrons, and the well-known
$X^{\pm}$ resonances appear at energies $\approx$20-35 meV below $X^{0}$. As
extensively described in previous works, $X^{\pm}$ are optically-active bound
states arising from the interaction of the photoexcited e-h pair with the
carriers in the Fermi sea that possess distinguishable quantum numbers from
those of the photoexcited e-h pair (i.e., different spin and/or valley). Thus,
only a single $X^{+}$ resonance appears on the hole-doped side, but two
conventional $X^{-}$ resonances appear on the electron-doped side (the so-
called singlet $X_{s}^{-}$ and triplet $X_{t}^{-}$ charged excitons) because
there are two distinct Fermi seas with which to interact. The different
energies of $X_{s}^{-}$ and $X_{t}^{-}$ stem from the different amplitude of
the short-range electron-hole exchange interaction Glazov:2020 ; Hichri:2020 ;
Courtade:2017 .
Most importantly, at higher $n_{e}$ the $X_{s}^{-}$ and $X_{t}^{-}$ resonances
disappear and a new strong absorption resonance appears at even lower energy
($\approx$15 meV below $X_{s}^{-}$), indicating the emergence of a new bound
excitonic ground state with large oscillator strength. First observed in gated
WSe2 monolayers in 2013 Jones:2013 and very clearly resolved in several
subsequent studies Wang:2017NL ; Wang:2017 ; Barbone:2018 ; SuFeiShi:2020 ;
Liu:2021 , this low-energy absorption resonance – occasionally called
$X^{-\prime}$ in earlier literature – completely dominates the absorption
spectrum of WSe2 at high $n_{e}$. Moreover, it does not appear in hole-doped
WSe2 at the A-exciton, and does not appear in the A-exciton absorption
spectrum of gated MoSe2 monolayers Wang:2017NL ; Smolenski:2019 . Recently,
this absorption resonance was identified as a qualitatively new type of many-
body composite (six-particle hexciton) state, arising from the simultaneous
interaction of the photoexcited e-h pair with both of the Fermi seas that
reside in the lower CBs of monolayer WSe2 Li:2022 ; vanTuanPRL:2022 ;
vanTuan:2022 . As depicted in Fig. 1d, each of these two Fermi seas has
quantum numbers that are distinguishable from those of the photoexcited
electron (one has opposite spin, the other has opposite valley), and –
extending the picture of $X^{\pm}$ being four-particle tetrons Suris:2003 ;
Rana:2020 – the hexciton bound state comprises the photoexcited e-h pair, an
electron from each of the two distinguishable Fermi seas, and the two Fermi
holes that are left behind in the Fermi seas. As described recently, hexcitons
are the stable ground state of this interacting exciton-Fermi sea system, and
the Fermi holes not only ensure overall charge neutrality but also provide the
‘glue’ that binds the complex vanTuanPRL:2022 ; vanTuan:2022 .
Figure 2: a,b) Magnetic field evolution of the hexciton absorption resonance
in monolayer WSe2 and WS2, respectively, for both $\sigma^{+}$ and
$\sigma^{-}$ circularly polarized light, showing clear valley Zeeman
splitting, and the appearance of higher-energy absorption lines corresponding
to optical transitions to higher Landau levels ($V_{g}$=2 V). c,d) The average
energy of the $\sigma^{+}$ and $\sigma^{-}$ absorption lines reveals the
diamagnetic shift of the hexciton absorption resonance, which does not follow
a purely quadratic dependence (even at low $B$). As a point of reference, the
dashed red curves depict a $\sigma B^{2}$ shift, using a diamagnetic
coefficient $\sigma=3~{}\mu$eV/T2, which is $\approx$10$\times$ larger than
the small diamagnetic shifts of the neutral exciton in WSe2 and WS2.
The ordering of the spin and valley polarized CBs in monolayer WS2 is similar
to that of WSe2 (i.e., $\Delta_{c}$ is also positive), and optical transitions
at the A-exciton couple to the upper CBs. Consequently, optical signatures of
composite hexcitons can therefore be anticipated at high $n_{e}$ in WS2
monolayers. The gate-dependent absorption map of Fig. 1c confirms this
prediction: Strong absorption from $X^{0}$ is plainly visible at $V_{g}\approx
0$, and $X_{s,t}^{-}$ charged excitons appear at low $n_{e}$. These
conventional $X_{s,t}^{-}$ resonances in WS2 have been clearly resolved in
several recent studies Kapuscinski:2020 ; Zipfel:2020 ; Zinkiewicz:2021 ;
Robert:2021NC . Most importantly, our dual-gated structure allows a smooth
tuning to a regime of high $n_{e}$, where Fig. 1c shows that $X_{s,t}^{-}$
disappear and a new strong absorption resonance emerges at even lower energy
($\approx$15 meV below $X_{s}^{-}$). The gate-dependent absorption of
monolayer WS2 is therefore qualitatively identical to that of WSe2, albeit
with broader linewidths that are likely due to the reduced material quality of
sulfur-based TMDs. Thus, we associate the emergence of the low-energy
absorption resonance with the stable formation of 6-particle hexciton states.
Note that we were unable to dope our WS2 monolayers with mobile holes; even at
large negative $V_{g}$, only the neutral $X_{0}$ exciton was visible, likely
due to strong mid-gap pinning of the Fermi level by the larger number of
defects in sulfur-based TMDs.
Additional evidence supporting a picture of hexcitons in monolayer WS2 is the
evolution of its optical resonance in applied magnetic fields $B$. Figure 2
shows circularly polarized magneto-absorption from both WSe2 and WS2, under
conditions of large $n_{e}$ where the hexciton absorption dominates. As shown
recently Li:2022 , the hexciton resonance in WSe2 monolayers splits and shifts
with increasing $B$, and additional absorption resonances appear at higher
energy that disperse linearly with $B$ and are related to the development of
Landau levels (LLs) in the conduction and valence bands. Figure 2 shows that a
qualitatively similar $B$-dependent evolution of the hexciton peak also exists
in WS2, where additional LL-like absorption features emerge for $B>30$ T. From
the separation of these peaks we can estimate a combined electron and hole
cyclotron resonance energy of $\approx$0.45 meV/T, which is slightly larger
than obtained from WSe2 Li:2022 but in line with expectation given the
slightly lighter carrier masses in WS2 Goryca:2019 .
Figure 3: a) Gate-dependent absorption of a charge-tunable MoSe2 monolayer at
4 K. Only conventional $X^{-}$ and $X^{+}$ are observed at the A-exciton
($\sim$1.63 eV), even at large electron or hole densities, because
$\Delta_{c}<0$ and therefore only a single distinguishable Fermi sea is
available – see diagrams in panels b) and d), respectively. But at the
B-exciton ($\sim$1.83 eV), photoexcited e-h pairs couple to the upper CBs, and
interaction with both Fermi seas in the lower CBs is possible, leading to
hexciton formation at higher $n_{e}$ – see diagram in panel c). Moreover,
hexciton formation at high hole densities is always possible at the B-exciton
(for all TMD monolayers, because $\Delta_{v}\gg 0$) – see diagram in panel e).
Blue/red colors in the diagrams denote spin-up/down bands. For clarity, only
optical transitions in the $K$ valley are depicted.
Moreover, from these spectra we can extract the diamagnetic shifts of the
hexciton resonance, given by the average of the $\sigma^{+}$ and $\sigma^{-}$
absorption energies. This analysis, shown in Figs. 2c and 2d, reveals that the
diamagnetic shifts of hexcitons – even at low $B$ – deviate from the purely
quadratic behavior that is known to exist for neutral excitons in WSe2 and WS2
Stier:2018 ; Goryca:2019 (especially for WS2, where atypical diamagnetic
shifts have also been observed for charged excitons Plechinger:2016 ). This
behavior may arise from non-zero angular momentum contributions from the
particles in the hexciton complex, and will be further investigated in future
studies. For reference, however, the dotted red curves depict purely quadratic
shifts ($\sigma B^{2}$) using a large diamagnetic coefficient
$\sigma=3~{}\mu$eV/T2, which is $10\times$ larger than the known diamagnetic
shifts of the very small and tightly-bound neutral excitons Stier:2018 ;
Goryca:2019 . To the extent that these curves approximately capture the
overall shifts of the hexciton resonances, these data are qualitatively
consistent with recent calculations indicating that composite hexcitons are
several times larger in size than neutral excitons vanTuanPRL:2022 ;
vanTuan:2022
Taken together, these data provide evidence for the emergence of composite
hexciton states in electron-doped monolayer WS2, which appear at the A-exciton
owing to the positive sign of $\Delta_{c}$ and consequent ordering of the
spin-orbit-split CBs in the $K$ and $K^{\prime}$ valleys.
In marked contrast, neither electron-doped nor hole-doped MoSe2 monolayers
show any indication of hexciton formation at the A-exciton, as shown in Fig.
3a. Rather – and as also observed in earlier studies Wang:2017NL ;
Smolenski:2019 ; Liu:2021 – charge-tunable MoSe2 monolayers exhibit only a
single $X^{-}$ and $X^{+}$ absorption that appears $\approx$25 meV below the
neutral exciton $X_{A}^{0}$. This observation is consistent with the negative
sign of $\Delta_{c}$ in MoSe2 Song:2013 ; Kormanyos:2015 , which dictates that
optical transitions at the A-exciton photoexcite electrons to the lower (not
upper) CBs. As such, when mobile electrons populate the lower CBs, only a
single type of distinguishable electron exists in the Fermi sea, and many-body
hexcitons cannot form (see diagrams in Fig. 3b,d).
The ordering of the CBs in MoSe2 does, however, permit hexciton formation at
the higher energy B-exciton, where optical transitions couple to the upper CBs
(see Fig. 3c). As Fig. 3a shows, when $n_{e}$ is small, the neutral B-exciton
($X_{B}^{0}$) at 1.84 eV disappears and only a faint and diffuse absorption
remains at energies where conventional charged excitons are expected (that is,
at about 25 meV below $X_{B}^{0}$, or $\approx$1.815 eV). However, at larger
$n_{e}$ ($\approx 5\times 10^{12}$ cm-2), a stronger absorption with increased
oscillator strength clearly emerges at $\approx$1.795 eV. Its separation from
$X_{B}^{0}$ is $\approx$45 meV, which is larger than the 25 meV expected for
conventional charged excitons, but is commensurate with the larger energy
separation between hexcitons and neutral excitons that we observed in WSe2 and
WS2 (cf. Fig. 1). Notably, Fig. 3 shows that this new absorption does not
emerge until a large $n_{e}$ comparable to where the conventional $X^{-}$
trion loses oscillator strength, suggesting that it is not a conventional
trion. Therefore, although past studies of charge-tunable MoSe2 monolayers
have associated similar spectral signatures with conventional
trions/tetrons/exciton-polarons of the B-exciton Wang:2017NL ; Liu:2021 , we
argue that the low energy of the emerging absorption resonance and
(especially) its dependence on $n_{e}$ are, in fact, more consistent with the
formation and emergence of many-body hexcitons in electron-doped MoSe2
monolayers.
Furthermore, analogous signatures of hexcitons appear when the MoSe2 monolayer
is doped with high concentrations of mobile holes, $n_{h}$ (see Fig. 3a). As
depicted in the diagram of Fig. 3e, e-h pairs excited at the B-exciton will
always have two distinguishable Fermi seas of mobile holes with which to
interact, and composite hexcitons can be anticipated. Figure 3a shows that as
$n_{h}$ starts to increase, both $X_{A}^{0}$ and $X_{B}^{0}$ disappear, and a
conventional $X^{+}$ resonance appears 25 meV below $X_{A}^{0}$ when
$V_{g}=-3$ V. However, the concomitant response at the B-exciton is much
weaker, until $V_{g}$ increases up to $\approx-4$ V, at which point a strong
absorption emerges at a much lower energy of $\approx$40 meV below
$X_{B}^{0}$. This new resonance actually gains oscillator strength with
increasing $n_{h}$, while in this same doping range the conventional $X^{+}$
fades away. Based on this $n_{h}$ dependence, we rule out the possibility that
the new resonance could be, e.g., a conventional trion associated with
$X_{B}^{0}$ or an excited Rydberg state of $X_{A}^{0}$. Moreover, this new
resonance redshifts with increasing $n_{h}$, similar to the redshift observed
for hexcitons in electron-doped WSe2 and WS2 (cf. Fig. 1). These data
therefore support a picture of robust hexciton formation at the B-exciton in
hole-doped MoSe2. The resonance features are rather broad, however, likely due
to the shorter lifetime of B-excitons.
Indeed, owing to the large 100s-of-meV spin-orbit splitting of the valence
bands ($\Delta_{v}$) that exists in all monolayer TMD semiconductors, robust
hexciton formation at the B-exciton should emerge when any TMD monolayer
semiconductor is doped with a high density of mobile holes (see diagram in
Fig. 3e). Furthermore, because $\Delta_{v}$ is large, the Fermi sea of holes
never occupies the valence bands from which the photoexcited e-h pair
originates, and the photohole always remains distinguishable from every hole
in the Fermi sea. Hexcitons under such conditions should therefore remain
robust and should redshift due to the primary effects of bandgap
renormalization Scharf:2019 up to very large values of $n_{h}$. While in this
work we have studied B-excitons only in MoSe2, we note that the seminal work
of Wang et al. Wang:2017NL clearly revealed a strong absorption below
$X_{B}^{0}$ in heavily hole-doped WSe2 monolayers, that redshifted with
increasing $n_{h}$ and did not broaden, consistent with hexciton formation.
An important insight gained from these various results is that the decay and
broadening of excitonic complexes is likely governed more by the quantum-
mechanical distinguishability of the photoexcited e-h pair (in relation to the
carriers in the Fermi sea), than by screening from the Fermi sea. For example,
in the case of electron-doped WSe2 and WS2 shown in Fig. 1, the redshifts of
the hexciton resonances in WSe2 and WS2 cease, and their decay/broadening
begins, only when electrons begin to populate the upper CBs. Beyond this
point, the photoexcited electron is no longer distinguishable from every
electron in the Fermi sea, and the Pauli exclusion principle dictates that
when it is introduced into the (now occupied) upper CB, it must scatter away
those mobile electrons having similar spin and valley quantum numbers
vanTuan:2022 . This exchange scattering process leads to the decay and
broadening of the hexciton resonance. Hexciton resonances therefore retain
their amplitude and narrow linewidth when the charge density increases, as
long as the photoexcited e-h pair remains distinguishable from all carriers in
the Fermi sea. In hole-doped TMD monolayers, the large $\Delta_{v}$ ensures
distinguishability of e-h pairs excited at the B-exciton, and therefore a
robust stability of hexcitons even to very large $n_{h}$.
Moreover, screening by mobile carriers does not seem to be the primary driving
force for broadening of the hexciton resonance, as inferred by the observation
that the conventional $X^{\pm}$ resonances begin to lose oscillator strength
at much smaller carrier density. For example, Fig. 3 shows that the
conventional trions of MoSe2 around 1.63 eV decay when $|V_{g}|\gtrsim 5$ V.
Concomitantly, however, the type-B hexciton in hole-doped conditions around
1.8 eV neither broadens nor decays at the same (and even higher) $n_{h}$. A
similar behavior was also measured for a wider $n_{h}$ range by Liu et al.
Liu:2021 , where the type-B resonance around 1.8 eV maintains a large
oscillator strength and continues to redshift long after the $X^{+}$ resonance
at the A-exciton decays. This suggests that screening is not the primary cause
of decay and broadening, because the Coulomb potential cannot selectively
weaken the attraction between particles of one complex species but not of
another.
In summary, composite hexcitons are expected in all members of the monolayer
TMD family. When doped with a high density of electrons, hexcitons can emerge
at the A-exciton resonance (as for the case of WSe2 and WS2) or the B-exciton
resonance (as for the case of MoSe2) depending on the sign of $\Delta_{c}$ and
the consequent ordering of the CBs. For hole-doped TMD monolayers, hexcitons
should always emerge at the B-exciton, as investigated here for MoSe2 and as
suggested by earlier spectroscopic data from both WSe2 and MoSe2 Wang:2017NL ;
Liu:2021 . To the best of our knowledge, this has not yet been studied in
hole-doped WS2 or MoS2. As a final point of discussion, we note that electron-
doped MoS2 monolayers represent an interesting case: While early theory
suggested that $\Delta_{c}$ was small and negative (implying a CB ordering
similar to MoSe2 Kormanyos:2015 ), more recent experimental work indicates an
opposite CB ordering Robert:2020 ; Park:2022 , particularly when exciton
effects are taken into account, making it more akin to that of WS2 and WSe2.
In this case, a hexciton resonance can be expected to emerge in electron-doped
MoS2 at the A-exciton, at an energy below that of the conventional
$X_{s,t}^{-}$ charged excitons. Indeed, various recent studies have revealed
additional optical resonances emerging in electron-doped MoS2 monolayers
Klein:2021 ; Klein2:2021 ; Roch:2019 , although in some cases it was
associated with exotic ferromagnetic order. Looking forward, we anticipate
that composite 6-particle hexcitons (and, in high magnetic fields, 8-particle
oxcitons) can provide a rich platform to study novel many-body effects and
inter-valley correlations within a strongly interacting exciton-Fermi sea.
We thank X. Marie for helpful discussions, and we acknowledge support from the
Los Alamos LDRD program and the US Department of Energy (DOE) “Science of 100
T” program. The National High Magnetic Field Lab is supported by National
Science Foundation DMR-1644779, the State of Florida, and the US DOE. Work at
the University of Rochester was supported by the DOE Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award No. DE-SC0014349.
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|
Cooling and heating regions of Joule-Thomson expansion for AdS black holes:
Maxwell-power-Yang-Mills and Kerr Sen black holes
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti†,⋆333Email:
<EMAIL_ADDRESS>
Mohammad Ali S. Afshar ⋆444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran
###### Abstract
In this paper, we explore the Joule-Thomson expansion (JTE) process for the
Einstein-Power-Young-Mills (EPYM) and the AdS Kerr Sen (AKS) black holes. We
study the effect of free parameters on the Joule-Thomson coefficient (JTC),
the inversion curve, and the $T_{i}^{min}/T_{c}$. The isenthalpic curves of
the AKS black hole show cooling or heating behavior depending on the inversion
curve, which is affected by the mass and the parameters $b$ and $a$ of the
black hole. If we assume the parameter $b$ to be zero, the results reduce to
the Kerr–AdS black holes[1]. In [2, 3], for the Einstein–Power–Yang–Mills AdS
black hole with $q>1$ and $n=2$, the $T_{i}^{min}/T_{c}$ is $1/2$. But in this
paper, for the AdS-Maxwell-power-Yang-Mills black hole, when $q>1$, the
$T_{i}^{min}/T_{c}$ is almost equal to $1/2$ for the increase of Maxwell’s
charge $C$, and when $q=1/2$, the $T_{i}^{min}/T_{c}$ is equal to $1/2$ for
all values of $C$. Also, when $1/2<q<1$, the $T_{i}^{min}/T_{c}$ is close to
the value of $1/2$, and finally when $0<q<1/2$, the values of
$T_{i}^{min}/T_{c}$ move away from the value of $1/2$, that is, they become
smaller. For the AKS black hole, we found that for free parameters $a=0.00951$
and $b=0.00475$, the $T_{i}^{min}/T_{c}$ is almost $1/2$. Finally, we compare
our findings with others in the literature and summarize our results in Tables
1-5.
Keywords: JTE, AdS-Maxwell-power-Yang-Mills black holes, AdS Kerr Sen black
holes
###### Contents
1. 1 Introduction
2. 2 Joule-Thomson expansion
3. 3 Case I: AdS-Maxwell-power-Yang-Mills black holes
4. 4 Case II: AdS-Kerr-Sen black hole
5. 5 Discussion and results
1. 5.1 Case I
2. 5.2 Case II
6. 6 Conclusion
## 1 Introduction
The insights provided by astronomy, astrophysics, and experimental cosmology
suggest that cosmic structures adhere to the same principles and foundations
observed on Earth, albeit with minor variations and necessary simplifications.
This structural and fractal similarity is a crucial tool for human ideation
and cognition, enabling us to decode more aspects of this global pattern
daily. Researchers leverage this understanding in hypothesizing theoretical
models, which may not yet be empirically verifiable. They construct various
models based on mathematical logic and physical laws, exploring all potential
scenarios to predict the most plausible ideas consistent with the formation of
the enigmatic, infinite universe. The existing black hole models, are actually
one of the most obvious examples of this form of inference. Despite being one
of the most elusive cosmic entities, they are studied based on this pattern.
The comparison of a black hole’s gravitational behavior with a thermodynamic
ensemble in the 1970s gave rise to a significant branch of black hole physics,
namely black hole thermodynamics. For instance, the four laws of black hole
mechanics bear a striking resemblance to the laws of thermodynamics [4, 5].
Similarly, a black hole’s surface area is analogous to entropy in
thermodynamics, and its surface gravity is comparable to temperature[4, 5].
The introduction of Maldacena duality, also known as the AdS/CFT
correspondence, established a deeper connection between black hole
thermodynamics and this duality, offering profound insights into quantum
gravity. In the context of AdS/CFT, a black hole’s entropy in the bulk AdS
space correlates with the entropy of the corresponding CFT on the boundary,
known as the Bekenstein-Hawking entropy [6]. Moreover, the black hole’s
temperature is related to the CFT’s temperature[6, 7, 8, 9].
This correspondence provides a powerful tool to study the quantum aspects of
gravity and black holes using the methods of quantum field theory. In AdS
space, there is a Hawking-Page phase transition between a stable large black
hole and a thermal gas[10]. This phase transition is a first-order transition
that occurs when the temperature of the system reaches a critical value, where
the free energy of the black hole becomes lower than that of the thermal
gas[10]. This phase transition can be interpreted as a confinement
deconfinement phase transition of a gauge field[11]. The Hawking-Page phase
transition can be seen as a transition from a deconfined phase in the thermal
gas to a confined phase in the black hole[11]. When the AdS black holes have
electric charge, they exhibit rich phase structures that were studied by
Chamblin et al[12, 13]. They found that the phase transition behavior of
charged AdS black holes resembles the liquid-gas phase transition in a van der
Waals system[14]. In the extended phase space where the cosmological constant
is treated as pressure[15], the P-V critical behavior of charged AdS black
holes was investigated and it was shown that they have a similar analogy to
the van der Waals liquid-gas system. In addition to the phase transition and
critical phenomena[16, 17, 18, 19, 20, 21, 22, 23], the analogy between the
black holes and the van der Waals system was also creatively applied to the
well-known JTE process[24] recently. This means that the JTC can be used to
study the thermodynamics of black holes as well. For example, the isenthalpic
expansion process is the analogue of the JTE process for black holes, where
the black hole mass is constant while the black hole pressure and volume are
changed. The black hole pressure is related to the cosmological constant, and
the black hole volume is related to the horizon radius.In this case, the
inversion curve is the curve that separates the regions where the black hole
temperature increases or decreases as the pressure decreases.One of the
intriguing features of the inversion curves for black hole systems is that
they have only positive slopes, unlike the van der Waals system, which has
both positive and negative slopes. For example, for charged AdS black holes
[24] and Kerr-AdS black holes [1], the isenthalpic expansion process and the
inversion curve have been analyzed and observed that the inversion curve was
found to have a positive slope, meaning that the black hole temperature always
decreases as the pressure decreases. Then the analysis was generalized to
other types of AdS black holes, such as quintessence charged AdS black
holes[25], holographic superfluids [26], charged AdS black holes in f(R)
gravity [27], AdS black holes with a global monopole [28], and AdS black holes
in Lovelock gravity [29]. For further study, you can see also [30, 31, 32, 33,
34, 35, 36, 37, 2, 38].
All the results showed that the inversion curves for all these black hole
systems have only positive slopes. We are interested in exploring whether this
feature is universal for all black hole systems, or whether there are other
effects that can alter it. To this end, we will focus on two types of AdS
black holes: AdS-Maxwell-power-Yang-Mills and AdS Kerr Sen. These black holes
have additional parameters that can affect their thermodynamic behavior and
phase transitions. AdS-Maxwell-power-Yang-Mills black holes are black holes
that have a non-linear electromagnetic field, which is described by a power-
law function of the field strength[39]. This field can be seen as a
generalization of the Maxwell field, which is the standard model of
electromagnetism. AdS Kerr Sen black holes are black holes that have electric
charge and angular momentum in a low-energy limit of heterotic string
theory[40]. This theory is a type of string theory that combines the features
of bosonic and supersymmetric strings. These black holes have different
properties and characteristics than the standard AdS black holes, such as the
existence of a dilaton field, which is a scalar field that couples to the
electromagnetic field and the curvature. We will investigate how these
parameters affect the inversion curves and the JTE process for these black
holes, and compare them with the previous results for other black hole
systems.
The structure of this paper is as follows. In Sec.II, we give a brief overview
of the JTE. In Sec.III and Sec.IV, We introduce briefly AdS-Maxwell-Power-
Yang-Mills and the AdS Kerr Sen black holes and their thermodynamic
properties.In Sec.V, we study the JTE process for these black holes, derive an
explicit expression for the JTC, and analyze and discuss the effect of the
parameters of each model on the inversion curves. In Sec.VI, we present our
conclusion and discussion.
## 2 Joule-Thomson expansion
In classical thermodynamics, a throttling process or the JTE process,
discovered in 1852, is a method of cooling or heating a system by changing its
pressure and volume, without adding or removing heat. In this process, the
high-pressure gas passes through a porous plug into a region with a low
pressure, while keeping the enthalpy constant. Since it is a constant-enthalpy
process, it can be used to experimentally measure the lines of constant
enthalpy (isenthalps) on the (p, T) diagram of a system. Combined with the
specific heat capacity at constant pressure, it allows the complete
measurement of the thermodynamic potential for the gas[41].
In this method, The main goal is to investigate the behavior of the
coefficient that describes the temperature change during the expansion or
compression of a system at constant enthalpy, which is denoted by $\mu$ and is
known as the JTC,
$\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{H}=\frac{1}{C_{P}}\big{[}T(\frac{\partial V}{\partial
T}\big{)}_{P}-V\big{]}.$
If the above coefficient is positive, as a result of pressure reduction, the
temperature will decrease. In other words, the expansion of the gas causes
cooling and the compression of the gas under investigation causes it to heat
up. In other words, the positive JTC indicates the same direction of
temperature and pressure. Whereas, if the JTC is negative, a decrease in
pressure causes an increase in temperature.
The JT inversion temperature,which determined by setting $\mu=0$, is the
temperature at which the sign of the JTC changes. Most real gases have an
inversion point. The temperature of this point depends on the gas pressure
before expansion.
$T_{i}=V\big{(}\frac{\partial T}{\partial V}\big{)}_{P}.$
The important point is that if you plot the JTC on the (p,T) diagram, a closed
parabolic curve is created. In simpler terms, the inversion temperature is
placed on the boundary of the curve of temperature changes in terms of
pressure. At this temperature, the JTC changes from negative to positive. At a
given pressure, the isopressure-temperature line intersects the drawn curve at
two different points. These two points are called low temperature and high
temperature of inversion. In fact, at temperatures higher and lower than these
two temperatures, the sign of the JTC is negative and between these two
temperatures, the sign of the JTC is positive. According to the above, the
interesting phenomenon in this process is that the (p, T) diagram has two
regions: one where the gas cools down and one where the gas heats up. These
regions are separated by the inversion curve,the curve that shows the points
where the system temperature does not change during the expansion process,
that divides the graph into two regions: the cooling region and the heating
region. The cooling region is where the gas temperature decreases as the
pressure decreases, and the heating region is where the gas temperature
increases as the pressure decreases. The inversion curve depends on the type
of the gas and its initial conditions. If with respect to zero coefficient for
ideal gases, we choose the van der Waals system, which is a more realistic
model than the ideal gas, and takes into account the finite size and the
attractive forces of the molecules, we find that for the van der Waals system,
the inversion curves have both positive and negative slopes, forming a circle
in the pressure axis. The inversion curve for the van der Waals system has a
negative slope in the low-pressure region, where the attractive forces
dominate, and a positive slope in the high-pressure region, where the
repulsive forces dominate.
## 3 Case I: AdS-Maxwell-power-Yang-Mills black holes
The AMPYM black holes are rooted in the study of black hole solutions in the
context of supergravity theories, especially in anti-de Sitter (AdS) space.
The study of black holes in AdS space is important for various reasons,
including its relevance to string theory and the AdS/CFT correspondence, which
is a duality between gravitational theories in AdS space and field theories
defined on its boundary. These black holes are solutions to the equations of
motion of supergravity theories with additional matter fields, such as Maxwell
fields and power-Yang-Mills fields, within the context of AdS space that
arises from a generalization of the Einstein-Maxwell theory with a negative
cosmological constant and a non-Abelian gauge field.The history of AMPYM black
holes can be traced back to the discovery of the first black holes in
Einstein-Yang-Mills theory, which were considered in the works of Yasskin [42]
and Kasuya [43]. It should be noted that in the non-Abelian case there are
various gauge groups, but to obtain black hole solutions it is necessary to
choose a specific form for the gauge potential. One of the simplest forms that
nevertheless allowed to derive interesting and important results is the so-
called Wu-Yang ansatz, which leads to magnetic-type solutions. [44, 45, 46,
47]. Of course, an interesting point that can be mentioned is that the primary
black holes with non-Abelian fields, which were studied in the late 80s [48,
49, 50, 51], were found to be unstable in the case of asymptotically flat
geometry [52, 53], but later black hole solutions were obtained in the AdS
case, which were shown to be stable [54, 55, 56, 57]. The AMPYM black holes
have been extensively studied in the context of theoretical physics,
particularly in the context of holography and its applications to
understanding strongly coupled field theories. In recent years, the study of
AMPYM black holes has continued to be an active area of research, with a focus
on understanding their thermodynamic properties, phase transitions, and
connections to gauge/gravity duality. These black holes have been studied as
models for strongly coupled systems in the dual field theories, providing
valuable insights into nonperturbative phenomena in quantum field theories.
Researchers have investigated various aspects of AMPYM black holes, such as
their stability, entropy, and critical behavior. Their thermodynamic
properties have been of particular interest, as they exhibit rich phase
structures and can undergo phase transitions similar to those observed in
condensed matter systems. Furthermore, the holographic interpretation of these
black holes has led to fruitful connections between gravitational physics and
field theory, shedding light on fundamental questions in quantum gravity and
strongly coupled systems[58, 59, 60]. This section provides a summary of the
thermodynamics of N-dimensional Einstein-Maxwell-power-Yang-Mills gravity with
a cosmological constant $\Lambda$. This theory of gravity is based on the
following action, which we will explain in more details. So we will have,
$\begin{split}I=\frac{1}{2}\int
dx^{N}\sqrt{-g}\bigg{(}R-\frac{(N-1)(N-2)\Lambda}{3}-F_{\mu\nu}F^{\mu\nu}-(Tr(F_{\mu\nu}^{(a)}F^{(a)\mu\nu}))^{q}\bigg{)}.\end{split}$
(1)
The trace element is represented by $Tr(.)=\Sigma_{a=1}^{(N-1)(N-2)/2}(.)$,
with $R$ as the Ricci scalar and $q$ as a real positive parameter. The
Yang–Mills and Maxwell fields are defined accordingly [4],
$\begin{split}F_{\mu\nu}^{(a)}=\partial_{\mu}A_{\nu}^{(a)}-\partial_{\nu}A_{\mu}^{(a)}+\frac{1}{2\sigma}C^{(a)}_{(b)(c)}A^{b}_{\mu}A^{c}_{\nu},\end{split}$
(2)
$\begin{split}F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\end{split}$
(3)
$C^{(a)}_{(b)(c)}$ represents the structure constants of the $(N-1)(N-1)/2$
parameter Lie group $G$, while $\sigma$ denotes the coupling
constant.$A^{(a)}_{\mu}$ refers to the $SO(N-1)$ gauge group Yang-Mills
potentials, with $A_{\mu}$ representing the conventional Maxwell potential
[39]. The metric solution corresponding to the $N$-dimensional spherically
symmetric line element is as follows [40],
$\begin{split}ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{n}^{2}.\end{split}$
(4)
The $d\Omega_{n}^{2}$ denotes the volume of the unit $n$-sphere. In this
study, we will direct our attention to the Einstein-Maxwell-power-Yang-Mills
theory ($EMPYM$) with $N(=n+2)\geq 4$ and $q\neq(n+1)/4$. The solution to the
$N$-dimensional $EMPYM$ black hole with a negative cosmological constant under
the condition of $q\neq\frac{n+1}{4}$ is provided [39],
$\begin{split}&f(r)=1-\frac{2m}{r^{n-1}}-\frac{\Lambda
r^{2}}{3}+\frac{2(n-1)C^{2}}{nr^{2(n-1)}}+\frac{Q}{r^{4q-2}}\\\
&Q=\frac{[n(n-1)Q_{1}^{2}]^{q}}{n(4q-n-1)}.\end{split}$ (5)
It is important to note that the parameter $m$ represents the mass of the
black hole, while $C$ and $Q_{1}$ denote the charges of the Maxwell field and
Yang-Mills field, respectively. In the extended phase space, the cosmological
constant is considered as a thermodynamic pressure $P=-\frac{\Lambda}{8\pi}$,
in this case, the Hawking temperature, mass and entropy of the black hole are
obtained as follows,
$\begin{split}T=-\frac{C^{2}(n-1)^{2}}{2\pi
nr_{+}^{2n-1}}+\frac{2}{3}(n+1)Pr_{+}-\frac{Q(-n+4q-1)}{4\pi
r_{+}^{4q-1}}+\frac{n-1}{4\pi r_{+}},\end{split}$ (6)
$\begin{split}M=\frac{n\omega_{n}}{48\pi}\bigg{(}\frac{6C^{2}(n-1)}{nr_{+}^{n-1}}+8\pi
Pr_{+}^{n+1}+3Qr_{+}^{n-4q+1}+3r_{+}^{n-1}\bigg{)},\end{split}$ (7)
$\begin{split}S=\frac{\omega_{n}r_{+}^{n}}{4},\qquad\omega_{n}=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)},\end{split}$
(8)
where $r_{+}$ and $\omega_{n}$ are the horizon radius and the volume of the
unit $n$-sphere respectively.
## 4 Case II: AdS-Kerr-Sen black hole
The Kerr-Sen-AdS black hole is a complex and fascinating topic in the field of
theoretical physics. It is a type of rotating, charged black hole that emerges
from heterotic string theory. It is a generalization of the Kerr-Newman-AdS
black hole, which is a solution of the Einstein-Maxwell equations with a
negative cosmological constant. The Kerr-Sen-AdS black hole also involves a
dilaton and an axion field, which are scalar and pseudoscalar fields that
appear in string theory [61]. The first exact solution of the Einstein field
equations, known as the Schwarzschild solution, was discovered by Karl
Schwarzschild in 1916. This solution describes a simple, non-rotating,
uncharged black hole. The next major advancement came in 1963, when Roy P.
Kerr found a solution to the Einstein field equations that describes a
rotating black hole. This was a significant development, as it is believed
that most black holes in the universe are rotating. In 1965, Ezra Newman and
his collaborators found a solution that describes a rotating, charged black
hole, known as the Kerr-Newman black hole. This solution was later extended by
Ashoke Sen to include a dilaton field and an axion field, resulting in the
Kerr-Sen black hole. The AdS form of this black hole was first derived by
Ashoke Sen in 1992, by applying a series of duality transformations to the
Kerr-Newman-AdS black hole. Sen showed that the Kerr-Sen-AdS black hole
retains some of the symmetries and properties of the Kerr-Newman-AdS black
hole, such as the existence of an event horizon, an ergosphere, and a Penrose
process. However, the Kerr-Sen-AdS black hole also has some unique features,
such as the dependence of the mass and angular momentum on the dilaton charge,
and the violation of the cosmic censorship conjecture for some values of the
parameters [61]. The Kerr-Sen-AdS black hole solution is a specific example of
a rotating black hole with additional fields, which is of particular interest
due to the role of angular momentum and the presence of nontrivial matter
content in the spacetime geometry and is a solution to the equations of motion
of supergravity theories with additional matter fields, such as the Sen-type
dilaton and antisymmetric tensor fields, within the context of AdS space.
The properties and thermodynamics of Kerr Sen-AdS black holes have been the
subject of intense research, given their relevance to understanding the
behavior of rotating black holes in the presence of nontrivial matter content
and their implications for the AdS/CFT correspondence[62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72]. The study of Kerr Sen-AdS black holes has been motivated
by theoretical developments in string theory, quantum gravity, and holography,
as well as by their potential implications for gravity dualities and the
behavior of strongly coupled systems in the dual field theories. This black
hole has been studied from various perspectives, including its thermodynamic
properties, stability, and connections to the dynamics of dual field theories.
In recent years, there has been growing interest in exploring the dynamical
behavior of Kerr Sen-AdS black holes, including their evolution, instability,
and the connections to chaos and information loss puzzles. Researchers have
investigated the behavior of these black holes under various perturbations and
have sought to understand their implications for fundamental questions about
black hole thermodynamics and the fate of information in quantum gravity.
Moreover, the holographic interpretation of Kerr Sen-AdS black holes has led
to fruitful connections between gravitational physics and field theory,
shedding light on fundamental questions in quantum gravity and strongly
coupled systems[62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72]. In summary, the
history and ongoing research on Kerr Sen-AdS black holes represent a rich and
interdisciplinary area of study that has fruitful connections to diverse
fields of theoretical physics, including string theory, quantum gravity,
holography, and nonlinear dynamics. The exploration of these black holes
continues to be an exciting frontier for probing the fundamental nature of
spacetime and its connections to quantum theory. In this section provides a
short overview of the Kerr-Sen black hole and its generalization to the anti-
de Sitter spacetimes. Sen [61] found a solution of the low-energy effective
action of the heterotic string theory, which describes a charged rotating
black hole, known as the Kerr-Sen black hole.The action is a modification of
the general relativity action with additional fields from the heterotic string
theory, given by[62, 78],
$\begin{split}S=\int
d^{4}x\sqrt{-\widetilde{g}}e^{-\Phi}\left[\mathcal{R}+(\nabla\Phi)^{2}-\frac{1}{8}F^{2}-\frac{1}{12}H^{2}\right],\end{split}$
(9)
where $\tilde{g}$ is the determinant of the metric tensor $g_{\mu\nu}$,
$\mathcal{R}$ is the Ricci scalar, $F=F_{\mu\nu}F^{\mu\nu}$ with $F_{\mu\nu}$
being the $U(1)$ Maxwell field strength tensor, $\Phi$ is a scalar dilaton
field, and $H=H_{\mu\nu\rho}H^{\mu\nu\rho}$ is the field strength for the
axion field. The action can be transformed to the Einstein frame by a
conformal transformation of the metric:
$\begin{split}ds^{2}_{E}=e^{-\Phi}d\widetilde{s}^{2}.\end{split}$ (10)
The action in the Einstein frame is given by:
$\begin{split}S=\int
d^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\nabla\Phi)^{2}-\frac{e^{-\Phi}}{8}F^{2}-\frac{e^{-2\Phi}}{12}H^{2}\right].\end{split}$
(11)
The metric of a Kerr-Sen-AdS black hole in Boyer-Lindquist coordinates is
given by[62, 78]:
$\begin{split}ds^{2}=-\frac{\Delta_{r}}{\rho^{2}}(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi)^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\sin^{2}\theta\Delta_{\theta}}{\rho^{2}}[-\frac{(r^{2}+2br+a^{2})}{\Xi}d\phi+adt]^{2},\end{split}$
(12)
where
$\begin{split}&\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+2br,\\\
&\Delta=(r^{2}+2br+a^{2})(1+\frac{r^{2}+2br}{\ell^{2}})-2mr,\\\
&\Xi=1-\frac{a^{2}}{\ell^{2}},\\\
&\Delta_{\theta}=1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta.\end{split}$ (13)
The parameter b is the dyonic charge of the black holes and is expressed as
$b=q^{2}/(2m)$, where q is the electric charge and m is the mass of the black
holes. In the limit of $\ell\to\infty$, the metric reduces to the usual Kerr-
Sen black holes. The non-rotating case ($a=0$) reduces to the Gibbons-Maeda-
Garfinkle-Horowitz-Strominger (GMGHS) solution. Gibbons and Maeda [73, 74]
found the black hole and black brane solutions in the dilaton theory, and
Garfinkle-Horowitz-Strominger obtained their charged version [75]. It is worth
mentioning that the Kerr-Sen-AdS black holes in four dimensions have been
explored from various aspects, such as the black hole shadows [76] and the
phase space thermodynamics in the extended phase space[77]. The ADM mass M,
the angular momentum J, and the charge Q in AdS spacetimes are related by,
$\begin{split}&M=\frac{m}{\Xi^{2}},\\\ &J=\frac{ma}{\Xi^{2}},\\\
&Q=\frac{q}{\Xi}.\end{split}$ (14)
Also, the entropy is given by,
$\begin{split}S=\frac{A}{4}=\frac{\pi(r_{+}^{2}+2br+a^{2})}{\Xi},\end{split}$
(15)
where $r_{+}$ is the radius of the event horizon, which is the largest root of
$\Delta=0$. When b=0, the thermodynamic quantities become the same as those of
Kerr-AdS black holes. Kerr-AdS black holes are solutions of the Einstein field
equations in four dimensions with negative cosmological constant and rotation.
## 5 Discussion and results
In this section, we investigate the JTE process for AdS black holes: AMPYM and
AdS Kerr Sen and derive a clear formula for the JTC. We also examine how the
charge and the parameters of AMPYM and Kerr-Sen theories affect the inversion
curves. We also compare the inversion curves for different scenarios
### 5.1 Case I
In this section, we explore the JTE of black hole systems in the extended
phase space, where the black hole mass M is the same as the enthalpy H. We
compare this process with the JT process of van der Waals gases with fixed
particle number, and we use the fixed charge Q for the black hole systems. We
also assume that the other parameters are constant. Using [2, 45] as a
reference, and considering the mass of black holes, we can write the pressure
P as a function of M and $r_{+}$, the mass and the horizon radius of a black
hole. By substituting this expression in the temperature formula, we can
obtain the temperature as a function of M and $r_{+}$ as well. Furthermore, we
can express the mass M and the temperature T in terms of the pressure P and
the radius $r_{+}$ of a black hole. Therefore, we get,
$P(M,r_{+})=-\frac{3\left(\frac{2C^{2}(n-1)r_{+}^{-2n}}{n}+Qr_{+}^{-4q}+\frac{1}{r_{+}^{2}}\right)}{8\pi}+\frac{6Mr_{+}^{-n-1}}{n\omega_{n}}.$
(16)
Also, we have T in terms of M and $r_{+}$ as follows,
$T(M,r_{+})=-\frac{2C^{2}(n-1)r_{+}^{2-2n}+2qQr_{+}^{2-4q}+1}{2\pi
r_{+}}+\frac{4M(n+1)r_{+}^{-n}}{n\omega_{n}}.$ (17)
By solving the above equations, one can obtain the function T(M, P), which is
lengthy and will not be shown here. For particular free parameters for this
model, the T(M, P) curve can be shown. According to the definition of the JTC
$\mu=\left(\frac{\partial T}{\partial P}\right)_{M}$, the inversion pressure
and temperature between the cooling and heating regions are
($\tilde{P}$-$\tilde{T}$), which are determined by $\mu=0$. Therefore, the
most important thing is to find the function expression of $\mu$. By setting
$\mu=0$, one can obtain the inversion points ($\tilde{P}$, $\tilde{T}$) for
different fixed enthalpy M. With respect to[24], the JTC is given by,
$\begin{split}\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{M}=\frac{1}{C_{P}}\big{[}T(\frac{\partial V}{\partial
T}\big{)}_{P}-V\big{]}.\end{split}$ (18)
This approach is elegant. However, in this paper, we will use more
straightforward methods by applying only mass and temperature to derive the
JTC $\mu$. we can see that temperature is a function of pressure and radius,
and radius is a function of pressure and mass. So, the JTC is given by,
$\begin{split}\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{M}=\frac{\partial_{,r_{+}}T}{\partial_{,r_{+}}P}.\end{split}$ (19)
Now, using the relationship $\mu=\frac{\partial T}{\partial
P}|_{M}=\frac{\frac{\partial T}{\partial r_{+}}}{\frac{\partial P}{\partial
r_{+}}}$, we calculate Joule Thomson coefficient as follows,
$\begin{split}&\mu=\frac{2nr_{+}\left(\omega_{n}\left(r_{+}^{4q}\left(2C^{2}\left(2n^{2}-3n+1\right)r_{+}^{2}+r_{+}^{2n}\right)+8q^{2}Qr^{2n+2}-2qQr_{+}^{2n+2}\right)-8\pi
M(n+1)r_{+}^{n+4q+1}\right)}{3\left(n\omega_{n}\left(r_{+}^{4q}\left(2C^{2}(n-1)r_{+}^{2}+r_{+}^{2n}\right)+2qQr_{+}^{2n+2}\right)-8\pi
M(n+1)r^{n+4q+1}\right)}.\end{split}$ (20)
When the value of the coefficient $\mu$ is positive during the expansion, it
means that the temperature decreases and therefore it is called a cooling
phenomenon. However, when $\mu$ is negative, the temperature increases, and
this is called a heating process.Using various equations, we can write the
mass M and the temperature T as functions of the pressure P and the radius
$r_{+}$, which are the properties of a black hole. For $\mu=0$, we can obtain
the inversion temperature, in which the process of the temperature changes
reverses. It can be obtained by the formula,
$T_{i}=V\big{(}\frac{\partial T}{\partial V}\big{)}_{P},$ (21)
$\begin{split}V=\frac{\partial M}{\partial
p}=\frac{1}{6}nr_{+}^{n+1}\omega_{n}.\end{split}$ (22)
At the inversion temperature, the value of $\mu$ is 0, and the inversion
temperature is determined by the following equation:
$\begin{split}T_{i}=V\frac{\partial T}{\partial V}=V\frac{\frac{\partial
T}{\partial r_{+}}}{\frac{\partial V}{\partial r_{+}}}.\end{split}$ (23)
This is beneficial for identifying the areas of heating and cooling in the
$T-P$ plane. We calculate t using the equations (17), (20), (21), (22) and
(23),
$\begin{split}T_{i}=\frac{r_{+}\left(\frac{6C^{2}(n-1)^{2}(2n-1)r_{+}^{-2n}}{n}+8\pi(n+1)P_{i}+3(1-4q)Q(n-4q+1)r_{+}^{-4q}+\frac{3-3n}{r_{+}^{2}}\right)}{12\pi(n+1)}.\end{split}$
(24)
We can also have from relation (17),
$\begin{split}T_{i}=\frac{-\frac{6C^{2}(n-1)^{2}r_{+}^{1-2n}}{n}+8\pi(n+1)r_{+}P_{i}+3Q(n-4q+1)r_{+}^{1-4q}+\frac{3(n-1)}{r_{+}}}{12\pi}.\end{split}$
(25)
Figure 1 displays the Hawking temperature as a function of the horizon. We
keep the free parameters constant for each plot. In each subfigure, we can
observe some zero points for different free parameters. These zero points
correspond to the divergence points of the JTC, as we can easily see in figure
1. According to the radius of the horizon and the values of free parameters of
a black hole, for small values for the radius of the event horizon, our
structural behavior is completely distinct, and for larger radii, the figures
almost converge.
Figure 1: It shows the plot of $(T-r_{+})$ for the AMPYM black hole with
respect to free parameters that are determined in each plot
We have plotted the isenthalpic curves and the inversion curve of the AMPYM
black hole for various values of the free parameter in each plot of Figure 2.
In every subfigure, two isenthalpic curves with different values of M, along
with the corresponding inversion curve that occurs at the highest point of the
isenthalpic curves. We denote the inversion temperature and pressure of each
isenthalpic curve as $T_{i}$ and $P_{i}$, respectively. The isenthalpic curves
are divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole
undergoes cooling during the expansion process. However, for $P>P_{i}$, the
isenthalpic curve has a negative slope, implying that the black hole
experiences heating during the expansion process. This behavior is consistent
for different values of the free parameter in Figures (2a-2b).
Figure 2: Isenthalpic curves and inversion curve of the AMPYM black hole with
respect to mentioned free parameters that determine in each plot.
We do not present the explicit expressions of the solutions here, as they are
too long and complex. However, we can illustrate the relation between the
inversion temperature $T_{i}$ and the inversion pressure $P_{i}$ by using the
equations given above. Figure 3 shows the inversion curves for different free
parameters. The inversion curve has only one branch. The inversion temperature
rises steadily with the inversion pressure, but the slope of the inversion
curves becomes smaller. We can also notice some fine structures in the cases
of subfigures 3. For low pressure, the inversion temperature varies with the
free parameters that are specified in each plot. It was interesting for us to
study the effect of parameters and we observed that the slope of the inversion
curve increases with the changes in the various values of each of the
parameters.
Figure 3: It shows the plot of The inversion curve for the AMPYM black hole
with respect to free parameters that are determined in each plot
Before the end of this section, if we want to talk purely about the effect of
Maxwell charges as a specific result of our work, we should say that in [3],
the authors found that for $q>1$ and $n=2$, the ratio $T_{i}^{min}/T_{c}$ is
fully established, but in the present work, by adding the Maxwell charge to
the mentioned black hole, we found that for $C\gg 1$ the Yang–Mills parameter
has no effect on the ratio, and in any case, the $T_{i}^{min}/T_{c}$ is equal
to $1/2$, which has the same behavior as a four-dimensional charged black
hole. For $q>1$, when $C$ increases, the $T_{min}/T_{c}$ approaches the value
of $1/2$. However, for $q<1$, the conditions are slightly different from the
previous situation, because when $q=1/2$, $n=2$, and $Q_{1}=1$, the
$T_{i}^{min}/T_{c}$ is equal to $1/2$ for all values of ”C”, which is the same
as a charged black hole in four dimensions. Also, according to Table 1 to 3,
it can be inferred that when $1/2<q<1$, with the increase of Maxwell charge,
i.e. $C$, the $T_{i}^{min}/T_{c}$ only approaches the value of $1/2$ and
exhibits a more similar behavior to a four-dimensional charged black hole. If
for $0<q<1/2$, the $T_{i}^{min}/T_{c}$ becomes further or smaller than the
value of $1/2$, which is contrary to a charged black hole in four dimensions.
In general, when $q>1$, for the increase of Maxwell’s charge $C$, the
$T_{i}^{min}/T_{c}$ is almost equal to $1/2$ and when $q=1/2$, for all values
of the Maxwell’s charge, the $T_{i}^{min}/T_{c}$ is equal to $1/2$. Also, when
$1<q<1/2$, it is close to the value of $1/2$ and finally when $1/2<q<0$, the
values of the $T_{i}^{min}/T_{c}$ become smaller than of $1/2$.
### 5.2 Case II
With respect to [62, 78] and the Mass of black holes, We can express pressure
P in terms of M and $r_{+}$ and replace it in the formula for the temperature,
which will also become in terms of M and $r_{+}$. We also rewrite the mass M
and temperature T as a function of the pressure P and the radius $r_{+}$ of a
black hole. So we will have,
$P(M,r_{+})=-\frac{3\left(a^{2}+r_{+}\left(2b-2\Xi^{2}M+r_{+}\right)\right)}{8\pi
r_{+}(2b+r_{+})\left(a^{2}+r_{+}(2b+r_{+})\right)}.$ (26)
Also, the temperature in terms of M and $r_{+}$ is as follows[62, 78],
$\begin{split}&\mathcal{X}=a^{4}(b+r_{+})+a^{2}r_{+}(4b^{2}+6br_{+}+r_{+}(2r_{+}-\Xi^{2}M))+r_{+}^{2}(2b+r_{+})(3r_{+}(b-\Xi^{2}M)+2b(b-\Xi^{2}M)+r_{+}^{2})\\\
&\mathcal{Y}=2\pi
r_{+}(2b+r_{+})(a^{2}+2br_{+}+r_{+})(a^{2}+r_{+}(2b+r_{+}))\\\
&T(M,r_{+})=-\frac{\mathcal{X}}{\mathcal{Y}}.\end{split}$ (27)
Therefore, by using the equation (19) on can obtain,
$\begin{split}&\mathcal{A}=a^{8}(2b^{2}+2br_{+}+r_{+}^{2})+a^{6}r_{+}\big{(}16b^{3}+2b^{2}(9r_{+}+2)+br_{+}(2e^{2}M+8r_{+}+5)+r_{+}^{2}(r_{+}+2)\big{)}+a^{4}r_{+}^{2}\\\
&\bigg{[}48b^{4}+4b^{3}(2\Xi^{2}M+16r_{+}+5)+4b^{2}r_{+}(8\Xi^{2}M+7r_{+}+8)+br_{+}^{2}(20\Xi^{2}M+2r_{+}+19)\\\
&-r_{+}^{2}\big{(}\Xi^{2}(M-4Mr_{+})+(r_{+}-4)r_{+}\big{)}\bigg{]}+a^{2}r_{+}^{3}(2b+r_{+})\bigg{[}32b^{4}+4b^{3}(9r_{+}+4)+2b^{2}r_{+}(8\Xi^{2}M+5r_{+}+11)\\\
&+br_{+}\big{(}4\Xi^{2}M(r_{+}+1)+r_{+}(11-2r_{+})\big{)}-(r_{+}-2)r_{+}^{3}\bigg{]}\\\
&+(2b+1)r_{+}^{4}(2b+r_{+})^{2}\bigg{[}4b^{2}(b-\Xi^{2}M)+r_{+}^{2}(b-3\Xi^{2}M)+4br_{+}(b-\Xi^{2}M)\bigg{]}\\\
&\mathcal{B}=\bigg{[}2\pi
r_{+}^{2}(2b+r_{+})^{2}(a^{2}+r_{+}+2br_{+})^{2}(a^{2}+r_{+}(2b+r_{+}))^{2}\bigg{]}\\\
&\mathcal{C}=3\bigg{[}a^{4}(b+r_{+})+a^{2}r_{+}\big{(}4b^{2}+6br_{+}+r_{+}(2r_{+}-\Xi^{2}M)\big{)}+r_{+}^{2}(2b+r_{+})\big{(}3r_{+}(b-\Xi^{2}M)+2b(b-\Xi^{2}M)+r_{+}^{2}\big{)}\bigg{]}\\\
&\mathcal{D}=4\pi
r_{+}^{2}(2b+r_{+})^{2}\big{(}a^{2}+r_{+}(2b+r_{+})\big{)}^{2}\\\
&\mu=\frac{\mathcal{A}/\mathcal{B}}{\mathcal{C}/\mathcal{D}}.\end{split}$ (28)
We express the mass M and temperature T in terms of the pressure P and the
radius $r_{+}$ of a black hole, using different equations,
$M(P,r_{+})=\frac{\left(a^{2}+r_{+}(2b+r_{+})\right)(8\pi
pr_{+}(2b+r_{+})+3)}{6\Xi^{2}r_{+}},$ (29) $T(P,r_{+})=\frac{a^{2}\left(8\pi
pr_{+}^{2}-3\right)+r_{+}^{2}(8\pi p(2b+r_{+})(2b+3r_{+})+3)}{12\pi
r_{+}\left(a^{2}+2br_{+}+r_{+}\right)}.$ (30)
The $V(P)$ for the AKS black hole is calculates as follows,
$V(P)=\frac{4\pi(2b+r_{+})\left(a^{2}+r_{+}(2b+r_{+})\right)}{3\Xi^{2}}.$ (31)
So, by using the equation (19), (21), (22) and (23) one can obtain,
$\begin{split}&T_{i}=(2b+r_{+})(a^{2}+r_{+}(2b+r_{+}))\\\
&\times\bigg{[}a^{4}(8\pi P_{i}r_{+}^{2}+3)+a^{2}r_{+}\bigg{(}32\pi
b^{2}P_{i}r_{+}+4b(32\pi P_{i}r_{+}^{2}+3)+3(24\pi
P_{i}r_{+}^{3}+r_{+}+2)\bigg{)}\\\
&+16\pi(2b+1)P_{i}r_{+}^{4}(4b+3r_{+})\bigg{]}\\\ &\bigg{/}12\pi
r_{+}^{2}(a^{2}+2br_{+}+r_{+})^{2}\big{(}a^{2}+(2b+r_{+})(2b+3r_{+})\big{)}\end{split}$
(32)
Also, with respect to equation (30) we will have,
$\begin{split}T_{i}=\frac{a^{2}\left(8\pi
r_{+}^{2}P_{i}-3\right)+r_{+}^{2}(8\pi P(2b+r_{+})(2b+3r_{+})+3)}{12\pi
r_{+}\left(a^{2}+2br_{+}+r_{+}\right)}.\end{split}$ (33)
We have drawn the isenthalpic curves and the inversion curve of the AdS Kerr
Sen black hole for various free parameter values in each plot of Figure 4. In
every subfigure, three isenthalpic curves with different M are visible, along
with the corresponding inversion curve that occurs at the highest point of the
isenthalpic curves. We denote each isenthalpic curve’s inversion temperature
and pressure as $T_{i}$ and $P_{i}$, respectively. The isenthalpic curves are
divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole
undergoes cooling during the expansion process. With the changes in the
parameters of the black hole, we found that the slope of the figures will
change, and these changes are visible in each subfigure. Because an analytical
solution is difficult to obtain, we used a numerical method to draw graphs. If
we assume the parameter $b$ to be zero, our equations and graphs will simplify
to the Kerr–AdS black holes, whose results are thoroughly discussed in [1].
Figure 4: Isenthalpic curves and inversion curve of the AdS-Ker-Sen black hole
with respect to mentioned free parameters that determine in each plot.
Due to the complexity of the calculations, we do not show the explicit
expressions here. However, we can demonstrate the relation between the
inversion temperature $T_{i}$ and the inversion pressure $P_{i}$ by using the
equations given above. Figure (5a-5d) displays the inversion curves for
different free parameters of the AKS black holes. The inversion curve has only
one branch. The inversion temperature increases gradually with the inversion
pressure, but the slope of the inversion curves becomes smaller. We can also
observe some fine structures in the cases of subfigures 5. Inversion curves
change with the free parameters that are specified in each plot. It was very
difficult to find the analytical solution for drawing the graphs, so we used
numerical solutions to draw them. Moreover, we note that due to the huge
difference in the values of the horizontal and vertical graphs corresponding
to the change of the free parameters, we have drawn the figures separately.
Figure 5: It shows the plot of The inversion curve for the AdS-Ker-Sen black
hole with respect to free parameters that are determined in each plot with
M=20
In Figure 6, we plot the Hawking temperature as a function of the horizon,
which is the boundary of the black hole. The horizon can be affected by the
presence of other fields or dimensions, which are represented by the free
parameters in our model. We consider the free parameters constant for each
plot, but we vary them across different plots to see how they influence the
Hawking temperature. In each subfigure, we can observe some zero points, where
the Hawking temperature becomes zero for different free parameters. This means
that the black hole became extremal at these points. These zero points
correspond to the divergence points of the JTC, which is a quantity that
describes the temperature change when a gas expands or compresses at constant
enthalpy.
Figure 6: It shows the plot of $(T-r_{+})$ for the AdS-Ker-Sen black hole with
respect to free parameters that are determined in each plot
In the study of Kerr black hole[1], it was shown that the rotation parameter
”a” does not have much effect on the Joule-Thomson representation. For
example, it is practically eliminated in the ratio of the $T_{i}^{min}/T_{C}$,
and the value of this ratio is always a constant value of 1/2. In this paper,
we found that the parameters related to AKS black hole, i.e. a and b, can play
a vital role in the representation of the value of the $T_{i}^{min}/T_{C}$.
The value of this ratio increases with the reduction of these parameters. For
this purpose, due to the structural complexities of this black hole, we could
not use the analytical method to obtain the critical points of the black hole,
so we resorted to the approximate method and the numerical calculations. In
the [77], the value of critical temperature and critical pressure has been
calculated. We calculated some values for this black hole according to Table
4. As evident, regarding various values of free parameters, the
$T_{i}^{min}/T_{C}$ is obtained, which shows the obvious effect of parameters
a and b.
## 6 Conclusion
In this article, we investigated the behavior of two structurally different
black holes under the JTE thermodynamic process. The JTE is a process that
cools or heats a system by changing its pressure and volume at constant
enthalpy, and its main goal is to study the behavior of the $\mu$ coefficient,
which describes the change in temperature during the expansion or compression
of a system. But before presenting the results, we must first explain the
motivations that led us to choose these particular black holes. Recently,
Einstein-Power-Young-Mills black hole with AdS structure under JTE process was
investigated in a research paper[2, 3]. This raised the question in our mind
that if a fields in the Maxwellian form are added to the elements in the
action of the above article, can this change and addition of a special form of
the field cause a different thermodynamic behavior in their JT representation?
Similarly, in the past, the behavior of various forms of rotating black holes
was also investigated in this thermodynamic process. But what effect can the
addition of Sen-type dilaton fields and antisymmetric tensor fields have on
its thermodynamic properties and JT representation? This was a question that
could be a good motivation for our investigation. For this purpose, we plotted
the isenthalpic curves and the inversion curves for each of the two black
holes, for different values of the free parameters, which can be referred to
the following results for each. For the AMPYM black hole, the Hawking
temperature has zero points that depend on the free parameters and the horizon
radius, and the JTC diverges at these points. The isenthalpic curves are
divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole is
cooling during the expansion process, and for $P>P_{i}$, the isenthalpic curve
has a negative slope, which means that the black hole experiences heating
during the expansion process. The inversion curves for different free
parameters show that the inversion curve has only one branch and the inversion
temperature increases steadily with the inversion pressure, but the slope of
the inversion curves becomes smaller. Also, the graphs showed that for low
pressure, the inversion temperature varies with the free parameters specified
in each graph. Studying the effect of parameters was interesting for us
because we observed that the slope of the inversion curve increases with
changes in different values of each parameter. But if we want to talk about
the effect of Maxwell’s charges as a special result of our work, we must state
that, we have found that the ratio of minimum to critical temperature for a
4-dimensional black hole with Yang-Mills hair depends on the values of $q$,
$M$, and $C$. For $q>1$, the ratio is almost equal to $1/2$ regardless of the
Yang-Mills parameter when $"C\gg 1"$. For $q=1/2$, the ratio is exactly equal
to $1/2$ for all values of $C$. For $1/2<q<1$, the ratio approaches $1/2$ as
$C$ increases. For $0<q<1/2$, the ratio deviates from $1/2$ as $C$ increases.
These results show that the Maxwell charge can affect the thermodynamic
behavior of the black hole and make it more or less similar to a 4-dimensional
charged black hole. Also, for the AKS black hole, we found that its
parameters, i.e. a and b, can play a vital role in the representation of the
value of the ratio of $T_{i}^{min}/T_{C}$, so that by reducing these
parameters, the value of the ratio increases. The results are shown in Figure
7 and are summarized in Tables 1 to 5. For the AdS Kerr Sen black, It was very
difficult to find the analytical solution for drawing the graphs, so we used
numerical solutions to draw them. For $P<P_{i}$, the isenthalpic curve has a
positive slope, indicating that the black hole is cooling during the expansion
process, but with the increases of the black hole parameters, we found that
the slope of the figures will change. Also, if we assume the parameter b (
dyonic charge) to be zero, our equations and graphs will simplify to the Kerr-
AdS black holes, whose results are thoroughly discussed in[1]. In general, it
can be said about this black hole the isenthalpic curves of the AKS black hole
show cooling or heating behavior depending on the inversion curve, which is
affected by the mass and the parameter $b,a$ of the black hole. The inversion
curve has a single branch with a positive slope that decreases with the free
parameters, and the inversion temperature and pressure are related by some
equations that are numerically solved. The Hawking temperature of the AKS
black hole has zero points that depend on the free parameters and the horizon
radius and lead the JTC to reflect divergence behavior.
Figure 7: It shows the plot of $(\frac{T_{i}^{min}}{T_{c}})$ in terms of $C$ with respect to free parameters mentioned in the plot. $q=0.3,n=2,Q_{1}=1$ | $T_{i}^{min}$ | $T_{c}$ | $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ | $0.171532$ | $0.355378$ | $0.482674$
$C=0.2$ | $0.068796$ | $0.148927$ | $0.461947$
$C=0.3$ | $0.035473$ | $0.082056$ | $0.432299$
$C=1$ | Not Exist | Not Exist | Not Exist
$C=5$ | Not Exist | Not Exist | Not Exist
$C=10$ | Not Exist | Not Exist | Not Exist
Table 1: Summary of the results for the AMPYM black hole for $q=0.3$ $q=0.5,n=2,Q_{1}=1$ | $T_{i}^{min}$ | $T_{c}$ | $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ | $0.034331$ | $0.068662$ | $0.5$
$C=0.2$ | $0.017165$ | $0.034330$ | $0.5$
$C=0.3$ | $0.011443$ | $0.022886$ | $0.5$
$C=1$ | $0.003433$ | $0.006866$ | $0.5$
$C=5$ | $0.000686$ | $0.001372$ | $0.5$
$C=10$ | $0.000343$ | $0.000686$ | $0.5$
Table 2: Summary of the results for the AMPYM black hole for $q=0.5$ $q=0.9,n=2,Q_{1}=1$ | $T_{i}^{min}$ | $T_{c}$ | $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ | $0.019146$ | $0.03874$ | $0.493657$
$C=0.2$ | $0.018873$ | $0.038208$ | $0493962$
$C=0.3$ | $0.018446$ | $0.037309$ | $0.494405$
$C=1$ | $0.013846$ | $0.027832$ | $0497502$
$C=5$ | $0.004141$ | $0.008286$ | $0.499762$
$C=10$ | $0.002133$ | $0.004267$ | $0.499921$
Table 3: Summary of the results for the AMPYM black hole for $q=0.9$ ($a,b$) | $T_{i}^{min}$ | $T_{c}$ | $T_{i}^{min}/T_{c}$
---|---|---|---
($0.0099$, $0.00499$) | $0.0787687$ | $0.335030$ | $0.235109$
($0.00972$, $0.00486$) | $0.0782007$ | $0.2302899$ | $0.33957$
($0.00951$, $0.00475$) | $0.078821$ | $0.156982$ | $0.5$
($0.00937$, $0.00468$) | $0.0788340$ | $0.116006$ | $0.67379$
($0.00892$, $0.00442$) | $0.776375$ | $0.0454247$ | $1.709$
Table 4: Summary of the results for the AKS black hole AMPYM BH | $q>1$, $C\gg 1$ | $T_{min}/T_{c}=\frac{1}{2}$ | This paper
---|---|---|---
AMPYM BH | $q=0.5$, all values of $"C"$ | $T_{min}/T_{c}\simeq\frac{1}{2}$ | This paper
AMPYM BH | $0.5<q\leq 1$ , $C\gg 1$ | $T_{min}/T_{c}\simeq\frac{1}{2}$ | This paper
AKS BH | ($a=0.00951$, $b=0.00475$) | $T_{min}/T_{c}\simeq\frac{1}{2}$ | This paper
Van der Waals fluid | Exist | $T_{min}/T_{c}=\frac{3}{4}$ | [24]
RN-AdS BH | Exist | $T_{min}/T_{c}=\frac{1}{2}$ | [25]
d-dimensional AdS BH | Exist | $T_{min}/T_{c}<\frac{1}{2}$ | [79]
Gauss-Bonnet BH | Exist | $T_{min}/T_{c}=0.4765$ | [80]
torus-like BH | Not Exist | $T_{min}/T_{c}$= Not Exist | [81]
BTZ BH | Not Exist | $T_{min}/T_{c}$= Not Exist | [82]
Table 5: Summary of the results for the AMPYM and AKS black holes compare with
other results
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killcontents
# Resilient Distributed Optimization Algorithms for Resource Allocation
César A. Uribe†, Hoi-To Wai†, Mahnoosh Alizadeh CAU and HTW have contributed
equally. CAU is with LIDS, MIT, Cambridge, MA, USA. HTW is with Dept. of SEEM,
CUHK, Shatin, Hong Kong. MA is with Dept. of ECE, UCSB, Santa Barbara, CA,
USA. This work is partially supported by UCOP Grant LFR-18-548175 and CUHK
Direct Grant #4055113. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Distributed algorithms provide flexibility over centralized algorithms for
resource allocation problems, e.g., cyber-physical systems. However, the
distributed nature of these algorithms often makes the systems susceptible to
man-in-the-middle attacks, especially when messages are transmitted between
price-taking agents and a central coordinator. We propose a resilient strategy
for distributed algorithms under the framework of primal-dual distributed
optimization. We formulate a robust optimization model that accounts for
Byzantine attacks on the communication channels between agents and
coordinator. We propose a resilient primal-dual algorithm using state-of-the-
art robust statistics methods. The proposed algorithm is shown to converge to
a neighborhood of the robust optimization model, where the neighborhood’s
radius is proportional to the fraction of attacked channels.
## 1 Introduction
Consider the following multi-agent optimization problem involving the average
of parameters in the constraints:
$\begin{array}[]{rl}\displaystyle\min_{\bm{\theta}_{i}\in\mathbb{R}^{d},\forall
i}&U(\bm{\theta})\mathrel{\mathop{:}}=\frac{1}{N}\sum_{i=1}^{N}U_{i}(\bm{\theta}_{i})\\\
{\rm s.t.}&g_{t}\left(\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}\right)\leq
0,~{}t=1,...,T,\vspace{.1cm}\\\
&\bm{\theta}_{i}\in{\mathcal{}C}_{i},~{}i=1,...,N,\end{array}$ (1)
where both $U_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ and
$g_{t}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ are continuously differentiable,
convex functions, and ${\mathcal{}C}_{i}$ is a compact convex set in
$\mathbb{R}^{d}$. We let ${\bm{0}}\in{\mathcal{}C}_{i}$ and
$\max_{\bm{\theta},\bm{\theta}^{\prime}\in{\mathcal{}C}_{i}}\|\bm{\theta}-\bm{\theta}^{\prime}\|\leq
R,~{}i=1,...,N,$ (2)
such that $R$ is an upper bound on the diameters of ${\mathcal{}C}_{i}$.
Problem (1) arises in many resource allocation problems with a set of
potentially _nonlinear_ constraints on the amount of allowable resources, see
Section 1.1 for a detailed exploration.
We consider a system where there exists a central coordinator and $N$ agents.
In this context, the function $U_{i}(\bm{\theta}_{i})$ and parameter
$\bm{\theta}_{i}$ are the utility of the $i$th agent and the resource
controlled by agent $i$, respectively. As the agents work independently, it is
desirable to design algorithms that allow the $N$ agents to solve (1)
cooperatively through communication with the central coordinator. Among
others, the primal-dual optimization methods [1] have been advocated as they
naturally give rise to decomposable algorithms that favor distributed
implementation [2]. In addition to their practical success, these methods are
supported by strong theoretical guarantees where fast convergence to an
optimal solution of (1) is well established. However, the distributed nature
of these methods also exposes the system to vulnerabilities not faced by
traditional centralized systems. Precisely, existing algorithms assume the
agents, and the communication links between central server and agents, to be
_completely trustworthy_. However, an attacker can take over a sub-system
operated by the agents, and deliberately edit the messages in these
communication links, i.e., a Byzantine attack. This might result in an
unstable system with possible damages to hardware and the system overall.
In this paper, we propose strategies for securing primal-dual distributed
algorithms, e.g., in [1], tailored to solving a relaxed version of the
resource allocation problem (1). A key observation is that the existing
algorithms depend on reliably computing the average of a set of parameter
vectors, $\\{\bm{\theta}_{i}\\}_{i=1}^{N}$, transmitted by the agents. As a
remedy, we apply robust statistics techniques as a subroutine, therefore
proposing a _resilient_ distributed algorithm that is proven to converge to a
neighborhood of the optimal solution of a robust version of (1).
Vulnerabilities of various types of distributed algorithms have been
identified and addressed in a number of recent studies. Relevant examples are
[3, 4, 5, 6, 7] which study secure decentralized algorithms on a general
network topology but consider consensus-based optimization models. Moreover,
[8, 9, 10] consider a similar optimization architecture as this paper, yet
they focus on securing distributed algorithms for machine learning tasks which
assumes i.i.d. functions, a fundamentally different setting from the current
paper. Our work is also related to the literature on robust statistics [11,
12], and particularly, with the recently rekindled research efforts on high
dimensional robust statistics [13, 14, 15]. These works will be the working
horse for our attack resilient algorithm.
Our contributions and organization are as follows. First, we derive a formal
model for attack resilient resource allocation via a conservative
approximation for the robust optimization problem [cf. Section 3]. Second, we
apply and derive new robust estimation results to secure distributed resource
allocation algorithms [cf. Section 4]. Third, we provide a non-asymptotic
convergence guarantee of the proposed attack resilient algorithm [cf. Section
4.1]. In particular, our algorithm is shown to converge to a
${\mathcal{}O}(\alpha^{2})$ neighborhood to the optimal solution of (1), where
$\alpha\in[0,\frac{1}{2})$ is the fraction of attacked links.
Notations. Unless otherwise specified, $\|\cdot\|$ denotes the standard
Euclidean norm. For any $N\in\mathbb{N}$, $[N]$ denotes the finite set
$\\{1,...,N\\}$.
### 1.1 Motivating Examples
Our set-up here can be employed in a wide range of optimization problems for
resource allocation and networked control in multi-agent systems, e.g., in the
pioneering example of congestion control in data networks [16, 17]; in
determining the optimal price of electricity and enabling more efficient
demand supply balancing (a.k.a. demand response) in smart power distribution
systems [18, 19]; in managing user transmit powers and data rates in wireless
cellular networks [20]; in determining optimal caching policies by content
delivery networks [21]; in optimizing power consumption in wireless sensor
networks with energy-restricted batteries [22, 23]; and in designing
congestion control systems in urban traffic networks [24]. These examples
would have different utility functions and constraint sets that can be handled
through our general formulation in (1). For example, in the power/rate control
problem in data networks, the cost functions are usually logarithmic functions
associated with rate $\theta_{i}$, e.g.,
$U_{i}(\theta_{i})=-\beta_{i}\log(\theta_{i})$. In demand response
applications in power distribution systems, the utilities capture the users’
benefits from operating their electric appliances under different settings.
For example, we can capture the cost function of temperature $\theta_{i}$
controlled by a price-responsive air conditioner as
$U_{i}(\theta_{i})=b_{i}(\theta_{i}-\theta_{\rm{comf}})^{2}-c_{i}$ [19]. In
terms of constraints, our general nonlinear constraint formulation can not
only capture common linear resource constraints such as link capacity in data
networks [16, 17], but can also handle important non-linear constraints
arising in many different applications. For example, in radial power
distribution systems, nonlinear convexified power flow constraints can be
included for distributed demand response optimization (to see a description of
distribution system power flow constraints, see, e.g., [25, 26]). This can
enable our algorithm to perform demand supply balancing in power disribution
systems in a distributed and resilient fashion.
## 2 Primal-dual Algorithm for Resource Allocation
This section reviews the basic primal-dual algorithm for resource allocation.
Let $\bm{\lambda}\in\mathbb{R}_{+}^{T}$ be the dual variable. We consider the
Lagrangian function of (1):
$\begin{split}&{\mathcal{}L}(\\{\bm{\theta}_{i}\\}_{i=1}^{N};\bm{\lambda})\mathrel{\mathop{:}}=\frac{1}{N}\sum_{i=1}^{N}U_{i}(\bm{\theta}_{i})+\sum_{t=1}^{T}\lambda_{t}\\!~{}g_{t}\Big{(}\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}\Big{)}.\end{split}\vspace{-.1cm}$
Assuming strong duality holds (e.g., under the Slater’s condition), solving
problem (1) is equivalent to solving its dual problem:
$\max_{\bm{\lambda}\in\mathbb{R}_{+}^{T}}~{}\min_{\bm{\theta}_{i}\in{\mathcal{}C}_{i},\forall
i}~{}{\mathcal{}L}(\\{\bm{\theta}_{i}\\}_{i=1}^{N};\bm{\lambda}).$ (P)
For a given $\bm{\lambda}$, the inner minimization of (P) is known as the
Lagrangian relaxation of (1), which can be interpreted as a _penalized_
resource allocation problem [19].
In a distributed setting, the goal is to solve (1) where the agents only
observe a _pricing signal_ received from the central coordinator, and this
pricing signal is to be updated iteratively at the central coordinator. As
suggested in [1], we apply the primal-dual algorithm (PDA) to a regularized
version of (P). Let us define
$\begin{split}&{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}\\}_{i=1}^{N};\bm{\lambda})\mathrel{\mathop{:}}=\\\
&\textstyle{\mathcal{}L}(\\{\bm{\theta}_{i}\\}_{i=1}^{N};\bm{\lambda})+\frac{\upsilon}{2N}\sum_{i=1}^{N}\|\bm{\theta}_{i}\|^{2}-\frac{\upsilon}{2}\|\bm{\lambda}\|^{2},\end{split}$
(3)
such that ${\mathcal{}L}_{\upsilon}(\cdot)$ is $\upsilon$-strongly convex and
$\upsilon$-strongly concave in $\\{\bm{\theta}_{i}\\}_{i=1}^{N}$ and
$\bm{\lambda}$, respectively. Let $k\in\mathbb{Z}_{+}$ be the iteration index,
$\gamma>0$ be the step sizes, the PDA recursion is described by:
$\displaystyle\bm{\theta}_{i}^{(k+1)}=$ (4a)
$\displaystyle~{}~{}~{}~{}{\mathcal{}P}_{{\mathcal{}C}_{i}}\big{(}\bm{\theta}_{i}^{(k)}-\gamma\\!~{}{\nabla}_{\bm{\theta}_{i}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N};\bm{\lambda}^{(k)})\big{)},\forall~{}i\in[N]$
$\displaystyle\bm{\lambda}^{(k+1)}=\big{[}\bm{\lambda}^{(k)}+\gamma\\!~{}{\nabla}_{\bm{\lambda}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N};\bm{\lambda}^{(k)})\big{]}_{+}$
(4b)
where ${\mathcal{}P}_{{\mathcal{}C}_{i}}(\cdot)$ is the Euclidean projection
operator, $[\cdot]_{+}$ denotes $\max\\{0,\cdot\\}$, and the gradients are:
$\begin{split}&{\nabla}_{\bm{\theta}_{i}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N};\bm{\lambda}^{(k)})=\textstyle\frac{1}{N}\Big{(}{\nabla}_{\bm{\theta}_{i}}U_{i}(\bm{\theta}_{i}^{(k)})+\upsilon\\!~{}\bm{\theta}_{i}^{(k)}\\\
&\hskip
45.52458pt\textstyle+\sum_{t=1}^{T}\lambda_{t}^{(k)}{\nabla}_{\bm{\theta}}g_{t}(\bm{\theta})\Big{|}_{\bm{\theta}=\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}^{(k)}}\Big{)},\\\\[-17.07182pt]
\end{split}$ (5)
$\begin{split}&\big{[}{\nabla}_{\bm{\lambda}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N};\bm{\lambda}^{(k)})\big{]}_{t}=g_{t}\Big{(}{\textstyle\frac{1}{N}\sum_{i=1}^{N}}\bm{\theta}_{i}^{(k)}\Big{)}-\upsilon\\!~{}\lambda_{t}^{(k)},\end{split}$
(6)
for all $i$, $t$. We denoted $[{\bm{x}}]_{t}$ as the $t$th element of
${\bm{x}}\in\mathbb{R}^{T}$. In particular, observe that (4) performs a
projected gradient descent/ascent on the primal/dual variables.
From the above, both gradients with respect to (w.r.t.) $\bm{\theta}_{i}$ and
$\lambda_{t}$ depend only on the average parameter
$\overline{\bm{\theta}}^{(k)}\mathrel{\mathop{:}}=\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}^{(k)}$.
We summarize the primal dual distributed resource allocation (PD-DRA)
procedure in Algorithm 1. In addition to solving the general problem (1),
Algorithm 1 also serves as a general solution method to popular resource
allocation problems [19].
Algorithm 1 PD-DRA Procedure.
1: for $k=1,2,...$ do
2: _(Message exchanges stage)_ :
1. (a)
Central coordinator receives $\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N}$ from
agents and computes $\overline{\bm{\theta}}^{(k)}$,
$\\{{\nabla}_{\bm{\theta}}g_{t}(\overline{\bm{\theta}}^{(k)})\\}_{t=1}^{T}$.
2. (b)
Central coordinator broadcasts the vectors $\overline{\bm{\theta}}^{(k)}$,
$\overline{\bm{g}}^{(k)}\mathrel{\mathop{:}}=\sum_{t=1}^{T}\lambda_{t}^{(k)}{\nabla}_{\bm{\theta}}g_{t}(\overline{\bm{\theta}}^{(k)})$
to agents.
3: _(Computation stage)_ :
1. (a)
Agent $i$ computes the update for $\bm{\theta}_{i}^{(k+1)}$ according to (4a)
using the received $\overline{\bm{\theta}}^{(k)}$.
2. (b)
The central coordinator computes the update for $\bm{\lambda}^{(k+1)}$
according to (4b).
4: end for
As the regularized primal-dual problem is strongly convex/concave in
primal/dual variables, Algorithm 1 converges linearly to an optimal solution
[1]. To study this, let us denote
${\bm{z}}^{(k)}=(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N},\bm{\lambda}^{(k)})$ as
the primal-dual variable at the $k$th iteration,
$\bm{\Phi}({\bm{z}}^{(k)})\mathrel{\mathop{:}}=\left(\begin{array}[]{c}{\nabla}_{\bm{\theta}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N},\bm{\lambda}^{(k)})\\\
{\nabla}_{\bm{\lambda}}{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i=1}^{N},\bm{\lambda}^{(k)})\end{array}\right).\vspace{-.1cm}$
(7)
###### Fact 1.
[1, Theorem 3.5] Assume that the map $\bm{\Phi}({\bm{z}}^{(k)})$ is $L_{\Phi}$
Lipschitz continuous. For all $k\geq 1$, we have
$\|{\bm{z}}^{(k+1)}-{\bm{z}}^{\star}\|^{2}\leq(1-2\gamma\upsilon+\gamma^{2}L_{\Phi}^{2})\\!~{}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}\;,$
(8)
where ${\bm{z}}^{\star}$ is a saddle point to the regularized version of (P).
Setting $\gamma=\upsilon/L_{\Phi}^{2}$ gives
$\|{\bm{z}}^{(k+1)}-{\bm{z}}^{\star}\|^{2}\leq\big{(}1-\upsilon^{2}/L_{\Phi}^{2}\big{)}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}$,
$\forall~{}k\geq 1$.
## 3 Problem Formulation
Despite the simplicity and the strong theoretical guarantee, the PD-DRA method
is susceptible to attacks on the channels between the central coordinator and
the agents, as described below.
…Central CoordinatorAgent $i$Agent $j$get $\overline{\bm{g}}^{(k)}$send
$\bm{\theta}_{i}^{(k)}$Attacked!
Figure 1: Illustrating the PD-DRA algorithm under attack. The uplink for agent
$j$ is compromised such that the correct $\bm{\theta}_{j}^{(k)}$ is not
transmitted to the central node. The up/downlink for agent $i$ are operating
properly.
Attack Model. We consider a situation when _uplink_ channels between agents
and the central coordinator are compromised [see Fig. 1]. Let
${\mathcal{}A}\subset[N]$ be the set of _compromised uplink channels_ , whose
identities are unknown to the central coordinator. We define
${\mathcal{}H}\mathrel{\mathop{:}}=[N]\setminus{\mathcal{}A}$ as the set of
trustworthy channels. At iteration $k$, instead of receiving
$\bm{\theta}_{i}^{(k)}$ from each agent $i\in[N]$ [cf. Step 2(a)], the central
coordinator receives the following messages:
${\bm{r}}_{i}^{(k)}=\begin{cases}\bm{\theta}_{i}^{(k)},&\text{if}~{}i\in{\mathcal{}H},\\\
{\bm{b}}_{i}^{(k)},&\text{if}~{}i\in{\mathcal{}A}.\end{cases}\vspace{-.2cm}$
(9)
We focus on a Byzantine attack scenario such that the messages,
${\bm{b}}_{i}^{(k)}$, communicated on the attacked channels can be arbitrary.
Under such scenario, if the central coordinator forms the naive average
$\widehat{\bm{\theta}}^{(k)}=1/N\sum_{i=1}^{N}{\bm{r}}_{i}^{(k)}$ and computes
the gradients ${\nabla}g_{t}(\widehat{\bm{\theta}}^{(k)})$ accordingly, this
may result in uncontrollable error since the deviation
$\widehat{\bm{\theta}}^{(k)}-(1/N)\sum_{i=1}^{N}\bm{\theta}_{i}^{(k)}$ can be
arbitrarily large. It is anticipated that the PD-DRA method would not provide
a solution to the regularized version of (P).
Robust Optimization Model. In light of the Byzantine attack, it is impossible
to optimize the original problem (P) since the contribution from
$U_{i}(\cdot):i\in{\mathcal{}A}$ becomes unknown to the central coordinator.
As a compromise, we focus on optimizing the cost function of agents with
trustworthy uplinks and the following robust optimization problem as our
target model:
$\displaystyle\min_{\begin{subarray}{c}\bm{\theta}_{i}\in{\mathcal{}C}_{i},i\in{\mathcal{}H}\end{subarray}}$
$\displaystyle~{}{\textstyle\frac{1}{|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}U_{i}(\bm{\theta}_{i})}$
(10a) $\displaystyle~{}{\rm s.t.}$
$\displaystyle\displaystyle\max_{\bm{\theta}_{j}\in{\mathcal{}C}_{j},j\in{\mathcal{}A}}~{}g_{t}\Big{(}{\textstyle\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}}\Big{)}\leq
0,~{}\forall~{}t,\vspace{.1cm}$ (10b)
note that $\\{\bm{\theta}_{j}\\}_{j\in{\mathcal{}A}}$ is taken away from the
decision variables and we have included (10b) to account for the _worst case_
scenario for the resource usage of the agents with compromised uplinks. This
is to ensure that the physical operation limit of the system will not be
violated under attack. Consider the following assumption which will be assumed
throughout the paper:
###### H 1.
For all $\bm{\theta}\in\mathbb{R}^{d}$, the gradient of $g_{t}$ is bounded
with $\|{\nabla}g_{t}(\bm{\theta})\|\leq B$ and is $L$-Lipschitz continuous.
We define
$\overline{g}_{t}(\bm{\theta})\mathrel{\mathop{:}}=g_{t}(\bm{\theta})+{\textstyle\frac{|{\mathcal{}A}|}{N}}\big{(}RB+{\textstyle\frac{1}{2}}LR^{2}\big{)},$
(11)
###### Lemma 1.
Under H1. The following problem yields a _conservative_ approximation of (10),
i.e., its feasible set is a subset of the feasible set of (10):
$\begin{array}[]{rl}\displaystyle\min_{\bm{\theta}_{i}\in{\mathcal{}C}_{i},i\in{\mathcal{}H}}&\frac{1}{|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}U_{i}(\bm{\theta}_{i})\vspace{.1cm}\\\
{\rm
s.t.}&\displaystyle\overline{g}_{t}\left({\textstyle\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}}\right)\leq
0,~{}\forall~{}t\in[T],\end{array}$ (12)
Similar to PD-DRA, we define the regularized Lagrangian function of (12) as:
$\begin{split}&\overline{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}\\}_{i\in{\mathcal{}H}};\bm{\lambda};{\mathcal{}H})\\\
&\mathrel{\mathop{:}}={\textstyle\frac{1}{|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}}U_{i}(\bm{\theta}_{i})+{\textstyle\sum_{t=1}^{T}}\lambda_{t}\\!~{}\overline{g}_{t}\left({\textstyle\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}}\right)\\\
&\hskip
14.22636pt\textstyle+\frac{\upsilon}{2|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}\|\bm{\theta}_{i}\|^{2}-\frac{\upsilon}{2}\|\bm{\lambda}\|^{2}.\end{split}$
(13)
Again, the regularized Lagrangian function is $\upsilon$-strongly convex and
concave in $\bm{\theta}$ and $\bm{\lambda}$, respectively.
Our main task is to tackle the following modified problem of (P) under
Byzantine attack on (some of) the uplinks:
$\max_{\bm{\lambda}\in\mathbb{R}_{+}^{T}}\min_{\bm{\theta}_{i}\in{\mathcal{}C}_{i},\forall
i\in{\mathcal{}H}}~{}\overline{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}\\}_{i\in{\mathcal{}H}};\bm{\lambda};{\mathcal{}H}),$
(P’)
and we let
$\widehat{\bm{z}}^{\star}=(\widehat{\bm{\theta}}^{\star},\widehat{\bm{\lambda}}^{\star})$
be the optimal solution to (P’). Notice that (P’) bears a similar form as (P)
and thus one may apply the PD-DRA method to the former naturally. However,
such application requires the central coordinator to compute the sample
average
$\textstyle\overline{\bm{\theta}}^{(k)}_{\mathcal{}H}\mathrel{\mathop{:}}=\frac{1}{|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}^{(k)},$
(14)
at each iteration. However, the above might not be computationally feasible
under the attack model, since the central coordinator is oblivious to the
identity of ${\mathcal{}H}$. This is the main objective in the design of our
scheme.
## 4 Robust Distributed Resource Allocation
In this section, we describe two estimators for approximating
$\overline{\bm{\theta}}^{(k)}_{\mathcal{}H}$ [cf. (14)] from the received
messages (9) without knowing the identity of links in ${\mathcal{}H}$. To
simplify notations, we define $\alpha\geq|{\mathcal{}A}|/N$ as a known upper
bound to the fraction of compromised channels and assume $\alpha<1/2$ where
less than half of the channels are compromised.
As discussed after (14), the problem at hand is _robust mean estimation_ ,
whose applications to robust distributed optimization has been considered in
the machine learning literature [9, 10, 14] under the assumption that the
‘trustworthy’ signals are drawn i.i.d. from a Gaussian distribution. Our
setting is different since the signals $\bm{\theta}_{i}^{(k)}$,
$i\in{\mathcal{}H}$ are variables from the previous iteration whose
distribution is non-Gaussian in general. Our analysis will be developed
without such assumption on the distribution.
We first consider a simple median-based estimator applied to each coordinate
$j=1,...,d$. First, define the coordinate-wise median as:
$\big{[}{\bm{\theta}}_{\sf med}^{(k)}\big{]}_{j}={\sf
med}\big{(}\\{[{\bm{r}}_{i}^{(k)}]_{j}\\}_{i=1}^{N}\big{)},$ (15)
where ${\sf med}(\cdot)$ computes the median of the operand. Then, our
estimator is computed as the mean of the nearest $(1-\alpha)N$ neighbors of
$\big{[}{\bm{\theta}}_{\sf med}^{(k)}\big{]}_{j}$. To formally describe this,
let us define:
${\mathcal{}N}_{j}^{(k)}=\\{i\in[N]:\big{|}\big{[}{\bm{r}}_{i}^{(k)}-{\bm{\theta}}_{\sf
med}^{(k)}\big{]}_{j}\big{|}\leq r_{j}^{(k)}\\},$ (16)
where $r_{j}^{(k)}$ is chosen as $|{\mathcal{}N}_{j}^{(k)}|=(1-\alpha)N$. Our
estimator is:
$\textstyle[\widehat{\bm{\theta}}^{(k)}_{{\mathcal{}H}}]_{j}=\frac{1}{(1-\alpha)N}\sum_{i\in{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}]_{j}.$
(17)
The following bounds the performance of (17).
###### Proposition 1.
Suppose that
$\max_{i\in{\mathcal{}H}}\big{\|}\bm{\theta}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\big{\|}_{\infty}\leq
r$, then for any $\alpha\in(0,\frac{1}{2})$, it holds that
$\big{\|}\widehat{\bm{\theta}}^{(k)}_{\mathcal{}H}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\big{\|}\leq\frac{\alpha}{1-\alpha}\Big{(}2+\sqrt{\frac{(1-\alpha)^{2}}{1-2\alpha}}\Big{)}r\sqrt{d}\;.\vspace{-.1cm}$
(18)
Under mild assumptions, the condition
$\max_{i\in{\mathcal{}H}}\big{\|}\bm{\theta}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\big{\|}_{\infty}\leq
r$ can be satisfied with $r=\Theta(R)$, as implied by the compactness of
${\mathcal{}C}_{i}$ [cf. (2)]. Moreover, for sufficiently small $\alpha$, the
right hand side on (18) can be approximated by ${\mathcal{}O}(\alpha
R\sqrt{d})$. However, this median-based estimator may perform poorly for large
$\alpha$ (especially when
$\alpha\rightarrow{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{1}/{2}}$)
or dimension $d$. For these situations, a more sophisticated estimator is
required, as detailed next.
Algorithm 2 Recovering the mean of a set [15].
1: Input: $\alpha$, $\bm{\theta}_{i}^{(k)}$, $c_{i}=1$ for all $i=1,\ldots,N$,
and $\mathcal{B}=\\{1,\ldots,N\\}.$
2: Set $X_{\mathcal{B}}=[\cdots\bm{\theta}_{j}^{(k)}\cdots]^{\top}$ for
$j\in\mathcal{B}$ as the concatenated data matrix.
3: Let $Y\in\mathbb{R}^{d\times d}$ and
$W\in\mathbb{R}^{\mathcal{B}\times\mathcal{B}}$ be the maximizer/minimizer of
the saddle point problem $\displaystyle\max_{\begin{subarray}{c}Y\succeq
0,\\\ \text{tr}(Y)\leq
1\end{subarray}}\min_{\begin{subarray}{c}0\leq{W_{ij}},\\\
W_{ij}\leq\frac{4{-}\alpha}{\alpha(2{+}\alpha){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n}},\\\
\sum_{j}W_{ji}=1\end{subarray}}\sum_{i\in\mathcal{B}}c_{i}(\bm{\theta}_{i}^{(k)}{-}X_{\mathcal{B}}w_{i})^{\top}Y(\bm{\theta}_{i}^{(k)}{-}X_{\mathcal{B}}w_{i})$
4: Let
$\tau_{i}^{*}=(\bm{\theta}_{i}^{(k)}-X_{\mathcal{B}}w_{i})^{\top}Y(\bm{\theta}_{i}^{(k)}-X_{\mathcal{B}}w_{i})$.
5: if $\sum_{i\in\mathcal{B}}c_{i}\tau_{i}^{*}>4n\sigma^{2}$ then
6: For $i\in\mathcal{B}$, replace $c_{i}$ with
$\big{(}1-\frac{\tau^{*}_{i}}{\max_{j\in\mathcal{B}}\tau^{*}_{j}}\big{)}c_{i}$.
7: For all $i$ with $c_{i}<\frac{1}{2}$, remove from $\mathcal{B}$.
8: Go back to Line 3.
9: end if
10: Set $W_{1}$ as the result of zeroing out all singular values of $W$ that
are greater than $0.9$.
11: Set $Z=X_{\mathcal{B}}W_{0}$ where
$W_{0}=(W{-}W_{1})(I{-}W_{1})^{\text{-}1}$.
12: if $\text{rank}(Z)=1$ then
13: Output: $\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}$ as average of the
columns of $X_{\mathcal{B}}$.
14: else
15: Output: $\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}$ as a column of $Z$ at
random.
16: end if
To derive the second estimator, we apply an auxiliary result from [15] which
provides an algorithm for estimating
$\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}$, as summarized in Algorithm 2. We
observe:
###### Proposition 2.
[15, Proposition 16] Suppose that
$\lambda_{\max}(\frac{1}{|{\mathcal{}H}|}\sum_{i\in{\mathcal{}H}}(\bm{\theta}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)})(\bm{\theta}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)})^{\top})\leq\sigma^{2}$.
For any $\alpha\in[0,\frac{1}{4})$, Algorithm 2 produces an output such that
$\|\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}-\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}\|={\mathcal{}O}(\sigma\sqrt{\alpha})$.
Again, similar to Proposition 1, the required condition above can be satisfied
with $\sigma=\Theta(R)$ under mild conditions. Thus, Proposition 2 states that
Algorithm 2 recovers $\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}$ up to an
error of ${\mathcal{}O}(\sqrt{\alpha}R)$. Note that this bound is dimension
free unlike the median estimator analyzed in Proposition 1.
The idea behind Algorithm 2 is to sequentially identify and remove the subset
of points that cannot be re-constructed from the mean of the data points. The
solution of the optimization problem in Line 3 measures how well can we
recover the data points as an average of the other $|{\mathcal{}H}|$ points.
The bounded sample variance assumption guarantees that one can re-construct
any element in the set ${\mathcal{}H}$ from its mean, thus, all such points
that introduce a large error, as quantified by $c_{i}$ can be safely removed.
Line 5 quantifies the magnitude of the optimal point of Line 3, and if such
value is large, such points that introduce a large error are down-weighted.
The process is repeated until the optimal solution of Line 3 is small enough
and a low rank approximation of the optimal $W$ can be used to return the
sample mean estimate.
Algorithm 3 Resilient PD-DRA
1: Input: Each agent has initial state $\bm{\theta}_{i}^{(0)}$.
2: for $k=1,2,...$ do
3: _(At the Central Coordinator)_ :
1. (a)
Receives $\\{{\bm{r}}_{i}^{(k)}\\}_{i=1}^{N}$, see (9), from agents.
2. (b)
Computes robust mean $\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}$ using the
estimator (17) or Algorithm 2.
3. (c)
Broadcasts the vectors $\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}$ and
$\widehat{\bm{g}}^{(k)}_{\mathcal{}H}\mathrel{\mathop{:}}=\sum_{t=1}^{T}\lambda_{t}^{(k)}{\nabla}_{\bm{\theta}}\overline{g_{t}}(\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)})$
to agents.
4. (d)
Computes the update for $\bm{\lambda}^{(k+1)}$ with (20).
4: _(At each agent $i$)_:
1. (a)
Agent receives $\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}$ and
$\widehat{\bm{g}}^{(k)}_{\mathcal{}H}$.
2. (b)
Agent computes update for $\bm{\theta}_{i}^{(k+1)}$ with (19).
5: end for
Attack Resilient PD-DRA method. The above section provides the enabling tool
for developing the resilient PD-DRA method, which we summarize in Algorithm 3.
The algorithm behaves similarly as Algorithm 1 applied to (P’), with the
exception that the central coordinator is oblivious to ${\mathcal{}H}$, and it
uses a robust mean estimator to find an approximate average for the signals
sent through the trustworthy links. This approximate value is used to compute
the new price signals, and sent back to agents. In particular, the primal-dual
updates are described by
$\bm{\theta}_{i}^{(k+1)}={\mathcal{}P}_{{\mathcal{}C}_{i}}\big{(}\bm{\theta}_{i}^{(k)}-{\textstyle\frac{\gamma}{N}}\big{(}\widehat{\bm{g}}^{(k)}_{\mathcal{}H}+{\nabla}U_{i}(\bm{\theta}_{i}^{(k)})+\upsilon\bm{\theta}_{i}^{(k)}\big{)}\big{)},$
(19)
$\lambda_{t}^{(k+1)}=\big{[}\lambda_{t}^{(k)}+\gamma\big{(}\overline{g_{t}}({\textstyle\frac{|{\mathcal{}H}|}{N}}\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)})-\upsilon\lambda_{t}^{(k)}\big{)}\big{]}_{+}.$
(20)
###### Lemma 2.
Algorithm 3 is a primal-dual algorithm [1] for (P’) with perturbed gradients:
$\displaystyle\widehat{\bm{g}}_{\bm{\theta}}^{(k)}$
$\displaystyle={\nabla}_{\bm{\theta}}\overline{\mathcal{}L}_{\upsilon}(\bm{\theta}^{(k)};\bm{\lambda}^{(k)};{\mathcal{}H})+{\bm{e}}_{\bm{\theta}}^{(k)},$
(21a) $\displaystyle\widehat{\bm{g}}_{\bm{\lambda}}^{(k)}$
$\displaystyle={\nabla}_{\bm{\lambda}}\overline{\mathcal{}L}_{\upsilon}(\bm{\theta}_{i}^{(k)};\bm{\lambda}^{(k)};{\mathcal{}H})+{\bm{e}}_{\bm{\lambda}}^{(k)},$
(21b)
where we have used concatenated variable as
$\bm{\theta}=(\bm{\theta}_{1},...,\bm{\theta}_{N})$ and
$\bm{\lambda}=(\lambda_{1},...,\lambda_{T})$. Under H1 and assuming that
$\lambda_{t}^{(k)}\leq\overline{\lambda}$ for all $k$, we have:
$\|{\bm{e}}_{\bm{\theta}}^{(k)}\|\leq\overline{\lambda}LT\|\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\|,$
(22) $\|{\bm{e}}_{\bm{\lambda}}^{(k)}\|\leq
BT\|\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\|.\vspace{-.1cm}$
(23)
The assumption $\lambda_{t}^{(k)}\leq\overline{\lambda}$ can be guaranteed
since
$\overline{g_{t}}({\textstyle\frac{|{\mathcal{}H}|}{N}}\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)})$
is bounded.
### 4.1 Convergence Analysis
Finally, based on Lemma 2, we can analyze the convergence of Algorithm 3. Let
$\widehat{\bm{z}}^{\star}=(\widehat{\bm{\theta}}^{\star},\widehat{\bm{\lambda}}^{\star})$
be a saddle point of (P’) and define
$\overline{\bm{\Phi}}({\bm{z}}^{(k)})\mathrel{\mathop{:}}=\left(\begin{array}[]{c}{\nabla}_{\bm{\theta}}\overline{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i\in{\mathcal{}H}},\bm{\lambda}^{(k)};{\mathcal{}H})\\\
-{\nabla}_{\bm{\lambda}}\overline{\mathcal{}L}_{\upsilon}(\\{\bm{\theta}_{i}^{(k)}\\}_{i\in{\mathcal{}H}},\bm{\lambda}^{(k)};{\mathcal{}H})\end{array}\right),$
(24)
###### Theorem 1.
Assume the map $\overline{\bm{\Phi}}({\bm{z}}^{(k)})$ is $L_{\Phi}$-Lipschitz
continuous. For Algorithm 3, for all $k\geq 0$ it holds
$\begin{split}&\|{\bm{z}}^{(k+1)}-\widehat{\bm{z}}^{\star}\|^{2}\leq\big{(}1-\gamma\upsilon+2\gamma^{2}L_{\Phi}^{2}\big{)}\|{\bm{z}}^{(k)}-\widehat{\bm{z}}^{\star}\|^{2}\\\
&+\big{(}\frac{4\gamma}{\upsilon}+2\gamma^{2}\big{)}E_{k}.\end{split}$ (25)
where
$E_{k}\mathrel{\mathop{:}}=\|{\bm{e}}_{\bm{\theta}}^{(k)}\|^{2}+\|{\bm{e}}_{\bm{\lambda}}^{(k)}\|^{2}$
is the total perturbation at iteration $k$. Moreover, if we choose
$\gamma<{\upsilon}/{2L_{\Phi}^{2}}$ and $E_{k}$ is upper bounded by
$\overline{E}$ for all $k$, then
$\limsup_{k\rightarrow\infty}\|{\bm{z}}^{(k)}-\widehat{\bm{z}}^{\star}\|^{2}\leq\frac{\frac{4}{\upsilon}+2\gamma}{\upsilon-2\gamma
L_{\Phi}^{2}}\overline{E}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(26)
Combining the results from the last subsection, the theorem shows the desired
result that the resilient PD-DRA method converges to a
${\mathcal{}O}(\alpha^{2}R^{2}d)$ neighborhood of the saddle point of (P’), if
the median-based estimator (17) is used [or ${\mathcal{}O}(\alpha R^{2})$ if
Algorithm 2 is used], where $\alpha$ is the fraction of attacked uplink
channels. Moreover, it shows that the convergence rate to the neighborhood is
linear, which is similar to the classical PDA analysis [1].
Interestingly, Theorem 1 illustrates a trade-off in the choice of the step
size $\gamma$ between convergence speed and accuracy. In specific, (25) shows
that the rate of convergence factor $1-\gamma\upsilon+2\gamma^{2}L_{\Phi}^{2}$
can be minimized by setting $\gamma=\upsilon/(4L_{\Phi}^{2})$. However, in the
meantime, the asymptotic upper bound in (26) is increasing with $\gamma$ and
it can be minimized by setting $\gamma\rightarrow 0$. This will be a design
criterion to be explored in practical implementations.
## 5 Conclusions
In this paper, we studied the strategies for securing a primal-dual algorithm
for distributed resource allocation. Particularly, we propose a resilient
distributed algorithm based on primal-dual optimization and robust statistics.
We derive bounds for the performance of the studied algorithm and show that it
converges to a neighborhood of a robustified resource allocation problem when
the number of attacked channels is small.
## Acknowledgement
The authors would like to thank the anonymous reviewers for feedback, and Mr.
Berkay Turan (UCSB) for pointing out typos in the original submission of this
paper.
## Appendix A Proof of Lemma 1
Since $g_{t}$ is $L_{t}$-smooth, the following holds
$\begin{split}&\textstyle
g_{t}\big{(}\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}\big{)}\leq
g_{t}\big{(}\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}\big{)}\\\
&\textstyle+\frac{1}{N}\sum_{j\in{\mathcal{}A}}\Big{\langle}\bm{\theta}_{j},{\nabla}g_{t}\big{(}\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}\big{)}\Big{\rangle}+\frac{L_{t}}{2N^{2}}\big{\|}\sum_{j\in{\mathcal{}A}}\bm{\theta}_{j}\big{\|}^{2}\end{split}$
(27)
Furthermore, observe that the gradient of $g_{t}$ is uniformly bounded by $B$
and the diameter of ${\mathcal{}C}_{j}$ is $R$, then the right hand side of
(27) can be upper bounded by
$g_{t}\big{(}{\textstyle\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}}\big{)}+{\textstyle\frac{1}{N}}\sum_{j\in{\mathcal{}A}}\big{(}RB+{\textstyle\frac{1}{2}}LR^{2}\big{)}$
(28)
As such, defining
$c_{t}\mathrel{\mathop{:}}=\frac{|{\mathcal{}A}|}{N}\big{(}RB+\frac{1}{2}LR^{2}\big{)}$,
it can be seen that
$g_{t}\big{(}{\textstyle\frac{1}{N}\sum_{i\in{\mathcal{}H}}\bm{\theta}_{i}}\big{)}+c_{t}\leq
0,~{}t=1,...,T$ (29)
implies the desired constraint in (10).
## Appendix B Proof of Proposition 1
Fix any $j\in[d]$. The assumption implies that for all $i\in{\mathcal{}H}$,
one has
$\big{|}[\bm{\theta}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}]_{j}\big{|}\leq
r.$ (30)
We observe that $|{\mathcal{}H}|\geq(1-\alpha)N$. Applying [8, Lemma 1] shows
that the median estimator111At each coordinate, the median is the geometric
median estimator of one dimension in [8]. satisfies
$\big{|}\big{[}{\bm{\theta}}_{\sf
med}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\big{]}_{j}\big{|}\leq(1-\alpha)\sqrt{\frac{1}{1-2\alpha}}r.$
(31)
The above implies that for all $i\in{\mathcal{}H}$, we have
$\big{|}[\bm{\theta}_{i}^{(k)}-{\bm{\theta}}_{\sf
med}^{(k)}]_{j}\big{|}\leq\Big{(}1+\sqrt{\frac{(1-\alpha)^{2}}{1-2\alpha}}\Big{)}r.$
(32)
This implies that
$r_{j}^{(k)}\leq\Big{(}1+\sqrt{\frac{(1-\alpha)^{2}}{1-2\alpha}}\Big{)}r$
since $|{\mathcal{}H}|\geq(1-\alpha)N$. We then bound the performance of
$\widehat{\bm{\theta}}^{(k)}$:
$\begin{split}&(1-\alpha)N[\widehat{\bm{\theta}}^{(k)}]_{j}=\sum_{i\in{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}]_{j}\\\
&=\sum_{i\in{\mathcal{}H}}[{\bm{r}}_{i}^{(k)}]_{j}-\sum_{i\in{\mathcal{}H}\setminus{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}]_{j}+\sum_{i\in{\mathcal{}A}\cap{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}]_{j},\end{split}$
(33)
thus
$\begin{split}&(1-\alpha)N[\widehat{\bm{\theta}}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}]_{j}\\\
&=-\sum_{i\in{\mathcal{}H}\setminus{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}]_{j}+\sum_{i\in{\mathcal{}A}\cap{\mathcal{}N}_{j}^{(k)}}[{\bm{r}}_{i}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}]_{j}.\end{split}$
Notice that $|{\mathcal{}A}\cap{\mathcal{}N}_{j}^{(k)}|\leq\alpha N$ and thus
$|{\mathcal{}H}\setminus{\mathcal{}N}_{j}^{(k)}|\leq\alpha N$. Gathering terms
shows
$|[\widehat{\bm{\theta}}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}]_{j}|\leq\frac{\alpha
N}{(1-\alpha)N}\Big{(}2+\sqrt{\frac{(1-\alpha)^{2}}{1-2\alpha}}\Big{)}r.$ (34)
The above holds for all $j\in[d]$. Applying the norm equivalence shows the
desired bound.
## Appendix C Proof of Lemma 2
Comparing the equations in (21) with (19), (20), we identify that
$\big{[}{\bm{e}}_{\bm{\theta}}^{(k)}\big{]}_{i}=\frac{1}{N}{\sum_{t=1}^{T}}\lambda_{t}^{(k)}\big{(}{\nabla}_{\bm{\theta}}\overline{g_{t}}({\textstyle\frac{|{\mathcal{}H}|}{N}}\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)})-{\nabla}_{\bm{\theta}}\overline{g_{t}}({\textstyle\frac{|{\mathcal{}H}|}{N}}\overline{\bm{\theta}}_{\mathcal{}H}^{(k)})\big{)}$
(35)
$\big{[}{\bm{e}}_{\bm{\lambda}}^{(k)}\big{]}_{t}=\overline{g}_{t}({\textstyle\frac{|{\mathcal{}H}|}{N}}\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)})-\overline{g}_{t}({\textstyle\frac{|{\mathcal{}H}|}{N}}\overline{\bm{\theta}}_{\mathcal{}H}^{(k)})$
(36)
where $\big{[}{\bm{e}}_{\bm{\theta}}^{(k)}\big{]}_{i}$ denotes the $i$th block
of ${\bm{e}}_{\bm{\theta}}^{(k)}$. Using H1 and the said assumptions, we
immediately see that
$\|\big{[}{\bm{e}}_{\bm{\theta}}^{(k)}\big{]}_{i}\|\leq\overline{\lambda}\frac{LT}{N}\|\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\|$
(37)
which then implies (22). H1 implies that $\overline{g}_{t}$ is $B$-Lipschitz
continuous, therefore
$|\big{[}{\bm{e}}_{\bm{\lambda}}^{(k)}\big{]}_{t}|\leq
B\|\widehat{\bm{\theta}}_{\mathcal{}H}^{(k)}-\overline{\bm{\theta}}_{\mathcal{}H}^{(k)}\|,$
(38)
which implies (23).
## Appendix D Proof of Theorem 1
Based on Proposition 2, our idea is to perform a perturbation analysis on the
PDA algorithm. Without loss of generality, we assume $N=1$ and denote
$\bm{\theta}=\bm{\theta}_{1}$. To simplify notations, we also drop the
subscript, denote the modified and regularized Lagrangian function as
${\mathcal{}L}=\overline{\mathcal{}L}_{\upsilon}$. Furthermore, we denote the
saddle point to (P’) as
${\bm{z}}^{\star}=(\bm{\theta}^{\star},\bm{\lambda}^{\star})$.
Using the fact that
$\bm{\theta}^{\star}={\mathcal{}P}_{{\mathcal{}C}}(\bm{\theta}^{\star})={\mathcal{}P}_{{\mathcal{}C}}\big{(}\bm{\theta}^{\star}-\gamma{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\big{)}$,
we observe that in the primal update:
$\displaystyle\|\bm{\theta}^{(k+1)}-\bm{\theta}^{\star}\|^{2}$
$\displaystyle\overset{(a)}{\leq}\|\bm{\theta}^{(k)}-\bm{\theta}^{\star}\|^{2}-2\gamma\langle\widehat{\bm{g}}_{\bm{\theta}}^{(k)}-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star}),\bm{\theta}^{(k)}-\bm{\theta}^{\star}\rangle$
$\displaystyle\hskip
22.76228pt+\gamma^{2}\|\widehat{\bm{g}}_{\bm{\theta}}^{(k)}-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\|^{2}$
where (a) is due to the projection inequality
$\|{\mathcal{}P}_{{\mathcal{}C}}({\bm{x}}-{\bm{y}})\|\leq\|{\bm{x}}-{\bm{y}}\|$.
Furthermore, using the Young’s inequality, for any $c_{0},c_{1}>0$, we have
$\displaystyle\|\bm{\theta}^{(k+1)}-\bm{\theta}^{\star}\|^{2}$
$\displaystyle\leq\|\bm{\theta}^{(k)}-\bm{\theta}^{\star}\|^{2}$
$\displaystyle\hskip
14.22636pt-2\gamma\langle{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star}),\bm{\theta}^{(k)}-\bm{\theta}^{\star}\rangle$
$\displaystyle\hskip
14.22636pt{+\gamma^{2}(1+c_{0})\|{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\|^{2}}$
$\displaystyle\hskip
14.22636pt{-2\gamma\langle{\bm{e}}_{\bm{\theta}}^{(k)},\bm{\theta}^{(k)}-\bm{\theta}^{\star}\rangle+\gamma^{2}\big{(}1+\frac{1}{c_{0}}\big{)}\|{\bm{e}}_{\bm{\theta}}^{(k)}\|^{2}}$
$\displaystyle\leq(1+2c_{1}\gamma)\\!~{}\|\bm{\theta}^{(k)}-\bm{\theta}^{\star}\|^{2}$
$\displaystyle\hskip
14.22636pt-2\gamma\langle{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star}),\bm{\theta}^{(k)}-\bm{\theta}^{\star}\rangle$
$\displaystyle\hskip
14.22636pt+\gamma^{2}(1+c_{0})\|{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\theta}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\|^{2}$
$\displaystyle\hskip
14.22636pt+\Big{(}\frac{2\gamma}{c_{1}}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}\|{\bm{e}}_{\bm{\theta}}^{(k)}\|^{2}.$
Similarly, in the dual update we get,
$\displaystyle\|\bm{\lambda}^{(k+1)}-\bm{\lambda}^{\star}\|^{2}$
$\displaystyle\leq\|\bm{\lambda}^{(k)}-\bm{\lambda}^{\star}\|^{2}+\gamma^{2}\|\widehat{\bm{g}}_{\bm{\lambda}}^{(k)}-{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\|^{2}$
$\displaystyle\hskip
14.22636pt+2\gamma\langle\widehat{\bm{g}}_{\bm{\lambda}}^{(k)}-{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star}),\bm{\lambda}^{(k)}-\bm{\lambda}^{\star}\rangle$
$\displaystyle\leq(1+2c_{1}\gamma)\\!~{}\|\bm{\lambda}^{(k)}-\bm{\lambda}^{\star}\|^{2}$
$\displaystyle\hskip
14.22636pt+2\gamma\langle{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star}),\bm{\lambda}^{(k)}-\bm{\lambda}^{\star}\rangle$
$\displaystyle\hskip
14.22636pt+\gamma^{2}(1+c_{0})\|{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{(k)},\bm{\lambda}^{(k)})-{\nabla}_{\bm{\lambda}}{\mathcal{}L}(\bm{\theta}^{\star},\bm{\lambda}^{\star})\|^{2}$
$\displaystyle\hskip
14.22636pt+\Big{(}\frac{2\gamma}{c_{1}}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}\|{\bm{e}}_{\bm{\lambda}}^{(k)}\|^{2}.$
Summing up the two inequalities gives:
$\begin{split}&\|{\bm{z}}^{(k+1)}-{\bm{z}}^{\star}\|^{2}\\\
&\leq(1+2c_{1}\gamma)\\!~{}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}+\Big{(}\frac{2\gamma}{c_{1}}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}E_{k}\\\
&\hskip
7.11317pt-2\gamma\langle\bm{\Phi}({\bm{z}}^{(k)})-\bm{\Phi}({\bm{z}}^{\star}),{\bm{z}}^{(k)}-{\bm{z}}^{\star}\rangle\\\
&\hskip
14.22636pt+\gamma^{2}(1+c_{0})\|\bm{\Phi}({\bm{z}}^{(k)})-\bm{\Phi}({\bm{z}}^{\star})\|^{2}\\\
&\overset{(a)}{\leq}\Big{(}1+2\gamma(c_{1}-\upsilon)+\gamma^{2}(1+c_{0})L_{\Phi}^{2}\Big{)}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}\\\
&\hskip
14.22636pt+\Big{(}\frac{2\gamma}{c_{1}}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}E_{k},\end{split}$
where (a) uses the strong monotonicity and smoothness of the map $\bm{\Phi}$.
Setting $c_{1}=\upsilon/2$ yields
$\begin{split}&\|{\bm{z}}^{(k+1)}-{\bm{z}}^{\star}\|^{2}\\\
&\leq\Big{(}1-\gamma\upsilon+\gamma^{2}(1+c_{0})L_{\Phi}^{2}\Big{)}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}\\\
&\hskip
14.22636pt+\Big{(}\frac{4\gamma}{\upsilon}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}E_{k}.\end{split}$
(39)
Observe that we can choose $\gamma$ such that
$1-\gamma\upsilon+\gamma^{2}(1+c_{0})L_{\Phi}^{2}<1$. Moreover, the above
inequality implies that $\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}$ evaluates to
$\begin{split}&\|{\bm{z}}^{(k+1)}-{\bm{z}}^{\star}\|^{2}\\\
&\leq(1-\gamma\upsilon+\gamma^{2}(1+c_{0})L_{\Phi}^{2})^{k}\|{\bm{z}}^{(0)}-{\bm{z}}^{\star}\|^{2}+\\\
&\sum_{\ell=1}^{k}(1-\gamma\upsilon+\gamma^{2}(1+c_{0})L_{\Phi}^{2})^{k-\ell}\Big{(}\frac{4\gamma}{\upsilon}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}\Big{)}E_{\ell}\end{split}$
If $E_{k}\leq\overline{E}$ for all $k$, then ${\bm{z}}^{(k)}$ converges to a
neighborhood of ${\bm{z}}^{\star}$ of radius
$\limsup_{k\rightarrow\infty}\|{\bm{z}}^{(k)}-{\bm{z}}^{\star}\|^{2}\leq\frac{\frac{4\gamma}{\upsilon}+\gamma^{2}+\frac{\gamma^{2}}{c_{0}}}{\gamma\upsilon-\gamma^{2}(1+c_{0})L_{\Phi}^{2}}\overline{E}$
(40)
Setting $c_{0}=1$ concludes the proof.
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|
# Benefits of non-adiabatic quantum control in quantum computation through
spin qubit systems
Nirupam Dutta<EMAIL_ADDRESS>Independent Researcher, 25B Shanti Ghosh
Street, Kolkata, West Bengal, India
(August 28, 2024)
###### Abstract
This is evident that the controllable quantum systems can be the reliable
building blocks for Quantum computation. In reality we are witnessing the
progress towards making the idea tractable enough, though optimistic but the
threshold is not very near to us. The dawn of quantum computation has begun.
In the future, we hope to see a full fledged operationally stable quantum
computer which can solve the problems beyond the scope of classical digital
computers. We may call it quantum supremacy. Nevertheless, we should not
forget that there are problems which demand classical computers to be in the
game for a better performance in comparison to the same through quantum
devices. In the current stage of computing technology, the most beneficial
area is nothing but an hybrid approach and that is for no doubt will reign the
market for the next five to ten years. This hybrid aspect has several
directions such as simulating quantum computation on a classical computer.
Keeping both the aspect, computation through real physical devices and
simulation on a classical computer by accessing available quantum computers
for cloud computing, some advantages have been discussed in this article which
will be elaborated as well in future articles. These advantages are inherent
if we can achieve proper non-adiabatic control over the spin system in the
laboratory. Otherwise these aspects can always be simulated by using quantum
algorithms to see whether they can be useful in comparison to a purely
classical computing machine. This is no doubt a new window for progress in the
direction of quantum computation.
We need controllable quantum systems in order to employ their quantum
mechanical properties to perform computational tasks zero ; one ; two that
are either beyond the scope of any classical computer or quicker in processing
information in comparison to the same performed on a classical computing
machine three ; four . This agenda is ruling the current state of the art in
the domain of computation. A large number of theorists and experimentalists at
various organisations are participating in this race for achieving the so
called quantum supremacy five ; six ; seven ; eight . Physical systems at the
atomic and subatomic scale are quite difficult to control in order to
manipulate them according to our computational needs. For example a two level
quantum system which is the most basic text book topic of quantum mechanics
can not be prepared easily in the laboratory. It has already been four decades
that scientists are advocating in favour of quantum computation as in many
ways it can revolutionise the subject of computation and perhaps can push the
current technological horizon. Theoretical understanding has already made us
able to realise that the quantum superposition, interference and entanglement
are three important pillars nine for making this dream a reality. In today’s
scenario, many have achieved creating qubits in various different systems
under sophisticated laboratory conditions.
Quantum computation, we all know is different from classical computation in
the following way. Instead of classical beats which are either on or off like
a switch in the classical circuit, here the basic component is a superposed
state of a two level system which we name as qubit. But unless there is
entanglement among the qubits, they are no where more efficient than classical
bits as it somehow creates the technical essence of parallel computing. With
the help of the qubits, like we do for a classical computing device, we can
create gates for performing basic operations through qubits and at the end a
measurement of this physical process though destroys the superposition but
strategically designing the circuit for constructive interference we assure
the desired results. This is the general process of gate based quantum
computation but this is not the only mode of quantum computation. There are
quantum algorithms written for collection of qubits which do not require
anything like a gate structure for computing complex problems like
optimisation problems or factorisation problems. Quantum annealing ten ;
nishi1 ; nishi2 is one of the examples which deals with adiabatic quantum
computation to solve optimisation problems. Almost decade later the D-WAVE
device king is successfully handling various tasks and has already been
updated to a controlled device dealing with more than thousand qubits.
There are few types of quantum mechanical systems which have been successfully
employed so far to create qubits in the laboratory. These are for example
superconducting qubits, trapped ions, laser controlled photons and controlled
spin qubits. In order to achieve operationally functional computing devices
which can run through different quantum algorithms, hundreds of thousands of
stable qubits are needed. In this context it is worth mentioning that this
number is not actually close tho the number of qubits needed for achieving
quantum supremacy of the computing device. On the other hand, these large
numbers of qubits have to be entangled with each other in order to become
clusters of logical qubits to be able to perform computation. It is evident
that to perform different types of tasks based on different algorithms, the
number of active quantum qubits are different but a sustainable correlation
among them can not be assured trivially. The reason is simple and that is
nothing but the measurement sensitivity of quantum systems.
The most important thing that we still should keep in mind in this context is
that these different types of protocols and schemes that I have discussed
mostly employ the adiabatic control over the quantum systems. The basics of
quantum annealing are strictly based on the adiabatic control over the system.
The Hamiltonian of the system is strictly maintained within the slow rate of
change such that it should not break the adiabatic condition. On the other
hand, gate based quantum computation has also been developed through adiabatic
control.
In this article I will talk about some benefits of nonadiabatic controls over
quantum systems. In my demonstration I will first talk about physical spin
systems in which formation of physical qubits can be tailored by non adiabatic
evolution and in the same context, I will additionally discuss how
nonadiabatic effects can give rise to the creation of several qubits almost in
no time from parent qubits. These are important in real physical systems when
they can be controlled within a confined region of space but this challenge
obviously can be tackled by the experimentalist as far as the viability of the
process is concerned.
Furthermore, a full fledged computer is still a matter of waiting for the
sufficient number of controllable qubits. Considering the state of the art,
the recipe we present here for non adiabatic control can be useful in
simulating various schemes of quantum computation through a hybrid approach
that combines the cloud computing facilities of available quantum computers
and simulation of various methods of solving complex problems through specific
quantum algorithms.
## I A. Qubit Multiplier
Figure 1: This is a schematic diagram of a Stern-Gerlach device which shows
that the inhomogeneous magnetic field actually separates two spin states in
the up and down direction.
Let me start with a two level system for example a physical spin qubit. A spin
qubit can always be prepared through any available mechanism. Now if we allow
the qubit $a_{0}|0\rangle+a_{1}|1\rangle$ to pass through an inhomogeneous
time dependent or independent magnetic field, the evolved state should be
determined through the nature of the evolution. We are using $0$, $1$ to
denote the up and down spin along the $z$ axis of the specified coordinate
system 1. If the system evolves through an adiabatic process the final state
remains one qubit but the state will be entangled with the wave functions in
the up and down directions. This phenomena is known as path spin entanglement
home . The final state $|\Psi(t)\rangle$ can be expressed as,
$|\Psi(t)\rangle=a_{0}e^{-i\omega_{0}t}|0\rangle\otimes\phi_{u}+a_{1}e^{i\omega_{0}t}|1\rangle\otimes\phi_{d}$
(1)
By path spin entanglement, I mean the spatial wave functions $\phi_{u}$ and
$\phi_{d}$ corresponding to upward and downward direction along the $z$ axis
are coupled to the up and down spin states. This is evident from the above
equation that spatial and spin part of the state are not product separable.
For an operationally and formally ideal situation of the Stern-Gerlach set up
1, the up and down part of the wave function can be well separated. In that
case, if an ensemble of $N$ number of spin half particles undergo this
magnetic field configuration, the up and down spins will be accumulated within
a small region of the space in the upward and downward directions
respectively. The probability of having the up spin in the spatially upward
direction is given by,
$P_{0,u}={a_{0}}^{2}\int{\phi_{u}}^{2}dX.$ (2)
Where $X$ represents spatial variables.
So the total number of accumulated particles $N_{0,u}$ in the state
$|0\rangle$ in the upward direction is $N$ times the probability $P_{0,u}$.
Similarly total number of accumulated particles $N_{0,d}$ having the down spin
$|1\rangle$ in the downward direction is $NP_{1,d}$ as the probability of
having down spin in the downward direction is given by,
$P_{1,d}={|{a_{1}}|}^{2}\int{\phi_{d}}^{2}dX$ (3)
Actually each of the $N_{0,u}$ number of particles are in the up state and at
the same time the cluster itself can be used as one logical bit $|0\rangle$.
The same argument applies to the cluster of particles which are confined in
the downward direction.
### I.1 A. 1. Converter Oracle
This adiabatic evolution and path spin entanglement has an interesting feature
in the context of computations through the qubits. Suppose we start with a
qubit state and with one step boosting through regular ideal S-G set up, we
can create well separated two streams of two different states. Now, among the
two streams, if we allow only the upward one to be released, when considered
as a final output, the manipulation results in conversion of a superposed
state to a pure state which is equivalent to a classical bit.
Figure 2: This is a schematic diagram of the converter which can transform a
superposed state toa pure state with the help of ideal S-G apparatus.
The entire process in this case serves like an oracle which changes an input
qubit to an output which serves as a classical bit. A reverse process can
always restore to the initial state and hence can keep the unitarity
unviolated through the process. This oracle can be used as a gate that
transforms quantum to classical beats and can be useful in various algorithms
as a component of an integrated circuit to solve specific computational
problems.
### I.2 A. 2. The non-adiabatic multiplier
A very different situation arises when the system evolves through a non-
adiabatic evolution. For this purpose, the Hamiltonian of the spin system
needs to be changing quite fast with respect to time. How fast it should be is
determined with respect to the a characteristic time scale $\tau$ of the
system which is the inverse of the energy gap between the Zeeman levels, I
mean the energy states of the two level system. If the characteristic time
$\tau$ is much less than the time needed for the unit amount change of the
Hamiltonian, the system evolves adiabatically otherwise it evolves non-
adiabatically. The precise criteria for non-adiabatic evolution is as follows
aharonov ; amin ; tong .
$\Big{|}\frac{\langle
0(t)|\dot{H}(t)|1(t)\rangle}{(E_{0}-E_{1})^{2}}\Big{|}<<1.$ (4)
$|0(t)\rangle$ and $|1(t)\rangle$ are the instantaneous eigenstates of $H(t)$
and $E_{0}$ and $E_{1}$ are the corresponding energy eigenvalues.
There can be many different possible magnetic field configurations, which can
violates this condition and serve our purpose. For a simple demonstration, let
me consider a very well known configuration which has been understood
previously in the context of spin states in other areas of physics. Here we
will be enjoying a few advantages for the purpose of quantum computing. The
considered field configuration is a rotating magnetic field and in order to
have the desired outcome, I will present a specific case when a rotating field
component will be added to a formally and operationally ideal Stern-Gerlach
setup. The rotating field is confined in the X-Y plane only and it rotates
about the Z axis. Another method of violating adiabatic condition was
presented in one of our previous articles in the context of evolution of spin
states when there is azimuthal inhomogeneity inside the S-G set up dutta . In
both the cases the mathematical construction of the problem is same. This
field strength has to be decided in such a way that the time scale of spin
states dynamics is much smaller than that of the spatial wave function. Then,
We can represent the Hamiltonian in the following way.
$\displaystyle H(t)=\gamma B(\sigma_{x}\sin\theta\cos\phi(t)$
$\displaystyle+\sigma_{y}\sin\theta\sin\phi(t)+\sigma_{z}\cos\theta)$
$\displaystyle=\omega_{0}(\sigma_{x}\sin\theta\cos\omega t$ (5)
$\displaystyle+\sigma_{y}\sin\theta\sin\omega t+\sigma_{z}\cos\theta).$
Where $\gamma$ is the gyromagnetic ratio and $\sigma_{x}$, $\sigma_{y}$,
$\sigma_{z}$ are three different Pauli spin matrices in the chosen orientation
of the total spin $S$. The rotating components makes the field rotating in the
x-y plane with a frequency $\omega$. $\theta$ is the angle between the
z-component of the magnetic moment ($\mu=\gamma S$) and the magnetic field
$\vec{B}$ and $\phi(t)$ is the azimuthal angle that magnetic field makes in
the $X-Y$plane considered here. The matrix form of the Hamiltonian is given
below.
$\displaystyle\begin{aligned}
H(t)=\frac{\omega_{0}}{2}\left(\begin{array}[]{cc}\cos\theta&e^{-i\omega
t}\sin\theta\\\ e^{i\omega t}\sin\theta&-\cos\theta\\\
\end{array}\right)\end{aligned}$
The quantity $\omega_{0}$ is the Larmor frequency of the magnetic moment of
the spin$1/2$ particle.
The instantaneous Eigenstates of the time dependent Hamiltonian can be
expressed as,
$|0(t)\rangle=\begin{pmatrix}e^{-i\omega t/2}\sin\theta/2\\\ e^{i\omega
t/2}\cos\theta/2\end{pmatrix}$
and
$|1(t)\rangle=\begin{pmatrix}e^{i\omega t/2}\sin\theta/2\\\ -e^{-i\omega
t/2}\cos\theta/2\end{pmatrix}$
.
Plugging these two states in the equation 4, we arrive to the condition for
adiabatic evolution as,
$\frac{\omega}{2\omega_{0}}\sin\theta<<1.$ (6)
This equation tells us that whether the evolution is adiabatic or not depends
on three different parameters: $\omega_{0}$, $\omega$ and $\theta$. One can
employ this freedom to break the adiabatic condition and in that case the
evolution of spin states will be determined by solving the Schrödinger
equation.
Figure 3: This is a schematic diagram of the oracle ”qubit multiplier” which
is based on the non-adiabatic control inside a formally and operationally
ideal Stern-Gerlach device.
Now let me come back to the story that we have started in this section.
Suppose the initial qubit is a superposition of two spin states $|0\rangle$
and $|1\rangle$. The moment they enter into the magnetic field, the up and
down states will be coupled to the spatial up and down component of the wave
function and will evolve according to the time dependent Hamiltonian. The
interaction Hamiltonian $H$ does not contain any spatial part hence the time
dependence will affect the spin part only. The spatial inhomogeneity of the
basic ideal S-G setup will help separating the wave functions in the upward
and downward direction along the Z axis. So we just need to study the
evolution of up and down spin states under the influence of the additional
rotating component.
We have to calculate the evolution of the up state which will be deflected in
the upward direction and for the down state which will be deflected in the
downward direction. For both purposes we have to solve the Schrödinger
equation for the time dependent Hamiltonian with two different initial
conditions with the help of the following instantaneous Eigenstates. Let’s
write down the evolved state in the upward direction in the following way,
$\Psi_{u,z}=|s_{zu}(t)\rangle\otimes\Phi_{u}.$ (7)
Where,
$|s_{zu}(t)\rangle=\alpha_{0}(t)|0(t)\rangle+\alpha_{1}(t)|1(t)\rangle$
Now, We see that the $N_{u}$ number of particles forms the cluster of qubits
which will be accumulated in the upward direction. On the other hand the
$N_{d}$ number of particles will form a cluster of qubits in the downward
direction. The state can be written as,
$\Psi_{d,z}=|s_{zd}(t)\rangle\otimes\Phi_{d}.$ (8)
Where,
$|s_{zd}(t)\rangle=\beta_{0}(t)|0(t)\rangle+\beta_{1}(t)|1(t)\rangle$
Now plugging this in the Schrödinger equation with the initial conditions
$a_{0}(t=0)=1$ for the upward stream and $a_{1}(t=0)=1$ for the down stream,
we are left with the following solutions for $\alpha_{0}$, $\alpha_{1}$,
$\beta_{0}$ and $\beta_{1}$ as follows.
$\displaystyle\begin{aligned}
\alpha_{0}(t)&=\cos\frac{\bar{\omega}t}{2}-i\frac{\omega_{0}-\omega\cos\theta}{\bar{\omega}}\sin\frac{\bar{\omega}t}{2}\\\
\alpha_{1}(t)&=i\frac{\omega\sin\theta}{\bar{\omega}}\sin\frac{\bar{\omega}t}{2}\\\
\beta_{0}(t)&=i\frac{\omega\sin\theta}{\bar{\omega}}\sin\frac{\bar{\omega}t}{2}\\\
\beta_{1}(t)&=\cos\frac{\bar{\omega}t}{2}+i\frac{\omega_{0}-\omega\cos\theta}{\bar{\omega}}\sin\frac{\bar{\omega}t}{2}\end{aligned}$
(9)
Where,
$\bar{\omega}=\sqrt{\omega_{0}^{2}+\omega^{2}-2\omega_{0}\omega\cos\theta}$
The first advantage of this nonadiabatic process is that we are able to create
two different logical qubits through two different clusters of physical
qubits. This is like an oracle which doubles the number of logical qubits just
by giving one order of non-adiabatic boosting through the time dependent
magnetic field inside the ideal S-G setup. With such $n$ number of non
adiabatic booster the logical qubits can be scaled by a factor of $2^{n}$.
The second advantage is that we can create many basic one qubit logic gates
through the non-adiabatic control. Let us discuss just a few among these gates
and the associated values of the control parameters. There are four different
parameters such as $\omega_{0}$, $\omega$, $\theta$ and $t$ which can be
adjusted to achieve the desired output. The scope of exploration is almost
unlimited but let me just show one particular choice of these free parameters
and how we can achieve three different one qubit quantum logic gates through
this.
Lets start with a particular choice which is
$\cos\theta=\frac{\omega_{0}}{\omega}$. The ratio of the two frequencies have
to be chosen in such a way so that it does not satisfy the adiabatic condition
6. It is very much possible as there is a big domain for which the ratio
$\frac{\omega_{0}}{\omega}$ can be fixed accordingly.
Case 1. Quantum Not Gate: Suppose the incoming state to the S-G setup is
$a_{0}|0\rangle+a_{1}|1\rangle$ The evolved state in the upward direction
becomes $a_{1}|0\rangle+a_{0}|1\rangle$ at some instant $t=\tau_{not}$ for
which
$\cos\frac{\bar{\omega}\tau_{not}}{2}=a_{1}\hskip 8.5359pt\text{and}\hskip
8.5359pti\sin\frac{\bar{\omega}\tau_{not}}{2}=a_{0}$ (10)
For this purpose the time dependent Hamiltonian should change up to
$H(\tau_{not})$ from $t=0$ to $t=\tau_{not}$ and should remain fixed
afterwards. The qubit at the final instant can be treated as the output of a
quantum NOT gate and can be utilised accordingly. Under this circumstance, the
qubits in the downward direction is the same as the input qubit. This
operation is equivalent to an Identity operation on the input state
Case 2. The Z Gate: With the same value of $\theta$, if we consider the final
state in the upward direction at some instant $\tau_{z}$ so that it satisfies
the following condition.
$\cos\frac{\bar{\omega}\tau_{z}}{2}=a_{0}\hskip 8.5359pt\text{and}\hskip
8.5359pti\sin\frac{\bar{\omega}\tau_{z}}{2}=-a_{1}$ (11)
In this case the final outcome in the upward direction is $a_{0}|0\rangle-
a_{1}|1\rangle$. Obviously, the Hamiltonian in this case shall remain
unchanged from $t=\tau_{z}$ with the final value $H(\tau_{z})$.
Case 3. The Hadamard Gate: For the same value of $\theta$, if we impose the
following condition,
$\cos\frac{\bar{\omega}\tau_{H}}{2}=\frac{a_{0}+a_{1}}{\sqrt{2}}\hskip
8.5359pt\text{and}\hskip
8.5359pti\sin\frac{\bar{\omega}\tau_{H}}{2}=\frac{a_{0}-a_{1}}{\sqrt{2}}$ (12)
The final state in the upward direction is
$a_{0}\frac{|0\rangle+|1\rangle}{\sqrt{2}}+a_{0}\frac{|0\rangle-|1\rangle}{\sqrt{2}}$.
This is the output of a single qubit Hadamard gate.
## II B. Non-adiabatic control over multi qu-bit entangled states
So far, we have talked regarding the non-adiabatic control through the
rotating time dependent magnetic field inside a formally and operationally
ideal S-G apparatus. In this section we will discuss the control we can have
over multi qubit entangled states which is one of the important ingredients
for quantum computation. I won’t be making this section prolonged as in an
upcoming article I will show preparation of various many qubit quantum gates
and specific controls for introducing algorithms to solve complex problems.
Let me give an introductory description along with a few important hints that
can create some ambition for future study in the domain of non-adiabatic
quantum control.
Instead of S-G set up, here we consider Mach-zehnder setup in conjunction with
an S-G device for the purpose of creating entanglement among different multi-
particle qubits pan . This is actually done through the mechanism of swapping
of path-spin entanglement to spin spin entanglement. The issue has been well
studied previously. I am not going into the details of the mechanism as we are
trying to emphasise on the improvement that we are bringing in through the
application of an additional time dependent rotating magnetic field. Through a
specific process first the mechanism creates two particle (particle $1$ and
$2$) states which are coupled to a pseudo path like spatial variable and later
a third particle 3 is introduced which previously has not interacted with the
two particles. The swapping then creates various entanglement between 2 and 3.
Intuitively, the process is the following. Particles ‘2’ and ‘3’ are kept
separated without allowing them to interact throughout the process. Now the
scheme allows the particle ‘1’ to interact independently with the particles
‘2’ and ‘3’, without the need of having ‘2’ and ‘3’ in the near vicinity of
each other.
Now the qubit undergoes through two different S-G apparatus and these devices
provide an unitary transformation /cite. Upon measurement mentioned on the
state of the particle $1$ we can create two different kinds of entangled
states which are given below.
$1)$. $a|00\rangle_{23}+b|11\rangle_{23}$
$2)$. $a|00\rangle_{23}-b|11\rangle_{23}$
But if we use the non adiabatic control over the outgoing state of the Mach-
Zehender state through a time dependent rotating magnetic field, we can create
four different types of entangled states in the two particle tensor product
space of particle $2$ and $3$. Furthermore, by adjusting the parameter values,
we can create various two qubits gates in this context. In this article we are
not going into the details of those gates and quantum circuits that can be
employed to create various quantum algorithms for solving complex issues. This
issue I leave for a different article in near future.
## III Discussion
My purpose in this article is to introduce the key idea of non adiabatic
control and the various benefits that it can offer for computing. Now
implementation of the control in the form of a physical device needs necessary
engineering and that might suffer from several difficulties in the laboratory.
Nevertheless the procedure opens up new possibilities for integrated devices
which can be useful in solving various complicated problems with the help of
proper algorithms. On the other hand, a mathematical simulation of non-
adiabatic control can be a good approach if it is implemented through quantum
algorithms running on basic quantum computers that are available for cloud
computing. In a different article we will show a few such algorithms and how
they can be useful in solving various computations in the simulation of many
body physics and chemistry as well as in the domain optimization problems.
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* (12) Kadowaki, Tadashi and Nishimori, Hidetoshi, “Quantum annealing in the transverse Ising model”, PhysRevE.58. 5355, (1998).
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|
Laser-induced alignment of nanoparticles and macromolecules for single-particle-imaging
Muhamed Amin
Jean-Michel Hartmann
Amit K. Samanta
Jochen Küpper
Laser-induced alignment of particles and molecules was long envisioned to support
three-dimensional structure determination using single-particle imaging with x-ray free-electron
lasers [PRL 92, 198102 (2004)]. However, geometric alignment of isolated macromolecules has not
yet been demonstrated. Using molecular modeling, we analyzed and demonstrated how the alignment
of large nanorods and proteins is possible with standard laser technology, and performed a
comprehensive analysis on the dependence of the degree of alignment on molecular properties and
experimental details. Calculations of the polarizability anisotropy of about 150,000 proteins
yielded a skew-normal distribution with a location of 1.2, which reveals that most of these
proteins can be aligned using appropriate, realistic experimental parameters. Moreover, we
explored the dependence of the degree of alignment on experimental parameters such as particle
temperature and laser-pulse energy.
§ INTRODUCTION
X-ray free-electron lasers (XFELs) promise the diffractive imaging of single molecules and
nanoparticles at atomic resolution <cit.>. Ultra-short, high intensity x-ray
pulses interact with individual molecules and using the “diffraction-before-destruction”
approach <cit.> large series of diffraction patterns are
collected. Two-dimensional diffraction patterns from randomly oriented samples would then be
computationally assembled to a diffraction volume to retrieve the three-dimensional
structure <cit.>.
In the standard single-particle imaging (SPI) method, the camera images do not contain a
priori information of the molecules' orientation and, so far, the uncertainty is attacked
in silico <cit.>. Significant efforts were made in improving the
reconstruction process and the achievable resolution, but it is still a highly challenging task,
especially for weakly scattering particles like individual proteins, where diffraction signals from
single molecules are generally not strong enough to allow for the averaging and sorting
approaches <cit.>. This is one of the major bottlenecks for
atomic-spatial-resolution SPI.
Imaging samples with controlled alignment or orientation <cit.> allows to
significantly mitigate this problem by allowing to sum the diffraction signals from many
identically aligned molecules to provide a much stronger signal, thus improving the image
reconstruction step and paving the way toward atomic-resolution SPI <cit.>. Careful analysis of simulated diffraction patterns of
laser-aligned proteins demonstrated that it is possible to observe the secondary structure of
proteins with only reasonably-strong degrees of alignment
$\costhreeD\geq0.9$ <cit.>.
The alignment of small molecules using external electric fields from moderately intense, nonresonant
light pulses was studied extensively using experimental and theoretical
methods <cit.>. Strong alignment was achieved for linear, symmetric
top, and asymmetric top molecules in the adiabatic <cit.>,
intermediate <cit.>, and impulsive
regimes <cit.>. Considerable efforts were made
for laser-induced alignment of large molecules <cit.> and even for some complex and floppy polyatomic
molecules <cit.> as well as for weakly bound molecular
complexes <cit.>. Such aligned-molecule samples were also studied by
electron <cit.> and x-ray diffractive imaging
experiments <cit.>. Possibilities to laser-align large biomolecules without
deterioration of the secondary structure were proposed <cit.>,
but, so far, no alignment for such systems has been reported.
For macromolecules, there are several challenges for achieving the required alignment. Theoretical
predictions supported by in silico analysis can be a very important step to guide the
experiments. However, atomistic molecular dynamics (MD) simulations are computationally very expensive
for large particles, especially when the interaction time extends to hundreds of
nanoseconds <cit.>. In addition, to accurately predict the “ensemble averaged”
single-particle diffraction images of macromolecules, simulation of a large distribution of
particles is essential rather than a single particle. Ensemble computations are also crucial for
studying the temperature effects, as this can be an important factor in preserving the secondary
structures of macromolecules, like proteins, and this step makes MD simulations even more
computationally expensive.
In this work, we predicted and analyzed the laser-induced alignment of
(bio-)nanoparticles. Nanoparticles and proteins were treated as rigid bodies, supported by previous
molecular-dynamics simulations of structural changes in strong electric
fields <cit.>. The key parameters for such simulations are the overall
polarizability, shape and their anisotropy, the temperature of the macromolecules, and the alignment
laser field. We disentangled how these parameters can be tuned for maximum alignment of gold-nanorod
model systems and how this can be exploited for the strong laser alignment of biological
macromolecules, , proteins.
§ COMPUTATIONAL METHODS
The response of the particles to a nonresonant external electric field was calculated based on their
polarizability tensors, which yielded the time-dependent induced dipole moments. For metallic
nanorods, the polarizabilities were directly obtained by solving Laplace’s equation with Dirichlet
boundary conditions and using Monte Carlo path integral methods <cit.>. For
proteins, the polarizabilities were derived from the same calculations using regression-based
scaling <cit.>.
Molecular ensembles were set up with random initial phase-space distributions, , with initial
angular velocities according to a Boltzmann distribution at the given temperature and with random
initial angular positions. Each run included 20,000 particles. The angular positions of particles
were stored in quaternions throughout the calculation. The inertial tensors of the artificial
nanorods were calculated for cylinders. However, for proteins the tensors were calculated based on
the atomic masses and coordinates as available from the protein data bank
(PDB) <cit.>.
Electric fields of the alignment-laser pulses were represented by Gaussian functions with variable
peak intensities. For simplicity, we used a temporal full-width at half maximum (FWHM) of 8 ns,
corresponding to the typical pulse duration of standard Q-switched Nd:YAG lasers, often used in
alignment experiments. We also assumed typical linearly polarized laser pulses with intensities of
$10^{10}\ldots10^{12}$ .
The particles' angular phase-space positions were propagated in time by integrating Euler's
equations of angular motion using a new, Python-based, openly available software package
CMIclassirot <cit.>, which was based on and checked against previous
classical-alignment computations <cit.>. All simulations propagated the
particles for 50 ns, except for the alignment of gold nanoparticles in Fig. <ref>, for
which the simulation time was extended to 200 ns. The effect of resonances was ignored in the
simulation, , the laser-field frequency was assumed to be far off resonance from the molecules
and nanoparticles absorption.
§ RESULTS
The computed time-dependent degrees of laser-induced alignment of nanorods of three different sizes
are shown in Fig. <ref> for 298 K and 4 K and alignment-laser peak intensities of
$10^{11}\ldots10^{12}~\Wpcmcm$. Throughout this manuscript, nanorod sizes are represented as
($height$/nm, $diameter$/nm), with the calculations performed for 102, 5010,
10020, respectively.
Degree of alignment of gold nanorods for different temperatures ($T$) and laser-field
intensities (I). a) $T=298$ K and $I=10^{11}~\Wpcmcm$. b) $T=298$ K and $I= 10^{12}~\Wpcmcm$.
and c) $T=4$ K and $I=10^{11}~\Wpcmcm$. The degree of alignment of 102 rods is
shown in red, 5010 in blue, and 10020 in black. The temporal laser
profile is indicated by the shaded gray area and the corresponding field intensities are
specified on the secondary axes. Please see the text for details and definitions of the
During the pulse and with particles initially at room temperature, the degree of alignment for all three
particles' sizes, Fig.<ref> a, b, follows the temporal laser profile on the rising
edge in a quasi-adiabatic fashion <cit.>. The smallest particle
102 exhibits the largest alignment of $\costhreeD=0.96$.
However, the laser turn-off dynamics show considerable differences: The two larger nanorods exhibit
permanent alignment after the laser pulse is turned off, whereas the small nanorod
quasi-adiabatically follows the temporal laser profile to an isotropic field-free angular
distribution, , $\costhreeD=1/3$. This can be rationalized by comparing the rotation periods of
the nanorods, , the average time needed for each nanorod to rotate around its center of mass by
360, to the alignment-laser-pulse duration. The rotation periods depend on the temperature,
because the initial phase space is assigned based on a corresponding Boltzmann distribution. For
large nanorods at room temperature, the rotational periods are within 10 (, $\omega\approx6.3\cdot10^{5}$ rad/s), which is three orders of
magnitude larger than the pulse duration. Thus, these particles exhibit non-adiabatic alignment,
resulting in a permanent alignment after the pulse is off, Fig. <ref> a, b. Already
at low intensity, $I=10^{11}~\Wpcmcm$, the particles are confined even after the laser is off to
fully rotate in a plane containing the laser polarization vector, corresponding to $\costhreeD=0.5$,
Fig. <ref> a. Thus, no increase in the permanent alignment is possible at
$I=10^{12}~\Wpcmcm$, Fig. <ref> b. For small nanorod, the rotational period reduces to
20 ns, which is comparable to the alignment-laser pulse duration. Thus, these particles exhibit
quasi-adiabatic alignment <cit.> and no permanent
alignment is observed after the pulse, Fig. <ref> a, b.
However, at 4 K, which is experimentally achievable <cit.>, the rotational
period increases to 400 ns for the 102 rods, resulting in the transition to the
non-adiabatic regime and thus a field-free permanent alignment is observed even for this smallest
rod after the pulse is off, see Fig. <ref> c.
For the larger nanorods (blue, black) the degree of permanent alignment after the pulse shows a
strong oscillation that decays with time, which is a result of the slow dephasing between the
rotations of the confined nanorods, which start to rotate with different angular velocities, but
significantly slower than for the small particles. This variation in the angular velocities causes the
decay of the pattern with time. Furthermore, to ensure that these oscillations were not a result of
undersampling, we performed a convergence study in which we increased the number of particles for
the large nanorods to 200,000 particles and we obtained the same oscillations.
Although all our simulations take into account the full polarizability tensor, it is instructive to
study the degree of alignment as a function of the nanorod shape described by the ratio of its
principal moments of polarizability, , the ratio of $\alpha_{\parallel}$ to $\alpha_{\perp}$,
which correlates with the polarizability anisotropy:
\begin{align}
\label{eqn:pol-anisotropy}
\alpha_\parallel &= \max\left(\alpha_{11},\alpha_{22},\alpha_{33}\right) \notag \\
\alpha_\perp &= \left(3\alpha-\alpha_\parallel\right)/2 \\
\alpha_r &= \alpha_\parallel / \alpha_\perp \notag
\end{align}
with the static-polarizability components in the principal axes of polarizability frame
$\alpha_{ii}$ and $\alpha=\sum\alpha_{ii}$. For $\alpha_r$ the scaling factor applied to the
elements of the polarizability tensors to account for the dielectric medium of proteins cancels out,
and does not have to be included in this discussion of shape.
a) Maximum degree of alignment, max, as a function of the polarizability ratio
$\alpha_r$ at different temperatures and intensities for a nonorod of $10\pi~\text{nm}^3$. At
low temperature a high degree of alignment is achieved even at very low intensity. b) The
$\alpha_r$ distribution of about 150,000 proteins in the protein data bank (PDB) approximately
follows a skew-normal distribution with a location of $1.2$, scale of $0.7$ and skewness
parameter of $6.5$.
Fig. <ref> a shows the dependence of the maximum degree of alignment , maxon
$\alpha_r$ for two different laser peak intensities of $I= 10^{10}~\Wpcmcm$ and $I= 10^{11}~\Wpcmcm$.
There is a quick rise of maxfor $\alpha_r$ in the range
$1.2\ldots2.5\cdot10^{10}~\Wpcmcm$, depending on temperature, confirming that with increasing
particle anisotropy there is a significant increase in the maximum degree of alignment, as could be
expected. Further increasing $\alpha_r$ led to a further slow increase in the maximum alignment
toward an asymptotic maximum. Furthermore, for higher intensity one observes stronger alignment,
especially for small values of $\alpha_r$. The same holds for lower temperatures, which enable
significantly increased alignment even at lower laser intensities <cit.>. Specifically, for $10^{10}~\Wpcmcm$ and room-temperature particles it
requires $\alpha_r>8$ to obtain a reasonable degree of alignment of $\costhreeD>0.8$, whereas at low
temperature (4 K) <cit.>, strong alignment of $\costhreeD=0.9$ is achieved
even at $\alpha_r=1.5$ .
Proteins and similar biological macromolecules are the principal target for single particle imaging.
Thus, we calculated the polarizability ratio $\alpha_r$ of about 150,000 proteins using the database
we built previously <cit.>, see Fig. <ref> b. These data were
obtained by calculating the polarizability tensors of proteins based on their PDB structures using
ZENO <cit.>. These tensors were diagonalized to put all
molecules in the polarizability frame, their $\alpha_r$ values were computed according to
(<ref>), and summarized in the histogram in Fig. <ref> b. The $\alpha_r$
distribution follows a skew-normal distribution with a location of $\ordsim1.2$. As $96~\%$ of the
data have an $\alpha_r$ larger than the location value, this distribution directly indicates that a
corresponding, significant fraction of these proteins have sufficiently anisotropic polarizabilities
that make them directly amenable to strong laser alignment.
To provide further understanding of the effect of shape of the object on the maximum achieved
alignment, we studied the effect of laser intensity and particle temperature for a gold nanorod
with $\alpha_r=1.2$ and size 3.63.2, see Fig. <ref>.
a) The effect of laser pulse intensity on the degree of alignment of a gold nanorod with
$\alpha_r=1.2$ and size 3.63.2 at 298 K (blue) and 4 K (orange). A good degree of
alignment $\max\costhreeD=0.9$ is achieved at 4 K with intensities as low as of
$10^{10}~\Wpcmcm$ (filled circles). However, a significantly stronger intensities (more than
$5\cdot10^{11}~\Wpcmcm$) are required to obtain a good degree of alignment at 298 K. b) The
effect of temperature on the degree of alignment of the same gold nanorod (polarizability ratio
of 1.2) for two different laser intensities, $10^{10}~\Wpcmcm$ (blue) and $10^{11}~\Wpcmcm$
(orange). Strong alignment is achieved for both intensities at very low temperatures. However,
a quick decay in the maxis observed for the low intensity with increasing the
Fig. <ref> a shows the effect of different laser intensities on the degree of alignment at
298 K and 4 K. The degree of alignment quickly increases at low intensities, depending on
temperature, and saturates toward the asymptotic limit (vide supra). Again, the achievable
degree of alignment at 4 K is significantly higher than at room temperature. In the latter case,
strong alignment of $\max\costhreeD=0.9$ is achieved for the highest intensities of
$10^{12}~\Wpcmcm$, whereas a 4 K sample enables very strong alignment of $\max\costhreeD=0.95$ for
intensities as low as $5\cdot10^{10}~\Wpcmcm$.
Similarly, Fig. <ref> b shows the degree of alignment for the same particle versus the
temperature at two different laser intensities, demonstrating the decrease of alignment with
increasing temperature, especially so for weaker laser intensities.
Thus, based on our model system, which is a nanorod with an anisotropy similar to that of most of
the proteins, Fig. <ref>, a good degree of alignment is achievable for most proteins at 298 K
using 10 ns alignment-laser pulses with peak intensities around $10^{11}~\Wpcmcm$. This alignment
can be improved significantly exploiting cryogenically cooled (4 K) <cit.>
proteins even with a much weaker laser pulse. Furthermore, the cooling will also reduce the chance
of structural damage of proteins by intense laser fields <cit.>.
Degree of alignment of the green fluorescent protein, depicted in the inset, at 298 K
(red) and 4 K (blue) using $10^{12}~\Wpcmcm$ pulse intensity (shaded area in gray). In
consistency with Fig. <ref>, a good degree of alignment of $\costhreeD=0.84$ is
achieved at 4 K even at very low intensities, , in the raising flank of the laser pulse at
$5\cdot10^{10}~\Wpcmcm$. However, a stronger intensity of $10^{12}~\Wpcmcm$ is required to
achieve similar alignment at room temperature.
To simulate the alignment of an actual protein, we have applied our method to simulate laser-induced
alignment of the prototypical green fluorescent protein (GFP), which has a cylindrical shape
($\alpha_r=1.5$) and thus strong alignment could be expected, see Fig. <ref>. We calculated the
inertia and polarizability tensors based on the PDB structure with PDBID
1GFL <cit.>. The elements of the polarizability tensors were scaled by a factor
of $0.4$ to account for the dielectric medium of proteins
($\epsilon_r=3.2$ <cit.>). Then, we applied a Gaussian laser pulse with a
FWHM of 8 ns. As shown in Fig. <ref>, moderate alignment of $\max\costhreeD=0.84$ was
obtained at room temperature for a peak intensity of $10^{12}~\Wpcmcm$. However, at 4 K
$\max\costhreeD=0.85$ was already achieved for $2\cdot10^{10}~\Wpcmcm$ and for $10^{12}~\Wpcmcm$ we
obtained $\max\costhreeD=0.94$.
In summary, using simulations based on the classical dynamics of rigid bodies we showed that
significant laser-induced alignment of nanorods and biological macromolecules, with typical
polarizability ratios $\alpha_{\parallel}/\alpha_{\perp}$, can be achieved. The dependence of the
degree of alignment on the alignment-laser intensity, sample temperature, and molecular size and
polarizability were analyzed. We showed that a very high degree of alignment can be achieved for
cryogenically-cooled <cit.> proteins at a moderate laser power of
$10^{10}~\Wpcmcm$, which should not cause radiation damage <cit.>. This high
degree of control of $\costhreeD\geq0.94$ paves the way for future atomic
resolution <cit.> single particle x-ray and electron diffractive imaging
experiments. Furthermore, the achievable atomic spatial and femtosecond temporal resolution provide
the prerequisites for future time resolved studies of ultrafast biochemical dynamics.
Our approach provides clear insight into the optical control of macromolecules and enables better
modeling of the experimental parameters for successful laser-induced alignment experiments, which are
currently in progress in our group. Envisioned future experiments plan to make use of our cryogenic
nanoparticle cooling setup <cit.> together with efficient laser control to
achieve a very high degree of alignment for shock-frozen proteins and, in turn, sub-nanometer
resolution in single particle x-ray imaging. We believe that this framework will prove useful for
furthering the field of single-particle x-ray imaging and would allow us to observe atomically
resolved snapshots of ultrafast chemical dynamics.
The achievable strong laser alignment of nanoscopic objects could have further applications in
nanoscience <cit.> as well as in nanoscale quantum optics or quantum
sensing <cit.>.
§ ACKNOWLEDGMENT
We acknowledge financial support by Deutsches Elektronen-Synchrotron DESY, a member of the Helmholtz
Association (HGF) and the use of the Maxwell computational resources operated at DESY. This work has
been supported by the European Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) through the Consolidator Grant COMOTION (614507) and the Cluster of Excellence
“Advanced Imaging of Matter” (AIM, EXC 2056, ID 390715994) of the Deutsche Forschungsgemeinschaft
§ CODE AVAILABILITY
Classical rotational dynamics simulations were performed using CMIclassrot, available at
|
# Great Britain’s Hydrogen Infrastructure Development – Investment Roadmap and
its Flexibility
Tatyana Dergunova Andrew Lyden<EMAIL_ADDRESS>
###### Abstract
Future pathways for Great Britain’s energy system decarbonization have
highlighted the importance of low-carbon hydrogen as an energy carrier and
demand flexibility support. However, the potential application within various
sectors (heating, industry, transport) and production capacity through
different technologies (methane reformation with carbon capture, biomass
gasification, electrolysis) is highly varying, introducing substantial
uncertainties for energy infrastructure development bottlenecking hydrogen
economy growth. This study addresses limitations of previous research by
providing comprehensive modelling of the entire hydrogen system of Great
Britain based on three Net Zero scenarios set out by National Grid utilizing
open-source software. The analysis outcome recommends prioritizing the
establishment of green hydrogen hubs aligning with future synthetic fuels
production, where hydrogen will act as feedstock, unlocking roadmap for up to
10GW (95TW/h) of production capacity. Additionally, low-carbon hydrogen
production for industry and power decarbonization shall follow localised
approach at early development stages to increase future decisions flexibility.
###### keywords:
Green hydrogen , Low-carbon hydrogen , Energy infrastructure , Decarbonization
pathways , Synthetic fuels , Open-source modelling
††journal: INTERNATIONAL JOURNAL OF HYDROGEN ENERGY
[label1]organization=School of Engineering, Institute for Energy Systems,
University of Edinburgh, addressline=Colin Maclaurin Road, city=Edinburg,
postcode=EH9 3DW, country=UK
Low-carbon hydrogen in Great Britain’s future energy system
Decarbonization strategies for ‘no-regret’ decisions
Open-source hydrogen network modelling
## 1 Introduction
Hydrogen is recognized globally as one of the important constitutes of energy
system decarbonization due to its ability to cover periods where renewable
energy sources cannot provide required flexibility to meet demands or
decarbonization cannot be achieved with direct electrification [1]. The 2022
global energy crisis increased the interest for low-carbon hydrogen economy
development in multiple countries with the aim not only to reduce the
emissions but also to decouple from fossil fuels dependency and diversify
energy supply mix [2]. Even though Liquefied Natural Gas (LNG) is currently
the primary solution to meet short-term energy security in the European Union
(EU) [3], hydrogen facilities development can be seen in multiple countries
around the world including export and import ports [4] with intend to
diversity energy supply sources in a long-term run.
The United Kingdom’s (UK’s) abundant onshore and offshore wind resources [5]
and proximity to the EU, the leader in hydrogen market development, could
facilitate the realization of the UK government’s hydrogen ambitions, which
includes 10GW of low-carbon hydrogen production by 2030 to meet local
decarbonization with potential export to continental Europe [6],[7]. However,
current lack of clarity in projections for hydrogen market development holds
back investment decisions which in turn slows down the growth of low-carbon
hydrogen supply chain.
The uncertainties in future local hydrogen market are substantial, with annual
demand forecast ranging from 120TWh per Consumer Transformation scenario to
446TWh per System Transformation scenario in 2050 according to Future Energy
Scenarios (FES) 2023 [8] issued by National Grid. The major driver of hydrogen
consumption difference is residential sector, which is the second largest
energy consumer in the UK after transport sector as of 2021 [9] and may
account for hydrogen consumption between 0TWh and 145TWh in 2050 [10].
Hydrogen application in decarbonization of the industrial sector can take
different directions and be responsible for hydrogen consumption from 11TWh to
88TWh in 2050 [10].
Further, the requirement of hydrogen infrastructure and its economic
feasibility will influence the extent of low-carbon hydrogen adaptation. To
support hydrogen infrastructure development, National Gas (the UK gas system
operator) announced Project Union launch under which about 2,000km of 100%
hydrogen backbone is planned by early 2030s to link production, storage, and
consumption clusters [11]. However, without in depth analysis of all potential
hydrogen adaptation scenarios and economic feasibility, phasing of Project
Union cannot be fully justified.
### 1.1 Key Drivers of Hydrogen Economy in the UK
To support smooth integration of low-carbon hydrogen into energy systems, the
UK government selected ‘twin-track’ approach which supports two major types of
hydrogen production at the early stage of development being: 1) electrolytic
hydrogen production; 2) methane reformation with carbon capture utilization
and storage (CCUS) [6]. At this stage of development, electrolysers have to be
connected directly to renewable energy sources (‘non-networked’ electrolysis)
as the power grid is yet to be decarbonized to introduce ‘networked’
electrolysis as highlighted by [12]. Another type of electrolytic hydrogen
production reviewed by FES-2022 [10] is ‘nuclear’ electrolysis, where more
uncertainties could be seen due to ageing nuclear reactors [13],[14] and
criticism from public on new nuclear power development [15]. Even though the
production side still requires certainties to align with long-term scenarios,
supply chain of any product is primarily dependent on demand; therefore,
hydrogen consumers and infrastructure will play a crucial role in defining the
future scale and pace of low-carbon hydrogen integration into the UK’s energy
system.
#### 1.1.1 Demand
Approximately 2.5GW of carbon-intensive hydrogen production capacity is
currently installed in the UK with about two-thirds coming as by-product from
industrial processes (mainly steelworks and refineries) and only one-third
being produced by methane reformation without carbon capture [16]. As of
future market opportunities, the following major sectors are reviewed in FES
scenarios as potential low-carbon hydrogen consumers: 1) heating; 2) industry;
3) power generation; 4) transport; 5) blending (temporary sink to support
hydrogen supply chain development); 6) exports.
_1) Heating: 0-145TWh H 2 in 2050 [10]_
Heating decarbonization has one of the most debatable paths, with extensive
competition between direct electrification (heat pumps) and ‘clean’ gas
(hydrogen boilers). From one hand, the process efficiency of hydrogen boilers
in heating sector is significantly lower than that of heat pumps, accounting
for about 65% and 90% respectively [17]. This fact creates an open debate for
delay in policymaking decision and indirect support of less attractive techno-
economic solution as hydrogen according to [18]. However, [19] highlights
potential constraints of multi-family house (apartment blocks) heating
decarbonization by ground-source heat pumps and economic downsides of air-
source heat pumps which could potentially be used in this application. Another
option for apartment blocks heating is district heating which could be fed by
ground-source heat pumps as summarized by [20], however, techno-economic
feasibility of this option is still to be confirmed. Therefore, heating
decarbonization may still be a mix of hydrogen and electrification.
_2) Industry: 11-88TWh H 2 in 2050 [10]_
The UK industries are concentrated in ‘clusters’ and its decarbonization
strategy was issued in March 2021 by BEIS (Department for Business, Energy &
Industrial Strategy) [21]. Under this strategy a review of potential
decarbonization pathways for each industrial sector is presented. The review
shows that even though hydrogen can be applied in decarbonization of each
industry to a certain extent, the highest potential of hydrogen application is
in clustered sites.
_3) Power Generation: 12-17TWh H 2 in 2050 [10]_
To cover variations in renewable electricity generation, a long-term energy
storage is critical. Different technologies were proposed for energy storage
to cover peak electricity demand and curtailment reduction; however, a single
type of storage cannot provide sufficient system flexibility as indicated by
[22]. According to evaluation by [23] performed for Climate Change Committee
(CCC), power generation sector will be the major hydrogen consumer in the
nearest future with the UK government commitment of electricity grid
decarbonization by 2035 (subject to energy security).
_4) Transport: 83-138TWh H 2 in 2050 [10]_
Transport sector covers road transport, shipping, aviation, and rail. Hydrogen
as fuel is not projected to receive widespread adaptation for light duty
vehicles as per [24], while decarbonization of shipping and aviation is
projected to utilise hydrogen equally in all FES decarbonization scenarios in
2050 accounting for about 70TWh and 10TWh respectively [10]. According to [25]
hydrogen will be used as feedstock for synthetic fuels production for shipping
and aviation industries rather than a fuels in pure form. As for rail sector,
is more likely to proceed with electrification and maximum annual consumption
is predicted to be limited to 2TWh in 2050 [10].
_5) Blending_
One of the short-term solutions for expansions of hydrogen market analysed by
the UK government is hydrogen blending into natural gas networks with up to
20% (by volume), with plans to announce the final decision on this approach in
2023 [6]. Implementation of this strategy may create flexible consumption that
is favourable condition to support low-carbon hydrogen production development
according to [26].
_6) Exports_
At this early stage, the UK exploring opportunities to export hydrogen to
Europe given its long relationships with energy trade and increasing hydrogen
demand [6]. Two potential export interconnectors were identified: 1) Bacton
industrial cluster to Netherlands; 2) South-West border of Scotland to Ireland
[8]. However, the development of low-carbon hydrogen export projects will
depend not only on production capability of the UK but also readiness of the
import countries and economic viability.
#### 1.1.2 Infrastructure
The existing gas infrastructure of the UK is not suitable for 100% hydrogen
use and substantial investments will be required in case of decision in its
favour. Project Union [11] is performing a study in which existing pipelines
infrastructure repurpose will be covered along with proposal of new pipelines
construction. One of the limitations for the existing infrastructure repurpose
is transition period where natural gas will still play a significant role and
rely on its historic infrastructure. Alongside with Project Union, feasibility
study for low-carbon hydrogen infrastructure development on a regional level
is also performed being: 1) North-East Network and Industrial Cluster
Development (Operator: SGN) [27]; 2) East-Coast Hydrogen (Operator: NGN,
Cadent) [28]; 3) Capital Hydrogen (Operator: SGN, Cadent) [29]; 4) HyNet North
West Hydrogen Pipeline (Operator: Cadent) [30]. Based on the above ongoing
studies review, it is concluded that early development of hydrogen economy and
phasing selection by specific region will depend not only on resource
availability (renewable energy or natural gas) but also convenience for
hydrogen storage and suitability of infrastructure for CO2 storage in case of
methane reformation with CCUS.
#### 1.1.3 Storage
Another obstacle on the way of large-scale hydrogen deployment is storage,
which will be vital to manage the difference in supply and demand patterns.
Evaluation of theoretical capacity of offshore depleted hydrocarbon fields was
done by [31],[32] with potential storage capacity estimated at not less than
2,500TWh considering solely gas reservoirs. Further, study by [33] concluded
that the UK salt caverns can accommodate about 2,000TWh theoretical hydrogen
storage capacity. With maximum hydrogen storage requirement of about 55TWh in
2050 according to ‘System Transformation’ scenario peak [10], the estimated
theoretical capacity of geological storage by far exceeding the requirements.
Even though, the relatively high theoretical storage capacity compared to
requirements could be taken as a positive sign, detailed research on hydrogen
storage in porous media considering biological and geochemical reactions is
still yet to be completed which may cause not only hydrogen contamination but
also losses as emphasized by [34].
### 1.2 Current State of Research
This section reviews the current state of research on hydrogen infrastructure
development in the UK, ranging from initial endeavors to the most recent
studies, with the aim of identifying gaps and promoting further advancements
in this area.
In 2013, an early attempt to model the spatial expansion of hydrogen
infrastructure in the UK within the context of broader energy system
interactions was undertaken by Balta-Ozkan and Baldwin [35]. This study
proposed a geographical infrastructure development plan with ten-year
intervals, spanning from 2020 to 2050. Limitations of this study include:
1) Demand side covers only heating and transport (road and aviation) and
mainly disaggregated based on geographic population density.
2) Scenarios are based on different levels of emission reduction by 2050 with
high reliance on liquid hydrogen import. The scenario with local hydrogen
production does not meet 2050 emission reduction targets.
Further research conducted by [36] in 2017, proposed hydrogen network
development in stages starting from Midlands and expanding across the UK by
2070. Additionally, the study incorporated the development of carbon dioxide
(CO2) pipeline infrastructure from methane reformation to offshore storage.
Limitations of this study include:
1) Hydrogen demand for road transport is covered only (in view of filling
stations).
2) Electrolysis as source of hydrogen production introduced only in 2060.
3) Scenarios are based on discount rate variations (3.5% and 10%).
In 2019, a study conducted by [19] covered crucial elements as electrolysers
location and whether energy transport shall be done via transmission grid or
hydrogen pipelines. This model also accounted for inter-seasonal and daily
demand fluctuations, as well as the intermittent nature of renewable power
generation. Furthermore, it assessed various subsidy levels for upgrading
heating devices within the system optimization framework. Limitations include:
1) Demand side covers consumption by heating only.
2) Hydrogen production from methane reformation, biomass gasification and
nuclear electrolysis is not covered.
3) Various decarbonization scenarios development are not included with heavy
reliance on energy system optimization by software.
One of the most recent studies is performed by [37] under ELEGANCY project.
The project focusing on the transportation and storage of hydrogen and carbon
dioxide from methane reformation process, with the proposal to phase out
investment in hydrogen infrastructure and adjust it based on results of the
first phase outcome. However, based on publicly available information, certain
limitations were identified, including:
1) Demand side covers consumption by heating and transport only.
2) Hydrogen produced mainly by methane reformation with carbon capture
including small share of biomass gasification.
3) Water electrolysis is suggested only to cover peak hydrogen demand in case
if hydrogen storage is not available.
4) Phasing of the development and basis is not provided.
It worth to add, that all the above studies [19],[35],[36] provide a good
basis for further research and developments, however the developed models
could not be found in open access and therefore new studies have to start
modelling from ‘scratch’ investing substantial amount of time for basic data
collection and validation. Even though ELEGANCY project [37] claims to use an
open-source software, the model could not be located in easy access to perform
a critical analysis of the assumptions made.
[38] conducted an extensive study covering the EU and the UK with open-source
model available to public. However, due to large-scale nature of the model, it
provides only a high-level overview of certain critical details, making it
insufficient for making decisions specific to the UK. Furthermore, the model
relies solely on optimization and lacks the flexibility to accommodate various
scenarios that align not only with general system perspectives but also with
individual government targets. In summary, [38] has established a solid
foundation and an openly accessible model that can serve as a basis for
further enhancements for each individual country.
### 1.3 Aim and Contributions
This study aims to provide independent evaluation of Great Britain’s (GB’s)
hydrogen infrastructure development roadmap accounting for flexibility of
future decisions on hydrogen adaptation in various sectors. While policies are
for the UK, this study looking for GB only, as Northern Ireland has a separate
electricity market connected to Republic of Ireland network.
The study performs future GB’s hydrogen system modelling incorporating all
producers and consumers according to FES-2022 scenarios [10] with other
essential inputs sourced from PyPSA-GB [39] \- a future power system model of
GB based on FES. This integration enhances the validity of the model and the
accuracy of the analysis results. The modelling was conducted utilizing
“Python for Power System Analysis” (PyPSA), an open-source energy flow
optimization software [40] providing an opportunity to review and further
improve the modelling.
Furthermore, this study addresses the gaps in the FES-2022 data to improve the
visibility of potential pathways for hydrogen economy development. It also
assesses the sensitivity of the FES-2022 scenarios and highlights potential
minimum and maximum boundaries of hydrogen adaptation in various sectors.
## 2 Methodology
The primary challenge in future energy systems modelling is its heavy
dependence on software-based network optimization, which often introduces
numerous uncertainties and limitations, as emphasized by [41]. In addition,
limited scenarios variety does not represent all potential pathways of energy
systems development making review of the whole system narrowed down to
boundaries set by multiple uncertainties. This section will provide outline of
the methodology utilized in this study, summarizing details of the software
used, scenarios selection, network modeling approach, and assumptions made for
hydrogen supply and demand modelling, while also addressing the study’s
limitations with potential areas for future improvements.
### 2.1 Software
PyPSA-GB-H2 [42] available at https://github.com/tatyanadergunova/PyPSA-
GB-H2.git, covering hydrogen production, consumption, and storage, was built
using open-source tool PyPSA [40]. PyPSA was originally created for modelling
power systems but can also model hydrogen systems. In hydrogen modelling, it
represents hydrogen flows as energy streams and includes components for
hydrogen production, storage, transport, and demand. This allows users to
analyse different energy system scenarios and optimize operation and
investments to minimize costs. The optimization of PyPSA-GB-H2 model will be
run by Gurobi Optimization package [43], which has a limitation being open-
source for academia only.
### 2.2 Scenarios
National Grid annual reports [8] provide potential pathways of GB’s energy
system decarbonization, drawing from ongoing research and development efforts
as well as inputs from various stakeholders. The energy flow estimates from
these reports are available to public and were used as the foundation for
setting hydrogen production and demand capacities in PyPSA-GB-H2 model. Only
FES-2022 workbook data [10] was run in this study and further studies can be
updated utilizing FES-2023 workbook as an input of low-carbon hydrogen
consumption in each scenario [8]. FES reports covering four different
scenarios; however, the scope of this study is limited to evaluation of
‘decarbonization’ scenarios only which are summarized in Table 1.
Table 1: Total hydrogen demand in 2050 (FES-2022) [10] Scenario | Annual Hydrogen Demand in 2050, TWh
---|---
Consumer Transformation | 113
Leading the Way | 244
System Transformation | 431
Early development of hydrogen infrastructure shall account for long-term run
therefore accommodate year 2050, when Net Zero operation is targeted to be
achieved. Once all scenarios evaluated for year 2050, the crossing load
sections on infrastructure can be identified. These sections will be utilized
in future regardless of selected scenario of hydrogen adaption and ‘early’
development of infrastructure around them could be considered as ‘no regret’
decision. Therefore, this study covered year 2050 only.
### 2.3 Network Modelling
The major limitation of FES-2022 scenarios [10] in judgement for hydrogen
infrastructure scale is share of hydrogen supply and demand by regions and
technologies, with 2050 ‘networked’ electrolysis data being the only
exception. The ’networked’ electrolysis is defined as hydrogen production via
water electrolysis with electricity sourced from power grid; therefore, the
location of this producer shall be assigned to a particular region only when
techno-economics of other less flexible hydrogen production technologies,
hydrogen infrastructure and power grid expansion are evaluated.
_Modelled Hydrogen Network (Industrial Cluster Level)_
Figure 2.1: _Industrial clusters and salt caverns plot on GB Distributed
Network Operator (DNO) License Areas map[44]. This Figure is for illustrative
purposes only._
This study introduced geospatial division of hydrogen supply and demand and
identified drivers for flexibility of sectors location on industrial clusters
level. The assumption of industrial cluster level, shown on Figure 2.1, serves
the purpose of this study to identify the ’flexible’ sectors to review minimum
infrastructure requirement. However, higher resolution may be required for
more detailed analysis [Limitation-1].
Hydrogen pipelines between industrial clusters were modelled as energy flows,
disregarding thermodynamic properties and loss of energy associated with
frictional pressure drop. This simplification limits possibility of hydrogen
infrastructure costs evaluation [Limitation-2] which shall include not only
pipelines installation, but also compression stations and operational costs of
electricity demand required for compression which can vary between 0.81-1.1
kWh/kgH2 according to [45]. Furthermore, estimated operating pressure
(average) of salt caverns by [33] is 132-278 barg, what will need additional
compression from hydrogen network operating at 40-80 barg [27],[28] and
therefore capital and operational costs would increase further.
Multiple models including [38], model hydrogen transport as star network.
However, according to hydrogen modelling evaluation performed by [46] star
network are not be representative for actual pipelines loads while tree
network will benefit the study at low computational costs. Therefore, tree
network model was adopted for this study, as shown on Figure 2.1. The network
simulation step of one hour was adopted to cover hourly fluctuations of supply
and demand sides.
### 2.4 Supply
#### 2.4.1 Efficiency and Availability
Every energy generation technology, including hydrogen production, has two
major parameters for evaluation of actual output capacity being efficiency and
availability. FES-2022 workbook [10] was reviewed and efficiency and
availability for each hydrogen production technology evaluated which is
summarized in Table 2.
Table 2: Efficiency and availability of hydrogen production technologies (FES-2022) [10] Technology | Availability | Efficiency
---|---|---
Methane Reformation with CCUS | 95% | 74-82%
Biomass Gasification | 90% | 71-78%
’Nuclear’ Electrolysis | 80% | 84-88%
’Non-networked’ Electrolysis | 30% | 84-88%
’Networked’ Electrolysis | 44-50% | 84-88%
_Methane Reformation with CCUS & Biomass Gasification._ Efficiency of hydrogen
production is constant throughout the years but varying in different
scenarios, increasing with higher share of this hydrogen type in total mix.
_’Nuclear’ Electrolysis, ’Non-networked’ Electrolysis & ’Networked’
Electrolysis._ Electrolysis efficiency is increasing through 2030-2050,
assuming wider adaptation and technology maturity increase.
_’Networked’ Electrolysis._ Availability of ‘networked’ electrolysis is
increasing through 2030-2050 assuming increase in flexibility and robustness
of energy supply sources to the power grid. However, it stays steady at 40%
throughout all years in ‘System Transformation’ scenario with high reliance on
hydrogen for heating and high share of methane reformation with CCUS.
#### 2.4.2 Geographical Limitations
To account for potential limitations of geographical locations, each
technology dependency on feedstock supply and other requirement were reviewed
with summary presented in Table 3.
Table 3: Geographical limitations of hydrogen production technologies by industrial clusters Technology | Limited to Cluster
---|---
Methane Reformation with CCUS | | St. Fergus, Grangemouth, Teesside, Humberside, Bacton,
---
Merseyside, Theddlethorpe
Biomass Gasification | | Bacton, Grain LNG, Southampton,
---
South Wales, Theddlethorpe
’Nuclear’ Electrolysis | Bacton, Grain LNG, South Wales, Merseyside
’Non-networked’ Electrolysis | St. Fergus, Grangemouth, Merseyside
’Networked Electolysis | Proposed location in FES-2022 [10]
Imports | Bacton, Grangemouth
_Methane Reformation with CCUS._ Offshore CO2 storage is assumed at similar
locations as hydrogen storage potential sites investigated by [32]. Therefore,
proximity to depleted oil and gas fields is assumed as beneficial location for
steam reformation with CCUS to reduce CO2 transport infrastructure.
_Biomass Gasification._ Limited to the south part of England and Wales, areas
of higher population density as no major pre-requisites were identified at
this stage.
_’Nuclear’ Electrolysis._ Civilian nuclear fleet of the UK is ageing, and
currently operated nuclear reactors are projected to be decommissioned in
coming years [13],[14]. Therefore, for the purpose of this study potential
location of proposed nuclear power plants was assumed as location of future
‘nuclear’ electrolysis.
_’Non-networked’ Electrolysis._ As PyPSA-GB and PyPSA-GB-H2 will be connected
at a later stage, at this stage potential locations of ‘non-networked’
electrolysis assumed as Scotland and North-West of England [Limitation-3].
_’Networked’ Electrolysis._ The given values in FES-2022 workbook [10]
represent the final installed capacity of ‘networked’ electrolysis by region
and include ‘nuclear’ electrolysis. It is assumed that all indicated capacity
is ‘networked’ electrolysis and generation is split between industrial cluster
based on share for each region. Optimization of ‘networked’ electrolysis
location shall be done in further studies after PyPSA-GB and PyPSA-GB-H2
interconnection [Limitation-3].
_Imports._ Based on Project Union export pipeline interconnections [11]. The
location of potential import points through seaports is excluded from this
study.
### 2.5 Demand
Heating creates substantial seasonal fluctuations and its crutial to model it
dynamically covering not only daily fluctuations but also hourly what will
allow evaluation of peak storage requirement and infrastructure sizing.
Natural gas is still the primary source for heating in the UK, accounting for
74% of households in 2021 according to [47]. Therefore, share of natural gas
consumed by each Local Distribution Zone (LDZ) [48] was split between
industrial clusters obtaining share of total consumption per each cluster.
These shares were further used to calculate annual demand of each cluster
based on total annual hydrogen consumption for heating (residential and
commercial) per FES-2022 workbook [10]. For hourly heat demand fluctuations
hourly ambient temperature fluctuations and type of buildings (single family
house, multifamily house or commercial) were used as an input to produce
annual heat demand per each type of building, see [49] for more details. 2010
temperature profile of the UK was assumed for this study. The proportion of
single ( 80%) and multi-family ( 20%) houses is adopted for 2021 from [50] and
provided as annual heat demand input per buildings type. As output data will
be used to obtain share only, exact heat demand is not considered as critical
and assumed similar to given in [49]. Annual temperature readings were taken
from Renewables Ninja [51].
To estimate energy consumption by industries, daily fluctuations of natural
gas consumption by industries from National Gas transmission data [48] were
used. The obtained share was then divided by 24 hours and multiplied by total
annual energy demand by industry sector from FES-2022 workbook [10]. Further,
all consumption were assigned per industrial clusters adopted for this study.
Even though the roadmap for industrial decarbonization is not defined yet and
may account for some share of electrification, the selected approach is deemed
to be acceptable for this level of study.
For hydrogen power generation hourly output from PyPSA-GB [39] was used and
power plant’s location split between clusters based on geographical proximity.
Share of road transport energy consumption by regions was taken from [52] and
multiplied by total energy consumption from FES-2022 workbook [10]. Hourly
hydrogen consumption assumed to be steady throughout the year considering
buffer storage of liquid hydrogen at consumer side.
For shipping and aviation, hydrogen consumption split among clusters based on
location of airports and major seaports is not accurate as future fuel for
this transportation may not be hydrogen itself but rather its derivatives in
form of synthetic fuels according to [25]. Therefore, hydrogen consumption
shall be related to synthetic fuel production areas which presumably could be
in existing industrial clusters. Based on this, shipping and aviation demand
was combined. Hourly hydrogen consumption assumed to be steady throughout the
year as chemical plants have controlled production and synthetic fuels buffer
storage can be provided for end users demand fluctuation. The same assumption
was adopted for rail as it has relatively low demand compared to other
consumers.
Another demand sector being considered in FES-2022 workbook [10] is blending.
Even though the hydrogen blending into natural gas network is sill pending the
UK government decision, FES-2022 indicate its phase-out by 2043 at the latest,
meaning natural gas network will not be acting as hydrogen sink anymore.
Therefore, this demand type was excluded from this study. This may be added in
future studies where years earlier than 2050 of hydrogen deployment will be
simulated.
FES-2022 does not include exports in demand side [10], therefore supply-demand
balance would not be achieved if exports included in PyPSA-GB-H2 model.
Further, FES-2023 [8] excluded hydrogen exports from all scenarios in
workbook, and there is only a mention of potential hydrogen exports. Based on
above, exports were excluded from PyPSA-GB-H2 model [Limitation-4] and this
shall be addressed in further studies.
Hydrogen consumption by Direct Air Carbon Capture and Storage (DACCS) is
present only in ‘Leading the Way’ scenario of FES-2022 standing at about 25TWh
in 2050 [10]. The purpose of DACCS described in FES reports [8],[17] is to
create a negative emissions source by capturing CO2 from air and storing it at
local storage sites. It is also suggested in FES reports that DACCS shall be
located close to industrial carbon capture and storage infrastructure,
therefore similar approach to location as of methane reformation with CCUS is
adopted.
### 2.6 Storage
Only salt caverns storage was assumed for this study considering potential
contamination and competition with CO2 storage for offshore depleted oil and
gas fields. As highlighted by [34] there is a risk of hydrogen contamination
and biochemical reactions in porous geological media. Further, [53] identified
that CO2 presence in depleted gas fields will cause higher level of hydrogen
conversion to CH4 via methanogenesis. In addition, report issued for CCC on
power system decarbonization [23] assumed only salt caverns and tanks as
hydrogen storage technology. It shall be noted that peak of required hydrogen
storage capacity per FES-2022 workbook [10] is about 55TWh which is by far
smaller than available theoretical storage capacity of the UK salt caverns
estimated by [33] which is summarized in Table 4.
Table 4: Theoretical storage capacity of the UK onshore salt caverns [33] Region | Combined Theoretical Storage of all caverns, TWh
---|---
Cheshire Basin | 129
East Yorkshire | 1465
Wessex Basin | 557
## 3 Results
This section describes PyPSA-GB-H2 modelling results for each scenario and
their comparison with FES-2022 trends. Further, analysis of hydrogen supply-
demand flexibility based on spacial constraints is outlined for each scenario.
### 3.1 Hourly Supply-Demand Trends
Output from PyPSA-GB-H2 for hourly supply-demand trends is presented in Figure
3.1 for each FES-2022 decarbonization scenario.
_(A) - ’Consumer Transformation (B) - ’Leading the Way’ (C) - ’System
Transformation’_
Figure 3.1: _Hourly hydrogen supply and demand by various sectors for three
FES-2022 Net Zero scenarios, year 2050, based on PyPSA-GB-H2 output. Supply-
demand patterns and cluster’s share per methodology of this study._
_’Consumer Transformation.’_ Hydrogen supply in this scenario is predominated
by ‘networked’ electrolysis at about 90% of total production in 2050 followed
by ‘nuclear’ electrolysis at about 7% [10]. Hydrogen production profile by
‘networked’ electrolysis resonates with offshore wind profile, while ‘nuclear’
electrolysis has a steady production. As for demand, it can be observed that
the consumption throughout the year is relatively steady, except of high peaks
in cold season (November-to-March). These sharp peaks are caused by hydrogen
demand for power generation required to cover seasonal gaps in renewable
energy production. Synthetic fuels (shipping and aviation) is the main
consumer in this scenario standing at about 70% in 2050 [10].
_’Leading the Way.’_ The largest share of hydrogen supply in this scenario is
‘networked’ electrolysis accounting for about 50%, followed by ‘non-networked’
electrolysis at about 20% of total annual production in 2050 [10]. ‘Networked’
electrolysis availability was adopted from offshore wind and ‘non-networked’
electrolysis from onshore wind profiles. Fluctuations of hydrogen consumption
throughout the year are more pronounced in this scenario compared to ‘Consumer
Transformation’ as can be seen from Figure 3.1 . This is caused by increased
demand of hydrogen by heating (residential and commercial) and industrial
sectors. Sharp consumption peaks by power generation are also observed in cold
season. Hydrogen demand in this scenario predominated by three sectors:
synthetic fuels production (shipping and aviation) about 30% of total annual
consumption in 2050, followed by heating (residential and commercial) and
industry standing at about 25% and 20% respectively [10].
_’System Transformation.’_ Unlike other FES scenarios, hydrogen supply in
‘System Transformation’ largely come from methane reformation with CCU with
about 50% of total supply in 2050; ‘networked’ electrolysis still plays an
important role with approximately 30% of total supply [10]. In addition,
‘System Transformation’ scenario has the highest fluctuations of demand
throughout the year in comparison to other FES scenarios. This is caused by
higher share of hydrogen for heating (residential) with about 30% of total
annual demand in 2050. Hydrogen consumption for synthetic fuels production
(shipping and aviation) stays the same throughout all scenarios, with hydrogen
to power generation slightly decreasing with total hydrogen consumption
increase, therefore, high peaks becoming less dominant.
### 3.2 Clusters’ Balance and Network Load
Figure 3.2 gives graphical representation of supply-demand spacial flexibility
and Figure 3.3 shows mean hydrogen flows between clusters for each FES
scenario in 2050.
_(A) - ’Consumer Transformation (B) - ’Leading the Way’ (C) - ’System
Transformation’_
Figure 3.2: _FES-2022 scenarios supply-demand geospatial flexibility, year
2050. With increased share of hydrogen adaptation in GB’s energy system demand
side flexibility is dropping bringing the requirement of complex hydrogen
infrastructure. This is mainly caused by dispersed demand of residential
heating and a high developed infrastructure will be required regardless the
increase in supply flexibility.‘Networked’ electrolysis was assigned to
’fixed’ locations as per FES-2022 workbook[10]; however, further evaluation
shall be done upon PyPSA-GB-H2 and PyPSA-GB [39] interconnection._
_‘Consumer Transformation.’_ Even though this scenario has the lowest annual
hydrogen consumption compared to other FES scenarios, hydrogen generation is
not equal to consumption for each cluster. This is mainly related to modelling
limitations in regards of spatial resolution and fixation of future ‘flexible’
demand such as synthetic fuels production. Demand flexibility in this scenario
is governed by synthetic fuels production for shipping and aviation
industries, while methane reformation with CCUS and ‘nuclear’ electrolysis
represent supply flexibility and limited to certain clusters based on
feedstock / additional infrastructure dependency. As for lines loading, a
substantial difference between mean and peak hydrogen flows can be observed in
pipelines connecting clusters and Battery Limits (B/Ls), the points where
hydrogen flow splits to two or more clusters as can be seen on Figure 3.3 .
This is caused by high instantaneous flows of hydrogen to power generation and
model resolution to industrial clusters rather than smaller regions like Grid
Supply Points (GSPs).
_(A) - ’Consumer Transformation (B) - ’Leading the Way’ (C) - ’System
Transformation’_
Figure 3.3: _PyPSA-GB-H2 output for mean and peak hydrogen energy flows
including demand per cluster for three FES-2022 Net Zero scenarios, year
2050._
_’Leading the Way’_ has the medium adaptation of hydrogen, compared to other
FES-2022 scenarios. Figure 3.2 shows the increased difference in supply-demand
balance within one node causing higher hydrogen flows between the clusters.
The higher share of ‘fixed’ supply and demand making ‘Leading the Way’
scenario less flexible compared to ‘Consumer Transformation’. Nonetheless,
strategic location of synthetic fuels production (shipping and aviation) and
DACCS may allow hydrogen infrastructure reduction to regional levels. The
lines loading of ‘Leading the Way’ scenario have similar trends as of
‘Consumer Transformation’ having extreme loading caused by instantaneous flows
to power generators what can be seen in Figure 3.3.
_’System Transformation’_ has the highest hydrogen adaptation particularly in
heating sector, accounting for about 45% of total hydrogen consumption by
commercial and residential sectors in 2050 [10]. This makes demand side less
flexible in view of geographical distribution. Overall, there is still a space
for infrastructure optimization with flexible consumers as synthetic fuels
(for shipping and aviation) production and hydrogen producers as methane
reformation with CCUS, biomass gasification and ’nuclear’ electrolysis.
‘System Transformation’ has the highest difference in mean and peak lines
loading caused not only by power generation peaking but also hydrogen flow for
residential heating due to low temperature peaks.
### 3.3 Storage
_’Consumer Transformation’._ The variation of hydrogen storage profile between
FES-2022 and PyPSA-GB-H2 shown in Figure 3.4 can be explained by the
difference in supply-demand fluctuations. The major differences of PyPSA-GB-H2
from FES-2022 are high peaks of hydrogen demand in cold season (November-to-
March) caused by hydrogen flow to power generation as modelling of this sector
is based on power demand and does not account technological specifics of this
process. In addition, hydrogen production curve in PyPSA-GB-H2 model has
sharper peaks throughout the year associated with ‘networked’ electrolysis
being major hydrogen producer and linked to offshore wind availability only.
_(A) - ’Consumer Transformation (B) - ’Leading the Way’ (C) - ’System
Transformation’_
Figure 3.4: _Hydrogen storage ’state of charge’ - FES-2022[10] vs PyPSA-GB-H2
[42], 2050. Even though a similar trends can be observed, both initial and
peak storage capacity is lower in PyPSA-GB-H2 model compared to FES-2022,
which is caused by difference in supply-demand modelling._
_’Leading the Way’._ Even though a similar trend can be observed, both initial
and peak stage of charge is almost twice lower in PyPSA-GB-H2 model compared
to FES-2022 (Figure 3.4) which is relevant to difference in supply-demand
profile. Hydrogen demand by synthetic fuels (shipping and aviation) and power
generation stays approximately at the same levels as of ‘Consumer
Transformation’ scenario. The consumption spikes in FES-2022 could be
associated with different approach to heating demand modelling: daily
temperature profile and combined electric-hydrogen heating with hydrogen used
for lower temperature peaks only.
_’System Transformation’._ Annual supply-demand profile for ‘System
Transformation’ scenario in PyPSA-GB-H2 modelling is relatively close to
FES-2022. FES-2022 demand curve is smoother compared to PyPSA-GB-H2 what could
be caused by difference in temperature fluctuations andd introduction of local
storage. Hydrogen storage state of charge is also showing high similarity
between FES-2022 and PyPSA-GB-H2 models, with only variation in peak storage
during cold season.
## 4 Discussions
Development of hydrogen infrastructure and hydrogen economy are two
interdependent aspects. From one side, the scale of infrastructure shall be
defined by hydrogen integration in the future energy mix. From another,
hydrogen supply chain growth may be constrained by the absence of facilities
to link potential supply and demand sites. Moreover, the early development
stages in energy sector are typically supported by fair competition therefore
creating progression at dispersed locations increasing the pressure for
infrastructure support.
The uncertainties in hydrogen economy development and the lack of open-source
modelling can lead to substantial investment in complex infrastructure which
may be largely underutilized in the future. Or even more, hydrogen adaptation
may have to be adjusted around the new infrastructure missing out
opportunities for more efficient energy systems development.
In this study, an open-source model, PyPSA-GB-H2 [42], was developed to
simulate Great Britain (GB) future energy scenarios in hydrogen sector,
covering the gaps of the previous studies and including all consumers and
production technologies projected in Future Energy Scenarios (FES) [8]
presented by National Grid. The open-source nature of model and the variety of
covered sectors and scenarios enhances accuracy and transparency in this
research area and laying a foundation for further studies.
This section will cover the study analysis and interpretation of the findings
related to GB’s hydrogen infrastructure development. Additionally, policy
recommendations and outline of the next steps for advancing this work will be
given.
### 4.1 FES Scenarios Analysis
This study evaluated geospatial flexibility of hydrogen producers and
consumers based on FES-2022 [10] workbook and analysis of technical
dependencies with summary presented in Figure 4.1.
Figure 4.1: _Share of geographically ’flexible’ supply and demand sectors,
with capacities per FES-2022 year 2050[10]. As the share of hydrogen
adaptation in GB’s energy system increases, demand side flexibility decreases,
enforcing a more complex hydrogen infrastructure. This is primarily due to
dispersed demand for residential heating, which requires advanced
infrastructure, regardless of increased supply flexibility._
The summary shows decrease of ’flexible’ demand share, which could potentially
be located in proximity to hydrogen production, with increased proportion of
hydrogen in total energy mix while the opposite trend can be seen for supply
side. The major highlights of analysis are summarized below based on specific
sub-area of research.
_1) Hydrogen Infrastructure Scale for FES-2022 Net Zero Scenarios:_
_‘Consumer Transformation’:_ Local hydrogen networks with strategic location
of ‘flexible’ consumers, as synthetic fuels (shipping and aviation) is deemed
to serve the purpose. _‘Leading the Way’:_ Regional networks as not all
demands can be met by local production and some residential heating is added.
Location planning of synthetic fuel production (shipping and aviation) and
Direct Air Carbon Capture and Storage (DACCS) will support reduction of future
hydrogen infrastructure, however it is expected that regional networks would
still be required to cover hydrogen distribution for residential heating.
_’System Transformation’:_ Country scale networks, high reliance on hydrogen
for heating. Overall, this scenario has the highest share of GB’s energy
system reliance on hydrogen and the largest geographical area of consumers to
be covered. Therefore, introduction of complex hydrogen infrastructure is
essential for ‘System Transformation’ scenario.
_2) Residential Heating Decarbonization Debate:_
Emissions reduction in residential heating sector is still encountering major
uncertainties. While ’Consumer Transformation’ scenario relies purely on
domestic heating electrification, ’Leading the Way’ adopts 25%
($\approx$42TWh) and ‘System Transformation’ 50% ($\approx$145TWh) of hydrogen
in total heating energy mix by 2050 [10]. Even though heating heat pumps are
by far more efficient compared to hydrogen boilers [17], temperature
requirement cannot always be met by heat pumps particularly in apartment
blocks [19]. There is a potential for large-scale district heating sourced by
heat pumps as highlighted by David et al. [20], however, further research is
required to justify its feasibility.
_3) Hydrogen in Industry Decarbonization:_
Review of industry sector decarbonization by BEIS [21] shows high range of
potential solutions for emissions reduction in this sector which can also be
indirectly observed in FES scenarios. As example, hydrogen adaptation in
’Consumer Transformation’ is less than 10% ($\approx$11TWh) and ’System
Transformation’ is about 45% ($\approx$ 87TWh) of total energy consumption by
industries in 2050 [10]. Even though future role of hydrogen in industry
sector decarbonization has lack of clarity, the UK industrial sites are mostly
concentrated in clusters which may support localised decarbonization approach
reducing requirement in extensive hydrogen infrastructure.
_4) Balancing Hydrogen Storage Needs:_
Peak hydrogen flows were observed during PyPSA-GB-H2 [42] model runs in each
FES scenario, which are associated with hourly temperature fluctuations and
power generation peaks. These flow peaks shall see a reduction with addition
of pipeline packing and increase in model resolution, however local storage of
hydrogen (potentially in liquefied form) shall be considered for all scenarios
to eliminate high-capacity short-term loads on infrastructure. As for
geological storage, strategy for infrastructure development shall include
geological storage starting from minimum hydrogen adaptation scenario and
feasibility of decentralised approach.
_5) GB’s Net Zero Scenarios Crossroads:_
Shipping and aviation sectors are reaching the same hydrogen consumption in
all FES scenarios by 2050, standing at about 70TWh and 10TWh respectively; in
addition, power generation having a similar requirements in hydrogen among all
FES scenarios with about 15TWh in 2050 [10].
Based on the above, hydrogen infrastructure development shall be phased
starting from localized hubs prioritizing decarbonization of power grid. This
will also allow expansion of low-carbon hydrogen production sector to include
‘networked’ electrolysis. In addition, advanced planning for synthetic fuels
for shipping and aviation shall be reviewed in early phase of low-carbon
hydrogen hubs development. As for the existing industrial sector, it is
evaluated that localized decarbonization without country-scale hydrogen
infrastructure will meet the requirements.
### 4.2 Policy Recommendations
The following areas of improvement are proposed to current policies as outcome
of research completed under this study:
(1) Set priorities for hydrogen production hubs support under government
schemes considering strategic location of future hydrogen power and synthetic
fuel generation facilities.
(2) Further government support to research centers on residential heating,
particularly multi-family house (apartment blocks).
(3) Project Union initiated by National Gas (the UK gas system operator), to
provide roadmap and phasing strategy with emphasis on self-sufficient clusters
decarbonization approach.
(4) If blending strategy is approved, ‘green light’ for hydrogen producers
tapping shall be strictly monitored accounting for future strategic planning
of potential final consumers.
### 4.3 Further Work
The following aspects shall be optimized in future studies of GB’s hydrogen
infrastructure development:
(1) ‘Networked’ electrolysis location suggested by FES-2022 shall be re-
evaluated after interconnection of PyPSA-GB-H2 [42] with PyPSA-GB [39] which
will allow understanding of grid constraints and economic optimization of the
GB’s energy system.
(2) Spatial resolution of PyPSA-GB-H2 model to be increased including
potential location of power generation plants rather than limitation of all
production and consumption to industrial cluster level.
(3) Pipeline packing and provision of local storage for peak loads for both
power generation and heating may be incorporated in PyPSA-GB-H2 model, with
re-evaluation of hydrogen power plants modelling.
### 4.4 Conclusions
FES scenarios presented by National Grid [8] on an annual basis provide
potential directions of hydrogen integration in the GB energy system, however,
the extent of hydrogen application in different sectors and production
capacity by each technology is highly varying between the scenarios. Moreover,
FES scenarios are not providing guarantee of proceeding with one or another
decarbonization scenario, in fact future energy mix will potentially land
somewhere in between. Nonetheless, the scenarios are setting the minimum and
maximum boundaries of potential hydrogen adaptation based on current state of
research and stakeholders involvement. Therefore, these boundaries could act
as a guideline for hydrogen infrastructure development.
This study provided evaluation of Great Britain’s hydrogen infrastructure
development roadmap, considering the adaptability and flexibility of future
hydrogen integration across different sectors. An open-source model PyPSA-
GB-H2 was developed, improving visibility of hydrogen adaptation across
various sectors building up a base for further studies. Furthermore, policy
recommendations were provided highlighting importance of early decisions to
enhance the development of more efficient energy systems.
## 5 Supplementary Material
More details on PyPSA-GB-H2 modelling approach can be found on a public GitHub
repository - https://github.com/tatyanadergunova/PyPSA-GB-H2.git.
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|
# On the Bundle of KMS state spaces for flows on a $\mathcal{Z}$-absorbing
C*-algebra
George A. Elliott, Yasuhiko Sato, and Klaus Thomsen Department of
Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
<EMAIL_ADDRESS>Graduate School of Mathematics, Kyushu University,
744 Motoka, Nishi-ku, Fukuoka, Japan<EMAIL_ADDRESS>Department of
Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark
<EMAIL_ADDRESS>
## 1\. Introduction
The recent classification results for simple $C^{*}$-algebras have provided
new tools for the construction of flows on $C^{*}$-algebras with a hitherto
unseen richness and complexity in the configuration of KMS state spaces. The
basic idea behind the construction of flows with specified KMS behaviour goes
back to work by Bratteli, Elliott, Herman, and Kishimoto in [BEH], [BEK1], and
[BEK2] where such flows were constructed on various simple unital
$C^{*}$-algebras. By using the classification results it is now possible to
construct such flows on many given $C^{*}$-algebras. This was done for simple
AF algebras in [Th3] and [ET] and it is the purpose here to do this with
arbitrary simple algebras that absorb the Jiang-Su algebra tensorially,
including in particular the purely infinite $C^{*}$-algebras that are
classified by the Kirchberg-Phillips classification result. As in [Th3] and
[ET] some of the key ingredients come from arguments developed by the second
author and Matui.
Although little was known about flows that are not approximately inner on an
AF-algebra, Kishimoto constructed a non approximately inner flow on a certain
AF-algebra based on H. Lin’s TAF classification theorem [Ki3]. In [Sa], [MS2],
and [MS3], a generalization of the classifiability permitted a construction of
a non approximately inner flow on any UHF-algebra, which is known as a
counter-example to the Powers-Sakai conjecture. The main result of this paper
widens these previous constructions in the directions of both classifiable
$C^{*}$-algebras and the variations of KMS spectra. As a consequence, we can
construct uncountably many flows which are not approximately inner on any
classifiable monotracial $C^{*}$-algebra up to weak cocycle conjugacy
(Corollary 4.2).
## 2\. Proper simplex bundles and flows on unital $C^{*}$-algebras
We start with an abstract characterization of bundles of KMS state spaces
introduced in [ET]. Fix a second countable locally compact Hausdorff space $S$
and let $\pi$ be a continuous map from $S$ to $\mathbb{R}$. We shall say that
the couple $(S,\pi)$ is a _simplex bundle_ if the inverse image $\pi^{-1}(t)$
is a compact metrizable Choquet simplex for any $t\in\mathbb{R}$ in the
relative topology from $S$. Here we remark that $\pi$ is not necessarily
surjective and the empty set is regarded as a simplex. When $(S,\pi)$ is a
simplex bundle we shall denote by $\mathcal{A}(S,\pi)$ the set of continuous
functions $f$ from $S$ to $\mathbb{R}$ such that the restriction
$f|_{\pi^{-1}(t)}$ of $f$ to $\pi^{-1}(t)$ is affine for any $t\in\mathbb{R}$.
###### Definition 2.1.
(See [ET] and [BEK2].) A simplex bundle $(S,\pi)$ is _proper_ , if
* (1)
$\pi$ is proper, i.e., $\pi^{-1}(K)$ is compact in $S$ for any compact subset
$K$ of $\mathbb{R}$,
* (2)
$\mathcal{A}(S,\pi)$ separates points of $S$, i.e., for any $x\neq y$ in $S$
there exists $f\in\mathcal{A}(S,\pi)$ such that $f(x)\neq f(y)$.
Note that when $S$ is compact, condition (1) is automatically satisfied and
that $(S,\pi)$ is then a compact simplex bundle over $\mathbb{R}$ in the sense
of [BEK2]. Since we only consider bundles over $\mathbb{R}$ in this paper we
shall call $(S,\pi)$ a _compact simplex bundle_ when $S$ is compact.
Two proper simplex bundles $(S,\pi)$ and $(S^{\prime},\pi^{\prime})$ are
_isomorphic_ when there is a homeomorphism $\phi:S\to S^{\prime}$ such that
$\pi^{\prime}\circ\phi=\pi$ and
$\phi:\pi^{-1}(\beta)\to{\pi^{\prime}}^{-1}(\beta)$ is affine for all
$\beta\in\mathbb{R}$.
The reason for introducing the concept of proper simplex bundle is that the
collection of KMS state spaces for a flow on a unital separable
$C^{*}$-algebra is a proper simplex bundle in a canonical way. To explain this
we emphasize that all $C^{*}$-algebras in this paper are assumed to be
separable and all traces and weights on a $C^{*}$-algebra are required to be
non-zero, densely defined and lower semicontinuous. A flow
$\theta=(\theta_{t})_{t\in\mathbb{R}}$ on a $C^{*}$-algebra $A$ is a
continuous representation of $\mathbb{R}$ by automorphisms of $A$. Let $A$ be
a $C^{*}$-algebra, $\theta$ a flow on $A$ and $\beta$ a real number. A
$\beta$-KMS weight for $\theta$ is a weight $\omega$ on $A$ such that
$\omega\circ\theta_{t}=\omega$ for all $t$, and
$\omega(a^{*}a)\ =\
\omega\left(\theta_{-\frac{i\beta}{2}}(a)\theta_{-\frac{i\beta}{2}}(a)^{*}\right),\
\ a\in D(\theta_{-\frac{i\beta}{2}}).$ (2.1)
In particular, a $0$-KMS weight for $\theta$ is a $\theta$-invariant trace. A
bounded $\beta$-KMS weight is called a $\beta$-KMS functional and a
$\beta$-KMS state when it is of norm one. For states alternative formulations
of the KMS condition can be found in [BR].
Assume that $A$ is unital. For each $\beta\in\mathbb{R}$ denote by
$S^{\theta}_{\beta}$ the (possibly empty) set of $\beta$-KMS states for
$\theta$, and by $E(A)$ the state space of $A$, a compact convex set in the
weak* topology. Set
$S^{\theta}=\left\\{(\omega,\beta)\in E(A)\times\mathbb{R}:\ \omega\in
S^{\theta}_{\beta}\right\\},$
and equip $S^{\theta}$ with the relative topology inherited from the product
topology of $E(A)\times\mathbb{R}$. Let $\pi^{\theta}:S^{\theta}\to\mathbb{R}$
denote the projection onto the second coordinate. It follows from general
facts about KMS states that $(S^{\theta},\pi^{\theta})$ is a proper simplex
bundle, _the KMS bundle of $\theta$_. See [ET].
## 3\. Compact simplex bundles as KMS bundles for flows on the Jiang-Su
algebra
While we would like to construct a flow on the Jiang-Su algebra such that its
bundle of KMS state spaces is an arbitrary proper simplex bundle for which the
fiber $\pi^{-1}(0)$ over $0$ contains exactly one point, at present we only
know how to do this when the bundle is compact. We fix therefore a compact
simplex bundle $(S,\pi)$ such that $\pi^{-1}(0)$ contains exactly one point
$\overline{o}$.
### 3.1. Construction of a simple ordered abelian group
In what follows we denote the closed support of a real-valued function $f$ by
$\operatorname{supp}f$. Let $\mathcal{A}_{0}(S,\pi)$ denote the set of
elements $f\in\mathcal{A}(S,\pi)$ for which
$\overline{o}\notin\operatorname{supp}f$. Since the topology of $S$ is second
countable we can choose a countable subgroup $G_{0}$ of
$\mathcal{A}_{0}(S,\pi)$ with the following density property:
###### Property 3.1.
For all $\epsilon>0$ and all $f\in\mathcal{A}(S,\pi)$ such that
$f(\overline{o})=0$, there is an element $g\in G_{0}$ such that $\sup_{x\in
S}|f(x)-g(x)|<\epsilon$.
Enlarging $G_{0}$, we may suppose that the functions
${e^{n\pi}}{(1-e^{-\pi})^{m}}f,\ \ n,m\in\mathbb{Z},$
all are in $G_{0}$ when $f$ is. Set
$G=\left(\bigoplus_{\mathbb{Z}}\mathbb{Z}\right)\oplus G_{0},$
and define ${L}:G\to\mathcal{A}(S,\pi)$ by
${L}\left(\xi,g\right)(x)=g(x)+\sum_{n\in\mathbb{Z}}z_{n}e^{n\pi(x)},$
where $\xi=(z_{n})_{n\in\mathbb{Z}}\in\bigoplus_{\mathbb{Z}}\mathbb{Z}$. Set
$G^{+}=\left\\{(\xi,g)\in G:\ L(\xi,g)(x)>0,\ x\in S\right\\}\cup\\{0\\}.$
The proof of the following lemma is straightforward. The last statement uses
that $S$ is compact.
###### Lemma 3.2.
$G=G^{+}-G^{+},\ G^{+}\cap(-G^{+})=\\{0\\}$, and every non-zero element of
$G^{+}$ is an order unit for $(G,G^{+})$.
In short, this lemma says that $(G,G^{+})$ is a simple ordered abelian group.
Note that
$\mathbb{Q}\otimes_{\mathbb{Z}}G=\left(\bigoplus_{\mathbb{Z}}\mathbb{Q}\right)\oplus\mathbb{Q}G_{0}$
and that $L$ extends to a $\mathbb{Q}$-linear map
$\overline{L}:\mathbb{Q}\otimes_{\mathbb{Z}}G\to\mathcal{A}(S,\pi)$. We set
$(\mathbb{Q}\otimes_{\mathbb{Z}}G)^{+}=\left\\{(\xi,g)\in\mathbb{Q}\otimes_{\mathbb{Z}}G:\
\overline{L}(\xi,g)(x)>0,\ x\in S\right\\}\cup\\{0\\}.$
###### Lemma 3.3.
$(\mathbb{Q}\otimes_{\mathbb{Z}}G,(\mathbb{Q}\otimes_{\mathbb{Z}}G)^{+})$ is a
simple ordered abelian group with the strong Riesz interpolation property: If
$g_{i},k_{j}\in\mathbb{Q}\otimes_{\mathbb{Z}}G$ and $g_{i}<k_{j}$ for
$i,j\in\\{1,2\\}$, then there is an element
$h\in\mathbb{Q}\otimes_{\mathbb{Z}}G$ such that $g_{i}<h<k_{j}$ for all
$i,j\in\\{1,2\\}$.
###### Proof.
Only the interpolation property is not straightforward, so assume that
$g_{i},k_{j}\in\mathbb{Q}\otimes_{\mathbb{Z}}G$ and $g_{i}<k_{j}$ for
$i,j\in\\{1,2\\}$. Choose $q\in\mathbb{Q}$ such that
$g_{i}(\overline{o})<q<k_{j}(\overline{o})$ for all $i,j$. It follows from
Lemma 2.2 of [BEK2] that there is an element $h_{0}\in\mathcal{A}(S,\pi)$ such
that $h_{0}(\overline{o})=q$ and $g_{i}(x)<h_{0}(x)<k_{j}(x)$ for all $i,j$
and all $x\in S$. Let $\delta>0$ be smaller than $k_{j}(x)-h_{0}(x)$ and
$h_{0}(x)-g_{i}(x)$ for all $i,j$ and all $x\in S$. Thanks to Property 3.1
there is $g\in G_{0}$ such that $\left|h_{0}(x)-q-g(x)\right|<\delta$ for all
$x\in S$. For $t\in\mathbb{Q}$ denote by $t^{(0)}$ the element of
$\bigoplus_{\mathbb{Z}}\mathbb{Z}$ given by
$t^{(0)}_{n}=\begin{cases}t,&\ n=0,\\\ 0,&\ n\neq 0.\end{cases}$
Then $h=(q^{(0)},g)\in G$ has the desired properties. ∎
In the notation of the last proof, set $v=(1^{(0)},0)\in G^{+}$. Then $v$ is
an order unit in both $(G,G^{+})$ and
$(\mathbb{Q}\otimes_{\mathbb{Z}}G,(\mathbb{Q}\otimes_{\mathbb{Z}}G)^{+})$.
Denote by $S(G)$ and $S(\mathbb{Q}\otimes_{\mathbb{Z}}G)$ the corresponding
state spaces. The restriction map
$S(\mathbb{Q}\otimes_{\mathbb{Z}}G)\to S(G)$ (3.1)
is clearly an affine homeomorphism and we conclude therefore from Lemma 3.3
that $S(G)$ is a Choquet simplex by Proposition 1.7 of [EHS].
Let $\mathcal{Z}$ denote the Jiang-Su algebra, [JS]. In what follows a
$C^{*}$-algebra $A$ will be called $\mathcal{Z}$-stable when
$A\otimes\mathcal{Z}\simeq A$. Let $F_{0},F_{1}$ be finite-dimensional
$C^{*}$-algebras and $\varphi_{0},\varphi_{1}:F_{0}\to F_{1}$ unital
$*$-homomorphisms. The $C^{*}$-algebra
$\left\\{(a,f)\in F_{0}\oplus(C[0,1]\otimes F_{1}):\ \varphi_{i}(a)=f(i),\
i=0,1\right\\}$
will be called a _building block_. In what follows we shall denote by $T(A)$
the tracial state space of a unital $C^{*}$-algebra $A$.
###### Proposition 3.4.
([E3],[Th1],[GLN1].) There is a simple unital $\mathcal{Z}$-stable
$C^{*}$-algebra $E$ which is an inductive limit of building blocks with the
following properties.
* (1)
$K_{1}(E)=0$.
* (2)
The canonical map $\tau:T(E)\to S(K_{0}(E),[1])$ is bijective.
* (3)
There is an isomorphism $\phi:K_{0}(E)\to G$ of ordered abelian groups such
that $\phi([1])=v$.
###### Proof.
This follows from Corollary 13.51 of [GLN1] since the quadruple
$((G,G^{+},v),0,S(G),\operatorname{id})$
is an Elliott invariant with the properties required in that statement. ∎
In the sequel we fix $E$ as in Proposition 3.4 and just write
$(K_{0}(E),K_{0}(E)^{+},[1],T(E))=(G,G^{+},v,S(G)).$
We shall denote by $\mathbb{K}$ the $C^{*}$-algebra of compact operators on an
infinite-dimensional separable Hilbert space and fix a non-zero minimal
projection $e_{11}\in\mathbb{K}$. Denote by
$\operatorname{Hom}_{+}(G,\mathbb{R})$ the cone of non-zero positive
homomorphisms from $G$ to $\mathbb{R}$ and by
$\mathcal{T}(E\otimes\mathbb{K})$ the cone of densely defined lower
semicontinuous traces on $E\otimes\mathbb{K}$. Recall that, as $E$ is unital
and simple, the restriction map $\tau\mapsto\tau|_{E\otimes e_{11}}$ is
bijective.
Define the automorphism $\alpha$ of $(G,G^{+})$ by
$\alpha(\xi,f)=(\sigma(\xi),e^{-\pi}f),$
where
$\sigma\in\operatorname{Aut}\left(\bigoplus_{\mathbb{Z}}\mathbb{Z}\right)$ is
the shift;
$\sigma(\xi)_{n}=\xi_{n+1}.$
###### Lemma 3.5.
There is an automorphism $\gamma\in\operatorname{Aut}E\otimes\mathbb{K}$ such
that $\gamma_{*}=\alpha$.
###### Proof.
Set $p_{1}=1\otimes e_{11}$. There is a projection $p_{2}\in
E\otimes\mathbb{K}$ such that $\alpha([p_{1}])=[p_{2}]$ in $K_{0}(E)$.
Consider the homomorphism $s:E\otimes\mathbb{K}\to
E\otimes\mathbb{K}\otimes\mathbb{K}$, $a\mapsto a\otimes e_{11}$. It is well
known that $s$ is homotopic in
$\operatorname{Hom}(E\otimes\mathbb{K},E\otimes\mathbb{K}\otimes\mathbb{K})$
to a $*$-isomorphism $\mu:E\otimes\mathbb{K}\to
E\otimes\mathbb{K}\otimes\mathbb{K}$. In particular, ${s}_{*}:K_{0}(E)\to
K_{0}(E\otimes\mathbb{K})$ is an isomorphism. It follows from [B] that there
are isometries $V_{i}\in M(E\otimes\mathbb{K}\otimes\mathbb{K})$ such that
$V_{i}V_{i}^{*}=p_{i}\otimes 1_{\mathbb{K}}$. Then
$\operatorname{Ad}V_{i}\circ s:E\otimes\mathbb{K}\to
p_{i}(E\otimes\mathbb{K})p_{i}\otimes\mathbb{K}$ is $*$-homomorphism homotopic
to a $*$-isomorphism and we get therefore an isomorphism
$\alpha^{\prime}:K_{0}(p_{1}(E\otimes\mathbb{K})p_{1})\to
K_{0}(p_{2}(E\otimes\mathbb{K})p_{2})$
of ordered groups when we set
$\alpha^{\prime}=(\operatorname{Ad}V_{2}\circ
s)_{*}\circ\alpha\circ(\operatorname{Ad}V_{1}\circ s)_{*}^{-1}.$
Note that $V_{i}(p_{i}\otimes e_{11})$ is a partial isometry in
$p_{i}(E\otimes\mathbb{K})p_{i}\otimes\mathbb{K}$ giving a Murray-von Neumann
equivalence $V_{i}(p_{i}\otimes e_{11})V_{i}^{*}\sim p_{i}\otimes e_{11}$.
Thus,
$(\operatorname{Ad}V_{i}\circ s)_{*}^{-1}[p_{i}\otimes e_{11}]=[p_{i}]$
and hence, on setting $i=1$, $\alpha^{\prime}([p_{1}\otimes
e_{11}])=[p_{2}\otimes e_{11}]$. Thus $\alpha^{\prime}$ is an isomorphism
$\displaystyle(K_{0}(p_{1}(E\otimes\mathbb{K})p_{1}),K_{0}(p_{1}(E\otimes\mathbb{K})p_{1})^{+},[1_{p_{1}(E\otimes\mathbb{K})p_{1}}])$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\to(K_{0}(p_{2}(E\otimes\mathbb{K})p_{2}),K_{0}(p_{2}(E\otimes\mathbb{K})p_{2})^{+},[1_{p_{2}(E\otimes\mathbb{K})p_{2}}])$
of ordered groups with order unit.
Note that the condition (2) of Proposition 3.4 is equivalent to the statement
that the canonical map $\mathcal{T}(E\otimes\mathbb{K})$ to positive
functionals on $K_{0}(E)=K_{0}(E\otimes\mathbb{K})$ is bijective. In other
words, traces on $E\otimes\mathbb{K}$ are determined by their values on
projections. Since $p_{i}(E\otimes\mathbb{K})p_{i}\otimes\mathbb{K}\simeq
E\otimes\mathbb{K}$, the same is true with $p_{i}(E\otimes\mathbb{K})p_{i}$ in
place of $E,\ i=1,2$. It follows immediately that there is a unique affine
homeomorphism $\chi:T(p_{1}(E\otimes\mathbb{K})p_{1})\to
T(p_{2}(E\otimes\mathbb{K})p_{2})$ compatible with $\alpha^{\prime}$ in the
natural sense, i.e.,
$\chi(\omega)_{*}(\alpha^{\prime}(x))=\omega_{*}(x)$
for all $\omega\in T(p_{1}(E\otimes\mathbb{K})p_{1})$ and all $x\in
K_{0}(p_{1}(E\otimes\mathbb{K})p_{1})$.
It follows from [B] that $p_{i}(E\otimes\mathbb{K})p_{i}$ is stably isomorphic
to $E$ and hence from (1) in Proposition 3.4 that
$K_{1}(p_{i}(E\otimes\mathbb{K})p_{i})=0$. So we see that the pair
$(\alpha^{\prime},\chi)$ is an isomorphism of the Elliott invariant of
$p_{1}(E\otimes\mathbb{K})p_{1}$ onto that of
$p_{2}(E\otimes\mathbb{K})p_{2}$.
We wish to apply Theorem 2.7 of [EGLN] to conclude that the isomorphism
$(\alpha^{\prime},\chi)$ arises from a $*$-isomorphism between the algebras.
This means showing that $p_{1}(E\otimes\mathbb{K})p_{1}$ and
$p_{2}(E\otimes\mathbb{K})p_{2}$ belong to the class $\mathcal{N}_{1}$ of
[EGLN], i.e., have rational generalized tracial rank at most one, and in
addition are amenable, satisfy the UCT, and are $\mathcal{Z}$-stable.
For convenience, set $p_{i}(E\otimes\mathbb{K})p_{i}=E_{i},\ i=1,2$. Note
that, by Lemma 3.19 of [GLN1], both $E_{1}$ and $E_{2}$ are simple inductive
limits of building blocks since $E$ is. Both $E_{1}$ and $E_{2}$ are
hereditary sub-$C^{*}$-algebras of $E\otimes\mathbb{K}$ and hence
$\mathcal{Z}$-stable by Corollary 3.1 of [TW], as $E$ is $\mathcal{Z}$-stable.
Hence by Theorem A of [ENST], $E_{1}$ and $E_{2}$ have finite decomposition
rank (in fact, $dr(E_{i})\leq 2,\ i=1,2$). By 22.3.5 (e) and 23.1.1 of [Bl],
the Universal Coefficient Theorem (UCT) holds for inductive limits of type I
algebras and so $E_{1}$ and $E_{2}$ satisfy the UCT. Hence by Theorem 1.1 of
[EGLN], $E_{1}$ and $E_{2}$ belong to $\mathcal{N}_{1}$ – and so the second
part of Theorem 2.7 of [EGLN] applies, and the isomorphism between the Elliott
invariants is determined by a $*$-isomorphism $\rho:E_{1}\to E_{2}$.
Having $\rho$, we set
$\gamma=\mu^{-1}\circ\operatorname{Ad}V_{2}^{*}\circ(\rho\otimes\operatorname{id}_{\mathbb{K}})\circ\operatorname{Ad}V_{1}\circ\mu\in\operatorname{Aut}E\otimes\mathbb{K}$.
Then
$\displaystyle\gamma_{*}=(\mu^{-1})_{*}\circ(\operatorname{Ad}V_{2}^{*})_{*}\circ\rho_{*}\circ(\operatorname{Ad}V_{1})_{*}\circ\mu_{*}$
$\displaystyle=(s_{*})^{-1}\circ(\operatorname{Ad}V_{2}^{*})_{*}\circ\alpha^{\prime}\circ(\operatorname{Ad}V_{1}\circ
s)_{*}$ $\displaystyle=(\operatorname{Ad}V_{2}\circ
s)_{*}^{-1}\circ\alpha^{\prime}\circ(\operatorname{Ad}V_{1}\circ
s)_{*}=\alpha.$
∎
The following lemma follows from [Sa], [MS2] and [MS3]; see also the second
step in the proof of Lemma 3.4 of [Th2].
###### Lemma 3.6.
The automorphism $\gamma$ can be chosen to have the following additional
properties.
* (A)
The restriction map $\mu\ \mapsto\ \mu|_{E\otimes\mathbb{K}}$ is a bijection
from traces $\mu$ in
$\mathcal{T}(E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z})$ onto the
$\gamma$-invariant traces in $\mathcal{T}(E\otimes\mathbb{K})$, and
* (B)
$(E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z}$ is $\mathcal{Z}$-stable; that
is
$((E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z})\otimes\mathcal{Z}\simeq(E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z}$
where $\mathcal{Z}$ denotes the Jiang-Su algebra, [JS].
We assume in what follows that $\gamma\in\operatorname{Aut}E\otimes\mathbb{K}$
is chosen such that $\gamma_{*}=\alpha$ and such that (A) and (B) of Lemma 3.6
hold. Set
$C=(E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z}\ .$
Since $\gamma_{*}^{k}=\alpha^{k}\neq\operatorname{id}$ when $k\neq 0$, no non-
trivial power of $\gamma$ is inner. Since $E\otimes\mathbb{K}$ is simple it
follows from Theorem 3.1 of [Ki1] that $C$ is simple. It follows from the
Pimsner-Voiculescu exact sequence, [PV], that we can identify $K_{0}(C)$, as a
group, with the quotient
$G/(\operatorname{id}-\alpha)(G),$
in such a way that the map $\iota_{*}:K_{0}(E)\to K_{0}(C)$ induced by the
inclusion $\iota:E\otimes\mathbb{K}\to C$ becomes the quotient map
$q:G\to G/(\operatorname{id}-\alpha)(G).$
Define $\Sigma_{0}:G\to\mathbb{Z}$ by
$\Sigma_{0}((z_{n})_{n\in\mathbb{Z}},f)=\sum_{n\in\mathbb{Z}}z_{n}.$
###### Lemma 3.7.
$\ker\Sigma_{0}=(\operatorname{id}-\alpha)(G)$.
###### Proof.
If $((z_{n})_{n\in\mathbb{Z}},f)\in\ker\Sigma_{0}$ it follows as in the proof
of Lemma 4.6 of [Th3] that there is an element
$\xi\in\bigoplus_{\mathbb{Z}}\mathbb{Z}$ such that
$(z_{n})_{n\in\mathbb{Z}}=(\operatorname{id}-\sigma)(\xi)$. It follows
immediately from the conditions imposed on $G_{0}$ that $f=(1-e^{-\pi})g$ for
some $g\in G_{0}$. Then
$(\operatorname{id}-\alpha)(\xi,g)=((z_{n})_{n\in\mathbb{Z}},f)$ showing that
$\ker\Sigma_{0}\subseteq(\operatorname{id}-\alpha)(G)$. The reverse inclusion
is trivial. ∎
It follows from Lemma 3.7 that $\Sigma_{0}$ induces an isomorphism
$\Sigma:K_{0}(C)=G/(\operatorname{id}-\alpha)(G)\to\mathbb{Z}\ $
such that $\Sigma\circ q=\Sigma_{0}$. Every point $s\in S$ defines by
evaluation a state $\operatorname{ev}_{s}$ on $\mathcal{A}(S,\pi)$. Note that
$\operatorname{ev}_{s}\circ L\circ\alpha=e^{-\beta}\operatorname{ev}_{s}\circ
L$
where $\beta=\pi(s)$, and $L$ is as defined in Section 3.1.
###### Lemma 3.8.
Let $\phi\in S(G)$ be a state of $(G,G^{+},v)$ with the property that
$\phi\circ\alpha=s\phi$ for some $s>0$. Set $\beta=-\log s$. There is a unique
point $s\in\pi^{-1}(\beta)$ such that $\phi=\operatorname{ev}_{s}\circ L$.
###### Proof.
Extend $\phi$ to a $\mathbb{Q}$-linear state $\overline{\phi}$ of
$(\mathbb{Q}\otimes_{\mathbb{Z}}G,(\mathbb{Q}\otimes_{\mathbb{Z}}G)^{+},v)$.
If $(\xi,g)\in\mathbb{Q}\otimes_{\mathbb{Z}}G$ and $\overline{L}(\xi,g)=0$ we
have that $\frac{1}{n}v\pm(\xi,g)\in(\mathbb{Q}\otimes_{\mathbb{Z}}G)^{+}$ and
hence
$-\frac{1}{n}\leq\overline{\phi}(\xi,g)\leq\frac{1}{n}$
for all $n\in\mathbb{N}$, i.e. $\overline{\phi}(\xi,g)=0$. It follows that
there is a $\mathbb{Q}$-linear map
$\psi:\overline{L}(\mathbb{Q}\otimes_{\mathbb{Z}}G)\to\mathbb{R}$ such that
$\psi\circ\overline{L}=\overline{\phi}$. If
$f\in\overline{L}(\mathbb{Q}\otimes_{\mathbb{Z}}G)$, $n,m\in\mathbb{N}$ and
$\left|f(x)\right|<\frac{n}{m}$ for all $x\in S$, there is an element
$w\in\mathbb{Q}\otimes_{\mathbb{Z}}G$ such that $\overline{L}(w)=f$ and
$-nv<mw<nv$ in $\mathbb{Q}\otimes_{\mathbb{Z}}G$, implying that
$\left|\psi(f)\right|\leq\frac{n}{m}.$
It follows that $\left|\psi(f)\right|\leq\left\|f\right\|$. It is an easy
consequence of Property 3.1 that
$\overline{L}(\mathbb{Q}\otimes_{\mathbb{Z}}G)$ is dense in
$\mathcal{A}(S,\pi)$ and it follows therefore that $\psi$ extends to a linear
norm-contractive map $\psi:\mathcal{A}(S,\pi)\to\mathbb{R}$. Using the Hahn-
Banach theorem we can extend $\psi$ further to a norm-contractive linear map
on the space $C_{\mathbb{R}}(S)$ of continuous real-valued functions on $S$.
Since $\psi(1)=\phi(v)=1$ this extension is positive and it follows that there
is a Borel probability measure $m$ on $S$ such that $\psi$ is given by
integration with respect to $m$. Then
$\displaystyle\int_{S}e^{-\pi(x)}L(\xi,g)(x)\
\mathrm{d}m(x)=\int_{S}L(\sigma(\xi),e^{-\pi}g)(x)\ \mathrm{d}m(x)$
$\displaystyle=\phi\circ\alpha(\xi,g)=s\phi(\xi,g)=s\int_{S}L(\xi,g)(x)\
\mathrm{d}m(x)$
for all $(\xi,g)\in G$. Using that
$\overline{L}(\mathbb{Q}\otimes_{\mathbb{Z}}G)=\mathbb{Q}{L}(G)$ is dense in
$\mathcal{A}(S,\pi)$ it follows that
$\int_{S}e^{-\pi(x)}f(x)\ \mathrm{d}m(x)=s\int_{S}f(x)\ \mathrm{d}m(x)$
for all $f\in\mathcal{A}(S,\pi)$. Let $g\in C_{\mathbb{R}}(\pi(S))$. Then
$g\circ\pi\in\mathcal{A}(S,\pi)$ and hence
$\displaystyle\int_{\pi(S)}e^{-t}g(t)\
\mathrm{d}m\circ\pi^{-1}(t)=\int_{S}e^{-\pi(x)}g\circ\pi(x)\ \mathrm{d}m(x)$
$\displaystyle=s\int_{S}g\circ\pi(x)\ \mathrm{d}m(x)=s\int_{\pi(S)}g(t)\
\mathrm{d}m\circ\pi^{-1}(t).$
Since this holds for all $g\in C_{\mathbb{R}}(\pi(S))$ it follows that $-\log
s\in\pi(S)$ and that $m\circ\pi^{-1}$ is concentrated at $\beta=-\log s$,
i.e., $m$ is concentrated on $\pi^{-1}(\beta)$. In follows that $\psi$
factorises through ${\operatorname{Aff}}\pi^{-1}(\beta)$, i.e.,
$\psi=\tilde{\psi}\circ r$ where
$\tilde{\psi}:{\operatorname{Aff}}\pi^{-1}(\beta)\to\mathbb{R}$ is linear and
$r:\mathcal{A}(S,\pi)\to{\operatorname{Aff}}\pi^{-1}(\beta)$ is given by
restriction. When $a\in{\operatorname{Aff}}\pi^{-1}(\beta)$ there is
$\tilde{a}\in\mathcal{A}(S,\pi)$ such that $\tilde{a}|_{\pi^{-1}(\beta)}=a$ by
Lemma 2.2 of [BEK2]. If, in addition, $a\geq 0$ and $\epsilon>0$ it follows
from Lemma 2.3 of [BEK2] that we can choose $\tilde{a}$ such that
$-\epsilon<\tilde{a}$ in $\mathcal{A}(S,\pi)$. It follows in this way that
$\tilde{\psi}$ is a state on ${\operatorname{Aff}}\pi^{-1}(\beta)$. Since
every state of ${\operatorname{Aff}}\pi^{-1}(\beta)$ is given by evaluation at
a point in $\pi^{-1}(\beta)$ we obtain a point $s\in\pi^{-1}(\beta)$ such that
$\tilde{\psi}=\operatorname{ev}_{s}$. It follows that
$\psi=\operatorname{ev}_{s}$ and hence $\phi=\operatorname{ev}_{s}\circ L$.
The point $s$ is unique because $\mathcal{A}(S,\pi)$ separates the points of
$S$ and $L(G)$ linearly spans a dense set in $\mathcal{A}(S,\pi)$. ∎
Reformulating Lemma 3.8 in terms of $E$, we obtain the following statement.
###### Lemma 3.9.
There is a unique $\gamma$-invariant trace $\tau^{0}$ on $E\otimes\mathbb{K}$
with the property that $\tau^{0}(1\otimes e_{11})=1$. It is determined by the
condition that $\tau^{0}_{*}=\operatorname{ev}_{\overline{o}}\circ L$.
###### Lemma 3.10.
$\Sigma(K_{0}(C)^{+})=\mathbb{N}\cup\\{0\\}$.
###### Proof.
In the sequel we shall denote by $P$ the canonical conditional expectation
$P:C\to E\otimes\mathbb{K}$. Let $n\in\mathbb{N}$. Then $(n^{(0)},0)\in
G^{+}$, $q((n^{(0)},0))\in K_{0}(C)^{+}$ and $\Sigma(q((n^{(0)},0)))=n$. It
follows that $\mathbb{N}\subseteq\Sigma(K_{0}(C)^{+})$. Let $x\in
K_{0}(C)^{+}$ and write $x=q((z_{n})_{n\in\mathbb{Z}},g)$ for some
$((z_{n})_{n\in\mathbb{Z}},g)\in G$. Since $x\in K_{0}(C)^{+}$,
$\left(\tau^{0}\circ P\right)_{*}(x)\geq 0,$
where $\tau^{0}$ is the $\gamma$-invariant trace on $E\otimes\mathbb{K}$ from
Lemma 3.9. Since
$\displaystyle\left(\tau^{0}\circ
P\right)_{*}(x)=\tau^{0}_{*}((z_{n})_{n\in\mathbb{Z}},g))=\operatorname{ev}_{\overline{o}}\circ
L((z_{n})_{n\in\mathbb{Z}},g))=\sum_{n\in\mathbb{Z}}z_{n}\in\mathbb{Z},$
it follows that $\Sigma(x)\in\mathbb{N}$. ∎
###### Lemma 3.11.
$K_{1}(C)=0$.
###### Proof.
To establish this from the Pimsner-Voiculescu exact sequence, [PV], we must
show that $\operatorname{id}-\alpha$ is injective, which is easy: If
$\xi\in\bigoplus_{\mathbb{Z}}\mathbb{Z}$ and $\sigma(\xi)=\xi$ it follows
immediately that $\xi=0$, and if $g\in G_{0}$ and $e^{-\pi}g=g$ it follows
that $g=0$ because $g$ is supported away from $\overline{o}$. ∎
Let $p=1\otimes e_{11}\in E\otimes\mathbb{K}$.
###### Lemma 3.12.
$pCp$ is $*$-isomorphic to the Jiang-Su algebra $\mathcal{Z}$.
###### Proof.
By combining (A) of Lemma 3.6 with Proposition 4.7 of [CP] and Lemma 3.9 above
it follows that $pCp$ has exactly one trace state. By Lemma 3.10 the
isomorphism $\Sigma:K_{0}(C)\to\mathbb{Z}$ is an isomorphism of ordered
groups. Note that $\Sigma([p])=1$. Since
$(K_{0}(pCp),K_{0}(pCp)^{+})=(K_{0}(C),K_{0}(C)^{+})$ and
$K_{1}(pCp)=K_{1}(C)=0$ by Lemma 3.11 we see that $pCp$ has the same Elliott
invariant as $\mathcal{Z}$. By (B) of Lemma 3.6 $C$ is $\mathcal{Z}$-stable
and so by 3.1 of [TW], $pCp$ is $\mathcal{Z}$-stable. Thus, $pCp$ and
$\mathcal{Z}$ are both simple, unital, nuclear, $\mathcal{Z}$-stable, and
satisfy the UCT (see 23.1.1 and 22.3.5 of [Bl]), and have a unique tracial
state. By Corollary 6.2 of [MS2] and Corollary 4.6 of [Ro2] they are
$*$-isomorphic. See Remark 4.12 of [Th3]. ∎
### 3.2. Flows on the Jiang-Su algebra
We consider the dual action on
$C=(E\otimes\mathbb{K})\rtimes_{\gamma}\mathbb{Z}$ as a $2\pi$-periodic flow
and we denote by $\theta$ the restriction of this flow to $pCp$.
###### Lemma 3.13.
The KMS bundle $(S^{\theta},\pi^{\theta})$ of $\theta$ is isomorphic to
$(S,\pi)$.
###### Proof.
Let $(\omega,\beta)\in S^{\theta}$. By Remark 3.3 of [LN] $\omega$ extends
uniquely to a $\beta$-KMS weight $\widehat{\omega}$ on $C$, and by Lemma 3.1
of [Th3] the restriction $\widehat{\omega}|_{E\otimes\mathbb{K}}$ is a trace
on $E\otimes\mathbb{K}$ such that
$\widehat{\omega}|_{E\otimes\mathbb{K}}\circ\gamma=e^{-\beta}\widehat{\omega}|_{E\otimes\mathbb{K}}$.
Since $\widehat{\omega}(1\otimes e_{11})=1$ it follows from Lemma 3.8 that
$\left(\widehat{\omega}|_{E\otimes\mathbb{K}}\right)_{*}=\operatorname{ev}_{s}\circ
L\in S(G)$ for some $s\in\pi^{-1}(\beta)$. This gives us a map
$\xi:S^{\theta}\to S$ such that $\xi(\omega,\pi)=s$. Note that
$\pi\circ\xi=\pi^{\theta}$. To see that the map is surjective, let $s\in S$
and set $\beta=\pi(s)$. Then $\operatorname{ev}_{s}\circ L\in S(G)$ and
$\operatorname{ev}_{s}\circ L\circ\alpha=e^{-\beta}\operatorname{ev}_{s}\circ
L$ so Lemma 3.9 gives us a trace $\rho$ on $E\otimes\mathbb{K}$ such that
$\rho_{*}=\operatorname{ev}_{s}\circ L$ and $\rho\circ\gamma=e^{-\beta}\tau$.
By Lemma 3.1 of [Th3] the weight $\rho\circ P$ is a $\beta$-KMS weight for the
dual action on $C$. Since $\rho\circ P(1\otimes
e_{11})=\rho_{*}(v)=\operatorname{ev}_{s}\circ L(v)=1$, the restriction of
$\rho\circ P$ to $pCp$ is $\beta$-KMS state for $\theta$ and it is clear that
$\xi(\rho\circ P|_{pCp},\beta)=s$. To see that the map is injective, let
$(\omega^{i},\beta^{i}),i=1,2$, be elements of $S^{\theta}$ such that
$\xi(\omega^{1},\beta^{1})=\xi(\omega^{2},\xi^{2})=s$. Since
$s\in\pi^{-1}(\beta^{1})\cap\pi^{-1}(\beta^{2})$ it follows that
$\beta^{1}=\beta^{2}=\beta$. Since
$\left(\widehat{\omega^{1}}|_{E\otimes\mathbb{K}}\right)_{*}=\operatorname{ev}_{s}\circ
L=\left(\widehat{\omega^{2}}|_{E\otimes\mathbb{K}}\right)_{*}$ it follows by
Proposition 3.4 that
$\widehat{\omega^{1}}|_{E\otimes\mathbb{K}}=\widehat{\omega^{2}}|_{E\otimes\mathbb{K}}$.
By Lemma 3.1 of [Th3] this implies that
$\widehat{\omega^{1}}=\widehat{\omega^{2}}$ and hence that
$\omega^{1}=\omega^{2}$. It follows that $\xi$ is a bijection. By Proposition
3.4 above and Lemma 3.1 of [Th3] we have the following formula for the inverse
$\xi^{-1}$:
$\xi^{-1}(s)=\left(\left(\tau^{-1}(ev_{s}\circ
L)\otimes\operatorname{Tr}_{\mathbb{K}}\right)\circ P|_{pCp},\ \pi(s)\right).$
It follows that $\xi^{-1}$ is continuous and therefore a homeomorphism.
∎
Combining Lemma 3.12 and Lemma 3.13 we obtain our main result:
###### Theorem 3.14.
Let $(S,\pi)$ be a compact simplex bundle such that $\pi^{-1}(0)$ consists of
exactly one point. There is a $2\pi$-periodic flow $\theta$ on the Jiang-Su
algebra whose KMS bundle is isomorphic to $(S,{\pi})$.
From Theorem 3.14 we obtain the following corollary; see Corollary 3.2 of
[ET].
###### Corollary 3.15.
Let $K$ be a compact subset of real numbers containing $0$.
* •
There is a $2\pi$-periodic flow on the Jiang-Su algebra whose KMS spectrum is
$K$ and such that there is a unique $\beta$-KMS state for all $\beta\in K$.
* •
There is a $2\pi$-periodic flow $\theta$ on the Jiang-Su algebra whose KMS
spectrum is $K$ and such that $S^{\theta}_{\beta}$ is not affinely
homeomorphic to $S^{\theta}_{\beta^{\prime}}$ when $\beta,\beta^{\prime}\in
K\backslash\\{0\\}$ and $\beta\neq\beta^{\prime}$.
## 4\. Flows that are not approximately inner
In this section we shall use Theorem 3.14 to show that flows that are not
approximately inner exist in abundance.
###### Theorem 4.1.
Let $A$ be a unital separable $C^{*}$-algebra with a unique trace state.
Suppose that $A$ absorbs the Jiang-Su algebra tensorially. Then for any
compact simplex bundle $(S,\pi)$ with $\pi^{-1}(0)$ consisting of exactly one
point, there exists a $2\pi$-periodic flow on $A$ whose KMS bundle is
isomorphic to $(S,\pi)$.
###### Proof.
Let $\mathcal{Z}$ be the Jiang-Su algebra. By Theorem 3.14 there is a flow
$\theta$ on $\mathcal{Z}$ whose KMS bundle $(S^{\theta},\pi^{\theta})$ is
isomorphic to $(S,\pi)$. Set
$\theta^{\prime}_{t}=\operatorname{id}_{A}\otimes\theta_{t}$
to get a flow $\theta^{\prime}$ on $A\otimes\mathcal{Z}$. Let $\tau$ be the
trace state of $A$. Then
$\omega\mapsto\tau\otimes\omega$ (4.1)
defines a continuous map $S^{\theta}\to S^{\theta^{\prime}}$ of KMS bundles
which is clearly injective. To show that it is surjective, let
$\omega^{\prime}$ be a $\beta$-KMS state for $\theta^{\prime}$. Let $a_{i}\in
A,i=1,2,\ b\in\mathcal{Z}$. Since $a_{1}\otimes 1$ is fixed by
$\theta^{\prime}$ we find that
$\omega^{\prime}(a_{1}a_{2}\otimes b)=\omega^{\prime}((a_{1}\otimes
1)(a_{2}\otimes b))=\omega^{\prime}((a_{2}\otimes b)(a_{1}\otimes
1))=\omega^{\prime}(a_{2}a_{1}\otimes b).$
Thus, $a\mapsto\omega^{\prime}(a\otimes b)$ is a bounded trace functional on
$A$ and hence $\omega^{\prime}(a\otimes b)=\tau(a)\omega(b)$ for all $a\in A$
and some $\omega(b)\in\mathbb{C}$. It is straightforward to see that
$b\mapsto\omega(b)$ is a state on $\mathcal{Z}$ and in fact a $\beta$-KMS
state for $\theta$ since $\omega^{\prime}$ is a $\beta$-KMS state for
$\theta^{\prime}$. This shows that the map (4.1) is surjective and hence a
homeomorphism of KMS bundles. By moving the flow $\theta^{\prime}$ onto $A$
via a $*$-isomorphism $A\simeq A\otimes\mathcal{Z}$, we obtain the desired
flow on $A$.
∎
For the statement of the following corollary we say that two flows $\alpha$
and $\mu$ on the same unital $C^{*}$-algebras $A$ are _weakly cocycle-
conjugate_ when there are an automorphism $\gamma\in\operatorname{Aut}A$, a
non-zero real number $\lambda\in\mathbb{R}\backslash\\{0\\}$ and continuous
path $\\{u_{t}\\}_{t\in\mathbb{R}}$ of unitaries in $A$ such that
* $u_{s}\alpha_{\lambda s}(u_{t})=u_{s+t}$ and
* $\gamma\circ\mu_{t}\circ\gamma^{-1}=\operatorname{Ad}u_{t}\circ\alpha_{\lambda t}$
for all $s,t\in\mathbb{R}$. Weak cocycle-conjugacy is an equivalence relation
on flows. Recall from [PS] that a flow $\alpha$ is _approximately inner_ when
there is a sequence $\\{h_{n}\\}$ of self-adjoint elements of $A$ such that
$\lim_{n\to\infty}\left\|\alpha_{t}(a)-\operatorname{Ad}e^{ith_{n}}(a)\right\|=0$
uniformly on compact subsets of $\mathbb{R}$ for all $a\in A$. It follows from
Corollary 1.4 of [Ki2] that a flow $\alpha$ is approximately inner if and only
if all flows weakly cocycle-conjugate to $\alpha$ are approximately inner.
###### Corollary 4.2.
Let $A$ be a unital separable $C^{*}$-algebra with a unique trace state.
Suppose that $A$ absorbs the Jiang-Su algebra tensorially. There are
uncountably many weak cocycle-conjugacy classes of flows on $A$ that are not
approximately inner.
###### Proof.
Note that weak cocycle-conjugacy, as defined in the preceding paragraph, is
the combination of cocycle-conjugacy as defined for example in [Ki2] and
scaling, where $\alpha_{t}$ is replaced by $\alpha_{\lambda t}$. It follows
from Proposition 2.1 of [Ki2] and Section 4.1 of [Th2] that cocycle-conjugate
flows $\alpha$ and $\mu$ have almost the same KMS bundles. More precisely, for
each $\beta\in\mathbb{R}$ the simplices $S^{\alpha}_{\beta}$ and
$S^{\mu}_{\beta}$ of $\beta$-KMS states for $\alpha$ and $\mu$, respectively,
are strongly affinely isomorphic in the sense of [Th2] when $\alpha$ and $\mu$
are cocycle-conjugate. It follows that if $\alpha$ and $\mu$ are weakly
cocycle-conjugate, there is a $\lambda\in\mathbb{R}\backslash\\{0\\}$ such
that $S^{\alpha}_{\beta}$ is strongly affinely homeomorphic to
$S^{\mu}_{\lambda\beta}$ for all $\beta\in\mathbb{R}$. Recall that by Theorem
3.2 of [PS] an approximately inner flow on $A$ has KMS spectrum equal to
$\mathbb{R}$, so to get uncountably many flows that are not mutually weakly
cocycle-conjugate and also not approximately inner, there are many ways to go;
for example the following. Let $X$ be a compact metric space. By Theorem 4.1
there is a flow $\theta^{X}$ on $A$ such that
$S^{\theta^{X}}_{\beta}=\emptyset$ when $\beta\notin\\{0,1\\}$ while
$S^{\theta^{X}}_{1}$ is affinely homeomorphic to the simplex $M(X)$ of Borel
probability measures on $X$. Since $M(X)$ is not strongly affinely isomorphic
to $M(Y)$ when $X$ is not homeomorphic to $Y$, the flow $\theta^{X}$ is not
weakly cocycle-conjugate to $\theta^{Y}$ when $X$ is not homeomorphic to $Y$.
Hence the cardinality of weak cocycle-conjugacy classes of flows on $A$ that
are not approximately inner exceeds the cardinality of homeomorphism classes
of compact metric spaces.∎
It is known that many simple monotracial $C^{*}$-algebras absorb the Jiang-Su
algebra tensorially and so are covered by Corollary 4.2. In particular, this
is the case when $A$ in addition is nuclear and has the strict comparison
property; cf. [MS1] and [MS2].
## 5\. KMS bundles for flows on a unital infinite $C^{*}$-algebra
In this section we prove the following statement which is a complement to the
main result of [ET].
###### Theorem 5.1.
Let $A$ be a separable, simple, unital, purely infinite and nuclear
$C^{*}$-algebra in the UCT class. Assume that $K_{1}(A)$ is torsion free. Let
$(S,\pi)$ be a proper simplex bundle such that $0\notin\pi(S)$. There is a
$2\pi$-periodic flow $\theta$ on $A$ whose bundle of KMS states is isomorphic
to $(S,\pi)$.
Since a flow on a unital $C^{*}$-algebra without trace states cannot have
$0$-KMS states and since in any case the KMS states form a proper simplex
bundle (see Section 2 above and [ET]), we have the following corollary.
###### Corollary 5.2.
Let $A$ be as in Theorem 5.1 and let $D$ be a separable unital $C^{*}$-algebra
without trace states and $\theta^{\prime}$ a flow on $D$. There is a
$2\pi$-periodic flow $\theta$ on $A$ such that
$(S^{\theta^{\prime}},\pi^{\theta^{\prime}})$ is isomorphic to
$(S^{\theta},\pi^{\theta})$.
### 5.1. The proof of Theorem 5.1
Let $(S,\pi)$ be a proper simplex bundle such that $0\notin\pi(X)$. If
$S=\emptyset$ we can take the $\theta$ to be the trivial flow; so we assume
that $S\neq\emptyset$. Choose a countable subgroup $\mathcal{C}$ of
$\mathcal{A}(S,\pi)$ with the following properties.
* (1)
The support of each element of $\mathcal{C}$ is compact.
* (2)
For every $N\in\mathbb{N}$, every $\epsilon>0$ and every
$f\in\mathcal{A}(S,\pi)$ such that $\operatorname{supp}f\in\pi^{-1}([-N,N])$,
there is an element $g\in\mathcal{C}$ such that
$\operatorname{supp}g\subseteq\pi^{-1}([-N,N])$ and
$\sup_{x\in S}\left|f(x)-g(x)\right|<\epsilon.$
* (3)
The function
$x\mapsto e^{n\pi(x)}{(1-e^{-\pi(x)})^{m}}f(x)$
is in $\mathcal{C}$ for all $f\in\mathcal{C}$ and all $n,m\in\mathbb{Z}$.
Let $1_{+}$ and $1_{-}$ denote the characteristic functions of $[0,\infty)$
and $(-\infty,0]$, respectively. Since $0\notin\pi(X)$ the functions
$1_{\pm}\circ\pi$ are both continuous on $S$. Let
$G_{0}\subseteq\mathcal{A}(S,\pi)$ denote the subgroup generated by
$\mathcal{C}$ and the functions
$x\mapsto(1_{\pm}\circ\pi)e^{n\pi(x)}{(1-e^{-\pi(x)})^{m}},\
n,m\in\mathbb{Z}.$
Set $G=\mathbb{Q}G_{0}$ and
$G^{+}=\left\\{f\in G:\ f(x)>0,\ x\in S\right\\}\cup\\{0\\}.$
###### Lemma 5.3.
The pair $(G,G^{+})$ has the following properties.
* (1)
$G^{+}\cap(-G^{+})=\\{0\\}$.
* (2)
$G=G^{+}-G^{+}$.
* (3)
$(G,G^{+})$ is unperforated, i.e., $n\in\mathbb{N}\backslash\\{0\\},\ g\in G,\
ng\in G^{+}\Rightarrow g\in G^{+}$.
* (4)
$(G,G^{+})$ has the strong Riesz interpolation property, i.e. if
$f_{i},g_{j}$, $i,j\in\\{1,2\\}$, are elements of $G$ and $f_{i}<g_{j}$ in $G$
for all $i,j\in\\{1,2\\}$, then there is an element $h\in G$ such that
$f_{i}<h<g_{j}$
for all $i,j\in\\{1,2\\}$.
###### Proof.
The first three items are easy to establish. To prove (4) let
$f_{1},f_{2},g_{1},g_{2}\in G$ such that $f_{i}<g_{j}$ in $G$ for all
$i,j\in\\{1,2\\}$. By definition of $G$ there is $N\in\mathbb{N}$ so large
that there are polynomials $p_{1},p_{2},q_{1},q_{2}$ with rational
coefficients such that
$\left(e^{\pi(x)}(e^{\pi(x)}-1)\right)^{N}f_{i}(x)=p_{i}(e^{\pi(x)})$
and
$\left(e^{\pi(x)}(e^{\pi(x)}-1)\right)^{N}g_{j}(x)=q_{j}(e^{\pi(x)})$
for all $i,j$ and all large $x$. It follows then from Lemma 4.7 of [ET] that
there is a polynomial $p_{+}$ with rational coefficients such that the
function
$h_{+}(x)=1_{+}\circ\pi(x)\left(e^{\pi(x)}(e^{\pi(x)}-1)\right)^{-N}p_{+}(e^{\pi(x)})$
is an element of $G$ with the property that there is $K_{+}>0$ such that
$h_{+}(x)=0$ when $\pi(x)\leq 0$ and
$f_{i}(x)<h_{+}(x)<g_{j}(x)$
for all $i,j$ and all $x\in S$ with $\pi(x)\geq K_{+}$. In the same way we get
also an element $h_{-}\in G$ and a $K_{-}>0$ such that $h_{-}(x)=0$ when
$\pi(x)\geq 0$ and
$f_{i}(x)<h_{-}(x)<g_{j}(x)$
for all $i,j$ and all $x\in S$ with $\pi(x)\leq-K_{-}$. Set
$K=\max\\{K_{+},K_{-}\\}$. It follows from (2) in Lemma 4.4 of [ET] that there
is $H\in\mathcal{A}(S,\pi)$ such that $H(x)=h^{-}(x)$ when $\pi(x)\leq-K$,
$H(x)=h^{+}(x)$ when $\pi(x)\geq K$ and $f_{i}(x)<H(x)<g_{j}(x)$ for all $x\in
S$ and all $i,j$. Let $\delta>0$ be smaller than $H(x)-f_{i}(x)$ and
$g_{j}(x)-H(x)$ for all $i,j$ and all $x\in\pi^{-1}([-K,K])$. Since
$H-h_{+}-h_{-}$ is supported in $[-K,K]$ it follows from the definition of
$\mathcal{C}$ that there is $g\in\mathcal{C}$ such that
$\operatorname{supp}g\subseteq\pi^{-1}([-K,K])$ and
$\left|g(x)-(H(x)-h_{+}(x)-h_{-}(x))\right|<\delta$ for all $s\in S$. Then
$h=g+h_{+}+h_{-}$
is an element of $G$ with the desired property.
∎
In short, Lemma 5.3 says that $(G,G^{+})$ is a dimension group with the strong
Riesz interpolation property. We define an automorphism $\alpha$ of
$(G,G^{+})$ such that
$\alpha(g)=e^{-\pi}g.$
Let $A$ be the algebra from Theorem 5.1. By Proposition 3.5 of [Ro1] there is
a countable abelian torsion free group $H$ and an automorphism $\kappa$ of $H$
such that $K_{1}(A)\simeq\ker(\operatorname{id}-\kappa)$ and
$\operatorname{coker}(\operatorname{id}-\kappa)\simeq K_{0}(A)$. We note that
we may assume that $H\neq\\{0\\}$; if not we exchange $H$ with $\mathbb{Q}$
and set $\kappa(x)=2x$. Set
$G_{\sharp}=H\oplus{G}\ .$
Let $p:G_{\sharp}\to G$ be the canonical projection and set
$G_{\sharp}^{+}=\left\\{x\in G_{\sharp}:\ p(x)\in
G^{+}\backslash\\{0\\}\right\\}\cup\\{0\\}.$
It follows from Lemma 3.2 of [EHS] and Lemma 5.3 above that
$(G_{\sharp},G^{+}_{\sharp})$ is dimension group.
Define $\alpha_{\sharp}\in\operatorname{Aut}G_{\sharp}$ by
$\alpha_{\sharp}=\kappa\oplus\alpha.$
It follows from [EHS] and [E1] that there is a stable AF algebra $B$ such that
$(K_{0}(B),K_{0}(B)^{+})=(G_{\sharp},G^{+}_{\sharp})$ and an automorphism
$\gamma\in\operatorname{Aut}B$ such that $\gamma_{*}=\alpha_{\sharp}$.
###### Lemma 5.4.
$(G_{\sharp},G_{\sharp}^{+})$ has large denominators; that is, for all $g\in
G_{\sharp}^{+}$ and $m\in\mathbb{N}$ there is an element $h\in G_{\sharp}^{+}$
and an $n\in\mathbb{N}$ such that $mh\leq g\leq nh$ in $G_{\sharp}$.
###### Proof.
It suffices to show that $(G,G^{+})$ has large denominators and this is
automatic because $\mathbb{Q}G=G$. ∎
It follows now from Lemma 3.4 of [Th3] that we may arrange for $\gamma$ to
have the following additional properties.
* (A)
The restriction map $\mu\ \mapsto\ \mu|_{B}$ is a bijection from traces $\mu$
on $B\rtimes_{\gamma}\mathbb{Z}$ onto the $\gamma$-invariant traces on $B$,
and
* (B)
$B\rtimes_{\gamma}\mathbb{Z}$ is $\mathcal{Z}$-stable; that is
$(B\rtimes_{\gamma}\mathbb{Z})\otimes\mathcal{Z}\simeq
B\rtimes_{\gamma}\mathbb{Z}$ where $\mathcal{Z}$ denotes the Jiang-Su algebra,
[JS].
###### Lemma 5.5.
The only order ideals $I$ in $G_{\sharp}$ such that $\alpha_{\sharp}(I)=I$ are
$I=\\{0\\}$ and $I=G_{\sharp}$.
###### Proof.
Recall that an order ideal $I$ in $G_{\sharp}$ is a subgroup such that
* (a)
$I=I\cap G_{\sharp}^{+}-I\cap G_{\sharp}^{+}$, and
* (b)
when $0\leq y\leq x$ in $G_{\sharp}$ and $x\in I$, then $y\in I$.
Let $I$ be a non-zero order ideal such that $\alpha_{\sharp}(I)=I$. Then
$p(I)$ is an order ideal in $G$ such that $\alpha(p(I))=p(I)$. Since $p(I)\cap
G^{+}\neq\\{0\\}$ there is an element $g\in p(I)\cap G^{+}\backslash\\{0\\}$.
By definition of $G$ there are natural numbers $n,m,K\in\mathbb{N}$ such that
the function
$g^{\prime}(x)=1_{-}\circ\pi e^{n\pi}\ +\ 1_{+}\circ\pi e^{-m\pi}$
has the property that
$0<g^{\prime}(x)<Kg(x),\ \ x\in S.$
It follows that $g^{\prime}\in p(I)$. Since $p(I)$ is $\alpha$-invariant it
follows that
$\alpha^{l}(g^{\prime})=1_{-}\circ\pi e^{(n-l)\pi}\ +\ 1_{+}\circ\pi
e^{-(m+l)\pi}\in p(I)$
for all $l\in\mathbb{Z}$. For an arbitrary element $g\in G^{+}$ we can find
$l_{1},l_{2}\in\mathbb{Z}$ and $M\in\mathbb{N}$ such that
$g(x)<M(\alpha^{l_{1}}(g^{\prime})(x)\ +\ \alpha^{l_{2}}(g^{\prime})(x))$
for all $x\in S$. This implies that $G^{+}\subseteq p(I)$ and hence that
$G=p(I)$. To conclude that $I=G_{\sharp}$ it only remains to show that
$H\oplus 0\subseteq G_{\sharp}$. For this note that there is an element
$h^{\prime}\in H$ such that $(h^{\prime},1)\in I$. Note that, for any $h\in
H$, $(h,0)=(h+h^{\prime},1)-(h^{\prime},1)$ and
$0<(h+h^{\prime},1)<2(h^{\prime},1)$ in $G_{\sharp}$. It follows that
$(h+h^{\prime},1)\in I$ and hence that $(h,0)\in I$. ∎
It follows from Lemma 5.5 and [E1] that there are no non-trivial
$\gamma$-invariant ideals in $B$. Since $\gamma_{*}^{k}=\alpha^{k}$ is non-
trivial for all $k\neq 0$, no non-trivial power of $\gamma$ is inner and it
follows therefore from [E2] that
$C=B\rtimes_{\gamma}\mathbb{Z}$
is simple. (See also [Ki1].)
Let $q:H\to H/(\operatorname{id}-\kappa)(H)=K_{0}(A)$ be the quotient map and
choose $w\in H$ such that $q(w)=[1]$. Let $v=(w,1)\in G_{\sharp}^{+}$.
###### Lemma 5.6.
Let $\phi:G_{\sharp}\to\mathbb{R}$ be a positive homomorphism and $s>0$ a
positive numbers such that $\phi(v)=1$ and $\phi\circ\alpha_{\sharp}=s\phi$.
Let $\beta=-\log s$. There is an element $\omega\in\pi^{-1}(\beta)$ such that
$\phi(h,g)=g(\omega)$ for all $(h,g)\in G_{\sharp}$.
###### Proof.
For $h\in H$, we find that $\pm n(h,0)+v\geq 0$ in $G_{\sharp}$ and hence $\pm
n\phi(h,0)+1\geq 0$ for all $n\in\mathbb{N}$, implying that $\phi(h,0)=0$. It
follows that $\phi$ factorises through $p$, i.e., there is a positive
homomorphism $\psi:G\to\mathbb{R}$ such that $\psi\circ p=\phi$. If $f\in G$,
$n,m\in\mathbb{N}$ and $|f(x)|<\frac{n}{m}$ for all $x\in S$, it follows that
$-n<mf<n$ in $G$. Since $\psi(1)=1$ this implies that
$\left|\psi(f)\right|\leq\frac{n}{m}$. It follows that $\psi$ is a
$\mathbb{Q}$-linear contraction. Let $\mathcal{A}_{\mathbb{R}}(S,\pi)$ be the
subspace of $\mathcal{A}(S,\pi)$ consisting of the elements of
$\mathcal{A}(S,\pi)$ that have a limit at infinity and denote by
$\mathcal{A}_{0}(S,\pi)$ the subset of $\mathcal{A}_{\mathbb{R}}(S,\pi)$
consisting of the elements of $\mathcal{A}_{\mathbb{R}}(S,\pi)$ that vanish at
infinity. Every element of $\mathcal{A}_{\mathbb{R}}(S,\pi)$ can approximated
in norm by elements of the form
$q1_{-}\circ\pi+q1_{+}\circ\pi+f$
where $q\in\mathbb{Q}$ and $f\in\mathcal{A}_{0}(S,\pi)$. It follows therefore
from the second condition on $\mathcal{C}$ that every element of
$\mathcal{A}_{\mathbb{R}}(S,\pi)$ can be approximated in norm by elements of
$G$, implying that $\psi$ extends by continuity to a linear contraction
$\psi:\mathcal{A}_{\mathbb{R}}(S,\pi)\to\mathbb{R}$. By the Hahn-Banach
theorem there is a contractive extension of $\psi$ to all continuous real-
valued functions on $S$ with a limit at infinity. Since $\psi(1)=1$ this
extension is positive and it follows therefore that there is a bounded Borel
measure $m$ on $S$ such that
$\psi(f)=\int_{S}f(x)\ \mathrm{d}m(x)$
for all $f\in\mathcal{A}_{0}(S,\pi)$. From here the argument from the last
part of the proof of Lemma 4.12 of [ET] can be repeated to obtain an element
$\omega\in\pi^{-1}(\beta)$ such that $\psi(f)=f(\omega)$ for all $f\in G$,
implying that $\phi(h,g)=g(\omega)$ for all $(h,g)\in G_{\sharp}$. ∎
Let $e\in B$ be a projection such that $[e]=v$ in $K_{0}(B)=G_{\sharp}$.
###### Lemma 5.7.
$eCe$ is $*$-isomorphic to $A$.
###### Proof.
It follows from the Pimsner-Voiculescu exact sequence, [PV], that $K_{0}(C)$
can be identified with the cokernel of $\operatorname{id}-\alpha_{\sharp}$ in
such a way that the map $\iota_{*}:K_{0}(B)\to K_{0}(C)$ induced by the
inclusion $\iota:B\to C$ becomes the quotient map $q:G_{\sharp}\to
G_{\sharp}/(\operatorname{id}-\alpha_{\sharp})(G_{\sharp})$. It follows from
the definition of $G$ that $\operatorname{id}-\alpha$ is an automorphism of
$G$ and hence
$G_{\sharp}/(\operatorname{id}-\alpha_{\sharp})(G_{\sharp})=H/(\operatorname{id}-\kappa)(H)=K_{0}(A).$
The resulting isomorphism $K_{0}(eCe)\to K_{0}(A)$ takes $[e]$ to $[1]$.
Similarly, the Pimsner-Voiculescu exact sequence shows that
$K_{1}(C)=\ker(\operatorname{id}-\alpha_{\sharp})=\ker(\operatorname{id}-\kappa)=K_{1}(A).$
We conclude that $K_{1}(eCe)\simeq K_{1}(A)$. The proof is then completed by
using the Kirchberg-Phillips result; see (iv) of Theorem 8.4.1 of [Ro3]. For
this we need to verify that both algebras are separable, simple, unital,
purely infinite, nuclear and in the UCT class. This is assumed for $A$ and
regarding $eCe$ most of these properties are well-known. (For the UCT see
23.1.1 and 22.3.5 (g) of [Bl].) To see that $eCe$ is purely infinite we note
that $eCe$ is $\mathcal{Z}$-stable thanks to [TW] and (B) above. Furthermore,
it follows from (A) above, in combination with Lemma 3.5 of [Th3] and Lemma
5.6 above, that there are no traces on $eCe$. By Corollary 5.1 of [Ro2] this
implies that $eCe$ is purely infinite. ∎
We consider the dual action on $C=B\rtimes_{\gamma}\mathbb{Z}$ as a
$2\pi$-periodic flow and we denote by $\theta$ the restriction of this flow to
$eCe$.
###### Lemma 5.8.
The KMS bundle $(S^{\theta},\pi^{\theta})$ of $\theta$ is isomorphic to
$(S,\pi)$.
###### Proof.
Let $(\omega,\beta)\in S^{\theta}$. By Remark 3.3 in [LN] there is a
$\beta$-KMS weight $\hat{\omega}$ for the dual action on $C$ which extends
$\omega$. By Corollary 4.2 of [ET] the map
$\omega\mapsto\left(\hat{\omega}|_{B}\right)_{*}$
is an affine homeomorphism from the simplex of $\beta$-KMS states for $\theta$
onto the set of elements $\phi$ from
$\operatorname{Hom}_{+}(G_{\sharp},\mathbb{R})$ such that
$\phi\circ\alpha=e^{-\beta}\phi$ and $\phi(v)=1$. By Lemma 5.6 there is a
point $\omega^{\prime}$ in $\pi^{-1}(\beta)$ such that
$\left(\hat{\omega}|_{B}\right)_{*}(h,g)=g(\omega^{\prime})$ for all $(h,g)\in
G_{\sharp}$. Define $\varphi:S^{\theta}\to S$ by
$\varphi(\omega,\beta)=\omega^{\prime}$. Note that
$\pi\circ\varphi=\pi^{\theta}$. To see that $\varphi$ is surjective, let
$\mu\in S$. Define $\operatorname{ev}_{\mu}:G_{\sharp}\to\mathbb{R}$ by
$\operatorname{ev}_{\mu}(h,g)=g(\mu)$. Then
$\operatorname{ev}_{\mu}\in\operatorname{Hom}_{+}(G_{\sharp},\mathbb{R})$,
$\operatorname{ev}_{\mu}\circ\alpha=e^{-\pi(\mu)}\operatorname{ev}_{\mu}$, and
$\operatorname{ev}_{\mu}(v)=1$. Using Corollary 4.2 of [ET] we get a
$\pi(\mu)$-KMS state $\omega$ for $\theta$ such that
$\left(\hat{\omega}|_{B}\right)_{*}=\operatorname{ev}_{\mu}$. Then
$\varphi(\omega,\pi(\mu))=\mu$. To see that $\varphi$ is injective, consider
$(\omega_{i},\beta_{i})\in S^{\theta}$ such that
$\varphi(\omega_{1},\beta_{1})=\varphi(\omega_{2},\beta_{2})$. Then
$\beta_{1}=\pi((\varphi(\omega_{1},\beta_{1}))=\varphi((\varphi(\omega_{2},\beta_{2}))=\beta_{2}$
and
$\left(\widehat{\omega_{1}}|_{B}\right)_{*}=\left(\widehat{\omega_{2}}|_{B}\right)_{*}.$
Since $B$ is AF it follows that
$\widehat{\omega_{1}}|_{B}=\widehat{\omega_{2}}|_{B}$; see Lemma 3.5 of [Th3].
By Lemma 3.1 of [Th3] it follows that
$\widehat{\omega_{1}}=\widehat{\omega_{2}}$ and hence
$\omega_{1}=\widehat{\omega_{1}}|_{eCe}=\widehat{\omega_{2}}|_{eCe}=\omega_{2}$.
It remains to show that $\varphi$ is a homeomorphism. But since $\varphi$ is a
bijection and $\pi\circ\varphi=\pi^{\theta}$ it suffices to show that
$\varphi^{-1}$ is continuous. Let therefore $\\{\mu^{n}\\}$ be a sequence in
$S$ such that $\lim_{n\to\infty}\mu^{n}=\mu$ in $S$. Set
$\beta_{n}=\pi(\mu^{n})$ and note that $\lim_{n\to\infty}\beta_{n}=\beta$,
where $\beta=\pi(\mu)$. It follows that
$\lim_{n\to\infty}\operatorname{ev}_{\mu^{n}}(x)=\operatorname{ev}_{\mu}(x)$
for all $x\in G_{\sharp}$. Let $\tau^{n}$ and $\tau$ be the traces on $B$
determined by the conditions that ${\tau^{n}}_{*}=\operatorname{ev}_{\mu^{n}}$
and $\tau_{*}=\operatorname{ev}_{\mu}$. Then
$\varphi^{-1}(\mu^{n})=(\tau^{n}\circ P|_{eCe},\beta_{n})$ and
$\varphi^{-1}(\omega)=(\tau\circ P|_{eCe},\beta)$. It suffices therefore to
show that
$\lim_{n\to\infty}\tau^{n}\circ P(exe)=\tau\circ P(exe)$ (5.1)
for all $x\in C$. Since $\tau^{n}\circ P(e)=\tau\circ P(e)=1$, the positive
functionals on $C$ given by $x\mapsto\tau^{n}\circ P(exe)$ and
$x\mapsto\tau\circ P(exe)$ are all of norm $\leq 1$. To establish (5.1) for
all $x\in C$ it suffices therefore to check the equality for $x$ in a dense
subset of $C$. If $u$ is the canonical unitary in the multiplier algebra of
$C$ coming from the construction of $C$ as a crossed product, it suffices to
show that $\lim_{n\to\infty}\tau^{n}\circ P(ebu^{k}e)=\tau\circ P(ebu^{k}e)$
for all $k\in\mathbb{Z}$ and all $b\in B$. Since $P(ebu^{k}e)=0$ when $k\neq
0$ it suffices to consider the case $k=0$; that is, it suffices to show that
$\lim_{n\to\infty}\tau^{n}(ebe)=\tau(ebe)$. On approximating $ebe$ by a sum of
projections from $eBe$ it suffices to show that
$\lim_{n\to\infty}\tau^{n}(p)=\tau(p)$ when $p$ is a projection in $eBe$. This
holds because
$\lim_{n\to\infty}\tau^{n}(p)=\lim_{n\to\infty}\operatorname{ev}_{\mu^{n}}([p])=\operatorname{ev}_{\mu}([p])=\tau(p).$
∎
Theorem 5.1 follows from Lemma 5.8 and Lemma 5.7.
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# Mass and decay width of $T_{ccs}$ from symmetries
Mitsuru Tanaka<EMAIL_ADDRESS>Department of Physics, Nagoya
University, Nagoya 464-8602, Japan Yasuhiro Yamaguchi
<EMAIL_ADDRESS>Department of Physics, Nagoya University,
Nagoya 464-8602, Japan Kobayashi-Maskawa Institute for the Origin of
Particles and the Universe, Nagoya University, Nagoya, 464-8602, Japan Meson
Science Laboratory, Cluster for Pioneering Research, RIKEN, Hirosawa, Wako,
Saitama 351-0198, Japan Masayasu Harada<EMAIL_ADDRESS>Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya, 464-8602, Japan Department of Physics, Nagoya
University, Nagoya 464-8602, Japan Advanced Science Research Center, Japan
Atomic Energy Agency, Tokai 319-1195, Japan
###### Abstract
We analyze the mass and width of the doubly heavy tetraquark $T_{ccs}$
composed of a heavy diquark and a light quark cloud with strangeness with
assuming that a color anti-triplet heavy diquark is a dominant component of
the doubly charmed tetraquarks $T_{cc}$ and $T_{ccs}$. We construct an
effective Lagrangian for masses of heavy hadrons based on the superflavor
symmetry between the doubly heavy tetraquarks and the singly heavy baryons
with including the terms which simultaneously break the heavy-quark and light
flavor symmetries, and predict the mass of $T_{ccs}$ as $M(T_{ccs})=4057\pm
40$ MeV. The comparison of this prediction with future experimental
observation will give a clue to understand the color structure of the heavy
diquark. We also predict the spin-averaged mass of $\Omega_{cc}$
($J^{P}=1/2^{+},3/2^{+})$ as $M(\Omega_{cc})=3760\pm 18\,$MeV. We next
calculate the decay width of $T_{ccs}$, based on solely the light flavor
symmetry, as $\Gamma(T_{ccs})=1.2\pm 0.3$ MeV.
Keywords
superflavor symmetry, heavy quark symmetry, flavor symmetry, diquark, doubly
heavy tetraquark
## I INTRODUCTION
The discovery of $X(3872)$ [1] marked the beginning of numerous exotic hadron
discoveries in the heavy quark sectors, yet their structure remains poorly
understood. Hadrons with exotic structures beyond ordinary baryons ($qqq$) and
mesons ($q\bar{q}$) were already indicated by Gell-Mann and Zweig in the 1960s
[2, 3, 4, 5, 6]. Possible structures of the multiquark state being a color
singlet have been discussed in many literatures (See, for a reviews, e.g.
Refs. [7, 8]). The compact multiquark has been investigated as a color singlet
state of few-body multiquark systems by the constituent quark model etc (See
e.g. Refs. [9, 10, 11, 12, 13, 14, 15, 16]). The emergence of the hadronic
molecules as a deuteron-like state, discussed as a deuson in Ref. [17], is
expected near the thresholds. In fact many candidates of a hadronic molecule
have been reported in the experimental studies as the $XYZ$ tetraquarks being
a meson-meson state and the $P_{c}$ pentaquarks being a meson-baryon one.
Investigating the exotic structures would lead to an understanding of the QCD
phenomena such as color confinement.
The doubly charmed tetraquark $T_{cc}^{+}$ was reported in the LHCb experiment
in 2021 [18, 19]. The reported state is consistent with a genuine exotic
hadron having a flavor structure $cc\bar{u}\bar{d}$. The spin and parity of
$T_{cc}^{+}$ are determined to be $J^{P}=1^{+}$, and the LHCb considers
$T_{cc}^{+}$ as an isoscalar. The mass of $T_{cc}^{+}$ is
$3874.817\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ close to the
$D^{0}D^{\ast+}$ threshold. The decay to $D^{0}D^{0}\pi^{+}$ has been
confirmed, with a decay width of $48\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
[18] or $410\,$keV [19]. Furthermore, the LHCb analysis supports that
$T_{cc}^{+}$ decays to $D^{0}D^{0}\pi^{+}$ via the intermediate state
$D^{*+}$.
Since the discovery of $T_{cc}^{+}$, research on the doubly heavy tetraquarks
(DHTs) has been actively conducted [7]. However at present, no clear answer
has been obtained regarding the structure of DHTs. For example, analyses based
on the hadronic molecular model, which assumes that $T_{cc}^{+}$ is a loosely
bound state with $D$ and $D^{*}$, have been conducted [20, 21]. This is due to
the fact that the mass of $T_{cc}^{+}$ resides in the vicinity of the
$DD^{\ast}$ threshold. On the other hand, a compact tetraquark structure of
DHT is considered [22], based on the diquark picture proposed by Jaffe [23].
Symmetries such as flavor symmetry and chiral symmetry play a crucial role in
the classification of hadronic states. For exotic hadrons including heavy
quarks, symmetries that emerge in the heavy quark limit, such as heavy quark
symmetry (HQS) (see e.g. Ref. [24]) and superflavor symmetry (SFS) [25, 26,
27], are considered potentially useful for understanding the structures of
exotics. The superflavor symmetry emerges in an exchange of an anti-heavy
quark and a heavy diquark with the same color configuration $\bar{\boldmath
3}$ in the heavy quark limit. Due to the sufficiently large mass of the heavy
diquark, the spin–dependent color magnetic force is negligible, and does not
contribute in the heavy quark limit. In this context, the heavy diquark and an
anti-heavy quark behave as the static color $\bar{3}$ source and are
equivalent in terms of color interaction. The property of hadrons remains
invariant under the interchange of the heavy diquark and anti heavy quark. We
will refer to hadrons related under this superflavor symmetry as superflavor
partners. For instance, an anti-heavy meson(HM) $\bar{Q}q$ and a doubly heavy
baryon(DHB) $QQq$ are superflavor partners. Another color representation for
diquarks, $6$, is also allowed in DHTs but not in ordinary hadrons. If
diquarks take color $6$ representation, the superflavor symmetry does not
arise. Hence, superflavor symmetry might shed light on the color configuration
of exotic hadrons. We think that understanding the color configuration of the
diquark in DHTs is important because the interaction changes according to the
color representation.
In order to analyze the mass spectrum of DHTs in terms of superflavor
symmetry, this study assumes that DHTs consist of a color heavy diquark,
treated as a spatially compact object, and a light quark cloud surrounding it.
Although it may exist in a mixed state within a DHT, we also assume that the
color anti-triplet state is the dominant state of the diquark because the
color $\bar{3}$ diquark is likely realized in the ground state [28, 29, 7].
Here, “being spatially compact” means that the heavy diquark can be
approximated as a point-like particle, with no radial excitation occurring
between two heavy quarks. The analyses of $T_{bb}$, being the bottom
counterpart of $T_{cc}$, based on the quark model [29, 30, 31] suggest that
the distance between the two bottom quarks is shorter compared to other quark
distances. This observation appears to support the notion that $T_{bb}$ is
composed of a heavy diquark and a light quark cloud. In the present analysis,
we assume that $T_{cc}$ also holds such heavy diquark structure. For
simplicity, this analysis focuses solely on diquarks where both heavy quarks
are of the same flavor. Under these conditions, a color $\bar{3}$ diquark
possesses spin one ($S_{QQ}=1$), whereas a color $6$ diquark has spin zero
($S_{QQ}=0$).
If $T_{cc}^{+}$ is the superflavor partner of $\Lambda_{c}^{+}$, we can
naturally expect the existence of $T_{ccs}$ as the superflavor partner of
$\Xi_{c}$, which belongs to the same flavor multiplet as $\Lambda_{c}^{+}$. We
consider that investigation of $T_{ccs}$ is useful to understand the nature
not only of $T_{ccs}$ itself but also of $T_{cc}^{+}$. We analyze $T_{ccs}$ by
using experimental result of $T_{cc}^{+}$ as an input. Thus if the results
obtained in this paper are eventually consistent with results in future
experiments, it can be interpreted that the color anti-triplet state is
dominant in DHTs $T_{cc}^{+}$ and $T_{ccs}$. In the following, we first derive
simple mass relations assuming that the heavy diquark is spatially compact and
color anti-triplet state together with superflavor symmetry. We note that the
relations agree with the ones derived in Ref. [32]. However, we find a
discrepancy between the prediction by the simple mass relation and the recent
experimental data of $T_{cc}^{+}$. Thus we invent the improved mass relations
with correction terms violating the symmetries and a mixing term of color
$\bar{3}$ and $6$ states of the $cc$ diquark. Then, we obtain new relations
among HMs, DHBs, SHBs and DHTs, and predict the mass of $T_{ccs}$.
Furthermore, we also predict the decay width of $T_{ccs}$ from the
$\rm{SU(3)}$ flavor symmetry.
This paper is organized as follows. In Sec, II, we derive the simple mass
relations among singly heavy and doubly heavy hadrons based on the heavy quark
and superflavor symmetries. We point out the existence of discrepancy between
the theoretical formulas and the experimental results. In Sec. III, we
construct the effective Lagrangians including corrections and obtain the
improved mass relations. The mass of $T_{ccs}$ and in addition, the one of
$\Omega_{cc}$ are predicted. In Sec. IV, the decay width of $T_{ccs}$ is
predicted by using the effective Lagrangian approach respecting the flavor
symmetry. Finally, Sec. V is devoted to the summary.
## II Simple mass relations from superflavor and light-flavor symmetries
In this section, we first introduce simple mass relations among superflavor
partners, primarily derived from superflavor, heavy quark, and light-flavor
symmetries. Then, we demonstrate that these mass relations are somewhat broken
among real hadrons, indicating the need for improvements to the mass
relations. Under the superflavor symmetry, a DHT is related to an anti-SHB,
and a DHB to an anti-HM. Considering that $T_{cc}^{+}$ has isospin $I=0$, the
superflavor partners of $T_{cc}^{+}$, $T_{ccs}^{+}$, and $T_{ccs}^{++}$ are
respectively the anti-baryons of $\Lambda_{c}^{+}$, $\Xi_{c}^{+}$, and
$\Xi_{c}^{0}$. Similarly, the superflavor partners of DHBs like $\Xi_{cc}$ and
$\Omega_{cc}$ are the heavy mesons (HMs), $\bar{D}$ and $\bar{D}_{s}$,
respectively.
We first study the relation between the masses of the DHBs and HMs in terms of
the superflavor symmetry, and then extend the analysis to the DHTs and SHBs.
Based on the HQS, we divide a heavy hadron into a heavy object and light-quark
cloud which includes the interaction between them. Therefore, the dynamics of
these hadrons are determined by the properties of the light quark cloud. As a
result, the masses of heavy hadrons treated in the present analysis are
expressed as a sum of the mass of heavy objects and the energy of light quark
cloud in the heavy quark limit. Let us first estimate the masses of HMs. Here,
we consider the spin average of the doublet under the HQS. We note that due to
the spin average, the first-order term in $1/m_{Q}$ expansion that breaks only
the heavy quark spin symmetry does not appear in the mass formulas. As a
result, the spin-averaged mass can be expressed as
$\displaystyle M_{\rm ave}\left(\bar{Q}q\right)$
$\displaystyle=M\left(\bar{Q}\right)+E\left(q\right),$ (1)
where $M(\bar{Q})$ ($\bar{Q}=\bar{c},\bar{b}$) is the mass of anti-heavy
quark, and $E(q)$ ($q=u,d,s$) denotes the contribution from the light-quark
cloud. From Eq. (1), we obtain the meson mass difference between flavor
partners as
$M_{\rm ave}(\bar{Q}s)-M_{\rm ave}(\bar{Q}n)=E(s)-E(n)\ ,$ (2)
where $n=u,d$.
In the DHBs, the heavy diquark $QQ$ takes the color $\bar{3}$ representation,
so that the anti-HMs and the DHBs include common light-quark cloud in the
heavy quark limit. Thus, the spin-averaged mass of the doublet members of DHBs
is expressed in a similar formula as for the HMs:
$\displaystyle M_{\rm ave}\left(QQq\right)$
$\displaystyle=M\left(QQ\right)+E\left(q\right)\ .$ (3)
We stress that the term $E\left(q\right)$ is common in Eqs. (1) and (3). The
mass difference between flavor partners is $E(\bar{q}_{1})-E(\bar{q}_{2})$,
where $q_{1},q_{2}=u,d,s$. This leads to the following mass relation [33]:
$\displaystyle M_{\rm ave}\left(QQq_{1}\right)-M_{\rm
ave}\left(QQq_{2}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(\bar{Q}q_{1}\right)-M_{\rm ave}\left(\bar{Q}q_{2}\right)\ .$ (4)
This implies that the mass differences between flavor partners are the same in
the superflavor partners.
Next, we consider the masses of anti-SHBs and DHTs. By a similar argument as
above, the mass of an anti-SHB ($\bar{Q}\bar{q}\bar{q}$) is expressed as
$\displaystyle M_{\rm ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)$
$\displaystyle=M\left(\bar{Q}\right)+E\left(\bar{q}_{1}\bar{q}_{2}\right)\ .$
(5)
We note that we generally employ the notation $M_{\rm ave}$ even though the
anti-SHB considered here belongs to a HQS singlet.
As we stated above, we assume that two charm quarks inside $T_{cc}$ form a
compact diquark and that the diquark belonging to the color $\bar{3}$
representation is dominated. Therefore, in the heavy quark limit, the DHT
shares a common light-quark cloud with the anti-SHB according to the
superflavor symmetry. Consequently, the mass of DHT is expressed as
$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)$
$\displaystyle=M\left(QQ\right)+E\left(\bar{q}_{1}\bar{q}_{2}\right)\ .$ (6)
From the mass formulas in Eqs. (5) and (6), we obtain the following mass
relation corresponding to the mass reletion (4):
$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(QQ\bar{q}_{3}\bar{q}_{4}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(\bar{Q}\bar{q}_{3}\bar{q}_{4}\right)\ .$ (7)
where $\bar{q}_{i}=\bar{u},\bar{d},\bar{s}$ ($i=1,2,3,4$).
We further combine Eqs. (1), (3), (5) and (6) to derive the following simple
mass relation:
$\displaystyle M_{\rm ave}\left(QQ\bar{q}_{1}\bar{q}_{2}\right)-M_{\rm
ave}\left(\bar{Q}\bar{q}_{1}\bar{q}_{2}\right)$ $\displaystyle\quad=M_{\rm
ave}\left(QQq\right)-M_{\rm ave}\left(\bar{Q}q\right)\ ,$ (8)
Now, we compare the obtained simple mass relations with existing experimental
data. We first note that the relation (2) implies that the mass difference
between $D$ and $D_{s}$ is equal to that between $B$ and $B_{s}$, namely
$\displaystyle M_{\rm ave}(D_{s})-M_{\rm ave}(D)=M_{\rm ave}(B_{s})-M_{\rm
ave}(B)\,,$ (9)
because the energy difference between light clouds, $E(s)-E(n)$, is
independent of the heavy flavor.
Hadrons | Mass (MeV) | Input Value (MeV)
---|---|---
$T_{cc}^{+}$ | 3874.74 | 3875
$\Xi_{cc}^{+}$ | 3623.0 | 3622
$\Xi_{cc}^{++}$ | 3621.55 | 3622
$D^{0}$ | 1864.84 | 1867
$D^{\pm}$ | 1869.66 | 1867
$D^{*0}$ | 2006.85 | 2009
$D^{*\pm}$ | 2010.26 | 2009
$D_{s}^{\pm}$ | 1968.35 | 1968
$D_{s}^{*\pm}$ | 2112.2 | 2112
$\Lambda_{c}^{+}$ | 2286.46 | 2286
$\Xi_{c}^{+}$ | 2467.71 | 2469
$\Xi_{c}^{0}$ | 2470.44 | 2469
$D^{*0}D_{s}^{\pm}$ | 3975.20 | 3979
$D^{*\pm}D_{s}^{\pm}$ | 3978.61 | 3979
$D^{0}D_{s}^{*\pm}$ | 3977.0 | 3980
$D^{\pm}D_{s}^{*\pm}$ | 3981.9 | 3980
$B^{\pm}$ | 5279.34 | 5280
$B^{0}$ | 5279.66 | 5280
$B^{*}$ | 5324.71 | 5325
$B_{s}^{0}$ | 5366.92 | 5367
$B_{s}^{*}$ | 5415.4 | 5415
Table 1: Experimental values [18, 19, 34] and input values in this study.
However, from the experimental values of masses shown in Table 1, the mass
differences are obtained as
$\displaystyle M_{\rm ave}\left(D_{s}\right)-M_{\rm
ave}\left(D\right)=$103\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$,$ (10)
$\displaystyle M_{\rm ave}\left(B_{s}\right)-M_{\rm
ave}\left(B\right)=$78\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$.$ (11)
This discrepancy implies the necessity of considering correction terms that
break both the heavy quark flavor symmetry and the light flavor symmetry.
Similarly, applying the mass relation (8) to $T_{cc}$, we obtain the following
mass relation:
$\displaystyle M_{\rm ave}\left(T_{cc}^{+}\right)-M_{\rm
ave}\left(\Lambda_{c}^{+}\right)=M_{\rm ave}\left(\Xi_{cc}\right)-M_{\rm
ave}\left(D\right)\ .$ (12)
Since $\Xi_{cc}^{*}$ has not experimentally confirmed yet, we estimate its
mass using the following mass relation obtained from the superflavor symmetry
[35, 27]:
$\displaystyle
M_{\Xi_{cc}^{*}}-M_{\Xi_{cc}}=\frac{3}{4}\left(M_{D^{*}}-M_{D}\right)\ ,$ (13)
leading to $M_{\Xi_{cc}^{*}}=$3728\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$,
and thus $M_{\rm ave}(\Xi_{cc})=3693$ MeV. Substituting this value into Eq.
(12), we get $M\left(T_{cc}^{+}\right)=3970\>\rm{MeV}$. This value is clearly
different from experimental value, $3875\>\rm{MeV}$, which motivates us to
consider the mixing between a state with a heavy diquark in the color $3$
representation and one in the color $6$ representation.
## III IMPROVED MASS RELATIONS
In this section, we construct effective Lagrangian terms for solving the
problems raised in the previous section for heavy mesons and the $T_{cc}$. As
we stated in the previous section, we need to include the terms which
simultaneously break the heavy quark flavor symmetry and SU(3)-flavor symmetry
for light quarks to cure the problem of masses of heavy mesons. For the
problem of the mass of $T_{cc}$, we include a term leading to the mixing
between the states constructed from the heavy-diquark in the color $\bar{3}$
representation and the one in the color $6$ representation.
At first, we define the effective DHB($QQq$) fields with quantum numbers
$J^{P}=\frac{1}{2}^{+}$ and $J^{P}=\frac{3}{2}^{+}$ in the heavy quark limit
by combining the heavy diquark with $J^{P}_{\rm heavy}=1^{+}$ to light quark
cloud with $J^{P}_{\rm light}=\frac{1}{2}^{+}$. This doubly-heavy baryon field
${\mathcal{B}}$ is expressed as
$\displaystyle\left[{\mathcal{B}}\right]_{hh^{\prime}l}=\left[P_{+}\gamma_{\mu}CP_{+}^{T}\right]_{hh^{\prime}}\psi^{\mu}_{l}\
,$ (14)
where $h$ and $h^{\prime}$ are spinor indices for heavy quarks and $l$ is the
spinor index for the light quark cloud, and $\psi^{\mu}$ is the field for the
heavy-quark spin doublet with $J^{P}=(1/2^{+},3/2^{+})$. We should note that
the field ${\mathcal{B}}$ carries the index to specify the heavy quark flavor
$Q$. But here and henceforce, we omit the index to avoid too many indices for
one field. The projection operator for heavy quark $P_{+}$ is defined as
$P_{+}=(1+v^{\mu}\gamma_{\mu})/2$, where $v^{\mu}$ is the velocity of the
heavy-diquark. The $\psi^{\mu}$ field is further decomposed into
$J^{P}=1/2^{+}$ field $\psi_{1/2}$ and $J^{P}=3/2^{+}$ field
$\psi_{3/2}^{\mu}$ field as
$\psi^{\mu}=\frac{1}{\sqrt{3}}\sigma^{\mu\nu}v_{\nu}\,\psi_{1/2}+\psi^{\mu}_{3/2}\
,$ (15)
where
$\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu}\,,\,\gamma^{\nu}\right]\ ,$
(16)
and the spin-$3/2$ Rarita-Schwinger field $\psi^{\mu}_{3/2}$ satisfies
$\displaystyle v^{\mu}\gamma_{\mu}\,\psi^{\mu}_{3/2}=\psi^{\mu}_{3/2}\ ,$
$\displaystyle v_{\mu}\psi^{\mu}_{3/2}=\gamma_{\mu}\psi^{\mu}_{3/2}=0\ .$ (17)
For later use, we define the conjugate field $\bar{\mathcal{B}}$ for DHB as
$\left[\bar{\mathcal{B}}\right]_{hh^{\prime}l}=\left[\gamma_{0}\right]_{hh_{1}}\left[{\mathcal{B}}^{\dagger}\right]_{h_{1}h_{2}l_{1}}\left[\gamma_{0}\right]_{h_{2}h^{\prime}}\left[\gamma_{0}\right]_{l_{1}l}\
.$ (18)
For realizing the superflavor symmetry, we take the effective DHB field
$\mathcal{B}$ and the effective anti-HM field $\bar{H}(\sim\bar{Q}q)$ into a
unified field $\bar{\Psi}$ as
$\displaystyle\bar{\Psi}=\begin{pmatrix}\bar{H}\\\ B\end{pmatrix}\ .$ (19)
The parity transformations are given by
$\displaystyle\bar{H}\rightarrow\gamma_{0}\bar{H}\gamma_{0},$ (20)
$\displaystyle[B]_{hh^{\prime}l}\rightarrow[\gamma_{0}]_{ll_{1}}[\gamma_{0}]_{hh_{1}}[B]_{h_{1}h_{2}l_{1}}[\gamma_{0}]_{h_{2}h^{\prime}},$
(21)
where $h,h^{\prime},h_{1},h_{2}$ are heavy quark spinor index and $l,l_{1}$
are spinor indices for light cloud. Since the field $\bar{\Psi}$ belongs to
$3$ representation of $\rm{SU(3)_{f}}$ light flavor, the $\rm{SU(3)_{f}}$
transformation is given by
$\displaystyle\bar{\Psi^{i}}\rightarrow U^{i}_{\>j}\bar{\Psi}^{j}\ ,$ (22)
where $i,j$ are the light flavor indices and $U\in\rm{SU(3)_{f}}$.
Next, we define the effective DHT($\sim\bar{Q}\bar{Q}qq$) fields. As we stated
in the previous section, we include two DHT fields: One is constructed from
the heavy diquark with color $\bar{3}$ representation carrying $J_{\rm
heavy}^{P}=1^{+}$, which is combined with the light-quark cloud with
$J^{P}_{\rm light}=0^{+}$ to make the DHT with $J^{P}=1^{+}$. Another one is
constructed from the heavy diquark carrying color $6$ representation and
$J^{P}_{\rm heavy}=0^{+}$ combined with $J^{P}_{\rm light}=1^{+}$ to make the
one with $J^{P}=1^{+}$. The former one is denoted as $T_{\mu}^{(\bar{3})}$ and
the latter as $T_{\mu}^{(6)}$. These fields are defined as
$\displaystyle\left[T_{\mu}^{(\bar{3})}\right]_{hh^{\prime}}=\left[P_{+}\gamma_{\mu}CP_{+}^{T}\right]_{hh^{\prime}}\,\phi\
,$ (23)
$\displaystyle\left[T_{\mu}^{(6)}\right]_{hh^{\prime}}=\left[P_{+}\gamma_{5}CP_{+}^{T}\right]_{hh^{\prime}}\,\varphi_{\mu}\
,$ (24)
where $h$ and $h^{\prime}$ are spinor-indices for heavy quarks, and the upper
indices of $T$, $(\bar{3})$ and $(6)$, represent the color representation of
heavy diquark in DHTs. We note that $\phi$ and $\varphi_{\mu}$ stand for the
annihilation operators and the same applies to $T$. We also note that these
fields have the index of light quark flavor. As we said in the introduction,
we consider heavy diquarks made from two heavy quarks with the same flavor.
The light quark cloud is made from two anti light quarks in the flavor anti-
symmetric representation. Then, the fields $T$ belong to $3$ representation of
$\rm{SU(3)_{f}}$ light flavor symmetry.
Based on the superflavor symmetry, the field $T_{\mu}^{(3)}$ and the effective
anti-SHB field $\bar{S}(\sim\bar{Q}\bar{q}\bar{q})$ are arranged into a
unified field $\Phi$ as
$\displaystyle\bar{\Phi}=\begin{pmatrix}\bar{S}\\\
T_{\mu}^{(\bar{3})}\end{pmatrix}\ .$ (25)
The parity transformations are given by
$\displaystyle\bar{S}$ $\displaystyle\rightarrow\bar{S}\gamma_{0}\ ,$ (26)
$\displaystyle[T]_{hh^{\prime}}$
$\displaystyle\rightarrow[\gamma_{0}]_{hh_{1}}[T]_{h_{1}h_{2}}[\gamma_{0}]_{h_{2}h^{\prime}}\
.$ (27)
The conjugate fields are defined as
$\displaystyle S=\gamma_{0}\bar{S}^{\dagger},$ (28)
$\displaystyle[\bar{T}]_{hh^{\prime}}=[\gamma_{0}]_{hh_{1}}[T^{\dagger}]_{h_{1}h_{2}}[\gamma_{0}]_{h_{2}h^{\prime}}\
.$ (29)
The $\rm{SU(3)_{f}}$ transformation is given by
$\displaystyle\bar{\Phi}_{ij}\rightarrow\left(U^{\ast}\right)_{i}^{\>k}\bar{\Phi}_{kl}\left(U^{\dagger}\right)_{\>j}^{l}$
(30)
Let us construct the effective Lagrangian for the DHT, SHB, DHB and HM. As we
said in the previous section, we need to include the terms which break
simultaneously the heavy-quark flavor symmetry and the light flavor symmetry.
We first note that the terms which break the heavy quark flavor symmetry are
inversely proportional to the heavy quark mass, i.e. $\propto 1/m_{Q}$. We
also include the term for generating the mixing between $\bar{T}^{(3)}_{\mu}$
and $\bar{T}^{(\bar{6})}_{\mu}$, which must include $1/m_{Q}$ since the mixing
vanishes at the heavy quark limit. For the light-flavor symmetry breaking, we
use the spurion field corresponding to the light quark mass matrix
${\mathcal{M}}_{q}$ as
${\mathcal{M}}_{q}=\begin{pmatrix}m_{u}&&\\\ &m_{d}&\\\ &&m_{s}\\\
\end{pmatrix}\ ,$ (31)
where $m_{u}$, $m_{d}$ and $m_{s}$ are the current quark masses of $u$, $d$
and $s$ quarks, respectively. This ${\mathcal{M}}_{q}$ transforms as
$\left({\mathcal{M}}_{q}\right)^{i}{}_{j}\to\left(U\right)^{i}{}_{k}\,\left({\mathcal{M}}_{q}\right)^{k}{}_{\ell}\,\left(U^{\dagger}\right)^{\ell}{}_{j}\
.$ (32)
Now, the kinetic and mass terms invariant under the heavy quark symmetry and
SU(3) light flavor symmetry are given by
$\displaystyle{\mathcal{L}}_{0}=$
$\displaystyle-\mbox{Tr}\left[\bar{\Psi}iv\cdot\partial\,\Psi+\bar{\Phi}iv\cdot\partial\,\Phi\right]$
$\displaystyle{}-\Lambda_{\Psi}\,\mbox{Tr}\left[\bar{\Psi}\Psi\right]-\Lambda_{\Phi}\,\mbox{Tr}\left[\bar{\Phi}\Phi\right]\
,$ (33)
where $\Lambda_{\Psi}$ and $\Lambda_{\Phi}$ are constants with mass dimension
one, and Tr implies that the traces in the light-flavor space in addition to
the spinor space and heavy-spin space are taken. In the following, we
explicitly write the indices for light flavor while we omit the indices for
heavy quark spin and flavor.
A possible term which breaks the SU(3) light flavor symmetry for $\Psi$ field
is written as
$\displaystyle{\mathcal{L}}_{\Psi{\rm-br}}=$ $\displaystyle-
c_{\Psi}\,\mbox{tr}\,\left[\bar{\Psi}^{i}\Psi_{j}\right]\left(\mathcal{M}_{q}\right)^{j}{}_{i}\
,$ (34)
where $c_{\Psi}$ is a constant with mass dimension zero and tr indicates that
the traces in spinor space and heavy-spin space are taken.
By noting that two light quarks in the $\Phi$ field are anti-symmetric in
flavor, the light flavor breaking term is expressed as
$\displaystyle{\mathcal{L}}_{\Phi{\rm-br}}=$ $\displaystyle-
c_{\Phi}\,\mbox{tr}\,\left[\bar{\Phi}_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}\Phi^{ki}\right]\
,$ (35)
where $c_{\Phi}$ is a constant with mass dimension zero.
We next construct possible terms which break both SU(3) light flavor symmetry
and the heavy quark flavor symmetry. We note that the terms also break the
superflavor symmetry, so that the terms should be written separately for
superflavor partners. The terms for $\bar{H}$ (anti-HM) and $B$ (DHB) included
in $\Psi$ are expressed as
$\displaystyle{\mathcal{L}}_{\rm HB-br}=$
$\displaystyle-\frac{\Lambda_{q}}{m_{Q}}\mbox{tr}\left[\bar{H}^{i}\,H_{j}\left(\mathcal{M}_{q}\right)^{j}{}_{i}\right]$
$\displaystyle{}-\frac{\Lambda_{q}^{\prime}}{2m_{Q}}\mbox{tr}\left[B^{i}\,\bar{B}_{j}\left(\mathcal{M}_{q}\right)^{j}{}_{i}\right]\
,$ (36)
where $\Lambda_{q}$ and $\Lambda_{q}^{\prime}$ are constants with mass
dimension one. We put an extra factor $1/2$ in the second line since the DHB
includes two heavy quarks. Similarly, possible breaking terms for $\bar{S}$
(anti-SHB) and $T$ (DHT) in $\Phi$ field are expressed as
$\displaystyle{\mathcal{L}}_{\rm ST-
br}=-\frac{\Lambda_{qq}}{m_{Q}}\,\mbox{tr}\left[\bar{S}_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}S^{ki}\right]$
$\displaystyle\quad{}-\frac{\Lambda_{qq}^{\prime}}{2m_{Q}}\,\mbox{tr}\left[\left(T_{\mu}^{(\bar{3})}\right)_{ij}\left(\mathcal{M}_{q}\right)^{j}{}_{k}\left(\bar{T}^{(\bar{3})\mu}\right)^{ki}\right]\
,$ (37)
where $\Lambda_{qq}$ and $\Lambda_{qq}^{\prime}$ are constants with mass
dimension one.
Finally, we consider a DHT field $T_{\mu}^{(6)}$ constructed from the heavy
diquark with color ${\boldmath 6}$ representation. The kinetic and mass terms
are written as
${\mathcal{L}}_{T^{(6)}}=-\mbox{Tr}\,\left[\bar{T}_{\mu}^{(6)}\left(iv\cdot\partial+\Lambda_{6}\right)T^{(6)\mu}\right]\
,$ (38)
where $\Lambda_{6}$ is a constant with mass dimension one. The term for the
mixing between two DHT fields $T_{\mu}^{(3)}$ and $T_{\mu}^{(6)}$ is expressed
as
$\displaystyle{\mathcal{L}}_{\rm mix}=$
$\displaystyle-\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\mbox{Tr}\Bigg{[}\left(\bar{T}^{(6)\mu}\right)\left(T_{\mu}^{(\bar{3})}\right)+\left(\bar{T}^{(\bar{3})\mu}\right)\left(T_{\mu}^{(6)}\right)\Bigg{]}\
.$ (39)
From the above Lagrangian terms, the masses of HM, DHB and SHB are modified
from Eqs. (1), (3) and (5) as
$\displaystyle M_{\rm ave}\left(\bar{Q}q\right)=$ $\displaystyle
M\left(\bar{Q}\right)+E\left(q\right)+\frac{m_{q}}{m_{Q}}\Lambda_{q},$ (40)
$\displaystyle M_{\rm ave}\left(QQq\right)=$ $\displaystyle
M\left(QQ\right)+E\left(q\right)+\frac{m_{q}}{2m_{Q}}\Lambda_{q}^{\prime},$
(41) $\displaystyle M_{\rm ave}\left(\bar{Q}\bar{q_{1}}\bar{q_{2}}\right)=$
$\displaystyle
M\left(\bar{Q}\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)+\frac{m_{q_{1}}+m_{q_{2}}}{m_{Q}}\Lambda_{qq},$
(42)
where
$\displaystyle E(q)$ $\displaystyle=\Lambda_{\Psi}+c_{\Psi}m_{q}\ ,$
$\displaystyle E(\bar{q}_{1}\bar{q}_{2})$
$\displaystyle=\Lambda_{\Phi}+c_{\Phi}\left(m_{q_{1}}+m_{q_{2}}\right)\ .$
(43)
The square mass matrix for two DHT fields is expressed as
$\begin{pmatrix}M_{3}^{2}&\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\sqrt{M_{6}M_{3}}\\\
\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\sqrt{M_{6}M_{3}}&M_{6}^{2}\\\
\end{pmatrix}\ ,$ (44)
where
$M_{3}=M\left(QQ\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)+\frac{m_{q_{1}}+m_{q_{2}}}{2m_{Q}}\Lambda_{qq}^{\prime}\
,$ (45)
and $M_{6}$ is the mass of $T_{\mu}^{(6)}$ field before mixing. By
diagonalizing this matrix up to $1/m_{Q}$ order, the mass of lightest DHT is
obtained as
$\displaystyle M_{\rm
ave}\left(QQ\bar{q_{1}}\bar{q_{2}}\right)=M\left(QQ\right)+E\left(\bar{q_{1}}\bar{q_{2}}\right)$
$\displaystyle\quad{}+\frac{m_{q_{1}}+m_{q_{2}}}{2m_{Q}}\Lambda_{qq}^{\prime}-\frac{M_{6}}{2(M_{6}^{2}-M_{3}^{2})}\left(\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\right)^{2}\
.$ (46)
Combining the above mass formulas, we obtain the following modified mass
relations:
$\displaystyle M_{\rm ave}\left(QQs\right)-M_{\rm
ave}\left(QQn\right)-\frac{m_{s}-m_{n}}{2m_{Q}}\Lambda_{q}^{\prime}$
$\displaystyle=M_{\rm ave}\left(\bar{Q}s\right)-M_{\rm
ave}\left(\bar{Q}n\right)-\frac{m_{s}-m_{n}}{m_{Q}}\Lambda_{q},$ (47)
$\displaystyle M_{\rm ave}\left(QQ\bar{s}\bar{n}\right)-M_{\rm
ave}\left(QQ\bar{u}\bar{d}\right)-\frac{m_{s}-m_{n}}{2m_{Q}}\Lambda_{qq}^{\prime}$
$\displaystyle=M_{\rm ave}\left(\bar{Q}\bar{s}\bar{n}\right)-M_{\rm
ave}\left(\bar{Q}\bar{u}\bar{d}\right)-\frac{m_{s}-m_{n}}{m_{Q}}\Lambda_{qq},$
(48) $\displaystyle M_{\rm ave}\left(QQ\bar{u}\bar{d}\right)-M_{\rm
ave}\left(\bar{Q}\bar{u}\bar{d}\right)$
$\displaystyle+\frac{M_{6}}{2\left(M_{6}^{\>2}-M_{3}^{\>2}\right)}\left(\frac{\Lambda_{\rm{mix}}^{2}}{2m_{Q}}\right)^{2}-\frac{m_{n}}{2m_{Q}}\left(\Lambda_{qq}^{\prime}-2\Lambda_{qq}\right)$
$\displaystyle=M_{\rm ave}\left(QQn\right)-M_{\rm
ave}\left(\bar{Q}n\right)-\frac{m_{n}}{2m_{Q}}\left(\Lambda_{q}^{\prime}-2\Lambda_{q}\right)\
,$ (49)
where $n=u,d$ and $m_{n}=(m_{u}+m_{d})/2$ is the isospin averaged mass of up
and down quarks. Furthermore, from Eq. (40), we obtain the following mass
relation:
$\displaystyle\left[M_{\rm ave}(D_{s})-M_{\rm ave}(D)\right]-\left[M_{\rm
ave}(B_{s})-M_{\rm ave}(B)\right]$
$\displaystyle\quad=\left(m_{s}-m_{n}\right)\left(\frac{1}{m_{c}}-\frac{1}{m_{b}}\right)\Lambda_{q}\
,$ (50)
where $n=u,d$.
Quarks | Mass (MeV)
---|---
$u,d$ | 3.415
$s$ | 93.40
$c$ | 1270
$b$ | 4180
Table 2: Mass of quarks [34] used as inputs.
Using the mass differences in Eqs. (10) and (11) together with the current
quark masses shown in Table 2, we determine the value of $\Lambda_{q}$ as
$\Lambda_{q}=506\,\mbox{MeV}\ .$ (51)
The other parameters $\Lambda^{\prime}_{q}$, $\Lambda_{qq}$ and
$\Lambda^{\prime}_{qq}$ cannot be determined due to lack of input hadron
masses. We therefore assume that they are of the same order as $\Lambda_{q}$.
Specifically, we set their median values as $0$ and assign an error margin of
$\pm 506\,\rm{MeV}$, resulting in
$\Lambda^{\prime}_{q}=\Lambda_{qq}=\Lambda^{\prime}_{qq}=0\pm 506\,\rm{MeV}$.
By applying the improved mass relation corresponding to Eq. (7) to
$T_{cc}^{+}$, we derive
$\displaystyle M_{\rm ave}\left(T_{ccs}\right)-M_{\rm
ave}\left(T_{cc}^{+}\right)+\frac{m_{s}-m_{n}}{2m_{c}}\Lambda_{qq}^{\prime}$
$\displaystyle\quad=M_{\rm ave}\left(\Xi_{c}\right)-M_{\rm
ave}\left(\Lambda_{c}^{+}\right)+\frac{m_{s}-m_{n}}{m_{c}}\Lambda_{qq}\ ,$
(52)
and we obtain the mass of $T_{ccs}$ as
$\displaystyle M_{\rm ave}\left(T_{ccs}\right)=4057\pm 40\>\rm{MeV},$ (53)
where $M_{\rm ave}(T_{ccs})$ denotes the isospin averaged mass of
$T_{ccs}^{+}$ and $T_{ccs}^{++}$.
Hadrons | Mass (MeV) |
---|---|---
$T_{ccs}$ | $4057\pm 40$ | Our result
$4106$ | NQM [36]
$T_{ccs}^{+}$ | $3969\pm 8$ | Lattice QCD [37]
Table 3: Theoretical predictions for the masses of $T_{ccs}$. NQM imply non-
relativistic quark model.
If this result agrees with future experimental data, it means that the color
anti-triplet state of the heavy diquark is dominant in $T_{cc}^{+}$ and
$T_{ccs}$. We compare our result with other results in Table 3.
From the mass relation in Eq. (47), we further obtain
$\displaystyle M_{\rm ave}\left(\Omega_{cc}\right)$ $\displaystyle-M_{\rm
ave}\left(\Xi_{cc}\right)+\frac{m_{s}-m_{n}}{2m_{c}}\Lambda_{q}^{\prime}$
$\displaystyle=M_{\rm ave}\left(D_{s}\right)-M_{\rm
ave}\left(D\right)+\frac{m_{s}-m_{n}}{m_{c}}\Lambda_{q}.$ (54)
From this relation, the mass of $\Omega_{cc}$ which has not been
experimentally reported so far is also predicted as
$\displaystyle M_{\rm ave}\left(\Omega_{cc}\right)=3760\pm 18\>\rm{MeV}.$ (55)
This result serves as another indicator for assessing whether our mass
relations are correct. If this result does not match future experimental
results, one possible reason might be that $\Lambda_{q}^{\prime}$ is larger
than $\Lambda_{q}$. Additionally, the second-order effects of $1/m_{Q}$ might
also contribute. We show comparison with other results in Table 4.
Hadrons | Mass (MeV) |
---|---|---
$\Omega_{cc}$ | $3760\pm 18$ | Our result
3811 | NQM [38]
3753 | RQM [39]
3841 | RQM [40]
3657 | NQM [41]
3856 | NQM [42]
3773 | Lattice QCD [43]
3754 | Lattice QCD [44]
Table 4: Theoretical predictions of the mass of $\Omega_{cc}$. NQM and RQM
imply non-relativistic and relativistic quark models, respctively.
## IV WIDTH OF $T_{ccs}^{+}$
In this section, we construct the Lagrangian solely from the $\rm{SU(3)}$
light flavor symmetry and predict the decay width of $T_{ccs}$ using the decay
width of $T_{cc}^{+}$ as an input. Our approach hinges on the fact that both
$T_{ccs}$ and $T_{cc}^{+}$ belong to the same light flavor representation.
Notably, this method is independent of both superflavor symmetry and color
representation.
Since $T_{cc}^{+}$ locates just below the $DD^{*}$ threshold, we consider the
decay process of $T_{cc}^{+}\rightarrow D^{0}D^{0}\pi^{+}$ and
$T_{cc}^{+}\rightarrow D^{0}D^{+}\pi^{0}$ with $D^{*}$ as an intermediate
state, as depicted in Fig. 1. While the decay $T_{cc}^{+}\rightarrow
D^{0}D^{+}\pi^{0}$ has not been observed in experiments [18, 19], it is not
prohibited kinematically. Hence, in this study, we also incorporate Fig.
1(b),(c). For $T_{ccs}$, we consider the decay processes $T_{ccs}\rightarrow
DD_{s}^{*}$ and $T_{ccs}\rightarrow D^{\ast}D_{s}$ as shown in Fig. 2, because
the mass of $T_{ccs}$ predicted in the previous section is above the
$DD_{s}^{*}$ and $D^{*}D_{s}$ thresholds.
Let us construct an effective Lagrangian for the interaction among the
tetraquarks and charmed mesons based on just light flavor symmetry. In the
following we use relativistic field $T^{\mu}$ for the mass eigenstates of
flavor triplet tetraquarks:
$T^{\mu}=\begin{pmatrix}0&T_{cc}^{+}&T_{ccs}^{++}\\\
-T_{cc}^{+}&0&T_{ccs}^{+}\\\ -T_{ccs}^{++}&-T_{ccs}^{+}&0\\\ \end{pmatrix}\ ,$
(56)
While the fields for charmed mesons with $J^{P}=\left(0^{-},1^{-}\right)$ are
represented by relativistic fields $\left(D,D^{*\mu}\right)$:
$D=\begin{pmatrix}D^{0}\\\ D^{+}\\\ D_{s}\\\ \end{pmatrix}\ ,\quad
D^{\ast}=\begin{pmatrix}D^{\ast 0}\\\ D^{\ast+}\\\ D_{s}^{\ast}\\\
\end{pmatrix}\ .$ (57)
From Eqs. (22) and (30), the $\rm{SU(3)}$ light flavor transformations of
these fields are given as
$\displaystyle
T_{ij}^{\mu}\rightarrow\left(U^{\ast}\right)_{i}^{\>k}T_{kl}^{\mu}\left(U^{\dagger}\right)_{\>j}^{l},$
$\displaystyle D_{i}\rightarrow
D_{j}\left(U^{{\dagger}}\right)^{j}_{\;i},\>D_{i}^{*\mu}\rightarrow
D_{j}^{*\mu}\left(U^{{\dagger}}\right)^{j}_{\;i}\ .$ (58)
The Lagrangian for $T_{cc(s)}$ and heavy mesons invariant under this
transformation is given by
$\displaystyle\mathcal{L}=g_{TDD}\bar{D}_{\mu}^{\ast
i}T_{ij}^{\mu}\bar{D}^{j}\ .$ (59)
| |
---|---|---
(a) | (b) | (c)
Figure 1: Feynmann diagrams of the $T_{cc}^{+}\to DD\pi$ decays.
As we said above, we determine the value of $g_{TDD}$ from the decay width of
$T_{cc}$ assuming that decay is dominated by the process shown in Fig. 1. By
using the value of $D^{\ast}D\pi$ coupling constant determined from the decay
of $D^{\ast}$ meson (see e.g. Ref [24]), the value of $g_{TDD}$ is calculated
as
$\displaystyle g_{TDD}=0.68\times 10^{3}\>\rm{MeV}\ .$ (60)
We note that this value is natural when nondimensionalized by the mass of the
charm quark.
|
---|---
(a) | (b)
Figure 2: Feynmann diagrams of the $T_{ccs}\to DD_{s}^{*}$ and $D^{*}D_{s}$
decays.
Now, based on the processes shown in Fig. 2. the formula of the decay width of
$T_{ccs}$ is calculated as
$\displaystyle\Gamma\left(T_{ccs}\right)=g_{TDD}^{2}\frac{\left|P_{1}\right|+\left|P_{2}\right|}{6\pi
M\left(T_{ccs}\right)^{2}}\ ,$ (61)
where $P_{1}$ and $P_{2}$ and the momenta for $D_{s}^{\pm}$ and $D^{0}$,
respectively. By using the mass of $T_{ccs}$ predicted in Eq. (53) together
with the value of $g_{TDD}$ in Eq. (60), the decay width is predicted as
$\displaystyle\Gamma\left(T_{ccs}\right)=1.2\pm 0.3\>\rm{MeV}\ .$ (62)
We consider this decay width to be sufficiently small as to be an
experimentally observable.
## V SUMMARY
In this study, we investigated the mass and decay width of the doubly heavy
tetraquark $T_{ccs}$ from the superflavor and $\rm{SU(3)}$ light flavor
symmetries. We assumed that doubly charmed tetraquarks are constructed from a
color anti-triplet $cc$ diquark and thus they are the superflavor partners of
the singly heavy baryons. First, we derived the simple mass relations under
heavy quark and superflavor symmetries. However, we found discrepancy between
predictions of the obtained mass relations and the experimental data. Then, we
constructed an effective Lagrangian based on these symmetries with including
correction terms violating simultaneously the heavy quark symmetry and the
light flavor symmetry. From the Lagrangian, we obtained the improved mass
relations among heavy mesons (HMs), doubly heavy baryons (DHBs), singly heavy
baryons (SHBs) and doubly heavy tetraquarks (DHTs). Based on the relations, we
predicted the mass of unobserved tetraquark $T_{ccs}$ as
$M\left(T_{ccs}\right)=4057\pm 40\>\rm{MeV}$. We also predicted the mass of
unobserved $\Omega_{cc}$ as $M(\Omega_{cc})=3760\pm 18\,\rm{MeV}$.
We then constructed an effective Lagrangian term for the decay of $T_{ccs}$
based on SU(3) light flavor symmetry. The unknown coupling constant was
determined by using the $T_{cc}$ decay data. Incorporating the predicted mass,
we derived the decay width of $T_{ccs}$ as $\Gamma\left(T_{ccs}\right)=1.2\pm
0.3\>\rm{MeV}$.
The obtained masses and widths will be useful to understand the color
configuration of DHTs. If these results agree with future experimental data,
it means that the color anti-triplet state in $T_{cc}^{+}$ and $T_{ccs}$ is
dominant.
In this study, we have primarily focused on the charm sector. This is because
hadrons containing two $b$ quarks have not yet been experimentally discovered,
and we cannot determine the value of $M(bb)-M(b)$. However, our results are
immediately applicable to the bottom sector once hadrons such as $\Xi_{bb}$,
$\Omega_{bb}$, $T_{bb}$, and $T_{bbs}$ are found.
This work was supported in part by JSPS KAKENHI Grant Nos. JP20K03927,
JP23H05439 (MH) and JP20K14478 (YY).
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|
# An information-theoretic upper bound on prime gaps
Aidan Rocke
<EMAIL_ADDRESS>
###### Abstract
Within the setting of rare event modelling, the method of level sets allows us
to define an equivalence relation over rare events with distinct rates of
entropy production. This method allows us to clarify the relation between the
empirical density of primes and their source distribution which then allows us
to address Cramér’s conjecture, an open problem in probabilistic number
theory. As a natural consequence, this analysis places strong epistemic limits
on the application of machine learning to analyse the distribution of primes.
## 1 Classifying rare events using level sets
Sequences of rare events that are not finite-state compressible are recurrent,
of variable frequency, and unpredictable relative to a finite-state machine
such as a machine learning model. For these reasons, they are of great
scientific interest. But, how might we classify rare events that satisfy these
criteria?
If we identify rare events using the function
$1_{X}:\mathbb{N}\rightarrow\\{0,1\\}$, we may analyse the rate of entropy
production of the computable binary sequence
$X_{n}=\\{x_{i}\\}_{i=1}^{n}\in\\{0,1\\}^{n}$. In order to qualify as a rare
event, a randomly sampled element $x_{z}$ of the sequence $X_{n}$ must
generally pass $\pi_{x}(n)$ tests. These tests $A_{i},A_{j\neq
i}\in\\{A_{i}\\}_{i=1}^{\pi_{x}(n)}$ are de-correlated for rare events as
these are assumed to occur independently of each other:
$\forall z\sim U([1,n]),P(z\in A_{i}\land z\in A_{j\neq i})=P(z\in A_{i})\cdot
P(z\in A_{j\neq i})$ (1)
where the $P(z\in A_{i})$ have a natural frequentist interpretation. We
generally assume that the tests are consistent so
$\\{A_{i}\\}_{i=1}^{\pi_{x}(n)}\subset\\{A_{i}\\}_{i=1}^{\pi_{x}(n+1)}$.
Furthermore, $\pi_{x}(n)$ counts the number of rare events in the discrete
time interval $[1,n]$ so it is monotonically increasing. This may be deduced
from the fact that each rare event that occurs in an interval defines one
information-theoretic constraint $A_{i}$ where $A_{i}\perp A_{j\neq i}$ due to
pairwise independence. Therefore, if $m$ rare events occur in an interval, a
uniformly sampled element $x_{z}$ must simultaneously satisfy $m$ independent
constraints in order to qualify as a rare event.
Given (1), we may define the average probability density using the inclusion-
exclusion principle:
$\forall z\sim
U([1,n]),P(x_{z}=1)=P(z\in\bigcap_{k\leq\pi_{x}(n)}A_{k})=\prod_{k=1}^{\pi_{x}(n)}P(z\in
A_{k})$ (2)
as well as the entropy of the uniformly distributed random variable:
$\forall z\sim U([1,n]),H(X_{n})=-\ln\prod_{k=1}^{\pi_{x}(n)}P(z\in A_{k})$
(3)
which is a measure of expected surprise, or the expected information gained
from observing a rare event.
Having defined the entropy (3), we may compare the entropies of distinct
sequences using the typical probabilities $q_{n}\in(0,1)$, a parameter which
allows us to define an equivalence relation over entropies:
$\mathcal{L}_{q_{n}}=\\{X_{n}\in\\{0,1\\}^{\infty}:H(X_{n})\sim\ln\big{(}\frac{1}{q_{n}}\big{)}\\}$
(4)
This parameter is chosen because it corresponds to the multiplicative inverse
of the exponential of entropy known hereafter as the entropy rate
$\frac{1}{q_{n}}\approx e^{H(X_{n})}$ which is a robust measure of the
expected waiting time before rare events. As these parameters are both
parameterisation invariant i.e. don’t depend upon the choice of base for
logarithms, they are an ideal choice for classifying rare events in terms of
their rate of entropy production.
Given the definition (4), we have:
$\forall X_{n}\in\mathcal{L}_{q_{n}},\lim_{n\to\infty}-\frac{\ln
q_{n}}{H(X_{n})}=1$ (5)
so the following holds asymptotically almost surely:
$\frac{\partial}{\partial q_{n}}H(X_{n})=\frac{\partial}{\partial q_{n}}-\ln
q_{n}\implies\forall z\sim U([1,N]),P(x_{z}=1)\sim q_{N}$ (6)
and therefore for large $N\in\mathbb{N}$:
$\pi_{x}(N)\sim N\cdot q_{N}$ (7)
This leads us naturally to consider statistical generalisations of the Prime
Number Theorem.
### 1.1 Statistical generalisations of the Prime Number Theorem
From statistical considerations, Gauss and Legendre inferred that the density
of primes at $n\in\mathbb{N}$ is on the order of $\sim\frac{1}{\ln n}$. This
led them to conjecture that the number of primes less than $N$ is on the order
of:
$\pi(N)\sim\int_{2}^{N}\frac{1}{\ln x}dx\sim N\cdot\frac{1}{\ln N}$ (8)
which is equivalent to the Prime Number Theorem.
However, given our assumptions we are more generally interested in the family
of density functions that satisfy:
$0<c\ll N,\int_{c}^{N}\frac{1}{f(x)}dx\sim N\cdot\frac{1}{f(N)}$ (9)
where $f$ is analytic and $\lim\limits_{N\to\infty}\frac{1}{f(N)}=0$. Using
integration by parts, this is equivalent to the criterion:
$\int_{c}^{N}g(x)dx=N\cdot g(N)-c\cdot g(c)-\int_{c}^{N}x\cdot
g^{\prime}(x)dx\sim N\cdot g(N)$ (10)
where $g(x)=\frac{1}{f(x)}$, and $g(x)$ dominates $x\cdot g^{\prime}(x)$. So
this family naturally includes density functions of the form:
$\exists a>0\forall x\in\mathbb{R}_{+},g(x)=(\ln x)^{-a}$ (11)
and it is worth adding that (11) admits an interpretation as an average
probability density which we may now make precise.
### 1.2 The typical probability as the average probability
Given that the density $g(x)=(\ln x)^{-a}$ is Riemann-Integrable on $[2,N]$
for $N<\infty$, by the Intermediate Value Theorem there exists a sequence
$x_{n}\in[n,n+1]$ such that:
$\int_{2}^{N}(\ln x)^{-a}dx=\sum_{n=2}^{N}\frac{1}{(\ln x_{n})^{a}}$ (12)
and since $\forall n\in\mathbb{N},1\leq\big{(}\frac{\ln x_{n}}{\ln
n}\big{)}^{a}\leq\big{(}\frac{\ln(n+1)}{\ln n}\big{)}^{a}$ where $\forall
a>0,\lim\limits_{n\to\infty}\big{(}\frac{\ln(n+1)}{\ln n}\big{)}^{a}=1$ so we
have:
$\int_{2}^{N}(\ln x)^{-a}dx\sim\sum_{n=2}^{N}\frac{1}{(\ln n)^{a}}\sim
N\cdot\frac{1}{(\ln N)^{a}}$ (13)
and therefore the typical probability is asymptotically equivalent to the
average probability:
$q_{N}\sim\frac{1}{N}\sum_{n=2}^{N}\frac{1}{(\ln n)^{a}}$ (14)
In fact, for large $n$ the average probability density in (2) may be expressed
as:
$\forall z\sim U([1,n]),P(x_{z}=1)\sim\frac{1}{n}\sum_{k=1}^{n}P(x_{k}=1)$
(15)
### 1.3 A remark on applications of the level set method
In practice, the empirical data will consist of a binary sequence of rare
event observations and the actual tests are generally unknown. These may be
modelled as latent variables that are a function of discrete time(i.e. the
integers) as well as hidden variables that provide us with initial conditions.
Thus, we shall generally assume that empirical binary sequences are the output
of a discrete dynamical system.
Assuming that deep neural networks may be used as universal data compressors,
we may determine whether the data is finite-state incompressible using a
machine-learning driven overfitting test provided in the appendix [12]. As a
concrete demonstration of the analytical power of the level set method, we
shall consider its application to an open problem in probabilistic number
theory.
However, we must first carefully go over the information-theoretic derivation
of the Prime Number Theorem [9].
## 2 Information-theoretic derivation of the Prime Number Theorem
If we know nothing about the distribution of primes, in the worst case we may
assume that each prime less than or equal to $N$ is drawn uniformly from
$[1,N]$. So our source of primes is:
$X\sim U([1,N])$ (16)
where $H(X)=\ln N$ is the Shannon entropy of the uniform distribution.
Now, we may define the prime encoding of $[1,N]$ as the binary sequence
$X_{N}=\\{x_{n}\\}_{n=1}^{N}$ where $x_{n}=1$ if $n$ is prime and $x_{n}=0$
otherwise. With no prior knowledge, given that each integer is either prime or
not prime, we have $2^{N}$ possible prime encodings in
$[1,N]\subset\mathbb{N}$.
If there are $\pi(N)$ primes less than or equal to $N$ then the average number
of bits per arrangement gives us the average amount of information gained from
correctly identifying each prime in $[1,N]$ as:
$S_{c}=\frac{\log_{2}(2^{N})}{\pi(N)}=\frac{N}{\pi(N)}$ (17)
and if we assume a maximum entropy distribution over the primes then we would
expect that each prime is drawn from a uniform distribution as in (16). So we
would have:
$S_{c}=\frac{N}{\pi(N)}\sim\ln N$ (18)
As for why the natural logarithm appears in (21), we may first note that the
base of the logarithm in the Shannon Entropy may be freely chosen without
changing its properties. Moreover, given the assumptions the expected
information gained from observing each prime in $[1,N]$ is on the order of:
$\sum_{k=1}^{N-1}\frac{1}{k}\cdot|(k,k+1]|=\sum_{k=1}^{N-1}\frac{1}{k}\approx\ln
N$ (19)
as there are $k$ distinct ways to sample uniformly from $[1,k]$ and a
frequency of $\frac{1}{k}$ associated with the event that $k\in\mathbb{P}$.
This implies that the average number of bits per prime number is given by
$\frac{N}{\pi(N)}\sim\ln N$. Rearranging, we find:
$\frac{\pi(N)}{N}\sim\frac{1}{\ln N}$ (20)
which is in complete agreement with the Prime Number Theorem. Moreover, this
derivation indicates that the prime numbers are empirically distributed as if
they were arranged uniformly so we may distinguish the empirical distribution
of the primes from their source distribution.
### 2.1 The Shannon source coding theorem and the algorithmic randomness of
prime encodings
By the Shannon source coding theorem, we may infer that $\pi(N)$ primes can’t
be compressed into fewer than $\pi(N)\cdot\ln N$ bits. Furthermore, as the
expected Kolmogorov Complexity equals the Shannon entropy for computable
probability distributions:
$\mathbb{E}[K(X_{N})]\sim\pi(N)\cdot\ln N\sim N$ (21)
where the identification of $\mathbb{E}[K(X_{N})]$ with $\pi(N)\cdot\ln N$ is
a direct consequence of the fact that $K(X_{N})$ measures the information
gained from observing $X_{N}$ and $\pi(N)\cdot\ln N$ measures the expected
information gained from observing $X_{N}$. From an information-theoretic
perspective, this implies that prime encodings are finite-state incompressible
as they have a maximum entropy distribution.
The only implicit assumption in this derivation is that all the information in
the Universe is conserved as it is only in such Universes that Occam’s razor
is generally applicable. The Law of Conservation of information and its
significance is discussed in more detail at the end of the Appendix.
## 3 Clarifying the relation between the empirical density of primes and
their source distribution
In the worst case, rare events $x_{k}\in X_{n}$ are arranged uniformly in the
discrete time interval $[1,n]$ so the expected information gained from
observing all the events in $X_{n}$ is on the order of:
$I_{N}=\sum_{n=1}^{N}q_{n}\cdot\ln n$ (22)
Furthermore, in order to model rare events it is crucial to consider the sum
of entropies (3):
$S_{N}=\sum_{n=1}^{N}H(X_{n})$ (23)
subject to the constraint $H(X_{n})\sim-\ln q_{n}$ so we may define the
Lagrangian function:
$\mathcal{L}(\lambda,q_{n})=\sum_{n=1}^{N}q_{n}\cdot\ln
n-\lambda\big{(}S_{N}+\sum_{n=1}^{N}\ln q_{n}\big{)}$ (24)
Thus, we find that:
$\frac{\partial\mathcal{L}}{\partial q_{n}}=\ln
n-\frac{\lambda}{q_{n}}=0\implies q_{n}=\frac{\lambda}{\ln n}$ (25)
which only makes sense if $\lambda=1$ so we have:
$q_{n}=\frac{1}{\ln n}$ (26)
and therefore among the rare-event densities, the empirical density of
primes(which we may observe) is a natural model for a uniform source
distribution which is not directly observable.
## 4 Application to Cramér’s random model, Part I
Using the method of level sets, we may demonstrate that the typical
probability that an integer in the interval $[1,n]\subset\mathbb{N}$ is prime
is given by:
$\forall z\sim U([1,n]),P(z\in\mathbb{P})\sim\frac{1}{\ln n}$ (27)
If $X_{n}=\\{x_{i}\\}_{i=1}^{n}\in\\{0,1\\}^{n}$ defines a prime encoding i.e.
a sequence where $x_{k}=1$ if $k\in\mathbb{P}$ and $x_{k}=0$ otherwise, then
we may define $\pi(\sqrt{n})$ primality tests as any composite integer in
$[1,n]$ has at most $\pi(\sqrt{n})$ distinct prime factors:
$\forall
A_{p}\in\\{A_{p_{k}}\\}_{k=1}^{\pi(\sqrt{n})},A_{p}=\\{z\in[1,n]:\text{gcd}(p,z)=1\\}\bigcup\\{p\\}$
(28)
and therefore:
$\forall z\sim U([1,n]),H(X_{n})=-\ln\prod_{k=1}^{\pi(\sqrt{n})}P(z\in
A_{p_{k}})$ (29)
In order to define $P(z\in A_{p_{k}})$ we shall implicitly use the
approximation that for
$p\leq\sqrt{n},\frac{1}{p}-\frac{1}{n}\approx\frac{1}{p}$ so we have:
$\forall z\sim U([1,n]),P(z\in
A_{p_{k}})=\big{(}1-\frac{1}{p_{k}}\big{)}+\frac{1}{n}\approx\big{(}1-\frac{1}{p_{k}}\big{)}$
(30)
which allows us to make (29) precise:
$H(X_{n})\sim-\ln\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}$ (31)
Using Mertens’ third theorem we have:
$\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\approx\frac{e^{-\gamma}}{\frac{1}{2}\cdot\ln
n}\approx\frac{0.9}{\ln n}$ (32)
so we may infer that:
$H(X_{n})\sim\ln\ln n$ (33)
and therefore $X_{n}\in\mathcal{L}_{q_{n}}$ where:
$\mathcal{L}_{q_{n}}=\\{X_{n}\in\\{0,1\\}^{\infty}:H(X_{n})\sim\ln\ln n\\}$
(34)
From (34), we may deduce (27):
$\forall z\sim U([1,n]),q_{n}=P(z\in\mathbb{P})\sim\frac{1}{\ln n}$ (35)
which means that the density of the primes at $x\in\mathbb{N}$ is on the order
of $\sim\frac{1}{\ln x}$ and therefore the expected number of primes less than
$n$ is given by:
$\pi(n)\sim\int_{2}^{n}\frac{1}{\ln x}dx\sim\frac{n}{\ln n}$ (36)
in complete agreement with the Prime Number Theorem.
## 5 Application to Cramér’s random model, Part II
Given the prime encoding $X_{n}$ with uniform source distribution, let’s
define the nth prime gap $G_{n}=p_{n+1}-p_{n}$ so we may consider the
cumulative probability:
$\sum_{k=1}^{G_{n}}P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)=1$ (37)
and its associated entropy:
$H_{n}=\sum_{k=1}^{G_{n}}-P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)\cdot\ln
P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)$ (38)
where $G_{n}<p_{n}$ due to Bertrand’s postulate and we’ll note that
$\exists k,P(p_{n+1}-p_{n}=k)=1\implies H_{n}=0$ (39)
so the entropy $H_{n}$ collapses if and only if we simultaneously measure the
values of both $p_{n}$ and $p_{n+1}$.
As the subsequence $\\{x_{p_{n+k}}\\}_{k=1}^{G_{n}}$ halts at $k\in[1,G_{n}]$
where $x_{p_{n}+k}=1$, there are at most $G_{n}$ possible ways for this
sequence to halt. Furthermore, in order to bound the halting probability
$P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)$ we may use the density formula (36) to
consider its components:
$\forall k\sim
U([1,G_{n}]),P(p_{n+1}=p_{n}+k)=P(p_{n}=p_{n+1}-k)\sim\frac{1}{\ln p_{n}}$
(40)
$\forall k\sim U([1,G_{n}]),P(x_{p_{n}+k}=1)\sim\frac{1}{\ln p_{n}}$ (41)
where
$P(x_{p_{n}+k}=1\land p_{n+1}-p_{n}=k)\geq P(x_{p_{n}+k}=1)\cdot
P(p_{n+1}=p_{n}+k)$ (42)
Using (40),(41) and (42) we may then derive the lower bound:
$\forall k\sim U([1,G_{n}]),P(x_{p_{n}+k}=1\land
p_{n+1}-p_{n}=k)\gtrsim\frac{1}{(\ln p_{n})^{2}}$ (43)
which implies:
$\frac{1}{(\ln
p_{n})^{2}}\lesssim\frac{1}{G_{n}}\sum_{k=1}^{G_{n}}P(x_{p_{n}+k}=1\land
p_{n+1}-p_{n}=k)=\frac{1}{G_{n}}$ (44)
and since $G_{n}=p_{n+1}-p_{n}$, we may conclude:
$p_{n+1}-p_{n}=\mathcal{O}((\ln p_{n})^{2})$ (45)
as conjectured by Harald Cramér in 1936 [1].
## 6 Details of the $\frac{1}{p}-\frac{1}{n}\approx\frac{1}{p}$ approximation
in Cramér’s model
If we define,
$\Lambda_{k}:=\text{set of }{\pi(\sqrt{n})\choose k}\text{ distinct subsets of
}\\{p_{k}\\}_{k=1}^{\pi(\sqrt{n})}$ (46)
then $\lvert\Lambda_{k}\rvert={\pi(\sqrt{n})\choose k}$ and we may derive the
absolute error:
$\Big{\lvert}\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}+\frac{1}{n})-\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\Big{\rvert}\leq\sum_{k=1}^{\pi(\sqrt{n})-1}\frac{1}{n^{k}}\sum_{\lambda\in\Lambda_{k}}\prod_{p\in\lambda}\big{(}1-\frac{1}{p}\big{)}+\frac{1}{n^{\pi(\sqrt{n})}}$
(47)
and given that:
$\pi(\sqrt{n})<\sqrt{n}\implies\frac{1}{n^{k}}\cdot{\pi(\sqrt{n})\choose
k}\leq\frac{1}{n^{k/2}}$ (48)
the absolute error satisfies the following inequality:
$\Delta_{n}=\Big{\lvert}\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}+\frac{1}{n}\big{)}-\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}\Big{\rvert}\leq\frac{1}{\sqrt{n}}+\frac{1}{n}$
(49)
so the relative error converges to zero:
$\lim_{n\to\infty}\frac{\Delta_{n}}{\prod_{p\leq\sqrt{n}}\big{(}1-\frac{1}{p}\big{)}}\leq\lim_{n\to\infty}\big{(}\frac{\ln
n}{\sqrt{n}}+\frac{\ln n}{n}\big{)}=0$ (50)
which provides us with the necessary justifications in our derivation of
Cramér’s random model, specifically formulas (32),(33) and (34).
## 7 The incompressibility of prime encodings as a fundamental law of
physics?
> Information is physical.-Rolf Landauer
Let’s suppose we get the world’s best theoretical physicists to design a
machine learning system whose aim is to predict the location of the Nth prime
given the locations of the first $N-1$ primes. What is the best case scenario
we can hope for?
Considering that the prime numbers have a maximum entropy distribution the
true positive rate of any such system can’t exceed 50%. This state of affairs
should hold true at all times regardless of technological progress, given what
is known about Turing Machines without access to Oracles. So this fundamental
limit might as well be a Physical Law.
This Law might even have Cosmological implications. In fact, let’s suppose
that the Universe emerged from a Singularity which would be consistent with
Big Bang Cosmology. At this precise instant, when the Universe and all of its
mathematical structure came into existence the most reasonable machine
learning system would have to assume a maximum entropy distribution over the
location of the primes. This would be equivalent to the application of Occam’s
razor, a principle that is generally applicable in Universes where Quantum
Information is conserved.
Thus, at this Singularity the Minimum Description Length of prime encodings
would have been defined relative to a Universal Turing Machine as follows:
$\mathbb{E}[K(X_{N})]\sim\pi(N)\cdot\ln N\sim N$ (51)
which is consistent with present-day observations i.e. the information-
theoretic analyses on pages 4 and 5. How might we explain this mysterious
coincidence?
In accordance with Big Bang Cosmology and the principle that all the Quantum
Information in the Universe is conserved we are led to a natural Cosmological
hypothesis concerning the distribution of primes. The maximum entropy
distribution of prime encodings may be understood as a mathematical signature
that time travelled from the earliest moments of the Big Bang.
The author concedes that such a Hypothesis would appear heretical to
scientists who haven’t considered the possibility that the Universe may be
simulated by a Universal Quantum Turing Machine.
## 8 Conclusion
The finite-state incompressibility of prime encodings is of general scientific
importance as it implies that it is not possible for any finite state machine
such as a machine learning model to infer the definition of prime numbers(i.e.
Unique Factorisation Theorem) from a prime encoding $X_{n}$ of any length $n$.
Equivalently, theoretical physicists can’t design a machine learning system
which has favourable odds of locating the next prime number i.e. a true-
positive rate greater than 50%. Therefore, we may conclude that machine
learning does not confer an advantage to a mathematician investigating the
distribution of primes.
Having said this, it is worth clarifying that this analysis does not preclude
the application of machine learning to derive number-theoretic insights. In
fact, Yang-Hui He recently found machine learning to be particularly effective
at identifying interesting structures in the setting of arithmetic geometry
and he suspects that it is for the following reason [11]:
> At the most basic level, every computation in algebraic geometry, be it a
> spectral sequence or a Gröbner basis, reduces to finding kernels and
> cokernels of sets of matrices (over $\mathbb{Z}$ or even over $\mathbb{C}$),
> albeit of quickly forbidding dimensions. Matrix/tensor manipulation is the
> heart of any neural network. Number theory, on the other hand, ultimately
> involves patterns of prime numbers which, as is well known, remain elusive.
which is both consistent with our findings and gives us reason for measured
optimism.
## Appendix A An invariance theorem for algorithmically random data
Let’s suppose we have a natural signal described by the process $X$:
$x_{n}\in\\{0,1\\},x_{n+1}=\varphi\circ x_{1:n}$ (52)
If we should use machine learning to approximate $\varphi$ given the datasets
$X_{N}^{\text{train}}=\\{x_{i}\\}_{i=1}^{N},X_{N}^{\text{test}}=\\{x_{i}\\}_{i=N+1}^{2N}$
such that for any $\hat{f}\in F_{\theta}$:
$\exists k\in[1,n-1],x_{n+1}=\hat{f}\circ
x_{n-k:n}\Rightarrow\delta_{\hat{f}(x_{n-k:n}),x_{n+1}}=1$ (53)
then $X_{N}$ is asymptotically incompressible if for large $N$ any solution to
the empirical risk minimisation problem:
$\hat{f}=\max_{f\in
F_{\theta}}\frac{1}{N-k}\sum_{n=k+1}^{N}\delta_{f(x_{n-k:n}),x_{n+1}}$ (54)
has an expected performance:
$\frac{1}{N-k}\sum_{n=N+k+1}^{2N-1}\delta_{\hat{f}(x_{n-k:n}),x_{n+1}}\leq\frac{1}{2}$
(55)
Furthermore, if the dataset is imbalanced i.e.
$\frac{1}{N}\sum_{i=1}^{N}x_{i}\neq\frac{1}{2}$ then we may generalise this
result by introducing the auxiliary definitions:
$y_{n}=x_{n+1}$ (56)
$\hat{y_{n}}=\hat{f}\circ x_{n-k:n}$ (57)
$\beta_{n}=\delta_{y_{n},\hat{y_{n}}}$ (58)
$N_{1}=\sum_{n=N+k+1}^{2N}\delta_{y_{n},1}$ (59)
$N_{0}=\sum_{n=N+k+1}^{2N}\delta_{y_{n},0}$ (60)
and so for large $N$, we have:
$\mathcal{L}_{N}[\hat{f}]=\min\Big{[}\frac{1}{N_{0}}\sum_{n=N+k+1}^{2N-1}\delta_{y_{n},0}\cdot\beta_{n},\frac{1}{N_{1}}\sum_{n=N+k+1}^{2N-1}\delta_{y_{n},1}\cdot\beta_{n}\Big{]}\leq\frac{1}{2}$
(61)
and therefore:
$\forall\hat{f}\in
F_{\theta},\lim_{N\to\infty}P(\mathcal{L}_{N}[\hat{f}]>\frac{1}{2})=0$ (62)
Finally, as these results are invariant to transformations that preserve the
phase-space dimension of $X$, this theorem may be used as an overfitting test
for binary sequences that are finite-state incompressible.
## Appendix B Revisiting the unreasonable effectiveness of mathematics
62 years since Eugene Wigner’s highly influential essay on the unreasonable
effectiveness of mathematics in the natural sciences, it may be time for a re-
appraisal. On balance, with important theoretical advances in algorithmic
information theory and Quantum Computation it appears that the remarkable
effectiveness of mathematics in the natural sciences is quite reasonable.
By effectiveness, I am specifically referring to Wigner’s observation that
mathematical laws have remarkable generalisation power.
### B.1 An information-theoretic perspective
An acute observer will note that the same mathematical laws with remarkable
generalisation power in the natural sciences are also constrained by Occam’s
razor. Given two computable theories, Einstein explicitly stated that a
physicist ought to choose the simplest theory that yields negligible
experimental error:
> It can be scarcely denied that the supreme goal of all theory is to make the
> irreducible basic elements as simple and as few as possible without having
> to surrender the adequate representation of a single datum of
> experience.-Einstein(1933)
In fact, from an information-theoretic perspective the remarkable
generalisation power of mathematical laws in the natural sciences is a direct
consequence of the effectiveness of Occam’s razor.
### B.2 The Law of Conservation of Information
From an information-theoretic perspective, a Universe where Occam’s razor is
generally applicable is one where information is generally conserved. This law
of conservation of information which dates back to von Neumann essentially
states that the von Neumann entropy is invariant to Unitary transformations.
This is meaningful within the framework of Everettian Quantum Mechanics as a
density matrix may be assigned to the state of the Universe. This way
information is conserved as we run a simulation of the Universe forward in
time.
Moreover, given that Occam’s razor has an appropriate formulation within the
context of algorithmic information theory as the Minimum Description Length
principle, this information-theoretic perspective generally presumes that the
Universe itself may be simulated by a Universal Turing Machine.
### B.3 The Physical Church-Turing thesis
The research of David Deutsch(and others) on the Physical Church-Turing thesis
explains how a Universal Quantum computer may simulate the laws of physics.
This is consistent with the general belief that Quantum Mechanics may be used
to simulate all of physics so the most important contributions to the Physical
Church-Turing thesis have been via theories of quantum computation.
More importantly, the Physical Church-Turing thesis provides us with a
credible explanation for the remarkable effectiveness of mathematics in the
natural sciences.
### B.4 What is truly remarkable
If we view the scientific method as an algorithmic search procedure then there
is no reason, a priori, to suspect that a particular inductive bias should be
particularly powerful. This much was established by David Wolpert in his No
Free Lunch theorems [22].
On the other hand, the history of the natural sciences indicates that Occam’s
razor is remarkably effective. The effectiveness of this inductive bias has
recently been used to explain the generalisation power of deep neural networks
when applied to data that is finite-state compressible [23].
## References
[1] Cramér, H. "On the Order of Magnitude of the Difference Between
Consecutive Prime Numbers." Acta Arith. 2, 23-46, 1936.
[2] Hardy, G. H.; Ramanujan, S. (1917), "The normal number of prime factors of
a number n", Quarterly Journal of Mathematics
[3] Turán, Pál (1934), "On a theorem of Hardy and Ramanujan", Journal of the
London Mathematical Society
[4] Yufei Zhao. The Probabilistic Method in Combinatorics. 2019.
[5] F. Mertens. J. reine angew. Math. 78 (1874)
[6] Olivier Rioul. This is IT: A Primer on Shannon’s Entropy and Information.
Séminaire Poincaré. 2018.
[7] E.T. Jaynes. Information Theory and Statistical Mechanics. The Physical
Review. 1957.
[8] Lance Fortnow. Kolmogorov Complexity. 2000.
[9] Aidan Rocke (https://mathoverflow.net/users/56328/aidan-rocke),
information-theoretic derivation of the prime number theorem, URL (version:
2021-04-08): https://mathoverflow.net/q/384109
[10] Aidan Rocke (https://mathoverflow.net/users/56328/aidan-rocke), Egyptian
number theory, URL (version: 2021-06-22): https://mathoverflow.net/q/395939
[11] Yang-Hui He. Deep-Learning the Landscape. Arxiv. 2018.
[12] Aidan Rocke (https://cstheory.stackexchange.com/users/47594/aidan-rocke),
An invariance theorem for algorithmically random data in statistical learning,
URL (version: 2021-02-22): https://cstheory.stackexchange.com/q/48452
[13] Eugene Wigner. The Unreasonable Effectiveness of Mathematics in the
Natural Sciences. 1960.
[14] David Deutsch. Quantum theory, the Church–Turing principle and the
universal quantum computer. 1985.
[15] Peter D. Grünwald. The Minimum Description Length Principle . MIT Press.
2007.
[16] A. N. Kolmogorov Three approaches to the quantitative definition of
information. Problems of Information and Transmission, 1(1):1–7, 1965
[17] G. J. Chaitin On the length of programs for computing finite binary
sequences: Statistical considerations. Journal of the ACM, 16(1):145–159,
1969.
[18] R. J. Solomonoff A formal theory of inductive inference: Parts 1 and 2.
Information and Control, 7:1–22 and 224–254, 1964.
[19] Michael Nielsen. Interesting problems: The Church-Turing-Deutsch
Principle. 2004. https://michaelnielsen.org/blog/interesting-problems-the-
church-turing-deutsch-principle/
[20] Marcus Hutter et al. (2007) Algorithmic probability. Scholarpedia,
2(8):2572.
[21] The Evolution of Physics, Albert Einstein and Leopold Infeld, 1938,
Edited by C.P. Snow, Cambridge University Press.
[22] Wolpert, D.H., Macready, W.G. (1997), "No Free Lunch Theorems for
Optimization", IEEE Transactions on Evolutionary Computation 1, 67.
[23] Guillermo Valle Pérez, Chico Camargo, Ard Louis. Deep Learning
generalizes because the parameter-function map is biased towards simple
functions. 2019.
|
# Thresh : A Unified, Customizable and Deployable Platform
for Fine-Grained Text Evaluation
David Heineman, Yao Dou, Wei Xu
School of Interactive Computing, Georgia Institute of Technology
{david.heineman<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Fine-grained, span-level human evaluation has emerged as a reliable and robust
method for evaluating text generation tasks such as summarization,
simplification, machine translation and news generation, and the derived
annotations have been useful for training automatic metrics and improving
language models. However, existing annotation tools implemented for these
evaluation frameworks lack the adaptability to be extended to different
domains or languages, or modify annotation settings according to user needs.
And the absence of a unified annotated data format inhibits the research in
multi-task learning. In this paper, we introduce Thresh , a unified,
customizable and deployable platform for fine-grained evaluation. By simply
creating a YAML configuration file, users can build and test an annotation
interface for any framework within minutes – all in one web browser window. To
facilitate collaboration and sharing, Thresh provides a community hub that
hosts a collection of fine-grained frameworks and corresponding annotations
made and collected by the community, covering a wide range of NLP tasks. For
deployment, Thresh offers multiple options for any scale of annotation
projects from small manual inspections to large crowdsourcing ones.
Additionally, we introduce a Python library to streamline the entire process
from typology design and deployment to annotation processing. Thresh is
publicly accessible at https://thresh.tools.
## 1 Introduction
As modern large language models are able to generate human-level quality text
Brown et al. (2020); OpenAI (2023), the evaluation of these models becomes
increasingly challenging. Recent work has shown traditional surface-level
evaluation methods such as pairwise comparison or Likert-scale ratings become
less reliable and robust (Clark et al., 2021; Maddela et al., 2023) due to the
close performance of these LLMs. To address this, several fine-grained human
evaluation frameworks have been proposed for various tasks such as open-ended
generation (Dou et al., 2022a), text simplification Heineman et al. (2023),
and machine translation Freitag et al. (2021). In these frameworks, annotators
identify and annotate specific spans corresponding to quality or errors in the
generated text.
Figure 1: Examples of fine-grained evaluation frameworks implemented on
Thresh. In order: SALSA (Heineman et al., 2023), MQM (Freitag et al., 2021),
Scarecrow (Dou et al., 2022a).
Framework | Task | Released
---|---|---
Evaluation | |
MQM (Freitag et al., 2021) | Translation | ✓
FRANK (Pagnoni et al., 2021) | Summarization | ✓
SNaC (Goyal et al., 2022b) | Narrative Summarization | ✓
Scarecrow (Dou et al., 2022a) | Open-ended Generation | ✓
SALSA (Heineman et al., 2023) | Simplification | ✓
ERRANT (Bryant et al., 2017) | Grammar Error Correction | ✗
FG-RLHF (Wu et al., 2023) | Fine-Grained RLHF | ✓
Inspection | |
MultiPIT (Dou et al., 2022b) | Paraphrase Generation | ✗
CWZCC | Zamboanga Chavacano | ✗
(Himoro and Pareja-Lora, 2020) | Spell Checking
Propaganda | Propaganda Analysis | ✓
(Da San Martino et al., 2019)
arXivEdits (Jiang et al., 2022) | Scientific Text Revision | ✓
Table 1: Existing typologies currently implemented on Thresh. Released
indicates whether the annotated data is released. Corresponding links on
Thresh for each framework can be found in Table 2 in the Appendix.
However, each of these evaluation frameworks releases its own dedicated
annotation interface that is difficult to modify or adapt to different
evaluation schemes, thus limiting the customizablility. For example,
Scarecrow’s typology Dou et al. (2022a), which is designed for open-ended text
generation for news, may require modifications when applied to other domains
such as story or scientific writing. Frameworks like MQM Freitag et al. (2021)
only allow selections of the spans in the target sentence, restricting the
ability to select the associated source spans in error categories such as
mistranslation. Furthermore, modern LLMs are ideally evaluated on multiple
tasks OpenAI (2023), but the lack of a unified annotation tool makes this
process inconvenient. The absence of a unified annotation format hinders the
research in multi-task learning with fine-grained human feedback, considering
the recent success of multi-task instruction fine-tuning for large language
models Wei et al. (2021); Sanh et al. (2021).
To this end, we present Thresh : a unified and customizable platform for
building, distributing and orchestrating fine-grained human evaluation for
text generation in an efficient and easy-to-use manner. Our platform allows
users to create, test and deploy an evaluation framework within minutes, all
in a single browser window. Thresh also serves as a community hub for fine-
grained evaluation frameworks and annotation data, all presented in a unified
format. Figure 1 displays three examples of evaluation frameworks built on
Thresh. The following are the design principles of Thresh:
* •
Unified: Thresh standardizes fine-grained evaluation into two key components:
span selection and span annotation. Users can easily implement any framework
by writing a YAML template file (see Figure 5), and Thresh will build the
corresponding annotation interface. All resulting annotations adhere to a
consistent JSON format.
* •
Customizable: Thresh offers extensive customization to meet a wide range of
user needs. This includes different span selection methods from subword to
word-level, diverse annotation options such as simple yes/no questions and
text boxes to handle arbitrary typologies, as well as customized interface
elements in any language.
* •
Deployable: Thresh supports a range of deployment options for annotation
projects of various scales. Small-scale linguistic inspections can be directly
hosted on the platform. For larger projects, users can host their template in
a GitHub repository and link it to Thresh. Thresh is also compatible with
crowdsourcing platforms such as Prolific111https://www.prolific.co and Amazon
MTurk222https://www.mturk.com.
* •
Contributive: Thresh also operates as a community hub where users can
contribute and access a wide variety of fine-grained evaluation frameworks and
their annotation data. As of now, it includes 11 frameworks as displayed in
Table 1.
* •
End-to-End: Beyond facilitating the creation and deployment of evaluation
frameworks, Thresh streamlines every step of the annotation process. It offers
functions for authors to publish their typologies as research artifacts and a
supplementary Python library, released under the Apache 2.0 license, to help
data collection.333https://www.pypi.org/project/thresh
## 2 Related Work
Fine-grained Text Evaluation. Given the limitations of traditional human
evaluation methods such as Likert-scale and pairwise comparison in the era of
LLMs, many recent studies have proposed fine-grained human evaluation
frameworks. Dou et al. (2022a) introduces Scarecrow to capture error spans in
open-ended text generation for news, MQM Freitag et al. (2021) identifies
errors in machine translation, and FRANK Pagnoni et al. (2021) captures
factual errors in abstractive text summarization. We list other evaluation and
inspection typologies in Table 1. However, these existing frameworks usually
develop their own annotation tools which lack customizability and
universality, making them difficult to adapt to other languages or domains, or
to new annotation settings. Recently, Goyal et al. (2022a) proposes FALTE,
customizable span-level error highlighting for long text evaluation, but it
only includes a subset of features offered by Thresh, limiting its ability to
implement complex typologies such as SALSA Heineman et al. (2023).
Specifically, FALTE only highlights errors without rating their severity or
efficacy, does not support multi-span or composite selection, and cannot
select overlapping spans. Moreover, its lack of a tree structure can make the
interface cluttered if there are more than a handful of categories. Thresh
instead builds unified and customizable support across task setups.
Annotation Tool. Accessible and replicable annotation tools have been a
persistent goal for NLP tasks. Stenetorp et al. (2012) introduces BRAT, the
first web browser-based annotation tool and Yimam et al. (2013) further
improves BRAT on speed and label configuration. In recent years, a new
generation of universal annotation tools have been introduced by academia and
industry, including Prodigy Montani and Honnibal (2018), Doccano Nakayama et
al. (2018), LightTag Perry (2021), and POTATO (Pei et al., 2022). Focusing on
universality, these tools allow authors to add custom UI elements such as
multiple choice questions, text boxes or pairwise comparison. However, these
surface-level annotation options are not sufficient to implement complex
typology setups demanded by fine-grained evaluation, which are typically
structured by decision trees Heineman et al. (2023). Thresh addresses this gap
by recursively building the interface, which allows for nested questions.
Besides, Thresh encourages sharing and reproducibility by providing a
community hub where users can upload their new or use existing fine-grained
frameworks and annotated data.
Span-level Annotation. Span-level annotation has a long history across NLP
tasks. In Named Entity Recognition (NER), spans are selected and labeled as
names of persons, organizations, locations, or other entities (Tjong Kim Sang
and De Meulder, 2003). Word alignment focuses on selecting aligned words or
phrases between two parallel corpora across languages (Och and Ney, 2003), or
within monolingual tasks (Lan et al., 2021). Span selection has also been used
for question answering such as in SQuAD Rajpurkar et al. (2016), where the
answer is defined by a span within the document context. Furthermore,
extractive text summarization (Hermann et al., 2015) highlights the spans that
summarizes a given document. With a goal of understanding where and how text
generation succeeds or fails, fine-grained text evaluation selects spans that
are either quality or error in generated text. These selected spans are then
annotated following a complex typology and rated on the severity of errors or
efficacy of high-quality content Freitag et al. (2021); Dou et al. (2022a);
Heineman et al. (2023).
## 3 Fine-Grained Text Evaluation
Thresh formulates fine-grained text evaluation as two components: span
selection and span annotation. During development, users define their
annotation typology and interface features using a YAML template (see Sec 4
and Fig 5 for more details). Based on the configuration, Thresh then
constructs an annotation interface that integrates both components, as
illustrated in Figures 2 and 3.
Figure 2: The span selection component of Thresh, customized with the SALSA
Heineman et al. (2023) typology as an example.
Figure 3: The span annotation component of Thresh, customized with the SALSA
Heineman et al. (2023) typology as an example.
Figure 4: The left figure shows a grammar error typology with 35 categories
for contemporary written Zamboangueño Chabacano, a variant of Philippine
Creole Spanish Himoro and Pareja-Lora (2020). The center figure shows its
annotation interface built on Thresh, highlighting the ability for Thresh to
support complex, recursive annotation trees. The right figure shows the Python
serialization for the annotation, generated by the Thresh library.
### 3.1 Span Selection
In each annotation instance, we have the source, target and context. For
example, in open-ended text generation Zellers et al. (2019), the source is a
starting sentence and the target is a model-generated continuation. In text
simplification Xu et al. (2016), the source would be a complex sentence or
paragraph, and the target would be the generated simplification. The context
holds additional relevant information, such as a prompt instruction, a
retrieved Wikipedia page, or a dialogue history. During the span selection
stage, annotators select relevant spans, which we refer to as Edits, in the
source and target, following the edit category definitions outlined in the
typology, as illustrated in Figure 2.
Selection Type. For each edit category, users can specify one of three
selection types: single-span, multi-span, or composite – the latter grouping
together multiple single-span or multi-span selections. Multi-span selection
is well-suited for edits that impact multiple parts of the source or target,
e.g., the “Redundant” error in Scarecrow Dou et al. (2022a), which requires
selecting both the repetitive spans and their antecedents. Composite
selections are ideal for high-level edits performed as a combination of
several low-level edits, e.g., the “Structure” edit in SALSA Heineman et al.
(2023). Users can also customize each edit category to be selectable not only
on the target, but also on the source (e.g., “Deletion” edit), or on both
(e.g., “Substitution” edit), useful for text revision tasks.
Selection Boundary. Many span-selection interfaces define selection boundaries
as each character, which can inadvertently lead to partial word selections and
slow the annotation process. Dou et al. (2022a) proposes a solution that
“snaps” the selection to the nearest whitespace, but this approach is limited
in: (1) punctuation gets selected with adjacent words, even when this is not
intended by annotators, (2) languages with no whitespace boundaries between
words (e.g., Chinese) cannot be supported and (3) the annotation data cannot
be perfectly translated to training data for token-level labeling tasks. We
therefore introduce sub-word boundaries as a third option, in which users can
use any LLMs tokenizer of their choice (such as RobertaTokenizer from
Transformers444https://www.github.com/huggingface/tokenizers) to tokenize the
data and specify a boundary: subword flag in the YAML configuration file.
### 3.2 Span Annotation
In the YAML file, users define the typology in a decision tree structure to
further categorize the selected spans into fine-grained types. Unlike previous
work which presents every fine-grained edit types on the same initial level,
Thresh recursively compiles the annotation interface based on a tree.
Annotators thus will answer a series of questions or follow-up questions under
each edit type, as shown in Figure 3. This tree structure enables support for
complex error typologies. An example of this can be seen in Figure 4, where we
implement a 35-category typology for a grammar error correction task. Thresh
supports binary, three and five-scale questions with customized label names,
as well as text boxes for tasks that require human post-editing or
explanations. With these features, our interface supports complex annotation
schemes in a flexible and easily extensible way.
We also give users the option of only enabling one of the two above
components. This allows annotation for word/span alignment tasks Sultan et al.
(2014) (where no annotation is needed) or two-stage annotation, where one set
of annotators selects spans and then another set labels them.
### 3.3 Additional Features
Adjudication View. Using the adjudication flag, users can deploy two or three
interfaces side-by-side, allowing adjudicators to inspect annotators’ quality
by comparing multiple candidate annotations simultaneously.
Multi-Language Support. Fine-grained evaluation has seen almost exclusive
attention to English tasks (Huidrom and Belz, 2022). To smoothen the
deployment barrier for multilingual fine-grained evaluation, all interface
elements can be overridden to suit any language. For our default interface
text, we support 14 translations which can be enabled out-of-the-box by adding
a language flag: zh, en, es, hi, pt, bn, ru, ja, vi, tr, ko, fr and ur.
Instructions. Users may write interface instructions with Markdown formatting,
which allows for links, pictures and inline code. They have the option to
display their instructions as a pop-up modal, or prepend the text above the
interface.
Paragraph-level Annotation. Users can specify an additional context_before or
context_after field to add paragraph-level context. By breaking evaluation
down to each individual sentence, authors can reduce the cognitive load
required for lengthy annotation tasks such as identifying errors in long-form
summarization (Goyal et al., 2022a).
## 4 Interactive Interface Builder
Figure 5: Thresh deployment workflow. Users build and test their template and
then deploy with one of 4 options.
To alleviate the time consuming process of customizing and hosting front-end
code — even building custom databases in some cases — Thresh implements an in-
browser interface builder, which allows users to create, test and deploy a
fine-grained interface within a single web browser page, as depicted in Figure
5. We design our builder to be intuitive to anyone familiar with popular
cloud-based text formatting tools such as Overleaf.555http://www.overleaf.com
Users use a YAML template to construct their interface and provide data with a
JSON file. The Compile button allows users to preview their interface, and the
Deploy button presents instructions for different deployment options, which
are described in §5.
Template Hub. As Thresh aims to facilitate easy use and distribution of fine-
grained evaluation frameworks, it provides a template hub that makes it simple
for any NLP practitioner to access a framework with their own data. Alongside
the 10 tutorial templates that explain each interface feature, the annotation
builder currently includes 11 widely used inspection and evaluation typologies
across major text generation tasks. Table 1 (on Page 2) lists each framework,
its associated task and link to our implementation.
To upload a framework to the hub, users can create a GitHub pull request with
their typology’s YAML file, which is merged publicly. We also include other
features to facilitate sharing and replication. Users can add a citation flag
along with a BibTex citation, which creates a Cite this Typology button in the
annotation builder, a paper_link flag, which adds a link to their research
paper in the builder and on deployment, and a demo_data_link flag which
creates a View Demo Data button to allow viewers to use the interface with
example data.
For testing, users can paste data into the interface builder interactively,
and for deployment can link to data files. Data can be blank or come with
existing annotations, in which case the annotations will be appropriately
parsed, verified and rendered.
Unified Data Model. As shown in Table 1 on Page 2, many existing frameworks
have released their annotated data, but in varied formats. To ensure
compatibility, we create conversion scripts that adapt these annotations to
our unified format. Our scripts are designed to be bidirectional, meaning data
published for these typologies can be converted to our format and back without
data loss. Our unified fine-grained data format allows smooth transfer of
analysis, agreement calculation and modeling code between different projects.
We believe this will support research in learning with multi-task fine-grained
training setups or model feedback. Like framework templates, users can upload
their annotated data to the hub via a GitHub pull request.
## 5 Deployment
Managing and collecting fine-grained annotations becomes bulky at scale, we
thus release supplementary tools to deploy interfaces quickly or
programmatically, and integrate loading annotations directly into Python. This
includes the thresh library666https://www.pypi.org/project/thresh, which is
useful for compiling interfaces and loading annotations. We support the
following 4 types of deployment as shown in Figure 5:
* •
Hosted: Best for small-scale inspection or data exploration, users can
download a file that bundles the data and template together. Then, users can
upload this file to thresh.tools/annotate to begin annotations immediately.
* •
Serverless: Users upload their YAML template to a public repository such as
GitHub or HuggingFace, and link their template to thresh.tools through a URL
parameter: gh or hf respectively. Users can also link data via the d
parameter. In addition, we release demo code for users to host their interface
on their own domain without cloning the Thresh repository.
* •
Python: For large scale projects, users can programmatically generate and
deploy templates using the create_template functionality provided in the
thresh library. This helps for projects with a large number of templates, such
as annotation in multiple languages. Additionally, integration with Python
allows a direct connection from model generation to annotation processing,
supporting the creation of workflows like fine-grained RLHF (Wu et al., 2023).
* •
Crowdsource: If the data collection process is mishandled, annotation by
crowdworkers can lead to poorly standardized or noisy data (Karpinska et al.,
2021; Veselovsky et al., 2023). To assist annotation quality control, we
publish tools to encourage best practices when using crowdsource platforms.
Our crowdsource deployment workflow includes example code for interactive,
multi-stage tutorials to create qualification tasks and the ability to disable
interface actions unless edits are fully annotated or selected. To guide users
through the process, we provide step-by-step tutorials for deployment on both
Prolific and Amazon Mechanical Turk.
Python Serialization. Compared to previous work that simply exports JSON
annotations, leaving the laborious step of parsing annotations into a useful
format to practitioners, our supplementary thresh library includes
functionality for loading and combining annotation files to simplify the data
ingestion process. For example, the load_annotations function merges multiple
data files, serializes the data into Python objects, and evaluates whether the
data collected is consistent with the configuration used to load the data.
## 6 Conclusion
We present Thresh , a unified, customizable, and deployable platform for fine-
grained text evaluation. Thresh offers extensive customization via a simple
YAML configuration file, and facilitates a community hub for sharing
frameworks and annotations. The platform also ensures seamless deployment for
any scale of annotation projects and introduces a Python library to further
ease the process from typology design to annotation processing.
## Ethical Considerations
We do not anticipate any ethical issues pertaining to the topics of fine-
grained evaluation supported by our interface. Nevertheless, as Thresh lowers
the barrier to fine-grained evaluation, vast ethical responsibility falls upon
practitioners using our platform to prevent the exploitation of crowdsource
workers, through fair pay (Fort et al., 2011) and safeguards against exposure
to harmful or unethical content (Shmueli et al., 2021). As task difficulty and
complexity scales with the granularity of data collected, increasing care must
be taken for training annotators adequately and to scale pay accordingly
(Williams et al., 2019).
## Acknowledgments
This research is supported in part by the NSF awards IIS-2144493 and
IIS-2112633, ODNI and IARPA via the HIATUS program (contract
2022-22072200004). The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies, either expressed or implied, of NSF, ODNI, IARPA, or the U.S.
Government. The U.S. Government is authorized to reproduce and distribute
reprints for governmental purposes notwithstanding any copyright annotation
therein.
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Framework | Task | Released | Link
---|---|---|---
Evaluation | | |
MQM (Freitag et al., 2021) | Translation | ✓ | thresh.tools/mqm
FRANK (Pagnoni et al., 2021) | Summarization | ✓ | thresh.tools/frank
SNaC (Goyal et al., 2022b) | Narrative Summarization | ✓ | thresh.tools/snac
Scarecrow (Dou et al., 2022a) | Open-ended Generation | ✓ | thresh.tools/scarecrow
SALSA (Heineman et al., 2023) | Simplification | ✓ | thresh.tools/salsa
ERRANT (Bryant et al., 2017) | Grammar Error Correction | ✗ | thresh.tools/errant
FG-RLHF (Wu et al., 2023) | Fine-Grained RLHF | ✓ | thresh.tools/fg-rlhf
Inspection | | |
MultiPIT (Dou et al., 2022b) | Paraphrase Generation | ✗ | thresh.tools/multipit
CWZCC (Himoro and Pareja-Lora, 2020) | Zamboanga Chavacano Spell Checking | ✗ | thresh.tools/cwzcc
Propaganda (Da San Martino et al., 2019) | Propaganda Analysis | ✓ | thresh.tools/propaganda
arXivEdits (Jiang et al., 2022) | Scientific Text Revision | ✓ | thresh.tools/arxivedits
Table 2: Existing typologies implemented on Thresh with their associated link.
Released indicates whether the annotated data is released.
|
simulates the overall proportion of heads in a run of $n$ separate coin flips.
###### Definition 8.1 (Binomial Distribution)
A random variable $X$ is said to follow the binomial distribution with
parameter $p\in(0,1)$ and $n\in\mathbb{N}$, denoted by
$X\sim\mathrm{Binom}(n,p)$ if
$f(x;n,p)=\binom{n}{x}p^{x}(1-p)^{n-x}.$
The mean and variance of $X\sim\mathrm{Binom}(a,b)$ are given by
$\mathrm{E}[X]=np,\qquad\mathrm{Var}[X]=np(1-p).$
Figure 8.1 compares different parameters of $p$ with $n=10$ for the binomial
distribution.
Figure 8.1: Binomial distribution probability mass functions for different
values of the parameters $p$ with $n=10$.
And we will see the prior under this model is the probability density function
of Beta distribution:
$\mathrm{prior}=\mathrm{Beta}(\theta|a,b)=p(\theta|a,b)=\frac{1}{B(a,b)}\theta^{a-1}(1-\theta)^{b-1}\mathds{1}(0\leq\theta\leq
1),$
where $B(a,b)$ is the Euler’s beta function and it can be seen as a
normalization term.
###### Definition 8.2 (Beta Distribution)
A random variable $X$ is said to follow the beta distribution with parameter
$a>0$ and $b>0$, denoted by $X\sim\mathrm{Beta}(a,b)$ if
$f(x;a,b)=\left\\{\begin{aligned}
&\frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1},&\mathrm{\,\,if\,\,}0\leq x\leq 1.\\\
&0,&\mathrm{\,\,otherwise\,\,},\end{aligned}\right.$
where $B(a,b)$ is the Euler’s beta function and it can be seen as a
normalization term. Equivalently, $B(a,b)$ can be obtained by
$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},$
where $\Gamma(\cdot)$ is the gamma function (Definition 3.1, p. 3.1). The mean
and variance of $X\sim\mathrm{Beta}(a,b)$ are given by
$\mathrm{E}[X]=\frac{a}{a+b},\qquad\mathrm{Var}[X]=\frac{ab}{(a+b+1)(a+b)^{2}}.$
Figure 8.2 compares different parameters of $a,b$ for beta distribution. When
$a=b=1$, the beta distribution is just a uniform distribution in the range of
0 and 1.
We put a Beta prior over the parameter $\theta$ of the Bernoulli distribution.
The posterior is obtained by
$\displaystyle\mathrm{posterior}=p(\theta|x_{1:n})$ $\displaystyle\propto
p(x_{1:n}|\theta)p(\theta|a,b)$ $\displaystyle=\theta^{\sum
x_{i}}(1-\theta)^{n-\sum
x_{i}}\times\frac{1}{B(a,b)}\theta^{a-1}(1-\theta)^{b-1}\mathds{1}(0<\theta<1)$
$\displaystyle\propto\theta^{a+\sum x_{i}-1}(1-\theta)^{b+n-\sum
x_{i}-1}\mathds{1}(0<\theta<1)$ $\displaystyle\propto Beta(\theta|a+\sum
x_{i},b+n-\sum x_{i}).$
Figure 8.2: Beta distribution probability density functions for different
values of the parameters $a,b$. When $a=b=1$, the beta distribution is the
uniform distribution in the range of 0 and 1.
We find that the posterior distribution shares the same form as the prior
distribution. When this happens, we call the prior conjugate prior. The
conjugate prior has a nice form such that it is easy to work with for
computing the posterior probability density function and its derivatives,
sampling from the posterior.
###### Remark 8.1 (Prior Information in Beta-Bernoulli Model)
A comparison of the prior and posterior formulation would find that the
hyperparameter $a$ is the prior number of $1$’s in the output and $b$ is the
prior number of 0’s in the output. And $a+b$ is the prior information about
the sample size.
###### Remark 8.2 (Bayesian Estimator)
From this example of the Beta-Bernoulli model, like maximum likelihood
estimator and method of moment (MoM, i.e., using the moment information to get
the model parameter.), the Bayesian model is also a kind of point estimator.
But Bayesian models output a probability of the parameter of interest
$p(\theta|x_{1:n})$.
When we want to predict for new coming data, we do not give out the prediction
by a direct model $p(x_{n+1}|\theta)$. But rather an integration:
$p(x_{n+1}|x_{1:n})=\int p(x_{n+1}|\theta)p(\theta|x_{1:n})d\theta.$
In another word, $x_{n+1}$ is dependent on $x_{1:n}$. $x_{1:n}$ provide
information on $\theta$, which in turn provides information on $x_{n+1}$
(i.e., $x_{1:n}\rightarrow\theta\rightarrow x_{n+1}$).
###### Example 8.2.1 (Amount of Data Matters)
Suppose we have three observations for the success in Bernoulli distribution:
1). 10 out of 10 are observed to be success (1’s);
2). 48 out of 50 are observed to be success (1’s);
3). 186 out of 200 are observed to be success (1’s).
So, what is the probability of success in the Bernoulli model? Normal answer
to case 1, 2, 3 are 100%, 96% and 93% respectively. But an observation of 10
inputs is rather a small amount of data and noise can make it less convincing.
Suppose we put a $\mathrm{Beta}(1,1)$ (a uniform distribution, see Figure 8.2)
prior over the Bernoulli distribution. The posterior probability of success
for each case would be $\frac{11}{12}=91.6\%$, $\frac{49}{52}=94.2\%$ and
$\frac{187}{202}=92.6\%$ respectively. Now we find the case 1 has less
probability of success compared to case 2.
A Bayesian view of the problem naturally incorporates the amount of data as
well as its average. This special case shown here is also called the Laplace’s
rate of succession (Ollivier, 2015). Laplace’s “add-one” rule of succession
modifies the observed frequencies in a sequence of successes and failures by
adding one to the observed counts. This improves prediction by avoiding zero
probabilities and corresponds to a uniform Bayesian prior on the parameter.
$\square$
This example above shows that Bayesian models consider prior information on
the parameters in the model making it particularly useful to regularize
regression problems where data information is limited. And this is why the
Bayesian approach gains worldwide attention for decades. The prior information
$p(\theta)$ and likelihood function $p(x|\theta)$ represent a rational
person’s belief, and then the Bayes’ rule is an optimal method of updating
this person’s beliefs about $\theta$ given new information from the data
(Fahrmeir et al., 2007; Hoff, 2009). The prior information given by
$p(\theta)$ might be wrong if it does not accurately represent our prior
beliefs. However, this does not mean that the posterior $p(\theta|x)$ is not
useful. A famous quote is “all models are wrong, but some are useful” (Box and
Draper, 1987). If the prior $p(\theta)$ approximates our beliefs, then the
posterior $p(\theta|x)$ is also a good approximation to posterior beliefs.
### 8.3 Bayesian Linear Model: Zero-Mean Prior
We now present the Bayesian methods for linear models. Assume
$\boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}$ where
$\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\boldsymbol{I})$
and $\sigma^{2}$ is fixed. As discussed in Section 2 (p. 2), this additive
Gaussian noise assumption gives rise to the likelihood. Let
$\mathcal{X}(\bm{x}_{1:n})=\\{\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{n}\\}$ be
the observations of $n$ data points,
$\mathrm{likelihood}=\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2}\sim\mathcal{N}(\bm{X}\boldsymbol{\beta},\sigma^{2}\bm{I}).$
Suppose we specify a Gaussian prior with zero-mean over the weight parameter
$\mathrm{prior}=\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0}).$
By the Bayes’ theorem
“$\mathrm{posterior}\propto\mathrm{likelihood}\times\mathrm{prior}$”, we get
the posterior
$\displaystyle\mathrm{posterior}$
$\displaystyle=p(\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2})$
$\displaystyle\propto
p(\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2})p(\boldsymbol{\beta}|\boldsymbol{\Sigma}_{0})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left(-\frac{1}{2\sigma^{2}}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right)$
$\displaystyle\,\,\,\,\,\,\,\times\frac{1}{(2\pi)^{n/2}|\boldsymbol{\Sigma}_{0}|^{1/2}}\exp\left(-\frac{1}{2}\boldsymbol{\beta}^{\top}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}\right)$
$\displaystyle\propto\exp\left(-\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_{1})^{\top}\boldsymbol{\Sigma}_{1}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{1})\right),$
where
$\boldsymbol{\Sigma}_{1}=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}$
and
$\boldsymbol{\beta}_{1}=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{y})$.
Therefore the posterior distribution is also a Gaussian distribution (same
form as the prior distribution):
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{1},\boldsymbol{\Sigma}_{1}).$
##### A word on the notation
Note that we use $\\{\boldsymbol{\beta}_{1},\boldsymbol{\Sigma}_{1}\\}$ to
denote the posterior mean and posterior covariance in the zero-mean prior
model. Similarly, the posterior mean and posterior covariance in semi-
conjugate prior and full-conjugate prior models will be denoted as
$\\{\boldsymbol{\beta}_{2},\boldsymbol{\Sigma}_{2}\\}$ and
$\\{\boldsymbol{\beta}_{3},\boldsymbol{\Sigma}_{3}\\}$ respectively (see
sections below).
In this case, we do not need to assume $\boldsymbol{X}$ has full rank
generally. Note further that if we assume $\bm{X}$ has full rank, in the
limit, when $\boldsymbol{\Sigma}_{0}\rightarrow\boldsymbol{0}$,
$\boldsymbol{\beta}_{1}\rightarrow\hat{\boldsymbol{\beta}}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}\bm{y}$,
in which case, maximum a posteriori (MAP) estimator from Bayesian model goes
back to ordinary least squares estimator. And the posterior is
$\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\hat{\boldsymbol{\beta}},\sigma^{2}(\boldsymbol{X}^{\top}\boldsymbol{X})^{-1})$,
which shares similar form as the OLS estimator
$\hat{\boldsymbol{\beta}}\sim\mathcal{N}(\boldsymbol{\beta},\sigma^{2}(\boldsymbol{X}^{\top}\boldsymbol{X})^{-1})$
under Gaussian disturbance (see Lemma 3.2, p. 3.2).
###### Remark 8.3 (Ridge Regression)
In least squares approximation, we use $\bm{X}\boldsymbol{\beta}$ to
approximate $\bm{y}$. Two issues arise: the model can potentially overfit and
$\bm{X}$ may not have full rank. In ridge regression, we regularize large
value of $\boldsymbol{\beta}$ and thus favor simpler models. Instead of
minimizing $||\bm{y}-\bm{X}\boldsymbol{\beta}||^{2}$, we minimize
$||\bm{y}-\bm{X}\boldsymbol{\beta}||^{2}+\lambda||\boldsymbol{\beta}||^{2}$,
where $\lambda$ is a hyper-parameter that can be tuned:
$\mathop{\arg\min}_{\boldsymbol{\beta}}{(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})+\lambda\boldsymbol{\beta}^{\top}\boldsymbol{\beta}}.$
By differentiating and setting the derivative to zero we get
$\hat{\boldsymbol{\beta}}_{ridge}=(\bm{X}^{\top}\bm{X}+\lambda\bm{I})^{-1}\bm{X}^{\top}\bm{y},$
in which case, $(\bm{X}^{\top}\bm{X}+\lambda\bm{I})$ is invertible even when
$\bm{X}$ does not have full rank. We leave more details about ridge regression
to the readers.
Realize that when we set $\boldsymbol{\Sigma}_{0}=\bm{I}$, we obtain
$\boldsymbol{\beta}_{1}=(\bm{X}^{\top}\bm{X}+\sigma^{2}\bm{I})^{-1}\bm{X}^{\top}\bm{y}$
and
$\boldsymbol{\Sigma}_{1}=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\bm{I})^{-1}$.
Since
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{1},\boldsymbol{\Sigma}_{1})$.
The MAP estimator of
$\boldsymbol{\beta}=\boldsymbol{\beta}_{1}=(\bm{X}^{\top}\bm{X}+\sigma^{2}\bm{I})^{-1}\bm{X}^{\top}\bm{y}$,
which shares the same form as ridge regression by letting
$\sigma^{2}=\lambda$. Thus we notice ridge regression is a special case of
Bayesian linear model with zero-mean prior. And ridge regression has a nice
interpretation from the Bayesian approach - finding the mode of the posterior.
An example is shown in Rasmussen (2003) where the “well determined” (i.e., the
distribution around the slope is more compact) slope of $\boldsymbol{\beta}$
is almost unchanged after the posterior process while the intercept (which is
more dispersed) shrunk towards zero. This is actually a regularization effect
on the parameter like ridge regression.
#### 8.3.1 Zeller’s $g$-Prior and Variable Transformation
As a specific example, suppose we want to analyze a person’s weight by human’s
characteristics and a variable in $\bm{X}$ denotes the height of a person
which is in meter. What if now the variable is in centimeter? The model is the
same if we divide the related parameter in $\boldsymbol{\beta}$ by 100
(transfer centimeter back into meter measurement). More generally, suppose
input matrix $\bm{X}$ is transferred by $\tilde{\bm{X}}=\bm{X}\bm{P}$ given
some $p\times p$ matrix $\bm{P}$ in which case the model parameter is
$\tilde{\boldsymbol{\beta}}$. Then, we have
$\displaystyle\bm{y}=\bm{X}\boldsymbol{\beta}=\tilde{\bm{X}}\tilde{\boldsymbol{\beta}}=\bm{X}\bm{P}\tilde{\boldsymbol{\beta}}.$
According to the principle of invariance, the posterior distributions of
$\boldsymbol{\beta}$ and $\bm{P}\tilde{\boldsymbol{\beta}}$ should be the
same. As shown previously, for $\bm{X}$, the posterior is
$\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}\left((\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{y}),(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}\right).$
Similarly, for $\tilde{\bm{X}}$, the posterior
$\bm{P}\tilde{\boldsymbol{\beta}}$ is
$\bm{P}\tilde{\boldsymbol{\beta}}|\bm{y},\tilde{\bm{X}},\sigma^{2}\sim\mathcal{N}\left({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{P}}(\frac{1}{\sigma^{2}}\tilde{\bm{X}}^{\top}\tilde{\bm{X}}+\tilde{\boldsymbol{\Sigma}_{0}}^{-1})^{-1}(\frac{1}{\sigma^{2}}\tilde{\bm{X}}^{\top}\bm{y}),{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{P}}(\frac{1}{\sigma^{2}}\tilde{\bm{X}}^{\top}\tilde{\bm{X}}+\tilde{\boldsymbol{\Sigma}_{0}}^{-1})^{-1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{P}^{\top}}\right).$
A simple calculation can show that this condition is met if
$\boldsymbol{\Sigma}_{0}=k(\bm{X}^{\top}\bm{X})^{-1}$ where $k>0$ is a
hyperparameter. A popular specification of $k$ is to relate it to the noise
variance $\sigma^{2}$ by $k=g\sigma^{2}$. This is called the Zeller’s
$g$-prior (Zellner, 1986). Following the Bayesian linear model with zero-mean
prior, the posterior of $\boldsymbol{\beta}$ is
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{1},\boldsymbol{\Sigma}_{1}).$
where
$\boldsymbol{\Sigma}_{1}=\frac{g\sigma^{2}}{g+1}(\bm{X}^{\top}\bm{X})^{-1}$
and
$\boldsymbol{\beta}_{1}=\frac{g}{g+1}(\bm{X}^{\top}\bm{X})^{-1}(\bm{X}^{\top}\bm{y})$.
### 8.4 Bayesian Linear Model: Semiconjugate Prior Distribution
We will use gamma distribution as the prior of the inverse variance
(precision) parameter in Gaussian distribution.
###### Definition 8.3 (Gamma Distribution)
A random variable $X$ is said to follow the gamma distribution with parameter
$r>0$ and $\lambda>0$, denoted by $X\sim\mathrm{Gamma}(r,\lambda)$ if
$f(x;r,\lambda)=\left\\{\begin{aligned}
&\frac{\lambda^{r}}{\Gamma(r)}x^{r-1}exp(-\lambda x),&\mathrm{\,\,if\,\,}x\geq
0.\\\ &0,&\mathrm{\,\,if\,\,}x<0.\end{aligned}\right.$
So if $X\sim\chi_{(p)}^{2}$, then $X\sim\mathrm{Gamma}(p/2,1/2)$, i.e., Chi-
square distribution is a special case of the gamma distribution. The mean and
variance of $X\sim\mathrm{Gamma}(r,\lambda)$ are given by
$\mathrm{E}[X]=\frac{r}{\lambda},\qquad\mathrm{Var}[X]=\frac{r}{\lambda^{2}}.$
Figure 8.3 compares different parameters for gamma distribution and Chi-square
distribution.
(a) Gamma probability density functions for different values of the parameters
$r,\lambda$.
(b) Chi-square probability density functions for different values of the
parameter $p$.
Figure 8.3: Comparison between the gamma distribution and Chi-square
distribution (Definition 3.1, p. 3.1).
As for the reason of using the gamma distribution as the prior for precision,
we quote the description from Kruschke (2014): Because of its role in
conjugate priors for normal likelihood function, the gamma distribution is
routinely used as a prior for precision (i.e., inverse variance). But there is
no logical necessity to do so, and modern MCMC methods permit more flexible
specification of priors. Indeed, because precision is less intuitive than
standard deviation, it can be more useful to give standard deviation a uniform
prior that spans a wide range.
Same setting as Section 8.3 (p. 8.3), but we assume now $\sigma^{2}$ is not
fixed. Again, we have the likelihood function by
$\mathrm{likelihood}=\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2}\sim\mathcal{N}(\bm{X}\boldsymbol{\beta},\sigma^{2}\bm{I}).$
We specify a non zero-mean Gaussian prior over the weight parameter
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathrm{prior:\,}}$
$\displaystyle\boldsymbol{\beta}\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\beta}_{0}},\boldsymbol{\Sigma}_{0})$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma=1/\sigma^{2}\sim\mathrm{Gamma}(a_{0},b_{0})},$
where we differentiate from previous descriptions by blue text.
(1). Then, given $\sigma^{2}$, by the Bayes’ theorem
“$\mathrm{posterior}\propto\mathrm{likelihood}\times\mathrm{prior}$”, we get
the posterior
$\displaystyle\mathrm{posterior}$
$\displaystyle=p(\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2})$
$\displaystyle\propto
p(\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2})p(\boldsymbol{\beta}|\boldsymbol{\beta}_{0},\boldsymbol{\Sigma}_{0})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left(-\frac{1}{2\sigma^{2}}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right)$
$\displaystyle\,\,\,\,\,\,\times\frac{1}{(2\pi)^{n/2}|\boldsymbol{\Sigma}_{0}|^{1/2}}\exp\left(-\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})^{\top}\boldsymbol{\Sigma}_{0}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\right)$
$\displaystyle\propto\exp\left(-\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_{2})^{\top}\boldsymbol{\Sigma}_{2}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{2})\right),$
where
$\boldsymbol{\Sigma}_{2}=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}$
and
$\boldsymbol{\beta}_{2}=\boldsymbol{\Sigma}_{2}(\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{y})=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}}+\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{y}).$
Therefore, the posterior density is also a Gaussian distribution:
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{2},\boldsymbol{\Sigma}_{2}).$
1\. $\boldsymbol{\Sigma}_{0}$ here is a fixed hyperparameter. 2\. We note that
$\boldsymbol{\beta}_{1}$ in Section 8.3 (p. 8.3) is a special case of
$\boldsymbol{\beta}_{2}$ when $\boldsymbol{\beta}_{0}=\boldsymbol{0}$. 3\. And
if we assume further $\bm{X}$ has full rank. When
$\boldsymbol{\Sigma}_{0}^{-1}\rightarrow\boldsymbol{0}$,
$\boldsymbol{\beta}_{2}\rightarrow\hat{\boldsymbol{\beta}}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}\bm{y}$
which is the OLS estimator. 4\. When $\sigma^{2}\rightarrow\infty$,
$\boldsymbol{\beta}_{2}$ is approximately approaching to
$\boldsymbol{\beta}_{0}$, the prior expectation of parameter. However, in
zero-mean prior, $\sigma^{2}\rightarrow\infty$ will make
$\boldsymbol{\beta}_{1}$ approach to $\boldsymbol{0}$. 5\. Weighted average:
we reformulate by $\displaystyle\boldsymbol{\beta}_{2}$
$\displaystyle=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{y})$
$\displaystyle=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}\frac{\bm{X}^{\top}\bm{X}}{\sigma^{2}}(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top}\bm{y}$
$\displaystyle=(\bm{I}-\bm{A})\boldsymbol{\beta}_{0}+\bm{A}\hat{\boldsymbol{\beta}},$
where $\hat{\boldsymbol{\beta}}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top}\bm{y}$
is the OLS estimator of $\boldsymbol{\beta}$ and
$\bm{A}=(\frac{1}{\sigma^{2}}\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}\frac{\bm{X}^{\top}\bm{X}}{\sigma^{2}}$.
We see that the posterior mean of $\boldsymbol{\beta}$ is a weighted average
of prior mean and OLS estimator of $\boldsymbol{\beta}$. Thus, if we set the
prior parameter $\boldsymbol{\beta}_{0}=\hat{\boldsymbol{\beta}}$, the
posterior mean of $\boldsymbol{\beta}$ will be exactly
$\hat{\boldsymbol{\beta}}$.
(2). Given $\boldsymbol{\beta}$, again, by Bayes’ theorem, we obtain the
posterior
$\displaystyle\mathrm{posterior}$
$\displaystyle=p(\gamma=\frac{1}{\sigma^{2}}|\bm{y},\bm{X},\boldsymbol{\beta})$
$\displaystyle\propto
p(\bm{y}|\bm{X},\boldsymbol{\beta},\gamma)p(\gamma|a_{0},b_{0})$
$\displaystyle=\frac{\gamma^{n/2}}{(2\pi)^{n/2}}\exp\left(-\frac{\gamma}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right)$
$\displaystyle\,\,\,\,\,\,\,\times\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\gamma^{a_{0}-1}\exp(-b_{0}\gamma)$
$\displaystyle\propto\gamma(a_{0}+\frac{n}{2}-1)\exp\left(-\gamma\left[b_{0}+\frac{1}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right]\right),$
and the posterior is a Gamma distribution:
$\mathrm{posterior\,\,of\,\,}\gamma\mathrm{\,\,given\,\,}\boldsymbol{\beta}=\gamma|\bm{y},\bm{X},\boldsymbol{\beta}\sim\mathrm{Gamma}\left(a_{0}+\frac{n}{2},[b_{0}+\frac{1}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})]\right).$
(8.1)
1\. We notice that the prior mean and posterior mean of $\gamma$ are
$\mathrm{E}[\gamma]=\frac{a_{0}}{b_{0}}$ and
$\mathrm{E}[\gamma|\boldsymbol{\beta}]=\frac{a_{0}+\frac{n}{2}}{b_{0}+\frac{1}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})}$
respectively. So the internal meaning of $2a_{0}$ is the prior sample size for
the noise variance parameter $\sigma^{2}=\frac{1}{\gamma}$. 2\. As we assume
$\bm{y}=\bm{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}$ where
$\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$, then
$\frac{(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})}{\sigma^{2}}\sim\chi_{(n)}^{2}$
and
$\mathrm{E}[\frac{1}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})]=\frac{n}{2}\sigma^{2}$.
So the internal meaning of $\frac{2b_{0}}{a_{0}}$ is the prior variance of the
noise. 3\. Some textbooks would write
$\gamma\sim\mathrm{Gamma}(n_{0}/2,n_{0}\sigma_{0}^{2}/2)$ to make this
explicit (in which case, $n_{0}$ is the prior sample size, and
$\sigma_{0}^{2}$ is the prior variance). But a prior in this form seems coming
from nowhere at a first glance.
#### 8.4.1 Gibbs Sampling with Two Variables
Gibbs sampling was introduced by Turchin (Turchin, 1971), and later introduced
by brothers Geman in the context of image restoration (Geman and Geman, 1984).
The Geman brothers named the algorithm after the physicist J. W. Gibbs, some
eight decades after his death, in reference to an analogy between the sampling
algorithm and statistical physics.
Gibbs sampling is applicable when the joint distribution is not known
explicitly or is difficult to sample from directly, but the conditional
distribution of each variable is known and easy to sample from. A Gibbs
sampler generates a draw from the distribution of each parameter or variable
in turn, conditional on the current values of the other parameters or
variables. Therefore, a Gibbs sampler is a componentwise algorithm. In our
example from Section 8.1, given some data $\mathcal{X}$ and a probability
distribution $p(\boldsymbol{\beta}|\mathcal{X},\boldsymbol{\alpha})$
parameterized by
$\boldsymbol{\beta}=\\{\beta_{1},\beta_{2},\ldots,\beta_{p}\\}$. We can
successively draw samples from the distribution by sampling from
$\beta_{i}^{(t)}\sim
p(\beta_{i}|\boldsymbol{\beta}_{-i}^{(t-1)},\mathcal{X},\boldsymbol{\alpha}),$
(8.2)
where $\boldsymbol{\beta}_{-i}^{(t-1)}$ is all current values of
$\boldsymbol{\beta}$ in $(t-1)^{th}$ iteration except for $\beta_{i}$. If we
sample long enough, these $\beta_{i}$ values will be random samples from the
distribution $p$.
In deriving a Gibbs sampler, it is often helpful to observe that
$p(\beta_{i}\,|\,\boldsymbol{\beta}_{-i},\mathcal{X})=\frac{p(\beta_{1},\beta_{2},\ldots,\beta_{p},\mathcal{X})}{p(\boldsymbol{\beta}_{-i},\mathcal{X})}\propto
p(\beta_{1},\beta_{2},\ldots,\beta_{p},\mathcal{X}).$ (8.3)
The conditional distribution is proportional to the joint distribution. We
will get a lot of benefits from this simple observation by dropping constant
terms from the joint distribution (relative to the parameters we are
conditioned on).
Shortly, as a simplified example, given a joint probability distribution
$p(\beta_{1},\beta_{2}|\mathcal{X})$, a Gibbs sampler would draw
$p(\beta_{1}|\beta_{2},\mathcal{X})$ , then
$p(\beta_{2}|\beta_{1},\mathcal{X})$ iteratively. The procedure defines a
sequence of realization of random variables $\beta_{1}$ and $\beta_{2}$
$(\beta_{1}^{0},\beta_{2}^{0}),(\beta_{1}^{1},\beta_{2}^{1}),(\beta_{1}^{2},\beta_{2}^{2}),...$
which converges to the joint distribution $p(\beta_{1},\beta_{2})$. More
details about Gibbs sampling can be found in Turchin (1971); Geman and Geman
(1984); Müller and Quintana (2004); Rencher and Schaalje (2008); Hoff (2009);
Gelman et al. (2013); Kruschke and Liddell (2018).
By this Gibbs sampling method, we can construct a Gibbs sampler for the
Bayesian linear model with semiconjugate prior in Section 8.4:
0\. Set initial values to $\boldsymbol{\beta}$ and
$\gamma=\frac{1}{\sigma^{2}}$;
1\. update $\boldsymbol{\beta}$:
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\gamma\sim\mathcal{N}(\boldsymbol{\beta}_{2},\boldsymbol{\Sigma}_{2})$;
2\. update $\gamma$:
$\mathrm{posterior}=\gamma|\bm{y},\bm{X},\boldsymbol{\beta}\sim\mathrm{Gamma}\left(a_{0}+\frac{n}{2},[b_{0}+\frac{1}{2}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})]\right)$.
#### 8.4.2 Zeller’s $g$-Prior
Similar to Section 8.3.1 (p. 8.3.1), suppose input $\bm{X}$ is transferred by
$\tilde{\bm{X}}=\bm{X}\bm{P}$ given some $p\times p$ matrix $\bm{P}$ in which
case, the model parameter is $\tilde{\boldsymbol{\beta}}$. Then, we have
$\bm{y}=\bm{X}\boldsymbol{\beta}=\tilde{\bm{X}}\tilde{\boldsymbol{\beta}}=\bm{X}\bm{P}\tilde{\boldsymbol{\beta}}.$
According to the principle of invariance, the posterior distributions of
$\boldsymbol{\beta}$ and $\bm{P}\tilde{\boldsymbol{\beta}}$ should be the
same. Simple calculation can show that this condition is met if
$\boldsymbol{\beta}_{0}=\boldsymbol{0},\boldsymbol{\Sigma}_{0}=g\sigma^{2}(\bm{X}^{\top}\bm{X})^{-1}$.
Following the Bayesian linear model with semi-conjugate prior, the posterior
of $\boldsymbol{\beta}$ is
$\mathrm{posterior}=\boldsymbol{\beta}|\bm{y},\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{2},\boldsymbol{\Sigma}_{2}).$
where
$\boldsymbol{\Sigma}_{2}=\frac{g\sigma^{2}}{g+1}(\bm{X}^{\top}\bm{X})^{-1}$
and
$\boldsymbol{\beta}_{2}=\frac{g}{g+1}(\bm{X}^{\top}\bm{X})^{-1}(\bm{X}^{\top}\bm{y})$.
#### Derivation of $p(\bm{y}|\bm{X},\sigma^{2})$
Under the $g$-prior specified above, we derive the conditional distribution
$p(\bm{y}|\bm{X},\sigma^{2})$ which will be very useful for the Bayesian
variable selection procedure. We realize that
$\displaystyle p(\bm{y},\boldsymbol{\beta}|\bm{X},\sigma^{2})$
$\displaystyle=p(\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2})p(\boldsymbol{\beta}|\bm{X},\sigma^{2})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left(-\frac{1}{2\sigma^{2}}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right)\qquad$
$\displaystyle(\text{$\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2}\sim\mathcal{N}(\bm{X}\boldsymbol{\beta},\sigma^{2}\bm{I})$})$
$\displaystyle\,\,\,\,\,\,\times\frac{1}{(2\pi)^{n/2}|\boldsymbol{\Sigma}_{0}|^{1/2}}\exp\left(-\frac{1}{2}\boldsymbol{\beta}^{\top}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}\right)\qquad$
$\displaystyle(\text{$\boldsymbol{\beta}|\bm{X},\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{0},\boldsymbol{\Sigma}_{0})$})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left(-\frac{1}{2\sigma^{2}}\bm{y}^{\top}\bm{y}\right)\frac{|\boldsymbol{\Sigma}_{2}|^{1/2}}{|\boldsymbol{\Sigma}_{0}|^{1/2}}\exp\left(\frac{1}{2}\boldsymbol{\beta}_{2}^{\top}\boldsymbol{\Sigma}_{2}^{-1}\boldsymbol{\beta}_{2}\right)$
$\displaystyle\,\,\,\,\,\,\times\frac{1}{(2\pi)^{n/2}|\boldsymbol{\Sigma}_{2}|^{1/2}}\exp\left(-\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}_{2})^{\top}\boldsymbol{\Sigma}_{2}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{2})\right),$
where $\boldsymbol{\beta}$ only appears in the third term. And the third term
is a multivariate normal with mean $\boldsymbol{\beta}_{2}$ and covariance
$\boldsymbol{\Sigma}_{2}$ and integrates to 1. And we notice that,
$\boldsymbol{\Sigma}_{2}=\frac{1}{g+1}\boldsymbol{\Sigma}_{0}$ such that
$\frac{|\boldsymbol{\Sigma}_{2}|^{1/2}}{|\boldsymbol{\Sigma}_{0}|^{1/2}}=\frac{1}{(g+1)^{p/2}}$.
Therefore, we obtain
$\displaystyle p(\bm{y}|\bm{X},\sigma^{2})$ $\displaystyle=\int
p(\bm{y},\boldsymbol{\beta}|\bm{X},\sigma^{2})d\boldsymbol{\beta}$ (8.4)
$\displaystyle=\int
p(\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2})p(\boldsymbol{\beta}|\bm{X},\sigma^{2})d\boldsymbol{\beta}$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left(-\frac{1}{2\sigma^{2}}\bm{y}^{\top}\bm{y}\right)\cdot\frac{1}{(g+1)^{p/2}}\exp\left(\frac{1}{2}\boldsymbol{\beta}_{2}^{\top}\boldsymbol{\Sigma}_{2}^{-1}\boldsymbol{\beta}_{2}\right)$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\frac{1}{(g+1)^{p/2}}\exp\left(-\frac{r}{2\sigma^{2}}\right)$
where
$r=\bm{y}^{\top}\bm{y}-\bm{y}^{\top}(\frac{g}{g+1}\bm{X}(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top})\bm{y}$.
#### 8.4.3 Bayesian Variable Selection
We previously introduce variable selection by $F$-test in Section 4.3 (p.
4.3). An alternative way can be achieved by Bayesian variable selection.
#### The Model
Suppose $\bm{z}=[z_{1},z_{2},\ldots,z_{p}]\in^{p}$ is a mask vector such that
$z_{j}\in\\{0,1\\}$ for all $j\in\\{1,2,\ldots,p\\}$. For each variable
$\beta_{j}$ in $\boldsymbol{\beta}$, we set $\beta_{j}=z_{j}\times b_{j}$
where $b_{j}$ can be understood as the original variable, and $\beta_{j}$ is
the final variable. That is $\boldsymbol{\beta}=\bm{z}\odot\bm{b}$ where
$\odot$ is the Hadamard product. Then the model can be expressed as
$\bm{y}=\bm{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}=\bm{X}(\bm{z}\odot\bm{b})+\boldsymbol{\epsilon},$
where $\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$.
Bayesian variable selection can be understood as obtaining a posterior
distribution for the mask vector $\bm{z}$. By Bayes’ theorem, we obtain the
posterior distribution by
$p(\bm{z}|\bm{y},\bm{X})\propto p(\bm{z})p(\bm{y}|\bm{X},\bm{z}).$
Alternatively, the ratio of two models $\bm{z}_{a}$, $\bm{z}_{b}$ is
$\displaystyle\text{odds}(\bm{z}_{a},\bm{z}_{b}|\bm{y},\bm{X})$
$\displaystyle=\frac{p(\bm{z}_{a}|\bm{y},\bm{X})}{p(\bm{z}_{b}|\bm{y},\bm{X})}$
$\displaystyle=$ $\displaystyle\,\,\,\,\frac{p(\bm{z}_{a})}{p(\bm{z}_{b})}$
$\displaystyle\times$
$\displaystyle\,\,\,\,\,\frac{p(\bm{y}|\bm{X},\bm{z}_{a})}{p(\bm{y}|\bm{X},\bm{z}_{b})}$
posterior odds $\displaystyle=$ prior odds $\displaystyle\times$ Bayes factor
where the Bayes factor can be understood as how much the data favor the model
$\bm{z}_{a}$ over the model $\bm{z}_{b}$.
#### Derivation of the Bayes Factor
Write out the Bayes factor by
$\displaystyle p(\bm{y}|\bm{X},\bm{z})$ $\displaystyle=\int\int
p(\bm{y},\boldsymbol{\beta},\sigma^{2}|\bm{X},\bm{z})d\boldsymbol{\beta}d\sigma^{2}$
$\displaystyle=\int\left(\int
p(\bm{y},\boldsymbol{\beta},|\bm{X},\bm{z},\sigma^{2})d\boldsymbol{\beta}\right)p(\sigma^{2})d\sigma^{2}$
$\displaystyle=\int
p(\bm{y}|\bm{X},\sigma^{2},\bm{z})p(\sigma^{2})d\sigma^{2},$
where $p(\bm{y}|\bm{X},\sigma^{2},\bm{z})=\left(\int
p(\bm{y},\boldsymbol{\beta},|\bm{X},\bm{z},\sigma^{2})d\boldsymbol{\beta}\right)$
can be obtained from Equation (8.4) (under Zeller’s $g$-prior) by substituting
$\bm{X}$ by $\bm{X}_{z}$ where we remove the variable $i$ if $z_{i}$=0.
We realize that $\gamma=\frac{1}{\sigma^{2}}$ and
$p(\sigma^{2})=p(\gamma|a_{0},b_{0})=\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\gamma^{a_{0}-1}\exp(-b_{0}\gamma).$
Then,
$\displaystyle p(\bm{y}|\bm{X},\sigma^{2},\bm{z})p(\sigma^{2})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\frac{1}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z}}/2}}\exp\left(-\frac{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z}}}{2\sigma^{2}}\right)\cdot\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\gamma^{a_{0}-1}\exp(-b_{0}\gamma)$
$\displaystyle=\frac{1}{(2\pi)^{n/2}}\frac{1}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z}}/2}}\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\gamma^{a_{0}+n/2-1}\exp\left(-(b_{0}+\frac{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z}}}{2})\gamma\right)$
$\displaystyle=\frac{1}{(2\pi)^{n/2}}\frac{1}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z}}/2}}\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\frac{\Gamma(a_{n})}{{b_{n}}^{a_{n}}}\cdot\frac{{b_{n}}^{a_{n}}}{\Gamma(a_{n})}\gamma^{a_{n}-1}\exp\left(-b_{n}\gamma\right)$
$\displaystyle=\frac{1}{(2\pi)^{n/2}}\frac{1}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z}}/2}}\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\frac{\Gamma(a_{n})}{{b_{n}}^{a_{n}}}\cdot\mathrm{Gamma}(\gamma|a_{n},b_{n}),$
where
$r_{z}=\bm{y}^{\top}\bm{y}-\bm{y}^{\top}(\frac{g}{g+1}\bm{X}_{z}(\bm{X}_{z}^{\top}\bm{X}_{z})^{-1}\bm{X}_{z}^{\top})\bm{y}$,
$p_{z}$ is the number of 1’s in $\bm{z}$, $a_{n}=a_{0}+n/2$,
$b_{n}=b_{0}+\frac{r_{z}}{2}$, and $\mathrm{Gamma}(\gamma|a_{n},b_{n})$ is the
probability density function of Gamma distribution w.r.t. $\gamma$ with
parameters $a_{n},b_{n}$. Since $\gamma$ only appears in the last term
$\mathrm{Gamma}(\gamma|a_{n},b_{n})$ which integrates to 1, we have
$p(\bm{y}|\bm{X},\bm{z})=\frac{1}{(2\pi)^{n/2}}\frac{1}{(g+1)^{p_{z}/2}}\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\frac{\Gamma(a_{n})}{{b_{n}}^{a_{n}}}.$
#### Same Prior Hyperparameters
Similarly, under models $\bm{z}_{a}$ and $\bm{z}_{b}$ where we assume the same
parameters of $a_{0}$ and $b_{0}$ in the two models, we have
$\frac{p(\bm{y}|\bm{X},\bm{z}_{a})}{p(\bm{y}|\bm{X},\bm{z}_{b})}=\frac{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z_{b}}}/2}}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z_{a}}}/2}}\left(\frac{2b_{0}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z_{b}}}}{2b_{0}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z_{a}}}}\right)^{a_{0}+\frac{n}{2}},$
where $p_{z_{a}}$ is the number of variables selected in the model
$\bm{z}_{a}$, and $p_{z_{b}}$ is the number of variables selected in the model
$\bm{z}_{b}$.
#### Different Prior Hyperparameters
We have previously mentioned that
1\. $2a_{0}$ is the prior sample size for the noise
$\sigma^{2}=\frac{1}{\gamma}$.
2\. $\frac{2b_{0}}{a_{0}}$ is the prior variance of the noise.
Suppose now, given the two models $\bm{z}_{a}$ and $\bm{z}_{b}$, we assume
$2a_{0}=1$ for both of the models (i.e., prior sample sizes for the noise are
both 1), and set the $\frac{2b_{0}}{a_{0}}$ to be the estimated residual
variance under the least squares estimate for each model, say maximum
likelihood estimators
$\hat{\sigma}^{2}_{z_{a}}=\frac{1}{n}||\bm{y}-\bm{X}_{z_{a}}\hat{\boldsymbol{\beta}}_{z_{a}}||^{2}$
and
$\hat{\sigma}^{2}_{z_{b}}=\frac{1}{n}||\bm{y}-\bm{X}_{z_{b}}\hat{\boldsymbol{\beta}}_{z_{b}}||^{2}$
(which are biased estimators for $\sigma^{2}$, see Section 2.2, p. 2.2). Or we
could choose the unbiased estimators which are divided by $n-p_{z_{a}}$ and
$n-p_{z_{b}}$ respectively rather than divided by $n$ (see Section 3.4, p.
3.4). Then, we have
$\frac{p(\bm{y}|\bm{X},\bm{z}_{a})}{p(\bm{y}|\bm{X},\bm{z}_{b})}=\frac{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z_{b}}}/2}}{(g+1)^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{z_{a}}}/2}}\left(\frac{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\hat{\sigma}^{2}_{z_{a}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\hat{\sigma}^{2}_{z_{b}}}}\right)^{\frac{1}{2}}\left(\frac{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\hat{\sigma}^{2}_{z_{b}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z_{b}}}}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\hat{\sigma}^{2}_{z_{a}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{z_{a}}}}\right)^{\frac{n+1}{2}}.$
(8.5)
Notice that the ratio of the marginal probabilities is essentially a balance
between the model complexity and goodness of fit:
* •
A larger value of $p_{z_{b}}$ means the model $\bm{z}_{b}$ has more selected
variables (more complexity) which will make the ratio larger and penalize
model $\bm{z}_{b}$.
* •
However, a more complex model will make $r_{z_{b}}$ smaller which in turn will
make the ratio smaller and penalize model $\bm{z}_{a}$.
#### Gibbs Sampler
Given a current value $\bm{z}=[z_{1},z_{2},\ldots,z_{p}]^{\top}$, a new value
of the $j$-th variable $z_{j}$ is generated by sampling from
$p(z_{j}|\bm{y},\bm{X},\bm{z}_{-j})$ where $z_{-j}$ refers to the values of
$\bm{z}$ except the $j$-th element $z_{j}$. Specifically, we define the
intermediate parameter222This intermediate parameter is quite useful in other
contexts, e.g., Bayesian inference for interpolative decomposition (Lu, 2022a,
b).
$o_{j}=\frac{p(z_{j}=1|\bm{y},\bm{X},\bm{z}_{-j})}{p(z_{j}=0|\bm{y},\bm{X},\bm{z}_{-j})}=\frac{p(z_{j}=1)}{p(z_{j}=0)}\times\frac{p(\bm{y}|\bm{X},\bm{z}_{-j},z_{j}=1)}{p(\bm{y}|\bm{X},\bm{z}_{-j},z_{j}=0)},$
where the last term can be obtained by Equation (8.5). Trivially, we can set
$p(z_{j}=1)=p(z_{j}=0)=0.5$. Then, using the intermediate parameter, the full
conditional probability of $z_{j}$ being equal to 1 can be obtained by
$p(z_{j}=1|\bm{y},\bm{X},\bm{z}_{-j})=\frac{o_{j}}{1+o_{j}}.$ (8.6)
Therefore, given the value of $\bm{z}^{(k)}$ in $k$-th step, we can generate
$\\{\bm{z}^{(k+1)},\gamma^{(k+1)},\boldsymbol{\beta}^{(k+1)}\\}$ by the
following steps:
0\. Set initial values to $\boldsymbol{\beta}$, $\gamma=\frac{1}{\sigma^{2}}$,
and $\bm{z}$ if $k$=1;
1\. Update $\bm{z}$: For $j\in\\{1,2,\ldots,p\\}$ in random order, replace
$z_{j}$ with a sample from $p(z_{j}=1|\bm{y},\bm{X},\bm{z}_{-j})$ (Equation
(8.6));
1\. Update $\boldsymbol{\beta}$:
$\boldsymbol{\beta}|\bm{y},\bm{X},\gamma,\bm{z}\sim\mathcal{N}(\boldsymbol{\beta}_{2},\boldsymbol{\Sigma}_{2})$,
where
$\boldsymbol{\Sigma}_{2}=\frac{g\sigma^{2}}{g+1}(\bm{X}_{z}^{\top}\bm{X}_{z})^{-1}$
and
$\boldsymbol{\beta}_{2}=\frac{g}{g+1}(\bm{X}_{z}^{\top}\bm{X}_{z})^{-1}(\bm{X}_{z}^{\top}\bm{y})$
(Equation (8.4.2), p. 8.4.2);
2\. Update $\gamma$:
$\gamma|\bm{y},\bm{X},\boldsymbol{\beta},\bm{z}\sim\mathrm{Gamma}\left(a_{0}+\frac{n}{2},[b_{0}+\frac{1}{2}||\bm{y}-\bm{X}_{z}\boldsymbol{\beta}_{z}||^{2}]\right)$
(Equation (8.1), p. 8.1).
### 8.5 Bayesian Linear Model: Full Conjugate Prior
To derive the full conjugate algorithm for the linear model, we place an
inverse-gamma prior over the variance parameter instead. Putting a gamma prior
over the inverse variance is equivalent to putting an inverse-gamma prior on
the variance.
###### Definition 8.4 (Inverse-Gamma Distribution)
A random variable $Y$ is said to follow the inverse-gamma distribution with
parameters $r>0$ and $\lambda>0$ if
$f(y;r,\lambda)=\left\\{\begin{aligned}
&\frac{\lambda^{r}}{\Gamma(r)}y^{-r-1}\exp(-\frac{\lambda}{y}),&\mathrm{\,\,if\,\,}y>0.\\\
&0,&\mathrm{\,\,if\,\,}y\leq 0,\end{aligned}\right.$
where again $\Gamma(\cdot)$ is the gamma function (Definition 3.1, p. 3.1).
And it is denoted by $Y\sim\mathrm{Inverse\mathchar 45\relax
Gamma}(r,\lambda)$. The mean and variance of inverse-gamma distribution are
given by
$\mathrm{E}[Y]=\left\\{\begin{aligned}
&\frac{\lambda}{r-1},\,&\mathrm{if\,}r\geq 1.\\\
&\infty,\,&\mathrm{if\,}0<r<1.\end{aligned}\right.\qquad\mathrm{Var}[Y]=\left\\{\begin{aligned}
&\frac{\lambda^{2}}{(r-1)^{2}(r-2)},\,&\mathrm{if\,}r\geq 2.\\\
&\infty,\,&\mathrm{if\,}0<r<2.\end{aligned}\right.$
Figure 8.4 compares different parameters for gamma distribution and inverse-
gamma distribution.
(a) Inverse-amma probability density functions for different values of the
parameters $r,\lambda$.
(b) Gamma probability density functions for different values of the parameters
$r,\lambda$.
Figure 8.4: Comparison between the inverse-gamma distribution and gamma
distribution (Definition 8.3, p. 8.3).
Note that the inverse-gamma density is not simply the gamma density with $x$
replaced by $\frac{1}{y}$. There is an additional factor of $y^{-2}$ from the
Jacobian in the change-of-variables formula. 333Which is from the Jacobian in
the change-of-variables formula. A short proof is provided here. Let
$y=\frac{1}{x}$ where $y\sim\mathrm{Inverse\mathchar 45\relax
Gamma}(r,\lambda)$ and $x\sim\mathrm{Gamma}(r,\lambda)$. Then,
$f(y)|dy|=f(x)|dx|$ which results in
$f(y)=f(x)|\frac{dx}{dy}|=f(x)x^{2}\xlongequal{\mathrm{y}=\frac{1}{x}}\frac{\lambda^{r}}{\Gamma(r)}y^{-r-1}exp(-\frac{\lambda}{y})$
for $y>0$.
Same setting as the semiconjugate prior distribution in Section 8.4 (p. 8.4).
We have the likelihood function:
$\mathrm{likelihood}=\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2}\sim\mathcal{N}(\bm{X}\boldsymbol{\beta},\sigma^{2}\bm{I}),$
which is the same as the likelihood density in the zero-mean model (Section
8.3, p. 8.3) and the semiconjugate model (Section 8.4, p. 8.4). But now we
specify a Gaussian prior (with unfixed covariance matrix) over the weight
parameter, and an inverse-gamma density over the variance parameter by
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathrm{prior:\,}}$
$\displaystyle\boldsymbol{\beta}|\sigma^{2}\sim\mathcal{N}(\boldsymbol{\beta}_{0},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}}\boldsymbol{\Sigma}_{0})$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\sim\mathrm{Inverse\mathchar
45\relax Gamma}(a_{0},b_{0})},$
where again we differentiate from previous descriptions by blue text. Note we
place an inverse-gamma density over the variance parameter in the full
conjugate model instead of a gamma density over the precision (inverse
variance) parameter in the semiconjugate case. However, it is demonstrable
that the two circumstances are identical (via the Jacobian in the change-of-
variables formula).
Equivalently, we can formulate the prior into one which is called the normal-
inverse-gamma (NIG) distribution:
$\displaystyle\mathrm{prior:\,}$
$\displaystyle\boldsymbol{\beta},\sigma^{2}\sim
NIG(\boldsymbol{\beta}_{0},\boldsymbol{\Sigma}_{0},a_{0},b_{0})$
$\displaystyle=\mathcal{N}(\boldsymbol{\beta}_{0},\sigma^{2}\boldsymbol{\Sigma}_{0})\cdot\mathrm{Inverse\mathchar
45\relax Gamma}(a_{0},b_{0}).$
Again by applying the Bayes’ theorem
“$\mathrm{posterior}\propto\mathrm{likelihood}\times\mathrm{prior}$”, we
obtain the posterior
$\displaystyle\mathrm{posterior}$
$\displaystyle=p(\boldsymbol{\beta},\sigma^{2}|\bm{y},\bm{X})$
$\displaystyle\propto
p(\bm{y}|\bm{X},\boldsymbol{\beta},\sigma^{2})p(\boldsymbol{\beta},\sigma^{2}|\boldsymbol{\beta}_{0},\boldsymbol{\Sigma}_{0},a_{0},b_{0})$
$\displaystyle=\frac{1}{(2\pi\sigma^{2})^{n/2}}\exp\left\\{-\frac{1}{2\sigma^{2}}(\bm{y}-\bm{X}\boldsymbol{\beta})^{\top}(\bm{y}-\bm{X}\boldsymbol{\beta})\right\\}$
$\displaystyle\,\,\,\,\,\,\times\frac{1}{(2\pi\sigma^{2})^{p/2}|\boldsymbol{\Sigma}_{0}|^{1/2}}\exp\left\\{-\frac{1}{2\sigma^{2}}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})^{\top}\boldsymbol{\Sigma}_{0}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\right\\}$
$\displaystyle\,\,\,\,\,\,\times\frac{{b_{0}}^{a_{0}}}{\Gamma(a_{0})}\frac{1}{(\sigma^{2})^{a_{0}+1}}\exp\\{-\frac{b_{0}}{\sigma^{2}}\\}$
$\displaystyle\propto\frac{1}{(2\pi\sigma^{2})^{p/2}}\exp\left\\{\frac{1}{2\sigma^{2}}(\boldsymbol{\beta}-\boldsymbol{\beta}_{3})^{\top}\boldsymbol{\Sigma}_{3}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{3})\right\\}$
$\displaystyle\,\,\,\,\,\,\times\frac{1}{(\sigma^{2})^{a_{0}+\frac{n}{2}+1}}\exp\left\\{-\frac{1}{\sigma^{2}}[b_{0}+\frac{1}{2}(\bm{y}^{\top}\bm{y}+\boldsymbol{\beta}_{0}^{\top}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}-\boldsymbol{\beta}_{3}^{\top}\boldsymbol{\Sigma}_{3}^{-1}\boldsymbol{\beta}_{3})]\right\\},$
where
$\boldsymbol{\Sigma}_{3}=(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}$
and
$\boldsymbol{\beta}_{3}=\boldsymbol{\Sigma}_{3}(\bm{X}^{\top}\bm{y}+\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0})=(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+\bm{X}^{\top}\bm{y}).$
Let $a_{n}=a_{0}+\frac{n}{2}+1$ and
$b_{n}=b_{0}+\frac{1}{2}(\bm{y}^{\top}\bm{y}+\boldsymbol{\beta}_{0}^{\top}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}-\boldsymbol{\beta}_{3}^{\top}\boldsymbol{\Sigma}_{3}^{-1}\boldsymbol{\beta}_{3})$.
The posterior admits conjugacy and is a NIG distribution:
$\displaystyle\mathrm{posterior}$
$\displaystyle=\boldsymbol{\beta},\sigma^{2}|\bm{y},\bm{X}\sim
NIG(\boldsymbol{\beta}_{3},\boldsymbol{\Sigma}_{3},a_{n},b_{n}).$
1\. $\boldsymbol{\Sigma}_{0}$ here is a fixed hyperparameter. 2\. If we assume
further $\bm{X}$ has full rank, when
$\boldsymbol{\Sigma}_{0}^{-1}\rightarrow\boldsymbol{0}$,
$\boldsymbol{\beta}_{3}\rightarrow\hat{\boldsymbol{\beta}}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}\bm{y}$
which is the OLS estimator. 3\. When $b_{0}\rightarrow\infty$, then
$\sigma^{2}\rightarrow\infty$ and $\boldsymbol{\beta}_{3}$ is approximately
$\boldsymbol{\beta}_{0}$, the prior expectation of parameter. Compared to
$\boldsymbol{\beta}_{2}$ in Section 8.4 (p. 8.4),
$\sigma^{2}\rightarrow\infty$ will make $\boldsymbol{\beta}_{2}$ approach to
$\boldsymbol{\beta}_{0}$ where $\sigma^{2}$ is a fixed hyperparameter. 4\.
Weighted average: we reformulate $\displaystyle\boldsymbol{\beta}_{3}$
$\displaystyle=(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+\bm{X}^{\top}\bm{y})$
$\displaystyle=(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\beta}_{0}+(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\bm{X}^{\top}\bm{X})(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top}\bm{y}$
$\displaystyle=(\bm{I}-\bm{C})\boldsymbol{\beta}_{0}+\bm{C}\hat{\boldsymbol{\beta}},$
where $\hat{\boldsymbol{\beta}}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top}\bm{y}$
is the OLS estimator of $\boldsymbol{\beta}$ and
$\bm{C}=(\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\bm{X}^{\top}\bm{X})$.
We see that the posterior mean of $\boldsymbol{\beta}$ is a weighted average
of the prior mean and the OLS estimator of $\boldsymbol{\beta}$. Thus, if we
set $\boldsymbol{\beta}_{0}=\hat{\boldsymbol{\beta}}$, the posterior mean of
$\boldsymbol{\beta}$ will be exactly $\hat{\boldsymbol{\beta}}$. 5\. From
$a_{n}=a_{0}+\frac{n}{2}+1$, we realize that $2a_{0}$ is the prior sample size
for $\sigma^{2}$. 6\.
$\boldsymbol{\Sigma}_{3}^{-1}=\bm{X}^{\top}\bm{X}+\boldsymbol{\Sigma}_{0}^{-1}$:
The posterior inverse covariance is equal to $\bm{X}^{\top}\bm{X}$ \+ prior
inverse covariance.
## Chapter 9 Beyond Bayesian Approach: Gaussian Process Regression
### 9.1 Beyond Bayesian Approach: Gaussian Process Regression
In some textbooks or college courses, people talk about the Gaussian process
without any introduction to the Bayesian linear model. This is quite
confusing. We will discover that the Bayesian linear model with zero-mean
prior is where the Gaussian process regression genuinely ”arises.”
### 9.2 Predictive Distribution of Bayesian Linear Model with Zero-Mean Prior
Following from Bayesian linear model with zero mean in Section 8.3 (p. 8.3).
The predictive distribution $g_{\ast}=g_{\ast}(\bm{x}_{\ast})$ for a new data
point $\bm{x}_{\ast}$ is again a Gaussian distribution
$\displaystyle p(g_{\ast}|\bm{x}_{\ast}\bm{X},\bm{y},\sigma^{2})$
$\displaystyle=\int
p(g_{\ast}|\bm{x}_{\ast},\boldsymbol{\beta})p(\boldsymbol{\beta}|\bm{X},\bm{y},\sigma^{2})d\boldsymbol{\beta}$
(9.1)
$\displaystyle=\mathcal{N}(\frac{1}{\sigma^{2}}\bm{x}_{\ast}^{\top}\boldsymbol{\Sigma}_{1}\bm{X}^{\top}\bm{y},\bm{x}_{\ast}^{\top}\boldsymbol{\Sigma}_{1}\bm{x}_{\ast})$
$\displaystyle=\mathcal{N}(\bm{x}_{\ast}^{\top}\boldsymbol{\beta}_{1},\bm{x}_{\ast}^{\top}\boldsymbol{\Sigma}_{1}\bm{x}_{\ast}),$
where the mean of the predictive distribution is the posterior mean of weight
parameter (i.e., $\boldsymbol{\beta}_{1}$) multiplied by the new input. And
the predictive variance is a quadratic form of the new input indicating that
the predictive uncertainties grow with the magnitude of the new input.
However, this Bayesian linear model suffers from limited expressiveness. One
solution to overcome this issue is to use a set of basis functions and
transfer from the $p\mathchar 45\relax$dimensional space to higher dimensional
space, say $q\mathchar 45\relax$dimensional space:
$\bm{x}\in^{p}\rightarrow\boldsymbol{\phi}(\bm{x})\in^{q}$ and
$\bm{X}\rightarrow\boldsymbol{\Phi}(\bm{X})\in^{n\times q}$. But another issue
arises that the computation grows quadratically, e.g.,
$\mathcal{O}(p^{2}+p)\rightarrow\mathcal{O}(q^{2}+q)$ for the computation of
predictive variance.
###### Remark 9.1 (Kernel Trick)
Kernel trick could help reduce computational complexity in high dimensional
space. The key point to use kernel trick is to make sure that the input space
only involves dot products.
###### Example 9.2.1 (Kernel Trick)
1\. The computation of
$\boldsymbol{\phi}(\bm{x}_{\ast})^{\top}\boldsymbol{\Phi}(\bm{X})^{\top}=\bm{z}\in^{1\times
n}$ only has the dot product from the input space. So we could use kernel
trick to do the computation. And each $z_{i}=k(\bm{x}_{\ast},\bm{x}_{i})$
where $k(\cdot,\cdot)$ is a kernel function. The computation of kernel
function is potentially less than $\mathcal{O}(q)$. 2\. The computation of
$\bm{y}^{\top}\boldsymbol{\Phi}(\bm{X})$ has dot product between the input
space and output space. So it cannot be reduced to kernel form.
Having this in mind, if we can reduce the computation of Equation (9.1)
involving input space to having only dot products, we could apply the kernel
trick.
Transfer the parameters in Equation (9.1) into high dimensional space. By
transforming from {$\bm{x},\bm{X}$} in $x$-space $\in^{p}$ into
{$\boldsymbol{\phi},\boldsymbol{\Phi}$} in $z$-space $\in^{q}$, we obtain the
predictive distribution in the $z$-space form:
$\boxed{\begin{aligned} z\mathchar 45\relax\mathrm{Space\,Form:}\\\
g_{\ast}|\bm{x}_{\ast}\bm{X},\bm{y},\sigma^{2}&\sim\mathcal{N}(\boldsymbol{\phi}(\bm{x}_{\ast})^{\top}\boldsymbol{\beta}_{1},\boldsymbol{\phi}(\bm{x}_{\ast})^{\top}\boldsymbol{\Sigma}_{1}\boldsymbol{\phi}(\bm{x}_{\ast}))\\\
&=\mathcal{N}\left(\boldsymbol{\phi}(\bm{x}_{\ast})^{\top}(\frac{1}{\sigma^{2}}\boldsymbol{\Phi}^{\top}\boldsymbol{\Phi}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}(\frac{1}{\sigma^{2}}\boldsymbol{\Phi}^{\top}\bm{y}),\boldsymbol{\phi}(\bm{x}_{\ast})^{\top}\boldsymbol{\Sigma}_{1}\boldsymbol{\phi}(\bm{x}_{\ast})\right),\end{aligned}}$
where
$\boldsymbol{\Sigma}_{1}=(\frac{1}{\sigma^{2}}\boldsymbol{\Phi}^{\top}\boldsymbol{\Phi}+\boldsymbol{\Sigma}_{0}^{-1})^{-1}$.
Let $\bm{K}=\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}$
and $\boldsymbol{\phi}_{\ast}=\boldsymbol{\phi}(\bm{x}_{\ast})$, we have
$\frac{1}{\sigma^{2}}\boldsymbol{\Phi}^{\top}(\bm{K}+\sigma^{2}\bm{I})=\frac{1}{\sigma^{2}}\boldsymbol{\Phi}^{\top}(\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}+\sigma^{2}\bm{I})=\boldsymbol{\Sigma}_{1}^{-1}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top},$
(9.2)
where we use the fact that
$\bm{A}\bm{B}\bm{A}+\bm{A}=\bm{A}(\bm{B}\bm{A}+\bm{I})=(\bm{A}\bm{B}+\bm{I})\bm{A}$.
Note that we do not differentiate the notations of $\boldsymbol{\Sigma}_{0}$,
$\boldsymbol{\Sigma}_{1}$ in $x$ and $z$-spaces. But only differentiate the
notation of $\bm{X}$ and $\boldsymbol{\Phi}$ in $x$ and $z$-spaces
respectively. Then by multiplying Equation (9.2) by $\boldsymbol{\Sigma}_{1}$
from left and $(\bm{K}+\sigma^{2}\bm{I})^{-1}$ from right, we transform the
predictive prediction into inner product form:
$\boxed{\begin{aligned} \mathrm{Inner\,Product\,Form:}\\\
g_{\ast}|\bm{x}_{\ast}\bm{X},\bm{y},\sigma^{2}\sim\mathcal{N}(&\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}(\bm{K}+\sigma^{2}\bm{I})^{-1}\bm{y},\\\
&\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast}-\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}(\bm{K}+\sigma^{2}\bm{I})^{-1}\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast}),\end{aligned}}$
where the input space only has the forms
$\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast},\,\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top},\,\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}$.
So we meet the kernel trick requirement.
Define the kernel variables:
$K(\bm{x}_{\ast},\bm{x}_{\ast})=\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast}$,
$K(\bm{x}_{\ast},\bm{X})=\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}$,
$K(\bm{X},\bm{X})=\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}$.
Then we transform the predictive distribution into the kernel form:
$\boxed{\begin{aligned} \mathrm{Kernel\,Form:}\\\
g_{\ast}|\bm{x}_{\ast}\bm{X},\bm{y},\sigma^{2}\sim\mathcal{N}(&K(\bm{x}_{\ast},\bm{X})(K(\bm{X},\bm{X})+\sigma^{2}\bm{I})^{-1}\bm{y},\\\
&K(\bm{x}_{\ast},\bm{x}_{\ast})-K(\bm{x}_{\ast},\bm{X})(K(\bm{X},\bm{X})+\sigma^{2}\bm{I})^{-1}K(\bm{X},\bm{x}_{\ast})).\end{aligned}}$
(9.3)
### 9.3 Kernels In A Nutshell
Kernel with basis functions transforms
$\bm{x}\in^{p}\rightarrow\boldsymbol{\phi}(\bm{x})\in^{q}$. Thus the inner
product in the $p\mathchar 45\relax$dimensional space
$\bm{x}^{\top}\bm{x}^{\prime}$ is transformed into the inner product in the
$q\mathchar 45\relax$dimensional space
$k(\bm{x},\bm{x}^{\prime})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\phi}(\bm{x}^{\prime})$.
Then the kernel matrix $K(\bm{X},\bm{X})$ has the following properties
1\. $K$ should be symmetric, i.e.,
$k(\bm{x},\bm{x}^{\prime})=k(\bm{x}^{\prime},\bm{x})$.
2\. The kernel matrix $K$ is positive semi-definite (PSD).
Proof [of Kernel Matrix $K$ is PSD] Let $K_{ij}=k(\bm{x}_{i},\bm{x}_{j})$,
$\forall i,j\in\\{1,2,...,n\\}$. And for any vector $\bm{t}$, we have
$\displaystyle\bm{t}^{\top}K\bm{t}$
$\displaystyle=\sum_{i,j=1}^{n}t_{i}t_{j}K_{ij}$
$\displaystyle=\sum_{i,j=1}^{n}t_{i}t_{j}\boldsymbol{\phi}(\bm{x}_{i})^{\top}\boldsymbol{\phi}(\bm{x}_{j})$
$\displaystyle=(\sum_{i=1}^{n}t_{i}\boldsymbol{\phi}(\bm{x}_{i}))^{\top}(\sum_{j=1}^{n}t_{j}\boldsymbol{\phi}(\bm{x}_{j}))$
$\displaystyle=||\sum_{i=1}^{n}t_{i}\boldsymbol{\phi}(\bm{x}_{i})||^{2}\geq
0.$
This completes the proof.
We may find $k(\bm{x},\bm{x}^{\prime})$ can be any function related to
$\bm{x},\bm{x}^{\prime}$ at a first glance. But from this PSD property of the
kernel matrix, this function is restricted to a small part in order to make
the $k(\bm{x},\bm{x}^{\prime})$ to be an inner-product form implicitly in
$q$-dimensional space.
###### Remark 9.2 (Some Specific Kernels)
1\. Linear Kernel: $k(\bm{x},\bm{x}^{\prime})=\bm{x}^{\top}\bm{x}^{\prime}$.
2\. Polynomial Kernel:
$k(\bm{x},\bm{x}^{\prime})=(\eta+\gamma\bm{x}^{\top}\bm{x}^{\prime})^{Q}$ with
$\gamma>0,\eta\geq 0$. 3\. Gaussian Kernel:
$k(\bm{x},\bm{x}^{\prime})=\exp(-\gamma||\bm{x}-\bm{x}^{\prime}||^{2})$. We
here prove that Gaussian kernel is of infinite-dimensional transformation.
Without loss of generality, let $\gamma=1$, then, $\displaystyle
k(\bm{x},\bm{x}^{\prime})$
$\displaystyle=\exp(-||\bm{x}-\bm{x}^{\prime}||^{2})$
$\displaystyle=\exp(-\bm{x}^{\top}\bm{x})\exp(-\bm{x}^{\prime\top}\bm{x}^{\prime})\exp(2\bm{x}^{\top}\bm{x}^{\prime})$
$\displaystyle\underset{\mathrm{expansion}}{\overset{\mathrm{Taylor}}{=}}\exp(-\bm{x}^{\top}\bm{x})\exp(-\bm{x}^{\prime\top}\bm{x}^{\prime})\exp\left(\sum_{i=0}^{\infty}\frac{(2\bm{x}^{\top}\bm{x}^{\prime})^{i}}{i!}\right)$
$\displaystyle=\sum_{i=0}^{\infty}\left(\exp(-\bm{x}^{\top}\bm{x})\exp(-\bm{x}^{\prime\top}\bm{x}^{\prime})\sqrt{\frac{2^{i}}{i!}}\sqrt{\frac{2^{i}}{i!}}(\bm{x})^{i}\cdot(\bm{x}^{\prime})^{i}\right)$
$\displaystyle=\sum_{i=0}^{\infty}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\exp(-\bm{x}^{\top}\bm{x})\sqrt{\frac{2^{i}}{i!}}(\bm{x})^{i}}\cdot{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\exp(-\bm{x}^{\prime\top}\bm{x}^{\prime})\sqrt{\frac{2^{i}}{i!}}(\bm{x}^{\prime})^{i}}\right)$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\boldsymbol{\phi}(\bm{x})^{\top}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\phi}(\bm{x}^{\prime})},$
where
$\boldsymbol{\phi}(\bm{x})=\sum_{i=0}^{\infty}\exp(-\bm{x}^{\top}\bm{x})\sqrt{\frac{2^{i}}{i!}}(\bm{x})^{i}$,
which is a sum of 1-dimensional input to infinite-dimensional input.
Similarly, we can get the infinite-dimensional transformation of Gaussian
kernel when $\gamma\neq 1$. 4\. Other Valid Kernels: A big advantage of using
kernels is that we do not need to specify $\boldsymbol{\phi}(\bm{x})$
explicitly, since we can work directly with $K$.
### 9.4 Gaussian Process from Zero-Mean Prior
We use Gaussian processes (GPs) to describe a distribution over functions.
Gaussian processes are natural generalizations of multivariate Gaussian random
variables to infinite (countably or continuous) index sets. Formally we define
Gaussian processes as follows:
###### Definition 9.1 (Gaussian Process)
A Gaussian process is a collection of random variables, any finite number of
which have a joint Gaussian distribution. The definition does not exclude
Gaussian processes with finite index sets which would be simply Gaussian
distributions.
#### 9.4.1 Noise-Free Observations
Following the Bayesian linear model with zero-mean prior in Section 8.3 (p.
8.3), we assume zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0})$
and define $g(\bm{x})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}$ for
each input observation $\bm{x}$. Then the mean and covariance for the prior
output are:
$\displaystyle\mathrm{E}[g(\bm{x})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}]=0,$
(9.4) $\displaystyle\mathrm{E}[g(\bm{x})g(\bm{x}^{\prime})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}\boldsymbol{\beta}^{\top}]\boldsymbol{\phi}(\bm{x}^{\prime})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}(\bm{x}^{\prime}),$
where the prior covariance matrix is specified manually and we could use a
kernel function instead:
$k(\bm{x},\bm{x}^{\prime})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}(\bm{x}^{\prime})$.
Suppose we have the training input design matrix $\bm{X}$, training output
vector $\bm{y}$, test input design matrix $\bm{X}_{\ast}$, and test output
vector $\bm{g}_{\ast}$. We obtain the joint distribution of the training
outputs $\bm{y}$ and the test outputs $\boldsymbol{g}_{\ast}$ by Equation
(9.4):
$\left[\begin{matrix}\bm{y}\\\
\boldsymbol{g}_{\ast}\end{matrix}\right]\sim\mathcal{N}\left(\boldsymbol{0},\left[\begin{matrix}K(\bm{X},\bm{X}),&K(\bm{X},\boldsymbol{X}_{\ast})\\\
K(\boldsymbol{X}_{\ast},\bm{X}),&K(\boldsymbol{X}_{\ast},\boldsymbol{X}_{\ast})\end{matrix}\right]\right).$
Therefore, we could use Gaussian identities to get the marginal distribution
of test outputs $\boldsymbol{g}_{\ast}$.
###### Lemma 9.3 (Marginal Distribution of Test Outputs)
Follow from Section 8.3 (p. 8.3), assume zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0})$
and define $g(\bm{x})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}$.
Given observed training inputs $\bm{X}$, training outputs $\bm{y}$ and test
inputs $\bm{x}_{\ast}$, the marginal distribution of test outputs
$\boldsymbol{g}_{\ast}$ is
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}\bm{y},$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}K(\bm{X},\bm{X}_{\ast})).$
Proof [of Lemma 9.3] Let $\bm{x}$ and $\bm{y}$ be jointly Gaussian random
vectors:
$\left[\begin{matrix}\bm{x}\\\
\bm{y}\end{matrix}\right]\sim\mathcal{N}\left(\left[\begin{matrix}\bm{u}_{x}\\\
\bm{u}_{y}\end{matrix}\right],\left[\begin{matrix}\bm{A},&\bm{C}\\\
\bm{C}^{\top},&\bm{B}\end{matrix}\right]\right).$ By Gaussian identity, the
marginal distribution of $\bm{x}$ and the conditional distribution of $\bm{x}$
given $\bm{y}$ are $\displaystyle\bm{x}$
$\displaystyle\sim\mathcal{N}(\bm{u}_{x},\bm{A}),$
$\displaystyle\bm{x}|\bm{y}$
$\displaystyle\sim\mathcal{N}(\bm{u}_{x}+\bm{C}\bm{B}^{-1}(\bm{y}-\bm{u}_{y}),\bm{A}-\bm{C}\bm{B}^{-1}\bm{C}^{\top})$
$\displaystyle\bm{y}$ $\displaystyle\sim\mathcal{N}(\bm{u}_{y},\bm{B})$
$\displaystyle\bm{y}|\bm{x}$
$\displaystyle\sim\mathcal{N}(\bm{u}_{y}+\bm{C}^{\top}\bm{A}^{-1}(\bm{x}-\bm{u}_{x}),\bm{B}-\bm{C}^{\top}\bm{A}^{-1}\bm{C}).$
Thus,
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}\bm{y},$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}K(\bm{X},\bm{X}_{\ast})),$
which completes the proof.
The marginal distribution of test outputs $\boldsymbol{g}_{\ast}$ agrees with
the kernel form of the Bayesian linear model with zero-mean prior in Equation
(9.3) except the noise term and the output are vectors now (i.e.,
$g_{\ast}\rightarrow\bm{g}_{\ast}$).
When given
$k(\bm{x},\bm{x}^{\prime})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}(\bm{x}^{\prime})$
which implies a zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0})$,
we have prior distribution over $\boldsymbol{g}_{\ast}$
$\begin{matrix}\boldsymbol{g}_{\ast}\end{matrix}\sim\mathcal{N}\left(\boldsymbol{0},\left[\begin{matrix}K(\boldsymbol{X}_{\ast},\boldsymbol{X}_{\ast})\end{matrix}\right]\right).$
Thus we can sample functions from
$p(\boldsymbol{g}_{\ast}|\boldsymbol{X}_{\ast})$. By sampling functions, we
mean the realizations of the output values given the inputs (possibly finite
or infinite number of samples) and the distribution of it. And this is the
meaning of distribution over functions. When given observed data $\bm{y}$ and
$\bm{X}$, we have prior distribution over $\bm{y},\boldsymbol{g}_{\ast}$
$\left[\begin{matrix}\bm{y}\\\
\boldsymbol{g}_{\ast}\end{matrix}\right]\sim\mathcal{N}\left(\boldsymbol{0},\left[\begin{matrix}K(\bm{X},\bm{X}),&K(\bm{X},\boldsymbol{X}_{\ast})\\\
K(\boldsymbol{X}_{\ast},\bm{X}),&K(\boldsymbol{X}_{\ast},\boldsymbol{X}_{\ast})\end{matrix}\right]\right).$
Thus we can compute the posterior by
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}\bm{y},$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})K(\bm{X},\bm{X})^{-1}K(\bm{X},\bm{X}_{\ast})),$
by which we can sample functions for the posterior distribution of
$p(\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y})$.
#### 9.4.2 Noisy Observations
Following again from Section 8.3 (p. 8.3]), we assume zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0})$.
And we define
$g(\bm{x})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\epsilon}}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\epsilon}\sim\mathcal{N}(0,\sigma^{2})}$.
Then the mean and covariance for the prior output are:
$\displaystyle\mathrm{E}[g(\bm{x})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}]+\mathrm{E}[\boldsymbol{\epsilon}]=\boldsymbol{0},$
$\displaystyle\mathrm{E}[g(\bm{x})g(\bm{x})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}\boldsymbol{\beta}^{\top}]\boldsymbol{\phi}(\bm{x})+2\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}]\mathrm{E}[\epsilon]+\mathrm{E}[\epsilon^{2}]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}(\bm{x})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}},$
$\displaystyle\mathrm{E}[g(\bm{x})g(\bm{x}^{\prime})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}\boldsymbol{\beta}^{\top}]\boldsymbol{\phi}(\bm{x}^{\prime})+\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}]\mathrm{E}[\epsilon]+\boldsymbol{\phi}(\bm{x}^{\prime})^{\top}\mathrm{E}[\boldsymbol{\beta}]\mathrm{E}[\epsilon]++\mathrm{E}[\epsilon^{2}]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}(\bm{x}^{\prime}).$
That is
$\mathrm{cov}(y_{i},y_{j})=k(\bm{x}_{i},\bm{x}_{j})\delta_{ij},$
where $\delta_{ij}$ is a Kronecker delta which is equal to 1 if and only if
when $i=j$ and 0 otherwise. Thus, we obtain the joint distribution of the
training outputs $\bm{y}$ and the test outputs $\boldsymbol{g}_{\ast}$
$\left[\begin{matrix}\bm{y}\\\
\boldsymbol{g}_{\ast}\end{matrix}\right]\sim\mathcal{N}\left(\boldsymbol{0},\left[\begin{matrix}K(\bm{X},\bm{X})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\bm{I}},&K(\bm{X},\boldsymbol{X}_{\ast})\\\
K(\boldsymbol{X}_{\ast},\bm{X}),&K(\boldsymbol{X}_{\ast},\boldsymbol{X}_{\ast})\end{matrix}\right]\right).$
Similarly, we could use Gaussian identities to get the marginal distribution
of test outputs $\boldsymbol{g}_{\ast}$.
###### Lemma 9.4 (Marginal Distribution of Test Outputs)
Follow from Section 8.3 (p. 8.3), assume zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{0})$
and define
$g(\bm{x})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\epsilon}$
where
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\epsilon\sim\mathcal{N}(0,\sigma^{2})}$.
Given observed training inputs $\bm{X}$, training outputs $\bm{y}$ and test
inputs $\bm{x}_{\ast}$, the marginal distribution of test outputs
$\boldsymbol{g}_{\ast}$ is
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle
K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\bm{I}})^{-1}\bm{y},$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\bm{I}})^{-1}K(\bm{X},\bm{X}_{\ast})).$
The marginal distribution of test outputs $\boldsymbol{g}_{\ast}$ agrees with
the form of Equation (9.3) except the output is a vector now (i.e.,
$g_{\ast}\rightarrow\bm{g}_{\ast}$).
#### 9.4.3 Further Extension, Generalized Gaussian Process
From the idea of the generalized linear model in Section 6.1 (p. 6.1), we
could set a different noise variance value for each observation. Suppose the
noise covariance matrix now is $\sigma^{2}\boldsymbol{\Lambda}$ where
$\boldsymbol{\Lambda}$ is a diagonal matrix. Then the marginal distribution of
test outputs $\bm{g}_{\ast}$ is
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle
K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\boldsymbol{\Lambda}})^{-1}\bm{y},$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma^{2}\boldsymbol{\Lambda}})^{-1}K(\bm{X},\bm{X}_{\ast})).$
Apparently, the noise covariance should not be dependent on the input matrix
$\bm{X}$. Otherwise, we will not be able to use the kernel trick.
(a) Use half data set
(b) Use all data set
Figure 9.1: Comparison of using half of the data and using all of the data.
(a) Realization 1
(b) Realization 2
(c) Realization 3
(d) Realization 4
Figure 9.2: Random realizations.
(a) Repeat the data set once
(b) Repeat the data set twice
(c) Repeat the data set three times
(d) Repeat the data set four times
Figure 9.3: Repeat data set to have larger data set number.
###### Example 9.4.1
For a data set with input being the house area and output being the rent of
the house. We use cross-validation to select the parameters in the Gaussian
kernel.
In Figure 9.1, the red dots are the training inputs, the blue line is the MAP
estimator via GP regressor and the shaded area are the ones with a confidence
interval of 95%.
In Figure 9.1(b), we use all the data. While in Figure 9.1(a), we only use the
data with house areas smaller than 100 $m^{2}$. We can see the estimator does
not predict well when the house area is larger than 100 $m^{2}$.
In Figure 9.2, we use all the data to predict and we draw four random
realizations from the posterior as shown in orange lines. Thus we can have a
further taste of what the distribution over functions means.
In Figure 9.3, we repeat the data set once, twice, three times, and four times
to see if the number of data sets matter. The yellow line is the predictor by
OLS. We find that the estimate from OLS does not change because it is exactly
the same data set. To see this, suppose the OLS estimator for data without
repeating is given by
$\hat{\boldsymbol{\beta}}_{1}=(\bm{X}^{\top}\bm{X})^{-1}\bm{X}^{\top}\bm{y}.$
The OLS estimator for data with repeating twice is given by
$\displaystyle\hat{\boldsymbol{\beta}}_{2}$
$\displaystyle=\left(\begin{bmatrix}\bm{X}\\\
\bm{X}\end{bmatrix}^{\top}\begin{bmatrix}\bm{X}\\\
\bm{X}\end{bmatrix}\right)^{-1}\begin{bmatrix}\bm{X}\\\
\bm{X}\end{bmatrix}^{\top}\begin{bmatrix}\bm{y}\\\ \bm{y}\end{bmatrix}$
$\displaystyle=\frac{1}{2}(\bm{X}^{\top}\bm{X})^{-1}\cdot
2\bm{X}^{\top}\bm{y}=\hat{\boldsymbol{\beta}}_{1}.$
which is exactly the same as $\hat{\boldsymbol{\beta}}_{1}$. Similarly, we can
also show that the OLS estimator for data with repeating three times and four
times are the same as well.
However, when we repeat the data set for the GP regressor, the estimate from
the GP regressor has a smaller variance when the data set number increases. In
Section 8.2 (p. 8.2), we mentioned that the Bayesian model considers prior
information on the parameters in the model making it particularly useful to
regularize regression problems where data information is limited. So the data
set number matters in the Bayesian approach where we also gave an example in
Section 8.2 (p. 8.2). This exact example shows the Bayesian thoughts behind
the GP.
### 9.5 Limitations of Gaussian Process from Non Zero-Mean Prior*
When the prior mean $\boldsymbol{\beta}_{0}$ is not zero and the model is
under Gaussian noise
$\boldsymbol{\epsilon}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$, we
transform the predictive prediction into
$\displaystyle g_{\ast}|\bm{x}_{\ast}\bm{X},\bm{y},\sigma^{2}\sim\mathcal{N}($
$\displaystyle\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}(\bm{K}+\sigma^{2}\bm{I})^{-1}\bm{y}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\phi}_{\ast}^{\top}(\frac{1}{\sigma^{2}}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}\boldsymbol{\Phi}+\bm{I})^{-1}\boldsymbol{\beta}_{0}},$
$\displaystyle\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast}-\boldsymbol{\phi}_{\ast}^{\top}\boldsymbol{\Sigma}_{0}\boldsymbol{\Phi}^{\top}(\bm{K}+\sigma^{2}\bm{I})^{-1}\boldsymbol{\Phi}\boldsymbol{\Sigma}_{0}\boldsymbol{\phi}_{\ast}),$
where we use the fact that $(\bm{A}\bm{B})^{-1}=\bm{B}^{-1}\bm{A}^{-1}$.
Following from Section 8.3 (p. 8.3), if we assume a non zero-mean prior
$\boldsymbol{\beta}\sim\mathcal{N}(\boldsymbol{\beta}_{0},\boldsymbol{\Sigma}_{0})$
and define $g(\bm{x})=\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}$.
Then, the mean and covariance for the prior output are:
$\displaystyle\mathrm{E}[g(\bm{x})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}]={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\phi}(\bm{x})^{\top}\boldsymbol{\beta}_{0}},$
$\displaystyle\mathrm{E}[g(\bm{x})g(\bm{x}^{\prime})]$
$\displaystyle=\boldsymbol{\phi}(\bm{x})^{\top}\mathrm{E}[\boldsymbol{\beta}\boldsymbol{\beta}^{\top}]\boldsymbol{\phi}(\bm{x}^{\prime})=\boldsymbol{\phi}(\bm{x})^{\top}(\boldsymbol{\Sigma}_{0}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boldsymbol{\beta}_{0}\boldsymbol{\beta}_{0}^{\top}})\boldsymbol{\phi}(\bm{x}^{\prime}),$
Therefore, we could get the joint distribution of the training outputs
$\bm{y}$ and the test outputs $\boldsymbol{g}_{\ast}$
$\left[\begin{matrix}\bm{y}\\\
\boldsymbol{g}_{\ast}\end{matrix}\right]\sim\mathcal{N}\left(\left[\begin{matrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi(\bm{x})^{\top}\boldsymbol{\beta}_{0}}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi(\boldsymbol{g}_{\ast})^{\top}\boldsymbol{\beta}_{0}}\end{matrix}\right],\left[\begin{matrix}K(\bm{X},\bm{X})+\sigma^{2}\bm{I},&K(\bm{X},\boldsymbol{X}_{\ast})\\\
K(\boldsymbol{X}_{\ast},\bm{X}),&K(\boldsymbol{X}_{\ast},\boldsymbol{X}_{\ast})\end{matrix}\right]\right).$
By Gaussian identities, the marginal distribution of test outputs is
$\displaystyle\bm{g}_{\ast}|\bm{X}_{\ast}\bm{X},\bm{y}\sim\mathcal{N}($
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi(\boldsymbol{g}_{\ast})^{\top}\boldsymbol{\beta}_{0}}+K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+\sigma^{2}\bm{I})^{-1}(\bm{y}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\phi(\bm{x})^{\top}\boldsymbol{\beta}_{0}}),$
$\displaystyle
K(\bm{X}_{\ast},\bm{x}_{\ast})-K(\bm{X}_{\ast},\bm{X})(K(\bm{X},\bm{X})+\sigma^{2}\bm{I})^{-1}K(\bm{X},\bm{X}_{\ast})).$
where we notice that we cannot find an explicit form in the $z$-dimensional
space for the mean of $g(\bm{x})$ and we cannot apply the kernel trick in this
sense. Thus we lose some flexibility from non zero-mean prior.
## Chapter 10 Linear Algebra
### A Dimension of Column Space and Row Space
In this appendix, we prove Lemma 10.1 that the dimension of the column space
of a matrix $\bm{X}\in^{n\times p}$ is equal to the dimension of its row
space, i.e., the row rank and the column rank of a matrix $\bm{X}$ are equal.
###### Lemma 10.1 (Dimension of Column Space and Row Space)
The dimension of the column space of a matrix $\bm{X}\in^{n\times p}$ is equal
to the dimension of its row space, i.e., the row rank and the column rank of a
matrix $\bm{X}$ are equal.
Proof [of Lemma 10.1] We first notice that the null space of $\bm{X}$ is
orthogonal complementary to the row space of $\bm{X}$:
$\mathcal{N}(\bm{X})\bot\mathcal{C}(\bm{X}^{\top})$ (where the row space of
$\bm{X}$ is exactly the column space of $\bm{X}^{\top}$), that is, vectors in
the null space of $\bm{X}$ are orthogonal to vectors in the row space of
$\bm{X}$. To see this, suppose $\bm{X}$ has rows
$\bm{a}_{1}^{\top},\bm{a}_{2}^{\top},\ldots,\bm{a}_{n}^{\top}$ and
$\bm{X}=[\bm{a}_{1}^{\top};\bm{a}_{2}^{\top};\ldots;\bm{a}_{n}^{\top}]$. For
any vector $\boldsymbol{\beta}\in\mathcal{N}(\bm{X})$, we have
$\bm{X}\boldsymbol{\beta}=\boldsymbol{0}$, that is,
$[\bm{a}_{1}^{\top}\boldsymbol{\beta};\bm{a}_{2}^{\top}\boldsymbol{\beta};\ldots;\bm{a}_{n}^{\top}\boldsymbol{\beta}]=\boldsymbol{0}$.
And since the row space of $\bm{X}$ is spanned by
$\bm{a}_{1}^{\top},\bm{a}_{2}^{\top},\ldots,\bm{a}_{n}^{\top}$. Then
$\boldsymbol{\beta}$ is perpendicular to any vectors from
$\mathcal{C}(\bm{X}^{\top})$ which means
$\mathcal{N}(\bm{X})\bot\mathcal{C}(\bm{X}^{\top})$.
Now suppose, the dimension of row space of $\bm{X}$ is $r$. Let
$\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}$ be a set of vectors in p and form a
basis for the row space. Then the $r$ vectors
$\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}$ are in the column
space of $\bm{X}$, furthermore, they are linearly independent. To see this,
suppose we have a linear combination of the $r$ vectors:
$\beta_{1}\bm{A}\bm{r}_{1}+\beta_{2}\bm{A}\bm{r}_{2}+\ldots+\beta_{r}\bm{A}\bm{r}_{r}=0$,
that is,
$\bm{X}(\beta_{1}\bm{r}_{1}+\beta_{2}\bm{r}_{2}+\ldots+\beta_{r}\bm{r}_{r})=0$
and the vector
$\bm{v}=\beta_{1}\bm{r}_{1}+\beta_{2}\bm{r}_{2}+\ldots+\beta_{r}\bm{r}_{r}$ is
in null space of $\bm{X}$. But since
$\\{\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}\\}$ is a basis for the row space
of $\bm{X}$, $\bm{v}$ is thus also in the row space of $\bm{X}$. We have shown
that vectors from null space of $\bm{X}$ is perpendicular to vectors from row
space of $\bm{X}$, thus $\bm{v}^{\top}\bm{v}=0$ and
$\beta_{1}=\beta_{2}=\ldots=\beta_{r}=0$. Then
$\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}$ are in the column
space of $\bm{X}$ and they are independent which means the dimension of the
column space of $\bm{X}$ is larger than $r$. This result shows that row rank
of $\bm{X}\leq$ column rank of $\bm{X}$.
If we apply this process again for $\bm{X}^{\top}$. We will have column rank
of $\bm{X}\leq$ row rank of $\bm{X}$. This completes the proof.
Further information can be drawn from this proof is that if
$\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}$ is a set of vectors in p that forms
a basis for the row space, then
$\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}$ is a basis for the
column space of $\bm{X}$. We formulate this finding into the following lemma.
###### Lemma 10.2 (Column Basis from Row Basis)
For any matrix $\bm{X}\in^{n\times p}$, let
$\\{\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}\\}$ be a set of vectors in p which
forms a basis for the row space, then
$\\{\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}\\}$ is a basis
for the column space of $\bm{X}$.
### B Fundamental Theorem of Linear Algebra
Proof [of Theorem 1.4, p. 1.4] Following the proof of Lemma 10.1 in Appendix
A. Let $\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}$ be a set of vectors in p that
form a basis for the row space, then
$\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}$ is a basis for the
column space of $\bm{X}$. Let $\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}\in^{p}$
form a basis for the null space of $\bm{X}$. Following again the proof of
Lemma 10.1, $\mathcal{N}(\bm{X})\bot\mathcal{C}(\bm{X}^{\top})$, thus,
$\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r}$ are perpendicular to
$\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}$. Then,
$\\{\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r},\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}\\}$
is linearly independent in p.
For any vector $\boldsymbol{\beta}\in^{p}$, $\bm{X}\boldsymbol{\beta}$ is in
the column space of $\bm{X}$. Then it can be expressed by a combination of
$\bm{X}\bm{r}_{1},\bm{X}\bm{r}_{2},\ldots,\bm{X}\bm{r}_{r}$:
$\bm{X}\boldsymbol{\beta}=\sum_{i=1}^{r}a_{i}\bm{X}\bm{r}_{i}$ which states
that $\bm{X}(\boldsymbol{\beta}-\sum_{i=1}^{r}a_{i}\bm{r}_{i})=\boldsymbol{0}$
and $\boldsymbol{\beta}-\sum_{i=1}^{r}a_{i}\bm{r}_{i}$ is thus in
$\mathcal{N}(\bm{X})$. Since $\\{\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}\\}$
is a basis for the null space of $\bm{X}$,
$\boldsymbol{\beta}-\sum_{i=1}^{r}a_{i}\bm{r}_{i}$ can be expressed by a
combination of $\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}$:
$\boldsymbol{\beta}-\sum_{i=1}^{r}a_{i}\bm{r}_{i}=\sum_{j=1}^{k}b_{j}\bm{n}_{j}$,
i.e.,
$\boldsymbol{\beta}=\sum_{i=1}^{r}a_{i}\bm{r}_{i}+\sum_{j=1}^{k}b_{j}\bm{n}_{j}$.
That is, any vector $\boldsymbol{\beta}\in^{p}$ can be expressed by
$\\{\bm{r}_{1},\bm{r}_{2},\ldots,\bm{r}_{r},\bm{n}_{1},\bm{n}_{2},\ldots,\bm{n}_{k}\\}$
and the set forms a basis for p. Thus the dimension sum to $p$: $r+k=p$, i.e.,
$dim(\mathcal{N}(\bm{X}))+dim(\mathcal{C}(\bm{X}^{\top}))=p$. Similarly, we
can prove $dim(\mathcal{N}(\bm{X}^{\top}))+dim(\mathcal{C}(\bm{X}))=n$.
### C Similar Matrices
###### Definition 10.1 (Similar Matrices)
For any nonsingular matrix $\bm{P}$, the matrices $\bm{A}$ and
$\bm{P}\bm{A}\bm{P}^{-1}$ are called similar matrices.
###### Lemma 10.3 (Eigenvalue and Rank of Similar Matrices)
Any eigenvalue of $\bm{A}$ is also an eigenvalue of $\bm{P}\bm{A}\bm{P}^{-1}$
given any nonsingular matrix $\bm{P}$. The converse is also true that any
eigenvalue of $\bm{P}\bm{A}\bm{P}^{-1}$ is also an eigenvalue of $\bm{A}$.
And also the rank of $\bm{A}$ is equal to the rank of matrix
$\bm{P}\bm{A}\bm{P}^{-1}$ given any nonsingular matrix $\bm{P}$.
Proof [of Lemma 10.3] For any eigenvalue $\lambda$ of $\bm{A}$, we have
$\bm{A}\bm{x}=\lambda\bm{x}$. Then
$\lambda\bm{P}\bm{x}=\bm{P}\bm{A}\bm{P}^{-1}\bm{P}\bm{x}$ such that
$\bm{P}\bm{x}$ is an eigenvector of $\bm{P}\bm{A}\bm{P}^{-1}$ corresponding to
$\lambda$.
Similarly, for any eigenvalue $\lambda$ of $\bm{P}\bm{A}\bm{P}^{-1}$, we have
$\bm{P}\bm{A}\bm{P}^{-1}\bm{x}=\lambda\bm{x}$. Then
$\bm{A}\bm{P}^{-1}\bm{x}=\lambda\bm{P}^{-1}\bm{x}$ such that
$\bm{P}^{-1}\bm{x}$ is an eigenvector of $\bm{A}$ corresponding to $\lambda$.
For the rank of $\bm{P}\bm{A}\bm{P}^{-1}$, we have
$trace(\bm{P}\bm{A}\bm{P}^{-1})=trace(\bm{A}\bm{P}^{-1}\bm{P})=trace(\bm{A})$,
where the first equality comes from the fact that the trace of a product is
invariant under cyclical permutations of the factors:
$\boxed{trace(\bm{A}\bm{B}\bm{C})=trace(\bm{B}\bm{C}\bm{A})=trace(\bm{C}\bm{A}\bm{B})},$
if all $\bm{A}\bm{B}\bm{C}$, $\bm{B}\bm{C}\bm{A}$, and $\bm{C}\bm{A}\bm{B}$
exist.
### D Cochran’s Theorem
In this appendix, we provide proof for Theorem 4.5 (p. 4.5). To prove the
Cochran’s theorem, we need the following lemma.
###### Lemma 10.4 (Idempotent Decomposition: Rank-Additivity)
For $n\times n$ square matrices $\bm{A}_{1},\bm{A}_{2},\ldots,\bm{A}_{k}$, and
$\bm{A}_{1}+\bm{A}_{2}+\ldots+\bm{A}_{k}=\bm{I}_{n}$, then the following three
conditions are equivalent:
i). $\bm{A}_{i}^{2}=\bm{A}_{i}$, for all $i\in\\{1,2,\ldots,k\\}$, i.e.,
$\bm{A}_{i}$’s are idempotent;
ii). $rank(\bm{A}_{1})+rank(\bm{A}_{2})+\ldots+rank(\bm{A}_{k})=n$;
iii). $\bm{A}_{i}\bm{A}_{j}=\boldsymbol{0}$ for all $i\neq j$ and
$i,j\in\\{1,2,\ldots,k\\}$.
Proof [of Lemma 10.4] From i) to ii), by Lemma 1.19 (p. 1.19), the trace and
rank of any idempotent matrix are the same. Then,
$\sum_{i=1}^{k}rank(\bm{A}_{i})=\sum_{i=1}^{k}trace(\bm{A}_{i})=trace(\bm{I}_{n})=n.$
From ii) to iii), we have the following block Gaussian elimination for a
$(k+1)\times(k+1)$ block matrix (that is, $(k+1)n\times(k+1)n$ matrix) where
$(2,2),(3,3),\ldots,(k+1,k+1)$ blocks are
$\bm{A}_{1},\bm{A}_{2},\ldots,\bm{A}_{k}$ respectively.
$\displaystyle\bm{X}=$ $\displaystyle\begin{bmatrix}\boldsymbol{0}_{n}&&&\\\
&\bm{A}_{1}&&\\\ &&\ddots&\\\ &&&\bm{A}_{k}\\\
\end{bmatrix}\stackrel{{\scriptstyle\bm{E}_{1}\times}}{{\rightarrow}}\begin{bmatrix}\boldsymbol{0}_{n}&\bm{A}_{1}&\ldots&\bm{A}_{k}\\\
&\bm{A}_{1}&\\\ &&\ddots&\\\ &&&\bm{A}_{k}\\\
\end{bmatrix}\stackrel{{\scriptstyle\bm{E}_{2}\times}}{{\rightarrow}}\begin{bmatrix}\bm{I}_{n}&\bm{A}_{1}&\ldots&\bm{A}_{k}\\\
\bm{A}_{1}&\bm{A}_{1}&\\\ \vdots&&\ddots&\\\ \bm{A}_{k}&&&\bm{A}_{k}\\\
\end{bmatrix}\stackrel{{\scriptstyle\bm{E}_{3}\times}}{{\rightarrow}}$
$\displaystyle\begin{bmatrix}\bm{I}_{n}&\bm{A}_{1}&\ldots&\bm{A}_{k}\\\
&\bm{A}_{1}-\bm{A}_{1}^{2}&&-\bm{A}_{1}\bm{A}_{k}\\\ &\vdots&\ddots&\vdots\\\
&-\bm{A}_{k}\bm{A}_{1}&&\bm{A}_{k}-\bm{A}_{k}^{2}\\\
\end{bmatrix}\stackrel{{\scriptstyle\bm{E}_{4}\times}}{{\rightarrow}}\begin{bmatrix}\bm{I}_{n}&&&\\\
&\bm{A}_{1}-\bm{A}_{1}^{2}&&-\bm{A}_{1}\bm{A}_{k}\\\ &\vdots&\ddots&\vdots\\\
&-\bm{A}_{k}\bm{A}_{1}&&\bm{A}_{k}-\bm{A}_{k}^{2}\\\ \end{bmatrix}=\bm{Y},$
where blank entries indicate zeros. And
$\bullet$ $\bm{E}_{1}$ is adding row-2, row-3, $\ldots$, row-(k+1) to row-1;
$\bullet$ $\bm{E}_{2}$ is adding column-2, column-3, $\ldots$, column-(k+1) to
column-1;
$\bullet$ $\bm{E}_{3}$ is subtracting the row-2 by $\bm{A}_{1}\times$(row-1),
subtracting the row-3 by $\bm{A}_{2}\times$(row-1), $\ldots$;
$\bullet$ $\bm{E}_{4}$ is subtracting the column-2 by
$\bm{A}_{1}\times$(column-1), subtracting column-3 by
$\bm{A}_{2}\times$(column-1), $\ldots$.
We notice that elementary operations/transformations will not change the rank
of the matrix. Since $\bm{X}$ is of rank $\sum_{i=1}^{k}r_{k}=n$, and
$\bm{I}_{n}$ in $\bm{Y}$ is of rank-$n$ as well. We must have
$\begin{bmatrix}\bm{A}_{1}-\bm{A}_{1}^{2}&&-\bm{A}_{1}\bm{A}_{k}\\\
\vdots&\ddots&\vdots\\\ -\bm{A}_{k}\bm{A}_{1}&&\bm{A}_{k}-\bm{A}_{k}^{2}\\\
\end{bmatrix}=\boldsymbol{0},$
which implies $\bm{A}_{i}\bm{A}_{j}=\boldsymbol{0}$ for all $i\neq j$ and
$i,j\in\\{1,2,\ldots,k\\}$.
From iii) to i). we have
$\displaystyle\bm{A}_{i}$ $\displaystyle=\bm{A}_{i}\bm{I}_{n}$
$\displaystyle=\bm{A}_{i}(\bm{A}_{1}+\bm{A}_{2}+\ldots+\bm{A}_{k})$
$\displaystyle=\bm{A}_{i}^{2}.$
This completes the proof.
##### Note on nomenclature
We say that the $\bm{A}_{i}$’s are orthogonal if iii) of Lemma 10.4 is
satisfied and the rank is additive if ii) is satisfied.
Now we are ready to prove the Cochran’s Theorem as follows:
Proof [of Theorem 4.5, p. 4.5] From Lemma 10.4, $\bm{A}_{i}$’s are idempotent.
Case 1, If $\bm{y}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$:
By Spectral Theorem 1.16 (p. 1.16) and Lemma 1.17 (p. 1.17, the only possible
eigenvalues of idempotent matrices are 0 and 1), we can rewrite the $q_{i}$ by
$q_{i}=\bm{y}^{\top}\bm{A}_{i}\bm{y}=\bm{y}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\bm{y}$,
where $\bm{A}_{i}=\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top}$ is the spectral
decomposition of $\bm{A}_{i}$. From the fact that rotations on the normal
distribution do not affect the distribution111Rotations on the Gaussian
distribution do not affect the distribution. That is for any orthogonal matrix
$\bm{Q}$ with $\bm{Q}\bm{Q}^{\top}=\bm{Q}^{\top}\bm{Q}=\bm{I}$, if
$\bm{X}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$, then
$\bm{Q}\bm{X}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I})$., we can define
$\boldsymbol{\eta}=\bm{Q}^{\top}\bm{y}\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I}).$
Thus,
$q_{i}=\boldsymbol{\eta}^{\top}\boldsymbol{\Lambda}\boldsymbol{\eta}\sim\sigma^{2}\chi^{2}_{\mathrm{rank(\bm{A}_{i})}}\sim\sigma^{2}\chi^{2}_{(r_{i})},$
Case 2, If $\bm{y}\sim\mathcal{N}(\boldsymbol{\mu},\sigma^{2}\bm{I})$, and
$\boldsymbol{\mu}^{\top}\bm{A}_{i}\boldsymbol{\mu}=0$:
Let
$p_{i}=(\bm{y}-\boldsymbol{\mu})^{\top}\bm{A}_{i}(\bm{y}-\boldsymbol{\mu})$,
similarly, we can rewrite the $p_{i}$ by
$p_{i}=(\bm{y}-\boldsymbol{\mu})^{\top}\bm{A}_{i}(\bm{y}-\boldsymbol{\mu})=(\bm{y}-\boldsymbol{\mu})^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})(\bm{y}-\boldsymbol{\mu})$,
where $\bm{A}_{i}=\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top}$ is the spectral
decomposition of $\bm{A}_{i}$. From the fact that rotations on the normal
distribution do not effect the distribution, we can define
$\boldsymbol{\eta}=\bm{Q}^{\top}(\bm{y}-\boldsymbol{\mu})\sim\mathcal{N}(\boldsymbol{0},\sigma^{2}\bm{I}).$
Thus,
$p_{i}=\boldsymbol{\eta}^{\top}\boldsymbol{\Lambda}\boldsymbol{\eta}\sim\sigma^{2}\chi^{2}_{\mathrm{rank(\bm{A}_{i})}}\sim\sigma^{2}\chi^{2}_{(r_{i})}.$
Then, we decompose the $p_{i}$ into
$\displaystyle p_{i}$
$\displaystyle=\bm{y}^{\top}\bm{A}_{i}\bm{y}-2\bm{y}^{\top}\bm{A}_{i}\boldsymbol{\mu}+\boldsymbol{\mu}^{\top}\bm{A}_{i}\boldsymbol{\mu}$
$\displaystyle=q_{i}-2\bm{y}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\boldsymbol{\mu}+\boldsymbol{\mu}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\boldsymbol{\mu}.$
Since we assume
$\boldsymbol{\mu}^{\top}\bm{A}_{i}\boldsymbol{\mu}=\boldsymbol{\mu}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\boldsymbol{\mu}=0$,
and $\boldsymbol{\Lambda}$ contains only 1 and 0 on the diagonal. We have
$\boldsymbol{\Lambda}=\boldsymbol{\Lambda}\boldsymbol{\Lambda}^{\top}$. That
is
$\boldsymbol{\mu}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\boldsymbol{\mu}=\boldsymbol{\mu}^{\top}(\bm{Q}\boldsymbol{\Lambda}\boldsymbol{\Lambda}^{\top}\bm{Q}^{\top})\boldsymbol{\mu}=||\boldsymbol{\Lambda}^{\top}\bm{Q}^{\top}\boldsymbol{\mu}||^{2}=0,$
which implies
$2\bm{y}^{\top}\bm{A}_{i}\boldsymbol{\mu}=2\bm{y}^{\top}(\bm{Q}\boldsymbol{\Lambda}\bm{Q}^{\top})\boldsymbol{\mu}=0$.
Thus $q_{i}=q_{i}\sim\sigma^{2}\chi^{2}_{(r_{i})}$.
Case 3: From Lemma 10.4, $\bm{A}_{i}\bm{A}_{j}=\boldsymbol{0}$ if $i\neq j$.
Let $\bm{A}_{i}=\bm{Q}_{i}\boldsymbol{\Lambda}_{i}\bm{Q}_{i}^{\top}$,
$\bm{A}_{j}=\bm{Q}_{j}\boldsymbol{\Lambda}_{j}\bm{Q}_{j}^{\top}$ be the
spectral decomposition of $\bm{A}_{i}$ and $\bm{A}_{j}$. Then, we have
$\displaystyle\bm{A}_{i}\bm{A}_{j}$
$\displaystyle=\bm{Q}_{i}\boldsymbol{\Lambda}_{i}\bm{Q}_{i}^{\top}\bm{Q}_{j}\boldsymbol{\Lambda}_{j}\bm{Q}_{j}^{\top}=\boldsymbol{0}$
(10.1) $\displaystyle\bm{Q}_{i}^{\top}\bm{A}_{i}\bm{A}_{j}\bm{Q}_{j}$
$\displaystyle=\boldsymbol{\Lambda}_{i}\bm{Q}_{i}^{\top}\bm{Q}_{j}\boldsymbol{\Lambda}_{j}=\boldsymbol{0}.$
Write out $q_{i}$ and $q_{j}$:
$\displaystyle q_{i}$
$\displaystyle=\bm{y}^{\top}\bm{Q}_{i}\boldsymbol{\Lambda}_{i}\bm{Q}_{i}^{\top}\bm{y}=\bm{y}^{\top}\bm{Q}_{i}\boldsymbol{\Lambda}_{i}\boldsymbol{\Lambda}_{i}^{\top}\bm{Q}_{i}^{\top}\bm{y}$
$\displaystyle q_{j}$
$\displaystyle=\bm{y}^{\top}\bm{Q}_{j}\boldsymbol{\Lambda}_{j}\bm{Q}_{j}^{\top}\bm{y}=\bm{y}^{\top}\bm{Q}_{j}\boldsymbol{\Lambda}_{j}\boldsymbol{\Lambda}_{j}^{\top}\bm{Q}_{j}^{\top}\bm{y}.$
Let $\bm{a}_{i}=\boldsymbol{\Lambda}_{i}^{\top}\bm{Q}_{i}^{\top}\bm{y}$ and
$\bm{a}_{j}=\boldsymbol{\Lambda}_{j}^{\top}\bm{Q}_{j}^{\top}\bm{y}$, we have
$\displaystyle\mathrm{Cov}[\bm{a}_{i},\bm{a}_{j}]$
$\displaystyle=\boldsymbol{\Lambda}_{i}^{\top}\bm{Q}_{i}^{\top}\mathrm{Cov}[\bm{y},\bm{y}]\bm{Q}_{j}\boldsymbol{\Lambda}_{j}=\sigma^{2}\boldsymbol{\Lambda}_{i}^{\top}\bm{Q}_{i}^{\top}\bm{Q}_{j}\boldsymbol{\Lambda}_{j}=\boldsymbol{0},$
where the last equality is from Equation (10.1). This implies
$\mathrm{Cov}[q_{i},q_{j}]=0$ since $q_{i}=\bm{a}_{i}^{\top}\bm{a}_{i}$ and
$q_{j}=\bm{a}_{j}^{\top}\bm{a}_{j}$. And we complete the proof.
Another proof is provided in Gut (2009b). But the author did not provide an
inductive case for $k>2$. Interesting readers can refer to it.
## Chapter 11 Matrix Decomposition
### A CR Decomposition
The CR decomposition is proposed in Strang (2021); Strang and Moler (2022). We
firstly give the result and we will discuss the existence and the origin of
this decomposition in the following sections.
###### Theorem 11.1 (CR Decomposition)
Any rank-$r$ matrix $\bm{A}\in^{n\times p}$ can be factored as
$\underset{n\times p}{\bm{A}}=\underset{n\times
r}{\bm{C}}\,\,\,\,\,\,\,\,\underset{r\times p}{\bm{R}}$
where $\bm{C}$ is the first $r$ linearly independent columns of $\bm{A}$, and
$\bm{R}$ is an $r\times p$ matrix to reconstruct the columns of $\bm{A}$ from
columns of $\bm{C}$. In particular, $\bm{R}$ is the row reduced echelon form
(RREF) of $\bm{A}$ without the zero rows.
The storage for the decomposition is then reduced or potentially increased
from $np$ to $r(n+p)$.
#### A.1 Existence of the CR Decomposition
Since matrix $\bm{A}$ is of rank $r$, there are some $r$ linearly independent
columns in $\bm{A}$. We then choose linearly independent columns from $\bm{A}$
and put them into $\bm{C}$:
Find $r$ linearly Independent Columns From $\bm{A}$ 1\. If column 1 of
$\bm{A}$ is not zero, put it into the column of $\bm{C}$; 2\. If column 2 of
$\bm{A}$ is not a multiple of column 1, put it into the column of $\bm{C}$;
3\. If column 3 of $\bm{A}$ is not a combination of columns 1 and 2, put it
into the column of $\bm{C}$; 4\. Continue this process until we find $r$
linearly independent columns (or all the linearly independent columns if we do
not know the rank $r$ beforehand).
When we have the $r$ linearly independent columns from $\bm{A}$, we can prove
the existence of CR decomposition by the column space view of matrix
multiplication.
##### Column space view of matrix multiplication
A multiplication of two matrices $\bm{D}\in^{n\times k},\bm{E}\in^{k\times p}$
is
$\bm{A}=\bm{D}\bm{E}=\bm{D}[\bm{e}_{1},\bm{e}_{2},\ldots,\bm{e}_{p}]=[\bm{D}\bm{e}_{1},\bm{D}\bm{e}_{2},\ldots,\bm{D}\bm{e}_{p}]$,
i.e., each column of $\bm{A}$ is a combination of columns from $\bm{D}$.
Proof [of Theorem 11.1] As the rank of matrix $\bm{A}$ is $r$ and $\bm{C}$
contains $r$ linearly independent columns from $\bm{A}$, the column space of
$\bm{C}$ is equivalent to the column space of $\bm{A}$. If we take any other
column $\bm{a}_{i}$ of $\bm{A}$, $\bm{a}_{i}$ can be represented as a linear
combination of the columns of $\bm{C}$, i.e., there exists a vector
$\bm{r}_{i}$ such that $\bm{a}_{i}=\bm{C}\bm{r}_{i}$, $\forall
i\in\\{1,2,\ldots,p\\}$. Put these $\bm{r}_{i}$’s into the columns of matrix
$\bm{R}$, we obtain
$\bm{A}=[\bm{a}_{1},\bm{a}_{2},\ldots,\bm{a}_{p}]=[\bm{C}\bm{r}_{1},\bm{C}\bm{r}_{2},\ldots,\bm{C}\bm{r}_{p}]=\bm{C}\bm{R},$
from which the result follows.
#### A.2 Reduced Row Echelon Form (RREF)
We write out the Gaussian elimination for a $4\times 4$ square matrix, where
$\boxtimes$ represents a value that is not necessarily zero, and boldface
indicates the value has just been changed:
Gaussian Elimination for a Square Matrix
$\mathop{{}\begin{bmatrix}\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes\end{bmatrix}}_{\textstyle\mathstrut\bm{A}}\stackrel{{\scriptstyle\bm{E}_{1}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\bm{0}&\bm{0}&\bm{\boxtimes}&\bm{\boxtimes}\\\
\bm{0}&\bm{\boxtimes}&\bm{\boxtimes}&\bm{\boxtimes}\\\
\bm{0}&\bm{\boxtimes}&\bm{\boxtimes}&\bm{\boxtimes}\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{P}_{1}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{\boxtimes}}&\bm{\boxtimes}&\bm{\boxtimes}\\\
\bm{0}&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{\boxtimes}}&\bm{\boxtimes}\\\
0&\boxtimes&\boxtimes&\boxtimes\end{bmatrix}}_{\textstyle\mathstrut\bm{P}_{1}\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{E}_{2}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boxtimes}&\boxtimes&\boxtimes\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\boxtimes}&\boxtimes\\\
0&\bm{0}&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{\boxtimes}}\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{2}\bm{P}_{1}\bm{E}_{1}\bm{A}}.$
Furthermore, the Gaussian elimination can also be applied to a rectangular
matrix, we give an example for a $4\times 5$ matrix as follows:
Gaussian Elimination for a Rectangular Matrix
$\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&10&9&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\boxtimes&\boxtimes&\boxtimes&\boxtimes&\boxtimes\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{A}}\stackrel{{\scriptstyle\bm{E}_{1}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&10&9&\boxtimes\\\
\bm{0}&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{5}}&\bm{6}&\bm{\boxtimes}\\\
\bm{0}&\bm{0}&\bm{2}&\bm{\boxtimes}&\bm{\boxtimes}\\\
\bm{0}&\bm{0}&\bm{\boxtimes}&\bm{\boxtimes}&\bm{\boxtimes}\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{E}_{2}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&10&9&\boxtimes\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}5}&6&\boxtimes\\\
0&0&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{3}}&\bm{\boxtimes}\\\
0&0&\bm{0}&\bm{0}&\bm{0}\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{2}\bm{E}_{1}\bm{A}},$
where the blue-colored numbers are pivots and we call the last matrix above
row echelon form. Note that we get the 4-th row as a zero row in this specific
example. Going further, we subtract each row by a multiple of the next row to
make the entries above the pivots to be zero:
Reduced Row Echelon Form: Get Zero Above Pivots
$\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&10&9&\boxtimes\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}5}&6&\boxtimes\\\
0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3}&\boxtimes\\\
0&0&0&0&0\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{2}\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{E}_{3}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&\bm{0}&\bm{-3}&\bm{\boxtimes}\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}5}&6&\boxtimes\\\
0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3}&\boxtimes\\\
0&0&0&0&0\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{3}\bm{E}_{2}\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{E}_{4}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&0&\bm{0}&\bm{\boxtimes}\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}5}&\bm{0}&\bm{\boxtimes}\\\
0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3}&\boxtimes\\\
0&0&0&0&0\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{4}\bm{E}_{3}\bm{E}_{2}\bm{E}_{1}\bm{A}},$
where $\bm{E}_{3}$ subtracts 2 times the $2$-nd row from the $1$-st row, and
$\bm{E}_{4}$ adds the $3$-rd row to the $1$-st row and subtracts 2 times the
$3$-rd row from the $2$-nd row. Finally, we get the full row reduced echelon
form by making the pivots to be 1:
Reduced Row Echelon Form: Make The Pivots To Be 1
$\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}&\boxtimes&0&0&\boxtimes\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}5}&0&\boxtimes\\\
0&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3}&\boxtimes\\\
0&0&0&0&0\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{4}\bm{E}_{3}\bm{E}_{2}\bm{E}_{1}\bm{A}}\stackrel{{\scriptstyle\bm{E}_{5}}}{{\longrightarrow}}\mathop{{}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{1}}&\bm{\boxtimes}&\bm{0}&\bm{0}&\bm{\boxtimes}\\\
\bm{0}&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{1}}&\bm{0}&\bm{\boxtimes}\\\
\bm{0}&\bm{0}&\bm{0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bm{1}}&\bm{\boxtimes}\\\
0&0&0&0&0\\\
\end{bmatrix}}_{\textstyle\mathstrut\bm{E}_{5}\bm{E}_{4}\bm{E}_{3}\bm{E}_{2}\bm{E}_{1}\bm{A}},$
where $\bm{E}_{5}$ makes the pivots to be 1. We call this final matrix the
reduced row echelon form of $\bm{A}$ where it has 1’s as pivots and zeros
above the pivots.
###### Lemma 11.2 (Rank and Pivots)
The rank of $\bm{A}$ is equal to the number of pivots.
###### Lemma 11.3 (RREF in CR)
The reduced row echelon form of the matrix $\bm{A}$ without zero rows is the
matrix $\bm{R}$ in the CR decomposition.
Proof [Informal Proof of Lemma 11.3 and Lemma 11.2] Following the steps in the
Gaussian elimination, the number of pivot elements indicates the number of
linearly independent columns in matrix $\bm{A}$, which is on the other hand,
exactly the rank of the matrix.
Following the example above, we have
$\bm{E}_{5}\bm{E}_{4}\bm{E}_{3}\bm{E}_{2}\bm{E}_{1}\bm{A}=\bm{R}\quad\longrightarrow\quad\bm{A}=(\bm{E}_{5}\bm{E}_{4}\bm{E}_{3}\bm{E}_{2}\bm{E}_{1})^{-1}\bm{R}.$
We notice that the columns 1, 3, 4 of $\bm{R}$ have only one element 1 which
means we could construct a matrix $\bm{C}$ (exactly the same column matrix in
CR decomposition) whose first 3 columns are the same as the 1-st, 3-rd, 4-th
columns of matrix $\bm{A}$ respectively, i.e.,
$\bm{C}=[\bm{a}_{1},\bm{a}_{3},\bm{a}_{4}]$. Furthermore, since the last row
of $\bm{R}$ is a zero vector, the 4-th column of $\bm{C}$ does not account for
any computation so that we can just ignore the 4-th column of $\bm{C}$ and the
4-th row of $\bm{R}$. And this $\bm{C}$ is the only matrix that can
reconstruct the 1-st, 3-rd, 4-th columns of $\bm{A}$ as the pivots of $\bm{R}$
are all 1. We obtain
$\bm{A}=\bm{C}\bm{R}.$
This completes the proof.
In short, we first compute the reduced row echelon form of the matrix $\bm{A}$
by $rref(\bm{A})$, Then $\bm{C}$ is obtained by removing from $\bm{A}$ all the
non-pivot columns (this can be determined by looking for columns in
$rref(\bm{A})$ which do not contain a pivot). And $\bm{R}$ is obtained by
eliminating zero rows of $rref(\bm{A})$. And this is actually a special case
of rank decomposition of matrix $\bm{A}$. However, CR decomposition is so
special that it involves the reduced row echelon form so we introduce it here
particularly.
$\bm{R}$ has a remarkable form whose $r$ columns containing the pivots form an
$r\times r$ identity matrix. Note again that we can just remove the zero rows
from the row reduced echelon form to obtain this matrix $\bm{R}$. In Strang
(2021), the authors give a specific notation for the row reduced echelon form
without removing the zero rows as $\bm{R}_{0}$:
$\bm{R}_{0}=rref(\bm{A})=\begin{bmatrix}\bm{R}\\\
\boldsymbol{0}\end{bmatrix}=\begin{bmatrix}\bm{I}_{r}&\bm{F}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\bm{P},$
where the $p\times p$ permutation matrix $\bm{P}$ puts the columns of $r\times
r$ identity matrix $\bm{I}_{r}$ into the correct positions, matching the first
$r$ linearly independent columns of the original matrix $\bm{A}$.
The important theorem in linear algebra that the row rank equals the column
rank of any matrix can be proved by the UTV framework (Theorem 10.1, p. 10.1,
see Lu (2021a) for using UTV to prove the theorem). The CR decomposition also
reveals the theorem.
Proof [of Theorem 10.1, p. 10.1, A Second Way] For CR decomposition of matrix
$\bm{A}=\bm{C}\bm{R}$, we have $\bm{R}=[\bm{I}_{r},\bm{F}]\bm{P}$, where
$\bm{P}$ is a $p\times p$ permutation to put the columns of the $r\times r$
identity matrix $\bm{I}_{r}$ into the correct positions as shown above. It can
be easily verified that the $r$ rows of $\bm{R}$ are linearly independent
because of the submatrix $\bm{I}_{r}$ (since $\bm{I}_{r}$ is nonsingular) such
that the row rank of $\bm{R}$ is $r$.
Firstly, from the definition of the CR decomposition, the $r$ columns of
$\bm{C}$ are from $r$ linearly independent columns of $\bm{A}$, and the column
rank of $\bm{A}$ is $r$. Further,
$\bullet$ Since $\bm{A}=\bm{C}\bm{R}$, all rows of $\bm{A}$ are combinations
of the rows of $\bm{R}$. That is, the row rank of $\bm{A}$ is no larger than
the row rank of $\bm{R}$;
$\bullet$ From $\bm{A}=\bm{C}\bm{R}$, we also have
$(\bm{C}^{\top}\bm{C})^{-1}\bm{C}^{\top}\bm{C}\bm{R}=(\bm{C}^{\top}\bm{C})^{-1}\bm{C}^{\top}\bm{A}$,
that is $\bm{R}=(\bm{C}^{\top}\bm{C})^{-1}\bm{C}^{\top}\bm{A}$.
$\bm{C}^{\top}\bm{C}$ is nonsingular since it has full column rank $r$. Then
all rows of $\bm{R}$ are also combinations of the rows of $\bm{A}$. That is,
the row rank of $\bm{R}$ is no larger than the row rank of $\bm{A}$;
$\bullet$ By “sandwiching”, the row rank of $\bm{A}$ is equal to the row rank
of $\bm{R}$ which is $r$.
Therefore, both the row rank and column rank of $\bm{A}$ are equal to $r$ from
which the result follows.
In the above proof, we apply CR decomposition to show that the row rank of a
matrix is equal to its column rank. Moreover, we will also discuss the special
form of pseudo-inverse from CR decomposition in Appendix A (p. A).
#### A.3 Rank Decomposition
We previously mentioned that the CR decomposition is a special case of rank
decomposition. Formally, we prove the existence of the rank decomposition
rigorously in the following theorem.
###### Theorem 11.4 (Rank Decomposition)
Any rank-$r$ matrix $\bm{A}\in^{n\times p}$ can be factored as
$\underset{n\times p}{\bm{A}}=\underset{n\times
r}{\bm{D}}\,\,\,\,\,\,\,\,\underset{r\times p}{\bm{F}},$
where $\bm{D}\in^{n\times r}$ has rank $r$, and $\bm{F}\in^{r\times p}$ also
has rank $r$, i.e., $\bm{D},\bm{F}$ have full rank $r$.
The storage for the decomposition is then reduced or potentially increased
from $np$ to $r(n+p)$.
Proof [of Theorem 11.4] By ULV decomposition in Theorem 1.8 (p. 1.8), we can
decompose $\bm{A}$ by
$\bm{A}=\bm{U}\begin{bmatrix}\bm{L}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\bm{V}.$
Let $\bm{U}_{0}=\bm{U}_{:,1:r}$ and $\bm{V}_{0}=\bm{V}_{1:r,:}$, i.e.,
$\bm{U}_{0}$ contains only the first $r$ columns of $\bm{U}$, and $\bm{V}_{0}$
contains only the first $r$ rows of $\bm{V}$. Then, we still have
$\bm{A}=\bm{U}_{0}\bm{L}\bm{V}_{0}$ where $\bm{U}_{0}\in^{n\times r}$ and
$\bm{V}_{0}\in^{r\times p}$. This is also known as the reduced ULV
decomposition as shown in Figure 1.6 (p. 1.6). Let {$\bm{D}=\bm{U}_{0}\bm{L}$
and $\bm{F}=\bm{V}_{0}$}, or {$\bm{D}=\bm{U}_{0}$ and
$\bm{F}=\bm{L}\bm{V}_{0}$}, we find such rank decompositions.
The rank decomposition is not unique. Even by elementary transformations, we
have
$\bm{A}=\bm{E}_{1}\begin{bmatrix}\bm{Z}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\bm{E}_{2},$
where $\bm{E}_{1}\in^{n\times n},\bm{E}_{2}\in^{p\times p}$ represent
elementary row and column operations, $\bm{Z}\in^{r\times r}$. The
transformation is rather general, and there are dozens of these
$\bm{E}_{1},\bm{E}_{2},\bm{Z}$. By similar construction on this decomposition
as shown in the above proof, we can recover another rank decomposition.
Analogously, we can find such $\bm{D},\bm{F}$ by SVD, URV, CR, CUR, and many
other decompositional algorithms. However, we may connect the different rank
decompositions by the following lemma.
###### Lemma 11.5 (Connection Between Rank Decompositions)
For any two rank decompositions of
$\bm{A}=\bm{D}_{1}\bm{F}_{1}=\bm{D}_{2}\bm{F}_{2}$, there exists a nonsingular
matrix $\bm{P}$ such that
$\bm{D}_{1}=\bm{D}_{2}\bm{P}\qquad\text{and}\qquad\bm{F}_{1}=\bm{P}^{-1}\bm{F}_{2}.$
Proof [of Lemma 11.5] Since $\bm{D}_{1}\bm{F}_{1}=\bm{D}_{2}\bm{F}_{2}$, we
have
$\bm{D}_{1}\bm{F}_{1}\bm{F}_{1}^{\top}=\bm{D}_{2}\bm{F}_{2}\bm{F}_{1}^{\top}$.
It is trivial that $rank(\bm{F}_{1}\bm{F}_{1}^{\top})=rank(\bm{F}_{1})=r$ such
that $\bm{F}_{1}\bm{F}_{1}^{\top}$ is a square matrix with full rank and thus
is nonsingular. This implies
$\bm{D}_{1}=\bm{D}_{2}\bm{F}_{2}\bm{F}_{1}^{\top}(\bm{F}_{1}\bm{F}_{1}^{\top})^{-1}$.
Let $\bm{P}=\bm{F}_{2}\bm{F}_{1}^{\top}(\bm{F}_{1}\bm{F}_{1}^{\top})^{-1}$, we
have $\bm{D}_{1}=\bm{D}_{2}\bm{P}$ and $\bm{F}_{1}=\bm{P}^{-1}\bm{F}_{2}$.
### B QR Decomposition
In this appendix, we provide the proof for Theorem 1.5 (p. 1.5), the existence
of QR decomposition. In many applications, we are interested in the column
space of a matrix
$\bm{X}=[\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{p}]\in^{n\times p}$. The
successive spaces spanned by the columns $\bm{x}_{1},\bm{x}_{2},\ldots$ of
$\bm{X}$ are
$\mathcal{C}([\bm{x}_{1}])\,\,\,\,\subseteq\,\,\,\,\mathcal{C}([\bm{x}_{1},\bm{x}_{2}])\,\,\,\,\subseteq\,\,\,\,\mathcal{C}([\bm{x}_{1},\bm{x}_{2},\bm{x}_{3}])\,\,\,\,\subseteq\,\,\,\,\ldots,$
where $\mathcal{C}([\ldots])$ is the subspace spanned by the vectors included
in the brackets. The idea of QR decomposition is the construction of a
sequence of orthonormal vectors $\bm{q}_{1},\bm{q}_{2},\ldots$ that span the
same successive subspaces. That is
$\mathcal{C}([\bm{q}_{1}])=\mathcal{C}([\bm{x}_{1}]),\qquad\mathcal{C}([\bm{q}_{1},\bm{q}_{2}])=\mathcal{C}([\bm{x}_{1},\bm{x}_{2}]),\qquad\mathcal{C}([\bm{q}_{1},\bm{q}_{2},\bm{q}_{3}])=\mathcal{C}([\bm{x}_{1},\bm{x}_{2},\bm{x}_{3}]),\ldots.$
To achieve this, we first discuss how to project a vector onto another vector,
based on which the Gram-Schmidt process is employed iteratively.
#### B.1 Project a Vector Onto Another Vector
Project a vector $\bm{a}$ to a vector $\bm{b}$ is to find the vector closest
to $\bm{a}$ on the line of $\bm{b}$. The projection vector $\hat{\bm{a}}$ is
some multiple of $\bm{b}$. Let $\hat{\bm{a}}=\hat{x}\bm{b}$ and
$\bm{a}-\hat{\bm{a}}$ is perpendicular to $\bm{b}$ as shown in Figure 11.1(a).
We then get the following result:
Project Vector $\bm{a}$ Onto Vector $\bm{b}$ $\bm{a}-\hat{\bm{a}}$ is
perpendicular to $\bm{b}$, so $(\bm{a}-\hat{x}\bm{b})^{\top}\bm{b}=0$:
$\hat{x}$ = $\frac{\bm{a}^{\top}\bm{b}}{\bm{b}^{\top}\bm{b}}$ and
$\hat{\bm{a}}=\frac{\bm{a}^{\top}\bm{b}}{\bm{b}^{\top}\bm{b}}\bm{b}=\frac{\bm{b}\bm{b}^{\top}}{\bm{b}^{\top}\bm{b}}\bm{a}$.
(a) Project onto a line
(b) Project onto a space
Figure 11.1: Project a vector onto a line and onto a space.
#### B.2 Project a Vector Onto a Plane
Project a vector $\bm{a}$ to a space spanned by
$\bm{b}_{1},\bm{b}_{2},\ldots,\bm{b}_{n}$ is to find the vector closest to
$\bm{a}$ on the column space of $[\bm{b}_{1},\bm{b}_{2},\ldots,\bm{b}_{n}]$.
The projection vector $\widehat{\bm{a}}$ is a combination of
$\bm{b}_{1},\bm{b}_{2},\ldots,\bm{b}_{n}$:
$\widehat{\bm{a}}=\widehat{x}_{1}\bm{b}_{1}+\widehat{x}_{2}\bm{b}_{2}+\ldots+\widehat{x}_{n}\bm{b}_{n}$.
This is actually a least squares problem. To find the projection, we just
solve the normal equation
$\bm{B}^{\top}\bm{B}\widehat{\bm{x}}=\bm{B}^{\top}\bm{a}$ where
$\bm{B}=[\bm{b}_{1},\bm{b}_{2},\ldots,\bm{b}_{n}]$ and
$\widehat{\bm{x}}=[\widehat{x}_{1},\widehat{x}_{2},\ldots,\widehat{x}_{n}]$.
For each vector $\bm{b}_{i}$, the projection of $\bm{a}$ in the direction of
$\bm{b}_{i}$ can be analogously obtained by
$\widehat{\bm{a}}_{i}=\frac{\bm{b}_{i}\bm{b}_{i}^{\top}}{\bm{b}_{i}^{\top}\bm{b}_{i}}\bm{a},\,\,\,\,\,\,\,\,\forall
i\in\\{1,2,\ldots,n\\}.$
Let $\widehat{\bm{a}}=\sum_{i=1}^{n}\widehat{\bm{a}_{i}}$, this results in
$\bm{a}^{\perp}=(\bm{a}-\widehat{\bm{a}})\perp\mathcal{C}(\bm{B}),$
i.e., $(\bm{a}-\widehat{\bm{a}})$ is perpendicular to the column space of
$\bm{B}=[\bm{b}_{1},\bm{b}_{2},\ldots,\bm{b}_{n}]$ as shown in Figure 11.1(b).
#### B.3 Existence of the QR Decomposition via the Gram-Schmidt Process
We then develop the Gram-Schmidt process based on the projection of a vector.
Given three independent vectors $\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}$ and the
space spanned by the three vectors
$\mathcal{C}{([\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}])}$, i.e., the column space of
matrix $[\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}]$. We intend to construct three
orthogonal vectors $\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}$ in which case
$\mathcal{C}{([\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}])}$ =
$\mathcal{C}{([\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}])}$. Then we divide the
orthogonal vectors by their length to normalize. This process produces three
orthonormal vectors $\bm{q}_{1}=\frac{\bm{b}_{1}}{||\bm{b}_{1}||}$,
$\bm{q}_{2}=\frac{\bm{b}_{2}}{||\bm{b}_{2}||}$,
$\bm{q}_{2}=\frac{\bm{b}_{2}}{||\bm{b}_{2}||}$.
For the first vector, we choose $\bm{b}_{1}=\bm{a}_{1}$ directly. The second
vector $\bm{b}_{1}$ must be perpendicular to the first one. This is actually
the vector $\bm{a}_{2}$ subtracting its projection along $\bm{b}_{1}$:
$\displaystyle\bm{b}_{2}$
$\displaystyle=\bm{a}_{2}-\frac{\bm{b}_{1}\bm{b}_{1}^{\top}}{\bm{b}_{1}^{\top}\bm{b}_{1}}\bm{a}_{2}=(\bm{I}-\frac{\bm{b}_{1}\bm{b}_{1}^{\top}}{\bm{b}_{1}^{\top}\bm{b}_{1}})\bm{a}_{2}\qquad$
$\displaystyle(\text{Projection view})$
$\displaystyle=\bm{a}_{2}-\underbrace{\frac{\bm{b}_{1}^{\top}\bm{a}_{2}}{\bm{b}_{1}^{\top}\bm{b}_{1}}\bm{b}_{1}}_{\hat{\bm{a}}_{2}},\qquad$
$\displaystyle(\text{Combination view})$
where the first equation shows $\bm{b}_{2}$ is a multiplication of a matrix
and $\bm{a}_{2}$, i.e., project $\bm{a}_{2}$ onto the orthogonal complement
space of $\mathcal{C}{([\bm{b}_{1}])}$. The second equation shows $\bm{a}_{2}$
is a combination of $\bm{b}_{1}$ and $\bm{b}_{2}$. Clearly, the space spanned
by $\bm{b}_{1},\bm{b}_{2}$ is the same space spanned by
$\bm{a}_{1},\bm{a}_{2}$. The siguation is shown in Figure 11.2(a) in which we
choose the direction of $\bm{b}_{1}$ as the $x$-axis in the Cartesian
coordinate system. $\hat{\bm{a}}_{2}$ is the projection of $\bm{a}_{2}$ onto
the line $\bm{b}_{1}$. It can be clearly shown that the part of $\bm{a}_{2}$
perpendicular to $\bm{b}_{1}$ is $\bm{b}_{2}=\bm{a}_{2}-\hat{\bm{a}}_{2}$ from
the figure.
For the third vector $\bm{b}_{3}$, it must be perpendicular to both the
$\bm{b}_{1}$ and $\bm{b}_{2}$ which is actually the vector $\bm{a}_{3}$
subtracting its projection along the plane spanned by $\bm{b}_{1}$ and
$\bm{b}_{2}$
$\displaystyle\bm{b}_{3}$
$\displaystyle=\bm{a}_{3}-\frac{\bm{b}_{1}\bm{b}_{1}^{\top}}{\bm{b}_{1}^{\top}\bm{b}_{1}}\bm{a}_{3}-\frac{\bm{b}_{2}\bm{b}_{2}^{\top}}{\bm{b}_{2}^{\top}\bm{b}_{2}}\bm{a}_{3}=(\bm{I}-\frac{\bm{b}_{1}\bm{b}_{1}^{\top}}{\bm{b}_{1}^{\top}\bm{b}_{1}}-\frac{\bm{b}_{2}\bm{b}_{2}^{\top}}{\bm{b}_{2}^{\top}\bm{b}_{2}})\bm{a}_{3}\qquad$
$\displaystyle(\text{Projection view})$ (11.1)
$\displaystyle=\bm{a}_{3}-\underbrace{\frac{\bm{b}_{1}^{\top}\bm{a}_{3}}{\bm{b}_{1}^{\top}\bm{b}_{1}}\bm{b}_{1}}_{\hat{\bm{a}}_{3}}-\underbrace{\frac{\bm{b}_{2}^{\top}\bm{a}_{3}}{\bm{b}_{2}^{\top}\bm{b}_{2}}\bm{b}_{2}}_{\bar{\bm{a}}_{3}},\qquad$
$\displaystyle(\text{Combination view})$
where the first equation shows $\bm{b}_{3}$ is a multiplication of a matrix
and $\bm{a}_{3}$, i.e., project $\bm{a}_{3}$ onto the orthogonal complement
space of $\mathcal{C}{([\bm{b}_{1},\bm{b}_{2}])}$. The second equation shows
$\bm{a}_{3}$ is a combination of $\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}$. We will
see this property is essential in the idea of QR decomposition. Again, it can
be shown that the space spanned by $\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}$ is the
same space spanned by $\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}$. The situation is
shown in Figure 11.2(b), in which we choose the direction of $\bm{b}_{2}$ as
the $y$-axis of the Cartesian coordinate system. $\hat{\bm{a}}_{3}$ is the
projection of $\bm{a}_{3}$ onto the line $\bm{b}_{1}$, $\bar{\bm{a}}_{3}$ is
the projection of $\bm{a}_{3}$ onto the line $\bm{b}_{2}$. It can be shown
that the part of $\bm{a}_{3}$ perpendicular to both $\bm{b}_{1}$ and
$\bm{b}_{2}$ is $\bm{b}_{3}=\bm{a}_{3}-\hat{\bm{a}}_{3}-\bar{\bm{a}}_{3}$ from
the figure.
Finally, we normalize each vector by dividing their length which produces
three orthonormal vectors $\bm{q}_{1}=\frac{\bm{b}_{1}}{||\bm{b}_{1}||}$,
$\bm{q}_{2}=\frac{\bm{b}_{2}}{||\bm{b}_{2}||}$,
$\bm{q}_{2}=\frac{\bm{b}_{2}}{||\bm{b}_{2}||}$.
(a) Project $\bm{a}_{2}$ onto the space perpendicular to $\bm{b}_{1}$.
(b) Project $\bm{a}_{3}$ onto the space perpendicular to
$\bm{b}_{1},\bm{b}_{2}$.
Figure 11.2: Gram-Schmidt process.
This idea can be extended to a set of vectors rather than only three. And we
call this process as Gram-Schmidt process. After this process, matrix $\bm{X}$
will be triangularized. The method is named after Jørgen Pedersen Gram and
Erhard Schmidt, but it appeared earlier in the work of Pierre-Simon Laplace in
the theory of Lie group decomposition.
The Gram–Schmidt process is not the only algorithm for QR decomposition.
Several other QR decomposition algorithms exist such as Householder
reflections and Givens rotations which are more reliable in the presence of
round-off errors. These QR decomposition methods may also change the order in
which the columns of $\bm{X}$ are processed.
#### B.4 Orthogonal vs Orthonormal
The vectors $\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{p}\in^{n}$ are orthogonal
when their dot products $\bm{q}_{i}^{\top}\bm{q}_{j}$ are zero whenever $i\neq
j$. When each vector is divided by its length, the vectors become orthogonal
unit vectors. Then the vectors $\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{p}$ are
orthonormal. We put the orthonormal vectors into a matrix $\bm{Q}$.
When $n\neq p$: the matrix $\bm{Q}$ is easy to work with because
$\bm{Q}^{\top}\bm{Q}=\bm{I}\in^{p\times p}$.
When $n=p$: the matrix $\bm{Q}$ is square, $\bm{Q}^{\top}\bm{Q}=\bm{I}$ means
that $\bm{Q}^{\top}=\bm{Q}^{-1}$, i.e., the transpose of $\bm{Q}$ is the
inverse of $\bm{Q}$. Then we also have $\bm{Q}\bm{Q}^{\top}=\bm{I}$, i.e.,
$\bm{Q}^{\top}$ is the two-sided inverse of $\bm{Q}$. We call this $\bm{Q}$ an
orthogonal matrix. Note here we use the term orthogonal matrix to mean the
matrix $\bm{Q}$ has orthonormal columns. The term orthonormal matrix is not
used for historical reasons.
#### B.5 Property of the QR Decomposition
Given any matrix $\bm{X}$, we have the property: $\mathcal{N}(\bm{X}^{\top})$
is the orthogonal complement of the column space $\mathcal{C}(\bm{X})$ in n:
$dim(\mathcal{N}(\bm{X}^{\top}))+dim(\mathcal{C}(\bm{X}))=n$; This is called
rank-nullity theorem. And the proof can be found in Appendix B (p. B). In
specific, from QR decomposition, we can find a basis for this subspace. In
singular value decomposition, we will also find the basis for
$\mathcal{N}(\bm{X})$ and $\mathcal{C}(\bm{X}^{\top})$.
###### Lemma 11.6 (Orthonormal Basis in n)
For full QR decomposition, we have the following property:
$\bullet$ $\\{\bm{q}_{1},\bm{q}_{2}\ldots,\bm{q}_{p}\\}$ is an orthonormal
basis of $\mathcal{C}(\bm{X})$;
$\bullet$ $\\{\bm{q}_{p+1},\bm{q}_{p+2},\ldots,\bm{q}_{n}\\}$ is an
orthonormal basis of $\mathcal{N}(\bm{X}^{\top})$.
Proof [of Lemma 11.6] From the Gram-Schmidt process, it is trivial that
$span\\{\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{k}\\}$ is equal to
$span\\{\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{k}\\}$ for all
$k\in\\{1,2,\ldots,p\\}$. Thus
$\mathcal{C}(\bm{X})=span\\{\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{p}\\}=span\\{\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{p}\\}$,
and $\\{\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{p}\\}$ is an orthonormal basis
for the column space of $\bm{X}$. As
$\mathcal{N}(\bm{X}^{\top})\bot\mathcal{C}(\bm{X})$,
$dim(\mathcal{N}(\bm{X}^{\top}))=n-dim(\mathcal{C}(\bm{X}))=n-p$. And the
space spanned by $\\{\bm{q}_{p+1},\bm{q}_{p+2},\ldots,\bm{q}_{n}\\}$ is also
$\bot\mathcal{C}(\bm{X})$ with dimension $n-p$. Thus,
$\\{\bm{q}_{p+1},\bm{q}_{p+2},\ldots,\bm{q}_{n}\\}$ is an orthonormal basis
for $\mathcal{N}(\bm{X}^{\top})$.
#### B.6 Computing the Reduced QR Decomposition via the Gram-Schmidt Process
We write out this form of QR Decomposition
$\bm{X}=\left[\begin{matrix}\bm{x}_{1}&\bm{x}_{2}&\ldots&\bm{x}_{p}\end{matrix}\right]=\left[\begin{matrix}\bm{q}_{1}&\bm{q}_{2}&\ldots&\bm{q}_{p}\end{matrix}\right]\begin{bmatrix}r_{11}&r_{12}&\dots&r_{1p}\\\
&r_{22}&\dots&r_{2p}\\\ &&\ddots&\vdots\\\
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The orthogonal matrix $\bm{Q}$ can be easily calculated by the Gram-Schmidt
process. To see why we have the upper triangular matrix $\bm{R}$, we write out
these equations:
$\displaystyle\bm{x}_{1}$ $\displaystyle=r_{11}\bm{q}_{1}$
$\displaystyle=\sum_{i=1}^{1}r_{i1}\bm{q}_{1},$ $\displaystyle\bm{x}_{2}$
$\displaystyle=r_{12}\bm{q}_{1}+r_{22}\bm{q}_{2}$
$\displaystyle=\sum_{i=1}^{2}r_{i2}\bm{q}_{2},$ $\displaystyle\vdots$
$\displaystyle\bm{x}_{p}$
$\displaystyle=r_{1p}\bm{q}_{1}+r_{2p}\bm{q}_{2}+\ldots+r_{pp}\bm{q}_{p}$
$\displaystyle=\sum_{i=1}^{p}r_{ip}\bm{q}_{p},$
which coincide with the second equation of Equation (11.1) and make the shape
of an upper triangular matrix $\bm{R}$. And if we extend the idea of Equation
(11.1) into the $p$-th term, we get
$\displaystyle\bm{x}_{p}$
$\displaystyle=\sum_{i=1}^{p-1}(\bm{q}_{i}^{\top}\bm{x}_{p})\bm{q}_{i}+\bm{b}_{p}$
$\displaystyle=\sum_{i=1}^{p-1}(\bm{q}_{i}^{\top}\bm{x}_{p})\bm{q}_{i}+||\bm{b}_{p}||\bm{q}_{p},$
which implies we can gradually orthonormalize $\bm{X}$ to an orthonormal set
$\bm{Q}=[\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{p}]$ as in Algorithm 2.
Algorithm 2 Reduced QR Decomposition
1:matrix $\bm{X}$ has independent columns with size $n\times p$ and $n\geq p$;
$\displaystyle 1:\bm{q}_{1}$ $\displaystyle=\bm{x}_{1}/r_{11},$ $\displaystyle
r_{11}$ $\displaystyle=||\bm{x}_{1}||,$ $\displaystyle 2:\bm{q}_{2}$
$\displaystyle=(\bm{x}_{2}-r_{12}\bm{q}_{1})/r_{22},$ $\displaystyle r_{12}$
$\displaystyle=\bm{q}_{1}^{\top}\bm{x}_{2},r_{22}=||\bm{x}_{2}-r_{12}\bm{q}_{1}||,$
$\displaystyle 3:\bm{q}_{3}$
$\displaystyle=(\bm{x}_{3}-r_{13}\bm{q}_{1}-r_{12}\bm{q}_{2})/r_{33},$
$\displaystyle r_{13}$
$\displaystyle=\bm{q}_{1}^{\top}\bm{x}_{3},r_{23}=\bm{q}_{2}^{\top}\bm{x}_{3},r_{33}=||\bm{x}_{3}-r_{13}\bm{q}_{1}-r_{23}\bm{q}_{2}||,$
$\displaystyle\vdots$ $\displaystyle=\vdots$ $\displaystyle\vdots$
$\displaystyle n:\bm{q}_{p}$
$\displaystyle=(\bm{x}_{p}-\sum_{i=1}^{p-1}r_{ip}\bm{q}_{i})/r_{pp},$
$\displaystyle r_{ip}$
$\displaystyle=\bm{q}_{i}^{\top}\bm{x}_{p},r_{pp}=||\bm{x}_{p}-\sum_{i=1}^{p-1}r_{ip}\bm{q}_{i}||,\forall
i\in\\{1,2,\ldots,p-1\\},$
It can be shown Algorithm 2 requires $\sim 2np^{2}$ flops to compute a reduced
QR decomposition of an $n\times p$ matrix with independent columns and $n\geq
p$. See Lu (2021a) for more details on the complexity of QR decomposition.
#### B.7 Full QR Decomposition via the Gram-Schmidt Process
A full QR decomposition of an $n\times p$ matrix with independent columns goes
further by appending additional $n-p$ orthonormal columns to $\bm{Q}$ so that
it becomes an $n\times n$ orthogonal matrix. In addition, rows of zeros are
appended to $\bm{R}$ so that it becomes an $n\times p$ upper triangular
matrix. We refer to the additional columns in $\bm{Q}$ as silent columns and
additional rows in $\bm{R}$ as silent rows. The comparison of reduced QR
decomposition and full QR decomposition is shown in Figure 1.4 (p. 1.4) where
silent columns in $\bm{Q}$ are denoted in grey, blank entries are zero and
blue entries are elements that are not necessarily zero.
#### B.8 Dependent Columns
Previously, we assumed matrix $\bm{X}$ has independent columns. However, this
is not always necessary. Suppose in step $k$ of Algorithm 2, $\bm{x}_{k}$ is
in the plane spanned by $\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{k-1}$ which is
equivalent to the space spanned by
$\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{k-1}$, i.e., vectors
$\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{k}$ are dependent. Then $r_{kk}$ will be
zero and $\bm{q}_{k}$ does not exist because of the zero division. At this
moment, we simply pick $\bm{q}_{k}$ arbitrarily to be any normalized vector
that is orthogonal to
$\mathcal{C}([\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{k-1}])$ and continue the
Gram-Schmidt process. Again, for matrix $\bm{X}$ with dependent columns, we
have both reduced and full QR decomposition algorithms. We reformulate the
$k$-th step in the algorithm as follows:
$\bm{q}_{k}=\left\\{\begin{aligned}
&(\bm{x}_{k}-\sum_{i=1}^{k-1}r_{ik}\bm{q}_{i})/r_{kk},\qquad
r_{ik}=\bm{q}_{i}^{\top}\bm{x}_{k},r_{kk}=||\bm{x}_{k}-\sum_{i=1}^{k-1}r_{ik}\bm{q}_{i}||,&\mathrm{if\,}r_{kk}\neq
0,\\\
&\mathrm{pick\,one\,in\,}\mathcal{C}^{\bot}([\bm{q}_{1},\bm{q}_{2},\ldots,\bm{q}_{k-1}]),\qquad&\mathrm{if\,}r_{kk}=0.\end{aligned}\right.$
This idea can be further extended that when $\bm{q}_{k}$ does not exist, we
just skip the current steps. And add the silent columns in the end. In this
sense, QR decomposition for a matrix with dependent columns is not unique.
However, as long as you follow a systematic process or methodical procedure,
QR decomposition for any matrix is unique.
This finding can also help to decide whether a set of vectors are independent
or not. Whenever $r_{kk}$ in Algorithm 2 is zero, we report the vectors
$\bm{x}_{1},\bm{x}_{2},\ldots,\bm{x}_{k}$ are dependent and stop the algorithm
for “independent checking”.
#### B.9 Existence of the QR Decomposition via the Householder Reflector
In this section, we provide another view to find the QR decomposition via the
Householder reflector. We first give the formal definition of a Householder
reflector and we will discuss its properties.
###### Definition 11.1 (Householder Reflector)
Let $\bm{u}\in^{p}$ be a vector of unit length (i.e., $||\bm{u}||=1$). Then
$\bm{H}=\bm{I}-2\bm{u}\bm{u}^{\top}$ is said to be a Householder reflector,
a.k.a., Householder transformation. We call this $\bm{H}$ the Householder
reflector associated with the unit vector $\bm{u}$.
Then we have the following corollary from this definition.
###### Corollary 11.7 (Unreflected by Householder)
Any vector $\bm{v}$ that is perpendicular to $\bm{u}$ is left unchanged by the
Householder transformation.
The proof is trivial that
$(\bm{I}-2\bm{u}\bm{u}^{\top})\bm{v}=\bm{v}-2\bm{u}\bm{u}^{\top}\bm{v}=\bm{v}$.
Suppose $\bm{u}$ is a unit vector, and a vector $\bm{v}$ is perpendicular to
$\bm{u}$. Then any vector $\bm{x}$ on the plane can be decomposed into two
parts $\bm{x}=\bm{x}_{\bm{v}}+\bm{x}_{\bm{u}}$: the first one
$\bm{x}_{\bm{u}}$ is parallel to $\bm{u}$ and the second one $\bm{x}_{\bm{v}}$
is perpendicular to $\bm{u}$ (i.e., parallel to $\bm{v}$). Following from
Section B.1 (p. B.1), $\bm{x}_{\bm{u}}$ can be computed by
$\bm{x}_{\bm{u}}=\frac{\bm{u}\bm{u}^{\top}}{\bm{u}^{\top}\bm{u}}\bm{x}=\bm{u}\bm{u}^{\top}\bm{x}$.
We then transform this $\bm{x}$ by the Householder reflector associated with
$\bm{u}$,
$\bm{H}\bm{x}=(\bm{I}-2\bm{u}\bm{u}^{\top})(\bm{x}_{\bm{v}}+\bm{x}_{\bm{u}})=\bm{x}_{\bm{v}}-\bm{u}\bm{u}^{\top}\bm{x}=\bm{x}_{\bm{v}}-\bm{x}_{\bm{u}}$.
That is, the space perpendicular to $\bm{u}$ acts as a mirror and any vector
$\bm{x}$ is reflected by the Householder reflector associated with $\bm{u}$.
The situation is shown in Figure 11.3.
Figure 11.3: Demonstration of Householder reflector
If we know two vectors are reflected to each other, the next corollary tells
us how to find the associated Householder reflector.
###### Corollary 11.8 (Finding the Householder Reflector)
Suppose $\bm{x}$ is reflected to $\bm{y}$ by a Householder reflector, then the
Householder reflector is
$\bm{H}=\bm{I}-2\bm{u}\bm{u}^{\top},\text{ where
}\bm{u}=\frac{\bm{x}-\bm{y}}{||\bm{x}-\bm{y}||}.$
###### Remark 11.9
If $\bm{H}$ is a Householder reflector, then it has the following properties:
$\bullet$ $\bm{H}\bm{H}=\bm{I}$;
$\bullet$ $\bm{H}=\bm{H}^{\top}$;
$\bullet$ $\bm{H}^{\top}\bm{H}=\bm{H}\bm{H}^{\top}=\bm{I}$ such that
Householder reflector is an orthogonal matrix.
We note from the Gram-Schmidt section that QR decomposition is to use an
orthogonal to triangularize a matrix $\bm{X}$. The further idea is that, if we
have a set of orthogonal matrices that can make $\bm{X}$ to be triangular step
by step, then we can also recover the QR decomposition. Specifically, if we
have an orthogonal matrix $\bm{Q}_{1}$ that can introduce zeros to the 1-st
column of $\bm{X}$ except the entry (1,1), and an orthogonal matrix
$\bm{Q}_{2}$ that can introduce zeros to the 2-nd column except the entries
(2,1), (2,2), $\ldots$. Then, we can also find the QR decomposition.
For the way to introduce zeros, we could reflect the columns of the matrix to
a basis vector $\bm{e}_{1}$ whose entries are all zero except the first entry.
|
# Finite Blocklength Regime Performance of Downlink Large Scale Networks
Nourhan Hesham, Student Member, IEEE, Anas Chaaban, Senior Member, IEEE,
Hesham ElSawy, Senior Member, IEEE, Jahangir Hossain, Senior Member, IEEE
N. Hesham, A. Chaaban, and J. Hossain are with the School of Engineering,
University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail:
{nourhan.soliman,anas.chaaban,jahangir.hossain}@ubc.ca), and N. Hesham is on
leave from the Department of Electronics and Electrical Engineering, Cairo
University, Cairo, Egypt. H. ElSawy is with the School of Computing, Queen’s
University, Kingston, ON K7L 2N8, Canada. (e-mail: [email protected]).
This publication is based upon work supported by King Abdullah University of
Science and Technology (KAUST) under Award No. OSR-2018-CRG7-3734. Part of
this work was presented in the IEEE International Conference on Communications
(ICC 2021) [1]
###### Abstract
Some emerging 5G and beyond use-cases impose stringent latency constraints,
which necessitates a paradigm shift towards finite blocklength performance
analysis. In contrast to Shannon capacity-achieving codes, the codeword length
in the finite blocklength regime (FBR) is a critical design parameter that
imposes an intricate tradeoff between delay, reliability, and information
coding rate. In this context, this paper presents a novel mathematical
analysis to characterize the performance of large-scale downlink networks
using short codewords. Theoretical achievable rates, outage probability, and
reliability expressions are derived using the finite blocklength coding theory
in conjunction with stochastic geometry, and compared to the performance in
the asymptotic regime (AR). Achievable rates under practical modulation
schemes as well as multilevel polar coded modulation (MLPCM) are investigated.
Numerical results provide theoretical performance benchmarks, highlight the
potential of MLPCM in achieving close to optimal performance with short
codewords, and confirm the discrepancy between the performance in the FBR and
that predicted by analysis in the AR. Finally, the meta distribution of the
coding rate is derived, providing the percentiles of users that achieve a
predefined target rate in a network.
###### Index Terms:
Average Coding Rate, Finite Blocklength, Meta Distribution, Multilevel Polar-
Coded Modulation, Stochastic Geometry.
## I Introduction
The proliferating applications and diverse use-cases of beyond 5G (B5G)
networks enforce stringent key performance indicators that cannot be realized
via conventional coding schemes [2]. For instance, mission critical
applications require sub-$1\ \text{msec}$ latency that cannot be achieved with
conventional long codes [3]. Moreover, many Internet of Things (IoT) devices
have an extreme low-power consumption profile that cannot support the
complexity and long transmission intervals of conventional long codes [4].
Consequently, there is a paradigm shift towards using short codes to comply
with the stringent latency and power consumption constraints of the foreseen
B5G networks.111In a network, there are four dominant delays: transmission
delay, propagation delay, processing delay, and queuing delay. As the network
becomes denser, the transmission delay can dominate the propagation delay
which motivates using short codes. Such reduced code length comes at the
expense of inevitable errors and lower information coding rates. The tolerance
to the induced errors differs with the application, which can be lower than
$10^{-9}$ for ultra-reliable low-latency communications (URLLC) [5]. This
motivates the study of the finite blocklength coding theory [6, 7], which
characterizes the information coding rate as a function of the code length and
frame error rate (FER). Therefore, for a context-aware short code design in
B5G networks, it is of primary importance to extend finite blocklength coding
theory to interference-prone scenarios that account for the intrinsic large-
scale and dense deployments of future networks.
Performance of large-scale cellular networks is well investigated using the
classical Shannon capacity [8, 9, 10, 11, 12, 13], defined as the maximum
achievable rate such that the error probability vanishes as the code length
increases [14]. Such idealistic scenario is hereafter denoted as the
asymptotic regime (AR). The AR analysis in [8, 9, 10, 11, 12, 13] is not
applicable for networks operating in the finite blocklength regime (FBR).
Short codes (e.g., 128 symbols) violate the AR assumptions and lead to
inevitable errors as a cost for constraints on delay and/or power consumption.
Thus, the code length $n$ of the FBR is a critical design parameter that
imposes a delicate trade-off between FER ($\epsilon$) and information coding
rate. To characterize such trade-off, the work of Polyanskiy et al. in [6, 7]
finds the maximum achievable rate as function of the code length $n$ and FER
$\epsilon$ for a point-to-point link.
Motivated by the need to migrate towards the FBR, the work in [6, 7] is
extended for several use cases in [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. For instance, the work in [15] extends the FBR analysis to relaying
channels. The achievable rate for non-orthogonal multiple access in the FBR is
studied in [16]. The impact of the FBR on rate splitting is characterized in
[17]. The performance of multi-user MIMO, massive-MIMO, and cell-free MIMO in
the FBR are investigated in [18], [19], and [20], respectively. The authors in
[21, 22] optimize the blocklength and other network parameters (e.g., transmit
power or transmitter location) to minimize the decoding error probability
subject to a latency constraint. Works was also done in [23, 24, 25] to
maximize the achievable rate, improve latency, enhance reliability, and/or
provide a secure communication in the FBR. However, the models in [15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25] are limited to small scale networks. To the
best of the authors’ knowledge, performance characterization of large-scale
networks in the FBR is still an open problem.
Using the coding theory in the FBR in conjunction with stochastic geometry
tools, this paper intends to contribute to the aforementioned research gap, as
an extension of [1]. In particular, this paper develops novel mathematical
analysis to characterize the trade-off between the code length $n$, the FER
$\epsilon$, and the information coding rate $R$ in large-scale downlink (DL)
networks using orthogonal multiple access (OMA). To this end, achievable rates
and outage in the FBR are characterized for Gaussian codebooks as well as
practical modulation schemes. Multi-level polar-coded modulation (MLPCM) [26,
27, 28, 29], is presented as a practical validation for the obtained
theoretical achievable rates.222Polar codes is standardized for 5G in 3GPP
Release 16 Specification $\\#$ 38.212[30], and MLPCM was proposed to be used
in the future generations in [31, 32] The contributions of this paper are
summarized as follows.
The paper analyzes the performance of a large-scale DL network in the FBR as a
function of $n$ and $\epsilon$ in terms of:
* •
the average coding rate under a random distance between the user and its
serving BS, under Gaussian codebooks,
* •
the average coding rate using QAM constellations,
* •
the rate outage probability and reliability bounds and approximations, and
* •
the approximation of the coding rate meta distribution.
Additionally, the paper simulates the average coding rate achieved by MLPCM
for different QAM modulation orders in comparison with the obtained
theoretical benchmarks. All results are validated via Monte Carlo simulations.
It is shown that MLPCM achieves rates close to the derived theoretical
benchmarks. It is also shown that the derived outage probability bounds are
tight and coincide with simulation results. Additionally, it is shown that the
proposed approximation of the coding rate meta-distribution is fairly tight
under moderate and high SINR. Throughout the paper, we conduct qualitative and
quantitative comparisons with the performance in the AR to motivate the
importance of analyzing the performance in the FBR and demonstrate the
discrepancy with AR results. We also investigate the effect of various network
parameters in the FBR.
This paper is organized as follows. In Sec. II, the system model and
assumptions of the analysis are presented. In Sec. III, the average coding
rate of a large-scale DL network in the FBR is derived, under Gaussian
codebooks and under a constraint of QAM constellations. Then, the theoretical
results are numerically evaluated and compared with the performance of MLPCM
as a practical scheme. In Sec. IV, bounds on the rate outage probability are
derived, followed by the characterization of the reliability, the meta
distribution of the coding rate, numerical evaluations, and a detailed
discussion. Finally, the paper is concluded in Sec. V.
## II System Model
Consider a single-tier OMA large-scale DL network with universal frequency
reuse and no intra-cell interference. The base stations (BSs) are located
according to a Poisson point process (PPP) $\Psi\subset\mathbb{R}^{2}$ with
intensity $\lambda\ \text{BS}/\text{km}^{2}$. The network applies universal
frequency reuse with one user equipment (UE) served per BS at a given resource
block. Each UE is served by its geographically closest BS, where the intended
distance between a UE and its serving BS is denoted as $r_{0}$. The distances
to the interfering BSs ordered with respect to the intended UE are denoted as
$r_{1},\ r_{2},\dots,\ r_{i},\dots$ such that $r_{i+1}>r_{i}$. For the sake of
simple presentation, the set $\tilde{\Psi}\in\mathbb{R}$ is defined as the BSs
distances to the desired UE. Hence, the received signal at the desired UE is
given by
$y=\sqrt{\mathcal{P}}h_{0}r_{0}^{-\eta/2}s_{0}+\underset{I_{agg}}{\underbrace{\sum_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\sqrt{\mathcal{P}}h_{i}r_{i}^{-\eta/2}s_{i}}}+w,$
(1)
where $\mathcal{P}$ is the transmit power of the BSs, $h_{0}$ (resp. $h_{i}$)
represents the channel fading of the intended (resp. $i^{th}$ interfering)
channel, $\eta$ is the path loss exponent, $s_{0}$ (resp. $s_{i}$) is a unit
average power codeword symbol transmitted by the serving (resp. $i^{th}$
interfering) BS, $I_{agg}$ is the aggregate interference from all other BSs,
and $w\sim\mathcal{CN}(0,\sigma_{w}^{2})$ is circularly symmetric complex
Gaussian noise with zero mean and variance $\sigma_{w}^{2}$. We model $h_{0}$
and $h_{i}$ to be circularly symmetric complex Gaussian with zero mean and
unit variance (modeling Rayleigh fading). An independent and identically
distributed block fading channel model is assumed where the channel
coefficients $h_{0}$ and $h_{i}$ remain constant for a duration of $L$
consecutive symbols and change to independent realizations at the end of each
symbol interval.
The serving BS wants to send information to the UE using codewords from a code
with the rate $R$ and length $n$ symbols, where $n$ can be defined following a
latency and/or energy consumption constraint. To ensure that the channel
remains constant during the transmission of $n$ symbols, we require $n=L/l$
for some integer $l$ [33]. This ensures that the channels remain constant
during the transmission, but makes consecutive transmission blocks have
identical channels. Nonetheless, by considering the sequence of blocks
consisting of the first block (of $n$ symbols) in each coherence interval (of
$L$ symbols), we have independent channels and we can derive the average
performance over all transmissions. This is because the remaining sequences of
blocks (such as the second frame in each coherence interval) have identical
statistics.
To this end, two modes of operation are considered. The first mode assumes
knowledge of channel-state information (CSI) at the transmitter and the
receiver in the form of SINR availability.333Assuming SINR knowledge
benchmarks the performance of the network. Under this mode, we investigate the
average coding rate $\mathbb{E}\\{R\\}$ as a function of the FER and
codelength (see Sec. III). The second mode assumes that the CSI is unknown at
the transmitter but known at the receiver. Under this scenario, we derive the
outage probability and utilize the coding-rate meta distribution to
characterize the percentile of users achieving a given transmission
reliability (See Sec. IV).
## III Average Rate Analysis
Considering the first mode of operation, we assume that the SINR is available
at the transmitter and the receiver. Hence, the coding rate can be adapted
based on the SINR in each transmission block so that the transmission using
codelength $n$ is successful with a desired FER $\epsilon$. This rate
adaptation can be realized using the following lemma, which characterizes the
maximum achievable rate over an additive white Gaussian noise (AWGN) channel
in the FBR [6].
###### Lemma 1.
([6]) For an AWGN channel with signal to noise power ratio (SNR) $\alpha$,
blocklength $n$, and FER $\epsilon\in(0,0.5)$, the maximum coding rate is
approximated as444This approximation is shown to be tight at blocklengths
($>100$) and FER $>10^{-6}$ in [6].
$R_{n,\epsilon}(\alpha)=C_{\infty,0}(\alpha)-\frac{\sqrt{V(\alpha)}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(2)
where $C_{\infty,0}(\alpha)=\log_{2}(1+\alpha)$ is the AWGN channel capacity
in the AR, $V(\alpha)=\frac{\alpha(\alpha+2)}{(\alpha+1)^{2}}\log_{2}^{2}(e)$
is known as the channel dispersion, and $Q^{-1}(\cdot)$ is the inverse of the
Q-function.
In this mode, due to the random changes of the channels $h_{0}$ and $h_{i}$,
we characterize performance by the average coding rate subject to a target FER
$\epsilon$ and codelength $n$. However, Lemma 1 is derived for an AWGN
channel, and hence, cannot be directly applied to a large-scale network due to
the additional non-Gaussian interference term $I_{agg}$ in (1). To overcome
this, we utilize the equivalence in distribution (EiD) approach to express the
interference as a conditionally Gaussian random variable [34, 35, 12], as
described next.
### III-A Conditional Gaussian Representation
Using the EiD approach, we represent the aggregate interference for a given
$r_{0}$ as follows
$\displaystyle I_{agg}{\overset{\text{eid}}{=}}\sqrt{\mathcal{B}}{g},$ (3)
where $\overset{\text{eid}}{=}$ denotes the EiD, ${g}\sim\mathcal{CN}(0,1)$
and $\mathcal{B}>0$ is a positive random variable independent of $g$ with a
probability density function (PDF) whose Laplace transform (LT) is given by
[12],
$\displaystyle{\mathcal{L}_{\mathcal{B}}(u)=\exp\left\\{\sum_{k=1}^{\infty}(-1)^{k}2\pi\lambda
r_{0}^{2}\left(\frac{\mathcal{P}}{r_{0}^{\eta}}\right)^{k}\frac{\mathbb{E}\big{\\{}|s|^{2k}\big{\\}}u^{k}}{(\eta
k-2)k!}\right\\}.}$ (4)
where $s$ is the transmitted symbol from an arbitrary codebook by an arbitrary
BS and follows the same distribution as $s_{0}$ and $s_{i}$.
The EiD in (3) is proved by showing that the product $\sqrt{\mathcal{B}}{g}$
has the same characteristic function as $I_{agg}$ [12]. Using (3), the
equivalent system model is expressed as
$\displaystyle
y=\sqrt{\mathcal{P}}h_{0}r_{0}^{-\eta/2}s_{0}+\sqrt{\mathcal{B}}{g}+w,$
where for a given $\mathcal{B}$, the lumped term $\sqrt{\mathcal{B}}{g}+w$ is
Gaussian with zero mean and variance $\mathcal{B}+\sigma_{w}^{2}$. Hence, the
resulting conditional SINR, given $\mathcal{B}$ and $h_{0}$, is given by
${\Upsilon}=\frac{|h_{0}|^{2}\mathcal{P}r_{0}^{-\eta}}{\mathcal{B}+\sigma_{w}^{2}}.$
(5)
The random variable $\mathcal{B}$ in (5) augments the noise power with the
aggregate network interference power, for any constellation of $s_{0}$ and
$s_{i}$. If $s_{0}$ and $s_{i}$ is Gaussian distributed (Gaussian codebooks),
it can be shown that (4) simplifies to [12]
$\displaystyle\mathcal{L}_{\mathcal{B}_{g}}(u)=\exp\left\\{\frac{-2\pi\lambda
u\mathcal{P}r_{0}^{2-\eta}}{\eta-2}{}_{2}F_{1}\left(1,1-\frac{2}{\eta};2-\frac{2}{\eta};-\frac{u\mathcal{P}}{r_{0}^{\eta}}\right)\right\\}.$
(6)
Since at this point, interference is modeled as conditionally Gaussian, and
since Lemma 1 provides a tight approximation when noise is Gaussian, then the
tightness of the approximation provided in Lemma 1 holds here too. Thus, from
this point onward, we focus on analyzing performance using $R_{n,\epsilon}$ as
a surrogate for the maximum achievable rate in the FBR given that this is a
tight approximation. Under this framework, the maximum average coding rate is
defined next.
###### Definition 1.
Given a DL large-scale network with a distance $r_{0}$ between the desired UE
and the serving BS as defined in (1), assuming the SINR knowledge is available
at the transmitter and the receiver, we define the maximum average coding rate
in the FBR with blocklength $n$ and FER $\epsilon$ with
$\alpha_{0}=\frac{\mathcal{P}r_{0}^{-\eta}}{\sigma_{w}^{2}}$, as
$R_{n,\epsilon}(\alpha_{0})=\mathbb{E}_{h_{0},\mathcal{B}}\\{R_{n,\epsilon}(\Upsilon)\\}$,
where $\Upsilon$ is the SINR defined in (5) and $R_{n,\epsilon}(\Upsilon)$ is
as defined in Lemma 1.
Next, we use Lemma 1 and Def. 1 to characterize the average coding rate for a
large-scale network in the FBR, first using Gaussian codebooks, and then under
finite constellations (QAM). Then, we present a practical MLPCM to validate
the achievability of the derived theoretical benchmarks.
### III-B Average Coding Rate in the Finite Block-Length Regime
By virtue of the EiD, the AWGN results in Lemma 1 can be now extended to
large-scale networks. For a given intended UE distance $r_{0}$, the FBR
average coding rate is characterized in the following theorem.
###### Theorem 1.
The maximum average coding rate of the large-scale network modeled by (1) with
Gaussian codebooks, blocklength $n$, FER $\epsilon$, and distance $r_{0}$
between the UE and its serving BS is given by
$\displaystyle\mathcal{R}_{n,\epsilon}({\alpha_{0}})=\mathcal{C}_{\infty,0}({\alpha_{0}})-\frac{\mathcal{V}({\alpha_{0}})Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(7)
where
$\mathcal{C}_{\infty,0}({\alpha_{0}})=\int_{0}^{\infty}\exp\left(-\frac{2^{c}-1}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{c}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}dc,$
$\mathcal{V}({\alpha_{0}})=\int_{0}^{\log_{2}(e)}e^{\frac{-z(v)}{{\alpha_{0}}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}z(v)\right\\}d{v}$,
$\alpha_{0}=\frac{\mathcal{P}r^{-\eta}_{0}}{\sigma_{w}^{2}}$,
$z(v)=\sqrt{\frac{1}{1-\frac{{v^{2}}}{\log_{2}^{2}(e)}}}-1$, and
$\mathcal{L}_{\mathcal{B}_{g}}\\{\cdot\\}$ is given in (6).
###### Proof.
By virtue of the EiD approach, Lemma 1 is applicable to large-scale networks
by replacing $\alpha$ by $\Upsilon$ where $\Upsilon$ is defined in (2), which
leads to a maximum coding rate $R_{n,\epsilon}(\Upsilon)$ under a given
$h_{0}$ and $\mathcal{B}$. The average rate is then
$\mathcal{R}_{n,\epsilon}({\alpha_{0}})=\mathbb{E}\\{R_{n,\epsilon}(\Upsilon)\\}$,
where the averaging is over $h_{0}$ and $\mathcal{B}$. Thus, we need to
average $C_{\infty,0}(\Upsilon)$ and $\sqrt{V(\Upsilon)}$ over $\mathcal{B}$
and $h_{0}$. The term
$\mathcal{C}_{\infty,0}({\alpha_{0}})=\mathbb{E}\\{C_{\infty,0}(\Upsilon)\\}$
is the average capacity in a large-scale network in the AR with CSI available
at the BSs, which was derived in [36]. It remains to find
$\mathbb{E}\\{\sqrt{{V}(\Upsilon)}\\}$ which is shown to be equal to
$\mathcal{V}(\alpha_{0})$ in App. A. ∎
Theorem 1 governs the maximum average rate at a fixed distance $r_{0}$ between
the UE and its serving BS, considering the stochastic locations of the
interfering BSs and the Rayleigh fading environment. The following corollary
generalizes Theorem 1 by accounting for the randomness of $r_{0}$.
###### Corollary 1.
The average coding rate of the large-scale network modeled by (1) with
blocklength $n$, FER $\epsilon$, and Gaussian codebooks is given by
$\displaystyle\mathcal{\bar{R}}_{n,\epsilon}=\mathcal{\bar{C}}_{\infty,0}-\frac{\mathcal{\bar{V}}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
where
$\mathcal{\bar{C}}_{\infty,0}=\int_{0}^{\infty}\mathcal{{C}}_{\infty,0}({\alpha_{0}})f_{r_{0}}(r_{0})dr_{0}$
and
$\mathcal{\bar{V}}=\int_{0}^{\infty}\mathcal{{V}}({\alpha_{0}})f_{r_{0}}(r_{0})dr_{0}$.
###### Proof.
The average capacity and average channel dispersion are derived by directly
averaging (7) with respect to $r_{0}$ which has the following probability
density function[36]
$\displaystyle f_{r_{0}}(r_{0})=2\pi\lambda r_{0}e^{-\pi\lambda r_{0}^{2}},\ \
0<r_{0}<\infty.$ (8)
∎
As can be seen in Theorem 1 and Corollary 1, the average coding rate of the
large-scale network in the FBR is the average capacity in the AR, plus a
penalty term (loss) which is a function of the blocklength $n$, and FER
$\epsilon$, in addition to the average SNR $\alpha_{0}$ and the stochastic
geometry of the network, manifested in the Laplace transform term. This
penalty term implies a trade-off between the reliability and the average rate:
the higher the reliability requirement (low FER), the lower the rate. However,
for a given average achievable rate, the reliability can be increased
($\epsilon$ decreased) by increasing the blocklength, converging to the AR
performance as $n$ grows to infinity.
Theorem 1 expresses the average coding rate under Gaussian codebooks, which is
of theoretical relevance. Characterizing the performance in the FBR when using
a finite constellation is of practical interest and is discussed next.
### III-C Average Rate Using Finite Constellations
This section focuses on achievable rates in the FRB under transmission using
standard finite constellations. The maximum achievable rate using an $M$-ary
constellation over an AWGN channel in the FBR was given in [37]. Following the
same methodology as in Sec. III-B, we use the EiD approach to extend the
results of [37] to large-scale networks.
Consider an encoder that maps messages from a message set into a length $n$
sequence of symbols chosen from an $M$-ary constellation consisting of $M$
complex-valued symbols $s_{1},s_{2},...,s_{M}$. Using an $M$-ary QAM
constellation, denote by
$\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$ the average capacity
achieved in the AR, by ${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$ the
average square root channel dispersion, and by
$\mathcal{R}^{\text{\tiny(M)}}_{n,\epsilon}(\alpha_{0})$ the average coding
rate under a blocklength $n$ and FER $\epsilon$. To characterize the average
capacity of the $M$-ary QAM under FBR, the conditional mutual information
$I(s_{0};y|h_{0},\mathcal{B},r_{0})$ has to be averaged with respect to the
aggregate interference power $\mathcal{B}$ and the channel gain $h_{0}$.
However, the distribution of the aggregate interference power $\mathcal{B}$ is
unknown which leads to an intractable expression. Although such an expression
can be evaluated using Monte Carlo simulations, an expression that is amenable
to numerical integration is preferable. A similar argument applies for
${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$. Hence, to obtain tractable
integral expressions, we adopt an approximation for the distribution of the
interference power $\mathcal{B}$ proposed in [38, 39] which was shown to be a
tight approximation. In particular, in [38, 39], the distribution of the
interference power $\mathcal{B}$ was approximated by a Gamma distribution
$\displaystyle
f(x;q,\theta)=\frac{x^{q-1}e^{\frac{-x}{\theta}}}{\Gamma(q)\theta^{q}},\ \ \ \
\ \ \ \ \ x>0$ (9)
where $q=\frac{4\pi\lambda r_{0}^{2}(\eta-1)}{(\eta-2)^{2}}$ is the shape
parameter, and $\theta=\frac{(\eta-2)\mathcal{P}}{2(\eta-1)r_{0}^{\eta}}$ is
the scale parameter. The approximation is based on a moment-matching approach.
The first two moments of $\mathcal{B}$ can be obtained from (4) as
$\displaystyle\mathbb{E}\\{\mathcal{B}\\}$
$\displaystyle=\left.\frac{d\mathcal{L}_{\mathcal{B}}(u)}{du}\right|_{u=0}=\frac{2\pi\lambda
r_{0}^{2-\eta}\mathcal{P}}{(\eta-2)}$
$\displaystyle\mathbb{E}\\{\mathcal{B}^{2}\\}$
$\displaystyle=\left.\frac{d^{2}\mathcal{L}_{\mathcal{B}}(u)}{du^{2}}\right|_{u=0}=\left(\frac{2\pi\lambda
r_{0}^{2-\eta}\mathcal{P}}{(\eta-2)}\right)^{2}+\frac{\pi\lambda
r_{0}^{2-2\eta}\mathcal{P}^{2}}{(\eta-1)}.$
The scale and shape parameters of the gamma distribution are then obtained as
$q=\frac{\mathbb{E}\\{\mathcal{B}\\}^{2}}{\mathbb{V}ar\\{\mathcal{B}\\}}$ and
$\theta=\frac{\mathbb{V}ar\\{\mathcal{B}\\}}{\mathbb{E}\\{\mathcal{B}\\}}$,
where
$\mathbb{V}ar\\{\mathcal{B}\\}=\mathbb{E}\\{\mathcal{B}^{2}\\}-\mathbb{E}\\{\mathcal{B}\\}^{2}$.
Fig. 1 numerically demonstrates the accuracy of this approximation by plotting
the approximate CDF of $\mathcal{B}$ (using (9)) and its exact CDF obtained
from a numerical inversion of (4). The figure shows that the Gamma
approximation in (9) provides a fairly tight approximation for the CDF of
$\mathcal{B}$. Note that the Gamma approximation is crucial to maintain the
tractability of the analysis. Using this approximation, the average coding
rate of a large-scale network in the FBR, at a given distance $r_{0}$ and
using $M$-ary QAM constellation, is characterized.
Figure 1: CDF of the interference power $\mathcal{B}$ and its approximation at
different values of $\mathcal{P}$, $\lambda$ and $r_{0}$
The maximum achievable rate for a finite constellation is provided in [37].
However, the channel dispersion in [37] is defined as the variance of the
information density ($V=Var(i(x;y))$). This definition is considered to be
imprecise according to [6], where the channel dispersion is defined as the
conditional variance of the information density for finite constellation and
is defined as the unconditional variance of the information density in the
case of using Gaussian signals. Therefore, in this paper, we derive the
maximum achievable rate using this definition for large-scale networks in the
FBR in the following theorem.
###### Theorem 2.
For a DL large-scale network with BS density $\lambda\
\mathrm{BS}/\mathrm{km}^{2}$, blocklength $n$, FER $\epsilon$, distance
$r_{0}$ between the UE and the serving BS, and an $M$-ary constellation with
symbols $\\{\underline{s}_{m}\\}_{m=1}^{M}$ where
$\underline{s}_{m}=[\mathcal{R}e\\{s_{m}\\},\mathcal{I}m\\{s_{m}\\}]^{T}$ and
$\mathcal{R}e\\{s_{m}\\}$ and $\mathcal{I}m\\{s_{m}\\}$ are the real and
imaginary parts of $s_{m}$, respectively, if the interference power
$\mathcal{B}$ follows a Gamma distribution (9), then the average achievable
coding rate can be expressed by
$\displaystyle\mathcal{R}^{\text{\tiny(M)}}_{n,\epsilon}(\alpha_{0})=\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})-\frac{\mathcal{{V}}^{\text{\tiny{(M)}}}(\alpha_{0})Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(10)
where $\alpha_{0}=\frac{\mathcal{P}r_{0}^{-\eta}}{\sigma_{w}^{2}}$,
$\displaystyle\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$
$\displaystyle=\log_{2}(M)-\frac{1}{M\pi}\sum_{m=1}^{M}\int_{0}^{\infty}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}{g_{m}(\underline{t})}f_{{\Upsilon}}(\Upsilon|r_{0})d\underline{t}\
d\Upsilon$ (11) $\displaystyle{\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$
$\displaystyle=\int_{0}^{\infty}\sqrt{V_{\text{\tiny
M}}(\Upsilon)}f_{{\Upsilon}}(\Upsilon|r_{0})\ d\Upsilon$ (12)
$\displaystyle{V}_{\text{\tiny{M}}}(\Upsilon)$
$\displaystyle=\sum_{m=1}^{M}\left(\int_{{\mathbb{R}^{2}}}\frac{e^{-\|\underline{t}\|^{2}}}{M\pi}{g_{m}(\underline{t})^{2}}d\underline{t}-\left(\int_{\mathbb{R}^{2}}\frac{e^{-\|\underline{t}\|^{2}}}{M\pi}{g_{m}(\underline{t})}d\underline{t}\right)^{2}\right)$
(13)
where
$g_{m}(\underline{t})=\log_{2}\left(\sum_{l=1}^{M}e^{-2\sqrt{{\Upsilon}}\underline{t}^{T}(\underline{s}_{m}-\underline{s}_{l})-{\Upsilon}\|\underline{s}_{m}-\underline{s}_{l}\|^{2}}\right)$,
$\underline{t}=[t_{1},t_{2}]^{T}$ is a 2-D real-valued vector, $\Upsilon$ is
the SINR given in (5), and
$\displaystyle
f_{{\Upsilon}}(\Upsilon|r_{0})=\frac{\exp\left\\{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}\right\\}}{\mathcal{P}r_{0}^{-\eta}}\frac{(1+\theta
r_{0}^{\eta}\Upsilon/\mathcal{P})\sigma_{w}^{2}+q\theta}{(1+\theta
r_{0}^{\eta}\Upsilon/\mathcal{P})^{q+1}},0\leq\Upsilon\leq\infty.$ (14)
###### Proof.
The average capacity of an $M$-ary constellation
($\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$) at a fixed $r_{0}=r$
is derived by averaging the conditional mutual information between the
transmitted symbol $s_{0}\in\\{s_{m}\\}_{m=1}^{M}$ and the received signal
$y$, i.e., $I(s_{0};y|\Upsilon,r_{0}=r)$, provided in [37], with respect to
the SINR $\Upsilon$ which is a function of interference power $\mathcal{B}$
and fading channel $h_{0}$ statistics. Similarly, the average square-root
channel dispersion (${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$) is
derived by averaging the channel dispersion in (13) with respect to
$\Upsilon$. The derivation of (13) is provided in App. B. However, instead of
averaging over the distribution of $\Upsilon$ which is unknown, we average
over $f_{{\Upsilon}}(\Upsilon|r_{0})$ in (14) which is obtained using the
Gamma approximation provided in (9) (See App. C). ∎
The following corollary generalizes Theorem 2 by accounting for the randomness
of $r_{0}$.
###### Corollary 2.
The average achievable coding rate of the large-scale network modeled by (1)
with blocklength $n$, FER $\epsilon$, and using an $M$-ary constellation with
symbols $\\{s_{m}\\}_{m=1}^{M}$ is given by
$\displaystyle\bar{\mathcal{R}}^{\text{\tiny(M)}}_{n,\epsilon}\approx\bar{\mathcal{C}}^{\text{\tiny(M)}}_{\infty,0}-\frac{\mathcal{\bar{V}}^{\text{\tiny{(M)}}}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(15)
where
${\bar{\mathcal{C}}^{\text{\tiny(M)}}_{\infty,0}=\int_{0}^{\infty}{\mathcal{C}}^{\text{\tiny{(M)}}}_{\infty,0}(\alpha_{0})f_{r_{0}}(r_{0})dr_{0}}$,
${\bar{\mathcal{V}}^{\text{\tiny{(M)}}}=\int_{0}^{\infty}{\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})f_{r_{0}}(r_{0})dr_{0}}$,
and $f_{r_{0}}(r_{0})$ as defined in (8).
###### Proof.
This is derived by averaging (10) with respect to $r_{0}$. ∎
The average achievable rate depends on the distances between symbols in the
constellation set, i.e.
$\|\underline{d}_{ml}\|=\|\underline{s}_{m}-\underline{s}_{l}\|$ for $m\neq
l$, where neighboring symbols contribute more to error than far ones. Such
distances are in turn affected by the constellation choice. Moreover, the
average achievable rate is affected by the network parameters $\lambda,\
r_{0}$ which will be investigated later in the numerical evaluations section.
The achievable rates provided so far are theoretical. In the next section, we
use MLPCM, which was proposed in [31, 32] to be used in future generations, to
validate the achievability of the obtained theoretical benchmarks.
### III-D Multilevel Polar-Coded Modulation
$\mathbb{X}$
Demultiplexer
$x_{1}$$x_{2}$$x_{\log_{2}M}$EncoderEncoderEncoderEncoder
M-ary Modulator
$u_{1}$$u_{2}$$u_{\log_{2}M}$${s_{0}}$Channel$\mathbb{y}$Decoder$\widetilde{x}_{1}$DelayDecoder$\widetilde{x}_{2}$DelayDecoder$\widetilde{x}_{\log_{2}M-1}$$\widetilde{x}_{\log_{2}M-2}$$\widetilde{x}_{1}$DelayDecoder$\widetilde{x}_{\log_{2}M}$$\widetilde{x}_{1}$
Multiplexer
$\widetilde{\mathbf{X}}$ Figure 2: Multilevel Coded Modulation Block Diagram
To validate the theoretical results in Sec. III-B and III-C, we use the MLPCM
scheme shown in Fig. 2. The information bits (message)
$\mathbf{X}\in\mathbb{F}_{2}^{k\log_{2}(M)}$ are fed into a demultiplexer that
splits them into $\log_{2}(M)$ sub-messages $\\{x_{i}\\}_{i=1}^{\log_{2}(M)}$
where $x_{i}\in\mathbb{F}_{2}^{k}$. Each sub-message $x_{i}$ is encoded using
a polar encoder with rate $R_{c}=k/n$ to obtain codewords
$\mathbf{U}\in\mathbb{F}_{2}^{\log_{2}(M)\times n}$, where
$u_{i}\in\mathbb{F}_{2}^{n}$ is the $i^{th}$ row of $\mathbf{U}$ ($i^{th}$
codeword). Then, the transmitter collects one bit from each of the codewords
$u_{i}$ to form a $\log_{2}(M)$-bit symbol which is then mapped to a symbol
$s_{0}$ from an $M$-ary constellation such as the $M$-QAM constellation. The
symbols are then scaled according to the power constraint in Sec. II. The
result of repeating this process for all $n$ codeword symbols of all
$\log_{2}(M)$ sub-messages is one MLPCM codeword of length $n$, which is then
transmitted through the channel. At the receiver side, multistage decoding
takes place to efficiently recover the message. It breaks the decoding process
into $\log_{2}(M)$ stages.
In each stage $i=1,\dots,\log_{2}(M)$, the receiver feeds the demapper of the
$i^{th}$ stage with the received signal $y$ and the submessages recovered from
previous stages $(\tilde{x}_{1},\dots,\tilde{x}_{i-1})$, (which is defined as
$\emptyset$ for $i=1$), and the demapper demaps the symbols into bits to
obtain $\tilde{u}_{i}$, using the multilevel decoding criteria [26]. Then, the
demapped bits $\tilde{u}_{i}$ are decoded using a polar decoder to obtain
$\tilde{x}_{i}$ using the successive cancellation algorithm provided in [40].
Finally, all sub-messages are fed to a multiplexer to form the message
$\mathbf{\tilde{X}}$. This MLPCM will be incorporated in the large-scale DL
network Monte Carlo simulator to validate the achievability of the theoretical
rates obtained in Sec. III-B and III-C.
(a) The maximum average coding rate at $r_{0}=250\ m$ for different values of
$n$ and $\epsilon$
(b) The average coding rate for M-QAM at $r_{0}=150\ m$, $n=128$, and
$\epsilon=10^{-2}$.
Figure 3: Average coding rate analysis of a large-scale network with
$\lambda=1\ \mathrm{BS/km^{2}}$ for fixed $r_{0}$
### III-E Average Rate Numerical Results
This subsection presents illustrative numerical results for the theoretical
rate analysis provided in Sec. III-B and III-C, which are also validated via
Monte Carlo simulation. The results presented investigate the effect of the
blocklength $n$ and the FER $\epsilon$, the tightness of the proposed
theoretical approximations, and the performance comparison between the FBR and
the AR. Then, the performance of the MLPCM is investigated in comparison with
the theoretical average coding rate.
Fig. 3 plots the maximum average coding rate and the average coding rate for
M-QAM versus the average SNR $\alpha_{0}$ for a fixed $r_{0}$. In Fig. 3(a),
the average coding rate is plotted for $n\in\\{128,2048\\}$, FER
$\epsilon\in\\{10^{-2},\ 10^{-5},\ 10^{-6}\\}$, and $r_{0}=250$ m. Given a FER
$\epsilon$, a gap exists between the average coding rate in the FBR and the
AR. The gap is around $1$ dB for $n=2048$ and is between $3$ and $5$ dB for
$n=128$ at $\epsilon=10^{-5}$ and $10^{-6}$, respectively, which shows the
severity of the SNR penalty in the FBR compared to the AR. The figure shows
the trade-off between reliability (represented by FER) and maximum average
coding rate at a given $n$, where the maximum average coding rate
decreases/increases as the FER decreases/increases. For instance, to achieve a
given maximum average coding rate at $n=128$ and FER $\epsilon=10^{-5}$, we
need around $2$ dB more power compared with the same blocklength $n$ under the
FER $\epsilon=10^{-2}$. Furthermore, the impact of small $n$ becomes more
severe when the FER $\epsilon$ is smaller, resulting in rate degradation.
Overall, this shows that it is impractical to use results from the AR in the
FBR.
Second, the maximum achievable coding rate for different modulation schemes,
presented in Sec. III-C, is simulated for $r_{0}=150\ m$, $\lambda=1\
\text{BS}/{\text{km}}^{2}$, blocklength $n=128$, and FER $\epsilon=10^{-2}$ in
Fig. 3(b). Fig. 3(b) demonstrates the approximated maximum achievable coding
rate using the Gamma approximation for various modulation orders,
${M}=\\{2,4,8,16\\}$, (Theorem 2), along with the maximum achievable rate
under Gaussian codebooks (Theorem 1) versus $\alpha_{0}$. The Monte Carlo
simulated maximum achievable coding rate for QAM is provided to show the
tightness of the approximation. The approximated maximum achievable rate is
very close to the simulated maximum achievable rate. Thus, the Gamma
approximation in (9) leads to a fairly tight approximation. It is also
observed that as the QAM modulation order increases, the performance improves
towards the maximum average coding rate (without Gaussian codebooks) in a
large-scale network in the FBR (Theorem 1).
The spatially averaged coding rate is depicted in Fig 4(a) versus the transmit
signal to noise power ratio $\frac{\mathcal{P}}{\sigma_{w}^{2}}$.555The figure
is plotted for a wide range of $\frac{\mathcal{P}}{\sigma_{w}^{2}}$ for
illustrative purposes, bearing in mind that some values of
$\frac{\mathcal{P}}{\sigma_{w}^{2}}$ in the plot may be impractical. The
figure shows that for high SNR (above $0$ dB), one cannot achieve the same
rate and FER achieved at a given $n$ by decreasing $n$ and paying an SNR
penalty. For instance, the rate achieved at $n=2048$, $\epsilon=10^{-5}$,
$\text{SNR}=10$ dB cannot be achieved at $n=128$ and $\epsilon=10^{-5}$. The
only way to achieve such a rate at $n=128$ is by paying a FER penalty, which
is impractical for applications that need high reliability. Hence, it is
important to select $n$ that balances the trade-off between reliability and
coding rate. Similar to Fig. 3(a), Fig. 4(a) shows the importance of the
results in this paper for characterizing and optimizing the performance of
large-scale networks with latency-sensitive applications.
(a) Maximum Average Coding Rate for different values of $n$ and $\epsilon$.
(b) Average coding rate for M-QAM at $n=512$, and $\epsilon=10^{-2}$.
Figure 4: Average coding rate analysis for random $r_{0}$ at $\lambda=1\
\mathrm{BS/km^{2}}$
Fig. 4(b) plots, the average achievable rate versus
$\frac{\mathcal{P}}{\sigma_{w}^{2}}$ under a Rayleigh distributed $r_{0}$, QAM
constellation with $M=2,4,8,$ and $16$, blocklenth $n=512$, and FER
$\epsilon=10^{-2}$, showing a similar trend as for fixed $r_{0}$. It is
observed that the average achievable rate saturates at a rate of $\approx 2\
\text{bits/transmission}$ for this choice of $\lambda$, $n$, and $\epsilon$,
which is low compared to the case $r_{0}=150\ m$. This performance is a
consequence of averaging the performance of nearby UEs which have relatively
strong received desired signals (high SINR) and far away UEs which suffer from
more interference (low SINR).
Finally, considering the same setting in Fig. 3(b), we simulate MLPCM with a
QAM constellation for modulation orders $M=2,4,8,$ and $16$ in Fig. 5. The
simulations are performed by fixing the SNR and increasing the coding rate
until reaching the FER $\epsilon$. The achievable rates are plotted along with
the theoretical expressions proposed in (7) and (10). The figure shows that
the achievable rates of the MLPCM scheme with 2-QAM or 4-QAM approaches the
theoretical rate of the corresponding QAM constellation. However, for 8-QAM
and 16-QAM, a gap of $\approx 2.5$ dB exists between the theoretical results
and the simulation results. Nonetheless, this SNR gap translates to a small
rate gap of fractions of bits/transmission due to the slow increase of the
rate versus SNR in the large-scale network which is caused by interference.
Fig. 5 thus validates that the theoretical performance under finite
constellations can be approached using MLPCM, which validates the potential of
MLPCM for future technologies.
Figure 5: The average coding rate versus $r_{0}^{-\eta}\alpha$ at $\lambda=1\
\mathrm{BS/km^{2}}$, $r_{0}=150\ m$, and $n=128$.
## IV Rate Outage Probability & Meta Distribution
In the previous section, we studied the case where the SINR is known at the
transmitter. However, if the BS does not track the SINR, then the rate
adaptation investigated in the previous section is not possible. Considering
the second mode of operation, where the SINR is unknown at the transmitter but
known at the receiver, the transmitter encodes at a constant (target) rate
$R_{t}$ and blocklength $n$. As a result, the resulting (conditional) FER
conditioned on SINR $\Upsilon$ may or may not satisfy a desired design FER
$\epsilon$ due to the unknown stochastic variation of the intended channel
fading and aggregate interference. In this case, it is important to keep this
conditional FER below a threshold. This is especially relevant bearing in mind
that $n=L/l$, i.e., a large conditional FER means that there are coherence
intervals that produce large FERs for all blocks within these coherence
intervals. This can lead to bursts of frame errors when the channel undergoes
a period of coherence intervals with weak channels, which is undesired.
Thus, we define an FER threshold $\bar{\epsilon}$, and it is desired to
maintain the conditional FER below $\bar{\epsilon}$. If the conditional FER
exceeds $\bar{\epsilon}$, we say that we have an outage. This equivalently
means that rate $R_{n,\bar{\epsilon}}(\Upsilon)$ supported by the channel at
the given $n$ and FER threshold $\bar{\epsilon}$ decreases below $R_{t}$. This
is defined formally as follows.
###### Definition 2.
Given a target rate $R_{t}$ and a DL large-scale network with a distance
$r_{0}$ between the desired UE and the serving BS as defined in (1), we define
the outage probability in the FBR with blocklength $n$ and FER threshold
$\bar{\epsilon}$ as
$\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})=P(R_{n,\bar{\epsilon}}(\Upsilon)<R_{t})$,
where $R_{n,\bar{\epsilon}}(\Upsilon)$ is as defined in Lemma 1.
This outage probability is studied in this section.
### IV-A DL Outage Analysis in the Finite Blocklength Regime
We start by reviewing the outage probability of a large-scale network in the
AR. In this case, the outage is defined as the probability that the channel
capacity (defined as the maximum rate such that the error probability vanishes
as $n\to\infty$) is lower than a target rate $R_{t}$. The rate outage
probability under a given $r_{0}$ is given as [41]
$\displaystyle\mathcal{O}(r_{0},R_{t})=1-e^{-\frac{(2^{R_{t}}-1)\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{R_{t}}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}.$
(16)
Studying the rate outage probability under blocklength $n$ and FER threshold
$\bar{\epsilon}$ is generally difficult to simplify as it is a function of the
fading distribution as well as the network geometry. Instead, we provide
bounds which are fairly tight in the following theorem.
###### Theorem 3.
For an average SNR $\alpha_{0}$, the rate outage probability defined in Def. 2
satisfies
$\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$,
where
$\displaystyle\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathcal{O}(r_{0},R_{t}),$ (17)
$\displaystyle\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathcal{O}(r_{0},R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}),$ (18)
where $\mathcal{O}(r_{0},R_{t})$ is defined in (16),
$a_{n,\bar{\epsilon}}=\sqrt{\frac{\log_{2}^{2}(e)}{n}}Q^{-1}(\bar{\epsilon})$,
and ${b_{n}=\frac{1}{2n}\log_{2}n}$.
###### Proof.
From Def. 2, we have
$\displaystyle\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})=\mathbb{P}(R_{n,\bar{\epsilon}}(\Upsilon)<R_{t})=\mathbb{P}\Bigg{(}\log_{2}(1+\Upsilon)-{a_{n,\bar{\epsilon}}}\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}{+b_{n}}<R_{t}\Bigg{)}.$
(19)
Noting that $\mathcal{K}(\Upsilon)=\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}$ is
monotonically increasing in $\Upsilon$ as shown in Fig. 6, satisfying
$\mathcal{K}(\Upsilon)\in[0,1]$, we conclude that
$\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$,
where the lower and upper bounds are obtained by setting
$\mathcal{K}(\Upsilon)$ to $0$ and $1$, respectively. ∎
Figure 6: $\mathcal{K}(\Upsilon)$ versus $\Upsilon$ and the PDF of the SINR at
$r_{0}\in\\{150,250\\}\ m$
The outage probability lower bound confirms that the outage probability in the
FBR is larger than that in the AR and the upper bound provides a guaranteed
performance. By observing Fig. 6, it can be seen that $\mathcal{K}(\Upsilon)$
quickly approaches one as $\Upsilon$ grows. The figure also shows that the
density $f(\Upsilon|r_{0})$ in the interval of SINR, where the approximation
$\mathcal{K}(\Upsilon)\approx 1$ is tight ($\Upsilon>5$ e.g. ), increases as
$r_{0}$ decreases (which is the case for dense networks). Hence, the upper
bound $\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$ can be used to evaluate
guaranteed outage performance in general and can be used as a tight
approximation for dense networks. This is verified in Fig. 7 & 8. Next, the
spatially averaged outage probability is presented.
###### Corollary 3.
The average rate outage probability defined in Def. 2 with blocklength $n$,
FER $\bar{\epsilon}$, and target rate $R_{t}$ can be upper bounded as
$\displaystyle\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})=1-\int_{0}^{\infty}2\pi\lambda
r_{0}$ $\displaystyle
e^{-\frac{2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1}{\alpha_{0}}-\pi\lambda
r_{0}^{2}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}dr_{0}$
(20)
###### Proof.
Using the law of iterated expectation, the average outage probability is the
average of (18) with respect to $r_{0}$, i.e.,
$\mathbb{P}(R>R_{t})=\mathbb{E}_{r_{0}}\\{\mathbb{P}(R>R_{t}|r_{0})\\}$. The
result then follows since $r_{0}$ follows a Rayleigh distribution. ∎
For a shadowed urban cellular network, the path-loss exponent $\eta$ lies in
the range $3-5$ [42], where $\eta=4$ is a common value of practical relevance
that is widely utilized in the literature. For the case $\eta=4$, the outage
probability in (20) can be simplified as given next.
###### Corollary 4.
The average rate outage probability of the large-scale network modeled by (1)
with blocklength $n$, FER $\bar{\epsilon}$, target rate $R_{t}$, and path loss
exponent $\eta=4$, can be upper bounded as
$\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})=1-\pi\lambda\sqrt{\frac{\pi\mathcal{P}}{\sigma_{w}^{2}\mu}}e^{\frac{\sigma_{w}^{2}\mu
z^{2}}{4\mathcal{P}}}Q\left(\sqrt{\frac{z^{2}\mu\sigma_{w}^{2}}{2\mathcal{P}}}\right)$
(21)
where $\mu=2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1$ and
$z=\frac{\lambda\pi\mathcal{P}}{\sigma_{w}^{2}\mu}\left(\sqrt{\mu}\arctan(\sqrt{\mu})+1\right)$.
###### Proof:
A closed-form expression for the upper bounded outage probability at $\eta=4$
is obtained as follows
$\displaystyle\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})$
$\displaystyle=1-\int_{0}^{\infty}e^{-\frac{\mu
r_{0}^{4}\sigma_{w}^{2}}{\mathcal{P}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{\mu
r_{0}^{4}}{\mathcal{P}}\right\\}f_{r_{0}}(r_{0})\ dr_{0}$ (22)
$\displaystyle\overset{(i)}{=}1-\int_{0}^{\infty}2\pi\lambda
r_{0}e^{-\frac{\sigma_{w}^{2}\mu}{\mathcal{P}}\left(r_{0}^{4}+z\
r_{0}^{2}\right)}dr_{0}$ (23)
$\displaystyle\overset{(ii)}{=}1-\pi\lambda\sqrt{\frac{\pi\
\mathcal{P}}{\sigma_{w}^{2}\mu}}e^{\frac{z^{2}\mu\sigma_{w}^{2}}{4\mathcal{P}}}Q\left(\sqrt{\frac{z^{2}\mu\sigma_{w}^{2}}{2\mathcal{P}}}\right)$
(24)
where we used
${\mathcal{L}_{\mathcal{B}_{g}}\\{s\\}}_{\eta=4}=\exp\left\\{-\pi\lambda\sqrt{s\mathcal{P}}\text{tan}^{-1}\left(\sqrt{\frac{s\mathcal{P}}{r_{0}^{4}}}\right)\right\\}$,
$(i)$ is obtained using a change of variable
$z=\frac{\lambda\pi\mathcal{P}}{\mu\sigma_{w}^{2}}\left(\sqrt{\mu}\arctan(\sqrt{\mu})+1\right)$,
and $(ii)$ is obtained using the change of variables and the CDF of the
Gaussian distribution with mean $\left(\frac{z}{2}\right)$ and variance
$\left(\frac{\mathcal{P}}{2\mu\sigma_{w}^{2}}\right)$. ∎
The rate outage probability of a large-scale network in the FBR is upper
bounded, with guaranteed performance at high SINR, in a generic form in (18)
and (20) and in a closed-form expression for $\eta=4$ in (24), as a function
of BS density $\lambda$, FER $\bar{\epsilon}$, blocklength $n$, and average
SNR $\alpha_{0}$. Next, the reliability of the network under the FBR.
### IV-B Reliability of a Large-Scale Network in the Finite Blocklength
Regime
We define a metric to evaluate the overall network reliability for a fixed
$r_{0}$, defined as the probability of correctly decoding a codeword. A
codeword is decoded correctly in two cases in the FBR. The first case is when
$\mathcal{R}_{n,\bar{\epsilon}}(\Upsilon)>R_{th}$, in which case the codeword
is decoded correctly with a probability larger than or equal to
$(1-\bar{\epsilon})$. The second case is when
$\mathcal{R}_{n,\bar{\epsilon}}(\Upsilon)<R_{th}$, in which case few codewords
may still be decoded correctly. Since in practice, it is desirable to operate
at low outage probability, hence the second case does not contribute much to
reliability. Thus, to simplify the analysis, we will lower bound the
probability of decoding a codeword correctly in the second case by $0$ (i.e.,
its contribution to reliability is neglected) and we will only consider the
first case. Hence, the reliability is lower bounded by the following
expression, providing a guaranteed performance for a given $r_{0}$,
$\mathcal{T}_{n,\bar{\epsilon}}(r_{0},R_{t})=(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))(1-\bar{\epsilon}).$
(25)
The guaranteed reliability averaged over $r_{0}$ is given by
$\bar{\mathcal{T}}_{n,\bar{\epsilon}}(R_{t})=(1-\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon}))(1-\bar{\epsilon}),$
(26)
Also, for the sake of comparison, the reliability of the AR can be defined as
${\mathcal{T}}_{\infty,0}(r_{0},R_{t})=1-\mathcal{O}(r_{0},R_{t}),$ (27)
since the error probability is assumed to be vanishing as $n\to\infty$, and
hence failure occurs when there is an outage. The reliability will be assessed
numerically in Sec. IV-D.
Studying the rate outage probability and the overall system reliability gives
more insight into the performance of the network. Moreover, studying the
coding rate meta distribution, which provides the percentiles of users
achieving a rate $R_{t}$, will provide a complete picture of the network
performance. Hence, the meta distribution of the coding rate is derived next.
### IV-C Coding Rate Meta Distribution in the Finite Blocklength Regime
The meta distribution provides fine-grained information about network
performance, in the form of the fraction of users that can achieve a minimum
data rate $R_{t}$ at a FER threshold $\bar{\epsilon}$ with probability at
least $p_{t}$. The definition and analysis of this quantity are provided next.
###### Definition 3.
Given a target rate $R_{t}$ and a DL large-scale network with a distance
$r_{0}$ between the desired UE and the serving BS as defined in (1), we define
the coding rate meta distribution in the FBR with blocklength $n$ and FER
threshold $\bar{\epsilon}$ as
$F_{R_{t}}(p_{t})={P}(P_{s}(R_{t},n,\bar{\epsilon})<p_{s})$ where
$P_{s}(R_{t},n,\bar{\epsilon})=P(R_{n,\bar{\epsilon}}(\Omega)>R_{t}|\tilde{\Psi})$
is the probability of success, $\Omega$ is the signal-to-interference ratio
(SIR) and $R_{n,\bar{\epsilon}}(\Omega)$ is as defined in Lemma 1.
For simplicity, we assume an interference-limited scenario where noise is
neglected and the SIR is given by
$\Omega=\frac{\mathcal{P}r_{o}^{-\eta}|h_{0}|^{2}}{\sum_{r_{i}\in\tilde{\Psi}/r_{o}}\mathcal{P}r_{i}^{-\eta}|h_{i}|^{2}}$.
Then, $P_{s}(R_{t},n,\bar{\epsilon})$ is given by
$\displaystyle
P_{s}(R_{t},n,\bar{\epsilon})=\mathbb{P}^{!}(R_{n,\bar{\epsilon}}(\Omega)>R_{t}|\tilde{\Psi})=\mathbb{P}^{!}\left(\log_{2}(1+\Omega)-\frac{\sqrt{V(\Omega)}Q^{-1}(\bar{\epsilon})}{\sqrt{n}}{+b_{n}}>R_{t}|\tilde{\Psi}\right)$
(28)
where $\mathbb{P}^{!}(\cdot)$ is the reduced Palm measure of the point process
[43],[44, Def. 8.8]. Note that $P_{s}(R_{t},n,\bar{\epsilon})$ is a random
variable which is a function of the stochasticity of the network. Its moments
are defined as
$\displaystyle\mathcal{M}_{d}=\mathbb{E}\left\\{P_{s}(R_{t},n,\bar{\epsilon})^{d}\right\\},$
(29)
and are used to calculate the meta distribution as follows [43]
$\displaystyle
F_{R_{t}}(p_{t})=\mathbb{P}(P_{s}(R_{t},n,\bar{\epsilon})>p_{t}){=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathcal{I}m(e^{-t\log(p_{t})}\mathcal{M}_{jt})}{t}dt},$
(30)
where $\mathcal{I}m(\cdot)$ is the imaginary part of a complex number and
$\mathcal{M}_{jt}$ is obtained at $d=jt$ in (29).
In [43], a simple yet tight approximating distribution of
$P_{s}(R_{t},n,\bar{\epsilon})$ is proposed for the AR, which is the beta
distribution given by
$\displaystyle
f(P_{s};R_{t})\approx\frac{P_{s}^{\frac{\vartheta(\beta+1)-1}{1-\vartheta}}(1-P_{s})^{\beta-1}}{\text{B}\left(\frac{\vartheta\beta}{1-\vartheta},\beta\right)},$
(31)
where $\text{B}(\cdot,\cdot)$ is the beta function,
$\vartheta=\mathcal{M}_{1}$ and the
$\beta=\frac{(\mathcal{M}_{1}-\mathcal{M}_{2})(1-\mathcal{M}_{1})}{\mathcal{M}_{2}-\mathcal{M}_{1}^{2}}$.
Hence, the meta distribution could be computed as the CCDF of the beta
distribution. However, $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ provided in
(29) are difficult to evaluate in the FBR, and hence, an approximation is
proposed in this paper. At high SIR, where the term
$\mathcal{K}(\Upsilon){=\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}}$ is approximately
equal to 1, the approximated probability of success is given by
(a) $n=128$
(b) $n=2048$
Figure 7: The outage probability versus $r_{0}$ at $\lambda\in\\{1,9\\}\
\mathrm{BS/km^{2}}$, $\bar{\epsilon}\in\\{10^{-2},10^{-6}\\}$, and $\alpha=0\
\text{dB}$.
$\displaystyle\hat{P}_{s}(R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathbb{P}^{!}\left(\log_{2}(1+\Omega)-a_{n,\bar{\epsilon}}{+b_{n}}>R_{t}|\tilde{\Psi}\right).$
Therefore, the approximated probability of success can be expressed as a
function of the SIR instead of the coding rate, as follows,
$\displaystyle\hat{P}_{s}(R_{t},n,\bar{\epsilon})=\mathbb{P}^{!}\left(\Omega>2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1|\tilde{\Psi}\right)=\prod_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\frac{1}{1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)\left(\frac{r_{0}}{r_{i}}\right)^{\eta}},$
(32)
The expression provided in (32) is obtained by averaging over the channel
gains. Then, its moments are given by
$\displaystyle\hat{\mathcal{M}}_{d}$
$\displaystyle=\mathbb{E}\left\\{\hat{P}_{s}(R_{t},n,\bar{\epsilon})^{d}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\prod_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\frac{1}{\left(1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)\left(\frac{r_{0}}{r_{i}}\right)^{\eta}\right)^{d}}\right\\}$
$\displaystyle=\exp\left\\{-2\pi\lambda\int_{r_{0}}^{\infty}r\left(1-\frac{1}{(1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)(r_{0}/r)^{\eta})^{d}}\right)dr\right\\},$
(33)
Thus, the meta distribution can be evaluated using the approximated moments in
(IV-C) as follows
$\displaystyle\hat{F}_{R_{t}}(p_{t})=\mathbb{P}(\hat{P}_{s}(R_{t},n,\bar{\epsilon})>p_{t})=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathcal{I}m(e^{-t\log(p_{t})}\hat{\mathcal{M}}_{jt})}{t}dt.$
(34)
Using the beta distribution approximation, the meta-distribution can be
approximated as
$\displaystyle\hat{F}_{R_{t}}(p_{t})\approx
1-I_{p_{t}}\left(\frac{\hat{\mathcal{M}}_{1}(\hat{\mathcal{M}}_{1}-\hat{\mathcal{M}}_{2})}{(\hat{\mathcal{M}}_{2}-\hat{\mathcal{M}}_{1}^{2})},\frac{(1-\hat{\mathcal{M}}_{1})(\hat{\mathcal{M}}_{1}-\hat{\mathcal{M}}_{2})}{(\hat{\mathcal{M}}_{2}-\hat{\mathcal{M}}_{1}^{2})}\right),$
(35)
where $I_{z}(x,y)=\frac{1}{\text{B}(x,y)}\int_{0}^{z}t^{x-1}(1-t)^{y-1}dt$ is
the regularized incomplete beta function.
(a) $\bar{\epsilon}=10^{-2}$
(b) $\bar{\epsilon}=10^{-6}$
Figure 8: The outage probability versus the rate threshold $R_{t}$ at
$\lambda=1\ \mathrm{BS/km^{2}}$, $n\in\\{128,\ 2048\\}$, and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, averaged with respect to
Rayleigh distributed $r_{0}$.
### IV-D Numerical Results
Now, we provide numerical results for the bounds and approximations of the
rate outage probability, reliability, and the coding rate meta-distribution.
We investigate the tightness of these approximations and compare them with
results from the literature for the AR.
First, in Fig. 7, the rate outage probability is plotted versus $r_{0}$ with
$\lambda\in\\{1,\ 9\\}\ \mathrm{BS/km^{2}}$, $\eta=4$,
$\bar{\epsilon}\in\\{10^{-2},\ 10^{-6}\\}$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, and $R_{t}=1$
bit/transmission. The PDF of $r_{0}$ is plotted to illustrate the probability
of occurrence of the $r_{0}$ values. The figure shows the Monte Carlo
simulated outage probability, in addition to the upper bound in (18) and the
outage probability in the AR (16) which serves as a lower bound (cf. Thm. 3).
It shows that the rate outage in the AR underestimates the rate outage
probability in the FBR and the gap to the actual outage probability increases
as the FER threshold decreases. This gap can be as large as $30\%$ in some
cases, such as with $r_{0}=200$ m, $\lambda=1$ BS/km2, $n=128$, and
$\bar{\epsilon}=10^{-6}$, where the actual outage probability is 0.13, but
(16) underestimates it as 0.1. Fig. 7 also shows that the upper bound of the
outage probability expression in (18) is fairly accurate to provide a
convenient approximation for a broad range of $n,\lambda$, and
$\bar{\epsilon}$. The outage probability is seen to increase as the FER
threshold $\bar{\epsilon}$ decreases because the term
$\sqrt{\frac{V(\Upsilon)}{n}}Q^{-1}(\bar{\epsilon})$ in (7) increases as
$\bar{\epsilon}$ decreases. However, at a specific $\bar{\epsilon}$, as $n$
increases, the outage probability approaches the outage probability in the AR
as expected. Thus, while the results in [12] can be a good representative of
the performance when $n$ is relatively large (Fig. 7(a)), this is not the case
when $n$ is small in which case our derived bounds and approximation are more
accurate. Moreover, in Fig. 8, the rate outage probability approximation for a
Rayleigh distributed $r_{0}$, provided in (24), is plotted versus the target
rate $R_{t}$ and compared to the simulated outage probability for BS density
$\lambda=1\ \text{BSs}/\text{km}^{2}$, blocklengths $n\in\\{128,\ 2048\\}$,
FER threshold $\bar{\epsilon}\in\\{10^{-2},\ 10^{-6}\\}$, and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$. This plot also proves the
tightness of the upper bound and the inaccuracy of the AR.
In Fig. 9(a), the rate outage probability under QAM constellations is shown
for various modulation orders versus $r_{0}$, where the rate outage for a
specific modulation scheme is defined as
$\mathcal{O}^{\text{\tiny(M)}}(r_{0},R_{t},n,\bar{\epsilon})=\mathbb{P}(\mathcal{R}_{n,\bar{\epsilon}}^{\text{\tiny(M)}}(\Upsilon)<R_{t}|r_{0})$.
This expression is computed using Monte Carlo simulation. The figure is
plotted for $n=128$, $\bar{\epsilon}=10^{-2}$, $\lambda=1$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, and
$R_{t}\in\\{0.825,1.85\\}$. Similar to the previous plots, the rate outage
probability $\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})$ acts as the lower
bound of the outage probability of all constellation sets. Also, the higher
order modulation (i.e. $M=16$) yields an outage probability lower than the
lower order modulation (i.e. $M=2$) at a fixed $R_{t}$ as the modulation order
limits the maximum transmission rate $(\log_{2}(M))$. However, at lower
$R_{t}$ as such $R_{t}=0.825$, most of the high order modulation schemes
provide a performance close to the theoretical lower bound
$\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})$. Fig. 9(b) shows the outage
probability for random $r_{0}$, i.e.,
$\bar{\mathcal{O}}^{\text{\tiny(M)}}(R_{t},n,\bar{\epsilon})=\mathbb{P}(\mathcal{R}_{n,\bar{\epsilon}}^{\text{\tiny(M)}}(\Upsilon)<R_{t})$
versus $R_{t}$. As shown, the rate outage probability increases as the rate
threshold $R_{t}$ increases, and it jumps to 1 at a certain $R_{t}$ for each
modulation order, which is when $R_{t}$ exceeds the maximum rate achievable by
the modulation order ($\log_{2}(M)$).
(a) Outage probability versus $r_{0}$, for $R_{t}\in\\{0.825,1.85\\}$
(b) Outage probability versus $R_{t}$
Figure 9: Outage probability using M-QAM at $\lambda=1\
\text{BS}/\text{km}^{2}$, $\bar{\epsilon}=10^{-2}$, $n=128$ and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$
The overall network reliability versus the FER threshold $\bar{\epsilon}$ is
evaluated in Fig. 10 for fixed and random $r_{0}$ for $n\in\\{128,2048\\}$ and
$R_{t}\in\\{0.1375,1,3.46\\}$. It can be observed that the reliability of the
network increases as the FER threshold increases until it reaches a maximum
around $\bar{\epsilon}=10^{-2}$ then it rapidly decays. This is because the
increase of $(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ with respect to
$\bar{\epsilon}$ is faster than the decrease of $(1-\bar{\epsilon})$ for small
$\bar{\epsilon}$. However, for large $\bar{\epsilon}$,
$(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ saturates to $1$ and
$(1-\bar{\epsilon})$ dominates and contributes to the decay of the reliability
at high $\bar{\epsilon}$. In other words, at $\bar{\epsilon}<10^{-2}$, the
rate of increase of $(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ is higher
than the rate of decay of $(1-\bar{\epsilon})$. However, at
$\bar{\epsilon}>10^{-2}$, the rate of increase of
$(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ is nearly zero and the rate of
decay of $(1-\bar{\epsilon})$ dominates. Therefore, the reliability rapidly
decays. It is shown that for a fixed $R_{t}$, a gap exists between curves
simulated under FBR and AR. This gap grows wider up to $0.15$ as FER threshold
$\bar{\epsilon}$ decreases because the short blocklengths (e.g. $n=128$) are
less reliable. Also, it is obvious that the reliability of the network at
random $r_{0}$ is lower than that at fixed $r_{0}$ as it averages the
performance of users which are close and those that are far away from the BS.
Moreover, as $R_{t}$ increases the approximated reliability becomes close to
the exact one. This is because the outage upper bound provided in Corollary 3
is tight for high SINR. Hence, from Def. 2, as the $R_{t}$ increases, only
high SINRs can achieve the desired rate and FER threshold at the given $n$,
which is where the approximation becomes tighter. Whereas, for low $R_{t}$,
low SINRs can also achieve the desired performance which loosens the
approximation.
(a) Fixed $r_{0}$: $\lambda=1$ $\text{BS}/\text{km}^{2}$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$ and $r_{0}=250\ m$
(b) Random $r_{0}$: $\lambda=0.1$ $\text{BS}/\text{km}^{2}$ and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=10\ \text{dB}$
Figure 10: Reliability versus the FER threshold $\bar{\epsilon}$
Finally, we validate the coding rate meta distribution in Fig. 11(a). The
coding rate meta distribution provided in (30) is simulated via Monte-Carlo
simulations at $\lambda=1\ \text{BSs}/\text{km}^{2}$, $r_{0}=\ 150\ m$, an
area of $500\ \text{km}^{2}$, a blocklength $n=128$, FER threshold
$\bar{\epsilon}=10^{-5}$, and a rate threshold
$R_{t}=\\{0.1375,0.3964,1,2.0574,3.4594\\}$. It is observed that there is a
significant gap between the performance in the AR and the FBR, showing that
the meta distribution of the coding rate in the AR overestimates the
performance of the network in the FBR. This gap can reach up to $0.09$ at
$R_{t}=3.46$ and $p_{t}=0.9$, which means $9\%$ of the users predicted to
achieve the required performance in the AR will not achieve it in the FBR. It
can also be observed that the gap increases as $R_{t}$ or $p_{t}$ increases.
Thus, the AR analysis in [43, 45, 46] cannot be used to provide precise
results in the FBR. For the sake of preciseness of the results, we have
simulated the coding rate meta distribution including the noise term with
noise power $N_{0}=-90\ \rm dBm$ [47]. The results show that the network is an
interference-limited network and the noise can be neglected for the simplicity
of the analysis. In Fig. 11(b), the approximation proposed in (32) is shown to
be very tight compared to the exact coding rate meta distribution in (30) for
different $r_{0}$. The tightness of the approximation increases by decreasing
$r_{0}$ or alternatively increasing SIR as the term
$\sqrt{1-\frac{1}{(1+\Omega)^{2}}}$ converges to $1$ as the SIR increases.
Using this approximation it is easier to find an expression for the moments of
the success probability and use it to evaluate the meta distribution using the
beta distribution provided in (35). Hence, the beta approximation of the meta-
distribution provides a close performance to the exact and approximated meta-
distribution.
(a) The FBR versus the AR at $n=128$ and $\bar{\epsilon}=10^{-5}$
(b) Exact versus Approximation at $r_{0}\in\\{150,\ 250,\ 500\\}m$ at
$R_{t}=1$, $n=128$, and $\bar{\epsilon}=10^{-2}$
Figure 11: The meta distribution of the coding rate versus $p_{t}$
Thus, we saw that our proposed expressions provide an accurate representation
of the rate outage probability and the coding rate meta distribution in the
FBR, which is more accurate than using AR expressions from the literature.
## V Conclusion
This paper investigates different aspects of the performance of a large-scale
DL network in the FBR using the theory of coding in the FBR in conjunction
with stochastic geometric tools. We start with rate analysis where we derive
the average coding rate using Gaussian codebooks and $M$-ary QAM
constellation, and investigate the practical scheme of MLPCM in the FBR
showing that its performance approaches the theoretical benchmarks. We also
study the rate outage probability, reliability, and the coding rate meta
distribution, where we provide bounds and approximations. From the results, we
conclude that the performance analysis provided under an AR provides a
misleading (overestimated) performance and cannot be used to characterize the
performance in the FBR. The performance in the FBR provides accurate rate and
FER characterization, and approaches the AR performance at sufficiently large
blocklengths $>2048$ and high FER $>10^{-2}$. The developed expressions for
the average coding rate, the rate outage probability, and the coding rate meta
distribution are fairly tight as shown by our numerical results. In
conclusion, the results are relevant for the theoretical analysis of large-
scale networks in the FBR, and the theoretical performance can be approached
using practical transmission schemes such as MLPCM. The results in this paper
can be extended to characterize the performance of networks under different
multiple access schemes as in [48, 18, 17].
## Appendix A The Average Square Root Channel Dispersion
$\mathcal{V}(\alpha_{0})$
From the definition of channel dispersion provided in (2), the average square
root channel dispersion $\mathcal{V}(\alpha_{0})$ can be written as follows
$\displaystyle\mathcal{V}({\alpha_{0}})=\mathbb{E}\left\\{\sqrt{V(\Upsilon)}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\log_{2}(e)\sqrt{\left(1-\frac{1}{(\Upsilon+1)^{2}}\right)}\right\\}$
$\displaystyle=\int_{0}^{\infty}(1-\mathbb{F}_{\sqrt{V}}({v}))d{v},$ (36)
where $\mathbb{F}_{\sqrt{V}}(v)$ is the cumulative distribution function (CDF)
of $\sqrt{V(\Upsilon)}$, and the last step follows as an application of
Fubini’s theorem [49]. The CDF of $\sqrt{V(\Upsilon)}$ for a given
$\mathcal{B}_{g}$ is given by
$\displaystyle\mathbb{F}_{\sqrt{V}}({v}|\mathcal{B}_{g})$
$\displaystyle=\mathbb{P}(\sqrt{V(\Upsilon)}<{v}\big{|}\mathcal{B}_{g})$
$\displaystyle=\mathbb{P}\left(|h_{0}|^{2}<\left(\frac{1}{{\alpha_{0}}}+\frac{\mathcal{B}_{g}}{\mathcal{P}r_{0}^{-\eta}}\right){z(v)}\bigg{|}\mathcal{B}_{g}\right)$
$\displaystyle=1-\exp\left(-\left(\frac{1}{{\alpha_{0}}}+\frac{\mathcal{B}_{g}}{\mathcal{P}r_{0}^{-\eta}}\right){z(v)}\right),$
(37)
where the last step follows from the exponential distribution of
$|h_{0}|^{2}$. Averaging with respect to the interference term
$\mathcal{B}_{g}$ yields
$\mathbb{E}_{\mathcal{B}_{g}}\\{\mathbb{F}_{\sqrt{V}}({v}|\mathcal{B}_{g})\\}=\mathbb{E}_{\mathcal{B}_{g}}\left\\{1-\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\exp\left(-\frac{\mathcal{B}_{g}r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right)\right\\}$.
Hence, the CDF of the square root of the channel dispersion is given as
$\displaystyle\mathbb{F}_{\sqrt{V}}({v})=1-\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right\\},\
\ \ \ \ \ \ \ 0\leq v<\log_{2}(e),$ (38)
which is obtained using the definition of Laplace transform
$\left(\mathcal{L}_{x}\\{u\\}=\mathbb{E}\left\\{e^{-ux}\right\\}\right)$. By
substituting (38) in (36), we obtain
$\displaystyle\mathcal{V}(\alpha_{0})=\int_{0}^{\log_{2}(e)}\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right\\}d{v},$
as defined in Theorem 1.
## Appendix B The Channel Dispersion for A Large-Scale Network Under a Finite
Constellation
The channel dispersion is defined as the conditional variance of the
information density conditioned on the input distribution as follows
$\displaystyle V_{\text{\tiny M}}(\Upsilon)$
$\displaystyle=\mathbb{E}_{S}\left\\{\mathbb{V}ar\left\\{\log_{2}\frac{P_{Y|S}(y|s)}{P_{Y}(y)}\mid
S\right\\}\right\\}$
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\\{\log_{2}^{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}-\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\\{\log_{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}^{2}.$
where
$\underline{s}_{m}=[\mathcal{R}e\\{s_{m}\\},\mathcal{I}m\\{s_{m}\\}]^{T}$ is
the 2-D real-valued vector form of the complex-valued symbol $s_{m}$ of an
$M$-ary constellation.
Define
$I_{1}=\mathbb{E}\left\\{\log_{2}^{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}$
and
$I_{2}=\mathbb{E}\left\\{\log_{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}^{2}$.
The first term $I_{1}$ is derived as follows
$\displaystyle
I_{1}=\mathbb{E}\left\\{\log_{2}^{2}P_{Y|S}(y|\underline{s}_{m})\right\\}-2\mathbb{E}\left\\{\log_{2}P_{Y|S}(y|\underline{s}_{m})\log_{2}P_{Y}(y)\right\\}+\mathbb{E}\left\\{\log_{2}^{2}P_{Y}(y)\right\\}=I_{11}-2I_{12}+I_{13}.$
The term $I_{11}$ is given by
$\displaystyle
I_{11}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}\|^{2}}{N_{0}+\mathcal{B}}}}{{\pi(N_{0}+\mathcal{B})}}\log_{2}^{2}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}\|^{2}}{N_{0}+\mathcal{B}}}}{{\pi(N_{0}+\mathcal{B})}}\
d{\underline{y}},$
where $\underline{y}=[y_{1},y_{2}]^{T}$. Define
${\underline{t}}=\frac{{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}}{\sqrt{N_{0}+\mathcal{B}}}$,
hence by substitution, $I_{11}$ is given by
$\displaystyle I_{11}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\|{\underline{t}}\|^{2}}}{\pi}\left(\log_{2}^{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+2\|{\underline{t}}\|^{2}\log_{2}\left(e\right)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+\log_{2}^{2}e^{\|{\underline{t}}\|^{2}}\right)d{\underline{t}}$
$\displaystyle=\log_{2}^{2}\frac{1}{\pi(N_{0}+\mathcal{B})}-2\log_{2}(e)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+3\log_{2}^{2}(e).$
The term $I_{12}$ is given by
$\displaystyle
I_{12}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\left(\frac{1}{M}\sum_{l=1}^{M}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B)}}}}{\pi(N_{0}+\mathcal{B})}\right)\log_{2}\left(\frac{e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B)}}}}{\pi(N_{0}+\mathcal{B})}\right)d{\underline{y}},$
and is divided into four terms as
$I_{12}=I_{12}^{(1)}+I_{12}^{(2)}+I_{12}^{(3)}+I_{12}^{(4)}$, where
$\displaystyle I_{12}^{(1)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}d{\underline{y}}\hskip
170.71652pt$ $\displaystyle=\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})},$
$\displaystyle I_{12}^{(2)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}d{\underline{y}}\hskip
56.9055pt\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle=-\log_{2}(e)\log_{2}\frac{1}{\pi M(N_{0}+\mathcal{B})},$
$\displaystyle I_{12}^{(3)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}\hskip 170.71652pt$
$\displaystyle\stackrel{{\scriptstyle\text{(i)}}}{{=}}\frac{1}{\pi}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\|\underline{t}\|^{2}-2\underline{t}^{T}(\underline{s}_{m}-\underline{s}_{l})\frac{h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}}{\sqrt{N_{0}+\mathcal{B}}}-\frac{|h_{0}|^{2}{\mathcal{P}}r_{0}^{-\eta}}{{N_{0}+\mathcal{B}}}\left\|\underline{s}_{m}-\underline{s}_{l}\right\|^{2}}d\underline{t}$
$\displaystyle\stackrel{{\scriptstyle\text{(ii)}}}{{=}}\frac{1}{\pi}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\left(\|\underline{t}\|^{2}+2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}+\Upsilon\left\|\underline{d}_{ml}\right\|^{2}\right)}d\underline{t}$
$\displaystyle=-\log_{2}(e)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+\mathbb{E}\left\\{\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}-\Upsilon\left\|\underline{d}_{ml}\right\|^{2}}\right\\},$
where $(\rm i)$ follows from a change of variable
${\underline{t}}=\frac{{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}}{\sqrt{N_{0}+\mathcal{B}}}$,
and $(\rm ii)$ follows by using
$\underline{d}_{ml}=\underline{s}_{m}-\underline{s}_{l}$ and
$\Upsilon=\frac{|h_{0}|^{2}\mathcal{P}r_{0}^{-\eta}}{{N_{0}+\mathcal{B}}}$,
and $I_{12}^{(4)}$ is given by
$\displaystyle I_{12}^{(4)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}\log_{2}\sum_{l=1}^{M}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}d{\underline{y}}\hskip
85.35826pt$
$\displaystyle\stackrel{{\scriptstyle\text{(i)}}}{{=}}\int_{{\mathbb{R}^{2}}}\frac{e^{-\|\underline{t}\|^{2}}}{\pi}\log_{2}e^{-\|\underline{t}\|^{2}}\log_{2}\sum_{l=1}^{M}e^{-\left(\|\underline{t}\|^{2}+2\underline{t}^{T}\underline{d}_{ml}\sqrt{\Upsilon}+\Upsilon\left\|\underline{d}_{ml}\right\|^{2}\right)}d{\underline{t}}$
$\displaystyle=3\log_{2}^{2}(e)-\log_{2}(e)\mathbb{E}_{\underline{t}}\left\\{\|\underline{t}\|^{2}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$
where $(\rm i)$ is obtained by the substitution of $\underline{t}$,
$\underline{d}_{ml}$, and $\Upsilon$.
The term $I_{13}$ is given by
$\displaystyle
I_{13}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\left(\log_{2}\frac{1}{M\pi(N_{0}+\mathcal{B})}+\log_{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\right)^{2}d{\underline{y}},$
and is divided into three terms as
$I_{13}=I_{13}^{(1)}+I_{13}^{(2)}+I_{13}^{(3)}$, where
$\displaystyle I_{13}^{(1)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}^{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\ d{\underline{y}}\hskip 85.35826pt\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle=\log_{2}^{2}\frac{1}{\pi M(N_{0}+\mathcal{B})},$ $\displaystyle
I_{13}^{(2)}$
$\displaystyle={2}\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}$ $\displaystyle=-2\log_{2}(e)\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}+2\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$
$\displaystyle I_{13}^{(3)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{1}{\pi(N_{0}+\mathcal{B})}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}\log_{2}^{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}$ $\displaystyle=3\log_{2}^{2}(e)-2\log_{2}(e)\
\mathbb{E}\left\\{\|{\underline{t}}\|^{2}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\}+\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$
By adding $I_{11}$, $I_{12}$, and $I_{13}$, $I_{1}$ is given by
$\displaystyle
I_{1}=\log_{2}^{2}\frac{1}{M}+2\log_{2}\frac{1}{M}\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}+\mathbb{E}\left\\{\log_{2}^{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\},$
The second term $I_{2}$ is the square of the relative entropy given as
$\displaystyle
I_{2}=\log_{2}^{2}(M)-2\log_{2}(M)\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}+\mathbb{E}^{2}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}.$
To compute the average square-root channel dispersion, $I_{2}$ is subtracted
from $I_{1}$ and then averaged over all constellation symbols, leading to the
following expression as indicated in (13)
$\displaystyle V_{\text{\tiny
M}}(\Upsilon)=\frac{1}{M}\sum_{m=1}^{M}\left\\{\mathbb{E}_{\underline{t}}\left\\{\log_{2}^{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}-\mathbb{E}_{\underline{t}}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}^{2}\right\\}.$
## Appendix C The SINR distribution at a Fixed $r_{0}$
The distribution of the SINR $\Upsilon$ at a fixed $r_{0}$ is derived by
defining the CDF as follows
$\displaystyle\mathbb{F}_{\Upsilon}(\Upsilon|r_{0})$
$\displaystyle=\mathbb{P}\left(|h_{0}|^{2}<\frac{\Upsilon(\mathcal{B}+\sigma_{w}^{2})}{\mathcal{P}r_{0}^{-\eta}}\right)$
$\displaystyle=1-\mathbb{E}\left\\{e^{\frac{-\Upsilon\mathcal{B}}{\mathcal{P}r_{0}^{-\eta}}}\right\\}e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}$
$\displaystyle=1-\mathcal{L}_{{\mathcal{B}}}\left\\{\frac{\Upsilon}{\mathcal{P}r_{0}^{-\eta}}\right\\}e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}},$
using the Gamma approximation provided in (9). The PDF of the SINR $\Upsilon$
at a fixed $r_{0}$ is given by differentiating the CDF as follows
$\displaystyle
f(\Upsilon|r_{0})=\frac{e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}}{\mathcal{P}r_{0}^{-\eta}}\frac{(1+\frac{\theta
r_{0}^{\eta}\Upsilon}{\mathcal{P}})\sigma_{w}^{2}+q\theta}{(1+\frac{\theta
r_{0}^{\eta}\Upsilon}{\mathcal{P}})^{q+1}},$ (39)
this proves (14).
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|
mycases $\displaystyle{##}$$\displaystyle{##}$ {. ††thanks: These two authors
contributed equally
††thanks: These two authors contributed equally
# Fate of the non-Hermitian skin effect in many-body fermionic systems
Faisal Alsallom Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Loïc Herviou
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland Oleg V. Yazyev Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Marta Brzezińska<EMAIL_ADDRESS>Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
###### Abstract
We revisit the fate of the skin modes in many-body non-Hermitian fermionic
systems. Contrary to the single-particle case, the many-body ground state
cannot exhibit an exponential localization of all eigenstates due to the Pauli
exclusion principle. However, asymmetry can still exist in the density
profile, which can be quantified using the imbalance between the two halves of
the system. Using the non-Hermitian Su-Schrieffer-Heeger (SSH) chain as an
illustration, we show the existence of two distinct scaling regimes for the
imbalance. In the first one, the imbalance grows linearly with the system
size, as generically expected. In the second one, the imbalance saturates to a
finite value. By combining high-precision exact diagonalization calculations
and analytical arguments, we observe that the imbalance does not scale when
the occupied bands can be deformed to their Hermitian limit. This suggests a
direct connection between the corresponding bulk topological invariants and
the skin effect in many-body systems. Importantly, this relation also holds
for interacting systems.
Over the years, the interplay between non-Hermiticity and topology has
gathered an immense interest, both from experimental and theoretical
perspectives. Non-Hermitian (nH) Hamiltonians can serve as an effective
description of open systems, accurately modeling various non-conservative
classical and quantum platforms, including metamaterials Barton _et al._
(2018); Brandenbourger _et al._ (2019); Ghatak _et al._ (2020); Scheibner
_et al._ (2020); Zhou and Zhang (2020); Rao _et al._ (2021); Chen _et al._
(2021), optics and photonics Regensburger _et al._ (2012); Peng _et al._
(2014); Zeuner _et al._ (2015); Feng _et al._ (2017); Zhou _et al._ (2018);
Harari _et al._ (2018); Bandres _et al._ (2018); Pan _et al._ (2018); El-
Ganainy _et al._ (2018); Miri and Alù (2019), or electric circuits Ezawa
(2019a, b, c); Hofmann _et al._ (2020); Helbig _et al._ (2020). Within the
framework of topological band theory, non-Hermiticity gives rise to exotic
phenomena without Hermitian counterparts. This includes exceptional points
Heiss (2004); Leykam _et al._ (2017); Martinez Alvarez _et al._ (2018) –
degeneracies where eigenstates coalesce – and novel topological phases
characterized by a winding of the eigenspectrum in the complex plane Rudner
and Levitov (2009); Esaki _et al._ (2011); Lee (2016); Shen _et al._ (2018);
Liu _et al._ (2019); Zhou and Lee (2019); Kawabata _et al._ (2019); Zhang
_et al._ (2020a).
Another unique feature of nH systems is the anomalous localization of all
eigenstates dubbed the non-Hermitian skin effect Lee (2016); Yao and Wang
(2018); Lee _et al._ (2019); Lee and Thomale (2019); Borgnia _et al._
(2020). All single-particle eigenstates become exponentially localized at the
boundaries of the system. Generally, skin modes arise due to the breakdown of
reciprocity, i.e., introducing asymmetric hoppings. The conventional bulk-
boundary correspondence is then broken, obscuring the prediction of universal
boundary phenomena from the periodic bulk properties Xiong (2018); Kawabata
_et al._ (2018); Gong _et al._ (2018); Yao _et al._ (2018); Kunst _et al._
(2018); Herviou _et al._ (2019a); Jin and Song (2019); Song _et al._ (2019);
Yokomizo and Murakami (2019); Ozcakmakli Turker and Yuce (2019); Edvardsson
_et al._ (2019); Zirnstein _et al._ (2021). Indeed, the phase diagram of the
single particle Hamiltonian with periodic boundary conditions (PBC) can
significantly differ from the one with open boundary conditions (OBC). The
skin effect is directly linked to the structure of the single-particle
spectrum with PBC Okuma _et al._ (2020); Zhang _et al._ (2020a) . A non-zero
winding of the PBC energy bands around any point in the complex plane
guarantees that the related OBC modes are exponentially localized. The
corresponding OBC eigenenergies are (nearly all) located within these non-zero
winding regions. The existence of the skin effect was confirmed experimentally
Brandenbourger _et al._ (2019); Song _et al._ (2020); Scheibner _et al._
(2020); Helbig _et al._ (2020); Xiao _et al._ (2020); Zhang _et al._
(2021a); Liu _et al._ (2021). Recently, the notion of skin effect has been
expanded by introducing different types of skin states Yi and Yang (2020) such
as symmetry-enforced variants Rui _et al._ (2019); Okuma _et al._ (2020);
Yoshida _et al._ (2020), reciprocal skin effect Hofmann _et al._ (2020) in
two or higher dimensions, higher-order skin effect Kawabata _et al._ (2020a);
Okugawa _et al._ (2020); Fu _et al._ (2021); Palacios _et al._ (2021);
Zhang _et al._ (2021b); Zou _et al._ (2021), or the critical skin effect Li
_et al._ (2020); Yokomizo and Murakami (2021).
So far, studies of the skin effect have largely focused on the single-particle
Hamiltonian. Most of the experiments described by these effective nH
Hamiltonians are either weakly interacting or non-interacting bosonic systems
(e.g. quantum optics) or classical systems (e.g. the various types of circuits
and active media). Moreover, in Hermitian non-interacting systems, the
properties of the many-body states can be straightforwardly deduced from those
of the single-particle Hamiltonian; this is not the case for nH models. Due to
the non-orthogonality of the single-particle eigenmodes, the Pauli principle
no longer acts trivially. The fermionic repulsion reshapes the occupied
orbitals so that no more than one fermion (equivalently one hard-core boson)
populates each physical site. Hence, the exponential localization of all
fermions at a boundary becomes impossible Lee _et al._ (2020); Liu _et al._
(2020). Recently, several authors have investigated the fate of the skin modes
in the many-body context Mu _et al._ (2020); Lee _et al._ (2020); Shen and
Lee (2021); Zhang _et al._ (2021c); Yoshida (2021); Xi _et al._ (2021). For
instance, for interacting nH systems such as topological Mott insulators, the
skin effect was observed in fermionic systems Liu _et al._ (2020); Cao _et
al._ (2021), but not in the bosonic case Zhang _et al._ (2020b); Xu and Chen
(2020). Nonetheless, no general criterion for the survival of the skin effect
has been derived so far.
In this Letter, we investigate the connection between topological properties
of non-interacting nH systems and the skin effect in a many-body wave-
function. The exponential localization of the skin modes naturally translates
to an asymmetry of the many-body density profile that grows linearly with the
system size. Through numerical simulations, using high-precision exact
diagonalization (ED) and density matrix renormalization group (DMRG), we show
that the many-body eigenstates of OBC systems exhibit a transition from a
regime with an infinite asymmetry in the thermodynamic limit to one where it
saturates. The critical anisotropy corresponds to a change in the topology of
the _periodic_ single-particle spectrum, even though the open system remains
gapped at all times. In addition, we show that both scaling regimes survive
disorder and interactions.
Conventions and model: To exemplify our findings, we consider the nH chiral
SSH model Lieu (2018); Kunst _et al._ (2018):
$\mathcal{H}_{\mathrm{SSH}}\\{g\\}=-t_{1}\sum\limits_{j}\left(e^{g}c^{\dagger}_{j,B}c_{j,A}+e^{-g}c^{\dagger}_{j,A}c_{j,B}\right)\\\
-t_{2}\sum\limits_{j}\left(e^{g}c^{\dagger}_{j+1,A}c_{j,B}+e^{-g}c^{\dagger}_{j,B}c_{j+1,A}\right),$
(1)
where $c^{(\dagger)}$ are fermionic annihilation (creation) operators, $j$
denotes the lattice position, $A/B$ corresponds to the sublattice, and
$t_{1/2}$ is the hopping amplitude. $g\geq 0$ is a hopping anisotropy that
breaks Hermiticity, see Fig. 1(a). We consider three boundary conditions: PBC,
OBC (with $2L$ sites) or ABA-BC (OBC with $2L+1$ sites). ABA-BC has an
explicit analytical solution (see Appendix A and Ref. Kouachi, 2006). For PBC,
the single-particle gap closes for $|t_{1}e^{\pm 2g}|=|t_{2}|$, with two
phases adiabatically connected to the trivial and topological Hermitian
phases, and an intermediate phase where the system has no line gap. In all
cases, the single-particle spectrum has a non-trivial winding, and therefore
the nH skin effect is always present. Indeed, for OBC and ABA-BC, all single-
particle eigenstates are exponentially localized to the right side of the
system. The model defined in Eq. (1) is related to the Hermitian SSH chain via
the similarity transformation
$\mathcal{U}_{g}=\exp\left(g\sum\limits_{j}(2j-1)n_{j,A}+g\sum\limits_{j}2jn_{j,B}\right)$
(2)
such that
$\mathcal{H}_{\mathrm{SSH}}\\{g\\}=\mathcal{U}_{g}\,\mathcal{H}_{\mathrm{SSH}}\\{0\\}\,\mathcal{U}_{g}^{-1}.$
(3)
Note that the OBC and ABA-BC phase diagrams are therefore independent of $g$
with a gap closing at $t_{1}=t_{2}$, and their spectrum is always real.
In the following, we focus on the ground state properties of the OBC and ABA-
BC systems. We define the ground state as the right eigenstate that minimizes
the real part of the energy:
$\mathcal{H}_{\mathrm{SSH}}\\{g\\}\ket{\Psi\\{g\\}}=E_{\mathrm{GS}}\ket{\Psi\\{g\\}}\propto
E_{\mathrm{GS}}\,\mathcal{U}_{g}\ket{\Psi\\{0\\}}.$ (4)
Our results generalize straightforwardly to other right eigenstates and to the
left eigenvectors. The observables are taken to be:
$\braket{O}_{g}=\braket{\Psi\\{g\\}}{O}{\Psi\\{g\\}}.$ (5)
This expectation value is relevant when considering nH Hamiltonians obtained
from post-selection Dalibard _et al._ (1992); Mølmer _et al._ (1993). The
quantum state described by such an approach remains a proper Hermitian density
matrix. As $\mathcal{H}_{\mathrm{SSH}}\\{g\\}$ is non-interacting, observables
can be obtained efficiently from the diagonalization of the single-particle
Hamiltonian Herviou _et al._ (2019b) (see Appendix B). We numerically
diagonalize the Hermitian system with high precision before applying the
similarity transformation to obtain the eigenstates of the nH models.
Imbalance as a marker of the skin effect: As can be seen from the
transformation $\mathcal{U}_{g}$, all the single-particle orbitals are
exponentially localized at the right boundary of the chain. We expect the
asymmetry of the orbitals to transfer to the many-body density distribution
$\braket{n_{j}}_{g}=\braket{n_{j,A}+n_{j,B}}_{g}$. Due to the Pauli principle,
only one fermion can be located at a given site and therefore the many-body
version of the skin effect is not a sum of the exponential orbitals. To
quantify the degree of asymmetry of the density distribution, we use the
imbalance
$\mathcal{I}=\sum_{j\in\text{right
half}}\braket{n_{j}}_{g}-\sum_{j\in\text{left half}}\braket{n_{j}}_{g}.$ (6)
Working at fixed charge density
$N_{F}=\sum\limits_{j}\braket{n_{j}}_{g}/2L\leq 0.5$111$N_{F}>0.5$ can be
deduced from $N_{F}<0.5$, the imbalance can at most scale linearly with system
size. We consider this linear scaling to be the manifestation of the skin
effect in many-body systems. In Fig. 2, we show the scaling of the imbalance
in the limit $t_{1}=t_{2}$ where the SSH model maps to the Hatano-Nelson model
Hatano and Nelson (1996); Mu _et al._ (2020). We observe a linear scaling of
the imbalance at all fillings. At small $g$, the slope increases with the
anisotropy $g$, but saturates at a value equal to $N_{F}$. The large $g$ limit
corresponds to a situation where all particles are located in the right half
of the system. We note that the imbalance can be generalized to the higher
moments of the density with similar results.
Figure 1: (a) Pictorial representation of the non-Hermitian SSH model. (b)
Imbalance $\mathcal{I}$ as a function of the system size for $t_{1}/t_{2}=2$
and three representative values of $g$. At $g<g_{c}$, $\mathcal{I}$ saturates
to a finite value. At $g>g_{c}$, the scaling becomes linear. It also exhibits
a series of damped oscillations whose frequency diverges when $g\rightarrow
g_{c}^{+}$. The inset shows the corresponding PBC spectra in the complex
plane. Figure 2: (a) Scaling of the imbalance in the Hatano-Nelson model with
$t_{1}=t_{2}=1$ and $g=0.25$ at three different fillings $N_{F}$. Due to the
structure of the PBC spectrum (see inset), the band is always partially
occupied, which leads to a linear scaling with system size. (b) Slopes of the
imbalance as a function of $g$. For small values of $g$, the slope does not
depend on $N_{F}$. As $g$ is increased, all particles localize on the right
boundary.
Skin effect and band topology: We now turn to the general case where
$t_{1}\neq t_{2}$ and focus on half-filled systems ($L/(2L+1)$ filling for
ABA-BC). Fig. 1 summarizes our results for ABA-BC. Below and at the critical
value $g_{c}=0.5\,|\log t_{2}/t_{1}|$, the imbalance saturates with system
size. For $g<g_{c}$, the saturating value is reached exponentially fast. At
the critical point, the convergence becomes algebraic, with an exponent which
depends on the $t_{1}/t_{2}$ ratio and appears to peak at $t_{1}=t_{2}$ (see
Appendix C for more details). Above $g_{c}$, the imbalance scales linearly
with $L$, marking a fundamental anisotropy in the thermodynamic limit. We also
observe slowly decaying oscillations around the linear regime. The relative
amplitude of the oscillations, as well as their period, increases with
decreasing $g$. The limit $g\rightarrow g_{c}^{+}$ appears to correspond to a
divergence of the period of the oscillation, as well as a vanishing of the
linear term. Due to the instability of the non-normal Hamiltonians, or
alternatively the high amplitude of the coefficients in the similarity
transformation $\mathcal{U}_{g}$, we stress the importance of using high-
precision libraries to distinguish between finite-size Chen _et al._ (2019)
and finite-precision effects. The system sizes shown here require a precision
of up to several hundred of digits, hence the use of the ABA-BC where we can
bypass diagonalization of the single-particle Hamiltonian. The same results
were obtained for OBC.
Remarkably, the critical value $g_{c}$ for the _open_ system corresponds to
the gap closing of the _periodic_ system. The single-particle Hamiltonian with
OBC or ABA-BC remains gapped when crossing $g_{c}$222In fact the energy
spectrum is independent of $g$, and its eigenvectors vary smoothly. The abrupt
change in the imbalance (discontinuous in the thermodynamic limit) directly
arises from the Pauli principle.
To understand the source of the discontinuity, we summarize here how the many-
body state is obtained from the single-particle orbitals. Let us denote
$\\{\ket{R_{j}}\\}$ the right eigenvectors of the single-particle Hamiltonian,
and $c^{\dagger}_{R,j}=\vec{c}\,^{\dagger}\ket{R_{j}}$, the corresponding
orbitals. The eigenvectors of the many-body Hamiltonian are given by
$\ket{\Psi}\propto c^{\dagger}_{R,i_{1}}c^{\dagger}_{R,i_{2}}...\ket{0}.$ (7)
Due to the non-orthonormality of the $\\{\ket{R_{j}}\\}$’s, the orbitals do
not respect the conventional fermionic anticommutation relation and the right
side of Eq. (7) is not normalized. It is convenient to instead work with
$\\{\ket{Q_{i_{j}}}\\}$ — an orthonormal family generating the occupied
$\\{\ket{R_{i_{j}}}\\}$ — and the corresponding orbitals
$\\{q^{\dagger}_{i_{j}}\\}$. They can be obtained by Gram-Schmidt
orthonormalization. We then have
$\ket{\Psi}=q^{\dagger}_{i_{1}}q^{\dagger}_{i_{2}}...\ket{0}.$ (8)
The $q_{i_{j}}$ now obey standard anticommutation relations. Due to the
orthonormalization, the smooth deformation of the original eigenvectors when
$g$ crosses $g_{c}$ is amplified as the norm of the intermediate states can be
arbitrarily close to zero.
We now focus on the saturating regime, $g<g_{c}$. Note that the saturation of
the imbalance depends on the fine structure of the orbitals. Taking the
corresponding PBC Hermitian state instead leads to a linear scaling of the
imbalance after applying $\mathcal{U}_{g}$. Generalized Brillouin zone (GBZ)
approaches Yao and Wang (2018); Yokomizo and Murakami (2019); Yang _et al._
(2020); Kawabata _et al._ (2020b) give the same results. It is informative to
briefly study a simple example of saturation. Consider $L$ independent
orbitals
$\ket{R_{n}}=\sum\limits_{j=1}^{L}a_{j,n}\ket{j}\otimes\ket{A}+b_{j,n}\ket{j}\otimes\ket{B}$
(9)
such that $a_{j,n}=\alpha b_{j,n}$ for $1\leq n\leq L$. By dimensional
arguments, we obtain
$\displaystyle\text{Span}(\ket{R_{1}},...,\ket{R_{L}})=\text{Span}(\ket{1}\otimes(\ket{A}$
(10)
$\displaystyle+\alpha\ket{B}),...,\ket{L}\otimes(\ket{A}+\alpha\ket{B})).$
The many-body state with $L$ occupied orbitals can therefore be rewritten as
$\ket{\Psi}=\frac{1}{(\sqrt{1+\alpha^{2}})^{L}}\prod_{j}(c^{\dagger}_{j,A}+\alpha
c^{\dagger}_{j,B})\Ket{0}$ (11)
independently of the exact form of $a_{j,n}$333We only require them to form an
independent family.. This also holds when the orbitals are all exponentially
localized at a boundary; the imbalance is constant and equal to zero. As all
orbitals have the same local structures, the Pauli principle forces electrons
to spread out through the system.
The suppression of the scaling is observed only at half-filling in two-band
models. If the low-energy band is only partially occupied (resp. the high-
energy band is partially occupied), the imbalance grows linearly with system
size. This can be immediately deduced from the toy model in Eq. (9): if we do
not occupy the full translation-invariant subspace in Eq. (10), the imbalance
survives. In the nH-SSH model, for $g<g_{c}$, the periodic single-particle
spectrum has a line gap, while for $g>g_{c}$, it forms a single band with no
line gap and therefore its many-body spectrum is gapless. Let
$\\{\mathcal{E}_{j}^{\mathrm{PBC/OBC}}\\}$ be the set of energy bands with PBC
or OBC, respectively, and the bands are separated by line gaps. Following Ref.
Okuma _et al._ , 2020, the OBC and PBC bands can be deformed into each other
(merging several OBC bands to form a single one if required), up to some small
set of states. We propose the following conjecture:
###### Conjecture 1
For non-interacting systems exhibiting single-particle skin effect, if the
sets of orbitals occupied in a many-body state $\ket{\Psi}$ can be mapped to
fully occupied PBC bands (up to a number of orbitals of measure 0 in the
thermodynamic limit), the many-body skin effect is suppressed. Otherwise, the
imbalance in $\ket{\Psi}$ will scale linearly with system size.
Beyond the SSH model: disorder, interactions and symmetries: Our conjecture
implies that the many-body skin effect has a topological origin that can be
predicted from considering the periodic system. It is generally not the case
for topological properties of nH systems which can be obtained from a GBZ
approach or from the bulk of the OBC system.
To further show the resilience of the skin effect, we investigate the effect
of disorder Hatano and Nelson (1996); Goldsheid and Khoruzhenko (1998);
Yusipov _et al._ (2017); Tzortzakakis _et al._ (2020); Huang and Shklovskii
(2020); Weidemann _et al._ (2021). For single-particle states, skin effect
and disorder compete to localize states either at the boundaries or within the
bulk of the system. When the disorder-induced localization length is larger
than the one induced by the breaking of reciprocity ($g^{-1}$ in our example),
the single-particle states are Anderson-localized. They then occupy arbitrary
positions in the lattice which therefore limits the scaling of the imbalance.
On the other hand, the saturation is highly sensitive to the structure of the
orbitals and therefore its survival is unclear. In Fig. 3, we show the
imbalance in a disordered nH-SSH model with a random chemical potential
$-\sum\limits_{j}\mu_{j}^{A}c^{\dagger}_{j,A}c_{j,A}+\mu_{j}^{B}c^{\dagger}_{j,B}c_{j,B},\quad\mu_{j}^{\alpha}\in[-W,W]$
(12)
for several values of the disorder strength $W$. We consider two different
disorder configurations: (a) $\mu_{j}^{B}=-\mu_{j}^{A}$, acting as a local
chemical potential, or (b) $\mu_{j}^{B}$ and $\mu_{j}^{A}$ independent.
Saturated and linear regimes are resilient to the presence of small disorder
in both setups. For configuration (a), the disorder increases the critical
value of $g$ by opening a line gap between the bands of the corresponding PBC
models (see Fig. 3(c, d)). At small $W$, similar results are obtained for
configuration (b). However, at large $W$, the disorder closes the line gap at
(and below) $g_{c}$ and the imbalance scales linearly accordingly.
Figure 3: (a, b) Scaling of the imbalance for $t_{1}/t_{2}=2$, $g=g_{c}$ and
$g=g_{c}\pm 0.01$ for the two disorder configurations defined in Eq. (12) with
W = $0.1,0.5,2$, averaged over $10\,000$ disorder realizations. (c, d) Typical
PBC energy spectrum for the two disorder configurations at $g=g_{c}$ and (c)
$W=0.5$ or (d) $W=2$. The imbalance remains robust at small $W$ as the
disorder opens a line gap. For (b), at $W=2$, the line gap closes and the
imbalance grows linearly.
In addition, we investigate the effect of interactions in the two scaling
regimes. We use DMRG Hauschild and Pollmann (2018) to access larger system
sizes with OBC before applying the similarity transformation which is a matrix
product operator of bond dimension $1$. We are mainly limited by the floating
point precision in the DMRG. We add to the Hamiltonian in Eq. (1) the
following interaction:
$\displaystyle
H_{\mathrm{int}}=U\sum\limits_{j}\left(n_{j,A}-\frac{1}{2}\right)\left(n_{j,B}-\frac{1}{2}\right)$
(13)
$\displaystyle+\left(n_{j,B}-\frac{1}{2}\right)\left(n_{j+1,A}-\frac{1}{2}\right).$
For $U$ real, the system remains gaugeable to a Hermitian model by the
similarity transformation given in Eq. (2). In the limit $t_{1}=t_{2}$ (and
$g=0$), the model is equivalent to the XXZ chain without transverse field. In
Fig. 4, we show the scaling of the imbalance for (a) attractive and (b)
repulsive interactions. Both linear and saturated regimes survive the presence
of small interactions. For attractive interactions ($U<0$), fermions prefer to
occupy neighbouring orbitals, and therefore the critical $g_{c}$ decreases.
Conversely, for repulsive interactions ($U>0$), the interactions favor the
electrons spreading throughout the system, so $g_{c}$ increases. The notion of
bands is no longer well-defined in the presence of interactions, but systems
without a single-particle line gap have a gapless many-body spectrum. We
verify that the variation of the critical $g_{c}$ matches a shift of the gap
closing point. We use the response of the periodic system to flux insertion to
detect the presence of a gap given the limited system sizes we have access to.
Figure 4: Imbalance scaling in the presence of (a) attractive and (b)
repulsive interactions with $t_{1}/t_{2}=\sqrt{2}$. A smaller $t_{1}/t_{2}$
ratio was used to reach larger sizes. The lines correspond to the DMRG data,
while the superimposed points are the ED data up to 24 sites. Due to
oscillations with a period four, we show the results only for every forth
length. When $U<0$, the critical $g_{c}$ decreases and the slope of
$\mathcal{I}$ increases. It is in contrast to positive $U$.
In the Appendices, we show that our results are also valid when the system is
no longer gaugeable to a Hermitian limit or when we break particle-hole
symmetry by introducing a third band.
Conclusions: In this Letter, we have investigated the skin effect in many-
body wavefunctions. Generically, for fermionic systems, the single-particle
skin effect translates into a linear scaling of the imbalance. However, we
have numerically shown that if the orbitals occupied in an OBC many-body state
correspond to fully occupied PBC bands, the skin effect is suppressed in the
thermodynamic limit. Remarkably, this sharp transition occurs while the OBC
spectrum remains completely gapped. We have demonstrated that the suppression
of the skin effect, despite being sensitive to the details of the orbitals,
survives both the presence of disorder and interactions. This strongly implies
a topological origin of this phenomenon, similar to the topological nature of
the nH skin effect in single-particle Hamiltonians. An analytical proof of
this connection to topology is left for future work. Our results pave the way
to a better understanding of the skin effect in higher dimensions, with the
interplay between the dimensionality of the Fermi surface and other forms of
skin effect.
###### Acknowledgements.
The authors thank Jens Bardarson, Jérémy Bensadon, Tomáš Bzdušek, Frédéric
Mila, Titus Neupert, Nicolas Regnault, and Songbo Zhang for the useful
discussions. Exact diagonalization studies were performed with QuSpin Weinberg
and Bukov (2017, 2019) and DMRG calculations were done using TeNPy Library
(version 0.8.4) Hauschild and Pollmann (2018). To achieve an arbitrary
precision, we employed mpmath library Johansson _et al._ (2021). This work
was supported by a grant from the Swiss National Supercomputing Centre (CSCS)
under project s1008. F.A. acknowledges support from the KAUST Gifted Student
Program and the EPFL Research Internship Program.
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## Appendix A Analytical solution for ABA-BC nH-SSH model
The nH-SSH model with ABA-BC takes the form:
$H=\begin{pmatrix}0&t_{1}e^{-g}&0&0&\cdots&0\\\
t_{1}e^{g}&0&t_{2}e^{-g}&0&\cdots&0\\\ 0&t_{2}e^{g}&0&\ddots&\ddots&\vdots\\\
0&0&\ddots&\ddots&\ddots&0\\\
\vdots&\vdots&\ddots&\ddots&\ddots&t_{2}e^{-g}\\\
0&\cdots&\cdots&0&t_{2}e^{g}&0\end{pmatrix}.$ (14)
Following Ref. Kouachi, 2006, we can write explicitly its eigenvalues and
eigenstates. Note that due to the choice of boundary conditions, there is
always a zero-energy edge mode. The eigenvalues are given by:
$E_{n}=\begin{cases}\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos\theta_{n}},\,&n=1,\ldots,L\\\
-\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos\theta_{n}},\,&n=L+1,\ldots,2L\\\
0,\,&n=2L+1\end{cases}$ (15)
and the corresponding eigenstates $u_{j}^{(n)}$ are
$u_{j}^{(n)}=\rho_{j}\begin{cases}t_{1}t_{2}\sin\left(\left[\frac{2L+1-j}{2}+1\right]\theta_{n}\right)+t_{1}^{2}\sin\left(\frac{2L+1-j}{2}\theta_{n}\right),\\\
\qquad\qquad\qquad\qquad\qquad\qquad\text{if }j\text{ is odd}\\\
-\sqrt{t_{1}t_{2}}E_{n}\sin\left(\left[\frac{2L+1-j}{2}+\frac{1}{2}\right]\theta_{n}\right),\\\
\qquad\qquad\qquad\qquad\qquad\qquad\text{if }j\text{ is even}\end{cases}$
(16)
with
$\rho_{j}=\mycases(\sqrt{t_{1}t_{2}})^{2L}\,e^{g(j-1)},\,&j\text{ is odd}\\\
-(\sqrt{t_{1}t_{2}})^{2L}\sqrt{\frac{t_{1}}{t_{2}}}\,e^{g(j-1)},\,j\text{ is
even}$ (17)
and
$\theta_{n}=\begin{cases}\frac{n\pi}{L+1},\,&1\leq n\leq L\\\
\frac{(n-L)\pi}{L+1},\,&L+1\leq n\leq 2L.\end{cases}$ (18)
The zero energy eigenstate is then given by
$u_{j}^{(2L+1)}=\mycases\left(t_{1}t_{2}e^{2g}\right)^{(j-1)/2}\left(-t_{2}^{2}\right)^{(2L+1-j)/2},\,&j\text{
is odd}\\\ 0,\,j\text{ is even.}$ (19)
## Appendix B QR decomposition for right eigenvectors
We summarize here the main arguments of Ref. Herviou _et al._ , 2019b. We
focus on the properties of the right eigenstates of the many-body Hamiltonian.
All observables are taken as in Eq. (4). For non-interacting Hamiltonians, we
can determine the properties of the ground states (and all eigenstates) by
computing the correlation matrix Peschel (2003). The correlation matrix can be
obtained from the eigenvectors of the single-particle Hamiltonian. Let us
consider a Hamiltonian of the form:
$\mathcal{H}=\vec{c}\,^{\dagger}H\vec{c},$ (20)
with $\vec{c}=(c_{1},c_{2},...)$ the vector of annihilation operators whose
index run over all degrees of freedom. If $H$ is diagonalizable, with
$H=\sum\limits_{n}E_{n}\ket{\psi_{n}^{R}}\bra{\psi_{n}^{L}},$ (21)
then we define the operators:
$\vec{r}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\psi^{R}_{n}},\qquad\vec{l}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\psi^{L}_{n}}.$
(22)
They verify the biorthogonal fermionic anticommutation relations:
$\\{r_{m}^{\dagger},l_{n}\\}=\delta_{m,n},$ (23)
$\\{r_{m},l_{n}\\}=\\{r_{m},r_{n}\\}=\\{l_{m},l_{n}\\}=0.$ (24)
The many-body Hamiltonian $\mathcal{H}$ can be rewritten as
$\mathcal{H}=\sum\limits_{n}E_{n}r_{n}^{\dagger}l_{n}.$ (25)
With this expression, we see directly that its right eigenstates are
proportional to
$\ket{\Psi}\propto\prod\limits_{j}r^{\dagger}_{i_{j}}\ket{0}.$ (26)
Due to the non-orthogonality of the diagonalizing basis of nH operators, these
states are not normalized. Moreover, contrary to the Hermitian case,
observables cannot be directly computed from the expression above. Instead,
one needs to orthonormalize the family of occupied orbitals. If
$\\{\ket{\phi_{j}}\\}$ is an orthonormal basis of
$\mathrm{Span}(\\{\ket{\psi^{R}_{i_{j}}}\\}$, and we define
$\vec{q}_{n}^{\dagger}=\vec{c}\,^{\dagger}\ket{\phi_{n}},$ (27)
we see that the $q_{i_{j}}$’s are conventional fermionic operators. The
eigenstate in the form
$\ket{\Psi}=\prod\limits_{j}q^{\dagger}_{j}\ket{0}\propto\prod\limits_{j}r^{\dagger}_{i_{j}}\ket{0}$
(28)
is indeed normalized. $\ket{\Psi}$ is a conventional fermionic Gaussian state,
on which we can apply all the standard tricks.
## Appendix C Scaling analysis
As discussed in the main text, the imbalance as a function of the system size
exhibits intriguing features depending on the value of $g$. To underline the
universality of our results, we perform a scaling analysis for three $g$
regimes. Below $g_{c}$, the imbalance saturates to a finite and constant
value. Away from the dimerized limits ($t_{1}=0$ or $t_{2}=0$) and from the
gapless point $t_{1}=t_{2}$, the imbalance $\mathcal{I}$ can be well
approximated by an exponential $\mathcal{I}(L)=\mathcal{I}_{\infty}+\lambda
e^{-L/\xi}$, as shown in Fig. 5(a). At $g=g_{c}$, the correlations become
algebraic with an exponent that depends on $t_{1}/t_{2}$, as presented in Fig.
5(b).
Figure 5: Scaling of the imbalance as a function of the system size $L$. (a)
Below $g_{c}$, the imbalance saturates exponentially fast with a correlation
length that varies with both $g$ and $t_{1}/t_{2}$. (b) At $g_{c}$, the
convergence becomes algebraic with an exponent which depends on $t_{1}/t_{2}$.
Finally, in the $g>g_{c}$ regime, we observe oscillations of the imbalance
around the linear scaling (see Fig. 6(a)). To analyze these oscillations, we
study the derivative $d\mathcal{I}/dL$, which clearly shows distinct peaks as
demonstrated in Fig. 6(b). The amplitude of the oscillations slowly decays
with $L$, while their period appears to diverge as $(g-g_{c})^{-1}$: the
saturation can be seen as an infinitely long plateau (see Fig. 6(c)).
Figure 6: (a) Imbalance scaling for several $g>g_{c}$. Each curve exhibits a
sequence of plateaus, with a period depending on the exact value of $g$. To
quantify this dependence, we compute the numerical derivative
$d\mathcal{I}/dL$ and perform a spline interpolation to estimate the positions
of the peaks, shown in (b). The distance between peaks $T$ diverges as
$\sim(g-g_{c})^{-1}$.
## Appendix D Requirements for numerical precision
Apart from the errors originating from too small system sizes, i.e., finite
size effects, numerical instabilities in nH matrices pose additional
difficulties. 64-bits floating point precision becomes insufficient due to a
combination of large system sizes and large values of the reciprocity-breaking
terms. It is therefore necessary to systematically study how numerical
accuracy affects the results, as it is known that nH matrices are
exponentially unstable Krause (1994); Herviou _et al._ (2019a). In Fig. 7, we
compare the many-body ground state densities obtained from Gram-Schmidt
orthonormalization at different numerical precisions. At low precision (up to
$150$ digits), the local density close to the edge exceeds $1$, which is
mathematically incorrect for normalized fermionic states. The minimal number
of required decimal digits scales roughly as $gL$.
Figure 7: Many-body densities $|\Psi_{RR}|^{2}$ obtained from the analytical
solutions for ABA-BC with $g=g_{c}+0.1$ and $L=400$ ($801$ sites) at different
numerical accuracies in the Gram-Schmidt procedure. The numerical inaccuracies
lead to incorrect density profiles at the edges of the system. For the given
system size, at least 200 digits precision is required to faithfully represent
the density. A standard double type variable uses 64 bits to store the value,
which translates to around 16 decimal digits.
## Appendix E Non-gaugeable models
So far, we showed that the relation between occupied orbitals and the scaling
imbalance works for gaugeable models, i.e., models with real OBC spectrum on
which we can apply a local similarity transformation. The single-particle
Hamiltonian (represented in momentum space)
$H(k)=H_{\mathrm{SSH}}(k)+\begin{pmatrix}0&t_{p}e^{g_{p}}e^{2\mathrm{i}k}\\\
t_{p}e^{g_{p}}e^{-2\mathrm{i}k}&0\\\ \end{pmatrix}$ (29)
is not gaugeable when $g_{p}\neq 3g$. The bulk gap closes under the condition
$t_{p}=(t_{1}e^{\pm g}-t_{2}e^{\mp g})e^{\mp g_{p}}$. We verified that the
imbalance indeed changes scaling at the critical value above.
## Appendix F Three-band models
The existence of two distinct scaling regimes of $\mathcal{I}$ is not
restricted to two-band models. In this Appendix, we build a three-band model
that can have either three bands separated by line gaps, or two asymmetric
bands with a different number of orbitals. These regimes non-trivially break
particle-hole symmetry. To do so, we consider two elementary three-band
models. $H_{1}$ corresponds to a generalized nH-SSH model with three sites per
unit cell:
$H_{1}(k)=\begin{pmatrix}0&t_{1}e^{g}&t_{3}e^{-g}e^{-\mathrm{i}k}\\\
t_{1}e^{-g}&0&t_{2}e^{g}\\\
t_{3}e^{g}e^{\mathrm{i}k}&t_{2}e^{-g}&0\end{pmatrix}.$ (30)
$H_{2}$ is a conventional nH-SSH model to which we attach an extra decoupled
site within a unit cell:
$H_{2}(k)=\begin{pmatrix}\mu_{1}&\tilde{t}_{1}e^{g}+\tilde{t}_{2}e^{-2g}e^{-2\mathrm{i}k}&0\\\
\tilde{t}_{2}e^{2g}e^{2\mathrm{i}k}+\tilde{t}_{1}e^{-g}&\mu_{2}&0\\\
0&0&\mu_{3}\end{pmatrix}.$ (31)
We then study the interpolation $H=(1-\alpha)H_{1}+\alpha H_{2}$. The models
and their eigenspectra are represented in Fig. 8. Whether we consider $1/3$,
$1/2$ or $2/3$-filled ground states, the imbalance follows the conjecture in
the main text for all parameter values. In particular, we verified that the
transition from saturated to linear regime occurs when the system goes from
three separate bands (with PBC, $L$ orbitals per band) to two bands (one with
$2L$ orbitals and another one with $L$ orbitals) in the $1/3$-filled case.
Figure 8: (a) A sketch of real-space hoppings for the two three-band models,
$H_{1}$ and $H_{2}$, together with (b) their spectra. We set $t_{1}=t_{2}=1$,
$t_{3}=0.5$ for $H_{1}$ and $\tilde{t}_{1}=\tilde{t}_{2}=0.5$,
$\mu_{1}=\mu_{2}=-1$, $\mu_{3}=0.5$ for $H_{2}$; $g=0.2$. By linear
interpolation between $H_{1}$ and $H_{2}$, we obtain a model for which the
eigenvalues go from three circles to two.
## Appendix G Flux insertion in the nH periodic Hamiltonians
To check whether our interacting periodic systems have a many-body gap, we use
flux insertion. Due to the limited system sizes and the lack of similarity
transformation to map the systems back to Hermitian equivalents, scalings of
the gap are unreliable. Instead, we introduce a flux $\phi$ at the boundary of
the system, and study the evolution of the lowest energies when we tune $\phi$
from $0$ to $2\pi$. If the system is gapped, the lowest energy remains
detached from the upper one (and its trajectory is a closed contour in the
complex plane). In the gapless case, the trajectory merges with those
corresponding to higher energy and is no longer closed. In Fig. 9, we show the
trajectories of the two lowest energies in the model given in Eqs. (1) and
(13) with PBC for several values of $U$. The change of behavior (from gapped
to gapless, and vice-versa) matches the change in the scaling of the imbalance
(from saturated to linear regime).
Figure 9: Evolution under flux insertion of the ground energy (dots) and first
excited energy (crosses) in the model given in Eqs. (1) and (13) with PBC for
(a) $g<g_{c}$, (b) $g=g_{c}$, and (c) $g>g_{c}$. The computations were
performed for $U=-1$, $0$ and $1$ and at fixed system size $L=12$ ($24$
orbitals).
## Appendix H Many-body computation of the imbalance
In this Appendix, we briefly introduce some convenient formulas for computing
the imbalance and its variants in many-body systems. The imbalance
$\mathcal{I}$ can be expressed as:
$\mathcal{I}=\frac{\braket{\Psi(0)}{I\mathcal{U}_{g}^{2}}{\Psi(0)}}{\braket{\Psi(0)}{\mathcal{U}_{g}^{2}}{\Psi(0)}},$
(32)
where $\ket{\Psi(0)}$ is the Hermitian ground state. From this structure, it
is mathematically more convenient to work with
$\tilde{I}_{1}=\frac{1}{L}\sum\limits_{j}\left(2j-1n_{j,A}+2jn_{j,B}\right)-(2L-1).$
(33)
This is a minor generalization of the imbalance introduced in the main text,
with essentially the same properties. It allows us to write:
$\tilde{\mathcal{I}}_{1}=\braket{\tilde{I}_{1}}=\frac{1}{2gL}\partial_{g}\log\braket{\Psi(0)}{\exp(2gL\tilde{I}_{1})}{\Psi(0)}.$
(34)
$\tilde{\mathcal{I}}_{1}$ is therefore proportional to the derivative of the
cumulant generating function of $L\tilde{I}_{1}$. We numerically check that in
the non-interacting case, the probability distribution is
$p(\tilde{I}_{1}=i_{1})\propto\exp\left(-\beta|Li_{1}|\right)\exp\left(-\alpha
i_{1}^{2}\right)$ (35)
with
$\beta=|\log t_{1}/t_{2}|,$ (36)
which explains our results.
|
$r\leq|x-z|\leq\frac{\sqrt{5}}{2}r\leq\frac{\sqrt{5}}{4}|x-y|,\qquad(1-\frac{\sqrt{5}}{4})|x-y|\leq|y-z|\leq(1+\frac{\sqrt{5}}{4})|x-y|,$
(6.9)
we see that
$\displaystyle A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,z)$
$\displaystyle\asymp\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{r}\bigg{)}^{\beta_{1}}\bigg{(}1\wedge\frac{(x_{d}+r)\vee
t^{1/\alpha}}{r}\bigg{)}^{\beta_{2}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{((x_{d}+r)\vee
t^{1/\alpha})\wedge r}{(x_{d}\vee t^{1/\alpha})\wedge
r}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{r}{((x_{d}+r)\vee
t^{1/\alpha})\wedge r}\bigg{)}$
$\displaystyle\asymp\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{r}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{r}{(x_{d}\vee
t^{1/\alpha})\wedge r}\bigg{)}$ (6.10) $\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x(r))$
and
$\displaystyle A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,z,y)$
$\displaystyle\asymp\bigg{(}1\wedge\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\bigg{(}1\wedge\frac{x_{d}+r}{|x-y|}\bigg{)}^{\beta_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{x_{d}+r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{|x-y|}{x_{d}+r}\bigg{)}$
(6.11) $\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x(r),y).$
Thus, using the semigroup property, Proposition 6.2 and (6.9), we get
$\displaystyle p(t,x,y)\geq\int_{K}p(t/2,x,z)p(t/2,z,y)dz$ $\displaystyle\geq
c_{2}^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\int_{K}\frac{tA_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,z)}{|x-z|^{d+\alpha}}\,\frac{tA_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,z,y)}{|y-z|^{d+\alpha}}dz$
$\displaystyle\geq
c_{3}t^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}$
$\displaystyle\quad\times\int_{(y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}\frac{A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x(r))}{r^{d+\alpha}}\frac{A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x(r),y)}{|x-y|^{d+\alpha}}\int_{K(r)}d\widetilde{z}dr$
$\displaystyle\geq
c_{4}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\frac{t}{|x-y|^{d+\alpha}}$
$\displaystyle\quad\times t\int_{(y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x(r))\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x(r),y)\frac{dr}{r^{\alpha+1}}.$
$\Box$
Note that, for any $x,y\in{\mathbb{R}}^{d}_{+}$ and $|x-y|/4\leq
r\leq|x-y|/2$,
$\displaystyle
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})$
$\displaystyle\asymp\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{(x_{d}\vee
t^{1/\alpha})\wedge|x-y|}\bigg{)}$ $\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+\frac{|x-y|}{2}{\mathbf{e}}_{d})$
(6.12)
and, since $y_{d}\leq x_{d}+4r$,
$\displaystyle
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)$
$\displaystyle\asymp\bigg{(}1\wedge\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{(y_{d}\vee
t^{1/\alpha})\wedge|x-y|}\bigg{)}$ $\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+\frac{|x-y|}{2}{\mathbf{e}}_{d},y)$
$\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,y,y+\frac{|x-y|}{2}{\mathbf{e}}_{d}).$
(6.13)
In particular, we have
$\displaystyle
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+\frac{|x-y|}{2}{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+\frac{|x-y|}{2}{\mathbf{e}}_{d},y)$
$\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+\frac{|x-y|}{2}{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,y,y+\frac{|x-y|}{2}{\mathbf{e}}_{d})$
$\displaystyle\asymp
A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$ (6.14)
By (6)–(6), we have that for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle\int_{|x-y|/4}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+\frac{|x-y|}{2}{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+\frac{|x-y|}{2}{\mathbf{e}}_{d},y)\int_{|x-y|/4}^{|x-y|/2}\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp|x-y|^{-\alpha}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$ (6.15)
###### Remark 6.5.
Here we give a proof of the comparability of (1.6) and (1.7). Let $t>0$ and
$x,y\in{\overline{\mathbb{R}}}^{d}_{+}$ be such that $|x-y|>6t^{1/\alpha}$.
For any $z\in B(x+2^{-1}|x-y|{\mathbf{e}}_{d},4^{-1}|x-y|)$, by using the
triangle inequality several times, we have
$\displaystyle z_{d}-t^{1/\alpha}\asymp z_{d}\asymp x_{d}\wedge
y_{d}+|x-y|\asymp x_{d}\vee y_{d}+|x-y|$ (6.16)
and
$\displaystyle|x+t^{1/\alpha}{\mathbf{e}}_{d}-z|\asymp|y+t^{1/\alpha}{\mathbf{e}}_{d}-z|\asymp|x-z|\asymp|y-z|\asymp|x-y|.$
(6.17)
For any $t>0$ and $x,y\in{\overline{\mathbb{R}}}^{d}_{+}$ with
$|x-y|>6t^{1/\alpha}$, if $x_{d}\wedge y_{d}\geq|x-y|/4$, then by (A3), (6.3),
(6.4), (6.16) and (6.17),
$\displaystyle
t|x-y|^{d+\alpha}\int_{B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)}J(x+t^{1/\alpha}{\mathbf{e}}_{d},z)\,J(z,y+t^{1/\alpha}{\mathbf{e}}_{d})dz$
$\displaystyle\asymp\frac{t}{|x-y|^{d+\alpha}}\int_{B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)}dz\asymp\frac{t}{|x-y|^{\alpha}}$
$\displaystyle\asymp\bigg{(}\\!1\wedge\frac{t}{|x-y|^{\alpha}}\\!\bigg{)}B_{\beta_{1},\beta_{1},0,\beta_{3}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\\!\bigg{(}\\!e\\!+\\!\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)},$
Otherwise, if $x_{d}\wedge y_{d}<|x-y|/4$, then $x_{d}\vee y_{d}\leq
x_{d}\wedge y_{d}+|x-y|<5|x-y|/4$ so that
$z_{d}-t^{1/\alpha}\asymp z_{d}\asymp|x-y|\quad\text{for $z\in
B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)$}$
by (6.16). Using this, (6.17) and (6.3), we get that for $z\in
B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)$,
$\displaystyle
B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},z)\asymp
B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},z+t^{1/\alpha}{\mathbf{e}}_{d})$
$\displaystyle\asymp A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,z)\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+2^{-1}|x-y|{\mathbf{e}}_{d})$
(6.18)
and
$\displaystyle
B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(z,y+t^{1/\alpha}{\mathbf{e}}_{d})\asymp
B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(z+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})$
$\displaystyle\asymp A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,z,z)\asymp
A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+2^{-1}|x-y|{\mathbf{e}}_{d},y).$
(6.19)
By (A3), (6.17), (6.5), (6.5) and (6), we arrive at
$\displaystyle
t|x-y|^{d+\alpha}\int_{B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)}J(x+t^{1/\alpha}{\mathbf{e}}_{d},z)\,J(z,y+t^{1/\alpha}{\mathbf{e}}_{d})dz$
$\displaystyle\asymp\frac{tA_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+2^{-1}|x-y|{\mathbf{e}}_{d})A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+2^{-1}|x-y|{\mathbf{e}}_{d},y)}{|x-y|^{d+\alpha}}\int_{B(x+2^{-1}|x-y|{\mathbf{e}}_{d},\,4^{-1}|x-y|)}\\!\\!\\!\\!\\!\\!\\!\\!dz$
$\displaystyle\asymp\frac{t}{|x-y|^{\alpha}}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}$
$\displaystyle\asymp\bigg{(}\\!1\wedge\frac{t}{|x-y|^{\alpha}}\\!\bigg{)}B_{\beta_{1},\beta_{1},0,\beta_{3}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\\!\bigg{(}\\!e\\!+\\!\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)}.$
Hence, (1.6) and (1.7) are comparable.
Combining (6), Proposition 6.2 and Lemma 6.4, we get the following result.
###### Proposition 6.6.
There exists a constant $C>0$ such that for all $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle p(t,x,y)$ $\displaystyle\geq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$
$\displaystyle\quad+\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+\frac{|x-y|}{2}{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+\frac{|x-y|}{2}{\mathbf{e}}_{d},y)\bigg{]}$
$\displaystyle\asymp\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$
$\displaystyle\quad+\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}\bigg{]}.$
###### Remark 6.7.
If $\beta_{2}<\alpha+\beta_{1}$, then there is a constant $C>0$ such that for
all $t>0$ and $x,y\in{\mathbb{R}}^{d}$,
$\displaystyle\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}\leq
CA_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y).$
Indeed, for ${\varepsilon}:=(\alpha+\beta_{1}-\beta_{2})/2>0$, using (10.1),
we get that for all $t>0$ and $x,y\in{\mathbb{R}}^{d}$,
$\displaystyle\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}$ $\displaystyle\leq
c_{1}\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}-{\varepsilon}}\bigg{(}1\wedge\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}-{\varepsilon}}$
$\displaystyle=c_{1}\bigg{(}1\wedge\frac{t^{1/\alpha}}{|x-y|}\bigg{)}^{{\varepsilon}}\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}-{\varepsilon}}\bigg{(}1\wedge\frac{t^{1/\alpha}}{|x-y|}\bigg{)}^{-\beta_{1}+\beta_{2}+{\varepsilon}}\bigg{(}1\wedge\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}-{\varepsilon}}$ $\displaystyle\leq
c_{1}\bigg{(}1\wedge\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\bigg{(}1\wedge\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{2}}\leq
c_{1}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y).$
Therefore, in view of Proposition 6.2, Proposition 6.6 is relevant only if
$\beta_{2}\geq\alpha+\beta_{1}$.
###### Lemma 6.8.
Suppose that $\beta_{2}=\alpha+\beta_{1}$. Then for all $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle\big{(}|x-y|^{\alpha}\wedge t\big{)}\int_{(x_{d}\vee y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}+\beta_{4}+1}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$
Proof. Without loss of generality, we assume $x_{d}\leq y_{d}$.
Assume first that $y_{d}\vee t^{1/\alpha}<|x-y|/4$. Using
$\beta_{2}=\alpha+\beta_{1}$ and (6)–(6), since $x_{d}\leq y_{d}\vee
t^{1/\alpha}$, we get
$\displaystyle\int_{y_{d}\vee
t^{1/\alpha}}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp\int_{y_{d}\vee
t^{1/\alpha}}^{|x-y|/2}\bigg{(}\frac{x_{d}\vee
t^{1/\alpha}}{r}\bigg{)}^{\beta_{1}}\bigg{(}\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\bigg{(}\frac{r}{|x-y|}\bigg{)}^{\alpha+\beta_{1}}$
$\displaystyle\qquad\times\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{|x-y|}{r}\bigg{)}\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp\bigg{(}\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\bigg{(}\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}|x-y|^{-\alpha}$
$\displaystyle\qquad\times\int_{y_{d}\vee
t^{1/\alpha}}^{|x-y|/2}\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{|x-y|}{r}\bigg{)}\frac{dr}{r}.$
(6.20)
By a change of the variables and Lemma 10.11, we see that
$\displaystyle\int_{y_{d}\vee
t^{1/\alpha}}^{|x-y|/2}\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{|x-y|}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle=\int_{4(y_{d}\vee
t^{1/\alpha})/|x-y|}^{2}\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|s}{4(x_{d}\vee
t^{1/\alpha})}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|s}{4(y_{d}\vee
t^{1/\alpha})}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{4}{s}\bigg{)}\frac{ds}{s}$
$\displaystyle\asymp\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{4(x_{d}\vee
t^{1/\alpha})}\bigg{)}\log^{\beta_{3}+\beta_{4}+1}\bigg{(}e+\frac{|x-y|}{4(y_{d}\vee
t^{1/\alpha})}\bigg{)}.$ (6.21)
By (6)–(6), since $y_{d}\vee t^{1/\alpha}<|x-y|/4$, we obtain
$\displaystyle\big{(}|x-y|^{\alpha}\wedge t\big{)}\int_{y_{d}\vee
t^{1/\alpha}}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}\bigg{(}\frac{x_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}\bigg{(}\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\bigg{)}^{\beta_{1}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{4(x_{d}\vee
t^{1/\alpha})}\bigg{)}\log^{\beta_{3}+\beta_{4}+1}\bigg{(}e+\frac{|x-y|}{4(y_{d}\vee
t^{1/\alpha})}\bigg{)}.$ (6.22)
If $y_{d}\vee t^{1/\alpha}\geq|x-y|/4$, then
$\log\big{(}e+\frac{|x-y|}{(y_{d}\vee t^{1/\alpha})\wedge|x-y|}\big{)}\asymp
1$. Thus, by (6),
$\displaystyle\big{(}|x-y|^{\alpha}\wedge
t\big{)}\int_{|x-y|/4}^{|x-y|/2}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}$
$\displaystyle\asymp\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}+\beta_{4}+1}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{(x_{d}\vee
t^{1/\alpha})\wedge|x-y|}\bigg{)}.$ (6.23)
The proof is now complete. $\Box$
Combining Proposition 6.2, Lemma 6.4 and Lemma 6.8. we get the following
###### Proposition 6.9.
Suppose that $\beta_{2}=\alpha+\beta_{1}$. There exists a constant $C>0$ such
that for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle p(t,x,y)$ $\displaystyle\geq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$
$\displaystyle\quad+\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}A_{\beta_{1},\beta_{1},0,\beta_{3}+\beta_{4}+1}(t,x,y)\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}\bigg{]}.$
###### Remark 6.10.
In this section, for $\overline{Y}$, (A3)(II) is only used to get (IUBS).
Therefore, the results of Propositions 6.6 and 6.9 hold under weaker
assumptions (A1), (A3)(I), (A4) and (IUBS).
## 7\. Sharp upper bounds of heat kernels
In this section we prove the sharp upper bounds of heat kernels. The key
results are Theorem 7.5 and its Corollary 7.9 which deals with the case
$\beta_{2}<\alpha+\beta_{1}$, and Theorem 7.10 which deals with the case
$\beta_{2}\geq\alpha+\beta_{1}$.
Recall that
$U(r)=\\{x=(\widetilde{x},x_{d})\in{\mathbb{R}}^{d}:\,|\widetilde{x}|<r/2,\,0\leq
x_{d}<r/2\\}$ for $r>0$, $d\geq 2$ and $U(r)=[0,r/2)$ for $d=1$, and that the
function $A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$ is defined by
(6). We also recall that we use $p(t,x,y)$ for both $\overline{p}(t,x,y)$ and
$p^{\kappa}(t,x,y)$. We remind the readers that, for an open set
$D\subset{\overline{\mathbb{R}}}^{d}_{+}$ relative to the topology on
${\overline{\mathbb{R}}}^{d}_{+}$,
$\tau_{D}=\overline{\tau}_{D}=\inf\\{t>0:\overline{Y}_{t}\notin D\\}$ when we
consider $\overline{Y}$, and
$\tau_{D}=\tau^{\kappa}_{D}=\inf\\{t>0:Y^{\kappa}_{t}\notin
D\cap{\mathbb{R}}^{d}_{+}\\}$ when we consider $Y^{\kappa}$.
###### Lemma 7.1.
Let $b_{1},b_{3}\geq 0$ be constants with $b_{1}>0$ if $b_{3}>0$. Suppose that
there exists a constant $C_{0}>0$ such that for all $t>0$ and
$z,y\in{\mathbb{R}}^{d}_{+}$,
$p(t,z,y)\leq
C_{0}\left(1\wedge\frac{z_{d}}{t^{1/\alpha}}\right)^{q}A_{b_{1},0,b_{3},0}(t,z,y)\left(t^{-d/\alpha}\wedge\frac{t}{|z-y|^{d+\alpha}}\right).$
(7.1)
Then there exists a constant $C=C(C_{0})>0$ such that for all $t>0$ and
$x=(\widetilde{0},x_{d})\in{\mathbb{R}}^{d}_{+}$ with $x_{d}\leq 2^{-5}$,
${\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)\leq
Ct\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.2)
Proof. We first note that (7.1) implies that for all $t>0$ and
$y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle{\mathbb{P}}_{y}(|Y_{t}-y|>2^{-3},\,t<\zeta)=\int_{z\in{\mathbb{R}}^{d}_{+},\,|z-y|>2^{-3}}p(t,y,z)dz$
$\displaystyle\leq
c_{1}t\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\int_{z\in{\mathbb{R}}^{d}_{+},\,|z-y|>2^{-3}}\bigg{(}\frac{y_{d}\vee
t^{1/\alpha}}{|z-y|}\wedge
1\bigg{)}^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{|z-y|}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\frac{dz}{|z-y|^{d+\alpha}}$ $\displaystyle\leq
c_{2}t\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(y_{d}\vee
t^{1/\alpha})^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{2^{-3}}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\int_{z\in{\mathbb{R}}^{d}_{+},\,|z-y|>2^{-3}}\frac{dz}{|z-y|^{d+\alpha/2+b_{1}}}$
$\displaystyle\leq
c_{3}t\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(y_{d}\vee
t^{1/\alpha})^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)},$ (7.3)
where in the first inequality above we used (7.1) and Lemma 10.2, and in the
second we used (10.2).
By Proposition 2.7 (see also Remark 3.12), we have
$\sup_{s\leq t,\,y\in{\mathbb{R}}^{d}_{+}}{\mathbb{P}}_{y}\big{(}|Y_{s}-y|\geq
2^{-2},\,s<\zeta\big{)}\leq c_{1}\sup_{s\leq
t,\,y\in{\mathbb{R}}^{d}_{+}}\int_{z\in{\mathbb{R}}^{d}_{+},\,|z-y|\geq
2^{-2}}\frac{s}{|z-y|^{d+\alpha}}dz\leq c_{2}t.$ (7.4)
If $t\geq 1/(2c_{2})$, then (7.2) follows from Corollary 5.2.
Let $t<1/(2c_{2})$. By the strong Markov property and (7.4), we have
$\displaystyle{\mathbb{P}}_{x}\big{(}\tau_{U(1)}<t<\zeta,\,|Y_{t}-Y_{\tau_{U(1)}}|\geq
2^{-2}\big{)}={\mathbb{E}}_{x}\left[{\mathbb{P}}_{Y_{\tau_{U(1)}}}\left(|Y_{t-\tau_{U(1)}}-Y_{0}|\geq
2^{-2}\right):\tau_{U(1)}<t<\zeta\right]$
$\displaystyle\leq{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)\sup_{s\leq
t,\,y\in{\mathbb{R}}^{d}_{+}}{\mathbb{P}}_{y}(|Y_{s}-y|\geq
2^{-2},\,s<\zeta)\leq 2^{-1}{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta).$
Thus,
$\displaystyle{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)$
$\displaystyle=2\big{(}{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)-2^{-1}{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)\big{)}$
$\displaystyle\leq
2\big{(}{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)-{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta,\,|Y_{t}-Y_{\tau_{U(1)}}|\geq
2^{-2})\big{)}$
$\displaystyle=2{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta,\,|Y_{t}-Y_{\tau_{U(1)}}|<2^{-2}).$
(7.5)
Note that by the triangle inequality, for any
$y\in{\mathbb{R}}^{d}_{+}\setminus U(1)$ and $z\in B(y,2^{-2})$, we have
$|z-x|\geq|y-x|-|y-z|>15/32-1/4>2^{-3}$. Therefore using (7) and (7), we have
$\displaystyle{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta)\leq
2{\mathbb{P}}_{x}(\tau_{U(1)}<t<\zeta,\,|Y_{t}-Y_{\tau_{U(1)}}|<2^{-2})$
$\displaystyle\leq 2{\mathbb{P}}_{x}(|Y_{t}-x|>2^{-3},\,t<\zeta)\leq
c_{4}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}.$
The proof is complete. $\Box$
Note that for any $t,k,r>0$ and $a\geq 0$,
$\bigg{(}1\wedge\frac{r}{t^{1/\alpha}}\bigg{)}^{a}(r\vee
t^{1/\alpha})^{k}=r^{k}\bigg{(}1\wedge\frac{r}{t^{1/\alpha}}\bigg{)}^{a-k}.$
(7.6)
###### Lemma 7.2.
There exists a constant $C>0$ such that for all $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,
$p(t,x,y)\leq C\left(1\wedge\frac{x_{d}\wedge
y_{d}}{t^{1/\alpha}}\right)^{q}A_{\beta_{1},0,\beta_{3},0}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$
Proof. By Proposition 5.1, the lemma holds for $\beta_{1}=0$.
We assume $\beta_{1}>0$ and set $a_{n}=\beta_{1}\wedge\frac{n\alpha}{2}$ for
$n\geq 0$. Below, we prove by induction that for any $n\geq 0$, there exists a
constant $C>0$ such that for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle p(t,x,y)\leq C\left(1\wedge\frac{x_{d}\wedge
y_{d}}{t^{1/\alpha}}\right)^{q}A_{a_{n},0,\beta_{3},0}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).$
(7.7)
The lemma is a direct consequence of (7.7).
By Proposition 5.1 and the fact that the logarithmic term in
$A_{\beta_{1},0,\beta_{3},0}(t,x,y)$ is always larger than 1, (7.7) holds for
$n=0$. Suppose (7.7) holds for $n-1$. By symmetry and (3.30), we can assume
without loss of generality that $x_{d}\leq y_{d}$, $\widetilde{x}=0$ and
$|x-y|=5$. If $t>1$ or $x_{d}>2^{-5}$, then (7.7) follows from Proposition 5.1
and (6.4).
Let $t\leq 1$ and $x_{d}\leq 2^{-5}$. Then $y_{d}\leq x_{d}+|x-y|\leq
4+2^{-5}$ by the triangle inequality. Our goal is to show that there exists a
constant $c_{1}>0$ independent of $t,x,y$ such that
$p(t,x,y)\leq
c_{1}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)}.$ (7.8)
Set $V_{1}=U(1)$, $V_{3}=B(y,2)\cap{\overline{\mathbb{R}}}^{d}_{+}$ and
$V_{2}={\overline{\mathbb{R}}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. Similarly
to (5.10) and (5.11), we get from Proposition 2.7 and the triangle inequality
that
$\sup_{s\leq t,\,z\in V_{2}}p(s,z,y)\leq c_{2}\sup_{s\leq
t,\,z\in{\mathbb{R}}^{d}_{+},|z-y|\geq 2}\frac{s}{|z-y|^{d+\alpha}}\leq
2^{-d-\alpha}c_{2}t$ (7.9)
and
${\rm dist}(V_{1},V_{3})\geq\sup_{u\in V_{1},\,w\in V_{3}}(4-|x-u|-|y-w|)\geq
1.$ (7.10)
By the induction hypothesis, condition (7.1) in Lemma 7.1 holds with
$b_{1}=a_{n-1}$ and $b_{3}=\beta_{3}$. Thus, since
$a_{n}-a_{n-1}\leq\alpha/2$, we get from Lemma 7.1 and (7.9) that
$\displaystyle{\mathbb{P}}_{x}(\tau_{V_{1}}<t<\zeta)\sup_{s\leq t,\,z\in
V_{2}}p(s,z,y)$ $\displaystyle\leq
c_{3}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{a_{n-1}}(t^{1/\alpha})^{\alpha/2}t^{1/2}\log^{\beta_{3}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}\,$
$\displaystyle\leq
c_{3}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{a_{n-1}}(t^{1/\alpha})^{a_{n}-a_{n-1}}\,\left(\sup_{s\leq
1}s^{1/2}\log^{\beta_{3}}\bigg{(}e+\frac{1}{s^{1/\alpha}}\bigg{)}\right)$
$\displaystyle\leq
c_{4}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{a_{n}}.$ (7.11)
In order to apply Lemma 3.15 and get the desired result, it remains to bound
$\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{V_{1}}(t-s,x,u){\cal
B}(u,w)p(s,y,w)dudwds$. We consider the following two cases separately.
(Case 1) $q\geq\alpha+a_{n}$ and $10x_{d}<t^{1/\alpha}$.
Pick ${\varepsilon}\in(0,\beta_{1})$ such that
$q<\alpha+\beta_{1}-{\varepsilon}$. Using (A3)(II), Lemmas 10.1(i)–(ii), 10.2
(see Remark 10.3), and (7.10), we see that for all $u\in V_{1}$ and $w\in
V_{3}$,
$\displaystyle{\cal B}(u,w)\leq
c_{5}B_{\beta_{1}-{\varepsilon},0,0,0}(u,w)\leq
c_{6}\left(\frac{u_{d}}{|u-w|}\right)^{\beta_{1}-{\varepsilon}}\leq
c_{6}u_{d}^{\beta_{1}-{\varepsilon}}.$ (7.12)
By (7.12) and Lemma 5.12, we have
$\displaystyle\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{V_{1}}(t-s,x,u){\cal
B}(u,w)p(s,y,w)dudwds$ $\displaystyle\leq
c_{6}\int_{0}^{t}\int_{V_{1}}p^{V_{1}}(t-s,x,u)u_{d}^{\beta_{1}-{\varepsilon}}du\int_{V_{3}}p(s,y,w)dwds\leq
c_{6}\int_{0}^{\infty}\int_{V_{1}}p^{V_{1}}(s,x,u)u_{d}^{\beta_{1}-{\varepsilon}}duds$
$\displaystyle\leq
c_{7}x_{d}^{q}=c_{7}t\left(\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(t^{1/\alpha})^{q-\alpha}\leq
c_{8}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{a_{n}}.$ (7.13)
In the last inequality above, we used the facts that $q-\alpha\geq a_{n}$,
$10x_{d}<t^{1/\alpha}$ and $x_{d}\vee t^{1/\alpha}\leq 1$. Now, using Lemma
3.15, (7) and (7), we get (7.8) in this case.
(Case 2) $q<\alpha+a_{n}$ or $10x_{d}\geq t^{1/\alpha}$.
By (A3)(II), (7.10), Lemma 10.1(ii) and Lemma 10.2 (see Remark 10.3), it holds
that for all $u\in V_{1}$ and $w\in V_{3}$,
$\displaystyle{\cal B}(u,w)$ $\displaystyle\leq
c_{9}B_{\beta_{1},0,\beta_{3},0}(u,w)\leq
c_{10}\left(\frac{u_{d}}{|u-w|}\right)^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}\wedge|u-w|}{u_{d}\wedge|u-w|}\bigg{)}$
$\displaystyle\leq
c_{10}u_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}.$
(7.14)
By (10.2) (or using that $u\mapsto u^{\beta_{1}}\log^{\beta_{3}}(e+t/u)$ is
almost increasing) and Corollary 5.2, since $\beta_{1}>0$ and
$a_{n}\leq\beta_{1}$, we get that for any $0<s<t$ and $w\in V_{3}$,
$\displaystyle\int_{u\in
V_{1}:u_{d}<x_{d}}p(s,x,u)u_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}du\leq\int_{u\in
V_{1}:u_{d}<x_{d}\vee
s^{1/\alpha}}p(s,x,u)u_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}du$
$\displaystyle\leq c_{11}(x_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\int_{u\in V_{1}:u_{d}<x_{d}\vee s^{1/\alpha}}p(s,x,u)du$
$\displaystyle\leq c_{11}{\mathbb{P}}_{x}(\zeta>s)(x_{d}\vee
s^{1/\alpha})^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
s^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{12}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q}(x_{d}\vee
s^{1/\alpha})^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
s^{1/\alpha}}\bigg{)}.$ (7.15)
Next, using the induction hypothesis and Lemma 10.10, since
$a_{n}\leq\beta_{1}$ and $a_{n}<\alpha+a_{n-1}$, we get that for any $0<s<t$
and $w\in V_{3}$,
$\displaystyle\int_{u\in V_{1}:u_{d}\geq
x_{d}}p(s,x,u)u_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}du$
$\displaystyle\leq
c_{13}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q}\int_{u\in
V_{1}:u_{d}\geq x_{d}}\bigg{(}\frac{x_{d}\vee s^{1/\alpha}}{|x-u|}\wedge
1\bigg{)}^{a_{n-1}}\log^{\beta_{3}}\bigg{(}e+\frac{|x-u|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-u|}\bigg{)}$ $\displaystyle\hskip
156.49014pt\times\left(s^{-d/\alpha}\wedge\frac{s}{|x-u|^{d+\alpha}}\right)u_{d}^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}du$
$\displaystyle\leq
c_{14}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q}(x_{d}\vee
s^{1/\alpha})^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
s^{1/\alpha}}\bigg{)}.$ (7.16)
Similarly, again splitting the integration into two parts $w_{d}<y_{d}$ and
$w_{d}\geq y_{d}$, and using the induction hypothesis and Lemma 10.10 again,
we also get that for any $0<s<t$,
$\displaystyle\int_{V_{3}}p(s,y,w)\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}dw$
$\displaystyle\leq
c_{15}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{15}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}.$ (7.17)
By (7)–(7) and (7.6), we have
$\displaystyle\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{V_{1}}(t-s,x,u){\cal
B}(u,w)p(s,y,w)dudwds$ $\displaystyle\leq
c_{10}\int_{0}^{t}\int_{V_{3}}p(s,y,w)\int_{V_{1}}p(t-s,x,u)u_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}dudwds$
$\displaystyle\leq
c_{16}\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q}(x_{d}\vee(t-s)^{1/\alpha})^{a_{n}}\int_{V_{3}}p(s,y,w)\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}dwds$
$\displaystyle\leq
c_{17}x_{d}^{a_{n}}\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}ds=:I.$ (7.18)
When $q<\alpha+a_{n}$, we get from Lemma 10.4 that
$\displaystyle I\leq
c_{18}tx_{d}^{a_{n}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)}.$ (7.19)
When $10x_{d}\geq t^{1/\alpha}$, we also get from Lemma 10.4 that
$\displaystyle I$ $\displaystyle\leq
c_{17}x_{d}^{a_{n}}\int_{0}^{t}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}ds\leq
c_{19}tx_{d}^{a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
10^{q-a_{n}}c_{19}tx_{d}^{a_{n}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-a_{n}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)}.$ (7.20)
Now, (7.8) follows from (7.6), (7), (7), (7) and Lemma 3.15. The proof is
complete. $\Box$
Now we use Lemma 7.2 to improve the bound in (7).
###### Lemma 7.3.
Let $0<t\leq 1$ and $x,y\in{\mathbb{R}}^{d}_{+}$ be such that
$\widetilde{x}=\widetilde{0}$, $x_{d}\leq 2^{-5}$ and $|x-y|=5$. Set
$V_{1}=U(1)$, $V_{3}=B(y,2)\cap{\overline{\mathbb{R}}}^{d}_{+}$ and
$V_{2}={\overline{\mathbb{R}}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. There
exists a constant $C>0$ independent of $t,x$ and $y$ such that
$\displaystyle{\mathbb{P}}_{x}(\tau_{V_{1}}<t<\zeta)\sup_{s\leq t,\,z\in
V_{2}}p(s,z,y)$ $\displaystyle\leq
Ct^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee t^{1/\alpha})^{\beta_{1}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
Proof. Note that, since $y_{d}\leq x_{d}+|x-y|\leq 5+2^{-5}<6$, for any
$0<s\leq 1$ and $z\in{\mathbb{R}}^{d}_{+}$ with $|z-y|\geq 2$,
$\displaystyle(y_{d}\vee s^{1/\alpha})\wedge|z-y|\geq\frac{1}{3}(y_{d}\vee
s^{1/\alpha}).$ (7.21)
By Lemmas 7.2 and 10.2 and applying (7.21),
$\displaystyle\sup_{s\leq t,\,z\in V_{2}}p(s,z,y)\leq c_{1}\sup_{s\leq
t,\,z\in{\mathbb{R}}^{d}_{+},|z-y|\geq 2}\left(1\wedge\frac{z_{d}\wedge
y_{d}}{s^{1/\alpha}}\right)^{q}A_{\beta_{1},0,\beta_{3},0}(s,z,y)\frac{s}{|z-y|^{d+\alpha}}$
$\displaystyle\leq c_{1}\sup_{s\leq t,\,z\in{\mathbb{R}}^{d}_{+},|z-y|\geq
2}\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}\bigg{(}\frac{y_{d}\vee
s^{1/\alpha}}{|z-y|}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{3|z-y|}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{s}{|z-y|^{d+\alpha}}.$ (7.22)
Using that $u\mapsto u^{\beta_{1}}\log^{\beta_{3}}(e+1/u)$ is almost
increasing, we have that for any $0<s\leq t\leq 1$ and
$z\in{\mathbb{R}}^{d}_{+}$ with $|z-y|\geq 2$,
$\displaystyle\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}\bigg{(}\frac{y_{d}\vee
s^{1/\alpha}}{|z-y|}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{3|z-y|}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{s}{|z-y|^{d+\alpha}}$ $\displaystyle\leq
c_{2}\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}(y_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{s}{|z-y|^{d+\alpha}}$ $\displaystyle\leq
2^{-d-\alpha}c_{2}s\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}(y_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}.$ (7.23)
Let ${\varepsilon}>0$ be such that $q<\alpha+\beta_{1}-{\varepsilon}$. Using
(7.6), we see that
$\displaystyle s\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}(y_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}$
$\displaystyle=sy_{d}^{\beta_{1}-{\varepsilon}}\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q-\beta_{1}+{\varepsilon}}(y_{d}\vee
s^{1/\alpha})^{{\varepsilon}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}$
$\displaystyle=s^{(\alpha+\beta_{1}-q-{\varepsilon})/\alpha}y_{d}^{\beta_{1}-{\varepsilon}}(y_{d}\wedge
s^{1/\alpha})^{q-\beta_{1}+{\varepsilon}}(y_{d}\vee
s^{1/\alpha})^{{\varepsilon}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}.$
Thus, the map $s\mapsto
s\big{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\big{)}^{q}(y_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\big{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\big{)}$ is almost increasing on $(0,t]$ by (10.2). Using this
and (7)–(7), we get that
$\displaystyle\sup_{s\leq t,\,z\in V_{2}}p(s,z,y)$ $\displaystyle\leq
c_{3}\sup_{s\leq
t}s\left(1\wedge\frac{y_{d}}{s^{1/\alpha}}\right)^{q}(y_{d}\vee
s^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{4}t\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.24)
Note that (7.1) is satisfied with $a_{1}=\beta_{1}$ and $a_{3}=\beta_{3}$ by
Lemma 7.2. Thus, by Lemma 7.1 and (7) we obtain
$\displaystyle{\mathbb{P}}_{x}(\tau_{V_{1}}<t<\zeta)\sup_{s\leq t,\,z\in
V_{2}}p(s,z,y)$ $\displaystyle\leq
c_{4}t^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee t^{1/\alpha})^{\beta_{1}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
$\Box$
###### Lemma 7.4.
Let $\eta_{1},\eta_{2},\gamma\geq 0$. There exists a constant $C>0$ such that
for any $x\in{\mathbb{R}}^{d}_{+}$ and any $s,k,l>0$,
$\displaystyle\int_{B_{+}(x,2)}p(s,x,z)z_{d}^{\gamma}\log^{\eta_{1}}\bigg{(}e+\frac{k}{z_{d}}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{z_{d}}{l}\bigg{)}dz$
$\displaystyle\leq
Cx_{d}^{\gamma}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q-\gamma}\log^{\eta_{1}}\bigg{(}e+\frac{k}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{x_{d}\vee
s^{1/\alpha}}{l}\bigg{)}$ $\displaystyle\quad+C{\bf
1}_{\\{\gamma>\alpha+\beta_{1}\\}}sx_{d}^{\beta_{1}}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q-\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{2}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}(e+k)\log^{\eta_{2}}\bigg{(}e+\frac{1}{l}\bigg{)}$
$\displaystyle\quad+C{\bf 1}_{\\{\gamma=\alpha+\beta_{1},\,x_{d}\vee
s^{1/\alpha}<2\\}}sx_{d}^{\beta_{1}}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q-\beta_{1}}$
$\displaystyle\qquad\times\int_{x_{d}\vee
s^{1/\alpha}}^{2}\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}\bigg{(}e+\frac{k}{r}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{r}{l}\bigg{)}\frac{dr}{r}.$
Proof. For any $x,z\in{\mathbb{R}}^{d}_{+}$ and $s>0$, by Lemmas 7.2 and 10.2,
$\displaystyle p(s,x,z)\leq
c_{1}\left(1\wedge\frac{x_{d}}{s^{1/\alpha}}\right)^{q}A_{\beta_{1},0,\beta_{3},0}(s,x,z)\left(s^{-d/\alpha}\wedge\frac{s}{|x-z|^{d+\alpha}}\right)$
$\displaystyle\leq
c_{2}\left(1\wedge\frac{x_{d}}{s^{1/\alpha}}\right)^{q}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{\beta_{1}}\log^{\beta_{3}}\left(e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\right)\left(s^{-d/\alpha}\wedge\frac{s}{|x-z|^{d+\alpha}}\right).$
Now combining (7.6) and Lemma 10.10, we get the desired result. $\Box$
We now state the first main result of this section.
###### Theorem 7.5.
For any ${\varepsilon}\in(0,\alpha/2]$, there exists a constant $C>0$ such
that
$\displaystyle p(t,x,y)$ $\displaystyle\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}A_{\beta_{1},\beta_{2}\wedge(\alpha+\beta_{1}-{\varepsilon}),\beta_{3},\beta_{4}}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),$
for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$.
This theorem will proved by using several lemmas. We first introduce some
additional notation: For $b_{1},b_{2},b_{3},b_{4}\geq 0$, $t>0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$, define
$\displaystyle{\cal I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})$
$\displaystyle:=\int_{0}^{t/2}\int_{B_{+}(y,2)}\int_{B_{+}(x,2)}{\bf
1}_{\\{u_{d}\leq w_{d}\\}}p(t-s,x,u)p(s,y,w)$ $\displaystyle\hskip
56.9055pt\times
u_{d}^{b_{1}}w_{d}^{b_{2}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}\,du\,dw\,ds,$
(7.25) $\displaystyle{\cal I}_{t,2}(x,y;b_{1},b_{2},b_{3},b_{4})$
$\displaystyle:=\int_{0}^{t/2}\int_{B_{+}(y,2)}\int_{B_{+}(x,2)}{\bf
1}_{\\{w_{d}\leq u_{d}\\}}p(t-s,x,u)p(s,y,w)$ $\displaystyle\hskip
56.9055pt\times
u_{d}^{b_{2}}w_{d}^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{u_{d}}{w_{d}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{u_{d}}\bigg{)}\,du\,dw\,ds.$
(7.26)
###### Lemma 7.6.
Let $b_{1},b_{2},b_{3},b_{4}\geq 0$. Let $b_{1}^{\prime}\in[0,b_{1}]$ and
$b_{2}^{\prime}:=b_{1}+b_{2}-b_{1}^{\prime}$. Then
${\cal I}_{t,i}(x,y;b_{1},b_{2},b_{3},b_{4})\leq{\cal
I}_{t,i}(x,y;b_{1}^{\prime},b_{2}^{\prime},b_{3},b_{4}),\quad i=1,2.$
Proof. If $u_{d}\leq w_{d}$, then $u_{d}^{b_{1}}w_{d}^{b_{2}}\leq
u_{d}^{b_{1}^{\prime}}w_{d}^{b_{2}^{\prime}}$, implying that ${\cal
I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})\leq{\cal
I}_{t,1}(x,y;b_{1}^{\prime},b_{2}^{\prime},\linebreak b_{3},b_{4}).$ The other
inequality is analogous. $\Box$
###### Lemma 7.7.
Let $b_{1},b_{2},b_{3},b_{4}\geq 0$ with $b_{1}\vee b_{2}<\alpha+\beta_{1}$,
$x,y\in{\mathbb{R}}^{d}_{+}$ with $|x-y|=5$, and $t\in(0,1]$.
(i) If $b_{2}>q-\alpha$ or $y_{d}\geq t^{1/\alpha}$, then there exists a
constant $C>0$ independent of $t,x,y$ such that
$\displaystyle{\cal I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
Ctx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
(ii) If $b_{1}>q-\alpha$ or $y_{d}\geq t^{1/\alpha}$, then there exists a
constant $C>0$ independent of $t,x,y$ such that
$\displaystyle{\cal I}_{t,2}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
Ctx_{d}^{b_{2}}y_{d}^{b_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\log^{b_{3}}\bigg{(}e+\frac{x_{d}\vee
t^{1/\alpha}}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}.$
Proof. We give the proof for (i). (ii) can be proved similarly.
By using Lemma 7.4 together with the fact that $t-s\asymp t$ if $0\leq s\leq
t/2$, we see that for $0\leq s\leq t/2$,
$\displaystyle\int_{B_{+}(x,2)}p(t-s,x,u)u_{d}^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}du\leq
c_{1}x_{d}^{b_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}.$
Thus, using Lemma 7.4 again, we get that for $0\leq s\leq t/2$,
$\displaystyle\int_{B_{+}(y,2)}\int_{B_{+}(x,2)}p(t-s,x,u)p(s,y,w)u_{d}^{b_{1}}w_{d}^{b_{2}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}\,du\,dw$
$\displaystyle\leq
c_{1}x_{d}^{b_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\int_{B_{+}(y,2)}p(s,y,w)w_{d}^{b_{2}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}dw$
$\displaystyle\leq
c_{2}x_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}$
From this and Lemma 10.5, we get that
$\displaystyle{\cal I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
c_{2}x_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\int_{0}^{\frac{t}{2}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$ $\displaystyle\leq
c_{2}x_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\int_{0}^{t}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$ $\displaystyle\leq
c_{3}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
$\Box$
###### Lemma 7.8.
Let $b_{1},b_{2},b_{3},b_{4}\geq 0$ be such that $b_{1}>q-\alpha$ and
$b_{1}\vee b_{2}<\alpha+\beta_{1}$. Assume that $b_{2}>0$ if $b_{4}>0$. Then,
there exists a constant $C>0$ such that for all $x,y\in{\mathbb{R}}^{d}_{+}$
with $x_{d}\leq y_{d}$ and $|x-y|=5$, and all $t\in(0,1]$,
$\displaystyle{\cal I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})+{\cal
I}_{t,2}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
Ctx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ (7.27)
and
$\displaystyle{\cal I}_{t,1}(y,x;b_{1},b_{2},b_{3},b_{4})+{\cal
I}_{t,2}(y,x;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
Ctx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.28)
Proof. Let $\delta\in(0,1)$ be such that
$\displaystyle b_{1}\vee b_{2}+(1-\delta)(b_{1}\wedge
b_{2})<\alpha+\beta_{1},$ (7.29)
and let $b_{1}^{\prime}:=\delta(b_{1}\wedge b_{2})$ and
$b_{2}^{\prime}:=b_{1}+b_{2}-b_{1}^{\prime}$. Then we see that
$b_{1}^{\prime}\in[0,b_{1}\wedge b_{2}]$, $b_{2}^{\prime}\geq b_{1}>q-\alpha$
and
$\displaystyle b_{1}^{\prime}\leq b_{2}^{\prime}=b_{1}\vee
b_{2}+(1-\delta)(b_{1}\wedge b_{2})<\alpha+\beta_{1}$ (7.30)
by (7.29). Moreover, since $x_{d}\leq y_{d}$ and
$b_{2}^{\prime}-b_{1}=b_{2}-b_{1}^{\prime}$, we see that
$\displaystyle(x_{d}\vee t^{1/\alpha})^{b_{2}^{\prime}-b_{1}}(y_{d}\vee
t^{1/\alpha})^{b_{1}^{\prime}-b_{2}}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{-b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\leq c_{1}.$ (7.31)
Indeed, when $b_{4}=0$, (7.31) clearly holds with $c_{1}=1$. If $b_{4}>0$,
then $b_{2}>0$ so that $b_{2}>\delta b_{2}\geq b_{1}^{\prime}$. Hence, we get
(7.31) from (10.2).
(i) We first prove (7.8). For this, we distinguish between two cases:
$y_{d}\geq t^{1/\alpha}$ and $y_{d}<t^{1/\alpha}$.
Assume first that $y_{d}\geq t^{1/\alpha}$. The desired bound for ${\cal
I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})$ follows from Lemma 7.7(i). On the other
hand, by using Lemma 7.6 in the first inequality, Lemma 7.7(ii) in the second
inequality (which uses (7.30) and $y_{d}\geq t^{1/\alpha}$), (7.6) in the
equality, and (7.31) in the last inequality, we get that
$\displaystyle{\cal I}_{t,2}(x,y;b_{1},b_{2},b_{3},b_{4})\leq{\cal
I}_{t,2}(x,y;b_{1}^{\prime},b_{2}^{\prime},b_{3},b_{4})$ $\displaystyle\leq
c_{2}tx_{d}^{b_{2}^{\prime}}y_{d}^{b_{1}^{\prime}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}^{\prime}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}^{\prime}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}$
$\displaystyle=c_{2}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\quad\;\times(x_{d}\vee
t^{1/\alpha})^{b_{2}^{\prime}-b_{1}}(y_{d}\vee
t^{1/\alpha})^{b_{1}^{\prime}-b_{2}}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{-b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{1}c_{2}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
Assume now that $y_{d}<t^{1/\alpha}$. Using Lemma 7.6 and Lemma 7.7(i) (which
uses $b_{2}^{\prime}>q-\alpha$), we get
$\displaystyle{\cal I}_{t,1}(x,y;b_{1},b_{2},b_{3},b_{4})\leq{\cal
I}_{t,1}(x,y;b_{1}^{\prime},b_{2}^{\prime},b_{3},b_{4})$ $\displaystyle\leq
c_{3}tx_{d}^{b_{1}^{\prime}}y_{d}^{b_{2}^{\prime}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}^{\prime}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}^{\prime}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.32)
Also, since $b_{1}>q-\alpha$, we get from Lemma 7.7(ii) that
$\displaystyle{\cal I}_{t,2}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
c_{4}tx_{d}^{b_{2}}y_{d}^{b_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\\!\log^{b_{3}}\bigg{(}e+\frac{x_{d}\vee
t^{1/\alpha}}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.33)
Since $x_{d}\leq y_{d}<t^{1/\alpha}$ and
$b_{1}^{\prime}+b_{2}^{\prime}=b_{1}+b_{2}$, it holds that $x_{d}\vee
t^{1/\alpha}=y_{d}\vee t^{1/\alpha}=t^{1/\alpha}$ and
$\displaystyle
x_{d}^{b_{1}^{\prime}}y_{d}^{b_{2}^{\prime}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}^{\prime}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}^{\prime}}=x_{d}^{b_{2}}y_{d}^{b_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}$
$\displaystyle=\frac{x_{d}^{q}y_{d}^{q}}{t^{(2q-b_{1}-b_{2})/\alpha}}=x_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}.$
Thus, (7.8) follows from (7) and (7).
(ii) Now, we prove (7.8). By using Lemma 7.6 in the first inequality, Lemma
7.7(i) in the second inequality (which uses (7.30) and
$b_{2}^{\prime}>q-\alpha$), (7.6) together with the fact that $x_{d}\leq
y_{d}$ in the third inequality, (7.31) in the last inequality, we obtain
$\displaystyle{\cal I}_{t,1}(y,x;b_{1},b_{2},b_{3},b_{4})\leq{\cal
I}_{t,1}(y,x;b_{1}^{\prime},b_{2}^{\prime},b_{3},b_{4})$ $\displaystyle\leq
c_{5}tx_{d}^{b_{2}^{\prime}}y_{d}^{b_{1}^{\prime}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}^{\prime}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}^{\prime}}\log^{b_{3}}\bigg{(}e+\frac{x_{d}\vee
t^{1/\alpha}}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{5}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\quad\times(x_{d}\vee
t^{1/\alpha})^{b_{2}^{\prime}-b_{1}}(y_{d}\vee
t^{1/\alpha})^{b_{1}^{\prime}-b_{2}}\log^{b_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{-b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\leq
c_{1}c_{5}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
On the other hand, by Lemma 7.7(ii), it holds that
$\displaystyle{\cal I}_{t,2}(y,x;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
c_{3}tx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$
The proof is complete. $\Box$
Proof of Theorem 7.5. Set
$\widehat{\beta}_{2}:=\beta_{2}\wedge(\alpha+\beta_{1}-{\varepsilon})$. As in
the proof of Lemma 7.2, by symmetry, Proposition 5.1, (3.30), and (6.4), we
can assume without loss of generality that $x_{d}\leq y_{d}\wedge 2^{-5}$,
$\widetilde{x}=0$ and $|x-y|=5$, and then it is enough to show that there
exists a constant $c_{1}>0$ independent of $x$ and $y$ such that for any
$t\leq 1$,
$\displaystyle p(t,x,y)$ $\displaystyle\leq
c_{1}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee t^{1/\alpha})^{\widehat{\beta}_{2}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\,\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.34)
Let $t\leq 1$. Set $V_{1}=U(1)$,
$V_{3}=B(y,2)\cap{\overline{\mathbb{R}}}^{d}_{+}$ and
$V_{2}={\overline{\mathbb{R}}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. By Lemma
7.3,
$\displaystyle{\mathbb{P}}_{x}(\tau_{V_{1}}<t<\zeta)\sup_{s\leq t,\,z\in
V_{2}}p(s,z,y)$ $\displaystyle\leq
c_{2}t^{2}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{2\beta_{3}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}$
$\displaystyle=c_{2}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee
t^{1/\alpha})^{\alpha+\beta_{1}-{\varepsilon}}(y_{d}\vee
t^{1/\alpha})^{-\alpha+{\varepsilon}}t\log^{2\beta_{3}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}$
$\displaystyle\leq
c_{2}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee
t^{1/\alpha})^{\widehat{\beta}_{2}}t^{{\varepsilon}/\alpha}\log^{2\beta_{3}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}$
$\displaystyle\leq
c_{3}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee t^{1/\alpha})^{\widehat{\beta}_{2}},$
(7.35)
where in the last inequality above we used (10.1).
Next, we show that there exists a constant $C^{\prime}>0$ such that
$\displaystyle I$
$\displaystyle:=\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{V_{1}}(t-s,x,u){\cal
B}(u,w)p(s,y,w)du\,dw\,ds$ $\displaystyle\leq
C^{\prime}tx_{d}^{\beta_{1}}y_{d}^{\widehat{\beta}_{2}}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q-\beta_{1}}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q-\widehat{\beta}_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.36)
Once we get (7), by (7.6) and (7), we can apply Lemma 3.15 to get (7) and
finish the proof.
By (A3)(II), since $|u-w|\asymp 1$ and $u_{d}\vee w_{d}\leq y_{d}+2\leq
7+2^{-5}$ for $u\in V_{1}$ and $w\in V_{3}$, we have
$\displaystyle I$ $\displaystyle\leq
c_{4}\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}{\bf 1}_{\\{u_{d}\leq
w_{d}\\}}p(t-s,x,u)p(s,y,w)u_{d}^{\beta_{1}}w_{d}^{\widehat{\beta}_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}du\,dw\,ds$
$\displaystyle\;+c_{4}\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}{\bf
1}_{\\{w_{d}\leq
u_{d}\\}}p(t-s,x,u)p(s,y,w)u_{d}^{\widehat{\beta}_{2}}w_{d}^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{u_{d}}{w_{d}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{u_{d}}\bigg{)}du\,dw\,ds$
$\displaystyle=c_{4}\bigg{(}\big{(}\int_{0}^{t/2}+\int_{t/2}^{t}\big{)}\int_{V_{3}}\int_{V_{1}}{\bf
1}_{\\{u_{d}\leq
w_{d}\\}}\dots+\big{(}\int_{0}^{t/2}+\int_{t/2}^{t}\big{)}\int_{V_{3}}\int_{V_{1}}{\bf
1}_{\\{w_{d}\leq u_{d}\\}}\dots\bigg{)}.$
Thus, by the change of variable $\widetilde{s}=t-s$ in integrals
$\int_{t/2}^{t}$,
$\displaystyle I\leq c_{4}\sum_{i=1}^{2}{\cal
I}_{t,i}(x,y;\beta_{1},\widehat{\beta}_{2},\beta_{3},\beta_{4})+c_{4}\sum_{i=1}^{2}{\cal
I}_{t,i}(y,x;\beta_{1},\widehat{\beta}_{2},\beta_{3},\beta_{4}),$
where the functions ${\cal
I}_{t,i}(x,y;\beta_{1},\widehat{\beta}_{2},\beta_{3},\beta_{4})$, $1\leq i\leq
2$, are defined in (7.25)–(7.26). Then by using Lemma 7.6(i) and Lemma 7.8, we
conclude that (7) holds. The proof is complete. $\Box$
As an immediate consequence of Theorem 7.5, we obtain the following sharp
upper bound of the heat kernel for $\beta_{2}<\alpha+\beta_{1}$.
###### Corollary 7.9.
Suppose that $\beta_{2}<\alpha+\beta_{1}$. There exists a constant $C>0$ such
that
$\displaystyle p(t,x,y)$ $\displaystyle\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right),$
for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$.
Here is the second main result of the section.
###### Theorem 7.10.
Suppose that $\beta_{2}\geq\alpha+\beta_{1}$. There exists a constant $C>0$
such that
$\displaystyle p(t,x,y)$ $\displaystyle\leq
C\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)\bigg{[}A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,y)$
$\displaystyle\quad+\bigg{(}1\wedge\frac{t}{|x-y|^{\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})+t^{1/\alpha})\wedge|x-y|}\bigg{)}$
$\displaystyle\quad\quad\times\bigg{(}{\bf
1}_{\\{\beta_{2}>\alpha+\beta_{1}\\}}A_{\beta_{1},\beta_{1},0,\beta_{3}}(t,x,y)+{\bf
1}_{\\{\beta_{2}=\alpha+\beta_{1}\\}}A_{\beta_{1},\beta_{1},0,\beta_{3}+\beta_{4}+1}(t,x,y)\bigg{)}\bigg{]}$
for all $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$.
Again, we first introduce additional notation and prove a lemma. For
$b_{1},b_{2},b_{3},b_{4}\geq 0$, $t>0$ and $x,y\in{\mathbb{R}}^{d}_{+}$,
define
$\displaystyle{\cal I}_{t}(x,y;b_{1},b_{2},b_{3},b_{4})$
$\displaystyle:=\int_{0}^{t}\int_{B_{+}(y,2)}\int_{B_{+}(x,2)}p(t-s,x,u)p(s,y,w)$
$\displaystyle\qquad\times
u_{d}^{b_{1}}w_{d}^{b_{2}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{u_{d}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}\,du\,dw\,ds.$
(7.37)
###### Lemma 7.11.
Let $b_{1},b_{2},b_{3},b_{4}\geq 0$ be such that $b_{1}\wedge b_{2}>q-\alpha$
and $b_{1}<\alpha+\beta_{1}$. There exists a constant $C>0$ such that for all
$x,y\in{\mathbb{R}}^{d}_{+}$ with $|x-y|=5$, and all $t\in(0,1]$,
$\displaystyle{\cal I}_{t}(x,y;b_{1},b_{2},b_{3},b_{4})$ $\displaystyle\leq
Ctx_{d}^{b_{1}}y_{d}^{b_{2}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\\!\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e\\!+\\!\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\\!\log^{b_{4}}\bigg{(}e\\!+\\!\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle+{\bf
1}_{\\{b_{2}>\alpha+\beta_{1}\\}}Ct^{2}x_{d}^{b_{1}}y_{d}^{\beta_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\\!\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-\beta_{1}}\\!\log^{b_{3}}\bigg{(}e\\!+\\!\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\\!\log^{b_{3}}\bigg{(}e\\!+\\!\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle+{\bf
1}_{\\{b_{2}=\alpha+\beta_{1},y_{d}<2\\}}Ctx_{d}^{b_{1}}y_{d}^{\beta_{1}}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\\!\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-\beta_{1}}$
$\displaystyle\quad\times\int_{y_{d}}^{2}(r^{\alpha}\wedge
t)\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}.$
Proof. By using Lemma 7.4 in the first inequality (integration with respect to
$u$; note that $b_{1}<\alpha+\beta_{1}$) and the second inequality
(integration with respect to $w$), we get
$\displaystyle{\cal I}_{t}(x,y;b_{1},b_{2},b_{3},b_{4})\leq
c_{1}x_{d}^{b_{1}}\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}$
$\displaystyle\hskip
71.13188pt\times\int_{B_{+}(y,2)}p(s,y,w)w_{d}^{b_{2}}\log^{b_{3}}\bigg{(}e+\frac{w_{d}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{w_{d}}\bigg{)}dwds$
$\displaystyle\leq c_{2}\left(x_{d}^{b_{1}}y_{d}^{b_{2}}I_{1}+{\bf
1}_{\\{b_{2}>\alpha+\beta_{1}\\}}x_{d}^{b_{1}}y_{d}^{\beta_{1}}I_{2}+{\bf
1}_{\\{b_{2}=\alpha+\beta_{1},y_{d}<2\\}}x_{d}^{b_{1}}y_{d}^{\beta_{1}}I_{3}\right),$
where
$I_{1}:=\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\\!\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds,$
$I_{2}:=\int_{0}^{t}s\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-\beta_{1}}\\!\log^{b_{3}}\bigg{(}e+\frac{2}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{1}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}ds,$
and
$\displaystyle I_{3}:=$
$\displaystyle\int_{0}^{t}s\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-\beta_{1}}$
$\displaystyle\hskip 36.98866pt\times\int_{y_{d}\vee
s^{1/\alpha}}^{2}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}\,ds.$
Applying Lemma 10.5 to $I_{1}$ and $I_{2}$, Lemma 10.6 to $I_{3}$, and (10.2),
wee see that
$I_{1}\leq
c_{3}t\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)},$ $I_{2}\leq
c_{3}t^{2}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-\beta_{1}}\log^{b_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)},$
and
$\displaystyle I_{2}\leq$ $\displaystyle
c_{3}t\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-\beta_{1}}$
$\displaystyle\quad\times\int_{y_{d}}^{2}(r^{\alpha}\wedge
t)\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}.$
This proves the lemma. $\Box$
Proof of Theorem 7.10. As in the proof of Lemma 7.2, by symmetry, (6),
Proposition 5.1, (3.30) and (6.4), we can assume without loss of generality
that $x_{d}\leq y_{d}\wedge 2^{-5}$, $\widetilde{x}=0$ and $|x-y|=5$, and then
it is enough to show that there exists a constant $c_{1}>0$ independent of $x$
and $y$ such that for any $t\leq 1$,
$\displaystyle p(t,x,y)\leq
c_{1}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}$ $\displaystyle\qquad\times\bigg{[}(y_{d}\vee
t^{1/\alpha})^{\beta_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\,\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ (7.38) $\displaystyle\quad\qquad+{\bf
1}_{\\{\beta_{2}>\alpha+\beta_{1}\\}}t(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\quad\qquad+{\bf
1}_{\\{\beta_{2}=\alpha+\beta_{1}\\}}t(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}+\beta_{4}+1}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\bigg{]}.$ (7.39)
Let $t\leq 1$. Set $V_{1}=U(1)$,
$V_{3}=B(y,2)\cap{\overline{\mathbb{R}}}^{d}_{+}$ and
$V_{2}={\overline{\mathbb{R}}}^{d}_{+}\setminus(V_{1}\cup V_{3})$. By Lemmas
3.15 and 7.3 it remains to prove that
$I:=\int_{0}^{t}\int_{V_{3}}\int_{V_{1}}p^{V_{1}}(t-s,x,u){\cal
B}(u,w)p(s,y,w)dudwds$ is bounded above by the right-hand side of (7.39).
By (A3)(II), since $|u-w|\asymp 1$ for $u\in V_{1}$ and $w\in V_{3}$, using
the change of variables $\widetilde{s}=t-s$ we have
$\displaystyle I\leq c_{2}\left({\cal
I}_{t}(x,y;\beta_{1},\beta_{2},\beta_{3},\beta_{4})+{\cal
I}_{t}(y,x;\beta_{1},\beta_{2},\beta_{3},\beta_{4})\right),$ (7.40)
where the functions ${\cal
I}_{t}(x,y;\beta_{1},\beta_{2},\beta_{3},\beta_{4})$ is defined in (7.37). By
Lemma 7.11 and (7.6), the right hand side of (7.40) is less than or equal to
$c_{3}t(1\wedge\frac{x_{d}}{t^{1/\alpha}})^{q}(1\wedge\frac{y_{d}}{t^{1/\alpha}})^{q}$
times
$\displaystyle(x_{d}\vee t^{1/\alpha})^{\beta_{1}}(y_{d}\vee
t^{1/\alpha})^{\beta_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\,\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ (7.41) $\displaystyle\quad+(x_{d}\vee
t^{1/\alpha})^{\beta_{2}}(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{x_{d}\vee
t^{1/\alpha}}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}$ (7.42) $\displaystyle\quad+{\bf
1}_{\\{\beta_{2}>\alpha+\beta_{1}\\}}t(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ (7.43) $\displaystyle\quad+{\bf
1}_{\\{\beta_{2}=\alpha+\beta_{1}\\}}(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}(y_{d}\vee t^{1/\alpha})^{\beta_{1}}$
$\displaystyle\qquad\;\times\int_{x_{d}}^{2}(r^{\alpha}\wedge
t)\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}.$
(7.44)
In the last line we used $x_{d}\leq y_{d}\wedge 2$ to get that ${\bf
1}_{\\{y_{d}<2\\}}\int_{y_{d}}^{2}\leq\int_{x_{d}}^{2}$.
Since $\beta_{2}>\beta_{1}$ and $x_{d}\leq y_{d}$, clearly
$\displaystyle(x_{d}\vee
t^{1/\alpha})^{\beta_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{x_{d}\vee
t^{1/\alpha}}{y_{d}\vee t^{1/\alpha}}\bigg{)}\leq(x_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)},$ (7.45)
and by using (10.2) (with $\varepsilon=(\beta_{2}-\beta_{1})/\beta_{4})$), we
get
$\displaystyle(y_{d}\vee
t^{1/\alpha})^{\beta_{1}}\log^{\beta_{4}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\leq c_{4}(y_{d}\vee
t^{1/\alpha})^{\beta_{2}}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.46)
Applying (7.45) and (7.46) to (7.42) and combining it with (7.41) and (7.43),
we arrive at the result in case $\beta_{2}>\alpha+\beta_{1}$.
Assume now that $\beta_{2}=\alpha+\beta_{1}$. From the above argument, to
prove the result, in view of (7.38), (7.39) and (7.44), it suffices to show
that
$\displaystyle\int_{x_{d}}^{2}(r^{\alpha}\wedge
t)\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\leq c_{5}t\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}+\beta_{4}+1}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle\quad+c_{5}(y_{d}\vee
t^{1/\alpha})^{\beta_{2}-\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.47)
By Lemma 10.11 (with $b_{1}=\beta_{3}$, $b_{2}=\beta_{4}$, $k=x_{d}\vee
t^{1/\alpha}$ and $l=y_{d}\vee t^{1/\alpha}$), it holds that
$\displaystyle\int_{y_{d}\vee t^{1/\alpha}}^{2}(r^{\alpha}\wedge
t)\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\leq t\int_{y_{d}\vee
t^{1/\alpha}}^{2}\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\leq c_{6}t\log^{\beta_{3}}\bigg{(}e+\frac{1}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}+\beta_{4}+1}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.48)
On the other hand, using (10.2), we see that
$\displaystyle\int_{x_{d}}^{y_{d}\vee t^{1/\alpha}}(r^{\alpha}\wedge
t)\log^{\beta_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\leq\log^{\beta_{3}}(e+1)\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee t^{1/\alpha}}\bigg{)}\int_{x_{d}}^{y_{d}\vee
t^{1/\alpha}}\log^{\beta_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r^{1-\alpha}}$
$\displaystyle\leq c_{7}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\int_{x_{d}}^{y_{d}\vee
t^{1/\alpha}}\bigg{(}\frac{y_{d}\vee
t^{1/\alpha}}{r}\bigg{)}^{\alpha/2}\frac{dr}{r^{1-\alpha}}$ $\displaystyle\leq
c_{8}(y_{d}\vee
t^{1/\alpha})^{\alpha}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$ $\displaystyle=c_{8}(y_{d}\vee
t^{1/\alpha})^{\beta_{2}-\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}.$ (7.49)
Combining (7)–(7), we show that (7) holds true. The proof is complete. $\Box$
## 8\. Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1. Since $\overline{p}(t,x,y)$ is jointly continuous (see
Remark 3.12), it suffices to prove that (1.1)–(1.1) hold for
$(t,x,y)\in(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$.
We first note that by (A3) and (6.4),
$\displaystyle
t^{-d/\alpha}\wedge\big{(}tJ(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\big{)}\asymp
t^{-d/\alpha}\wedge\frac{tB_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})}{|x-y|^{d+\alpha}}$
$\displaystyle\asymp\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right)B_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d}),$
(8.1)
which implies the second comparison in (1.1).
(i) Using (6.3), we get the lower heat kernel estimate in the first comparison
in (1.1) from Proposition 6.2 and the upper heat kernel estimate from
Corollary 7.9.
(ii) For (1.4), using (6.3), we get the lower heat kernel estimate from
Proposition 6.6 (see Remark 6.7) and the upper heat kernel estimate from
Theorem 7.10.
(iii) Using (6.3), we get the lower heat kernel estimate in (1.1) from
Proposition 6.9. The upper heat kernel estimate in (1.1) follows from Theorem
7.10.
From the comparisons in (i)-(iii) and (6.4), we have that
$\overline{p}(t,x,y)\asymp t^{-d/\alpha}$ when $t^{1/\alpha}\geq|x-y|/8$ and
this implies that (1.1) holds for $t^{1/\alpha}\geq|x-y|/8$. Moreover, (1.1)
for $\beta_{2}<\alpha+\beta_{1}$ follows from (1.1), (6.3) and (8).
By (A3) and (6.3), we have that when $t^{1/\alpha}<|x-y|/8$,
$\displaystyle t\int_{(x_{d}\vee y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}J(x+t^{1/\alpha}{\mathbf{e}}_{d},x+r{\mathbf{e}}_{d})J(x+r{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})r^{d-1}dr$
$\displaystyle\asymp\frac{t}{|x-y|^{d+\alpha}}\int_{(x_{d}\vee y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x,x+r{\mathbf{e}}_{d})\,A_{\beta_{1},\beta_{2},\beta_{3},\beta_{4}}(t,x+r{\mathbf{e}}_{d},y)\frac{dr}{r^{\alpha+1}}.$
(8.2)
Thus, we see that, for $\beta_{2}=\alpha+\beta_{1}$ and
$t^{1/\alpha}<|x-y|/8$, (1.1) follows from (1.1), (8) and Lemma 6.8, and that,
for $\beta_{2}>\alpha+\beta_{1}$ and $t^{1/\alpha}<|x-y|/8$, the lower bound
in (1.1) follows from (8) and Proposition 6.2 and Lemma 6.4.
We have from (8), (6) and (6.3) that, when $t^{1/\alpha}<|x-y|/8$,
$\displaystyle t\int_{(x_{d}\vee y_{d}\vee
t^{1/\alpha})\wedge(|x-y|/4)}^{|x-y|/2}J(x+t^{1/\alpha}{\mathbf{e}}_{d},x+r{\mathbf{e}}_{d})J(x+r{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})r^{d-1}dr$
$\displaystyle\geq\frac{c_{1}t}{|x-y|^{d+2\alpha}}B_{\beta_{1},\beta_{1},0,\beta_{3}}(x+t^{1/\alpha}{\mathbf{e}}_{d},y+t^{1/\alpha}{\mathbf{e}}_{d})\log^{\beta_{3}}\bigg{(}e+\frac{|x-y|}{((x_{d}\wedge
y_{d})\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$ (8.3)
Now, for $\beta_{2}>\alpha+\beta_{1}$ and $t^{1/\alpha}<|x-y|/8$, the upper
bound in (1.1) follows from (8), (8) and the upper bound in (1.4).
Finally, from the joint continuity of $\overline{p}(t,x,y)$ and upper heat
kernel estimates, we deduce that $\overline{Y}$ is a Feller process and finish
the proof by Remark 3.12. $\Box$
Proof of Theorem 1.2. The second comparison in (1.9) follows from Corollary
6.3. By (6.3), Theorem 1.1 and Propositions 6.2, 6.6 and 6.9,
$p^{\kappa}(t,x,y)\geq
c_{1}\big{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\big{)}^{q_{\kappa}}\big{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\big{)}^{q_{\kappa}}\overline{p}(t,x,y)$
for all
$(t,x,y)\in(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$. On
the other hand, by (6.3), Theorem 1.1, Corollary 7.9, and Theorem 7.10,
$p^{\kappa}(t,x,y)\leq
c_{2}\big{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\big{)}^{q_{\kappa}}\big{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\big{)}^{q_{\kappa}}\overline{p}(t,x,y)$
for all
$(t,x,y)\in(0,\infty)\times{\mathbb{R}}^{d}_{+}\times{\mathbb{R}}^{d}_{+}$ and
hence (1.9) holds true.
Note that for each $(t,y)\in(0,\infty)\times{\mathbb{R}}^{d}_{+}$, the map
$x\mapsto p^{\kappa}(t,x,y)$ vanishes continuously on
$\partial{\mathbb{R}}^{d}_{+}$. Hence, using the joint continuity of
$p^{\kappa}(t,x,y)$ and upper heat kernel estimates, we deduce that
$Y^{\kappa}$ is a Feller process. By Remark 3.12, the proof is complete.
$\Box$
## 9\. Green function estimates
In this section, we give proofs of Theorems 1.3 and 1.4.
Proof of Theorem 1.3. When $d>\alpha$, we get the upper bound of (1.11) from
Corollary 3.13. On the other hand, by Lemma 3.7 and Remark 3.12, we have
$\displaystyle\overline{G}(x,y)\geq\int_{|x-y|^{\alpha}}^{\infty}\overline{p}(t,x,y)dt\geq
c_{1}\int_{|x-y|^{\alpha}}^{\infty}t^{-d/\alpha}dt=\begin{cases}\frac{c_{1}\alpha}{d-\alpha}|x-y|^{-d+\alpha}\;\;&\mbox{if
}d>\alpha;\\\\[4.0pt] \infty\;\;&\mbox{if }d\leq\alpha.\end{cases}$
The proof is complete. $\Box$
In the remainder of this section, we assume the setting of Theorem 1.4 holds
and denote by $q$ the constant $q_{\kappa}$ in (1.8), which is strictly
positive.
Let $x,y\in{\mathbb{R}}^{d}_{+}$ be such that $x_{d}\leq y_{d}$ and $|x-y|=1$.
From Theorem 7.5, Proposition 6.2 and (6.4), we have for $t\leq 1$,
$\displaystyle p^{\kappa}(t,x,y)$ $\displaystyle\leq
Ct\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\big{(}(x_{d}\vee
t^{1/\alpha})\wedge 1\big{)}^{\beta_{1}}\big{(}(y_{d}\vee
t^{1/\alpha}\big{)}\wedge 1\big{)}^{\beta_{2}\wedge(\alpha/2+\beta_{1})}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{(y_{d}\vee
t^{1/\alpha})\wedge 1}{(x_{d}\vee t^{1/\alpha})\wedge
1}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{(y_{d}\vee t^{1/\alpha})\wedge
1}\bigg{)},$ (9.1) $\displaystyle p^{\kappa}(t,x,y)$ $\displaystyle\geq
ct\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}\big{(}(x_{d}\vee
t^{1/\alpha})\wedge 1\big{)}^{\beta_{1}}\big{(}(y_{d}\vee
t^{1/\alpha}\big{)}\wedge 1\big{)}^{\beta_{2}}$
$\displaystyle\quad\times\log^{\beta_{3}}\bigg{(}e+\frac{(y_{d}\vee
t^{1/\alpha})\wedge 1}{(x_{d}\vee t^{1/\alpha})\wedge
1}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{(y_{d}\vee t^{1/\alpha})\wedge
1}\bigg{)},$ (9.2)
and for $t>1$,
$\displaystyle p^{\kappa}(t,x,y)\asymp
t^{-d/\alpha}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}.$
(9.3)
###### Lemma 9.1.
Let $x,y\in{\mathbb{R}}^{d}_{+}$ be such that $x_{d}\leq y_{d}$ and $|x-y|=1$.
Set $\widehat{q}:=2\alpha+\beta_{1}+\beta_{2}-q$. Then we have
$\int_{0}^{1}p^{\kappa}(t,x,y)dt\asymp\begin{cases}(x_{d}\wedge
1)^{q}(y_{d}\wedge 1)^{q}&\mbox{ if }q<\widehat{q},\\\\[5.0pt]
\displaystyle(x_{d}\wedge 1)^{q}(y_{d}\wedge
1)^{q}\log^{\beta_{4}+1}\left(e+\frac{1}{y_{d}\wedge 1}\right)&\mbox{ if
}q=\widehat{q},\\\\[8.0pt] \displaystyle(x_{d}\wedge 1)^{q}(y_{d}\wedge
1)^{\widehat{q}}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}\wedge 1}\right)&\mbox{
if }q>\widehat{q},\end{cases}$
where the comparison constant is independent of $x,y$.
Proof. Set $G_{1}:=\int_{0}^{1}p^{\kappa}(t,x,y)dt$ and
$\widehat{\beta}_{2}:=\beta_{2}\wedge(\alpha/2+\beta_{1})$.
(Case 1) $x_{d}\geq 1$: We have from (9) and (9) that
$G_{1}\asymp\int_{0}^{1}tdt\asymp 1.$
(Case 2) $y_{d}\geq 1>x_{d}$: By (9) and (7.6),
$\displaystyle G_{1}$ $\displaystyle\geq
c_{1}x_{d}^{\beta_{1}}\int_{1/2}^{1}t\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q-\beta_{1}}dt\geq
2^{-1}c_{1}x_{d}^{\beta_{1}}\int_{1/2}^{1}(1\wedge
x_{d})^{q-\beta_{1}}dt=2^{-2}c_{1}x_{d}^{q}.$
Besides, since $\alpha+\beta_{1}-q>0$, we get from (9) and (7.6) that
$\displaystyle G_{1}$ $\displaystyle\leq
c_{2}x_{d}^{\beta_{1}}\int_{0}^{1}t\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-\beta_{1}}\log^{\beta_{3}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}dt$
$\displaystyle\leq
c_{3}x_{d}^{q}\int_{0}^{1}t^{(\alpha+\beta_{1}-q)/\alpha}\log^{\beta_{3}}\left(e+\frac{1}{t^{1/\alpha}}\right)dt\leq
c_{4}x_{d}^{q}.$
(Case 3) $1>y_{d}\geq x_{d}$: Note that
$\displaystyle\int_{y_{d}^{\alpha}}^{1}t^{-1+(\widehat{q}-q)/\alpha}\log^{\beta_{4}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}dt\asymp\begin{cases}1&\mbox{
if }\,q<\widehat{q};\\\\[2.0pt]
\displaystyle\log^{\beta_{4}+1}(e+1/y_{d})&\mbox{ if
}\,q=\widehat{q};\\\\[2.0pt] \displaystyle
y_{d}^{\widehat{q}-q}\log^{\beta_{4}}(e+1/y_{d})&\mbox{ if
}\,q>\widehat{q}.\end{cases}$ (9.4)
For the lower bound, we get from (9) and (9.4) that
$\displaystyle G_{1}$ $\displaystyle\geq
c_{5}x_{d}^{q}y_{d}^{q}\int_{y_{d}^{\alpha}}^{1}t^{(\alpha+\beta_{1}+\beta_{2}-2q)/\alpha}\log^{\beta_{4}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}dt$
$\displaystyle=c_{5}x_{d}^{q}y_{d}^{q}\int_{y_{d}^{\alpha}}^{1}t^{-1+(\widehat{q}-q)/\alpha}\log^{\beta_{4}}\bigg{(}e+\frac{1}{t^{1/\alpha}}\bigg{)}dt$
$\displaystyle\asymp x_{d}^{q}y_{d}^{q}\times\begin{cases}1&\mbox{ if
}\,q<\widehat{q};\\\\[2.0pt] \displaystyle\log^{\beta_{4}+1}(e+1/y_{d})&\mbox{
if }\,q=\widehat{q};\\\\[2.0pt] \displaystyle
y_{d}^{\widehat{q}-q}\log^{\beta_{4}}(e+1/y_{d})&\mbox{ if
}\,q>\widehat{q}.\end{cases}$
For the upper bound, we see from (9) that
$\displaystyle G_{1}$ $\displaystyle\leq
c_{6}x_{d}^{\beta_{1}}y_{d}^{\widehat{\beta}_{2}}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}}{x_{d}}\bigg{)}\log^{\beta_{4}}\bigg{(}e+\frac{1}{y_{d}}\bigg{)}\int_{0}^{x_{d}^{\alpha}}tdt$
$\displaystyle\quad+c_{6}x_{d}^{q}y_{d}^{\widehat{\beta}_{2}}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right)\int_{x_{d}^{\alpha}}^{y_{d}^{\alpha}}t^{(\alpha+\beta_{1}-q)/\alpha}\log^{\beta_{3}}\bigg{(}e+\frac{y_{d}}{t^{1/\alpha}}\bigg{)}dt$
$\displaystyle\quad+c_{6}x_{d}^{q}y_{d}^{q}\int_{y_{d}^{\alpha}}^{1}t^{(\alpha+\beta_{1}+\widehat{\beta}_{2}-2q)/\alpha}\log^{\beta_{4}}\left(e+\frac{1}{t^{1/\alpha}}\right)dt$
$\displaystyle=:c_{6}(I+II+III).$
Using (10.1), we obtain
$\displaystyle I\leq
c_{7}x_{d}^{2\alpha+\beta_{1}}y_{d}^{\widehat{\beta}_{2}}\bigg{(}\frac{y_{d}}{x_{d}}\bigg{)}^{2\alpha+\beta_{1}-q}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right)=c_{7}x_{d}^{q}y_{d}^{2\alpha+\beta_{1}+\widehat{\beta}_{2}-q}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right).$
(9.5)
Since $\alpha+\beta_{1}>q$, the map $t\mapsto
t^{(\alpha+\beta_{1}-q)/\alpha}\log^{\beta_{3}}(e+y_{d}/t^{1/\alpha})$ is
almost increasing. Therefore,
$\displaystyle II\leq
c_{8}x_{d}^{q}y_{d}^{\alpha+\beta_{1}+\widehat{\beta}_{2}-q}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right)\int_{x_{d}^{\alpha}}^{y_{d}^{\alpha}}dt\leq
c_{8}x_{d}^{q}y_{d}^{2\alpha+\beta_{1}+\widehat{\beta}_{2}-q}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right).$
(9.6)
We consider the cases $q<\widehat{q}$ and $q\geq\widehat{q}$ separately.
First, suppose that $q<\widehat{q}$, which is equivalent to
$q<\alpha+(\beta_{1}+\beta_{2})/2$. Since $q<\alpha+\beta_{1}$, it follows
that $q<\alpha+(\beta_{1}+\widehat{\beta}_{2})/2$. Using (10.1), since
$y_{d}<1$, we see from (9.5)–(9.6) that
$\displaystyle I+II\leq
c_{9}x_{d}^{q}y_{d}^{q}y_{d}^{2\alpha+\beta_{1}+\widehat{\beta}_{2}-2q}\log^{\beta_{4}}\left(e+\frac{1}{y_{d}}\right)\leq
c_{10}x_{d}^{q}y_{d}^{q}.$
Moreover, since $(\alpha+\beta_{1}+\widehat{\beta}_{2}-2q)/\alpha>-1$, we get
$\displaystyle III\leq
c_{7}x_{d}^{q}y_{d}^{q}\int_{0}^{1}t^{(\alpha+\beta_{1}+\widehat{\beta}_{2}-2q)/\alpha}\log^{\beta_{4}}\left(e+\frac{1}{t^{1/\alpha}}\right)dt\leq
c_{11}x_{d}^{q}y_{d}^{q}.$
Therefore, we arrive at the result in this case.
Suppose that $q\geq\widehat{q}$. In this case, we have
$2\alpha+\beta_{1}+\beta_{2}=q+\widehat{q}\leq 2q<2\alpha+2\beta_{1}$. Thus,
$\beta_{2}<\beta_{1}$ and $\widehat{\beta}_{2}=\beta_{2}$. Then we deduce the
desired upper bound from (9.4)–(9.6). The proof is complete. $\Box$
###### Lemma 9.2.
Let $x,y\in{\mathbb{R}}^{d}_{+}$ be such that $x_{d}\leq y_{d}$ and $|x-y|=1$.
Then we have
$\int_{1}^{\infty}p^{\kappa}(t,x,y)dt\asymp\begin{cases}(x_{d}\wedge
1)^{q}(y_{d}\wedge 1)^{q}&\mbox{ if }d>\alpha,\\\\[5.0pt] (x_{d}\wedge
1)^{q}(y_{d}\wedge 1)^{q}\log\big{(}e+(x_{d}\vee 1)\big{)}&\mbox{ if
}d=1=\alpha,\\\\[5.0pt] \displaystyle(x_{d}\wedge 1)^{q}(y_{d}\wedge
1)^{q}(x_{d}\vee 1)^{\alpha-1}&\mbox{ if }d=1<\alpha,\end{cases}$
where the comparison constant is independent of $x,y$.
Proof. By (9.3), we have
$\int_{1}^{\infty}p^{\kappa}(t,x,y)dt\asymp\int_{1}^{\infty}t^{-d/\alpha}\left(1\wedge\frac{x_{d}}{t^{1/\alpha}}\right)^{q}\left(1\wedge\frac{y_{d}}{t^{1/\alpha}}\right)^{q}dt=:G_{2}.$
If $d>\alpha$, then by Lemma 10.12, $G_{2}\asymp(x_{d}\wedge
1)^{q}(y_{d}\wedge 1)^{q}$.
If $d=1=\alpha$, then using Lemma 10.12, we get
$\displaystyle G_{2}$ $\displaystyle\asymp\int_{1}^{x_{d}\vee
1}t^{-1}dt+x_{d}^{q}\int_{x_{d}\vee
1}^{\infty}t^{-1-q}\left(1\wedge\frac{y_{d}}{t}\right)^{q}dt$
$\displaystyle\asymp{\bf 1}_{\\{x_{d}>1\\}}\log(x_{d})+x_{d}^{q}(x_{d}\vee
1)^{-q}\left(1\wedge\frac{y_{d}}{x_{d}\vee 1}\right)^{q}$
$\displaystyle\asymp\begin{cases}\log(e+x_{d})&\text{ if }x_{d}>1\\\
x_{d}^{q}(y_{d}\wedge 1)^{q}&\text{ if }x_{d}\leq
1\end{cases}\asymp(x_{d}\wedge 1)^{q}(y_{d}\wedge 1)^{q}\log(e+(x_{d}\vee
1)).$
If $d=1<\alpha$, then since $y_{d}>2$ implies $x_{d}\geq y_{d}-|x-y|\geq
y_{d}/2>1$, and $\alpha-q-1\leq 0$, we get
$\displaystyle G_{2}$ $\displaystyle\asymp{\bf
1}_{\\{y_{d}>2\\}}\bigg{(}\int_{1}^{x_{d}^{\alpha}}t^{-1/\alpha}dt+x_{d}^{q}\int_{x_{d}^{\alpha}}^{(2y_{d})^{\alpha}}t^{-(q+1)/\alpha}dt+x_{d}^{q}y_{d}^{q}\int_{(2y_{d})^{\alpha}}^{\infty}t^{-(2q+1)/\alpha}dt\bigg{)}$
$\displaystyle\quad+{\bf 1}_{\\{y_{d}\leq
2\\}}x_{d}^{q}y_{d}^{q}\int_{1}^{\infty}t^{-(2q+1)/\alpha}dt$
$\displaystyle\asymp{\bf
1}_{\\{y_{d}>2\\}}\big{(}x_{d}^{\alpha-1}+x_{d}^{q+\alpha-(q+1)}+x_{d}^{q}y_{d}^{\alpha-q-1}\big{)}+{\bf
1}_{\\{y_{d}\leq 2\\}}x_{d}^{q}y_{d}^{q}$ $\displaystyle\asymp{\bf
1}_{\\{y_{d}>2\\}}x_{d}^{\alpha-1}+{\bf 1}_{\\{y_{d}\leq
2\\}}x_{d}^{q}y_{d}^{q}\asymp(x_{d}\wedge 1)^{q}(y_{d}\wedge 1)^{q}(x_{d}\vee
1)^{\alpha-1}.$
The proof is complete. $\Box$
Proof of Theorem 1.4. From scaling property (3.30), we obtain
$\displaystyle
G^{\kappa}(x,y)=|x-y|^{-d+\alpha}\,G^{\kappa}(x/|x-y|,y/|x-y|),\quad
x,y\in{\mathbb{R}}^{d}_{+}.$ (9.7)
Using (9.7), symmetry and Lemmas 9.1 and 9.2, we deduce the desired result.
$\Box$
## 10\. Appendix
Note that for any ${\varepsilon}>0$,
$\log(e+r)<(2+{\varepsilon}^{-1})r^{\varepsilon}\quad\text{for all}\;r\geq 1,$
(10.1)
$\frac{\log(e+ar)}{\log(e+r)}<(1+{\varepsilon}^{-1})a^{{\varepsilon}}\quad\quad\text{for
all }a\geq 1\text{ and }r>0.$ (10.2)
Recall the definition of $A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)$ from (6).
###### Lemma 10.1.
Let $b_{1},b_{2},b_{3},b_{4}\geq 0$.
(i) If $b_{1}>0$, then for any ${\varepsilon}\in(0,b_{1}]$, there exists
$c_{1}>0$ such that
$A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)\leq
c_{1}A_{b_{1}-{\varepsilon},b_{2},0,b_{4}}(t,x,y)\quad\text{ for all }t\geq
0,\,x,y\in{\mathbb{R}}^{d}_{+}.$
(ii) If $b_{2}>0$, then for any ${\varepsilon}\in(0,b_{2}]$, there exists
$c_{2}>0$ such that
$A_{b_{1},b_{2},b_{3},b_{4}}(t,x,y)\leq
c_{2}A_{b_{1},b_{2}-{\varepsilon},b_{3},0}(t,x,y)\quad\text{ for all }t\geq
0,\,x,y\in{\mathbb{R}}^{d}_{+}.$
Proof. The results follow from (10.1). $\Box$
###### Lemma 10.2.
Let $b_{1},b_{3}\geq 0$. Suppose that $b_{1}>0$ if $b_{3}>0$. Then there
exists a constant $C>0$ such that for all $t\geq 0$ and
$x,y\in{\mathbb{R}}^{d}_{+}$,
$\displaystyle A_{b_{1},0,b_{3},0}(t,x,y)$ $\displaystyle\leq
C\,\bigg{(}\frac{x_{d}\vee t^{1/\alpha}}{|x-y|}\wedge
1\bigg{)}^{b_{1}}\log^{b_{3}}\bigg{(}e+\frac{(y_{d}\vee
t^{1/\alpha})\wedge|x-y|}{(x_{d}\vee t^{1/\alpha})\wedge|x-y|}\bigg{)}.$
Proof. When $x_{d}\leq y_{d}$, the result is clear. Assume that $x_{d}>y_{d}$.
Set $r=\frac{x_{d}\vee t^{1/\alpha}}{|x-y|}\wedge 1$ and $s=\frac{y_{d}\vee
t^{1/\alpha}}{|x-y|}\wedge 1$. Then $0<s\leq r\leq 1$ and the desired
inequality is equivalent to
$\log^{b_{3}}(e+r/s)\log^{-b_{3}}(e+s/r)\leq C(r/s)^{b_{1}}.$ (10.3)
If $b_{3}=0$, then (10.3) clearly holds with $C=1$. If $b_{3}>0$ and
$b_{1}>0$, then we get from (10.1) that
$\log^{b_{3}}(e+r/s)\log^{-b_{3}}(e+s/r)\leq\log^{b_{3}}(e+r/s)\leq
c_{1}(r/s)^{b_{1}}.$
This proves the lemma. $\Box$
###### Remark 10.3.
Recall that
$B_{b_{1},b_{2},b_{3},b_{4}}(x,y)=A_{b_{1},b_{2},b_{3},b_{4}}(0,x,y)$ for
$x,y\in{\mathbb{R}}^{d}_{+}$. Therefore, the results of Lemmas 10.1 and 10.2
hold true with $B_{b_{1},0,b_{3},0}(x,y)$ instead of
$A_{b_{1},0,b_{3},0}(t,x,y)$.
###### Lemma 10.4.
Let $\gamma\in{\mathbb{R}}$, $b\geq 0$ and $t,k,l>0$. Suppose that either
$\gamma<\alpha$ or $k\geq t^{1/\alpha}$. Then we have
$\int_{0}^{t}\bigg{(}1\wedge\frac{k}{s^{1/\alpha}}\bigg{)}^{\gamma}\log^{b}\bigg{(}e+\frac{l}{k\vee
s^{1/\alpha}}\bigg{)}ds\leq
Ct\bigg{(}1\wedge\frac{k}{t^{1/\alpha}}\bigg{)}^{\gamma}\log^{b}\bigg{(}e+\frac{l}{k\vee
t^{1/\alpha}}\bigg{)},$
where $C>0$ is a constant which depends only on $\gamma$ and $b$.
Proof. If $k\geq t^{1/\alpha}$, then the desired inequality holds since the
left hand side is $t\log^{b}(e+l/k)$.
Suppose that $k<t^{1/\alpha}$ and $\gamma<\alpha$. Let ${\varepsilon}>0$ be
such that $\gamma+b{\varepsilon}<\alpha$. Then the left hand side is
$\displaystyle\log^{b}\bigg{(}e+\frac{l}{k}\bigg{)}\int_{0}^{k^{\alpha}}ds+k^{\gamma}\int_{k^{\alpha}}^{t}\frac{1}{s^{\gamma/\alpha}}\log^{b}\bigg{(}e+\frac{l}{s^{1/\alpha}}\bigg{)}ds$
$\displaystyle\leq
k^{\alpha}\log^{b}\bigg{(}e+\frac{l}{k}\bigg{)}+c_{1}k^{\gamma}\log^{b}\bigg{(}e+\frac{l}{t^{1/\alpha}}\bigg{)}\int_{k^{\alpha}}^{t}\frac{t^{b{\varepsilon}/\alpha}}{s^{\gamma/\alpha+b{\varepsilon}/\alpha}}ds$
$\displaystyle\leq
c_{2}k^{\alpha}\bigg{(}\frac{t^{1/\alpha}}{k}\bigg{)}^{\alpha-\gamma}\log^{b}\bigg{(}e+\frac{l}{t^{1/\alpha}}\bigg{)}+c_{2}k^{\gamma}t^{1-\gamma/\alpha}\log^{b}\bigg{(}e+\frac{l}{t^{1/\alpha}}\bigg{)}$
$\displaystyle=2c_{2}t\bigg{(}\frac{k}{t^{1/\alpha}}\bigg{)}^{\gamma}\log^{b}\bigg{(}e+\frac{l}{t^{1/\alpha}}\bigg{)}=2c_{2}t\bigg{(}1\wedge\frac{k}{t^{1/\alpha}}\bigg{)}^{\gamma}\log^{b}\bigg{(}e+\frac{l}{t^{1/\alpha}}\bigg{)}.$
We used (10.2) in both inequalities above. $\Box$
###### Lemma 10.5.
Let $b_{1},b_{2}\in{\mathbb{R}}$, $b_{3},b_{4}\geq 0$ and $t,x_{d},y_{d}>0$.
Suppose that (1) either $b_{1}>q-\alpha$ or $x_{d}\geq t^{1/\alpha}$, and (2)
either $b_{2}>q-\alpha$ or $y_{d}\geq t^{1/\alpha}$. Then we have
$\displaystyle\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$ $\displaystyle\leq
Ct\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}$
where $C>0$ is a constant which depends only on $b_{1},b_{2},b_{3}$ and
$b_{4}$.
Proof. Observe that
$\displaystyle\int_{0}^{\frac{t}{2}}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$ $\displaystyle\leq
c_{1}\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\int_{0}^{\frac{t}{2}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$
and
$\displaystyle\int_{\frac{t}{2}}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
s^{1/\alpha}}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
s^{1/\alpha}}\bigg{)}ds$ $\displaystyle\leq
c_{2}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}\log^{b_{4}}\bigg{(}e+\frac{1}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\int_{0}^{\frac{t}{2}}\bigg{(}1\wedge\frac{x_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{1}}\log^{b_{3}}\bigg{(}e+\frac{y_{d}\vee
t^{1/\alpha}}{x_{d}\vee s^{1/\alpha}}\bigg{)}ds.$
Using Lemma 10.4 twice, we arrive at the result. $\Box$
###### Lemma 10.6.
Let $b_{1},b_{2}\in{\mathbb{R}}$, $b_{3},b_{4}\geq 0$ and $t,x_{d},y_{d}>0$.
Suppose that (1) either $b_{1}>q-\alpha$ or $x_{d}\geq t^{1/\alpha}$, and (2)
either $b_{2}>q-\alpha$ or $y_{d}\geq t^{1/\alpha}$. Then we have that for
$y_{d}\vee t^{1/\alpha}<2$,
$\displaystyle\int_{0}^{t}s\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\int_{y_{d}\vee
s^{1/\alpha}}^{2}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}\,ds$
$\displaystyle\leq
Ct\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\int_{y_{d}}^{2}(r^{\alpha}\wedge
t)\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r},$
where $C>0$ is a constant which depends only on $b_{1},b_{2},b_{3}$ and
$b_{4}$.
Proof. Using Fubini’s theorem and Lemma 10.4 twice as in the proof of Lemma
10.5, we get
$\displaystyle\int_{0}^{t}s\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\int_{y_{d}\vee
s^{1/\alpha}}^{2}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}\,ds$
$\displaystyle=$ $\displaystyle\int_{y_{d}}^{2}\int_{0}^{r^{\alpha}\wedge
t}s\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\,ds\frac{dr}{r}$
$\displaystyle\leq$ $\displaystyle\int_{y_{d}}^{2}(r^{\alpha}\wedge
t)\int_{0}^{t}\bigg{(}1\wedge\frac{x_{d}}{(t-s)^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{s^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee(t-s)^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\,ds\frac{dr}{r}$
$\displaystyle\leq$
$\displaystyle\,Ct\bigg{(}1\wedge\frac{x_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{1}}\bigg{(}1\wedge\frac{y_{d}}{t^{1/\alpha}}\bigg{)}^{q-b_{2}}$
$\displaystyle\quad\times\int_{y_{d}}^{2}(r^{\alpha}\wedge
t)\log^{b_{3}}\bigg{(}e+\frac{r}{x_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{3}}\bigg{(}e+\frac{r}{y_{d}\vee
t^{1/\alpha}}\bigg{)}\log^{b_{4}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}.$
$\Box$
###### Lemma 10.7.
There is a constant $C>0$ such that for all $x\in{\mathbb{R}}^{d}_{+}$ and
$A>0$,
$\displaystyle\int_{B_{+}(x,A)}z_{d}^{-1/2}dz\leq CA^{d}(x_{d}\vee A)^{-1/2}.$
Proof. We have
$\displaystyle\int_{B_{+}(x,A)}z_{d}^{-1/2}dz\leq\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{x}-\widetilde{z}|<A}d\widetilde{z}\int_{(x_{d}-A)\vee
0}^{x_{d}+A}z_{d}^{-1/2}dz_{d}\leq c_{1}A^{d-1}\int_{(x_{d}-A)\vee
0}^{x_{d}+A}z_{d}^{-1/2}dz_{d}.$ (10.4)
If $x_{d}\geq 2A$, then
$\displaystyle\int_{(x_{d}-A)\vee
0}^{x_{d}+A}z_{d}^{-1/2}dz_{d}\leq\frac{1}{(x_{d}/2)^{1/2}}\int_{x_{d}-A}^{x_{d}+A}dz_{d}=2^{3/2}Ax_{d}^{-1/2}.$
(10.5)
If $x_{d}<2A$, then
$\displaystyle\int_{(x_{d}-A)\vee
0}^{x_{d}+A}z_{d}^{-1/2}dz_{d}\leq\int_{0}^{4A}z_{d}^{-1/2}dz_{d}=4A^{1/2}.$
(10.6)
Combining (10.4) with (10.5)–(10.6), we arrive at the result. $\Box$
###### Lemma 10.8.
(i) There is a constant $C>0$ such that for all $x\in{\mathbb{R}}^{d}_{+}$ and
$0<A<x_{d}$,
$\displaystyle\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>A}\frac{dz}{z_{d}^{1/2}|x-z|^{d+\alpha}}\leq
Cx_{d}^{-1/2}A^{-\alpha}.$
(ii) Let ${\varepsilon}\in(0,1)$ and $\delta>0$. There is a constant
$C^{\prime}>0$ such that for all $x\in{\mathbb{R}}^{d}_{+}$ and $A\geq x_{d}$,
$\displaystyle\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>A}\frac{dz}{z_{d}^{{\varepsilon}}\,|x-z|^{d+\delta}}\leq
C^{\prime}A^{-{\varepsilon}-\delta}.$
Proof. Without loss of generality, we assume $x=(\widetilde{0},x_{d})$.
(i) Note that
$\displaystyle\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>A}\frac{dz}{z_{d}^{1/2}|x-z|^{d+\alpha}}$
$\displaystyle=\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>A,\,|\widetilde{z}|\leq|x_{d}-z_{d}|}\frac{dz}{z_{d}^{1/2}|x-z|^{d+\alpha}}+\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>A,\,|\widetilde{z}|>|x_{d}-z_{d}|}\frac{dz}{z_{d}^{1/2}|x-z|^{d+\alpha}}$
$\displaystyle=:I+II.$
First, we have
$\displaystyle I$
$\displaystyle\leq\int_{x_{d}\geq|x_{d}-z_{d}|>\frac{A}{2}}\frac{1}{z_{d}^{1/2}|x_{d}-z_{d}|^{d+\alpha}}\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{z}|\leq|x_{d}-z_{d}|}d\widetilde{z}dz_{d}$
$\displaystyle\leq
c_{1}\bigg{(}\int_{0}^{\frac{x_{d}}{2}}\frac{dz_{d}}{z_{d}^{1/2}|x_{d}-z_{d}|^{1+\alpha}}+\int_{\frac{x_{d}}{2}}^{x_{d}-\frac{A}{2}}\frac{dz_{d}}{z_{d}^{1/2}|x_{d}-z_{d}|^{1+\alpha}}+\int_{x_{d}+\frac{A}{2}}^{2x_{d}}\frac{dz_{d}}{z_{d}^{1/2}|x_{d}-z_{d}|^{1+\alpha}}\bigg{)}$
$\displaystyle\leq
c_{1}\bigg{(}\frac{2^{1+\alpha}}{x_{d}^{1+\alpha}}\int_{0}^{\frac{x_{d}}{2}}{z_{d}^{-1/2}}dz_{d}+\frac{1}{(x_{d}/2)^{1/2}}\int_{\frac{x_{d}}{2}}^{x_{d}-\frac{A}{2}}\frac{dz_{d}}{|x_{d}-z_{d}|^{1+\alpha}}+{x_{d}^{-1/2}}\int_{x_{d}+\frac{A}{2}}^{2x_{d}}\frac{dz_{d}}{|x_{d}-z_{d}|^{1+\alpha}}\bigg{)}$
$\displaystyle\leq
c_{2}\left(x_{d}^{-1/2-\alpha}+x_{d}^{-1/2}A^{-\alpha}+x_{d}^{-1/2}A^{-\alpha}\right)\leq
c_{3}x_{d}^{-1/2}A^{-\alpha}.$
On the other hand, we see that for any $z\in{\mathbb{R}}^{d}_{+}$ with
$x_{d}\geq|x-z|>A$ and $|\widetilde{z}|>|x_{d}-z_{d}|$,
$\displaystyle z_{d}\geq x_{d}-|x_{d}-z_{d}|\geq
x_{d}-\frac{1}{2}(|\widetilde{z}|+|x_{d}-z_{d}|)\geq
x_{d}-\frac{1}{\sqrt{2}}|x-z|\geq\left(1-\frac{1}{\sqrt{2}}\right)x_{d}.$
Hence, we also have that
$\displaystyle II$ $\displaystyle\leq
c_{4}x_{d}^{-1/2}\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>A}\frac{dz}{|x-z|^{d+\alpha}}\leq
c_{5}x_{d}^{-1/2}A^{-\alpha}.$
(ii) Observe that
$\displaystyle\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>A}\frac{dz}{z_{d}^{{\varepsilon}}\,|x-z|^{d+\delta}}$
$\displaystyle\leq\int_{z\in{\mathbb{R}}^{d}_{+},\,|x_{d}-z_{d}|\geq|\widetilde{z}|\vee\frac{A}{2}}\frac{d\widetilde{z}\,dz_{d}}{z_{d}^{{\varepsilon}}\,|x_{d}-z_{d}|^{d+\delta}}+\int_{z\in{\mathbb{R}}^{d}_{+},\,|\widetilde{z}|\geq|x_{d}-z_{d}|\vee\frac{A}{2}}\frac{d\widetilde{z}\,dz_{d}}{z_{d}^{{\varepsilon}}\,|\widetilde{z}|^{d+\delta}}=:I+II.$
Using Fubini’s theorem, since $A\geq x_{d}$, we see that
$\displaystyle II$
$\displaystyle\leq\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{z}|\geq\frac{A}{2}}\frac{1}{|\widetilde{z}|^{d+\delta}}\int_{0}^{|\widetilde{z}|+x_{d}}z_{d}^{-{\varepsilon}}dz_{d}\,d\widetilde{z}\leq
c_{1}\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{z}|\geq\frac{A}{2}}\frac{(|\widetilde{z}|+A)^{1-{\varepsilon}}}{|\widetilde{z}|^{d+\delta}}d\widetilde{z}$
$\displaystyle\leq
c_{2}\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{z}|\geq\frac{A}{2}}\frac{d\widetilde{z}}{|\widetilde{z}|^{d-1+{\varepsilon}+\delta}}\leq
c_{3}A^{-{\varepsilon}-\delta}.$
On the other hand, using the fact that
$\int_{\widetilde{z}\in{\mathbb{R}}^{d-1},\,|\widetilde{z}|\leq|x_{d}-z_{d}|}d\widetilde{z}\leq
c_{4}|x_{d}-z_{d}|^{d-1}$, we also see that
$\displaystyle I$ $\displaystyle\leq
c_{5}\bigg{(}\int_{0}^{(x_{d}-\frac{A}{2})\vee
0}\frac{dz_{d}}{z_{d}^{{\varepsilon}}\,|x_{d}-z_{d}|^{1+\delta}}+\int_{x_{d}+\frac{A}{2}}^{\infty}\frac{dz_{d}}{z_{d}^{{\varepsilon}}\,|x_{d}-z_{d}|^{1+\delta}}\bigg{)}$
$\displaystyle\leq
c_{6}\bigg{(}\frac{1}{A^{1+\delta}}\int_{0}^{(x_{d}-\frac{A}{2})\vee
0}z_{d}^{-{\varepsilon}}dz_{d}+\frac{1}{A^{{\varepsilon}}}\int_{x_{d}+\frac{A}{2}}^{\infty}\frac{dz_{d}}{|x_{d}-z_{d}|^{1+\delta}}\bigg{)}$
$\displaystyle\leq
c_{7}\bigg{(}\frac{1}{A^{1+\delta}}\Big{(}\big{(}x_{d}-\frac{A}{2}\big{)}\vee
0\Big{)}^{1-{\varepsilon}}+\frac{1}{A^{{\varepsilon}+\delta}}\bigg{)}\leq\frac{c_{6}}{A^{{\varepsilon}+\delta}},$
where we used the fact that $A\geq x_{d}$ in the last inequality. The proof is
complete. $\Box$
For $\gamma,\eta_{1},\eta_{2}\geq 0$ and $k,l>0$, define
$f_{\gamma,\eta_{1},\eta_{2},k,l}(r):=r^{\gamma}\log^{\eta_{1}}\bigg{(}e+\frac{k}{r}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{r}{l}\bigg{)}.$
###### Lemma 10.9.
Let $\gamma,\eta_{1},\eta_{2}\geq 0$.
(i) For any ${\varepsilon}>0$, there exist constants $C,C^{\prime}>0$ such
that for any $k,l,r>0$ and any $a\geq 1$,
$Ca^{\gamma-{\varepsilon}}\leq\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(ar)}{f_{\gamma,\eta_{1},\eta_{2},k,l}(r)}\leq
C^{\prime}a^{\gamma+{\varepsilon}}.$ (10.7)
(ii) Assume that $\gamma>0$. Then there exists a constant $C>0$ such that for
any $k,l,r>0$ and any $a\geq 1$,
$\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(ar)}{f_{\gamma,\eta_{1},\eta_{2},k,l}(r)}\geq
C.$
Proof. (i) If $\eta_{2}=0$, the second inequality in (10.7) is true with any
${\varepsilon}\geq 0$. In case $\eta_{2}>0$, for any given ${\varepsilon}>0$,
let ${\varepsilon}^{\prime}:={\varepsilon}/\eta_{2}$. We get from (10.2) that
for all $a\geq 1$ and $r>0$,
$\displaystyle\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(ar)}{f_{\gamma,\eta_{1},\eta_{2},k,l}(r)}\leq
a^{\gamma}\left(\frac{\log(e+ar/l)}{\log(e+r/l)}\right)^{\eta_{2}}\leq
a^{\gamma}\big{(}(1+1/{\varepsilon}^{\prime})a^{{\varepsilon}^{\prime}}\big{)}^{\eta_{2}}=c({\varepsilon},\eta_{2})a^{\gamma+{\varepsilon}}.$
The first inequality can be proved by a similar argument.
(ii) The desired result follows from the first inequality in (10.7) with
${\varepsilon}=\gamma$. $\Box$
###### Lemma 10.10.
Let $b_{1},b_{2},\eta_{1},\eta_{2},\gamma\geq 0$. There exists a constant
$C>0$ such that for any $x\in{\mathbb{R}}^{d}_{+}$ and $s,k,l>0$,
$\displaystyle\int_{B_{+}(x,2)}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}$ $\displaystyle\hskip
51.21504pt\times\left(s^{-d/\alpha}\wedge\frac{s}{|x-z|^{d+\alpha}}\right)z_{d}^{\gamma}\log^{\eta_{1}}\bigg{(}e+\frac{k}{z_{d}}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{z_{d}}{l}\bigg{)}dz$
$\displaystyle\leq C(x_{d}\vee
s^{1/\alpha})^{\gamma}\log^{\eta_{1}}\bigg{(}e+\frac{k}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{x_{d}\vee
s^{1/\alpha}}{l}\bigg{)}$ $\displaystyle\quad+C{\bf 1}_{\\{x_{d}\vee
s^{1/\alpha}<2\\}}s(x_{d}\vee s^{1/\alpha})^{b_{1}}\bigg{[}{\bf
1}_{\\{\gamma>\alpha+b_{1}\\}}\log^{b_{2}}\bigg{(}e+\frac{2}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}(e+k)\log^{\eta_{2}}\bigg{(}e+\frac{1}{l}\bigg{)}$
$\displaystyle\hskip 88.2037pt+{\bf
1}_{\\{\gamma=\alpha+b_{1}\\}}\int_{x_{d}\vee
s^{1/\alpha}}^{2}\log^{b_{2}}\bigg{(}e+\frac{r}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}\bigg{(}e+\frac{k}{r}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{r}{l}\bigg{)}\frac{dr}{r}\bigg{]}.$
Proof. Using the triangle inequality, we see that for any
$z\in{\mathbb{R}}^{d}_{+}$,
$z_{d}\leq x_{d}+|x-z|\leq 2(x_{d}\vee|x-z|).$ (10.8)
Therefore, using Lemma 10.9(i)-(ii) and Lemma 10.7, we get that
$\displaystyle\int_{B_{+}(x,s^{1/\alpha})}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}\,s^{-d/\alpha}f_{\gamma,\eta_{1},\eta_{2},k,l}(z_{d})dz$
$\displaystyle=(\log^{b_{2}}(e+1))s^{-d/\alpha}\int_{B_{+}(x,s^{1/\alpha})}z_{d}^{-1/2}f_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(z_{d})dz$
$\displaystyle\leq
c_{1}s^{-d/\alpha}f_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(2(x_{d}\vee
s^{1/\alpha}))\int_{B_{+}(x,s^{1/\alpha})}z_{d}^{-1/2}dz$ $\displaystyle\leq
c_{2}(x_{d}\vee
s^{1/\alpha})^{-1/2}f_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha})=c_{2}f_{\gamma,\eta_{1},\eta_{2},k,l}(x_{d}\vee s^{1/\alpha}).$
When $x_{d}>s^{1/\alpha}$, we get from (10.8), Lemma 10.9(i)-(ii) and Lemma
10.8(i) that
$\displaystyle
s\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>s^{1/\alpha}}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(z_{d})}{|x-z|^{d+\alpha}}dz$
$\displaystyle=(\log^{b_{2}}(e+1))s\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>s^{1/\alpha}}\frac{f_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(z_{d})}{z_{d}^{1/2}|x-z|^{d+\alpha}}dz$
$\displaystyle\leq
c_{3}sf_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(2x_{d})\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\geq|x-z|>s^{1/\alpha}}\frac{1}{z_{d}^{1/2}|x-z|^{d+\alpha}}dz$
$\displaystyle\leq
c_{4}x_{d}^{-1/2}f_{\gamma+\frac{1}{2},\eta_{1},\eta_{2},k,l}(x_{d})=c_{4}f_{\gamma,\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha}).$
It remains to bound the integral over $\\{z\in{\mathbb{R}}^{d}_{+}:x_{d}\vee
s^{1/\alpha}<|x-z|<2\\}$ under the assumption $x_{d}\vee s^{1/\alpha}<2$. For
this, we consider the following three cases separately.
(i) Case $\gamma<\alpha+b_{1}$: Fix ${\varepsilon}\in(0,1)$ such that
$\gamma+3{\varepsilon}<\alpha+b_{1}$. Using (10.8), (10.1), Lemma 10.9(i)-(ii)
and Lemma 10.8(ii), we get
$\displaystyle s\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>x_{d}\vee
s^{1/\alpha}}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(z_{d})}{|x-z|^{d+\alpha}}dz$
$\displaystyle=s(x_{d}\vee
s^{1/\alpha})^{b_{1}}\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>x_{d}\vee
s^{1/\alpha}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{f_{\gamma+{\varepsilon},\eta_{1},\eta_{2},k,l}(z_{d})}{z_{d}^{{\varepsilon}}|x-z|^{d+\alpha+b_{1}}}dz$
$\displaystyle\leq c_{5}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>x_{d}\vee
s^{1/\alpha}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{f_{\gamma+{\varepsilon},\eta_{1},\eta_{2},k,l}(2|x-z|)}{z_{d}^{{\varepsilon}}|x-z|^{d+\alpha+b_{1}}}dz$
$\displaystyle\leq c_{6}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>x_{d}\vee
s^{1/\alpha}}\bigg{(}\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}^{{\varepsilon}+\gamma+2{\varepsilon}}\frac{f_{\gamma+{\varepsilon},\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha})}{z_{d}^{{\varepsilon}}|x-z|^{d+\alpha+b_{1}}}dz$
$\displaystyle=c_{6}s(x_{d}\vee
s^{1/\alpha})^{b_{1}-\gamma-3{\varepsilon}}f_{\gamma+{\varepsilon},\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha})\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|>x_{d}\vee
s^{1/\alpha}}\frac{1}{z_{d}^{\varepsilon}|x-z|^{d+\alpha+b_{1}-\gamma-3{\varepsilon}}}dz$
$\displaystyle\leq c_{7}s(x_{d}\vee s^{1/\alpha})^{-\alpha}(x_{d}\vee
s^{1/\alpha})^{-{\varepsilon}}f_{\gamma+{\varepsilon},\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha})\leq c_{7}f_{\gamma,\eta_{1},\eta_{2},k,l}(x_{d}\vee
s^{1/\alpha}).$
(ii) Case $\gamma>\alpha+b_{1}$: Fix ${\varepsilon}>0$ such that
$\gamma-{\varepsilon}>\alpha+b_{1}$. Using (10.8), Lemma 10.9(i)-(ii) and
(10.2), we get
$\displaystyle s\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\vee
s^{1/\alpha}<|x-z|<2}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(z_{d})}{|x-z|^{d+\alpha}}dz$
$\displaystyle\leq c_{8}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\vee
s^{1/\alpha}<|x-z|<2}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(2|x-z|)}{|x-z|^{d+\alpha+b_{1}}}dz$
$\displaystyle\leq c_{9}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}f_{\gamma,\eta_{1},\eta_{2},k,l}(4)\log^{b_{2}}\bigg{(}e+\frac{2}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\int_{z\in{\mathbb{R}}^{d}_{+},\,|x-z|<2}\frac{dz}{|x-z|^{d+\alpha+b_{1}-\gamma+{\varepsilon}}}$
$\displaystyle\leq c_{10}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}f_{\gamma,\eta_{1},\eta_{2},k,l}(1)\log^{b_{2}}\bigg{(}e+\frac{2}{x_{d}\vee
s^{1/\alpha}}\bigg{)}.$
(iii) Case $\gamma=\alpha+b_{1}$: In this case, we see that
$\displaystyle s\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\vee
s^{1/\alpha}<|x-z|<2}\bigg{(}1\wedge\frac{x_{d}\vee
s^{1/\alpha}}{|x-z|}\bigg{)}^{b_{1}}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{(x_{d}\vee
s^{1/\alpha})\wedge|x-z|}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(z_{d})}{|x-z|^{d+\alpha}}dz$
$\displaystyle\leq c_{11}s(x_{d}\vee
s^{1/\alpha})^{b_{1}}\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\vee
s^{1/\alpha}<|x-z|<2}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\frac{f_{\gamma,\eta_{1},\eta_{2},k,l}(|x-z|)}{|x-z|^{d+\alpha+b_{1}}}dz$
$\displaystyle=c_{11}s(x_{d}\vee s^{1/\alpha})^{b_{1}}$
$\displaystyle\times\int_{z\in{\mathbb{R}}^{d}_{+},\,x_{d}\vee
s^{1/\alpha}<|x-z|<2}\log^{b_{2}}\bigg{(}e+\frac{|x-z|}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}\bigg{(}e+\frac{k}{|x-z|}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{|x-z|}{l}\bigg{)}\frac{dz}{|x-z|^{d}}$
$\displaystyle\leq c_{12}s(x_{d}\vee s^{1/\alpha})^{b_{1}}\int_{x_{d}\vee
s^{1/\alpha}}^{2}\log^{b_{2}}\bigg{(}e+\frac{r}{x_{d}\vee
s^{1/\alpha}}\bigg{)}\log^{\eta_{1}}\bigg{(}e+\frac{k}{r}\bigg{)}\log^{\eta_{2}}\bigg{(}e+\frac{r}{l}\bigg{)}\frac{dr}{r}.$
The proof is complete. $\Box$
###### Lemma 10.11.
Let $b_{1},b_{2}\geq 0$. For any $0<k\leq l<1$,
$\int_{l}^{2}\log^{b_{1}}\bigg{(}e+\frac{r}{k}\bigg{)}\log^{b_{1}}\bigg{(}e+\frac{r}{l}\bigg{)}\log^{b_{2}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}\asymp\log^{b_{1}}\bigg{(}e+\frac{1}{k}\bigg{)}\log^{b_{1}+b_{2}+1}\bigg{(}e+\frac{1}{l}\bigg{)},$
with comparison constants independent of $k$ and $l$.
Proof. Note that
$\log\bigg{(}e+\frac{1}{r}\bigg{)}\asymp\log\bigg{(}\frac{2e}{r}\bigg{)}\asymp\log\bigg{(}e+\frac{1}{\sqrt{r}}\bigg{)},\quad
0<r<2.$ (10.9)
Hence, we get
$\displaystyle\int_{l}^{2}\log^{b_{1}}\bigg{(}e+\frac{r}{k}\bigg{)}\log^{b_{1}}\bigg{(}e+\frac{r}{l}\bigg{)}\log^{b_{2}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\geq
c_{1}\int_{\sqrt{l}}^{2}\log^{b_{1}}\bigg{(}\frac{2er}{k}\bigg{)}\log^{b_{1}}\bigg{(}\frac{2er}{l}\bigg{)}\log^{b_{2}}\bigg{(}\frac{2e}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\geq
c_{1}\log^{b_{1}}\bigg{(}\frac{2e}{\sqrt{k}}\bigg{)}\log^{b_{1}}\bigg{(}\frac{2e}{\sqrt{l}}\bigg{)}\int_{\sqrt{l}}^{2}\log^{b_{2}}\bigg{(}\frac{2e}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle=\frac{c_{1}}{b_{2}+1}\log^{b_{1}}\bigg{(}\frac{2e}{\sqrt{k}}\bigg{)}\log^{b_{1}}\bigg{(}\frac{2e}{\sqrt{l}}\bigg{)}\bigg{(}\log^{b_{2}+1}\bigg{(}\frac{2e}{\sqrt{l}}\bigg{)}-1\bigg{)}$
$\displaystyle\geq
c_{2}\log^{b_{1}}\bigg{(}e+\frac{1}{k}\bigg{)}\log^{b_{1}+b_{2}+1}\bigg{(}e+\frac{1}{l}\bigg{)}.$
On the other hand, using (10.9) and (10.2), we get
$\displaystyle\int_{l}^{2}\log^{b_{1}}\bigg{(}e+\frac{r}{k}\bigg{)}\log^{b_{1}}\bigg{(}e+\frac{r}{l}\bigg{)}\log^{b_{2}}\bigg{(}e+\frac{1}{r}\bigg{)}\frac{dr}{r}$
$\displaystyle\leq
c_{3}\log^{b_{1}}\bigg{(}e+\frac{2}{k}\bigg{)}\log^{b_{1}}\bigg{(}e+\frac{2}{l}\bigg{)}\int_{l}^{2}\log^{b_{2}}\bigg{(}\frac{2e}{r}\bigg{)}\frac{dr}{r}\leq
c_{4}\log^{b_{1}}\bigg{(}e+\frac{1}{k}\bigg{)}\log^{b_{1}+b_{2}+1}\bigg{(}e+\frac{1}{l}\bigg{)}.$
$\Box$
###### Lemma 10.12.
Let $\gamma>1$ and $b_{1},b_{2}\geq 0$. For any $a,k,l>0$,
$\int_{a}^{\infty}t^{-\gamma}\bigg{(}1\wedge\frac{k}{t}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{t}\bigg{)}^{b_{2}}dt\asymp
a^{1-\gamma}\bigg{(}1\wedge\frac{k}{a}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{a}\bigg{)}^{b_{2}},$
with comparison constants independent of $a,k$ and $l$.
Proof. We have
$\displaystyle\int_{a}^{\infty}t^{-\gamma}\bigg{(}1\wedge\frac{k}{t}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{t}\bigg{)}^{b_{2}}dt\leq\bigg{(}1\wedge\frac{k}{a}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{a}\bigg{)}^{b_{2}}\int_{a}^{\infty}t^{-\gamma}dt\leq\frac{a^{1-\gamma}}{\gamma-1}\bigg{(}1\wedge\frac{k}{a}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{a}\bigg{)}^{b_{2}}$
and
$\displaystyle\int_{a}^{\infty}t^{-\gamma}\bigg{(}1\wedge\frac{k}{t}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{t}\bigg{)}^{b_{2}}dt$
$\displaystyle\geq\bigg{(}1\wedge\frac{k}{2a}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{2a}\bigg{)}^{b_{2}}\int_{a}^{2a}t^{-\gamma}dt$
$\displaystyle\geq\frac{(1-2^{1-\gamma})a^{1-\gamma}}{2^{b_{1}+b_{2}}(\gamma-1)}\bigg{(}1\wedge\frac{k}{a}\bigg{)}^{b_{1}}\bigg{(}1\wedge\frac{l}{a}\bigg{)}^{b_{2}}.$
$\Box$
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Let $f, g\in \RR^{\RR_\infty}$ with $f(x) \asymp g(x) \ (x \to \infty)$, and let $n$ be a nonnegative integer. If $g$ has exact logexponential degree to order $n$, then so does $f$, and one has $\dege_k f = \dege_k g$ for all $k \leq n$.
\end{proposition}
\begin{proof}
Let $\underline{g} = g|_{\dom f \cap \dom g}$.
By Proposition \ref{aspropexoexp}(3), one has $f_{(k)}(x) \asymp \underline{g}_{(k)}(x) \ (x \to \infty)$ for all $k$. Moreover, by Proposition \ref{arithbridge2}, the function $\underline{g}_{(k)} = g_{(k)}|_{\dom f_{(k)} \cap \dom g_{(k)}}$ has exact degree, whence, by Proposition \ref{firstprop1}(3), the function $f_{(k)}$ also has exact degree, for all $k \leq n$. Finally, again by Propositions
\ref{arithbridge2} and \ref{firstprop1}(3), one has $\dege_k f = \dege_k \underline{g} = \dege_k g$ for all $k \leq n$.
\end{proof}
Let $f \in \RR^{\RR_\infty}$, and define $f_{\{k\}}$ and $\underline{\dege}_k f =\underline{ \deg } \, f_{\{k\}} \in \overline{\RR}$ for all nonnegative integers $k$, recursively, as follows. Let $f_{\{0\}} = f$. Suppose that $f_{\{k\}}$ is defined, and set $d_k = \underline{\dege}_k f =\underline{\deg} \, f_{\{k\}}.$ We then let
$$f_{\{k+1\}}(x) = \left.
\begin{cases}
f_{\{k\}}(e^x) e^{-d_k x}& \text{if } d_k \neq \pm \infty \\
\log |f_{\{k\}}(x)| & \text{if } d_k = \infty \\
\displaystyle -\frac{1}{\min(\log |f_{\{k\}}(x)|,0)} & \text{if } d_k =- \infty.
\end{cases}
\right.$$
We also set $$\underline{\dege}\, f = (\underline{\dege}_0 f, \underline{\dege}_1 f, \underline{\dege}_2 f, \ldots) \in \prod_{n = 0}^\infty \overline{\RR}.$$ We call $\underline{\dege}\, f$ the {\bf lower logexponential degree of $f$}\index{lower logexponential degree $\underline{\dege}$}\index{lower exponential degree $\underline{\dege}$}\index[symbols]{.g g@$\underline{\dege}\, f$} and $\underline{\dege}_k f$ the {\bf lower logexponential degree of $f$ of order $k$}.
For all $f \in \RR^{\RR_\infty}$, one has
$$\limsup_{x \to \infty}\, (- f(x)) = -\liminf_{x \to \infty} f(x)$$
$$\liminf_{x \to \infty}\, (- f(x)) = -\limsup_{x \to \infty} f(x),$$
and $\lim_{x \to \infty} f(x)$ exists or is $\pm \infty$ if and only if $\limsup_{x \to \infty} f(x) = \liminf_{x \to \infty} f(x)$. From these basic facts, and from the definitions of degree, lower degree, $f_{(k)}$, and $f_{\{k\}}$, one readily deduces the following.
\begin{proposition}
Let $f \in \RR^{\RR_\infty}$. If $f$ is not eventually nonzero, then $\underline{\dege}\, f = (-\infty, -\infty, -\infty,\ldots)$. Suppose, on the other hand, that $f$ is eventually nonzero. Then one has the following.
\begin{enumerate}
\item $f_{\{k\}} = 1/(1/f)_{(k)}$ for all $k$.
\item $\underline{\dege}\, f = - \dege(1/f)$.
\item $\underline{\dege}\, f \leq \dege f$, and equality holds if and only if $f$ has exact logexponential degree.
\item $f$ has exact logexponential degree to order $n \geq 0$ if and only if $\dege_k f = \underline{\dege}_k f$ for all $k \leq n$.
\end{enumerate}
\end{proposition}
\begin{corollary}\label{exactloginverse}
Let $r \in \RR^{\RR_{\infty}}$, and let $n$ be a nonnegative integer.
\begin{enumerate}
\item If $r$ has exact logexponential degree to order $n$ and is eventually nonzero, then, for all nonnegative integers $k \leq n$, one has
$$r_{(k+1)}(x) = \left.
\begin{cases}
r_{(k)}(e^x) e^{-(\deg r_{(k)}) x}& \text{if } \deg r_{(k)} \neq \pm \infty \\
\log r_{(k)}(x) & \text{if } \deg r_{(k)} = \infty \\
\displaystyle -\frac{1}{\log r_{(k)}(x)} & \text{if } \deg r_{(k)} =- \infty
\end{cases}
\right.$$ and $(1/r)_{(k)} = 1/r_{(k)}$, and therefore $\dege_k (1/r) = -\dege_k r$.
\item $r$ has exact logexponential degree to order $n$ if and only if $\dege_k r = -\infty$ for all $k \leq n$ or $r$ is eventually nonzero and $\dege_k (1/r) = -\dege_k r$ for all $k \leq n$.
\item $r$ has exact logexponential degree if and only if $\dege r = (-\infty, -\infty,-\infty, \ldots)$ or $r$ is eventually nonzero and $\dege(1/r) = -\dege r$.
\end{enumerate}
\end{corollary}
\begin{corollary}\label{aspropexoexpunder}
Let $f, g \in \RR^{\RR_\infty}$, and let $\underline{g} = g|_{\dom f \cap \dom g}$.
\begin{enumerate}
\item If $f(x) \gg g(x) \ (x \to \infty)$, then $\underline{\dege} \, f \geq \underline{\dege} \, \underline{g} \geq \underline{\dege} \, g$.
\item If $f(x) = O(g(x)) \ (x \to \infty)$, then $\underline{\dege} \, f \leq \underline{\dege} \, \underline{g}$.
\item If $f(x) \asymp g(x) \ (x \to \infty)$, then $f_{\{k\}}(x) \asymp \underline{g}_{\{k\}}(x) \ (x \to \infty)$ for all $k$ and $\underline{\dege} \, f = \underline{\dege} \, \underline{g}$.
\item If $f$ is eventually bounded away from $0$, then $\underline{\dege} \, f \geq (0,0,0,\ldots)$.
\item If $\underline{\dege} \, f > (0,0,0,\ldots)$, then $\displaystyle \lim_{x \to \infty} |f(x) | = \infty$.
\end{enumerate}
\end{corollary}
\begin{example}
By \cite[Theorem 328]{har}, one has
$$ \liminf_{n \to\infty} \frac{\phi(n) \log \log n}{n} =e^{-\gamma },$$
where $\phi(n)$ is Euler's totient. It follows that
$$\underline{\dege} \, \phi(n) = (1,0,-1,0,0,0,\ldots).$$
Moreover, it is easy to check that
$$ \limsup_{n \to\infty} \frac{\phi(n)}{n} =1,$$
and therefore
$$\dege \phi(n) = (1,0,0,0,\ldots).$$
\end{example}
The following proposition yields, in the subsequent corollary, a useful property of functions of exact logexponential degree.
\begin{proposition}\label{oexpprop}
Let $f,g \in \RR^{\RR_\infty}$, where $\dom g$ contains the intersection of $\dom f$ with some neighborhood of $\infty$, and let $n$ be a nonnegative integer.
\begin{enumerate}
\item If the smallest nonnegative integer $N$ such that $\dege_N f \neq \underline{\dege}_N g$ exists and is at most $n$, where also $\dege_N f < \underline{\dege}_N g$, then $f(x) = o(g(x)) \ (x \to \infty)$.
\item $\dege f <\underline{\dege}\, g$, then $f(x) = o(g(x)) \ (x \to \infty)$.
\end{enumerate}
\end{proposition}
\begin{proof}
It suffices to prove statement (1). Note that the function $g_{(k)}$ is eventually nonzero, for each $k \leq n$. For all $k < N$, one has $d_k := \dege_k f = \underline{\dege}_k g$. Moreover, one has $\deg f_{(N)} < \underline{\deg}\, g_{\{N\}}$, so that $f_{(N)}(x) = o(g_{\{N\}}(x)) \ (x \to \infty)$, by Proposition \ref{firstprop2}(7). Let $m$ be least such that $f_{(m)}(x) = o(g_{\{m\}}(x)) \ (x \to \infty)$, so that $m \leq N$.
We claim that $m = 0$ and therefore $f(x) = o(g(x)) \ (x \to \infty)$. Suppose to obtain a contradiction that $m \geq 1$. Suppose first that $d_{m-1} \neq \pm \infty$. Then $g_{\{m-1\}}(x) = x^{d_{m-1}}g_{\{m\}}(\log x)$ and therefore
$$f_{(m-1)}(x)= x^{d_{m-1}}f_{(m)}(\log x) = o( g_{\{m-1\}}(x)) \ (x \to \infty).$$ Suppose, on the other hand, that $d_{m-1} = -\infty$. Then $\lim_{x \to \infty} \frac{\log |f_{(m-1)}(x)|}{\log x} = -\infty$ and thus
$$\lim_{x \to \infty} \frac{1}{ f_{(m)}(x)} = - \lim_{x \to \infty} \log |f_{(m-1)}(x)| = \infty,$$
so that
\begin{align*}
\frac{|f_{(m-1)}(x)|}{|g_{\{m-1\}}(x)|} \leq |f_{(m-1)}(x)|\max\left(\left| \frac{1}{g_{\{m-1\}}(x)} \right |,1\right) & = \exp\left(-\frac{1}{f_{(m)}(x)}+\frac{1}{g_{\{m\}}(x)} \right) \\
& = \exp\left(\left(\frac{f_{(m)}(x)}{g_{\{m\}}(x)}-1 \right) \frac{1}{f_{(m)}(x)}\right) \\ & \to \exp(-1 \cdot \infty) = 0 \\
\end{align*}
as $x \to \infty$, and therefore $f_{(m-1)}(x)= o( g_{\{m-1\}}(x)) \ (x \to \infty)$.
Finally, suppose that $d_{m-1}= \infty$. Then $\lim_{x \to \infty} \frac{\log |g_{\{m-1\}}(x)|}{ \log x} = \infty$ and $g_{\{m\}}(x) = \log |g_{\{m-1\}}(x)|$
and thus
$$\lim_{x \to \infty} g_{\{m\}}(x) = \lim_{x \to \infty} \log |g_{\{m-1\}}(x)|= \infty,$$
so that
\begin{align*}
\frac{|f_{(m-1)}(x)|}{|g_{\{m-1\}}(x)|} \leq \frac{\max(|f_{(m-1)}(x)|,1)}{|g_{\{m-1\}}(x)|} & = \exp\left(f_{(m)}(x)-g_{\{m\}}(x) \right) \\
& = \exp\left(\left(\frac{f_{(m)}(x)}{g_{\{m\}}(x)}-1 \right) g_{\{m\}}(x)\right) \\ & \to \exp(-1 \cdot \infty) = 0, \\
\end{align*}
whence $f_{(m-1)}(x)= o( g_{\{m-1\}}(x)) \ (x \to \infty)$. Thus, we have shown that $$f_{(m-1)}(x)= o( g_{\{m-1\}}(x)) \ (x \to \infty),$$ in all three cases. This contradicts the minimality of $m$. It follows that $m = 0$, as claimed.
\end{proof}
\begin{corollary}\label{oexp}
Let $f,g \in \RR^{\RR_\infty}$, where $\dom g$ contains the intersection of $\dom f$ with some neighborhood of $\infty$, and let $n$ be a nonnegative integer.
\begin{enumerate}
\item If $g$ has exact logexponential degree to order $n$, and if the smallest nonnegative integer $N$ such that $\dege_N f \neq \dege_N g$ exists and is at most $n$, where also $\dege_N f < \dege_N g$, then $f(x) = o(g(x)) \ (x \to \infty)$.
\item If $g$ has exact logexponential degree and $\dege f < \dege g$, then $f(x) = o(g(x)) \ (x \to \infty)$.
\end{enumerate}
\end{corollary}
\begin{corollary}\label{oexpcor0}
Let $f, g \in \RR^{\RR_\infty}$, where $\dom g$ contains the intersection of $\dom f$ with some neighborhood of $\infty$, and let $\underline{g} = g|_{\dom f \cap \dom g}$. Suppose that $ f(x) = O(g(x)) \ (x \to \infty)$ but $ f(x) \neq o(g(x)) \ (x \to \infty)$. Then one has
$$\underline{\dege} \, f \leq \underline{\dege} \, \underline{g} \leq \dege f \leq \dege \, g.$$
Consequently, if $g$ has exact logexponential degree, then $\dege f = \dege g$. Likewise, if $f$ has exact logexponential degree, then $\dege f = \underline{\dege} \, \underline{g}$.
\end{corollary}
\begin{corollary}\label{oexpcor}
Let $f, g \in \RR^{\RR_\infty}$, where $g$ has exact logexponential degree and $\dom g$ contains the intersection of $\dom f$ with some neighborhood of $\infty$. One has
\begin{align*}
\degl f < \degl g \quad & \Longrightarrow \quad \dege f < \dege g \\
\quad & \Longrightarrow \quad f(x) = o(g(x)) \ (x \to \infty) & \\
\quad & \Longrightarrow \quad f(x) = O(g(x)) \ (x \to \infty) \\
\quad & \Longrightarrow \quad \dege f \leq \dege g \\
\quad & \Longrightarrow \quad \degl f \leq \degl g.
\end{align*}
In particular, if $ f(x) = O(g(x)) \ (x \to \infty)$ but $ f(x) \neq o(g(x)) \ (x \to \infty)$, then $\dege f = \dege g$.
\end{corollary}
Note that none of the implications in the corollary above are reversible.
\begin{remark}[Leading coefficient and logexponential degree]
Assume the notation of Remark \ref{lc}. Let $f \in \RR^{\RR_\infty}$ with $\deg f = d \neq \pm \infty$. If $\operatorname{lc} f < \infty$, that is, if $f(x) = O(x^{d}) \ (x \to \infty)$, then $\dege f \leq (d,0,0,0,\ldots)$. On the other hand, if $\operatorname{lc} f > 0$, that is, if $f(x) \neq o(x^{d}) \ (x \to \infty)$, then $\dege f \geq (d,0,0,0,\ldots)$. Consequently, if $0< \operatorname{lc} f < \infty$, then $\dege f = (d,0,0,0,\ldots)$.
\end{remark}
\subsection{Relationships with logarithmico-exponential functions}
The following proposition provides constraints on what sequences in $\prod_{n = 0}^\infty\overline{\RR}$ can be of the form $\dege f$ for some $f \in \RR^{\RR_\infty}$. By Theorem \ref{degeequiv} below, these constraints are exhaustive.
\begin{proposition}\label{degeprop}
Let $f \in \RR^{\RR_\infty}$, and let the $f_{(k)}$ be defined recursively as in the definition of $\dege$, so that $\dege_k f = \deg f_{(k)}$ for all $k$. For all $n \geq 0$, one has the following.
\begin{enumerate}
\item Suppose that $\dege_n f = \infty$. Then one has $f_{(n+1)}(x) \neq O(\log x) \ (x \to \infty)$ and therefore
$$\dege_{(n+1)} f \geq (0,1,0,0,0,\ldots)$$
\begin{align*}
\dege f \geq (\dege_0 f, \ldots, \dege_{n} f, 0,1, 0,0,0,\ldots).
\end{align*}
\item Suppose that $\dege_n f = -\infty$. Then one has $f_{(n+1)}(x) = o\left(\frac{1}{\log x}\right) \ (x \to \infty)$ and therefore
$$\dege_{n+1} f \leq (0,-1,0,0,0,\ldots)$$
\begin{align*}
\dege f \leq (\dege_0 f, \ldots, \dege_{n} f, 0,-1, 0,0,0,\ldots).
\end{align*}
\item If $\dege_n f$ is finite and $\dege_{n+1} f= \infty$, then one has $f_{(n+2)}(x) = o(x) \ (x \to \infty)$, so that $\dege f_{n+2} \leq (1,0,0,0,\ldots)$ and therefore
\begin{align*}
(\dege_0 f, \ldots, \dege_n f, \infty, 0,1, 0,0,0\ldots) & \leq \dege f \\ & \leq (\dege_0 f, \ldots, \dege_n f, \infty, 1,0,0,\ldots).
\end{align*}
\item If $\dege_n f$ is finite and $\dege_{n+1} f= -\infty$, then one has $f_{(n+2)}(x) \neq O\left(\frac{1}{x} \right) \ (x \to \infty)$, so that $\dege_{n+2} f \geq (-1,0,0,0,\ldots)$ and therefore
\begin{align*}
(\dege_0 f, \ldots, \dege_n f, -\infty, 0,-1, 0,0,0\ldots) & \geq \dege f \\ & \geq (\dege_0 f, \ldots, \dege_n f,- \infty, -1,0,0,\ldots).
\end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose that $\dege_n f = \infty$. Then $$\limsup_{x \to \infty} \frac{\log |f_{(n)}(x)|}{\log x} = \infty$$ and therefore $f_{(n+1)}(x) = \max(\log |f_{(n)}(x)|,0) \neq O(\log x) \ (x \to \infty)$, so that $\dege f_{(n+1)} \geq (0,1,0,0,0,\ldots)$. Statement (1) follows.
Suppose that $\dege_n f = -\infty$. Then
$\lim_{x \to \infty} \frac{-1}{\log |f_{(n)}(x)|} = 0$ and $$\lim_{x \to \infty} \frac{\log |f_{(n)}(x)|}{\log x} = -\infty$$ and therefore
$$\lim_{x \to \infty} \frac{\frac{-1}{\log |f_{(n)}(x)|}}{\frac{1}{\log x}} = 0,$$
so that
$$f_{(n+1)}(x) = \frac{-1}{\log |f_{(n)}(x)|} = o \left(\frac{1}{\log x} \right) \ (x \to \infty)$$
and therefore $\dege f_{(n+1)} \leq (0,-1,0,0,0,\ldots)$. Statement (2) follows.
Suppose now that $d = \dege_n f = \deg f_{(n)}$ is finite. For any $t >d$, one has $|f_{(n)}(x)| \leq x^t$ for all $x\gg 0$, and one has $f_{(n+1)}(x) = f_{(n)}(e^x)e^{-dx}$. If $\dege_{n+1} f = \infty$, then one has
\begin{align*}
0 \leq f_{(n+2)}(x) & = \log \max (|f_{(n+1)}(x)|,1) \\
& = \log \max( |f_{(n)}(e^x)e^{-dx}|,1) \\
& \leq \log \max (e^{(t-d)x},1) \\
& = (t-d) x
\end{align*}
for all $t >d$ and all $x \gg 0$, and therefore $0 \leq f_{(n+2)}(x) = o(x) \ (x \to \infty)$. Statement (3) follows. Suppose, on the other hand, that $\dege_{n+1} f = -\infty$. Then for any $t < d$, one has $|f_{(n)}(x)|> x^t$ on an unbounded set of $x >0$. It follows that $$0 \leq -\log |f_{(n+1)}(x)| = -\log | f_{(n)}(e^x)| + dx < (d-t)x,$$ so that
$$f_{(n+2)}(x) = -\frac{1}{\log |f_{(n+1)}(x)|} \geq \frac{1}{(d-t)x} > 0,$$
on an unbounded set of $x> 0$, and therefore $f_{(n+2)}(x) \neq O\left(\frac{1}{x} \right) \ (x \to \infty)$. Statement (4) follows.
\end{proof}
Note that the various upper and lower bounds in the proposition are attained by examples (19)(e) and (19)(f) of Example \ref{degeex}.
Let $$\prod_{n = 0}^{ \infty *}\overline{\RR} \subsetneq \prod_{n = 0}^{\infty}\overline{\RR} \index[symbols]{.g h@$\prod_{n = 0}^ { \infty *} \overline{\RR}$}\index{restricted product $\prod_{n = 0}^{ \infty *}\overline{\RR}$}$$ (to be contrasted with $\prod_{n = 0}^{* \infty}\overline{\RR}$) denote the {\bf restricted product} consisting of all sequences $\dd$ in $\prod_{n = 0}^{\infty}\overline{\RR}$ satisfying the following four conditions for all nonnegative integers $n$.
\begin{enumerate}
\item If $\dd_n = \infty$, then $\dd \geq (\dd_0, \dd_1, \ldots, \dd_n, 0,1,0,0,0,\ldots).$
\item If $\dd_n = -\infty$, then $\dd \leq (\dd_0, \dd_1, \ldots, \dd_n, 0,-1,0,0,0,\ldots).$
\item If $\dd_n$ is finite and $\dd_{n+1} = \infty$, then $\dd \leq (\dd_0, \dd_1, \ldots, \dd_{n+1},1,0,0,0,\ldots).$
\item If $\dd_n$ is finite and $\dd_{n+1} = -\infty$, then $\dd \geq (\dd_0, \dd_1, \ldots, \dd_{n+1}, -1,0,0,0,\ldots).$
\end{enumerate}
By Proposition \ref{degeprop}, if $\dd = \dege f$ for some $f \in \RR^{\RR_\infty}$, then $\dege f \in \prod_{n = 0}^{ \infty *}\overline{\RR}$. In fact the converse holds.
The following theorem is proved in Section 7.5. (We do not require the theorem in Parts 2 or 3.)
\begin{theorem}\label{degeequiv}
For any sequence $\dd \in \prod_{n= 0}^{\infty}\overline{\RR}$, one has $\dd = \dege f$ for some $f \in \RR^{\RR_\infty}$ if and only if $\dd \in \prod_{n = 0}^{ \infty *}\overline{\RR}$, if and only if $\dd = \dege f$ for some positive, monotonic, infinitely differentiable function $f$ on $\RR_{>0}$ of exact logexponential degree.
\end{theorem}
Let $$\mathbb{L} = \operatorname{Def}(\RR,\id,\exp, \log,+,\cdot,/, \circ)\index[symbols]{.i k@$\mathbb{L}$}$$ denote the ring of all {\bf logarithmico-exponential functions}\index{logarithmico-exponential functions} [117] [118], that is, the ring of all (germs of) real functions defined on a neighborhood of $\infty$ that can be can be built from all real constants and the functions $\id$, $\exp$, and $\log$ using the operations $+$, $\cdot$, $/$, and $\circ$. The ring $\mathbb{L}$ contains the field $\RR(\mathfrak{L})$ as a subfield, and, as a consequence of Theorem \ref{infpropexp} below, it is the proper analogue of the field $\RR(\mathfrak{L})$ for $\dege$.
Note that, by \cite[Theorem, p.\ 24]{har3}, any function in the ring $\mathbb{L}$ is positive, $0$, or negative, for all $x \gg 0$, and thus $\mathbb{L}$ is a field.
\begin{proposition}\label{Kfield}
Let $r \in \mathbb{L}$. One has the following.
\begin{enumerate}
\item $r$ is positive, $0$, or negative, for all $x \gg 0$.
\item $r$ is infinitely differentiable on its domain, and all of its derivatives lie in $\mathbb{L}$.
\item Each of the functions $r_{(k)}$ as in the definition of $\dege r$ has exact degree and is equal to some function in $\mathbb{L}$ on some neighborhood of $\infty$.
\item $r$ has exact logexponential degree.
\end{enumerate}
\end{proposition}
For any real functions $f$ and $g$ defined on a neighborhood of $\infty$, we write $f \leq_\infty g$ if $g-f$ is eventually positive or eventually $0$, we write $f <_\infty g$ if $g-f$ is eventually positive, and we write $f =_\infty g$ if $g-f$ is eventually $0$. Thus, for all $f, g\in \mathbb{L}$, one has either $f\leq_\infty g$ or $g \leq_\infty f$, and $f =_\infty g$ holds if and only if both $f\leq_\infty g$ and $g \leq_\infty f$. The ordering $\leq_\infty $ makes $\mathbb{L}$ into a (totally) ordered field, provided that one identifies any two functions $f$ and $g$ that are eventually equal, i.e., that satisfy $f =_\infty g$. This totally ordered field is a subfield of the ring of germs of real functions at $\infty$. Let us also define $$\mathbb{L}_{> 0} = \{f \in \mathbb{L}: 0<_\infty f\}.$$
Note that, if $r \in \mathbb{L}^*$, then $|r| \in \mathbb{L}_{> 0} $ and $r_{(k)} \in \mathbb{L}^*$ for all $k$, and, if also $\dege_n r = \pm \infty$ for some $n$, then $r_{(k)} \in \mathbb{L}_{> 0}$ for all $k > n$, where, again, one identifies functions that are eventually equal.
For any $f,g \in \RR^{\RR_\infty}$, we let $\mathcal{E}(f,g) = \mathcal{S}(\dege f, \dege g)$, that is, we let $\mathcal{E}(f,g)$ denote the infimum of the set of all nonnegative integers $n$ such that $\dege_n f \neq \dege_n g$ (which is $\infty$ if there is no such nonnegative integer $n$).\index[symbols]{.g p@$\mathcal{E}(f,g)$} We say that $g$ {\bf approximates $f$ to logexponential order $n$}\index{approximates $f$ to logexponential order $n$} if $\mathcal{E}(f,g) > n$, that is, if $\dege_k f = \dege_k g$ for all $k \leq n$. The following proposition implies that any function in $\RR^{\RR_\infty}$ can be approximated to any logexponential order by some function in $\mathbb{L}_{> 0}$.
\begin{proposition}\label{degerprop}
Let $\dd \in \prod_{n = 0}^{ \infty}\overline{\RR}$. Then $\dd \in \prod_{n = 0}^{ \infty*}\overline{\RR}$ if and only if, for every nonnegative integer $n$, there exists an $r \in \mathbb{L}_{> 0}$ such that $\dege_k r = \dd_k$ for all $k \leq n$. Consequently, for any $f \in \RR^{\RR_\infty}$ and any nonnegative integer $n$, there exists an $r \in \mathbb{L}_{> 0}$ such that $\mathcal{E}(f,r) > n$.
\end{proposition}
\begin{proof}
By Proposition \ref{degeprop} and the definition of $\prod_{n = 0}^{ \infty *}\overline{\RR}$, if for every nonnegative integer $n$ there exists an $r \in \mathbb{L}_{> 0}$ such that $\dege_k r = \dd_k$ for all $k \leq n$, then $\dd \in \prod_{n = 0}^{ \infty*}\overline{\RR}$. Conversely, let $\dd \in \prod_{n= 0}^{\infty*}\overline{\RR}$, and let $n$ be a nonnegative integer. Then the shifted sequence $(\dd_1, \dd_2, \dd_3, \ldots)$ also lies in $\prod_{n= 0}^{\infty*}\overline{\RR}$. Therefore, by induction we may assume that we have an $s \in \mathbb{L}_{> 0}$ such that $\dege_k s = \dd_{k+1}$ for all $k$ with $1 \leq k \leq n$, so that, for example, $\deg s = \dd_1$. We wish to find an $r \in \mathbb{L}_{> 0}$ such that $\dege_k r = \dd_k$ for all $k \leq n$. Equivalently, we wish to find an $r \in \mathbb{L}_{> 0}$ such that $\deg r= \dd_0$ and $\dege_k r = \dege_{k-1} s$ for all $k$ with $1 \leq k \leq n$.
Suppose first that $\dd_0 \neq \pm \infty$. If $\dd_1 \neq \pm \infty$, then $r(x) = x^{\dd_0}s(\log x) \in \mathbb{L}_{> 0}$ satisfies our requirement for $r$. Indeed, in this case, one has
$$\deg s(\log x) = \lim_{x \to \infty} \frac{\log s(\log x)}{\log x} = \lim_{x \to \infty} \frac{\log s(x)}{x} = \lim_{x \to \infty} \frac{\log s(x)}{\log x} \frac{\log x}{x} = 0$$
and therefore $r_{(1)}(x) =(s(\log x))_{(1)} = s(x)$, so that $$\dege s(\log x) = (0,\dege_0 s , \dege_1 s, \dege_2 s, \ldots)$$
and thus
$$\dege r = (\dd_0, \dege_0 s , \dege_1 s, \dege_2 s, \ldots),$$
whence $r$ fulfills our requirement that $\dege_k r = \dd_k$ for all $k \leq n$ (and in fact $\dege_k r = \dege_{k-1} s$ for all $k \geq 1$).
Suppose, instead, that $\dd_1 = \infty$, so that
$$ \dd \leq (\dd_0, \infty,1,0,0,0,\ldots).$$
Then $s(x) = e^{s_{(1)}(x)}$ for all $x \gg 0$, where $\lim_{x \to \infty} \frac{s_{(1)}(x)}{\log x} = \infty$ and
$$\dege s_{(1)} \leq (1,0,0,0,\ldots).$$ If $ \dege s_{(1)} < (1,0,0,0,\ldots)$, then $s_{(1)}(x) = o(x) \ (x \to \infty)$, so that
$$\deg s(\log x) = \lim_{x \to \infty} \frac{s_{(1)}(\log x)}{\log x} = \lim_{x \to \infty} \frac{s_{(1)}(x)}{x} = 0$$
and therefore $(s(\log x))_{(1)} = s(x)$ and thus
$$\dege s(\log x) = (0,\dege_0 s, \dege_1 s, \dege_2 s, \ldots).$$ Therefore, in this case, the function
$r(x) = x^{\mathbf{d_0}} s(\log x)$ satisfies $r_{(1)}(x) = s(x)$ and fulfills our requirement for $r$. Suppose, on the other hand, that $\dege s_{(1)} = (1,0,0,0,\ldots)$,
so that
$$\dege s = (\infty, 1,0,0,0,\ldots).$$
Then $r(x) = x^{\mathbf{d_0}} x^{1/\log^{\circ(n+1) }x} = x^{\mathbf{d_0}} e^{\log x/\log^{\circ(n+1) }x} \in \mathbb{L}_{> 0}$ has
$$\dege r = (\mathbf{d_0},\infty,1,0,0,0,\ldots, 0,-1,0,0,0,\ldots),$$
where the $-1$ is preceded by $n-1$ zeros, and so $\dege_k r= \dd_k$ for all $k \leq n$.
Suppose, on the other hand, that $\dd_1 = -\infty$, so that
$$\dd \geq (\dd_0, -\infty,-1,0,0,0,\ldots).$$
Then $s(x) = e^{-1/s_{(1)}(x)}$ for all $x \gg 0$, where $\lim_{x \to \infty} \frac{1/s_{(1)}(x)}{\log x} = \infty$ and
$$\dege s_{(1)} \geq (-1,0,0,0,\ldots).$$ If $ \dege s_{(1)} > (-1,0,0,0,\ldots)$, then $1/s_{(1)}(x) = o(x) \ (x \to \infty)$, so that
$$\deg s(\log x) = \lim_{x \to \infty} \frac{-1/s_{(1)}(\log x)}{\log x} = \lim_{x \to \infty} \frac{-1/s_{(1)}(x)}{x} = 0$$
and therefore $(s(\log x))_{(1)} = s(x)$ and thus
$$\dege s(\log x) = (0,\dege_0 s, \dege_1 s, \dege_2 s, \ldots).$$ Therefore, in this case, the function
$r(x) = x^{\mathbf{d_0}} s(\log x)$ satisfies $r_{(1)}(x) = s(x)$ and fulfills our requirement. Suppose, on the other hand, that $\dege s_{(1)} = (-1,0,0,0,\ldots)$,
so that
$$\dege s = (-\infty, -1,0,0,0,\ldots).$$
Then $r(x) = x^{\mathbf{d_0}} x^{-1/\log^{\circ(n+1) }x} = x^{\mathbf{d_0}} e^{-\log x/\log^{\circ(n+1) }x} \in \mathbb{L}_{> 0}$ has
$$\dege r = (\mathbf{d_0},-\infty,-1,0,0,0,\ldots, 0,1,0,0,0,\ldots),$$
where the $1$ is preceded by $n-1$ zeros, and so $\dege_k r= \dd_k$ for all $k \leq n$.
The proves the proposition in the case where $\dd_0 \neq \pm \infty$. Suppose now that $\dd_0 = \infty$, so that
$$\dd \geq (\infty,0,1,0,0,0,\ldots)$$
and therefore
$$\dege s \geq (0,1,0,0,0,\ldots).$$
Suppose that $ \dege s > (0,1,0,0,0,\ldots)$, so that $s(x) \neq O(\log x) \ (x \to \infty)$ and therefore $\lim_{x \to \infty} \frac{s(x)}{\log x} = \infty$, since $\frac{s(x)}{\log x} \in \mathbb{L}_{> 0}$ is positive and unbounded. Let $r(x) = e^{s(x)}$. Then
$$\dege r = \lim_{x \to \infty} \frac{s(x)}{\log x} = \infty$$ and therefore $r_{(1)}(x) = \log e^{s(x)} = s(x)$, so that
$$\dege r = (\infty, \dege_0 s, \dege_1 s, \dege_2 s, \ldots),$$
whence $\dege_k r = \dd_k$ for all $k\leq n$. Suppose, on the other hand, that $\dege s = (0,1,0,0,0,\ldots)$.
Then $r(x) = x^{\log^{\circ(n+1) }x} = e^{(\log x) (\log^{\circ(n+1) }x)} \in \mathbb{L}_{> 0}$ has
$$\dege r = (\infty,0,1,0,0,0,\ldots, 0,1,0,0,0,\ldots),$$
where the second $1$ is preceded by $n-1$ zeros, and so $\dege_k r= \dd_k$ for all $k \leq n$.
Finally, suppose that $\dd_0 = -\infty$, so that
$$\dd \leq (-\infty,0,-1,0,0,0,\ldots)$$
and therefore
$$\dege s \leq (0,-1,0,0,0,\ldots).$$
Suppose that $ \dege s < (0,-1,0,0,0,\ldots)$. Then $1/s(x) \neq O(\log x) \ (x \to \infty)$ and therefore $\lim_{x \to \infty} \frac{1/s(x)}{\log x} = \infty$. Let $r(x) = e^{-1/s(x)}$. Then
$$\dege r = \lim_{x \to \infty} \frac{-1/s(x)}{\log x} =- \infty$$ and therefore $r_{(1)}(x) =- \frac{1}{\log e^{-1/s(x)}} = s(x)$ for all $x \gg 0$, so that
$$\dege r = (-\infty, \dege_0 s, \dege_1 s, \dege_2 s, \ldots),$$
whence $\dege_k r = \dd_k$ for all $k\leq n$. On the other hand, suppose that $\dege s = (0,-1,0,0,0,\ldots)$.
Then $r(x) = x^{-\log^{\circ(n+1) }x} = e^{-(\log x) (\log^{\circ(n+1) }x)} \in \mathbb{L}_{> 0}$ has
$$\dege r = (-\infty,0,-1,0,0,0,\ldots, 0,-1,0,0,0,\ldots),$$
where the second $-1$ is preceded by $n-1$ zeros, and so $\dege_k r= \dd_k$ for all $k \leq n$. This completes the proof.
\end{proof}
In Section 7.6, we use transseries to prove the analogue of Theorem \ref{degeequiv} for the logarithmico-exponential functions, namely, Theorem \ref{chartheorem2}, by computing the set of all $\dd \in \prod_{n = 0}^{ \infty*}\overline{\RR}$ such that $\dd = \dege f$ for some logarithmico-exponential function $f$.
Our next theorem is an analogue of Proposition \ref{infprop} for logexponential degree. The theorem has many applications, both to proving various properties of logexponential degree (in Subsection 6.3.4) and to applying logexponential degree to analytic number theory (in Part 3).
\begin{lemma}\label{compllem}
The totally ordered set $\prod_{n = 0}^{ \infty*}\overline{\RR}$ is complete.
\end{lemma}
\begin{proof}
Let $\mathcal{S}$ be a subset of $\prod_{n = 0}^{ \infty*}\overline{\RR}$, and let $\dd$ be the infimum of $ \mathcal{S}$ in $\prod_{n = 0}^{ \infty}\overline{\RR}$. If $\dd$ lies in $\prod_{n = 0}^{ \infty*}\overline{\RR}$, then clearly $\dd$ is the infimum of $ \mathcal{S}$ in $\prod_{n = 0}^{ \infty*}\overline{\RR}$. Thus, we may suppose that $\dd$ does not lie in $\prod_{n = 0}^{ \infty*}\overline{\RR}$. Let $n$ be the smallest nonnegative integer violating conditions (1)--(4) of the definition of $\prod_{n = 0}^{ \infty*}\overline{\RR}$. There are four cases to consider.
Suppose first that $\dd_n = \infty$ and $ \dd < (\dd_0, \dd_1, \ldots, \dd_n, 0,1,0,0,0,\ldots)$. Then there exists some $\dd' \in \mathcal{S}$ such that
$$\dd \leq (\dd_0, \dd_1 \ldots, \dd_{n-1},\infty, \dd_{n+1}',\dd_{n+1}',\ldots) = \dd' < (\dd_0, \dd_1, \ldots, \dd_{n-1},\infty, 0,1,0,0,0,\ldots).$$
But this contradicts $\dd' \in \prod_{n = 0}^{ \infty*}\overline{\RR}$. Therefore this case is impossible.
Suppose now that $\dd_n = -\infty$ and $ \dd > (\dd_0, \dd_1, \ldots, \dd_n, 0,-1,0,0,0,\ldots)$.
Then one has $$\dd' > (\dd_0, \dd_1, \ldots, \dd_n, 0,-1,0,0,0,\ldots) \in \prod_{n = 0}^{ \infty*}\overline{\RR}$$
for all $\dd' \in \mathcal{S}$. Suppose that $\dd''$ is some lower bound of $\mathcal{S}$ in $\prod_{n = 0}^{ \infty*}\overline{\RR}$ such that $\dd'' > (\dd_0, \dd_1, \ldots, \dd_n, 0,-1,0,0,0,\ldots)$. Then
$$(\dd_0, \dd_1 \ldots, \dd_{n-1},-\infty, \dd_{n+1},\dd_{n+1},\ldots) = \dd \geq \dd'' >(\dd_0, \dd_1, \ldots, \dd_{n-1},-\infty, 0,-1,0,0,0,\ldots),$$
which contradicts $\dd'' \in \prod_{n = 0}^{ \infty*}\overline{\RR}$. Therefore $(\dd_0, \dd_1, \ldots, \dd_n, 0,-1,0,0,0,\ldots)$ is the greatest lower bound of $\mathcal{S}$ in $\prod_{n = 0}^{ \infty*}\overline{\RR}$. Thus, $\mathcal{S}$ has an infimum in $\prod_{n = 0}^{ \infty*}\overline{\RR}$ in this case.
Suppose now that $\dd_n \neq \pm \infty$, $\dd_{n+1} = \infty$ and $ \dd > (\dd_0, \dd_1, \ldots, \dd_{n+1},1,0,0,0,\ldots)$. Then one has $$\dd' > (\dd_0, \dd_1, \ldots, \dd_{n+1}, 1,0,0,0,\ldots) \in \prod_{n = 0}^{ \infty*}\overline{\RR}$$
for all $\dd' \in \mathcal{S}$. Suppose that $\dd''$ is some lower bound of $\mathcal{S}$ in $\prod_{n = 0}^{ \infty*}\overline{\RR}$ such that $\dd'' > (\dd_0, \dd_1, \ldots, \dd_{n+1}, 1,0,0,0,\ldots)$. Then
$$(\dd_0, \dd_1 \ldots, \dd_{n},\infty, \dd_{n+2},\dd_{n+3},\ldots) = \dd \geq \dd'' >(\dd_0, \dd_1, \ldots, \dd_{n},\infty, 1,0,0,0,\ldots),$$
which contradicts $\dd'' \in\prod_{n = 0}^{ \infty*}\overline{\RR}$. Therefore $(\dd_0, \dd_1, \ldots, \dd_{n+1}, \infty,1,0,0,0,\ldots)$ is the greatest lower bound of $\mathcal{S}$ in $\prod_{n = 0}^{ \infty*}\overline{\RR}$. Thus, $\mathcal{S}$ has an infimum in $\prod_{n = 0}^{ \infty*}\overline{\RR}$ in this case.
Finally, suppose that $\dd_n \neq \pm \infty$, $\dd_{n+1} =- \infty$ and $ \dd < (\dd_0, \dd_1, \ldots, \dd_{n+1},-1,0,0,0,\ldots)$. Then there exists some $\dd' \in \mathcal{S}$ such that
$$\dd \leq (\dd_0, \dd_1 \ldots, \dd_{n},-\infty, \dd_{n+2}',\dd_{n+3}',\ldots) = \dd' < (\dd_0, \dd_1, \ldots, \dd_{n},-\infty, -1,0,0,0,\ldots).$$
But this contradicts $\dd' \in \prod_{n = 0}^{ \infty*}\overline{\RR}$. Therefore this case is impossible.
\end{proof}
\begin{theorem}\label{infpropexp}
Let $f \in \RR^{\RR_\infty}$. One has
\begin{align*}
\dege f & = \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \dege f \leq \dege r \} \\
& = \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) = O(r(x)) \ (x \to \infty) \} \\
& = \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \forall x \gg 0\, |f(x)|\leq r(x) \} \\
& = \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) = o(r(x)) \ (x \to \infty) \} \\
& = \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \dege f < \dege r \},
\end{align*}
where the infima (exist and) are computed in $\prod_{n = 0}^ {\infty*}\overline{\RR}$.
\end{theorem}
\begin{proof}
The given infima exist by Lemma \ref{compllem}. By Corollary \ref{oexp}, if $\dege f < \dege r$, where $r \in \mathbb{L}_{> 0}$, then one has $f(x) = o(r(x)) \ (x \to \infty)$. Moreover, by Proposition \ref{aspropexoexp}, if $f(x) = O(r(x)) \ (x \to \infty)$, then $\dege f \leq \dege r$. It follows that
\begin{align*}
\dege f & \leq \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \dege f \leq \dege r \} \\
& \leq \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) = O(r(x)) \ (x \to \infty) \} \\
& \leq \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \forall x \gg 0\ |f(x)|\leq r(x) \} \\
& \leq \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) = o(r(x)) \ (x \to \infty) \} \\
& \leq \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \dege f < \dege r\},
\end{align*}
Suppose to obtain a contradiction that
$$\dege f < \inf\{\dege r:r \in \mathbb{L}_{> 0}, \, \dege f < \dege r\}.$$
Then one has $\dege f < \dd$ for some $\dd \in \prod_{n = 0}^{ \infty*}\overline{\RR}$ such that $\dd \leq \dege r$ for all $r \in \mathbb{L}_{> 0}$ such that $\dege f < \dege r$. Let $N$ be least such that $\dege_{N} f < \dd_{N}$, so that $\dege_{k} f = \dd_k$ for all $k < N$. Let $t \in \RR$ be such that $\dege_{N} f < t< \dd_{N}$. We claim that there is an $r \in \mathbb{L}_{> 0}$ such that $\dege_k r = \dege_k f$ for all $k < N$ and $\dege_N r= t$. Assume that this claim is true. Then one has $\dege f < \dege r < \dd$, which our desired contradiction.
Thus, we must prove the claim above. Note first that, by the definition of $\prod_{n = 0}^ {\infty*}\overline{\RR}$, one has
$$( \dd_0, \dd_1, \dd_2, \ldots, \dd_{N-1}, t, 0,0,0,0,\ldots) \in \prod_{n = 0}^ {\infty*}\overline{\RR}$$
for some $t$ with $\dege_{N} f < t< \dd_{N}$ provided that it is not the case that $ \dd_{N-1} = -\infty$ and $\dege_N f = 0$ and it is not the case that
$ \dd_{N-2} \neq \pm\infty$, $ \dd_{N-1} = \infty$, and $\dege_N f = 1$. Thus, in this case, our claim holds by Proposition \ref{degerprop}.
We may suppose, then, that either $ \dd_{N-1} = -\infty$ and $\dege_N f = 0$, or
$ \dd_{N-2} \neq \pm\infty$, $ \dd_{N-1} = \infty$, and $\dege_N f = 1$. Suppose the first case.
\begin{align*}
\dege f & < \dd = (\dd_0, \dd_1, \ldots, \dd_{N-2},-\infty, \dd_{N}, \dd_{N+1},\dd_{N+2},\ldots) \\ & \leq (\dd_0, \dd_1, \ldots, \dd_{N-2}, -\infty,0,-1,0,0,0,\ldots),
\end{align*}
which implies $0 = \dege_N f < \dd_{N} \leq 0$, a contradiction. Likewise, the second case is impossible, since
\begin{align*}
\dege f & < \dd = (\dd_0, \dd_1, \ldots, \dd_{N-2},\infty, \dd_{N}, \dd_{N+1},\dd_{N+2},\ldots) \\ & \leq (\dd_0, \dd_1, \ldots, \dd_{N-2}, \infty,1,0,0,0,\ldots),\end{align*}
which implies $1 = \dege_N f < \dd_{N} \leq 1$, again a contradiction. This completes the proof.
\end{proof}
\begin{corollary}\label{infpropexplow}
Let $f \in \RR^{\RR_\infty}$. One has
\begin{align*}
\underline{\dege}\, f & = \sup\{\dege r:r \in \mathbb{L}_{> 0}, \, \underline{\dege}\, f\geq \dege r \} \\
& = \sup\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) \gg r(x) \ (x \to \infty) \} \\
& = \sup\{\dege r:r \in \mathbb{L}_{> 0}, \, \forall x \gg 0\ |f(x)|\geq r(x) \} \\
& = \sup\{\dege r:r \in \mathbb{L}_{> 0}, \, f(x) \ggg r(x) \, (x \to \infty) \} \\
& = \sup\{\dege r:r \in \mathbb{L}_{> 0}, \, \underline{\dege}\, f > \dege r \},
\end{align*}
where the suprema (exist and) are computed in $\prod_{n = 0}^ {\infty*}\overline{\RR}$.
\end{corollary}
\begin{corollary}
Let $f,g \in \RR^{\RR_\infty}$. One has $\dege f \leq \dege g$ if and only if $\dege f \leq \dege r$ for all $r \in \mathbb{L}_{> 0}$ such that $g(x) \ll r(x) \ (x \to \infty)$. Likewise, one has $\underline{\dege}\, f \leq \underline{\dege}\, g$ if and only if $\dege r \leq \underline{\dege}\, g$ for all $r \in \mathbb{L}_{> 0}$ such that $f(x) \gg r(x) \ (x \to \infty)$.
\end{corollary}
The following proposition is the analogue of Proposition \ref{fresupperlower} for logexponential degree.
\begin{proposition}
Let $f \in \RR^{\RR_\infty}$. For all $X \subseteq \dom f$ with $\sup X = \infty$, one has
$$\underline{\dege}\, f \leq \underline{\dege}\, f|_X \leq \dege f|_X \leq \dege f.$$
Moreover, one has
$$\underline{\dege}\, f = \inf\left\{\dege f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}$$
$$\dege f = \sup\left\{\underline{\dege}\, f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}.$$
\end{proposition}
\begin{proof}
The first statement is clear, as then are the inequalities
$$\underline{\dege}\, f \leq \inf\left\{\dege f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}$$
$$\dege f \geq \sup\left\{\underline{\dege}\, f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}.$$
We prove that the second inequality above is an equality; the proof for the other is similar. Suppose, to obtain a contradiction, that
$$\dege f > \sup\left\{\underline{\dege}\, f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}.$$
By Proposition \ref{degerprop} (or by Proposition \ref{oexppropstrong} of Section 7.5), there exists an $r \in \mathbb{L}_{>0}$ such that
$$\dege f > \dege r > \sup\left\{\underline{\dege}\, f|_X: X \subseteq \dom f \text{ and } \sup X = \infty \right\}.$$
It follows that, for every set $X \subseteq \dom f$ with $\sup X = \infty$, one has
$\dege r > \underline{\dege}\, f|_X$, whence it is not the case that $f|_X(x) \gg r(x) \ (x \to \infty)$, i.e., one has $$\liminf_{x \to \infty} \frac{|f|_X(x)|}{r(x)} = 0.$$ Since this holds for every set $X \subseteq \dom f$ with $\sup X = \infty$, one has $\limsup_{x \to \infty} \frac{|f(x)|}{r(x)} = 0$ and therefore $\lim_{x \to \infty} \frac{|f(x)|}{r(x)} = 0$, i.e., $f(x) = o(r(x)) \ (x \to \infty)$. However, this contradicts $\dege f > \dege r$.
\end{proof}
For any functions $f$ and $g$ on $[0,\infty)$ with $f(x) \geq g(x)$ for all $x$, there is a function $h$ on $[0,\infty)$ with $f(x) \geq h(x) \geq g(x)$ for all $x$ and for which $h(x) = f(x)$ on some unbounded subset of $[0,\infty)$ and $h(x) = g(x)$ on some unbounded subset of $[0,\infty)$. From the proposition, then, it follows that $\dege h = \dege f$ and $\underline{\dege} \, h = \underline{\dege}\, g$. Moreover, if $f$ and $g$ are continuous and increasing (resp., decreasing), then one can find a corresponding $h$ with the same properties.
\subsection{Relationships with operations on functions}
The next proposition relates $\dege$ to the operations of addition, multiplication, and division of functions. For all $\dd, \ee \in \prod_{n = 0}^\infty\overline{\RR}$, we define
$$\dd\oplus \ee = \dd+ \ee\index[symbols]{.g m@$\dd \oplus \ee$} $$
if $\dd_k$ and $\ee_k$ are finite for all $k$, and
$$\dd\oplus \ee = (\dd_0+\ee_0, \ldots, \dd_{n-1}+\ee_{n-1},f_{0},f_{1},f_{2},\ldots)
if $n$ is the least nonnegative integer such that $\dd_n$ and $\ee_n$ are not both finite, where
(f_0,f_1,f_2,\ldots) = \begin{cases} \max( (\dd_{n},\dd_{n+1},\ldots),(\ee_{n},\ee_{n+1},\ldots)) & \text{if } \dd_{n} = \infty \text{ or } \ee_{n} = \infty \\
\min( (\dd_{n},\dd_{n+1},\ldots),(\ee_{n},\ee_{n+1},\ldots)) & \text{otherwise}
\end{cases}$$
as computed in $\prod_{n = 0}^\infty\overline{\RR}$. The operation $\oplus$ is a binary operation on $\prod_{n = 0}^\infty\overline{\RR}$ and restricts to a binary operation on $\prod_{n = 0}^{\infty*}\overline{\RR}$.
\begin{lemma}\label{difflemexp}
Let $f,g \in \RR^{\RR_\infty}$ with $\dom f = \dom g$. Then one has
$$\max(\dege f,\dege g) = \dege \max(|f(x)|,|g(x)|).$$
\end{lemma}
\begin{proof}
Since $0 \leq |f(x)| \leq \max(|f(x)|,|g(x)|)$ and $0 \leq |g(x)| \leq \max(|f(x)|,|g(x)|)$ for all $x$ (in $\dom f = \dom g$), the inequality $\max(\dege f,\dege g) \leq \dege \max(|f(x)|,|g(x)|)$ follows from Proposition \ref{aspropexoexp}(1). To prove the reverse inequality, we may suppose without loss of generality that $\dege f \geq \dege g$. Let $r \in \mathbb{L}_{> 0}$ with $\dege r > \dege f$. By Corollary \ref{oexp}, since $\dege f < \dege r$ and $\dege g < \dege r$, one has $|f(x)| \leq r(x)$ and $|g(x)| \leq r(x)$, and therefore $\max(|f(x)|,|g(x)|) \leq r(x)$, for all $x \gg 0$, whence
$\dege\max(|f(x)|,|g(x)|) \leq \dege r$. Taking the infimum over all $r$ as chosen and applying Theorem \ref{infpropexp}, we conclude that $\dege \max(|f(x)|,|g(x)|) \leq \dege f = \max(\dege f,\dege g)$.
\end{proof}
\begin{theorem}\label{diffpropexp}
Let $f$ and $g$ be real functions such that both $\underline{f} = f|_{\dom f \cap \dom g}$ and $\underline{g} = g|_{\dom f \cap \dom g}$ are in $\RR^{\RR_\infty}$, and let $n$ be a nonnegative integer. One has the following.
\begin{enumerate}
\item $\dege(f + g) \leq \max (\dege \underline{f},\dege \underline{g})$, with equality if $\dege \underline{f} \neq \dege \underline{g}$.
\item If $\dege(f+g) \neq \max(\dege \underline{f}, \dege \underline{g})$, then $\dege \underline{f} = \dege \underline{g}$.
\item $\dege(fg) \leq \dege \underline{f} \oplus \dege \underline{g}$.
\item If both $\underline{f}$ and $\underline{g}$ have finite logexponential degree to order $n$, then $\dege_k (fg) \leq \dege_k \underline{f}+ \dege_k \underline{g}$ for the smallest $k \leq n$, if any, for which equality does not hold.
\item Suppose that (a) $\underline{g}$ has finite logexponential degree to order $n-1$ and exact logexponential degree to order $n$; (b) $\underline{f}$ has finite logexponential degree to order $n-1$; (c) exactly one of $\dege_n \underline{g}$ and $\dege_n \underline{f}$ is finite; and (d) if $\dege_n \underline{g} = \pm \infty$, then $\underline{g}$ has exact logexponential degree. Then one has $\dege(fg) = \dege \underline{f} \oplus \dege \underline{g}$.
\item If $\underline{g}$ has exact logexponential degree to order $n$, and if one of $\dege_k \underline{f}$ and $\dege_k \underline{g}$ is finite for the smallest $k \leq n$ such that $\dege_k \underline{f}$ and $\dege_k \underline{g}$ are not both finite, should such a $k$ exist, then $\mathcal{S}(\dege(fg), \dege \underline{f} \oplus \dege \underline{g}) > n$.
\item If $\underline{g}$ has exact logexponential degree, and if one of $\dege_k \underline{f}$ and $\dege_k \underline{g}$ is finite for the smallest $k$ such that $\dege_k \underline{f}$ and $\dege_k \underline{g}$ are not both finite, should such a $k$ exist, then $\dege(fg) = \dege \underline{f} \oplus \dege \underline{g}$.
\item If $\underline{g}$ has finite and exact logexponential degree, then
$\dege(fg) = \dege \underline{f} \oplus \dege \underline{g}$.
\item $\dege (f^k) = \dege f \oplus \dege f \oplus\cdots \oplus \dege f$ for any positive integer $k$. More generally, for any $a > 0$, if $f$ has finite logexponential degree to order $n$, then $\dege_k (|f|^a) = a \dege_k f$ for all $k \leq n$, and, if one also has $\dege_{n+1} f = \pm \infty$, then $\dege_k (|f|^a) = \dege_k f$ for all $k > n$.
\end{enumerate}
\end{theorem}
\begin{proof}
We may suppose without loss of generality that $f = \underline{f}$ and $g = \underline{g}$. Since $$|f(x)+g(x)| \leq |f(x)|+|g(x)| \leq 2 \max(|f(x)|,|g(x)|)$$ for all $x \in \dom f = \dom g$, by Lemma \ref{difflemexp} one has
$$\dege (f+g) \leq \dege \max(|f(x)|,|g(x)|) = \max(\dege f,\dege g).$$ This proves statement (1), and statement (2) is an immediate consequence of (1).
We now prove statement (3). By Proposition \ref{diffprop}, we may assume that $\dege_{k} f$ and $\dege_{k} g$ are not both finite for all $k$.
Let $n$ be least so that $\dege_{n} f$ and $\dege_{n} g$ are not both finite. We may also suppose that $\degl_k (fg) = \degl_k f + \degl_k g$ for all $k < n$, for otherwise $\dege (fg) < \dege f\oplus \dege g$ by Proposition \ref{diffprop}. It follows, then, that $(fg)_{(k)} = f_{(k)} g_{(k)}$ for all $k \leq n$.
First, suppose that $\deg f_{(n)} = \deg g_{(n)} = -\infty$, so that $\deg \, (fg)_{(n)} =- \infty$. Then $f_{(n+1)}(x) = -\frac{1}{\log |f_{(n)}(x)|}$ and $g_{(n+1)}(x) = -\frac{1}{\log|g_{(n)}(x)|}$ and
\begin{align*}
0 \leq (fg)_{(n+1)}(x) & = -\frac{1}{\log|f_{(n)}(x)g_{(n)}(x)|} \\
& = -\frac{1}{\log|f_{(n)}(x)|+\log |g_{(n)}(x)|} \\
& = \frac{1}{\frac{1}{f_{(n+1)}(x)}+\frac{1}{g_{(n+1)}(x)}} \\
& \leq \min(f_{(n+1)}(x),g_{(n+1)}(x)).
\end{align*}
for all $x \gg 0$. It follows, then, that
\begin{align*}
\dege \, (fg)_{(n+1)} \leq \dege \min(f_{(n+1)}(x),g_{(n+1)}(x)) \leq \min (\dege f_{(n+1)}, \dege g_{(n+1)}),
\end{align*}
whence statement (3) holds in this case.
Next, suppose that $\deg f_{(n)} = -\infty$ and $-\infty < \deg g_{(n)} < \infty$, so that $\deg \, (fg)_{(n)} = \deg (f_{(n)}g_{(n)}) = -\infty$. One has
\begin{align*}
0 \leq (fg)_{(n+1)}(x) = -\frac{1}{\log|f_{(n)}(x)|+\log |g_{(n)}(x)|}
\end{align*}
and therefore
\begin{align*}
\limsup_{x \to \infty} \frac{(fg)_{(n+1)}(x) }{f_{(n+1)}(x)} & = \limsup_{x \to \infty} \frac{\log|f_{(n)}(x)|}{\log|f_{(n)}(x)|+\log |g_{(n)}(x)|} \\ & = \limsup_{x \to \infty} \frac{1}{1+\frac{\log |g_{(n)}(x)|}{\log|f_{(n)}(x)|}} \\
& = \frac{1}{1+ \liminf_{x \to \infty} \frac{\log |g_{(n)}(x)|}{\log|f_{(n)}(x)|}} \\
& = \frac{1}{1- \limsup_{x \to \infty} \frac{\log |g_{(n)}(x)|/\log x}{-\log|f_{(n)}(x)|/\log x}} \\
& = \frac{1}{1- (\deg g_{(n)}) \left(\frac{1}{\infty} \right) }\\
& = 1,
\end{align*}
$$(fg)_{(n+1)}(x) = O(f_{(n+1)}(x)) \ (x \to \infty).$$
It follows that $$ \dege \, (fg)_{(n+1)} \leq \dege f_{(n+1)}.$$
Thus, statement (3) holds in this case.
Next, suppose that $\deg f_{(n)} = \deg g_{(n)} = \infty$. If $\deg \, (fg)_{(n)} < \infty$, then $\dege \, (fg)_{(n)} < \max(\dege f_{(n)}, \dege g_{(n)})$. If, on the other hand, one has $\deg \, (fg)_{(n)} = \infty$, then $f_{(n+1)}(x) = \max(\log |f_{(n)}(x)|,0)$ and $g_{(n+1)}(x) = \max(\log |g_{(n)}(x)|,0)$, and thus $$0\leq (fg)_{(n+1)}(x) = \max(\log |f_{(n)}(x)g_{(n)}(x)|,0) \leq f_{(n+1)}(x)+g_{(n+1)}(x),$$ so that $\dege \, (fg)_{(n+1)} \leq \max (\dege f_{(n+1)}, \dege g_{(n+1)})$. Thus, statement (3) holds in either case.
To complete the proof of (3), suppose that $\deg f_{(n)} = \infty$ and $\deg g_{(n)} < \infty$. Let $F = f_{(n)}$ and $G = g_{(n)}$. Note that
$$\dege(FG) = \dege((F+G)^2-F^2-G^2) \leq \max(\dege(F+G)^2, \dege(F^2),\dege( G^2)),$$
$$\dege(F^2) \leq \dege F \oplus \dege F = \dege F,$$
$$\deg(G^2) = 2 \deg G < \infty,$$
$$\dege(F+G)^2 \leq \dege (F+G) \oplus \dege (F+G) = \dege (F+G) = \dege F,$$
$$\dege(FG) \leq \max(\dege(F+G)^2, \dege(F^2),\dege( G^2)) \leq \dege F.$$
Thus, in this case, we have $\dege \, (fg)_{(n)} \leq \dege f_{(n)}$. This completes the proof of (3).
Statement (4) follows from Propositions \ref{diffprop} and \ref{degeprop0}.
Assume the hypotheses of statement (5). By Proposition \ref{mmm}(1), one has $\dege_k(fg) = \dege_k f + \dege_k g$ and $(fg)_{(k)} = f_{(k)} g_{(k)}$, where $g_{(k)}$ has exact degree $\dege_k g$, for all $k \leq n$. Suppose that $d:=\dege_n g$ is finite, so that $\dege_n f = \pm\infty$. Suppose first that $\dege_n f = -\infty$. Then one has
$$\lim_{x \to \infty} \frac{\log |g_{(n)}(x)|}{\log |f_{(n)}(x)|} = \lim_{x \to \infty} \frac{d\log x}{\log |f_{(n)}(x)|} = \frac{d}{-\infty}= 0,$$
and therefore
$$\lim_{x \to \infty} \frac{f_{(n+1)}(x)}{(fg)_{(n+1)}(x)} = \lim_{x \to \infty} \frac{\log |f_{(n)}(x)g_{(n)}(x)|}{\log |f_{(n)}(x)|}= \lim_{x \to \infty} \left(1+\frac{\log|g_{(n)}(x)|}{\log |f_{(n)}(x)|}\right) = 1.$$
It follows that $(fg)_{(n+1)}(x) \sim f_{(n+1)}(x) \ (x \to \infty)$, whence $\dege \, (fg)_{(n)} = \dege f_{(n)}$.
Suppose, on the other hand, that $\dege_n f = \infty$. Let $N$ be any real number with $N > -\deg g_{(n)}$ and $N \geq 0$, and let
\begin{align*}
F(x) & = \frac{\max(\log |f_{(n)}(x)|,2N \log x)}{\max(\log |f_{(n)}(x)g_{(n)}(x)|,2N \log x+\log|g_{(n)}(x)|)} \\
& = \frac{\max\left(\frac{\log |f_{(n)}(x)|}{\log x},2N\right)}{\max\left(\frac{\log |f_{(n)}(x)|}{\log x}+\frac{\log|g_{(n)}(x)|}{\log x},2N+\frac{\log|g_{(n)}(x)|}{\log x}\right)}.
\end{align*}
Note that $F(x) \geq 0$ for all $x \gg 0$. If $\deg g_{(n)} > 0$, then we may let $N = 0$, and then $F(x) \leq 1$ for all $x \gg 0$. Suppose, on the other hand, that
$\deg g_{(n)} \leq 0$, so that $N > -\deg g_{(n)} \geq 0$. We claim that $F(x) \leq 2$ for all $ x \gg 0$. One has
$$-N< \frac{\log|g_{(n)}(x)|}{\log x}$$ for all $x \gg 0$. If $\frac{\log |f_{(n)}(x)|}{\log x} \geq 2N$, then one has
$$F(x) = \frac{\frac{\log |f_{(n)}(x)|}{\log x}}{\frac{\log |f_{(n)}(x)|}{\log x}+\frac{\log |g_{(n)}(x)|}{\log x}} < \frac{\frac{\log |f_{(n)}(x)|}{\log x}}{\frac{\log |f_{(n)}(x)|}{\log x}-N} \leq 2$$
for all $x \gg 0$.
On the other hand, if $\frac{\log |f_{(n)}(x)|}{\log x} \leq 2N$, then
$$F(x) = \frac{2N}{2N+ \frac{\log |g_{(n)}(x)|}{\log x}} < \frac{2N}{2N-N} = 2$$
for all $x \gg 0$. Thus, we have shown that $0\leq F(x) \leq 2$ for all $x \gg 0$. It follows that
$$\max(\log |f_{(n)}(x)|,2N \log x) = O\left(\max(\log |f_{(n)}(x)g_{(n)}(x)|,2N \log x+\log|g_{(n)}(x)|) \right) \ (x \to \infty).$$
Therefore, since $$\dege \max(\log |f_{(n)}(x)|,0) \geq (0,1,0,0,0,\ldots) =\dege (2N \log x)$$ and likewise $$\dege \max(\log |f_{(n)}(x)g_{(n)}(x)|,0) \geq \dege(2N \log x+\log|g_{(n)}(x)|),$$
by Lemma \ref{difflemexp} and Proposition \ref{aspropexoexp}(1) one has
\begin{align*}
\dege f_{(n+1)} & =
\dege \max(\log |f_{(n)}(x)| ,0) \\ & = \dege \max(\log |f_{(n)}(x)|,2N \log x)
\\ & \leq \dege \max(\log |f_{(n)}(x)g_{(n)}(x)|,2N \log x+\log|g_{(n)}(x)|) \\
& = \dege \max(\log |f_{(n)}(x)g_{(n)}(x)|,0) \\
& = \dege \, (fg)_{(n+1)}.
\end{align*}
Since the reverse inequality was proved in the proof of statement (3), equalities hold. This proves (5) in the case where $\dege_n g$ is finite.
Suppose, now, that $\dege_n g = \pm \infty$, so that $\dege_n f$ is finite and $g$ has exact logexponential degree. It follows that $\dege_n(fg) = \deg (f_{(n)} g_{(n)}) = \dege_n g$. Suppose that $\dege_n g = -\infty$.
The proof of (3) shows that
$$\limsup_{x \to \infty} \frac{(fg)_{(n+1)}(x) }{g_{(n+1)}(x)} = 1,$$
whence $(fg)_{(n+1)}(x) =O( g_{(n+1)}(x)) \ (x \to \infty)$ and $(fg)_{(n+1)}(x)\neq o( g_{(n+1)}(x)) \ (x \to \infty)$. Therefore, since $g_{(n+1)}$ has exact logexponential degree, it follows from Corollary \ref{oexpcor} that $\dege \, (fg)_{(n+1)} = \dege g_{(n+1)}$.
Suppose, on the other hand, that $\dege_n g = \infty$, so that
$\lim_{x \to \infty} |g_{(n)}(x)| = \infty$ and $g_{(n+1)}(x) = \log |g_{(n)}(x)|$ for all $x \gg 0$, while also
$$\limsup_{x \to \infty} \frac{\log |f_{(n)}(x)|}{\log |g_{(n)}(x)|} = \limsup_{x \to \infty} \frac{\log |f_{(n)}(x)|/\log x}{\log |g_{(n)}(x)|/\log x} = \frac{\dege_n f }{\infty} = 0.$$ It follows that
\begin{align*}
\limsup_{x \to \infty} \frac{(fg)_{(n+1)}(x)}{g_{(n+1)}(x)} & = \limsup_{x \to \infty} \frac{\max(\log |f_{(n)}(x)g_{(n)}(x)|,0)}{\log |g_{(n)}(x)|} \\
\\ & = \limsup_{x \to \infty} \max\left(1+\frac{\log |f_{(n)}(x)|}{\log |g_{(n)}(x)|},0\right) \\
& = 1,
\end{align*}
whence $\dege \, (fg)_{(n+1)} = \dege g_{(n+1)}$ also in that case. Statement (5) follows.
Note that statement (7) is an immediate consequence of statement (5) and Proposition \ref{mmm}(1), and statement (8) is an immediate consequence of statement (7). Statement (9) is an easy consquence of the fact that $\log |f|^a = a \log |f|$. Finally, to prove (6), note that the proof of (5) implies that $\dege_l (fg) = \dege_l f + \dege_l g$ and $(fg)_{(l)} = f_{(l)}g_{(l)}$ for all $l \leq k$, while also $(fg)_{(k+1)}(x) =O( g_{(k+1)}(x)) \ (x \to \infty)$ and $(fg)_{(k+1)}(x)\neq o( g_{(k+1)}(x)) \ (x \to \infty)$, where by hypothesis $g_{(k+1)}(x)$ has exact logexponential degree to order $n-(k+1)$, so that, by Proposition \ref{aspropexoexp}(1) and Corollary \ref{oexp}(1), one has $$\mathcal{L}((fg)_{(k+1)}, g_{(k+1)}) > n-(k+1),$$
$$\mathcal{S}(\dege(fg), \dege \underline{f} \oplus \dege \underline{g}) > n.$$ This completes the proof.
\end{proof}
\begin{example} \
\begin{enumerate}
\item Let $f(x) = e^{x^2}$ and $g(x) = e^{-x}$, so that $\deg f = \infty$ and $\deg g = -\infty$. Then $f$ and $g$ have exact logexponential degree, and one has
$$\dege (fg) = (\infty,2,0,0,0,\ldots) = \dege f \oplus \dege g.$$
\item Let $f(x) = e^x$ and $g(x) = e^{-x^2}$, so that $\deg f = \infty$ and $\deg g = -\infty$. Then $f$ and $g$ have exact logexponential degree, yet one has
$$\dege (fg) = (-\infty,-2,0,0,0,\ldots) < \dege f \oplus \dege g = (\infty,1,0,0,0,\ldots).$$
\item Let $f(x)$ be defined on $\RR$ so that $f(x) = 1$ for $x \in [2N,2N+1)$ and $f(x) = 0$ for $x \in [2N-1,2N)$ for all integers $N$. Let $g(x) = 1-f(x)$. Then $f(x) + g(x) = 1 = \max(f(x),g(x))$ has exact logexponential degree $(0,0,0,0\ldots)$, while also $\underline{\dege} \, f = \underline{\dege} \, g = (-\infty,-\infty,\infty, \ldots)$. This example shows that, in general, one need not have $\underline{\dege}(f+g) \leq \max (\underline{\dege}\, f,\underline{\dege} \, g)$, and the obvious inequality $\max(\underline{\dege}\, f , \underline{\dege}\, g) \leq \underline{\dege} \max (|f|,|g|)$ need not be an equality. Nevertheless, since $|f+g| \leq 2\max ( |f|,|g|)$, in general one has $\underline{\dege}(f+g) \leq \underline{\dege} \max (|f|,|g|)$. Moreover, one has $\min(\underline{\dege}\, f, \underline{\dege}\, g) = \underline{\dege} \min (|f|,|g|)$: the proof is similar to that of Lemma \ref{difflemexp}.
\end{enumerate}
\end{example}
\begin{problem}
Generalize statements (3)--(8) of Theorem \ref{diffpropexp} to the extent possible.
\end{problem}
Next, we relate $\dege$ to the operation of composition of functions. The most general case possible seems to be rather complicated, so we restrict ourselves to the special cases below, which are sufficient for our purposes.
\begin{theorem}\label{fgie}
Let $f$ and $g$ be real functions defined on a neighborhood of $\infty$. Suppose that $f$ has finite degree $d$ and exact logexponential degree, $g$ has positive degree and is eventually positive, and $g(x) \asymp r(x) \ (x \to \infty)$ for some $r \in \mathfrak{L}$. Then one has
$$\dege( f \circ g )= \dege f \oplus d (\dege g + (-1,0,0,0,\ldots)),$$
and $f \circ g$ has exact logexponential degree.
\end{theorem}
\begin{proof}
Let $e_k = \dege_k g= \dege_k r$ for all $k$.
Note that
$$\log g(x) \sim \log r(x) \sim e_0 \log x \ (x \to \infty)$$
and therefore
$$\log g(e^x) \sim e_0 x \ (x \to \infty).$$
It follows by induction, then, that
$$\log^{\circ k} g(x) \sim \log^{\circ k } x \ (x \to \infty)$$ for all $k \geq 2$.
Thus, for any nonnegative integer $n$, one has
$$h_n(x) := \log^{\circ (n+1)} g(\exp^{\circ (n+1)}) \sim a_n x \ (x \to \infty),$$
where $a_0 = e_0$ and $a_n = 1$ if $n \geq 1$.
Replacing $f$ with $|f|$, we may assume that $f$ is nonnegative and eventually positive. Since $f$ has finite degree $d$, we may suppose that $f$ has finite logarithmic degree to order $n \geq 0$. Then, by Proposition \ref{fgi}, one has
$\deg (f \circ g)= de_0$ and
$$\degl_1 (f \circ g) = \degl_1 f + d \degl_1 g= \degl_1 f + de_1,$$
and, provided that $n \geq 1$, also
$$\degl_k (f \circ g) = \degl_k f + d \degl_k g = \degl_k f, \quad \forall k = 2,3,\ldots,n.$$
Moreover, since $f$ and $g$ have exact logexponential degree, all of the limits superior in the proof of Proposition \ref{fgi} can be replaced with limits, and thus $f \circ g$ has exact logexponential degree to order $n+1$. Thus, the conclusion of the theorem holds if $f$ has finite logarithmic degree. We may suppose, then, without loss of generality, that $\dege_{n+1} f = \deg f_{(n+1)} = \pm \infty$.
Let $d_k = \dege_k f$ for all $k$.
One has $$f_{(n+1)}(\log^{\circ (n+1)} x) = f(x)x^{-d}(\log x)^{-d_1} \cdots (\log^{\circ n}x)^{-d_ {n}},$$
so that
$$f_{(n+1)}(\log^{\circ (n+1)} g(x)) = f(g(x))g(x)^{-d}(\log g(x))^{-d_1} \cdots (\log^{\circ n}g(x))^{-d_ {n}},$$
and therefore
\begin{align*}
(f\circ g)_{(n+1)}(\log^{\circ (n+1)} x) & = f(g(x))(\log x)^{-d_1} \cdots (\log^{\circ n}x)^{-d_ {n}} \cdot x^{-de_0} (\log x)^{-de_1}\cdots (\log^{\circ n} x)^{-de_n} \\
& = f_{(n+1)}(\log^{\circ (n+1)} g(x))\left( \frac{g(x)}{r_n(x)}\right)^{d} \left( \frac{\log g(x)}{\log x}\right)^{d_1} \cdots\left( \frac{\log^{\circ n} g(x)}{\log^{\circ n}x}\right)^{d_{n}} \\
& \sim f_{(n+1)}(\log^{\circ (n+1)} g(x))\left( \frac{g(x)}{r_n(x)}\right)^{d}e_0^{d_1} \\
& = f_{(n+1)}(\log^{\circ (n+1)} g(x)) k_n(x),
\end{align*}
$$r_n(x) = x^{e_0}(\log x)^{e_1} \cdots (\log^{\circ n} x)^{e_n}$$
$$k_n(x) = \left( \frac{g(x)}{r_n(x)}\right)^{d}e_0^{d_1}.$$
It follows that
$$(f\circ g)_{(n+1)}(x) \sim f_{(n+1)}(h_n(x))\, k_n(\exp^{\circ(n+1)}x) \ (x \to \infty),$$
where, as before,
$$h_n(x) = \log^{\circ (n+1)} g(\exp^{\circ (n+1)}) \sim a_n x \ (x \to \infty).$$
Note that, since $g(x) \asymp r(x) \ (x \to \infty)$, one has
$$k_n(x) \asymp \prod_{k > n}(\log^{\circ k} x)^{de_k} \ (x \to \infty),$$
and therefore
$$\log k_n(x) = O( \log^{\circ (n+2)} x) \ (x \to \infty),$$ whence
$$\log k_n(\exp^{\circ(n+1)}x) = O(\log x) \ (x \to \infty).$$
Since $\overline{\underline{\deg}} \, f_{(n+1)} = \pm \infty$, one has
$$\lim_{x \to \infty} \frac{\log x}{ \log f_{(n+1)}(h_n(x)) } = \lim_{x \to \infty} \frac{\log h_n(x)}{ \log f_{(n+1)}(h_n(x)) } = 0$$
and therefore
$$\log k_n(\exp^{\circ(n+1)}x) = O(\log x) = o( \log f_{(n+1)}(h_n(x)) ) \ (x \to \infty)$$
If $\deg f_{(n+1)} = \infty$, then also $\deg \, (f\circ g)_{(n+1)} = \infty$ and therefore
\begin{align*}
(f\circ g)_{(n+2)}(x) & = \log \, (f\circ g)_{(n+1)}(x) \\
& \sim \log f_{(n+1)}(h_n(x)) + \log k_n(\exp^{\circ(n+1)}x) \\
& \sim \log f_{(n+1)}(h_n(x)) \\
& = f_{(n+2)}(h_n(x))\ (x \to \infty),
\end{align*}
where $f_{(n+2)}$ has finite degree (by Proposition \ref{degeprop}(3)). On the other hand, if $\deg f_{(n+1)} = -\infty$, then also $\deg \, (f\circ g)_{(n+1)} = -\infty$ and therefore
\begin{align*}
(f\circ g)_{(n+2)}(x) & = -\frac{1}{\log \, (f\circ g)_{(n+1)}(x)} \\
& \sim -\frac{1}{ \log f_{(n+1)}(h_n(x)) + \log k_n(\exp^{\circ(n+1)}x) } \\
& \sim -\frac{1}{ \log f_{(n+1)}(h_n(x)) } \\
& = f_{(n+2)}(h_n(x))\ (x \to \infty),
\end{align*}
where $f_{(n+2)}$ has finite degree (by Proposition \ref{degeprop}(4)).
Thus, we have shown that
$$(f\circ g)_{(n+2)}(x) \sim (f_{(n+2)} \circ h_n)(x) \ (x \to \infty),$$
and therefore
$$\dege \, (f\circ g)_{(n+2)} = \dege (f_{(n+2)} \circ h_n)$$
for some function $h_n$ with $h_n(x) \sim a_n x \ (x \to \infty)$ for some $a_n > 0$, where $f_{(n+2)}$ has finite degree and exact logexponential degree. The argument may then be repeated {\it ad infinitum} to show that
$$\dege (f_{(n+2)} \circ h_n) = \dege f_{(n+2)}.$$
Therefore, since
$$\dege \, (f\circ g)_{(n+2)} = \dege f_{(n+2)},$$
the identity for $\dege \, (f\circ g)$ stated in the theorem follows from the definition of the operation $\oplus$, and the claim that $f \circ g$ has exact logexponential degree follows by repeated application of Proposition \ref{exactas}.
\end{proof}
\begin{lemma}\label{fgie00lemma}
Let $\dd,\dd',\ee \in \prod_{n = 0}^{ \infty*}\overline{\RR}$ with $\ee_n \neq \pm \infty$ for all $n$, and let
$\mathcal{S}$ be a subset of $\prod_{n = 0}^{ \infty*}\overline{\RR}$. One has the following.
\begin{enumerate}
\item If $\dd \leq \dd'$, then $\dd \oplus \ee \leq \dd' \oplus \ee$.
\item $(\dd\oplus \ee) \oplus (-\ee) = \dd$.
\item $(\inf \mathcal{S}) \oplus \ee = \inf\{\dd\oplus\ee: \dd \in \mathcal{S}\}.$
\end{enumerate}
\end{lemma}
\begin{proof}
Statements (1) and (2) are clear, and statement (3) is an easy consequence of statements (1) and (2).
\end{proof}
\begin{theorem}\label{fgie00}
Let $f$ and $g$ be real functions defined on a neighborhood of $\infty$. Suppose that $f$ has finite degree $d$, that $g$ has positive degree and is eventually positive, continuous, and increasing, and that $g(x) \asymp r(x) \ (x \to \infty)$ for some $r \in \mathfrak{L}$. Then one has
$$\dege( f \circ g )= \dege f \oplus d (\dege g + (-1,0,0,0,\ldots)).$$
\end{theorem}
\begin{proof}
Let $e_k = \dege_k g= \dege_k r$ for all $k$. Note that $g$ has exact logexponential degree, by Proposition \ref{exactas}. Thus, by Proposition \ref{circle1}(2), one has $\deg(f \circ g) = \deg f \deg g$. Note also that the compositional inverse $g^{-1}$ of $g$ is eventually continuous and increasing, and one has
$$g^{-1}(x) \asymp x^{1/e_0} (\log x)^{-e_1/e_0}(\log^{ \circ 2} x)^{-e_2/e_0} \cdots (\log^{ \circ n} x)^{-e_n/e_0} \ (x \to \infty).$$
Suppose first that $\dege f < (d,\infty,1,0,0,0,\ldots)$.
Then there exists an $s \in \mathbb{L}_{> 0}$ with $f(x) = O(s(x)) \ (x \to \infty)$ and $\deg s = d$. For any such $s$, one has
$f(g(x)) = O(s(g(x))) \ (x \to \infty)$, and therefore
$$\dege (f \circ g) \leq \dege(s \circ g) = \dege s \oplus d (\dege g + (-1,0,0,0,\ldots)),$$
by Theorem \ref{fgie}. Taking the infimum over all $s$ as chosen, by Theorem \ref{infpropexp} and Lemma \ref{fgie00lemma} we have
$$\dege (f \circ g) \leq \dege f \oplus d ( \dege g + (-1,0,0,0,\ldots)) < (de_0,\infty,1,0,0,0,\ldots).$$
From the inequality above, it follows that $(f \circ g)(x) = O(t(x)) \ (x \to \infty)$ for some $t \in \mathbb{L}_{> 0}$ with $\deg t = \deg(f \circ g) = de_0$. For any such $t$, one has $f(x) = O(t(g^{-1}(x))) \ (x \to \infty)$, and therefore
\begin{align*}
\dege f & \leq \dege t(g^{-1}(x)) \\ & = \dege t\oplus de_0(1/e_0-1,-e_1/e_0, -e_2/e_0,-e_3/e_0,\ldots) \\
& = \dege t\oplus (d-de_0,-de_1, -de_2,-de_3,\ldots) \\
& = \dege t \oplus -d(\dege g+ (-1,0,0,0,\ldots)),
\end{align*}
again by Theorem \ref{fgie}. Taking the infimum over all $t$ as chosen, we see that
\begin{align*}
\dege f \leq \dege(f \circ g)\oplus -d(\dege g + (-1,0,0,0,\ldots)).
\end{align*}
Thus, by Lemma \ref{fgie00lemma}, we have
\begin{align*}
\dege (f \circ g) & \leq \dege f \oplus d (\dege g + (-1,0,0,0,\ldots)) \\
& \leq (\dege(f \circ g)\oplus -d(\dege g + (-1,0,0,0,\ldots)))\oplus d (\dege g + (-1,0,0,0,\ldots)) \\
& = \dege(f \circ g).
\end{align*}
This proves the theorem in the case where $\dege f < (d,\infty,1,0,0,0,\ldots)$.
Suppose now that $\dege f = (d,\infty,1,0,0,0,\ldots)$. Then, since $\deg(f \circ g) = de_0$, one has
$$\dege (f \circ g) \leq (de_0,\infty,1,0,0,0,\ldots).$$
On the other hand, if $\dege (f \circ g) < (de_0,\infty,1,0,0,0,\ldots)$, then
\begin{align*}
\dege f & = \dege((f \circ g)\circ g^{-1}) \\ &
= \dege (f \circ g) \oplus de_0(\dege g^{-1} + (-1,0,0,0,\ldots)) \\
& < (d, \infty,1,0,0,0,\ldots),
\end{align*}
which is a contradiction. It follows that
$$\dege (f \circ g) = (de_0,\infty,1,0,0,0,\ldots) = \dege f \oplus d(\dege g + (-1,0,0,0,\ldots)).$$
This completes the proof.
\end{proof}
Theorems \ref{fgie} and \ref{fgie00} combine as follows.
\begin{theorem}\label{fgieth}
Let $f$ and $g$ be real functions defined on a neighborhood of $\infty$. Suppose that the following conditions hold.
\begin{enumerate}
\item $f$ has finite degree $d$.
\item $g$ has positive degree and is eventually positive.
\item Either $f$ has exact logexponential degree or $g$ is
eventually continuous and increasing.
\item $g(x) \asymp r(x) \ (x \to \infty)$ for some $r \in \mathfrak{L}$.
\end{enumerate}
Then one has
$$\dege( f \circ g )= \dege f \oplus d (\dege g + (-1,0,0,0,\ldots)).$$
\end{theorem}
\begin{corollary}\label{fgie2}
Let $f$ and $g$ be real functions defined on a neighborhood of $\infty$. Suppose that $g$ is eventually positive, $g(x) \asymp x \ (x \to \infty)$, and either $f$ has exact logexponential degree or $g$ is eventually continuous and increasing.
Then one has
$$\dege( f \circ g )= \dege f.$$
\end{corollary}
\begin{proof}
Let $h = f \circ g$.
If $d \neq \pm \infty$, then by Theorem \ref{fgieth} one has
$$\dege h = \dege f.$$
On the other hand, if $d = \pm \infty$, then $\deg h = \deg f$ and $h_{(1)} = f_{(1)} \circ g$, by Proposition \ref{circle1}(2).
An obvious inductive argument, then, yields $$\dege_k h = \deg h_{(k)} = \deg f_{(k)} = \dege_k f$$ for all nonnegative integers $k$.
\end{proof}
\begin{problem}
Generalize Theorem \ref{fgieth} to the extent possible.
\end{problem}
\begin{remark}[Base independence of logexponential degree]\label{altbase}
Let $f \in \RR^{\RR_{\infty}}$, and let $b> 1$ and $d \in \RR$. Then $$f(b^x)b^{-dx} = f(e^x)e^{-dx} \circ (x \log b)$$
$$\max( \log_{b} |f(x)|, 0) = \frac{1}{\log b} \max( \log |f(x)|, 0)$$
$$-\frac{1}{\log_{b} |f(x)|} = (\log b) \left(-\frac{1 }{\log |f(x)|}\right).$$
Let $b_k > 1$ for all nonnegative integers $k$. Let $f_0 = f$, and supposing that $f_k$ is defined for some nonnegative integer $k$, let $d_k = \deg f_k$ and
$$f_{k+1}(x) = \left.
\begin{cases}
f_{k}(b_k^x) b_k^{-d_k x} & \text{if } d_k \neq \pm \infty \\
\max( \log_{b_k} |f_{k}(x)|, 0) & \text{if } d_k = \infty \\
\displaystyle -\frac{1}{\log_{b_k} |f_{k}(x)|} & \text{if } d_k =- \infty.
\end{cases}
\right.$$
By Corollary \ref{fgie2}, Proposition \ref{aspropexoexp}(3), and induction, one has
$$ \dege_k f = d_k = \deg f_k$$
for all $k$. This shows that $\dege$ is independent of choice of bases. Our convention to use $b_k = e$ for all $k$ is the most natural choice for functions arising in analytic number theory.
\end{remark}
\section{Further results on degree and logexponential degree}
The following proposition can be useful for studying the logexponential degree of an arithmetic function. (For example, it is used in the proof of Proposition \ref{lipiprop}.)
\begin{proposition}\label{arithb}
Let $f$ be a real-valued arithmetic function. One has
$$\dege f = \dege f(\lfloor x \rfloor) = \dege f(\lceil x \rceil).$$
\end{proposition}
\begin{proof}
By Lemma \ref{reslemma}, one has
$$\dege f \leq \dege f(\lfloor x \rfloor)$$
$$\dege f \leq \dege f(\lceil x \rceil).$$
Let $r \in \mathbb{L}_{>0}$ with $f(n) = O(r(n)) \ (n \to \infty)$. Then one has
$$f(\lfloor x \rfloor) = O(r(\lfloor x \rfloor)) \ (x \to \infty),$$ whence by Corollary \ref{fgie2} one has
$$\dege f(\lfloor x \rfloor) \leq \dege r(\lfloor x \rfloor) = \dege r(x).$$
Taking the infimum over all $r$ as chosen and applying Theorem \ref{infpropexp}, we find that
$$\dege f(\lfloor x \rfloor) \leq \dege f.$$
Similarly, one has
$$\dege f(\lceil x \rceil) \leq \dege f.$$
This completes the proof.
\end{proof}
Next, we show that all functions of finite degree in $\mathbb{L}$ are regularly varying. This is quite useful in combination with Theorem \ref{infpropexp} and Karamata's integral theorem: for example, see the proofs of Theorem \ref{lithetapsi} in Section 9.3 and Theorem \ref{mertenssecond} in Section 10.1.
\begin{proposition}\label{kregvar}
A function $r \in \mathbb{L}$ is regularly varying if and only if it is of finite degree, and, if those equivalent conditions hold, then $r$ is regularly varying of index $$\overline{\underline{\deg}} \, r = \lim_{x \to \infty}\frac{xr'(x)}{r(x)}.$$
\end{proposition}
\begin{proof}
Let $r \in \mathbb{L}^*$ with $\deg r \neq \pm \infty$. Then $r$ is continuously differentiable on $[N,\infty)$ for some real number $N$. Since $\frac{xr'(x)}{r(x)} \in \mathbb{L}$, the limit $d = \lim_{x \to \infty}\frac{xr'(x)}{r(x)}$
exists or is $\pm \infty$. But then $d = \deg r$, so that $d \neq \pm \infty$. Therefore, by Corollary \ref{regvarcor}, the function $r$ is regularly varying of index $d$.
\end{proof}
Next, we note the following.
\begin{proposition}\label{supprop}
Let $f \in \RR^{\RR_\infty}$ with $f$ unbounded on $X = \dom f$ at $\infty$. Let $N \geq 0$, and suppose that $f$ is bounded on $[N,x] \cap X$ for all $x \geq N$, and let
$$\widetilde{f}(x) = \sup_{f \in [N,x] \cap X} |f(t)|, \quad \forall x \in [N,\infty) \cap X.$$ Then $\widetilde{f}$ is nonnegative and nondecreasing on $[N,\infty) \cap X$ with $\lim_{x \to \infty} \widetilde{f}(x) = \infty$, and one has
$ \dege \widetilde{f} = \dege f $.
\end{proposition}
\begin{proof}
Since $0 \leq |f(x)| \leq \widetilde{f}(x)$ for all $x \in [N,\infty) \cap X$, one has $\dege f \leq \dege \widetilde{f}$. To prove the reverse inequality, let $r \in \mathbb{L}_{>0}$ with $|f(x)| \leq r(x)$ for all $x \in [N',\infty) \cap X$. Since $f(x)$ is unbounded on $[N',\infty) \cap X$, the function $r$ must be eventually increasing, say, increasing for $x \geq M \geq \max(N,N')$. Then, for all sufficiently large $x \in [M,\infty) \cap X$, one has
\begin{align*}
\widetilde{f}(x) & = \max\left(\sup_{t \in [N,M] \cap X} |f(t)|, \sup_{t \in [M,x] \cap X}| f(t) |\right)
\\ & \leq \max\left(\sup_{t \in [N,M] \cap X} |f(t)|, \sup_{t \in [M,x] \cap X} r(t) \right) \\
& = \max\left(\sup_{t \in [N,M] \cap X} |f(t)|, r(x) \right) \\
& = r(x).
\end{align*}
Therefore, one has $\dege \widetilde{f} \leq \dege r$.
Taking the infimum over all $r$ as chosen, we find that $\dege \widetilde{f} \leq \dege f$. This completes the proof.
\end{proof}
The corresponding result for lower logexponential degree is as follows.
\begin{proposition}\label{suppropinf}
Let $f \in \RR^{\RR_\infty}$ with $\liminf_{x \to \infty} f(x)= 0$, and let $X = \dom f$. Let $N \geq 0$, and suppose that $f$ is bounded away from $0$ on $[N,x] \cap X$ for all $x \geq N$, and let
$$\widetilde{f}(x) = \inf_{f \in [N,x] \cap X} |f(t)|, \quad \forall x \in [N,\infty) \cap X.$$ Then $\widetilde{f}$ is nonnegative and nonincreasing on $[N,\infty) \cap X$ with $\lim_{x \to \infty} \widetilde{f}(x) = 0$, and one has
$ \underline{\dege}\, \widetilde{f} = \underline{\dege}\, f $.
\end{proposition}
Regarding nondecreasing nonnegative step functions, we have the following.
\begin{proposition}\label{stepdeg}
Let $f$ be a nondecreasing nonnegative step function on $[0,\infty)$ that is continuous from the right, and whose set of points of discontinuity is an unbounded subset of $X = \{x_1,x_2,\ldots\}$, where the sequence $x_n$ is increasing without bound. Let $U = f|_X$, and let $L$ be defined on $X\backslash\{x_1\}$, with $L(x_n) = f(x_n^-)$, and therefore $L(x_n)= f(x_{n-1})\leq f(x_n) = U(x_n)$, for all $n \geq 1$. Then one has
$$\underline{\dege} \, f = \underline{\dege}\, L \leq\dege U = \dege f .$$
\end{proposition}
\begin{proof}
Since $U$ is a restriction of $f$, one has $\dege U \leq \dege f$. Let $r \in \mathbb{L}_{>0}$ with $U(x_n) \leq r(x_n)$ for all $n$. Since $U$ is nondecreasing, we may suppose without loss of generality that $r$ is eventually nondecreasing. Since $r(x)$ is nondecreasing on $[x_n,x_{n+1})$, one has $f(x) = U(x_n) \leq r(x_n) \leq r(x)$, for all $n \gg 0$ and all $x \in[x_n,x_{n+1})$. It follows that $f(x) \leq r(x)$ for all $x \gg 0$, whence $\dege f \leq \dege U$, and equality holds.
Now, let $r \in \mathbb{L}_{>0}$ with $L(x_n) \geq r(x_n)$ for all $n \gg 0$. Again, we may suppose without loss of generality that $r$ is eventually nondecreasing. Since $r(x)$ is nondecreasing on $[x_{n-1},x_{n})$, one has $f(x) = L(x_n) \geq r(x_n) \geq r(x)$, for all $n \gg 0$ and all $x \in[x_{n-1},x_{n})$. It follows that $\underline{\dege} \, f \geq \underline{\dege}\, L$. Finally, let $r \in \mathbb{L}_{>0}$ with $f(x) \geq r(x)$ for all $x \gg 0$. Then, by the eventual continuity of $r$, one has $L(x_n) = f(x_n^-) \geq r(x_n)$ for all $n \gg 0$, and therefore $\underline{\dege} \, f \leq \underline{\dege}\, L$. This completes the proof.
\end{proof}
Thus, for example, if $f$ is any nonnegative arithmetic function, then $\dege S_f = \dege S_f|_{\ZZ_{>0}}$ and $\underline{\dege} \, S_f = \underline{\dege} \, (S_f-f)$.
Finally, we note the following.
\begin{proposition}\label{strongdegg}
Let $f$ be a real function that is Riemann integrable on $[N,x]$ for all $x>N$, where $N >0$ and $\deg f \in \RR\backslash\{-1\}$. Let $F(x) = \int_N^x f(t) \, dt$ or $F(x) = \int_x^\infty f(t) \, dt$ on $(N,\infty)$ according to whether $\deg f > -1$ or $\deg f < -1$. Let $s \in \RR$.
\begin{enumerate}
\item One has
$$\dege F \leq \dege f + (1,0,0,0,\ldots).$$
\item If $\deg f >-1$ and $s < \deg F$, then
$$\dege \int_N^x \frac{f(t)}{t^s} \, dt \leq \dege F + (-s,0,0,0,\ldots),$$
with equality if $\deg F > 0$.
\item If $\deg f >-1$ and $s > \deg f+1$, then
$$\dege \int_x^\infty \frac{f(t)}{t^s} \, dt \leq \dege F + (-s,0,0,0,\ldots),$$
with equality if $\deg F > 0$.
\item If $\deg f < -1$ and $s < \deg F$, then
$$\dege \int_N^x \frac{f(t)}{t^s} \, dt = \dege F + (-s,0,0,0,\ldots).$$
\item If $\deg f < -1$ and $s > \deg f + 1$, then
$$\dege \int_x^\infty \frac{f(t)}{t^s} \, dt = \dege F + (-s,0,0,0,\ldots).$$
\end{enumerate}
\end{proposition}
\begin{proof}
To prove (1), we suppose that $\deg f > -1$. The proof when $\deg f < -1$ is similar.
Suppose that $f(x) = O(r(x)) \ (x \to \infty)$, where $r$ is regularly varying of index $d$ and both positive and continuous on $[N,\infty)$. Since $d \geq \deg f >-1$, Karamata's integral theorem implies that
$$\frac{1}{x}\int_N^x f(t)\, dt \ll \frac{1}{x} \int_N^x r(t)\, dt \asymp r(x) \ (x \to \infty),$$
and therefore
$$\dege \int_N^x f(t)\, dt + (-1,0,0,0,\ldots) \leq \dege r.$$
Statement (1) follows by taking the infimum of $\dege r$ over all $r$ as chosen.
Suppose, now, that $\deg f > -1$ and $s < \deg F$. Let
$$F_s(x) = \int_N^x \frac{f(t)}{t^s} \, dt.$$
By Riemann--Stieltjes integration by parts \cite[Section 1.1.3]{borg} (along with \cite[Chapter I Theorem 6a]{widd}), one has
$$F_s(x) = \frac{F(x)}{x^s} + s \int_N^x \frac{F(t)}{t^{s+1}} \, dt.$$
Let $r$ be regularly varying of index $d$ and both positive and continuous on $[N,\infty)$. Suppose that $F(x) \ll r(x) \ (x \to \infty)$.
Then $d-s-1 \geq \deg F-s-1 > -1$, so that, by Karamata's integral theorem, one has
$$F_s(x) \ll \frac{r(x)}{x^s} + |s| \int_N^x \frac{r(t)}{t^{s+1}} \, dt \ll \frac{r(x)}{x^s} \ (x \to \infty),$$
and therefore
$$\dege F_s \leq \dege r+(-s,0,0,0,\ldots).$$
Taking the infimum over all $r$ as chosen, we see that $$\dege F_s \leq \dege F+(-s,0,0,0,\ldots).$$
Again by Riemann--Stieltjes integration by parts, one has
$$F(x) = x^s F_s(x) - s \int_N^x t^{s-1}F_s(x) \, dt.$$
Therefore, if $\deg F_s>-s$, that is, if $\deg F_s+s-1 >-1$, then, by similar reasoning as above, one has
$$\dege F \leq \dege F_s + (s,0,0,0,\ldots).$$ On the other hand,
$\deg F_s \leq -s$ implies $\deg F \leq 0$. Statement (2) follows.
Suppose, now, that $\deg f > -1$ and $s > \deg f+1$, so that $s > \deg F$. Let
$$G_s(x) = \int_x^\infty \frac{f(t)}{t^s} \, dt.$$
By Riemann--Stieltjes integration by parts, since $\lim_{x \to \infty} \frac{F(x)}{x^s} = 0$, one has
$$G_s(x) = -\frac{F(x)}{x^s} + s \int_x^\infty \frac{F(t)}{t^{s+1}} \, dt.$$
Let $r$ be regularly varying of index $d$ and both positive and continuous on $[N,\infty)$. Suppose that $F(x) \ll r(x) \ (x \to \infty)$, with $\deg F \leq d < s$. Then, by Karamata's integral theorem, one has
$$G_s(x) \ll \frac{r(x)}{x^s} + |s| \int_N^x \frac{r(t)}{t^{s+1}} \, dt \ll \frac{r(x)}{x^s} \ (x \to \infty),$$
and therefore
$$\dege G_s \leq \dege r+(-s,0,0,0,\ldots).$$
Taking the infimum over all $r$ as chosen, we see that $$\dege G_s \leq \dege F+(-s,0,0,0,\ldots).$$
Again by Riemann--Stieltjes integration by parts, one has
$$F(x) = N^sG_s(N)-x^s G_s(x) + s \int_N^x t^{s-1}G_s(x) \, dt.$$
Therefore, if $\deg G_s>-s$, that is, if $\deg G_s+s-1 >-1$, then, by similar reasoning as above, one has
$$\dege F \leq \dege G_s + (s,0,0,0,\ldots).$$ On the other hand,
$\deg G_s \leq -s$ implies $\deg F \leq 0$. Statement (3) follows.
Suppose, now, that $\deg f < -1$ and $s< \deg F$, so that $\deg F <0$. Let
$$F_s(x) = \int_N^x \frac{f(t)}{t^s} \, dt.$$
By Riemann--Stieltjes integration by parts, one has
$$F_s(x) = \frac{F(N)}{N^s}-\frac{F(x)}{x^s} - s \int_N^x \frac{F(t)}{t^{s+1}} \, dt.$$
Let $r$ be regularly varying of index $d$ and both positive and continuous on $[N,\infty)$. Suppose that $F(x) \ll r(x) \ (x \to \infty)$.
Then $d-s \geq \deg F-s > 0$ and $d-s-1>-1$, so that
$$F_s(x) \ll \left| \frac{F(N)}{N^s} \right|+\frac{r(x)}{x^s} + |s| \int_N^x \frac{r(t)}{t^{s+1}} \, dt \ll 1+\frac{r(x)}{x^s} \ (x \to \infty)$$
and therefore
$$\dege F_s \leq \max((0,0,0,\ldots), \dege r+(-s,0,0,0,\ldots)) = \dege r+(-s,0,0,0,\ldots).$$
Taking the infimum over all $r$ as chosen, we see that $$\dege F_s \leq \dege F+(-s,0,0,0,\ldots).$$
Thus, one has $\deg F_s \leq \deg F-s < -s$, so that $\deg F_s+s-1 <-1$.
Again by Riemann--Stieltjes integration by parts, one has
$$F(x) = -x^s F_s(x) + s \int_x^\infty t^{s-1}F_s(x) \, dt.$$
Therefore, by similar reasoning as above, one has
$$\dege F \leq \dege F_s + (s,0,0,0,\ldots).$$
Statement (4) follows.
Finally, suppose that $\deg f < -1$ and $s> \deg f+1$, so that $\deg f-s <-1 $ and $\deg F \leq \deg f+1 <s$. Let
$$G_s(x) = \int_x^\infty \frac{f(t)}{t^s} \, dt.$$
By Riemann--Stieltjes integration by parts, since $\lim_{x \to \infty} \frac{F(x)}{x^s} = 0$, one has
$$G_s(x) = \frac{F(x)}{x^s} - s \int_x^\infty \frac{F(t)}{t^{s+1}} \, dt.$$
Let $r$ be regularly varying of index $d$ and both positive and continuous on $[N,\infty)$. Suppose that $F(x) \ll r(x) \ (x \to \infty)$,
with $\deg F \leq d < s$.
Then $d-s-1< -1$, so that
$$G_s(x) \ll \frac{r(x)}{x^s} + |s| \int_x^\infty \frac{r(t)}{t^{s+1}} \, dt \ll \frac{r(x)}{x^s} \ (x \to \infty).$$
It follows that $$\dege G_s \leq \dege F+(-s,0,0,0,\ldots).$$
Thus, one has $\deg G_s \leq \deg F-s < -s$, so that $\deg G_s+s-1 <-1$.
Again by Riemann--Stieltjes integration by parts, since $\lim_{x \to \infty} x^s G_s(x) = 0$, one has
$$F(x) = x^s G_s(x) + s \int_x^\infty t^{s-1}G_s(x) \, dt.$$
Therefore, by similar reasoning as above, one has
$$\dege F \leq \dege G_s + (s,0,0,0,\ldots).$$
Statement (5) follows. This completes the proof.
\end{proof}
\chapter{Asymptotic algebra}
In this chapter, we provide applications of asymptotic differential algebra (e.g., Hardy fields) to the study of logexponential degree, and vice versa. Nearly all of this chapter is not used in later chapters, the only exceptions being the definitions in Section 7.3 of a {\it Hardian} function and of the ordered field $\mathbb{H} \supsetneq \mathbb{L}$ of all {\it universally Hardian} functions, along with Theorem \ref{hardintth} and Propositions \ref{hardianexactlog} and \ref{oexppropstrong}.
\section{Lattice-ordered rings}
In this section, we generalize the various asymptotic relations studied in Section 2.1 to the setting of {\it lattice-ordered rings}, providing a modest supplement to the well-developed abstract approach to asymptotic relations taken by asymptotic differential algebra, e.g., in [11], and by the theory of ordered exponential fields, e.g., in [161]. We describe the abstract approach to asymptotic relations, with or without differentials, as {\it asymptotic algebra}.
A {\bf partially ordered abelian group}\index{partially ordered abelian group} is an abelian group $G$ (which we write additively) equipped with a partial ordering $\leq$ on $G$ such that $a \leq a'$ and $b \leq b'$ implies $a+ b \leq a'+b'$ for all $a,a',b,b' \in G$. A {\bf (totally) ordered abelian group}\index{ordered abelian group} is a partially ordered abelian group $G$ such that the partial ordering on $G$ is a total ordering. A {\bf lattice-ordered abelian group}\index{lattice-ordered abelian group} is a partially ordered abelian group $G$ such that the partial ordering on $G$ is a lattice ordering, i.e., such that the least upper bound $a \vee b$ and greatest lower bound $a \wedge b$ of $a$ and $b$ exist in $G$ for all $a, b \in G$. Note that, if least upper bounds exists in a partially ordered abelian group $G$, then so do greatest lower bounds, since then $a \wedge b = -((-a)\vee (-b))$ for all $a, b \in G$. The same holds with $\wedge$ and $\vee$ interchanged. Less obviously, by \cite[Chapter XIV Theorem 2]{birkh}, a partially ordered abelian group $G$ is lattice-ordered if and only if $a \vee 0$ exists for all $a \in G$.
Let $G$ be a lattice-ordered abelian group. For any $a \in G$, let $|a| = a \vee (-a)$ denote the {\bf absolute value} of $a$, and let $a^+ = a \vee 0$ and $a^- = (-a) \vee 0 = -(a \wedge 0)$ denote the {\bf positive part} and {\bf negative part} of $a$, respectively. One has the following elementary result.
\begin{proposition}[{\cite[Proposition 1.3]{johnson}}]
Let $G$ be a lattice-ordered abelian group, and let $a,b,c \in G$, and let $n$ be a positive integer. One has the following.
\begin{enumerate}
\item $a+(b \vee c) = (a+b) \vee (a+c)$ and $a+(b \wedge c) = (a+b) \wedge (a+c)$.
\item $a+b = (a \vee b)+(a \wedge b)$.
\item $|a+b| \leq |a|+|b|$ and $|a-b| \geq ||a|-|b||$.
\item $n(a \vee b) = (na) \vee (nb)$ and $n(a \wedge b) = (na) \wedge (nb)$.
\item $|na| =n|a|$.
\item If $na \geq 0$, then $a \geq 0$.
\item $a$ has infinite order.
\item $a = a^+-a^-$.
\item $|a| = a^++a^- \geq 0$.
\item $a^+ \wedge a^- = 0$.
\item $|a| = 0$ if and only if $a= 0$.
\end{enumerate}
\end{proposition}
All rings in this chapter are assumed commutative with identity.
A {\bf partially ordered ring}\index{partially ordered ring} is a ring $R$ equipped with a partial ordering $\leq$ on $R$ with $1 \geq 0$ such that
$R$ is a partially ordered abelian group under addition and $a,b \geq 0$ implies $ab \geq 0$
for all $a,b \in R$ (or, equivalently, $a \geq b$ implies $ac \geq bc$ for all $a,b,c \in R$ with $c \geq 0$. A {\bf totally ordered ring}\index{totally ordered ring} (resp., {\bf lattice-ordered ring})\index{lattice-ordered ring} is a partially ordered ring whose partial ordering is a total ordering (resp., a lattice ordering). Note that, if $R$ is a lattice-ordered ring, then
$$|ab| \leq |a||b|$$
for all $a,b \in R$. If $R$ is a totally ordered ring (hence also a lattice-ordered ring), then
$$|a|= \max(a,-a),$$
$$|ab| = |a||b|,$$
$$a^2 \geq 0,$$
and $a \wedge b = 0$ implies $a = 0$ or $b = 0$, for all $a, b \in R$. Moreover, none of these four properties need hold in a lattice-ordered ring.
The theories of {\it lattice-ordered groups}, or {\it $\ell$-groups}, and {\it lattice-ordered rings}, or {\it $\ell$-rings}---not necessarily abelian or commutative, respectively---are quite developed. See [273], for example.
Let $R$ be a lattice-ordered ring containing a totally ordered subfield $k$. Let us define the following {\bf asymptotic relations on $R$},\index{asymptotic relations} all defined relative to $k$: for all $a, b \in R$, we let
\begin{enumerate}
\item $a \preceq b$ if $|a| \leq r|b|$ for some $r \in k_{>0}$;\index[symbols]{.e hh1@$\preceq$}
\item $a \asymp b$ if $a \preceq b$ and $b \preceq a$;\index[symbols]{.e ee@$\asymp$}
\item $a \prec b$ if $|a| \leq r|b|$ for all $r \in k_{>0}$;\index[symbols]{.e hh2@$\prec$}
\item $a \sim b$ if $a-b \prec b$.\index[symbols]{.e hh@$\sim$}
\end{enumerate}
Note that, although these four relations depend on the choice of $k$, different choices of $k$ can yield the same relations, e.g., any subfield of $k = \RR$ yields the same asymptotic relations as does $\RR$ itself. Also note that, as long as $R$ contains a field, $R$ contains a unique minimal totally ordered subfield, necessarily isomorphic to $\QQ$. Thus, the four relations can be defined as long as $R$ contains a field, and then its minimal subfield is a natural choice for $k$. We call the four asymptotic relations on $R$ defined with respect to its minimal subfield the {\bf standard asymptotic relations on $R$}. Note that, if $k \subseteq k'$ are totally ordered subfields of $R$, then the relations $\preceq$ and $\asymp$ with respect to $k$ are finer than those with respect to $k'$, while the relations $\prec$ and $\sim$ with respect to $k$ are coarser than those with respect to $k'$ (because of properties of the existential and universal quantifiers, respectively).
The following proposition notes several properties of these four relations on $R$ and their relationships to one another.
\begin{proposition}\label{lor1}
Let $R$ be a lattice-ordered ring containing a totally ordered subfield $k$, and let $a,b,c \in R$. One has the following.
\begin{enumerate}
\item $a \preceq b$ if and only if $|a| \preceq |b|$, and likewise for $\prec$ and for $\asymp$.
\item $a \preceq a$, and likewise for $\prec$ and for $\asymp$.
\item If $a \preceq b$ and $b \preceq c$, then $a \preceq c$.
\item If $a \preceq b$ and $b \prec c$, then $a \prec c$.
\item If $a \prec b$ and $b \preceq c$, then $a \prec c$.
\item $a \asymp b$ if and only if $b \asymp a$.
\item $a \sim b$ if and only if $b \sim a$.
\item If $a \prec b$ or $a \asymp b$, then $a \preceq b$.
\item If $a \sim b$, then $a \asymp b$.
\item If $a \sim b$ and $b \sim c$, then $a \sim c$.
\item $0 \prec a$.
\item $a \preceq 0$ if and only if $a \prec 0$, if and only if $a \prec a$, if and only if $a \sim 0$, if and only if $a = 0$.
\end{enumerate}
\end{proposition}
The following proposition relates the asymptotic relations on $R$ to the ring operations on $R$.
\begin{proposition}\label{lor2} Let $R$ be a lattice-ordered ring containing a totally ordered subfield $k$, let $a_1, a_2, b_1, b_2, a, b \in R$, and let $r_1, r_2 \in k$. One has the following.
\begin{enumerate}
\item $|a| \vee |b| \leq |a|+|b| \leq 2( |a| \vee |b|)$, and therefore $|a|+|b| \asymp |a| \vee |b|$.
\item If $ a_{1} \preceq b_{1} $ and $a_{2}\preceq b_{2},$ then
$ a_{1}+a_{2} \preceq |b_{1}| + |b_{2}| $ and
$a_{1}a_{2} \preceq |b_{1}||b_{2}|.$
\item If
$ a_{1} \preceq b \text{ and } a_{2} \preceq b$,
then $r_1 a_1 + r_2 a_2 \preceq b.$
\item If $ a_{1} \prec b_{1} $ and $a_{2}\prec b_{2}$, then
$ a_{1}+a_{2} \prec |b_{1}| + |b_{2}| $ and
$a_{1}a_{2} \prec |b_{1}||b_{2}|.$
\item If
$ a_{1} \prec b \text{ and } a_{2} \prec b$,
then $r_1 a_1 + r_2 a_2 \prec b.$
\item If $ a_{1} \prec b_{1} $ and $a_{2}\preceq b_{2}$, then
$a_{1}a_{2} \prec |b_{1}||b_{2}|.$
\item If $a_{1} \sim b_{1}$, $a_{2}\sim b_{2}$ and $|b_{1} b_{2}| = |b_{1}||b_{2}|$, then
$a_{1}a_{2} \sim b_{1}b_{2}.$
\item If $a_1 \sim b$, then $a_1 + a_2 \sim b$ if and only if $a_2 \prec b$.
\end{enumerate}
\end{proposition}
Note that, by statement (1) of the proposition, one can replace $|b_{1}| + |b_{2}| $ in statements (2) and (4) with
$|b_{1}| \vee |b_{2}|$.
Assume now an additional condition on the ring $R$, namely, that $|ab| = |a||b|$ for all $a,b \in R$. This holds, for example, if $R$ is totally ordered, or, more generally, if $R$ is an {\it f-ring} [145] [273]. Then we may define an addition $[a]+[b] = [ab]$ on $\asymp$-equivalence classes $[a],[b] \in R/{\asymp}$, as well as a partial ordering $[a]\leq [b]$ on $R/{\asymp}$, defined to hold precisely when $b \preceq a$. Then $R/{\asymp}$ is a lattice-ordered commutative monoid with identity element $0 = [1]$, that is, $R/{\asymp}$ is a commutative monoid with identity element $[1]$ and $\leq$ is a lattice ordering on $R/{\asymp}$ such that $[a_1]\leq [b_1]$ and $[a_2]\leq [b_2]$ implies $[a_1]+[a_2]\leq [b_1]+[b_2]$ for all $a_1,a_2,b_1,b_2 \in R$. Note that $[|a| \vee |b|] = [a] \vee [b]$ for all $a,b \in R$. Now, let
$$v: R \longrightarrow R/{\asymp}$$
act by $$v: a \longmapsto [a].$$ Then the map $v$ is surjective and satisfies the following conditions for all $a,b \in R$.
\begin{enumerate}
\item $v(1) = 0$.
\item $v(ab) = v(a)+v(b)$.
\item $v(a+b) \geq v(a) \wedge v(b)$.
\end{enumerate}
Indeed, (1) and (2) are obvious, and (3) follows immediately from
$$|a+b| \leq |a|+|b| \asymp |a| \vee |b|.$$
We denote $[0]$ by $\infty$, since $[0]+[a]=[0]$ for all $a\in R$, and we let $\Gamma_R = (R/{\asymp}) \backslash \{\infty\}$, so that $v$ restricts to a map
$$v: R\backslash\{0\} \longrightarrow \Gamma_R.$$ Note that $R$ is an integral domain if and only if $R\backslash\{0\}$ is a submonoid of $R^\times$, if and only if $0 \neq \infty$ and $[a]+[b] = \infty$ implies $[a]= \infty$ or $[b] = \infty$ for all $a,b \in R$, if and only if $\Gamma_R$ is a submonoid of $R/{ \asymp}$. Moreover, if $R$ is a lattice-ordered field, or {\bf $\ell$-field} [273], then $\Gamma_R$ is an abelian group. Likewise, if $R$ is totally ordered, then $\Gamma_R$ is totally ordered and $$v(a+b) \geq v(a) \wedge v(b) = \min(v(a),v(b))$$ for all $a,b \in R$.
An important special case, employed to a great extent in the next section, is where $R$ is a totally ordered field. In that case, the map $v: R \longrightarrow R/{\asymp} \ = \Gamma_R \cup \{\infty\}$ is a {\it valuation on $R$} and the relation $\preceq$ is a {\it dominance relation on $R$} \cite[Section 3.1]{asch}. Let $K$ be any field. A {\bf valuation on $K$}\index{valuation on a field} is a surjective map $v: K \longrightarrow \Gamma \cup \{\infty\}$, where $\Gamma$ is a (totally) ordered abelian group, and where one sets $\gamma <\infty = \infty+\infty = \gamma+\infty = \infty+\gamma$ for all $\gamma \in \Gamma$, such that $v(a) = 0$ if and only if $a = \infty$, and such that $v(ab) = v(a)+v(b)$ and $v(a+b) \geq \min(v(a) , v(b))$ for all $a,b \in K$. Moreover, the {\bf dominance relation $\preceq$ on $K$ associated to $v$}, defined by $a \preceq b$ if $v(a) \geq v(b)$, coincides with our asymptotic relation $\preceq$ defined earlier. The same holds of the relations $\asymp$, $\prec$, and $\sim$ associated to $v$ as defined in \cite[Section 3.1]{asch}, where, specifically, one writes $a \asymp b$ if $a \preceq b$ and $b \preceq a$, one writes $a \prec b$ if $a \preceq b$ and $b \not \preceq a$, and one writes $a \sim b$ if $a-b \prec b$.
A {\bf valued field}\index{valued field} is a field $K$ equipped with a valuation $v: K \longrightarrow \Gamma \cup \{\infty\}$. The ordered abelian group $\Gamma_v = \Gamma$ is called the {\bf value group of $v$}. \index{value group of a valuation} Let $K$ be a valued field with valuation $v$. The subset $$K_v = \{a \in K: v(a) \geq 0\}$$ of $K$ is a subring of $K$ such that $a \in K_v$ or $1/a \in K_v$ for all $a \in K^*$ and is called the {\bf valuation ring of $v$}. \index{valuation ring of a valuation} It follows that the ring $K_v$ is an integral domain with unique maximal ideal
$$\mm_v = \{a \in K: v(a) > 0\}.$$
It is well known that there are natural one-to-one correspondences between the valuations on $K$ up to {\it equivalence}, the valuation rings of the valuations on $K$, the {\it valuation rings of $K$}, and the dominance relations on $K$ \cite[Section 3.1]{asch}.
To summarize, we have shown that any totally ordered field $R$ equipped with a totally ordered subfield $k$ induces a valuation $v: R \longrightarrow \Gamma_R \cup \{\infty\} = R/{ \asymp}$ on $R$, which is canonical for $k \cong \QQ$. Moreover, the map $v$ and the asymptotic relations $\preceq$, $\asymp$, $\prec$, and $\sim$ associated to $v$ as in \cite[Section 3.1]{asch} generalize to any lattice-ordered ring $R$ containing a distinguished totally ordered subfield $k$, and they share many of the properties as they do when $R$ is a totally ordered field, e.g., those properties expressed in Propositions \ref{lor1} and \ref{lor2}.
\section{The ring of germs of real functions at $\infty$}
If $R$ is a ring (commutative with identity) and $X$ is a set, then we denote by $R^X = \prod_{x \in X}R$ the ring of all functions from $X$ to $R$ under pointwise addition and multiplication. If $R$ is a partially ordered ring, then $R^X$ is a partially ordered ring when ordered by the relation $\leq$, where $f \leq g$ if $f(x) \leq g(x)$ for all $x \in X$. Moreover, if $R$ is a lattice-ordered ring, then $R^X$ is also a lattice-ordered ring, with $(f \vee g)(x) = f(x) \vee g(x)$ for all $x \in X$, and likewise for $\wedge$. In particular, the ring $\RR^X$ is lattice-ordered.
Also when $R = \RR$, one has $f \leq g$ if and only if $g-f = h^2$ for some $h \in \RR^X$.
We extend any function $f \in \RR^{\RR_\infty}$ to a function on $\RR$ by defining it to be $0$ at all $x \in \RR\backslash \dom f$. Thus we have a natural surjection $ \RR^{\RR_\infty} \longrightarrow \RR^{\RR} = \prod_{x \in \RR} \RR$.
The ring $\mathcal{R}\index[symbols]{.i ka@$\mathcal{R}$}$ of germs at $\infty$ of all functions in $\RR^\RR$, that is, the ring $\mathcal{R} = \RR^{\RR}/{ =_\infty}$, is a lattice-ordered ring under the relation $\leq \ = \ \leq_{\infty}$. Another way to construct $\mathcal{R}$ is as the quotient ring $(\RR^{\RR})_\infty = \RR^{\RR}/(0)_\infty$ by the (radical) ideal $(0)_\infty$ in $\RR^\RR$ of all functions that vanish on a neighborhood of $\infty$. Note that $f \leq g$ in $\mathcal{R}$ if and only if $g-f = h^2$ for some $h \in \mathcal{R}$. The group of units $\mathcal{R}^*$ of the ring $\mathcal{R}$ is equal to the set of all germs of functions that are eventually nonzero, and it contains $\mathcal{R}_{>0}$ and $\mathcal{R}_{>0} \cup \mathcal{R}_{<0} \cong \mathcal{R}_{>0} \oplus \{1,-1\}$ as proper subgroups. Moreover, the monoid of nonzerodivisors of $\mathcal{R}$ is equal to $\mathcal{R}^*$, that is, $\mathcal{R}$ is a total quotient ring.
A (commutative) ring $R$ is {\bf von Neumann regular}\index{von Neumann regular ring} if $I^2 = I$ for all principal ideals (or equivalently, all ideals) $I$ of $R$ [105]. A ring $R$ is von Neumann regular if and only if $R$ is a reduced total quotient ring in which every prime ideal is maximal. The von Neumann regular rings are a natural generalization of the fields to rings with zerodivisors, since, for example, a field is equivalently a von Neumann regular ring that is an integral domain. The class of von Neumann regular rings is closed under arbitrary direct products and quotient rings. Thus, the rings $\RR^\RR$ and $\mathcal{R}$ are von Neumann regular.
The absolute value on the lattice-ordered ring $\mathcal{R}$, as described in the previous section, coincides with the usual absolute value on $\mathcal{R}$ and therefore satisfies $|fg| = |f||g|$ for all $f,g \in \mathcal{R}$. Moreover, since $\mathcal{R}$ is a lattice-ordered ring containing the totally ordered field $\RR$, the results of the previous section yield (standard) asymptotic relations $\preceq$, $\asymp$, $\prec$, and $\sim$ on $\mathcal{R}$, and it is clear that these relations coincide with the usual $O$, $\asymp$, $o$, and $\sim$ relations on $\mathcal{R}$. In other words, for all $f, g \in \mathcal{R}$, one has $f \preceq g$ if and only if $f(x) = O(g(x)) \ (x \to \infty)$, and so on.
(Note that, on $\RR^X$, the former asymptotic relations must hold globally, e.g., one has $f \preceq g$ in $\RR^X$ if and only if there exists an $M \in \RR_{>0}$ such that $|f(x)| \leq M |g(x)|$ for all $x \in X$. Unfortunately, then, one has $f \prec g$ in $\RR^X$ if and only if $f = 0$.) Consequently, from Propositions \ref{lor1} and \ref{lor2}, we deduce many of the the well-known properties of the asymptotic relations $O$, $\asymp$, $o$, and $\sim$ on $\mathcal{R}$ stated in Section 2.1. Moreover, if $g \in \mathcal{R}^*$, that is, if $g$ is eventually nonzero, then $f \preceq g$ is equivalent to $\limsup_{x \to \infty} \frac{|f(x)|}{|g(x)|} < \infty$, while $f \prec g$ is equivalent to $\lim_{x \to \infty} \frac{|f(x)|}{|g(x)|} = 0$ and $f \sim g$ is equivalent to $\lim_{x \to \infty} \frac{|f(x)|}{|g(x)|} = 1$. Likewise, if $f \in \mathcal{R}^*$, then $f \preceq g$ is equivalent to $\liminf_{x \to \infty} \frac{|g(x)|}{|f(x)|} > 0$, and $f \prec g$ is equivalent to $\lim_{x \to \infty} \frac{|g(x)|}{|f(x)|} = \infty$. If $R$ is any subring of $\mathcal{R}$, then we let $\preceq$, $\asymp$, $\prec$ and $\sim$ denote the relations on $\mathcal{R}$ restricted to $R$.
Now, one has $e^f \in \mathcal{R}_{>0}$ if $f \in \mathcal{R}$, and $\log f \in \mathcal{R}$ if $f \in \mathcal{R}_{>0}$, and $e^f \leq e^g$ if and only if $f \leq g$, while also $e^{f+g} = e^f e^g$, for all $f, g \in \mathcal{R}$. Thus, the map $$\exp \circ -: \mathcal{R}^+ \longrightarrow \mathcal{R}_{>0}$$ acting by $f \longmapsto e^f$ is an order isomorphism from the additive partially ordered group $\mathcal{R}^+$ of $\mathcal{R}$ onto the multiplicative partially ordered group $\mathcal{R}_{>0}$, with inverse acting by $f \longmapsto \log f$.
If $R$ is a partially ordered ring, then an {\bf exponential on $R$}\index{exponential on a partially ordered ring} is an isomorphism $E$ from the partially ordered additive group $R^+$ onto the partially ordered group $R^* \cap R_{>0}$. This condition means that $E$ is a group isomorphism and both $E$ and its inverse $E^{-1}$ are increasing. Let us say that a {\bf (partially) ordered exponential ring}\index{ordered exponential ring} is a partially ordered ring $R$ equipped with an exponential on $R$. Thus, the pair $(\mathcal{R}, \exp \circ -)$ is an exponential ring. Note that, if $R$ is a partially ordered field, then $R^* \cap R_{>0} = R_{>0}$. A {\bf (totally) ordered exponential field}\index{ordered exponential field} is an ordered exponential ring that is a totally ordered field, that is, it is a totally ordered field $K$ equipped with an exponential $K \longrightarrow K_{>0}$ on $K$ \cite[p.\ 22]{kuhl}.
Let $(R,E)$ be an ordered exponential ring. Then $f^\wedge a := E(aE^{-1}(f))$ behaves much like ``$f^a$'' for all $a \in R$ and all $f \in R^* \cap R_{>0}$ in that it satisfies the obvious laws of exponents: for all $f,g \in R^* \cap R_{>0}$ and all $a,b \in R$, one has the following.
\begin{enumerate}
\item $f^\wedge a \in R^* \cap R_{>0}$.
\item $f^\wedge 0 =1$.
\item $1^\wedge a = 1$.
\item $f^\wedge(a+b) = (f^\wedge a)(f^\wedge b)$.
\item $f^\wedge (ab) = (f^\wedge a)^\wedge b$.
\item $(fg)^\wedge a = (f^\wedge a)( g^\wedge a)$.
\item $f^\wedge a = f^a$ if $a \in \ZZ$.
\item $(f^a)^\wedge b = f^{\wedge} (ab)= (f^{\wedge}b )^a$ if $a \in \ZZ$.
\item If $a < b$, then $f^\wedge a < f^\wedge b$ if $f >1$ and $f^\wedge a > f^\wedge b$ if $f < 1$.
\item If $f <g$, then $f^\wedge a < g^\wedge a$ if $a > 0$, and $f^\wedge a > g^\wedge a$ if $a < 0$.
\end{enumerate}
It follows that, for any fixed $g \in R^* \cap R_{>0}$, the map $g^\wedge- = E( E^{-1}(g) \cdot -)$ acting by $a \longmapsto g^\wedge a$ is an exponential on $R$ with $1 \longmapsto g$. If $F: R \longrightarrow R^* \cap R_{>0}$ is another exponential on $R$, then $$(E^{-1}\circ F)(r+s) = E^{-1}(F(r)F(s)) = (E^{-1}\circ F)(r)+(E^{-1}\circ F)(s)$$ for all $r,s \in R$, and therefore $E^{-1}\circ F$ is an automorphism of the ordered additive group $R^+$ of $R$. Conversely, if $\phi$ is any automorphism of the ordered group $R^+$, then $E \circ \phi$ is an exponential on $R$. Thus, if a partially ordered ring $R$ admits an exponential, then the set of all exponentials on $R$ is a group, isomorphic to the group of all automorphisms of the ordered group $R^+$, and it contains (a group isomorphic to) the group $R^* \cap R_{>0} \cong R^+$ as a subgroup, since dilation $x \longmapsto rx$ is an automorphism of the ordered group $R^+$ for some $r \in R$ if and only if $r \in R^* \cap R_{>0}$. In particular, if $E$ is an exponential on $R$, then $E \circ (r\cdot -)$ is an exponential on $R$ for all $r \in R^* \cap R_{>0}$.
For the ordered exponential ring $(\mathcal{R}, \exp \circ -)$, one has $f^\wedge a = e^{a \log f} = f^a$ for all $f \in \mathcal{R}$ and all $a \in \mathcal{R}_{>0}$. Thus, applying the remarks in the previous paragraph to $g = \id$, and to $r = \log$, we see in two distinct ways that the map
$$E_- = \id^- = \exp \circ (\log \cdot -): \mathcal{R}^+ \longrightarrow \mathcal{R}_{>0}$$ acting by $$f(x) \longmapsto E_f(x) = x^{f(x)} = e^{(\log x)f(x)}$$ is a ``base independent'' exponential on $\mathcal{R}$, with inverse $$L_- = (E_-)^{-1} = \frac{\log \circ -}{\log}: \mathcal{R}_{>0} \longrightarrow \mathcal{R}^+,$$ which is also base independent, acting by
$$L_-: f(x) \longmapsto L_f(x) = \frac{\log f(x)}{\log x}.$$
We extend $\mathcal{R}$ to the set $\mathcal{R}_{\pm \infty}$ of germs of all functions from $\RR$ to $\overline{\RR}$, and we extend the map $L_-$ to a map $\mathcal{R}_{\geq 0} \longrightarrow \mathcal{R}_{\pm \infty}$ by setting $L_f(x) = -\infty$ if $f(x) = 0$. Note that $\limsup$ and $\liminf$ define maps
$$\limsup_{x \to \infty}: \mathcal{R}_{\pm \infty} \longrightarrow \overline{\RR}$$
$$\liminf_{x \to \infty}: \mathcal{R}_{\pm \infty} \longrightarrow \overline{\RR},$$
and the (upper) degree $\deg$ and lower degree $\underline{\deg}$ maps on $\mathcal{R}$ are given respectively by
the composition
$$\mathcal{R} \overset{|-|}{\longrightarrow} \mathcal{R}_{\geq 0} \overset{L_-}{\longrightarrow} \mathcal{R}_{\pm \infty} \overset{\underset{x \to \infty}{\limsup}}{\longrightarrow} \overline{\RR}$$
$$\mathcal{R} \overset{|-|}{\longrightarrow} \mathcal{R}_{\geq 0} \overset{L_-}{\longrightarrow} \mathcal{R}_{\pm \infty} \overset{\underset{x \to \infty}{\liminf}}{\longrightarrow} \overline{\RR}.$$
This is our entry point into understanding the degree map from a ``universal'' perspective, a topic that is taken up further in Section 7.4.
\section{Real asymptotic differential algebra: Hardy rings and Hardy fields}
Let $\mathcal{C}$\index[symbols]{.i kaa@$\mathcal{C}$} be the subring of $\mathcal{R}$ consisting of the germs of continuous functions.
The group of units $\mathcal{C}^*$ of the ring $\mathcal{C}$ is the group of all functions in $\mathcal{C}$ that are eventually positive or eventually negative, that is, that are comparable to $0$, and thus $\mathcal{C}^* = \mathcal{C}_{>0} \cup \mathcal{C}_{<0}$.
\begin{proposition}[{\cite[Section 2]{bos}}]
Let $R$ be a subring of $\mathcal{C}$ containing $\RR$. The following conditions are equivalent.
\begin{enumerate}
\item $R$ is contained in some subfield of $\mathcal{C}$.
\item $R$ is an integral domain.
\item $R$ is contained in $\mathcal{C}^* \cup \{0\}$.
\item $R$ is contained in $\mathcal{C}_{>0} \cup \mathcal{C}_{<0} \cup \{0\}$.
\item $R$ is totally ordered with respect to the relation $\leq$ on $\mathcal{C}$.
\end{enumerate}
Suppose that these conditions hold. Then the quotient field of $R$ is the smallest subfield of $\mathcal{C}$ containing $R$. Suppose also that $R$ contains $\RR$. Then, for all $f \in R$, the ring $\RR(f)$ is the quotient field of $\RR[f]$ and is the smallest subfield of $R$ containing $\RR$ and $f$. Moreover, an $f \in \mathcal{C}$ belongs to some subfield of $\mathcal{C}$ if and only if $f$ is $\leq$-comparable to all $c \in \RR$, if and only if $\RR[f]$ is totally ordered by $\leq$.
\end{proposition}
\begin{corollary}
Let $K$ be a subfield of $\mathcal{C}$. Then $$K\backslash\{0\} = K^* \subseteq \mathcal{C}^* = \mathcal{C}_{>0} \cup \mathcal{C}_{<0}.$$
Thus, every function in $K$ is (eventually) positive, (eventually) negative, or (eventually) $0$, whence $K$ is a totally ordered field under the ordering $\leq \ = \ \leq_\infty$.
\end{corollary}
For any nonnegative integer $n$, let $\mathcal{C}^{(n)}$ denote the ring of germs of functions from $\RR$ to $\RR$ that are $n$-times continuously differentiable on a neighborhood of $\infty$, and let $\mathcal{D} = \mathcal{C}^{(\infty)} = \bigcap_{n = 1}^\infty \mathcal{C}^{(n)}\index[symbols]{.i kb@$\mathcal{D}$}$. The group of units of $\mathcal{D}$ is given by $\mathcal{D}^* = \mathcal{D}_{>0} \cup \mathcal{D}_{<0}$. Note that a function in $\mathcal{D}$ need not be infinitely differentiable on a neighborhood of $\infty$. Equivalently, one has $\mathcal{D} = \mathcal{C}^{(\infty)} \supsetneq \mathcal{C}^\infty$, where $\mathcal{C}^\infty$ is the ring of germs of all infinitely differentiable functions on $\RR$. Differentiation $D = \frac{d}{dx} = (-)'$ is a {\it derivation} $D: \mathcal{D} \longrightarrow \mathcal{D}$ on $\mathcal{D}$ and makes the ring $\mathcal{D}$ into a {\it differential $\RR$-algebra}. ({\it Differential algebra}, introduced by J.\ Ritt in 1950, is the study of differential rings and their applications to the algebraic study of differential equations.) Let us say that a {\bf Hardy ring}\index{Hardy ring} is a subring of $\mathcal{D}$ that is closed under differentiation. In other words, a Hardy ring is a subring $R$ of $\mathcal{D}$ such that $D(R) \subseteq R$, i.e., such that $f' \in R$ for all $f \in R$. Thus, $\mathcal{D}$ is the largest Hardy ring, and $\mathcal{C}^\infty$ is also a Hardy ring. Two obvious classes of examples of Hardy rings are $C$ and $C[x] \cong C[X]$ (generated as a subring of $\mathcal{D}$), where $C$ is any subring of $\RR$ and where $x = \id$ is the identity function. Every Hardy ring $R$ is a {\it differential $C_R$-algebra}, where $C_R = \ker D|_R$ is the subring $C_R = R \cap \RR$ of $\RR$ of all constants in $R$. Note that the intersection of a collection of Hardy rings is a Hardy ring, and therefore every subring of $\mathcal{D}$ is contained in a smallest Hardy ring. For convenience, hereafter we assume that all subrings $R$ of $\mathcal{R}$ contain $\RR$, or, equivalently, have ring of constants $R \cap \RR = \RR$.
A {\bf Hardy field}\index{Hardy field} is a Hardy ring that is a field. Equivalently, a Hardy field is a subfield $K$ of $\mathcal{D}$ containing $\RR$ such that $f' \in K$ for all $f \in K$. Since every subfield of $\mathcal{C}$ is a totally ordered field ordered by $\leq$, so is every Hardy field. Examples of Hardy fields include the ordered fields $$\RR \subsetneq \RR(x) \subsetneq \RR(x^a: a \in \RR) \subsetneq \RR(\mathfrak{L})\subsetneq \mathbb{L}$$ employed in Chapter 6. Note that, if $G$ is any subgroup of $\RR$ containing $\ZZ$, then $\RR(x^a: r \in G) \cong \RR(G)$ is a Hardy field containing $\RR(x)$, and, if $G'$ is another such group, distinct from $G$, then $\RR(x^a: r \in G) \neq \RR(x^a: r \in G')$.
Let $K$ be a Hardy field. If $f \in K$, then $f' \in K$, so that $f'$ is (eventually) positive, negative, or $0$, so that $f$ is (eventually) (strictly) increasing, (strictly) decreasing, or constant. It follows that $f \preceq g$ is equivalent to $g \not \prec f$, as long as $f \neq 0$. Indeed, if $g \neq 0$, so that $f/g \in K$, then $f \preceq g$ is equivalent to
$\lim_{x \to \infty} \frac{f(x)}{g(x)} \in \RR$, while $f \asymp g$ is equivalent to $\lim_{x \to \infty} \frac{f(x)}{g(x)} \in \RR^*$, and $f \prec g$ is equivalent to $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$.
Since any subfield $K$ of $\mathcal{C}$ is totally ordered with totally ordered subfield $\QQ$, any such field $K$ is equipped with a natural valuation on $K$, as described in the previous section: the set $K^*/{\asymp}$ of all $\asymp$-equivalence classes $[f]$ of germs $f \in K^*$ is a totally ordered abelian group $\Gamma_K\index[symbols]{.i kg@$\Gamma_K$}$ under the ordering $\leq$ on $\asymp$-equivalence classes $[f]$, where $[f] \leq [g]$ if $g \preceq f$, and the map $v: K \longrightarrow \Gamma_K \cup \{\infty\}$ acting by $f \longmapsto v(f) = [f]$, where $0 \longmapsto \infty$, is a valuation on the field $K$. The valuation ring $$\mathcal{O}_K = K_v = \{f \in K: v(f) \geq 0\} = \{f \in K: f \preceq 1\}\index[symbols]{.i kh@$\mathcal{O}_K$}$$ of the valued field $K$ is the ring of all bounded functions $f$ in $K$, or, equivalently, all functions $f$ in $K$ such that $\lim_{x \to \infty} f(x) \in \RR$. The unique maximal ideal $\mm_K$ of the valuation ring $\mathcal{O}_K$ is the ideal
$$\mm_K = \mm_v = \{f \in K: v(f) > 0\} = \{f \in K: f \prec 1\}\index[symbols]{.i ki@$\mm_K$}$$
of all functions $f$ in $K$ with $\lim_{x \to \infty} f(x) = 0$. Thus, one has a canonical isomorphism
$$\mathcal{O}_K \cong \RR\oplus \mm_K$$
of abelian groups, and the map to the residue field $\mathcal{O}_K/\mm_K$ is the projection
$$\lim_{x \to \infty} : \mathcal{O}_K \longrightarrow \RR \cong \mathcal{O}_K/\mm_K$$
onto $\RR$, which has a section given by the inclusion $ \RR\longrightarrow \mathcal{O}_K$, and which extends to the map
$$\lim_{x \to \infty} : K \longrightarrow \overline{\RR}$$
with $\mathcal{O}_K$ equal to the preimage of $\RR \subsetneq \overline{\RR}$.
Moreover, the group of units of the valuation domain $\mathcal{O}_K$ is the group
$$\mathcal{O}_K^* = \mathcal{O}_K\backslash \mm_K = \{f \in K: v(f) = 0\} = \{f \in K: f \asymp 1\}$$
of all functions $f$ in $K$ with $\lim_{x \to \infty} f(x) \in \RR^*$. Note, then, that $v(f)= v(g)$ if and only if $f/g \in \mathcal{O}_K^*$, if and only if $(f) = (g)$ as principal fractional ideals in $\mathcal{O}_K$. Thus, $\Gamma_K$ is order-isomorphic to the group of all nonzero principal fractional ideals of $\mathcal{O}_K$ ordered by $\supseteq$. It follows that, as groups, one has $\Gamma_K \cong K^*/\mathcal{O}_K^*$. Moreover, the set $K_{>0}$ of all positive elements of $K$ is a subgroup of $K^*$, and one has $K^* \cong K_{>0} \oplus \{1,-1\}$.
We say that a subring $R$ of $\mathcal{R}$ is {\bf logexponentially closed}\index{logexponentially closed} if $e^f \in R$ for all $f \in R$ and and $\log |f| \in R$ for all $f \in R^*$, that is, if the exponential $\exp \circ -$ on $\mathcal{R}$ restricts/corestricts to an exponential on $R$. If a subring $R$ of $\mathcal{R}$ is logexponentially closed, then $R$ inherits from $\mathcal{R}$ the structure of an ordered exponential ring $(R, \exp \circ ( a \cdot -))$ for any $a \in R$. In this case, $R$ also has the structure of a ``base independent'' ordered exponential ring with exponential $E_- = \id^- = \exp\circ (\log \cdot -)$ if and only if $R \supseteq \RR(x)$, i.e., if and only if $\id \in R$. Note that the Hardy field $\mathbb{L}$ of all logarithmico-exponential functions is the smallest logexponentially closed subfield of $\mathcal{R}$ containing $\RR(x)$. Similarly, the Hardy field $\RR(\mathfrak{L})$ is the smallest subfield $K$ of $\mathcal{R}$ containing $\RR(x^a : a \in \RR)$ such that $f \circ \log \in K$ for all $f \in K$, and the Hardy field $\RR(x^a : a \in \RR)$ is the smallest subfield $K$ of $\mathcal{R}$ containing $\RR(x)$ such that $f \circ x^a \in K$ for all $f \in K$ and all $a>0$.
The largest Hardy ring $\mathcal{D}$ has several nice closure properties: it is closed under differentiation, closed under antidifferentiation, and logexponentially closed. The rings $\mathcal{R}$ and $\mathcal{C}$ are also logexponentially closed, and $\mathcal{C}$ is also closed under antidifferentiation. The operation $$\operatorname{Dlog}: \mathcal{D}^* \longrightarrow \mathcal{D}^+\index[symbols]{.i ke@$\operatorname{Dlog}$}$$
of {\bf logarithmic differentiation},\index{logarithmic differentiation} acting by $$\operatorname{Dlog}: f \longmapsto \frac{f'}{f} = (\log |f|)',$$ is a group homomorphism. If $R$ is a Hardy ring, then $\operatorname{Dlog}$ restricts to a group homomorphism $R^* \longrightarrow R^+$.
Since the intersection of a collection of Hardy subfields of a Hardy field is a Hardy field, every subset of a Hardy field $K$ is contained in a smallest Hardy subfield of $K$. However, if $K$ and $L$ are Hardy fields, then there need not be a Hardy field containing both $K$ and $L$. A {\bf maximal Hardy field}\index{maximal Hardy field} is a Hardy field that is not a proper subfield of any other Hardy field. By Zorn's lemma, every Hardy field is contained in a maximal Hardy field. The intersection $\mathbb{H}\index[symbols]{.i kc@$\mathbb{H}$}$ of all maximal Hardy fields is a Hardy field (denoted by $E$ in [35]). As we show later, one has $\mathbb{L} \subsetneq \mathbb{H}$. If $K$ is a Hardy field, then, following [35], we let $\operatorname{E}(K)\index[symbols]{.i kca@$\operatorname{E}(K)$}$ denote the intersection of all maximal Hardy fields containing $K$. A Hardy field $K$ is {\bf perfect}\index{perfect Hardy field} if it is the intersection of some collection of maximal Hardy fields, or, equivalently, if $\operatorname{E}(K) = K$ [36]. Thus, $\mathbb{H} = \operatorname{E}(\RR)$ is the smallest perfect Hardy field. Moreover, if $K$ is a Hardy field, then $\operatorname{E}(K)$ is the smallest perfect Hardy field containing $K$.
A function $f \in \mathcal{D}$ is {\bf Hardian}\index{Hardian function} if there exists a Hardy field containing $f$.
If $R$ is a Hardy ring and $f \in \mathcal{D}$, then the $R$-subalgebra $R[f,f',f'',\ldots]$ of $\mathcal{D}$ generated by $f$ and all of its derivatives is the smallest Hardy ring containing $f$, and there is a unique (surjective) ring homomorphism $$D_{f,R}: R[X_0,X_1,X_2,\ldots] \longrightarrow R[f,f',f'',\ldots]\index[symbols]{.i ki1@$D_{f,R}$}$$ sending $X_n$ to $f^{(n)}$ for all $n$. If $K$ is a Hardy field, then a (Hardian) function $f \in \mathcal{D}$ is {\bf Hardy adjoinable to $K$}\index{Hardy adjoinable} if there exists a Hardy field containing both $K$ and $f$. Thus, $f \in \mathcal{D}$ is Hardian if and only if $f$ is Hardy adjoinable to $\RR$, if and only if $f$ is Hardy adjoinable to the Hardy field $\mathbb{H}$. If $f$ is Hardy adjoinable to a Hardy field $K$, then there is a smallest Hardy field containing $K$ and $f$, denoted $K\{f\}$.
The following result is clear.
\begin{proposition}[{[35]}]
Let $f \in \mathcal{D}$, and let $K$ be a Hardy field. The following conditions are equivalent.
\begin{enumerate}
\item $f$ is Hardy adjoinable to $K$.
\item The image of the map $D_{f,K}$ is contained in $\mathcal{D}^* \cup \{0\}$.
\item For all $p(X_0, X_1,X_2,\ldots) \in K[X_0,X_1,X_2,\ldots]$, the function $p(f,f',f'',\ldots)$ is either eventually positive, eventually negative, or eventually $0$, i.e., $p(f,f',f'',\ldots)$ lies in $\mathcal{D}^* \cup \{0\}$.
\item The ring $K[f,f',f'',\ldots]$ is an integral domain.
\item The kernel $D_{f,K}^{-1}(0)$ of $D_{f,K}$ is prime, as an ideal of $K[X_0,X_1,X_2,\ldots]$.
\end{enumerate}
When these conditions hold, one has
\begin{align*}
K\{f\} & = K(f,f',f'',\ldots) \\
& = \{p(f,f',f'',\ldots)/q(f,f',f'',\ldots): p,q \in K[X_0,X_1,X_2,\ldots], \, q(f,f',f'',\ldots) \neq 0\}
\end{align*}
where $K(f,f',f'',\ldots) $ (generated as a subfield of $ \mathcal{D}$) is the quotient field of the integral domain $K[f,f',f'',\ldots]$, and it is isomorphic to the quotient field of $K[X_0,X_1,X_2,\ldots]/D_{f,K}^{-1}(0)$, i.e., the residue field of the domain $K[X_0,X_1,X_2,\ldots]$ at the prime ideal $D_{f,K}^{-1}(0)$.
\end{proposition}
We say that a subset $S$ of $ \mathcal{D}$ is {\bf Hardy adjoinable to $K$}\index{Hardy adjoinable} if there exists a Hardy field containing both $K$ and $S$, and we let $K\{S\}\index[symbols]{.i kl@$K\{S\}$}$ denote the smallest Hardy field containing $K$ and $S$. If $K$ and $L$ are Hardy fields, then $L$ is Hardy adjoinable to $K$ if and only if $K$ is Hardy adjoinable to $L$, and when both conditions hold one has $K\{L\} = L\{K\}$.
\begin{example} Let $K$ be a Hardy field. Since $x$, $\exp x$, and $\log x$ are in $\mathbb{H}$, the set $\{x, \exp x, \log x\}$ is Hardy adjoinable to $K$, and one has $K\{x\} = K(x)$, $K\{\exp x\} = K(\exp x)$, $K\{\log x \} = K(x,\log x)$, and $$K\{\exp x, \log x\} = (K\{\exp x\})\{\log x\} = (K(\exp x))\{\log x\} = K(\exp x,x,\log x).$$
\end{example}
As we have noted, the Hardy field $\mathbb{H}$ is equal to the set of all functions in $ \mathcal{D}$ that are Hardy adjoinable to every Hardy field. Thus, we call elements of $\mathbb{H}$ {\bf universally Hardian}, \index{universally Hardian function} and $\mathbb{H}$ is the Hardy field of all universally Hardian functions (or, rather, germs at $\infty$). If $K$ is any Hardy field, then an obvious transfinite inductive argument shows that $\mathbb{H}$ is Hardy-ajoinable to $K$, and therefore $K$ is Hardy adjoinable to $\mathbb{H}$. Thus, every Hardy field is Hardy adjoinable to $\mathbb{H}$. (Indeed, if $K$ is a Hardy field, then $K$ is contained in some maximal Hardy field $M$, and then $M$ contains both $K$ and $\mathbb{H}$.) Moreover, $\mathbb{H}$ is the largest Hardy field with this property: if $K$ is a Hardy field such that every Hardy field is Hardy adjoinable to $K$, then $K$ is Hardy adjoinable to every Hardy field, so that all elements of $K$ are Hardy adjoinable to every Hardy field, and thus $K \subseteq \mathbb{H}$. Thus, the Hardy field $ \mathbb{H}$ has the universal property that it is the largest Hardy field $K$ such that every Hardy field is Hardy adjoinable to $K$. Equivalently, the Hardy field $ \mathbb{H}$ is the largest Hardy field that is Hardy adjoinable to every Hardy field. Thus, we call $ \mathbb{H}$ the {\bf universally Hardy adjoinable Hardy field}.\index{universally Hardy adjoinable Hardy field $\mathbb{H}$}
Maximal Hardy fields are precisely the Hardy fields $M$ such that $f \in M$ for all Hardian $f \in \mathcal{D}$ that are Hardy adjoinable to $M$. Moreover, for any Hardy field $K$, the intersection $\operatorname{E}(K)$ of all maximal Hardy fields containing $K$ is equal to the set of all $f \in \mathcal{D}$ that are Hardy adjoinable to every Hardy field containing $K$.
Let $K$ be a subfield of $\mathcal{D}$. The algebraic closure $K^o\index[symbols]{.i kn@$K^o$}$ of $K$ in $\mathcal{D}$ is a {\it real closed} subfield of $\mathcal{D}$, and in fact it is isomorphic to the {\it real closure of $K$},
and one has $(K^o)^o = K^o$. Thus, $K$ is real closed if and only if $K^o = K$. Moreover, if $K$ is a Hardy field, then $K^o$ is a (real closed) Hardy field. It follows that $K$ is real closed if $K$ is a perfect Hardy field. For proofs of these facts, see \cite[Section 3]{bos}.
\begin{theorem}[{\cite[Theorem 2]{rosenl}}]\label{rosenl}
Let $K$ be a Hardy field, let $p(X),q(X) \in K[X]$, and let $y$ be the (germ of) an eventually differentiable function such that $q(y)$ is eventually nonzero and such that $y' = p(y)/q(y)$ on some neighborhood of $\infty$. Then $K(y)$ is a Hardy field. Thus, $y$ is Hardy adjoinable to $K$ and $K\{y\} = K(y)$. Consequently, for all $f \in K$, one has the following.
\begin{enumerate}
\item $K(e^f)$ is a Hardy field, since $y = e^f$ satisfies $y' = fy$.
\item $K(\log |f|)$ is a Hardy field if $f \neq 0$, since $y = \log |f|$ satisfies $y' = f'/f$ and $f'/f \in K$.
\item $K(\int f(x) \, dx)$ is a Hardy field for any eventual antiderivative $y = \int f(x) \, dx$ of $f$, since $y$ satisfies $y' = f$.
\end{enumerate}
\end{theorem}
\begin{corollary}\label{Hsim}
Let $K$ be a perfect Hardy field. Let $p(X),q(X) \in K[X]$, and let $y$ be the (germ of) an eventually differentiable function such that $q(y)$ is eventually nonzero and such that $y' = p(y)/q(y)$ on some neighborhood of $\infty$. Then $y \in K$. Thus, if $f \in \mathcal{D}$ is Hardian (resp., lies in $K$), then $e^f$, any eventual antiderivative of $f$, and $\log |f|$ if $f$ is eventually nonzero, are also Hardian (resp., lie in $K$).
\end{corollary}
\begin{corollary}
Any Hardy field is contained in some logexponentially closed Hardy field.
\end{corollary}
Note that, if $f \in \mathcal{D}$ is Hardian and unbounded, then the compositional inverse $f^{-1} \in \mathcal{D}$ of $f$ exists and is also Hardian \cite[Corollary 6.5]{bos}.
\begin{example}
Let $W$ denote the Lambert $W$ function. One has
$$W'(x) = \frac{W(x)}{x(1+W(x))}, \quad \forall x \in (-1/e,0)\cup(0,\infty).$$
It follows that $W \in \RR\{W(x)\} = \RR(x,W(x)) \subseteq \mathbb{H}$, and thus $e^{x/W(x)} \in \mathbb{H}$. The function $e^{x/W(x)}$ is the compositional inverse of $\log x \log \log x \in \mathfrak{L}$. It is known (and originally conjectured by Hardy in [118]) that there is no $f \in \mathbb{L}$ such that $f (x) \sim e^{x/W(x)}$ [67] \cite[Theorem 6.49]{kuhl}. However, one has
$$e^{x/\log x} \prec e^{x/W(x)} \prec (e^{x/\log x})^a$$
for all $a > 1$, and thus $\dege e^{x/W(x)} = \dege e^{x/\log x} = (\infty,1,-1,0,0,0,\ldots)$. Note also that $W^{-1}(x) = xe^x$ is in $\RR(x,\exp x)$. Moreover, one has $\int_0^x W(t) \, dt = x\left(W(x)-1 + \frac{1}{W(x)}\right)$, so any Hardy field containing $W(x)$ contains $x$ and $\int_0^x W(t) \, dt$. In fact, any subfield of $\mathcal{D}$ containing $x$ and $W(x)$ contains the $n$-fold derivative and every $n$-fold antiderivative of $W(x)$ for all $n$, since these lie in $\RR(x,W(x))$.
\end{example}
\begin{remark}[The asymptotic equivalence class of $\log$]
Let $K$ be a Hardy field, let $f \in K_{>0}$ with $f \succ 1$. \cite[Proposition 6]{rosenl1} states that there exists a $g \in K_{>0}$ with $g \sim \log f$ if and only if there exists a $g \in K_{>0}$ with $g \succ 1$ and $f \succ g^n$ for every positive integer $n$. This provides a ``universal property'' of the asymptotic equivalence class of $\log$; more precisely, it provides a characterization of the $\sim$-equivalence class of $\log f$ for any $f$ in a Hardy field $K$ with $f > 0$ and $f \succ 1$.
\end{remark}
Let $R$ be a subring of $\mathcal{D}$. A function $f \in \mathcal{D}$ is {\bf differentially algebraic over $R$}\index{differentially algebraic} if there exists a nonzero polynomial $p(X_0,X_1,X_2,\ldots) \in R[X_0,X_1,X_2,\ldots]$ such that $p(f,f',f'',\ldots) = 0$, or, equivalently, if the kernel $D_{f,R}^{-1}(0)$ of the ring homomorphism $D_{f,R}$ is nontrivial. A function $f \in \mathcal{D}$ is {\bf differentially algebraic} if it is differentially algebraic over $\RR$. Suppose that $f \in \mathcal{D}$ is differentially algebraic over $R$. Then for some nonnegative integer $d$ there exists a nonzero polynomial $p(X_0,X_1,\ldots,X_d) \in R[X_0,X_1,\ldots,X_d]$ in the kernel $D_{f,R}^{-1}(0)$ of minimal degree $m > 0$ in $X_d$, and thus the kernel $D_{f,R}^{-1}(0) \cap R[X_0,X_1,X_2,\ldots,X_d]$ of the restricted map $\RR[X_0,X_1,X_2,\ldots,X_d] \longrightarrow S = R[f,f',f'',\ldots,f^{(d)}]$ is a nonzero ideal of $R[X_0,X_1,X_2,\ldots,X_d]$. The least $d = d_{f,R}\index[symbols]{.i kp@$d_{f,R}$}$ possible is called the {\bf differential degree of $f$ over $R$}\index{differential degree $d_{f,R}$ of $f$ over $R$}. We set $d_{g,R} = \infty$ for any $g \in \mathcal{D}$ that is not differentiably algebraic over $R$. Differentiating the equation $p = D_{f,R}(p) = p(f,f',f'',\ldots, f^{(d)}) = 0$, we obtain
$$D(p) = \sum_{k = 0}^d p_k f^{(k+1)} = 0,$$
where $p_k = \frac{\partial p}{\partial f^{(k)}} = D_{f, R} \frac{\partial p}{\partial X_k} \in S$ for all $k \leq d$.
Note then that $p_d \neq 0$ since the polynomial $p_d$ in $f,f',f'',\ldots,f^{(d)}$ has degree $m-1 < m$ in $f^{(d)}$. Suppose that $p_d \in \mathcal{D}^*$ (which holds, for example, if $R$ is a Hardy field and $f$ is Hardy adjoinable to $R$). Then
$$f^{(d+1)} = -\frac{1}{p_d}\sum_{k = 0}^{d-1} p_k f^{(k+1)} \in R[f,f',f'',\ldots,f^{(d)}][1/p_d]$$
and thus
$$p_k' \in R[f,f',f'',\ldots,f^{(d+1)}] \subseteq R[f,f',f'',\ldots,f^{(d)}][1/p_d]$$
for all $k \leq d$. By induction, then, one has
$$f^{(k)} \in R[f,f',f'',\ldots,f^{(d)}][1/p_d]$$
for all $k$, i.e.,
$$R[f,f',f'',\ldots] \subseteq R[f,f',f'',\ldots,f^{(d)}][1/p_d].$$ Conversely, if there exists a $p \in R[f,f',f'',\ldots, f^{(d)}] \cap \mathcal{D}^*$ such that $$R[f,f',f'',\ldots] \subseteq R[f,f',f'',\ldots,f^{(d)}][1/p]$$ for some $d$, then $p^n f^{(d+1)} -q = 0$ for some $q \in R[f,f',f'',\ldots]$ and some nonnegative integer $n$, so there exist polynomials
$P$ and $Q$ in $R[X_0,X_1,\ldots, X_d]$ with $P \neq 0$ such that $0 \neq P X_{d+1}-Q \in R[X_0,X_1,X_2,\ldots]$ lies in the kernel of $D_{f,R}$, and therefore $f$ is differentially algebraic over $R$. Thus, we have the following.
\begin{proposition}
Let $R$ be a subring of $\mathcal{D}$, let $f \in \mathcal{D}$, and let $d$ be a nonnegative integer. The following conditions are equivalent.
\begin{enumerate}
\item There exists a $p \in R[f,f',f'',\ldots,f^{(d)}] \cap \mathcal{D}^*$ such that $$R[f,f',f'',\ldots] \subseteq R[f,f',f'',\ldots,f^{(d)}][1/p].$$
\item There exist $P,Q \in R[X_0,X_1,\ldots, X_d]$ with $P \neq 0$ such that $P X_{d+1}-Q$ lies in the kernel of $D_{f,R}$ and $D_{f,R}(P) \in \mathcal{D}^*$.
\end{enumerate}
Moreover, if the conditions above hold, then $f$ is differentially algebraic over $R$.
\end{proposition}
\begin{proposition}\label{diffeq}
Let $K$ be a Hardy field, let $f \in \mathcal{D}$ be Hardy adjoinable to $K$, and let $d$ be a nonnegative integer. The following conditions are equivalent.
\begin{enumerate}
\item $f$ is differentially algebraic over $K$ with $d_{f,K} \leq d$.
\item There exists a nonzero $p \in K[f,f',f'',\ldots,f^{(d)}]$ such that $$K[f,f',f'',\ldots] \subseteq K[f,f',f'',\ldots,f^{(d)}][1/p].$$
\item There exist $P,Q \in K[X_0,X_1,\ldots, X_d]$ with $P \neq 0$ such that $P X_{d+1}-Q$ lies in the kernel of $D_{f,R}$ but $P$ does not.
\item There exists an $F \in K(X_0,X_1,\ldots, X_d)$ such that $f^{(d+1)} \in F(f,f',f'',\ldots,f^{(d)})$.
\item $K\{f\} \subseteq K(f,f',f'',\ldots,f^{(d)})$.
\item $K\{f\} = K(f,f',f'',\ldots,f^{(d)})$.
\end{enumerate}
\end{proposition}
\begin{corollary}[{\cite[Section 14]{bos2}}]\label{diffeqcor}
Let $f \in \mathcal{D}$ be Hardian, and let $d$ be a nonnegative integer. The following conditions are equivalent.
\begin{enumerate}
\item $f$ is differentially algebraic of differential degree at most $d$ over $\RR$.
\item There exists a nonzero $p \in \RR[f,f',f'',\ldots,f^{(d)}]$ such that $$\RR[f,f',f'',\ldots] \subseteq \RR[f,f',f'',\ldots,f^{(d)}][1/p].$$
\item There exist $P,Q \in \RR[X_0,X_1,\ldots, X_d]$ with $P \neq 0$ such that $P X_{d+1}-Q$ lies in the kernel of $D_{f,R}$ but $P$ does not.
\item There exists an $F \in \RR(X_0,X_1,\ldots, X_d)$ such that $f^{(d+1)} \in F(f,f',f'',\ldots,f^{(d)})$.
\item $\RR\{f\} \subseteq \RR(f,f',f'',\ldots,f^{(d)})$.
\item $\RR\{f\} = \RR(f,f',f'',\ldots,f^{(d)})$.
\end{enumerate}
\end{corollary}
We say that a subring $S$ of $\mathcal{D}$ containing a ring $R$ is {\bf differentially algebraic over $R$}\index{differentially algebraic} if every element of $S$ is differentially algebraic over $R$. We say that a subring $R$ of $\mathcal{D}$ is {\bf differentially algebraic} if every element of $R$ is differentially algebraic, i.e., if $R$ is differentially algebraic over $\RR$. A subring $R$ of $\mathcal{D}$ differentially algebraic over a Hardy field $K$ (resp., over $\RR$) if and only if its elements satisfy the conditions of Prospotion \ref{diffeq} (resp., Corollary \ref{diffeqcor}). By \cite[Theorem 14.3]{bos2}, the Hardy field $\mathbb{H}$ is differentially algebraic. Moreover, by Theorem \ref{rosenl}, if $f$ is Hardian and $d_{f,\RR} \leq 1$, then $f \in \mathbb{H}$. \cite[Conjecture 2]{bos} states that, if $f\in \mathcal{D}$ is Hardian and differentially algebraic, then $f \in \mathbb{H}$. If this conjecture is true, then $\mathbb{H}$ is precisely the set of all differentially algebraic Hardian functions, and thus $\mathbb{H}$ is the largest differentially algebraic Hardy field. Note that the Hardian elements of $\mathcal{D}$ are those elements $f$ of $\mathcal{D}$ such that $p(f,f',f'',\ldots) \in \mathcal{D}^* \cup\{0\}$ for every $p \in \RR[X_0,X_1,X_2,\ldots]$, and the differentially algebraic elements of $\mathcal{D}$ are those elements $f $ of $\mathcal{D}$ such that $p(f,f',f'',\ldots) \in \{0\}$ for some nonzero $p \in \RR[X_0,X_1,X_2,\ldots]$. Thus, \cite[Conjecture 2]{bos} states that $\mathbb{H}$ is precisely the set of all $f \in \mathcal{D}$ for which both conditions are true. For any subring $R$ of $\mathcal{R}$, we say that $R$ is {\bf bounded by some continuous function} if there exists a $u \in \mathcal{C}$ such that $f \leq u$ for all $f \in R$. By \cite[Theorem 14.4]{bos2}, if $K$ is a Hardy field bounded by some continuous function, then $\operatorname{E}(K)$ is differentially algebraic over $K$. It is also conjectured that $\operatorname{E}(K)$ is is differentially algebraic over $K$ for any Hardy field $K$.
Any Hardian function that is differentially algebraic is analytic on some punctured neighborhood of $\infty$. Thus $\mathbb{H}$ is a subfield of the ring $\mathcal{A}$ of germs of functions that are analytic on some punctured neighborhood of $\infty$. Furthermore, any function that is analytic at $\infty$ is Hardian. However, if a function $r(x) = \sum_{n = 0}^\infty a_n/x^n$ analytic at $\infty$ is differentially algebraic (or in $\mathbb{H}$), then the (real) coefficients $a_0,a_1,a_2,\ldots$ must be algebraically dependent over $\QQ$. See [35] and [36] for proofs of these facts.
The following is a corollary of Proposition \ref{diffeq}.
\begin{corollary}[{cf., \cite[Theorem 14.8]{bos2}}]\label{diffalgprop}
Let $K$ be a Hardy field and $f \in \mathcal{D}$, and let $\aaa$ be the kernel of the ring homomorphism $D_{f,R}: R[X_0,X_1,X_2,\ldots] \longrightarrow R[f,f',f'',\ldots]$. Then $f$ is Hardy adjoinable to $K$ if and only if $\aaa$ is a prime ideal, and $f$ is differentially algebraic over $K$ if and only if $\aaa$ is nonzero. Suppose that $f$ is Hardy adjoinable to $K$, so that $\aaa = \ppp$ is a prime ideal.
\begin{enumerate}
\item The ring $K\{f\}$ is isomorphic to the quotient field of $K[X_0,X_1,X_2,\ldots]/\ppp$, that is, $K\{r\}$ is isomorphic to the residue field of the domain $K[X_0,X_1,X_2,\ldots]$ at $\ppp$.
\item If $f$ is not differentially algebraic over $K$, then $\ppp = 0$ and thus $K\{f\}$ is isomorphic to the field $K(X_0,X_1,X_2,\ldots)$ and has infinite transcencence degree over $K$, with transcendence basis $f, f',f'',\ldots$.
\item Suppose that $f$ is differentially algebraic over $K$ with $d = d_{f,K}$. Then
|
# FUSEE: A Fully Memory-Disaggregated Key-Value Store
(Extended Version)
Jiacheng Shen Pengfei Zuo Huawei Cloud Xuchuan Luo Fudan University
Tianyi Yang The Chinese University of Hong Kong
Yuxin Su Sun Yat-sen University Yangfan Zhou Fudan University Michael R.
Lyu The Chinese University of Hong Kong
###### Abstract
Distributed in-memory key-value (KV) stores are embracing the disaggregated
memory (DM) architecture for higher resource utilization. However, existing KV
stores on DM employ a _semi-disaggregated_ design that stores KV pairs on DM
but manages metadata with monolithic metadata servers, hence still suffering
from low resource efficiency on metadata servers. To address this issue, this
paper proposes FUSEE, a FUlly memory-diSaggrEgated KV StorE that brings
disaggregation to metadata management. FUSEE replicates metadata, i.e., the
index and memory management information, on memory nodes, manages them
directly on the client side, and handles complex failures under the DM
architecture. To scalably replicate the index on clients, FUSEE proposes a
client-centric replication protocol that allows clients to concurrently access
and modify the replicated index. To efficiently manage disaggregated memory,
FUSEE adopts a two-level memory management scheme that splits the memory
management duty among clients and memory nodes. Finally, to handle the
metadata corruption under client failures, FUSEE leverages an embedded
operation log scheme to repair metadata with low log maintenance overhead. We
evaluate FUSEE with both micro and YCSB hybrid benchmarks. The experimental
results show that FUSEE outperforms the state-of-the-art KV stores on DM by up
to $4.5$ times with less resource consumption.
## 1 Introduction
Traditional in-memory key-value (KV) stores on monolithic servers have
recently been ported to the disaggregated memory (DM) architecture for better
resource efficiency [73, 60]. Compared with monolithic servers, DM decouples
the compute and memory resources into independent network-attached compute and
memory pools [56, 55, 25, 65, 39, 48, 23, 3]. KV stores on DM can thus enjoy
efficient resource pooling and have higher resource efficiency.
However, constructing KV stores on DM is challenging because the memory pool
generally lacks the compute power to manage data and metadata. Existing work
[60] proposes a semi-disaggregated design that stores KV pairs in the
disaggregated memory pool but retains metadata management on monolithic
servers. In such a design, the KV pair storage enjoys high resource
utilization due to exploiting the DM architecture, but the metadata management
does not. Many additional resources are exclusively assigned to the metadata
servers in order to achieve high overall throughput [13, 69, 54].
To achieve full resource utilization, it is critical to bring disaggregation
to the metadata management, i.e., building a fully memory-disaggregated KV
store. The metadata, i.e., the index and memory management information, should
be stored in the memory pool and directly managed by clients rather than
metadata servers. However, it is non-trivial to achieve a fully memory-
disaggregated KV store due to the following challenges incurred from handling
complex failures and the weak compute power in the memory pool.
1) Client-centric index replication. To tolerate memory node failures, clients
need to replicate the index on memory nodes in the memory pool and guarantee
the consistency of index replicas. In existing replication approaches, e.g.,
state machine replication [51, 47, 34, 62] and shared register protocols [44,
7, 5], the replication protocols are executed by server-side CPUs. These
protocols cannot be executed on DM due to the weak compute power in the memory
pool. Meanwhile, if clients simply employ consensus protocols [51, 37, 47] or
remote locks [60], the KV store suffers from poor scalability due to the
explicit serialization of conflicting requests [70, 4, 11, 64].
2) Remote memory allocation. Existing semi-disaggregated KV stores manage
memory spaces with monolithic metadata servers. However, in the fully memory-
disaggregated setting, such a server-centric memory management scheme is
infeasible. Specifically, memory nodes cannot handle the compute-heavy fine-
grained memory allocation for KV pairs due to their poor compute power [60,
25]. Meanwhile, clients cannot efficiently allocate memory spaces because
multiple RTTs are required to modify the memory management information stored
on memory nodes [39].
3) Metadata corruption under client failures. In semi-disaggregated KV stores,
client failures do not affect metadata because the CPUs of monolithic servers
exclusively modify metadata. However, clients directly access and modify
metadata on memory nodes in the fully memory-disaggregated setting. As a
result, client failures can leave partially modified metadata accessible by
others, compromising the correctness of the entire KV store.
To address these challenges, we propose FUSEE, a fully memory-disaggregated
key-value store that has efficient index replication, memory allocation, and
fault-tolerance on DM. First, to maintain the strong consistency of the
replicated index in a scalable manner, FUSEE proposes the SNAPSHOT replication
protocol. The key to achieving scalability is to resolve write conflicts
without involving the expensive request serialization [7]. SNAPSHOT adopts
three simple yet effective conflict-resolution rules on clients to allow
conflicts to be resolved collaboratively among clients instead of
sequentially. Second, to achieve efficient remote memory management, FUSEE
employs a two-level memory management scheme that splits the server-centric
memory management process into compute-light and compute-heavy tasks. The
compute-light coarse-grained memory blocks are managed by the memory nodes
with weak compute power, and the compute-heavy fine-grained objects are
handled by clients. Finally, to deal with the problem of metadata corruption,
FUSEE adopts an embedded operation log scheme to resume clients’ partially
executed operations. The embedded operation log reuses the memory allocation
order and embeds log entries in KV pairs to reduce the log-maintenance
overhead on DM.
We implement FUSEE from scratch and evaluate its performance using both micro
and YCSB benchmarks [15]. Compared with Clover and pDPM-Direct [60], two
state-of-the-art KV stores on DM, FUSEE achieves up to $4.5$ times higher
overall throughput and exhibits lower operation latency with less resource
consumption. The code of FUSEE is available at
https://github.com/dmemsys/FUSEE.
In summary, this paper makes the following contributions:
* •
A fully memory-disaggregated KV store with disaggregated metadata and data
that is resilient to failures on DM.
* •
A client-centric replication protocol that uses conflict resolution rules to
enable clients to resolve conflicts collaboratively. The protocol is formally
verified with TLA+ [36] for safety and the absence of deadlocks under crash-
stop failures.
* •
A two-level memory management scheme that leverages both memory nodes and
clients to efficiently manage the remote memory space.
* •
An embedded operation log scheme to repair the corrupted metadata with low log
maintenance overhead.
* •
The implementation and evaluation of FUSEE to demonstrate the efficiency and
effectiveness of our design.
## 2 Background and Motivation
### 2.1 The Disaggregated Memory Architecture
The disaggregated memory architecture is proposed to address the resource
underutilization issue of traditional datacenters composed of monolithic
servers [56, 55, 25, 39, 65, 48]. DM separates CPUs and memory of monolithic
servers into two independent hardware resource pools containing compute nodes
(CNs) and memory nodes (MNs) [56, 73, 64, 60]. CNs have abundant CPU cores and
a small amount of memory as local caches [64]. MNs host various memory media,
e.g., DRAM and persistent memory, to accommodate different application
requirements with weak compute power. CPUs in CNs directly access memory in
MNs with fast remote-access interconnect techniques, such as one-sided RDMA
(remote direct memory access), Omni-path [16], CXL [43], and Gen-Z [14]. Each
MN provides READ, WRITE, and atomic operations, i.e., compare-and-swap (CAS)
and fetch-and-add (FAA), for CNs to access memory data. Besides, MNs own
limited compute power (e.g., 1-2 CPU cores) to manage local memory and
establish connections from CNs, providing CNs with the ALLOC and FREE memory
management interfaces. Without loss of generality, in this paper, we consider
CNs accessing MNs using one-sided RDMA verbs.
(a) Clover
(b) FUSEE
Figure 1: Two architectures of memory-disaggregated KV stores. (a) The semi-
disaggregated architecture (Clover [60]). (b) The fully disaggregated
architecture proposed in this paper.
### 2.2 KV Stores on Disaggregated Memory
Clover [60] is a state-of-the-art KV store built on DM. It adopts a semi-
disaggregated design that separates data and metadata to lower the ownership
cost and prevent the compute power of data nodes from becoming the performance
bottleneck. As shown in Figure 1a, Clover deploys clients on CNs and stores KV
pairs on MNs. It adopts additional monolithic metadata servers to manage the
metadata, including memory management information (MMI) and the hash index.
For SEARCH requests, clients look up the addresses of the KV pairs from
metadata servers and then fetch the data on MNs using RDMA_READ operations.
For INSERT and UPDATE requests, clients allocate memory blocks from metadata
servers with RPCs, write KV pairs to MNs with RDMA_WRITE operations, and
update the hash index on the metadata servers through RPCs. To prevent
clients’ frequent requests from overwhelming the metadata servers, clients
allocate a batch of memory blocks one at a time and cache the hash index
locally. As a result, Clover achieves higher throughput under read-intensive
workloads with less resource consumption.
However, the semi-disaggregated design of Clover cannot fully exploit the
resource efficiency of the DM architecture due to its monolithic-server-based
metadata management. On the one hand, monolithic metadata servers consume
additional resources, including CPUs, memory, and RNICs. On the other hand,
many compute and memory resources have to be reserved and assigned to the
metadata server of Clover to achieve good performance due to the CPU-intensive
nature of metadata management [13, 69, 54]. To show the resource utilization
issue of Clover, we evaluate its throughput with 2 MNs, 64 clients, and a
metadata server with different numbers of CPU cores. We control the number of
CPU cores by assigning different percentages of CPU time with cgroup [10]. As
shown in Figure 3, Clover has a low overall throughput with a small number of
CPU cores assigned to its metadata server. At least six additional cores have
to be assigned until the metadata server is no longer the performance
bottleneck.
To attack the problem, FUSEE adopts a fully memory-disaggregated design that
enables clients to directly access and modify the hash index and manage memory
spaces on MNs, as shown in Figure 1b. Compared with the semi-disaggregated
design, resource efficiency can be improved because client-side metadata
management eliminates the additional metadata servers. The overall throughput
can also be improved because the computation bottleneck of metadata management
no longer exists.
## 3 Challenges
This section introduces the three challenges of constructing a fully memory-
disaggregated KV store, i.e., index replication, remote memory allocation, and
metadata corruption.
### 3.1 Client-Centric Index Replication
The index must be replicated to tolerate MN failures. Strong consistency,
i.e., linearizability [26], is the most commonly adopted correctness standard
for data replication because it reduces the complexity of implementing upper-
level applications [7, 1, 12]. Linearizability requires that operations on an
object appear to be executed in some total order that respects the operations’
real-time order [26]. The key challenge of achieving a linearizable replicated
hash index under the fully memory-disaggregated setting comes from the client-
centric computation nature of DM.
Figure 2: The throughput of Clover with an increasing number of metadata
server CPUs.
Figure 3: The throughput of Derecho [28] and lock-based approaches.
First, existing replication methods are not applicable in the fully memory-
disaggregated setting due to their server-centric nature. State machine
replication (SMR) [51, 47, 34, 62, 59, 45, 50] and shared register protocols
[7, 44] are two major replication approaches that achieve linearizability.
However, both approaches are designed with a server-centric assumption that a
data replica is exclusively accessed and modified by the CPU that manages the
data. First, the SMR approaches consider the CPU and the data replica as a
state machine and achieve strong consistency by forcing the state machines to
execute deterministic KV operations in the same global order [51, 50]. Server
CPUs are extensively used to reach a consensus on a global operation order and
apply state transitions to data replicas. Second, shared register protocols
view the CPU and the data replica as a shared register with READ and WRITE
interfaces. Linearizability is achieved with a last-writer-wins conflict
resolution scheme [44] that forces a majority of shared registers to always
hold data with the newest timestamps. Shared register protocols also heavily
rely on server-side CPUs to compare timestamps and apply data updates. The
challenge with the server-centric approaches is that in the fully memory-
disaggregated scenario, there is no such management CPU because all clients
directly access and modify the hash index with one-sided RDMA verbs.
Second, naively adopting consensus protocols or remote locks among clients
results in poor throughput due to the expensive request serialization. To show
the performance issues of consensus protocols and remote locks, we store and
replicate a shared object on two MNs and vary the number of concurrent
clients. We use a state-of-the-art consensus protocol Derecho [28] and an RDMA
CAS-based spin lock to ensure the strong consistency of the replicated object.
As shown in Figure 3, both Derecho and lock-based approaches exhibit poor
overall throughput and cannot scale with the growing number of concurrent
clients.
### 3.2 Remote Memory Allocation
The key challenge of managing DM is where to execute the memory-management
computation. There are two possible DM management approaches [39], i.e.,
compute-centric ones and memory-centric ones. The compute-centric approaches
store the memory management metadata on MNs and allow clients to allocate
memory spaces by directly modifying the on-MN metadata. Since the memory
management metadata are shared by all clients, clients’ accesses have to be
synchronized. As a result, compute-centric approaches suffer from the high
memory allocation latency incurred by the expensive and complex remote
synchronization process on DM [39]. The memory-centric approaches maintain all
memory management metadata on MNs with their weak compute power. Such
approaches are also infeasible because the poor memory-side compute power can
be overwhelmed by the frequent fine-grained KV allocation requests from
clients. Although there are several approaches that conduct memory management
on DM, they all target page-level memory allocation and rely on special
hardware, i.e., programmable switches [39] and SmartNICs [25], which are
orthogonal to our problem.
### 3.3 Metadata Corruption
In fully memory-disaggregated KV stores, crashed clients can leave partially
modified metadata accessible by other healthy clients. Since the metadata
contains important system state, metadata corruption compromises the
correctness of the entire KV store. First, crashed clients may leave the index
in a partially modified state. Other healthy clients may not be able to access
data or even access wrong data with the corrupted index. Second, crashed
clients may allocate memory spaces but not use them, causing severe memory
leakage. Hence, in order to ensure the correctness of the KV store, the
corrupted metadata has to be repaired under client failures.
## 4 The FUSEE Design
### 4.1 Overview
As shown in Figure 4, FUSEE consists of clients, MNs, and a master. Clients
provide SEARCH, INSERT, DELETE, and UPDATE interfaces for applications to
access KV pairs. MNs store the replicated memory management information (MMI),
hash index, and KV pairs. The master is a cluster management process
responsible only for initializing clients and MNs and recovering data under
client and MN failures.
Figure 4: The FUSEE overview (MMI, Index, and KV pairs have multiple replicas,
i.e., $R_{0}$, $R_{1}$, and $R_{2}$. $R_{0}$ is the primary replica.). Figure
5: The structure of an index replica.
FUSEE replicates both the hash index and KV pairs to tolerate MN failures. We
adopt RACE hashing (Section 4.2) to index KV pairs and propose the SNAPSHOT
replication protocol to enforce the strong consistency of the replicated hash
index (Section 4.3). A two-level memory management scheme is adopted to
efficiently allocate and replicate variable-sized KV pairs (Section 4.4).
FUSEE uses logs to handle the corrupted metadata under client failures and
adopts an embedded operation log scheme to reduce the log maintenance overhead
(Section 4.5). Other optimizations are introduced in Section 4.6 to further
improve the system performance.
### 4.2 RACE Hashing
RACE hashing is a one-sided RDMA-friendly hash index. As shown in Figure 5, it
contains multiple 8-byte slots, with each storing a pointer referring to the
address of a KV pair, an 8-bit fingerprint (Fp), i.e., a part of the key’s
hash value, and the length of the KV pair (Len) [73]. For SEARCH requests, a
client reads the slots of the hash index according to the hash value of the
target key and then reads the KV pair on MNs according to the pointer in the
slot. For UPDATE, INSERT, and DELETE requests, RACE hashing adopts an _out-of-
place modification_ scheme. It first writes a KV pair to MNs and then modifies
the corresponding slot in the hash index to the address of the KV pair
atomically with an RDMA_CAS. Nevertheless, the RACE hashing only supports a
single replica.
Figure 6: The SNAPSHOT replication protocol.
### 4.3 The SNAPSHOT Replication Protocol
In FUSEE, multiple clients concurrently read or write the same slot in the
replicated hash index to execute SEARCH or UPDATE requests, as shown in Figure
6. To efficiently maintain the strong consistency of slot replicas in the
replicated hash index, FUSEE proposes the SNAPSHOT replication protocol, a
client-centric replication protocol that achieves linearizability without the
expensive request serialization.
There are two main challenges to efficiently achieving linearizability under
the fully memory-disaggregated setting. First, how to protect readers from
reading incomplete states during read-write conflicts. Second, how to resolve
write-write conflicts without expensively serializing all conflicting
requests. To address the first challenge, SNAPSHOT splits the replicated hash
index into a single primary replica and multiple backup replicas and uses
backup replicas to resolve write conflicts. Hence, incomplete states during
write conflicts only appear on backup replicas and the primary replica always
contains the correct and complete value. Readers can simply read the contents
in the primary replica without perceiving the incomplete states. To address
the second challenge, SNAPSHOT adopts a last-writer-wins conflict resolution
scheme similar to shared register protocols. SNAPSHOT leverages the _out-of-
place modification_ characteristic of RACE hashing that conflicting writers
always write different values into the same slot because the values are
pointers referring to KV pairs at different locations. Three conflict-
resolution rules are thus defined based on the values written by conflicting
writers in backup replicas, which enable clients collaboratively to decide on
a single last writer under write conflicts.
1:procedure READ($slot$)
2: $v=\texttt{RDMA\\_READ\\_primary}(slot)$
3: if $v=\texttt{FAIL}$ then deal with failure
4: return $v$
5:procedure WRITE($slot,v_{new}$)
6: $v_{old}=\texttt{RDMA\\_READ\\_primary}(slot)$
7: $v\\_list=\texttt{RDMA\\_CAS\\_backups}(slot,v_{old},v_{new})$
8: // Change all the $v_{old}$s in the $v\\_list$ to $v_{new}$s.
9: $v\\_list=\texttt{change\\_list\\_value}(v\\_list,v_{old},v_{new})$
10: $win=\texttt{EVALUATE\\_RULES}(v\\_list)$ $\triangleright$ The last writer
returns the winning rule while other writers return LOSE.
11: if $win=\texttt{Rule\\_1}$ then
12: $\texttt{RDMA\\_CAS\\_primary}(slot,v_{old},v_{new})$
13: else if $win\in\\{\texttt{Rule\\_2},\texttt{Rule\\_3}\\}$ then
14: $\texttt{RDMA\\_CAS\\_backups}(slot,v\\_list,v_{new})$
15: $\texttt{RDMA\\_CAS\\_primary}(slot,v_{old},v_{new})$
16: else if $win=\texttt{LOSE}$ then
17: repeat
18: sleep a little bit
19: $v_{check}=\texttt{RDMA\\_READ\\_primary}(slot)$
20: if notified failure then goto Line 24
21: until $v_{check}\neq v_{old}$
22: if $v_{check}=\texttt{FAIL}$ then goto Line 24
23: else if $win=\texttt{FAIL}$ then
24: deal with failure
25: return
Algorithm 1 The SNAPSHOT replication protocol
1:procedure evaluate_rules($v\\_list,slot,v_{new},v_{old}$)
2: $v_{maj}=$ The majority value in $v\\_list$
3: $cnt_{maj}=$ The number of $v_{maj}$ in $v\\_list$
4: if $\texttt{FAIL}\in v\\_list$ then
5: return FAIL
6: else if $cnt_{maj}=\texttt{Len}(v\\_list)$ then
7: return $\texttt{Rule 1}\text{ if }v_{maj}=v_{new}\text{ else
}\texttt{LOSE}$
8: else if $2*cnt_{maj}>\texttt{Len}(v\\_list)$ then
9: return $\texttt{Rule 2}\text{ if }v_{maj}=v_{new}\text{ else
}\texttt{LOSE}$
10: else if $v_{new}\not\in v\\_list$ then
11: return LOSE
12: $v_{check}=\texttt{RDMA\\_READ}(slot)$
13: if $v_{check}=\texttt{FAIL}$ then
14: return FAIL
15: else if $v_{check}\neq v_{old}$ then
16: return FINISH
17: else if $min(v\\_list)=v_{new}$ then
18: return Rule 3
19: return LOSE
Algorithm 2 The rule evaluation procedure of SNAPSHOT
Algorithm 1 shows the READ and WRITE processes of the SNAPSHOT replication
protocol. Here we focus on the execution of SNAPSHOT when no failure occurs
and leave the discussion of failure handling in Section 5. We call the slots
in the primary and backup hash indexes primary slots and backup slots,
respectively.
For READ operations, clients directly read the values in the primary slots
using RDMA_READ. For WRITE operations, SNAPSHOT first resolves write conflicts
by letting conflicting writers collaboratively decide on a last writer with
three conflict resolution rules and then let the decided last writer modify
the primary slot. Figure 6 shows the process that two clients simultaneously
WRITE the same slot. The corresponding algorithms are shown in Algorithms 1
and 2. Clients first read the value in the primary slot as $v_{old}$
(\footnotesize1⃝). Then each client modifies all backup slots by broadcasting
RDMA_CAS operations (\footnotesize2⃝) with $v_{old}$ as the expected value and
$v_{new}$ as the swap value. On receiving an RDMA_CAS, the RNICs on MNs
atomically modify the value in the target slot only if $v_{old}$ matches the
current value in the slot. Since all writers initiate RDMA_CAS operations with
the same $v_{old}$ and different $v_{new}$s and all backup slots initially
hold $v_{old}$, the atomicity of RDMA_CAS ensures that each backup slot can
only be modified once by a single writer. As a result, the values in all
backup slots will be fixed after each of them has received one RDMA_CAS from
one writer 111That the process that all conflicting clients broadcast
RDMA_CASes to modify backup slots is just like taking a snapshot, which is why
the replication protocol is named SNAPSHOT.. Meanwhile, since an RDMA_CAS
returns the value in the slot before it is modified, all clients can perceive
the new values in the backup slots (\footnotesize3⃝) through the return values
of the broadcast of RDMA_CAS operations. The return values are denoted as
$v\\_list$ in Algorithm 1.
With $v\\_list$, SNAPSHOT defines the following three rules to let conflicting
clients collaboratively decide on a last writer:
* Rule 1: A client that has successfully modified all the backup slots is the last writer.
* Rule 2: A client that has successfully modified a majority of backup slots is the last writer.
* Rule 3: If no last writer can be decided with the former two rules, the client that has written the minimal target value ($v_{new}$) is considered as the last writer.
The three rules are evaluated sequentially as shown in Algorithm 2. Rule 1
provides a fast path when there are no conflicting modifications. Rule 2
preserves the most successful CAS operations to minimize the overhead of
executing atomic operations on RNICs when conflicts are rare [30]. Finally,
Rule 3 ensures that the protocol can always decide on the last writer. To
ensure the uniqueness of the last write, a client issues another RDMA_READ to
check if the primary slot has been modified (Line 12, Algorithm 2) before
evaluating Rule 3. If the primary slot has not been modified, then the
RDMA_CAS_backups (Line 7, Algorithm 1) of the client must happen before the
last writer modifies the primary slot. Hence, it is safe to evaluate Rule 3
because the $v\\_list$ must contain the value of the last writer if it has
already been decided. Otherwise, Rule 3 will not be evaluated because the
modification of the primary slot means the decision of a last writer. Relying
on the three rules, a unique last writer can be decided without any further
network communications. For example, in Figure 6, Client 1 is the last writer
according to Rule 2. Client 1 then modifies the backup slots that do not yet
contain its proposed value using RDMA_CASes and then modifies the primary
slot. Other conflicting clients iteratively READ the value in the primary slot
and return success after finding the change in the primary slot. The primary
slot may remain unmodified only under the situation when the last writer
crashed, which will be discussed in Section 5.
Correctness. The SNAPSHOT replication protocol guarantees linearizability of
the replicated hash indexes with last-writer-wins conflict resolution like
shared register protocols [7, 44]. We briefly demonstrate the correctness of
SNAPSHOT using the notion of the linearizable point of KV operations. A formal
proof is shown in Appendix A. A linearizable point is a point when an
operation atomically takes effect in its invocation and completion [26]. For
READ, the linearizable point happens when it gets the value in the primary
slot. For WRITE operations, the linearizable point of the last writer happens
when it modifies the primary slot. Linearizable points of other conflicting
writers appear instantly before the last writer modifies the primary slot.
Conflicts between readers and the last writer are resolved by RNICs because
the last writer atomically modifies the primary slot using RDMA_CAS operations
and readers access the primary slot using RDMA_READ operations.
Performance. SNAPSHOT guarantees a bounded worst-case latency when clients
WRITE the hash index. Under the situation when Rule 1 is triggered, 3 RTTs are
required to finish a WRITE operation. Under situations when Rule 2 or Rule 3
is triggered, 4 or 5 RTTs are required, respectively.
Figure 7: The two-level memory management scheme.
### 4.4 Two-Level Memory Management
Memory management is responsible for allocating, replicating, and freeing
memory spaces for KV pairs on MNs. As discussed in Section 3.2, the key
challenge of DM management is that conducting the management tasks solely on
clients or on MNs cannot satisfy the performance requirement of frequent
memory allocation for KV requests. FUSEE addresses this issue via a two-level
memory management scheme. The key idea is to split the server-centric memory
management tasks into compute-light coarse-grained management and compute-
heavy fine-grained management run on MNs and clients.
FUSEE first replicates and partitions the 48-bit memory space on multiple MNs.
Similar to FaRM [18], FUSEE shards the memory space into 2GB memory regions
and maps each region to $r$ MNs with consistent hashing [33], where $r$ is the
replication factor. Specifically, consistent hashing maps a region to a
position in a hash ring. The replicas are then stored at the $r$ MNs
successively following the position and the primary region is placed on the
first of the $r$ MN.
Figure 7 shows the two-level memory allocation of FUSEE. Allocating a memory
space for a KV pair happens before writing the KV pair, as introduced in
Section 4.1. The first level is the coarse-grained MN-side memory block
allocation with low computation requirements. Each MN partitions its local
memory regions into coarse-grained memory blocks, e.g., 16 MB, and maintains a
block allocation table ahead of each region. For each memory block, the block
allocation table records a client ID (CID) that allocates it. Clients allocate
memory blocks by sending ALLOC requests to MNs. On receiving an ALLOC request,
an MN allocates a memory block from one of its primary memory regions, records
the client ID in the block allocation tables of both primary and backup
regions, and replies with the address of the memory block to the client. The
coarse-grained memory allocation information is thus replicated on $r$ MNs and
can survive MN failures. The second level is the fine-grained client-side
object allocation that allocates small objects to hold KV pairs. Specifically,
clients manage the blocks allocated from MNs exclusively with slab allocators
[6]. The client-side slab allocators split memory blocks into objects of
distinct size classes. A KV pair is then allocated from the smallest size
class that fits it.
The allocated objects can be freed by any client. To efficiently reclaim freed
memory objects on client sides, FUSEE stores a free bit map ahead of each
memory block, as shown in Figure 7, where each bit indicates the allocation
state of one object in the memory block. The free bit map is initialized to be
all zeros when a block is allocated. To free an object, a client sets the
corresponding bit to ‘1’ in the free bit map with an RDMA_FAA operation. By
reading the free bit map, clients can easily know the freed objects in their
memory blocks and reclaim them locally. FUSEE frees and reclaims memory
objects periodically using background threads in a batched manner to avoid the
additional RDMA operations on the critical paths of KV accesses. The
correctness of concurrently accessing KV pairs and reclaiming memory spaces is
guaranteed by RACE hashing [73], where clients check the key and the CRC of
the KV pair on data accesses.
(a) The embedded log entry.
(b) The organization of the embedded operation log.
Figure 8: The embedded operation log.
### 4.5 Embedded Operation Log
Operation logs are generally adopted to repair the partially modified hash
index incurred by crashed clients. Conventional operation logs record a log
entry for each KV request that modifies the hash index. The log entries are
generally written in an append-only manner so that the order of log entries
reflects the execution order of KV requests. The recovery process can thus
find the crashed request and fix the corrupted metadata by scanning the
ordered log entries. However, constructing operation logs incurs high log
maintenance overhead on DM because writing log entries adds remote memory
accesses on the critical paths of KV requests.
To reduce the log maintenance overhead on DM, FUSEE adopts an _embedded
operation log_ scheme that embeds log entries into KV pairs. The embedded log
entry is written together with its corresponding KV pair with one RDMA_WRITE
operation. The additional RTTs required for persisting log entries are thus
eliminated. However, by embedding log entries in KV pairs, the execution order
of KV requests cannot be maintained because the log entries are no longer
continuous. To address this problem, the embedded operation log scheme
maintains per-size-class linked lists to organize the log entries of a client
in the execution order of KV requests. As shown in Figure 8b, a per-size-class
linked list is a doubly linked list that links all allocated objects of the
size class in the order of their allocations. The object allocation order
reflects the execution order of KV requests because all KV requests that
modify the hash index, e.g., INSERT and UPDATE, need to allocate objects for
new KV pairs. For DELETE, FUSEE allocates a temporary object recording the log
entry and the target key and reclaims the object on finishing the DELETE
request. FUSEE stores the list heads on MNs during the initialization of
clients, which will be accessed during the recovery process of clients
(Section 5).
Figure 9: The workflows of different KV requests. INSERT: \scriptsize1⃝ write
the KV pair to all replicas and read the primary index slot. \scriptsize2⃝ CAS
all backup slots. \scriptsize3⃝ write the old value to the log header.
\scriptsize4⃝ CAS the primary slot. UPDATE & DELETE: \scriptsize1⃝ write the
KV pair, read the primary slot, and read the KV pair according to the index
cache. \scriptsize2⃝ CAS backup slots. \scriptsize3⃝ write the old value to
the log header. \scriptsize4⃝ CAS the primary slot. SEARCH: \scriptsize1⃝ read
the primary slot and the KV pair according to the index cache. \scriptsize2⃝
read the KV pair on cache misses.
An embedded log entry is a 22-byte data structure stored behind KV pairs, as
shown in Figure 8a. It contains a 6-byte next pointer, a 6-byte prev pointer,
an 8-byte old value, a 1-byte CRC, a 7-bit opcode, and a used bit. The next
pointer points to the next object of the size class that will be allocated and
the prev pointer points to the object allocated before the current one. The
old value records the old value of the primary slot for recovery proposes,
which will be discussed in Section 5. The 1-byte CRC is adopted to check the
integrity of the old value under client failures. The operation field records
the operation type, i.e., INSERT, UPDATE, or DELETE, so that the crashed
operation can be properly retried during recovery. The used bit indicates if
an object is in-use or free. Storing the used bit at the end of the entire
object can be used to check the integrity of an entire object. This is because
the order-preserving nature of RDMA_WRITE operations ensures that the used bit
is written only after all other contents in the object have been successfully
written.
FUSEE efficiently organizes per-size-class linked lists by co-designing the
linked list maintenance process with the memory allocation process. As shown
in Figure 8b, for each size class, a client organizes the addresses of remote
free objects locally as a free list. Since an object is always allocated from
the head of a local free list, the allocation order of each size class is pre-
determined. Based on the pre-determined order, for each allocation, a client
pre-positions the next pointer to point to the free object in the head of the
local free list and the prev pointer to point to the last allocated object of
the size class. Both the next pointer and the prev pointer are thus known
before each allocation and the entire log entry can be written to MNs with the
KV pair in a single RDMA_WRITE.
Combined with the SNAPSHOT replication protocol, the execution process is
shown as follows. First, for each writer, a log entry with an empty old value
and CRC is written with the KV pair in a single RDMA_WRITE. Then, for the last
writer of the SNAPSHOT replication protocol, the old value is modified to
store the old value of the primary slot before the primary slot is modified.
For other non-last writers, the used bits in their corresponding KV log
entries are reset to ‘0’ after finding the modification of the primary slot.
### 4.6 Optimizations
Adaptive index cache. Index caching is widely adopted on RDMA-based KV stores
to reduce request RTTs [68, 67, 66, 60]. For a key, the index cache caches the
remote addresses of the replicated index slots and the addresses of the KV
pairs locally. With the cached KV pair addresses, UPDATE, DELETE, and SEARCH
requests can read KV pairs in parallel with searching the hash index, reducing
an RTT on cache hits. To guarantee cache coherence, an invalidation bit is
stored together with each KV pair, which is used by clients to check whether
the KV pair is valid or invalid. However, by accessing the index cache,
invalid KV pairs (e.g., outdated) can be fetched into clients, causing read
amplification.
To attack the read amplification issue, FUSEE adaptive bypasses the index
cache by distinguishing read-intensive and write-intensive keys. For each
cached key, FUSEE maintains an access counter and an invalid counter which
increases by 1 each time the key is accessed or found to be invalid. A client
calculates an invalid ratio $I=\frac{\text{invalid counter}}{\text{access
counter}}$ for each cached key. The index cache is bypassed when accessing a
key with $I>threshold$ because the key is write-intensive and the cached key
address points to an invalid KV pair with high probability. The invalid ratio
can adapt to workload changes, i.e., a write-intensive key becomes read-
intensive, because the access counter of the key keeps increasing while the
invalid counter stops. Besides, the adaptive scheme does not affect the SEARCH
latency for most cases since only write-intensive keys bypass the cache.
RDMA-related optimizations. KV requests require multiple remote memory
accesses. FUSEE adopts doorbell batching and selective signaling [30] to
reduce RDMA overhead. Figure 9 shows the procedures for executing different KV
requests. Each request consists of multiple phases with multiple network
operations. For each phase, FUSEE adopts doorbell batching [30] to reduce the
overhead of transmitting network operations from user space to RNICs and
selective signaling to reduce the overhead of polling RDMA completion queues.
Consequently, each phase only incurs 1 network RTT. For INSERT, DELETE, and
UPDATE requests, four RTTs are required in general cases. For SEARCH requests,
at most two RTTs are required and only one RTT is required in the best case
due to the index cache.
## 5 Failure Handling
Similar to existing replication protocols [34, 62, 59], FUSEE relies on a
fault-tolerant master with a lease-based membership service [24] to handle
failures. The master maintains a membership lease for both clients and MNs so
that clients always know alive MNs by periodically extending their leases. The
failures of both clients and MNs can be detected by the master when they no
longer extend their leases. Master crashes are handled by replicating the
master with state machine replication [24, 59, 62]. We formally verify FUSEE
in TLA+ [36] for safety and absence of deadlocks under MN failures and the
detailed illustration can be found in Appendix A.
### 5.1 Failure Model
We consider a partially synchronous system where processes, i.e., clients and
MNs, are equipped with loosely synchronized clocks [20, 24, 34]. FUSEE assumes
crash-stop failures, where processes, i.e., clients and MNs, may fail due to
crashing and their operations are non-Byzantine.
Under this failure model, FUSEE guarantees linearizable operations, i.e., each
KV operation is atomically committed in a time between its invocation and
completion [26]. All the objects of FUSEE are durable and available under an
arbitrary number of client crashes and at most $r-1$ MN crashes, where $r$ is
the replication factor.
### 5.2 Memory Node Crashes
MN crashes lead to failed accesses to KV pairs and hash slots. For accesses to
KV pairs, clients can access the backup replicas according to the consistent
hashing scheme.
The complication comes from the unavailable primary and backup slots that
affect the normal execution of index READ and WRITE operations. FUSEE relies
on the fault-tolerant master to execute operations on clients’ behalves under
MN failures. We first introduce how clients READ/WRITE the replicated slots
and then introduce the master’s operations.
When executing index WRITE under MN crashes, FUSEE allows the last writer
decided by the SNAPSHOT replication protocol to continue modifying all _alive_
slots to the same value. Other writers send RPC requests to the master and
wait for the master to reply with a correct value in the replicated slots.
Under situations when no last writer can be decided, the master decides the
last writer and modifies all the index slots on behalf of clients. For READ
operations, executions are not affected under the following two cases. First,
if the primary slot is still alive, clients can read the primary slot
normally. Second, if the primary slot crashes, clients read all alive backup
slots. If all alive backup slots contain the same value, reading this value is
safe because there are no write conflicts. Otherwise, clients use RPCs and
rely on the master to return a correct value for the crashed slot. Since READ
operations are only affected under write conflicts, most READ can continue
under the read-intensive workloads that dominate in real-world situations [9,
71].
On detecting MN crashes, the master first blocks clients from further
modifying the crashed slots with the lease expiration. The master then acts as
a representative last writer that modifies all _alive_ slots to the same
value. Specifically, the master selects a value $v$ in an _alive_ backup slot
and modifies all _alive_ slots to $v$. Since the SNAPSHOT protocol modifies
the backup slots before the primary slot, the values in the backup slots are
always newer than the primary slot. Hence, the master choosing a value from a
backup slot is correct because it proceeds the conflicting write operations.
In cases where all backup slots crash, the master selects the value in the
primary slot. Clients that receive old values from the master retry their
write operations to guarantee that their new value is written. The master then
writes the old value in the operation log header to prevent clients from
redoing operations when recovering from crashed clients (Section 5.3).
Finally, the master reconfigures new primary and backup slots and returns the
selected value to all clients that wait for a reply. After the reconfiguration
of the primary and backup slots, all KV requests can be executed normally
without involving the master. During the whole process, only accesses to the
crashed slots are affected and the blocking time can be short thanks to the
microsecond-scale membership service [24].
### 5.3 Client Crashes
Crashed clients may result in two issues. First, their allocated memory blocks
remain unmanaged, causing memory leakage. Second, other clients may be unable
to modify a replicated index slot if the crashed client is the last writer.
The master uses embedded operation logs to address these two issues.
The recovery process is executed in the compute pool and consists of two
steps, i.e., memory re-management and index repair. Memory re-management
restores the coarse-grained memory blocks allocated by the client and the
fine-grained object usage information of the client. The recovery process
first gets all memory blocks managed by the crashed client by letting MNs
search for their local block allocation tables. Then the recovery process
traverses the per-size-class linked lists to find all used objects and log
entries. With the used objects and the allocated memory blocks, the recovery
process can easily restore the free object lists of the crashed client. Hence,
all the memory spaces of the crashed client are re-managed.
The index repair procedure then fixes the partially modified hash index. FUSEE
deems all requests at the end of per-size-class linked lists as potentially
crashed requests. For incomplete log entries, i.e., the used bit at the end of
the log entry is not set, the client must crashes during writing the KV pair
(c0 in Figure 9). The object is directly reclaimed without further operation
since the writing of the object has not been completed. For a log entry with
an incomplete old value according to the CRC field, FUSEE redoes the request
according to the operation field and the KV pair. Under this situation, either
the request belongs to the last writer that crashed before committing the log
(c1 in Figure 9), or it belongs to other non-last writers. In the first case,
the values in the backup slots may not be consistent and the primary slot has
not been modified to a new value. Redoing the request can make the backup and
primary slots consistent. In the second case, since the request of crashed
non-last writers has not returned to clients, redoing the request does not
violate the linearizability. For a request with a complete old value, the
request must belong to a last writer. However, the request may finish (c3) or
crash before the primary slot is modified (c2). The recovery process checks
the value in the primary slot ($v_{p}$) and the value in the old value
($v_{old}$) to distinguish c2 from c3. If $v_{p}=v_{old}$, the request crashed
before the primary was modified because $v_{old}$ records the value before
index modification. Since all backup slots are consistent, the recovery
process modifies the primary slot to the new value and finishes the recovery.
Otherwise, the request is finished and no further operation is required. After
recovering the request, the master asynchronously checks content in the
$v_{old}$s in log entries of the crashed client to recover its batched free
operations.
### 5.4 Mixed Crashes
In situations where clients and MNs crash together, FUSEE recovers the
failures separately. FUSEE first lets the master recover all MN crashes and
then starts the recovery processes for failed clients. KV requests can proceed
because the master acts as the last writer for all blocked KV requests. No
request is committed twice because the master commits the operation logs on
clients’ behalves.
(a) INSERT latency CDF.
(b) UPDATE latency CDF.
(c) SEARCH latency CDF.
(d) DELETE latency CDF.
Figure 10: The CDFs of different KV request latency under the microbenchmark.
## 6 Evaluation
### 6.1 Experiment Setup
Implementation. We implement FUSEE from scratch in C++ with 13k LOC. We
implement RACE hashing carefully according to the paper due to no available
open-source implementations. Coroutines are employed on clients to hide the
RDMA polling overhead, as suggested in [31, 73]. The design of FUSEE is
agnostic to the lower-level memory media of memory nodes, i.e., any memory
node with either persistent memory (PM) or DRAM that provides READ, WRITE, and
8-byte CAS interfaces is compatible. We adopt monolithic servers with RNICs
and DRAM to serve as MNs like Clover [60] since we do not have access to
smartNICs and PM. Specifically, we start an MN process on a monolithic server
to register RDMA memory regions and serve memory allocation RPCs with a UDP
socket. MN processes serve memory allocation requests with UDP sockets. Since
the socket receive is a blocking system call, the process will be in the
blocked state with no CPU usage most of the time.
Testbed. We run all experiments on 22 physical machines (5 MNs and 17 CNs) on
the APT cluster of CloudLab [19]. Each machine is equipped with an 8-core
Intel Xeon E5-2450 processor, 16GB DRAM, and a 56Gbps Mellanox ConnectX-3 IB
RNIC. These machines are interconnected with 56Gbps Mellanox SX6036G switches.
Comparison. We compare FUSEE with two state-of-the-art KV stores on DM, i.e.,
pDPM-Direct and Clover [60]. pDPM-Direct stores and manages the KV index and
memory space on the clients. It uses a distributed consensus protocol to
ensure metadata consistency and locks to resolve data access conflicts. We
extend the open-source version of pDPM-Direct to support string keys for fair
comparison in our evaluation. Clover is a semi-disaggregated KV store that
adopts monolithic servers to manage memory spaces and a hash index. All UPDATE
and INSERT requests have to go through the metadata server, requiring
additional compute power. For both pDPM-Direct and Clover, client-side caches
are enabled following their default settings. To show the effectiveness of
SNAPSHOT and the adaptive index cache, we implement FUSEE-CR and FUSEE-NC, two
alternative versions of FUSEE. FUSEE-CR replicates index modifications by
sequentially CASing all replicas to enforce sequential accesses. FUSEE-NC is
the version of FUSEE without a client-side cache. For all these methods, we
evaluate their throughput and latency with both micro and YCSB [15]
benchmarks.
Since the open-source version of Clover and pDPM-Direct only support one index
replica, we compare FUSEE with these two approaches with a single index
replica and two data replicas in the microbenchmark (Section 6.2) and YCSB
performance (Section 6.3) evaluations. When evaluating FUSEE with a single
index replica, the embedded log is constructed, but the commit of the log is
skipped since committing the log is used to ensure the consistency of multiple
index replicas. The performance of FUSEE with multiple replicas is evaluated
in the fault-tolerance evaluation (Section 6.4).
### 6.2 Microbenchmark Performance
We use microbenchmarks to evaluate the operation throughput and latency of the
three approaches. For FUSEE and pDPM-Direct, we use 16 CNs and 2 MNs. For
Clover, we use 17 CNs and 2 MNs because it needs an additional metadata
server, consuming 8 more CPU cores and an additional RNIC. We do not use
multiple metadata servers for Clover because the current open-source
implementation of Clover only supports a single metadata server. We run 128
client processes on the 16 CNs, where each CN holds 8 clients. The DELETE of
Clover is not tested because Clover does not support it.
Figure 11: The throughputs of microbenchmark.
Figure 12: The throughput of FUSEE under different KV sizes.
Latency. To evaluate the latency of KV requests, we use a single client to
iteratively execute each operation $10,000$ times. Figure 10 shows the
cumulative distribution functions (CDFs) of the request latency. FUSEE
performs the best on INSERT and UPDATE, since the SNAPSHOT replication
protocol has bounded RTTs. FUSEE has a little higher SEARCH latency than
Clover since FUSEE reads the hash index and the KV pair in a single RTT, which
is slower than only reading the KV pair in Clover. FUSEE has slightly higher
DELETE latency than pDPM-Direct because FUSEE writes a log entry and reads the
hash index in a single RTT, which is slower than just reading the hash index
in pDPM-Direct.
Throughput. Figure 12 shows the throughput of the three approaches. The
throughput of pDPM-Direct is limited by its remote lock, which causes
extensive lock contention as the number of clients grows. For Clover, even
though it consumes more hardware resources, i.e., 8 additional CPU cores and
an RNIC, the scalability is still lower than FUSEE. This is because the CPU
processing power of the metadata server bottlenecks its throughput. On the
contrary, FUSEE improves the overall throughput by eliminating the computation
bottleneck of the metadata server and efficiently resolving conflicts with the
SNAPSHOT replication protocol.
(a) A (SEARCH:UPDATE = 0.5:0.5).
(b) B (SEARCH:UPDATE = 0.95:0.05).
(c) C (100% SEARCH).
(d) D (SEARCH:INSERT = 0.95:0.05).
Figure 13: The scalability of FUSEE under different YCSB workloads.
(a) YCSB-A throughput.
(b) YCSB-C throughput.
Figure 14: The throughput with different numbers of MNs.
Figure 15: Throughput under different SEARCH-UPDATE ratios.
Figure 16: Throughput under different adaptive cache thresholds.
### 6.3 YCSB Performance
For YCSB benchmarks [15], we generate $100,000$ keys with the Zipfian
distribution ($\theta=0.99$). We use $1024$-byte KV pairs, which is
representative of real-world workloads [15, 9, 17]. The hardware setup is the
same as microbenchmarks.
YCSB Throughput. Figure 13 shows the throughput of three approaches with
different numbers of clients. Clover performs the best under a small number of
clients since adopting the metadata server simplifies KV operations. Compared
with Clover, pDPM-Direct and FUSEE require more RDMA operations to resolve
index modification conflicts. As the number of clients grows, the throughput
of Clover and pDPM-Direct does not increase because the throughput is
bottlenecked by the metadata server and the lock contention, respectively.
Compared with Clover, FUSEE scales better with the growing number of clients
while consuming fewer resources. Compared with pDPM-Direct, FUSEE improves the
throughput by avoiding lock contention. When the number of clients reaches
128, the throughput of FUSEE is $4.9\times$ and $117\times$ higher than Clover
and pDPM-Direct, respectively.
Figure 14 shows the throughput of the three approaches with a write-intensive
workload (YCSB-A) and a read-intensive workload (YCSB-C) when varying numbers
of MNs from 2 to 5 using 128 clients. The throughput of pDPM-Direct and Clover
does not increase due to being limited by lock contention and the limited
compute power of the metadata server, respectively. As for FUSEE, the
throughput improves as the number of memory nodes increases from 2 to 3. There
is no further throughput improvement because the total throughput is limited
by the number of compute nodes.
Figure 17: The throughput of different memory allocation methods.
Figure 18: YCSB throughput under different replication factors.
(a) UPDATE median latency.
(b) DELETE median latency.
(c) INSERT median latency.
(d) SEARCH median latency.
Figure 19: Median operation latency of FUSEE, FUSEE-NC and FUSEE-CR under
different replication factors.
Figure 12 shows the throughput of FUSEE under smaller KV sizes. Since the
throughput of FUSEE is limited by the bandwidth of MN-side RNICs, the YCSB-C
throughput of FUSEE increases by $44.1\%$ and $55.9\%$ with 512B and 256B KV
pairs, respectively. The performance of FUSEE is not affected by the dataset
size because the performance depends only on the number of RTTs of KV
requests, which is deterministic as presented in Section 4.
Read-write performance. Figure 16 shows the throughput of the three approaches
under different SEARCH-UPDATE ratios. As the portion of UPDATE grows, the
throughput of all three methods decreases because UPDATE requests involve more
RTTs. However, FUSEE exhibits the best throughput due to eliminating the
computation bottleneck of metadata servers.
Adaptive index cache performance. Figure 16 shows the YCSB-A throughput of
FUSEE with different adaptive index cache thresholds. The throughput of FUSEE
decreases with the increasing thresholds because more bandwidth is wasted on
fetching invalidated KV pairs with a high threshold.
Two-level memory allocation performance. To show the effectiveness of the two-
level memory allocation scheme, we compare FUSEE with an MN-centric memory
allocation scheme, as shown in Figure 18. The YCSB-A throughput drops $90.9\%$
due to the limited compute power on MNs, while the YCSB-C throughput remains
the same since no memory allocation is involved in the read-only setting.
Figure 20: YCSB-C throughput under a crashed memory node.
Figure 21: The elasticity of FUSEE.
### 6.4 Fault Tolerance & Elasticity
SNAPSHOT Replication Protocol. Figure 19 shows the median latency of FUSEE,
FUSEE-NC, and FUSEE-CR with different replication factors under
microbenchmarks. We set both the numbers of index replicas and data replicas
to $r$ where $r$ is the replication factor. The latency of FUSEE-CR on INSERT,
UPDATE, and DELETE grows linearly as the replication factor because it
modifies index replicas sequentially, and the number of RTTs equals the
replication factor. Differently, the latency of FUSEE grows slightly with the
replication factor because SNAPSHOT has a bounded number of RTTs. For SEARCH
requests, FUSEE and FUSEE-CR have comparable latency since they execute SEARCH
similarly. Compared with FUSEE-NC, FUSEE has lower latency for UPDATE, DELETE,
and SEARCH due to fewer RTTs. The INSERT latency is slightly higher than that
of FUSEE-NC because FUSEE spends additional time to maintain the local cache.
Figure 18 shows the throughput of FUSEE under different replication factors.
For YCSB-A and YCSB-B, the throughput drops as the replication factor grows.
The YCSB-D throughput slightly drops from 8.8 Mops to 8.6 Mops due to the
read-intensive nature of YCSB-D. The YCSB-C throughput remains the same due to
no index modifications.
Search under Crashed MNs. FUSEE allows SEARCH requests to continue when MNs
crash under read-intensive workloads. Figure 21 shows the throughput of 9
seconds of execution, where memory node 1 crashes at the 5th second. The
overall throughput drops to half of the peak throughput because all data
accesses come to one MN. The throughput is then limited by the network
bandwidth of a single RNIC.
Recover from Crashed Clients. To evaluate the efficiency of a client
recovering from failures, we crash and recover a client after UPDATE $1,000$
times. As shown in Table 1, FUSEE takes 177 milliseconds to recover from a
client failure. The memory registration and connection re-establishment
account for $92\%$ of the total recovery time. The log traversal and KV
request recovery only account for $4\%$ of the recovery time, which implies
the affordable overhead of log traversal.
Elasticity. FUSEE supports dynamically adding and shrinking clients. We show
the elasticity of FUSEE by dynamically adding and removing 16 clients when
running the YCSB-C workload. As shown in Figure 21, the throughput increases
when the number of clients increases from 16 to 32 and resumes to the previous
level after removing 16 clients.
Table 1: Client recovery time breakdown. Step | Time (ms) | Percentage
---|---|---
Recover connection & MR | 163.1 | 92.1%
Get Metadata | 0.3 | 0.2%
Traverse Log | 3.5 | 2.0%
Recover KV Requests | 3.5 | 2.0%
Construct Free List | 6.6 | 3.7%
Total | 177.0 | 100%
## 7 Related Work
Disaggregated Memory. Existing approaches can be classified into software-
based, hardware-based, and co-design-based memory disaggregation. Software-
based approaches hide the DM abstraction by modifying operating systems [56,
61, 23, 3, 48], virtual machine monitors [42], or runtimes [55, 63]. Hardware-
based ones design memory buses [41, 14] to enable efficient remote memory
access. Co-design-based approaches co-design software and hardware [25, 65,
39, 8] to gain better application throughput and latency on DM. The design of
FUSEE is agnostic to the low-level implementations of all these DM approaches.
Disaggregated Memory Management. MIND [39] and Clio [25] are the two state-of-
the-art memory management approaches on DM. But they both rely on special
hardware to manage memory spaces. The two-level memory management of FUSEE
resembles the hierarchical memory management of The Machine [35, 21]. The
difference is that FUSEE focuses on fine-grained KV allocation with commodity
RNICs, while The Machine relies on special SoCs and directly manages physical
memory devices.
Memory-disaggregated KV stores. Clover [60] and Dinomo [38] are the most
related memory-disaggregated KV stores. Compared with Clover [60], FUSEE
brings disaggregation to metadata management and gains better resource
efficiency and scalability. Finally, Dinomo [38] is a fully-disaggregated KV
store that was developed concurrently with our system. Dinomo proposes
ownership partitioning to reduce coordination overheads of managing
disaggregated metadata. However, it assumes that the disaggregated memory pool
is fault-tolerant, and hence its design does not consider MN failures. In
contrast, FUSEE addresses the challenges of handling MN failures with the
SNAPSHOT replication protocol. There are many related RDMA-based KV stores
[31, 32, 49, 60, 57, 29, 52, 66, 68, 67, 18, 46]. They are infeasible on DM
since they rely on server-side CPUs to execute KV requests. Besides, there are
emerging approaches that use SmartNICs to construct KV stores [40, 53]. FUSEE
can also benefit from the additional compute power by offloading the memory
management to SmartNICs.
Replication. Both traditional [62, 59, 2, 37, 44, 47, 51, 22] and RDMA-based
[34, 72, 58] replication protocols are designed to ensure data durability.
However, all these approaches are server-centric replication protocols
designed for monolithic servers. Differently, SNAPSHOT is a client-centric
replication protocol designed for the DM architecture and achieves high
scalability with collaborative conflict resolution.
## 8 Conclusion
This paper proposes FUSEE, a fully memory-disaggregated KV store, that
achieves both resource efficiency and high performance by disaggregating
metadata management. FUSEE adopts a client-centric replication protocol, a
two-level memory management scheme, and an embedded log scheme to attack the
challenges of weak MN-side compute power and complex failure situations on DM.
Experimental results show that FUSEE outperforms the state-of-the-art
approaches by up to $4.5\times$ with less resource consumption.
## Acknowledgments
We sincerely thank our shepherd Kimberly Keeton and the anonymous reviewers
for their constructive comments and suggestions. This work was supported by
the National Natural Science Foundation of China (Nos. 62202511 & 61971145),
the Research Grants Council of the Hong Kong Special Administrative Region,
China (No. CUHK 14210920 of the General Research Fund), and Huawei Cloud.
Pengfei Zuo is the corresponding author ([email protected]).
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## Appendix A Proof of Correctness
Since SNAPSHOT focuses on the linearizability of replicated index slots, it is
equivalent to proving that SNAPSHOT is correct on a single replicated slot.
The proof consists of four parts. First, we formally define the system model
and the correctness standard, i.e., linearizability. We then specify the
behavior of a legal replicated slot. After that, we formally prove that
SNAPSHOT is correct under failure-free executions. Finally, we illustrate that
SNAPSHOT is correct when failures occur with a fault-tolerant management
master.
### A.1 Formal System Model
Formally, the system is comprised of a set $C$ of clients
$\\{c_{1},...,c_{n}\\}$ and a set $S$ of replicated slots
$\\{s_{1},...,s_{r}\\}$, where $n$ is the number of clients and $r$ is the
replication factor. Each slot $s_{i}$ stores a value denoted by $v_{s_{i}}$.
Without loss of generality, we assume $s_{1}$ is the primary slot and
$\\{s_{2},...,s_{r}\\}$ are backup slots. Clients can access the contents in
the replicated slots via $\texttt{RDMA\\_READ}(s_{i})$, and
$\texttt{RDMA\\_CAS}(s_{i},v_{old},v_{new})$ operations.
$\texttt{RDMA\\_READ}(s_{i})$ returns the content in slot $s_{i}$.
$\texttt{RDMA\\_CAS}(s_{i},v_{old},v_{new})$ atomically compares $v_{s_{i}}$
with $v_{old}$ and updates $v_{s_{i}}^{\prime}=v_{new}$ if
$v_{s_{i}}=v_{old}$. All these operations are synchronous and reliable, i.e.,
client-initiated RDMA operations will not be re-ordered and operations reply a
FAIL state only if the corresponding slot crashes.
Both clients and slots may fail according to the crash-stop model: a failed
client stops executing instructions and a failed slot returns FAIL to clients
when performing RDMA operations. Slots are shared memory that provide only
RDMA_READ and RDMA_CAS operations for clients to manipulate its content.
Clients are state machines that deterministically transition between states
when events occur. Each operation $op$ contains two events, $inv(op)$
indicating the invocation of the operation and $resp(op)$ indicating the
completion of the operation.
We use the definitions of history from Herlihy and Wing [27] to facilitate our
proof. A history $h$ of an execution $e$ is an infinite sequence of operation
invocation and response events in the same order as they appear in $e$. We
denote by $ops(h)$ the set of all operations whose invocations appear in $h$.
A history $h$ is sequential if (1) the first event of $h$ is an invocation and
(2) each invocation, except the last is immediately followed by a matching
response and each response is immediately followed by an invocation. We use
$h|o$ to denote an object subhistory, where all events in $h$ occurred at
object $o$. A set $S$ of histories is prefix-closed if whenever $h$ is in $S$,
every prefix of $h$ is also in $S$. A sequential specification of an object
$o$ is a prefix-closed set of single-object sequential histories for $o$. A
sequential history $h$ is legal if $\forall o\in O:h|o$ belongs to the
sequential specification for $o$. A history induces an irreflexible partial
order on $ops(h)$, denoted as $<_{h}$, where $op_{1}<_{h}op_{2}$ if and only
if $resp(op_{1})<inv(op_{2})$ in h.
### A.2 Replicated Slot
A replicated slot is a data type that supports the following operations:
* •
READ(): returns the value of the replicated slot.
* •
WRITE(v): updates the value of the slot to v.
Different from traditional replicated objects, the values different clients
write to a replicated slot are not duplicated, which is guaranteed by the out-
of-place modification scheme of FUSEE. We use $reads(h)$ and $writes(h)$ to
denote the set of all operations that reads or writes in $ops(h)$
respectively.
###### Definition 1 (Replicated Slot Specification).
A sequential object history $h|o$ belongs to the sequential specification of a
replicated slot if for each $op\in reads(h|o)$ such that $resp(op)\in h|o$,
$resp(op)$ contains the value of the latest preceding operation $u\in
write(h|o)$ or if there is no preceding update, then $resp(op)$ contains the
initial value of $o$.
### A.3 Proof of Linearizability
In this subsection, we prove that the failure-free executions of SNAPSHOT
satisfy linearizability. The proof is based on Algorithm 1 and Algorithm 2.
Linearizability [26] is defined as follows.
###### Definition 2 (Linearizability).
A complete history $h$ satisfies linearizability if there exists a legal total
order $\tau$ of $ops(h)$ such that $\forall op_{1},op_{2}\in
ops(h),op_{1}<_{h}op_{2}\implies op_{1}<_{\tau}op_{2}$.
Proof Sketch. We prove the linearizability of SNAPSHOT by first attaching a
virtual label to the replicated slot. Then we formally define the rules to
update the slot virtual label. We assign each READ or WRITE operation an
operation label according to the slot virtual label. Finally, we define a
total order relationship on the operation virtual label and prove that the
label total order satisfies $\tau$ in Definition 2.
Notations Without further specification, we use $var^{op}$ and $func^{op}$ to
denote the variable $var$ and the function call $func$ in the execution of the
algorithms of $op$.
More Definitions. We first give more definitions to facilitate our proof.
###### Definition 3.
A complete operation $op\in reads(h)$ observes an update $u\in writes(h)$ if
the value returned in $resp(op)$ was written by $u$.
###### Definition 4 (Write Winner).
An $op\in writes(h)$ is a write winner, denoted by $Win(op)=True$, if
$win^{op}\in\\{\texttt{Rule\\_1},\texttt{Rule\\_2},\texttt{Rule\\_3}\\}$ in
Line 10 of Algorithm 1.
###### Definition 5 (Slot Virtual Label).
A virtual label of a replicated slot is of the form
$(u,v,w)\in\mathbb{N}^{3}$. The initial virtual slot label is $(0,0,0)$.
###### Definition 6 (Slot Virtual Label Update Rules).
For a label with value $(u,v,w)$ its update rules are defined as follows:
* •
$(u,v,w)\leftarrow(u,v,w+1)$ for each RDMA_READ operation in Line 2 of
Algorithm 1.
* •
$(u,v,w)\leftarrow(u+1,0,0)$ for each RDMA_CAS that modifies the primary slot
in Line 12 and Line 15 of Algorithm 1.
###### Definition 7 (Operation Virtual Label).
For each operation $op\in ops(h)$, its virtual label
$L_{op}=(u,v,w)\in\mathbb{N}^{3}$ is assigned as follows:
* •
If $op\in reads(h)$, $L_{op}=(u,0,w+1)$, where $(u,v,w)$ is the slot virtual
label when it initiates the RDMA_READ in Line 2 of Algorithm 1.
* •
If $op\in writes(h),Win(op)=True$, then $L_{op}=(u+1,0,0)$, where $(u,v,w)$ is
the slot virtual label when it initiates the RDMA_READ in Line 6 of Algorithm
1.
* •
If $op\in writes(h),Win(op)=False$, then $L_{op}=(u,1,c_{i})$, where $(u,v,w)$
is the slot virtual label when it initiates the RDMA_READ in Line 6 of
Algorithm 1 and $c_{i}$ is the client that performs the operation.
###### Definition 8 (Operation Label Order).
Two operation virtual labels
$L_{1}=(u_{1},v_{1},w_{1}),L_{2}=(u_{2},v_{2},w_{2})$, $L_{1}<_{l}L_{2}$ if
and only if $u_{1}<u_{2}\bigvee(u_{1}=u_{2}\bigwedge
v_{1}<v_{2})\bigvee(u_{1}=u_{2}\bigwedge v_{1}=v_{2}\bigwedge w_{1}<w_{2})$.
###### Definition 9 (Write Round).
For an $op\in writes(h)$, its write round $R_{op}=u$, where $u$ is the label
$(u,v,w)$ when Line 6 of Algorithm 1 is executed.
###### Definition 10 (Round Start Time).
The start time of a write round $k$ is defined as
$t_{start}^{k}=inv(op):\forall op_{j}\in writes(h):R(op)=R(op_{j})\implies
inv(op)\leq inv(op_{j})$.
###### Lemma 1.
Values in the primary slot is uniquely associated with a write round.
###### Proof.
By Definition 6 and Line 12 and Line 15 of Algorithm 1. The primary slot is
modified with different values and with a strictly monotonic $u$, where
$(u,v,w)$ is its virtual label. ∎
###### Lemma 2 (At least one winner).
$\forall$ write round $k$ if $v_{s_{1}}=...=v_{s_{r}}$ at $t_{start}^{k}$,
then $\exists op\in writes(h):R(op)=k\bigwedge win(op)=True$.
###### Proof.
Since $v_{s_{1}}=...=v_{s_{r}}$, by Lemma 1 $\forall op\in
writes(h):R(op)=k,v_{old}^{op}=v_{s_{1}}$. Since values in the backup slots
all equals $v_{s_{1}}$ and different $op$ proposes different $v_{new}^{op}$,
the atomicity of RDMA_CAS (Line 7, Algorithm 1) ensures that each backup can
only be modified once and all backups must have be modified once after some
$op$ reaches Line 9. Suppose $\forall op\in writes(h):R(op)=k\bigwedge
Win(op)=False$. The primary slot will not be modified as indicated by the if
condition in Line 16 of Algorithm 1, which implies that all $op$ has
$v_{check}^{op}=v_{old}$ in Line 12 of Algorithm 2. The if condition of Line
17 (Algorithm 2) must be false, and thus $\forall op:v_{new}^{op}\neq
min(v\\_list)$, which is impossible because all the backup slot must have be
modified by some $op$ and $v_{new}^{op}$ are distinct. ∎
###### Lemma 3 (At most one winner).
$\forall$ write round $k$ if $v_{s_{1}}=...=v_{s_{r}}$ at $t_{start}^{k}$,
then $\forall op_{1},op_{2}\in writes(h):R(op_{1})=R(op_{2})=k\bigwedge
Win(op_{1})=Win(op_{2})=True\implies op_{1}=op_{2}$.
###### Proof.
1\. No two client can both win by Rule 1 and Rule 2 because (1) all backup
slots can only be modified once and (2) majority cannot be overlapped because
different $op$ modifies with different $v_{new}$.
2\. Suppose there are two winners $op_{1}$ and $op_{2}$.
CASE 1: $win^{op_{1}}\in\\{\texttt{Rule\\_1},\texttt{Rule\\_2}\\}\bigwedge
win^{op_{2}}=\texttt{Rule\\_3}$.
If the finish of RDMA_CAS_backups (Line 7, Algorithm 1) of $op_{2}$ happens
before $op_{1}$ executes RDMA_CAS_primary (Line 12 or Line 15, Algorithm 1),
then guaranteed by the atomicity of RDMA_CAS, $op_{2}$ knows the majority and
cannot win. If the finish of RDMA_CAS_backups of $op_{2}$ happens after
$op_{1}$ executes RDMA_CAS_primary, then the RDMA_READ (Line 12, Algorithm 2)
must return $v_{check}^{op_{2}}\neq v_{orig}^{op_{2}}$. Then
$win^{op_{2}}=\texttt{FINISH}$ by Line 16 of Algorithm 2.
CASE 2: $win^{op_{1}}=win^{op_{2}}=\texttt{Rule 3}$.
Since there is no majority, Line 17 of Algorithm 2 indicates that
$v_{new}^{op_{1}}=v_{new}^{op_{2}}$, contradicting with the assumption that no
clients write duplicated value. ∎
###### Definition 11 (Round finish time).
The round finish time $t_{fini}^{k}$ of a write round $k$ is the time when its
only winner initiates RDMA_CAS_primary in Line 12 or Line 15 of Algorithm 1.
###### Lemma 4.
$\forall k$ at $t_{start}^{k}$, we have $v_{s_{1}}=...=v_{s_{r}}$.
###### Proof.
We prove this by induction.
1\. Initially, when $k=0$, all slots have the same value.
2\. Suppose all slots have the same value at $t_{start}^{k}$. By Lemma 2 and
Lemma 3, there must be a single write winner $op$. Guaranteed by the if-
condition in Line 11 and Line 13 of Algorithm 1, only the $op$ with
$Win(op)=True$ can further modify the replicated slot. If
$win^{op}=\texttt{Rule\\_1}$, then Line 7 of Algorithm 1 guarantees that all
backups $v_{s_{2}}=...=v_{s_{r}}=v_{new}^{op}$ before the initiation of
RDMA_CAS_primary (Line 12). If
$win^{op}\in\\{\texttt{Rule\\_2},\texttt{Rule\\_3}\\}$, then the
RDMA_CAS_backups (Line 14, Algorithm 1) guarantees that all backups
$v_{s_{2}}=...=v_{s_{r}}=v_{new}^{op}$ before the RDMA_CAS_primary (Line 15).
Since $op$ modifies the primary slot also to $v_{new}^{op}$ (Line 12 or Line
15), after $t_{fini}^{0}$ we have $v_{s_{1}}=...=v_{s_{r}}=v_{new}^{op}$. By
Definition 10, all modifications happens after $t_{start}^{k+1}$, which
implies $v_{s_{1}}=...=v_{s_{r}}$ at $t_{start}^{k+1}$. ∎
###### Lemma 5 (Exactly one write winner).
$\forall k\in\mathbb{N}$, we have:
* •
$\exists op\in writes(h):R(op)=k,Win(op)=True$.
* •
$\forall op_{1},op_{2}\in writes(h):R(op_{1})=R(op_{2})=k\bigwedge
Win(op_{1})=Win(op_{2})=True\implies op_{1}=op_{2}$.
###### Proof.
By Lemma 2 \+ Lemma 3 \+ Lemma 4. ∎
###### Lemma 6 (Label order respects history).
$\forall op_{1},op_{2}\in ops(h),op_{1}<_{h}op_{2}\implies op_{1}<_{l}op_{2}$.
###### Proof.
Suppose $\exists op_{1},op_{2}\in ops(h),op_{1}<_{h}op_{2}\bigwedge
op_{2}<_{l}op_{1}$. Their labels are $l_{1}=(u_{1},v_{1},w_{1})$ and
$l_{2}=(u_{2},v_{2},w_{2})$.
CASE 1: $op_{1},op_{2}\in reads(h)$.
If $u_{1}=u_{2}$, then $op_{1}<_{h}op_{2}$ implies that the RDMA_READ (Line 2,
Algorithm 1) of $op_{1}$ happens before the RDMA_READ of $op_{2}$. By
Definition 6, RDMA_READs in Line 2 of Algorithm 1 monotonically increase $w$
of the virtual slot label. Then we must have $w_{1}<w_{2}$. Since
$u_{1}=u_{2},v_{1}=v_{2}=0,w_{1}<w_{2}$, $op_{1}<_{l}op_{2}$, contradicts with
the assumption. If $u_{1}>u_{2}$, then $\exists
t_{fini}^{u_{2}}:resp(op_{1})<t_{fini}^{u_{2}}<inv(op_{2})$. By Definition 6,
$u_{1}\leq u_{2}$, contradicts with $u_{1}>u_{2}$.
CASE 2: $op_{1}\in reads(h),op_{2}\in writes(h)$.
If $Win(op_{2})=True$, then $op_{1}<_{h}op_{2}$ implies that
$resp(op_{1})<t_{fini}^{u_{2}-1}$. Then $u_{1}\leq u_{2}-1$ contradicts with
$op_{2}<_{l}op_{1}$. Otherwise $Win(op_{2})=False$, then $op_{1}<_{h}op_{2}$
implies that $resp(op_{1})<t_{fini}^{u_{2}}$. Since $u_{1}\leq u_{2}$ and
$0=v_{1}<v_{2}=1$ by Definition 7, there is a contradiction with
$op_{2}<_{l}op_{1}$.
CASE 3: $op_{1}\in writes(h),op_{2}\in reads(h)$.
Line 17-22 of Algorithm 1 ensures that $\forall op\in
writes(h),R(op)=k\implies resp(op)>t_{fini}^{k}$. If $Win(op_{1})=True$, then
$op_{1}<_{h}op_{2}$ implies that $t_{fini}^{u_{1}-1}<inv(op_{2})$ and
$l_{1}=(u_{1},0,0)$. Consequently, $u_{1}\leq u_{2}\bigwedge
v_{1}=v_{2}=0\bigwedge 0=w_{1}<1\leq w_{2}$ contradicts with
$op_{2}<_{l}op_{1}$. Otherwise $Win(op_{1})=False$, then $op_{1}<_{h}op_{2}$
implies that $t_{fini}^{u_{1}}<inv(op_{2})$. Then $u_{1}+1\leq u_{2}$ which
contradicts with $op_{2}<_{l}op_{1}$.
CASE 4: $op_{1},op_{2}\in writes(h)$.
If $Win(op_{1})=True$, then $op_{1}<_{h}op_{2}\implies
t_{fini}^{u_{1}-1}<inv(op_{2})\implies u_{1}\leq u_{2}$. Also
$Win(op_{1})=True\implies l_{1}=(u_{1},0,0)$. Then $u_{1}\leq
u_{2},0=v_{1}<v_{2}=1$ contradicts with $op_{2}<_{l}op_{1}$. Otherwise
$Win(op_{1})=False$, then $op_{1}<_{h}op_{2}\implies
t_{fini}^{u_{1}}<inv(op_{2})\implies u_{1}+1\leq u_{2}\implies u_{1}<u_{2}$,
which contradicts with $op_{2}<_{l}op_{1}$. ∎
###### Lemma 7 (Label order is a legal order).
The label order $<_{l}$ is a legal total order of $ops(h)$.
###### Proof.
By Definition 1, let $r\in reads(h)$ be an operation that observes a write
$k\in writes(h)$, $r$ is completed. Suppose $\exists k^{\prime}\in writes(h)$
such that $k<_{l}k^{\prime}<_{l}r$. Let
$l_{k}=(u_{k},v_{k},w_{k}),l_{k^{\prime}}=(u_{k}^{\prime},v_{k}^{\prime},w_{k}^{\prime}),l_{r}=(u_{r},v_{r},w_{r})$.
By Lemma 1, $r$ observes a value if and only if it returns the corresponding
virtual label in Line 2 of Algorithm 1. By Definition 6 and Lemma 5, the label
is exclusively updated by a single writer, which implies
$Win(k)=True,t_{fini}^{u_{k}-1}<inv(r)$ and $u_{k}=u_{r}$. Since
$k<_{l}k^{\prime}<_{l}r$ and $u_{k}=u_{r}$, $u_{k}=u_{k}^{\prime}=u_{r}$. If
$Win(k^{\prime})=True$, then $Win(k)=Win(k^{\prime})=True\bigwedge
R(k)=R(k^{\prime})=u_{k}-1$, which contradicts with Lemma 5. Otherwise
$Win(k^{\prime})=False$ then by Definition 7 $v_{k}^{\prime}=1$.
$u_{k}^{\prime}=u_{r},v_{k}^{\prime}=1>0=v_{r}$ contradicts with
$k^{\prime}<_{l}r$. As a result, there is no $k^{\prime}$ such that
$k<_{l}k^{\prime}<_{l}r$. ∎
###### Theorem 1.
FUSEE implements a replicated slot with linearizability.
###### Proof.
By Lemma 7 and Lemma 6. ∎
1:procedure Master
2: if MN failed or received client fail_query then
3: send member_prepare_change to all clients
4: wait all clients reply or membership lease expires
5: select a slot $s$ randomly in alive backups
6: modify all slots $s_{i}=s$
7: commit the operation log of $write(s)$
8: select new primary and backup
9: reply all clients’ fail_query with a new value $s$
10: send member_commit_change to all clients with the new membership
11: if Clients failed then
12: start RecoverClient($log$) thread.
13:procedure RecoverClient($log$)
14: if The log is committed then
15: return
16: $slot$ is the slot in the log
17: $v_{old}$ is the old value in the log
18: $v_{new}$ is the new value in the log
19: $s_{0}=\texttt{RDMA\\_READ\\_primary}(slot)$
20: if $\texttt{FAIL}\in\\{s_{0},bk\\_list\\}$ then
21: send FailReq to master.
22: else if $v_{old}$ is incomplete then
23: retry the operation
24: else if $v_{old}=s_{0}$ then
25: $\texttt{RDMA\\_CAS\\_primary}(slot,s_{0},v_{new})$
26: return
Algorithm 3 The master process.
1:procedure READ($slot$)
2: $v=\texttt{RDMA\\_READ\\_primary}(slot)$
3: if $v=\texttt{FAIL}$ then
4: $v\\_list=\texttt{RDMA\\_READ\\_backups}(slot)$
5: if All backups have the same value $v^{\prime}$ then
6: $v=v^{\prime}$
7: else
8: $v=\texttt{RPC\\_fail\\_query}(master,slot)$
9: wait for membership change
10: return $v$
11:procedure WRITE($slot,v_{new}$)
12: $v_{old}=\texttt{RDMA\\_READ\\_primary}(slot)$
13: if $v_{old}=\texttt{FAIL}$ then
14: wait for membership change
15: retry WRITE
16: $v\\_list=\texttt{RDMA\\_CAS\\_backups}(slot,v_{old},v_{new})$
17: // Change all the $v_{old}$s in the $v\\_list$ to $v_{new}$s.
18: $v\\_list=\texttt{change\\_list\\_value}(v\\_list,v_{old},v_{new})$
19: $win=\texttt{evaluate\\_rules}(v\\_list)$
20: if $win=\texttt{Rule\\_1}$ then
21: $\texttt{RDMA\\_CAS\\_primary}(slot,v_{old},v_{new}$
22: else if $win\in\\{\texttt{Rule\\_2},\texttt{Rule\\_3}\\}$ then
23: $\texttt{RDMA\\_CAS\\_backups}(slot,v\\_list,v_{new})$
24: $\texttt{RDMA\\_CAS\\_primary}(slot,v_{old},v_{new})$
25: else if $win=\texttt{LOSE}$ then
26: repeat
27: sleep a little bit
28: $v_{check}=\texttt{RDMA\\_READ\\_primary}(slot)$
29: if receive member_prepare_change() then
30: goto Line 35
31: until $v_{check}\neq v_{old}$
32: if $v_{check}=\texttt{FAIL}$ then
33: goto Line 35
34: else if $win=\texttt{FAIL}$ then
35: $v_{RPC}=\texttt{RPC\\_fail\\_query}(master,slot)$
36: wait for membership change
37: if $v_{RPC}=v_{old}$ then
38: retry WRITE
39: return
Algorithm 4 SNAPSHOT with failure handling
### A.4 Correctness of Failure Recovery
The recovery of FUSEE relies on a fault-tolerant management master, which is a
common assumption on membership-based replication protocols like Chain
Replication [62, 59] and Hermes [34]. The master adopts a lease-based
membership service [24] for clients and MNs. The lease-based membership
service provides a view of all alive MNs to clients. Clients check and extend
their leases before performing each read and write. The master can detect the
failures of clients and MNs when a client or MN no longer extend their leases.
The pseudo-code of the master is shown in Algorithm 3, and the full process of
clients is shown in Algorithm 4. The key to guaranteeing correctness under
failures is that the single-winner condition is not violated.
#### A.4.1 Handling MN Crashes
The failure handling process consists of three phases, i.e., a disconnection
phase (Line 2-4, Algorithm 3), a slot-modification phase (Line 5-7, Algorithm
3), and a notification phase (Line 8-10, Algorithm 3). The disconnection phase
starts with the master sending a member_prepare_change to all clients and
waiting for a lease expiration. On receiving the member_prepare_change,
clients stop their future writes to the failed slot. After the lease expires,
no clients can further modify the failed slot because the failed slot is
excluded from the membership view. The currently executing write operations
will also be stopped and wait for the final value to be decided by the master
(Line 35-28, Algorithm 4). Hence, all clients are disconnected to the crashed
slot.
During the modification phase, the master randomly selects a value in an alive
backup slot and uses the selected value to make all alive slots consistent.
Choosing values from backup slots is always safe because backup slots contain
no older values than the latest committed value (value in the primary slot
before it crashed). This is guaranteed by the SNAPSHOT replication protocol
that the write conflicts are resolved in the backup slots and then written to
the primary slot. If there are no alive backup slots, then there must have
only one survivor as the primary slot. In both cases, either a latest
committed value or a fresh uncommitted value (value written by conflicting
writers) is chosen. When a fresh uncommitted value is chosen, the master is
the representative last writer and finishes the last writer’s job on a
client’s behalf. Since no other client can modify the slot, the master is the
only last writer. When a committed value is chosen, the value will be replied
to clients and clients will retry their newer WRITEs (Line 38, Algorithm 4) on
receiving an old value. In this case, the new write round of clients is
considered not started. The retry of writes guarantees that the clients’ newer
write operations are executed before they are returned. The single last write
will then be decided among clients through the normal execution of the
SNAPSHOT replication protocol. After modifying all the values in the
replicated slots, the master commits the operation by writing a special value
($1$) to the old value field of the log header. This guarantees that clients
will never retry an operation finished by the master.
The attribute of exactly one winner on each write round is not violated. Line
4 of Algorithm 2 ensures that a winner can only be decided when no backup slot
crashes. Line 29 and Line 35-38 of Algorithm 4 ensure that all losers and
failed operations wait for the decided value returned by the master. If there
is a winner and the winner finishes execution before the disconnection phase
of the master, then the master can only choose no older value than the
winner’s committed value because all old values are modified to the winner’s
proposed new value. If the winner does not finish execution before the
disconnection phase of the master, then it will be disconnected and deemed
also as a loser. In this case, the master becomes a representative winner that
decides a unique winner value to all other waiting losers. If there is no
winner due to the backup slot failures, then the master will also become a
representative winner that decides a unique winner.
During MN crashes, reads can also get the latest committed value in the slot,
thus ensuring linearizability. The latest committed value is defined as the
last value stored in the primary slot before it crashes. Since the primary
slot is lastly modified in a write round (Line 21 and Line 24, Algorithm 4),
it always contains the latest committed value. If the primary slot is alive,
then reads execute normally by reading the content in the primary slot (Line
2, Algorithm 4). Linearizability is guaranteed by the SNAPSHOT replication
protocol. Otherwise, reads use RDMA_READs to read all backup slots and return
a value only if all the values in the backup slot are the same. This reflects
the situation when there are no write conflicts, and hence the value must be
committed. Under the situation when values in backup slots are not the same,
clients send RPCs to the master to let the master commit a value and return it
to clients.
#### A.4.2 Handling Client Crashes
FUSEE uses per-client operation logs to recover crashed client operations. The
$v_{old}$ is written to the log header by the write winner before the
execution of RDMA_CAS_primary (Line 21 and Line 24, Algorithm 4). $0$ is
written to the used bit of the log headers before the losers return their
operations. During the recovery of a crashed client, memory objects with unset
used bits are reclaimed because they are either incomplete data or free data.
Operations in the operation log with an incomplete $v_{old}$ are redone. These
operations may belong to a loser or a winner. Redoing a crashed operation of a
loser is safe because the losers’ operations have not been returned to clients
and cannot be observed by other clients. Redoing such operations of a winner
is also safe because the winner’s operation also cannot be observed by other
clients due to the unmodified primary slot. Under this situation, the retried
operation will also become a winner due to Lemma 5.
If the old value field is set, then the crashed client must be a write winner
since only the winner commits the log 4.5. However, for a winner, it is
possible that it crashed before the primary slot was modified. Under this
situation, all backup slots contain $v_{new}$, and the primary slot contains
the $v_{old}$. Hence, FUSEE checks if the primary slot equals $v_{old}$ and
modifies the primary slot to be $v_{new}$ if it is the case. Otherwise, the
operation must have been finished because the primary slot is modified in a
new write round.
#### A.4.3 Handling Client and MN Crashes
When handling concurrent client and MN failures, the master reconfigures MNs
to use a decreased replication factor. After the reconfiguration, the master
recovers the crashed client. The only conflict between recovering MNs and
clients is the case that the master decides the value written by a client, and
the client is crashed. In this situation, a winner client is crashed and
ignorant of its winning. The operations may be retired, causing old values to
be written to the slots. FUSEE addresses this issue by letting the master
commit the operation by writing a $v_{old}=0$ to the old value field of the
log header. As a result, when recovering the winner client using the log, it
will not execute the operation twice.
|
# Globalization of partial inverse semigroupoids actions on sets
Paulinho Demeneghi111Universidade Federal de Santa Catarina., Felipe Augusto
Tasca222Instituto Federal do Paraná.
###### Abstract
We prove that every partial action of an inverse semigroupoid on a set admits
a universal globalization. Moreover, we show that our construction gives a
reflector from the category of partial actions on the full subcategory of
global actions. Finally, we investigate if the mediating function given by the
universal property of our construction is injective.
## 1 Introduction
The notion of partial action has appeared for the first time in the literature
at [12], in a work developed by Exel in the context of C*-algebras where the
concept of crossed product of a C*-algebra by a partial action of the infinite
cyclic group was introduced. The work of Exel was generalized afterward by
McClanahan in [21] where a formal definition of crossed product of a
C*-algebra by a partial action of a discrete group was presented. Since then,
partial actions of groups were extensively explored and studied in many other
areas.
The notion of partial action of groups on sets was introduced by Exel in [13]
motivated by the great interest in the concept of partial actions of groups on
C*-algebras. Since then, the concept of group partial actions have been
appeared in many contexts.
Global actions naturally induce partial actions via a restriction process and
it is an important problem to know when a partial action can be obtained
through the restriction of a global one. Explicitly, given a partial action of
a group $G$ on a set $X$, one can obtain a new action by restricting the
action of $G$ to a subset $Y\subseteq X$. The resultant action is a partial
action of $G$ on $Y$ which is not, in general, a global action.
On the other side, given a partial action of $G$ on a set $Y$, one may ask if
there exists a global action of $G$ on a set $X$ containing $Y$ such that the
original partial action of $G$ on $Y$ can be obtained by the restriction of
the global action of $G$ on $X$. This question motivates the notion of
globalization of partial actions. A globalization of a partial action $\theta$
of a group $G$ on a set $Y$ is (isomorphic to) a global action
$\tilde{\theta}$ of $G$ on a set $X\supseteq Y$ such that $\theta$ may be
obtained by the restriction of $\tilde{\theta}$ to $Y$.
The problem of the existence of globalizations was initially studied by Abadie
at [1, 2] and later by Kellendonk and Lawson in [17] in an independent way. At
[1], Abadie proved that every partial action of a group on a set admits a
universal globalization. In this same paper, Abadie has also studied
enveloping actions of groups on topological spaces and C*-algebras. At [17],
in addition to the context of sets and topological spaces, Kellendonk and
Lawson also studied partial action on semi-lattices.
Motivated by the problem of the existence of globalizations for partial
actions, many works were developed in many other contexts exploring: partial
actions of groups on objects like topological spaces, rings, C*-algebras;
partial actions of monoids, semigroups, inverse semigroups, Hopf algebras,
ordered groupoids, groupoids and small categories; and even contexts involving
twisted actions. We may cite [3, 4, 5, 6, 10, 11, 15, 18, 19, 22, 23, 24] as
example. Furthermore, we indicate Dokuchaev extensive survey on the subject
[9].
In this paper we are interested in studying the problem of globalization of
inverse semigroupoid partial actions on sets. The notion of semigroupoids
includes both the notions of categories and semigroups. A semigroupoid may be
thought as a set equipped with a partially defined associative operation.
Actually, there are two notions of semigroupoids in the literature and both
notions are obtained by weakening category axioms. The first notion of
semigroupoid is presented by Tilson in [25] in order to treat category ideals
independently of their parent categories. Roughly speaking, a Tilson
semigroupoid is a small category which may fail to have identity arrows. The
second notion is presented by Exel in [14] with the goal of providing a
unified treatment for C*-algebras associated to combinatorial objects. As a
matter of fact, Exel’s notion is more general than the Tilson’s one, since
Tilson semigroupoids comes with an underlying graph structure, peculiar from
categories, which completely characterizes which pairs of morphisms can be
composed, meanwhile Exel semigroupoids comes equipped with a set of composable
pairs which may not admit a compatible underlying graph structure. Roughly
speaking, in Exel semigroupoids it may not be possible to compose morphisms
even though the codomain of the first agrees with the domain of the second.
The inverse semigroupoids framework provides a unified treatment for inverse
semigroups and groupoids and, it turns out that, despite the difference
between the two existing notions of semigroupoids, when the unique pseudo-
inverse axiom is added, every Exel inverse semigroupoid can be realized as a
Tilson inverse semigroupoid in a unique way, as proved by Cordeiro in [8]. So
that, for a better fit in our approach, we have opted by Tilson’s definition.
Partial actions of inverse semigroupoids were introduced by Cordeiro in [7,
8], where an extensive study on semigroupoids is developed. Based on groupoid
partial actions, in this work we present an adaptation of this definition,
which is equivalent to Cordeiro’s definition in this context.
We prove that, just as group partial actions on sets, every inverse
semigroupoid partial action on a set admits a universal globalization. This
result generalizes simultaneously Theorem 4.23 of [1] for group partial
actions on sets, Theorem 6.10 of [16] for inverse semigroups partial actions
on sets and Theorem 4 (Remark 29) of [23] for groupoids partial actions on
sets. Moreover, we prove that our constructions gives a reflector from the
category of partial actions on the full subcategory of global actions.
The paper is structured as follows: in the second section we first present the
definitions of semigroupoid and inverse semigroupoid adopted, as well as
basics facts about these structures that we are going to need. Then we proceed
to define inverse semigroupoid partial actions on sets and introduce the
necessary machinery for the development of this work. In the third section we
fix a partial action $\theta$ of an inverse semigroupoid
$\operatorname{\mathcal{S}}$ on a set $X$ and then construct a globalization
of $\theta$ which we further prove that is a reflector of $\theta$ in the full
subcategory of global actions (Theorem 3.13) and, hence, is universal among
all globalizations of $\theta$ (Theorem 3.14). Finally, we explore some
examples, one of them, showing that the mediating function given by the
universal property of our construction may not be injective. However, we show
that it is injective on some special subsets (Proposition 3.17).
## 2 Inverse Semigroupoids and Partial Actions
In this section we present the necessary machinery involving inverse
semigroupoids and inverse semigroupoid partial actions on sets. For further
references, we recommend [8, 20].
We first present the definition of semigroupoid we shall adopt.
###### Definition 2.1.
A _semigroupoid_ is a quintuple
$(\operatorname{\mathcal{S}},\operatorname{\mathcal{S}}^{(0)},d,c,\mu)$, in
which
* •
$\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{S}}^{(0)}$ are sets
called _arrow_ and _object_ sets, respectively;
* •
$d,c:\operatorname{\mathcal{S}}\to\operatorname{\mathcal{S}}^{(0)}$ are maps
called _domain_ and _codomain_ maps, respectively;
* •
$\mu:\operatorname{\mathcal{S}}^{(2)}\to\operatorname{\mathcal{S}}$ is a map
called _multiplication_ map (denoted by juxtaposition $\mu(s,t)=st$) defined
in the set
$\operatorname{\mathcal{S}}^{(2)}\subseteq\operatorname{\mathcal{S}}\times\operatorname{\mathcal{S}}$
of _composable pairs_ consisting of all pairs
$(s,t)\in\operatorname{\mathcal{S}}\times\operatorname{\mathcal{S}}$ such that
$d(s)=c(t)$.
satisfying
1. (i)
$d(st)=d(t)$ and $c(st)=c(s)$ for any composable pair
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$;
2. (ii)
$(ps)t=p(st)$ whenever $(p,s),(s,t)\in\operatorname{\mathcal{S}}^{(2)}$.
Notice that, if $p,s,t\in\operatorname{\mathcal{S}}$ are elements such that
$(p,s)$ and $(s,t)$ lies in $\operatorname{\mathcal{S}}^{(2)}$, then by axiom
(i) above $(ps,t)$ and $(p,st)$ also lies in
$\operatorname{\mathcal{S}}^{(2)}$, and hence axiom (ii) does make sense.
By an abuse of notation, whenever there is no chance of confusion, we shall
use just $\operatorname{\mathcal{S}}$ to refer to semigroupoid
$(\operatorname{\mathcal{S}},\operatorname{\mathcal{S}}^{(0)},d,c,\mu)$.
Notice that the class of semigroupoids includes semigroups and small
categories. Semigroups are precisely the semigroupoids with a single object
(and not necessarily an identity element) and small categories are precisely
the semigroupoids containing one identity for each object.
To be precise, by an identity element we mean an element
$e\in\operatorname{\mathcal{S}}$ in a semigroupoid
$\operatorname{\mathcal{S}}$ satisfying $es=s$ and $te=t$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that
$(e,s),(t,e)\in\operatorname{\mathcal{S}}^{(2)}$. If necessary, we may say
that $e\in\operatorname{\mathcal{S}}$ is an identity over an object
$u\in\operatorname{\mathcal{S}}^{(0)}$ to emphasize that $e$ is an identity
element such that $d(e)=c(e)=u$.
Another sort of elements which play an essential role in this work are
idempotent elements. By an idempotent element in a semigroupoid
$\operatorname{\mathcal{S}}$, we mean an element
$e\in\operatorname{\mathcal{S}}$ such that
$(e,e)\in\operatorname{\mathcal{S}}^{(2)}$ and $ee=e$. Again, we may emphasize
that $d(e)=c(e)=u$ by saying that $e$ is an idempotent over
$u\in\operatorname{\mathcal{S}}^{(0)}$. We denote by
$E(\operatorname{\mathcal{S}})$ the set of idempotent elements of
$\operatorname{\mathcal{S}}$, which evidently includes the identity elements.
As mentioned before, we are mainly interested in inverse semigroupoids, as
follows.
###### Definition 2.2.
A semigroupoid $\operatorname{\mathcal{S}}$ is an _inverse semigroupoid_ if
for every $s\in\operatorname{\mathcal{S}}$ there exists a unique
$s^{*}\in\operatorname{\mathcal{S}}$, called the _inverse_ of $s$, such that
$(s,s^{*}),(s^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$ and
$ss^{*}s=s\quad\text{ and }\quad s^{*}ss^{*}=s^{*}.$
Just like the class of semigroupoids includes small categories and semigroups,
the class of inverse semigroupoids includes groupoids and inverse semigroups.
We shall see now that many inverse semigroup properties keep working for this
more general setting. For example, usual rules for inverses are preserved:
$(s^{*})^{*}=s$ for every $s\in\operatorname{\mathcal{S}}$ and if
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$ then
$(t^{*},s^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and $(st)^{*}=t^{*}s^{*}$.
For every element $s\in\operatorname{\mathcal{S}}$ we have $s^{*}s,ss^{*}\in
E(\operatorname{\mathcal{S}})$ and for every $e\in
E(\operatorname{\mathcal{S}})$ we have $e^{*}=e$. Moreover, the idempotent
elements commute in the sense that, if $e,f\in E(\operatorname{\mathcal{S}})$
and $(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, then
$(f,e)\in\operatorname{\mathcal{S}}^{(2)}$ and $ef=fe$. This implies that for
every $s\in\operatorname{\mathcal{S}}$ and $e\in
E(\operatorname{\mathcal{S}})$ such that
$(s,e)\in\operatorname{\mathcal{S}}^{(2)}$ we have $ses^{*}\in
E(\operatorname{\mathcal{S}})$.
The commutativity of $E(\operatorname{\mathcal{S}})$ enables us to define a
canonical order relation on any inverse semigroupoid
$\operatorname{\mathcal{S}}$ just like in the inverse semigroup case. More
precisely, if $s,t\in\operatorname{\mathcal{S}}$, we say that $s\leq t$ if
$d(s)=d(t)$, $c(s)=c(t)$ and one of the following equivalent statements holds:
1. (i)
$s=ts^{*}s$;
2. (ii)
$s=te$ for some $e\in E(\mathcal{S})$ such that
$(t,e)\in\operatorname{\mathcal{S}}^{(2)}$;
3. (iii)
$s=ss^{*}t$;
4. (iv)
$s=ft$ for some $f\in E(\mathcal{S})$ such that
$(f,t)\in\operatorname{\mathcal{S}}^{(2)}$.
Notice that, if $s,t\in\operatorname{\mathcal{S}}$, then $s\leq t$ if and only
if $s^{*}\leq t^{*}$ and that, if
$s_{1},s_{2},t_{1},t_{2}\in\operatorname{\mathcal{S}}$ are such that
$(s_{1},s_{2})\in\operatorname{\mathcal{S}}^{(2)}$, $s_{1}\leq t_{1}$ and
$s_{2}\leq t_{2}$, then $(t_{1},t_{2})\in\operatorname{\mathcal{S}}^{(2)}$ and
$s_{1}s_{2}\leq t_{1}t_{2}$. Moreover, it is immediate from the definition
that, if $e,f\in E(\operatorname{\mathcal{S}})$ and
$(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, than $ef\leq e,f$. As a
consequence of the latter, $ses^{*}\leq ss^{*}$ whenever
$s\in\operatorname{\mathcal{S}}$, $e\in E(\operatorname{\mathcal{S}})$ and
$(s,e)\in\operatorname{\mathcal{S}}^{(2)}$.
Before we proceed to the definition of partial action, we present a inverse
semigroupoid example which is not an inverse semigroup, nor a groupoid (not
even a category).
###### Example 2.3.
As a set, $\operatorname{\mathcal{S}}$ is given by
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$.
The object set is a two element set and the graph structure may be viewed in
the following picture.
$a$$a^{*}$$a^{*}a$$b$$b^{*}$$b^{*}b$$bb^{*}$$aa^{*}$
Finally, the multiplication table is given by:
$\cdot$ | $a$ | $a^{*}$ | $b$ | $b^{*}$ | $a^{*}a$ | $aa^{*}$ | $b^{*}b$ | $bb^{*}$
---|---|---|---|---|---|---|---|---
$a$ | $-$ | $aa^{*}$ | $a$ | $a$ | $a$ | $-$ | $a$ | $a$
$a^{*}$ | $a^{*}a$ | $-$ | $-$ | $-$ | $-$ | $a^{*}$ | $-$ | $-$
$b$ | $-$ | $a^{*}$ | $a^{*}a$ | $bb^{*}$ | $a^{*}a$ | $-$ | $b$ | $a^{*}a$
$b^{*}$ | $-$ | $a^{*}$ | $b^{*}b$ | $a^{*}a$ | $a^{*}a$ | $-$ | $a^{*}a$ | $b^{*}$
$a^{*}a$ | $-$ | $a^{*}$ | $a^{*}a$ | $a^{*}a$ | $a^{*}a$ | $-$ | $a^{*}a$ | $a^{*}a$
$aa^{*}$ | $a$ | $-$ | $-$ | $-$ | $-$ | $aa^{*}$ | $-$ | $-$
$b^{*}b$ | $-$ | $a^{*}$ | $a^{*}a$ | $b^{*}$ | $a^{*}a$ | $-$ | $b^{*}b$ | $a^{*}a$
$bb^{*}$ | $-$ | $a^{*}$ | $b$ | $a^{*}a$ | $a^{*}a$ | $-$ | $a^{*}a$ | $bb^{*}$
in which the symbol “$-$” means that the product is undefined.
Notice that, the idempotent set is given by
$E(\operatorname{\mathcal{S}})=\\{a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$.
We now proceed to present the definition of a partial action of an inverse
semigroupoid on a set.
###### Definition 2.4.
A _partial action_ $\theta$ of an inverse semigroupoid $\mathcal{S}$ on a set
$X$ is a pair
$\theta=\big{(}\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}\big{)}$
consisting of a collection $\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ of
subsets of $X$ and a collection
$\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ of maps
$\theta_{s}\colon X_{s^{*}}\to X_{s}$
such that
1. (P1)
$\theta_{e}=\operatorname{id}_{X_{e}}$ for all $e\in
E(\operatorname{\mathcal{S}})$. Moreover, for all $x\in X$, there exists $e\in
E(\operatorname{\mathcal{S}})$ such that $x\in X_{e}$;
2. (P2)
$X_{s}\subseteq X_{ss^{*}}$ for every $s\in\operatorname{\mathcal{S}}$;
3. (P3)
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})=X_{(st)^{*}}\cap X_{t^{*}}$ for all
$(s,t)\in\mathcal{S}^{(2)}$ and, moreover,
$\theta_{s}\big{(}\theta_{t}(x)\big{)}=\theta_{st}(x)$ for all $x\in
X_{(st)^{*}}\cap X_{t^{*}}$.
We say that $\theta$ is a _global action_ if moreover it satisfies:
1. (P4)
$X_{s}=X_{ss^{*}}$ for every $s\in\operatorname{\mathcal{S}}$ (i.e., equality
holds in (P2)).
By an $\operatorname{\mathcal{S}}$-_set_ we shall mean a pair $(X,\theta)$ in
which $X$ is a set, $\operatorname{\mathcal{S}}$ is an inverse semigroupoid
and $\theta$ is a partial action of $\operatorname{\mathcal{S}}$ on $X$. We
shall, moreover, say that $(X,\theta)$ is a _global_
$\operatorname{\mathcal{S}}$-_set_ if $\theta$ is a global action.
We should also point out that, for an $\operatorname{\mathcal{S}}$-_set_
$(X,\theta)$, unless explicitly said otherwise, we will use the same symbols
$X$ and $\theta$, indexed by elements of $\operatorname{\mathcal{S}}$, to
denote the members of the family of subsets of $X$ and the members of the
family of maps that constitute $\theta$, respectively. That is, we will
implicitly assume that $\theta$ is given by
$(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$.
Before we proceed, let us point some observations about the last definition.
###### Remark 2.5.
First, notice that $\bigcup_{e\in E(\operatorname{\mathcal{S}})}X_{e}=X$ by
(P1). This is not a very restrictive requirement since, in its absence, we may
replace $X$ by the union of the domains $X_{e}$, leading to an action of
$\operatorname{\mathcal{S}}$ on $\bigcup_{e\in
E(\operatorname{\mathcal{S}})}X_{e}$ satisfying all the desired properties.
Actually, some authors refer to partial actions with this extra requirement as
_non-degenerate_ partial actions. We have chosen to not follow this approach
and ask for it directly in the definition.
Replacing $s$ by $s^{*}$ in (P2), we deduce that the domain $X_{s^{*}}$ of
$\theta_{s}$ is contained in the domain $X_{s^{*}s}$ of $\theta_{s^{*}s}$ and,
moreover, by (P4), $\theta_{s}$ and $\theta_{s^{*}s}$ share domains if the
action is global.
Given $s,t\in\operatorname{\mathcal{S}}$, we may not be able to compose
$\theta_{s}$ and $\theta_{t}$ in the traditional way, since the image of
$\theta_{t}$ may not be contained in the domain of $\theta_{s}$. However, we
will use the notation $\theta_{s}\circ\theta_{t}$ to refer to the map defined
on the largest domain in which the expression $\theta_{s}(\theta_{t}(x))$ does
make sense. Explicitly, the domain and codomain of $\theta_{s}\circ\theta_{t}$
are considered to be $\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ and
$\theta_{s}(X_{s^{*}}\cap X_{t})$, respectively.
Hence, under the convention explained in the last paragraph, we can deduce
from (P3) that, whenever $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$, the
domain $\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$
is contained in the domain $X_{(st)^{*}}$ of $\theta_{st}$ and
$\theta_{s}\circ\theta_{t}$ coincides with $\theta_{st}$ in the domain of
$\theta_{s}\circ\theta_{t}$. In other words, from (P3) we deduce that
$\theta_{st}$ is an extension333We shall use the term extension even though
the codomain of the functions involved may not agree with each other. of
$\theta_{s}\circ\theta_{t}$ for $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$,
which we shall denote by $\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$. That
is, we shall use the symbol “$\subseteq$” to express that the function on the
right-hand side is an extension of the one in the left-hand side.
###### Proposition 2.6.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. For every
$s\in\operatorname{\mathcal{S}}$, $\theta_{s}$ is a bijection from $X_{s^{*}}$
onto $X_{s}$ and, moreover, $\theta_{s}^{-1}=\theta_{s^{*}}$.
###### Proof.
Let $s\in\operatorname{\mathcal{S}}$. By (P3) we deduce that $\theta_{s^{*}s}$
is an extension of $\theta_{s^{*}}\circ\theta_{s}$. By (P1), it follows that
$\theta_{s^{*}s}$ is the identity map on $X_{s^{*}s}$. Hence,
$\theta_{s^{*}}\circ\theta_{s}$ is the identity map on its domain, which is
clearly $X_{s^{*}}$. Similarly, $\theta_{s}\circ\theta_{s^{*}}$ is the
identity map on $X_{s}$. ∎
With this result in hands we can rewrite the domain $\theta_{t}^{-1}(X_{t}\cap
X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ as $\theta_{t^{*}}(X_{t}\cap
X_{s^{*}})$. This enables us to obtain a characterization for the codomain
$\theta_{s}(X_{s^{*}}\cap X_{t})$ of $\theta_{s}\circ\theta_{t}$ in the same
way (P3) provides one for the domain of $\theta_{s}\circ\theta_{t}$.
###### Proposition 2.7.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$, then
$\theta_{s}(X_{s^{*}}\cap X_{t})=X_{s}\cap X_{st}.$
###### Proof.
Let $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. Hence,
$(t^{*},s^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and, by (P3),
$\theta_{s^{*}}^{-1}(X_{s^{*}}\cap X_{t})=X_{st}\cap X_{s}$. The result now
follows from the comment immediately before the statement. ∎
We now compare $\theta_{s}$ and $\theta_{t}$ when $s\leq t$.
###### Proposition 2.8.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If $s\leq t$, then $X_{s}\subseteq X_{t}$
and $\theta_{s}\subseteq\theta_{t}$.
###### Proof.
Let $s,t\in\operatorname{\mathcal{S}}$ such that $s\leq t$. Since $s\leq t$,
we have that $s=ss^{*}t$. Let $x\in X_{s}$. First, we shall prove
$X_{s}\subseteq X_{t}$. By (P2), $x\in X_{ss^{*}}$. Hence, by 2.7 and (P1), we
have
$x\in X_{ss^{*}}\cap X_{s}=X_{ss^{*}}\cap
X_{ss^{*}t}\overset{\ref{ran_composition}}{=}\theta_{ss^{*}}(X_{ss^{*}}\cap
X_{t})\overset{\ref{p1}}{=}X_{ss^{*}}\cap X_{t}\subseteq X_{t}.$
For the extension, we also have $s^{*}\leq t^{*}$ and, analogously to the
previous argument, we have $X_{s^{*}}\subseteq X_{t^{*}}$. Then, for every
$x\in X_{s^{*}}$, by (P1) and (P3), we have that $x$ lies in the domain
$X_{s^{*}}\cap X_{t^{*}}$ of $\theta_{ss^{*}}\circ\theta_{t}$ and
$\theta_{s}(x)\overset{\ref{p3}}{=}\theta_{ss^{*}}\big{(}\theta_{t}(x)\big{)}\overset{\ref{p1}}{=}\theta_{t}(x).$
Hence, $\theta_{t}$ is an extension of $\theta_{s}$ as stated. ∎
If $e,f$ are composable idempotent elements, the next result shows that
$\theta_{e}\circ\theta_{f}$ and $\theta_{ef}$ share domains and, hence,
coincide.
###### Proposition 2.9.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and $(X,\theta)$
an $\operatorname{\mathcal{S}}$-set. If $e,f\in E(\operatorname{\mathcal{S}})$
and $(e,f)\in\operatorname{\mathcal{S}}^{(2)}$, then $X_{ef}=X_{e}\cap X_{f}$
and $\theta_{ef}=\theta_{e}\circ\theta_{f}$.
###### Proof.
Let $e,f\in E(\operatorname{\mathcal{S}})$ such that $(e,f)\in S^{(2)}$.
Since, $ef\leq e$, by 2.8, we have $X_{ef}\subseteq X_{e}$. Hence, by (P1) and
2.7, we have
$X_{ef}=X_{e}\cap X_{ef}\overset{\ref{ran_composition}}{=}\theta_{e}(X_{e}\cap
X_{f})\overset{\ref{p1}}{=}X_{e}\cap X_{f}.$
Since the domains (and codomains) of $\theta_{ef}$ and
$\theta_{e}\circ\theta_{f}$ coincide, we must have
$\theta_{ef}=\theta_{e}\circ\theta_{f}$, as desired. ∎
We now present one alternative for the definition of partial action. The
reader is invited to compare the next proposition with Definition 1.7 [7].
###### Proposition 2.10.
Let $X$ be a set and $\operatorname{\mathcal{S}}$ an inverse semigroupoid. Let
$\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ be a collection of subsets of
$X$ and $\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}$ a collection of
maps $\theta_{s}\colon X_{s^{*}}\to X_{s}$. Then, the pair
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a partial action of $\operatorname{\mathcal{S}}$ on $X$ if, and only if, it
satisfies
1. (E1)
For every $s\in\mathcal{S}$, $\theta_{s}$ is a bijection and
$\theta_{s}^{-1}=\theta_{s^{*}}$. Moreover,
$X=\cup_{s\in\operatorname{\mathcal{S}}}X_{s}$;
2. (E2)
$\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that $(s,t)\in\mathcal{S}^{(2)}$;
3. (E3)
$X_{s}\subseteq X_{t}$ for every $s,t\in\operatorname{\mathcal{S}}$ such that
$s\leq t$.
Furthermore, if $\theta$ is a partial action of $\operatorname{\mathcal{S}}$
on $X$, then it is a _global action_ if, and only if
1. (E4)
$\theta_{st}=\theta_{s}\circ\theta_{t}$ for every
$s,t\in\operatorname{\mathcal{S}}$ such that $(s,t)\in\mathcal{S}^{(2)}$ (i.e.
equality holds in (E2)).
###### Proof.
If $\theta$ is a partial action, the first part of (E1) follows from
Proposition 2.6 and the second part follows immediately from the second part
of (P1). From the comments after Definition 2.4 we deduce (E2), and (E3) is a
direct consequence of Proposition 2.8.
Assume now that $\theta$ satisfies (E1), (E2) and (E3). Our goal is to prove
that $\theta$ is a partial action of $\operatorname{\mathcal{S}}$ on $X$.
By (E1) and (E2) we deduce (P2) since
$\operatorname{id}_{X_{s}}\overset{\ref{e1}}{=}\theta_{s}\circ\theta_{s^{*}}\overset{\ref{e2}}{\subseteq}\theta_{ss^{*}},$
and, hence, the domain $X_{s}$ of $\operatorname{id}_{X_{s}}$ is contained in
the domain $X_{ss^{*}}$ of $\theta_{ss^{*}}$. This also enables us to infer
the second part of (P1) since, by (E1), for any $x\in X$, there exists
$s\in\operatorname{\mathcal{S}}$ such that $x\in X_{s}$. Hence, $x\in
X_{s}\subseteq X_{ss^{*}}$ and $ss^{*}$ is an idempotent.
For the first part of (P1) we use again (E1) and (E2). In one hand we have
$\operatorname{id}_{X_{e}}\overset{\ref{e1}}{=}\theta_{e}\circ\theta_{e}\overset{\ref{e2}}{\subseteq}\theta_{e}$
and, since $\operatorname{id}_{X_{e}}$ and $\theta_{e}$ share domains, we must
have $\theta_{e}=\operatorname{id}_{X_{e}}$.
It remains to argue (P3). For that task, let
$(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. By (E2),
$\theta_{s}\circ\theta_{t}\subseteq\theta_{st}$ and, hence, the domain
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ is
contained in the domain $X_{(st)^{*}}$ of $\theta_{st}$. Combining this with
the fact that the domain of $\theta_{t}$ is $X_{t^{*}}$ we obtain
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})\subseteq X_{(st)^{*}}\cap X_{t^{*}}.$
(1)
To deduce the reverse inclusion, notice that we must also have
$(st,t^{*})\in\operatorname{\mathcal{S}}^{(2)}$ and using (E2) and a similar
argument we obtain
$\theta_{t^{*}}^{-1}(X_{t^{*}}\cap X_{(st)^{*}})\subseteq
X_{(stt^{*})^{*}}\cap X_{t}\overset{\ref{e3}}{\subseteq}X_{s^{*}}\cap X_{t},$
(2)
in which the last inclusion follows from (E3), since
$(stt^{*})^{*}=tt^{*}s^{*}\leq s^{*}$. Combining the fact that $X_{s^{*}}\cap
X_{t}$ is contained in the domain of $\theta_{t^{*}}$ and (E1), we obtain from
(2) that
$X_{t^{*}}\cap X_{(st)^{*}}\subseteq\theta_{t}^{-1}(X_{s^{*}}\cap X_{t}).$ (3)
Joining (1) and (3) we obtain the desired equality. Finally, if $x\in
X_{(st)^{*}}\cap X_{t^{*}}$, then $x$ lies in the domain
$\theta_{t}^{-1}(X_{t}\cap X_{s^{*}})$ of $\theta_{s}\circ\theta_{t}$ and,
hence, by (E2), we have that
$\theta_{s}\big{(}\theta_{t}(x)\big{)}=\theta_{st}(x),$
completing the verification of (P3).
Suppose now that $\theta$ is a global action of $\operatorname{\mathcal{S}}$
on $X$. Hence, by (P4) and Proposition 2.9, we have
$X_{(st)^{*}}\cap X_{t^{*}}\overset{\ref{pglob}}{=}X_{(st)^{*}(st)}\cap
X_{t^{*}t}\overset{\ref{ef_domain}}{=}X_{(st)^{*}(st)t^{*}t}=X_{(st)^{*}(st)}=X_{(st)^{*}},$
which means that the domain of $\theta_{s}\circ\theta_{t}$ coincides with the
domain of $\theta_{st}$ and, hence, concluding (E4).
Conversely, if $\theta$ is partial action that satisfies (E4), then for any
$s\in\operatorname{\mathcal{S}}$ we would have
$\theta_{ss^{*}}=\theta_{s}\circ\theta_{s^{*}}=\operatorname{id}_{X_{s}}$ and,
consequently, $X_{ss^{*}}=X_{s}$ concluding (P4). ∎
We end this section with two examples of partial actions that we shall explore
again in the next section.
###### Example 2.11.
Let $X=\\{1,2,3,4\\}$ and let
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$
be the inverse semigroupoid presented in Example 2.3. Consider the following
family of subsets of $X$:
$X_{b^{*}}=\\{1,2\\}$,
$X_{b^{*}b}=\\{1,2\\}$,
$X_{b}=\\{1,4\\}$,
$X_{bb^{*}}=\\{1,4\\}$,
$X_{a^{*}}=\\{1\\}$
$X_{a^{*}a}=\\{1\\}$,
$X_{a}=\\{3\\}$,
$X_{aa^{*}}=\\{3,4\\}$,
and the following family of bijections:
$\begin{array}[]{cccc}\theta_{b}\colon&X_{b^{*}}&\rightarrow&X_{b}\\\
&1&\mapsto&1\\\
&2&\mapsto&4\end{array},\qquad\begin{array}[]{cccc}\theta_{a}\colon&X_{a^{*}}&\rightarrow&X_{a}\\\
&1&\mapsto&4\\\ &&&\end{array},$
$\theta_{a^{*}}=\theta_{a}^{-1}$, $\theta_{b^{*}}=\theta_{b}^{-1}$ and
$\theta_{e}=\operatorname{id}_{X_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. It is
routine to check that
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is, indeed, a partial action of $\operatorname{\mathcal{S}}$ on $X$.
###### Example 2.12.
Let $Y=\\{1,2,3\\}$ and let
$\operatorname{\mathcal{S}}=\\{a,a^{*},b,b^{*},a^{*}a,aa^{*},b^{*}b,bb^{*}\\}$
be the inverse semigroupoid presented in Example 2.3. Consider the following
family of bijections:
$\begin{array}[]{cccc}\theta^{Y}_{a}\colon&Y&\rightarrow&Y\\\ &1&\mapsto&2\\\
&2&\mapsto&3\\\ &3&\mapsto&1\end{array},$
$\theta^{Y}_{a^{*}}={\theta^{Y}_{a}}^{-1}$ and
$\theta^{Y}_{s}=\operatorname{id}_{Y}$ if $s\neq a,a^{*}$. Setting $Y_{s}=Y$
for every $s\in\operatorname{\mathcal{S}}$, it is routine to check that
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta^{Y}_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a global action of $\operatorname{\mathcal{S}}$ on $Y$.
## 3 Restriction and Globalization of partial actions
It is well known that we can produce interesting examples of partial actions
via a process of restriction of global actions. Let us explain this process in
the present context: suppose
$\theta=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
is a partial action of an inverse semigroupoid $\operatorname{\mathcal{S}}$ on
a set $Y$. If $X$ is a subset of $Y$, we may try to restrict the action of
$\operatorname{\mathcal{S}}$ to $X$ by restricting the domain of each
$\theta_{s}$ to $X\cap Y_{s^{*}}$. The main problem in this process is that,
even though $x\in X\cap Y_{s^{*}}$, we may have $\theta_{s}(x)\notin X$.
However, we can turn around this problem by restricting each $\theta_{s}$ to
the set formed by all $x\in X\cap Y_{s^{*}}$ such that $\theta_{s}(x)$ also
lies in $X$. Explicitly, setting $X_{s}=\theta_{s}(X\cap Y_{s^{*}})\cap X$ and
noticing that $\theta_{s}(X_{s^{*}})=X_{s}$, we may let
${(\left.\theta\right|_{X})}_{s}\colon X_{s^{*}}\to X_{s}$
to be the restriction of $\theta_{s}$ to $X_{s^{*}}$ onto $X_{s}$. It is then
easy to verify that
$\left.\theta\right|_{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{{(\left.\theta\right|_{X})}_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
satisfies the axioms of Definition 2.4 and, hence, is a partial action of
$\operatorname{\mathcal{S}}$ on $X$ which, from now on, we shall call the
_restriction_ of the action $\theta$ to $X$.
Let us point out that, even if the original action $\theta$ of
$\operatorname{\mathcal{S}}$ on $Y$ is global, there is no guarantee that the
restriction $\left.\theta\right|_{X}$ is a global action. Indeed, for any
$x\in X\cap Y_{s^{*}}$ such that $\theta_{s}(x)$ does not lie in $X$, we would
have $x\in X_{s^{*}s}$, but $x\notin X_{s^{*}}$.
The observation in the last paragraph raises an interesting problem: if we
start with a partial action $\theta$ of an inverse semigroupoid
$\operatorname{\mathcal{S}}$ on a set $X$, does there exist a set $Y$
containing $X$ and a global action $\eta$ of $\operatorname{\mathcal{S}}$ on
$Y$ such that $\theta$ may be obtained by the restriction of the action of
$\eta$ to $X$? The main goal of this work is to give an affirmative answer for
this question.
Before we proceed, let us introduce the appropriate morphisms to form the
category of partial actions.
###### Definition 3.1.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and let
$\theta^{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{X}\\}_{s\in\operatorname{\mathcal{S}}})$
and
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{Y}\\}_{s\in\operatorname{\mathcal{S}}})$
be partial actions of $\operatorname{\mathcal{S}}$ on sets $X$ and $Y$,
respectively. A function $\varphi\colon X\rightarrow Y$ is said to be an
$\operatorname{\mathcal{S}}$_-function_ if $\varphi(X_{s})\subseteq Y_{s}$ for
every $s\in\operatorname{\mathcal{S}}$ and, moreover,
$\varphi\big{(}\theta_{s}^{X}(x)\big{)}=\theta_{s}^{Y}\big{(}\varphi(x)\big{)}$
for every $x\in X_{s^{*}}$.
Given an inverse semigroupoid $\operatorname{\mathcal{S}}$, the class of
$\operatorname{\mathcal{S}}$-sets together with the class of
$\operatorname{\mathcal{S}}$-functions form a category, which we denote by
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$. Likewise, the class of global
$\operatorname{\mathcal{S}}$-sets together with the class of
$\operatorname{\mathcal{S}}$-functions between then form a full subcategory of
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$, which we denote by
$\mathscr{A}(\operatorname{\mathcal{S}})$.
For example, if $(Y,\theta)$ is an $\operatorname{\mathcal{S}}$-set and
$X\subseteq Y$, it is easy to verify that the inclusion map $i\colon X\to Y$
is an $\operatorname{\mathcal{S}}$-function between the
$\operatorname{\mathcal{S}}$-sets $(X,\left.\theta\right|_{X})$ and
$(Y,\theta)$. Actually, $i$ is even more than an injective
$\operatorname{\mathcal{S}}$-function, it is an embedding in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$ as follows.
###### Definition 3.2.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid and let
$\theta^{X}=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{X}\\}_{s\in\operatorname{\mathcal{S}}})$
and
$\theta^{Y}=(\\{Y_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}^{Y}\\}_{s\in\operatorname{\mathcal{S}}})$
be partial actions of $\operatorname{\mathcal{S}}$ on sets $X$ and $Y$,
respectively. An injective $\operatorname{\mathcal{S}}$-function
$\varphi\colon X\rightarrow Y$ is said to be an _embedding_ if
$X_{s}=\varphi^{-1}\big{(}\theta_{s}^{Y}(\varphi(X)\cap Y_{s^{*}})\big{)}$
for every $s\in\operatorname{\mathcal{S}}$.
Notice that, if $\varphi\colon X\rightarrow Y$ is an embedding as in the
Definition above, then the action $\theta^{Y}$ of $\operatorname{\mathcal{S}}$
on $Y$ induces the action $\theta^{X}$ of $\operatorname{\mathcal{S}}$ on $X$
in the sense that, for given $x_{1},x_{2}\in X$ and
$s\in\operatorname{\mathcal{S}}$ such that $\varphi(x_{1})\in Y_{s^{*}}$ and
$\theta_{s}(\varphi(x_{1}))=\varphi(x_{2})$, then $x_{1}\in X_{s^{*}}$ and
$\theta_{s}(x_{1})=x_{2}$.
Actually, in order to an injective function $\varphi$ as in Definition 3.2 be
an embedding is necessary, and sufficient that, for given $x\in X$ and
$s\in\operatorname{\mathcal{S}}$, we have that $x\in X_{s^{*}}$ if, and only
if, $\varphi(x)\in Y_{s^{*}}$ and $\theta^{Y}_{s}(\varphi(x))\in\varphi(X)$
and, in the affirmative case,
$\theta^{Y}_{s}(\varphi(x))=\varphi(\theta_{s}^{X}(x))$. This amounts to say
that $\varphi$ co-restricts to an isomorphism $\varphi\colon X\to\varphi(X)$
between $(X,\theta^{X})$ and
$(\varphi(X),\left.\theta^{Y}\right|_{\varphi(X)})$ in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$.
This discussion gives the appropriate definition for globalizations of partial
actions.
###### Definition 3.3.
Let $(X,\theta^{X})$ and $(Y,\theta^{Y})$ be $\operatorname{\mathcal{S}}$-sets
and $\varphi\colon X\to Y$ a $\operatorname{\mathcal{S}}$-function. The triple
$(\varphi,Y,\theta^{Y})$ is said to be
1. (i)
a _globalization_ of $(X,\theta^{X})$ if $\varphi$ is an embedding and
$(Y,\theta^{Y})$ is a global $\operatorname{\mathcal{S}}$-set.
2. (ii)
an _universal globalization_ of $(X,\theta^{X})$ if it is a globalization of
$(X,\theta^{X})$ and, for any globalization $(\psi,Z,\theta^{Z})$ of
$(X,\theta^{X})$, there exists a unique _mediating_
$\operatorname{\mathcal{S}}$-function $\sigma\colon Y\to Z$ such that the
diagram
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Z}$
commutes.
In the context of Definition 3.3 above, if the
$\operatorname{\mathcal{S}}$-function $\varphi$ is implicit in the context, we
may say simply that $(Y,\theta^{Y})$ is a globalization of $(X,\theta^{X})$.
We shall prove that every partial action admits a universal globalization.
Moreover, we shall prove that our construction gives a reflector from
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$ on
$\mathscr{A}(\operatorname{\mathcal{S}})$ (Theorem 3.13), whose definition we
recall below.
###### Definition 3.4.
A full subcategory $\mathcal{D}$ of a category $\mathcal{C}$ is said to be
_reflective_ if, for any $\mathcal{C}$-object $C$, there exists a
$\mathcal{D}$-object $R(C)$ and a $\mathcal{C}$-morphism $\varphi_{C}\colon
C\to R(C)$ such that, for every $\mathcal{D}$-object $D$ and
$\mathcal{C}$-morphism $\psi\colon C\to D$, there exists a unique mediating
$\mathcal{D}$-morphism $\sigma\colon R(C)\to D$ such that the diagram
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}}$$\scriptstyle{\psi}$$\textstyle{R(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{D}$
commutes. Moreover, in this case, the pair $(\varphi_{C},R(C))$ is said to be
a $\mathcal{D}$-_reflector_ of $C$.
In the context of Definition 3.4 above, if the morphism $\varphi_{C}$ is
implicit in the context, we may say simply that $R(C)$ is a
$\mathcal{D}$-reflector of $C$.
If $\mathcal{D}$ is a reflective subcategory of a category $\mathcal{C}$, then
the map $C\mapsto R(C)$ induces a _reflector_ functor
$R\colon\mathcal{C}\to\mathcal{D}$ which is right adjoint to the inclusion
functor $\mathcal{D}\to\mathcal{C}$.
From now on we fix an inverse semigroupoid $\operatorname{\mathcal{S}}$, a set
$X$ and a partial action
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}}).$
Our goal now is to construct a global $\operatorname{\mathcal{S}}$-set
$(E,\eta)$ and an $\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such
that $(i,E,\eta)$ is a globalization of $(X,\theta)$ which is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$. Notice
that, a globalization of $(X,\theta)$ which is also a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$ is,
automatically, a universal globalization of $(X,\theta)$.
For reference, our construction is mainly inspired in [23]. We begin defining
a relation on a subset of $\operatorname{\mathcal{S}}\times X$ as follows.
###### Definition 3.5.
Let $D:=\\{(s,x)\in\operatorname{\mathcal{S}}\times X:x\in X_{s^{*}s}\\}$.
Define a relation $\sim$ on $D$ such that, for all $(s,x),(t,y)\in D$,
$(s,x)\sim(t,y)$ if either
1. (R1)
$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$;
2. (R2)
or $s,t\in E(\operatorname{\mathcal{S}})$ and $x=y$.
Now, we point some observations about the relation $\sim$ defined above.
###### Remark 3.6.
First, in the case in which item (R2) is checked, we have $x\in X_{s}$ and
$y\in X_{t}$ and $\theta_{s}(x)=x=y=\theta_{t}(y)$, by (P1).
Still in the case in which item (R2) is checked, if $s$ and $t$ are
composable, then by Proposition 2.9, $X_{st}=X_{s}\cap X_{t}$ and, hence,
$(t,s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{st}$ and
$\theta_{st}(x)=x$ by (P1). This means that $(s,x)\sim(t,y)$ is also checked
by item (R1). In other words, item (R2) in Definition 3.5 plays a relevant
role only in the case $s$ and $t$ are idempotent elements over different
objects.
The set $E$ that we are looking for is going to be the quotient of $D$ by an
equivalence relation which identifies pairs satisfying (R1) or (R2).
Unfortunately, the relation $\sim$ in Definition 3.5 may fail to be an
equivalence relation. Indeed, in Example 2.11, we may notice that
$(a,1)\sim(aa^{*},4)$, $(aa^{*},4)\sim(bb^{*},4)$ but $(a,1)\nsim(bb^{*},4)$.
At least, $\sim$ is reflexive and symmetric.
###### Proposition 3.7.
The relation $\sim$ defined in 3.5 is reflexive and symmetric.
###### Proof.
Let $(s,x)\in D$. Since $x\in X_{s^{*}s}$ and $s^{*}s\in
E(\operatorname{\mathcal{S}})$, we have $\theta_{s^{*}s}(x)=x$ by (P1). This
means that $(s,x)\sim(s,x)$ by item (R1) of Definition 3.5. Hence, $\sim$ is
indeed reflexive.
It remains to show that $\sim$ is symmetric. Let $(s,x),(t,y)\in D$ and
suppose that $(s,x)\sim(t,y)$. If $x=y$ and $s,t\in
E(\operatorname{\mathcal{S}})$ it is clear that $(t,y)\sim(s,x)$. Otherwise,
assume that $(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$
and $\theta_{t^{*}s}(x)=y$. Then, we must have
$(s^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{t^{*}s}$ and
$\theta_{s^{*}t}(y)=x$, which means that $(t,y)\sim(s,x)$ by (R2). ∎
Since $\sim$ may not be an equivalence relation, We turn around this problem
considering the equivalence relation generated by $\sim$, as follows.
###### Definition 3.8.
Let $\approx$ be the smallest equivalence relation on $D$ containing the
relation $\sim$ defined in 3.5. Explicitly, if $(s,x),(t,y)\in D$, then
$(s,x)\approx(t,y)$ if, and only if, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$. We denote by $E$ the quotient of $D$ by the
equivalence relation $\approx$. The equivalence class of an element $(s,x)\in
D$ is denoted by $[s,x]$.
The next step is to define an action of $\operatorname{\mathcal{S}}$ on $E$.
The idea here is that, if an element $p\in\operatorname{\mathcal{S}}$ acts in
the class of an element $(s,x)\in D$ such that
$(p,s)\in\operatorname{\mathcal{S}}^{(2)}$, we should obtain the class of the
element $(ps,x)$. However, in order to $(ps,x)$ be an element of $D$ we must
have $x\in X_{(ps)^{*}(ps)}$. So, we now prove some results with the aiming to
obtain a well defined action of $\operatorname{\mathcal{S}}$ on $E$ in this
fashion.
###### Lemma 3.9.
Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\sim(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Then $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$. Moreover, when the memberships are
verified, we have $(ps,x)\sim(pt,y)$.
###### Proof.
Notice, at first, that the membership $x\in X_{(ps)^{*}(ps)}=X_{s^{*}p^{*}ps}$
is necessary (and sufficient) to guarantee that $(ps,x)$ lies in $D$.
Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\sim(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Notice that, if $s,t\in
E(\operatorname{\mathcal{S}})$, since
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$, we must have
$(s,t),(t,s)\in\operatorname{\mathcal{S}}^{(2)}$. Hence, by the second
paragraph of Remark 2.5, we may assume that $(s,x)\sim(t,y)$ is checked by
item (R1) of Definition 3.5. Then, assume that
$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$. By symmetry of $\sim$, we have
$(s^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{t^{*}s}$ and
$\theta_{s^{*}t}(y)=x$.
Assume now that $x\in X_{s^{*}p^{*}ps}$ and notice that $y$ must lie in the
domain
$\theta_{s^{*}t}^{-1}\big{(}X_{s^{*}t}\cap\theta_{s^{*}p^{*}ps}^{-1}(X_{s^{*}p^{*}ps}\cap
X_{s^{*}t})\big{)}=\theta_{s^{*}t}^{-1}(X_{s^{*}t}\cap X_{s^{*}p^{*}ps})$
of $\theta_{t^{*}s}\circ\theta_{s^{*}p^{*}ps}\circ\theta_{s^{*}t}$ since $x$
lies in both $X_{s^{*}t}$ and $X_{s^{*}p^{*}ps}$ and $\theta_{s^{*}t}(y)=x$.
Now, observing that
$(t^{*}s)(s^{*}p^{*}ps)(s^{*}t)=t^{*}ss^{*}p^{*}pss^{*}t=t^{*}ss^{*}p^{*}pt\leq
t^{*}p^{*}pt$
and using (E2) and (E3), we have that $y\in X_{t^{*}ss^{*}p^{*}pt}\subseteq
X_{t^{*}p^{*}pt}$ as desired.
Conversely, if $y\in X_{t^{*}p^{*}pt}$, just notice that
$(s^{*}t)(t^{*}p^{*}pt)(t^{*}s)=s^{*}tt^{*}p^{*}ptt^{*}s=s^{*}tt^{*}p^{*}ps\leq
s^{*}p^{*}ps$
and use a similar argument to deduce that $x\in X_{s^{*}p^{*}ps}$.
Finally, we assume that $x\in X_{s^{*}p^{*}ps}$ and $y\in X_{t^{*}p^{*}pt}$
and prove that $(ps,x)\sim(pt,y)$. For that purpose, notice that
$(t^{*}p^{*},ps)\in\operatorname{\mathcal{S}}^{(2)}$ and that
$(pt)^{*}(ps)=t^{*}p^{*}ps=t^{*}p^{*}pss^{*}s=t^{*}ss^{*}p^{*}ps=(t^{*}s)(s^{*}p^{*}ps),$
from which we can deduce by (P3) that
$x\in X_{s^{*}t}\cap
X_{s^{*}p^{*}ps}=\theta_{s^{*}p^{*}ps}^{-1}(X_{s^{*}t}\cap
X_{s^{*}p*ps})\overset{\ref{p3}}{=}X_{(t^{*}p^{*}ps)^{*}}\cap
X_{s^{*}p^{*}ps}\subseteq X_{(ps)^{*}(pt)},$
and, moreover, that
$\theta_{(pt)^{*}(ps)}(x)=\theta_{t^{*}s}\big{(}\theta_{s^{*}p^{*}ps}(x)\big{)}=\theta_{t^{*}s}(x)=y.$
Therefore, by item (R1) of Definition 3.5, $(ps,x)\sim(pt,y)$ as desired. ∎
We may also slightly modify Lemma 3.9 by changing the relation $\sim$ by the
equivalence relation $\approx$ generated by $\sim$. But first, we will need an
auxiliary result.
###### Lemma 3.10.
Let $(s,x)$, $(t,y)\in D$ such that $(s,x)\approx(t,y)$. Then $x\in X_{s^{*}}$
if, and only if, $y\in X_{t^{*}}$. Moreover, when the memberships are
verified, we have $\theta_{s}(x)=\theta_{t}(y)$.
###### Proof.
Let $(s,x)$, $(t,y)\in D$ and assume, initially, that $(s,x)\sim(t,y)$. If
$(s,x)\sim(t,y)$ by (R2) of Definition 3.5, then the result follows direct
from the comment in the first paragraph after Definition 3.5. Otherwise,
$(s,x)\sim(t,y)$ by (R1) of Definition 3.5 and, then, we must have
$(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$.
If $x\in X_{s^{*}}$, then $x\in X_{s^{*}}\cap X_{s^{*}t}$ which is the domain
of $\theta_{t^{*}}\circ\theta_{s}$, by (P3). Hence, again by (P3) we have that
$y=\theta_{t^{*}s}(x)=\theta_{t^{*}}\big{(}\theta_{s}(x)\big{)}$
which lies in $X_{t^{*}}$. Moreover, applying $\theta_{t}$ in the above
equality, we obtain that $\theta_{t}(y)=\theta_{s}(x)$.
If $y\in X_{t^{*}}$ we may use the symmetry of $\sim$ and a similar argument
to argue that $x\in X_{s^{*}}$ as well.
For the general case, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$. Notice that, by the previous argument, $z_{i}\in
X_{r_{i}^{*}}$ if, and only if, $z_{i+1}\in X_{r_{i+1}^{*}}$ for all
$i\in\\{1\ldots,n-1\\}$ and, hence, $x\in X_{s^{*}}$ if, and only if, $y\in
X_{t^{*}}$. Moreover, assuming that $x\in X_{s^{*}}$ and $y\in X_{t^{*}}$,
then $z_{i}\in X_{r_{i}^{*}}$ for all $i\in\\{1\ldots,n\\}$ and, again by the
previous argument, $\theta_{r_{i}}(z_{i})=\theta_{r_{i+1}}(z_{i+1})$ for all
$i\in\\{1\ldots,n-1\\}$, which allow us to deduce that
$\theta_{s}(x)=\theta_{t}(y)$ as stated. ∎
###### Lemma 3.11.
Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\approx(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$. Then $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$. Moreover, when the memberships are
verified, we have $(ps,x)\approx(pt,y)$.
###### Proof.
Let $(s,x)$, $(t,y)\in D$ be such that $(s,x)\approx(t,y)$ and let
$p\in\operatorname{\mathcal{S}}$ be such that
$(p,s),(p,t)\in\operatorname{\mathcal{S}}^{(2)}$.
Assume initially that $s\in E(\operatorname{\mathcal{S}})$. Then, since
$(s,x)\in D$, we have $x\in X_{s^{*}}$. By Lemma 3.10 and (P1), we have that
$y\in X_{t^{*}}$ and $\theta_{t}(y)=\theta_{s}(x)=x$. Since
$(tt^{*},t)\in\operatorname{\mathcal{S}}^{(2)}$, we deduce from the previous
sentence that
$(t,y)\sim(tt^{*},x)$ (4)
by item (R1) of Definition 3.5. Notice that, indeed, $(tt^{*},x)\in D$, since
$x=\theta_{t}(y)\in X_{t}$ and $X_{t}\subseteq X_{tt^{*}}$ by (P2).
Since $tt^{*}$ and $s$ are idempotent elements, we have that
$(tt^{*},x)\sim(s,x)$ (5)
by item (R2) of Definition 3.5.
Combining (4) and (5) with Lemma 3.9, we deduce that $x\in X_{s^{*}p^{*}ps}$
if, and only if, $y\in X_{t^{*}p^{*}pt}$ and, in the affirmative case, that
$(ps,x)\sim(ptt^{*},x)\sim(pt,y)$.
We can use a similar argument if $t\in E(\operatorname{\mathcal{S}})$. Hence,
we now assume that neither $s\in E(\operatorname{\mathcal{S}})$, nor $t\in
E(\operatorname{\mathcal{S}})$. Then, there exist $n\in\mathbb{N}$ and
$(r_{1},z_{1}),\ldots,(r_{n},z_{n})\in D$ such that $(r_{1},z_{1})=(s,x)$,
$(r_{n},z_{n})=(t,y)$ and $(r_{i},z_{i})\sim(r_{i+1},z_{i+1})$ for all
$i\in\\{1,\ldots,n-1\\}$.
Assume $x\in X_{s^{*}p^{*}ps}$. We proceed by induction over $n$. If $n=2$,
Lemma 3.9 applies directly. If $n>2$, assume that our claim holds for $m<n$.
Since $s\notin E(\operatorname{\mathcal{S}})$, we must have
$(s,x)\sim(r_{2},z_{2})$ by item (R1) of Definition 3.5 and, hence,
$(r_{2}^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$. This implies that
$(p,r_{2})\in\operatorname{\mathcal{S}}^{(2)}$ and, then, Lemma 3.9 applies.
Hence $z_{2}\in X_{r_{2}^{*}p^{*}pr_{2}}$ and $(ps,x)\sim(pr_{2},z_{2})$. If
$r_{2}\in E(\operatorname{\mathcal{S}})$, then $(r_{2},z_{2})\approx(t,y)$ and
we can use the initial argument to deduce that $y\in X_{t^{*}p^{*}pt}$ and
$(ps,x)\sim(pr_{2},z_{2})\approx(pt,y)$. If $r_{2}\notin
E(\operatorname{\mathcal{S}})$, then we can use the induction hypothesis to
infer that $y\in X_{t^{*}p^{*}pt}$ and
$(ps,x)\sim(pr_{2},z_{2})\approx(pt,y)$. In any such case, we obtain the
desired result.
Assuming that $y\in X_{t^{*}p^{*}pt}$, we can use a similar argument (this
time from right to left) to establish that $x\in X_{s^{*}p^{*}ps}$ and
$(ps,x)\approx(pt,y)$ as desired. ∎
We now begin to construct an action of $\operatorname{\mathcal{S}}$ on $E$ by
first defining a collection of functions between subsets of $D$. For this,
given $s\in\operatorname{\mathcal{S}}$, we set
$\displaystyle\zeta_{s}\colon\quad D_{s^{*}}$ $\displaystyle\longrightarrow
D_{s}$ (6) $\displaystyle(p,x)$ $\displaystyle\longmapsto(sp,x),$
in which
$\displaystyle D_{s^{*}}=\\{(p,x)\in
D:(s,p)\in\operatorname{\mathcal{S}}^{(2)}\text{ and }x\in
X_{p^{*}s^{*}sp}\\}.$ (7)
Notice that, $\zeta_{s}$ is well defined since
$(s^{*},sp)\in\operatorname{\mathcal{S}}^{(2)}$ and
$X_{(s^{*}sp)^{*}(s^{*}sp)}=X_{p^{*}s^{*}sp}$, which implies that $(sp,x)$
indeed lies in $D_{s}$.
Moreover, if $(p,x),(q,y)\in D_{s^{*}}$ are such that $(p,x)\approx(q,y)$,
then, by Lemma 3.11, $\zeta_{s}(p,x)=(sp,x)\approx(sq,y)=\zeta_{s}(q,y)$.
Thus, setting $E_{s}$ as the image of $D_{s}$ by the quotient map of $D$ by
the equivalence relation $\approx$, we obtain a well defined function
$\eta_{s}\colon E_{s^{*}}\to E_{s}$ such that
$\eta_{s}\big{(}[p,x])=[sp,x]$
for any $(p,x)\in D_{s^{*}}$. We, thus, obtain a pair
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
(8)
which we now prove that is a global action of $\operatorname{\mathcal{S}}$ on
$E$. Before we proceed, just to emphasize, a point in $E$ lies in $E_{s}$ if,
and only if, it admits a representative in $D_{s}$.
###### Theorem 3.12.
The pair
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$,
as in (8), is a global action of $\operatorname{\mathcal{S}}$ on $E$.
###### Proof.
We shall argue (E1), (E2), (E3) and (E4) to use Proposition 2.10.
Let $s\in\operatorname{\mathcal{S}}$ and $[p,x]\in E_{s^{*}}$. Since $[p,x]\in
E_{s^{*}}$, there exist $(q,y)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$. We
have already argued that $\zeta_{s}(q,y)=(sq,y)\in D_{s}$. Moreover, since
$(q^{*},s^{*}sq)\in\operatorname{\mathcal{S}}^{(2)}$, $y\in X_{q^{*}s^{*}sq}$
and $\theta_{q^{*}s^{*}sq}(y)=y$, we have that $(s^{*}sq,y)\sim(q,y)$ by item
(R1) of Definition 3.5. Therefore,
$\eta_{s^{*}}\big{(}\eta_{s}([p,x])\big{)}=\eta_{s^{*}}\big{(}\eta_{s}([q,y])\big{)}=\eta_{s^{*}}([sq,y])=[s^{*}sq,y]=[q,y]=[p,x],$
from where we infer that
$\eta_{s^{*}}\circ\eta_{s}=\operatorname{id}_{E_{s^{*}}}$. Similarly, we
obtain that $\eta_{s}\circ\eta_{s^{*}}=\operatorname{id}_{E_{s}}$ and, thus,
$\eta_{s}^{-1}=\eta_{s^{*}}$.
Moreover, let $[p,x]\in E$. This means that $x\in X_{p^{*}p}$ and, since,
$(p^{*},p)\in\operatorname{\mathcal{S}}^{(2)}$, we have that $[p,x]\in E_{p}$.
This completes the proof that $E=\cup_{s\in\operatorname{\mathcal{S}}}E_{s}$
and item (E1) is verified.
Next, we prove that $\eta_{s}\circ\eta_{t}=\eta_{st}$ to verify (E2) and (E4)
simultaneously. Let $(s,t)\in\operatorname{\mathcal{S}}^{(2)}$. In one hand,
let $[p,x]\in E_{(st)^{*}}$. Hence, there exists $(q,y)\in D_{(st)^{*}}$ such
that $(p,x)\approx(q,y)$. Since $(st,q)\in\operatorname{\mathcal{S}}^{(2)}$
and $y\in X_{q^{*}t^{*}s^{*}stq}$, we have that
$(s,tq)\in\operatorname{\mathcal{S}}^{(2)}$ and, thus, $(tq,y)\in D_{s^{*}}$.
Moreover, $(t,q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}t^{*}tq}$, since $X_{q^{*}t^{*}s^{*}stq}\subseteq X_{q^{*}t^{*}tq}$ by
Proposition 2.8, from where we deduce that $(q,y)\in D_{t^{*}}$. Therefore,
$[q,y]\in E_{t^{*}}$ and $\eta_{t}([q,y])=[tq,y]\in E_{s^{*}}$ which allow us
to infer that
$[p,x]=[q,y]\in\eta_{t}^{-1}(E_{t}\cap E_{s^{*}})$
and
$\eta_{st}([p,x])=\eta_{st}([q,y])=[stq,y]=\eta_{s}([tq,y])=\eta_{s}\big{(}\eta_{t}([q,y])\big{)}=\eta_{s}\big{(}\eta_{t}([p,x])\big{)}.$
On the other hand, let $[p,x]\in\eta_{t}^{-1}(E_{t}\cap E_{s^{*}})$. Thus,
$[p,x]\in E_{t^{*}}$ and $\eta_{t}([p,x])\in E_{s^{*}}$, from where we obtain
$(q,y)\in D_{t^{*}}$ and $(r,z)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$
and $\zeta_{t}(q,y)\approx(r,z)$. Since $(r,z)\in D_{s^{*}}$ and
$(s,tq)\in\operatorname{\mathcal{S}}^{(2)}$, by Lemma 3.11, we have
$(tq,y)=\zeta_{t}(q,y)\in D_{s^{*}}$. This means that $y\in
X_{q^{*}t^{*}s^{*}stq}$ and, thus, $(q,y)\in D_{(st)^{*}}$. Hence,
$[p,x]=[q,y]\in E_{(st)^{*}}$ and
$\eta_{s}\big{(}\eta_{t}([p,x])\big{)}=\eta_{s}\big{(}\eta_{t}([q,y])\big{)}=\eta_{s}([tq,y])=[stq,y]=\eta_{st}([q,y])=\eta_{st}([p,x]).$
This completes the proof that $\eta_{s}\circ\eta_{t}=\eta_{st}$ and, hence,
(E2) and (E4) are verified.
It remains to verify (E3). For that purpose, let
$s,t\in\operatorname{\mathcal{S}}$ such that $s\leq t$ and let $[p,x]\in
E_{s}$. Thus, there exist $(q,y)\in D_{s}$ such that $(p,x)\approx(q,y)$.
Since, $(q,y)\in D_{s}$, we have that
$(s^{*},q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in X_{q^{*}ss^{*}q}$.
Since $s\leq t$, we have that $s^{*}\leq t^{*}$ and, hence,
$(t^{*},q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}ss^{*}q}\subseteq X_{q^{*}tt^{*}q}$, by Proposition 2.8. This means
that $(q,y)\in D_{t}$ and, hence, $[p,x]=[q,y]\in E_{t}$. This verifies (E3)
and closes this proof. ∎
We now prove that $(E,\eta)$ is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$.
###### Theorem 3.13.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid. The subcategory
$\mathscr{A}(\operatorname{\mathcal{S}})$ is reflective in
$\mathscr{A}_{p}(\operatorname{\mathcal{S}})$. More precisely, for any
$\operatorname{\mathcal{S}}$-set $(X,\theta)$, there exist a global
$\operatorname{\mathcal{S}}$-set $(E,\eta)$ and an
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such that $(i,E,\eta)$
is a $\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$.
###### Proof.
By (P1), for all $x\in X$, there exists $e\in E(\operatorname{\mathcal{S}})$
such that $x\in X_{e}$. So, define $i\colon X\to E$ by $x\mapsto[e,x]$. Notice
that $i$ is well defined. In fact, if $e,f\in E(\operatorname{\mathcal{S}})$
are such that $x\in X_{e}$ and $x\in X_{f}$, by item (R2) of Definition 3.5 we
have $(e,x)\sim(f,x)$ and so $[e,x]=[f,x]$.
Moreover, $i$ is $\operatorname{\mathcal{S}}$-function. In fact, suppose that
$x\in X_{s^{*}}$. Then, by (P2), $x\in X_{s^{*}s}$ and, thus,
$i(x)=[s^{*}s,x]$. Since $(s,s^{*}s)\in\operatorname{\mathcal{S}}^{(2)}$ and
$x\in X_{s^{*}s}$, we have $(s^{*}s,x)\in D_{s^{*}}$ and
$\eta_{s}([s^{*}s,x])=[s,x]$. On the other hand, since $\theta_{s}(x)\in
X_{s}\subseteq X_{ss^{*}}$, then $(ss^{*},\theta_{s}(x))\in D$. That said,
notice that $(s,x)\sim(ss^{*},\theta_{s}(x))$ and so
$[s,x]=[ss^{*},\theta_{s}(x)]$. Finally,
$\eta_{s}\big{(}i(x)\big{)}=\eta_{s}([s^{*}s,x])=[s,x]=[ss^{*},\theta_{s}(x)]=i(\theta_{s}(x)).$
Next we prove that $(i,E,\eta)$ is a
$\mathscr{A}(\operatorname{\mathcal{S}})$-reflector of $(X,\theta)$. Indeed,
we already know by Theorem 3.12 that $(E,\eta)$ is a global
$\operatorname{\mathcal{S}}$-set. So, let $(Z,\omega)$ be a global
$\operatorname{\mathcal{S}}$-set and $j\colon X\to Z$ be an
$\operatorname{\mathcal{S}}$-function and set
$\displaystyle\sigma_{D}\colon\qquad D$ $\displaystyle\longrightarrow Z$
$\displaystyle(s,x)$ $\displaystyle\longmapsto\omega_{s}(j(x)).$
Notice that $\sigma_{D}$ is a well defined function. Indeed, given $(s,x)\in
D$, we have $x\in X_{s^{*}s}$. Since $j$ is an
$\operatorname{\mathcal{S}}$-function, we have $j(x)\in j(X_{s^{*}s})\subseteq
Z_{s^{*}s}$. Since $\omega$ is a global action of $\operatorname{\mathcal{S}}$
on $Z$, we have that $j(x)\in Z_{s^{*}}$ by (P4).
Furthermore, $\sigma_{D}$ factors trough $E$. As a matter of fact, it is
enough to prove that $\sigma_{D}(s,x)=\sigma_{D}(t,y)$ for $(s,x),(t,y)\in D$
such that $(s,x)\sim(t,y)$. If $(s,x)\sim(t,y)$ by (R1) of Definition 3.5,
then $(t^{*},s)\in\operatorname{\mathcal{S}}^{(2)}$, $x\in X_{s^{*}t}$ and
$\theta_{t^{*}s}(x)=y$. Since $x\in X_{s^{*}t}$ and $j$ is an
$\operatorname{\mathcal{S}}$-function, we have that $j(x)\in
j(X_{s^{*}t})\subseteq Z_{s^{*}t}$ and
$\omega_{t^{*}s}(j(x))=j(\theta_{t^{*}s}(x))=j(y)$. Since $\omega$ is a global
action of $\operatorname{\mathcal{S}}$ on $Z$, by (E4), we have
$\omega_{t^{*}s}=\omega_{t^{*}}\circ\omega_{s}$ and, hence,
$\omega_{s}(j(x))=\omega_{t}(j(y))$. We, thus, have
$\sigma_{D}(s,x)=\omega_{s}(j(x))=\omega_{t}(j(y))=\sigma_{D}(t,y).$
If $(s,x)\sim(t,y)$ by (R2) of Definition 3.5, then $s,t\in
E(\operatorname{\mathcal{S}})$ and $x=y$. By (P1), we have
$\sigma_{D}(s,x)=\omega_{s}(j(x))=j(x)=j(y)=\omega_{t}(j(y))=\sigma_{D}(t,y).$
Therefore, we obtain a well-defined function
$\displaystyle\sigma\colon\qquad E$ $\displaystyle\longrightarrow Z$
$\displaystyle[s,x]$ $\displaystyle\longmapsto\omega_{s}(j(x)).$
We claim that $\sigma$ is an $\operatorname{\mathcal{S}}$-function. In fact,
let $s\in\operatorname{\mathcal{S}}$ and let $[p,x]\in E_{s^{*}}$. Thus, there
exists $(q,y)\in D_{s^{*}}$ such that $(p,x)\approx(q,y)$. Since $(q,y)\in
D_{s^{*}}$, we have that $(s,q)\in\operatorname{\mathcal{S}}^{(2)}$ and $y\in
X_{q^{*}s^{*}sq}$ and, moreover, $\eta_{s}([p,x])=\eta_{s}([q,y])=[sq,y]$.
Since $j$ is an $\operatorname{\mathcal{S}}$-function, we have that $j(y)\in
Z_{q^{*}s^{*}sq}$. By (P4), $Z_{q^{*}s^{*}sq}=Z_{q^{*}s^{*}}$ which is the
domain of $\omega_{sq}=\omega_{s}\circ\omega_{q}$ by (E4). Hence,
$\sigma([p,x])=\sigma([q,y])=\omega_{q}(j(y))\in Z_{s^{*}}.$
Moreover,
$\omega_{s}\big{(}\sigma([p,x])\big{)}=\omega_{s}\big{(}\omega_{q}(j(y))\big{)}=\omega_{sq}(j(y))=\sigma([sq,y])=\sigma\big{(}\eta_{s}([p,x])\big{)},$
and $\sigma$ is indeed an $\operatorname{\mathcal{S}}$-function.
Now, given $x\in X$, by (P1), choose $e\in E(\operatorname{\mathcal{S}})$ such
that $x\in X_{e}$. Notice that
$\big{(}\sigma\circ i\big{)}(x)=\sigma([e,x])=\omega_{e}(j(x))=j(x),$
which proves that the diagram
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{j}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Z}$
commutes.
It remains to prove the uniqueness part. For that, assume that
$\sigma^{\prime}:E\to Z$ is an $\operatorname{\mathcal{S}}$-function such that
$\sigma^{\prime}\circ i=j$. Let $[s,x]\in E$ and notice that $(s^{*}s,x)\in
D_{s^{*}}$, since $(s,s^{*}s)\in\operatorname{\mathcal{S}}^{(2)}$ and $x\in
X_{s^{*}s}$. Thus, $[s^{*}s,x]\in E_{s^{*}}$ and, being $\sigma^{\prime}$ an
$\operatorname{\mathcal{S}}$-function, we have that
$\sigma^{\prime}([s^{*}s,x])\in Z_{s^{*}}$ and moreover
$\sigma([s,x])=\omega_{s}(j(x))=\omega_{s}\big{(}(\sigma^{\prime}\circ
i)(x)\big{)}=\omega_{s}\big{(}\sigma^{\prime}([s^{*}s,x])\big{)}=\sigma^{\prime}\big{(}\eta_{s}([s^{*}s,x])\big{)}=\sigma^{\prime}([s,x]),$
from where we conclude that $\sigma^{\prime}=\sigma$, as desired. ∎
We finally prove that $(E,\eta)$ is a universal globalization of $(X,\theta)$.
###### Theorem 3.14.
Let $\operatorname{\mathcal{S}}$ be an inverse semigroupoid. For every
$\operatorname{\mathcal{S}}$-set $(X,\theta)$, there exist a global
$\operatorname{\mathcal{S}}$-set $(E,\eta)$ and an
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ such that $(i,E,\eta)$
is a universal globalization of $(X,\theta)$.
###### Proof.
Let $(i,E,\eta)$ be as in the proof of Theorem 3.13. So, it only remains to
prove that $i\colon X\to E$ is an embedding.
We first prove that $i$ is injective. Indeed, let $x,y\in X$ be such that
$i(x)=i(y)$. Let $e,f\in E(S)$ such that $x\in X_{e}$ and $y\in X_{f}$ and,
thus, we have $[e,x]=[f,y]$. Using Lemma 3.10, we infer that
$x=\theta_{e}(x)=\theta_{f}(y)=y$.
Finally, we prove that the action $\eta$ of $\operatorname{\mathcal{S}}$ on
$E$ induces the action $\theta$ of $\operatorname{\mathcal{S}}$ on $X$. Let
$x,y\in X$ and $s\in\operatorname{\mathcal{S}}$ such that $i(x)\in E_{s^{*}}$
and $\eta_{s}(i(x))=i(y)$. We need to show that $x\in X_{s^{*}}$ and
$\theta_{s}(x)=y$. Using (P1), choose $e,f\in E(\operatorname{\mathcal{S}})$
such that $x\in X_{e}$ and $y\in X_{f}$. Since $[e,x]=i(x)\in E_{s^{*}}$,
there exists $(r,z)\in D_{s^{*}}$ such that $(e,x)\approx(r,z)$. Hence,
$[f,y]=i(y)=\eta_{s}(i(x))=\eta_{s}([e,x])=\eta_{s}([r,z])=[sr,z].$
By Lemma 3.10, since $x\in X_{e}$, $y\in X_{f}$, $(e,x)\approx(r,z)$ and
$(f,y)\approx(sr,z)$, we have that $z\in X_{(sr)^{*}}\cap X_{r^{*}}$,
$\theta_{r}(z)=\theta_{e}(x)=x$ and $\theta_{sr}(z)=\theta_{f}(y)=y$. By (P3),
$x=\theta_{r}(z)\in X_{s^{*}}$ and
$\theta_{s}(x)=\theta_{s}(\theta_{r}(z))=\theta_{sr}(z)=y,$
which finishes the proof. ∎
We apply now our construction in the two examples in the end of section 2.
###### Example 3.15.
Let $\operatorname{\mathcal{S}}$ and $(X,\theta)$ be as in Example 2.11. In
this case, we have
$\displaystyle
D=\big{\\{}(b,1),(b,2),(b^{*}b,1),(b^{*}b,2),(b^{*},1),(b^{*},4),(bb^{*},1),(bb^{*},4),$
$\displaystyle(a,1),(a^{*}a,1),(a^{*},3),(a^{*},4),(aa^{*},3),(aa^{*},4)\big{\\}}.$
and five equivalence classes with respect to relation $\approx$:
$\displaystyle\overline{1}$
$\displaystyle=\\{(b,1),(a^{*},4),(b^{*},1),(a^{*}a,1),(b^{*}b,1),(bb^{*},1)\\},$
$\displaystyle\overline{2}$ $\displaystyle=\\{(b^{*}b,2),(b^{*},4)\\},$
$\displaystyle\overline{3}$ $\displaystyle=\\{(a^{*}a,3)\\},$
$\displaystyle\overline{4}$
$\displaystyle=\\{(b,2),(bb^{*},4),(a,1),(aa^{*},4)\\},$
$\displaystyle\overline{5}$ $\displaystyle=\\{(a^{*},3)\\}.$
The family $\\{D_{s}\\}_{s\in\operatorname{\mathcal{S}}}$, as in Equation (7)
is given by
$\displaystyle D_{a^{*}}$
$\displaystyle=D_{a^{*}a}=\\{(a^{*},3),(a^{*},4),(b,1),(b^{*},1),(a^{*}a,1),(bb^{*},1),(b^{*}b,1)\\},$
$\displaystyle D_{a}$
$\displaystyle=D_{aa^{*}}=\\{(a,1),(aa^{*},3),(aa^{*},4)\\},$ $\displaystyle
D_{b^{*}}$
$\displaystyle=D_{b^{*}b}=\\{(a^{*},3),(a^{*},4),(a^{*}a,1),(b,1),(b^{*}b,1),(b^{*}b,2),(b^{*},1),(b^{*},4),(bb^{*},1)\\},$
$\displaystyle D_{b}$
$\displaystyle=D_{bb^{*}}=\\{(a^{*},3),(a^{*},4),(b,1),(b,2),(b^{*},1),(a^{*}a,1),(bb^{*},1),(bb^{*},4),(b^{*}b,1)\\}.$
Thus, the globalization $(E,\eta)$ of $(X,\theta)$ is given by
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
in which
$\displaystyle E_{a^{*}}=E_{a^{*}a}=\\{\overline{1},\overline{5}\\},$
$\displaystyle E_{a}=E_{aa^{*}}=\\{\overline{3},\overline{4}\\},$
$\displaystyle
E_{b^{*}}=E_{b^{*}b}=\\{\overline{1},\overline{2},\overline{5}\\},$
$\displaystyle E_{b}=E_{bb^{*}}=\\{\overline{1},\overline{4},\overline{5}\\},$
$\eta_{b}\colon E_{b^{*}}\to E_{b}$ and $\eta_{a}\colon E_{a^{*}}\to E_{a}$
are given by
$\begin{array}[]{rcl}\eta_{b}(\overline{1})&=&\eta_{b}([b^{*}b,1])=[b,1]=\overline{1}\\\
\eta_{b}(\overline{2})&=&\eta_{b}([b^{*}b,2])=[b,2]=\overline{4}\\\
\eta_{b}(\overline{5})&=&\eta_{b}([a^{*},3])=[a^{*},3]=\overline{5}\end{array},\qquad\begin{array}[]{rcl}\eta_{a}(\overline{1})&=&\eta_{a}([a^{*}a,1])=[a,1]=\overline{4}\\\
\eta_{a}(\overline{5})&=&\eta_{a}([a^{*},3])=[aa^{*},3]=\overline{3}\\\
&&\end{array},$
and $\eta_{a^{*}}=\eta_{a}^{-1}$, $\eta_{b^{*}}=\eta_{b}^{-1}$ and
$\eta_{e}=\operatorname{id}_{E_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. The
$\operatorname{\mathcal{S}}$-function $i\colon X\to E$ is given by
$i(1)=\overline{1}$, $i(2)=\overline{2}$, $i(3)=\overline{3}$ e
$i(4)=\overline{4}$.
In the next example, we first restrict the global action of Example 2.12 and
then compare the globalization of the restricted action with the original
partial action.
###### Example 3.16.
Let $\operatorname{\mathcal{S}}$ and $(Y,\theta^{Y})$ be as in Example 2.12.
If we restrict the action $\theta^{Y}$ to $X=\\{1,2\\}$, we obtain a partial
action
$\theta=(\\{X_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\theta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
of $\operatorname{\mathcal{S}}$ on $X$ such that $X_{a^{*}}=\\{1\\}$,
$X_{a}=\\{2\\}$, $\theta_{a}\colon X_{a^{*}}\to X_{a}$ and
$\theta_{a^{*}}\colon X_{a}\to X_{a^{*}}$ are evident, $X_{s}=X$ and
$\theta_{s}=\operatorname{id}_{X}$ if $s\neq a,a^{*}$.
It is clear that $(j,Y,\theta^{Y})$, in which $j\colon X\to Y$ is the
inclusion map, is a globalization of $(X,\theta)$. However, this is not a
universal globalization of $(X,\theta)$.
Indeed, applying our construction in this case, we obtain
$D=\operatorname{\mathcal{S}}\times X$ and we have four equivalence classes
with respect to relation $\approx$:
$\displaystyle\overline{1}$
$\displaystyle=\\{(b^{*}b,1),(bb^{*},1),(a^{*}a,1),(aa^{*},1),(b,1),(b^{*},1),(a^{*},2)\\},$
$\displaystyle\overline{2}$
$\displaystyle=\\{(b^{*}b,2),(bb^{*},2),(a^{*}a,2),(aa^{*},2),(b,2),(b^{*},2),(a,1)\\},$
$\displaystyle\overline{3}$ $\displaystyle=\\{(a,2)\\},$
$\displaystyle\overline{4}$ $\displaystyle=\\{(a^{*},1)\\}.$
The family $\\{D_{s}\\}_{s\in\operatorname{\mathcal{S}}}$, as in Equation (7)
is simply given by
$D_{s^{*}}=\\{(p,x)\in D:(s,p)\in\operatorname{\mathcal{S}}^{(2)}\\}$
and, thus, the globalization $(E,\eta)$ of $(X,\theta)$ is given by
$\eta=(\\{E_{s}\\}_{s\in\operatorname{\mathcal{S}}},\\{\eta_{s}\\}_{s\in\operatorname{\mathcal{S}}})$
in which
$\displaystyle
E_{a^{*}}=E_{a^{*}a}=\\{\overline{1},\overline{2},\overline{4}\\},$
$\displaystyle E_{a}=E_{aa^{*}}=\\{\overline{1},\overline{2},\overline{3}\\},$
$\displaystyle
E_{b^{*}}=E_{b^{*}b}=\\{\overline{1},\overline{2},\overline{4}\\},$
$\displaystyle E_{b}=E_{bb^{*}}=\\{\overline{1},\overline{2},\overline{4}\\},$
$\eta_{b}\colon E_{b^{*}}\to E_{b}$ and $\eta_{a}\colon E_{a^{*}}\to E_{a}$
are given by
$\begin{array}[]{rcl}\eta_{b}(\overline{1})&=&\eta_{b}([b^{*}b,1])=[b,1]=\overline{1}\\\
\eta_{b}(\overline{2})&=&\eta_{b}([b^{*}b,2])=[b,2]=\overline{2}\\\
\eta_{b}(\overline{4})&=&\eta_{b}([a^{*},1])=[a^{*},1]=\overline{4}\end{array},\qquad\begin{array}[]{rcl}\eta_{a}(\overline{1})&=&\eta_{a}([a^{*}a,1])=[a,1]=\overline{2}\\\
\eta_{a}(\overline{2})&=&\eta_{a}([a^{*}a,2])=[a,2]=\overline{3}\\\
\eta_{a}(\overline{4})&=&\eta_{a}([a^{*},1])=[aa^{*},1]=\overline{1}\end{array},$
and $\eta_{a^{*}}=\eta_{a}^{-1}$, $\eta_{b^{*}}=\eta_{b}^{-1}$ and
$\eta_{e}=\operatorname{id}_{E_{e}}$ for $e\in
E(\operatorname{\mathcal{S}})=\\{b^{*}b,bb^{*},a^{*}a,aa^{*}\\}$. Moreover,
the $\operatorname{\mathcal{S}}$-function $i\colon X\to E$ is given by
$i(1)=\overline{1}$ and $i(2)=\overline{2}$.
Notice that, considering the globalization $(j,Y,\theta^{Y})$, the mediating
function $\sigma\colon E\to Y$ given by the universal property of $(i,E,\eta)$
is given by
$\begin{array}[]{rcl}\sigma(\overline{1})&=&\sigma([b^{*}b,1])=\theta_{b^{*}b}^{Y}(j(1))=\theta_{b^{*}b}^{Y}(1)=1\\\
\sigma(\overline{2})&=&\sigma([b^{*}b,2])=\theta_{b^{*}b}^{Y}(j(2))=\theta_{b^{*}b}^{Y}(2)=2\\\
\sigma(\overline{3})&=&\sigma([a,2])=\theta_{a}^{Y}(j(2))=\theta_{a}^{Y}(2)=3\\\
\sigma(\overline{4})&=&\sigma([a^{*},1])=\theta_{a^{*}}^{Y}(j(1))=\theta_{a^{*}}^{Y}(1)=3\end{array}$
and hence, $\sigma$ is not injective. Moreover, the reader can check that any
function $\sigma^{\prime}\colon Y\to E$ such that $\sigma^{\prime}\circ j=i$
is not an $\operatorname{\mathcal{S}}$-function.
The last example shows that the mediating function given by the universal
property of the globalization $(E,\eta)$ may not be injective (and, hence, may
not be an embedding). However, we can guarantee its injectivity in some
especial subsets of $E$.
Indeed, for each $u\in\operatorname{\mathcal{S}}^{(0)}$, set
$\displaystyle D_{u}=\\{(s,x)\in D:c(s)=u\\}.$ (9)
and let $E_{u}$ be the image of $D_{u}$ by the quotient map of $D$ by the
equivalence relation $\approx$.
We end this paper by showing that the mediating function given by the
universal property of $(i,E,\eta)$ is injective in each $E_{u}$.
###### Proposition 3.17.
Let $(X,\theta)$ be an $\operatorname{\mathcal{S}}$-set and $(j,Z,\omega)$ a
globalization of $(X,\theta)$. Given any
$u\in\operatorname{\mathcal{S}}^{(0)}$, the mediating function $\sigma\colon
E\to Z$ given by the universal property of the globalization $(i,E,\eta)$ is
injective in $E_{u}$.
###### Proof.
Let $[s,x],[t,y]\in E_{u}$ such that $\sigma([s,x])=\sigma([t,y])$. With no
loss in generality, we may assume $(s,x),(t,y)\in D_{u}$. This means that
$\omega_{s}(j(x))=\sigma([s,x])=\sigma([t,y])=\omega_{t}(j(y))$
and, hence, $\omega_{s}(j(x))\in Z_{t}\cap Z_{s}$. Since $c(s)=c(t)=u$, we
have $(t^{*},s)\in S^{(2)}$ and, since $\omega$ is a global action, we have
$\omega_{t^{*}s}=\omega_{t^{*}}\circ\omega_{s}$ by (E4). Thus,
$j(x)\in\omega_{s}^{-1}(Z_{t}\cap Z_{s})=Z_{s^{*}t}$ and
$\omega_{t^{*}s}(j(x))=\omega_{t^{*}}\big{(}\omega_{s}(j(x))\big{)}=\omega_{t^{*}}\big{(}\omega_{t}(j(y))\big{)}=j(y).$
Since $j$ is an embedding, $x\in X_{s^{*}t}$ and $\theta_{t^{*}s}(x)=y$, which
implies that $(s,x)\sim(t,y)$ by (R2). ∎
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|
# Discriminant Analysis in Contrasting Dimensions for Polycystic Ovary
Syndrome Prognostication
Abhishek M. Gupta
Bachelor of Engineering - EXTC
University of Mumbai
Mumbai, MH, IN
<EMAIL_ADDRESS>
Himanshu H. Soni
Bachelor of Engineering - EXTC
University of Mumbai
Mumbai, MH, IN
<EMAIL_ADDRESS>
Raunak M. Joshi
Mentor
Master of Engineering
Mumbai, MH, IN
<EMAIL_ADDRESS>
Ronald Melwin Laban
Assistant Professor - Dept of EXTC
St. John College of Engineering and Management
Palghar, MH, IN
<EMAIL_ADDRESS>
###### Abstract
A lot of prognostication methodologies have been formulated for early
detection of Polycystic Ovary Syndrome also known as PCOS using Machine
Learning. PCOS is a binary classification problem. Dimensionality Reduction
methods impact the performance of Machine Learning to a greater extent and
using a Supervised Dimensionality Reduction method can give us a new edge to
tackle this problem. In this paper we present Discriminant Analysis in
different dimensions with Linear and Quadratic form for binary classification
along with metrics. We were able to achieve good accuracy and less variation
with Discriminant Analysis as compared to many commonly used classification
algorithms with training accuracy reaching 97.37% and testing accuracy of
95.92% using Quadratic Discriminant Analysis. Paper also gives the analysis of
data with visualizations for deeper understanding of problem.
_K_ eywords Dimensionality Reduction $\cdot$ Linear Discriminant Analysis
$\cdot$ Quadratic Discriminant Analysis
## 1 Introduction
The Polycystic Ovary Syndrome also known as PCOS [1] is a symptom which causes
disorder in women ranging from reproductive age to their later stages in life.
The commonly associated disorder with PCOS is irregularity in menstrual cycles
and skin diseases. The high blood pressure and high cholesterol abide to the
PCOS symptoms making things worse. High blood sugar is also observed in the
later stages of the symptom. The early detection of this major disorder is a
necessary issue. Machine Learning can be leveraged to the advantage of early
detection. The medical expenses cannot be borne by all the women. Some also
feel insecurity for disclosing such a matter even to doctors. A functional
prognostication system can be very helpful in such a situation. Many people
have proved to be successful in using Machine Learning with the right data for
this problem. The PCOS is a binary classification problem. It clearly focuses
on the result of whether the woman is suffering from PCOS or not. There is no
distinction in the choices. Many machine learning algorithms can be used for
such a problem. The first algorithm taken into consideration is Logistic
Regression [2] being the binary classification problem. A lot of work has been
done on Logistic Regression so far now. Very brief implementation between
Logistic Regression and Bayesian Classifier has been performed [3]. The
results obtained were having a Bayesian Classifier which was able to beat the
performance of Logistic Regression with almost 2% improvement. Trying to give
more in depth implementation with machine learning inclination, various
algorithms like KNN, CART and SVM were used [4]. A detailed result section
gives the metrics for the best algorithm. The improvements simultaneously were
even extended with the boosting algorithms [5]. Many different boosting
algorithms were used in an implementation for PCOS where the CatBoost was able
to outrun all the boosting algorithms [6]. This was a compendious comparison
of all the advanced boosting algorithms like AdaBoost, LGBM, XGBoost and
CatBoost.
All methods that have been implemented by now are purely prognostication
oriented with machine learning. What we present in this paper is to use
Discriminant Analysis [7] for getting a deeper understanding of the problem.
Discriminant Analysis is an extension of dimensionality reduction with
classification. Dimensionality reduction has supervised as well as
unsupervised learning approaches [8]. The most commonly used dimensionality
reduction technique is Principal Component Analysis [9]. It is an unsupervised
learning algorithm. The most commonly used supervised dimensionality reduction
algorithm on the other hand is Linear Discriminant Analysis [10]. It maximizes
the separability between the classes. An increase in dimensions gives us the
Quadratic Discriminant Analysis [11]. It is a type of generative model.
There are some differences between Linear Discriminant Analysis and Quadratic
Discriminant Analysis. The observable difference is that Linear Discriminant
Analysis does not have class-specific covariance matrices, but one shared
covariance matrix among the classes. The shared covariance matrix is basically
the covariance of all the inputs given. The benefit of Discriminant Analysis
is that for the purpose of classification it uses the canonical variables.
## 2 Implementation
### 2.1 Linear Discriminant Analysis
The classification aspect of the Linear Discriminant Analysis works on maximum
separability. Working with higher dimensions is necessary to generate a better
understanding of the data for the final outcome. The higher dimensions not
necessarily can give better understanding of the data points. So in order to
maintain the feature dimensions and not increase them more than required,
Linear Discriminant Analysis minimizes the dimensions. It transforms the
features to one axis which is called Linear Discriminant. The common working
of the algorithm considers the maximum separability by maximizing the
difference between the means and minimizing the variance. This minimizing
variation is called scatter in LDA The covariance matrix is the underlying
source of the LDA. It approaches the problem by using conditional probability
density functions which fall under normal distribution mean parameters and
covariance parameters. For this the Bayesian Optimization [12] is used which
gives the log of likelihood for the ratios with threshold to classify the
right class.
### 2.2 Quadratic Discriminant Analysis
The QDA [11] is a variant of LDA [10] which considers an covariance matrix
that is individual. It is estimated for all the classes of observations. If
the individual classes orchestrates distinct covariances, in such a scenario
QDA can prove to be better as there is an involvement of the prior knowledge.
The effective parameters increase for QDA as compared to LDA. The most common
consideration for the binary classification based Quadratic Discriminant
Analysis or Linear Discriminant Analysis is given by a multivariate Gaussian
Distribution [13]. The formula is given as follows
$P(x|y=k)=\frac{1}{(2\pi)^{d/2}|\Sigma_{k}|^{1/2}}\exp\left(-\frac{1}{2}(x-\mu_{k})^{t}\Sigma_{k}^{-1}(x-\mu_{k})\right)$
(1)
The discriminant function for the LDA in Quadratic can be represented as
follows
$\delta_{k}(x)=-\frac{1}{2}\log|\Sigma_{k}|-\frac{1}{2}(x-\mu_{k})^{T}\Sigma_{k}^{-1}(x-\mu_{k})+\log\pi_{k}$
(2)
We try to estimate $\Sigma_{k}$ in 2 for each and every class instead of
assuming like we do in Linear Discriminant Analysis.
## 3 Results
This section constitutes our work with data. The implementation phase has many
facets of results. We have presented numerous results below.
### 3.1 Analysis
This section of the paper clearly focuses on the analysis of the entire data.
Many graphical representations given below provide an idea of different
features in the data. The analysis section was taken into consideration to see
the interaction of the different features with respect to age. The age is the
parameter that influences the entire other parameters. The entire data can be
checked for correlation using heat map. It gives a distinct distribution of
the dataset features with valuation.
Figure 1: Heatmap for all the features
The Figure 1 is a very larger overview of all the features. It gives you an
idea of how all the features are correlated from each other. More improvements
can be bought into the analysis section. We can perform analysis for intricate
observations. Boxplot [14] is known to be one of the most common form of
visual representation to give a better edge in analysis of data.
Figure 2: Categorical Plot for Exercise Level with respect to Age for Labels
The Figure 2 gives a very good analysis of the effect of exercise levels with
respect to age. The labels are taken into consideration. The visual
representation of the features gives inference that exercise is an influencing
factor taken into consideration for the prevention of the symptom. Another
very important consideration is the irregularities in the menstrual cycles. It
gives a very deep understanding about the problem. The Figure 3 gives a very
detailed analysis about the distribution of the occurrence and effect of
irregularities in menstrual cycles.
Figure 3: Categorical Plot for Periods with respect to Age for Labels
### 3.2 Train and Test Accuracy
The accuracies give the basic analogy of any algorithm. It the most primary
metric taken into consideration. It gives a baseline effect of how well the
model performs with the data.
Table 1: Comparison of LDA and QDA Training & Testing Accuracy Algorithm | Training Accuracy | Testing Accuracy
---|---|---
Linear Discriminant Analysis | 86.84 % | 81.63 %
Quadratic Discriminant Analysis | 97.37 % | 95.92 %
In Table 1 we can clearly see that the variation in the accuracy of Quadratic
Discriminant Analysis is comparatively less than Linear Discriminant Analysis.
### 3.3 Confusion Matrix
The Type-I and Type-II Error are the errors which give clear distinction of
interacting variables with the test data. The basic aspect of Confusion Matrix
[15] is observation of predicted class with target class. It gives the errors
which are targeted for specific problems. The Type-II Error works with False
Negatives which is a type of error we want to target. The importance of this
error is that if the person does have symptoms of PCOS, the system should
detect the symptom. The table 2 below gives a comparison of False Positives
and False Negatives between LDA and QDA.
Table 2: Confusion Matrix for LDA & QDA Algorithm | False Negatives | False Positives
---|---|---
Linear Discriminant Analysis | 6 | 3
Quadratic Discriminant Analysis | 0 | 0
### 3.4 Metric Scores
The two major metrics we are focusing in this paper are the Average Precision
Score [16] and Precision Recall Curve [17]. The Precision and Recall are
individually calculated. Precision is a measure of quality of predictions. The
Precision is given by formula
$Precision={\frac{TruePositives}{TruePositives+FalsePositives}}$ (3)
and Recall is given by formula
$Recall={\frac{TruePositives}{TruePositives+FalseNegatives}}$ (4)
Recall is a measure of the quantity of correctly classified predictions. Since
our problem works with binary classification problem [18], the values are
distinguished Yes/No and do not have any threshold values.
Average Precision considers a Precision-Recall curve as the weighted mean of
all the Precision achieved at each threshold, with the increase in recall from
the previous threshold used as the weight. The formula given below gives a
precise representation for the binary classification problem. If the data is
available with thresholds the graph considers an integral spread over the
points. The discrete data on the other hand gives the summation spread over
the points. The formula is given as
$AveragePrecision=\sum^{n}_{k=1}(Recall_{k}-Recall_{k-1}).Precision_{k}$ (5)
The below are given graphs that give clear distinction of above specified
metrics in the form of visualization. These graphs plot the precision and
recall score with respect to average precision. In Figure 4 we can see the
interaction of the Precision and Recall Curve. The sudden drop can be seen
after some points. Whereas in Figure 5 we can a sharp drop which goes beyond
the performance of the Linear Discriminant Analysis.
Figure 4: Precision Recall Curve with Average Precision for LDA Figure 5:
Precision Recall Curve with Average Precision for QDA
### 3.5 Receiver Operator Characteristic and Area Under Curve
Receiver Operator Characteristic Curve [19] also known as RoC Curve is used
for performance measurement of classification models. It outputs comprehensive
probability fluctuations throughout the predictions. This is computed on the
basis of two functions known as True Positive Rate (TPR) and False Positive
Rate (FPR). True Positive Rate is also known as Sensitivity and works much
similar to Recall. It is denoted by formula as
$TruePositiveRate={\frac{TruePositives}{TruePositives+FalseNegatives}}$ (6)
After True Positive Rate we have to work our way towards False Positive Rate,
but the issue is that the calculation cannot be done directly. One needs to
first calculate Specificity. It is a measure for finding negative classes
without anomalies. This is represented by formula as
$Specificity={\frac{TrueNegatives}{TrueNegatives+FalsePositives}}$ (7)
After this specificity can help compute the False Positive Rate. It can be
also denoted with a formula
$FalsePositiveRate={\frac{FalsePositives}{FalsePositives+TrueNegatives}}$ (8)
Apart from RoC Curve we require Area Under Curve which is also known as AUC
[20]. This AUC supports specifying threshold settings. It gives degree or
measure of separability. Higher the value of AUC, better is the prediction.
The higher the value on the scale, the more better algorithmic performance.
Figure 6: RoC Curve with AUC for LDA and QDA
In Figure 6 the Red Line indicates the boundary of threshold for the Area
Under Curve. The metrics for the models need to be above the Red Line. In our
paper, the Quadratic Discriminant Analysis performs better than Linear
Discriminant Analysis. The QDA covers much more area than LDA and the
inference can be derived that more area for classification is considered.
### 3.6 Separability
Considering Discriminant Analysis the representation of the points is given
over plots with dimensions.
Figure 7: The Separability of the Data with respect to Labels
In Figure 7 we can see the separation in both the labels of their respective
distributions. The blue ellipse represents the Yes labels and red ellipse
represents the No labels. The Linear Discriminant Analysis has ellipses that
are separated in Linear dimensional form while Quadratic Discriminant Analysis
has ellipses in quadratic form, with much more increased dimensions.
## 4 Conclusion
This paper presents a very intricate area of Machine Learning domain which is
not much worked with. Our aim in this paper was highlighting the point that
classification with dimensionality reduction is as important as other
classification parametric methods. We used supervised dimensionality reduction
algorithm Linear Discriminant Analysis. We worked on increasing the dimension
of the same algorithm which resulted in Quadratic Discriminant Analysis.
Definitely this paper promotes more ideas and new improvements can be done by
researchers in the future. With our best belief and knowledge we conclude this
paper.
## References
* [1] Gautam N Allahbadia and Rubina Merchant. Polycystic ovary syndrome and impact on health. Middle East Fertility Society Journal, 16(1):19–37, 2011.
* [2] Jan Salomon Cramer. The origins of logistic regression. 2002\.
* [3] Palak Mehrotra, Jyotirmoy Chatterjee, Chandan Chakraborty, Biswanath Ghoshdastidar, and Sudarshan Ghoshdastidar. Automated screening of polycystic ovary syndrome using machine learning techniques. In 2011 Annual IEEE India Conference, pages 1–5, 2011.
* [4] Amsy Denny, Anita Raj, Ashi Ashok, C Maneesh Ram, and Remya George. i-hope: Detection and prediction system for polycystic ovary syndrome (pcos) using machine learning techniques. In TENCON 2019 - 2019 IEEE Region 10 Conference (TENCON), pages 673–678, 2019.
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* [6] Abhishek Gupta, Sannidhi Shetty, Raunak Joshi, and Ronald Melwin Laban. Succinct differentiation of disparate boosting ensemble learning methods for prognostication of polycystic ovary syndrome diagnosis. arXiv preprint arXiv:2201.00418, 2022.
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|
11institutetext: Department of Physics and Astronomy, University of Turku,
Finland
11email<EMAIL_ADDRESS>22institutetext: Leibniz-Institut für
Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
22email<EMAIL_ADDRESS>33institutetext: Institut für Theoretische Physik,
IV, Ruhr-Universität Bochum, 44780 Bochum, Germany
33email<EMAIL_ADDRESS>44institutetext: Bay Area Environmental Research
Institute, NASA Research Park, Moffett Field, CA, USA 55institutetext: LESIA –
Observatoire de Paris, Univ. PSL, CNRS, Sorbonne Univ., Univ. de Paris, 5
place Jules Janssen, F- 92190 Meudon, France 66institutetext: University of
Minnesota, Minneapolis, MN, USA 77institutetext: European Space Agency (ESA),
European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ,
Noordwijk, The Netherlands 88institutetext: Institut für Experimentelle und
Angewandte Physik, Universität Kiel, 24118, Kiel, Germany
# Connecting solar flare hard X-ray spectra to in situ electron spectra
A comparison of RHESSI and STEREO/SEPT observations
N. Dresing 1177 A. Warmuth 22 F. Effenberger 3344 K.-L. Klein 55 S. Musset
6677 L. Glesener 66 M. Brüdern 88
###### Abstract
Aims. We aim to constrain the acceleration, injection, and transport processes
of flare-accelerated energetic electrons by comparing their characteristics at
the Sun with those injected into interplanetary space.
Methods. We have identified 17 energetic electron events well-observed with
the SEPT instrument aboard STEREO which show a clear association with a hard
X-ray (HXR) flare observed with the RHESSI spacecraft. We compare the spectral
indices of the RHESSI HXR spectra with those of the interplanetary electrons.
Because of the frequent double-power-law shape of the in situ electron
spectra, we paid special attention to the choice of the spectral index used
for comparison.
Results. The time difference between the electron onsets and the associated
type III and microwave bursts suggests that the electron events are detected
at 1 AU with apparent delays ranging from 9 to 41 minutes. While the parent
solar activity is clearly impulsive, also showing a high correlation with
extreme ultraviolet jets, most of the studied events occur in temporal
coincidence with coronal mass ejections (CMEs). In spite of the observed onset
delays and presence of CMEs in the low corona, we find a significant
correlation of about 0.8 between the spectral indices of the HXR flare and the
in situ electrons. The correlations increase if only events with significant
anisotropy are considered. This suggests that transport effects can alter the
injected spectra leading to a strongly reduced imprint of the flare
acceleration.
Conclusions. We conclude that interplanetary transport effects must be taken
into account when inferring the initial acceleration of solar energetic
electron events. Although our results suggest a clear imprint of flare
acceleration for the analyzed event sample, a secondary acceleration might be
present which could account for the observed delays. However, the limited and
variable pitch-angle coverage of SEPT could also be the reason for the
observed delays.
###### Key Words.:
– Sun –
## 1 Introduction
In solar flares, energy that is stored in nonpotential coronal magnetic fields
is released impulsively, presumably triggered by magnetic reconnection. In
response, the solar atmosphere emits electromagnetic radiation over the whole
wavelength range from radio to gamma rays (e.g., Fletcher et al., 2011).
Analyses of this emission recorded by remote-sensing instruments have revealed
key insights into the physics of solar flares. In particular, the observation
of nonthermal bremsstrahlung in the hard X-ray (HXR) range has shown that
electrons are efficiently accelerated in flares and carry a significant
fraction of the energy released (cf. Holman et al., 2011; Warmuth & Mann,
2020). While nonthermal HXRs are primarily produced by electrons that
precipitate into deeper layers of the solar atmosphere, electrons can also
propagate upward through the corona and into interplanetary space. This is
revealed by type III radio bursts, which are rapidly drifting structures
observed in dynamic radio spectra (cf. White et al., 2011; Reid & Ratcliffe,
2014). They are generated by escaping electron beams that excite Langmuir
turbulence in the ambient plasma, which is subsequently converted to
electromagnetic radiation.
Such interplanetary electron beams can be detected in situ by particle
instruments on spacecraft. However, one common feature of solar electron
events has challenged our understanding of the parent acceleration process for
decades, which is a frequently observed delay of 10-20 minutes between the
occurrence of the solar counterpart, for example flare or radio type III
burst, and the inferred injection time of the electrons based on their
observed onset times at the spacecraft (e.g., Haggerty & Roelof, 2002; Kahler
et al., 2007). These delays were often interpreted as an indication of a
different acceleration process, for instance a coronal or CME-driven shock
(e.g., Haggerty et al., 2003; Kahler et al., 2007). However, also scenarios
where flare-accelerated electrons are re-accelerated or energized by ongoing
reconnection in the solar corona, possibly driven by the uplifting CME, are
under discussion (Maia & Pick, 2004). Furthermore, magnetic trapping might
also be involved in delayed onsets and for the acceleration of electrons to
higher energies (e.g., Dresing et al., 2018; Li et al., 2020).
However, not all solar energetic electron events show such a delay and some
case studies even reported different behavior at different energies, that is a
prompt and a delayed component, within a single event (Krucker et al., 1999;
Wang et al., 2006a; Li et al., 2020). This suggests that some electron events
detected in situ may indeed be of the same particle distribution as the HXR
producing electrons, or at least still carry imprints of the flare
acceleration. It is therefore tempting to compare the energy spectra of the
HXR flare with the one of the in situ electrons. Depending on which
approximation is used to invert the nonthermal HXR spectrum, the expected
relation of the photon spectral index $\gamma$ with the in situ electron
spectral index $\delta$ is either $\delta=\gamma_{thick}+1$ for the thick-
target model or $\delta=\gamma_{thin}-1$ for the thin-target model (cf. Brown,
1971; Holman et al., 2011). When analyzing 16 impulsive and non-delayed
electron events observed by Wind/3DP (Lin et al., 1995) and their HXR
counterpart detected by the RHESSI spacecraft (Lin et al., 2002), Krucker et
al. (2007) found a good linear correlation between the spectral indices of
0.83. However, the value pairs were neither consistent with the thin nor with
the thick-target model but were all lying in between. Another sample of 15
events studied by Krucker et al. (2007), which was characterized by larger
onset delays of $>8$ minutes, showed no clear correlation and a shift of the
points toward the thin-target solution, which led Petrosian (2016) to conclude
that the electrons of these events may have experienced a further acceleration
after their initial energization in the flare. An additional argument for the
close association of HXR-producing and escaping electrons has recently been
provided by Xia et al. (2021) who reported on two events showing consistent
energy cutoffs in the two populations.
The spectra of electrons observed in situ can often show spectral breaks or
transitions that can relate to transport processes both near the original
acceleration site and in interplanetary space (e.g., Kontar & Reid, 2009;
Strauss et al., 2020). Dresing et al. (2020) studied the spectra of 781
electron events observed with the Solar Electron and Proton Telescope (SEPT,
Müller-Mellin et al., 2008) aboard the two STEREO spacecraft and found double
power-law shapes in 56% of the events. In this paper, we use this electron
event list as a starting point and investigate the relation between the
energetic electron population precipitating onto the Sun as constrained by
RHESSI, and the interplanetary electrons detected in situ with the SEPT
instruments on board of the two STEREO spacecraft. We put a particular focus
on the timing and spectral relation between both populations.
Figure 1: Time evolution of the X-ray and radio emission associated with the
electron event on 2011 Mar 24. From bottom to top: time profiles of the GOES
soft X-ray flux, of the RHESSI 25–50 keV count rates (corrected for
instrumental effects), and of the radio flux at microwave frequencies, and
dynamic spectra in the meter-wave and decameter-hectometer-wave radio range.
The interval used for the computation of the RHESSI spectrum shown in Fig. 3
is indicated by the pair of red lines in the two bottom panels. Figure 2:
Solar energetic electron observations by SEPT aboard STEREO B on 2011 Mar 24.
The top panel shows the electron intensity in all available energy channels of
one viewing direction with pre-event background subtraction, the second panel
shows the 65-75 keV intensity in the four viewing directions of SEPT, and the
third panel displays the corresponding pitch-angles based in the central
pointing direction of the four telescopes.
## 2 Observations and data selection
Event selection started with a list of Solar Energetic Electron (SEE) events
observed with the SEPT instruments on board of the two STEREO spacecraft
compiled by Dresing et al. (2020) and available
online111http://www2.physik.uni-
kiel.de/stereo/downloads/sept_electron_events.pdf. The version of the list
used in this study covers the time from 2007 to 2018 and includes in total 925
SEE events, 557 at STEREO A and 368 at STEREO B. In the next step, the RHESSI
flare list222https://hesperia.gsfc.nasa.gov/rhessi3/data-access/rhessi-
data/flare-list/index.html was searched for solar flares that had a start time
within a one-hour window before the observed onset of the SEE event at the
STEREO spacecraft. This yielded 64 SEE event candidates covered by SEPT as
well as RHESSI. The low percentage of SEPT events recorded by RHESSI results
from the fact that the STEREO spacecraft were magnetically not well-connected
to the Earth-facing side of the Sun during the years of solar maximum, when
the majority of interplanetary electron events was observed. This was
especially the case for STEREO A flying ahead of Earth leading its magnetic
connection quickly to the backside of the Sun as seen from Earth. Therefore,
all except the first two events in 2007 were detected by STEREO B only.
In the next step, the association between the selected RHESSI flares and the
SEPT events was ascertained by the geometric consideration of whether the
flare was occurring at the correct hemisphere so that a magnetic connection
was at least remotely possible, followed by the inspection of radio
spectrograms provided by the WAVES instrument on the Wind satellite (Bougeret
et al., 1995) and the SWAVES instruments on the two STEREO spacecraft
(Bougeret et al., 2008). Type III bursts showing close temporal association
with the RHESSI flare were taken as strong evidence for an actual association.
Some events had to be discarded due to issues with the RHESSI data, for
example missing observations of the impulsive flare peak due to RHESSI
nighttime. In this manner, we finally obtained 17 events, for which we then
compared the spectral characteristics of the in situ electrons with the HXR
photon spectra. Two events on 2007 Jan 24 were observed by both STEREO
spacecraft, which at that time were separated by only 0.5 degrees. However,
because the lowest energy channels were corrupted in early 2007 for STEREO B
data we only use the observations of STEREO A for these two events in our
correlation. Section A.1 in the appendix discusses the spectral variation
between the closely spaced spacecraft for these two events and its implication
for our study.
Table 1: Event list including the basic parameters of the correlated flare and SEE events. For details, see main text. HXR | GOES | flare | | | | $E_{\mathrm{b}}$ | | $\Delta t$ | $L$ | $\Delta\Phi$, $\Delta\Theta$
---|---|---|---|---|---|---|---|---|---|---
peak time | class | location | $\gamma$ | $\delta_{1}$ | $\delta_{2}$ | [keV] | s/c | [min] | [AU] | [deg]
2007/01/24 00:31:24 | B5.1 | S05W61 | $3.2\pm 0.09$ | $3.0\pm 1.0$ | $3.8\pm 0.6$ | 79 | A | $42\pm 10$ | 2.37 | 1, 1
2007/01/24 00:31:24 | B5.1 | S05W61 | $3.2\pm 0.09$ | $2.7\pm 1.0$ | $3.6\pm 0.9$ | 107 | B | $43\pm 1$ | 2.49 | 1, 0
2007/01/24 05:16:09 | B6.8 | S05W64 | $4.1\pm 0.26$ | $3.6\pm 0.3$ | – | – | B | $37\pm 1$ | 2.09 | 3, 0
2007/01/24 05:16:09 | B6.8 | S05W64 | $4.1\pm 0.26$ | $3.2\pm 0.6$ | $3.9\pm 1.3$ | 98 | A | $42\pm 10$ | 2.37 | 4, 1
2009/12/22 04:56:10 | C7.2 | S28W47 | $2.7\pm 0.04$ | $2.0\pm 0.1$ | $3.0\pm 0.2$ | 122 | B | $31\pm 1$ | 1.75 | 35, 34
2010/02/08 03:12:24 | C6.2 | N22E00 | $3.4\pm 0.10$ | $2.4\pm 0.2$ | – | – | B | $34\pm 1$ | 1.92 | 7, 21
2010/11/12 03:53:41 | C1.0 | S21E02 | $3.7\pm 0.06$ | $3.6\pm 0.3$ | – | – | B | $37\pm 1$ | 2.04 | 29, 28
2010/11/12 08:01:54 | C1.5 | S23W01 | $3.9\pm 0.08$ | $2.9\pm 0.3$ | $4.5\pm 0.5$ | 118 | B | $36\pm 1$ | 2.04 | 27, 30
2010/11/17 04:36:44 | B7.8 | S34E21 | $4.3\pm 0.11$ | $1.6\pm 1.7$ | $3.4\pm 1.0$ | 69 | B | $47\pm 1$ | 2.6 | 10, 41
2011/03/24 17:04:34 | C9.1 | S15E41 | $4.1\pm 0.05$ | $2.9\pm 0.6$ | $5.1\pm 2.3$ | 195 | B | $40\pm 1$ | 2.21 | 14, 14
2012/01/12 00:51:58 | C1.5 | N21E20 | $4.9\pm 0.53$ | $4.4\pm 0.4$ | $6.4\pm 2.0$ | 87 | B | $29\pm 1$ | 1.64 | 30, 14
2012/03/25 00:27:54 | C3.0 | N20E26 | $3.4\pm 0.03$ | $2.7\pm 0.4$ | $4.0\pm 0.9$ | 110 | B | $50\pm 1$ | 2.83 | 23, 18
2012/04/16 00:26:14 | C1.8 | N13E89 | $4.9\pm 0.10$ | $4.8\pm 0.5$ | – | – | B | $39\pm 1$ | 2.21 | 10, 14
2012/05/07 03:21:58 | C2.7 | N13E67 | $4.4\pm 0.03$ | $2.9\pm 0.2$ | $4.9\pm 0.6$ | 106 | B | $61\pm 10$ | 3.45 | 15, 17
2012/06/27 12:36:00 | C3.4 | N15E64 | $3.7\pm 0.03$ | $2.4\pm 0.3$ | $4.9\pm 0.6$ | 90 | B | $29\pm 1$ | 1.64 | 1, 22
2012/06/28 02:12:24 | C2.6 | N17E56 | $4.3\pm 0.09$ | $2.8\pm 0.6$ | $3.7\pm 0.4$ | 90 | B | $32\pm 1$ | 1.81 | 20, 24
2012/07/01 07:14:44 | C5.4 | N16E11 | $4.3\pm 0.14$ | $3.6\pm 0.3$ | $5.5\pm 0.7$ | 104 | B | $31\pm 1$ | 1.7 | 54, 23
2014/03/19 16:26:14 | C3.3 | S12E81 | $5.9\pm 0.20$ | $3.1\pm 0.7$ | $4.0\pm 0.7$ | 101 | B | $35\pm 5$ | 1.98 | 23, 18
2014/06/09 17:05:04 | C8.8 | S19E90 | $3.5\pm 0.05$ | $2.6\pm 0.5$ | $4.0\pm 1.4$ | 90 | B | $43\pm 1$ | 2.43 | 28, 17
## 3 Analysis
Figures 1 and 2 show remote-sensing and in situ observations of an example
event in the analyzed sample occurring on 24 Mar 2011. We present the analysis
of both types of spectra in the following.
Figure 3: Left: Background-subtracted RHESSI photon flux spectrum (black) of
the flare of 2011 Mar 24 fitted with an isothermal component (red) and a
nonthermal photon power-law (blue). The spectrum was derived using all RHESSI
detectors except for number 2 and 3. For comparison, the pre-event background
(brown) is plotted as well. The fit results for emission measure EM,
temperature $T$, and spectral index $\gamma$ are indicated. Right: Background-
subtracted in situ electron spectrum of the 2011 Mar 24 event. The spectrum is
constructed by the peak intensities of each energy channel (marked by circles
in the top panel of Fig. 2). The line represents a broken-power law fit to the
data.
### 3.1 X-ray and radio observations
The time histories of X-ray and radio emission in Fig. 1 show a C9-class flare
preceded and followed by distinct minor events. During the rise phase of the
soft X-ray flux, bursts produced by nonthermal electrons are observed in hard
X-rays and microwaves (second and third panels from bottom). The hard X-ray
emission is bremsstrahlung produced by electrons of several tens of keV, the
microwave emission is gyrosynchrotron radiation from electrons of several tens
to hundreds of keV. The microwave data are from the Sagamore Hill (SGMR)
Station (USA) of the Radio Solar Telescope Network
(RSTN)333https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-
features/solar-radio/rstn-1-second/. The two top panels display dynamic
spectrograms in the (180-25) MHz range (SGMR444Data from
https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-
radio/rstn-spectral/) and the (13.6-1) MHz
range555https://cdaweb.gsfc.nasa.gov/pub/data/wind/waves/ (Wind/WAVES;
Bougeret et al., 1995). They show several groups of type III bursts, produced
by electron beams at a few tens of keV propagating outward from the low corona
along open magnetic field lines. The hard X-ray and microwave bursts in the
low solar atmosphere occur at the times of well-identified and rather strong
type III bursts. Besides at times of the main HXR and microwave emission there
are tiny events, which accompany the other type III bursts. But the event
clearly has one main episode of energetic electron acceleration in the low
corona, lasting a few minutes, and the type III bursts demonstrate that
simultaneously electrons escape along open field lines to the high corona and
the interplanetary space.
The event is in many respects representative of our data set. Firstly, the
electron events are accompanied by hard X-ray and microwave bursts and by
metric-to-hectometric type III bursts. Secondly, the microwave bursts of 14
out of 17 events show time profiles and flux density spectra typical of
gyrosynchrotron emission666We note that while it is in principle possible to
constrain the electron spectrum from the optically thin gyrosynchrotron
spectrum, a simple relationship between the two indices exists only in the
extremely relativistic case, which generally does not apply to solar microwave
bursts. In the case of the analyzed events, the microwave bursts are weak (a
few tens of sfu at most), only a few percent of the quiet-Sun background.
Although the nonthermal part can be identified, it is likely superposed on a
thermal bremsstrahlung component. Under these circumstances, the microwave
emission does not offer a valuable quantitative constraint of the electron
spectral index., which suggests that electrons are accelerated to energies
above 100 keV. In two other events the nonthermal microwave emission seems to
be plasma emission, in one event only thermal bremsstrahlung is observed. A
third general feature is the gyrosynchrotron emission that lasts from 10 s to
5 min, and the nonthermal HXR emission lasting from 30 s to 2 min, in temporal
coincidence with type III emission at m-$\lambda$. Finally, in most events
(13/17) several groups of type III bursts are observed, and only one is
accompanied by a clear HXR and microwave burst.
### 3.2 In situ observations of electrons
The top panel of Fig. 2 displays the intensity measured by SEPT in all energy
channels and the circles at the intensity maxima mark the peak intensities
used to construct the background-subtracted peak intensity spectrum, which is
shown in Fig. 3 (right). The second panel from top shows the intensity of
65-75 keV electrons in all four viewing directions of SEPT with the
corresponding pitch angles of the telescope center axes plotted below. We
note, that in case of anisotropic events (like shown in Fig. 2), we always use
the spectrum observed in the viewing direction measuring the highest
intensity. In case of isotropy, the Sun-facing telescope was used. A poor
pitch-angle coverage due to non-nominal magnetic field configurations can
seemingly reduce or even hide the real anisotropy of a particle beam. As in
such cases the center of the particle beam with the highest electron
intensities is not well observed, this can potentially also affect the
determined spectrum and spectral shape (see also section 3.5).
### 3.3 Analysis of photon and electron spectra
For the associated HXR bursts, the peak time of the nonthermal emission was
determined from RHESSI lightcurves by using the highest energy range in which
a flare signature was observed. In most cases, this was in the range of 25-50
keV. In all cases, the used lightcurves showed a more impulsive behavior as
seen in the 6-12 keV band, which is always dominated by thermal emission for
medium to large flares. In all events, the duration of the nonthermal peak was
around 30 sec. A RHESSI count spectrum was then obtained by integrating over a
30 sec time interval centered at this HXR peak time as marked by the vertical
lines in Fig. 1, using a combination of all detectors that were well-
functioning according to the RHESSI detector health database. Inspecting the
background-subtracted spectra, we found the nonthermal part to be consistent
with a single power-law in all events. The spectra were then fitted with the
combination of an isothermal plasma component, a broken power-law with a fixed
slope of -1.5 below the break that reproduces the nonthermal component
(Emslie, 2003). A photospheric albedo component was taken into account using
the standard RHESSI OSPEX spectral analysis
package777https://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex_explanation.htm
(based on Kontar et al. (2006)), assuming an isotropic electron distribution,
which is more consistent with observations than a strongly beamed distribution
(cf. Kontar & Brown, 2006). We have additionally performed the spectral
fitting without an albedo component and found only small differences of the
spectral indices (smaller than 0.1) and conclude that the albedo component
does not influence our results.
The nonthermal emission is thus characterized by the spectral index $\gamma$
above the break. The upper energy limit of the fit range was determined by the
counting statistics, which implied that several events could only be fitted up
to 30 keV (the limit was 60 keV on average). Since photons of a given energy
are mainly produced by electrons of about twice this energy, we do have an
energy overlap with SEPT even in these cases. The resulting spectrum and
corresponding fits for the example event on 24 Mar 2011 are shown in Fig. 3
(left). Additionally, we also performed thick-target fits to the spectra,
which we use to determine the number of accelerated electrons (see Sect. 3.6).
Using the resulting spectral indices from these thick-target fits in our
correlation analysis (not shown) does not change the correlation but as it
involves extra model assumptions, we decided not to use these values. We
furthermore refrained from using fits applying the thin-target model since we
see no evidence for thin-target emission, as discussed in Section 3.5.
Figure 4: Left: Histogram of the time delay $\Delta t$ between the nonthermal
HXR flare peak and the onset of the SEE event at STEREO. The black histogram
refers to all events, while the green one shows the distribution for the
events that are potentially associated with CMEs. The dotted lines indicate
the median of the distributions. Middle: Histogram of the longitude separation
$\Delta\Phi$ between the flare location and the extrapolated footpoint of the
magnetic field line connecting STEREO to the Sun. Right: Correlation plot of
the time delay $\Delta t$ and the longitude separation $\Delta\Phi$. The
colors denote the degree of anisotropy of the in situ electrons. $C$ indicates
the linear correlation coefficient.
The right-hand side shows the corresponding spectrum of the in situ electron
event and a broken-power law fit. For each event, the electron observations of
SEPT were corrected from potential contamination due to ions or higher energy
electrons as described in Dresing et al. (2020). To determine the spectrum of
the event we use the peak intensities observed individually in each available
energy channel (marked by the circles in the top panel of Fig. 2) after a pre-
event background subtraction has been applied. We then fit each spectrum with
single- and double-power law functions and chose the better fit based on the
reduced chi square of the fits. The fits take into account the uncertainties
of the peak intensities caused by counting statistics as well as the energy
bin widths representing an uncertainty in energy. The uncertainties of the fit
parameters were determined using 95%-confidence intervals. For details see
Dresing et al. (2020) and Strauss et al. (2020). We note that determining
electron fluence spectra with STEREO/SEPT data is complicated due to the issue
of ion contamination that is usually more dominant during the later phase of
the events. Determining reliable fluence values and, especially,
uncertainties, which are required by the fit, is difficult. We therefore
decided not to present an analysis of the in situ fluence spectra here but
comment on potential differences between the peak flux and the fluence spectra
in Section 4.2.
Table 1 shows the basic parameters for all 19 flare and SEE events, including
the HXR peak time (as detected at the spacecraft), GOES class, flare location,
HXR photon spectral index $\gamma$, in situ electron spectral indices
$\delta_{1}$ and $\delta_{2}$888If $\delta_{2}$ and $E_{b}$ are missing, a
single power law fit was used, spectral break energy $E_{\mathrm{b}}$, which
STEREO spacecraft detected the SEE event, the time delay $\Delta t$ between
the nonthermal HXR flare peak (corrected for light travel time) and the onset
of the SEE event at STEREO, propagation path length $L$ of 55-85 keV electrons
corresponding to $\Delta t$ (see section 3.4), and the longitude and
latitudinal separations $\Delta\Phi$ and $\Delta\Theta$ between the flare
location and the ballistically extrapolated footpoint of the magnetic field
line connecting the spacecraft to the Sun.
### 3.4 Timing and magnetic connectivity
The onset times of the SEPT electron events were determined using the
3$\sigma$ method, which can only be considered as an upper limit for the real
onset time. An earlier onset could be masked by the background noise of the
detector (Laitinen et al., 2010). Furthermore, especially in the case of more
gradual intensity increases, which can also be caused by non-nominal magnetic
field orientations resulting in a poor pitch-angle coverage at the SEPT
instrument, the 3$\sigma$ method can yield onset times that are too late.
Larger time averaging is often used to overcome these issues, and the time
averaging applied for each event has been used as a measure for the
uncertainty of the onset time (Dresing et al., 2020). This uncertainty has
been propagated to $\Delta t$ in table 1 in cases when higher averaging was
used. Otherwise, the uncertainty represents the time resolution of the
STEREO/SEPT data of 1 minute. The time delays $\Delta t$ in our sample, which
range from 29 to 61 minutes, with a median of $\Delta t$ = 37 minutes, are the
delays between the HXR peak time at the Sun and the detection of the $55-85$
keV electron onset. Assuming that the in situ measured electrons were injected
at the time of the HXR peak, $\Delta t$ represents the propagation time of the
first arriving particles. A scatter-free propagation along a nominal Parker
spiral would take about 20 minutes at these energies. Comparing $\Delta t$
with this nominal and scatter-free propagation time we find that all events in
our sample arrive delayed.
The events with medium to large anisotropy at their onset (see Table 3), where
the spacecraft was likely well-connected to the electron source, have delays
between 29 and 40 minutes. While these delays are among the shorter ones in
our sample, they still exceed expectation for scatter-free propagation by 9-20
minutes.
Therefore, our event sample does not seems to contain any true prompt events
as defined by Krucker et al. (2007) with a delay between the flare and the
inferred injection time of the in situ measured electrons of $<8$ minutes.
There can be several reasons for the observed delays. On the one hand, it
could be due to instrumental effects leading to a delayed onset determination
as described above. Furthermore, we note, that the pitch-angle coverage was
not ideal in the majority of the events, which could cause apparent onset
delays especially if the electron beam was very narrow during the early phase
of the event. In this case, the delay would not represent a physical process
related to the acceleration, injection, or transport of the particles. On the
other hand, the actual propagation path of the electrons could be longer than
the nominal Parker spiral, either due to large-scale distortions of the field
or due to pitch-angle scattering. Field-line random walk caused by footpoint
motion tied to the solar convection or by the turbulent solar wind evolution
can cause significant lengthening of the actual path that particles need to
follow along the magnetic field. Recent studies find that the effective field
line length can be up to twice as long as the nominal Parker connection (see,
e.g., Laitinen et al., 2016; Laitinen & Dalla, 2019; Chhiber et al., 2020),
which can cause equivalent delays in the observed particle onsets. Assuming
that such effects or scattering cause the delays, i.e. the electrons were
injected at the flare peak time, the column $L$ in the table provides the path
lengths corresponding to $\Delta t$ for 55-85 keV electrons. Finally, an
actually delayed injection of the electrons at the Sun can not be ruled out a
priori. Different effects or a mixture of them being dominant for different
events in our sample are, of course, also possible.
Figure 4 shows histograms of $\Delta t$ and the longitudinal separations
$\Delta\Phi$. Despite the partially significant delays, the majority of the
events is magnetically well-connected in longitude to STEREO, with
$\Delta\Phi$ between 0.5 and 54 degrees (median: 15 degrees). There is no
correlation between onset delay and longitude separation (see Fig. 4 right).
We also do not find any correlation with the latitudinal separation
$\Delta\Theta$ (listed in Table 1). Furthermore, the degree of anisotropy
(marked by the color in the right panel) does not correlate with the onset
delays suggesting that interplanetary pitch-angle scattering is not the main
reason for the observed delays. The green histograms in Fig. 4 denote those
events associated with CMEs that are located close to the flaring region and
have heights below one solar radius at the time of the flare (see appendix A).
They may therefore potentially influence the injection and coronal propagation
of the electrons. While the events associated with these CMEs are not
outstanding in other parameters of our analysis they seem to be among those
events showing larger separation angles (middle panel of Fig. 4) suggesting
that the CME may be involved in enlarging the injection region. The fact that
many of the analyzed events are observed in different magnetic polarity
sectors than that of the flaring active region at the Sun (see Table 3) could
furthermore suggest an injection that covers a wider angular range even across
the neutral line. Such a scenario was also suggested by Kallenrode (1993) who,
however, reported that a current sheet crossing can also lead to a decrease in
SEP intensities. On the other hand, pitch-angle scattering in the IP medium
may also be involved in filling both sides of the heliospheric current sheet
with the electron population. The majority of our events are also accompanied
by coronal EUV jets (see appendix A) but no particular connection with the
time delay or separation angle was found.
### 3.5 Spectral correlations
The results of correlating the spectral indices of the HXR photons and the in
situ electron measurements are shown in Figures 5 and 6. As discussed above
several energy-dependent effects may alter the spectrum observed in situ
eventually leading to spectral breaks, which requires to make a choice between
the two different spectral indices. For the events in our sample this choice
is often not straightforward as breaks are often found below or around 100
keV, which is in between the mean locations of the two different spectral
breaks (60 keV and 120 keV) as reported for instance by, Krucker et al.
(2007); Dresing et al. (2020). We therefore start by using always the lower
($\delta_{1}$) or the upper spectral index ($\delta_{2}$) as observed by SEPT
in case of broken power-laws and compare these with the HXR spectral index
$\gamma$. The corresponding correlation plots are shown in figures 5 and 6.
Note that in case of single power-law shapes the same in situ spectral index
has been used in both plots. The panels on the left of the figures include all
17 events, while the ones on the right consider only the nine events that
showed a distinct anisotropy (medium to large; see appendix A and last column
of table 3).
Figure 5: Correlation plots of the HXR photon spectral index $\gamma$ observed
by RHESSI and the electron spectral index $\delta$ obtained from STEREO/SEPT,
plotted for all events (left) and only the events with distinct anisotropy of
the in situ electron flux (right). In the case of events with broken power-
laws, we use here the spectral index below the break, $\delta_{1}$. For three
events, we consider either $\delta_{1}$ (red), or $\delta_{2}$ (blue). The
dotted lines indicate linear fits to the data. The dashed lines indicate the
relationships between $\gamma$ and $\delta$ expected for thick and thin-target
bremsstrahlung, respectively. The fit parameters are indicated, as well as the
linear correlation coefficients $C$.
Figure 6: As in Fig. 5, but showing the correlation of the HXR photon spectral
index $\gamma$ with the electron spectral index $\delta_{2}$ (characterizing
the high-energy part of the SEE spectrum) instead of $\delta_{1}$ in the cases
of a broken power-law. We consider either all events including the three
special cases (red), or we omit these events (black).
Three events appear to behave differently as their spectral indices
$\delta_{1}$ and $\delta_{2}$ lie systematically below the rest of the
distributions (marked by colored points in figures 5 and 6). We suspect that
these three cases represent events showing a spectral break caused by
Langmuir-wave generation (expected to occur at lower energies than the break
due to pitch-angle scattering) while the rest of the broken-power-law events
corresponds to breaks caused by pitch-angle scattering (see appendix A.2 for
more discussion on these events). Fig. 7 helps to illustrate this. It shows a
broken power-law spectrum containing two breaks and therefore three different
spectral indices $\delta_{0}$, $\delta_{1}$ and $\delta_{2}$. The first break,
marked by $\mbox{E}_{\mbox{b\\_L}}$ is assumed to correspond to Langmuir-wave
generation and the second break ( $\mbox{E}_{\mbox{b\\_trans}}$) to pitch-
angle scattering during interplanetary transport, respectively. According to
the description above, the three special events would cover the spectral part
of $\delta_{0}$ and $\delta_{1}$ while the other events correspond to
$\delta_{1}$ and $\delta_{2}$ in this sketch. To correctly handle the spectral
indices of these three events we therefore have to treat their $\delta_{2}$
values as $\delta_{1}$ values. Their $\delta_{1}$ would correspond to
$\delta_{0}$ in the sketch (not covered by the other events) and the
$\delta_{2}$ range of the other events is not covered by these three events.
We therefore have to use the blue points instead of red points in Fig. 5 and
remove the $\delta_{2}$ values (red points) from the correlation in Fig. 6.
Figure 5 shows that $\gamma$ and $\delta_{1}$ are correlated. For the three
events discussed above, both spectral indices $\delta_{1}$ (red, suspected
wrong values) and $\delta_{2}$ (blue, suspected correct values) are included.
The differently colored legends represent the Pearson correlation coefficient
and linear fit results using either $\delta_{1}$ (red) or $\delta_{2}$ (blue)
of these three events. Similarly, for Fig. 6 the correlation in red
corresponds to using all $\delta_{2}$ values in the sample, and the one in
black excludes the three red points as discussed above assuming that these
events do not provide a corresponding $\delta_{2}$ observation. The
correlations in both plots improve significantly when treating the three
events as discussed. Although the correlations with the lower and higher in
situ spectral indices are very similar, the implications for the relation
between the photon and in situ spectral indices are very different: The value
pairs in Fig. 5 clearly align along the thin-target approximation, while the
points in Fig. 6 shift to the range between thick and thin-target lines with
some points even lying above the thick target model.
Taking the correlation of $\gamma$ and $\delta_{1}$ at face value, one could
conclude that the nonthermal photon spectrum is actually generated by thin-
target emission, and that indeed both the remote-sensing and the in situ
observations detect the same particle population. However, thin-target
emission is generally believed to dominate nonthermal flare spectra only in
those cases where the HXR footpoints are occulted by the solar limb (cf.
Krucker et al., 2008). The footpoints are the locations where accelerated
electrons are stopped collisionally in the denser chromosphere, and
consequently they emit thick-target radiation, which will usually dominate any
thin-target contribution from the corona. We have performed HXR imaging with
RHESSI for all flares in order to ascertain whether HXR footpoints are present
or could be potentially obscured. In 7 flares, footpoint pairs could be
unambiguously detected, while in another 7 events, only possible indications
for footpoints could be found due to the limited image fidelity caused by the
small number of nonthermal counts. In the rest of the events, nonthermal and
thermal sources were found to be contiguous. Only one flare was located so
close to the limb that an occultation of the footpoints is possible. We find
no systematic differences in the spectral correlations for the events with
clearly visible footpoints as compared to the other flares. We thus conclude
that the HXR spectrum is actually dominated by thick-target emission.
Table 2 summarizes the determined correlation coefficients between the photon
spectral index $\gamma$ and the two electron spectral indices $\delta_{1}$ and
$\delta_{2}$, also including further subsamples based on a rough anisotropy
classification of the events (see appendix A and last column of table 3). The
correlations clearly improve if only anisotropic events are taken into
account, however the number of events with large anisotropies is unfortunately
very low. We also investigated the quality of the correlations with respect to
the onset delay, the longitudinal, and latitudinal separation angles between
flare and spacecraft magnetic footpoint at the Sun as well as the presence of
CMEs or EUV jets but did not find any dependence on these parameters.
Table 2: Pearson correlation coefficients $C$ between photon spectral index
$\gamma$ and in situ electron spectral index $\delta$. Also given are the
uncertainties on $C$ based on a bootstrapping approach.
| all | medium & large | large
---|---|---|---
| events | anisotropy | anisotropy
no. of events | 17 | 9 | 4
$\gamma$ vs. $\delta_{1}$ | $0.50\pm 0.19$ | $0.69\pm 0.20$ | $0.61\pm 0.55$
$\gamma$ vs. $\delta_{2}$ | $0.49\pm 0.19$ | $0.27\pm 0.44$ | $0.28\pm 0.77$
$\gamma$ vs. $\delta_{1}$11$1$I | $0.80\pm 0.08$ | $0.86\pm 0.09$ | $0.96\pm 0.06$
$\gamma$ vs. $\delta_{2}$22$2$I | $0.79\pm 0.09$ | $0.88\pm 0.18$ | $1.0\pm 10^{-4}$
999 n situ electron spectral index $\delta_{2}$ has been used for the three
special events (see text). n situ electron spectral index $\delta_{2}$ has
been excluded for the three special events (see text).
Figure 7: Sketch illustrating a broken-power law spectrum with two breaks
$\mbox{E}_{\mbox{b\\_L}}$ and $\mbox{E}_{\mbox{b\\_trans}}$ forming three
spectral indices $\delta_{0}$, $\delta_{1}$ and $\delta_{2}$.
Figure 8: Correlation plots of the number of nonthermal electrons above 45 keV
in the flares derived from thick-target fits of HXR spectra and the number of
electrons escaping into interplanetary space, plotted for all events (left)
and only the events with distinct anisotropy of the in situ electron flux
(right).
### 3.6 Total number of escaping electrons
Figure 8 shows the total number of escaping electrons inferred from the in
situ observations ($>45$ keV and $<425$ keV) as a function of the number of
electrons $>45$ keV accelerated in the flare assuming thick-target emission
and integrating the derived peak flux over 30 seconds. The number of in situ
electrons has been determined by integrating the fluence spectrum of the
events observed in the telescope showing the highest intensity increase, which
was also used to construct the peak intensity spectrum. The fluence is the
integrated flux over the duration of the event, with the duration being
defined by the onset of the event and an individual end time. This end time
was determined for each energy channel individually by the time when the flux
decreased to $1/e$ of the peak flux. In the same manner as for the peak
intensity spectra, analyzed in this work, the pre-event background was
subtracted from the flux before calculating the fluence and the contamination
correction was applied (c.f. Section 3.3) assuming a constant contamination
over the duration of the event.
Following Krucker et al. (2007) we assume an electron beam that is emitted
into a cone with a width of $30^{\circ}$ to derive the total number of in situ
electrons, and we use cones of $15^{\circ}$ and $60^{\circ}$ to estimate the
uncertainties shown in Fig. 8. Propagating this cone along the nominal Parker
spiral to a distance of 1 AU results in a spherical cap over which the
electrons are spread at the spacecraft position. To infer the total number of
electrons of the event, the number of electrons detected in the small SEPT
instrument are therefore scaled to the area of this spherical surface. We
note, that this approach assumes a constant energetic electron density over
the cone angle, which is only a first-order approximation. Secondly, the
sectored measurements by SEPT show that the observed particle distribution at
the spacecraft is much more isotropic compared to the assumed cone, which is
expected in case of non-negligible particle scattering during interplanetary
transport. However, without proper transport simulations and multi-spacecraft
observations a more accurate approach demanding fewer assumptions to determine
the total number of in situ electrons is not feasible.
The left hand plot of Fig. 8 shows all events in our sample, while the right
hand figure shows only events with significant anisotropy. The legends provide
the results of linear fits and the Pearson correlation coefficients. A
reasonable correlation of 0.75 is found for all events, and a slightly lower
correlation of 0.6 for the anisotropic events. Although one would rather
expect a larger correlation for anisotropic events, i.e. in the case of less
pitch-angle scattering, an effect that can reduce the determined number of
escaping electrons, the anisotropic sample does indeed contain on average
higher numbers of escaping in situ electrons. In contrast to results by James
et al. (2017), who used data by ACE/EPAM (Gold et al., 1998) and determined a
fraction of 6 to 148% of escaping electrons compared to the HXR-producing
electrons our results confirm the findings by Krucker et al. (2007). We find
very small fractions of escaping electrons with mean ratios of only 0.18% for
all events and 0.24% for the anisotropic events. For comparison, Krucker et
al. (2007) found a ratio of 0.2%.
## 4 Discussion
### 4.1 Modification of electron spectra between the acceleration region and
the spacecraft
Transport processes in the flare and in interplanetary space can alter the
initial spectral distribution of nonthermal electrons. One such process that
causes energy loss is due to Coulomb collisions (cf. Brown, 1971) when
energetic electrons propagate through a background plasma. For the escaping
electrons, this effect is negligible in interplanetary space, but it can play
a role near the acceleration site. Reid & Kontar (2013) concluded that coulomb
collisions are negligible as an energy-loss process above an energy near 40
keV, which is in the relevant range for our analysis. However, because
collisional energy loss will lead to a progressive flattening of the electron
flux spectrum toward lower energies, it could thus potentially account for the
fact that the in situ spectra are harder than the ones inferred from HXR
observations (see Fig. 5). We have, therefore, considered a model of the
ambient electron density consisting of a 2.5-fold Newkirk model (Newkirk,
1961) in the lower corona, which then transitions to the Mann model (Mann et
al., 1999) that is more appropriate for the upper corona and IP space. We
inferred the electron density in the corona above the acceleration region from
the starting frequency of the type III bursts, as observed by the worldwide
e-Callisto network (www.e-callisto.org). We found the median start frequency
near 330 MHz. Under the usual assumption of harmonic plasma emission this
corresponds to an ambient electron density of $3.4\cdot 10^{8}$ cm-3 and an
acceleration region below 0.1 R⊙ above the photosphere. Taking this value,
integrating from this height to 1 AU, and computing the effect of the spectral
flattening according to Brown (1971), we find that Coulomb collisions have
indeed a negligible effect on the spectral index in the energy range we have
considered here, and thus cannot account for the spectral differences between
precipitating and escaping electrons.
While Coulomb collisions may be neglected in our analysis, other processes
such as the generation of Langmuir waves or pitch-angle scattering may cause
spectral changes both in the flare and in the interplanetary medium (as
discussed in Section 3.5). Kontar et al. (2014) showed for example that strong
pitch-angle scattering in coronal loops can potentially cause a flattening of
the HXR spectrum and lead to spectral breaks. However, several previous
studies have found significant correlations between the HXR and in situ
electron spectral index (e.g., Kallenrode et al., 1987; Dröge, 1996; Krucker
et al., 2007) and it was suggested that 1) both spectra belong to the same
accelerated population and 2) that transport effects do only play a minor role
in altering the spectra. Many of such past studies, which analyzed solar
energetic electron spectra have used instruments, which covered only energies
$\gtrsim 100$ keV and which lacked a fine energy resolution in the lower
energy part that is needed to resolve spectral breaks caused by interplanetary
transport processes, i.e. the generation of Langmuir turbulence and pitch-
angle scattering, as discussed in this manuscript. Furthermore, both of the
mentioned transport processes are dominant at lower or near-relativistic
energies (Dröge, 2003; Agueda et al., 2014) making it difficult to detect
these with instrumentation covering mainly $\gtrsim 100$ keV. Instruments such
as the Wind/3DP and STEREO/SEPT mainly measure near-relativistic electrons
where these spectral breaks occur. However, because of the slightly different
energy coverage and resolution of the Wind/3DP and STEREO/SEPT instruments
they are likely sensitive to different spectral breaks with 3DP usually
detecting the break caused by Langmuir-wave generation and SEPT the one by
pitch-angle scattering (see Krucker et al., 2009; Kontar & Reid, 2009; Dresing
et al., 2020; Strauss et al., 2020) so that the spectral index above the break
$\delta_{2}$ detected by 3DP is likely covered by the spectral index below the
break $\delta_{1}$ as seen in SEPT data. However, an unambiguous
identification of the type of the spectral break in a single event can be
difficult because of the overlapping energy ranges of these effects. Three
events in our sample showed a systematic shift with respect to the spectral
indices of the rest of the events suggesting that these were indeed events
covering the Langmuir wave-related spectral break different to the other
events. Therefore, these events have been re-categorized accordingly (see
section 3.5 and figures 5 and 6) leading to significantly improved
correlations.
### 4.2 The spectral correlation of HXR-producing and in situ electrons
We find correlations between the HXR spectral index and the one of the in situ
electron spectrum of about 0.8 for both sets of value pairs either using the
spectral index below or above the break (see Table 2), however when using
$\delta_{1}$ the points align along the thin-target solution (Fig. 5), while
they are shifted more toward the thick-target line (lying between thick and
thin-target lines) when using $\delta_{2}$ (Fig. 6). A preliminary analysis of
the corresponding SEPT fluence spectra showed overall no change for the
$\delta_{1}$ values, but the $\delta_{2}$ values showed a trend of shifting
toward softer spectral indices causing the data points to align more along the
thick-target solution or lying even above it in the correlation plot (not
shown). However, due to the issue of ion contamination these fluence spectra
suffer large uncertainties and are less reliable.
We assume that our value pairs using $\delta_{1}$ should be compared to the
values shown by Krucker et al. (2007) who compared HXR spectra observed by
RHESSI with Wind/3DP electron spectra above the break ($\delta_{2}$). They
found a similar correlation of about 0.8, however, their value pairs were
lying between the thick and thin target solutions. The alignment of our value
pairs using $\delta_{1}$ along the thin-target model is not expected since
imaging of nonthermal footpoints shows that in most events thick-target
emission has to be dominant. After excluding also the role of energy loss due
to Coulomb collisions (see Section 4.1) we therefore suspect that another
systematic effect causes this shift. Krucker et al. (2007) found only a good
correlation for non-delayed events suggesting that the delays, frequently
observed during solar energetic electron events (e.g., Haggerty & Roelof,
2002; Kahler et al., 2007), may be caused by another or an additional
acceleration process. This is different for the events studied in this work:
all of our events show delays of at least 9 minutes with respect to the flare.
However, as discussed in Section 3.4 instrumental effects and especially
nonideal pitch-angle coverage could often be the cause for apparent delays.
Although Krucker et al. (2007) found only a low correlation of 0.43 for their
delayed event sample, those value pairs did also show a shift toward the thin-
target line, i.e. toward harder in situ electron spectra like our
$\gamma-\delta_{1}$-value pairs. Petrosian (2016) suggested that a further
acceleration process acting on the flare-accelerated electron distribution
could be the reason for this shift. However, given the still significant
correlation coefficient for our value pairs we note that such a further
acceleration should either only be of minor importance or scale with the flare
itself, so that only a systematic shift of the spectral index is caused, and
the overall correlation is preserved. We note that Krucker et al. (2007) also
suggested the possibility of further acceleration in the flare due to trapping
in closed and shrinking field lines. This would, however, only affect the
downward moving electrons, which produce the HXR spectrum.
### 4.3 Anisotropic electron events and the number of escaping electrons
We find a clear improvement of our correlations when only taking into account
anisotropic events suggesting that pitch-angle scattering can lead to a
vanishing imprint of the acceleration. We note that the lack of anisotropy can
either be caused by strong scattering during interplanetary transport but also
due to poor pitch-angle coverage of SEPT caused by non-nominal magnetic field
configurations. See table 3, which lists the strength of the anisotropy for
each event and marks if poor pitch-angle coverage was present. Consequently,
even in the case of low scattering conditions, the limitations of the
measurement can lead to vanishing correlations, i.e. to a change of the
observed spectral indices. We note, that Wind/3DP does not suffer from such a
limitation because of its unique directional observations covering the
complete 4$\pi$ space, which is for example not the case for the ACE/EPAM
instrument (Gold et al., 1998). Although EPAM/LEMS provides eight different
sectors determined through the spacecraft’s spin, there exist magnetic field
directions that are perpendicular to the measurement plane of EPAM/LEMS
reducing the pitch-angle coverage to only one point in pitch-angle space for
all sectors.
The anisotropy is also expected to influence the correlation between the
number of escaping electrons with the number of flare electrons as shown in
Fig. 8 because larger anisotropies imply weaker interplanetary scattering
conditions and therefore less dispersal of the injected population. However,
we do not find an improvement of the correlations when only taking into
account the anisotropic events. But poor pitch-angle coverage, which can lead
to the underestimation of the anisotropy as well as to an underestimation of
the total number of electrons, may again affect the results. Furthermore, the
strong assumptions, which have to be made to determine the number of escaping
electrons likely play an even larger role for the weak correlation. These
assumptions are i) the same size of the cone ($30^{\circ}$) filled with
electrons for each event, which is very likely not true due to varying
injection cone sizes, i.e. the angular width of open magnetic field lines. ii)
A constant distribution of electrons over the $30^{\circ}$ cone, which is only
a first-order approximation because of the observed presence of weak or
missing anisotropies depicting the presence of interplanetary scattering.
Furthermore, an angular-dependent injection function is possible. iii) A
nominal Parker field with a fixed Parker spiral length for all events, not
taking into account large-scale structures or field line meandering, which
will further introduce event to event variations leading to a vanishing
correlation. Nevertheless, our results confirm the findings by Krucker et al.
(2007) that only a very small fraction ($\sim 0.2$%) of accelerated electrons
are finally injected into interplanetary space when compared to the number of
the HXR producing electrons.
### 4.4 The timing of electron release in the corona
A simple picture of the relationship between electrons detected at 1 AU and
radio or X-ray emitting electrons in the solar atmosphere is a common short
acceleration and the release into magnetic structures in the corona and onto
open field lines to the heliosphere. This picture is supported by a number of
detailed timing studies of the start times of impulsive electron events (see
the discussion of the 2000/05/01 case in Klein, 2021). But in other events the
first electrons were reported to arrive later at the spacecraft than expected.
Instruments with limited pitch-angle coverage, such as the STEREO detectors,
might just miss the first arriving electrons, creating an apparent delay.
Linhua Wang and coworkers (Wang et al., 2006a, 2016) exploited the excellent
pitch-angle coverage of the Wind/3DP instrument in systematic analyses of the
onset time and the duration of release of electrons across the energy range
between a few keV and some tens or hundreds of keV. They found that electrons
detected with energies below 20 keV at the spacecraft were released since
earlier times and over longer durations than the electrons above about 30 keV.
Kontar & Reid (2009) ascribe such inferred earlier releases to the energy loss
of the Langmuir-wave generating (low-energy) electrons. As an alternative, or
an addition, our observations suggest a scenario where electron beams are
accelerated to different energies in several successive episodes: we find that
the solar counterparts of the electron events are in general groups of type
III bursts, discernible at frequencies above $\sim$1 MHz. As illustrated in
Fig. 1, the hard X-ray and microwave emissions, which are produced by
electrons at tens to hundreds of keV, accompany one of these groups of type
III bursts, but other groups occur before and afterward. Such additional type
III bursts without prominent microwave counterpart are observed in 13/17 of
our events. This is also shown by the timing of the microwave bursts and the
DH type III bursts (Table 3, cols. 2 and 4). Since type III bursts at 1 AU are
produced by electrons with energies below 20 keV (e.g., Ergun et al., 1998),
this timing implies that repeated episodes of impulsive electron acceleration
to relatively low energies (a few tens of keV) occur in the corona, and that
one such episode is also accompanied by the acceleration of near-relativistic
electrons. The low-energy electrons are hence accelerated over longer times
than those of higher energies that are seen through their hard X-ray and
microwave emission, consistent with Wang and coworkers. But the higher time
resolution offered by the radio and HXR analysis shows that the observations
cannot be ascribed to a single, time-extended acceleration episode of low-
energy electrons, and a different, shorter and delayed, episode at the higher
energies. The electron fluxes at 1 AU rather result from multiple injections
of electrons in the corona. Each release produces electrons up to a few tens
of keV that are able to emit type III bursts, but the near-relativistic
electrons are only accelerated during part of these acceleration episodes.
This particular interval actually consists of several successive events, too,
as seen by the multiple microwave peaks. Vlahos & Raoult (1995) argued indeed
that apparently individual type III bursts actually result from multiple
energy releases, and Chen et al. (2013) confirmed this idea by the Karl G.
Jansky Very Large Array (VLA) observations with high temporal and spectral
resolution.
### 4.5 Association with jets and CMEs
The association of the electron events in the present study with various types
of EUV activities, including jets and large-scale mass motions, is in line
with earlier studies on 3He-rich SEP events (Wang et al., 2006b; Pick et al.,
2006; Nitta et al., 2006, 2015). However, we do not find a unique association
with EUV jets. Y-M Wang and coworkers (Wang et al., 2006b; Pick et al., 2006)
used SoHO observations to identify EUV jets associated with 3He-rich SEP
events (13/21 events with suitable observations). They suggested that EUV
imaging with higher cadence would reveal more jets, and proposed a model where
the EUV jet was the signature of magnetic reconnection between closed and open
coronal magnetic field lines, which also led to the particle acceleration of
electrons and ions. But high-cadence EUV imaging from STEREO and especially
SDO used in the present work, as in Nitta et al. (2015), does not confirm the
expected systematic association between impulsive SEP events and coronal
plasma jets. In some events the eruptive coronal activity may hide the plasma
jet, but in some others jets are definitely not detected with SDO/AIA.
However, there are also events with a clear timing correspondence, within a
few minutes, of the electron acceleration in the corona revealed by hard X-ray
and microwave emission, and the plasma jet. There is no a priori reason to
exclude magnetic reconnection on the sole reason that no EUV jet is observed.
The association with type III bursts and the PFSS extrapolations show the
existence of open field lines in the parent active region. The scenario of
interchange reconnection (e.g., Shibata et al., 1994) as the origin of
impulsive electron events therefore remains attractive, although the
observations do not provide a unique simple picture where impulsive electron
events would be exclusively associated with EUV jets, rather than with
eruptive activity on larger scales.
The association of electron events with CMEs gives another hint to a more
diversified picture of the origin of impulsive particle events than the
historic two-class picture of either impulsive (flare-associated) or gradual
(CME-associated) SEP-events (Reames, 1999). CMEs are not an occasional
counterpart of the impulsive electron events studied here, but some white-
light signature is observed with virtually all our electron events. In the
only case where we did not identify a CME in the vicinity of the position
angle of the parent flare, 2007 Jan 24, a faint signature might be hidden by a
broad CME from a distinct active region. In the other events CMEs are seen in
the corona. Their morphologies range from jet-like, which on occasion (2010
Nov 17) are clearly the extension of an EUV jet to the higher corona, to
extended, as already reported by Wang et al. (2006b). The combination of
coronographic observations from SoHO and STEREO offered us the possibility to
have in most events one spacecraft, which saw the parent activity close to the
solar limb, so that the identification of a CME was much easier than in the
earlier SoHO observations. The enhanced cadence of the STEREO coronagraphs
also allows for a better timing identifications. In three out of 14 events
where adequate coronographic observations were available, the CME came from
previous erupting activity. They show that the CME is not a necessary
condition for the electrons to achieve near-relativistic energies. But in nine
cases the extrapolated height of the CME at the time of the hard X-ray and
microwave burst, i.e. at the time of acceleration of near-relativistic
electrons, was a few fractions of a solar radius above the photosphere. The
presence of these low-altitude CMEs seems to be associated with those events
at larger longitudinal separation angles (see Fig. 4). These CMEs might hence
play a role in widening the injection region, for example due to field-line
spreading or deflection in the corona, or to electron acceleration on open
field lines remote from the parent active region (e.g., Salas-Matamoros et
al., 2016).
## 5 Summary and conclusions
For 17 different solar flare events we correlated the HXR spectral
characteristics measured by RHESSI with the corresponding spectra of electron
events observed in situ with STEREO/SEPT. Most of the in situ electron events
show broken power-law spectra presumably caused by transport effects. At least
two processes during interplanetary transport have been identified that are
capable of causing such spectral breaks: i) the generation of Langmuir
turbulence and ii) pitch-angle scattering (see discussion above). However, the
range of the potential positions of these different breaks overlap at
$\sim$$100\text{\,}\mathrm{keV}$. This and the limited energy range and
resolution of energetic particle instruments are likely the reasons why
usually only a single break is identified in solar energetic electron spectra
(e.g., Lin et al., 1982; Reames et al., 1985; Krucker et al., 2009; Dresing et
al., 2020). It can therefore be not straightforward to identify which part of
the spectrum is least influenced by the above effects and best suited to infer
the acceleration spectrum.
We investigated the correlation of the HXR spectral index with both spectral
indices (in case of broken-power-law shapes) observed in the in situ spectra.
We find a good correlation of $\sim 0.8$ for both sets of value pairs with an
alignment along the thin-target solution, i.e. a shift toward harder in situ
electron spectral indices, when using the spectral index below the break
$\delta_{1}$ or a shift toward the thick-target solution when using
$\delta_{2}$. All of our events would fall into the class of delayed events as
defined by Krucker et al. (2007) who found only a low correlation of $\sim
0.4$ for these events. Although it cannot be ruled out that many of the
observed onset delays are only apparent delays caused by instrumental effects,
such as occasional poor pitch-angle coverage of STEREO/SEPT, the majority of
our events are accompanied not only by EUV jets but also by CMEs. While the
jets are likely a sign of interchange reconnection providing the flare-
accelerated electrons with a connection to open field lines, a potential role
of the CMEs in the acceleration process as suggested by Petrosian (2016),
which could also cause the observed delays, cannot be ruled out. However, we
do not find an effect of the CMEs on the correlation of the spectral indices.
Nevertheless, the CMEs could have perturbed the transport of electrons through
the lower corona, which might occasionally have caused a wider or shifted
injection into interplanetary space as suggested by the correlation of CMEs
with events observed at larger longitudinal separation angles.
We find clearly improving correlations when only considering events, which
show significant anisotropies in the in situ electron observations. This
suggests that transport effects such as pitch-angle scattering can reduce the
spectral imprint of the acceleration and need to be taken into account when
inferring the accelerated electron spectrum from spacecraft measurements.
Analysis of the starting frequencies of the associated type III radio bursts
suggests that the acceleration height of most of our events was below 0.1 R⊙
above the photosphere. Furthermore, a detailed inspection of radio and
microwave observations suggests that the electron fluxes at 1 AU could result
from multiple injections of electrons in the corona as the majority of our
events is accompanied by groups of type III bursts. However, higher energy,
i.e. near-relativistic electrons are only accelerated during part of these
acceleration episodes as an indicated by the shorter hard X-ray and microwave
emission.
in situ measurements of solar energetic electrons will remain a key observable
to study acceleration, injection, and transport processes of SEP events.
Furthermore, as their propagation time from the Sun to Earth is significantly
shorter than that of associated solar energetic ions, one main application of
solar energetic electrons is space weather forecast as used for example in the
Relativistic Electron Alert System for Exploration (REleASE, Posner et al.,
2009). Electron event observations during the upcoming solar cycle taken by
new space missions such as Parker Solar Probe or Solar Orbiter open up new
opportunities to understand solar energetic electrons events. Much
advantageous over the study presented here, Solar Orbiter will allow to detect
the HXR flare and the in situ electrons at the same spacecraft, with the
Spectrometer/Telescope for Imaging X-rays (STIX; Krucker et al., 2020) and the
Energetic Particle Detector (EPD; Rodríguez-Pacheco et al., 2020).
Furthermore, both new space missions will take measurements at much smaller
radial distance than 1 AU allowing to tackle the effect of interplanetary
transport especially when combined with 1 AU baseline observations provided
for instance by, STEREO A, SOHO, ACE, and Wind. The new state-of-the art
instruments also provide energetic electron measurements over a wider energy
range with very fine energy resolution, which might finally allow to separate
the imprints of acceleration and transport in the energy spectra.
###### Acknowledgements.
The work of A. W. was supported by DLR under grant No. 50 QL 1701. F.E. and
N.D. acknowledge support from NASA grant NNX17AK25G and F.E. from DFG grant EF
98/4-1. N.D. acknowledges financial support by DLR under grant 50OC1702. We
thank the International Space Science Institute (ISSI) for hosting our team on
“Solar flare acceleration signatures and their connection to solar energetic
particles.” L.G. acknowledges the NASA DRIVE SolFER Science Center grant
80NSSC20K0627. This study has received funding from the European Union’s
Horizon 2020 research and innovation program under grant agreement No.
101004159 (SERPENTINE).
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## Appendix A Additional information on the analyzed events
Table 3 provides additional information on the events under study. Column one
lists the HXR date and peak time. Columns 2 and 3 list the characteristics of
the associated microwave bursts: the start and end time, the highest frequency
where the nonthermal microwave signature was observed, and the type of the
predominant microwave emission, thermal (th), gyrosynchrotron (gs) or plasma
emission (p). Column 4 lists the start and end time of the decametric type III
burst, read from the dynamic spectra observed by Wind or STEREO near 10 MHz,
using data with 1 min integration. Column 5 indicates if the event was
associated with a type II radio burst and provides its start and end time.
Columns 6 and 7 provide the magnetic polarity of the associated active region
in the corona and locally at the spacecraft (indicated by column 8) during the
SEE event. The last column provides a rough classification for the strength of
the observed electron anisotropy during the early phase of the electron
events.
The polarity of the magnetic field in the corona was inferred from the
potential field source surface (PFSS) extrapolation of the SoHO/MDI
measurements in the photosphere. The method of Schrijver & DeRosa (2003)
implemented in the PFSS tool of the Solarsoft package was used. The polarity
in the Table is the one of the open field lines rooted in the flaring active
region. The in situ magnetic field polarity was determined by separating
between inward our outward pointing magnetic field vectors as measured by
STEREO/MAG (Acuña et al. 2008) during the time of each electron event. The
table shows, that the in situ electron events were often observed in opposite
polarity sectors than coronal polarity of the corresponding flare. We do,
however, not see any influence of the polarity on other parameters such as the
onset delays or the quality of the spectral correlations.
In two events (2009 Dec 22 and 2012 Apr 16) type II bursts were observed at
meter wavelengths, without counterpart in the Wind/WAVES spectrum below 14
MHz. Only the first was reported by NOAA/SWPC. On 2009 Dec 22 the type II
burst started with the intense microwave emission and HXR burst, and it lasted
a few minutes longer. DH type III bursts were observed by Wind/WAVES during
its entire duration. On 2012 Apr 16 the type II emission occurred during the
decay of the microwave and HXR burst. DH type III bursts accompanied the
microwave/HXR burst, but not the type II burst. No clear difference of the
energy spectra of the electrons was found when compared to the rest of the
events.
For the anisotropy classification, presented in the last column of table 3, we
first determined the anisotropies using the method described by Brüdern et al.
(2018), and categorized anisotropies $A<1$ as small, $1>A>2$ as medium, and
$A>2$ as large. Mixed classifications are provided for events with
anisotropies close to the limiting values of 1 or 2. The asterisks in the
anisotropy column mark events with limited or poor pitch-angle coverage, which
can lead to an underestimation of the anisotropy.
Table 3: Complementary radio, magnetic polarity and anisotropy information of
the analyzed event
HXR | Microwave burst | $\nu_{\rm max}$ | DH III | II | Magnetic polarity | |
---|---|---|---|---|---|---|---
peak time | start - end | [GHz]/ | start-end | | corona | in situ | s/c | anisotropy
| | type | | | | | |
2007/01/24 00:31:24 | 00:31:10-00:32:00 | 9/gs | 00:29-00:33 | – | neg | mixed | A | small/medium
2007/01/24 00:31:24 | – | – | – | – | neg | mixed | B | small
2007/01/24 05:16:09 | 05:15:50-05:16:30 | 5/gs | 05:12-05:19 | – | neg | mixed | B | small**$*$L
2007/01/24 05:16:09 | – | – | – | – | neg | mixed | A | small/medium
2009/12/22 04:56:10 | 04:53:00-04:57:00 | 17/gs | 04:52-05:04 | 04:57-05:03 | pos | pos | B | large
2010/02/08 03:12:24 | 03:12:20–03:14:00 | 17/gs | 03:13-03:16 | – | neg | neg | B | small
2010/11/12 03:53:41 | 03:53:25–03:56:00 | 1/p | 03:45-03:54 | – | neg | neg | B | small/medium**$*$L
2010/11/12 08:01:54 | 08:01:00–08:04:50 | 15/gs | 07:55-08:11 | – | neg | neg | B | medium/large **$*$L
2010/11/17 04:36:44 | 04:36:30-04:37:10 | 3/gs | 04:31-04:37 | – | pos | neg | B | small/medium **$*$L
2011/03/24 17:04:34 | 17:03:00-17:06:50 | 15/gs | 16:53-17:16 | – | pos | neg | B | large
2012/01/12 00:51:58 | $\sim$00:49–00:53 | 5/gs | 00:49-00:53 | – | neg | pos | B | small**$*$L
2012/03/25 00:27:54 | 00:27:05–00:28:30 | 9/gs | 00:22-00:28 | – | neg | neg | B | small**$*$L
2012/04/16 00:26:14 | $\sim$00:25-00:28 | 4/gs | 00:24-00:30 | 00:29-00:40 | uncertain | neg | B | small/medium**$*$L
2012/05/07 03:21:58 | 03:21:00–03:26 | 17/gs | 03:20-03:29 | – | neg | neg | B | medium**$*$L
2012/06/27 12:36:00 | 12:36:00–12:36:30 | 9/gs | 12:34-12:40 | – | neg | neg | B | medium/large
2012/06/28 02:12:24 | 02:12:00–02:13:30 | 2/p | 02:10-02:14 | – | neg | pos | B | medium**$*$L
2012/07/01 07:14:44 | 07:14:40–07:14:50 | 9/gs | 07:09-07:25 | – | neg | pos | B | large
2014/03/19 16:26:14 | 16:29:00–16:32 | 15/th | 16:24-16:30 | – | uncertain | pos | B | large
2014/06/09 17:05:04 | 17:04:30–17:06:20 | 15/gs | 17:03-17:07 | – | uncertain | pos | B | medium**$*$L
101010 imited or poor pitch-angle coverage in SEPT measurements due to non-
nominal magnetic field configuration.
Many of the analyzed flares were accompanied by coronal jets seen in EUV.
Table 4 summarizes the observations of these jets using EUV images. For the
first 4 events, EUV observations from the STEREO-A EUVI (Wuelser 2004) at 195
Å were used, because they provide the best available time cadence for these
dates, which are before the launch of the Solar Dynamic Observatory (SDO,
Pesnell et al. 2012) mission in 2010. However, the time cadence (5 minutes in
average) and spatial resolution are not sufficient to clearly confirm the
absence of jets, for instance during the first event. Despite these
instrumental limitations, jets and eruptions are still clearly found for the
other three events. For the later events, data from the Atmospheric Imaging
Assembly (AIA, Lemen et al. 2012) on board SDO was available. AIA has a
cadence of 12 second and provides images of the Sun in 7 EUV filters. The 304
Å filter was used to look for coronal jets within one hour of the X-ray peak
time of the flare, as jets are typically bright in this wavelength. In the 13
events observed in the SDO era, only two events are not associated with a
coronal jet. Most of the observed jets are observed at the time and location
of the flare, as reported in the third column of table 4. The events for which
no jet was found, or with a jet delayed or at a different location in the
active region than a flare, are not particularly associated with delayed SEP
electron events.
Most of the analyzed flares were accompanied by coronal mass ejections (CMEs).
Table 4 summarizes the observations using the coronagraphs aboard SoHO
(LASCO/C2, Brueckner et al. 1995) or STEREO (SECCHI/COR1, Howard et al. 2008).
In order to avoid spurious associations due to projection effects, we
restricted the sample to events where the parent active region was within 30∘
of the limb. The third column displays the central meridian distance (CMD) for
the chosen spacecraft. The fifth column gives the time of first appearance in
the field of view of the coronograph, together with the estimated heliocentric
distance as identified in the images111111Movies provided by JHelioviewer or
https://cdaw.gsfc.nasa.gov/stereo/daily_movies/. Whenever possible a rough
estimate of the speed in the plane of the sky is also provided, as well as the
height of the CME front at the time of the HXR peak, which is inferred from
linear back projection. Column six provides comments for specific events.
CMEs are found during all events where adequate coronographic observations are
available. The relationship with the flare and electron events is not always
clear, however. The 2007 Jan 24 events are not related with the partial halo
CME observed at the time of the flare. Since the CME appeared in LASCO images
at the east limb before the west limb, it likely originated from a region in
the eastern solar hemisphere. No active region was on the eastern solar disk
this day, but a candidate crossed the east limb three days later. We therefore
conclude that the partial halo CME was a backside event. In three cases the
CME was already high above the solar limb at the time of the flare (2012 Apr
16 and Jun 27, 2014 Mar 19). The starting frequencies of the type III bursts
associated with these events show that the electron acceleration must have
occurred at lower height. In nine events the extrapolated height of the CME
front at the time of the HXR burst is within 1 R⊙ above the photosphere, so
that its early rise may have been related with the electron acceleration as
traced by the HXR and radio emissions. The CMEs are in general not large, and
some are reminiscent of jets.
Table 4: Jets and CMEs possibly related with the electron events and flares.
| EUV jet | Coronal mass ejection
---|---|---
HXR peak time | Instr | First/ | Instrument/ | Height-time evolution | Comment
| | Distance | CMD [deg] | (first/$r_{0}$/$V$/$r$(HXR)) |
(1) | (2) | (3) | (4) | (5) | (6)
2007/01/24 00:31 | STA | -/- [1] [2] | SoHO W61 | 01:32/2.3 R⊙/-/- | E backside
2007/01/24 05:16 | STA | 05:15/$<5^{\prime\prime}$ | SoHO W64 | 06:06/2.5 R⊙/ -/- | E backside
2009/12/22 04:56 | STA | 04:55/$<5^{\prime\prime}$ | STB W114 | 05:11/1.7 R⊙/470/1.0 R⊙ |
2010/02/08 03:12 | STA | 03:20/$<5^{\prime\prime}$ [3] | STB W71 | 03:41/1.6 R⊙/200/1.1 R⊙ | jet
2010/11/12 03:53 | AIA | 04:05/$<5^{\prime\prime}$ [4] | STA W86 | 03:55/1.7 R⊙/580/1.6 R⊙ | faint diffuse
2010/11/12 08:02 | AIA | 08:55/$10^{\prime\prime}$ | STA W83 | 08:05/? R⊙/700/1.4 R⊙ | faint diffuse
2010/11/17 04:37 | AIA | flare/$<5^{\prime\prime}$ | STA E105 | 04:45/1.8 R⊙/930/1.2 R⊙ | jet
2011/03/24 17:05 | AIA | flare/$<5^{\prime\prime}$ | CMD $>30^{\circ}$ | |
2012/01/12 00:52 | AIA | 01:12/$<5^{\prime\prime}$ [3] [4] | STB W92 | 01:01/1.7R⊙/520/1.3 R⊙ |
2012/03/25 00:28 | AIA | -/- [1] | STB W93 | 00:26/1.9R⊙/860/1.7 R⊙ |
2012/04/16 00:26 | AIA | -/- [1] | SoHO E89 | 00:36/3.5 R⊙/$\sim 490$/3.1 R⊙ | narrow
2012/05/07 03:22 | AIA | -/$<5^{\prime\prime}$ [2] [4] | CMD $>30^{\circ}$ | |
2012/06/27 12:36 | AIA | flare/$20^{\prime\prime}$ | SoHO E64 | 12:10/3.2 R⊙/$\sim 760$/4.3 R⊙ | pre-existing
2012/06/28 03:22 | AIA | flare/$<5^{\prime\prime}$ [2] | CMD $>30^{\circ}$ | |
2012/07/01 07:15 | AIA | flare/$<5^{\prime\prime}$ | STB W105 | 07:25/2.2R⊙ /700/1.6 R⊙ |
2014/03/19 16:26 | AIA | flare/$15^{\prime\prime}$ | SoHO E81 | 15:24/4 R⊙/- /- | pre-existing
2014/06/09 17:05 | AIA | 17:20/$50^{\prime\prime}$ | SoHO E90 | 17:24/2.8 R⊙/$\sim 680$/1.7 R⊙ |
121212 Columns: (1) Date (see Table 1), (2)-(3): EUV jet (instrument (2),
start time/distance from HXR emission site (3), ); (4)-(6): CME (instrument
and central meridian distance of the flare (4), first detection/heliocentric
distance/ speed in the plane of the sky (km s-1)/extrapolated heliocentric
distance at the HXR peak) (5), comment (6) Comments in column (3): $[1]$ jet
not detected in the data; $[2]$ numerous jets in this active regions over a
few hours; $[3]$ possible filament eruption; $[4]$ faint ejection.
### A.1 Implications of the spectral variation in the two multi-spacecraft
events of 24 Jan 2007
Two events on 24 Jan 2007 were observed by both STEREO spacecraft when they
were still separated by less than one degree in longitude and latitude. As
expected the peak intensities and onset times are very similar as detected by
the two observers. However, both events were observed during changing magnetic
field conditions leading to variable and partly nonoptimal pitch-angle
coverage of the SEPT instruments. The pitch-angle coverage was always better
at STEREO A, which might be the reason for the slightly larger anisotropies
compared to STEREO B. For the first event, the determined spectral indices of
the peak spectra are similar at the two spacecraft but for the second event
STEREO B only observes a single power law while STEREO A detects a broken
power law. Although the spectral values observed at the two spacecraft agree
within their uncertainties for the two events this illustrates how the
magnetic field configuration and variation can not only influence onset
determinations but also the determined spectra. It is expected that the most
reliable spectra are those where the instruments detect the electron
population propagating anti-sunward along the magnetic field, i.e. at pitch
angle 0 or 180 (depending on the magnetic field polarity). If these directions
are not covered by the instrument or if strong scattering has led to a
vanished anisotropy, the determined spectra may carry systematic changes.
Unfortunately, periods of non-ideal pitch-angle coverage occur regularly in
SEPT measurements so that several of the analyzed events may be subject to
this limitation.
### A.2 The events on 17 Nov 2010, 28 June 2012, and 19 Mar 2014
An important issue discussed in this manuscript is that it is not
straightforward to choose the appropriate spectral index out of the usually
observed double power law spectrum for a comparison with its solar
counterpart. Observations of near-relativistic electrons usually show only one
spectral break or transition (e.g., using Wind/3DP (Krucker et al. 2007) or
STEREO/SEPT (Dresing et al. 2020)). However, due to the limited overall energy
range and energy resolution of these instruments their ability to resolve
which of the different effects caused the spectral break or if an overlap of
effects determines the spectral shape is also limited. The correlation with
the solar counterpart spectral index may even help here: When correlating the
photon spectral index $\gamma$ with both the lower $\delta_{1}$ and upper
$\delta_{1}$ spectral indices of the in situ electron spectra (Fig. 5 & 6) we
find that three events (17 Nov 2010, 28 June 2012, and 19 Mar 2014) do not fit
the rest of the distributions. As described in section 3 we suspect that these
three events show a spectral transition, which can rather be attributed to
Langmuir-wave generation while the spectral transitions of the rest of the
events are likely caused by pitch-angle scattering (Dresing et al. 2020;
Strauss et al. 2020). Indeed, when treating these three events accordingly in
the correlation plots (Fig. 5 & 6) the overall correlations increase
significantly. One of these three events (17 Nov 2010) is indeed the event
with the lowest break energy in our whole sample ($E_{b}=69$ keV), which
supports the assumption that this break can be attributed to Langmuir-wave
generation, which is expected to yield a break around 60 keV (Krucker et al.
2009). The break energies of the other two special events (90 and 101 keV) are
rather low with respect to the mean spectral break value of 120 keV found by
Dresing et al. (2020), which was attributed to pitch-angle scattering, however
many other events analyzed here show similar low break energies. The two
latter events are both anisotropic. While the pitch-angle coverage during the
28 June 2012 event is not optimal, likely leading to an underestimation of the
anisotropy, the pitch-angle coverage during the 19 Mar 2014 event is ideal and
shows a very strong anisotropy (not shown). Consultation of the STEREO level3
Interplanetary Coronal Mass Ejection (ICME) list131313https://stereo-
ssc.nascom.nasa.gov/data/ins_data/impact/level3/STEREO_Level3_ICME.pdf reveals
that STEREO B was embedded inside an ICME when the 19 Mar event occurred. The
usually very quiet magnetic field conditions inside ICMEs may have contributed
to very weak scattering conditions during this event and consequently high
anisotropy. The pitch-angle distribution during the 28 June event is peculiar
as the highest intensities are observed in SEPT’s north and ani-sun sectors,
suggesting that also during this event, the spacecraft was embedded inside a
non-nominal magnetic field configuration. However, an ICME passage is only
reported for about six hours after the event onset. The solar origin of this
electron event is, however, not doubted given the good temporal correlation
with the flare and the associated type III radio burst, and the notable
anisotropy. In spite of the peculiarities of these two latter events, they are
otherwise not outstanding when comparing their characteristics such as peak
intensities or energies, strength of the anisotropy, onset delays or radio
features with the rest of our sample. The reason why the spectral break due to
Langmuir-wave generation is dominant in the spectra of these events, is
therefore not conclusively understood.
|
# Applications of Information Theory:
statistics and statistical mechanics
Khizar Qureshi Prof. Peter Shor provided generous amounts of feedback
Department of Mathematics
Massachusetts Institute of Technology
18.424: Seminar in Information Theory
###### Abstract
The method of optimizing entropy is used to (i) conduct Asymptotic Hypothesis
Testing and (ii) determine the energy distribution for which Entropy is
maximized. This paper focuses on two related applications of information
theory: Statistics and Statistical Mechanics.
## Introduction
Entropy is one measure of uncertainty within a system, and is often used to
describe the disorder of sequences of quantized random variables. However,
entropy can also be extended to methods within optimization, in which the
disorder of a system of interest may be maximized or minimized. Such methods
are prevalent within statistics, the physical sciences, and econometrics.
#####
The concerns of a statistician observing a sequence of outcomes include the
validity of an explanatory hypothesis, its degree of significance, and any
assumptions underlying the statistical tests. While linear hypothesis testing
is often sufficient, larger sequences exhibit large deviations in behavior
that should receive separate treatment. Traditional linear hypothesis testing
trivially assigns a constant multiple of an explanatory parameter $\beta$ to
an observation when forming a hypothesis ($H_{0}=k\beta=0$). Optimal entropy,
in which disorder is locally minimized or maximized, can be used to construct
asymptotic, non-linear hypothesis tests. Unlike linear hypothesis testing,
error probability can be minimized.
Optimizing entropy extends to thermodynamic systems. The Third Law of
Thermodynamics states that the entropy of a closed system, i.e. one in which
no mass or energy is added or removed, must be bounded from below by zero.
Achieving a non-entropic system is nearly impossible, except within a perfect
crystal lattice. A more probable state is one for which the entropy of a
system is maximized, and observations follow a Boltzmann distribution.
In our study of optimal entropy, we will use classical Statistics and
Statistical Mechanics as a lens. We will demonstrate the concept of minimal
entropy through two statistical tests: the univariate optimality test defined
by Stein’s Lemma, and a multivariate optimality test, as defined by the
Chernoff Bounds. We will see that there exists a distribution, the Boltzmann
distribution, that approaches maximum entropy as temperature goes to infinity.
Finally, we will apply the concept of asymptotic hypothesis testing to
Statistical Mechanics. In particular, we will test and observe the evolution
of error probability with a growing sample size. We will also compare
Q-function error probability with that of the Chernoff bound. The remainder of
the paper is organized as follows. To better understand the atypicality of
sequences, we will study the method of types. We will then learn how such
sequences behave through the large deviation theory. The focus of the paper
will then shift to hypothesis testing, in which we will develop tools for
recognizing the asymptotic optimality of entropy. Illustrative examples of
this will include Stein’s Lemma and Chernoff bounds. For the interest of the
physical sciences, we will rigorously derive the Boltzmann distribution, for
which entropy is nearly maximized. Finally, we will converge the
aforementioned topics through simulations of asymptotic statistical testing.
## Hypothesis Testing
Statisticians are often concerned with not just observed data, but the several
possible underlying explanations. A few examples include: testing for the
effectiveness of a drug, determining whether or not a coin is biased, and the
effect of gender on wage growth. We begin with a simple case in which we
decide between two hypothesis, each of which is represented by an independent
and identical distribution, or i.i.d. Let $X_{1},X_{2},\ldots,X_{n}$ be i.i.d.
$\sim Q(x)$. For an observed outcome, we have two possible explanations:
* •
$H_{1}:Q=P_{1}$
* •
$H_{2}:Q=P_{2}$
We now define a general decision function, whose value reflects the acceptance
and rejection of the above hypothesis. Namely, for general decision
$g(x_{1},x_{2},\ldots,x_{n})$, $g(x)=i$ indicates that $H_{i}$ is accepted. In
the binary case, the set $A$ over which $g(x)=i$ is complemented by the set
$A^{c}$, over which $g(x)\neq i$.
#####
Quite often, statisticians are concerned with accepting incorrect hypotheses
and rejecting correct ones. Such occurrences, recognized as Type I/II errors,
often occur when sequences exhibit atypicality and large deviating behavior
(See appendix). Error probabilities are reflected through the decision
function using weights $\alpha,\beta$:
$\begin{split}\alpha=P(g(x)=2|H_{1}\text{ true})=P^{n}_{1}(A^{c})\\\
\beta=P(g(x)=1|H_{2}\text{ true})=P^{n}_{1}(A)\end{split}$ (1)
Notice that the general decision function takes on values contradicting those
implied by the conditional hypothesis. The first implies that $H_{2}$ was
accepted even though $H_{1}$ was true, and the second implies that $H_{1}$ was
accepted even though $H_{2}$ was true. Type I (reject true) and Type II
(accept false) errors similarly prove detrimental to experiments, and so we
wish to minimize probabilities $\alpha$ and $\beta$. Minimizing $\alpha$
increases $\beta$, and minimizing $\beta$ increases $\alpha$. We will now
explore methodology to minimize the overall probability of error by optimizing
entropy as a weighted sum of $\alpha$ and $\beta$.
### Stein’s Lemma
We first fix either $\alpha$ or $\beta$, and manipulate the other to minimize
the probability of error.
###### Theorem 1 (Stein’s Lemma).
Let $X_{1},X_{2},\ldots,X_{n}$ be i.i.d. $\sim Q$. Further, let
$D(P_{1}\|P_{2})$ represent the Kullback-Leibler distance, or relative entropy
between the probability densities. Consider the hypothesis test between two
alternatives $Q=P_{1}$ and $Q=P_{2}$ where $D(P_{1}\|P_{2})<\infty$. Let
$A_{n}\subseteq\mathcal{H}^{n}$ be an acceptance region for hypothesis 1. Let
the probabilities of error be
$\begin{split}\alpha_{n}=P^{n}_{1}(A^{c}_{n})\\\
\beta_{n}=P^{n}_{2}(A_{n})\end{split}$ (2)
and for $0<\epsilon<\frac{1}{2}$, define
$\beta^{\epsilon}_{n}=\text{min}_{A_{n}\subseteq X^{n}}\beta_{n}$ (3)
Then,
$\text{lim}_{\epsilon\rightarrow 0}\
\text{lim}_{n\rightarrow\infty}\frac{1}{n}\text{log}\beta^{\epsilon}_{n}=-D\left(P_{1}\|P_{2}\right)$
(4)
###### Proof.
See Appendix ∎
Thus, no sequence of sets $B_{n}$ has an exponent better than
$D\left(P_{1}\|P_{2}\right)$. But the sequence $A_{n}$ achieves the exponent
$D\left(P_{1}\|P_{2}\right)$. Thus $A_{n}$ is asymptotically optimal, and the
best error exponent is $D\left(P_{1}\|P_{2}\right)$.
### Chernoff Bound
Thus far, $\alpha$ and $\beta$ have been treated separately. The approach
underlying Stein’s Lemma was to set one error probability to be
infinitesimally small, and measure the effect on the resulting probability. We
saw that setting $\alpha\leq\epsilon$ achieved $\beta_{n}=2^{-nD}$. However,
the distribution of error amongst $\alpha$ and $\beta$ may be highly
asymmetrical, in which case univariate optimization may not suffice. We now
explore methodology for a bivariate optimization.
#####
An alternative approach is to minimize the weighted sum of $\alpha$ and
$\beta$. The resulting error exponent is known as the Chernoff Information.
Consider a distribution of i.i.d. random variables: $X_{1},X_{2},\ldots,X_{n}$
representative of the decision function. We assign $P_{1}$ to Q with
probability $\pi_{1}$ and $P_{2}$ to Q with probability $\pi_{2}$. Upholding
the definition of $\alpha$ and $\beta$, the overall probability of error is
$P^{n}_{\epsilon}=\pi_{1}\alpha_{n}+\pi_{2}\beta_{n}$ (5)
###### Theorem 2 (Chernoff).
The best achievable exponent in the Bayesian probability of error is $D^{*}$,
where
$D^{*}=\\\
text{lim}_{n\rightarrow\infty}\text{min}_{A_{n}\subseteq\mathcal{H}^{n}}-\frac{1}{n}\text{log}P^{n}_{\epsilon}=D\left(P_{\lambda^{*}}\|P_{1}\right)=D\left(P_{\lambda^{*}}\|P_{2}\right)$
(6)
where
$P_{\lambda}=\frac{P^{\lambda}_{1}(x)P^{1-\lambda}_{2}(x)}{\sum_{a\in
X}P^{\lambda}_{1}(a)P^{1-\lambda}_{2}(a)}$ (7)
and $\lambda^{*}$ the value of $\lambda$ such that
$D\left(P_{\lambda^{*}}\|P_{1}\right)=D\left(P_{\lambda^{*}}\|P_{2}\right).$
(8)
###### Proof.
See Appendix ∎
## Physical Chemistry
Claude Shannon first proposed that the uncertainty due to possible errors in a
message could be encapsulated by
$U(W)=\text{log}W$ (9)
where W is the number of possible ways (state space) of encoding random
information. Intuitively, the uncertainty increases with increasing W, and is
zero if W=1. The concept of entropy provides a deep-rooted link between
information theory and statistical mechanics. The state with the least
information available, or greatest entropy occurs when the set of all states
are equiprobable. This is also the state with maximum uncertainty. An
information theoretic perspective dictates that explicit knowledge of various
probabilities associated with the system constitutes greater information.
Similarly, the thermodynamics of a system of isolated particles indicate that
entropy is directly correlated with expected energy level. Below is a
molecular orbital diagram that illustrates the possible energy states, all of
which depend on the position an electron occupies.
Figure 1: The figure above is a molecular orbital diagram. When two atoms
interact, and possibly share their valence (outermost) electrons, the
electrons must occupy a particular orbital. Once occupying an orbital, the
atoms are able to form a bond. Here, we are not concerned with the type of the
bond, but rather, paired occupancy. The empty orbitals are indistinguishable,
and will be occupied with equal probability.
#####
Entropy may also be observed in macroscopic states. The second law of
thermodynamics states that in equilibrium, changes in entropy are proportional
to changes in system heat per unit temperature. We have
$dW=\frac{dQ_{sys}}{T}$. We can better understand this through an
illustration. Consider a system of gas particles that may be expanded or
compressed. We can study the system under various entropic regimes. The
diagram below illustrates how available work decreases (increases) for gaseous
expansion (compression) under bivariate states of pressure and volume.
Figure 2: The figure above is a pressure-volume diagram for a system of Argon
gas particles. Expanding or compressing a gas requires energy, the extent of
which depends on the state of the system. Notice that for an adiabatic system
$\left(dQ_{sys}=0\right)$, compressing the gas requires the least relative
energy. That is, when when the change in entropy in minimized, a system can be
most naturally expanded/compressed. To reach minimum work available, we move
down the gradient of steepest descent until entropy is globally minimized.
#####
Consider a perfectly structured crystal lattice structure, in which the
positions of each contributing molecule is fixed. If we observe such a system,
depart, and return after $n$ periods, the position of each molecule within the
crystal will have remained the same almost surely. If the particles did not
displace, then they also carried zero kinetic energy, which is representative
of a zero temperature system. This near certainty of a thermodynamic system is
an example of an optimization in which entropy is minimized. If instead, the
system consisted of a fair coin toss, with no extra information, entropy would
be maximized.
### The Boltzmann Distribution
The distribution that maximizes the state space for a fixed energy level is
the Boltzmann distribution. We will now derive such a distribution, and show
that it uniquely maximizes entropy on each energy level. Consider a crystal
containing $N$particles, each of which has available energy levels,
$\epsilon_{n}$. The state space, W, is the number of ways the total energy
$E=\sum_{n}N_{n}\epsilon_{n}$ can be distributed amongst the the particles in
each energy level, across all energy levels, $N=\sum_{n}N_{n}$. The expected
number of particles in each energy level, $N_{n}$, is the product of the
probability that a particle is at an energy level, $P_{n}$, and the total
number of particles, N. The only consideration for such a distribution is the
number of particles in each energy levels, not necessarily the amount of
energy allocated to each particle. Energy is conserved amongst states and
across energy levels. While there exist several ways of assigning the number
of particles in each energy level $\epsilon_{n}$, we wish to find the state
with the distribution achievable in the most number of ways for fixed energy
levels. Our first constraint is that the total state space W is the sum of
individual states occupied, $W_{i}$ for all possible distributions
$W=\sum W_{i}$ (10)
Amongst all possible distributions of particles, there exists one that can be
achieved in more ways than any other. A distribution that approaches maximum
entropy for fixed energy levels is the Boltzmann distribution.
$\text{log}W\cong\text{log}W_{B}$ (11)
To find the most probable distribution that maximizes W, we first note that
each particle in the crystal can be distinguished from the others because it
occupies a defined position in space. Therefore, such a setting allows us to
number the particles $1,2,\ldots,N$. We assume a large N, to maintain
consistency with typical non-deficient states. S A particular microstate of
the crystal will place particle 1 in energy level $\epsilon_{i}$, particle 2
in energy level $\epsilon_{j}$, and so on. Initially, we seek the number of
$W_{i}$ microstates in a distribution for which there are $N_{1}$ particles in
$\epsilon_{1}$, $N_{2}$ particles in $\epsilon_{2},$, and so on. We choose, at
random, particles from the crystal, $N_{i}$, and assign them to energy levels,
$\epsilon_{i}$. The number of ways this can be done is equal to the number of
different orders in which the particles can be chosen from the crystal. The
first particle can be chosen from a group of N. With $N-1$ particles
remaining, the second can be chosen in $N-1$ ways. We see that the number of
ways for selecting the first two particles is $N(N-1)$. Following this
procedure, we then we see that the number of ways for selecting the $N$
particles is $N(N-1)(N-2)(N-3)\dots(3)(2)(1)$, or $N!$.
#####
We have over counted the ways of achieving a given distribution, and have
assumed that all states are distinguishable. Consider the placement of the
first two particles into energy level $\epsilon_{1}$. It makes no difference
whether the first particle is placed into $\epsilon_{1}$ prior to or following
particle 2. That is, the states are indistinguishable. This relaxes the
strictness on order, and so permutation are ignored. Thus, the state space,
$W_{i}$, for a given distribution, is $N!$ divided by the product of all $N!$
$W_{i}=\frac{N!}{\prod_{n}N_{n}!}$ (12)
To find the distribution that maximizes $W_{i}$, we note Stirling’s
approximation for log N!
$\text{log}\ N!\approx N\ \text{log}\ N-N$ (13)
Finding the maximum of $W_{i}$ is equivalent to finding the maximum of
$\text{log}\ W_{i}$, so we combine equations (13) and (14), and re-arrange as
follows
$\text{log}\ W_{i}\approx\text{log}\ N!-\sum_{n}\text{log}N_{n}!=N\
\text{log}\ N-\sum_{n}N_{n}\ \text{log}N_{n}=\left(\sum_{n}N_{n}\right)\
\text{log}\sum_{n}N_{n}-\sum_{n}N_{n}\ \text{log}N_{n}$ (14)
However, the set of particles is conserved. Moreover, the net energy within
the system is conserved. This provides the following two constraints
$\begin{split}\sum_{j}N_{j}=N\\\ \sum_{j}\epsilon_{j}N_{j}=E\end{split}$ (15)
We now use Lagrange’s method of undetermined multipliers. When $W_{i}$ is
maximized, its differential log must be zero
$d\ \text{log}\ W_{i}=\sum_{j}\frac{\partial\text{log}W_{i}}{\partial
N_{j}}_{j}dN_{j}=0$ (16)
We multiply the constraints on particle count and energy by constants
$\alpha,\beta$, and then take the differential to obtain
$\begin{split}\alpha\sum_{j}dN_{j}=0\\\
\beta\sum_{j}\epsilon_{j}dN_{j}=0\end{split}$ (17)
Subtracting these two constraints from the log-entropy, we obtain
$\sum_{j}\left(\frac{\partial\text{log}W_{i}}{\partial
N_{j}}-\alpha-\beta\epsilon_{j}\right)dN_{j}=0$ (18)
Through use of log-properties and algebraic manipulation (See Appendix), the
expression above is reduced to
$\text{log}N_{j}=\text{log}N-\alpha-\beta\epsilon_{j}$ (19)
which, after exponentiating both sides is
$N_{j}=Ne^{-\alpha}e^{-\beta\epsilon_{j}}$ (20)
The significance of this result is that it shows the occupancy of en energy
state $\epsilon_{j}$ is proportional to $e^{-\beta\epsilon_{j}}$.
#### Key Result
To account for energy states, thermodynamicists often make use of temperature,
an intrinsic quantity. Temperature is equivalent to the average kinetic energy
of a system of particles. Because this varies across systems, we normalize.
For a temperature T, and the Boltzmann constant, $k_{B}$,
$\beta=\frac{1}{k_{B}T}.$ (21)
Inducting on the one particle case, in which,
$e^{\alpha}=\sum_{j}e^{\frac{-\epsilon_{j}}{k_{B}T}}$, and combining the
preceding three expressions, providing the desired result
$\begin{split}P_{n}=\frac{N_{n}}{N}=\frac{e^{z}}{\sum_{N}e^{z}}\\\ \text{where
}z=\frac{-\epsilon_{j}}{k_{B}T}\end{split}$ (22)
We have now found the Boltzmann distribution, for which entropy is maximized.
The Boltzmann probability above expresses the fraction of particles placed in
each quantum state $n$ to maximize entropy $W_{i}$ of the distribution over
each energy level, $\epsilon_{j}$.
## Simulation: Asymptotic Hypothesis Testing
Thus far, we have studied various methods of statistical testing, highlighting
the importance of asymptotic tests such as the Chernoff Information Bound. We
have also (briefly) explored Statistical Mechanics, in which we show that
entropy is maximized for a Boltzmann distribution. We now demonstrate the
importance of our learnings through a representative example.
#####
Robust methods of signal interpretation allow for communication, and involve
the separation of signal and noise. A simple signal will follow a Gaussian
distribution, for which entropy is maximized. A hypothesis consists of
assigning observations as either signals, or as noise. Such hypotheses carry
error probabilities, and should be studied with both linear testing, as well
as asymptotic testing.
### Example: Binary Detection
The classical binary detection problem involves the reception of finite-length
signals realized as a random process $r[n],n=1,2,\ldots,$ [3]. The signal can
be attributed to either Gaussian white noise $n[i]$ or a deterministic signal
$s[i]$. Basic studies involve the interpretation of the signal-to-noise ratio,
a measure of quality. Consider a binary detection problem:
$r=s_{i}+n,,\ i\in\\{1,2\\}$ (23)
* •
Detections are composed of signals and noise
* •
n: N-dimensional noise vector
* •
i.i.d. Gaussian random variables and $\sim\left(0,1\right)$
* •
$s_{1}=\left(m,m,\ldots,m\right)$
* •
$s_{2}=\left(0,0,\ldots 0\right)$
Evaluate the error probability for both $N=1$ and $N=4$ when
$m\in\\{1,2,3,4,5,6\\}$.
Traditionally, error probability is evaluated through the Q-function, which
represents the probability that a normal random variable will obtain a value
larger than $x$ standard deviations above the mean. It can also be thought of
as the "tail" probability of the standard normal distribution, and is useful
for linear hypothesis testing.
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}exp\left(\frac{-t^{2}}{2}\right)dt=\frac{1}{2}erf\left(\frac{x}{\sqrt{2}}\right)$
(24)
Given a signal-to-noise Ratio, the Q-function can be used to determine the
error probability
$P^{e}=Q\left(\frac{\rho}{\sigma}\right)\text{where }\
\frac{\rho^{2}}{\sigma^{2}}=\frac{Nm^{2}}{4}$ (25)
We also know that any error probability is bounded from above by the Chernoff
Information bound. For the Q-function,
$Q(x)\leq exp\left(-\frac{x^{2}}{2}\right)\text{erf}$ (26)
And so,
$P^{e}=Q\left(\frac{\sqrt{N}m}{2}\right)\leq
exp\left(-\frac{m^{2}N}{8}\right)$ (27)
The figure below illustrates the growth of error in both forms of testing.
Figure 3: The figure above shows the evolution of error probability with an
increasing signal length (m). Error probability decreases as: (i) the number
of bits in the signal increases, and (ii) the number of elements in the noise
vector increases.
## Concluding Remarks
Optimizing entropy demonstrates the applicability of information theory beyond
computing. Asymptotic testing captures error probability in atypical
sequences, and a Boltzmann distribution of particles approaches maximum
entropy, as temperature goes to infinity.
## Appendix
Motivation for asymptotic testing arises from atypicality and large deviations
in sequences. We briefly review this, and encourage the ambitious reader to
study further.
### The Method of Types
The Asymptotic Equipartition Property formalizes that although there exist
several possible outcomes of a stochastic process, there exists a set from
which sequences are typical , or most frequently observed. The centric
approach underlying the AEP involves defining an almost sure convergence in
probability between the expectation of a sequence to its entropy. Similarly,
the Method of Types defines strong bounds on the number of sequences of a
particular distribution, as well as the probability of each such sequence
being observed.
## Large Deviation Theory
Recall that type of a sequence $x^{n}_{i}\in A^{n}$ is representative of its
empirical distribution $\hat{P}=\hat{P}_{x^{n}_{i}}$ where:
$\hat{P}(a)=\frac{|\\{i:x_{i}=a\\}|}{n},a\in A.$ (28)
A distribution P on A is called an n-type if it is the type of some
$x^{n}_{1}\in A^{n}$. The set of all $x^{n}_{1}\in A^{n}$ of type P is called
the type class of the n-type P and is denoted by $\mathcal{T}^{n}_{p}$.
###### Lemma 3.
The number of possible n-types is
$\binom{n+|A|-1}{|A|-1}$ (29)
###### Proof.
$\mathcal{T}(P)=\\{x\in\mathcal{X}^{n}:P_{x}=P\\}$ (30)
The combinatoric cardinality of $\mathcal{T}(P)$ provides the result ∎
###### Lemma 4.
For any n-type P,
$\binom{n+|A|-1}{|A|-1}^{-1}2^{nH\left(P\right)}\leq|\mathcal{T}^{n}_{p}|\leq
2^{nH\left(P\right)}$ (31)
###### Proof.
First, we prove the upper bound using $P(\mathcal{T}(P)\leq 1$.
$1\geq P^{n}(\mathcal{T}(P))=\sum_{x\ inT(P)}P^{n}(x)=\sum_{x\
inT(P)}2^{-nH(P)}=\|T(P)\|2^{-nH(P)}$ (32)
Consequently, $\|T(P)\|\leq 2^{nH(P)}$. For the lower bound, using the fact
that $T(P)$ has the highest probability amongst all type classes in P, we can
bound the ratio of probabilities
$\frac{P^{n}(T(P))}{P^{n}(T(\hat{P}))}\\\
=\frac{\|T(P)\|\prod_{a\in\mathcal{X}P(a)^{nP(a)}}}{\|T(\hat{P})\|\prod_{a\in\mathcal{X}P(a)^{n\hat{P}(a)}}}=\prod_{a\in\mathcal{X}}\frac{(n\hat{P}(a))!}{(n{P}(a))!}P(a)^{n(P(a)-\hat{P}(a))}\\\
$ (33)
Using the identity $\frac{m!}{n!}\geq n^{m-n}$, we see
$\frac{P^{n}(\mathcal{T}(P))}{P^{n}(\mathcal{T}(\hat{P}))}\geq\prod_{a\in\mathcal{X}}n^{n(P(a)-\hat{P}(a))}=n^{n(1-1)}=1$
(34)
So $P^{n}(T(P))\geq P^{n}(T(\hat{P}))$. The lower bound can now be found as
$\displaystyle
1=\sum_{Q\in\mathcal{P}_{n}}P^{n}(T(Q))\leq\sum_{Q\in\mathcal{P}_{n}}$ (35)
$\displaystyle=max_{Q}P^{n}(T(Q))$
$\displaystyle=\sum_{Q\in\mathcal{P}_{n}}P^{n}(T(P))$
$\displaystyle\leq(n+1)^{\|\mathcal{X}\|}P^{n}(T(P))=(n+1)^{\|\mathcal{X}\|}\sum_{x\in
T(P)}P^{n}(x)$ $\displaystyle=(n+1)^{\|\mathcal{X}\|}\sum_{x\in
T(P)}2^{-nH(P)}$ $\displaystyle=(n+1)^{\|\mathcal{X}\|}\|T(P)\|2^{-nH(P)}$
∎
To connect the theory of types with general probability theory, we must
develop a sense of relative entropy. For any distribution P on A, let $P^{n}$
denote the distribution of n independent drawings from P, that is,
$P^{n}\left(x^{n}_{1}\right)=\prod_{i=1}^{n}P\left(x_{i}\right),x^{n}_{1}\in
A^{n}$.
###### Lemma 5.
For any distribution P on A and any n-type Q
$\begin{split}\frac{P^{n}\left(x^{n}_{1}\right)}{Q^{n}\left(x^{n}_{1}\right)}=2^{-nD\left(Q\|P\right)},ifx^{n}_{1}\in\mathcal{T}^{n}_{Q}\\\
\binom{n+|A|-1}{|A|-1}^{-1}2^{-nD\left(Q\|P\right)}\leq
P\left(\mathcal{T}^{n}_{p}\right)\leq 2^{-nD\left(Q\|P\right)}\end{split}$
(36)
###### Proof.
For probability $P\in P_{n}$, distribution $Q$, the probability of type class
$T(P)$ under $Q^{n}$ is $2^{-nD(P\|Q)}$. We see
$\displaystyle Q^{n}(T(P))=\sum_{x\in T(P)}Q^{n}(x)$ (37)
$\displaystyle=\sum_{x\in T(P)}2^{-n(D(P\|Q)+H(P))}$
$\displaystyle=\|T(P)\|2^{-n(D(P\|Q)+H(P))}$
Replacing $\|T(P)\|$ with the result from Lemma 3, we see the result. ∎
###### Corollary 6.
Let $\hat{P}_{n}$ denote the empirical distribution (type) of a random sample
of size n drawn from P. Then
$P\left(D\left(\hat{P}_{n}\|P\right)\geq\delta\right)\leq\binom{n+|A|-1}{|A|-1}2^{-n\delta},\forall\delta>0$
(38)
###### Proof.
Given an $\epsilon>0$, we can define a typical set
$\mathcal{T}^{\epsilon}_{Q}$ of sequences for the distribution Q as
$\mathcal{T}^{\epsilon}_{Q}=\\{x^{n}:D(P_{x^{n}}\|Q)\leq\epsilon\\}$. Then the
probability of an atypical sequence is
$\begin{split}1-Q^{n}(\mathcal{T}^{\epsilon}_{Q})=\sum_{P:D(P\|Q)\geq\epsilon}Q^{n}(T(P))\\\
\leq\sum_{P:D(P\|Q)\geq\epsilon}2^{-nD(P\|Q)}\\\
\leq\sum_{P:D(P\|Q)\geq\epsilon}2^{-n\delta}\\\
\leq\binom{n+|A|-1}{|A|-1}2^{-n\delta},\forall\delta\geq 0\end{split}$ (39)
∎
###### Theorem 7.
Sanov’s Theorem Let $\Pi$ be a set of distributions on A whose closure is
equal to the closure of its interior. Then for the empirical distribution of a
sample from a strictly positive distribution P on A,
$-\frac{1}{n}\text{\text{log}}P\left(\hat{P}\in\Pi\right)\rightarrow
D\left(\Pi\|P\right)$ (40)
###### Proof.
Sanov’s Theorem Let $\mathcal{P}_{n}$ be the set of possible n-types and let
$\Pi_{n}=\Pi\cap\mathcal{P}_{n}$. The previous lemma implies that
$\begin{split}Prob\left(\hat{P}_{n}\in\Pi_{n}\right)=P^{n}\left(\cup_{Q\in\Pi_{n}}\mathcal{T}^{n}_{Q}\right)\
\text{is upper bounded by}\\\
\binom{n+|A|-1}{|A|-1}2^{-nD\left(\Pi_{n}\|P\right)}\ \text{and lower bounded
by}\\\ \binom{n+|A|-1}{|A|-1}^{-1}2^{-nD\left(\Pi_{n}\|P\right)}\end{split}$
(41)
Since $D\left(Q\|P\right)$ is continuous in Q, the hypothesis on $\Pi$ implies
that $D\left(\Pi_{n}\|P\right)$ is arbitrarily close to $D\left(\Pi\|P\right)$
if n is large. ∎
### Proofs
###### Stein’s Lemma.
To prove the theorem, we construct a sequence of acceptance regions
$A_{n}\subseteq X^{n}$ such that $A_{n}<\epsilon$ and
$\beta_{n}=2^{-nD(P_{1}\|P_{2})}$. We then show that no other sequence of
tests has an asymptotically better exponent.
First, we define
$A_{n}=\left\\{x\in
X^{n}:2^{+n(D(P_{1}\|P_{2})^{-\delta})}\leq\frac{P_{1}x}{P_{2}x}\leq
2^{+n(D(P_{1}\|P_{2})^{+\delta}}\right\\}$ (42)
Then, we have the following properties:
1. 1.
$P^{n}_{1}(A_{n})\rightarrow 1$. This follows from:
$P^{n}_{1}(A_{n})=P^{n}_{1}\left(\frac{1}{n}\sum_{i=1}^{n}\text{log}\frac{P_{1}(X_{i})}{P_{2}(X_{i})}\in\left(D(P_{1}\|P_{2}\right)-\delta_{1}D\left(P_{1}\|P_{2}\right)+\delta\right)$
(43)
by the strong LLN, since
$D\left(P_{1}\|P_{2}\right)=E_{P_{1}}\left(\text{log}\frac{P_{1}(X)}{P_{2}(X)}\right)$.
Hence, for sufficiently large n, $A_{n}<\epsilon$.
2. 2.
$P^{n}_{2}(A_{n})\leq^{+n(D(P_{1}\|P_{2})^{-\delta})}$. Using the definition
of $A_{n}$, we have
$\begin{split}P^{n}_{2}(A_{n})=\sum_{A_{n}}P_{2}(x)\leq\sum_{A_{n}}P_{1}(x)2^{+n(D(P_{1}\|P_{2})^{-\delta})}\\\
=2^{+n(D(P_{1}\|P_{2})^{-\delta})}\sum_{A_{n}}P_{1}(x)\\\
=2^{+n(D(P_{1}\|P_{2})^{-\delta})}(1-\alpha_{n}).\end{split}$ (44)
Similarly, $P^{n}_{2}(A_{n})\geq
2^{+n(D(P_{1}\|P_{2})^{+\delta})}(1-\alpha_{n})$. And so,
$\frac{1}{n}\text{log}\beta_{n}\leq-D\left(P_{1}\|P_{2}\right)+\delta+\frac{\text{log}(1-\alpha_{n})}{n}$,
and
$\frac{1}{n}\text{log}\beta_{n}\geq-D\left(P_{1}\|P_{2}\right)-\delta+\frac{\text{log}(1-\alpha_{n})}{n}$,
hence $\
\text{lim}_{n\rightarrow\infty}\frac{1}{n}\text{\text{log}}\beta_{n}=-D\left(P_{1}\|P_{2}\right).$
∎
###### Chernoff Bound.
The optimum hypothesis test is a likelihood ratio test, which follows the
form:
$D\left(P_{X^{n}}\|P_{2}\right)-D\left(P_{X^{n}}\|P_{1}\right)>T$ (45)
The test divides the probability simplex into regions corresponding to
hypothesis 1 and hypothesis 2, respectively. This is illustrated below:
Figure 4: The figure above shows the probability simplex and the Chernoff
bound. Notice that for error probabilities $P_{1}$ and $P_{2}$, there exists
an optimal error probability $P_{\lambda}$. This is determined as a weighted
argmin.
Let A be the set of types associated with hypothesis 1. From the preceding
discussions, it follows that the closest point in the set $A^{c}$ to $P_{1}$
is on the boundary of A, and is of the form given by $[8]$. Then, it is clear
that $P_{\lambda}$ is the distribution in A that is closest to $P_{2}$. It is
also the distribution in $A^{c}$ that is closest to $P_{1}$. By Sanov’s
theorem, we can calculate the associated probabilities of error:
$\begin{split}\alpha_{n}=P^{n}_{1}(A^{c})=2^{-nD\left(P_{\lambda^{*}}\|P_{1}\right)}\\\
\beta_{n}=P^{n}_{2}(A)=2^{-nD\left(P_{\lambda^{*}}\|P_{2}\right)}\end{split}$
(46)
In the Bayesian case, the overall probability of error is the weighted sum of
the two probabilities of error,
$P_{e}=\pi_{1}2^{-nD\left(P_{\lambda^{*}}\|P_{1}\right)}+\pi_{2}2^{-nD\left(P_{\lambda^{*}}\|P_{2}\right)}=2^{-n\
\text{min}\\{{-D\left(P_{\lambda^{*}}\|P_{1}\right),-D\left(P_{\lambda^{*}}\|P_{2}\right)\\}}}$
(47)
since the exponential rate is determined by the worst exponent. Since
$D\left(P_{\lambda}\|P_{1}\right)$ increases with $\lambda$ and
$D\left(P_{\lambda}\|P_{2}\right)$ decreases with $\lambda$, the maximum value
of the minimum of
$\\{D\left(P_{\lambda}\|P_{1}\right),D\left(P_{\lambda}\|P_{2}\right)\\}$ is
attained when they are equal. We choose $\lambda$ so that
$D\left(P_{\lambda}\|P_{1}\right)=D\left(P_{\lambda}\|P_{2}\right)=C(P_{1},P_{2})$
(48)
Thus $C\left(P_{1},P_{2}\right)$ is the highest achievable exponent for the
probability of error, and is called the Chernoff information.
The closest point in the set $A^{c}$ to $P_{1}$ is on the boundary of A, and
is of the form given by []. Then from the previous discussion, it is clear
that $P_{\lambda}$ is the distribution in A that is closest to $P_{2}$; it is
also the distribution in $A^{c}$ that is closest to $P_{1}$. By Sanov’s
theorem, we can calculate the associated probabilities of error:
$\begin{split}\alpha_{n}=P^{n}_{1}\left(A^{c}\right)=2^{-nD\left(P_{\lambda^{*}}\|P_{1}\right)}\\\
\beta_{n}=P^{n}_{2}\left(A^{c}\right)=2^{-nD\left(P_{\lambda^{*}}\|P_{2}\right)}\end{split}$
(49)
In the Bayesian case, the overall probability of error is the weighted sum of
the individual two probabilities of error,
$P_{e}=\pi_{1}2^{-nD\left(P_{\lambda}\|P_{1}\right)}+\pi_{2}2^{-nD\left(P_{\lambda}\|P_{2}\right)}=2^{-n\
\text{min}\left(D(P_{\lambda}\|P_{1}\right),D\left(P_{\lambda}\|P_{2}\right)}$
(50)
since the exponential rate is determined by the worst exponent. Since
$D\left(P_{\lambda}\|P_{1}\right)=D\left(P_{\lambda}\|P_{1}\right)$ increases
with $\lambda$ and $D\left(P_{\lambda}\|P_{2}\right)$ decreases with
$\lambda$, the maximum value of the minimum of
$D\left(P_{\lambda}\|P_{1}\right),D\left(P_{\lambda}\|P_{2}\right)$ is
attained when they are equal.
This is illustrated below:
Figure 5: The figure above shows the relative entropy for each error
probability as a function of $\lambda$.
We choose $\lambda$ so that
$D\left(P_{\lambda}\|P_{1}\right)=D\left(P_{\lambda}\|P_{2}\right)=C(P_{1},P_{2})$
(51)
Thus $C\left(P_{1},P_{2}\right)$ is the highest achievable exponent for the
probability of error, and is called the Chernoff information.
∎
## References
* [1] T.M. Cover, J.A. Thomas, _Elements of Information Theory_ , Wiley, 2nd edition, 2006\.
* [2] D. Eisenberg, D. Crothers, _Physical Chemistry with Applications to the Life Sciences_ , Benjamin Cummings, 3rd edition, 1979\.
* [3] P. Gopych, _Sensitivity and Bias within the Binary Signal Detection Theory, BSDT_ , Information Theories & Applications, Vol. 11, 2004\.
|
11institutetext: Department of Electrical and Computer Engineering, National
University of Singapore 22institutetext: Department of Computer and
Information Science, University of Mississippi 33institutetext: CtrsVision
33email: {yihao<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>{xinchao<EMAIL_ADDRESS>
# Domain-Adaptive 2D Human Pose Estimation via Dual Teachers in Extremely Low-
Light Conditions
Yihao Ai Equal contribution.11 0009-0005-2336-4813 Yifei Qi ⋆ 11
0009-0007-4197-947X Bo Wang 2233 0000-0002-0127-2281 Yu Cheng 11
0000-0002-9830-0081
Xinchao Wang 11 0000-0003-0057-1404 Robby T. Tan 11 0000-0001-7532-6919
###### Abstract
Existing 2D human pose estimation research predominantly concentrates on well-
lit scenarios, with limited exploration of poor lighting conditions, which are
a prevalent aspect of daily life. Recent studies on low-light pose estimation
require the use of paired well-lit and low-light images with ground truths for
training, which are impractical due to the inherent challenges associated with
annotation on low-light images. To this end, we introduce a novel approach
that eliminates the need for low-light ground truths. Our primary novelty lies
in leveraging two complementary-teacher networks to generate more reliable
pseudo labels, enabling our model achieves competitive performance on
extremely low-light images without the need for training with low-light ground
truths. Our framework consists of two stages. In the first stage, our model is
trained on well-lit data with low-light augmentations. In the second stage, we
propose a dual-teacher framework to utilize the unlabeled low-light data,
where a center-based main teacher produces the pseudo labels for relatively
visible cases, while a keypoints-based complementary teacher focuses on
producing the pseudo labels for the missed persons of the main teacher. With
the pseudo labels from both teachers, we propose a person-specific low-light
augmentation to challenge a student model in training to outperform the
teachers. Experimental results on real low-light dataset (ExLPose-OCN) show,
our method achieves 6.8% (2.4 AP) improvement over the state-of-the-art (SOTA)
method, despite no low-light ground-truth data is used in our approach, in
contrast to the SOTA method. Our code is available at: DA-LLPose.
###### Keywords:
Human pose estimation Low-light Domain adaptation
## 1 Introduction
Human 2D pose estimation is a fundamental task in computer vision and
important for many downstream tasks like human activity recognition [59],
virtual tryon [12], motion capture [11], augmented reality [55], or Metaverse
[21]. The existing methods and benchmarks [34, 60, 31] primarily focus on
well-illuminated scenarios, where human subjects exhibit both clear visibility
and high recognizability in the majority of cases. However, poor lighting
conditions such as low light or nighttime are a prevalent aspect of daily
life, which signifies the importance of low-light human pose estimation
research. Despite the significance of this research area, its exploration has
been constrained by the lack of suitable low-light datasets or benchmarks.
A previous dataset for human pose estimation under poor lighting conditions
primarily revolves around nighttime scenarios [10, 9], where human subjects
generally remain recognizable due to the presence of light sources such as
lamps or car lights. Thus, these nighttime datasets do not focus on extremely
low-light conditions, which pose severe challenges to existing human pose
estimation methods. Recently, Lee et al. [29] proposed a new benchmark for
human pose estimation in extremely low-light conditions, which covers
different levels of low-light conditions, and a low-light pose estimation
method, which requires paired well-lit and low-light images with ground truths
for training. However, the scarcity of paired images with ground truths in
practical scenarios makes the application of this method impractical. To this
end, we aim to develop an innovative domain adaptive method for low-light
human pose estimation, utilizing well-lit ground-truth data only. Our goal is
to achieve performance on par with the SOTA methods that rely on both low-
light and well-lit ground truths.
To the best of our knowledge, we are the first to propose a solution to
estimate human poses in extremely low-light conditions utilizing well-lit
ground-truth data only. This is an extremely challenging task because the
visibility of human subjects is poor and low-light images are corrupted due to
severe low-light noise. The intensity and contrast are much lower in the low-
light images (e.g., average pixel intensity of well-lit and low-light images
are 90.5 and 2.0). Without paired well-lit and low-light images with ground-
truth data, two types of existing methods could be used for low-light pose
estimation: low-light image enhancement [52, 1] and domain adaptation [46,
26]. However, these two groups of methods have their own limitations. The
former is difficult to restore the extremely low-light images faithfully and
the latter still lacks a specific solution for bridging the gap between well-
lit and low-light conditions.
In response to these challenges, we introduce an innovative domain-adaptive
dual-teacher framework for human pose estimation that facilitates knowledge
transfer from well-lit to low-light domains. This innovation enables our model
to exceed state-of-the-art (SOTA) methods by eliminating the requirement for
training with low-light ground truths, which are often impractical to obtain.
Ideally, cross-domain knowledge transfer is achievable using a single teacher,
as demonstrated by existing methods [44, 26, 25]. However, a single center-
based teacher, designed to predict the entire human pose based on each
person’s center, may encounter difficulties in detecting individuals in low-
light images due to inherent architectural limitations that focus solely on
the detection of human centers. Consequently, this approach may fail to
identify individuals when their centers cannot be detected, resulting in a
limited number of pseudo labels. To generate a more comprehensive set of human
pose predictions for use as pseudo labels, we propose the incorporation of a
complementary teacher: a keypoint-based bottom-up human pose estimation model.
This model is capable of predicting partial human poses by grouping detected
keypoints, offering a complementary approach to the primary center-based
teacher.
Our method comprises two stages: pre-training and dual-teacher knowledge
acquisition. During the first stage, we perform supervised training with
labeled well-lit data alone. This initial step equips both teachers to
recognize human poses in well-lit images and their corresponding low-light
augmentations. In the subsequent second stage, our focus shifts to harnessing
knowledge directly from unlabeled low-light images. To challenge the student
model during the second stage, we introduce a novel augmentation technique
called Person-specific Degradation Augmentation (PDA). PDA is selectively
applied to images with pseudo labels generated by the teachers. Given that
low-light conditions pose a significant challenge in pose estimation, our
degradation augmentations primarily darken the individuals, making them less
distinguishable from their backgrounds to mimic realistic low-light scenarios.
Such a simple, flexible and effective degradation neither necessitates the
camera calibration as various physical-based augmentations did [53, 54], nor
relies on an additional network, which would cost extra computational
resources [39]. And it facilitates the student model’s exploration of
knowledge in the low-light domain, thereby encouraging it to surpass the
performance of the teachers in the second stage.
LABEL:fig:representative_result shows three images from ExLPose-OCN [29] and
the human pose estimation results of the existing methods [52, 1, 26, 46] and
our proposed method. Compared with the existing methods, our method can
effectively estimate human poses in extremely low-light conditions.
Specifically, our method is capable of handling low-contrast (last row) and
partially visible (second row) persons. For the relatively visible persons
(first row), our method can produce more accurate pose estimation. In summary,
our major contributions are listed as follows.
* •
We investigate a challenging yet practical task, i.e., estimating human poses
in extremely low-light conditions with well-lit ground-truth data only. To the
best of our knowledge, we are the first to propose a solution in this
challenging setting.
* •
We introduce a novel domain-adaptive dual-teacher framework, utilizing both
center-based and keypoint-based teachers to generate enriched pseudo labels
for effective student model training, eliminating the need for low-light
ground truths, which are often impractical to obtain. To our knowledge, this
represents a significant advancement in low-light human pose estimation.
* •
Our proposed method outperforms the SOTA method by 6.8% (2.4 AP) on the real
low-light dataset, ExlPose-OCN. Notably, our method utilizes well-lit ground-
truth data alone, while the SOTA method is trained with paired well-lit and
low-light images with ground truths.
## 2 Related Work
2D Human Pose Estimation Two primary paradigms exist within 2D human pose
estimation: top-down and bottom-up approaches. In general, top-down methods,
benefiting from applying human detection first [17, 5, 56, 45, 43], tend to
outperform their bottom-up counterparts [3, 37, 27, 6]. Early approaches of
bottom-up estimation [3, 37] are keypoint-based, detecting keypoints for each
individual in an image before grouping them to construct individual human
poses. Recently, Geng et al. propose a center-based bottom-up method [14],
which predicts human centers instead of grouping individual keypoints, and
regresses offsets between a human center and the keypoints for each
individual. This method sets a new trend among the bottom-up methods [58, 47,
48]. However, existing 2D pose estimation methods focus on well-lit
conditions, primarily due to the prevalence of well-lit images within the
benchmark datasets [34, 60, 31]. Additionally, since existing methods are
fully supervised, they demonstrate poor performance when applied directly to
low-light images after being trained on well-lit data.
Domain Adaptive Pose Estimation Domain adaptive pose estimation is potentially
useful for low-light conditions. A number of domain adaptation methods are
proposed in human pose [26, 41, 38], animal pose [36, 2, 30] and hand pose
estimations [22, 16, 23]. However, these methods are not designed for human
pose estimation in extremely low-light conditions. Source-free domain adaptive
human pose methods are developed [41, 38] to transfer knowledge from synthetic
to real data in the absence of the source data. This is in opposition to low-
light scenarios, where well-lit (source) domain data is typically available.
Kim et al. propose a unified framework for domain adaptive human pose
estimation [26], where the student is challenged to learn the target-domain
knowledge by applying affine transform-based augmentations. However, this
augmentation is not tailored for the severe lighting conditions.
Low-light Human Pose Estimation Apart from the domain adaptation-based
methods, fully-supervised approaches are developed as well. Crescitelli et al.
propose a nighttime human pose estimation method that combines images from
both RGB and infrared cameras [10, 9]. However, the reliance on infrared
cameras restricts the practical applicability of this method. Recently, Lee et
al. [29] propose a top-down human pose estimation approach for extremely low-
light conditions. Nevertheless, this method depends on the use of paired well-
lit and low-light images with ground-truth data during training, which are
impractical for real-world applications due to the scarcity of the paired data
and the inherent challenges associated with annotating low-light images. In
the context of semi-supervised learning for human pose estimation, the
utilization of multiple teachers, especially in dual-teacher configurations
[57, 19], is related to our work. However, semi-supervised learning typically
leverages both labeled and unlabeled data under the assumption that they come
from similar distributions. This assumption differs significantly from the
low-light setting, underscoring the necessity for our innovative domain-
adaptive dual-teacher framework.
Low-light Image Enhancement Low-light image enhancement might be used to
enhance low-light images prior to human pose estimation. Early approaches
primarily rely on gamma correction [40] or histogram equalization [7, 4]. More
advanced methods restore the low-light images based on the Retinex theory [32,
51]. Unfortunately, these methods tend to introduce additional noise with
color distortion. In the deep learning era, a number of convolutional neural
network (CNN)-based [35, 42, 50] and GAN-based [24, 20] methods are developed.
Recently, Wang et al. introduce an invertible network to capture the
distribution of the well-lit images [52]. Cai et al. propose a transformer-
based approach to restore the hidden corruption in low-light images [1].
However, these image enhancement methods still face difficulties in faithfully
restoring extremely low-light images. Thus, even with enhanced images of these
methods, low-light pose estimation is still challenging.
Figure 1: The overview of our method. The grey uni-directional arrows indicate
the supervised workflow in the pre-training stage. The grey and pink arrow
indicates the workflow of the supervised and unsupervised losses in the
knowledge acquisition stage. The low-light images, $I_{\rm fake}$, $I_{\rm
real}$ and $I_{\rm pda}$ are all brightened for improved visibility. NMS
stands for non-maximum suppression.
## 3 Proposed Method
Given an image $I\in{\mathbb{R}^{3\times{h}\times{w}}}$ which contains persons
in the low-light conditions, 2D human pose estimation is to obtain the human
skeletons from an image by locating $K$ types of human keypoints for each
person. An overview of our proposed framework is shown in Fig. 1, where we
present a novel domain-adaptive dual-teacher framework, utilizing both center-
based and keypoint-based teachers to generate enriched pseudo labels for
effective student model training, as shown in the yellow boxes.
### 3.1 Main Teacher and Complementary Teacher
#### Main Teacher
We build our main teacher based on DKER [14], where a person’s pose is
represented as the center and offset in addition to the heatmap for each
keypoint. In particular, the center location of the $i$-th person is
$\textbf{c}^{i}=\frac{1}{K}\sum_{k=1}^{K}\textbf{p}^{i}_{k}$ and the offsets
from the human center to each keypoints are
$\textbf{o}^{i}=\\{\textbf{p}^{i}_{1}-\textbf{c}^{i},\textbf{p}^{i}_{2}-\textbf{c}^{i},...,\textbf{p}^{i}_{k}-\textbf{c}^{i}\\}$,
where $\textbf{p}^{i}_{k}\in{\mathbb{R}^{2}}$ indicates the 2D location of the
$k$-th keypoints for the $i$-th person. The model is supervised by both
heatmap and offset losses:
$L_{\rm sup}^{m}=L_{H}^{m}+\lambda_{m}L_{O}^{m},$ (1)
where $L_{H}^{m}$ denotes a MSE loss of the main teacher’s heatmaps, and
$L_{O}^{m}$ is a smooth L1 loss of the offset map.
#### Complementary Teacher
The main drawback of the center-based representation is that the pose
estimation fully relies on the center’s predictions. If the center of a person
is failed to be detected, the whole person is missed, especially for the low-
contrast and partially visible persons in the low-light conditions. Therefore,
we build a complementary teacher to predict human poses for partially visible
persons following HigherHRNet-style design [6]. Specifically, the network is
trained to estimate the keypoints’ heatmap $\hat{H}_{c}$ and the corresponding
tag map $\hat{T}_{c}$. A person’s pose is grouped based on all the identity
information via Hungarian algorithm [28]. The method is supervised by
following losses:
$L_{\rm sup}^{c}=L_{H}^{c}+\lambda_{c}L_{\rm tag}^{c},$ (2)
where, $L_{H}^{c}$ is a MSE loss of the complementary teacher’s heatmaps, and
$L_{\rm tag}^{c}$ consists of push and pull losses of the tag maps. More
details are provided in the supplementary.
### 3.2 Pre-Training Stage
Firstly, the two teachers are trained on the well-lit dataset [29] to gain the
common knowledge of the human bodies. They are supervised by the losses
defined in Eq. 1 and Eq. 2 respectively. After the supervised training with
the labeled well-lit data, teachers can predict human poses on well-lit
images, but they cannot process low-light images due to the substantial domain
gap between well-lit and low-light conditions. As our framework relies solely
on well-lit ground-truth data, we propose Extreme Low-Light Augmentation
(ELLA) to augment low-light images from well-lit ones. This process involves
incorporating a set of low-light characteristics into the well-lit images.
The distinguishing characteristics of low-light images include extreme
darkness, high levels of noise, and low contrast. To simulate these low-light
specific features, ELLA applies a series of random augmentations sequentially
to well-lit images, including gamma correction, brightness adjustment,
contrast reduction, and Gaussian noise. The first two augmentations aim to
intensify the darkness of the images, while contrast reduction is employed to
decrease image contrast, and Gaussian noise is added to introduce noise into
the images. It is important to note that in ELLA, we do not explicitly
simulate light sources. This is because our primary focus is on low-light
conditions, where no light source is typically available. An example of the
augmented images using ELLA is shown in Fig. 2. The following equations, Eq.
3, Eq. 4, Eq. 5, and Eq. 6, mathematically represent the four ELLA
augmentations:
$\displaystyle I_{\rm out}$
$\displaystyle=255\cdot\left(I_{in}/255\right)^{\gamma},$ (3) $\displaystyle
I_{\rm out}$ $\displaystyle=bI_{\rm in},$ (4)
$\displaystyle I_{\rm out}$ $\displaystyle=cI_{\rm in}+(1-c)I_{\rm grey},$ (5)
$\displaystyle I_{\rm out}$ $\displaystyle=I_{\rm in}+{\rm Gaussian}(0,var),$
(6)
where $I_{\rm in}$ is the input image, $I_{\rm grey}$ is the greyscale image
of $I_{\rm in}$, and $I_{\rm out}$ stands for the output image.
$\gamma\in[2,5]$, $b\in[0.01,0.05]$, $c\in[0.2,1.0]$ and $var\in[0,40]$ are
the parameters uniformly sampled from their respective ranges.
The augmented well-lit images are named fake low-light images, denoted as
$I_{\rm fake}$. The augmentation is added randomly so that both teachers are
trained with a mixture of well-lit images and fake low-light images and
supervised by the losses in Eq. 1 and Eq. 2 respectively (each of the ELLA
augmentations has a 0.5 probability to be added). It is important to use both
well-lit images and fake low-light images in the training because it is
difficult for the model to transfer the knowledge to low-light domain if all
inputs suddenly change to the fake low-light images.
### 3.3 Dual-Teacher Knowledge Acquisition Stage
After the pre-training stage, the teachers possess the preliminary capability
to perform human pose estimation for extremely low-light images. In this
stage, our primary objective is to enhance the student model’s ability for
detecting and localizing human keypoints by leveraging real low-light images
and guiding the student to surpass both teachers. In particular, given a non-
paired fake low-light image $I_{\rm fake}$, a real low-light image $I_{\rm
real}$, and the pre-trained two teachers $T_{m}$, $T_{c}$, we train the
student $S$, who shares the same architecture of the main teacher at the
beginning, by a supervised loss and an unsupervised loss. The workflow of the
supervised and unsupervised losses is illustrated in Fig. 1 with grey and pink
arrows respectively.
#### Supervised Loss
The supervised loss is a continuation of the pre-training stage. To strengthen
the student’s knowledge on the low-light domain, a larger proportion of fake
low-light images is expected to be added in this stage. Ideally, all well-lit
images that are used in the student’s training should be augmented in the low-
light fashion by ELLA. However, this would trigger the knowledge forgetting
problem [33, 49] which is an inherent problem in domain adaptation methods. To
strengthen the learning in the target domain while keeping the source
knowledge, we make a few adjustment to the ELLA as follows.
#### Adjust ELLA
On the one hand, we set ELLA to do gamma correction and brightness adjustment
as Eq. 4 and Eq. 3 to every input well-lit images. This aims to augment more
samples in the low-light domain. On the other hand, to avoid the forgetting
problem, we randomly crop a few image patches in an augmented image and
restore it to the well-lit domain. The cropping probability for an image is
0.15 to guarantee the majority of the inputs are still fully-augmented images.
The supervised loss is computed by the following equations (shown in Fig. 1
with grey arrows). First, well-lit images $I_{\rm well}$ are augmented to fake
low-light images $I_{\rm fake}$ with adjusted ELLA as:
$I_{\rm fake}={\rm AdjustELLA}(I_{\rm well}).$ (7)
Second, student $S$ produces the prediction for $I_{\rm fake}$:
$\hat{H}_{f},\hat{O}_{f}=S(I_{\rm fake}),$ (8)
where $\hat{H}$ and $\hat{O}$ refer to the heatmaps and offset maps, which are
used in the calculation of the supervised loss following Eq. 2.
(a) Gamma
(b) Brightness
(c) Contrast
(d) Noise
(e) All Aug.
(f) Bri. All
Figure 2: Examples of our ELLA. (a)-(d) are the results of the individual
augmentation in ELLA. (e) is the image with all the augmentations, and Aug.
stands for augmentations. (e) is brightened up for improved visibility as
shown in (f), where Bri. All denotes brightened images with all the
augmentations.
#### Unsupervised Loss
Apart from the supervised loss, we teach the student with the pseudo labels
selected and fused by the predictions from both teachers. As shown in the Fig.
1, the two teachers firstly generate their own pose predictions $P_{m}$ and
$P_{c}$ by $P_{m}=T_{m}(I_{\rm real})$ and $P_{c}=T_{c}(I_{\rm real})$. Since
the two teachers score the poses in different manner (the main teacher
directly uses the center’s activation, $C_{m}$, while the complementary
teacher averages the heatmap activation of all keypoints as the confidence
score, $C_{c}$), we use different thresholds $s_{m}$ and $s_{c}$ to select out
the high-confidence poses. Then the poses from both teachers are concatenated
by and applied non-maximum suppression (NMS) [18] to rule out the redundant
poses predictions to obtain the pseudo labels $P_{\rm all}$ defined as:
$P_{\rm all}={\rm NMS}(P_{m}[C_{m}>s_{m}]\oplus P_{c}[C_{c}>s_{c}]),$ (9)
where $\oplus$ indicates concatenation operation.
#### Person-specific Degradation Augmentation
The student needs to gain low-light domain-specific knowledge in order to
outperform the teachers in challenging scenarios under low illumination. For
example, the right person with black shirt in the last row of
LABEL:fig:representative_result is largely indistinguishable from the dark
background, which demonstrates that low contrast is a major challenge for low-
light human pose estimation. Therefore, the network has to learn the specific
human structure in a texture-less condition. To this end, we propose a Person-
specific Degradation Augmentation (PDA) to realistically mimic the extremely
low-light conditions at individual-level. Specifically, after getting the
selected poses $P_{\rm all}$ from the teachers, a set of bounding boxes is
generated to crop the individuals. The cropped image patches are augmented
based on Eq. 4, which are then used to replace the original ones from input
$I_{\rm real}$ to generate output $I_{\rm pda}$.
The last step of the unsupervised loss computation is to generate the pseudo
labels based on $P_{\rm all}$ and to supervise the output of the student:
$\hat{H}_{r},\hat{O}_{r}=S({\rm PDA}(I_{\rm real},P_{\rm all})).$ (10)
The unsupervised loss $L_{\rm unsup}^{m}=L_{H}^{m}+\lambda_{m}L_{O}^{m}$ is
similar to Eq. 1. Combining both supervised and unsupervised losses, the loss
of the student $S$ is defined as follows:
$L{s}=\lambda_{\rm sup}L_{\rm sup}^{m}+\lambda_{\rm unsup}L_{\rm unsup}.$ (11)
## 4 Experiments
### 4.1 Implementations and Dataset
#### Dataset
ExLPose datasets [29] are used in validating our method, which is a new
dataset that is specifically designed for evaluating 2D human pose estimation
under extremely low-light conditions. The training set of ExLPose provides
$2,065$ pairs of well-lit and low-light images and the ground-truth definition
follows the CrowdPose [31] format. These images were taken from 251 indoor and
outdoor scenes in the daytime and the paired low-light images are artificially
produced by using a dual-camera system with different settings to ensure a
wide coverage of diverse low-light conditions.
There are two different testing sets in ExLPose [29]. One is ExLPose-OCN , in
which 360 extremely low-light images are captured at night with two different
cameras, A7M3 and RICOH3. Another one is ExLPose-test, which consists of 491
man-made low-light images captured and produced similarly as those in the
training set. Low-light All (LL-A) is used to denote the whole ExLPose-test,
which is further divided into three subsets according to their difficulty
level including Low-Light Normal (LL-N), Hard (LL-H), and Extreme (LL-E).
Among the two testing sets, ExLPose-OCN is captured in real low-light
environments, while the low-light images in the ExLPose-test are obtained by
artificially reducing the amount of light by 100 times [29]. In addition,
ExLPose-OCN is a test-only dataset, which is used to evaluate the
generalization capability of a model trained on ExLPose-test. Therefore, the
evaluation on ExLPose-OCN provides a more objective measurement of the
performance of low-light human pose estimation. In our method, we only use the
unpaired labeled well-lit images and unlabeled low-light images in the
training set to train our models. Low-light ground-truth data is never used in
training our model.
Table 1: Evaluation on ExLPose-OCN with top-down baselines (lower part of the
table) and the bottom-up baselines (upper part of the table) tested in our
paper. Our method is a bottom-up method. The reported top-down baselines are
all fully-supervised methods. LL indicates low-light labels. WL indicates
well-lit labels. The best is bold. The second best is underlined.
Methods | Training Labels | AP↑@0.5:0.95
---|---|---
| LL | WL | A7M3 | RICOH3 | Avg.
Base-low [14] | ✓ | | 27.1 | 15.9 | 21.5
Base-well [14] | | ✓ | 5.3 | 7.4 | 6.3
RFormer [1] | | ✓ | 18.9 | 17.7 | 18.3
LLFlow [52] | | ✓ | 23.7 | 19.1 | 21.4
UDA-HE [26] | | ✓ | 6.5 | 7.2 | 6.9
AdvEnt [46] | | ✓ | 9.1 | 11.2 | 10.1
CPN-low [5] | | ✓ | 23.7 | 23.9 | 23.8
CPN-well [5] | ✓ | | 15.2 | 15.6 | 15.4
CPN-all [5] | ✓ | ✓ | 32.8 | 31.7 | 32.2
LLFlow [52] | ✓ | ✓ | 25.6 | 28.2 | 27.0
LIME [15] | ✓ | ✓ | 33.2 | 28.4 | 30.7
DANN [13] | ✓ | ✓ | 27.9 | 30.6 | 29.3
AdvEnt [46] | ✓ | ✓ | 28.2 | 29.0 | 28.6
LSBN+LUPI [29] | ✓ | ✓ | 35.3 | 35.1 | 35.2
Ours | | ✓ | 39.1 | 36.2 | 37.6
#### Evaluation Protocol
We follow the COCO protocol [34], as used in LSBN+LUPI [29], to validate on
ExLPose-test and ExLPose-OCN. Average precision at various thresholds (i.e.,
[email protected]:0.95$) is reported for every subset.
#### Implementation Details
DEKR-W32 [14] is adopted as our backbone network for the main teacher and the
student. HigherHRNet [6] is modified to be the complementary teacher. In the
pre-training stage, the input image of our framework is cropped to $512\times
512$. The data augmentations follow the DEKR paper setting, which include
random rotation $[-30^{\circ},30^{\circ}]$, random scaling $[0.75,1.5]$,
random translation $[-40,40]$ and random flipping. The training strategy of
both teachers is the same with DEKR. The only difference is the loss weighting
factors, the $\lambda_{m}=0.03$ and $\lambda_{c}=0.001$.
In the knowledge acquisition stage, the confidence score thresholds are set to
be $s_{m}=0.9$ and $s_{c}=0.5$ respectively for dual teachers. In training, we
use Adam optimizer. The learning rate is set as $1e-4$. $\lambda_{\rm sup}$
and $\lambda_{unsup}$ are set to be 1.0. Our framework is trained on 4 RTX3090
GPUs (24GB of VRAM each) with a per-GPU batch size of 9 for the pre-training
and 8 for the dual teacher knowledge acquisition stage. The training time for
the dual-teacher setup is 139 seconds per epoch, which is comparable to the
single-teacher setting at 103 seconds per epoch. Additional information is
provided in supplementary material.
#### Baselines
To compare with existing image enhancement and domain adaptation methods, two
SOTA image enhancement methods (RFormer, LLFlow) [1, 52] and two SOTA domain
adaptation methods (UDA-HE, AdvEnt) [26, 46] are used as baselines. To ensure
a fair comparison, we employ the same base human pose estimator (DEKR) [14]
for all the aforementioned methods, as well as for our method, utilizing only
well-lit ground-truth data during training. Following the setup in ExLPose
[29], we replace AdvEnt’s semantic segmentation backbone with the same 2D pose
estimator used in our method. UDA-HE is a model-agnostic domain-adaptive human
pose estimation method. We upgrade the backbone network (ResNet) in UDA-HE by
HRNet, the same backbone used by our method and re-train AdaIN to ensure a
fair comparison. Note, UDA-HE is configured under a bottom-up paradigm, which
is included as a bottom-up baseline in Tab. 3. For a comprehensive comparison,
we add the top-down baselines reported in LSBN+LUPI [29], which are trained
with both well-lit and low-light ground-truth data (unpaired). CPN [5] is used
as the base human pose estimator for these top-down baselines. Among the
additional baselines [29], there is an additional domain adaptation method
(DANN) [13] and another image enhancement method (LIME) [15]. Note that the
training data of LSBN+LUPI [29] is different from all the baselines because it
relies on paired well-lit and low-light images with ground truths. All top-
down baselines employ the same top-down method (CPN) [5] as the base human
pose estimator.
Table 2: Evaluation on ExLPose-test with a bottom-up baseline [14], image
enhancement [1, 52], and domain adaptation [26, 46] methods. All the methods
are fully-supervised by well-lit labels. The best is bold. The second best is
underlined.
Methods | AP↑@0.5:0.95
---|---
LL-N | LL-H | LL-E | LL-A | WL
Base-well [14] | 1.3 | 0.0 | 0.0 | 0.8 | 60.3
RFormer [1] | 16.9 | 2.3 | 0.1 | 5.7 | 60.0
LLFlow [52] | 11.0 | 0.8 | 0.9 | 4.5 | 60.1
UDA-HE [26] | 16.4 | 3.6 | 0.1 | 7.4 | 53.7
AdvEnt [46] | 4.6 | 0.7 | 0.0 | 2.0 | 56.4
Ours | 35.6 | 18.6 | 5.0 | 21.1 | 61.6
Table 3: Evaluation on ExLPose-test with top-down methods trained on both low-
light and well-lit labels (denoted by ${\dagger}$). Differently, our method is
a bottom-up method trained with well-lit labels alone. The best is bold. The
second best is underlined.
Methods | AP↑@0.5:0.95
---|---
LL-N | LL-H | LL-E | LL-A | WL
CPN-low† [5] | 25.4 | 18.2 | 5.0 | 17.2 | 1.2
CPN-well† [5] | 15.3 | 2.7 | 0.4 | 6.7 | 59.9
CPN-all† [5] | 26.4 | 18.2 | 6.1 | 17.6 | 52.3
LLFlow† [52] | 28.7 | 15.7 | 5.3 | 17.4 | 60.7
LIME† [15] | 31.9 | 21.2 | 7.6 | 21.1 | 57.7
DANN† [13] | 28.0 | 17.5 | 5.3 | 17.8 | 52.0
Ours | 35.6 | 18.6 | 5.0 | 21.1 | 61.6
### 4.2 Performance on ExLPose-OCN
Quantitative comparison on ExLPose-OCN is provided in Tab. 1. The upper part
of the table is the bottom-up baselines and the lower part is top-down. Note
that our method is a bottom-up one. The top-down baselines include CPN fully
supervised on low-light images (CPN-low), well-lit images (CPN-well), or both
of them (CPN-all), CPN integrated with low-light image enhancement methods
[52, 15], and domain adaptation methods [13, 46]. It is worth mentioning all
the top-down baselines (except the CPN-well) are fully supervised by both the
well-lit and low-light ground truths and the reported performance, all using
ground-truth person detections. The bottom-up baselines include DEKR trained
on labeled low-light or well-lit images, denoted as Base-low and Base-well,
respectively. Also, we integrate SOTA image enhancement methods, including
RFormer [1] and LLFlow [52], as well as domain adaptation methods, including
output-level method UDA-HE [26] and feature-level method AdvEnt [46]. The
performance of the image enhancement methods is tested by applying the Base-
well to the enhanced images. Therefore, all the bottom-up baselines including
ours are trained with the well-lit ground-truth data alone with the training
set of ExLPose and we only use ExLPose-OCN for evaluation.
Our method outperforms the SOTA method [29] by 6.8% (2.4 AP) on average. As
ExLPose-OCN captures actual low-light images, the superior performance
validates the effectiveness of our method in addressing the low-light
challenges, where our method surpasses all the bottom-up and top-down
baselines without using low-light ground truths or paired well-lit and low-
light data. This highlights the potential of our method in real-world
applications, where we achieve performance surpassing the SOTA methods without
using low-light ground truths.
### 4.3 Performance on ExLPose-test
We compare our method with various bottom-up baselines on ExLPose-test,
including a fully-supervised baseline [14] and four baselines using well-lit
ground-truth data only for training based on image enhancement [52, 1] or
domain adaptation [26, 46], as shown in Tab. 3. As expected, the fully-
supervised baseline, Base-well, is heavily biased towards their training data
due to the lack of cross-domain generalization. Its poor performance clearly
demonstrates the substantial gap between well-lit and extremely low-light
domains. Our method outperforms the baselines using image enhancement or
domain adaptation. In particular, our performance beats the second best method
on the LL-N subset by 18.7 AP which almost doubled the second-best. For LL-H
and LL-E subsets, all methods except ours struggled to achieve reasonable
performance. Our method also maintains the best performance on the WL (well-
lit) subset compared with DEKR-well.
To make a comprehensive comparison, we further compare with top-down baselines
incorporated with image enhancement and domain adaptation methods in Tab. 3.
Note that the comparison is not fair to our approach as the top-down baselines
reported in [29] were all trained using both low-light and well-lit ground
truths. Furthermore, top-down human pose estimation methods generally
outperform their bottom-up counterparts due to their utilization of human
detection, as well as the reduction in human scale variation achieved through
detection cropping and human patch normalization [8].
However, we still achieve the highest performance on the LL-N, LL-A, and WL
subsets in Tab. 3, while obtaining comparable performance on other subsets
without utilizing the low-light ground-truth data. Note LSBN+LUPI [29] is not
included because its training data is different from the baselines in Tab. 3.
LSBN+LUPI utilizes paired well-lit and low-light data and the ground truths,
while the baselines included in the table use unpaired well-lit and low-light
datasets in training. With the extra paired training data and ground truths,
LSBN+LUPI manages to attain competitive performance on ExLPose-test, where its
performance on the WL subset (61.5) is still worse than ours, but it is better
than ours in LL-A (25.0). However, when evaluating the generalization ability
of each method on the real low-light dataset (ExLPose-OCN), where there is no
training and only testing, our method surpasses all the baselines, including
LSBN+LUPI, as demonstrated in Tab. 1.
Table 4: Ablation Study of the proposed modules in our method over ExLPose-
test. PT stands for Pre-Training stage. Step 1 is the fully-supervised
training on well-lit images. Step 2 is the fully supervised training with fake
low-lit images after adding ELLA. KA indicates the Knowledge Acquisition
stage. Single indicates the training with single teacher. Dual indicates our
dual-teacher framework. The best is bold. The second best is underlined.
| Our Design | AP↑@0.5:0.95
---|---|---
ELLA | PDA | LL-N | LL-H | LL-E | LL-A | WL
PT Step 1 | | | 1.3 | 0.0 | 0.0 | 0.8 | 60.3
PT Step 2 | ✓ | | 29.4 | 13.6 | 1.6 | 15.8 | 62.1
KA | ✓ | | 29.5 | 14.6 | 2.9 | 16.8 | 60.5
Single | ✓ | ✓ | 33.4 | 16.6 | 3.1 | 19.0 | 59.4
Dual | ✓ | ✓ | 35.6 | 18.6 | 5.0 | 21.1 | 61.6
Table 5: Number of the additional pseudo labels provided by the complementary
teacher in ExLPose training set. OKS Thres., referring to OKS threshold, is
used to determine the validity of pseudo labels by comparing the OKS between
pseudo labels and the ground truth with them. PL indicates pseudo labels. Main
indicates the mean teacher. Comp. indicates complementary teacher.
OKS Thres. | # of PL in Main | # of PL in Comp. | # of additional PL
---|---|---|---
0.3 | 10203 | 12155 | 4870 (+47.7%)
0.5 | 5056 | 8735 | 2136 (+49.8%)
0.7 | 3293 | 5435 | 1180 (+35.8%)
0.9 | 1463 | 1574 | 501 (+34.2%)
### 4.4 Ablation Studies
Ablation study is performed to validate the effectiveness of the individual
modules in our method is shown in Tab. 4. We use our framework in the pre-
training stage without any newly proposed components as the baseline, and
subsequently add each new module in separate experiments.
#### ELLA
We evaluate our method’s performance with and without the supervision of fake
low-light images generated from WL. The results are in the first two rows of
Tab. 4. The inclusion of ELLA significantly boosts the performance on LL-N,
LL-H, and LL-A test subsets from almost zero to a reasonable level.
#### PDA
We investigate our method’s performance with and without PDA in Stage 2
training. When comparing the third row (single teacher without PDA) to the
fourth row (single teacher with PDA), we observe a $20.2\mathcal{\%}$ (3.4 AP)
improvement on LL-A, accompanied by a marginal 0.4 AP drop on WL. This
highlights the effectiveness of PDA in enabling the student to surpass the
teachers.
#### Dual Teacher
Comparing the last two rows, we observe that the model’s performance with dual
teachers (last row) surpasses that of a single teacher (second last row) in
every split. Notably, The complementary teacher achieves a $11\mathcal{\%}$
(2.1 AP) improvement, which is comparable to the $13\mathcal{\%}$ (2.2 AP)
improvement obtained by the main teacher on LL-A (all subsets), underscoring
the effectiveness of our dual-teacher framework. Additionally, with the help
of the dual teachers, the student model exhibits enhanced performance on the
well-lit split.
To validate that the complementary teacher provides more valid pseudo labels,
quantitative evaluation is provided in Table 5. We measure the validity of
pseudo labels by computing their OKS with corresponding ground truths and
count the number of non-overlapping pseudo labels by thresholding the OKS at
0.5 between the dual teachers, following [34, 31]. The evaluation demonstrates
that the complementary teacher consistently produces additional valid pseudo
labels across different thresholds of quality, providing an average of 41.8%
more valid pseudo labels compared to using the main teacher alone. The
increased number of valid pseudo labels leads to further improvement in the
student’s training, highlighting the effectiveness of our dual-teacher
framework.
## 5 Conclusion
We introduce the first human pose estimation method for extremely low-light
conditions, utilizing well-lit ground-truth data exclusively. Our novelty lies
in leveraging center-based and keypoint-based teachers to generate enriched
pseudo labels, enabling the student model to achieve competitive performance
on extremely low-light images without the need for training with low-light
ground truths. To our knowledge, this represents a significant advancement in
the field of low-light human pose estimation.
## Acknowledgements
This research is in part supported by the Singapore Ministry of Education
Academic Research Fund Tier 1 (WBS: A-8001229-00-00), a project titled
“Towards Robust Single Domain Generalization in Deep Learning”.
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|
# Verification of C++ Programs with VeriFast
Niels Mommen imec-DistriNet Research Group, KU Leuven, Belgium Bart Jacobs
imec-DistriNet Research Group, KU Leuven, Belgium
(December 2022)
###### Abstract
VeriFast is a prototype tool based on separation logic for modular
verification of C and Java programs. We are in the process of adding support
for C++. In this report, we describe the features of C++ for which we added
support so far, as well as the proof obligations we generate for these
features. At this point, VeriFast has basic support for most object-oriented
programming features of C++: member functions, member function and operator
overloading, implicit and explicit conversions, constructors and initializer
lists, destructors, reference types, allocation and deallocation on the stack
or on the heap (using new and delete), inheritance (including multiple
inheritance but not virtual base classes), and virtual member functions and
overriding. To support specification of inheritance hierarchies, we added
support for instance predicates, which can be introduced in a base class and
overridden in derived classes. The main missing feature at this point is
support for C++ templates, which we plan to work on next.
###### Contents
1. 1 Introduction
2. 2 C++ basic features
3. 3 Classes and objects
1. 3.1 Construction of objects
2. 3.2 Destruction of objects
4. 4 Inheritance
1. 4.1 Upcasts
2. 4.2 Multiple inheritance
3. 4.3 Construction and destruction
5. 5 Virtual methods
1. 5.1 Object types at run time
2. 5.2 Construction and destruction
3. 5.3 Behavioural subtyping
6. 6 Instance predicates
7. 7 Limitations
8. 8 Future work
## 1 Introduction
VeriFast111https://github.com/verifast/verifast is a prototype tool that
performs modular symbolic execution for modular verification of C and Java
programs, based on separation logic [2]. Currently, support for verification
of programs written in C++ is being added to VeriFast. We use LibTooling, a
library to write tools based on Clang, to retrieve a typechecked abstract
syntax tree for well-typed C++ programs
222https://clang.llvm.org/docs/LibTooling.html. This report describes the C++
features that are currently verifiable by VeriFast and the changes that were
needed in order to verify these features. The added features include
verification of member functions, constructors and destructors, and virtual
member functions in the presence of multiple inheritance. To conclude, present
limitations and future work is discussed. A snapshot of binaries and the
source code of VeriFast with C++ support at the time of writing is available
as a Zenodo drop at https://zenodo.org/record/7486648.
## 2 C++ basic features
A first extension added to VeriFast is support for lvalue references and the
new and delete operator for primitive types.
#### Lvalue references
VeriFast treats lvalue references similar to pointers. By default, an lvalue
reference evaluates to the address the reference points to. When an lvalue to
rvalue conversion is needed to retrieve to value of the object the reference
points to, VeriFast implicitly dereferences the pointer. Lvalue reference
types are currently not supported in ghost code. It is however possible to
reason about an object that is referenced by an lvalue reference by taking the
address of that reference, which is equivalent to a pointer to that object.
#### new and delete
The new and delete operators in C++ are the counterparts of malloc and free in
C. However, it is not allowed to use them interchangeably: an object allocated
on the heap through new cannot be destroyed by passing the address returned by
new to free. Therefore, VeriFast respectively produces and consumes a
new_block when calling new and delete, instead of malloc_block chunks.
## 3 Classes and objects
Classes in C++ are an extended form of structs in C. First, optional default
expressions can be specified for data members. Such an expression is evaluated
to initialize the data member when an instance of the class, an object, is
created.
During object creation of class type S at address addr, VeriFast traverses
each field of S in their order of declaration and produces a corresponding
field chunk S_field_name(addr,value)333A points-to chunk a->field_name |-> v
can alternatively be used to refer to a field chunk at address a with value
v., where field_name is the name of the class field and value either is the
evaluation result of field_name’s default member initializer or represents an
unspecified value in case no initializer is present.
Next to data members, static and non-static member functions can be defined. A
non-static member function always has an implicit this parameter, which is a
pointer to an instance of the declaring class: it points to the target of the
function call.
When verifying a non-static member function, VeriFast produces a fresh symbol
that represents its implicit this argument, which is assumed to not be zero,
i.e. not a null pointer. Verification of a non-static member function
continues as usual: first the precondition is produced, next the compound
statement of the function body is verified, and finally the postcondition of
the function is consumed and a check is performed to verify that no chunks are
leaked. Calling a member function on a null pointer results in undefined
behaviour. Therefore, a member function cannot be called when it not possible
to prove that the target object of the member function call is not zero.
### 3.1 Construction of objects
A constructor in C++ is a special member function of a class that cannot be
called directly; it is automatically called when an instance of that class is
created. Multiple constructors can be defined within a class. An applicable
constructor is selected using overload resolution when an object is created.
A constructor consists of an optional initializer list and a body consisting
of a compound statement. During construction, all fields are initialized in
their declaration order prior to executing the destructor’s body. An
initializer list optionally defines data member initializer expressions which
take precedence over the default member initializers declared in its class.
Therefore, during field initialization of data field m of an object of class
type S, VeriFast selects the appropriate expression from the initalizer list
if it mentions m. Otherwise, the default member initializer is used if
available or the field is initialized using its default constructor when the
field type is a class type, or it is initialized with an unspecified value if
the field type is a primitive type. Accordingly, verification of a constructor
consists of the following steps:
1. 1.
Initialize fields in order of declaration, where entries in the initializer
list take precedence over default member initializers;
2. 2.
Symbolically execute the constructor body.
An initializer list of constructor C of class type S can alternatively consist
of exactly one call to another constructor C’. This skips field initialization
and delegates construction to the applicable constructor C’ selected using
overload resolution, in which case VeriFast verifies a call to C’. Afterwards,
construction continues by verifying the body of C.
An object can either be allocated on the stack or on the heap. A stack-
allocated object of class type S is created when a variable of type S is
declared by automatically verifying a call to an applicable constructor.
Objects can be allocated on the heap by invoking a new expression. This
expression invokes a call to an applicable constructor and returns a pointer
to the object that has been allocated on the heap. Verification of a new
expressions for an object of class type S first verifies a call to the
constructor and additionally produces a new_block_S(addr) chunk analogously to
a malloc_block_S(addr) chunk in C, where addr is the address in the heap at
which the object was created.
### 3.2 Destruction of objects
Destructors are the counterpart of constructors: they are called automatically
when the lifetime of an object ends. Contrary to constructors, destructors do
not have an equivalent counterpart of an initializer list.
Verification of a destructor D for an object of class type S at address addr
happens in reverse order of verifying a constructor of S. First, the
destructor body is verified. Next, all fields of the object are destroyed in
reverse order of their declaration. For a field m of primitive type, its
corresponding field chunk S_m(addr, _) is consumed, where _ represents a dummy
pattern which matches any value. Fields that are of class type S’ are
destroyed by verifying a call to the destructor of S’. Verification of a
destructor performs following the steps:
1. 1.
Symbolically execute the destructor body;
2. 2.
Destruct fields in reverse order of declaration.
Objects that are allocated on the stack are automatically destroyed at the end
of their scope by verifying a call to their destructor. Explicitly calling a
destructor on a stack-allocated object is not allowed by VeriFast. This would
lead to undefined behaviour when an object gets destroyed automatically at the
end of its scope when it was already explicitly destroyed.
For an object that was allocated on the heap at address addr, the delete
operator can be used to destruct its operand. Verification of a delete addr
expression, where addr is a pointer to an object of class type S, takes place
in two steps. First, a call to the destructor of S is verified. Afterwards, a
new_block_S(addr) chunk is consumed. This guarantees that an object allocated
on the heap is never destroyed twice, which would otherwise lead to undefined
behaviour.
## 4 Inheritance
A class in C++ can extend another class, inheriting all accessible members
from the class it extends. When class D extends class B, B is called a
(direct) base class of D and D is a derived class of B.
VeriFast models a base object of class type B as a subobject in a derived
object of class type D. For an object d of class type D created at address
d_addr, its base object of class type B can be accessed through a field
pointer field_ptr(d_addr, D_B_offset), where D_B_offset represents an offset,
$\geq 0$, from d to its base object of class type B.
### 4.1 Upcasts
Upcasting a derived object to its base object is supported by VeriFast both
implicitly and explicitly. Such a cast is first required when accessing a base
field member. Second, a cast is needed when passing a derived object as an
argument to a function or method parameter that expects a base object, base
object pointer or reference. Lastly, casting a derived object to its base
object is required when calling a base member function on a derived object.
The evaluation of an upcast from a derived object of class type D at address
d_addr to a base object of class type B is executed in two steps. First a
check is performed to make sure that D derives from B. Next, a field pointer
field_ptr(d_addr, D_B_offset) is computed to retrieve the address of the base
object.
Note that implicit upcasts are limited in ghost code. The type checker of
VeriFast automatically inserts them when it is able to deduce the need for
such a cast. This is currently not supported when a points-to notation is used
to reason about base fields in a derived object. E.g., B_m(d_addr, ?m)
implicitly involves an upcast to access base field m from derived object d,
while ((B *) d_addr)->m |-> ?m requires an explicit upcast to access the same
field.
### 4.2 Multiple inheritance
C++ offers the feature to derive from multiple base classes. Treating base
classes as subobjects through field pointers in VeriFast includes support for
multiple inheritance. Fields of any base object can be accessed by first
upcasting the derived object to the base, prior to accessing the field in that
base. Similarly, visible base member functions from any base can be used
through a derived object by first implicitly or explicitly performing an
upcast to the base target.
In the presence of multiple inheritance, an implicit upcast from a derived
object to a base object might be ambiguous when multiple base classes derive
from the same class. Assume a diamond scenario where class D derives from both
class B and class C, and classes B and C in turn both derive from class A. In
this scenario, an object of class type D at address d_addr has two subobjects:
one object of class type B at address field_ptr(d_addr, D_B_offset) and
another object of class type C at address field_ptr(d_addr, D_C_offset). These
subobjects both have one subobject of class type A themselves at respectively
address field_ptr(field_ptr(d_addr, D_B_offset), B_A_offset) and address
field_ptr(field_ptr(d_addr, D_C_offset), C_A_offset). Accessing a field m of
class A from d_addr is ambiguous: it not clear whether this refers to field m
reachable through the subobject of class type B or the subobject of class type
C. Hence, an explicit cast is required to render the upcast unambiguous. E.g.,
A_m((B *) d_addr, ?m_val) can be used to reason about field m in the subobject
at address field_ptr(field_ptr(d_addr, D_B_offset), B_A_offset).
### 4.3 Construction and destruction
Verification of constructors and destructors in the presence of inheritance
requires additional steps in the verification process. Verifying a constructor
first starts with constructing all base classes in declared derivation order
if the constructor being verified is not a delegating constructor. Next,
verification proceeds as described before in Section 3.1.
VeriFast verifies the construction of bases by verifying a call to a base
constructor of each direct base class. An appropriate base constructor is
selected by overload resolution if the initializer list of the constructor
mentions a base class initialization, otherwise the default base constructor
is used. The value of the implicit this argument passed to the base
constructor is calculated by performing an upcast from the current
constructor’s implicit this parameter to the base object that will be
constructed by verifying the selected constructor call. E.g., when verifying a
base constructor call of class B in the constructor of derived class D for an
object at address d_addr, a value field_ptr(d_addr, D_B_offset) is passed as
the implicit this argument to the constructor call of class B.
Verification of a destructor accordingly accounts for inheritance. After
following the verification steps for destruction listed in Section 3.2,
VeriFast first verifies a call to each destructor of all direct bases in
reverse order of derivation if the class object to be destroyed has any base
objects. Analogously to constructors, an upcast to the base object is
performed when calculating the value passed as the implicit this argument to
the base destructor verification call.
Care has to be taken during construction and destruction of objects in the
presence of inheritance. Member functions can be called directly and
indirectly during construction or destruction, but this results in undefined
behaviour if not all bases have been fully constructed. Therefore, an
S_bases_constructed(S *s_addr) chunk is introduced to determine whether all
bases for an object of class type S at address s_addr have been constructed.
This chunk is required when verifying a member function call where the target
object derives from at least one class, disallowing calling any member
function when this chunk is not available.
Verification of a constructor can now be summarized by the following steps:
1. 1.
Verify a constructor call for each base class in order of derivation;
2. 2.
Produce an S_bases_constructed(s_addr) chunk if S derives from at least one
class, where S is the class type of the object that is currently being
constructed and s_addr is the address of the object;
3. 3.
Initialize fields in order of declaration, where entries in the initializer
list take precedence over default member initializers;
4. 4.
Symbolically execute the constructor body.
Verification of a destructor now additionally destructs base classes and
optionally consumes a bases_constructed chunk:
1. 1.
Symbolically execute the destructor body;
2. 2.
Destruct fields in reverse order of declaration;
3. 3.
Consume an S_bases_constructed(s_addr) chunk if S derives from at least one
class, where S is the class type of the object that is currently being
destructed and s_addr is the address of the object;
4. 4.
Verify a destructor call for each base class in reverse order of derivation.
## 5 Virtual methods
A Member function can be declared virtual with the virtual keyword. This
allows to override the member function in derived classes. Dynamic dispatch is
used to select the appropriate member function implementation when an
unqualified member function call is evaluated: the dynamic type of the target
object is inspected at run time in order to select a member function
implementation. That is, if a virtual member function is called on a base
object, dynamic dispatch selects the final overrider. A virtual member
function in a base class is a final overrider if no derived class declares a
member function that overrides it. Qualified member function calls are
interpreted as statically bound calls, not resulting in dynamic dispatch at
run time. A class (object) that has at least one virtual member function, will
be referred to as a polymorphic class (object).
### 5.1 Object types at run time
In order to reason about virtual member functions and polymorphic classes,
VeriFast first introduces a typeid construct that can be used in ghost code.
This construct acts similar to the typeid operator in C++: it takes one
argument, a type expression, and returns a reference to an std::type_info
object which uniquely represents that type. The difference is that typeid in
VeriFast only accepts type expressions as its argument, whereas the typeid
operator in C++ accepts any expression. A typeid(S) evaluates to value
S_type_info, where S is a type expression referring to class type S.
In addition, for each polymorphic class S, VeriFast introduces a predicate
S_vtype(S *s_addr; std::type_info *s_info). This predicate allows to reason
about the run-time type or its most derived type. In case class S has at least
one polymorphic base, this predicate is defined as
⬇
/*@ predicate S_vtype(S *s_addr, std::type_info *s_info) =
B0_vtype(s_addr, s_info) &*&
B1_vtype(s_addr, s_info) &*&
... &*&
Bn_vtype(s_addr, s_info);
@*/
where $B_{0},\ldots,B_{n}$ are polymorphic direct base classes of S with
$n>0$. Otherwise, the vtype predicate is opaque and cannot be opened or
closed.
This allows to open a vtype chunk of a polymorphic class in order to retrieve
all vtype chunks of its polymorphic direct bases. Analogously, all vtype
chunks of polymorphic direct base objects are required to close a vtype chunk
of a polymorphic derived object.
When verifying a virtual member function call on a target object at address
s_addr with static class type S, VeriFast first checks that (some fraction of)
an S_vtype(s_addr, _) chunk is available. Otherwise, the member function call
is not allowed. In order to call a base member function on a derived target
object, the vtype chunk of the derived object can be opened to obtain a vtype
chunk for the base object.
### 5.2 Construction and destruction
To be able to call virtual member functions, vtype chunks for polymorphic
objects have to be produced at some point during construction. These chunks
then have to be consumed during destruction of an object.
Like regular member functions, virtual member functions can be called during
construction and destruction of an object. However, in presence of multiple
inheritance, care has to be taken. First, during construction or destruction
of an object, the virtual function called is the final overrider in the
constructor or destructor class. Hence, potential overriders in derived
classes are not taken into account during the construction or destruction of a
base class.
A second point of attention is the presence of multiple inheritance. During
construction and destruction, virtual base member function calls on the object
under construction or destruction are only allowed for bases that belong to
the inheritance sub hierarchy of that object. Otherwise, if a virtual member
function is called on a subobject belonging to another branch of the
inheritance hierarchy, the virtual member call would result in undefined
behaviour.
⬇
struct A {
virtual void foo() {}
};
struct B {
B(A *a) {
a->foo(); // undefined behaviour
}
virtual void bar() {}
};
struct C : public A, public B {
C() : A(), B(this) {
foo();
bar();
}
};
Listing 1: Virtual member call during construction, resulting in undefined
behaviour.
LABEL:lst:_vcall_ub illustrates an example where virtual base member functions
are called in the constructor of C. These calls are allowed because at the
time of the call, both base A and B are fully constructed and they belong to
the inheritance hierarchy of C. However, the constructor of B that is called
through the constructor of class C would lead to undefined behaviour because
it calls member function foo of base class A: the subobject of class type A is
fully constructed, but it does not belong to the inheritance hierarchy of the
subobject of class type B.
In order to address these instances where calling virtual member functions
would lead to undefined behaviour, VeriFast produces the D_vtype(d_addr,
D_type_info) chunk in the constructor of polymorphic class D after all its
bases have been constructed, where d_addr is the address of the object that is
constructed. Right after a constructor of a polymorphic base class B has
finished, the derived constructor of class D consumes the
B_vtype(field_ptr(d_addr, D_B_offset), B_type_info) that was produced by the
base constructor call. This disallows other base objects that still have to be
constructed in other branches of the inheritance hierarchy from calling
virtual methods in branches that would lead to undefined behaviour.
Verification of a constructor goes as follows:
1. 1.
Verify a constructor call for each base class in order of derivation;
1. (a)
If the base class is polymorphic, consume a B_vtype(field_ptr( d_addr
D_B_offset), B_vtype_info) chunk, where d_addr is the address of the derived
object of class type D and B represents the class type of the base object;
2. 2.
Produce a D_vtype(d_addr, D_type_info) chunk if the object under construction
is polymorphic, where D is the class type of the object under construction;
3. 3.
Produce a D_bases_constructed(d_addr) chunk if D derives from at least one
class, where D is the class type of the object that is currently being
constructed and d_addr is the address of the object;
4. 4.
Initialize fields in order of declaration, where entries in the initializer
list take precedence over default member initializers;
5. 5.
Symbolically execute the constructor body.
Verification of a destructor again changes in a similar fashion. After
destructing all member fields of class D, VeriFast consumes a vtype(d_addr,
D_type_info) chunk if it is polymorphic, where d_addr is the address of the
object under destruction. This disables polymorphic base objects from calling
virtual member functions of its derived object and calling member functions of
bases that are are part of another branch in the inheritance hierarchy. Prior
to verifying a polymorphic base destructor call, VeriFast produces a
B_vtype(b_addr, B_type_info) chunk, where B is the class type of the base
object at address b_addr. This allows the polymorphic base object to call
virtual member functions in its inheritance sub hierarchy. Hence, verification
of a destructor can be performed through the following steps:
1. 1.
Symbolically execute the destructor body;
2. 2.
Destruct fields in reverse order of declaration;
3. 3.
If the object of class type D at address d_addr under destruction is
polymorphic, consume a D_vtype(d_addr, D_type_info) chunk;
4. 4.
Consume a D_bases_constructed(d_addr) chunk if D derives from at least one
class, where D is the class type of the object that is currently being
constructed and d_addr is the address of the object;
5. 5.
Verify a destructor call for each base class in reverse order of derivation;
1. (a)
Prior to verifying a base destructor call of a base object of class type B
that is polymorphic, produce a vtype(field_ptr(d_addr, D_B_offset),
B_type_info), where D is the class type of the derived object at address
d_addr.
### 5.3 Behavioural subtyping
When verifying a virtual member function call, VeriFast uses the same contract
that would be used during a statically bound call. However, the run-time type
of the target object might be a derived type of the statically known target
type. The static type acts as an upper bound for the objects’s dynamic type
[3]. Therefore, VeriFast performs a behavioural subtyping check for each
virtual member function that is overridden to make sure that the overriding
member function can statically call the member function that is overridden
[4]. First, the precondition of the overridden member function is produced.
Next, VeriFast consumes the precondition of the overriding member function and
produces the postcondition of the overriding member function. Lastly, VeriFast
consumes the postcondition of the overridden member function.
This behavioural subtyping requirement makes verification of virtual member
functions harder and sometimes impossible. It often also requires changes to
the specification of the member function that is overridden and reverification
of the overridden member function.
⬇
struct A {
virtual int foo()
//@ requires true;
//@ ensures result >= 0;
{
return 0;
}
};
struct B : public A {
int m = 10;
int foo() override
/*@ requires
B_bases_constructed(this) &*&
this->m |-> ?m_val &*&
m_val >= 0;
@*/
/*@ ensures
B_bases_constructed(this) &*&
this->m |-> m_val &*&
result == m_val;
@*/
{
return m;
}
};
Listing 2: Virtual member function override, violating behavioural subtyping.
Take as an example the code snippet in LABEL:lst:_behavioral_subt_violation.
Class B overrides member function foo from its base class A. The specification
in class B clearly violates the behavioural subtyping check: the precondition
of the overriden member function does not imply the precondition of the
overriding member function. In order to be able to verify this program, the
precondition of foo in A has to be changed to require the chunks mentioned in
the precondition of foo in B. However, these chunks would only be available
when the run time class type of the target object would be B. Changing the
specification requires reverification of the overridden method, making modular
verification harder. In order to tackle this problem, VeriFast supports the
use of instance predicates.
## 6 Instance predicates
VeriFast already supports dynamically bound instance predicates for Java
programs [5]. An instance predicate can be defined in a class body and does
not have a single definition: each subclass defines its own definition of the
predicate. We also added this feature for classes in C++.
⬇
struct Shape {
//@ predicate valid() = true;
Shape()
//@ requires true;
//@ ensures valid() &*& Shape_vtype(this, thisType);
{
//@ close valid();
}
virtual int calcArea() const = 0;
//@ requires valid();
//@ ensures valid() &*& result >= 0;
};
class Square : public Shape {
int m_width;
public:
/*@ predicate valid() =
this->valid(&typeid(Shape))() &*&
this->m_width |-> ?w &*&
w >= 0 &*&
Square_bases_constructed(this);
@*/
Square(int width) : m_width(width)
//@ requires width >= 0;
//@ ensures valid() &*& Square_vtype(this, thisType);
{
//@ close valid();
}
int calcArea() const override
//@ requires valid();
//@ ensures valid() &*& result >= 0;
{
//@ open valid();
return m_width * m_width;
//@ mul_mono_l(0, this->m_width, this->m_width);
//@ close valid();
}
};
Listing 3: Shape and Square example with instance predicates.
An instance predicate has, similar to C++ member functions, an implicit this
parameter that is bound to the target object. Each derived class can override
an instance predicate that is declared in one of its base classes. Hence, an
instance predicate chunk contains an index argument to distinguish between the
different versions of the predicate. This index is a pointer to an
std::type_info object in VeriFast when verifying C++ programs. For example, an
index with value S_type_info determines the instance predicate defined in
class S. Every type has a unique std::type_info object, which allows to
differentiate overridden instance predicate chunks by their index.
The example in LABEL:lst:_inst_pred_ex defines an abstract Shape class that
declares an instance predicate valid. This class is inherited by the Square
class, which also overrides the definition of valid. An instance predicate
chunk of valid in Shape has signature Shape#valid(addr, Shape_type_info),
where addr is a pointer to the target object and Shape_type_info is the
symbolic value that represents a pointer to the std::type_info object of class
Shape. Remember that this value can be obtained by evaluating &typeid(Shape).
Both the constructor of Shape and Square in LABEL:lst:_inst_pred_ex mention a
thisType variable: a ghost variable that exists to make verification feasible.
This variable is implicitly available in each non-static member function of a
class and represents the value that would be obtained from evaluating
&typeid(S), where S is the class type of the target object that is bound the
member function call. The thisType ghost variable is implicitly used by
VeriFast as the index of an instance predicate where the target object is
implicit this. Hence, the chunk produced by valid() in the postcondition of
constructor Shape() is Shape#valid(addr, thisType), where addr is the address
of the Shape object that would be constructed by the constructor.
The value of thisType depends on whether a member function call is dynamically
or statically bound. This allows to use a different interpretation of the
contract for statically bound member function calls and dynamically bound
member function calls, only requiring one specification from the programmer.
I.e., the value of thisType depends on the binding of a member function call.
The value of thisType is assumed to be equal to S_type_info during
verification of a non-static member function, where S is the static type of
the target object. On the other hand, during verification of a dynamically
bound member function call, thisType is assumed to be a pointer to the
std::type_info object for the most derived class type of the target. As
explained in Section 5.1, a check is performed for the existence of (some
fraction of) an S_vtype(addr, ?info) before verifying a virtual member
function call, where S is the static class type of the target expression and
addr its address. VeriFast now assumes that thisType has value info during
verification of the virtual member function call. The interpretation of a
specification must be the same during verification at the call site and during
verification of the callee. Therefore, VeriFast checks that a derived class
overrides all virtual member functions in all of its polymorphic direct bases
to meet this requirement [1].
The target object of a predicate instance assertion can also be mentioned
explicitly, as LABEL:lst:_inst_pred_ex shows in the instance predicate
definition of valid in class Square. In this instance,
this->valid(&typeid(Shape))() refers to the instance predicate definition of
Shape and is represented by a Shape#valid(this, Shape_type_info) chunk.
Evaluation of an instance predicate assertion with an explicit target object
that does not mention an explicit index depends on the nature of its target.
For a class S that is not polymorphic and defines an instance predicate
s_pred($a_{0}$,$\ldots$,$a_{n}$), and an object of class type S at address
s_addr, s_addr->s_pred($a_{0}$,$\ldots$,$a_{n}$) evaluates to S#s_pred(s_addr,
S_type_info, $a_{0}$,$\ldots$,$a_{n}$). However, when S is polymorphic,
VeriFast treats s_addr->s_pred($a_{0}$,$\ldots$,$a_{n}$) as syntactic sugar
for S_vtype(s_addr, ?s_type) &*&
s_addr->s_pred(s_type)($a_{0}$,$\ldots$,$a_{n}$).
⬇
struct A {
//@ predicate valid() = true;
virtual int foo()
//@ requires valid();
//@ ensures valid() &*& result >= 0;
{
return 0;
}
};
struct B : public A {
int m = 10;
/*@ predicate valid() =
this->m |-> ?m_val &*&
m_val >= 0 &*&
B_bases_constructed(this);
@*/
int foo() override
//@ requires valid();
//@ ensures valid() &*& result >= 0;
{
//@ open valid();
return m;
//@ close valid();
}
};
Listing 4: Verifiable version of LABEL:lst:_behavioral_subt_violation with
instance predicates, while preserving behavioural subtyping.
Using instance predicates, we can now annotate and verify the example from
LABEL:lst:_behavioral_subt_violation while preserving behavioural subtyping as
illustrated in LABEL:lst:_inst_pred_beh_subt.
## 7 Limitations
It is currently only possible to pass objects by reference or by pointers. No
support has been added yet to pass objects by value. Therefore, all function
parameters and function return types are either primitive types, reference to
objects or pointers to objects.
Explicit destructor calls are currently not allowed by VeriFast. This avoids
explicitly destroying a stack-allocated object, when it might later be
destroyed automatically when the object goes out of scope, potentially leading
to undefined behaviour. However, explicit constructor calls are required once
placement new is supported in order to construct an object in preallcoated
memory.
Verification for virtual destructors is infeasible when a polymorphic class
inherits from multiple polymorphic base classes that have a virtual
destructor. The overriding virtual destructor in the derived class can only
meet the requirement of behavioural subtyping for one of its base classes,
because it is currently not supported to declare an instance predicate in a
derived class that overrides multiple instance predicates defined in its base
classes.
Furthermore, heap-allocated objects that inherit from at least one base object
cannot be destroyed with the new operator through a pointer to one of its base
objects, even if it has a virtual destructor. Verification of the delete
operator on a pointer to a base object of a derived object requires a
new_block chunk that was allocated for the base object.
## 8 Future work
Support for verification of C++ programs with VeriFast is still in
development. We are first planning to address the limitations discussed in
Section 7. In addition, we plan to add support for virtual inheritance and
rvalue references. Next, our goal is support for verification of programs with
C++ templates, which is one of the main missing features in VeriFast.
Constraints and concepts, language features added in C++20, are interesting
aspects when looking into verification of programs with templates. Once these
features have been added to VeriFast, we plan to verify some critical
industrial C++ programs. This would give insight in C++ features that are
commonly used and might involve interesting and new verification challenges.
## References
* [1] Michael Barnett, Robert DeLine, Manuel Fähndrich, K Rustan M Leino, and Wolfram Schulte. Verification of object-oriented programs with invariants. J. Object Technol., 3(6):27–56, 2004.
* [2] Bart Jacobs, Jan Smans, Pieter Philippaerts, Frédéric Vogels, Willem Penninckx, and Frank Piessens. VeriFast: A powerful, sound, predictable, fast verifier for C and Java. In NASA formal methods symposium, pages 41–55. Springer, 2011.
* [3] Gary T Leavens, Krishna Kishore Dhara, and Krishna Kishore Dhara. Concepts of behavioral subtyping and a sketch of their extension to component-based systems. 2000\.
* [4] Matthew J Parkinson. Local reasoning for Java. Technical report, University of Cambridge, Computer Laboratory, 2005.
* [5] Jan Smans, Bart Jacobs, and Frank Piessens. VeriFast for Java: A tutorial. Aliasing in Object-Oriented Programming. Types, Analysis and Verification, pages 407–442, 2013.
|
# $\Gamma$-Convergence for plane to wrinkles transition problem
Peter Bella111 Technische Universität Dortmund Fakultät für Mathematik, 44227
Dortmund, Germany. E-mail<EMAIL_ADDRESS>Roberta
Marziani222Dipartimento di Ingegneria e Scienze dell’Informazione e
Matematica, 67100 L’Aquila, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract
We consider a variational problem modeling transition between flat and
wrinkled region in a thin elastic sheet, and identify the $\Gamma$-limit as
the sheet thickness goes to $0$, thus extending the previous work of the first
author [Bella, ARMA 2015]. The limiting problem is scalar and convex, but
constrained and posed for measures. For the $\Gamma-\liminf$ inequality we
first pass to quadratic variables so that the constraint becomes linear, and
then obtain the lower bound using Reshetnyak’s theorem. The construction of
the recovery sequence for the $\Gamma-\limsup$ inequality relies on
mollification of quadratic variables, and careful choice of multiple
construction parameters. Eventually for the limiting problem we show existence
of a minimizer and equipartition of the energy for each frequency.
## 1 Introduction
This paper is about fine analysis of minimizers of a nonconvex variational
problem which describes wrinkling of thin elastic sheets.
Motivated by some physical experiments with thin elastic sheets [22, 23, 24],
the first author, in his PhD thesis [8] (see also [5]), considered a specific
variational problem describing deformations of a thin elastic sheet
$\Omega_{h}\subset\mathbb{R}^{3}$ of thickness $h$ and cross section of
annular shape. The elastic energy, corresponding to a deformation
$v\colon\Omega_{h}\to\mathbb{R}^{3}$, consists of a membrane term, measuring
stretching and compression of the sheet, and of a bending term, which
penalizes curvature. As a proxy for the energy one can think of
$E_{h}(v)=\int_{\Omega_{h}}\mathrm{dist}^{2}(\nabla v,{\mathrm{SO}}(3))\,{\rm
d}x+h^{2}\int_{\Omega_{h}}|\nabla^{2}v|^{2}\,{\rm d}x.$ (1.1)
The membrane part is non-convex, possibly giving rise to oscillations. In
contrast, the latter bending part is convex and of higher-order, thus
regularizing the problem. Since the bending resistance is related to the sheet
thickness $h$, the magnitude of this contribution asymptotically vanishes in
the limit $h\searrow 0$.
The physics approach to tackle these problems consists of a specific choice of
an ansatz (guess) for the form (shape) of a minimizer. In other words, one
restricts the analysis to a class of competitors having specific
characteristics, and look for a minimizer of the energy within that class. On
the other hand, the rigorous analytical approach does not make any assumptions
on the form of a minimizer, i.e., the energy is minimized over all possible
deformations. The problem in (1.1) being non-convex, hence possibly possessing
many (local) minimizers or critical points, the first step is to understand
the minimal value of the energy, with possibly learning some clues by which
deformations is this minimal value, at least approximately, achieved.
Figure 1: Elastic annular membrane stretched in the radial direction. The blue
dotted curve represents the free boundary that separates the outer stretched
region from the inner wrinkled one.
Hence, we first try to identify the minimal value of the energy. Precisely, in
the present situation, the goal is to understand its dependence on the (small)
sheets thickness $h$. It turns out that the minimal value $\min_{v}E_{h}(v)$
consists of a leading zeroth-order term $\mathcal{E}_{0}$ (coming from the
stretching of the sheet) plus a linear correction in $h$, which corresponds to
the cost of _wrinkling_ of the sheet [8, 5]. More precisely, there exist two
constants $0<C_{0}<C_{1}<\infty$ such that for any thickness $0<h<1$ there
holds
$\mathcal{E}_{0}+C_{0}h\leq\min_{v}E_{h}(v)\leq\mathcal{E}_{0}+C_{1}h.$ (1.2)
The wrinkling serves as a mechanism to relieve compressive stresses, which are
caused by specific geometrical effects. An alternative to wrinkling would be
simply compression, which contributes to the membrane part at the order
$O(1)$. Hence, in the case of small thickness (present situation) compression
is much less energetically favorable ($O(1)$ vs $O(h^{2})$), and thus not
expected.
The identified linear scaling law (1.2) in $h$ for the minimal value of the
energy raised a lot of discussion among the physics community, having improved
their ansatz-based prediction by a factor of $\log h$ (i.e. $h$ in [5] vs
$h{(|\log h|+1)}$ in [22]). It turns out that this discrepancy is related to a
suboptimal choice of the ansatz close to the interface between the wrinkled
and the flat region. Moreover, the upper bound in (1.2) is achieved through a
complex construction involving branching effects, a pattern which was not
observed experimentally.
To shed light on this discrepancy, the first author considered a variational
problem modelling the transition region [9] with the aim of better
understanding the behavior of the minimizer in that region. Working at the
level of the energy, this means to consider the quantity
$\frac{\min_{u}E_{h}(u)-\mathcal{E}_{0}}{h}$, which not only is bounded away
from $0$ and $\infty$ (see (1.2)), but as $h\searrow 0$ it actually converges
to some value $\sigma$ (as proven in [9]). Even though the value $\sigma$ is
characterized as a limit of minima of simpler scalar and convex variational
problems, it _does not_ provide any information on the form of sequence of
minimizers.
In that respect, the goal of this paper is to overcome this shortcoming by
showing $\Gamma$-convergence of the functionals
$\frac{E_{h}-\mathcal{E}_{0}}{h}$ as $h\searrow 0$. As usual, a consequence of
$\Gamma$-convergence is convergence of minimizers $u_{h}$ of $E_{h}$ to a
minimizer of the limiting problem, hence providing information on $u_{h}$, at
least for $0<h\ll 1$. Denoting by $\mathcal{F}_{\infty}$ the $\Gamma$-limit
functional (see (2.9) below), it turns out that as expected from [9],
$\mathcal{F}_{\infty}$ is scalar and convex, thus possibly much easier to
analyze than the original $E_{h}$. Nevertheless, the study of minimizers of
$\mathcal{F}_{\infty}$ is still far from trivial and we postpone it to a
future work – except for some preliminary results collected in Section 6.
There are many areas of material science, most of them falling within a class
of energy-driven pattern formation [29], where the idea to study energy
scaling laws for variational problems turned out to be very fruitful. The
common features of these problems is the presence of a nonconvex term in the
energy, which is regularized by a higher-order term with a small prefactor.
This small parameter (for now denoted $\varepsilon$) has different meanings:
thickness in the case of elastic films, inverse Ginzburg-Landau parameter in
the theory of superconductors, strength of the interfacial energy for models
of shape-memory alloys or micromagnetics, to name just few. As
$\varepsilon\searrow 0$, the oscillations caused by the nonconvexity are less
penalized, giving the energy more freedom to form patterns/microstructure.
The first paper in this direction, in the context of shape memory alloys, is a
seminal work of Kohn and Müller [28], where they studied a toy problem to
model the interface in the austenite-martensite phase transformation. They
showed that the energy minimum scales like $\epsilon^{2/3}$, which was in
contrast with the scaling $\varepsilon^{1/2}$, widely accepted in the physics
community. More precisely, the physics arguments were based on an ansatz of
“one-dimensional” structure of minimizers, whereas Kohn and Müller used a
branching construction to achieve lower energy. While they did not show the
form of minimizers, they provided localized (in one direction) estimates on
the energy distribution for the minimizer – thus providing hints on scales
used for branching. Subsequently, Conti [20] used an intricate upper bound
construction to show localized energy bounds (in both directions), which in
particular implies asymptotical self-similarity of the minimizer close to the
interface. The analysis of the toy model was later generalized in several
directions, for example analysis based on energy scalings laws for the cubic-
to-tetragonal phase transformation - e.g. rigidity of the microstructure [17,
18] or study of the energy barrier for the nucleation in the bulk [27] and at
the boundary [3]. In that respect it is worth to also mention recent works of
Rüland and Tribuzio [44, 43], where a novel use of Fourier Analysis allows to
obtain sharp lower bounds on the energy on a more advanced model.
The work of Kohn and Müller initiated many developments in other areas of
material science to study pattern formation driven by the energy minization,
for example in micromagnetics [37, 16, 41], island growth on epitaxially
strained films [4], diblock copolymers [19], optimal design [33, 32], or
superconductors [46]. Picking one of them as an example, the Ginzburg-Landau
model describes behavior of superconductors in different regimes of the
applied magnetics field. While for extreme values of the magnetic field (very
small or very large) there is only one (normal or superconducting) phase, for
intermediate values of the field the mixed states consisting of many vortices
are observed. There the leading order energy characterizes the number of
vortices, and the next order in the energy describes interaction between them
(see [46] for a survey, [42] for analysis in three spatial dimensions, and
[39] for a similar work in the context of $2d$ Coulomb gases).
The models for wrinkling of thin elastic films have similar feature, with the
leading order term in the energy expansion encoding the wrinkled regions while
the next term in the energy expansion being related to the form (e.g.
lengthscale) of wrinkling. The relevant physical object being a two
dimensional (thickened) surface in $\mathbb{R}^{3}$, the local energy expense
of a deformation $u$ is encoded using two principal values of a $3\times 2$
matrix $\nabla u$ – heuristically, singular value greater or smaller than $1$
corresponds to a tension or a compression, respectively. Wrinkling being an
energetically efficient alternative to a compression, we expect it to appear
in the case of (at least) one singular value being less than one.
A compressed elastic sheet can feel the compression in one (“tensile
wrinkling”) or both directions (“compressive wrinkling”). A class of problems
falling into the latter category for which the energy scaling laws were
identified include for instance blistering/delamination problem (with [30, 2,
14, 38] or without [26, 13, 12] substrate effects), crumpling of elastic
sheets [21, 50], or analysis of conical singularities in elastic sheets [36,
35]. A common feature of this problem is degeneracy of the relaxed energy: the
minimum of the relaxed energy $\mathcal{E}_{0}$ equals zero, and more
importantly it is achieved by many different minimizers, making the analysis
of the next order expansion of the energy often difficult.
In contrast, tensile wrinkling problems usually have relaxed problem with
unique minimizer, making the analysis of the next order term (which describes
wrinkling) more accessible. The need for compression usually comes from the
prescribed boundary conditions (as for example in the raft problem [15, 25],
twisted ribbon [31], hanging drapes [49, 6], or compressed cylinder [47]),
through prescribed incompatible strain [10, 34] or curvature effects [7, 11,
48].
The model we consider here is a mixture of the first and the second case,
i.e., it is driven both by the boundary conditions as well as prescribed
nontrivial metric (prestrain). The latter should mimic the need to “waste the
length” in one direction, this need coming from geometric effects in our
original motivation [5]. More precisely, in [5] an elastic annulus is
stretched radially with stronger inner loads, forcing some of the concentric
circles of material to move closer to the center. Pushing some circles into
less space naturally force compression or wrinkling out of plane, while the
circles towards outer boundary stay planar (and are actually stretched in the
azimuthal direction). As we will see, it is crucial that the amount of
arclength grows linearly in the distance from the free boundary (between the
wrinkled and planar region) and not slower (e.g. quadratically) – the latter
case is expected to be quite boring with the minimizer using only _one_
frequency. In contrast, the present problem requires infinitely many
frequencies, in particular near the transition higher and higher frequencies
are needed.
The rest of this section will provide an overview of our results and
organization of the paper. As in [9], we consider a specific thickness
dependent energy $E_{h}$ (see (2.1) for its precise definition), a model
problem describing transition between planar and wrinkled region in thin
elastic sheet, and are interested to understand structure of minimizers of the
energies as $h\searrow 0$. We consider a thin elastic sheet of thickness $h$
and cross section of rectangular shape
$[-1,1]\times[-1,1]\subset\mathbb{R}^{2}$, which represents a piece of the
elastic annulus depicted above in Figure 1 by a green region, and assume the
sheet is i) stretched in the $x$-direction, and ii) stretched/compressed in
the $y$-direction proportional to $x$ (i.e., it is unstrained for $x=0$,
stretched in the $y$-direction in the left half and compressed in the right
half of the domain). The streching/compression in the $y$-direction is modeled
via prescribed metric together with periodic boundary conditions at the top
and bottom boundary.
To relax the compression in the region $\\{x>0\\}$ we expect the sheet to
wrinkle, with the lengthscale of wrinkles of order $h^{1/2}$ [5]. In order to
analyze the limit of $\frac{E_{h}-\mathcal{E}_{0}}{h}$ as $h\searrow 0$, we
rescale the $y$-variable by $h^{-1/2}$ so that the wrinkles lengthscales stay
of order $1$, and the out-of-plane displacement has chance to converge to some
limiting shape. Though the rescaling cause changes of the domain to
$[-1,1]\times[-L,L]$ for $L:=h^{-1/2}\to\infty$, in particular the
$\Gamma$-limit of the functionals is not clear due to changing domain, we pass
to the Fourier space to avoid these complications. More precisely, we rewrite
the energy using Fourier expansion in $y$, with $L$ appearing through the
summation set $\frac{\pi\mathbb{Z}}{L}$. Heuristically, as $L\to 0$ the
Fourier sum will turn into an integral, hence there is a hope for a limiting
functional to make sense.
This strategy was successfully pursued by the first author in [9], by
observing i) the out-of-plane displacement $u$ being the only relevant
quantity to be monitored in this limit, and ii) for fixed (large) $L=h^{-1/2}$
the minimum of the excess energy $\frac{E_{h}-\mathcal{E}_{0}}{h}$ is well
approximated by minimum of a _scalar, convex, and constrained_ variational
problem for $u$ of the form
$\mathcal{S}_{L}(u):=\int_{0}^{1}\fint_{-L}^{L}u_{,x}^{2}+u_{,yy}^{2}\,{\rm
d}x\,{\rm d}y\quad\textrm{ subject to
}\quad\fint_{-L}^{L}u_{,y}^{2}(x,y)\,{\rm d}y=2x+o(1)\quad\textrm{for a.e.
}x\in(0,1)$ (1.3)
Denoting by $a_{k}(x)$ the Fourier coefficients in $y$ of $u(x,\cdot)$, we can
rewrite
$\mathcal{S}_{L}(u)=\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}(\dot{a}_{k}^{2}(x)+a_{k}^{2}(x)k^{4})\,{\rm
d}x,\quad\text{ and }\quad\fint_{-L}^{L}u_{,y}^{2}(x,y)\,{\rm
d}y=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a_{k}^{2}(x)k^{2}\,,$
where “dot” denotes the derivative. The main achievements of [9] was to show
that minima of $\mathcal{S}_{L}$ converge, and then to construct a recovery
sequence for the original energy $E_{h}$, including construction of the in-
plane displacement. Since the elastic energy $E_{h}$ includes all second
derivatives of $u$, and not only $u_{,yy}$ which appears in $\mathcal{S}_{L}$,
regularity statement for the minimizers of $\mathcal{S}_{L}$’s played a
crucial role for the construction of the recovery sequence.
The analysis of minima of $\mathcal{S}_{L}$ from [9] completely avoided the
notion of convergence of minimizers, which needs to be an integral part of a
$\Gamma$-convergence which we study here. To avoid the issue of nonlinear
constraint we use quadratic variables (i.e. monitoring $b_{k}:=a_{k}^{2}$
instead of $a_{k}$), which turns the constraint into a linear one. The second
term in the energy $\mathcal{S}_{L}$ becomes also linear, while the first term
can be rewritten as $\frac{(\dot{b}_{k})^{2}}{4b_{k}}$. One disadvantage of
this approach is the $L^{1}$-framework, which naturally leads the limit
functional to be defined on the space of measures. However, the constraint
provides a good pointwise control in $x$, in particular the limiting measure
can be written as a product of $\,{\rm d}x$ and $x$-dependent measures in $k$.
The lower bound argument (Proposition 4.2) is obtain using Reshetnyak Theorem.
The upper bound (construction of a recovery sequence) is much more tricky
since it needs to be done for any “limiting” measure with finite energy, in
contrast with [9], where it was done just for one (more regular) minimizer.
The proof of the upper bound (Proposition 5.1) consists of several steps:
1. 1.
Given a limiting measure, to obtain $a_{k}$’s we will “discretize” the measure
in the $k$-variable (Lemma 5.3). Moreover, using smoothing of $a_{k}$’s (more
precisely of $a_{k}^{2}$), for which we need to extend the coefficients
$a_{k}$ from $[0,1]$ via dilation into larger interval $[0,1+]$, we get a good
starting point for the construction.
2. 2.
Careful choice of the smoothing scale $\varepsilon(L)$ together with few other
parameters allow for definition of the out-of-plane displacement (see Lemma
5.4), which is then basis for the construction of the in-plane displacement as
well as estimates on the excess energy (Proposition 5.1).
The paper is organized as follows: in the next Chapter we provide a derivation
of the energy, including the functional-analytical framework in form of
measures with well-behaved distributional derivatives in $x$, as well as
rewriting the energy to a form compatible also with this framework. Finally,
we state the main $\Gamma$-convergence result of this paper Theorem 2.5. In
Chapter 3 we show how to disintegrate the limiting measures, while in the
subsequent Chapter we show the compactness for a sequence with excess energy
(Proposition 4.1). Subsequently we also show the lower bound (Proposition
4.2). The upper bound construction is content of Chapter 5. Eventually, in the
last Chapter we state and prove existence of minimizer (as a measure) for the
limiting energy as well as pointwise (in $k$) equipartion of the energy for
this minimizer (see Theorem 6.2). A finer analysis of this minimizer will be
pursued in a future work of the authors.
## 2 Setting of the problem and main results
We start by collecting some notation we will use throughout the paper.
Notation.
1. $(a)$
$\chi_{A}$ denotes the characteristic function of the set $A$;
2. $(b)$
$\mathcal{L}^{1}$ denotes the 1-dimensional Lebesgue measure;
3. $(c)$
$\delta_{k}$ denotes the Dirac measure on $k\in\mathbb{R}$;
4. $(d)$
$\mathcal{M}_{b}(A)$ denotes the space of bounded Radon measures on $A$ with
$A\subset\mathbb{R}^{2}$ Borel measurable;
5. $(e)$
$\mathcal{M}^{+}_{b}(A)$ denotes the subspace of $\mathcal{M}_{b}(A)$ of
positive bounded Radon measures;
6. $(f)$
For a function $f\colon A\subset\mathbb{R}\to\mathbb{R}$ we denote by
$\dot{f}(x)$ and $\ddot{f}(x)$ the first and the second derivative,
respectively;
7. $(g)$
For a function $u\colon A\subset\mathbb{R}^{2}\to\mathbb{R}$ we denote by
$u_{,\small\underbrace{x\dots x}_{i\text{ times}}\small\underbrace{y\dots
y}_{j\text{ times}}}$ its partial derivative
$D^{i+j}u(x,y)=\frac{\,{\rm d}^{j}}{\,{\rm d}y^{j}}\frac{\,{\rm d}^{i}}{\,{\rm
d}x^{i}}u(x,y)\,,\quad i,j\in\mathbb{N},\,1\leq i+j\leq 3\,;$
8. $(h)$
For a measure $\mu\in\mathcal{M}_{b}(A)$ we denote by $\mu_{,x}$ its
distributional derivative with respect to the first variable;
9. $(i)$
For a measure $\mu\in\mathcal{M}_{b}(A)$ we denote by
$|\mu|\in\mathcal{M}^{+}_{b}(A)$ its total variation;
10. $(j)$
For $\tilde{\mu}=(\mu_{1},\mu_{2})\in(\mathcal{M}_{b}(A))^{2}$ we analogously
denote by $|\tilde{\mu}|\in\mathcal{M}^{+}_{b}(A)$ its total variation;
11. $(k)$
For $\mu_{1}\in\mathcal{M}_{b}(A)$, $\mu_{2}\in\mathcal{M}^{+}_{b}(A)$ we
write $\mu_{1}\ll\mu_{2}$ if $\mu_{1}$ is absolute continuous with respect to
$\mu_{2}$ and we indicate by $\frac{\,{\rm d}\mu_{1}}{\,{\rm d}\mu_{2}}\in
L^{1}(A,\mu_{2})$ the associated density (Radon-Nikodým derivative).
The Model. Let us now describe the model (energy) for the transition between
the flat and wrinkled region, which the first author already analyzed in [9].
Instead of considering the annular elastic sheet as in [5], we consider only a
rectangular piece (cut off from the sheet) near the transition region, in
particular simplifying the problem by avoiding the need to work in the radial
geometry. The annular sheet in [5] is stretched in the radial direction and
the concentric circles close to the transition region are stretched/compressed
proportional to the distance from the free boundary. We will model the radial
stretching by dead tension loads in the horizontal direction, while the
stretching/compression in the vertical direction will be modeled by
prescribing a non-euclidean metric of the form $\,{\rm d}x+(1+x)\,{\rm d}y$.
Moreover, the rectangle being part of the annulus, we prescribe periodic
boundary conditions in the vertical direction.
It is physically natural [22] and mathematically convenient to use “small-
slope” geometrically linear Föppl-von Kármán theory. In the membrane part of
the energy the in-plane displacement is represented via the _linear_ strain
while the out-of-plane displacement is kept non-linear (quadratic). The
bending part is modeled by simply $L^{2}$ norm of the Hessian of the out-of-
plane displacement instead of $L^{2}$ norm of the second fundamental form.
Denoting by $w=(w_{1},w_{2})$ and $u$ the in-plane and out-of-plane
displacement respectively, the elastic energy $E_{h}$ (normalized per unit
thickness) has the form
$\begin{split}E_{h}(w,u):=&\frac{1}{2}\int_{-1}^{1}\int_{-1}^{1}|e(w)+\frac{1}{2}\nabla
u\otimes\nabla u-xe_{2}\otimes e_{2}|^{2}\,{\rm d}x\,{\rm d}y\\\
&+\frac{1}{2}\int_{-1}^{1}\int_{-1}^{1}h^{2}|\nabla^{2}u|^{2}\,{\rm d}x\,{\rm
d}y-\int_{-1}^{1}(w_{1}(1,y)-w_{1}(-1,y))\,{\rm d}y.\end{split}$ (2.1)
Here $e(w):=(\nabla w+\nabla^{T}w)/2$ denotes the symmetric gradient of $w$
and $xe_{2}\otimes e_{2}$ is the deviation of the prescribed metric from the
euclidean one. The third integral models the applied tensile dead loads in the
horizontal direction. The factor $1/2$ in front of the elastic energy is
chosen for convenience, and can be changed to any factor using simple
rescaling of $w$ and $u$. Finally, we assume the displacement $(w,u)$ is
$2$-periodic in the second variable.
The behavior of $E_{h}$ as $h\to 0$ at the leading order is well understood
using relaxation techniques [40] (also called tension-field theory in the
mechanics community). Applied to $E_{h}$ from (2.1), in the limit $h\to 0$ the
bending term simply disappears, and the integrand in the membrane term changes
to $\min_{A\geq 0}|e(w)-xe_{2}\otimes e_{2}+A|^{2}$. Hence, one can explicitly
compute the (unique) minimizer of the relaxed energy ($w_{2}=0$ and $w_{1}=x$)
and its minimum $-2+\frac{1}{3}=-\frac{5}{3}=:\mathcal{E}_{0}$.
From [5] we know that the next term in the energy $E_{h}$ scales linearly in
$h$, hence the right quantity to look at is the rescaled _excess energy_
$\frac{E_{h}-\mathcal{E}_{0}}{h}$. For $x>0$ one expects that the sheet
wrinkles out-of-plane in the $y$-direction, in order to offset $-xe_{2}\otimes
e_{2}$ with $u_{,y}^{2}$. The linear scaling in $h$ predicts
$h^{2}|\nabla^{2}u|^{2}\sim h$, in particular $u_{,yy}$ (its largest
component) to be of order $h^{-1/2}$. In particular, the scale of wrinkles in
the bulk should be reciprocal of this value, i.e., $h^{1/2}$. Not
surprisingly, this is also the scale used in the upper bound construction in
[5].
In order to analyze limiting form of the wrinkles as $h\to 0$ we rescale the
$y$-variable by a factor $L:=h^{-1/2}$, so that the characteristic lengthscale
of wrinkles becomes $1$. Precisely, after performing the change of variables
$\hat{w}_{1}(x,y):=w_{1}(x,L^{-1}y),\quad\hat{w}_{2}(x,y):=Lw_{2}(x,L^{-1}y),\quad\hat{u}(x,y):=Lu(x,L^{-1}y),$
the energy $E_{h}$ becomes (see [9, page 630] for a straightforward algebraic
manipulation)
$\displaystyle\mathcal{E}_{L}(w,u):=$
$\displaystyle\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}\biggl{(}w_{1,x}+\frac{u_{,x}^{2}}{2L^{2}}-1\biggr{)}^{2}-1\biggr{)}\,{\rm
d}x\,{\rm
d}y+\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w_{2,y}+\frac{u_{,y}^{2}}{2}-x\biggr{)}^{2}\,{\rm
d}x\,{\rm d}y$
$\displaystyle+\frac{1}{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}{\Big{(}L^{2}w_{1,y}+w_{2,x}+u_{,x}u_{,y}\Big{)}^{2}}\,{\rm
d}x\,{\rm
d}y+\frac{1}{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}\big{(}u^{2}_{,x}+u^{2}_{,yy}\big{)}\,{\rm
d}x\,{\rm d}y$ (2.2)
$\displaystyle+\frac{1}{L^{4}}\fint_{-L}^{L}\int_{-1}^{1}\Big{(}2u^{2}_{,xy}+\frac{1}{L^{2}}u^{2}_{,xx}\Big{)}\,{\rm
d}x\,{\rm d}y\,.$ (2.3)
Thus, the functional is defined as $\mathcal{E}_{L}\colon\mathcal{A}_{L}^{\rm
in}\times\mathcal{A}_{L}^{\rm out}\to[0,+\infty]$, where the function spaces
describing admissible deformations have the form
$\mathcal{A}_{L}^{\rm in}:=\Big{\\{}w=(w_{1},w_{2})\in
W^{1,2}((-1,1)\times\mathbb{R};\mathbb{R}^{2})\colon w(x,\cdot)\text{ is
$2L$-periodic $\forall x\in(-1,1)$}\Big{\\}},$ (2.4) $\mathcal{A}_{L}^{\rm
out}:=\Big{\\{}u\in W^{2,2}((-1,1)\times\mathbb{R})\colon u(x,\cdot)\text{ is
$2L$-periodic $\forall x\in(-1,1)$}\Big{\\}}.$ (2.5)
Furthermore, the $\frac{E_{h}-\mathcal{E}_{0}}{h}$ turns into
$\mathcal{F}_{L}\colon\mathcal{A}_{L}^{\rm in}\times\mathcal{A}_{L}^{\rm
out}\to\mathbb{R}$ defined as
$\mathcal{F}_{L}(w,u):=L^{2}(\mathcal{E}_{L}(w,u)-\mathcal{E}_{0})\,,$ (2.6)
where $\mathcal{E}_{0}=-\frac{5}{3}$ is as above the minimum of the relaxed
energy, so that
$\begin{split}\mathcal{F}_{L}(w,u)=L^{2}&\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w_{1,x}+\frac{u_{,x}^{2}}{2L^{2}}-1\biggr{)}^{2}\,{\rm
d}x\,{\rm
d}y-\frac{L^{2}}{3}+L^{2}\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w_{2,y}+\frac{u_{,y}^{2}}{2}-x\biggr{)}^{2}\,{\rm
d}x\,{\rm d}y\\\
&+\fint_{-L}^{L}\int_{-1}^{1}\Big{(}L^{2}w_{1,y}+w_{2,x}+u_{,x}u_{,y}\Big{)}^{2}\,{\rm
d}x\,{\rm
d}y+\fint_{-L}^{L}\int_{-1}^{1}\big{(}u^{2}_{,x}+u^{2}_{,yy}\big{)}\,{\rm
d}x\,{\rm d}y\\\
&+\frac{1}{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}\Big{(}2u^{2}_{,xy}+\frac{1}{L^{2}}u^{2}_{,xx}\Big{)}\,{\rm
d}x\,{\rm d}y\,.\end{split}$
Before we rigorously proceed further, let us discuss heuristically the form of
functional $\mathcal{F}_{L}$ and its implications. Most of the terms in the
energy are of quadratic nature, and since in addition we are dealing with
oscillatory objects defined on longer and longer intervals, it is natural to
look at the problem in the Fourier space.
Expecting the limit of $\mathcal{F}_{L}$ to exists (in particular having the
minimizing sequence bounded as $L\to\infty$), both integrals on the first line
needs to (quickly) converge to $0$. The first integral can easily achieve that
by simply choosing $w_{1}\sim x+o(L^{-1})$ and $u_{,x}$ not too big, the
smallness of the second integral (after integration in $y$ and using
periodicity of $w$) implies the constrain $\fint_{-L}^{L}u_{,y}^{2}\,{\rm
d}y=2x+o(L^{-1})$.
In order to have continuity of the constraint in the limit $L\to\infty$, and
also for other reasons which will be apparent later, we will work with squares
of the Fourier coefficients and suitably defined measures as primary objects
of studies. In the following we denote by $k\in\mathbb{R}$ the variable
corresponding to the Fourier transform in the $y$-variable. Moreover, we use
the same notation to denote the second variable when working with measures.
###### Definition 2.1 (Measures $\mu^{L}$ and $\mu^{L}_{,x}$).
Let $u\in\mathcal{A}_{L}^{\rm out}$. We denote by
$\mu^{L}(u)\in\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})$ the measure given
by
$\mu^{L}(u):=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(-1,1)}\times\delta_{k}\,,$
with
${a}(x,k):=k^{2}a_{k}^{2}(x)\,,$
and $a_{k}\in W^{2,2}(-1,1)$ being the $k$-th Fourier coefficient of
$u(x,\cdot)$ for all $x\in(-1,1)$, that is
$a_{k}(x):=\begin{cases}\displaystyle\sqrt{2}\fint_{-L}^{L}u(x,y)\sin(ky)\,{\rm
d}y&k\in\dfrac{\pi\mathbb{Z}}{L},k>0\,,\\\\[10.00002pt]
\displaystyle\sqrt{2}\fint_{-L}^{L}u(x,y)\cos(ky)\,{\rm
d}y&k\in\dfrac{\pi\mathbb{Z}}{L},k<0\,,\\\\[10.00002pt]
\displaystyle\fint_{-L}^{L}u(x,y)\,{\rm d}y&k=0\,.\end{cases}$ (2.7)
Moreover we denote by $\mu^{L}_{,x}(u)$ the distributional $x$-derivative of
$\mu^{L}(u)$.
###### Remark 2.2.
1. $(i)$
The distributional $x$-derivative of a measure
$\mu\in\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})$ is defined as follows: for
all $\varphi\in C^{\infty}_{c}((-1,1)\times\mathbb{R})$ we have
$\langle\mu_{,x},\varphi\rangle:=-\int_{(-1,1)\times\mathbb{R}}\varphi_{,x}\,{\rm
d}\mu\,.$
Moreover by a density argument $\mu_{,x}$ can be extended to functions
$\varphi(x,k)=\phi(x)\mathbbm{1}_{A}(k)$ with $\phi\in C^{\infty}_{c}(-1,1)$
and $A\subset\mathbb{R}$ bounded and measurable as
$\langle\mu_{,x},\phi(x)\mathbbm{1}_{A}(k)\rangle:=-\int_{(-1,1)\times
A}\dot{\phi}(x)\,{\rm d}\mu\,;$
2. $(ii)$
Let $\mu\in\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})$ be of the form
$\mu=\sum_{k\in K}a(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times\delta_{k}\,,$
with $K\subset\mathbb{R}$ countable and $a(\cdot,k)\in W^{1,1}(-1,1)$ for all
$k\in K$. Then
$\mu_{,x}=\sum_{k\in K}a_{,x}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times\delta_{k}\,.$
Moreover as $a_{,x}(\cdot,k)=0$ a.e. in $\\{x\in(-1,1)\colon{a}(x,k)=0\\}$ it
follows $\mu_{,x}\in\mathcal{M}((-1,1)\times\mathbb{R})$ and $\mu_{,x}\ll\mu$.
###### Definition 2.3 (Convergence).
For $L>0$ let $(w^{L},u^{L})\in\mathcal{A}_{L}^{\rm
in}\times\mathcal{A}_{L}^{\rm out}$. We say a sequence $(w^{L},u^{L})$
converges as $L\to\infty$ to
$\mu\in\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})$, if
$(\mu^{L}(u^{L}),\mu^{L}_{,x}(u^{L}))$ weakly-* converge to $(\mu,\mu_{,x})$.
We introduce the class of measures
$\begin{split}\mathcal{M}_{\infty}:=\bigg{\\{}\mu\in&\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})\colon\mu((-1,0]\times\mathbb{R})=0,\
\mu_{,x}\in\mathcal{M}_{b}((-1,1)\times\mathbb{R})\,,\\\
&\mu_{,x}\ll\mu\,,\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu(x,k)=\int_{0}^{1}2x\phi(x)\,{\rm d}x\quad\forall\phi\in
C_{c}^{\infty}(0,1)\bigg{\\}}\,,\end{split}$ (2.8)
and the functional
$\mathcal{F}_{\infty}\colon\mathcal{M}_{\infty}\to[0,+\infty]$
$\mathcal{F}_{\infty}(\mu)=\int_{(0,1)\times\mathbb{R}}\biggl{[}k^{2}+\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\Bigr{)}^{2}\biggr{]}\,{\rm d}\mu\,.$ (2.9)
Here $\frac{\,{\rm d}\mu_{,x}}{\,{\rm d}\mu}$ denotes the Radon-Nikodym
derivative, existence of which follows from absolute continuity of $\mu_{,x}$
w.r.t. $\mu$. Moreover, if $\mu_{L}$ from definition 2.1 would be supported in
$(0,1]\times\mathbb{R}$, then $\mathcal{F}_{\infty}(\mu_{L})$ is simply equal
to
$\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\int_{0}^{1}\dot{a}_{k}^{2}(x)+a_{k}^{2}(x)k^{4}\,{\rm
d}x$, i.e. via Plancherel equality (see equation (3.3)) it equals
$\mathcal{S}_{L}$ from (1.3).
###### Remark 2.4.
1. $(i)$
When convenient we will identify the class $\mathcal{M}_{\infty}$ with the
class of measures
$\begin{split}\bigg{\\{}\mu\in\mathcal{M}_{b}^{+}((0,1)\times\mathbb{R})&\colon\mu_{,x}\in\mathcal{M}_{b}((0,1)\times\mathbb{R})\,,\
\mu_{,x}\ll\mu\,,\\\ &\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu(x,k)=\int_{0}^{1}2x\phi(x)\,{\rm d}x\quad\forall\phi\in
C_{c}^{\infty}(0,1)\bigg{\\}}\,;\end{split}$ (2.10)
2. $(ii)$
For later convenience we observe that $\mathcal{F}_{\infty}$ can be rewritten
as follows
$\mathcal{F}_{\infty}(\mu)=\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu+\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{2}\,{\rm d}|\tilde{\mu}|\,,$
(2.11)
where $\tilde{\mu}=(\mu,\mu_{,x})$ and $|\tilde{\mu}|$ denote its total
variation. Indeed, since $\mu_{,x}\ll\mu\ll|\tilde{\mu}|$, we have
$\frac{\,{\rm d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}=\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\frac{\,{\rm d}\mu}{\,{\rm d}|\tilde{\mu}|}\,,$
from which we deduce
$\begin{split}\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\Bigr{)}^{2}\,{\rm
d}\mu&=\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\Bigr{)}^{2}\frac{\,{\rm d}\mu}{\,{\rm
d}|\tilde{\mu}|}\,{\rm d}|\tilde{\mu}|\\\
&=\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{-2}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{2}\frac{\,{\rm d}\mu}{\,{\rm
d}|\tilde{\mu}|}\,{\rm d}|\tilde{\mu}|\\\
&=\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{2}\,{\rm
d}|\tilde{\mu}|\,.\end{split}$
We are now ready to state our main result.
###### Theorem 2.5 ($\Gamma$-convergence).
Let $\mathcal{F}_{L}$ and $\mathcal{F}_{\infty}$ be as in (2.6) and (2.9)
respectively. Then the following holds:
* $a)$
$($Compactness$)$. For $L>0$ let $(w^{L},u^{L})\in\mathcal{A}_{L}^{\rm
in}\times\mathcal{A}_{L}^{\rm out}$ be such that
$\sup_{L}\mathcal{F}_{L}(w^{L},u^{L})<+\infty$. Then there exists a
subsequence (not relabeled) and $\mu\in\mathcal{M}_{\infty}$ such that
$(w^{L},u^{L})$ converges as $L\to+\infty$ in the sense of Definition 2.3 to
$\mu$;
* $b)$
$($$\Gamma$-convergence$)$. As $L\to+\infty$ the functionals $\mathcal{F}_{L}$
$\Gamma$-converge, with respect to the convergence in Definition 2.3, to the
functional $\mathcal{F}_{\infty}$.
## 3 Preliminaries
Let $u\in\mathcal{A}_{L}^{\rm out}$ and let $a_{k}(x)$ be defined as in (2.7).
Then we have
$\displaystyle u(x,y)$
$\displaystyle=a_{0}(x)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k>0}a_{k}(x)\sqrt{2}\sin(ky)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k<0}a_{k}(x)\sqrt{2}\cos(ky)$
(3.1) $\displaystyle=a_{0}(x)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k>0}{\rm
sign}(a_{k}(x))\frac{\sqrt{a(x,k)}}{k}\sqrt{2}\sin(ky)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k<0}{\rm
sign}(a_{k}(x))\frac{\sqrt{a(x,k)}}{-k}\sqrt{2}\cos(ky)\,.$
Then Plancherel equality yields
$\fint_{-L}^{L}u^{2}\,{\rm
d}y=a^{2}_{0}(x)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}a^{2}_{k}(x)=a^{2}_{0}(x)+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{{a}(x,k)}{k^{2}}\,.$ (3.2)
The same holds for partial derivatives of $u$, that is
$\begin{split}\fint_{-L}^{L}(D^{\alpha}u)^{2}\,{\rm
d}y&=(D^{\alpha}a_{0}(x))^{2}+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\Big{(}\frac{\,{\rm d}^{\alpha_{1}}}{\,{\rm
d}x^{\alpha_{1}}}a_{k}(x)k^{\alpha_{2}}\Big{)}^{2}\\\
&=(D^{\alpha}a_{0}(x))^{2}+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\Big{(}\frac{\partial^{\alpha_{1}}}{\partial
x^{\alpha_{1}}}\big{(}\sqrt{a(x,k)}\big{)}k^{\alpha_{2}-1}\Big{)}^{2}\,,\end{split}$
(3.3)
with $\alpha=(\alpha_{1},\alpha_{2})$ multi-index with $|\alpha|\leq 2$. In
case $u$ has higher regularity, i.e., $u\in W^{k,2}((-1,1)\times\mathbb{R})$
with $k>2$, then the same applies for the higher derivatives, i.e., for
$|\alpha|\leq k$. For later convenience we also note that
$\frac{\partial}{\partial
x}\big{(}\sqrt{a(x,k)}\big{)}=\frac{a_{,x}(x,k)}{2\sqrt{a(x,k)}}\,,\quad\frac{\partial^{2}}{\partial
x^{2}}\big{(}\sqrt{a(x,k)}\big{)}=\frac{a_{,xx}(x,k)}{2\sqrt{a(x,k)}}-\frac{(a_{,x}(x,k))^{2}}{4\sqrt{a^{3}(x,k)}}\,.$
(3.4)
We now recall the definition of disintegration of measures only in a specific
case that we will be used throughout the paper, and we refer to [1] for a
complete treatment of the subject.
###### Definition 3.1 (Disintegration of measures in the $x$-variable).
Let $I\subset\mathbb{R}$ be an interval and let
$\mu\in\mathcal{M}_{b}(I\times\mathbb{R})$. We say that the family
$(\nu_{x},g(x))_{x\in I}\subset\mathcal{M}_{b}(\mathbb{R})\times\mathbb{R}$
is a disintegration of $\mu$ $($in the $x$-variable$)$ if $x\mapsto\nu_{x}$ is
Lebesgue measurable, $|\nu_{x}|(\mathbb{R})=1$ for every $x\in I$, $g\in
L^{1}(I)$, and
$\int_{I\times\mathbb{R}}f(x,k)\,{\rm
d}\mu=\int_{I}\int_{\mathbb{R}}f(x,k)\,{\rm d}\nu_{x}(k)g(x)\,{\rm d}x\,,$
(3.5)
for every $f\in L^{1}(I\times\mathbb{R};|\mu|)$.
Formally it simply means $\,{\rm d}\mu(x,k)=\,{\rm d}\nu_{x}(k)g(x)\,{\rm
d}x$.
###### Lemma 3.2.
Let $I\subset\mathbb{R}$ be an interval and let
$\mu\in\mathcal{M}^{+}_{b}(I\times\mathbb{R})$. Then
$\int_{I\times\mathbb{R}}\phi(x)\,{\rm d}\mu=\int_{I}g(x)\phi(x)\,{\rm
d}x\quad\forall\phi\in C_{c}^{\infty}(I)\,,$ (3.6)
for some non-negative $g\in L^{1}(I)$, if and only if there exists
$x\mapsto\nu_{x}\in\mathcal{M}^{+}_{b}(\mathbb{R})$ Lebesgue measurable such
that $(\nu_{x},g(x))_{x\in I}$ is a disintegration of $\mu$.
###### Proof.
Let $(\nu_{x},g(x))_{x\in
I}\subset\mathcal{M}^{+}_{b}(\mathbb{R})\times\mathbb{R}^{+}$ be a
disintegration of $\mu$. Then (3.5) holds with $f(x,k)=\phi(x)\in
C_{c}^{\infty}(I)$ and since $|\nu_{x}|(\mathbb{R})=\nu_{x}(\mathbb{R})=1$ we
readily deduce (3.6).
Assume instead that (3.6) holds true. Let $\pi_{1}\colon I\times\mathbb{R}\to
I$ be the canonical projection and let
$(\pi_{1})_{\sharp}\mu\in\mathcal{M}^{+}_{b}(I)$ be the push-forward of $\mu$
with respect to $\pi_{1}$. By the Disintegration Theorem (cf. [1, Theorem
2.28]) there exists $x\mapsto\nu_{x}\in\mathcal{M}^{+}_{b}(\mathbb{R})$
measurable with $\nu_{x}(\mathbb{R})=1$ such that
$\int_{I\times\mathbb{R}}f(x,k)\,{\rm
d}\mu(x,k)=\int_{I}\int_{\mathbb{R}}f(x,k)\,{\rm d}\nu_{x}(k)\,{\rm
d}(\pi_{1})_{\sharp}\mu(x)\,,$
for all $f\in L^{1}(I\times\mathbb{R};\mu)$. On the other hand (3.6) implies
that $(\pi_{1})_{\sharp}\mu(x)=g(x)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits I$ and therefore
$(\nu_{x},g(x))_{x\in I}$ is a disintegration of $\mu$.
∎
###### Corollary 3.3 (Disintegration of $\mu\in\mathcal{M}_{\infty}$ in the
$x$-variable).
Let $\mu\in\mathcal{M}_{\infty}$. Then there exists
$x\mapsto\nu_{x}\in\mathcal{M}^{+}_{b}(\mathbb{R})$ measurable such that
$(\nu_{x},2x)_{x\in(0,1)}$ is a disintegration of $\mu$.
###### Proof.
The proof follows by Lemma 3.2 and from the fact that
$\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm d}\mu=\int_{0}^{1}2x\phi(x)\,{\rm
d}x\quad\forall\phi\in C_{c}^{\infty}(0,1)\,.$ (3.7)
∎
## 4 Compactness and lower bound
In this section we prove compactness and the $\Gamma-\liminf$ inequality.
###### Proposition 4.1 (Compactness).
Let for $L>0$ be $(w^{L},u^{L})\in\mathcal{A}_{L}^{\rm
in}\times\mathcal{A}_{L}^{\rm out}$ such that
$\sup_{L}\mathcal{F}_{L}(w^{L},u^{L})<+\infty$. Then there exist a, not
relabeled, subsequence and $\mu\in\mathcal{M}_{\infty}$ such that
$(w^{L},u^{L})$ converges to $\mu$, as $L\to+\infty$, in the sense of
Definition 2.3.
###### Proof.
Let $(w^{L},u^{L})$ be as in the statement. Let $\mu^{L}:=\mu^{L}(u^{L})$ and
$\mu^{L}_{,x}:=\mu^{L}_{,x}(u^{L})$ be defined accordingly to Definition 2.1,
i.e., there exist ${a}^{L}(x,k)$ such that $a(\cdot,k)\in W^{1,1}(-1,1)$ and
$\mu^{L}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(-1,1)}\times\delta_{k}\,,\quad\mu^{L}_{,x}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a_{,x}^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(-1,1)}\times\delta_{k}\,.$
Step 1: we show that there exists
$\mu\in\mathcal{M}_{b}^{+}((-1,1)\times\mathbb{R})$ with
$\mu_{,x}\in\mathcal{M}_{b}((-1,1)\times\mathbb{R})$ and such that
$(\mu^{L},\mu^{L}_{,x})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu,\mu_{,x})$.
To this aim we observe that by taking
$0<C_{0}:=\sup_{L}\mathcal{F}_{L}(w^{L},u^{L})<+\infty$ we have
$\mathcal{F}_{L}(w^{L},u^{L})\leq C_{0}\,,$
so that in particular
$\begin{split}C_{0}\geq\mathcal{F}_{L}(w^{L},u^{L})\geq{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}&\bigg{(}w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x\bigg{)}^{2}\,{\rm
d}x\,{\rm d}y-\frac{L^{2}}{3}\\\
&+\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2}\,{\rm
d}x\,{\rm d}y\,.\end{split}$ (4.1)
By Fubini’s theorem, Jensen’s inequality and the fact that $w^{L}(x,\cdot)$ is
$2L$-periodic we get
$\begin{split}{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}&\bigg{(}w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x\bigg{)}^{2}\,{\rm
d}x\,{\rm d}y\geq
L^{2}\int_{-1}^{1}\bigg{(}\fint_{-L}^{L}\Big{(}w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x\Big{)}\,{\rm
d}y\bigg{)}^{2}\,{\rm d}x\\\
&=L^{2}\int_{0}^{1}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y-x\bigg{)}^{2}\,{\rm
d}x+L^{2}\int_{-1}^{0}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y-x\bigg{)}^{2}\,{\rm d}x\\\ &\geq
L^{2}\int_{0}^{1}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y-x\bigg{)}^{2}\,{\rm
d}x+L^{2}\int_{-1}^{0}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y\bigg{)}^{2}\,{\rm d}x+L^{2}\int_{-1}^{0}x^{2}\,{\rm d}x\,,\end{split}$
(4.2)
where the last inequality follows by using that $(a+b)^{2}\geq a^{2}+b^{2}$
provided that $ab>0$ with $a=\frac{1}{2}\fint_{-L}^{L}(u_{,x}^{2})\,{\rm d}y$
and $b=-x$ for $x\in(-1,0)$. Combining (4.1) with (4.2) and using that
$\int_{-1}^{0}x^{2}\,{\rm d}x=\frac{1}{3}$ we find
$\begin{split}C_{0}\geq\mathcal{F}_{L}(w^{L},u^{L})\geq&L^{2}\int_{0}^{1}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y-x\bigg{)}^{2}\,{\rm d}x\\\
&+L^{2}\int_{-1}^{0}\bigg{(}\fint_{-L}^{L}\frac{(u_{,y}^{L})^{2}}{2}\,{\rm
d}y\bigg{)}^{2}\,{\rm
d}x+\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2}\,{\rm
d}x\,{\rm d}y\,.\end{split}$ (4.3)
Thus from (3.3) it follows
$\begin{split}\frac{C}{L^{2}}\geq\int_{0}^{1}\bigg{(}\fint_{-L}^{L}\frac{(u^{L}_{,y})^{2}}{2}\,{\rm
d}y-x\bigg{)}^{2}\,{\rm
d}x&=\int_{0}^{1}\bigg{(}\frac{1}{2}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)-x\bigg{)}^{2}\,{\rm
d}x\\\ &\geq
C\Bigg{(}\int_{0}^{1}\bigg{(}\frac{1}{2}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\bigg{)}^{2}\,{\rm
d}x-\frac{1}{3}\Bigg{)}\,,\end{split}$ (4.4)
and
$\frac{C}{L^{2}}\geq\int_{-1}^{0}\bigg{(}\fint_{-L}^{L}\frac{(u^{L}_{,y})^{2}}{2}\,{\rm
d}y\bigg{)}^{2}\,{\rm
d}x=\int_{-1}^{0}\bigg{(}\frac{1}{2}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\bigg{)}^{2}\,{\rm
d}x\,.$ (4.5)
Hence we obtain
$|\mu^{L}|((-1,1)\times\mathbb{R})=\mu^{L}((-1,1)\times\mathbb{R})=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(k,x)\,{\rm
d}x\leq C\,,$
from which we deduce the existence of a (not relabeled) subsequence and
$\mu\in\mathcal{M}^{+}_{b}((-1,1)\times\mathbb{R})$ such that
$\mu^{L}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$. In addition (4.3)
together with (3.3) and (3.4) yield
$\begin{split}C&\geq\int_{-1}^{1}\fint_{-L}^{L}(u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2}\,{\rm
d}y\,{\rm d}x\\\
&\geq\int_{-1}^{1}\Bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)k^{2}+\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{({a}^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\Bigg{)}\,{\rm
d}x\\\
&\geq\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}|{a}^{L}_{,x}(x,k)|\,{\rm
d}x=|\mu^{L}_{,x}|((-1,1)\times\mathbb{R})\,,\end{split}$ (4.6)
where the last inequality follows by Young’s inequality. Hence, up to
subsequence, we may deduce that there exists
$\tilde{\mu}\in\mathcal{M}_{b}(\times(-1,1)\times\mathbb{R})$ such that
$\mu^{L}_{,x}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\tilde{\mu}$.
Moreover given any $\varphi\in C^{\infty}_{c}((-1,1)\times\mathbb{R})$, it
holds
$\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\tilde{\mu}=\lim_{L\to+\infty}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}_{,x}=-\lim_{L\to+\infty}\int_{(-1,1)\times\mathbb{R}}\varphi_{,x}\,{\rm
d}\mu^{L}=-\int_{(-1,1)\times\mathbb{R}}\varphi_{,x}\,{\rm d}\mu\,,$
which in turn implies $\tilde{\mu}=\mu_{,x}$.
Step 2: we show that $\mu_{,x}\ll\mu$. By Remark 2.2 $(ii)$ we have that
$\mu^{L}_{,x}\ll\mu^{L}$. Now let $N\in\mathbb{N}$ be fixed and let
$\mu^{L}_{N}:=\mu^{L}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times(-N,N)$ and
$\mu_{N}:=\mu\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times(-N,N)$. Then the
following properties hold:
$\mu^{L}_{N,x}:=\mu_{,x}^{L}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times(-N,N)\,,\quad\mu_{N,x}:=\mu_{,x}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times(-N,N)\,,$
$\mu^{L}_{N,x}\ll\mu^{L}_{N}\,,\quad(\mu^{L}_{N},\mu^{L}_{N,x})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu_{N},\mu_{N,x})\,,$
(4.7)
and
$\frac{\,{\rm d}\mu^{L}_{N,x}}{\,{\rm d}\mu^{L}_{N}}(x,k)=\frac{\,{\rm
d}\mu^{L}_{,x}}{\,{\rm d}\mu^{L}}(x,k)\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times(-N,N)\,.$
Moreover recalling the definition of $\mu^{L}$ and (4.6) we have
$\begin{split}\int_{(-1,1)\times(-N,N)}\frac{1}{4N^{2}}\Big{(}\frac{\,{\rm
d}\mu^{L}_{N,x}}{\,{\rm d}\mu^{L}_{N}}(x,k)\Big{)}^{2}\,{\rm
d}\mu^{L}_{N}&\leq\int_{(-1,1)\times(-N,N)}\frac{1}{4k^{2}}\Big{(}\frac{\,{\rm
d}\mu^{L}_{,x}}{\,{\rm d}\mu^{L}}(x,k)\Big{)}^{2}\,{\rm d}\mu^{L}\\\
&\leq\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{({a}^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\,{\rm d}x\leq
C\,.\end{split}$ (4.8)
From (4.7), (4.8) and [1, Example 2.36 pg. 67, and discussion at pg. 66] we
deduce that $\mu_{N,x}\ll\mu_{N}$ for every $N\in\mathbb{N}$ and hence
$\mu_{,x}\ll\mu$.
Step 3: we show that $\mu\in\mathcal{M}^{+}_{b}((0,1)\times\mathbb{R})$, that
is, $\mu((-1,0]\times\mathbb{R})=0$, and that
$\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm d}\mu=\int_{0}^{1}2x\phi(x)\,{\rm
d}x,$ (4.9)
for all $\phi\in C_{c}^{\infty}((0,1))$. To this purpose for fixed
$\delta\in(0,1)$ by (4.1) we have
$\begin{split}\mu^{L}((-1,\delta)\times\mathbb{R})&=\int_{-1}^{\delta}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\,{\rm
d}x\\\ &\leq
C\int_{-1}^{0}\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\bigg{)}^{2}\,{\rm
d}x+C\int_{0}^{\delta}\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)-x\bigg{)}^{2}\,{\rm
d}x+C\int_{0}^{\delta}x^{2}\,{\rm d}x\\\
&\leq\frac{C}{L^{2}}+C\delta^{3}.\end{split}$
This together with the lower semicontinuity with respect to the weak*
convergence give
$\mu((-1,0]\times\mathbb{R})\leq\mu((-1,\delta)\times\mathbb{R})\leq\liminf_{L\to\infty}\mu^{L}((-1,\delta)\times\mathbb{R})\leq
C\delta^{3}.$
By sending $\delta\to 0$ we deduce $\mu((-1,0]\times\mathbb{R})=0.$ It remains
to show (4.9). Given $\phi\in C_{c}^{\infty}(0,1)$ it holds
$\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu^{L}=\int_{0}^{1}\phi(x)\Big{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)-2x\Big{)}\,{\rm
d}x+\int_{0}^{1}2x\phi(x)\,{\rm d}x.$
From (4.4) it follows that
$\int_{0}^{1}\Big{|}\phi(x)\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)-2x\Big{|}\,{\rm
d}x\leq
C\|\phi\|_{\infty}\bigg{(}\int_{0}^{1}\Big{(}\frac{1}{2}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)-x\Big{)}^{2}\,{\rm
d}x\bigg{)}^{1/2}\leq\frac{C}{L}\to 0,$
as $L\to+\infty$, so that
$\lim_{L\to+\infty}\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu^{L}=\int_{0}^{1}2x\phi(x)\,{\rm d}x.$ (4.10)
Next we fix $R\geq 1$ and take $\psi_{R}\in C^{\infty}_{c}(\mathbb{R})$ such
that $0\leq\psi_{R}\leq 1$, $\psi_{R}(k)\equiv 1$ if $|k|<R$ and
$\psi_{R}(k)\equiv 0$ if $|k|>R+1$. We have
$\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu^{L}=\int_{(0,1)\times\mathbb{R}}\phi(x)\psi_{R}(k)\,{\rm
d}\mu^{L}+\int_{(0,1)\times\mathbb{R}}\phi(x)(1-\psi_{R}(k))\,{\rm
d}\mu^{L}\,.$ (4.11)
The weak* convergence yields
$\lim_{L\to+\infty}\int_{(0,1)\times\mathbb{R}}\phi(x)\psi_{R}(k)\,{\rm
d}\mu^{L}=\int_{(0,1)\times\mathbb{R}}\phi(x)\psi_{R}(k)\,{\rm d}\mu\,,$
whereas for the second term on the right hand-side of (4.11) we get
$\begin{split}\int_{(0,1)\times\mathbb{R}}|\phi(x)(1-\psi_{R}(k))|\,{\rm
d}\mu^{L}&\leq\int_{0}^{1}|\phi(x)|\Big{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L},|k|>R}a^{L}(x,k)\Big{)}\,{\rm
d}x\\\
&\leq\frac{\|\phi\|_{\infty}}{R^{2}}\int_{0}^{1}\Big{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L},|k|>R}a^{L}(x,k)k^{2}\Big{)}\,{\rm
d}x\\\
&\leq\frac{\|\phi\|_{\infty}}{R^{2}}\int_{0}^{1}\fint_{-L}^{L}(u_{,yy})^{2}\,{\rm
d}y\,{\rm d}x\leq\frac{C}{R^{2}}\,,\end{split}$
where the last to inequalities follow from (3.3) and (4.1). Thus passing to
the limit as $L\to+\infty$ in (4.11) we obtain
$\int_{(0,1)\times\mathbb{R}}\phi(x)\psi_{R}(k)\,{\rm
d}\mu-\frac{C}{R^{2}}\leq\lim_{L\to+\infty}\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu^{L}\leq\int_{(0,1)\times\mathbb{R}}\phi(x)\psi_{R}(k)\,{\rm
d}\mu+\frac{C}{R^{2}}\,.$
Eventually by letting $R\to+\infty$ we deduce
$\lim_{L\to+\infty}\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu^{L}=\int_{(0,1)\times\mathbb{R}}\phi(x)\,{\rm d}\mu\,,$
which together with (4.10) yield (4.9). ∎
###### Proposition 4.2 (Lower bound).
Let $\mathcal{F}_{L}$ and $\mathcal{F}_{\infty}$ be as in (2.6) and (2.9)
respectively. Let for $L>0$ be $(w^{L},u^{L})\subset\mathcal{A}_{L}^{\rm
in}\times\mathcal{A}_{L}^{\rm out}$ a sequence converging to
$\mu\in\mathcal{M}_{\infty}$ in the sense of Definition 2.3. Then there holds
$\liminf_{L\to\infty}\mathcal{F}_{L}(w^{L},u^{L})\geq\mathcal{F}_{\infty}(\mu).$
(4.12)
###### Proof.
Let $(w^{L},u^{L})$ be as in the statement and let $\mu^{L}:=\mu^{L}(u^{L})$
and $\mu^{L}_{,x}:=\mu^{L}_{,x}(u^{L})$ be defined accordingly to Definition
2.1, that is,
$\mu^{L}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(-1,1)}\times\delta_{k}\,,\quad\mu^{L}_{,x}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a_{,x}^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(-1,1)}\times\delta_{k}\,.$
Recalling (4.3), (3.3) and (3.4) we have that
$\begin{split}\mathcal{F}_{L}(w^{L},u^{L})&\geq\int_{-1}^{1}\fint_{-L}^{L}(u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2}\,{\rm
d}y\,{\rm d}x\\\
&\geq\int_{0}^{1}\Big{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{({a}^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}+\sum_{k\in\frac{\pi\mathbb{Z}}{L}}a^{L}(x,k)k^{2}\Big{)}\,{\rm
d}x\\\
&=\int_{(0,1)\times\mathbb{R}}\bigg{(}k^{2}+\frac{1}{4k^{2}}\Big{(}\frac{\,{\rm
d}\mu_{,x}^{L}}{\,{\rm d}\mu^{L}}(x,k)\Big{)}^{2}\bigg{)}\,{\rm d}\mu^{L}\\\
&=\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu^{L}+\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu^{L}}{\,{\rm d}|\tilde{\mu}^{L}|}(x,k)\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu^{L}_{,x}}{\,{\rm d}|\tilde{\mu}^{L}|}(x,k)\Bigr{)}^{2}\,{\rm
d}|\tilde{\mu}^{L}|\,,\end{split}$ (4.13)
where $\tilde{\mu}^{L}:=(\mu^{L},\mu^{L}_{,x})$, and the last equality follows
from Remark 2.4 $(ii)$. By Reshetnyak Theorem (cf. [1, Theorem 2.38]) there
hold
$\liminf_{L\to+\infty}\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu^{L}\geq\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm d}\mu\,,$ (4.14)
and
$\begin{split}\liminf_{L\to+\infty}&\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu^{L}}{\,{\rm d}|\tilde{\mu}^{L}|}(x,k)\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu^{L}_{,x}}{\,{\rm d}|\tilde{\mu}^{L}|}(x,k)\Bigr{)}^{2}\,{\rm
d}|\tilde{\mu}^{L}|\\\
&\geq\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}(x,k)\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}(x,k)\Bigr{)}^{2}\,{\rm
d}|\tilde{\mu}|\,,\end{split}$ (4.15)
with $\tilde{\mu}:=(\mu,\mu_{,x})$. Gathering together (4.13), (4.14) and
(4.15) we find
$\begin{split}\liminf_{L\to\infty}\mathcal{F}_{L}(w^{L},u^{L})\geq&\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu+\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}(x,k)\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}(x,k)\Bigr{)}^{2}\,{\rm
d}|\tilde{\mu}|=\mathcal{F}_{\infty}(\mu)\,.\end{split}$
∎
## 5 Upper bound
In this section we prove the $\Gamma-\limsup$ inequality.
###### Proposition 5.1 (Upper bound).
Let $\mu\in\mathcal{M}_{\infty}$. Then for $L>0$ there exists a sequence
$(w^{L},u^{L})\in\mathcal{A}_{L}^{\rm in}\times\mathcal{A}_{L}^{\rm out}$ that
converges to $\mu\in\mathcal{M}_{\infty}$ in the sense of Definition 2.3 and
such that
$\limsup_{L\to\infty}\mathcal{F}_{L}(w^{L},u^{L})\leq\mathcal{F}_{\infty}(\mu)\,,$
with $\mathcal{F}_{L}$ and $\mathcal{F}_{\infty}$ defined as in (2.6) and
(2.9) respectively.
We divide the proof of Proposition 5.1 into a number of steps. For any
$\lambda\geq 1$ we define the following class of measures
$\begin{split}\mathcal{M}_{\infty}^{\lambda}:=\bigg{\\{}\mu\in&\mathcal{M}_{b}^{+}((0,\lambda)\times\mathbb{R})\colon\mu_{,x}\in\mathcal{M}_{b}((0,\lambda)\times\mathbb{R})\,,\quad\mu_{,x}\ll\mu\,,\\\
&\int_{0}^{\lambda}2x\phi(x)dx=\int_{(0,\lambda)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu(k,x)\quad\forall\phi\in
C_{c}^{\infty}(0,\lambda)\bigg{\\}}\,,\end{split}$ (5.1)
and the functional
$\mathcal{F}_{\infty}^{\lambda}\colon\mathcal{M}_{\infty}^{\lambda}\to[0,+\infty]$
$\mathcal{F}_{\infty}^{\lambda}(\mu)=\int_{(0,\lambda)\times\mathbb{R}}\biggl{[}k^{2}+\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\Bigr{)}^{2}\biggr{]}\,{\rm d}\mu\,.$ (5.2)
Then we have $\mathcal{M}_{\infty}=\mathcal{M}_{\infty}^{1}$ and
$\mathcal{F}_{\infty}=\mathcal{F}_{\infty}^{1}$.
###### Lemma 5.2 (Dilation of $\mu$).
Let $\mu\in\mathcal{M}_{\infty}$. Then for each $\lambda\in(1,2)$ there exists
$\mu_{\lambda}\in\mathcal{M}_{\infty}^{\lambda}$ such that
$(\mu_{\lambda},\mu_{\lambda,x})\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits((0,1)\times\mathbb{R})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu,\mu_{,x})\quad\text{as
}\lambda\searrow 1\,,$ (5.3)
and
$\int_{(0,\lambda)\times\mathbb{R}}k^{2}\,{\rm
d}\mu_{\lambda}=\lambda^{2}\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu\,,\quad\int_{(0,\lambda)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}\Bigr{)}^{2}\,{\rm
d}\mu_{\lambda}=\int_{(0,1)\times\mathbb{R}}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}\Bigr{)}^{2}\,{\rm d}\mu$ (5.4)
so that, in particular
$\lim_{\lambda\searrow
1}\mathcal{F}_{\infty}^{\lambda}\big{(}\mu_{\lambda}\big{)}=\mathcal{F}_{\infty}(\mu)\,.$
(5.5)
Moreover $(\nu_{\frac{x}{\lambda}},2x)_{x\in(0,\lambda)}$ is a disintegration
of $\mu_{\lambda}$ where $\nu_{x}\in\mathcal{M}_{b}^{+}(\mathbb{R})$ for
$x\in(0,1)$ is the measure given by Corollary 3.3.
###### Proof.
Let $\mu_{\lambda}\in\mathcal{M}_{b}^{+}((0,\lambda)\times\mathbb{R})$ be
defined via duality as follows
$\int_{(0,\lambda)\times\mathbb{R}}\psi(x,k)\,{\rm
d}\mu_{\lambda}=\lambda^{2}\int_{(0,1)\times\mathbb{R}}\psi(\lambda x,k)\,{\rm
d}\mu\,,$ (5.6)
for every $\psi\in C((0,\lambda)\times\mathbb{R})$. Notice that
$\mu_{\lambda,x}\in\mathcal{M}_{b}((0,\lambda)\times\mathbb{R})$ and is given
by
$\int_{(0,\lambda)\times\mathbb{R}}\psi(x,k)\,{\rm
d}\mu_{\lambda,x}=\lambda\int_{(0,1)\times\mathbb{R}}\psi(\lambda x,k)\,{\rm
d}\mu_{,x}\,.$ (5.7)
Hence $\mu_{\lambda,x}\ll\mu_{\lambda}$ and
$\frac{\,{\rm d}\mu_{\lambda,x}}{\,{\rm
d}\mu_{\lambda}}(x,k)=\frac{1}{\lambda}\frac{\,{\rm d}\mu_{,x}}{\,{\rm
d}\mu}\Big{(}\frac{x}{\lambda},k\Big{)}\,.$
Moreover (5.6) together with the change of variable $x=\lambda\hat{x}$ imply
$\int_{0}^{\lambda}2x\phi(x)\,{\rm
d}x=\int_{(0,\lambda)\times\mathbb{R}}\phi(x)\,{\rm
d}\mu_{\lambda}\quad\forall\phi\in C_{c}^{\infty}(0,\lambda)\,.$
It follows that $\mu_{\lambda}\in\mathcal{M}_{\infty}^{\lambda}$. Moreover
from (5.6) and (5.7) we deduce that
$(\mu_{\lambda},\mu_{\lambda,x})\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(\mathbb{R}\times(0,1))\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu,\mu_{,x})\quad\text{as
}\lambda\to 1\,,$
and (5.4) which implies (5.5). The fact that
$(\nu_{\frac{x}{\lambda}},2x)_{x\in(0,\lambda)}$ is a disintegration of
$\mu_{\lambda}$ follows again by a change of variable. ∎
###### Lemma 5.3 (Discretisation of $\mu$).
Let $\mu\in\mathcal{M}_{\infty}$ with $\mathcal{F}_{\infty}(\mu)<+\infty$ and
let $\lambda=\lambda(L)\searrow 1$ as $L\to\infty$. Then there exists
$(\mu^{L})\subset\mathcal{M}_{\infty}^{\lambda}$ with the following
properties:
1. $(i)$
$\mu^{L}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\overline{b}^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(0,\lambda)}\times\delta_{k}$
with
$\overline{b}^{L}(\cdot,k)\in
W^{1,1}(0,\lambda)\quad\text{and}\quad\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\overline{b}^{L}(x,k)=2x\,,\quad\forall
x\in(0,\lambda)\,,$ (5.8)
$\mathcal{F}_{\infty}^{\lambda}(\mu^{L})=\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}k^{2}\overline{b}^{L}(x,k)\,{\rm
d}x+\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{(\overline{b}^{L}_{,x}(x,k))^{2}}{\overline{b}^{L}(x,k)}\,{\rm
d}x\,;$ (5.9)
2. $(ii)$
$(\mu^{L},\mu^{L}_{,x})\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits((0,1)\times\mathbb{R})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu,\mu_{,x})$;
3. $(iii)$
$\limsup_{L\to\infty}\mathcal{F}_{\infty}^{\lambda}(\mu^{L})\leq\mathcal{F}_{\infty}\big{(}\mu\big{)}$.
###### Proof.
For each $L>1$ let $\mu_{\lambda}\in\mathcal{M}_{\infty}^{\lambda}$ be the
measure given by Lemma 5.2 and let
$(\nu_{\frac{x}{\lambda}},2x)_{x\in(0,\lambda)}$ be the corresponding
disintegration. We then define
$\mu^{L}\in\mathcal{M}_{b}^{+}((0,\lambda)\times\mathbb{R})$ as
$\mu^{L}:=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\overline{b}^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(0,\lambda)}\times\delta_{k}\,,$
(5.10)
where for $(x,k)\in(0,\lambda)\times\frac{\pi\mathbb{Z}}{L}$ we set
$\overline{b}^{L}(x,k):=\begin{cases}0&\text{if }k=0\,,\\\\[10.00002pt]
2x\nu_{\frac{x}{\lambda}}(I_{k}^{L})&\text{if }k\neq
0\,,\end{cases}\quad\text{and}\quad
I^{L}_{k}:=\begin{cases}(k-\frac{\pi}{L},k]&\text{if }k>0\,,\\\\[10.00002pt]
[k,k+\frac{\pi}{L})&\text{if }k<0\,.\end{cases}$ (5.11)
Now, for each $k\in\frac{\pi\mathbb{Z}}{L}$, $\overline{b}^{L}(\cdot,k)\in
W^{1,1}(0,\lambda)$ with
$\overline{b}_{,x}^{L}(x,k)=\begin{cases}0&\text{if }k=0\,,\\\\[10.00002pt]
\displaystyle 2x\int_{I_{k}^{L}}\frac{\,{\rm d}\mu_{\lambda,x}}{\,{\rm
d}\mu_{\lambda}}(x,\hat{k})\,{\rm d}\nu_{\frac{x}{\lambda}}(\hat{k})&\text{if
}k\neq 0\,.\end{cases}$
Indeed if $k=0$ there is nothing to prove. If instead $k\neq 0$, for $\phi\in
C^{\infty}_{c}(0,\lambda)$ from the definition of $\nu_{\frac{x}{\lambda}}$
and recalling Remark 2.2 we get
$\begin{split}\int_{0}^{\lambda}&\overline{b}^{L}(x,k)\dot{\phi}(x)\,{\rm
d}x=\int_{0}^{\lambda}\int_{\mathbb{R}}\mathbbm{1}_{I_{k}^{L}}(\hat{k})\dot{\phi}(x)\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\,2x\,{\rm d}x\\\
&=\int_{(0,\lambda)\times\mathbb{R}}\mathbbm{1}_{I_{k}^{L}}(\hat{k})\dot{\phi}(x)\,{\rm
d}\mu_{\lambda}=-\int_{(0,\lambda)\times\mathbb{R}}\mathbbm{1}_{I_{k}^{L}}(\hat{k})\phi(x)\,{\rm
d}\mu_{\lambda,x}\\\
&=-\int_{(0,\lambda)\times\mathbb{R}}\mathbbm{1}_{I_{k}^{L}}(\hat{k})\phi(x)\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\,{\rm
d}\mu_{\lambda}=-\int_{0}^{\lambda}\int_{I_{k}^{L}}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\,{\rm
d}\nu_{\frac{x}{\tilde{\lambda}}}(\hat{k})\,2x\phi(x)\,{\rm d}x\,;\end{split}$
furthermore by Young’s inequality and (5.5)
$\begin{split}\int_{0}^{\lambda}|\overline{b}_{,x}^{L}(x,k)|\,{\rm
d}x\leq\int_{I_{k}^{L}\times(0,\lambda)}\left|\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}\right|\,{\rm
d}\mu_{\lambda}\leq\frac{1}{2}\int_{I_{k}^{L}\times(0,\lambda)}\left(k^{2}+\frac{1}{k^{2}}\left(\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}\right)^{2}\right)\,{\rm
d}\mu_{\lambda}\leq C\,.\end{split}$
Thus in particular
$\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathcal{Z}}{L}}|\overline{b}_{,x}^{L}(x,k)|\,{\rm
d}x\leq\mathcal{F}_{\infty}^{\lambda}(\mu_{\lambda})\leq C\,.$
As a consequence we have that
$\mu_{,x}^{L}\in\mathcal{M}_{b}((0,\lambda)\times\mathbb{R})$, and
$\mu_{,x}^{L}=\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\overline{b}_{,x}^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(0,\lambda)}\times\delta_{k}\,,$
and by Remark 2.2 $(ii)$ $\tilde{\mu}_{,x}^{L}\ll\tilde{\mu}^{L}$. Moreover,
as $\nu_{\frac{x}{\lambda}}$ is a probability measure, there holds
$\sum_{k\in\frac{\pi\mathbb{Z}}{L}}\overline{b}^{L}(x,k)=2x\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\nu_{\frac{x}{\lambda}}(I_{k}^{L})\bigg{)}=2x\,,\quad\forall
x\in(0,\lambda)\,.$
Note in particular that $\mu^{L}\in\mathcal{M}_{\infty}^{\lambda}$. Moreover
(5.9) readily follows and $(i)$ is proved. We next show $(ii)$. Take
$\varphi\in C^{\infty}_{c}((0,1)\times\mathbb{R})$, so that from (5.11) we
obtain
$\begin{split}\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}&=\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\varphi(x,k)\nu_{\frac{x}{\lambda}}(I_{k}^{L})\,2x\,{\rm d}x\\\
&=\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\int_{I_{k}^{L}}\big{(}\varphi(x,k)-\varphi(x,\hat{k})\big{)}\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\,2x\,{\rm
d}x+\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm d}\mu_{\lambda}\,.\end{split}$
(5.12)
Since $\varphi$ is uniformly continuous for every $\varepsilon>0$ there is
$L_{0}>1$ such that for all $L\geq L_{0}$
$|\varphi(x,k)-\varphi(x,\hat{k})|<\varepsilon\quad\forall x\in(0,\lambda)\,,\
\forall k\in\frac{\pi\mathbb{Z}}{L},\ \forall\hat{k}\in I_{k}^{L}\,,$
from which we readily deduce that
$\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\int_{I_{k}^{L}}\big{|}\varphi(x,k)-\varphi(x,\hat{k})\big{|}\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\,2x\,{\rm
d}x\leq\mu_{\lambda}((0,\lambda)\times\mathbb{R})\varepsilon=\lambda^{2}\mu((0,1)\times\mathbb{R})\varepsilon\,.$
(5.13)
From (5.12), (5.13),(5.3) and the arbitrariness of $\varepsilon$ we infer
$\mu^{L}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits((0,1)\times\mathbb{R})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$
as $L\to+\infty$. By analogous arguments we get
$\mu_{,x}^{L}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits((0,1)\times\mathbb{R})\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu_{,x}$
as $L\to+\infty$. It remains to prove $(iii)$. We start by observing that for
all $\delta>0$ and all $\hat{k}\in I_{k}^{L}$ we have
$k^{2}\leq\left(\hat{k}+\frac{\pi}{L}\right)^{2}\leq(1+\delta)\hat{k}^{2}+(1+\delta^{-1})\frac{\pi^{2}}{L^{2}}\,,$
so that
$\begin{split}\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L}}k^{2}\overline{b}^{L}(x,k)\,{\rm
d}x&=\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\int_{I_{k}^{L}}k^{2}\,{\rm d}\nu_{\frac{x}{\lambda}}(\hat{k})\,2x\,{\rm
d}x\\\ &=\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq 0}\int_{(0,\lambda)\times
I_{k}^{L}}k^{2}\,{\rm d}\mu_{\lambda}(x,\hat{k})\\\
&\leq\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq 0}\int_{(0,\lambda)\times
I_{k}^{L}}\bigg{(}(1+\delta)\hat{k}^{2}+(1+\delta^{-1})\frac{\pi^{2}}{L^{2}}\bigg{)}\,{\rm
d}\mu_{\lambda}(x,\hat{k})\\\
&=(1+\delta)\int_{(0,\lambda)\times\mathbb{R}}\hat{k}^{2}\,{\rm
d}\mu_{\lambda}+\mu_{\lambda}((0,\lambda)\times\mathbb{R})(1+\delta^{-1})\frac{\pi^{2}}{L^{2}}\,.\end{split}$
(5.14)
Moreover there holds
$\begin{split}\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{(\overline{b}^{L}_{,x}(x,k))^{2}}{\overline{b}^{L}(x,k)}\,{\rm
d}x=\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\bigg{(}\frac{\overline{b}_{,x}^{L}(x,k)}{\overline{b}^{L}(x,k)}\bigg{)}^{2}\overline{b}^{L}(x,k)\,{\rm
d}x\,.\end{split}$ (5.15)
Since $1/|k|\leq 1/|\hat{k}|$ for $\hat{k}\in I_{k}^{L}$ the following
inequality follows
$\frac{1}{2|k|}\frac{\overline{b}_{,x}^{L}(x,k)}{\overline{b}^{L}(x,k)}=\frac{1}{2|k|}\fint_{I_{k}^{L}}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\leq\fint_{I_{k}^{L}}\frac{1}{2|\hat{k}|}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\,.$ (5.16)
Now combining (5.15) with (5.16) we obtain
$\begin{split}\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4k^{2}}\frac{(\overline{b}^{L}_{,x}(x,k))^{2}}{\overline{b}^{L}(x,k)}\,{\rm
d}x&=\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\bigg{(}\fint_{I_{k}^{L}}\frac{1}{2|\hat{k}|}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\bigg{)}^{2}\nu_{\frac{x}{\lambda}}(I_{k}^{L})\,2x\,{\rm
d}x\\\ &\leq\int_{0}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\int_{I_{k}^{L}}\frac{1}{4\hat{k}^{2}}\bigg{(}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\bigg{)}^{2}\,{\rm
d}\nu_{\frac{x}{\lambda}}(\hat{k})\,2x\,{\rm d}x\\\
&=\int_{(0,\lambda)\times\mathbb{R}}\frac{1}{4\hat{k}^{2}}\bigg{(}\frac{\,{\rm
d}\mu_{\lambda,x}}{\,{\rm d}\mu_{\lambda}}(x,\hat{k})\bigg{)}^{2}\,{\rm
d}\mu_{\lambda}\\\
&=\int_{(0,1)\times\mathbb{R}}\frac{1}{4\hat{k}^{2}}\bigg{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}\mu}(x,\hat{k})\bigg{)}^{2}\,{\rm d}\mu\,,\end{split}$
(5.17)
where the inequality follows by Jensen’s inequality. Finally gathering
together (5.14) and (5.17) and recalling (5.5) we infer
$\limsup_{L\to\infty}\mathcal{F}_{\infty}^{\lambda}(\mu^{L})\leq(1+\delta)\limsup_{L\to\infty}\mathcal{F}_{\infty}^{\lambda}\big{(}\mu_{\lambda}\big{)}\leq(1+\delta)\mathcal{F}_{\infty}(\mu)\,.$
By arbitrariness of $\delta$, $(iii)$ follows by letting $\delta\to 0$.
∎
###### Lemma 5.4 (Construction of $u$).
Let $\mu\in\mathcal{M}_{\infty}$ be such that
$\mathcal{F}_{\infty}(\mu)<+\infty$. Let $\varepsilon=\varepsilon(L)>0$ and
$n=n(L)\in\mathbb{N}$ be such that
$\lim_{L\to+\infty}\varepsilon(L)=0\,,\quad\lim_{L\to+\infty}n(L)=\lim_{L\to+\infty}\frac{L}{n(L)}=+\infty\,.$
Then there exists $\hat{u}^{L}\in\mathcal{A}_{L}^{\rm
out}\cap\mathcal{A}_{L_{0}}^{\rm out}$ with $L_{0}:=L/n(L)$ that satisfies the
following properties: let
$A^{L}(x):=\frac{1}{2}\fint_{-L}^{L}(\hat{u}^{L}_{,y}(x,\cdot))^{2}\,{\rm
d}y\quad\text{ and }\quad f^{L}(x):=\sqrt{\frac{x}{A^{L}(x)}}\,.$
Then for all $x\in(0,1)$ and $N\in\mathbb{N}$ there holds
$\max\\{x,\varepsilon\\}\lesssim A^{L}(x)\lesssim\max\\{x,\varepsilon\\}\,;$
(5.18)
$(f^{L}(x))^{2}\lesssim\frac{x}{\max\\{x,\varepsilon\\}}\,,\quad(f^{L}(x))^{2}\leq
1+o_{L}(1)\,;$ (5.19) $(f^{L}(x))^{2}\geq 1+o_{N}(1)\ \text{ if
}x\in(N\varepsilon,1)\,;$ (5.20)
$(\dot{f}^{L}(x))^{2}\lesssim\frac{\max\\{e^{-\frac{x}{\varepsilon}},e^{-\frac{1}{\sqrt{\varepsilon}}}\\}}{x\varepsilon}\,,\quad(\ddot{f}^{L}(x))^{2}\lesssim\frac{1}{x^{3}\varepsilon}\,;$
(5.21)
there exists a continuous increasing function
$\omega\colon[0,+\infty)\to[0,+\infty)$ with $\omega(0)=0$ such that
$\fint_{-L}^{L}(\hat{u}^{L}(x,\cdot))^{2}\,{\rm
d}y\lesssim{\begin{cases}\max\\{x,\varepsilon\\}(\omega(2N\varepsilon)+Ne^{-N})&\text{
if }x\in(0,N\varepsilon)\\\ x&\text{ if
}x\in(N\varepsilon,1)\end{cases}\quad}\,;$ (5.22)
$\fint_{-L}^{L}(\hat{u}^{L}(x,\cdot))^{4}\,{\rm d}y\lesssim
L_{0}^{2}(\max\\{\varepsilon,x\\})^{2}\,;$ (5.23)
$\fint_{-L}^{L}\left((\hat{u}_{,yy}^{L}(x,\cdot))^{2}+(\hat{u}_{,x}^{L}(x,\cdot))^{2}\right)\,{\rm
d}y\lesssim\frac{1}{\varepsilon}\,.$ (5.24)
Moreover
$(\mu^{L}(\hat{u}^{L}),\mu^{L}_{,x}(\hat{u}^{L}))\stackrel{{\scriptstyle*}}{{\rightharpoonup}}(\mu,\mu_{,x})\quad\text{in
}\mathcal{M}_{b}((-1,1)\times\mathbb{R})^{2}\,;$ (5.25)
$\limsup_{L\to\infty}\fint_{-L}^{L}\int_{-1}^{1}\big{(}(\hat{u}^{L}_{,x})^{2}+(\hat{u}^{L}_{,yy})^{2}\big{)}\,{\rm
d}x\,{\rm d}y\leq\mathcal{F}_{\infty}(\mu)\,;$ (5.26)
$\fint_{-L}^{L}\int_{-1}^{1}\left((\hat{u}^{L}_{,xy})^{2}+(\hat{u}^{L}_{,xx})^{2}+(\hat{u}_{,yyx}^{L})^{2}\right)\,{\rm
d}x\,{\rm d}y\lesssim\frac{1}{\varepsilon^{2}}\,;$ (5.27)
$\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,x})^{4}\,{\rm d}x\,{\rm
d}y\lesssim\frac{L_{0}^{2}}{\varepsilon^{2}}\,.$ (5.28)
Finally since $\hat{u}^{L}$ and all its derivatives are $2L_{0}$-periodic in
the $y$-variable the above estimates still hold true if we replace the average
integral on $[-L,L]$ with the average integral on $[-L_{0},L_{0}]$.
###### Proof.
Let $\mu\in\mathcal{M}_{\infty}$, $\varepsilon=\varepsilon(L)$, $n=n(L)$ and
$L_{0}$ be as in the statement. We set
$\lambda=\lambda(L):=(1+\sqrt{\varepsilon})\searrow 1\quad\text{as
}L\to+\infty\,,$
hence in particular $\lambda\searrow 1$ as $L_{0}\to+\infty$. We construct
$\hat{u}^{L}\in\mathcal{A}_{L_{0}}^{\rm out}$ and then we extend it
periodically in $[-1,1]\times[-L,L]$, without relabelling it. In this way,
since $L=n(L)L_{0}$ with $n(L)\in\mathbb{N}$, we have
$\hat{u}^{L}\in\mathcal{A}_{L}^{\rm out}$. The main idea is that of
discretizing the measure $\mu$ in the variable $k$ to get a measure
concentrated on lines $\mathbb{R}\times\\{k\\}$ with
$k\in\frac{\pi\mathbb{Z}}{L_{0}}$ where the weight on each line is a
coefficient $b^{L_{0}}(x,k)$. Afterwards we define $\hat{u}^{L}$ as in such a
way that its Fourier coefficients $a^{L}_{k}(x)$ are as close as possible to
${\sqrt{b^{L_{0}}(x,k)}}/k$ but at the same time have better regularity to
ensure $\hat{u}^{L}\in\mathcal{A}_{L}^{\rm out}$. In order to do that we first
dilate $\mu$ with a factor $\lambda$ in the $x$-variable as in Lemma 5.2, then
we discretize using Lemma 5.3, and finally we mollify $b^{L_{0}}(\cdot,k)$ at
scale $\varepsilon$ after a suitable extension in $\mathbb{R}$.
To this purpose we let $(\mu^{L_{0}})\subset\mathcal{M}_{\infty}^{\lambda}$ be
the sequence of Lemma 5.3 for the parameter $L_{0}$, which is of the form
$\mu^{L_{0}}=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits{(0,\lambda)}\times\delta_{k}\,.$
By the mean value theorem, for each $\lambda$, we can find
$\bar{\lambda}\in(\frac{\lambda+1}{2},\lambda)$ such that
$\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(\bar{\lambda},k)k^{2}\leq\fint_{\frac{\lambda+1}{2}}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x\,.$ (5.29)
By truncating $\bar{b}^{L_{0}}$ at $x=\bar{\lambda}$ we can define
$b^{L_{0}}\colon\mathbb{R}\times\frac{\pi\mathbb{Z}}{L_{0}}\to\mathbb{R}$ as
$b^{L_{0}}(k,x):=\begin{cases}0&\text{if }x\leq 0,\\\
\bar{b}^{L_{0}}(x,k)&\text{if }0<x<\bar{\lambda},\\\
\bar{b}^{L_{0}}(\bar{\lambda},k)&\text{if }x\geq\bar{\lambda}\,.\end{cases}$
(5.30)
Let
$\rho_{\varepsilon}(x):=\frac{1}{2\varepsilon}e^{\frac{-|x|}{\varepsilon}}$
and note that in particular
$|\dot{\rho}_{\varepsilon}(x)|=\frac{1}{\varepsilon}\rho_{\varepsilon}(x)\,.$
(5.31)
Finally we let
$a^{L}\colon\mathbb{R}\times\frac{\pi\mathbb{Z}}{L_{0}}\to\mathbb{R}$ be
defined as
$a^{L}(x,k):=b^{L_{0}}(\cdot,k)*\rho_{\varepsilon}(x)\,,$
and $\hat{u}^{L}\in\mathcal{A}_{L_{0}}^{\rm out}$ be the function
$\hat{u}^{L}(x,y):=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k>0}\frac{\sqrt{a^{L}(x,k)}}{k}\sqrt{2}\sin(ky)+\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k<0}\frac{\sqrt{a^{L}(x,k)}}{k}\sqrt{2}\cos(ky)\,.$
Eventually we extend $\hat{u}^{L}$, without relabelling it, periodically in
$[-1,1]\times[-L,L]$.
Step 1: in this step we show (5.18)–(5.21). By (3.3) and (5.8) we have that
$\begin{split}2A^{L}(x)&=\fint_{-L}^{L}(\hat{u}_{,y})^{2}\,{\rm
d}y=\fint_{-L_{0}}^{L_{0}}(\hat{u}_{,y})^{2}\,{\rm d}y\\\
&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}{a}^{L}(x,k)=\Big{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}{b}^{L_{0}}(\cdot,k)\Big{)}*\rho_{\varepsilon}(x)\\\
&=2\big{(}x\chi_{(0,\bar{\lambda})}+\bar{\lambda}\chi_{(\bar{\lambda},+\infty)}\big{)}*\rho_{\varepsilon}(x)=2x\chi_{\\{x\geq
0\\}}+\varepsilon\Big{(}e^{\frac{-|x|}{\varepsilon}}-e^{\frac{x-\bar{\lambda}}{\varepsilon}}\Big{)}\,,\end{split}$
(5.32)
for all $x\in[-1,1]$. Then for $\varepsilon$ small enough we have
$\frac{\varepsilon}{2e}\leq 2A^{L}(x)\leq{3\varepsilon}\quad\text{ if
}x\in(0,\varepsilon)\,,$ $x\leq
2x+\varepsilon\big{(}e^{\frac{-1}{\varepsilon}}-e^{\frac{1-\bar{\lambda}}{\varepsilon}}\big{)}\leq
2A^{L}(x)\leq{3x}\quad\text{ if }x\in(\varepsilon,1)\,,$
so that
$\frac{1}{2e}\max\\{x,\varepsilon\\}\leq
2A^{L}(x)=\fint_{-L}^{L}(\hat{u}_{,y})^{2}\,{\rm d}y\leq
3\max\\{x,\varepsilon\\}\,.$
We have that
$(f^{L}(x))^{2}=\frac{x}{A^{L}(x)}\,,$
and by estimates above
$(f^{L}(x))^{2}\lesssim\frac{x}{\max\\{x,\varepsilon\\}}\lesssim\frac{x}{\varepsilon}\quad\text{
if }x\in(0,\varepsilon)\,.$
Observing that
$(e^{\frac{-x}{\varepsilon}}-e^{\frac{x-\bar{\lambda}}{\varepsilon}})\geq 0$
if and only if $x\in(0,\bar{\lambda}/2)$ we have
$(f^{L}(x))^{2}=\frac{x}{x+\frac{\varepsilon}{2}(e^{\frac{-x}{\varepsilon}}-e^{\frac{x-\bar{\lambda}}{\varepsilon}})}\leq
1\quad\text{ in }(0,\bar{\lambda}/2)\,,$
and
$(f^{L}(x))^{2}\leq\frac{x}{x+\frac{\varepsilon}{2}(e^{\frac{-1}{\varepsilon}}-e^{\frac{1-\bar{\lambda}}{\varepsilon}})}\leq
1+\frac{\frac{\varepsilon}{2}|e^{\frac{-1}{\varepsilon}}-e^{\frac{1-\bar{\lambda}}{\varepsilon}}|}{\frac{1}{2}+\frac{\varepsilon}{2}(e^{\frac{-1}{\varepsilon}}-e^{\frac{1-\bar{\lambda}}{\varepsilon}})}=1+\frac{\varepsilon}{8}\quad\text{
for }x\in(\bar{\lambda}/2,1)\,.$
Moreover we have that
$(f^{L}(x))^{2}\geq\frac{x}{x+\frac{\varepsilon}{2}e^{{-N}}}=1-\frac{\frac{\varepsilon}{2}e^{{-N}}}{x+\frac{\varepsilon}{2}e^{{-N}}}\geq
1-\frac{\frac{\varepsilon}{2}e^{{-N}}}{N\varepsilon}=1+o_{N}(1)\quad\text{ for
}x\in[N\varepsilon,1)\,.$
A direct computation shows that
$(\dot{f}^{L}(x))^{2}=\frac{\left(A^{L}(x)-x\dot{A}^{L}(x)\right)^{2}}{4x(A^{L}(x))^{3}}\lesssim\frac{(\max\\{x,\varepsilon\\})^{2}\max\\{e^{-\frac{x}{\varepsilon}},e^{\frac{1-\bar{\lambda}}{\varepsilon}}\\}}{x(\max\\{x,\varepsilon\\})^{3}}\lesssim\frac{\max\\{e^{-\frac{x}{\varepsilon}},e^{\frac{1}{\sqrt{\varepsilon}}}\\}}{x\max\\{x,\varepsilon\\}}\lesssim\frac{\max\\{e^{-\frac{x}{\varepsilon}},e^{\frac{1}{\sqrt{\varepsilon}}}\\}}{x\varepsilon}\,,$
where the second inequality follows from
$\frac{1-\bar{\lambda}}{\varepsilon}\leq\frac{1-\frac{\lambda+1}{2}}{\varepsilon}=\frac{1}{2\sqrt{\varepsilon}}$.
Furthermore we have
$\begin{split}(\ddot{f}^{L}(x))^{2}&\lesssim\frac{x(\ddot{A}^{L}(x))^{2}}{(A^{L}(x))^{3}}+\frac{1}{x^{3}A^{L}(x)}+\frac{(\dot{A}^{L}(x))^{2}}{x(A^{L}(x))^{3}}+\frac{x(\dot{A}^{L}(x))^{4}}{(A^{L}(x))^{5}}\\\
&\lesssim\frac{x\varepsilon^{-2}e^{-\frac{x}{\varepsilon}}}{(\max\\{x,\varepsilon\\})^{3}}+\frac{1}{x^{3}\max\\{x,\varepsilon\\}}+\frac{1}{x(\max\\{x,\varepsilon\\})^{3}}+\frac{x}{(\max\\{x,\varepsilon\\})^{5}}\,.\end{split}$
(5.33)
Hence
$(\ddot{f}^{L}(x))^{2}\lesssim\frac{1}{\varepsilon^{4}}+\frac{1}{x^{3}\varepsilon}+\frac{1}{x\varepsilon^{3}}\lesssim\frac{1}{x^{3}\varepsilon}\quad\text{
if }x\in(0,\varepsilon)\,,$
and
$(\ddot{f}^{L}(x))^{2}\lesssim\frac{e^{-\frac{x}{\varepsilon}}}{x^{2}\varepsilon^{2}}+\frac{1}{x^{4}}\lesssim\frac{1}{x^{3}\varepsilon}\quad\text{
if }x\in(\varepsilon,1)\,.$
Step 2: in this step we show (5.22). By (3.2) it holds
$\begin{split}\fint_{-L}^{L}(\hat{u}^{L}(x,\cdot))^{2}\,{\rm
d}y=\fint_{-L_{0}}^{L_{0}}(\hat{u}^{L}(x,\cdot))^{2}\,{\rm
d}y&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{{a^{L}(x,k)}}{k^{2}}\\\
&=\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{{b^{L_{0}}(\cdot,k)}}{k^{2}}\bigg{)}*\rho_{\varepsilon}(x)\\\
&=\int_{0}^{+\infty}\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{{b^{L_{0}}(z,k)}}{k^{2}}\bigg{)}\rho_{\varepsilon}(x-z)\,{\rm
d}z\,.\end{split}$ (5.34)
Moreover by the fundamental theorem of calculus and Hölder’s inequality we
have
$b^{L_{0}}(z,k)=\left(\sqrt{b^{L_{0}}(z,k)}\right)^{2}=\left(\int_{0}^{z}\frac{b^{L_{0}}_{,x}(\hat{z},k)}{2\sqrt{b^{L_{0}}(\hat{z},k)}}\,{\rm
d}\hat{z}\right)^{2}\leq
z\int_{0}^{z}\frac{(b^{L_{0}}_{,x}(\hat{z},k))^{2}}{4{b^{L_{0}}(\hat{z},k)}}\,{\rm
d}\hat{z}\,.$ (5.35)
Combining (5.34) with (5.35) we find
$\begin{split}\fint_{-L}^{L}(\hat{u}^{L}(x,\cdot))^{2}\,{\rm
d}y&\leq\int_{0}^{+\infty}\bigg{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\int_{0}^{z}\frac{(b^{L_{0}}_{,x}(\hat{z},k))^{2}}{4k^{2}{b^{L_{0}}(\hat{z},k)}}\,{\rm
d}\hat{z}\bigg{)}z\rho_{\varepsilon}(x-z)\,{\rm d}z\end{split}$
By definition of $b^{L_{0}}$ it follows that
$\begin{split}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\int_{0}^{z}\frac{(b^{L_{0}}_{,x}(\hat{z},k))^{2}}{4k^{2}{b^{L_{0}}(\hat{z},k)}}\,{\rm
d}\hat{z}&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\int_{0}^{z\wedge\bar{\lambda}}\frac{(\bar{b}^{L_{0}}_{,x}(\hat{z},k))^{2}}{4k^{2}{\bar{b}^{L_{0}}(\hat{z},k)}}\,{\rm
d}\hat{z}\\\
&\leq\int_{(0,z\wedge\bar{\lambda})\times\mathbb{R}}\frac{1}{4k^{2}}\bigg{(}\frac{\,{\rm
d}\mu^{L_{0}}_{,x}}{\,{\rm d}\mu^{L_{0}}}\bigg{)}^{2}\,{\rm
d}\mu^{L_{0}}=:\omega(z)\,,\end{split}$ (5.36)
where the last inequality can be obtained by arguing exactly as in (5.17).
Therefore we deduce that
$\fint_{-L_{0}}^{L_{0}}(\hat{u}^{L}(x,\cdot))^{2}\,{\rm
d}y\leq\int_{0}^{+\infty}\omega(z)z\rho_{\varepsilon}(x-z)\,{\rm d}z\,.$
Note that $\omega(z)\to 0$ as $z\to 0$ and
$\omega(z)\leq\omega(\bar{\lambda})\leq(\lambda^{2})\mathcal{F}_{\infty}(\mu)\leq
C$. Let $N\geq 2$ be a natural number. Assume $x\in(0,N\varepsilon]$. Since
$\omega$ is increasing we have
$\begin{split}\int_{0}^{+\infty}\omega(z)z\rho_{\varepsilon}(x-z)\,{\rm
d}z&\leq\omega(2N\varepsilon)\int_{0}^{2N\varepsilon}z\rho_{\varepsilon}(x-z)\,{\rm
d}z+\omega(\bar{\lambda})\int_{2N\varepsilon}^{+\infty}z\rho_{\varepsilon}(x-z)\,{\rm
d}z\\\
&\leq\omega(2N\varepsilon)\max\\{x,\varepsilon\\}+CNe^{-N}\varepsilon\\\
&\lesssim\max\\{x,\varepsilon\\}(\omega(2N\varepsilon)+Ne^{-N})\,.\end{split}$
(5.37)
If instead $x\in(N\varepsilon,1)$, we get
$\begin{split}\int_{0}^{+\infty}\omega(z)z\rho_{\varepsilon}(x-z)\,{\rm
d}z&\leq\omega(\bar{\lambda})\int_{0}^{+\infty}z\rho_{\varepsilon}(x-z)\,{\rm
d}z\lesssim x.\end{split}$ (5.38)
Step 3: in this step we show (5.23). By the mean value theorem and the fact
that $u^{L}(x,\cdot)$ is $2L_{0}$-periodic, for fixed $x$, we can find
$y_{0}=y_{0}(x)\in[-L_{0},L_{0}]$ such that
$\hat{u}^{L}(x,y_{0})=\fint_{-L_{0}}^{L_{0}}\hat{u}^{L}(x,\hat{y})\,{\rm
d}\hat{y}=0\,,$
where the second equality follows by the definition of $\hat{u}^{L}$ and the
fact that
$\begin{split}&\fint_{-L_{0}}^{L_{0}}\sin(k\hat{y})\,{\rm
d}\hat{y}=\frac{k}{2L_{0}}(-\cos(kL_{0})+\cos(kL_{0}))=0\,,\\\
&\fint_{-L_{0}}^{L_{0}}\cos(k\hat{y})\,{\rm
d}\hat{y}=\frac{k}{2L_{0}}(\sin(kL_{0})-\sin(kL_{0}))=0\,,\end{split}$
for all $k\in\frac{\pi\mathcal{Z}}{L_{0}}$. Thus by the fundamental theorem of
calculus, Hölder’s inequality and Plancherel it holds
$\begin{split}|\hat{u}^{L}(x,y)|=\left|\int_{y_{0}}^{y}\hat{u}_{,y}(x,y^{\prime})\,{\rm
d}y^{\prime}\right|&\leq\sqrt{L_{0}}\left(\int_{-L_{0}}^{L_{0}}(u^{L}_{,y})^{2}\,{\rm
d}y^{\prime}\right)^{\frac{1}{2}}\\\ &=L_{0}\sqrt{2A^{L}(x)}\,.\end{split}$
This together with steps 1 and 2 yield
$\fint_{-L}^{L}\int_{-1}^{1}{(\hat{u}^{L})^{4}}\,{\rm d}x\,{\rm
d}y=\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}{(\hat{u}^{L})^{4}}\,{\rm d}x\,{\rm
d}y\lesssim
L_{0}^{2}A^{L}(x)\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}{(\hat{u}^{L})^{2}}\,{\rm
d}x\,{\rm d}y\lesssim L_{0}^{2}(\max\\{x,\varepsilon\\})^{2}\,.$
Step 4: in this step we show (5.24). By (3.3) the definition of $a^{L}$ and
(5.29)
$\begin{split}\fint_{-L}^{L}(u_{,yy}^{L})^{2}\,{\rm
d}y&=\fint_{-L_{0}}^{L_{0}}(u_{,yy}^{L})^{2}\,{\rm
d}y=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}k^{2}a^{L}(x,k)=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}k^{2}b^{L_{0}}(x,\cdot)*\rho_{\varepsilon}(x)\\\
&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\biggl{(}k^{2}\int_{0}^{\bar{\lambda}}\bar{b}^{L_{0}}(z,k)\rho_{\varepsilon}(x-z)\,{\rm
d}z\biggr{)}+\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\bar{b}^{L_{0}}(\bar{\lambda},k)k^{2}\int_{\bar{\lambda}}^{+\infty}\rho_{\varepsilon}(x-z)\,{\rm
d}z\\\
&\leq\|\rho_{\varepsilon}\|_{\infty}\int_{0}^{\bar{\lambda}}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x+e^{\frac{x-\bar{\lambda}}{\varepsilon}}\fint_{\frac{\lambda+1}{2}}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x\\\
&\lesssim\left(\frac{1}{\varepsilon}+\frac{1}{\lambda-1}e^{\frac{x-\bar{\lambda}}{\varepsilon}}\right)\int_{0}^{{\lambda}}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x\\\
&\lesssim\left(\frac{1}{\varepsilon}+\frac{1}{\lambda-1}e^{\frac{x-\bar{\lambda}}{\varepsilon}}\right)\mathcal{F}_{\infty}^{\lambda}(\mu^{L_{0}})\lesssim\frac{1}{\varepsilon}\,.\end{split}$
(5.39)
Since the function $(z_{1},z_{2})\mapsto{z_{1}^{2}}/{z_{2}}$ is convex by
Jensen’s inequality we have
$\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}=\frac{\big{(}b^{L_{0}}_{,x}(\cdot,k)*\rho_{\varepsilon}(x)\big{)}^{2}}{b^{L_{0}}(\cdot,k)*\rho_{\varepsilon}(x)}\leq\frac{(b^{L_{0}}_{,x}(\cdot,k))^{2}}{b^{L_{0}}(\cdot,k)}*\rho_{\varepsilon}(x)\,.$
This together with (3.3) and (3.4) imply
$\begin{split}\fint_{-L}^{L}(u_{,x}^{L})^{2}\,{\rm
d}y=\fint_{-L_{0}}^{L_{0}}(u_{,x}^{L})^{2}\,{\rm
d}y&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\leq\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(b^{L_{0}}_{,x}(\cdot,k))^{2}}{b^{L_{0}}(\cdot,k)}*\rho_{\varepsilon}(x)\\\
&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\bigg{(}\frac{1}{4k^{2}}\int_{0}^{\bar{\lambda}}\frac{(b^{L_{0}}_{,x}(z,k))^{2}}{b^{L_{0}}(z,k)}\rho_{\varepsilon}(x-z)\,{\rm
d}z\bigg{)}\\\
&\leq\|\rho_{\varepsilon}\|_{\infty}\int_{0}^{\bar{\lambda}}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(b^{L_{0}}_{,x}(x,k))^{2}}{b^{L_{0}}(x,k)}\,{\rm
d}x\leq\frac{1}{\varepsilon}\mathcal{F}_{\infty}^{\lambda}(\mu^{L_{0}})\lesssim\frac{1}{\varepsilon}\,.\end{split}$
(5.40)
Thus combining (5.39) with (5.40) we infer (5.24).
Step 5: in this step we show (5.25). By recalling Definition 2.1 we have
$\hat{\mu}^{L}:=\mu^{L}(\hat{u}^{L})=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}a^{L}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times\delta_{k}\,,$
and
$\hat{\mu}^{L}_{,x}=\mu^{L}_{,x}(\hat{u}^{L})=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}a^{L}_{,x}(x,k)\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times\delta_{k}\,,$
Let $\varphi\in C^{\infty}_{c}((-1,1)\times\mathbb{R})$. Then
$\begin{split}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\hat{\mu}^{L}&=\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\hat{\mu}^{L}-\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L_{0}}+\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L_{0}}\,.\end{split}$
By Lemma 5.3 we have that
$\lim_{L\to\infty}\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L_{0}}=\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm d}\mu\,,$
hence it suffices to show that
$\lim_{L\to+\infty}\biggl{(}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\hat{\mu}^{L}-\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L_{0}}\biggr{)}=0\,.$
Indeed by (5.32) and (5.8) we have
$\begin{split}\biggl{|}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\hat{\mu}^{L}-&\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L_{0}}\biggr{|}\leq\|\varphi\|_{\infty}\left|\int_{(-1,1)\times\mathbb{R}}\,{\rm
d}\hat{\mu}^{L}-\int_{(0,1)\times\mathbb{R}}\,{\rm d}\mu^{L_{0}}\right|\\\
&=\|\varphi\|_{\infty}\Bigg{|}\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}a^{L}(x,k)\,{\rm
d}x-\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\bar{b}^{L}(x,k)\,{\rm
d}x\Bigg{|}\\\
&=\|\varphi\|_{\infty}\left|\int_{-1}^{1}\varepsilon\Big{(}e^{-\frac{|x|}{\varepsilon}}-e^{\frac{x-\bar{\lambda}}{\varepsilon}}\Big{)}\,{\rm
d}x\right|\leq C\varepsilon^{2}\to 0\quad\text{as }L\to+\infty\,.\end{split}$
Moreover we have
$\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\hat{\mu}^{L}_{,x}=-\int_{(-1,1)\times\mathbb{R}}\varphi_{,x}\,{\rm
d}\hat{\mu}^{L}\to-\int_{(-1,1)\times\mathbb{R}}\varphi_{,x}\,{\rm
d}\mu=\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm d}\mu_{,x}\,.$
Step 6: in this step we show (5.26). By (3.3), (5.30) we have
$\begin{split}&\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\hat{u}^{L}_{,yy})^{2}\,{\rm
d}x\,{\rm
d}y=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}a^{L}(x,k)k^{2}\,{\rm
d}x=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}b^{L_{0}}(\cdot,k)*\rho_{\varepsilon}(x)k^{2}\,{\rm
d}x\\\
&=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\biggl{(}k^{2}\int_{0}^{\bar{\lambda}}\bar{b}^{L_{0}}(z,k)\rho_{\varepsilon}(x-z)\,{\rm
d}z\biggr{)}\,{\rm
d}x+\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\bar{b}^{L_{0}}(\bar{\lambda},k)k^{2}\int_{-1}^{1}\int_{\bar{\lambda}}^{+\infty}\rho_{\varepsilon}(x-z)\,{\rm
d}z\,.\end{split}$ (5.41)
Fubini’s theorem yields
$\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\biggl{(}k^{2}\int_{0}^{\bar{\lambda}}\bar{b}^{L_{0}}(z,k)\rho_{\varepsilon}(x-z)\,{\rm
d}z\biggr{)}\,{\rm
d}x\leq\int_{0}^{\bar{\lambda}}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\bar{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x\,.$ (5.42)
while from (5.29) we deduce
$\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\bar{b}^{L_{0}}(\bar{\lambda},k)k^{2}\int_{-1}^{1}\int_{\bar{\lambda}}^{+\infty}\rho_{\varepsilon}(x-z)\,{\rm
d}z\,{\rm
d}x\leq\frac{\varepsilon}{2}\left(e^{\frac{1-\bar{\lambda}}{\varepsilon}}-e^{\frac{-1-\bar{\lambda}}{\varepsilon}}\right)\fint_{\frac{\lambda+1}{2}}^{\lambda}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\overline{b}^{L_{0}}(x,k)k^{2}\,{\rm
d}x\,.$ (5.43)
Analogously from (3.3), (3.4) and Jensen’s inequality it holds
$\begin{split}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y&=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\,{\rm d}x\\\
&\leq\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(b^{L_{0}}_{,x}(\cdot,k))^{2}}{b^{L_{0}}(\cdot,k)}*\rho_{\varepsilon}(x)\,{\rm
d}x\\\
&\leq\int_{0}^{\bar{\lambda}}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(\bar{b}^{L_{0}}_{,x}(x,k))^{2}}{\bar{b}^{L_{0}}(x,k)}\,{\rm
d}x\,.\end{split}$ (5.44)
Gathering together (5.41)–(5.44) we obtain
$\fint_{-L}^{L}\int_{-1}^{1}\big{(}(\hat{u}^{L}_{,x})^{2}+(\hat{u}^{L}_{,yy})^{2}\big{)}\,{\rm
d}x\,{\rm
d}y\leq\biggl{(}1+\frac{C\varepsilon(e^{\frac{1-\bar{\lambda}}{\varepsilon}-e^{\frac{-1-\bar{\lambda}}{\varepsilon}}})}{1-\lambda}\biggr{)}\mathcal{F}^{\lambda}_{\infty}(\mu^{L_{0}})\,,$
(5.45)
and hence by letting $L\to+\infty$ and recalling Lemma 5.3 $(iii)$ we deduce
(5.26).
Step 7: in this step we show (5.27). By (3.3), (3.4)
$\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,xy})^{2}\,{\rm d}x\,{\rm
d}y=\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\hat{u}^{L}_{,xy})^{2}\,{\rm d}x\,{\rm
d}y=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\,{\rm d}x\,.$ (5.46)
Next we observe that (5.31) yields
${(a^{L}_{,x}(x,k))^{2}}={(b^{L}(\cdot,k)*\dot{\rho}_{\varepsilon}(x))^{2}}\leq{(b^{L}(\cdot,k)*|\dot{\rho}_{\varepsilon}|(x))^{2}}=\frac{1}{\varepsilon^{2}}{(b^{L}(\cdot,k)*\rho_{\varepsilon}(x))^{2}}\leq\frac{1}{\varepsilon^{2}}(a^{L}(x,k))^{2}\,,$
(5.47)
so that combining (5.46) with (5.47) and recalling (5.32) we obtain
$\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,xy})^{2}\,{\rm d}x\,{\rm
d}y\leq\frac{1}{4\varepsilon^{2}}\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}(x,k))^{2}\,{\rm
d}x\lesssim\frac{1}{\varepsilon^{2}}\,.$ (5.48)
In a similar way (3.3) and (3.4) give
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}_{,xx}^{L})^{2}\,{\rm
d}x\,{\rm
d}y&=\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\hat{u}_{,xx}^{L})^{2}\,{\rm
d}x\,{\rm d}y=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{k^{2}}\Big{[}\Big{(}\sqrt{a^{L}(x,k)}\Big{)}_{,xx}\Big{]}^{2}\,{\rm
d}x\\\ &\lesssim\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{k^{2}}\frac{(a^{L}_{,xx}(x,k))^{2}}{a^{L}(x,k)}\,{\rm
d}x+\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{k^{2}}\frac{(a^{L}_{,x}(x,k))^{4}}{(a^{L}(x,k))^{3}}\,{\rm d}x\\\
&\lesssim\frac{1}{\varepsilon^{2}}\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{1}{4k^{2}}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\,{\rm d}x\\\
&\lesssim\frac{1}{\varepsilon^{2}}\int_{-1}^{1}\fint_{-L_{0}}^{L_{0}}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}y\,{\rm
d}x\lesssim\frac{1}{\varepsilon^{2}}\mathcal{F}^{\lambda}_{\infty}(\mu^{L_{0}})\lesssim\frac{1}{\varepsilon^{2}}\,,\end{split}$
(5.49)
where the second inequality follows from
${(a^{L}_{,x}(x,k))^{4}}=(a^{L}_{,x}(x,k))^{2}{(b^{L}(\cdot,k)*\dot{\rho}_{\varepsilon}(x))^{2}}\leq\frac{1}{\varepsilon^{2}}(a^{L}_{,x}(x,k))^{2}(a^{L}(x,k))^{2}\,,$
and
$\begin{split}{(a^{L}_{,xx}(x,k))^{2}}={(b^{L}_{,x}(\cdot,k)*\dot{\rho}_{\varepsilon}(x))^{2}}\leq\frac{1}{\varepsilon^{2}}(a^{L}_{,x}(x,k))^{2}\,.\end{split}$
Moreover appealing again to (5.47) we find
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}_{,yyx}^{L})^{2}\,{\rm
d}x\,{\rm
d}y&=\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\hat{u}_{,yyx}^{L})^{2}\,{\rm
d}x\,{\rm d}y=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}{k^{2}}\Big{[}\Big{(}\sqrt{a^{L}(x,k)}\Big{)}_{,x}\Big{]}^{2}\,{\rm d}x\\\
&=\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}\frac{k^{2}}{4}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\,{\rm
d}x\lesssim\frac{1}{\varepsilon^{2}}\int_{-1}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k\neq
0}{k^{2}}{a^{L}(x,k)}\,{\rm d}x\\\
&\lesssim\frac{1}{\varepsilon^{2}}\int_{-1}^{1}\fint_{-L_{0}}^{L_{0}}(\hat{u}^{L}_{,yy})^{2}\,{\rm
d}y\,{\rm
d}x\lesssim\frac{1}{\varepsilon^{2}}\mathcal{F}^{\lambda}_{\infty}(\mu^{L_{0}})\lesssim\frac{1}{\varepsilon^{2}}\,.\end{split}$
(5.50)
Eventually gathering together (5.48)–(5.50) we deduce (5.27).
Step 8: in this step we show (5.28). Analogously to step 3 we can find
$y_{0}\in[-L_{0},L_{0}]$ such that
$\hat{u}^{L}_{,x}(x,y_{0})=\fint_{-L_{0}}^{L_{0}}\hat{u}^{L}_{,x}(x,\hat{y})\,{\rm
d}\hat{y}=0\,,$
so that by the fundamental theorem of calculus, Hölder’s inequality it holds
$\begin{split}|\hat{u}_{,x}^{L}(x,y)|=\left|\int_{y_{0}}^{y}\hat{u}_{,xy}(x,y^{\prime})\,{\rm
d}y^{\prime}\right|&\leq\sqrt{L_{0}}\left(\int_{-L_{0}}^{L_{0}}(u^{L}_{,xy})^{2}\,{\rm
d}y^{\prime}\right)^{\frac{1}{2}}\\\
&=\sqrt{2}L_{0}\Biggl{(}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}\frac{1}{4}\frac{(a^{L}_{,x}(x,k))^{2}}{a^{L}(x,k)}\Biggr{)}^{\frac{1}{2}}\\\
&\lesssim
L_{0}\Bigg{(}\frac{1}{4\varepsilon^{2}}\sum_{k\in\frac{\pi\mathbb{Z}}{L},k\neq
0}a^{L}(x,k)\Bigg{)}^{\frac{1}{2}}\lesssim\frac{L_{0}}{\varepsilon}\,,\end{split}$
(5.51)
where the last two inequalities follow from (5.47) and (5.32). Therefore
(5.51) gives
$\begin{split}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}{(\hat{u}^{L}_{,x})^{4}}\,{\rm
d}x\,{\rm
d}y&\lesssim\frac{L_{0}^{2}}{\varepsilon^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}{(\hat{u}^{L}_{,x})^{2}}\,{\rm
d}x\,{\rm d}y\lesssim\frac{L_{0}^{2}}{\varepsilon^{2}}\,,\end{split}$
and the proof is concluded.
∎
We are now in a position to prove Proposition 5.1.
###### Proof of Proposition 5.1.
Let $\mu\in\mathcal{M}_{\infty}$ be as in the statement. Let
$\varepsilon=\varepsilon(L)>0$ and $n=n(L)\in\mathbb{N}$ be such that
$\lim_{L\to+\infty}\varepsilon(L)=0\,,\quad\lim_{L\to+\infty}n(L)=\lim_{L\to+\infty}\frac{L}{n(L)}=+\infty\,,$
(5.52)
to be chosen later. Let $\hat{u}^{L}\in\mathcal{A}_{L}^{\rm
out}\cap\mathcal{A}_{L_{0}}^{\rm out}$ with $L_{0}:=L/n(L)$ be the function
given by Lemma 5.4. Recall that
$A^{L}(x)=\frac{1}{2}\fint_{-L}^{L}(\hat{u}^{L}_{,y})^{2}\,{\rm d}y\ \text{
and }\ f^{L}(x)=\sqrt{\frac{x}{A^{L}(x)}}\ \text{ for }\ x\geq 0\,.$
Furthermore we let $M=M(L)\in\mathbb{N}$, $M\geq 2$ to be chosen later such
that setting $\delta=\delta(L):=\frac{\varepsilon(L)}{M(L)}<\varepsilon(L)$ we
have
$\lim_{L\to+\infty}M(L)=+\infty\,,\quad\text{and}\quad\lim_{L\to+\infty}\delta(L)=\lim_{L\to+\infty}M^{2}(L)\delta(L)=0\,.$
(5.53)
We consider $\psi_{\delta}\in C^{\infty}(\mathbb{R})$ such that
$\psi_{\delta}\equiv 0\ \text{ in }(-\infty,\delta]\,,\quad\psi_{\delta}\equiv
1\ \text{ in }[2\delta,+\infty)\,,\quad|\dot{\psi}_{\delta}(x)|\leq
C\delta^{-1}\,,\quad|\ddot{\psi}_{\delta}(x)|\leq C\delta^{-2}\,.$
Note that $\dot{\psi}_{\delta}=\ddot{\psi}_{\delta}=0$ in
$(\delta,2\delta)^{c}$. We next define
$(w^{L},u^{L})=\big{(}(w^{L}_{1},w^{L}_{2}),u^{L}\big{)}$ as follows:
$\begin{split}&u^{L}(x,y):=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}(x,y)\,,\\\
&w_{2}^{L}(x,y):=\psi_{\delta}^{2}(x)xy+B^{L}(x)-\frac{1}{2}\int_{0}^{y}(u^{L}_{,y})^{2}\,{\rm
d}y^{\prime}\,,\\\
&w^{L}_{1}(x,y):=x-\frac{1}{L^{2}}\int_{0}^{y}(w^{L}_{2,x}+u^{L}_{,x}u^{L}_{,y})\,{\rm
d}y^{\prime}\,,\end{split}$
where
$B^{L}(x):=\frac{1}{2}\fint_{-L_{0}}^{L_{0}}\int_{0}^{y}(u^{L}_{,y})^{2}\,{\rm
d}y^{\prime}\,{\rm
d}y-\fint_{-L_{0}}^{L_{0}}\int_{0}^{x}u^{L}_{,x}u^{L}_{,y}\,{\rm
d}x^{\prime}\,{\rm d}y\,.$
Clearly $u^{L}\in\mathcal{A}_{L}^{\rm out}\cap\mathcal{A}_{L_{0}}^{\rm out}$.
We show that $w^{L}\in\mathcal{A}_{L}^{\rm in}\cap\mathcal{A}_{L_{0}}^{\rm
in}$. Precisely to see that $w^{L}(x,\cdot)$ is $2L_{0}$ periodic we use the
following fact:
A differentiable function $h$ is $T$-periodic if $h^{\prime}$ is $T$-periodic
and $h(t)=h(t+T)$ for some $t$.
The function $w^{L}_{2,y}(x,\cdot)$ is $2L_{0}$-periodic, since $u^{L}_{,y}$
is, and from (3.3) satisfies
$\begin{split}w_{2}^{L}(x,L_{0})-w_{2}^{L}(x,-L_{0})&=2L_{0}\psi_{\delta}^{2}(x)x-\int_{-L_{0}}^{L_{0}}(u^{L}_{,y})^{2}\,{\rm
d}y\\\ &=2L_{0}\psi_{\delta}^{2}(x)(x-(f^{L}(x))^{2}A^{L}(x))=0\,,\end{split}$
from which we deduce $w^{L}_{2}(x,\cdot)$ is $2L_{0}$-periodic. Using this
periodicity, and in particular also of $w^{L}_{2,x}$, we see that
$w^{L}_{1,y}(x,\cdot)$ is $2L_{0}$-periodic. Moreover we have
$\begin{split}w_{1}^{L}(x,L_{0})&-w_{1}^{L}(x,-L_{0})=-\frac{1}{L^{2}}\int_{-L_{0}}^{L_{0}}(w^{L}_{2,x}+u^{L}_{,x}u^{L}_{,y})\,{\rm
d}y\\\
&=-\frac{1}{L^{2}}\left(2L_{0}\dot{B}^{L}(x)-\int_{-L_{0}}^{L_{0}}\int_{0}^{y}u_{,y}^{L}u^{L}_{,xy}\,{\rm
d}y^{\prime}\,{\rm d}y+\int_{-L_{0}}^{L_{0}}u_{,x}^{L}u_{,y}^{L}\,{\rm
d}y\right)=0\,,\end{split}$
where the second and the third equalities follow from
$w^{L}_{2,x}=(\psi_{\delta}^{2}(x)x)^{\prime}y+\dot{B}^{L}(x)-\int_{0}^{y}u^{L}_{,y}u^{L}_{,xy}\,{\rm
d}y^{\prime}\,,$
and
$2L_{0}\dot{B}^{L}(x)=2L_{0}\left(\fint_{-L_{0}}^{L_{0}}u^{L}_{,y}u^{L}_{,xy}\,{\rm
d}y-\fint_{-L_{0}}^{L_{0}}u^{L}_{,x}u^{L}_{,y}\,{\rm d}y\right)\,.$
Thus we deduce that $w^{L}_{1,y}$ is $2L_{0}$-periodic. For the reader
convenience we divide the rest of the proof into several steps. We will
repeatedly use that the averaged integral over $(-L,L)$ of a $2L_{0}$-periodic
function is equal to the averaged integral over $(-L_{0},L_{0})$ of the same
function.
Step 1: we show that $(w^{L},u^{L})$ converges to $\mu$ in the sense of
Definition 2.3. We have that
$\hat{u}^{L}(x,y)=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k>0}a^{L}_{k}(x)\sin(ky)+\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k<0}a^{L}_{k}(x)\cos(ky)\,,$
so that
$u^{L}(x,y)=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k>0}\psi_{\delta}(x)f^{L}(x)a^{L}_{k}(x)\sin(ky)+\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}},k<0}\psi_{\delta}(x)f^{L}(x)a^{L}_{k}(x)\cos(ky)\,.$
Therefore we get
$\begin{split}\mu^{L}:=\mu^{L}(u^{L})&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\psi^{2}_{\delta}(x)(f^{L}(x))^{2}(a^{L}_{k}(x))^{2}k^{2}\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(-1,1)\times\delta_{k}\\\
&=\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\psi_{\delta}^{2}(x)(f^{L}(x))^{2}(a^{L}_{k}(x))^{2}k^{2}\mathcal{L}^{1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(\delta,1)\times\delta_{k}\,.\end{split}$
We show $\mu^{L}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$. We fix
$\varphi\in C_{c}^{\infty}((-1,1)\times\mathbb{R})$ and we write
$\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}=\left(\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}-\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}(\hat{u}^{L})\right)+\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}(\hat{u}^{L})\,.$
By Lemma 5.4 it holds
$\lim_{L\to+\infty}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}(\hat{u}^{L})=\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm d}\mu\,.$
Therefore it is sufficient to show that
$\lim_{L\to+\infty}\left(\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}-\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}(\hat{u}^{L})\right)=0\,.$
Recalling that $\psi_{\delta}=0$ in $(-1,\delta)$, $\psi_{\delta}=1$ in
$(2\delta,1)$ and $M\varepsilon=M\delta^{2}\geq 2\delta$, we have
$\begin{split}\bigg{|}\int_{(-1,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}-\int_{(0,1)\times\mathbb{R}}\varphi\,{\rm
d}\mu^{L}(\hat{u}^{L})\bigg{|}&=\bigg{|}\int_{0}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}\varphi(x,k)(a^{L}_{k}(x))^{2}k^{2}\Big{(}\psi^{2}_{\delta}(x)(f^{L}(x))^{2}-1\Big{)}\,{\rm
d}x\bigg{|}\\\
&\leq\|\varphi\|_{\infty}\bigg{|}\int_{0}^{M\varepsilon}\Big{(}\psi^{2}_{\delta}(x)(f^{L}(x))^{2}-1\Big{)}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}_{k}(x))^{2}k^{2}\,{\rm
d}x\bigg{|}\\\
&+\|\varphi\|_{\infty}\bigg{|}\int_{M\varepsilon}^{1}\Big{(}(f^{L}(x))^{2}-1\Big{)}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}_{k}(x))^{2}k^{2}\,{\rm
d}x\bigg{|}\,.\end{split}$
Since
$\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}_{k}(x))^{2}k^{2}=A^{L}(x)=\frac{1}{2}\fint_{-L}^{L}(\hat{u}_{,y}(x,\cdot))^{2}\,{\rm
d}y\,,$
by (5.18) and (5.19) we have
$\Big{(}\psi^{2}_{\delta}(x)(f^{L}(x))^{2}-1\Big{)}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}_{k}(x))^{2}k^{2}\leq\Big{(}\psi^{2}_{\delta}(x)(1+o_{L}(1))-1\Big{)}\max\\{x,\varepsilon\\}$
which together with (5.53) imply
$\|\varphi\|_{\infty}\bigg{|}\int_{0}^{M\varepsilon}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}(a^{L}_{k}(x))^{2}k^{2}(\psi^{2}_{\delta}(x)(f^{L}(x))^{2}-1)\,{\rm
d}x\bigg{|}\leq CM\varepsilon=CM^{2}\delta\to 0\quad\text{ as }L\to+\infty\,.$
Whereas (5.19), (5.20) with $N=M$ and the fact that $M=M(L)\to+\infty$ imply
$(f^{L}(x))^{2}=1+o_{L}(1)\quad\text{ in }\quad[M\varepsilon,1)\,,$
so that
$\|\varphi\|_{\infty}\bigg{|}\int_{M\varepsilon}^{1}\sum_{k\in\frac{\pi\mathbb{Z}}{L_{0}}}a^{L}_{k}(x)(f^{L}(x)-1)\,{\rm
d}x\bigg{|}\leq o_{L}(1)\to 0\quad\text{ as }L\to+\infty\,.$
Eventually by duality we have
$\int_{\mathbb{R}\times(-1,1)}\varphi\,{\rm
d}\mu^{L}_{,x}=-\int_{\mathbb{R}\times(-1,1)}\varphi_{,x}\,{\rm
d}\mu^{L}\to-\int_{\mathbb{R}\times(-1,1)}\varphi_{,x}\,{\rm
d}\mu=\int_{\mathbb{R}\times(-1,1)}\varphi\,{\rm d}\mu_{,x}\,,$
which in turn implies
$\mu^{L}_{,x}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu_{,x}$.
Step 2: we show that
$\limsup_{L\to+\infty}\fint_{-L}^{L}\int_{-1}^{1}((u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2})\,{\rm
d}x\,{\rm
d}y\leq\mathcal{F}_{\infty}(\mu)+C\lim_{L\to+\infty}\omega(2M^{2}\delta)\log
M\,.$ (5.54)
To this purpose we note that
$u^{L}_{,x}(x,y)=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,x}(x,y)+\dot{\psi}_{\delta}(x)f^{L}(x)\hat{u}^{L}(x,y)+\psi_{\delta}(x)\dot{f}^{L}(x)\hat{u}^{L}(x,y)\,.$
(5.55)
Therefore by Young’s inequality
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,x})^{2}\,{\rm d}x\,{\rm
d}y&\leq(1+\alpha)\fint_{-L}^{L}\int_{\delta}^{1}(f^{L}(x))^{2}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y\\\
&+2(1+\alpha^{-1})\frac{1}{\delta^{2}}\fint_{-L}^{L}\int_{\delta}^{2\delta}(f^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}x\,{\rm d}y\\\
&+2(1+\alpha^{-1})\fint_{-L}^{L}\int_{\delta}^{1}(\dot{f}^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}x\,{\rm d}y\,,\end{split}$ (5.56)
for any $\alpha>0$. In this way by recalling (5.19) (5.22) and (5.21) we have
$\fint_{-L}^{L}\int_{-1}^{1}(f^{L}(x))^{2}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm
d}y\leq(1+o_{L}(1))\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y\,,$ (5.57)
$\frac{1}{\delta^{2}}\fint_{-L}^{L}\int_{\delta}^{2\delta}(f^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{1}{\delta^{2}}{\delta}(\omega(2M\varepsilon)+Me^{-M})\int_{\delta}^{2\delta}\,{\rm
d}x\lesssim(\omega(2M\varepsilon)+Me^{-M})\,,$ (5.58)
and
$\begin{split}\fint_{-L}^{L}\int_{\delta}^{1}(\dot{f}^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}x\,{\rm
d}y&\lesssim\int_{\delta}^{\varepsilon}\frac{1}{x\varepsilon}\varepsilon(\omega(2M\varepsilon)+Me^{-M})\,{\rm
d}x\\\
&+\int_{\varepsilon}^{\sqrt{\varepsilon}}\frac{e^{-\frac{x}{\varepsilon}}}{x\varepsilon}x(\omega(2M\varepsilon)+Me^{-M})\,{\rm
d}x+\int_{\sqrt{\varepsilon}}^{1}\frac{e^{-\frac{1}{\sqrt{\varepsilon}}}}{x\varepsilon}x\,{\rm
d}x\\\
&\lesssim(\log\frac{\varepsilon}{\delta}+1)(\omega(2M\varepsilon)+Me^{-M})+\frac{e^{-\frac{1}{\sqrt{\varepsilon}}}}{\varepsilon}\,.\end{split}$
(5.59)
Gathering together (5.56)–(5.59) and recalling that $\delta=\varepsilon/M$ we
deduce
$\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,x})^{2}\,{\rm d}x\,{\rm
d}y\leq(1+\bar{\alpha})\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y+C(\log M+1)(\omega(2M^{2}\delta)+Me^{-M})+o_{L}(1)\,,$ (5.60)
with $\bar{\alpha}:=\alpha+\alpha o_{L}(1)+o_{L}(1)$. Moreover as
$u^{L}_{,yy}=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,yy}$ we get
$\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,yy})^{2}\,{\rm d}x\,{\rm
d}y\leq\fint_{-L}^{L}\int_{-1}^{1}(f^{L}(x))^{2}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y\leq\fint_{-L}^{L}\int_{-1}^{1}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}x\,{\rm d}y\,.$ (5.61)
By (5.60), (5.61), (5.26) and the fact that $M^{2}\delta\to 0$ we finally
deduce
$\limsup_{L\to+\infty}\fint_{-L}^{L}\int_{-1}^{1}((u^{L}_{,x})^{2}+(u^{L}_{,yy})^{2})\,{\rm
d}x\,{\rm
d}y\leq(1+\alpha)\mathcal{F}_{\infty}(\mu)+C\lim_{L\to+\infty}\omega(2M^{2}\delta)\log
M\,.$
Eventually by the arbitrariness of $\alpha$ we infer the desired estimate.
Step 3: we show that
$L^{2}\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w^{L}_{1,x}+\frac{(u^{L}_{,x})^{2}}{2L^{2}}-1\biggr{)}^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{3}M}\,.$
(5.62)
By Young’s inequality we have
$\begin{split}L^{2}\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w^{L}_{1,x}+\frac{(u^{L}_{,x})^{2}}{2L^{2}}-1\biggr{)}^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\fint_{-L}^{L}\int_{-1}^{1}\frac{(u^{L}_{,x})^{4}}{L^{2}}\,{\rm
d}x\,{\rm
d}y+L^{2}\fint_{-L}^{L}\int_{-1}^{1}\bigl{(}w^{L}_{1,x}-1\bigr{)}^{2}\,{\rm
d}x\,{\rm d}y\,.\end{split}$ (5.63)
We estimate the first term on the right hand-side of (5.63). By (5.55) it
follows
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}&{(u^{L}_{,x})^{4}}\,{\rm d}x\,{\rm
d}y\lesssim\fint_{-L}^{L}\int_{-1}^{1}(f^{L}(x))^{4}{(\hat{u}^{L}_{,x})^{4}}\,{\rm
d}x\,{\rm d}y\\\
&+\frac{1}{\delta^{4}}\fint_{-L}^{L}\int_{\delta}^{2\delta}(f^{L}(x))^{4}{(\hat{u}^{L})^{4}}\,{\rm
d}x\,{\rm
d}y+\fint_{-L}^{L}\int_{\delta}^{1}(\dot{f}^{L}(x))^{4}{(\hat{u}^{L})^{4}}\,{\rm
d}x\,{\rm d}y\,.\end{split}$ (5.64)
By (5.19) and (5.28) we have
$\fint_{-L}^{L}\int_{-1}^{1}(f^{L}(x))^{4}{(\hat{u}^{L}_{,x})^{4}}\,{\rm
d}x\,{\rm d}y\lesssim\frac{L_{0}^{2}}{\varepsilon^{2}}\,,$ (5.65)
whereas from (5.19), (5.23), and the fact that $x\in(\delta,2\delta)$ we get
$\begin{split}\frac{1}{\delta^{4}}\fint_{-L}^{L}\int_{\delta}^{2\delta}(f^{L}(x))^{4}{(\hat{u}^{L})^{4}}\,{\rm
d}x\,{\rm
d}y&\lesssim\frac{1}{\delta^{4}}\frac{\delta^{2}}{\varepsilon^{2}}L_{0}^{2}\int_{\delta}^{2\delta}(\max\\{x,\varepsilon\\})^{2}\,{\rm
d}x\lesssim\frac{L_{0}^{2}}{\delta}\,.\end{split}$ (5.66)
Finally by (5.21) and (5.23)
$\begin{split}\fint_{-L}^{L}\int_{\delta}^{1}(\dot{f}^{L}(x))^{4}{(\hat{u}^{L})^{4}}&\,{\rm
d}x\,{\rm d}y\lesssim
L_{0}^{2}\int_{\delta}^{1}\frac{1}{x^{2}\varepsilon^{2}}(\max\\{x,\varepsilon\\})^{2}\,{\rm
d}x\lesssim
L_{0}^{2}\left(\frac{1}{\delta}+\frac{1}{\varepsilon^{2}}\right)\,.\end{split}$
(5.67)
Thus gathering together (5.64)–(5.67) we infer
$\fint_{-L}^{L}\int_{-1}^{1}\frac{(u^{L}_{,x})^{4}}{L^{2}}\,{\rm d}x\,{\rm
d}y\lesssim\frac{L_{0}^{2}}{L^{2}}\left(\frac{1}{\delta}+\frac{1}{\varepsilon^{2}}\right)\,.$
(5.68)
We now pass to estimate the second term on the right hand side of (5.63). To
this aim we observe that integrating by parts it holds
$\begin{split}w_{2,x}^{L}+u^{L}_{,x}u^{L}_{,y}&=(x\psi_{\delta}^{2}(x))^{\prime}y+\dot{B}^{L}(x)-\int_{0}^{y}u^{L}_{,y}u^{L}_{,xy}\,{\rm
d}y^{\prime}+u^{L}_{,x}u^{L}_{,y}(x,y)\\\
&=(x\psi_{\delta}^{2}(x))^{\prime}y+\dot{B}^{L}(x)+\int_{0}^{y}u_{,yy}u_{,x}\,{\rm
d}y^{\prime}+u^{L}_{,x}u^{L}_{,y}|_{y=0}\\\
&=(x\psi_{\delta}^{2}(x))^{\prime}y+\int_{0}^{y}u_{,yy}u_{,x}\,{\rm
d}y^{\prime}+C^{L}(x)\,,\end{split}$ (5.69)
where
$C^{L}(x):=-\fint_{-L}^{L}\int_{0}^{y}u_{,yy}u_{,x}\,{\rm d}y^{\prime}\,{\rm
d}y=-\fint_{-L_{0}}^{L_{0}}\int_{0}^{y}u_{,yy}u_{,x}\,{\rm d}y^{\prime}\,{\rm
d}y\,,$
and the last equality is a consequence of the following identity
$\dot{B}^{L}(x)+u^{L}_{,x}u^{L}_{,y}|_{y=0}=\fint_{-L}^{L}\biggl{(}\int_{0}^{y}u^{L}_{,y}u^{L}_{,xy}\,{\rm
d}y^{\prime}-u^{L}_{,x}u^{L}_{,y}+u^{L}_{,x}u^{L}_{,y}|_{y=0}\biggr{)}\,{\rm
d}y=C^{L}(x)\,.$
Using (5.69) and the definition of $\omega$ we get
$\begin{split}w^{L}_{1,x}-1&=-\frac{1}{L^{2}}\int_{0}^{y}\left((x\psi_{\delta}^{2}(x))^{\prime\prime}y^{\prime}+\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}+\dot{C}^{L}(x)\right)\,{\rm d}y^{\prime}\\\
&=-\frac{1}{L^{2}}\left((x\psi_{\delta}^{2}(x))^{\prime\prime}\frac{y^{2}}{2}+\int_{0}^{y}\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}\,{\rm d}y^{\prime}+\dot{C}^{L}(x)y\right)\,.\end{split}$
This together with Young’s inequality give
$\begin{split}L^{2}\fint_{-L}^{L}\int_{-1}^{1}\bigl{(}w^{L}_{1,x}-1\bigr{)}^{2}\,{\rm
d}x\,{\rm
d}y&=L^{2}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}\bigl{(}w^{L}_{1,x}-1\bigr{)}^{2}\,{\rm
d}x\,{\rm d}y\\\
&\lesssim\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}((x\psi_{\delta}^{2}(x))^{\prime\prime})^{2}y^{4}\,{\rm
d}x\,{\rm d}y\\\
&+\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}\biggl{(}\int_{0}^{y}\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}\,{\rm d}y^{\prime}\biggr{)}^{2}\,{\rm d}x\,{\rm d}y\\\
&+\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\dot{C}^{L}(x))^{2}y^{2}\,{\rm
d}x\,{\rm d}y\,.\end{split}$ (5.70)
As
$(x\psi_{\delta}^{2}(x))^{\prime\prime}=4\psi_{\delta}(x)\dot{\psi}_{\delta}(x)+2x\psi_{\delta}(x)\ddot{\psi}_{\delta}(x)+2x(\dot{\psi}_{\delta}(x))^{2}$
in $(\delta,2\delta)$ and $(x\psi_{\delta}^{2}(x))^{\prime\prime}=0$
otherwise, we have
$\begin{split}\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}((x\psi_{\delta}^{2}(x))^{\prime\prime})^{2}y^{4}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{L_{0}^{4}}{L^{2}}\int_{\delta}^{2\delta}((x\psi_{\delta}^{2}(x))^{\prime\prime})^{2}\,{\rm
d}x\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta}\,.\end{split}$ (5.71)
We now estimate the second term on the right hand-side of (5.70). We first
observe that if $a,b,c,d$ are $2L_{0}$-periodic then by applying in order
Hölder, Young and Jensen inequalities we have
$\begin{split}\bigg{[}\int_{0}^{y}\int_{0}^{y^{\prime}}&(ab+cd)\,{\rm
d}y^{\prime\prime}\,{\rm
d}y^{\prime}\bigg{]}^{2}\leq\left[\int_{0}^{y}\|a\|_{L^{2}(0,y^{\prime})}\|b\|_{L^{2}(0,y^{\prime})}+\|c\|_{L^{2}(0,y^{\prime})}\|d\|_{L^{2}(0,y^{\prime})}\,{\rm
d}y^{\prime}\right]^{2}\\\
&\lesssim\left[\int_{0}^{y}\|a\|_{L^{2}(0,y^{\prime})}\|b\|_{L^{2}(0,y^{\prime})}\,{\rm
d}y^{\prime}\right]^{2}+\left[\int_{0}^{y}\|c\|_{L^{2}(0,y^{\prime})}\|d\|_{L^{2}(0,y^{\prime})}\,{\rm
d}y^{\prime}\right]^{2}\\\ &\lesssim
L_{0}^{2}\fint_{-L_{0}}^{L_{0}}\|a\|^{2}_{L^{2}(0,y^{\prime})}\|b\|^{2}_{L^{2}(0,y^{\prime})}\,{\rm
d}y^{\prime}+L_{0}^{2}\fint_{-L_{0}}^{L_{0}}\|c\|^{2}_{L^{2}(0,y^{\prime})}\|d\|^{2}_{L^{2}(0,y^{\prime})}\,{\rm
d}y^{\prime}\\\ &\lesssim
L_{0}^{4}\left[\bigg{(}\fint_{-L_{0}}^{L_{0}}a^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}b^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}+\bigg{(}\fint_{-L_{0}}^{L_{0}}c^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}d^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\right]\,.\end{split}$
Therefore it follows that
$\begin{split}\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}\biggl{(}\int_{0}^{y}&\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}\,{\rm d}y^{\prime}\biggr{)}^{2}\,{\rm d}x\,{\rm d}y\\\
&=\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{\delta}^{1}\biggl{(}\int_{0}^{y}\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}\,{\rm d}y^{\prime}\biggr{)}^{2}\,{\rm d}x\,{\rm d}y\\\
&\lesssim\frac{L_{0}^{4}}{L^{2}}\int_{\delta}^{1}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yy})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,xx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm d}x\\\
&+\frac{L_{0}^{4}}{L^{2}}\int_{\delta}^{1}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yyx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,x})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm d}x\,.\end{split}$ (5.72)
Next we show separately that:
$\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yy})^{2}\,{\rm
d}y^{\prime\prime}\lesssim\frac{1}{\varepsilon}\,,\qquad\qquad\fint_{-L_{0}}^{L_{0}}(u^{L}_{,x})^{2}\,{\rm
d}y^{\prime\prime}\lesssim\frac{1}{x\varepsilon}\max\\{x,\varepsilon\\}\,,$
(5.73) $\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,xx})^{2}\,{\rm
d}y^{\prime\prime}\,{\rm
d}x\lesssim\frac{1}{\delta^{2}}\,,\qquad\qquad\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yyx})^{2}\,{\rm
d}y^{\prime\prime}\,{\rm d}x\lesssim\frac{1}{\delta\varepsilon}\,.$ (5.74)
Since $u^{L}_{,yy}=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,yy}$ by (5.19) and
(5.24) we have
$\fint_{-L_{0}}^{L_{0}}(\hat{u}_{,yy}^{L})^{2}\,{\rm
d}y\lesssim\frac{1}{\varepsilon}\,.$
By (5.55), (5.19), (5.24), (5.22) and (5.21) we have
$\begin{split}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,x})^{2}\,{\rm
d}y\lesssim\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}^{L}_{,x})^{2}\,{\rm
d}y&+\frac{1}{\delta^{2}}\fint_{-L_{0}}^{L_{0}}\chi_{(\delta,2\delta)}(f^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}y+\fint_{-L_{0}}^{L_{0}}(\dot{f}^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm d}y\\\
&\lesssim\frac{1}{\varepsilon}+\frac{1}{\delta^{2}}\frac{\delta}{\varepsilon}\max\\{x,\varepsilon\\}\chi_{(\delta,2\delta)}+\frac{1}{x\varepsilon}\max\\{x,\varepsilon\\}\\\
&\lesssim\frac{1}{\varepsilon}+\frac{1}{\delta}\chi_{(\delta,2\delta)}+\frac{1}{x\varepsilon}\max\\{x,\varepsilon\\}\lesssim\frac{1}{x\varepsilon}\max\\{x,\varepsilon\\}\,.\end{split}$
From (5.55) it follows
$\begin{split}u^{L}_{,xx}(x,y)&=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,xx}(x,y)+\psi_{\delta}(x)\ddot{f}^{L}(x)\hat{u}^{L}(x,y)+\ddot{\psi}_{\delta}(x)f^{L}(x)\hat{u}^{L}(x,y)\\\
&+2\dot{\psi}_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,x}(x,y)+2\dot{\psi}_{\delta}(x)\dot{f}^{L}(x)\hat{u}^{L}(x,y)+2\psi_{\delta}(x)\dot{f}^{L}(x)\hat{u}^{L}_{,x}(x,y)\,.\end{split}$
Hence by (5.19), (5.27), (5.21), (5.22) and (5.26) we have
$\begin{split}\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(u_{,xx}^{L})^{2}\,{\rm
d}y\,{\rm
d}x&\lesssim\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}_{,xx}^{L})^{2}\,{\rm
d}y\,{\rm
d}x+\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(\ddot{f}^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}y\,{\rm d}x\\\
&+\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(\dot{f}^{L}(x))^{2}(\hat{u}_{,x}^{L})^{2}\,{\rm
d}y\,{\rm
d}x+\frac{1}{\delta^{4}}\int_{\delta}^{2\delta}\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}y\,{\rm d}x\\\
&+\frac{1}{\delta^{2}}\int_{\delta}^{2\delta}\fint_{-L_{0}}^{L_{0}}(\dot{f}^{L}(x))^{2}(\hat{u}^{L})^{2}\,{\rm
d}y\,{\rm
d}x+\frac{1}{\delta^{2}}\int_{\delta}^{2\delta}\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}_{,x}^{L})^{2}\,{\rm
d}y\,{\rm d}x\\\
&\lesssim\frac{1}{\varepsilon^{2}}+\int_{\delta}^{1}\frac{1}{x^{3}\varepsilon}\max\\{x,\varepsilon\\}\,{\rm
d}x+\frac{1}{\delta\varepsilon}+\frac{1}{\delta^{4}}\int_{\delta}^{2\delta}\frac{x}{\varepsilon}\max\\{x,\varepsilon\\}\,{\rm
d}x\\\
&+\frac{1}{\delta^{2}}\int_{\delta}^{2\delta}\frac{1}{x\varepsilon}\max\\{x,\varepsilon\\}\,{\rm
d}x+\frac{1}{\delta^{2}}\lesssim\frac{1}{\varepsilon^{2}}+\frac{1}{\delta^{2}}+\frac{1}{\delta\varepsilon}\lesssim\frac{1}{\delta^{2}}\,.\end{split}$
(5.75)
Analogously by (5.55) it follows
$u^{L}_{,yyx}(x,y)=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,yyx}(x,y)+\dot{\psi}_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,yy}(x,y)+\psi_{\delta}(x)\dot{f}^{L}(x)\hat{u}^{L}_{,yy}(x,y)\,,$
from which together with (5.27), (5.19), (5.24) and (5.21)
$\begin{split}\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(u_{,yyx}^{L})^{2}\,{\rm
d}y\,{\rm
d}x&\lesssim\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}_{,yyx}^{L})^{2}\,{\rm
d}y\,{\rm
d}x+\frac{1}{\delta^{2}}\int_{\delta}^{2\delta}\fint_{-L_{0}}^{L_{0}}(f^{L}(x))^{2}(\hat{u}_{,yy}^{L})^{2}\,{\rm
d}y\,{\rm d}x\\\
&+\int_{\delta}^{1}\fint_{-L_{0}}^{L_{0}}(\dot{f}^{L}(x))^{2}(\hat{u}_{,yy}^{L})^{2}\,{\rm
d}y\,{\rm
d}x\lesssim\frac{1}{\varepsilon^{2}}+\frac{1}{\delta^{2}}\frac{\delta}{\varepsilon}\frac{\delta}{\varepsilon}+\frac{1}{\delta\varepsilon}\lesssim\frac{1}{\delta\varepsilon}\,.\end{split}$
Now (5.73) and (5.74) yield
$\begin{split}\frac{L_{0}^{4}}{L^{2}}\int_{\delta}^{1}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yy})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,xx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm
d}x\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,,\end{split}$
(5.76)
and
$\begin{split}\frac{L_{0}^{4}}{L^{2}}\int_{\delta}^{1}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yyx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,x})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm
d}x\lesssim\frac{L_{0}^{4}}{L^{2}}\left(\frac{1}{\delta}+\frac{1}{\varepsilon}\right)\frac{1}{\delta\varepsilon}\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,.\end{split}$
(5.77)
Gathering together (5.72), (5.76) and (5.77) we find
$\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}\biggl{(}\int_{0}^{y}\int_{0}^{y^{\prime}}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime\prime}\,{\rm d}y^{\prime}\biggr{)}^{2}\,{\rm d}x\,{\rm
d}y\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,.$ (5.78)
It remains to estimate the third term on the right hand-side of (5.70). In a
similar way, see in particular (5.76) and (5.77), we have
$\begin{split}\frac{1}{L^{2}}\fint_{-L_{0}}^{L_{0}}\int_{-1}^{1}(\dot{C}^{L}(x))^{2}y^{2}\,{\rm
d}x\,{\rm
d}y&\lesssim\frac{L_{0}^{2}}{L^{2}}\int_{-1}^{1}\bigg{(}\fint_{-L_{0}}^{L_{0}}\int_{0}^{y}(u^{L}_{,yy}u^{L}_{,xx}+u^{L}_{,yyx}u^{L}_{,x})\,{\rm
d}y^{\prime}\,{\rm d}y\bigg{)}^{2}\,{\rm d}x\\\
&\lesssim\frac{L_{0}^{4}}{L^{2}}\int_{-1}^{1}\fint_{-L_{0}}^{L_{0}}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yy})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,xx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm d}y\,{\rm d}x\\\
&+\frac{L_{0}^{4}}{L^{2}}\int_{-1}^{1}\fint_{-L_{0}}^{L_{0}}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,yyx})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\bigg{(}\fint_{-L_{0}}^{L_{0}}(u^{L}_{,x})^{2}\,{\rm
d}y^{\prime\prime}\bigg{)}\,{\rm d}y\,{\rm d}x\\\
&\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,.\end{split}$
(5.79)
Gathering together (5.70), (5.71), (5.78) and (5.79) we infer
$L^{2}\fint_{-L}^{L}\int_{-1}^{1}\bigl{(}w^{L}_{1,x}-1\bigr{)}^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,,$
which together with (5.68) and (5.63) implies
$\begin{split}L^{2}\fint_{-L}^{L}\int_{-1}^{1}\biggl{(}w^{L}_{1,x}+\frac{(u^{L}_{,x})^{2}}{2L^{2}}-1\biggr{)}^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{L_{0}^{2}}{L^{2}}\left(\frac{1}{\delta}+\frac{1}{\varepsilon^{2}}\right)+\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\lesssim\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{2}\varepsilon}\,.\end{split}$
Step 4: we show that
$L^{2}\fint_{-L}^{L}\int_{-1}^{1}\Big{(}w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x\Big{)}^{2}\,{\rm
d}x\,{\rm d}y\leq L^{2}\left(\frac{1}{3}+C\delta^{3}\right)\,.$ (5.80)
Recalling the definition of $w_{2}^{L}$ it holds
$w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x=\psi_{\delta}^{2}(x)x-x$, so that
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}\Big{(}w^{L}_{2,y}+\frac{(u_{,y}^{L})^{2}}{2}-x\Big{)}^{2}\,{\rm
d}x\,{\rm d}y=\int_{-1}^{1}(\psi_{\delta}^{2}(x)x-x)^{2}\,{\rm
d}x\leq\int_{-1}^{2\delta}x^{2}\,{\rm
d}x=\frac{8}{3}\delta^{3}+\frac{1}{3}\,.\end{split}$
Step 5: we show that
$\fint_{-L}^{L}\int_{-1}^{1}\Big{(}L^{2}w_{1,y}^{L}+w_{2,x}^{L}+u_{,x}^{L}u_{,y}^{L}\Big{)}^{2}\,{\rm
d}x\,{\rm d}y=0\,.$ (5.81)
This is a trivial consequence of the following identity
$w^{L}_{1,y}=-\frac{1}{L^{2}}(w_{2,x}^{L}+u_{,x}^{L}u_{,y}^{L})\,,$
which follows from the definitions of $w_{1}^{L}$ and $w_{2}^{L}$.
Step 6: we show that
$\frac{1}{L^{2}}\fint_{-L}^{L}\int_{-1}^{1}\bigg{(}2(u^{L}_{,xy})^{2}+\frac{1}{L^{2}}(u^{L}_{,xx})^{2}\bigg{)}\,{\rm
d}x\,{\rm
d}y\lesssim\left(\frac{1}{L^{2}}+\frac{1}{L^{4}}\right)\frac{1}{\delta^{2}}\,.$
(5.82)
We have
$u^{L}_{,xy}(x,y)=\psi_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,xy}(x,y)+\dot{\psi}_{\delta}(x)f^{L}(x)\hat{u}^{L}_{,y}(x,y)+\psi_{\delta}(x)\dot{f}^{L}(x)\hat{u}^{L}_{,y}(x,y)\,.$
Thus by (5.18), (5.19), (5.27), (5.26) and (5.21)
$\begin{split}\fint_{-L}^{L}\int_{-1}^{1}(u^{L}_{,xy})^{2}\,{\rm d}x\,{\rm
d}y&\lesssim\fint_{-L}^{L}\int_{\delta}^{1}(f^{L}(x))^{2}(\hat{u}^{L}_{,xy})^{2}\,{\rm
d}x\,{\rm d}y\\\
&+\frac{1}{\delta^{2}}\fint_{-L}^{L}\int_{\delta}^{2\delta}(f^{L}(x))^{2}(\hat{u}^{L}_{,y})^{2}\,{\rm
d}x\,{\rm d}y\\\
&+\fint_{-L}^{L}\int_{\delta}^{1}(\dot{f}^{L}(x))^{2}(\hat{u}^{L}_{,y})^{2}\,{\rm
d}x\,{\rm
d}y\lesssim\frac{1}{\varepsilon^{2}}+1+\frac{1}{\varepsilon}+\frac{\varepsilon}{\delta}\lesssim\frac{1}{\delta\varepsilon}\,.\end{split}$
This together with (5.75) imply (5.82).
Conclusions. By Step 1 we have that $(\omega^{L},u^{L})$ converges to $\mu$ in
the sense of Definition 2.3. Moreover by collecting the estimates showed in
Steps 2–6, i.e., (5.54), (5.62), (5.80), (5.81) and (5.82) we find
$\begin{split}\limsup_{L\to+\infty}L^{2}(R_{L}(w^{L},u^{L})-\mathcal{E}_{0})&\leq\mathcal{F}_{\infty}(\mu)+C\lim_{L\to+\infty}\omega(2M^{2}\delta)\log
M\\\
&+C\lim_{L\to+\infty}\Big{(}\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{3}M}+L^{2}\delta^{3}+\frac{1}{L^{2}\delta^{2}}\Big{)}\,.\end{split}$
(5.83)
We now proceed with the choice of the parameters. We start by noticing that
for every $\overline{M}\in\mathbb{N}$ there exists
$L_{\overline{M}}\geq\overline{M}^{1/2}$ such that
$\omega(2\overline{M}^{2}L^{-2/3})\leq{\overline{M}}^{-1}\quad\forall L\geq
L_{\overline{M}}\,.$ (5.84)
Since $L_{\overline{M}+1}\geq L_{\overline{M}}$ we set
$M=M(L):=\overline{M}\quad\text{ if
}L\in[L_{\overline{M}},L_{\overline{M}+1})$
Next we define
$\delta:=\frac{\varepsilon}{M}=L^{-2/3}M^{-1/8}\,\iff\,\varepsilon=L^{-2/3}M^{7/8}\,,$
and we choose
$L_{0}:=\frac{L}{n}\in[M^{1/8},2M^{1/8})\,\iff\,n\in\Big{[}\frac{L}{2M^{1/8}},\frac{L}{M^{1/8}}\Big{)}\,.$
These choices ensures the validity of (5.52) and (5.53). Indeed we have
$\varepsilon=\frac{1}{L^{2/3}M^{-7/8}}=\frac{1}{L^{2/3}\overline{M}^{-7/8}}\leq\frac{1}{\overline{M}^{3}\overline{M}^{-7/8}}\quad\text{
if }L\in[L_{\overline{M}},L_{\overline{M}+1})\,,$
where the last inequality follows from the fact that $L\geq
L_{\overline{M}}\geq\overline{M}^{1/2}$. Hence $\varepsilon\to 0$ and
$\delta\to 0$ as $L\to+\infty$. In a similar way we have $M^{2}\delta\to 0$
and $L_{0},n\to+\infty$ as $L\to+\infty$.
Recalling that $\omega$ is monotone we find
$\omega(2M^{2}\delta)=\omega(2M^{2}L^{-2/3}M^{-1/8})\leq\omega(2M^{2}L^{-2/3})=\omega(2\overline{M}^{2}L^{-2/3})\quad\text{
if }L\in[L_{\overline{M}},L_{\overline{M}+1})\,,$
which together with (5.84) imply
$\log M\omega(2M^{2}\delta)\leq\log(\overline{M})\overline{M}^{-1}\quad\text{
if }L\in[L_{\overline{M}},L_{\overline{M}+1})\,.$ (5.85)
Moreover if $L\in[L_{\overline{M}},L_{\overline{M}+1})$ it holds
$L^{2}\delta^{3}=L^{2}L^{-2}M^{-3/8}=\overline{M}^{-3/8}\,,$ (5.86)
$\frac{1}{L^{2}\delta^{2}}=\frac{L^{4/3}M^{1/4}}{L^{2}}=\frac{M^{1/4}}{L^{2/3}}=\frac{\overline{M}^{1/4}}{L^{2/3}}\leq\frac{\overline{M}^{1/4}}{(\overline{M}^{1/2})^{2/3}}=\overline{M}^{-1/12}\,,$
(5.87)
and
$\frac{L_{0}^{4}}{L^{2}}\frac{1}{\delta^{3}M}\leq\frac{16M^{1/2}}{MM^{-3/8}}=16M^{-1/8}=16\overline{M}^{-1/8}\,,$
(5.88)
Eventually collecting (5.83)–(5.88) we infer
$\begin{split}\limsup_{L\to+\infty}L^{2}(R_{L}(w^{L},u^{L})-\mathcal{E}_{0})\leq\mathcal{F}_{\infty}(\mu)\,.\end{split}$
∎
## 6 Existence and regularity of minimizers of $\mathcal{F}_{\infty}$
In this section we address the existence of minimizers of the limiting
functional $\mathcal{F}_{\infty}$ and we discuss some properties such as
equipartition of the energy. In order to do that we need to introduce the
definition of disintegration of measures in the $k$-variable, which is
slightly different from the disintegration in the $x$-variable introduced in
Section 3.
In the following for a given interval $I\subset\mathbb{R}$ we denote by
$L^{0}(I)$ the space of functions $g\colon I\to\mathbb{R}$ that are Lebesgue
measurable. Moreover the map $\pi_{2}\colon I\times\mathbb{R}\to\mathbb{R}$
denotes the canonical projection, and for any
$\mu\in\mathcal{M}_{b}(I\times\mathbb{R})$ we indicate by
$(\pi_{2})_{\sharp}\mu\in\mathcal{M}_{b}^{+}(\mathbb{R})$ its push-forward
with respect to the map $\pi_{2}$.
###### Definition 6.1 (Disintegration of measures in the $k$-variable).
Let $I\subset\mathbb{R}$ be an interval and let
$\mu\in\mathcal{M}_{b}(I\times\mathbb{R})$. We say that the family
${\big{(}\lambda\,,\,(g_{k})_{k\in\mathbb{R}}\big{)}}\quad\text{ with
}\quad\lambda\in\mathcal{M}_{b}(\mathbb{R})\quad\text{ and }\quad g_{k}\in
L^{0}(I)\quad\forall k\in\mathbb{R}\,,$
is a disintegration of $\mu$ $($in the $k$-variable$)$ if $k\mapsto g_{k}$ is
$\lambda$-measurable, $\int_{0}^{1}g_{k}\,{\rm d}x=1$ for $\lambda$-a.e.
$k\in\mathbb{R}$ and
$\int_{I\times\mathbb{R}}f(x,k)\,{\rm
d}\mu=\int_{\mathbb{R}}\int_{I}f(x,k)g_{k}(x)\,{\rm d}x\,{\rm d}\lambda(k)\,,$
(6.1)
for every $f\in L^{1}(I\times\mathbb{R};|\mu|)$.
With this definition at hand we can state the main result of this section.
###### Theorem 6.2 (Minimizers of $\mathcal{F}_{\infty}$).
Let $\mathcal{M}_{\infty}$ and $\mathcal{F}_{\infty}$ be as in (2.8) and (2.9)
respectively. Then there exists $\hat{\mu}\in\mathcal{M}_{\infty}$ such that
$\mathcal{F}_{\infty}(\hat{\mu})=\inf_{\mu\in\mathcal{M}_{\infty}}\mathcal{F}_{\infty}(\mu)\,.$
Moreover, every minimizer $\hat{\mu}$ satisfies the following properties:
there exist a constant $C>0$ and a $(\pi_{2})_{\sharp}\hat{\mu}$-measurable
map $k\mapsto g_{k}$ with $g_{k}\in BV(0,1)$ for $(\pi_{2})_{\sharp}\hat{\mu}$
a.e. $k\in\mathbb{R}$, such that
${\big{(}(\pi_{2})_{\sharp}\hat{\mu}\,,\,(g_{k})_{k\in\mathbb{R}}\big{)}}\quad\text{
is a disintegration of }\ \hat{\mu}\,,$ $\int_{0}^{1}k^{2}g_{k}(x)\,{\rm
d}x=\int_{0}^{1}\frac{1}{4k^{2}}\left(\frac{\,{\rm d}\hat{\mu}_{,x}}{\,{\rm
d}\hat{\mu}}\right)^{2}g_{k}(x)\,{\rm d}x\quad\text{for }\
(\pi_{2})_{\sharp}\hat{\mu}\ \text{ a.e. $k\in\mathbb{R}$}\,,$ (6.2)
and
$(\pi_{2})_{\sharp}\hat{\mu}\Big{(}\big{\\{}|k|<C\big{\\}}\Big{)}=0\,.$ (6.3)
As a direct consequence minimizers of $\mathcal{F}_{\infty}$ satisfy
equipartition of the energy.
###### Corollary 6.3 (Equipartition of the energy).
Let $\hat{\mu}\in\mathcal{M}_{\infty}$ be a minimizers of
$\mathcal{F}_{\infty}$. Then it holds
$\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\hat{\mu}=\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Big{(}\frac{\,{\rm
d}\hat{\mu}_{,x}}{\,{\rm d}\hat{\mu}}\Big{)}^{2}\,{\rm d}\hat{\mu}\,.$
###### Proof.
By Theorem 6.2 it holds
$\begin{split}\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\hat{\mu}&=\int_{\mathbb{R}}\int_{0}^{1}k^{2}g_{k}(x)\,{\rm d}x\,{\rm
d}(\pi_{2})_{\sharp}\hat{\mu}\\\
&=\int_{\mathbb{R}}\int_{0}^{1}\frac{1}{4k^{2}}\left(\frac{\,{\rm
d}\hat{\mu}_{,x}}{\,{\rm d}\hat{\mu}}\right)^{2}g_{k}(x)\,{\rm d}x\,{\rm
d}(\pi_{2})_{\sharp}\hat{\mu}=\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Big{(}\frac{\,{\rm
d}\hat{\mu}_{,x}}{\,{\rm d}\hat{\mu}}\Big{)}^{2}\,{\rm
d}\hat{\mu}\,.\end{split}$
∎
We divide the proof of Theorem 6.2 into several steps. Precisely we need to
show that the functional $\mathcal{F}_{\infty}$ is convex and lower semi-
continuous and that the class of measures $\mathcal{M}_{\infty}$ admits a
disintegration in the $k$-variable of the form
${\big{(}(\pi_{2})_{\sharp}\hat{\mu}\,,\,(g_{k})_{k\in\mathbb{R}}\big{)}}$.
First of all we recall that by Remark 2.4 $(ii)$ we have
$\mathcal{F}_{\infty}(\mu)=\int_{(0,1)\times\mathbb{R}}k^{2}\,{\rm
d}\mu+\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{2}\,{\rm d}|\tilde{\mu}|\,,$
with $\tilde{\mu}=(\mu,\mu_{,x})$ and $|\tilde{\mu}|$ its total variation.
This alternative formulation turns out to be more convenient, in particular
the term
$\int_{(0,1)\times\mathbb{R}}\frac{1}{4k^{2}}\Bigl{(}\frac{\,{\rm
d}\mu}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{-1}\Bigl{(}\frac{\,{\rm
d}\mu_{,x}}{\,{\rm d}|\tilde{\mu}|}\Bigr{)}^{2}\,{\rm d}|\tilde{\mu}|$
is reminiscent of the Benamou-Brenier functional used in optimal transport
which enjoys nice properties such as lower semicontinuity and convexity. Here
we consider a specific case and we refer to [45, Section 5.3.1] for a general
treatment of this topic.
For any $\rho,E\in\mathcal{M}_{b}((0,1)\times\mathbb{R})$ the Benamou-Brenier
functional is defined as
$\mathscr{B}_{2}(\rho,E):=\sup\left\\{\int_{(0,1)\times\mathbb{R}}a(x,k)\,{\rm
d}\rho+\int_{(0,1)\times\mathbb{R}}b(x,k)\,{\rm d}E\colon(a,b)\in
C_{b}((0,1)\times\mathbb{R};K_{2})\right\\}\,,$ (6.4)
where
$K_{2}:=\left\\{(z_{1},z_{2})\in\mathbb{R}^{2}\colon
z_{1}+\frac{1}{2}z_{2}^{2}\leq 0\right\\}\,.$
We next recall some properties of $\mathscr{B}_{2}$ which follow from [45,
Proposition 5.18]. Then we state and prove two intermediate Lemmas (cf. Lemma
6.5 and Lemma 6.6) which will be used to show the validity of Theorem 6.2.
###### Proposition 6.4 (Properties of $\mathscr{B}_{2}$).
The functional $\mathscr{B}_{2}$ is convex and lower semi-continuous on the
space $(\mathcal{M}_{b}((0,1)\times\mathbb{R}))^{2}$. Moreover, the following
property hold: if both $\rho$ and $E$ are absolutely continuous w.r.t. a same
|
# Sampling and Filtering of Neural Machine Translation Distillation Data
Vilém Zouhar
Institute of Formal and Applied Linguistics, Charles University
<EMAIL_ADDRESS>
###### Abstract
In most of neural machine translation distillation or stealing scenarios, the
goal is to preserve the performance of the target model (teacher). The
highest-scoring hypothesis of the teacher model is commonly used to train a
new model (student). If reference translations are also available, then better
hypotheses (with respect to the references) can be upsampled and poor
hypotheses either removed or undersampled.
This paper explores the importance sampling method landscape (pruning,
hypothesis upsampling and undersampling, deduplication and their combination)
with English to Czech and English to German MT models using standard MT
evaluation metrics. We show that careful upsampling and combination with the
original data leads to better performance when compared to training only on
the original or synthesized data or their direct combination.
## 1 Introduction
Model distillation is a process of transferring the knowledge of one or more,
usually larger, model(s) into another, usually smaller, model Buciluǎ et al.
(2006). A variation of this is training a new model in a way that its
performance is similar to that of the already trained one. This is achieved by
making use of either teacher predictions (black-box) or other products of the
workings of the teacher, such as attention-score or decoder score (grey/glass-
box). Assuming we have access to a parallel corpus, we focus on sampling the
translation hypotheses and making use not only of the teacher scores but also
of their comparison to the reference.
There are various possible motivations for model distillation. The student
model can be much smaller than the teacher model, which has the benefit of
faster inference speed Germann et al. (2020). It can also be used for model
stealing, where an adversary tries to copy the teacher functionality. This is
a practical concern for production-level MT systems Wallace et al. (2020).
One of the approaches for knowledge distillation is to use the teacher model
to generate a new dataset for the student model to train on. Having access to
a trained teacher model, this approach does not require parallel data and can
leverage large monolingual corpora. Reference translations, however, help with
determining which of the teacher’s translations are good and which are of low
quality.
We focus on this approach and propose and compare several importance sampling
approaches to prepare training data for student models that leverage access to
reference translations. These include pruning, upsampling and undersampling,
deduplication and their combination. We show that a combination of these
methods improves the student performance over just using the reference or the
best hypothesis (by the decoder score), which is a common distillation
practice.
The experiment code is available open-source.111github.com/zouharvi/reference-
mt-distill
### 1.1 Related work
The general methodology for knowledge distillation in the form of teacher-
student has been proposed by Hinton et al. (2015). For the MT task
specifically, Tan et al. (2019) focus on vastly reducing the number of
parameters, while retaining the performance of a multi-lingual teacher. Wei et
al. (2019) and Gordon and Duh (2020) use distillation during training to
further improve the model performance.
The work of Kim and Rush (2016) shows that taking either the top sentence with
respect to the teacher decoder score or BLEU Papineni et al. (2002) improves
the performance. Germann et al. (2020) presented student models that distil
knowledge from a larger teacher model with a negligible loss in performance.
They manipulate the queried data based on target sentence quality, such as by
removing sentences that are not correctly recognized by a language identifier.
For the parallel part of the data, they extract the best BLEU scoring sentence
out of 8 hypotheses. Freitag et al. (2017) experiment with pruning sentences
that are below some TER Snover et al. (2006) threshold (lower is better). They
further document the effect of using an ensemble of teachers and also reducing
the student model size.
## 2 Methods
The evaluation of every sampling method follows the following three-step
process. First, the specific parallel corpus (Section 2.1) is translated by
the teacher model (Section 2.2) for the considered translation direction. New
datasets based on metrics are then created. The reference is taken into
consideration during the hypothesis selection. We train new models (students)
on these datasets and measure their performance. There are 12 hypotheses
(default in Marian NMT) provided by the teacher using beam search for every
source sentence which we consider when composing a new dataset.
Figure 1: Scheme of an example of hypothesis sampling with BLEU metric.
Figure 1 shows an example of the sampling process with BLEU. Twelve
translations are made of Source and each receives a score against the provided
reference. The new data contain Translation 2 three times, because of its high
score. Translation 12 is omitted because of its low score. This upsampling is
explained in detail in Section 2.3.
### 2.1 Data
We make use of the Europarl v10 parallel corpus Koehn (2005) for English-Czech
(0.6M sentences) and English-German (1.8M sentences). The sentences are longer
(23 target words per sentence on average) than in the WMT News Task domain
Barrault et al. (2020). To modern standards, this dataset is relatively small
and very domain restricted. This was chosen deliberately because of
computational limitations.222$\sim$4500 GPU hours in total for the whole
experiment Despite that it demonstrates the results of the different sampling
methods with respect to each other. These results may not be transferable to
large parallel corpora in which training data is abundant.
For every language pair, we randomly sample 15k sentences as development
dataset (used only for determining the best epoch and early stopping) and 15k
sentences for final test evaluation which is reported. The WMT News test
dataset is not used for student evaluation, because the students are trained
on a limited amount of data and on a different domain. Out of the WMT20 News
tokens, $0.18\%$ are not present in the Europarl training set. This would
introduce a higher variance into the WMT News test evaluation, which would be
largely dependent on the diversity of the teacher vocabulary.
### 2.2 Models
The teachers333Version student.base at github.com/browsermt/students in this
experiment are transformer-based Vaswani et al. (2017), speed optimized and
were themselves created by knowledge distillation from state-of-the-art models
Popel et al. (2020); Junczys-Dowmunt (2019), as proposed by Germann et al.
(2020). The Czech$\leftrightarrow$English model is described by Germann et al.
(2020) and the English$\rightarrow$German model by Bogoychev et al. (2020).
Our student models follow the teacher’s architecture with half the size of the
embedding vector (256 instead of 512) and half of the attention heads (4
instead of 8). Student models were trained with an early stopping of 20
evaluations on validation data with evaluation performed every 10k sentences.
Vocabularies were not shared from the teacher because they did not affect the
results, and not using them makes fewer assumptions regarding the level of
access to the teacher model. Marian NMT Junczys-Dowmunt et al. (2018) is used
for teacher decoding and student training.
Table 1 shows the teacher performance measured on WMT20 News and the test
subset of Europarl. Czech models performed better on the Europarl than on the
News task, while for the German model the trend was the opposite. This may be
caused by the fact that the models were distilled from a system that had
Europarl as part of the training data, CzEng 2.0 Kocmi et al. (2020).
Dataset: | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
BLEU: | | |
WMT20 News | $28.2$ | $35.8$ | $42.7$
Europarl | $46.1$ | $38.2$ | $32.1$
ChrF: | | |
WMT20 News | $0.57$ | $0.55$ | $0.66$
Europarl | $0.69$ | $0.64$ | $0.61$
TER: | | |
WMT20 News | $0.57$ | $0.71$ | $0.51$
Europarl | $0.41$ | $0.50$ | $0.61$
Table 1: Teacher models BLEU, ChrF and TER scores on WMT20 News Task dataset
and Europarl domain.
### 2.3 Sampling
Concerning the sampling metrics (always between the considered hypothesis and
the reference), we make use of BLEU, ChrF Popović (2015), TER (negative), the
difference (negative of absolute value) in subword unit counts by
SentencePiece Kudo and Richardson (2018) (SP) and decoder probability divided
by the number of output tokens (score). TER and SP are negative in Section 3
so that higher is always better. The motivation for SP is to capture the
difference in length of the hypotheses with respect to the reference. This is
a very naive metric, but we can use it to see the performance and the
behaviour of all the other metrics. Although BLEU is a document-level metric,
it can also be used to determine sentence similarity. Standard machine
translation metrics444 Sacrebleu metrics version strings:
BLEU+case.mixed+numrefs.1+smooth.exp+tok.13a+v1.4.14
ChrF2+numchars.6+space.false+v1.4.14
TER+tok.tercom-nonorm-punct-noasian-uncased+v1.4.14 are computed using
Sacrebleu Post (2018). Different sampling methods are used even though the
goal is to maximize the BLEU scores of the student models. There is no reason
to assume that sampling only based on BLEU will lead to the best results.
The number of training sentences differs for every method. We define the
following notation.
* •
$T$ \- top; $T_{\text{metric}}^{n}$ takes $n$ top translation hypotheses
according to metric; equal to $S_{\text{metric}}^{1,1,\ldots 1(n)}$. The
student model may benefit from seeing e.g. the second best hypothesis, even
though it’s not the best available. This results in $n$ times the number of
original sentences which are all different.
* •
$S$ \- skewed; $S_{\text{metric}}^{k_{1},k_{2},\ldots k_{n}}$ takes
$k_{1}\times$ the top translation hypotheses according to metric,
$k_{2}\times$ the second top translation, etc. As opposed to
$T_{\text{metric}}^{n}$, this method tries to preserve the information of the
ordering by setting $k_{1}\geq k_{2}\geq\ldots k_{n}$. This results in $(\sum
k_{i})$ times the number of original sentences but only $n$ times of which are
different sentences.
* •
$Dedup[X]$ deduplicates sentence pairs of $X$. It is used after joining the
results of other methods. This method is useful for emulating the or
operation: $Dedup[A+B]$ then means “all sentences in either $A$ or $B$.” The
output size is strictly dependent on their overlap.
* •
$G$ \- greater than; $G_{\text{metric}}^{m}$ takes all sentence translations
with metric at least $m$. This results in sentences that are close to the
reference according to the metric. The number of output sentences highly
dependent on the threshold and is discussed in the corresponding section.
Sampling methods can be combined: $T_{\text{bleu}}^{2}+G_{\text{score}}^{-10}$
joins the top 2 sentences measured by BLEU and adds them to the hypotheses
with decoder score of at least $-10$. Duplicates are intentionally not
removed; thus, hypotheses in both sampling methods are upsampled.
## 3 Results
#### Baseline.
Table 2 shows results for baseline sampling methods. Original corresponds to
training only to the provided parallel corpus (references).
$T^{1}_{\text{score}}$ takes only the highest-scoring hypothesis from the
decoder, which is related to the scenario where the reference is not
available, and the decoder score is the best measure for hypothesis
quality.555MT quality estimation tools could be used to approximate the
sentence translation quality or language models to use sentence fluency in
lieu of translation quality. The sampling method $T^{12}_{-}$ takes all
available hypotheses (metric does not matter).
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
Original | $41.6$ | $31.8$ | $25.1$
$T^{1}_{\text{score}}$ | $40.0$ | $31.2$ | $28.5$
$T^{12}_{-}$ | $41.1$ | $31.6$ | $28.4$
Table 2: BLEU scores for students trained on baseline datasets
Training on the original data leads to better results than training on the
best scoring hypotheses. Training on all hypotheses results in slightly lower
BLEU performance. This may be caused by the small amount of training data
available in which case taking all hypotheses just improves the vocabulary and
language modelling capacity.
#### Best hypotheses.
The results of datasets created by taking either the best one or the four best
hypotheses for every source sentence is shown in Table 3. In the case of
multiple hypotheses having the same score, the one with the highest decoder
score is chosen. The top one and top four hypotheses were chosen to show that
the optimum is neither the top one nor the top twelve (all) hypotheses.
On average, the hypothesis overlap666Overlap computed as
$\text{average}_{m1\neq m2}{|T^{1}_{m1}\cap T^{1}_{m2}|}/{n}$ and
$\text{average}_{m1\neq m2}{|T^{4}_{m1}\cap T^{4}_{m2}|}/{(4n)}$. Original
data size is $n$. in sampling between metrics is $29\%$ for $T^{1}$ and $51\%$
for $T^{4}$. This is expected and shows that when more top hypotheses are
taken into the new dataset, the individual metrics tend to matter less.
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{BLEU}}$ | $42.6$ | $34.4$ | $29.5$
$T^{1}_{\text{ChrF}}$ | $43.8$ | $33.9$ | $30.5$
$T^{1}_{\text{TER}}$ | $43.0$ | $36.1$ | $28.5$
$T^{1}_{\text{SP}}$ | $39.9$ | $29.5$ | $28.2$
$T^{4}_{\text{BLEU}}$ | $44.0$ | $33.3$ | $29.3$
$T^{4}_{\text{ChrF}}$ | $44.3$ | $34.9$ | $29.6$
$T^{4}_{\text{TER}}$ | $44.2$ | $32.0$ | $28.8$
$T^{4}_{\text{SP}}$ | $41.8$ | $32.3$ | $27.9$
$T^{4}_{\text{score}}$ | $44.2$ | $32.0$ | $28.8$
Table 3: BLEU scores for students trained on best-one and best-four hypotheses
datasets
Taking only the top-scoring hypothesis of reference-based metrics, $T^{1}$
showed better results than the baseline (training on the original data, taking
the highest decoder scoring hypothesis or taking all hypotheses). In all cases
the $T^{4}$ outperformed $T^{1}$. The main gains were on CS$\rightarrow$EN and
EN$\rightarrow$CS. Although the results on EN$\rightarrow$DE are only slightly
better than the baseline, they are systematic across all metrics except for
SP. The effect of choosing the metric for the top four hypotheses seems
marginal, even compared to sampling based on the decoder score. The only
exception is the SP difference, which leads to lower results.
#### Thresholding.
Determining a single threshold for all datasets leads to a vastly different
number of hypotheses being selected (the use of $G^{65}_{\text{BLEU}}$ results
in $1.3\times$ the original dataset for CS$\rightarrow$EN, but $0.6$ for
EN$\rightarrow$DE). Therefore, we establish different metric thresholds for
every dataset so that the new datasets are $1\times$ to $1.5\times$ the
original size for consistent results across language pairs.
Some of the source sentences were easier to translate, and more of their
hypotheses were put into the new dataset. Others had no hypothesis above a
given threshold and were not included in the new data at all. On average only
$25\%$ of original sentences were preserved for BLEU, ChrF, TER and SP. For
the decoder score metric, it is $46\%$. The high loss of source sentences is
expected since most of the hypotheses share large portions of the target
sentence and only differ in a few words. All of them will then behave
similarly with respect to the metric.
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$G_{\text{BLEU}}$ | $39.0_{\,65}$ | $30.2_{\,60}$ | $27.2_{\,55}$
$G_{\text{ChrF}}$ | $37.4_{\,0.82}$ | $29.2_{\,0.81}$ | $26.5_{\,0.80}$
$G_{\text{TER}}$ | $37.8_{\,-0.2}$ | $30.2_{\,-0.25}$ | $25.2_{\,-0.24}$
$G_{\text{SP}}$ | $32.5_{\,\text{--}1}$ | $19.6_{\,\text{--}2}$ | $23.0_{\,\text{--}1}$
$G_{\text{score}}$ | $39.0_{\,\text{--}0.08}$ | $32.0_{\,\text{--}0.09}$ | $27.6_{\,\text{--}0.11}$
Table 4: BLEU scores for students trained on datasets made of hypotheses above
threshold of different metrics. Metrics thresholds are in subscript.
The highest performance is achieved using $G_{\text{score}}$ which can be
explained by how much of the original sentences were preserved.
$G_{\text{score}}$ shows that it is possible to achieve a performance
comparable to $T^{1}_{\text{score}}$ with less than half of the source
sentences by only taking all hypotheses with a decoder score above a
threshold. $G_{\text{BLEU}}$ gets worse results (on average $-1.1$ BLEU), but
with only $27\%$ source sentences preserved.
Better performance could be achieved by lowering the threshold to allow more
source sentences and by intersecting the result with some of the other
sampling methods, thus eliminating only the very low-quality sentence pairs.
This is the approach (done with 5 hypotheses) done by Freitag et al. (2017):
$T^{1}_{score}\cap G^{-0.8}_{TER}$.
#### Upsampling.
In the first upsampling case, $S^{4,3,2,1}$, the best hypothesis is present
four times, the second-best three times, the third-best two times and the
fourth-best once. The reason for upsampling better hypotheses is that we want
to force the optimizer to make bigger steps for sentence pairs that are of
high quality, but at the same time, we want to present other hypotheses to
enlarge the vocabulary and improve the student’s language model. The most
straightforward approach is to put multiple copies of the high-quality example
into the dataset. We also experiment with $S^{2,2,1,1}$, because the
upsampling intensity for every hypothesis rank is an independent variable as
well. Both of these schemes are relatively conservative so that they can be
compared to each other and to $T^{4}$. Results for upsampling within a single
metric are shown in Table 5.
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$S^{4,3,2,1}_{\text{BLEU}}$ | $\boldsymbol{45.2}$ | $\boldsymbol{37.1}$ | $29.7$
$S^{4,3,2,1}_{\text{ChrF}}$ | $42.9$ | $36.6$ | $\boldsymbol{30.1}$
$S^{4,3,2,1}_{\text{TER}}$ | $44.4$ | $36.9$ | $29.8$
$S^{4,3,2,1}_{\text{SP}}$ | $41.8$ | $30.7$ | $28.5$
$S^{4,3,2,1}_{\text{score}}$ | $41.4$ | $33.7$ | $27.9$
$S^{2,2,1,1}_{\text{BLEU}}$ | $44.3$ | $36.5$ | $29.6$
$S^{2,2,1,1}_{\text{ChrF}}$ | $45.2$ | $36.1$ | $29.8$
$S^{2,2,1,1}_{\text{TER}}$ | $43.5$ | $33.4$ | $29.6$
$S^{2,2,1,1}_{\text{SP}}$ | $41.8$ | $33.3$ | $28.9$
$S^{2,2,1,1}_{\text{score}}$ | $43.5$ | $33.4$ | $29.6$
Table 5: BLEU scores for students trained on datasets made by upsampling top
hypotheses within a single metric using $S^{4,3,2,1}$ and $S^{2,2,1,1}$
Both versions of upsampling ($S^{4,3,2,1}$ and $S^{2,2,1,1}$) outperformed all
of the previous results. There seems to be no systematic difference between
$S^{4,3,2,1}$ and $S^{2,2,1,1}$. With the exception of SP and decoder score,
the metrics are comparable. A direct comparison can be made to
$T^{4}=S^{1,1,1,1}$ because both $T^{4}$ and the upsampling methods contain
all source sentences and even the same hypotheses. The only difference is that
in the upsampling case, the better hypothesis is upsampled. In this case
$S^{2,2,1,1}$ had higher results over $T^{4}$ with $p<0.005$ by Student’s
t-test.777Average was subtracted from the three directions so that $T^{4}$ and
$S^{2,2,1,1}$ could be treated as only two distributions.
#### Combination.
For the combination scenarios, the newly sampled datasets are joined together.
This is shown in Table 6. In the first four cases, the sampling methods were
joined with the original data. A baseline to this is
$T^{1}_{\text{score}}+\text{Original}$, which is commonly used for
distillation.
Deduplicating the top four hypotheses according to BLEU or decoder score and
adding them to the original data did not improve over the baseline. Combining
the upsampling according to the decoder score with the original data also did
not help. Replacing the decoder score with BLEU resulted in a significant
improvement. The original data is upsampled so that the ratio of synthetic to
original data is 4:1 in the first case and 2:1 in the second one.
For the rest of the cases, the methods are combined without the original data.
Baselines are shown in Table 2. The combination of the top four hypotheses
($T^{4}_{\text{BLEU}}$ or $T^{4}_{\text{score}}$) with all of the hypotheses,
$T^{12}_{-}$, improved over the baseline, including $T^{12}_{-}$, but
performed poorly with respect to the other methods. Taking hypotheses that are
in the top four according to either BLEU or decoder score leads to the best
results in this section. The top one hypothesis, according to BLEU, is
upsampled at least two and at most four times. This seems to work best for
EN$\rightarrow$DE where the training data were three times larger.
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{score}}+\text{Original}$ | $44.4$ | $36.4$ | $28.3$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+\text{Original}$ | $43.7$ | $35.3$ | $29.1$
$S^{4,3,2,1}_{\text{score}}+2\times\text{Original}$ | $43.9$ | $36.1$ | $28.3$
$S^{4,3,2,1}_{\text{BLEU}}+2\times\text{Original}$ | $\boldsymbol{45.5}$ | $\boldsymbol{37.3}$ | $28.8$
$S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$ | $\boldsymbol{45.5}$ $\star$ | $\boldsymbol{37.4}$ $\star$ | $28.9$
$T^{4}_{\text{score}}+T^{12}_{-}$ | $41.6$ | $33.2$ | $28.3$
$T^{4}_{\text{BLEU}}+T^{12}_{-}$ | $42.6$ | $33.9$ | $28.7$
$T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}$ | $43.3$ | $33.2$ | $28.9$
$Dedup[\sum T^{2}_{\text{metric}}]$ | $43.6$ | $34.7$ | $29.1$
$Dedup[\sum T^{2}_{\text{metrics}}]+T^{12}_{-}$ | $40.8$ | $32.0$ | $27.2$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+T^{1}_{\text{BLEU}}+T^{1}_{\text{score}}$ | $43.5$ | $34.7$ | $29.2$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+Dedup[T^{1}_{\text{BLEU}}+T^{1}_{\text{score}}]$ | $42.6$ | $34.9$ | $\boldsymbol{29.6}$ $\star$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]$ | $43.5$ | $35.0$ | $\boldsymbol{29.3}$
Table 6: BLEU scores for students trained on datasets made of combination of
sampling methods. $\sum_{\text{metric}}$ sums over all used metrics (BLEU,
ChrF, TER, SP, score).
#### Bigger student model.
To demonstrate the data sampling method behaviour on slightly larger models,
the common distillation baseline ($T^{1}_{\text{score}}+\text{Original}$) and
the best performing proposed sampling method
($S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$) were used to train a
student of the same size as the used teacher (embedding vector dimension 512
and 8 attention heads). The results are shown in Table 7. They are
systematically higher than for the smaller models, and the difference between
the baseline and the best sampling is preserved.
Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{score}}+\text{Original}$ | $44.7$ | $36.2$ | $28.3$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+$ Original | $44.3$ | $36.2$ | $28.5$
$S^{4,3,2,1}_{\text{BLEU}}+2\times\text{Original}$ | $\boldsymbol{46.9}$ | $\boldsymbol{38.5}$ | $\boldsymbol{28.8}$
$S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$ | $\boldsymbol{47.4}$ $\star$ | $\boldsymbol{38.9}$ $\star$ | $\boldsymbol{28.9}$
Table 7: BLEU scores for students trained on datasets made of combination of
top hypothesis and original data. Trained with parameters equal to the
teacher’s: embedding vector dimension 512 and 8 attention heads.
## 4 Summary
Although widely used, taking only the highest-scoring sentence (with respect
to the decoder score or any reference-based metrics, such as BLEU) does not
lead to the best results. In the context of the proposed experiments, these
are achieved by a combination of top hypotheses and the original data, such as
$S_{\text{BLEU}}^{4,3,2,1}+4\times\text{Original}$ (upsampling according to
BLEU and joining with the original data duplicated four times). Here, an
improvement of an average $+2$ BLEU points against $T^{1}_{\text{score}}$ \+
Original was achieved.
The choice of the sampling metric does not significantly influence the
results, especially in cases where more than the top one hypothesis is
sampled. Because of this, in most scenarios the decoder score can be used
instead, reducing the need for translation references.
#### Future work.
We worked with only two upsampling schemes: $S^{4,3,2,1}$ and $S^{2,2,1,1}$.
However, the two vectors are arbitrary and more of the vast vector space
should be explored, especially with more than the top four hypotheses
considered or more skewed towards the best hypothesis. More sophisticated
methods based on the value of the metric instead of just the ordering could
also be tried out.
The effects of large models (both teacher and student) and data access should
be explored to verify the transferability of the results of the current setup.
Specifically, the teacher model should not be a distilled model itself. The
robustness of the training should also be established.
Even though this paper focused solely on MT, the importance sampling methods
could also be applied and verified on other NLP tasks, possibly even on more
general machine learning problems.
## Acknowledgements
Sincere thanks to Philipp Zimmermann, Martin Popel and both PBML reviewers for
their helpful suggestions and comments.
This study was supported by the Czech Science Foundation (grant n. 19-26934X,
NEUREM3).
The work described also used services provided by the LINDAT/CLARIAH-CZ
Research Infrastructure (lindat.cz), supported by the Ministry of Education,
Youth and Sports of the Czech Republic (Project No. LM2018101).
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|
# Compact approach to the positivity of Brown-York mass and rigidity of
manifolds with mean-convex boundaries in flat and spherical contexts
Sebastián Montiel Departamento de Geometría y Topología
Universidad de Granada
18071 Granada
Spain<EMAIL_ADDRESS>
(Date: September, 2022)
###### Abstract.
In this article we develope a spinorial proof of the Shi-Tam theorem for the
positivity of the Brown-York mass without necessity of building non smooth
infinite asymptotically flat hypersurfaces in the Euclidean space and use the
positivity of the ADM mass proved by Schoen-Yau and Witten. This same compact
approach provides an optimal lower bound [HMZ] for the first non null
eigenvalue of the Dirac operator of a mean convex boundary for a compact spin
manifold with non negative scalar curvature, an a rigidity result for mean-
convex bodies in flat spaces. The same machinery provides analogous, but new,
results of this type, as far as we know, in spherical contexts, including a
version of Min-Oo’s conjecture.
###### Key words and phrases:
Dirac Operator, Spectrum, APS boundary condition, Brown-York mass, Mean-
convexity, Min-Oo conjecture
###### 1991 Mathematics Subject Classification:
Differential Geometry, Global Analysis, 53C27, 53C40, 53C80, 58G25
Declaration: The author declares to be free of conflicts of interests
competing interests of any type retive to this article.
Research partially supported by a Junta de Andalucia FQM-325
.
## 1\. Introduction
In a beautiful work by Shi and Tam [ST, Theorem 4.1], the positivity of the
ADM mass theorem, proved by Schoen and Yau [SY] and [Wi], independently, was
used in a new way to get nice results on the boundary behaviour of a compact
Riemannian manifold with non-negative scalar curvature. They proved that if
$\Omega$ is a three-dimensional compact Riemannian manifold with non-negative
scalar curvature and strictly convex boundary $\Sigma$, then
$\int_{\Sigma}H_{0}-\int_{\Sigma}H,$
that is, the so called Brown-York mass enclosed in $\Omega$, is greater than
or equal to zero, where $H$ is the inner mean curvature of $\Sigma$ in
$\Omega$ and $H_{0}$ is the Euclidean inner mean curvature of $\Sigma$ in
${\mathbb{R}}^{3}$ corresponding to the (unique up to rigid motions) Weyl
embedding of $\Sigma$ into ${\mathbb{R}}^{3}$ obtained by Pogorelov [Po] and
Nirenberg [Ni] independently. Moreover, the equality holds for some boundary
component $\Sigma$. The proof uses two fundamental facts. First, following an
idea by Bartnik [Bar], the construction of a suitable infinite asymptotically
flat extension of $\Omega$ with non negative scalar curvature by deforming the
exterior of $\Sigma$ in ${\mathbb{R}}^{3}$ in order to have the same mean
curvature as $\Sigma$ in $\Omega$ with the intention to glue it to manifold
$\Omega$ along its boundary $\Sigma$. This construction is equivalent to solve
a non linear parabolic equation in a $C^{1}$ context, because this is the
degree of differentiability after the glueing. The second point is that the
difference of integrals
$\int_{\Sigma_{r}}H_{0}-\int_{\Sigma_{r}}H_{r},$
where $\Sigma_{r}$ is an expansion to infinity of the original boundary
$\Sigma$ into ${\mathbb{R}}^{3}$, where $H_{0}$ is the mean curvature with
respect to the Euclidean metric and $H_{r}$ is the mean curvature with respect
to the Bartnik metric goes in a non decreasing way to the ADM mass of the
built asymptotically flat manifold and, so, one can use the non negativity of
its mass. This approach was successfully used to prove the positivity of other
quasi local masses proposed by Liu and Yau [LiY1, LiY2] and Wang and Yau
[WaY]. Along these comments one can see that positivity of quasi local masses
and rigidity of compact manifolds with non empty boundary are different,
although similar, aspects of a quasi same question.
It can be seen, since one starts to study elementary differential geometry,
for example, when one begins to prove the Cohn-Vossen rigidity of ovaloids in
${\mathbb{R}}^{3}$, that there is a close connexion between rigidity of these
manifolds and the fact that the integral its mean curvatures coincide [MR]
(pages 218-219). In this paper, we will obtain a compact approach of the
positivity of the Brown-York mass for the compact spin manifolds of arbitrary
dimension and its corresponding rigidity theorem for mean-convex bodies in the
Euclidean spaces. These two type of results are relied with an estimate for
the lower eigenvalues of the Dirac operator of these bodies and this, in turn,
to theorem type Min-Oo. In fact, in the flat case the corresponding Min-Oo
conjecture was obtained by Miao [Mi1] (see Remark 1), although it was an easy
consequence of the estimate obtained in [HMZ] for this eigenvalue of the Dirac
operator.
Also, in this article, for the sake of completeness, we explain the relation
between the spin structures of the Euclidean space or the sphere and their
hypersurfaces and we will see that their Dirac operator relate basically
through the mean curvature of these hypersurfaces and the scalar curvature of
the ambient spaces. This fact will allow us, for a $(n+1)$-dimensional compact
spin Riemannian manifold $\Omega$ with non negative scalar curvature and mean-
convex boundary $\Sigma$, to unify a compact proof to obtain a lower estimate
for the spectrum of the Dirac of $\Sigma$, the positivity of the Brown-York
mass for mean-convex non necessarily convex domains, the aforementioned
resolution of the flat version of Min-Oo’s conjecture and the rigidity of
these mean-convex bodies in the Euclidean spaces.
In the final section of the article, we will show that this same scheme, with
the important difference of establish the exact value $R=n(n+1)$ of the scalar
curvature of the bulk manifold, works to obtain exactly the same sequence of
results. In this case, all of them, as far as we know, are new.
## 2\. Riemannian spin manifolds and hypersurfaces
Consider an $(n+1)$-dimensional spin Riemannian manifold $\Omega$ with non-
empty boundary $\partial\Omega=\Sigma$ and denote by $\langle\;,\;\rangle$ its
scalar product and by $\nabla$ its corresponding Levi-Civita connection on the
tangent bundle $T\Omega$. We fix a spin structure (and so a corresponding
orientation) on the manifold $\Omega$ and denote by $\hbox{\bb S}\Omega$ the
associated spinor bundle, which is a complex vector bundle of rank
$2^{\left[\frac{n+1}{2}\right]}$. Then let
$\gamma:{\mathbb{C}}\ell(\Omega)\longrightarrow{\rm
End}_{{\mathbb{C}}}(\hbox{\bb S}\Omega)$
be the Clifford multiplication, which provides a fibre preserving irreducible
representation of the Clifford algebras constructed over the tangent spaces of
$\Omega$. When the dimension $n+1$ is even, we have the standard chirality
decomposition
(1) $\hbox{\bb S}\Omega=\hbox{\bb S}\Omega^{+}\oplus\hbox{\bb S}\Omega^{-},$
where the two direct summands are respectively the $\pm 1$-eigenspaces of the
endomorphism $\gamma(\omega_{n+1})$, with
$\omega_{n+1}=i^{\left[\frac{n+2}{2}\right]}e_{1}\cdots e_{n+1}$, the complex
volume form. It is well-known (see [LM]) that there are, on the complex spinor
bundle $\hbox{\bb S}\Omega$, a natural Hermitian metric $(\;,\;)$ and a
spinorial Levi-Civita connection, denoted also by $\nabla$, which is
compatible with both $(\;,\;)$ and $\gamma$ in the following sense:
(2) $\displaystyle X(\psi,\phi)=(\nabla_{X}\psi,\phi)+(\psi,\nabla_{X}\phi)$
(3)
$\displaystyle\nabla_{X}\left(\gamma(Y)\psi\right)=\gamma(\nabla_{X}Y)\psi+\gamma(Y)\nabla_{X}\psi$
for any tangent vector fields $X,Y\in\Gamma(T\Omega)$ and any spinor fields
$\psi,\phi\in\Gamma(\hbox{\bb S}\Omega)$ on $M$. Moreover, with respect to
this Hermitian product on $\hbox{\bb S}\Omega$, Clifford multiplication by
vector fields is skew-Hermitian or equivalently
(4) $(\gamma(X)\psi,\gamma(X)\phi)=|X|^{2}(\psi,\phi).$
Since the complex volume form $\omega_{n+1}$ is parallel with respect to the
spinorial Levi-Civita connection, when $n+1=\dim\Omega$ is even, the chirality
decomposition (1) is preserved by $\nabla$. Moreover, from (4), one sees that
it is an orthogonal decomposition.
In this setting, the (fundamental) Dirac operator $D$ on the manifold $M$ is
the first order elliptic differential operator acting on spinor fields given
locally by
$D=\sum_{i=1}^{n+1}\gamma(e_{i})\nabla_{e_{i}},$
where $\\{e_{1},\dots,e_{n+1}\\}$ is a local orthonormal frame in $T\Omega$.
When $n+1=\dim\Omega$ is even, $D$ interchanges the chirality subbundles
$\hbox{\bb S}\Omega^{\pm}$.
The boundary hypersurface $\Sigma$ is also an oriented Riemannian manifold
with the induced orientation and metric. If $\nabla^{\Sigma}$ stands for the
Levi-Civita connection of this induced metric we have the Gauss and Weingarten
equations
$\nabla_{X}Y=\nabla^{\Sigma}_{X}Y+\langle AX,Y\rangle
N,\qquad\nabla_{X}N=-AX,$
for any vector fields $X,Y$ tangent to $\Sigma$, where $A$ is the shape
operator or Weingarten endomorphism of the hypersurface $\Sigma$ corresponding
to the unit normal field $N$ compatible with the given orientation. As the
normal bundle of the boundary hypersurface is trivial, the Riemannian manifold
$\Sigma$ is also a spin manifold and so we will have the corresponding spinor
bundle $\hbox{\bb S}\Sigma$, the Clifford multiplication $\gamma^{\Sigma}$,
the spinorial Levi-Civita connection $\nabla^{\Sigma}$ and the intrinsic Dirac
operator $D^{\Sigma}$. It is not difficult to show (see [Bä2, BFGK, Bur, Tr,
Mo]) that the restricted Hermitian bundle
${\bf S}:=\hbox{\bb S}\Omega_{|\Sigma}$
can be identified with the intrinsic Hermitian spinor bundle $\hbox{\bb
S}\Sigma$, provided that $n+1=\dim\Omega$ is odd. Instead, if $n+1=\dim\Omega$
is even, the restricted bundle ${\bf S}$ could be identified with the sum
$\hbox{\bb S}\Sigma\oplus\hbox{\bb S}\Sigma$. With such identifications, for
any spinor field $\psi\in\Gamma({\bf S})$ on the boundary hypersurface
$\Sigma$ and any vector field $X\in\Gamma(T\Sigma)$, define on the restricted
bundle ${\bf S}$, the Clifford multiplication $\gamma^{{\bf S}}$ and the
connection $\nabla^{{\bf S}}$ by
(5) $\displaystyle\gamma^{{\bf S}}(X)\psi=\gamma(X)\gamma(N)\psi$ (6)
$\displaystyle\nabla^{{\bf S}}_{X}\psi=\nabla_{X}\psi-\frac{1}{2}\gamma^{{\bf
S}}(AX)\psi=\nabla_{X}\psi-\frac{1}{2}\gamma(AX)\gamma(N)\psi\,.$
Then it easy to see that $\gamma^{{\bf S}}$ and $\nabla^{{\bf S}}$ correspond
respectively to $\gamma^{\Sigma}$ and $\nabla^{\Sigma}$, for $n+1$ odd, and to
$\gamma^{\Sigma}\oplus-\gamma^{\Sigma}$ and
$\nabla^{\Sigma}\oplus\nabla^{\Sigma}$, for $n+1$ even. Then, $\gamma^{{\bf
S}}$ and $\nabla^{{\bf S}}$ satisfy the same compatibilty relations (2), (3)
and (4) and together with the following additional identity
$\nabla^{{\bf S}}_{X}\left(\gamma(N)\psi\right)=\gamma(N)\nabla^{{\bf
S}}_{X}\psi.$
As a consequence, the hypersurface Dirac operator ${\bf D}$ acts on smooth
sections $\psi\in\Gamma({\bf S})$ as
${\bf D}\psi:=\sum_{j=1}^{n}\gamma^{{\bf S}}(u_{j})\nabla^{{\bf
S}}_{u_{j}}\psi=\frac{n}{2}H\psi-\gamma(N)\sum_{j=1}^{n}\gamma(u_{j})\nabla_{u_{j}}\psi,$
where $\\{u_{1},\dots,u_{n}\\}$ is a local orthonormal frame tangent to the
boundary $\Sigma$ and $H=(1/(n)){\rm trace}\,A$ is its mean curvature
function, coincides with the intrinsic Dirac operator $D^{\Sigma}$ on the
boundary, for $n+1$ odd, and with the pair $D^{\Sigma}\oplus-D^{\Sigma}$, for
$n+1$ even. In the particular case where the field $\psi\in\Gamma({\bf S})$ is
the restriction of a spinor field $\psi\in\Gamma(\Sigma)$ on $\Omega$, this
means that
(7) ${\bf D}\psi=\frac{n}{2}H\psi-\gamma(N)D\psi-\nabla_{N}\psi.$
Note that we always have the anticommutativity property
(8) ${\bf D}\gamma(N)=-\gamma(N){\bf D}$
and so, when $\Sigma$ is compact, the spectrum of ${\bf D}$ is symmetric with
respect to zero and coincides with the spectrum of $D^{\Sigma}$, for $n+1$
odd, and with ${\rm Spec}(D^{\Sigma})\cup-{\rm Spec}(D^{\Sigma})$, for $n+1$
even [see HMR]. In fact, we and other authors also have remarked in several
papers that the spectrum Spec($\bf D$) is a ${\mathbb{Z}}$-symmetric sequence
$-\infty\swarrow\cdots\leq-\lambda_{k}\leq\cdots\leq-\lambda_{1}<\lambda_{0}=0<\lambda_{1}\leq\cdots\leq\cdots\leq\lambda_{k}\leq\cdots\nearrow+\infty$
(each eigenvalue repeated according to its corresponding multiplicity). This
is because $D$ is an elliptic operator of order one which is self-adjoint due
to the compacity of $\Sigma$ and the fact that $\gamma(N)$ maps the eigenvalue
$\lambda_{k}$ into $-\lambda_{k}$, being $\gamma$ the Clifford multiplication
and $N$ the inner unit normal of $\Sigma$ in $\Omega$. Let’s choose an
$L^{2}(\mathbb{S}\Sigma)$-orthonormal basis $\\{\psi_{k}\\}_{k\in{Z}}$ of the
Hilbert space $L^{2}(\mathbb{S}\Sigma)$ constituted by eigenspinors of ${\bf
D}$, that is, ${\bf D}\psi_{k}=\lambda_{k}\psi_{k}$ (and so $\psi_{k}\in
C^{\infty}(\Sigma))$, for all $k\in\mathbb{Z}$. We have to remark that the
presence of the eigenvalue $\lambda_{0}=0$ (repeated with its corresponding
multiplicity) is not compulsory. It is a standard fact (Perceval equality)
that, if $\phi$ is an $L^{2}$ spinor field on $\Sigma$, the series
$\sum^{+\infty}_{-\infty}\phi_{k}=\lim_{k\geq
0,k\rightarrow\infty}\sum^{k}_{-k}\phi_{k}=\phi,$
converges in the strong $L^{2}(\mathbb{S}\Sigma)$-topology, where each
$\phi_{k}$ is the projection of $\phi$ onto the eigenspace corresponding to
the eigenvalue $\lambda_{k}$. As a consequence of Hölder inequalities and
well-known facts, we have $L^{1}(\mathbb{S}\Sigma)$-convergence as well and
moreover, as an easy consequence of the completeness of
$L^{2}(\mathbb{S}\Sigma)$ or $L^{1}(\mathbb{S}\Sigma)$, pointwise convergence
almost everywhere (and so everywhere if $\phi$ is continous) for a suitable
subseries of $\sum^{+\infty}_{-\infty}\phi_{k}$.
## 3\. A spinorial Reilly inequality
A basic tool to relate the eigenvalues of the Dirac operator and the geometry
of the manifold $\Omega$ and those of its boundary $\Sigma$ will be, as in the
closed case (see [Fr1]), the integral version of the Schrödinger-Lichnerowicz
formula
(9) $D^{2}=\nabla^{*}\nabla+\frac{1}{4}R,$
where $R$ is the scalar curvature of $\Omega$. In fact, given a spinor field
$\psi$ on $\Omega$, taking into account the formula above, if we compute the
divergence of the one-form $\alpha$ defined by
$\alpha(X)=\left<\gamma(X)D\psi+\nabla_{X}\psi,\psi\right>,\qquad\forall X\in
T\Omega$
and integrate, one gets
$-\int_{\Sigma}\left<\gamma(N)D\psi+\nabla_{N}\psi,\psi\right>=\int_{\Omega}\left(|\nabla\psi|^{2}-|D\psi|^{2}+\frac{1}{4}R|\psi|^{2}\right),$
which by (7), could be written as
$\int_{\Sigma}\left(({\bf
D}\psi,\psi)-\frac{n}{2}H|\psi|^{2}\right)=\int_{\Omega}\left(|\nabla\psi|^{2}-|D\psi|^{2}+\frac{1}{4}R|\psi|^{2}\right).$
Finally, we will use the pointwise spinorial Schwarz inequality
$|D\psi|^{2}\leq(n+1)|\nabla\psi|^{2},\qquad\forall\psi\in\Gamma(\hbox{\bb
S}\Omega),$
where the equality is achieved only by the so-called twistor spinors, that is,
those satisfying the following over-determined first order equation
$\nabla_{X}\psi=-\frac{1}{n+1}\gamma(X)D\psi,\qquad\forall X\in T\Omega.$
Then we get the following integral inequality, called Reilly inequality [see
HMZ, for example], because of its similarity with the corresponding one
obtained in [Re] for the Laplace operator,
(10) $\int_{\Sigma}\left(({\bf
D}\psi,\psi)-\frac{n}{2}H|\psi|^{2}\right)\geq\int_{\Omega}\left(\frac{1}{4}R|\psi|^{2}-\frac{n}{n+1}|D\psi|^{2}\right),$
with equality only for twistor spinors on $\Omega$.
## 4\. The APS boundary condition
It is a well-known fact that the Dirac operator $D$ on a compact spin
Riemannian manifold $\Omega$ with boundary, $D:\Gamma(\hbox{\bb
S}\Omega)\rightarrow\Gamma(\hbox{\bb S}\Omega)$ has an infinite dimensional
kernel and a closed image with finite codimension. People has looked for
conditions $B$ to be imposed on the restrictions to the boundary $\Sigma$ of
the spinor fields on $\Omega$ so that this kernel becomes finite dimensional
and then the boundary problem
(BP) $\left\\{\begin{array}[]{lll}D\psi&=\Phi&\hbox{on $\Omega$}\\\
B\psi_{|\Sigma}&=\chi&\hbox{along $\Sigma$},\end{array}\right.$
for $\Phi\in\Gamma(\hbox{\bb S}\Omega)$ and $\chi\in\Gamma({\bf S})$, is of
Fredholm type. In this case, we will have smooth solutions for any data $\Phi$
and $\chi$ belonging to a certain subspace with finite codimension and these
solutions will be unique up to a finite dimensional kernel.
To our knowledge, the study of boundary conditions suitable for an elliptic
operator $D$ (of any order, although for simplicity, we only consider first
order operators) acting on smooth sections of a Hermitian vector bundle
$F\rightarrow\Omega$ has been first done in the fifties of past century by
Lopatinsky and Shapiro ([Hö, Lo]), but the main tool was discovered by
Calderón in the sixties: the so-called Calderón projector
${\mathcal{P}}_{+}(D):H^{\frac{1}{2}}(F_{|\Sigma})\longrightarrow\\{\psi_{|\Sigma}\,|\,\psi\in
H^{1}(F),D\psi=0\\}.$
This is a pseudo-differential operator of order zero (see [BW, Se]) with
principal symbol ${\mathfrak{p}}_{+}(D):T\Sigma\rightarrow{\rm
End}_{\mathbb{C}}(F)$ depending only on the principal symbol $\sigma_{D}$ of
the operator $D$ and can be calculated as follows
(11) ${\mathfrak{p}}_{+}(D)(X)=-\frac{1}{2\pi
i}\int_{\Gamma}\left[(\sigma_{D}(N))^{-1}\sigma_{D}(X)-\zeta
I\right]^{-1}\,d\zeta,$
for any $p\in\Sigma$ and $X\in T_{p}\Sigma$, where $N$ is the inner unit
normal along the boundary $\Sigma$ and $\Gamma$ is a positively oriented cycle
in the complex plane enclosing the poles of the integrand with negative
imaginary part. Although the Calderón projector is not unique for a given
elliptic operator $D$, its principal symbol is uniquely determined by
$\sigma_{D}$. One of the important features of the Calderón projector is that
its principal symbol detects the ellipticity of a boundary condition, or in
other words, if the corresponding boundary problem (BP) is a well-posed
problem (according to Seeley in [Se]). In fact (cfr. [Se] or [BW, Chap. 18]),
a pseudo-differential operator
$B:L^{2}(F_{|\Sigma})\longrightarrow L^{2}(V),$
where $V\rightarrow\Sigma$ is a complex vector bundle over the boundary, is
called a (global) elliptic boundary condition when its principal symbol
$b:T\Sigma\rightarrow{\rm Hom}_{\mathbb{C}}(F_{|\Sigma},V)$ satisfies that,
for any non-trivial $X\in T_{p}\Sigma$, $p\in\Sigma$, the restriction
$b(X)_{|{\rm image}\,{\mathfrak{p}}_{+}(D)(X)}:{\rm
image}\,{\mathfrak{p}}_{+}(D)(X)\subset F_{p}\longrightarrow V_{p}$
is an isomorphism onto ${\rm image}\,b(X)\subset V_{p}$. Moreover, if ${\rm
rank}\,V=\dim\\-{\rm image}\\-{\mathfrak{p}}_{+}(D)(X)$, we say that $B$ is a
local elliptic boundary condition. When $B$ is a local operator this
definition yields the so-called Lopatinsky-Shapiro conditions for ellipticity
(see for example [Hö]).
When these definitions and the subsequent theorems are applied to the case
where the vector bundle $F$ is the spinor bundle $\hbox{\bb S}\Omega$ and the
elliptic operator $D$ is the Dirac operator $D$ on the spin Riemannian
manifold $\Omega$, we obtain the following well-known facts in the setting of
the general theory of boundary problems for elliptic operators (see for
example [BrL, BW, GLP, Hö, Se]):
It is easy to see that the principal symbol $\sigma_{D}$ of the Dirac operator
$D$ on $\Omega$ is given by
$\sigma_{D}(X)=i\gamma(X),\qquad\forall X\in T\Omega.$
Then by (11), the principal symbol of the Calderón projector of the Dirac
operator is given by
${\mathfrak{p}}_{+}(D)(X)=-\frac{1}{2|X|}(i\gamma(N)\gamma(X)-|X|I)=\frac{1}{2|X|}(i\gamma^{\bf
S}(X)+|X|I),$
for each non-trivial $X\in T\Sigma$ and where $\gamma^{\bf S}$ is identified
in (5) as the intrinsic Clifford product on the boundary. As the endomorphism
$i\gamma(N)\gamma(u)=-i\gamma^{\bf S}(X)$ is self-adjoint and its square is
$|X|^{2}$ times the identity map, then it has exactly two eigenvalues, say
$|X|$ and $-|X|$, whose eigenspaces are of the same dimension
$\frac{1}{2}\dim\hbox{\bb S}\Omega_{p}=2^{[\frac{n+1}{2}]-1}$, since they are
interchanged by $\gamma(N)$. Hence the symbol ${\mathfrak{p}}_{+}(D)(X)$ is,
up to a constant, the orthogonal projection onto the eigenspace corresponding
to the eigenvalue $-|X|$ and so
$\displaystyle{\rm image}\,{\mathfrak{p}}_{+}(D)(X)=\\{\eta\in\hbox{\bb
S}\Omega_{p}\,|\,i\gamma(N)\gamma(X)\eta=-|X|\eta\\},$ $\displaystyle\dim{\rm
image}\,{\mathfrak{p}}_{+}(D)(X)=\frac{1}{2}\dim\hbox{\bb
S}\Omega_{p}=2^{[\frac{n+1}{2}]-1}.$
From these equalities and from the definition of ellipticity for the boundary
condition represented by the pseudo-differential operator $B$, we have that
the first equation in the statement above is equivalent to the injectivity of
the map $b(X)_{|{\rm image}\;{\mathfrak{p}}_{+}(D)(X)}$. The second one
implies that $\dim{\rm image}\;b(X)=\dim{\rm image}\;{\mathfrak{p}}_{+}(D)(X)$
and so, together with the injectivity, this means that $b(X)_{|{\rm
image}\;{\mathfrak{p}}_{+}(D)(X)}$ is surjective. So we have proved that the
two claimed conditions are equivalent to the ellipticity of a given boundary
condition $B$ for the Dirac operator $D$ on $\Omega$. Now, from this
ellipticity, one may deduce that the problem (BP) and the spectral
corresponding one are of Fredholm type and the remaining assertions on
eigenvalues and eigenspaces follow in a standard way (see [BW, Hö]).
At the risk of being too long, we will explain what is the Atiyah, Patodi and
Singer condition in this context. This well-known boundary condition was
introduced in [APS] in order to establish index theorems for compact manifolds
with boundary. Later, this condition has been used to study the positive mass
and the Penrose inequalities (see [He2, Wi]). Such a condition does not allow
to model confined particle fields since, from the physical point of view, its
global nature is interpreted as a causality violation. Although it is a well-
known fact that the APS condition is an elliptic boundary condition, we are
going to sketch the proof in the setting established above, for three reasons:
first, for completeness, second for pointing out that the APS condition for an
achiral Dirac operator covers both cases of odd and even dimension, although
the latter case is not referred to the spectral resolution of the intrinsic
Dirac operator $D^{\Sigma}$ but to the system $D^{\Sigma}\oplus-D^{\Sigma}$
and third, because we will use it with some little modifications which may not
familiar to some readers.
Precisely, this condition can be described as follows. Choose the
aforementioned Hermitian bundle $V$ over the boundary hypersurface $\Sigma$ as
the restricted spinor bundle ${\bf S}$ defined in Section 2, and define, for
each ${a\in\mathbb{Z}}$, as $B^{a}_{{\rm APS}\;}:=B_{\geq a}:L^{2}({\bf
S})\rightarrow L^{2}({\bf S})$ as the orthogonal projection onto the subspace
spanned by the eigenvalues of the self-adjoint intrinsic operator ${\bf D}$
greater or equal to $a$. Atiyah, Patodi and Singer showed in [APS] (see also
[BW, Prop. 14.2]) that $B_{{\rm APS}\;}^{a}$ is a zero order pseudo-
differential operator whose principal symbol $b_{{\rm APS}\;}$ (independent of
$a$) satisfies the following fact: for each $p\in\Sigma$ and $X\in
T_{p}\Sigma-\\{0\\}$, the map $b_{{\rm APS}\;}(X)$ is the orthogonal
projection onto the eigenspace of $\sigma_{{\bf D}}(X)=i\gamma^{\bf S}(X)$
corresponding to the positive eigenvalue $|X|$. That is
(12) $b_{{\rm APS}\;}(X)=\frac{1}{2}\left(i\gamma^{\bf
S}(X)+|X|I\right)=\frac{1}{2}\left(-i\gamma(N)\gamma(X)+|X|I\right),$
and so the principal symbol $b_{{\rm APS}\;}$ of the APS operator coincides,
up to a constant, with the principal symbol ${\mathfrak{p}}_{+}(D)$ of the
Calderón projector of $D$, for all $a\in{\mathbb{Z}}$. From this, it is
immediate to see that the two ellipticity required conditions are satisfied.
Analytically, this APS boundary condition can be formulated as follows. If
$\phi$ is a spinor on $\Sigma$ and $a\in\mathbb{Z}$ is an integer number, we
will denote by $b_{APS}^{a}(\phi)=\phi_{\geq a}$ the orthonormal projection
referred to the diagonalization of ${\bf D}$ given above just after (8)
$b_{APS}^{a}=\phi_{\geq a}=\sum^{+\infty}_{k\geq a}\phi_{k}=\lim_{l\geq
a}\sum^{l}_{m=a}\phi_{m},$
where the above subseries convergences remain to be valid, as we had already
commented for this truncated series, and moreover
$\int_{\Sigma}|\phi_{\geq a}|^{2}\leq\int_{\Sigma}|\phi|^{2}\qquad\forall
a\in{\mathbb{Z}}$
(Bessel inequality). It was long ago known the fact [see APS, Se, BB, HMR,
BäBa1,BäBa2] that this spectral projection $b_{APS}^{a}$ trough ${\bf D}$
provided an elliptic self-adjoint boundary condition on $\Omega$ for $D$ and
for all $a\in{\mathbb{R}}$, if we take the spectral projection on eigenspaces
corresponding to its non negative eigenvalues. (Be careful, in some papers, as
[APS and BäBa1,BäBa2], the adapted Dirac operator $\bf D$ on the boundary is
taken to be $-\bf D$ and, so, it is necessary to reverse positive and
negative). But, we strongly advise to study all those questions about first
order differential operators, existence of solutions and their regularity in
the two recent works by Ballmann and Bär [BäBa1,BäBa2]. They are much more
well clearly written and in a much more modern language. We have procured
attain at the maximum number of readers.
Anyway, we conclude that, for each integer number $a\leq 0$, we can pose the
elliptic boundary problem
(13) $\left\\{\begin{array}[]{ll}D\psi_{a}=\xi\\\ \psi_{\geq a}=\phi_{\geq
a},\end{array}\right.$
where $\psi$ and $\xi$ are spinor fields on the bulk manifold $\Omega$, $\phi$
is a spinor field on the boundary $\Sigma$ and the meaning of $\psi_{\geq a}$
should be already evident. This problem will be a Fredholm type problem.
Analogously, the corresponding eigenvalue problem
(14) $\left\\{\begin{array}[]{ll}D\psi_{a}=\lambda\psi_{a}\\\ \psi_{\geq
a}=0,\end{array}\right.$
[see HMR, BaBä1,BaBä2], is also well posed and has nice existence and
regularity properties.
## 5\. A compact approach spinor proof of the Shi-Tam theorem valid for mean-
convex domains
We assume that the scalar curvature $R$ of the compact spin Riemannian
manifold $\Omega$ is non-negative and consider a solution to the following
homogeneous problem on $\Omega$
(15) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\ \psi_{\geq
a}=0,\end{array}\right.$
for any $a\in{\mathbb{Z}}$. Then, if we put it in the inequality (10) and use
that $R\geq 0$. It only remains the terms on the boundary $\Sigma$ of
$\Omega$. We get
(16) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right),$
and the equality is attained if and only if $\psi_{a}$ is a parallel (twistor
plus harmonic) spinor on $\Omega$. Now, choose $a\leq 0$ in the problem above
and substitute the corresponding solution in the integral inequality above.
Since $\left<{\bf D}\psi_{<a},\psi_{<a}\right>$ and $\left<{\bf D}\psi_{\geq
a},\psi_{\geq a}\right>$ are clearly $L^{2}$-orthogonal, we have
$\int_{\Sigma}\left<{\bf D}\psi_{a},\psi_{a}\right>=\int_{\Sigma}\left<{\bf
D}\psi_{<a},\psi_{<a}\right>=\sum^{-\infty}_{k<a\leq
0}\lambda_{k}\int_{\Sigma}|\psi_{k}|^{2}\leq 0,$
for each integer $a\leq 0$. If, moreover, we add the hypothesis that the inner
mean curvature $H$ is non negative along the boundary $\Sigma$, (16) implies
(17) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right)\leq 0.$
Hence, the first term in the integral above implies $\psi_{<a}=0$ for all
$a\leq 0$. Since, from the boundary condition in (15), $\psi_{\geq a}=0$, for
all $a\in{\mathbb{Z}}$, $a\leq 0$, we conclude
$\psi_{a}=\psi_{<a}+\psi_{\geq a}=0.$
for all $a\in{\mathbb{Z}}$, $a\leq 0$ along $\Sigma$. So, the spinor
$\psi_{a}$ is identically zero along $\Sigma$. As $\psi_{a}$ was harmonic on
$\Omega$, its length must be identically zero because the Hausdorff measure of
$\Sigma$ is greater than two in the bulk manifold as was shown by Bär in
[Bä3]. Then, the unique solution to the homogeneous equation (15) is
$\psi_{a}=0$. Using that this equation is of type Fredholm, we need to study
its cokernel, that is, that the adjoint problem to (15) has also trivial
kernel. Thus, consider the homogeneous problem
(18) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\
\psi_{>a}=0,\end{array}\right.$
where it is easy to imagine what $\psi_{>a}$ denotes. Working as in (15), we
also obtain the same inequality (17) for all $a\leq 0$. But, in this case,
when the equality is attained, we are not sure that $\psi_{\leq a}=0$ for all
$a\leq 0$, because $\psi_{0}$ could be vanish, that is, $\psi$ could contain a
non trivial harmonic component $\psi_{0}$. So, the homogeneous problem (18)
could have non trivial harmonic solutions, unless $H>0$, that is, the boundary
$\Sigma$ be strictly mean convex. In any of these two cases, the unique
solutions to (15) and its adjoint (18) would be the null ones. Then, the non
homogeneous problems would always unique solutions.
###### Proposition 1.
Let $\Omega$ be a compact spin manifold with non negative scalar curvature and
non-empty boundary $\Sigma$ such that its inner mean curvature is non-
negative. Then, if we suppose that $\Sigma$ either does not admit harmonic
spinors or is strictly mean convex ($H>0$) in $\Omega$, both homogeneous
problems (15) and (18) have as unique solution the zero spinor. As a
consequence of this, the inhomogeneous problem
(19) $\left\\{\begin{array}[]{ll}D\psi_{a}=\xi\\\ \psi_{\geq a}=\phi_{\geq
a},\end{array}\right.$
has a unique solution $\psi_{a}\in H^{1}(\Omega)\cap H^{\frac{1}{2}}(\Sigma)$
for prescribed spinor fields $\xi\in L^{2}(\Omega)$ and $\phi\in
L^{2}(\Sigma)$ for each integer number $a\leq 0$.
We will apply this result à la Reilly, thought for solving equations on the
bulk manifold, to obtain results on its boundary.
###### Theorem 2.
Let $\Omega$ be a compact spin Riemannian manifold of dimension $n+1$ with
non-negative scalar curvature $R\geq 0$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is also non-negative (mean-
convex). Suppose also that $\Sigma$ either does not carry harmonic spinors or
$H>0$. Then, for every spinor $\phi\in H^{1}(\Sigma)$, we have
$\int_{\Sigma}|{\bf D}\phi||\phi|\geq\frac{n}{2}\int_{\Sigma}H|\phi|^{2}.$
The equality occurs if and only if $\phi$ is the restriction to $\Sigma$ of a
parallel spinor on $\Omega$ and so
${\bf D}\phi=\frac{n}{2}H\phi.$
Proof. Using the Proposition above, for all integer $a\leq 0$, solve the type
(19) equation
(20) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\ \psi_{\geq a}=\phi_{\geq
a}.\end{array}\right.$
By working in same manner as in the introduction of this section, we can
arrive to the inequality (16) because, until this moment, we did not use the
boundary conditions. On the other hand, by using the orthogonal projections
defined above, we know that
$\int_{\Sigma}\left<{\bf D}\psi_{a},\psi_{a}\right>\leq\int_{\Sigma}\left<{\bf
D}\psi_{\geq a},\psi_{\geq a}\right>,$
because the summand
$\int_{\Sigma}\left<{\bf D}\psi_{<a},{\bf\psi}_{<a}\right>$
only contains non positive eigenvalues of ${\bf D}$. So, since $\psi_{\geq
a}=\phi_{\geq a}$, using the pointwise Cauchy-Schwarz inequality, the unique
solution $\psi_{a}$ of (20) satisfies
(21) $\int_{\Sigma}\left<{\bf
D}\psi_{a},\psi_{a}\right>\leq\int_{\Sigma}\left<{\bf D}\phi_{\geq
a},\phi_{\geq a}\right>\leq\int_{\Sigma}|{\bf D}\phi_{\geq a}||\phi_{\geq
a}|,$
for all $a\in{\mathbb{Z}},a\leq 0$. Analogously, we have the $L^{2}$
decomposition
$0\leq\int_{\Sigma}|\psi_{a}|^{2}=\int_{\Sigma}|\psi_{\geq
a}|^{2}+\int_{\Sigma}|\psi_{<a}|^{2}.$
So, in a similar way as above,
(22) $\int_{\Sigma}|\psi_{a}|^{2}\leq\int_{\Sigma}|\psi_{\geq
a}|^{2}=\int_{\Sigma}|\phi_{\geq a}|^{2},$
for each non positive integer $a$. From the fact that $\psi_{\geq
a}=\phi_{\geq a}$ lies in $H^{\frac{1}{2}}(\Sigma)$, we know that this
sequence with index $a$ converges strongly in the $L^{2}(\Sigma)$ topology
and, so, in the $L^{1}$ topology as well. As the non negative mean curvature
function $H$ is smooth on the compact manifold $\Sigma$, $\sqrt{H}\in
L^{2}(\Sigma)$. Then,
$\lim_{a\rightarrow-\infty}\int_{\Sigma}H|\phi_{\geq
a}|^{2}\,d\Sigma=\int_{\Sigma}H|\phi|^{2}$
strongly in $L^{2}(\Sigma)$. From (16), (20) and this last limit, we have
$0\leq\lim_{a\rightarrow-\infty}\int_{\Sigma}\left(|{\bf D}\phi_{\geq
a}||\phi_{\geq a}|-\frac{n}{2}H|\phi_{\geq
a}|^{2}\right)=\int_{\Sigma}\left(|{\bf
D}\phi||\phi|-\frac{n}{2}H|\phi|^{2}\right).$
Hence, the inequality is true.
Let us see that the solution $\psi_{a}$ to the boundary problem (10) is smooth
on ${\bar{\Omega}}$ for all integer $a\leq 0$. In fact, the classical theory
for the Dirac operator [see BäBa1,BäBa2] assures the smoothness of solutions
in the interior $\Omega$. With respect to the regularity at $\Sigma$, we can
apply Theorems 6.11 and 7.17 and Corollary 7.18 in the first on the papers
cited above (using a bootstrapping process) and obtain smoothness on $\Sigma$
as well. Then
$\psi_{a}\in\cap_{k=1}^{\infty}H^{k}({\bar{\Omega}})=C^{\infty}({\bar{\Omega}})$.
For all non positive integer number $a$, the limit function $\psi$ will also
be smooth on the compact manifold ${\bar{\Omega}}$. This implies that the
limit function
$\psi=\lim_{a\rightarrow-\infty}\psi_{a}$
is also strongly harmonic on $\Omega$. Taking limit in (15) and taking into
account that we also have the equality in the right side of 10, we see that
$\psi$ is also a parallel spinor on the whole of ${\bar{\Omega}}$. And it is
clear that
$\phi=\lim_{a\rightarrow-\infty}\phi_{\geq a}=\psi_{|\Sigma}$
as well. But the relation between the operators $D$ and its restriction ${\bf
D}$ is
(23) ${\bf D}\phi=\frac{n}{2}H\phi-\gamma(N)D\psi-\nabla_{N}\psi.$
Then, [see Bä1,BFGK,Bur] the spinor $\phi$ has to satisfy on the whole
$\Sigma$ the equality
${\bf D}\phi=\frac{n}{2}H\phi.$
The converse is obvious.
###### Remark 1.
Suppose that, in Theorem 2 above, we choose the spinor $\phi$ on the boundary
$\Sigma$ as an eigenvalue corresponding to the first non negative eigenvalue
$\lambda_{1}({\bf D})$ of the intrinsic Dirac operator ${\bf D}$ of the
boundary. A direct application of Theorem 2 gives
$\int_{\Sigma}\left(\lambda_{1}({\bf D})-\frac{n}{2}H\right)|\phi|^{2}\geq 0.$
As a consequence, we obtain the following lower bound which we firstly found
in [HM]:
$\lambda_{1}({\bf D})\geq\frac{n}{2}\max_{\Sigma}H$
and that improves, in the immersed case, the well-known lower bound by
Fiedrich [F] for boundary manifolds in ${\mathbb{R}}^{n+1}$.
###### Remark 2.
It is also worthy to note that, as a consequence of this estimate, if $\Sigma$
is a compact boundary in an Euclidean space and we know that $\lambda_{1}({\bf
D})\leq n/2$ and $H\geq 1$, then have the equality and a fortiori and the mean
curvature $H$ of $\Sigma$ must be constant. Hence, from the Alexandrov
Theorem, $\Omega$ is a round disc. This is a Euclidean analogue to the Min-Oo
conjecture and was remarked to be true by Miao in the paper [Mi2], by
supposing that $\Sigma$ was isometric to a unit $n$-sphere.
###### Theorem 3 (Brown-York mass for mean-convex surfaces).
Let $\Omega$ be a compact spin Riemannian manifold of dimension $n+1$ with
non-negative scalar curvature $R\geq 0$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is strictly (strictly mean-
convex). Suppose that there is an isometric and isospin immersion from
$\Sigma$ into another spin manifold $\Omega_{0}$ carrying on a non-trivial
parallel spinor field and let $H_{0}$ its mean curvature with respect to any
of its orientations. Then, we have
$\int_{\Sigma}H\leq\int_{\Sigma}|H_{0}|.$
The equality implies that $H=|H_{0}|=H_{0}$. Then, if $n=2$, $\Omega_{0}$ is a
domain in ${\mathbb{R}}^{3}$ and the two embeddings differ by a direct rigid
motion
###### Remark 3.
Note that, in the original Shi-Tam original result, the authors assume that
the boundary $\Sigma$ is strictly convex. Then a well-known by Pogorelov [Po]
and Nirenberg [Ni] guarantees the existence of an isometric embedding into the
Euclidean space. Here, we need suppose the existence of this second isometric
immersion. Moreover we have $H_{0}>0$ a fortiori.
Proof. Denote by $\psi$ the parallel spinor on $\Omega_{0}$ and let
$\phi=\psi_{|\Sigma}$ its restriction onto $\Sigma$ through the existent
immersion. Let’s recall that the parallelism of $\psi$ (see (22)) gives
${\bf D}\phi=\frac{n}{2}H_{0}\phi\qquad\hbox{and}\qquad|\phi|=1.$
Now, we apply Theorem 2 and have the desired inequality
$\int_{\Sigma}H\leq\int_{\Sigma}|H_{0}|\,d\Sigma.$
If the equality is attained, then
$\frac{n}{2}H_{0}=D\phi=\frac{n}{2}H$
and so $H=H_{0}>0$. Then, the immersion of $\Sigma$ into the second ambient
space $\Omega_{0}$ is strictly mean-convex as well (with respect to the choice
of inner normal to $\Omega$).
When $n=2$, from this equality and the fact that $K=K_{\phi}$ (because the two
embeddings are isometric and preserve the Gauss curvatures), we deduce that
the two second fundamental forms coincide. The Fundamental Theorem of the
Local Theory of Surfaces allows us to conclude that the two boundaries differ
by a direct rigid motion of the Euclidean space.
###### Remark 4.
It is obvious that the result above is a generalization of the positivity
theorem for the Brown-York mass previously proved for strictly convex surfaces
by Shi and Tam. In their proof, the solution of difficult boundary equations
and the positivity of the ADM-mass obtained by Shoen-Yau and Witten [SY, Wi]
in the context of asymptotically flat manifolds are essential components.
Here, these difficulties are avoided and, as we have already remarked
somewhere above, this compact version of the theorem implies the
asymptotically flat version for the ADM mass (see [HMRa]).
###### Corollary 4.
Let $\Omega$ be a compact spinor manifold of dimension $3$ with non-negative
scalar curvature $R\geq 0$ and having a mean-convex boundary $\Sigma$
isometric to a sphere of any radius. Then, we have
$\int_{\Sigma}H\leq\sqrt{\pi A(\Sigma)}$
where $A(\Sigma)$ is the area of $\Sigma$. It the equality holds, then the two
boundaries are spheres of the same radius.
Proof. It is clear that the boundary ${\mathbb{S}}^{2}$ of $\Omega$ admits and
isometric and isospin (the sphere supports a unique spin stricture) embedding
into the Euclidean space ${\mathbb{R}}^{3}$ with $|H_{0}|=1/r$ and area
$A(\Sigma)=\pi r^{2}$, where $r>0$ is the radius of the sphere. The fact that
the two embeddings of ${\mathbb{S}}^{2}$ are isometric allows us to finish.
###### Corollary 5 (Cohn-Vossen rigidity theorem for mean-convex domains).
Two isometric and isospin strictly mean-convex compact surfaces in the
Euclidean space ${\mathbb{R}}^{3}$ must be congruent.
Proof. Let $\Omega$ and $\Omega_{0}$ be the two domains determined in
${\mathbb{R}}^{3}$ by two corresponding surfaces identified by means of an
isometry. Then, we can apply Theorem 3 interchanging the roles of $\Omega$ and
$\Omega_{0}$ and applying the case of the equality.
###### Remark 5.
The integral inequality in Corollary 4, for strictly convex surfaces of
${\mathbb{R}}^{3}$, is attributed to Minkowski (1901), although its very
probable that it were previously known to Alexandrov and Fenchel. Recently
have proved that the Minkowski inequality is not valid for any compact
surface, although they proved it is for the axisymmetric ones. Its is also
worthy to remark the following conjecture by Gromov: If $\Sigma$ is the
boundary of a compact Riemannian manifold $\Omega$, then, if $R\geq\sigma$,
for a certain constant $\sigma$, where $R$ is the scalar function of $\Omega$,
then there exists a constant $\Lambda(\Sigma,\sigma)$ such that
$\int_{\Sigma}H\,d\Sigma\leq\Lambda(\Sigma,\sigma).$
###### Remark 6.
Much more recently, in the context [SWWZ] of fill in problems posed firstly by
Bartnik, proved that, if $\Omega$ is the hemisphere $B^{n+1}$ and $\gamma$ is
a metric on the boundary ${\mathbb{S}}^{n}$ isotopic to the standard one with
mean curvature $H>0$, then there is a constant $h_{0}=h_{0}(\gamma)$ such that
$\int_{\Sigma}H\leq h_{0}.$
It is clear that this result and our Corollary 4 belong to a same family.
## 6\. Ambients with positive scalar curvature
Until now we have suppose that the scalar curvature of our compact spin
Riemannian manifold $\Omega$ satisfied $R\geq 0$ (Euclidean context). Let’s
enhance this positivity assumption to $R\geq n(n+1)$ (spherical context). This
lower bound is just the constant value of the scalar curvature of the
$(n+1)$-dimensional unit sphere. Then, by putting this assumption and the
Schwarz inequality
(24) $|D\psi|^{2}\leq(n+1)|\nabla\psi|$
(already used in Section 3) into the right side of the Weitzenbök-Lichnerowicz
formula, we obtain
(25) $\int_{\Sigma}\left(\left<{\bf
D}\psi,\psi\right>-\frac{n}{2}H|\psi|^{2}\right)\geq\int_{\Omega}\left(-\frac{1}{n+1}|{D}\phi|^{2}+\frac{n+1}{4}|\psi|^{2}\right),$
for all compact spin manifold $\Omega$, with equality only for the twistor
spinor fields on $\Omega$. From now on, by using this integral inequality, we
will work in a similar, but a few more elaborated, way as in Theorem 2, and
will get the following result.
###### Theorem 6.
Let $\Omega$ be a $(n+1)$-dimensional compact spin Riemannian manifold whose
scalar curvature is constant $R=n(n+1)$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is non negative (mean-convex).
Suppose also that $\Sigma$ does not admit harmonic spinors. Then, for every
spinor $\phi\in H^{1}(\Sigma)$, we have
$\int_{\Sigma}|{\bf
D}\phi||\phi|\geq\frac{n}{2}\int_{\Sigma}|\phi|^{2}\sqrt{H^{2}+1}.$
The equality holds if and only if $\phi$ is a spinor field coming from a real
Killing spinor $\psi$ defined on $\Omega$, and so
${\bf D}\phi=\frac{n}{2}H\phi-\frac{n}{2}\gamma(N)\phi.$
###### Remark 7.
Note that our hypotheses impose the equality $R=n(n+1)$ and not the maybe
expected inequality $R\geq n(n+1)$, conjectured by Min-Oo and based on the
flat case. Obviously the condition $R\geq n(n+1)$ is necessary, because one
can otherwise perturb the hemisphere at an interior point so that $R\geq
n(n+1)-\varepsilon$, for some small $\varepsilon>0$, without changing the
assumptions on the boundary [HW1]. Moreover the other possibility $R>n(n+1)$
had to be invalidated by the counterexamples built by Brendle, Marques and
Neves [BMN], since all of them require at least one point with this strict
inequality in the bulk manifold. Hence, we are brought to suppose the
equality.
Proof. Given $a\in{\mathbb{Z}}$, $a\leq 0$, consider now the self-adjoint
eigenvalue problem for the Dirac operator $D$ on the bulk manifold $\Omega$
(26) $\left\\{\begin{array}[]{ll}D\psi_{a}=\mu_{a}\,\psi_{a}\\\ \psi_{\geq
a}=0,\end{array}\right.$
corresponding to the Dirac operator on $\Omega$ subjected to the usual APS
elliptic boundary condition as in the section above. Since the boundary does
not carry harmonic spinors, then the corresponding $D$ is symmetric in
$\Omega$ and its eigenvalues are real numbers. If $\psi_{a}$ is a solution to
(25), from Schwarz inequality (23), we get from the spinorial Reilly
inequality
$0\geq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right)\geq\int_{\Omega}\left(-\frac{\mu^{2}_{a}}{n+1}+\frac{n+1}{4}\right)|\psi_{a}|^{2},$
Since we have the $L^{2}$-orthogonal decomposition $\psi_{|\Sigma}=\psi_{\geq
a}+\psi_{<a}=0$ on $\Sigma$, $\psi_{\geq a}=0$, from (20) and $H\geq 0$ in the
hypotheses, the integral on the right side is non positive. Then
$\mu_{a}^{2}\geq\frac{(n+1)^{2}}{4},$
with equality if and only $\psi_{a}$ is a twistor spinor and, in this case, a
real Killing spinor on $\Omega$ with
${\nabla_{X}}\psi_{a}=-{\mu_{a}}\gamma(X)\psi_{a},\qquad\forall X\in T\Omega.$
The equality would imply as well
$\int_{\Sigma}\left(\left<{\bf
D}\psi_{<a},\psi_{<a}\right>-\frac{n}{2}H|\psi|^{2}\right)=0,$
and then ${\psi_{a}}_{|\Sigma}=0$. But a real Killing spinor has constant
length and so $\psi_{a}$ would identically null on the whole of $\Omega$. As a
consequence of this, the first eigenvalue $\mu_{1}(a)$ of (25) satisfies
$\mu_{1}(a)>\frac{n+1}{2},$
for all $a$ integer non positive. Henceforth, there exists a unique solution
with $\psi_{a}\in H^{1}(\Omega)\cap H^{\frac{1}{2}}(\Sigma)$ for the problem
(27) $\left\\{\begin{array}[]{ll}D\psi_{a}=\frac{n+1}{2}\,\psi_{a}\\\
\psi_{\geq a}=\phi_{\geq a},\end{array}\right.$
for each $a\in{\mathbb{Z}}$, $a\leq 0$ and for all $\phi\in L^{2}(\Sigma)$.
Take a such solution to (26) and put it in (24). Then
(28) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right),\qquad\forall
a\in{\mathbb{Z}},a\leq 0,$
which is exactly the same expression that we obtained in the flat ambient
case, but here the equality would attain only in the case where $\psi_{a}$ is
a real Killing spinor (and not a parallel one)
$\nabla_{X}\psi_{a}=-\frac{1}{2}\gamma(X)\psi_{a},\qquad\forall X\in T\Omega.$
From this point, by working in the very exact way as in the proof of Theorem
2, we arrive at
(29) $0\leq\lim_{a\rightarrow-\infty}\int_{\Sigma}\left(|{\bf D}\phi_{\geq
a}||\phi_{\geq a}|-\frac{n}{2}H|\phi_{\geq a}|^{2}\right).$
But, with this choice of $\psi_{a}$ as a solution to (26), the left side in
the integral Reilly inequality after (25) vanishes. Hence, $\psi_{a}$ is a
twistor spinor and also
$D\psi_{a}=\frac{n+1}{2}\psi_{a}$
on the whole $\Omega$. That is, $\psi_{a}$ is a real Killing spinor. More
precisely, we arrive just to the expression which had already rejected some
times to avoid several reductionis ad absurdum, more precisely,
(30) $\nabla_{X}\psi_{a}=-\frac{1}{2}\gamma(X)\psi_{a},\qquad\forall X\in
T\Omega.$
By using (7) and the equality above, we have
(31) ${\bf D}\psi_{a}=\frac{n}{2}H\psi_{a}-\frac{n}{2}\gamma(N)\psi_{a},$
along the boundary $\Sigma$. Taking into account the orthogonal $L^{2}$
decomposition $\psi_{a}=\psi_{\geq a}+\psi_{<a}$, the fact that ${\bf D}$
preserves this decomposition and that $\gamma(N)$ reverses it pointwise, if we
multiply scalarly by the spinor $\gamma(N)\psi_{a}$ and integrate on $\Sigma$,
we get
$\int_{\Sigma}\left<{\bf
D}\psi_{a},\gamma(N)\psi_{a}\right>=-\frac{n}{2}\int_{\Sigma}|\psi_{a}|^{2}.$
By working in an analogous, but not exactly, equal way as in (20), we obtain
(32) $\lim_{a\rightarrow-\infty}\int_{\Sigma}|{\bf D}\phi_{\geq a}||\phi_{\geq
a}|\geq\lim_{a\rightarrow-\infty}\frac{n}{2}\int_{\Sigma}|\phi_{\geq a}|^{2},$
for all $a\in{\mathbb{Z}}$, $a\leq 0$ and the convergence is in
$H^{\frac{1}{2}}(\Sigma)$ and, so, strongly in $H^{2}(\Sigma)$.
In order to obtain the inequality in the Theorem, note that (28) and (31)
provide us integral sequences which converge strongly in $L^{2}(\Sigma)$.
Then, we can extract from each one of them a subsequence converging a.e. (and,
so, since $\phi$ is continuous converging on the whole) to the corresponding
function limit. These two subsequences will be (probably different), but have
been extracted from a same converging sequence, so, we will label them with
the same indices as the original ones, and have
$\lim_{a\rightarrow-\infty}|D\phi_{\geq a}||\phi_{\geq
a}|\geq\frac{n}{2}H|\phi|^{2}\quad\mbox{and}\quad\lim_{a\rightarrow-\infty}|D\phi_{\geq
a}||\phi_{\geq a}|\geq\frac{n}{2}|\phi|^{2}.$
As a consequence, we obtain
$\lim_{a\rightarrow-\infty}|D\phi_{\geq a}||\phi_{\geq
a}|\geq\frac{n}{2}\sqrt{1+H^{2}}|\phi_{\geq a}|^{2},$
which was the inequality we are looking for.
For the case of the equality, as in the proof of Theorem 2, we can use the
standard and well written results of regularity results in [BaBä1,BaBä2] to
show that the solution $\psi_{a}$ to the boundary problem (26) is smooth on
${\bar{\Omega}}$ for all integer $a\leq 0$. As before, with respect to the
boundary $\Sigma$, we can apply Theorem 7.17 and Corollary 7.18 in [BaBä]
repeatedly in a bootstraping process and obtain smoothness on $\Sigma$ as
well. Then
$\psi_{a}\in\cap_{k=1}^{\infty}H^{k}({\bar{\Omega}})=C^{\infty}({\bar{\Omega}})$.
For all non positive integer number $a$, the limit function $\psi$ is also
smooth on the compact manifold ${\bar{\Omega}}$. This implies that the limit
function
$\psi=\lim_{a\rightarrow-\infty}\psi_{a}$
satisfies (15) on $\Omega$ as well. That is, $D\psi=(n+1)/\psi$. Taking limit
in (29) and substituting in the integral Reilly inequality we have, from the
left side, that $\psi$ is a parallel spinor on the whole of ${\bar{\Omega}}$.
Also it is clear that
$\phi=\lim_{a\rightarrow-\infty}\phi_{\geq a}=\psi_{|\Sigma}.$
But the relation between the operators $D$ and its restriction ${\mathcal{D}}$
is
(33) ${\bf D}\phi=\frac{n}{2}H\phi-\gamma(N)D\psi-\nabla_{N}\psi$
and the spinor $\phi$ comes from a real Killing spinor on $\Omega$ with
constant $-1/2$. Hence, it has to satisfy on the whole $\Sigma$ the equality
${\bf D\phi}=\frac{n}{2}H\phi-\frac{n}{2}\phi.$
The converse is obvious.
As far as we know, we obtain from Theorem 6 a lower estimate for the first
eigenvalue of the Dirac operator in a hypersurface in a context on scalar
curvature positive.
###### Corollary 7.
Consider a compact spin Riemannian manifold $\Omega$ of dimension $n+1$ and
scalar curvature $R=n(n+1)$ and mean-convex boundary. Suppose that $\Sigma$
does not support harmonic spinors and let $\phi$ be the eigenspinor
corresponding to the first eigenvalue $\lambda_{1}({\bf D})$ positive of the
intrinsic Dirac operator ${\bf D}$ of the boundary. A direct application of
Theorem 6 gives
$\int_{\Sigma}\left(\lambda_{1}({\bf
D})-\frac{n}{2}\sqrt{1+H^{2}}\right)|\phi|^{2}\geq 0.$
As a consequence, we obtain the following lower bound:
$\lambda_{1}({\bf D})\geq\frac{n}{2}\max_{\Sigma}\sqrt{1+H^{2}}$
that improves the well-known lower bound by Fiedrich [F] for non immersed
manifolds boundary manifolds (the Gauß for the curvature in the unit
$(n+1)$-dimensional sphere gives
$\sqrt{\frac{nR}{4(n-1}}\leq\frac{n}{2}\sqrt{1+H^{2}}$
and the equality is attained only by the umbilical hypersurfaces).
###### Remark 8.
It is also worthy to note that, as a consequence of this estimate, if $\Sigma$
is a compact boundary in an Euclidean space and we know that $\lambda_{1}({\bf
D})\leq n/2$ and $H\geq 0$, then we have the equality and a fortiori the mean
curvature $H$ of $\Sigma$ must be identically null. Moreover, $\Omega$
supports the existence of a non trivial real Killing spinor (see [Bä]). Hence,
$\Omega$ is close to being a spherical domain bounded by a minimal
hypersurface. This would be the most similar to the solution to the spherical
analogue to the Min-Oo conjecture.
As in the flat case, we can obtain a kind of positivity for a possible quasi-
local mass in this new context.
###### Theorem 8 (Brown-York in mean-convex and spherical case).
Let $\Omega$ be a spin Riemannian manifold of dimension $n+1$ with scalar
curvature $R=n(n+1)$ and having a non-empty boundary $\Sigma$ whose inner mean
curvature $H\geq 0$ is also non-negative (mean-convex). Suppose that there is
an isometric and isospin immersion from $\Sigma$ into another spin manifold
$\Omega_{0}$ carrying on a non-trivial real Killing spinor field and let
$H_{0}$ its mean curvature with respect to any of its orientations. Then, we
have
$\int_{\Sigma}\sqrt{1+H^{2}}\leq\int_{\Sigma}\sqrt{1+H_{0}^{2}},$
provided that the boundaries do not admit harmonic spinors. The equality
implies that $H=H_{0}$. Then, if $n=2$, $\Omega_{0}$ is a domain in
${\mathbb{S}}^{3}$ and the two embeddings differ by a direct rigid motion.
Proof. Denote by $\psi$ the spinor on $\Omega_{0}$ and let
$\phi=\psi_{|\Sigma}$ its restriction onto $\Sigma$ through the existent
immersion. Let’s recall that the parallelism of $\psi$ gives
${\bf D}\phi=\frac{n}{2}\sqrt{1+H_{0}}\,\phi\qquad\hbox{and}\qquad|\phi|=1.$
Now, we apply Theorem 2 and have the desired inequality
$\int_{\Sigma}\sqrt{1+H^{2}}\leq\int_{\Sigma}\sqrt{1+H_{0}}.$
If the equality is attained, then
$\frac{n}{2}\sqrt{1+H_{0}^{2}}={\bf D}\phi=\frac{n}{2}\sqrt{1+H^{2}}$
and so $H=H_{0}\geq 0$. Then, the immersion of $\Sigma$ into the second
ambient space $\Omega_{0}$ is mean-convex as well (with respect to the choice
of inner normal to $\Omega$).
When $n=2$, from this equality and the fact that $K=K_{\phi}$ (because the two
embeddings are isometric and preserve the Gauss curvatures), we deduce that
the two second fundamental forms coincide. The Fundamental Theorem of the
Local Theory of Surfaces allows us to conclude that the two boundaries differ
by a direct rigid motion of the Euclidean space.
###### Corollary 9 (Cohn-Vossen rigidity theorem in the sphere).
Two isometric and isospin mean-convex compact surfaces embedded in a sphere
${\mathbb{S}}^{3}$ must be congruent, provided they do not admit harmonic
spinors.
Proof. Let $\Omega$ and $\Omega_{0}$ be the two domains determined in
${\mathbb{S}}^{3}$ by two corresponding surfaces identified by means of an
isometry. Then, we can apply Theorem 8 interchanging the roles of $\Omega$ and
$\Omega_{0}$ and applying the case of the equality.
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|
# Spin fractionalization and zero modes in the spin-$\frac{1}{2}$ XXZ chain
with boundary fields
Parameshwar R. Pasnoori Department of Physics, University of Maryland,
College Park, MD 20742, United States of America Laboratory for Physical
Sciences, 8050 Greenmead Dr, College Park, MD 20740, United States of America
Yicheng Tang Department of Physics and Astronomy, Center for Materials
Theory, Rutgers University, Piscataway, NJ 08854, United States of America
Junhyun Lee Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854, United States of America J. H.
Pixley Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854, United States of America Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York,
NY 10010, USA Natan Andrei Department of Physics and Astronomy, Center for
Materials Theory, Rutgers University, Piscataway, NJ 08854, United States of
America<EMAIL_ADDRESS><EMAIL_ADDRESS>Patrick Azaria
Laboratoire de Physique Thórique de la Matière Condensée, Sorbonne Université
and CNRS, 4 Place Jussieu, 75252 Paris, France
###### Abstract
In this work we argue that the antiferromagnetic spin $\frac{1}{2}$ XXZ chain
in the gapped phase with boundary magnetic fields hosts fractional spin
$\frac{1}{4}$ at its edges. Using a combination of Bethe ansatz and the
density matrix renormalization group we show that these fractional spins are
sharp quantum observables in both the ground and the first excited state as
the associated fractional spin operators have zero variance. In the limit of
zero edge fields, we argue that these fractional spin operators once projected
onto the low energy subspace spanned by the ground state and the first excited
state, identify with the strong zero energy mode discovered by P. Fendley
Fendley (2016).
## I Introduction
Since the discovery of solitons carrying half of the electron charge Jackiw
and Rebbi (1976); Su _et al._ (1979) it has been widely recognized Goldstone
and Wilczek (1981); Kivelson and Schriefer (1982); Jackiw _et al._ (1983)
that some states of matter can be characterized by fractional quantum numbers.
Maybe the most celebrated example is the fractional quantum Hall state where
quasiparticles carry fractional charges Laughlin (1983); Tsui _et al._
(1982). Other prominent examples coming from topological phases with short
range topological order, such as symmetry protected topological (SPT) systems
in one dimension, include spin-$1/2$ edge states in the spin one Haldane chain
Haldane (1983); Affleck _et al._ (1987) as well as spin-$1/4$ zero energy
modes (ZEM) localized at the edges of one dimensional spin triplet
superconductors Keselman and Berg (2015); Pasnoori _et al._ (2020, 2021). In
higher dimensions, surface states in topological insulators as well as
disordered magnetic systems like spin ice Castelnovo _et al._ (2012) and spin
liquids also exhibits signatures of fractionalization Banerjee _et al._
(2016).
In this work we shall demonstrate that fractionalization can also occur in
more conventional magnetic systems which exhibit long range magnetic order. To
this end we shall consider the paradigmatic XXZ spin $1/2$ chain with magnetic
fields at its two edges, that we solve exactly with the Bethe ansatz and
numerically using the density matrix renormalization group (DMRG), and show
that in the low energy sector it hosts quantum spin-$1/4$ states localized at
the edges. We shall further argue that this fractional quarter spins are sharp
quantum observables. We believe that this result might have some impact in
understanding dynamics Liu and Andrei (2014); Lancaster and Mitra (2010);
Pozsgay _et al._ (2014); Mestyán _et al._ (2015); Foster _et al._ (2011);
Joel _et al._ (2013); Misguich _et al._ (2017), heat and spin transport
Bertini _et al._ (2016); De Luca _et al._ (2017); Bulchandani _et al._
(2018).
Figure 1: Ground state phase diagram of the XXZ model with edge fields smaller
than the critical field $|h_{L,R}|<h_{c}=\Delta-1$ and for an odd number of
sites. In each of the two phases separated by the line $h_{L}+h_{R}=0$ we show
the ground state as well as the first excited state which is a midgap state
below the continuum. Both states host fractional quarter spins ${\cal
S}_{L,R}=\pm 1/4$ at both edges of the chain. These fractional spins are sharp
quantum observables which reconstruct the total spin of each state
$S^{z}={\cal S}_{L}+{\cal S}_{R}=\pm 1/2$. The $\pm 1/4$ quarter spins are
depicted by triangles pointing upward and downward respectively. On the
separatrix $h_{L}+h_{R}=0$ there is spontaneous symmetry breaking and the edge
spin operator becomes a zero energy mode.
We consider the XXZ hamiltonian with boundary magnetic fields $(h_{L},h_{R})$
at the left and the right edges of an open chain
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N-1}\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}+\Delta(\sigma^{z}_{j}\sigma^{z}_{j+1}-1)$
(1) $\displaystyle+$ $\displaystyle h_{L}\sigma^{z}_{1}+h_{R}\sigma^{z}_{N}$
where $\sigma_{j}^{x,y,z}$ are the Pauli matrices and $\Delta>1$ is the
anisotropy parameter. In the limit where the boundary fields are zero, on top
of being $U(1)$ symmetric, (1) is space parity $\mathbb{P}$ and time reversal
$\mathbb{T}$ invariant. It is also invariant under the
$\mathbb{Z}_{2}=\\{1,\tau\\}$ spin flip symmetry, i.e: $[H,\tau]=0$ where
$\tau=\Pi_{1}^{N}\sigma_{j}^{x}$. For generic non zero boundary fields
$h_{L,R},\neq 0$, both $\mathbb{P}$ and $\mathbb{Z}_{2}$ symmetries are
explicitly broken. However, on the two lines $h_{L}=\pm h_{R}$ the hamiltonian
(1) displays $\mathbb{P}$ and $\mathbb{P}\circ\mathbb{Z}_{2}$ symmetries
respectively.
The Hamiltonian in Eq. (1) is integrable by the method of the Bethe ansatz for
arbitrary boundary fields $h_{L,R}$ and $\Delta$, which is used in the present
paper to determine the low energy eigenstates analytically. The system with
periodic boundary conditions was first solved by Bethe Bethe (1931) in the
isotropic limit, $\Delta\rightarrow 1$. The solution was later extended to
include anisotropy along the z-direction Orbach (1958); Walker (1959); Yang
and Yang (1966a, b, c); Babelon _et al._ (1983). In the gapped regime
($\Delta>1$) it exhibits a continuous $U(1)$ symmetry and also a discrete
$\mathbb{Z}_{2}$ spin flip symmetry. The discrete $\mathbb{Z}_{2}$ symmetry is
spontaneously broken Syljuåsen (2003) and in the thermodynamic limit the
system exhibits two degenerate symmetry broken ground states Takahashi (1999).
The Bethe ansatz method to include the boundaries was developed in Alcaraz
_et al._ (1987); Cherednik (1984); Sklyanin (1988) and the ground state and
boundary excitations in various bulk phases exhibited by the XXZ spin chain
were found in Skorik and Saleur (1995); S.Skorik and A.Kapustin (1995);
Grijalva _et al._ (2019); Nassar and Tirkkonen (1998). An independent method
to diagonalize the Hamiltonian using vertex operators was developed in Davies
_et al._ (1993); Jimbo _et al._ (1992), and was later extended to include the
boundary fields in Jimbo _et al._ (1995) where the boundary S-matrix and the
integral formula for correlation functions have been found. Recently new band
structures in the spectrum at large anisotropies have been found Sharma and
Haque (2014).
Numerically we solve the model using DMRG, implemented through the TeNPy
software Hauschild and Pollmann (2018), that allows access to the ground state
and midgap state with arbitrary precision thanks to the gapped nature of both
of these states. We take a maximum bond dimension of 400, with a minimal
singular value decomposition cut off of $10^{-10}$, and converge the energy up
to a maximal energy error on the order of $\sim 10^{-10}$.
When $\Delta>1$ the ground state $|g\rangle$ displays antiferromagnetic order
with non zero staggered magnetization
$\sigma=\lim_{N\rightarrow\infty}N^{-1}\sum_{j=1}^{N}(-1)^{j}\langle
g|\sigma_{j}^{z}|g\rangle$ and is gapped. Indeed, for all values of the edge
fields, there is a gap ($m$) in the spectrum to single particle spin $1/2$
spinon excitations
$m=\sinh\gamma\sum_{n\in\mathbb{Z}}\frac{(-1)^{n}}{\cosh\gamma
n},\;\Delta=\cosh{\gamma}.$ (2)
However at low fields, i.e: $|h_{L,R}|<\Delta-1$, the lowest excited state is
a midgap state $|e\rangle$ which lies below the continuum. We obtain the bound
state energies 111Note that a different expression was obtained in S.Skorik
and A.Kapustin (1995); Grijalva _et al._ (2019). which are given by
$\displaystyle
m_{\alpha}=h_{\alpha}+\sinh\gamma\sum_{n\in\mathbb{Z}}(-1)^{n}\;\frac{\sinh(\gamma{\epsilon}_{\alpha}|n|)}{\cosh\gamma
n}e^{-\gamma|n|}$ (3)
where $\alpha=(L,R)$. This midgap state is reminiscent of the existence of
spin $1/2$ boundary bound states, localized at the two left and right edges.
The spin quantum numbers and energies of the ground state as well as the
midgap state depend on both the parity of the number of sites, $(-1)^{N}$, as
well as on the boundary fields $h_{L,R}$. When $N$ is odd, the two states
$|g\rangle$ and $|e\rangle$ have opposite total spins $S^{z}=\pm 1/2$. Taking
as a reference state the $|-\frac{1}{2}\rangle$ state with energy $E_{0}$, the
$|+\frac{1}{2}\rangle$ state is obtained by adding a localized bound state at
each edge. This state has energy $E_{0}+m_{L}+m_{R}$. Depending on the edge
magnetic fields, and hence on the sign of $m_{L}+m_{R}$, the ground state and
the midgap state ($|g\rangle$,$|e\rangle$) are
($|-\frac{1}{2}\rangle,|+\frac{1}{2}\rangle$) when $h_{L}+h_{R}>0$ and
($|\frac{1}{2}\rangle,|-\frac{1}{2}\rangle$) when $h_{L}+h_{R}<0$. Notice that
on the line $h_{L}+h_{R}=0$, the two states $|\pm\frac{1}{2}\rangle$ are
degenerate. In the $N\rightarrow\infty$ limit, there is spontaneous symmetry
breaking (SSB) of the $\mathbb{P}\circ\mathbb{Z}_{2}$ symmetry. In the
particular case of zero edge fields, both $\mathbb{P}$ and $\mathbb{Z}_{2}$
symmetries are spontaneously broken. For $N$ even both
($|g\rangle$,$|e\rangle$) states have total spins $S^{z}=0$ and the bound
state construction is presented in the Appendix. We display in Fig.(1) the
phase diagram for low fields for an odd number of sites $N$.
## II Spin Profiles
Due to the open boundaries and the presence of the edge fields
$(h_{L},h_{R})$, the spin profiles $S^{z}_{j}=\langle\sigma_{j}^{z}\rangle/2$
in both the ground state and the midgap state differ from the bulk
antiferromagnetic order close to the boundaries. For large enough $N$ we may
write
$S^{z}_{j}=(-1)^{j}\frac{\sigma}{2}+\Delta S^{z}(j),$ (4)
where
$\sigma=\pm\left(\Pi_{n=1}^{\infty}(\frac{1-q^{2n}}{1+q^{2n}})\right)^{2},\;q=e^{-\gamma},$
(5)
is the exact staggered magnetization of the XXZ chain in the thermodynamical
limit and $\Delta S^{z}(j)$ is the relative deviation with respect to the AF
bulk profile. Due to the gap in the bulk these deviations are expected to be
localized close to both the left and the right edges
$\Delta S^{z}(j)=\Delta S^{z}_{L}(j)+\Delta S^{z}_{R}(j),$ (6)
where $\Delta S^{z}_{L,R}(j)$ are localized close to $j=1$ and $j=N$
respectively (i.e: $\Delta S^{z}_{L,R}({N/2})\sim e^{-N/2}$). This is indeed
what we find. We plot in Fig. 2 our DMRG results for $\Delta S^{z}_{L}(j)$ in
both the ground state and the midgap state. We clearly observe an exponential
localization of the relative spin accumulation for various values of $\Delta$
at constant boundary fields $h_{L}=h_{R}=0.2$ (see insets of Fig. 2).
Figure 2: Spin profile $\Delta S^{z}(j)$ and the fitting of ansatz (a) in the
ground state $|g\rangle$ with total spin $S^{z}=-\frac{1}{2}$ (b) in the
midgap state $|e\rangle$ with total spin $S^{z}=\frac{1}{2}$ for model
parameters $N=101$, $h_{L}=0.1,h_{R}=0.5$ and $\Delta=3$. The insets show that
the relative spin are indeed localized exponentially on the edge, with red
dashed lines representing excellent linear fits on the log-scale.
## III Spin fractionalization
The above spin accumulations, or depletion, do not come as a surprise and are
expected due to the open boundaries and the presence of the edge fields. What
is non trivial is that they correspond to a genuine spin fractionalization in
both the ground state and the midgap state. As we shall now demonstrate, in
the thermodynamical limit and for all $\Delta>1,h_{{L},{R}}$, there exist
fractionalized quarter spin operators associated with each edge,
${\cal\hat{S}}^{z}_{L}$ and ${\cal\hat{S}}^{z}_{R}$, which have well defined
fractional eigenvalues
${\cal\hat{S}}^{z}_{{L},{R}}|g(e)\rangle={\cal
S}_{{L},{R}}|g(e)\rangle,\;{\cal S}^{z}_{L,R}=\pm\frac{1}{4}.$ (7)
In the basis $(|g\rangle,|e\rangle)$ the above fractional spin operators
commute with each other, and anticommute with the spin flip operator, i.e:
$[{\cal\hat{S}}^{z}_{L},{\hat{S}}^{z}_{R}]=0$,
$\\{\tau,{\cal\hat{S}}^{z}_{L,R}\\}=0$. Together, they reconstruct the
z-component of the total spin $\hat{S}^{z}=\sum_{i=1}^{N}\hat{S}_{i}^{z}$,
namely
$\hat{S}^{z}={\cal\hat{S}}^{z}_{L}+{\cal{\hat{S}}}^{z}_{R}.$ (8)
Since the edge spin operators have fractional spin $\pm 1/4$ one may verify
that the $\hat{S}^{z}$ have eigenvalues $0$ or $\pm 1/2$ depending on whether
$N$ is even or odd. For the fractional spin operators (7) to describe sharp
quantum observables in the subspace spanned by $(|g\rangle,|e\rangle)$, not
only they have to average to $\pm 1/4$ in both states, but also their variance
must vanish in the thermodynamical limit, i.e:
$\displaystyle\langle{\hat{S}}^{z}_{{L},{R}}\rangle$ $\displaystyle=$
$\displaystyle{\cal S}_{{L},{R}},$ (9)
and
$\displaystyle\delta{\cal S}^{2}_{{L},{R}}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\equiv}$
$\displaystyle\langle({\cal\hat{S}}^{z}_{{L},{R}})^{2}\rangle-({\cal
S}_{{L},{R}})^{2}=0,$ (10)
where the average $\langle...\rangle$ is taken in each of the two states
$(|g\rangle$ and $|e\rangle)$.
Following the authors of Refs.Kivelson and Schriefer (1982); Jackiw _et al._
(1983) we define the fractional spin operators as their convolution with a
decaying function $f(x)$, here we take $f(x)=e^{-\alpha x}$ to write
$\displaystyle{\cal\hat{S}}^{z}_{L}$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow
0}\lim_{N\rightarrow\infty}\sum_{j=1}^{N}f(j)\frac{\sigma^{z}_{j}}{2},$ (11)
$\displaystyle{\cal\hat{S}}^{z}_{R}$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow
0}\lim_{N\rightarrow\infty}\sum_{j=1}^{N}f(N+1-j)\frac{\sigma^{z}_{j}}{2},$
(12)
which takes the limit $\alpha\rightarrow 0$ after the limit
$N\rightarrow\infty$. We stress that the order of limits in (12) is important
since by taking the limit $\alpha\rightarrow 0$ first, both ${\cal
S}^{z}_{{L},{R}}$ would identify with the total magnetization $S^{z}$.
Due to the AF long range order it is convenient to distinguish between the
contributions of the staggered part of the spin profile and that of the
exponentially localized contributions
${\cal\hat{S}}^{z}_{{L}}=-\frac{\sigma}{4}+{\Delta\cal\hat{S}}^{z}_{{L}},\;{\cal\hat{S}}^{z}_{{R}}=-\frac{\sigma}{4}(-1)^{N}+{\Delta\cal\hat{S}}^{z}_{{R}}.$
(13)
where the relative accumulation operators are given by
$\displaystyle{\Delta\cal\hat{S}}^{z}_{{L}}$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{2}\sum_{j=1}^{N}[\sigma_{j}^{z}-\sigma(-1)^{j}]e^{-\alpha
j},$ $\displaystyle{\Delta\cal\hat{S}}^{z}_{{R}}$ $\displaystyle=$
$\displaystyle\lim_{\alpha\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{2}\sum_{j=1}^{N}[\sigma_{j}^{z}-\sigma(-1)^{j}]e^{-\alpha(N+1-j)}.$
(14)
We have used the identity $\lim_{\alpha\rightarrow
0}\sum_{j=1}^{\infty}(-1)^{j}e^{-\alpha j}=-\frac{1}{2}$. Numerically (14) are
much easier to investigate since the relative spin accumulations (2) are
exponentially localized. In practice one may set $\alpha=0$ in (14) provided
the summation over $j$ extends to the middle of the chain $j_{\rm max}=N/2$.
Convergence is then expected to be of order $e^{-N/2}$. Before going further,
it is worthy to point out that although the relative accumulations defined in
Eq.(14) have the same variance as the fractional spin operators (12) they do
not qualify as spin operators in the sense that they do not anticommute with
the spin flip operator $\tau$, i.e:
$\\{\Delta{\cal\hat{S}}_{{L},{R}},\tau\\}\neq 0$ 222For instance it would not
couple to an external magnetic field penetrating smoothly near the left edge
whereas ${\cal\hat{S}}_{{L}}$ would.. These relative accumulations would have
fractional eigenvalues which depend on the anisotropy parameter $\Delta$.
Taking into account the AF long range order in the bulk is essential for the
spin accumulations to have fractional eigenvalues $\pm 1/4$ independently of
the model parameters as we shall see.
##### Spin $\pm\frac{1}{4}$ accumulations.
We have computed, using extensive DMRG calculations, the edge spin
accumulations ${\cal S}_{L}=\langle{\cal\hat{S}}^{z}_{L}\rangle$ and ${\cal
S}_{R}=\langle{\cal\hat{S}}^{z}_{R}\rangle$ in both the ground state and the
midgap state for a wide range of boundary fields $|h_{L,R}|<\Delta-1$ and
parameters $\Delta>1$. All together our results are consistent with an
accumulation of a spin ${\cal S}_{L,R}=\pm 1/4$ at the two edges of the system
in both the ground state and the midgap state. Furthermore we verify
explicitly that these quarter spins reconstruct the total spin $S^{z}$, as
given by Eq.(8), of the ground state and the midgap state for both $N$ even
and $N$ odd. We show here our results for an odd number of sites fixing
$\Delta=3$ and an anisotropic edge fields configuration $h_{L}=0$ with varying
$h_{R}$ in the Figs.(3). To check that the quarter spins observed so far do
not depend on the value of $\Delta>1$, we also show the spin accumulations
fixing $h_{L}=h_{R}=0.2$ (in this case ${\cal S}_{L}={\cal S}_{R}$ thanks to
the $\mathbb{P}$ symmetry) and varying $\Delta$. More results are given in the
Supplementary Materials.
Figure 3: Edge spin accumulation ${\cal
S}_{L/R}=\langle{\cal\hat{S}}^{z}_{L/R}\rangle$ in the ground state
$|g\rangle$, with total spin $S^{z}=-\frac{1}{2}$, and in the midgap state
$|e\rangle$, with total spin $S^{z}=\frac{1}{2}$. The model parameters used
are (a) $N=1001$, $h_{R}=h_{L}=0.2$ with varying $\Delta$. (b) $N=1001$,
$h_{L}=0$ and $\Delta=3$ with varying $h_{R}$. The dashed grey lines are on
the curve $S=\pm 1/4$ to represent the expected value. Insets show the
difference of numerical values from the expected value, which is on the order
of the DMRG accuracy $\sim 10^{-12}$.
##### Variance.
We also calculated the spin variance to directly verify that the quarter spins
found so far are sharp quantum observables. To this end we define the spin
variance at, say, the left edge for a finite system $N$ and cutoff $\alpha$ as
$\delta{\cal S}^{2}_{L}(N,\alpha)=\langle{\cal
S}_{L}^{z}(N,\alpha)^{2}\rangle-\langle S_{L}^{z}(N,\alpha)\rangle^{2},$ (15)
where the average is taken in either the ground state or the midgap state
$|g\rangle$ and $|e\rangle$. In the thermodynamic limit, the variance as
defined in Eq.(10), is then obtained as
$\delta S_{L}^{2}\equiv\lim_{\alpha\rightarrow 0}\lim_{N\to\infty}\delta
S^{2}(N,\alpha).$ (16)
Taking the $N\rightarrow\infty$ is challenging and we circumvent this issue by
assuming an ansatz relating $\delta S^{2}_{L}(N,\alpha)$ and $\delta
S^{2}_{L}(\infty,\alpha)$
$\delta S^{2}_{L}(N,\alpha)=\delta S^{2}(\infty,\alpha)-\frac{A}{\Delta}\alpha
e^{-B\alpha N}.$ (17)
With the above ansatz $\delta S^{2}_{L}=\lim_{\alpha\rightarrow
0}S(\infty,\alpha)=0$, and we can hence calculate $\delta S^{2}_{L}$ without
taking explicitly the thermodynamic limit. We have verified this ansatz by
taking the difference of $\delta S^{2}_{L}(N,\alpha)$ for different $N$’s.
This is shown in Fig.4, where one can see that the ansatz fits the data very
well. The fitted parameter $B\approx 2$ is nearly independent of the boundary
fields, while $A$ takes a non-universal value.
In summary we find that in the low energy subspace spanned by the ground state
$|g\rangle$ and the midgap state $|e\rangle$ one can assign to the left and
the right edges a fractional spin state with eigenvalues ${\cal
S}_{{L},{R}}=\pm\frac{1}{4}$. On the basis of our results we find it safe to
expect that this is to be the case irrespective of the anisotropy parameter
$\Delta>1$ and the values of the edge fields $[h_{L,R}|<\Delta-1$. Due to the
zero variance of the fractional spin operators (12), the quarter spins ${\cal
S}_{{L},{R}}$ are not simple quantum averages of half-integers spins at
different sites but rather sharp quantum observables. The orientations of
these quarter spins depend on the boundary fields and on the parity of the
number of sites $N$ in such a way that (8) is satisfied in all ground states.
Since the fractional spins at each edge are good quantum numbers we may then
label the ground state and the midgap state as $|g(e)\rangle=|{\cal
S}_{{L}},{\cal S}_{{R}}\rangle$. For odd $N$ spin chains these states are
given by $|\pm 1/4,\pm 1/4\rangle$ whereas for even chains they are given by
$|\pm 1/4,\mp 1/4\rangle$. One can easily verify that the total spin is
$S^{z}=\pm 1/2$ and $S^{z}=0$ for the odd and even cases.
Figure 4: Edge spin variance $\delta S_{L}^{2}(N,\alpha)$ in (a) the ground
state (GS) $S^{z}=-\frac{1}{2}$ and (b) the midgap state (ES)
$S^{z}=\frac{1}{2}$ with model parameters $\Delta=3$, $h_{L}=h_{R}=0.2$ and
varying $N$.
We want to point out that the existence of a gap above the ground state and
the midgap state seems to be crucial for the quarter spins to be sharp quantum
observables. Indeed, in the limit $\Delta\rightarrow 1$ where the mass gap
goes to zero, we end up with the $XXX$ Heisenberg chain. In this case it was
found in Pasnoori _et al._ (2023) that although the fractional spin $\pm 1/4$
exist in the ground state, their variance is not zero and hence the fractional
spins are not genuine quantum observables.
## IV Discussion
The first natural question that arises is whether or not the quarter spins
found so far survive in the higher excited states of the spectrum of the XXZ
chain. However, excited states above the midgap state contain propagating
spinons. In such case, even if a quarter spin can be defined on average, we do
not expect its variance to be zero as found for the XXX spin chain with edge
fields Pasnoori _et al._ (2023). Another related question is whether these
quarter spins survive edge fields higher than the critical value
$h_{c}=\Delta-1$. In this regimes there are no midgap states Skorik and Saleur
(1995); S.Skorik and A.Kapustin (1995); Grijalva _et al._ (2019); Nassar and
Tirkkonen (1998) but we believe that sharp quarter spins exist in the ground
state due to the existence of the spectral gap.
We shall end by commenting about the relation between the quarter spins found
in this work with spontaneous symmetry breaking of the $\mathbb{Z}_{2}$
symmetry in the case of zero edge fields, i.e: $h_{L}=h_{R}=0$. In the limit
of zero edge fields the two states $|g(e)\rangle$ become degenerate in the
thermodynamical limit as the bound state energies (3) vanish. Without loss of
generality one may then choose $|g\rangle=|-1/4,-1/4\rangle$ for $N$ odd and
$|g\rangle=|1/4,-1/4\rangle$ for $N$ even. The two linear combinations
$|\pm\rangle=(|g\rangle\pm|e\rangle)/\sqrt{2}$ are eigenstates of $\tau$, i.e:
$\tau|\pm\rangle=\pm|\pm\rangle$. Since, in the same limit each of the two
states $|g(e)\rangle$ are eigenstates of ${\hat{S}}^{z}_{L,R}$, the fractional
spin operators map the two states $|\pm\rangle$ onto each other, i.e:
${\hat{S}}^{z}_{L,R}|\pm\rangle=(-1)^{N}\frac{1}{4}|\mp\rangle$. Hence the
fractional spin operators are zero energy modes (ZEM) in the basis $|g\rangle$
and $|e\rangle$. Notice that, since in this subspace ${\hat{S}}^{z}_{L,R}$ are
not independent as ${\hat{S}}^{z}_{L}=\pm{\hat{S}}^{z}_{R}$ in both states,
there exits only one ZEM say ${\hat{S}}^{z}_{L}$. At this point it is worth
mentioning that the hamiltonian (1) displays the remarkable property,
discovered by P. FendleyFendley (2016), of having a strong zero energy mode
$\Psi_{F}$ in the thermodynamical limit satisfying the following properties
$\displaystyle[\Psi_{F},H]=0,\,\,\,\\{\Psi_{F},\tau\\}=0,\,\,\,\Psi_{F}^{2}=1.$
(18)
The existence of the later operator insure that in the $N\rightarrow\infty$
limit the Hilbert space associated with the XXZ spin chain fractionalizes into
two degenerated towers with eigenvalues $\tau=\pm 1$ which are mapped onto
each other by the action of $\Psi_{F}$. We may therefore conclude that when
projected in the low energy subspace spanned by the ground state and the
midgap state the Fendley operator identifies with the fractional spin operator
$\Psi_{F}\equiv 4{\hat{S}}^{z}_{L}.$ (19)
Of course, since we do not expect a quarter fractional spin to be sharp in all
the excited states, ${\hat{S}}^{z}_{L}$ is not a strong ZEM in contrast with
the Fendley operator $\Psi_{F}$ but rather a soft ZEM. We finally notice that,
following the same lines of arguments as given above, the fractional spin
${\hat{S}}^{z}_{L}$ is also a soft ZEM on the two lines $h_{L}=h_{R}$ for $N$
even and $h_{L}=-h_{R}$ for $N$ odd where the symmetries
$\mathbb{P}\circ\mathbb{Z}_{2}$ and $\mathbb{P}$ are spontaneously broken. It
would interesting to know if a strong zero mode similar to (18) exists on
these two symmetric lines. We hope that the quarter spins found in this work
could be probed in experiments using ultra cold atoms in optical lattices Duan
_et al._ (2003).
###### Acknowledgements.
This work is is partially supported by the Air Force Office of Scientific
Research under Grant No. FA9550-20-1-0136 (J.L.,J.H.P.), NSF Career Grant No.
DMR-1941569 (J.H.P.), and the Alfred P. Sloan Foundation through a Sloan
Research Fellowship (J.H.P.).
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## Appendix A Even N DMRG
Figure 5: Ground state phase diagram of the XXZ model with edge fields smaller
than the critical field $|h_{L,R}|<h_{c}=\Delta-1$ and for an even number of
sites. The ground state as well as the first excited state which is a midgap
state have total spin $S^{z}=0$. Both states host fractional quarter spins
${\cal S}_{L,R}=\pm 1/4$ at both edges of the chain. These fractional spins
are sharp quantum observables which reconstruct the total spin of each state
$S^{z}={\cal S}_{L}+{\cal S}_{R}=0$. The $\pm 1/4$ quarter spins are depicted
by triangles pointing upward and downward respectively. On the separatrix
$h_{L}-h_{R}=0$ the two states are degenerate and the edge spin operator
become a zero energy mode.
### A.1 Edge spin accumulation
The lowest-energy state of the XXZ Hamiltonian defined in Eqn.(1) can be
solved numerically in a matrix product (MPS) form by the density matrix
renormalization group (DMRG) method. The magnetization at site $j$ can be
computed as $S^{z}_{j}=\frac{1}{2}\langle\psi|\sigma^{z}_{x}|\psi\rangle$.
Also, we define an Ansatz for the spin profile as
$\begin{split}S^{z}_{j}=&\Delta
S^{z}(j)+\frac{1}{2}\sigma(\Delta)(-1)^{x}\end{split}$ (20)
with $\sigma(\Delta)$ is the bulk staggered magnetization for a periodic $XXZ$
chain with anisotropy $\Delta$. The numerical results for the validity of
fitting in the lowest two excited states are shown in Figs.(6,7) for even
cites and in Figs.(2) for the odd sites.
With the definition of the edge spin accumulation as
${\cal S}_{L}=\lim_{\alpha\to 0}\lim_{N\to\infty}\sum_{j}e^{-\alpha
x}S^{z}(j,N)=\lim_{\alpha\to 0}\lim_{N\to\infty}S^{z}_{L}(N,\alpha),$ (21)
then ${\cal S}_{L/R}=\pm\frac{1}{4}$ in both of the two lowest energy states,
and they sum up to ${\cal S}_{L}+{\cal S}_{R}=S^{z}$. Noticing, that the non-
converging part is the bulk magnetization if we want to define the edge spin
accumulation on a finite system, we introduce the quantity
$S_{\alpha}=\pm\frac{1}{4}=\sum_{x}\Delta
S_{\alpha}^{z}(x)\pm\frac{1}{2}\sigma(\Delta)$ with $\alpha=L,R$. The edge
spin accumulation is shown in Fig.(8) for a chain with even number of sites
and Fig.(3) for a chain with odd number of sites.
Figure 6: Spin profile $\Delta S^{z}(j)$ and the fitting of ansatz (a) in the
ground state $|e\rangle$ (b) in the midgap state $|g\rangle$ with model
parameters $N=100$, $h_{L}=0.1,h_{R}=0.5$ and for $\Delta=3$. Figure 7: Spin
profile on the left hand side $\Delta S^{z}_{L}(j)$ of the two degenerated
ground states without the presence of magnetic field $h_{L}=h_{R}=0$ for
system size $N=1001$ and (a,b) $\Delta=2$ and (c,d) $\Delta=3$. Spin profile
on the left hand side $\Delta S^{z}_{L}(j)$ of the two degenerated ground
states without the presence of magnetic field $h_{L}=h_{R}=0$ for system size
$N=1000$ and (e,f) $\Delta=2$ and (g,h) $\Delta=3$.
### A.2 Edge Spin Variance
Defining the spin variance operator at the edge for a finite system $N$ and
cutoff $\alpha$ as
$\delta S^{2}_{L}(N,\alpha)=\langle S_{L}^{z}(N,\alpha)^{2}\rangle-\langle
S_{L}^{z}(N,\alpha)\rangle^{2}.$ (22)
The thermodynamic spin variance is defined through the same limit as in
Eq.(21), and the condition that the fractional spin ${\cal S}_{L/R}$ is a
sharp quantum observable is that the variance vanishes:
$\delta S^{2}_{L}\equiv\lim_{\alpha\to\infty}\lim_{N\to\infty}\delta
S_{L}^{2}(N,\alpha)=0.$ (23)
Taking the limit $N\rightarrow\infty$ is challenging, and we circumvent this
issue by assuming an ansatz relating $\delta S^{2}_{L}(N,\alpha)$ and $\delta
S^{2}_{L}(\infty,\alpha)$
$\delta S^{2}_{L}(N,\alpha)=\delta
S_{L}^{2}(\infty,\alpha)-\frac{A}{\Delta}\alpha e^{-B\alpha L}.$ (24)
Then, $\delta S^{2}_{L}=\lim_{\alpha\rightarrow 0}S(\infty,\alpha)$, and we
can calculation the value of $\delta S^{2}_{L}$ in the thermodynamic limit. We
verify this ansatz by taking the difference of $\delta S^{2}_{L}(N,\alpha)$
for different $N$’s. This is shown in Fig.(9) for even chain and Fig.(4) for
odd chain, where one can see that the ansatz fits the data very well.
Therefore, the thermodynamic limit of the variance does vanish, $\delta
S^{2}=0$, and the edge spin is indeed a well-defined quantum observable.
Figure 8: Edge spin accumulation ${\cal\hat{S}}_{\cal L/\cal
R}=\langle{\cal\hat{S}}^{z}_{\cal L/\cal R}\rangle$ in the ground state
$|g\rangle$, and in the midgap state $|e\rangle$. The model parameters used
are $N=1000$, $h_{\cal L}=0$ and $\Delta=3$ with varying $h_{\cal R}$. Figure
9: Edge spin variance $\delta S_{\cal L}^{2}(N,\alpha)$ in (a) the ground
state $S^{z}=-\frac{1}{2}$ and (b) the midgap state $S^{z}=\frac{1}{2}$ with
model parameters $\Delta=3$, $h_{\cal L}=h_{\cal R}=0.2$ and varying $N$.
## Appendix B Bethe Ansatz
### B.1 Hamiltonian
Recalling the Hamiltonian of the system:
$\displaystyle
H=\sum_{j=1}^{N-1}\left[\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}+\Delta(\sigma^{z}_{j}\sigma^{z}_{j+1}-1)\right]$
$\displaystyle+h_{L}\sigma^{z}_{1}+h_{R}\sigma^{z}_{N},$ (25)
where $h_{L},h_{R}$ are magnetic fields at the left and the right edges
respectively. We can introduce new parameters $\gamma$, $h_{c1}$, $h_{c2}$
such that
$\displaystyle\Delta=\cosh\gamma,\gamma>0,\;\;\;\;\;h_{c1}=\Delta-1,\;\;\;\;\;h_{c2}=\Delta+1$
(26)
The Bethe equations can be obtained by following the method of coordinate or
algebraic Bethe ansatz Alcaraz _et al._ (1987); Sklyanin (1988); Wang _et
al._ (2015). One obtains the following Bethe equations for the reference state
with all spin up
$\displaystyle\left(\frac{\sin\frac{1}{2}(\lambda_{j}-i\gamma)}{\sin\frac{1}{2}(\lambda_{j}+i\gamma)}\right)^{2N}\prod_{\alpha}^{L,R}\left(\frac{\sin\frac{1}{2}(\lambda_{j}+i\gamma(1+\epsilon_{\alpha}))}{\sin\frac{1}{2}(\lambda_{j}+i\gamma(1+\epsilon_{\alpha}))}\right)$
$\displaystyle=\prod_{\sigma=\pm}\prod_{k=1}^{M}\left(\frac{\sin\frac{1}{2}(\lambda_{j}+\sigma\lambda_{k}-2i\gamma)}{\sin\frac{1}{2}(\lambda_{j}+\sigma\lambda_{k}+2i\gamma)}\right)$
(27)
where
$\displaystyle
h_{\alpha}=-\sinh\gamma\coth(\frac{\epsilon_{\alpha}\gamma}{2}),\;\;\;\epsilon_{\alpha}=\tilde{\epsilon}_{\alpha}+i\delta_{\alpha}\frac{\pi}{\gamma},$
$\displaystyle\delta_{\alpha}=\begin{cases}1&|h_{\alpha}|<\sinh\gamma\\\
0&|h_{\alpha}|>\sinh\gamma\end{cases}$ (28)
Note that $h_{c1}<\sinh\gamma<h_{c2}$. The Bethe equations for reference state
with all spin down can be obtained by the transformation
$\epsilon_{\alpha}\rightarrow-\epsilon_{\alpha}$ Sklyanin (1988). The energy
of a state described by the set of Bethe roots $\lambda_{j}$ is given by
$\displaystyle E=\frac{1}{2}\left[(N-1)\cosh\gamma+h_{L}+h_{R}\right]$
$\displaystyle-2\sinh\gamma\sum_{j=1}^{M}\frac{\sinh\gamma}{\cosh\gamma-\cos\lambda_{j}}$
(29)
The boundary magnetic fields break the $\mathbb{Z}_{2}$ spin flip symmetry.
Under the spin flip of all the sites, the bulk remains invariant but the
boundary terms remain invariant only after the direction of both the magnetic
fields is reversed, hence we have the following isometry
$\displaystyle\prod_{i=1}^{N}\sigma^{x}_{i}H\sigma^{x}_{i},\;\;h_{L}\rightarrow-
h_{L},\;\;h_{R}\rightarrow-h_{R}.$ (30)
### B.2 Bethe Solution
In this section we construct the ground states and the boundary excitations
with the lowest energy in each of the four sub-phases $A_{j=(1,2,3,4)}$,
corresponding to the domains of the boundary fields $(0\leq h_{L}\leq
h_{c1},0\leq h_{R}\leq h_{c1}),(0\geq h_{L}\geq-h_{c1},0\leq h_{R}\leq
h_{c1}),(0\geq h_{L}\geq-h_{c1},0\geq h_{R}\geq-h_{c1})$ and $(0\leq h_{L}\leq
h_{c1},0\geq h_{R}\geq-h_{c1})$ respectively.
#### B.2.1 Region $A_{1}$: odd number of sites
The region $A_{1}$ corresponds to the following values of the boundary
magnetic fields: $0<h_{L},h_{R}<h_{c1}$. This corresponds to
$\epsilon_{\alpha}=-\tilde{\epsilon}_{\alpha}+i\pi$, with
$\tilde{\epsilon}_{\alpha}<1$, $\alpha=L,R$.
First consider the state with all real $\lambda$, which take values between
$(-\pi,\pi]$. Applying logarithm to B.1 we obtain
$\displaystyle
2N\varphi(\lambda_{j},1)-\sum_{\alpha=L,R}\varphi(\lambda_{j},1-\tilde{\epsilon}_{\alpha})+\varphi(\lambda_{j},1)+\varphi^{\prime}(\lambda_{j},1)=2\pi
I_{j}+\sum_{\sigma=\pm}\sum_{k\neq
j}\varphi(\lambda_{j}+\sigma\lambda_{k},2).$
where
$\displaystyle\varphi(x,y)=\ln\left(\frac{\sin\frac{1}{2}(x-i\gamma
y)}{\sin\frac{1}{2}(x+i\gamma
y)}\right),\;\;\;\;\varphi^{\prime}(x,y)=\ln\left(\frac{\cos\frac{1}{2}(x-i\gamma
y)}{\cos\frac{1}{2}(x+i\gamma y)}\right).$ (32)
We define the counting function $\nu(\lambda)$ such that
$\nu(\lambda_{j})=I_{j}$. Differentiating B.2.1 and using
$\frac{d}{d\lambda}\nu(\lambda)=\rho(\lambda)$, we obtain
$\displaystyle(2N+1)a(\lambda,1)-\sum_{\alpha=L,R}a(\lambda-\pi,1-\tilde{\epsilon}_{\alpha})+a(\lambda-\pi,1)-2\pi\delta(\lambda)-2\pi\delta(\lambda-\pi)=2\pi\rho(\lambda)+\sum_{\sigma=\pm}\int
a(\lambda+\sigma\mu,2)\rho(\mu)d\mu,$
where we have removed the solutions $\lambda=0,\pi$ as they lead to a
vanishing wavefunction Skorik and Saleur (1995). Here
$\displaystyle a(x,y)=\frac{\sinh(\gamma y)}{\cosh(\gamma y)-\cos(\lambda)}.$
(34)
The above equation can be solved by applying Fourier transform
$\displaystyle f(x)=\sum_{k=-\infty}^{\infty}\hat{f}(\omega)e^{i\omega
x},\;\;\;\;\;\;\;\;\hat{f}(\omega)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-i\omega
x}dx.$ (35)
Using $\hat{a}(\omega,y)=e^{-\gamma y|\omega|}$, we obtain the following
density distribution for the state with all real roots
$\displaystyle\hat{\rho}_{\left|\frac{1}{2}\right>_{A_{1}}}(\omega)=\frac{(2N+1)e^{-\gamma|\omega|}+(-1)^{\omega}e^{-\gamma|\omega|}-(1+(-1)^{\omega})-\sum_{\alpha=L,R}(-1)^{\omega}e^{-\gamma(1-\tilde{\epsilon}_{\alpha})|\omega|}}{4\pi(1+e^{-2\gamma|\omega|})}.$
(36)
The reason for the subscripts will become evident when we find the spin
$S^{z}$ of the state. The number of Bethe roots can be obtained by using the
relation
$\displaystyle M=\int_{-\pi}^{\pi}\rho(\lambda)d\lambda.$ (37)
The total spin $S^{z}$ of the state can be found using the relation
$S^{z}=\frac{N}{2}-M$. Using 36 in the above relations we find that the total
spin $S^{z}$ of the state described by the distribution
$\hat{\rho}_{\left|\frac{1}{2}\right>_{A_{1}}}(\omega)$ is
$S^{z}=\frac{1}{2}$. We denote this state by
$\left|\frac{1}{2}\right>_{A_{1}}$.
By starting with the Bethe equations corresponding to all spin down reference
state we have
$\displaystyle(2N+1)a(\lambda,1)-\sum_{\alpha=L,R}a(\lambda-\pi,1+\tilde{\epsilon}_{\alpha})+a(\lambda-\pi,1)-2\pi\delta(\lambda)-2\pi\delta(\lambda-\pi)=2\pi\rho(\lambda)+\sum_{\sigma=\pm}\int
a(\lambda+\sigma\mu,2)\rho(\mu)d\mu$
Following the same procedure as above, we obtain the following distribution
for a state with all real $\lambda$
$\displaystyle\hat{\rho}_{\left|-\frac{1}{2}\right>_{A_{1}}}(\omega)=\frac{(2N+1)e^{-\gamma|\omega|}+(-1)^{\omega}e^{-\gamma|\omega|}-(1+(-1)^{\omega})-\sum_{\alpha=L,R}(-1)^{\omega}e^{-\gamma(1+\tilde{\epsilon}_{\alpha})|\omega|}}{4\pi(1+e^{-2\gamma|\omega|})}.$
(39)
The total spin $S^{z}$ of this state is $S^{z}=-\frac{1}{2}$. We denote this
state by $\left|-\frac{1}{2}\right>_{A_{1}}$.
Using B.1 we can calculate the energy difference between the two states
$\left|\frac{1}{2}\right>_{A_{1}}$ and $\left|-\frac{1}{2}\right>_{A_{1}}$.
We have
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{1}}}-E_{\left|-\frac{1}{2}\right>_{A_{1}}}=h_{L}+h_{R}-2\sinh\gamma\sum_{\alpha=L,R}\int_{-\pi}^{\pi}a(\lambda,1)\;\delta\rho_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\lambda)d\lambda,$
(40)
where
$\delta\rho_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\lambda)$
is the difference in the density distributions of the states
$\left|\frac{1}{2}\right>_{A_{1}}$ and $\left|-\frac{1}{2}\right>_{A_{1}}$.
The expression 40 can be written as
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{1}}}-E_{\left|-\frac{1}{2}\right>_{A_{1}}}=h_{L}+h_{R}+4\pi\sinh\gamma\sum_{\omega=-\infty}^{\infty}\hat{a}(\omega,1)\Delta\hat{\rho}_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\omega).$
(41)
Using 36 and 39 in the above expression we obtain
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{1}}}-E_{\left|-\frac{1}{2}\right>_{A_{1}}}=h_{L}+h_{R}+\sinh\gamma\sum_{\alpha=L,R}\sum_{\omega=-\infty}^{\infty}(-1)^{\omega}\;\frac{\sinh(\gamma\tilde{\epsilon}_{\alpha}|\omega|)}{\cosh(\gamma\omega)}e^{-\gamma|\omega|},$
(42)
which can be written as
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{1}}}-E_{\left|-\frac{1}{2}\right>_{A_{1}}}=m_{L}+m_{R},$
(43)
where
$\displaystyle
m_{\alpha}=h_{\alpha}+\sinh\gamma\sum_{\omega=-\infty}^{\infty}(-1)^{\omega}\;\frac{\sinh(\gamma\tilde{\epsilon}_{\alpha}|\omega|)}{\cosh(\gamma\omega)}e^{-\gamma|\omega|}.$
(44)
The ground state is $\left|-\frac{1}{2}\right>_{A_{1}}.$
#### B.2.2 Region $A_{1}$: Even number of sites
The Bethe equations corresponding to all spin up reference state have two
boundary string solutions $\lambda_{bs\alpha}=\pi+\pm
i\gamma(1-\tilde{\epsilon}_{\alpha})$, $\alpha=L,R$ . Adding either of these
two boundary strings to the Bethe equations B.1 and taking logarithm we obtain
$\displaystyle
2N\varphi(\lambda_{j},1)-\sum_{\alpha=L,R}\varphi(\lambda_{j}-\pi,1-\tilde{\epsilon}_{\alpha})+\varphi(\lambda_{j},1)+\varphi^{\prime}(\lambda_{j},1)-\varphi(\lambda,(3-\tilde{\epsilon}_{\beta}))-\varphi(\lambda,(1+\tilde{\epsilon}_{\beta}))$
$\displaystyle=2\pi I_{j}+\sum_{\sigma=\pm}\sum_{k\neq
j}\varphi(\lambda_{j}+\sigma\lambda_{k},2),$ (45)
where $\beta$ is either $L$ or $R$. Differentiating the above equation with
respect to $\lambda$ and taking the Fourier transform we obtain
$\displaystyle\tilde{\rho}_{\left|0\right>_{\beta
A_{1}}}(\omega)=\tilde{\rho}_{\left|\frac{1}{2}\right>_{A_{1}}}(\omega)+\Delta\tilde{\rho}_{\beta}(\omega),$
(46)
where
$\displaystyle\Delta\tilde{\rho}_{\beta}(\omega)=-\frac{1}{4\pi}(-1)^{\omega}\frac{e^{-\gamma(3-\tilde{\epsilon}_{\beta})|\omega|}+e^{-\gamma(1+\tilde{\epsilon}_{\beta})|\omega|}}{1+e^{-2\gamma|\omega|}}.$
(47)
The spin of the state containing this boundary string can be calculated using
$S^{z}=\frac{N}{2}-M$, where
$\displaystyle M=1+\int_{-\pi}^{\pi}\rho_{\left|0\right>_{\beta
A_{1}}}(\lambda)d\lambda.$ (48)
We obtain $S^{z}_{\left|0\right>_{\beta A_{1}}}=0$, $\beta=L,R$. Hence there
are two states with $S^{z}=0$ that correspond to the presence of the boundary
strings $\lambda_{bsL}$ and $\lambda_{bsR}$.
The energy of the boundary string can be calculated using B.1. We have
$\displaystyle
E_{\lambda_{bs\beta}}=-\frac{2\sinh^{2}\gamma}{\cosh\gamma+\cosh\gamma(1-\tilde{\epsilon}_{\beta})}-2\sinh\gamma\int_{-\pi}^{\pi}a(\lambda-\pi,1)\Delta\rho_{\beta}(\lambda)d\lambda.$
(49)
Using 46 and evaluating the integral one obtains,
$\displaystyle
E_{\lambda_{bs\beta}}=-\frac{2\sinh^{2}\gamma}{\cosh\gamma+\cosh\gamma(1-\tilde{\epsilon}_{\beta})}+\sinh\gamma\sum_{\omega}(-1)^{\omega}e^{-2\gamma|\omega|}\frac{\cosh(\gamma(1-\tilde{\epsilon}_{\beta})\omega)}{\cosh(\gamma\omega)}=-m_{\beta}.$
(50)
Hence the ground state is either $\left|0\right>_{L,A_{1}}$ or
$\left|0\right>_{R,A_{1}}$ depending on the values of $h_{L},h_{R}$.
#### B.2.3 $A_{2}$: Odd number of sites
The region $A_{2}$ corresponds to the following values of the boundary
magnetic fields: $0<h_{R}<h_{c1}$, $-h_{c1}<h_{L}<0$. In this region the
logarithmic form of the Bethe equations can be obtained from B.2.1 by the
transformation $\tilde{\epsilon}_{L}\rightarrow-\tilde{\epsilon}_{L}$. We have
$\displaystyle(2N+1)a(\lambda,1)-a(\lambda-\pi,1-\tilde{\epsilon}_{R})-a(\lambda-\pi,1+\tilde{\epsilon}_{L})+a(\lambda-\pi,1)-2\pi\delta(\lambda)-2\pi\delta(\lambda-\pi)$
$\displaystyle=2\pi\rho(\lambda)+\sum_{\sigma=\pm}\int
a(\lambda+\sigma\mu,2)\rho(\mu)d\mu.$ (51)
Taking Fourier transform we obtain
$\displaystyle\hat{\rho}_{\left|\frac{1}{2}\right>_{A_{2}}}(\omega)=\frac{(2N+1)e^{-\gamma|\omega|}+(-1)^{\omega}e^{-\gamma|\omega|}-(1+(-1)^{\omega})-(-1)^{\omega}e^{-\gamma(1-\tilde{\epsilon}_{R})|\omega|}-(-1)^{\omega}e^{-\gamma(1+\tilde{\epsilon}_{L})|\omega|}}{4\pi(1+e^{-2\gamma|\omega|})}.$
(52)
The number of Bethe roots can be obtained by using the relation
$\displaystyle M=\int_{-\pi}^{\pi}\rho(\lambda)d\lambda.$ (53)
The total spin $S^{z}$ of the state can be found using the relation
$S^{z}=\frac{N}{2}-M$. Using 52 in the above relations we find that the total
spin $S^{z}$ of the state described by the distribution
$\hat{\rho}_{\left|\frac{1}{2}\right>_{A_{2}}}(\omega)$ is
$S^{z}=\frac{1}{2}$. We denote this state by
$\left|\frac{1}{2}\right>_{A_{2}}$.
By starting with the Bethe equations corresponding to all spin down reference
state we have
$\displaystyle(2N+1)a(\lambda,1)-a(\lambda-\pi,1+\tilde{\epsilon}_{R})-a(\lambda-\pi,1-\tilde{\epsilon}_{L})+a(\lambda-\pi,1)-2\pi\delta(\lambda)-2\pi\delta(\lambda-\pi)$
$\displaystyle=2\pi\rho(\lambda)+\sum_{\sigma=\pm}\int
a(\lambda+\sigma\mu,2)\rho(\mu)d\mu.$ (54)
Following the same procedure as above, we obtain the following distribution
for a state with all real $\lambda$
$\displaystyle\hat{\rho}_{\left|-\frac{1}{2}\right>_{A_{1}}}(\omega)=\frac{(2N+1)e^{-\gamma|\omega|}+(-1)^{\omega}e^{-\gamma|\omega|}-(1+(-1)^{\omega})-(-1)^{\omega}e^{-\gamma(1+\tilde{\epsilon}_{R})|\omega|}-(-1)^{\omega}e^{-\gamma(1-\tilde{\epsilon}_{L})|\omega|}}{4\pi(1+e^{-2\gamma|\omega|})}.$
(55)
The total spin $S^{z}$ of this state is $S^{z}=-\frac{1}{2}$. We denote this
state by $\left|-\frac{1}{2}\right>_{A_{2}}$.
Using B.1 we can calculate the energy difference between the two states
$\left|\frac{1}{2}\right>_{A_{2}}$ and $\left|-\frac{1}{2}\right>_{A_{2}}$.
We have
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=-h_{L}+h_{R}-2\sinh\gamma\sum_{\alpha=L,R}\int_{-\pi}^{\pi}a(\lambda,1)\;\delta\rho_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\lambda)d\lambda,$
(56)
where
$\delta\rho_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\lambda)$
is the difference in the density distributions of the states
$\left|\frac{1}{2}\right>$ and $\left|-\frac{1}{2}\right>$. The expression 56
can be written as
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=-h_{L}+h_{R}+4\pi\sinh\gamma\sum_{\omega=-\infty}^{\infty}\hat{a}(\omega,1)\Delta\hat{\rho}_{\tiny{\left|\frac{1}{2}\right>,\left|-\frac{1}{2}\right>}}(\omega).$
(57)
Using 52 and 55 in the above expression we obtain
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=-h_{L}+h_{R}+\sinh\gamma(-1)^{\omega}\;\frac{\sinh(\gamma\tilde{\epsilon}_{R}|\omega|)}{\cosh(\gamma\omega)}e^{-\gamma|\omega|}-\sinh\gamma(-1)^{\omega}\;\frac{\sinh(\gamma\tilde{\epsilon}_{L}|\omega|)}{\cosh(\gamma\omega)}e^{-\gamma|\omega|},$
(58)
which can be written as
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=m_{R}-m_{L}.$
(59)
Hence the ground state for odd number of sites is
$\left|\pm\frac{1}{2}\right>_{A_{2}}$ depending on the values of
$h_{L},h_{R}$.
#### B.2.4 $A_{2}$: Even number of sites
The Bethe equations corresponding to all spin up reference state have two
boundary string solutions $\lambda_{bsR}=\pi\pm
i\gamma(1-\tilde{\epsilon}_{R})$, $\lambda_{bsL^{\prime}}=\pi\pm
i\gamma(1+\tilde{\epsilon}_{L})$. Adding $\lambda_{bsR}$ to the state
$\left|\frac{1}{2}\right>_{A_{2}}$ leads to the state with following root
distribution
$\displaystyle\tilde{\rho}_{\left|0\right>_{\beta
A_{2}}}(\omega)=\tilde{\rho}_{\left|\frac{1}{2}\right>_{A_{2}}}(\omega)+\Delta\tilde{\rho}_{R}(\omega),$
(60)
where $\Delta\tilde{\rho}_{R}$ is given by 47 with $\beta=R$. The spin of the
state containing this boundary string can be calculated using
$S^{z}=\frac{N}{2}-M$, where
$\displaystyle
M=1+\int_{-\pi}^{\pi}\rho_{\left|0\right>_{RA_{2}}}(\lambda)d\lambda.$ (61)
We obtain $S^{z}_{\left|0\right>_{RA_{2}}}=0$. The energy of the boundary
string is given by 50, which is $-m_{R}$.
Adding the boundary string $\lambda_{bsL^{\prime}}$ to the state
$\left|\frac{1}{2}\right>_{A_{2}}$, we obtain
$\displaystyle\tilde{\rho}_{\left|0\right>_{L^{\prime}A_{2}}}(\omega)=\tilde{\rho}_{\left|\frac{1}{2}\right>_{A_{2}}}(\omega)+\Delta\tilde{\rho}_{L^{\prime}}(\omega),$
(62)
where
$\displaystyle\Delta\tilde{\rho}_{L^{\prime}}(\omega)=-\frac{1}{4\pi}(-1)^{\omega}\frac{e^{-\gamma(3+\tilde{\epsilon}_{L})|\omega|}+e^{-\gamma(1-\tilde{\epsilon}_{L})|\omega|}}{1+e^{-2\gamma|\omega|}}.$
(63)
The spin of the state containing this boundary string can be calculated using
$S^{z}=\frac{N}{2}-M$, where
$\displaystyle
M=1+\int_{-\pi}^{\pi}\rho_{\left|0\right>_{L^{\prime}A_{2}}}(\lambda)d\lambda.$
(64)
We obtain $S^{z}_{\left|0\right>_{L^{\prime}A_{2}}}=0$. The energy of the
boundary string $\lambda_{bsL^{\prime}}$ is given by
$\displaystyle
E_{\lambda_{bs\beta^{\prime}}}=-\frac{2\sinh^{2}\gamma}{\cosh\gamma+\cosh\gamma(1+\tilde{\epsilon}_{\beta})}+\sinh\gamma\sum_{\omega}(-1)^{\omega}e^{-2\gamma|\omega|}\frac{\cosh(\gamma(1+\tilde{\epsilon}_{\beta})\omega)}{\cosh(\gamma\omega)}=m_{\beta},$
(65)
with $\beta=L$. The energy difference between the states
$\left|0\right>_{L^{\prime}A_{2}}$ and $\left|0\right>_{RA_{2}}$ can be
calculated similar to the previous section, we obtain
$\displaystyle
E_{\left|0\right>_{L^{\prime}A_{2}}}-E_{\left|0\right>_{RA_{2}}}=m_{L}+m_{R}.$
(66)
Hence the ground state for even number of sites is $\left|0\right>_{RA_{2}}$.
#### B.2.5 $A_{3}$ and $A_{4}$ sub-phases
In constructing a state in the phase $A_{3}$ or $A_{4}$, we can use the
construction of the respective state in the phase $A_{1}$ or $A_{2}$
respectively, and use the following transformation:
$\displaystyle|\uparrow\uparrow...\uparrow\rangle\leftrightarrow|\downarrow\downarrow...\downarrow\rangle,\hskip
14.22636pth_{L}\rightarrow-h_{L},\hskip 11.38109pth_{R}\rightarrow h_{R},$
(67)
where the all spin up and all spin down reference states are interchanged and
the boundary magnetic fields change sign.
### B.3 Summary of Bethe Solution
In this section we summarize the construction of the Bethe solution obtained
above
#### B.3.1 Odd number of sites
##### The $A_{1}$ and $A_{3}$ sub-phases.
In these cases both boundary magnetic fields point towards the same direction:
along the positive $z$ axis for the $A_{1}$ sub-phase and negative $z$ axis
for the $A_{3}$ sub-phase. Both cases are related by the isometry (30).
Qualitatively speaking, in the sub-phases $A_{1,3}$ and for $N$ odd, the
boundary magnetic fields are not frustrating in the sense that in the Ising
limit of (B.1) the ground-state would exhibit perfect antiferromagnetic order.
1) In the $A_{1}$ phase we find that the ground-state is unique and has a
total spin $S^{z}=-\frac{1}{2}$. It is constructed by starting with all spin
down reference state and contains $\frac{N-1}{2}$ real roots. This state is
labelled by $\left|-\frac{1}{2}\right>_{A1}$.
2) In the $A_{1}$ phase, there exists an excited state with total spin
$S^{z}=+\frac{1}{2}$, which does not contain any spinons. It is constructed by
starting with all spin up reference state and contains $\frac{N-1}{2}$ real
roots. This state is labelled by $\left|+\frac{1}{2}\right>_{A1}$.
3)The energy difference between these two states in the $A_{1}$ phase is
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{1}}}-E_{\left|-\frac{1}{2}\right>_{A_{1}}}=m_{L}+m_{R}.$
(68)
4) All the states in the $A_{3}$ phase can be obtained by using the symmetry
67 described above.
##### The $A_{2}$ and $A_{4}$ sub-phases.
In these cases the boundary fields are frustrating for $N$ odd in the sense
discussed above.
1) In the $A_{2}$ phase, for $|h_{L}|<|h_{R}|$, the ground state has total
spin $S^{z}=-\frac{1}{2}$. It is constructed by starting with all spin down
reference state and contains $\frac{N-1}{2}$ real roots. This state is
labelled as $\left|-\frac{1}{2}\right>_{A2}$.
2) In the $A_{2}$ phase, there exists an excited state with total spin
$S^{z}=+\frac{1}{2}$, which does not contain any spinons. It is constructed by
starting with all spin up reference state and contains $\frac{N-1}{2}$ real
roots. This state is labelled by $\left|+\frac{1}{2}\right>_{A2}$.
3)The energy difference between these two states in the $A_{2}$ phase is
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=m_{R}-m_{L}.$
(69)
4) From the above expression, one can infer that in the $A_{2}$ phase, for
$|h_{L}|>|h_{R}|$, the state $\left|+\frac{1}{2}\right>_{A2}$ is the ground
state and the state $\left|-\frac{1}{2}\right>_{A2}$ is an excited state.
5) Using the symmetry 30, we can obtain all the states in the sub-phase
$A_{4}$ from the states in the sub-phase $A_{2}$.
##### Phase diagram
Having obtained the solution in the region where $|h_{L,R}|<h_{c1}$ for odd
number of sites chain, we can now obtain the phase diagram shown in the main
text 1. The region which corresponds to $h_{R}+h_{L}>0$, can be divided into
three regions: (a) $h_{R},h_{L}>0$ (b) $h_{R}>0,h_{L}<0,|h_{R}|>|h_{L}|$ (c)
$h_{R}<0,h_{L}>0,|h_{R}|<|h_{L}|$.
1) The region (a) is just the phase $A_{1}$ described previously. The ground
state and the first excited states are $\left|-\frac{1}{2}\right>$ and
$\left|\frac{1}{2}\right>$ respectively.
2) The region (b) is contained within the phase $A_{2}$. From the energy
difference between the states $\left|-\frac{1}{2}\right>$ and
$\left|\frac{1}{2}\right>$ 69, we can infer that the ground state and the
first excited states are $\left|-\frac{1}{2}\right>$ and
$\left|\frac{1}{2}\right>$ respectively.
3) The region (c) is contained within the phase $A_{4}$. The energy difference
between the states $\left|-\frac{1}{2}\right>$ and $\left|\frac{1}{2}\right>$
can be obtained by using 30, 69:
$\displaystyle
E_{\left|\frac{1}{2}\right>_{A_{2}}}-E_{\left|-\frac{1}{2}\right>_{A_{2}}}=-m_{R}+m_{L}$
(70)
Hence, the ground state and the first excited states are again
$\left|-\frac{1}{2}\right>$ and $\left|\frac{1}{2}\right>$ respectively.
4) Using 30, one can obtain all the states on the other side of the separatrix
$h_{R}+h_{L}=0$, which corresponds to $h_{L}+h_{R}<0$. We find that the ground
state and the first excited states in this region are
$\left|\frac{1}{2}\right>$ and $\left|-\frac{1}{2}\right>$ respectively.
5) On the separatrix, using 69, 70, we find that the two states
$\left|-\frac{1}{2}\right>$ and $\left|\frac{1}{2}\right>$ are degenerate.
#### B.3.2 Even number of sites
##### The $A_{1}$ and $A_{3}$ sub-phases.
1)In the $A_{1}$ phase, for $h_{R}<h_{L}$, the ground state has total spin
$S^{z}=0$. It is constructed by starting with all spin up reference state and
contains $\frac{N-2}{2}$ real roots and the boundary string corresponding to
the left edge $\lambda_{bsL}=\pi+\pm i\gamma(1-\tilde{\epsilon}_{L})$. This
state is represented by $\left|0\right>_{L,A_{1}}$.
2) In the $A_{1}$ phase, for $h_{R}<h_{L}$, there exists an excited state
which does not contain any spinons. This state has total spin $S^{z}=0$ and is
constructed by starting with all spin up reference state and contains
$\frac{N-2}{2}$ real roots and the boundary string corresponding to the right
edge $\lambda_{bsR}=\pi+\pm i\gamma(1-\tilde{\epsilon}_{R})$. This state is
represented by $\left|0\right>_{R}$.
3) The energy difference between these two states is
$\displaystyle
E_{\left|0\right>_{L,A_{1}}}-E_{\left|0\right>_{R,A_{1}}}=m_{R}-m_{L}.$ (71)
4) One can infer from the above expression that, for $h_{R}>h_{L}$, the state
$\left|0\right>_{R,A_{1}}$ is the ground state and the state
$\left|0\right>_{L,A_{1}}$ is the excited state which does not contain any
spinons.
5) Using the symmetry 30, we can obtain all the states in the sub-phase
$A_{3}$ from the states in the sub-phase $A_{1}$.
##### The $A_{2}$ and $A_{4}$ sub-phases.
1) In the sub-phase $A_{2}$, the ground state has total spin $S^{z}=0$. It is
constructed by starting with all spin up reference state and contains
$\frac{N-2}{2}$ real roots and the boundary string corresponding to the right
edge $\lambda_{bsR}=\pi+\pm i\gamma(1-\tilde{\epsilon}_{R})$. This state is
represented by $\left|0\right>_{R,A_{2}}$.
2) In the sub-phase $A_{2}$, there exists an excited state which does not
contain any spinons, and has total spin $S^{z}=0$. It is constructed by
starting with all spin up reference state and contains $\frac{N-2}{2}$ real
roots and the boundary string corresponding to the left edge
$\lambda_{bsL^{\prime}}=\pi+\pm i\gamma(1+\tilde{\epsilon}_{L})$. This state
is represented by $\left|0\right>_{L^{\prime},A_{2}}$.
3) The energy difference between these two states is
$\displaystyle
E_{\left|0\right>_{L^{\prime}A_{2}}}-E_{\left|0\right>_{RA_{2}}}=m_{L}+m_{R}.$
(72)
4) Using the symmetry 30, we can obtain all the states in the sub-phase
$A_{4}$ from the states in the sub-phase $A_{2}$.
##### Phase diagram
Having obtained the solution in the region where $|h_{L,R}|<h_{c1}$ for even
number of sites chain, we can now obtain the phase diagram for even number of
sites 5. The region which corresponds to $h_{R}>h_{L}$, can be divided into
three regions: (d) $h_{R}>h_{L}>0$ (e) $h_{R}>0,h_{L}<0$ (f)
$h_{R}<0,h_{L}<0,|h_{R}|<|h_{L}|$.
1) The region (e) is just the phase $A_{2}$ described previously. The ground
state and the first excited states have total spin $S^{z}=0$ containing
boundary strings $\lambda_{bsR}$, $\lambda_{bsL^{\prime}}$ respectively as
discussed above.
2) The region (d) is contained within the phase $A_{1}$. From the energy
difference between the two states 71, we can infer that the ground state and
the first excited states have total spin $S^{z}=0$ and contain the boundary
strings $\lambda_{bsR}$, $\lambda_{bsL}$ respectively as described above.
3) The region (f) is contained within the phase $A_{3}$. The ground state and
the first excited states and their energies can be obtained by using 30, 71,
and we find that the ground state and the first excited states are again same
as in the region (e) described above.
4) Using 30, one can obtain all the states on the other side of the separatrix
$h_{R}=h_{L}$, which corresponds to $h_{L}>h_{R}$. We find that the ground
state and the first excited states in this region are respectively the first
excited state and the ground state corresponding to the region $h_{L}<h_{R}$
discussed above.
5) On the separatrix, using 71, 72, we find that these two states are
degenerate.
|
# Photon Conversion and Interaction on Chip
Jia-Yang Chen These authors contributed equally Zhan Li These authors
contributed equally Zhaohui Ma Chao Tang Heng Fan Yong Meng Sua Yu-Ping
Huang Electronic address [1]<EMAIL_ADDRESS>
Electronic address [2]<EMAIL_ADDRESS>Department of Physics, Stevens
Institute of Technology, 1 Castle Point Terrace, Hoboken, New Jersey, 07030,
USA Center for Quantum Science and Engineering, Stevens Institute of
Technology, 1 Castle Point Terrace, Hoboken, New Jersey, 07030, USA
###### Abstract
The conversion and interaction between quantum signals at a single-photon
level are essential for scalable quantum photonic information technology.
Using a fully-optimized, periodically-poled lithium niobate microring, we
demonstrate ultra-efficient sum-frequency generation on chip. The external
quantum efficiency reaches $(65\pm 3)\%$ with only $(104\pm 4)$ $\mu$W pump
power, improving the state-of-the-art by over one order of magnitude. At the
peak conversion, $3\times 10^{-5}$ noise photon is created during the cavity
lifetime, which meets the requirement of quantum applications using single-
photon pulses. Using pump and signal in single-photon coherent states, we
directly measure the conversion probability produced by a single pump photon
to be $10^{-5}$—breaking the record by 100 times—and the photon-photon
coupling strength to be 9.1 MHz. Our results mark a new milestone toward
quantum nonlinear optics at the ultimate single photon limit, creating new
background in highly integrated photonics and quantum optical computing.
Unlike electrons, atoms, or any other material particles, photons do not
interact with each other in vacuum. Even when mixed in optical media of the
best known nonlinearities, their interaction is so weak that high optical
intensities are needed to produce an appreciable effect. This inefficiency
accounts for significant difficulties facing practical implementations of
quantum transduction [1], faithful entanglement swapping [2, 3], and heralded
entanglement generation [4, 5], to name a few. It also prohibits the
construction of nonlinear photon-photon gates, thus forming a bottleneck for
the development of scalable quantum computers at room temperature [6].
The recent advances in nanophotonics bring hopes to overcome this challenge,
by providing tight optical confinement, strong mode overlap, and extended
interaction length. Encouragingly, optical processes with improved efficiency
have now been demonstrated in nanophotonic circuits made of silicon nitride
[7, 8], aluminum nitride [9], gallium arsenide [10], gallium phosphide [11],
aluminium gallium arsenide[12] and lithium niobate [13, 14]. Among various
candidates, thin-film lithium niobate (TFLN) on insulator has quickly arisen
to a material platform of choice, due to its favorable ferro-electricity, wide
optical transparency window, strong second-order nonlinearity $\chi^{(2)}$,
and outstanding electro-optical responses. Compared with third-order nonlinear
($\chi^{(3)}$) materials, TFLN is centro-asymmetric and possesses
exceptionally large second-order nonlinearity ($\chi^{(2)}$) to produce orders
of magnitude stronger effects, as desirable to optical nonlinearities at a
single photon level. Thus far, a variety of TFLN microresonators have been
demonstrated with impressive nonlinearities [15, 16, 17, 13, 14, 18]. However,
their performance has been capped by the use of relatively small $\chi^{(2)}$
susceptibilities (e.g., $d_{31}\sim$ 4.7 pm/V) [15], poor mode overlapping
[16, 18], and/or low photon-extraction efficiency [15, 16, 17, 13]. For photon
conversion and interaction, while their high efficiency has been predicted
from, e.g., second-harmonic generation (SHG) [15, 16, 17] or parametric
downconversion experiments [13], there has been no direct demonstration.
Here, we present a TFLN resonator that overcomes all aforementioned
shortcomings and delivers its promised high efficiency for photon conversion
and interaction. As illustrated in Fig.1(a), it is an overly coupled, triply
resonant, periodically poled lithium niobate (PPLN) microring resonator for
sum-frequency generation (SFG). All interacting light waves are in the low-
loss fundamental modes with nearly perfect overlap and interact through TFLN’s
largest $\chi^{(2)}$ susceptibility tensor element (e.g., $d_{33}\sim$ 27
pm/V). It achieves an impressive photon-photon coupling strength of $g$ = 9.1
MHz (angular frequency). Crucially, by strongly overcoupling the cavity to
minimize the extraction loss of the sum-frequency (SF) photons, we demonstrate
photon conversion at a record-high external efficiency of $65\%$ with only
about 100 $\mu$W pump power, marking orders of magnitude improvement over the
state of the art across all existing photonic platforms; see Table 1. At the
peak conversion, the on-chip noise photon flux is only $3\times 10^{-5}$
photons per 100-ps cavity lifetime, despite small-detuning pumping. This
ultrahigh external efficiency yet low noise create new opportunities in
various applications like quantum frequency conversion, optical squeezing, and
phase sensitive amplification.
Material Platform | Structure | Nonlinear Process | $Q_{l}$ ($\times 10^{5}$) | $\eta_{\mathrm{con}}$ | Pump Power | Ref.
---|---|---|---|---|---|---
Si3N4 | microring | FWM-BS | 1.5/1.5/2.4 | 60% | 50 & 8 mW | [7]
Si3N4 | microring | DFWM | 2.8/3.0/1.2 | 13% | 0.33 mW | [8]
AlN | microring | SFG | 3.0/3.0/1.4 | 42% | 35 mW | [9]
PPLN | microring | SHG | 1.3/3.0 | 10% | 0.3 mW | [13]
PPLN | microring | SHG | 1.5/0.6 | 26% | 1.8 mW | [14]
PPLN | millimeter disk | SHG | 120/80 | 18% | 9 mW | [19]
PPLN | microring | SFG | 1.3/1.3/2.6 | 65% | 0.1 mW | this work
Table 1: The state-of-the-art frequency conversion in $\chi^{(2)}$ and
$\chi^{(3)}$ cavities. $\eta_{\mathrm{con}}$: conversion efficiency by photon
number; FWM-BS: four-wave mixing Bragg scattering; DFWM: degenerate four-wave
mixing. $Q_{l}$ lists the loaded quality factors for the pump, signal and SF
waves in the case of FWM-BS, DFWM, and SFG, and the pump and second-harmonic
waves in the case of SHG. This table only includes those whose conversion
efficiency is over $10\%$. Note: a recent arxiv preprint reported SHG of
$\eta_{\mathrm{con}}=33\%$ [18]. According to its reported normalized
efficiency of $602\%$/mW, the required pump power shall be
$P_{p}=16~{}\frac{66\%}{602\%/\mathrm{mW}}=1.75~{}\mathrm{mW}$ [15, 14], which
is over 16 times larger than its claimed value. We have not included it in
this table due to this apparent inconsistence. Figure 1: Integrated TFLN
circuits for photon conversion and interaction. (a) Schematic of the Z-cut
periodically poled microring, where the pump of $\omega_{p}$ and signal of
$\omega_{s}$ couple into the microring and generate the SF light of
$\omega_{f}$ via a $\chi^{(2)}$ process. A pulley coupler is designed for over
coupling all light waves, for high photon extraction efficiency. Insets
illustrate triply resonant and quasi-phase matching conditions. (b) and (c)
are the SEM images of the etched pulley coupler before poling process and the
periodic poled microring after removing the poling electrodes, respectively.
Beside the large coupling strength, our device’s high cavity quality for all
interacting waves provides extended interaction length to boost optical
nonlinearities towards the single photon regime. To assess this prospect, we
further perform photon interaction between two single-photon signals in weak
coherent states. Our measurement directly shows that with only one pump photon
in the microring, the external quantum efficiency for signal photon conversion
is $\sim 10^{-5}$, while the best efficiency reported hitherto is $10^{-7}$
[20]. The internal Rabi oscillation angle produced by the pump photon is 0.01,
which can be improved to approach unity by further reducing the cavity loss.
As a whole, our results constitute a new milestone along the long pursuit of
nonlinear optics in its ultimate quantum limit, where a single photon is
enough to produce significant nonlinear effects. While there is still a good
journey to take before hitting the finish line, the fact that we can attain
the theoretical performance of this device is critical for us to take steps
forward. By further improving the cavity quality, strong photon-photon
interaction is within sight. Based on the current nonlinear parameters, a
cavity qualify factor of $10^{8}$—which has been demonstrated in lithium
niobate microdisks [18]—will enable C-NOT gate between single photons.
Meanwhile, the demonstrated photon conversion and interaction efficiency can
already elevate the performance of nonlinear-optical devices for heralded
entanglement generation, faithful entanglement swapping, and so on.
Device design: In $\chi^{(2)}$ cavity, the effective Hamiltonian describing
SFG between photons in their single modes is
$\hat{H}_{\textrm{eff}}=\hbar\mathnormal{g}(\hat{a}_{p}\hat{a}_{s}\hat{a}_{f}^{\dagger}+\hat{a}_{p}^{\dagger}\hat{a}_{s}^{\dagger}\hat{a}_{f}),\\\
$ (1)
where $\\{\hat{a}_{j}\\}$ are the annihilation operators with $j=p,s,f$
standing for the pump, signal and SF light, respectively, each with angular
frequency $\omega_{j}$. ${g}$ is the photon-photon coupling strength, which
can be interpreted as the effective Rabi frequency produced by a pump photon
(see Supplementary Material.1). It is given by
$g\propto\frac{d_{\textrm{eff}}\xi}{\sqrt{V_{\textrm{eff}}}}\times\delta(m_{f}-m_{p}-m_{s}-M)$
(2)
where $d_{\textrm{eff}}$ is the effective nonlinear susceptibility. $\xi$ is
the mode overlapping factor. $V_{\textrm{eff}}$ is the effective mode volume.
$m_{j}$ is the azimuthal order of the cavity modes, and $M$ is the azimuthal
poling grating number, so that $\delta(m_{f}-m_{p}-m_{s}-M)$ accounts for
quasi-phase matching (QPM) by periodic poling.
In this work, we use a microring with a radius of 80 $\mu$m and a cross-
section of 600 nm in height and 1700 nm in top-width. The pump and signal are
both in the telecom C-band and their SF is in the visible band, chosen so with
repeater-based quantum communications in mind. To maximize $g$, all three
waves are in the fundamental quasi-transverse-magnetic (quasi-TM) modes and
interact through TFLN’s largest nonlinear tensor $d_{33}$ with over 90% mode
overlap. As shown in Fig.1, concentric periodic poling is applied to the
microring for QPM. To ensure triple resonances for all waves, fine temperature
tuning ($\sim$ 0.01 ∘C) is applied to compensate for any resonant mismatch due
to any fabrication error.
For photon conversion and interaction, the device figure of merit is the
external quantum efficiency (QE) defined as
$\eta_{\mathrm{QE}}=\frac{N_{f}}{N_{s}},$ (3)
where $N_{s}$ is the number of input signal photons to the cavity and $N_{f}$
is that of converted SF photons at the cavity output (thus accounting for any
internal cavity loss). This is in contrast to previous demonstrations, where
critical coupling was adopted to maximize the intracavity optical power for
high conversion efficiency [15, 16, 17, 13]. However, most of the input power
and about half of the converted photons are lost inside the cavity, rendering
a rather low QE while prohibiting cascaded operations. For practical
applications, one instead needs to over-couple the cavity so that the photons
can be extracted out before significantly lost in the cavity.
Under QPM and triple resonance (see Supplementary Material.1), the maximum QE
is given by [21]:
$\displaystyle\eta_{\mathrm{QE}}^{\mathrm{max}}\approx\frac{Q_{s,l}}{Q_{s,c}}\frac{Q_{f,l}}{Q_{f,c}},$
(4)
where $Q_{j,o}$ is the quality factor with $o=c,l$ denoting the coupling and
loaded Q, respectively. It is reached with an optimal pump power
$\displaystyle P_{p}^{\mathrm{opt}}\approx
8~{}\frac{\eta_{\mathrm{QE}}^{\mathrm{max}}}{\eta_{\mathrm{tran}}^{\mathrm{nor}}},$
(5)
where $\eta_{\mathrm{tran}}^{\mathrm{nor}}=P_{f}/(P_{s}P_{p})$ is the
normalized power transduction efficiency with $P_{j}$ the optical power of the
$j$-th wave. In our case, $\eta_{\mathrm{QE}}^{\mathrm{max}}\approx 65\%$ and
$\eta_{\mathrm{tran}}^{\mathrm{nor}}\approx 4.5\%/\mu$W, so that the optimal
power is around 115 $\mu$W.
We use a pulley coupler in optimized dimensions to achieve proper over-
coupling for both the signal and SF modes. This is done by first determining
its top width by requiring $n_{p}R_{p}=n_{r}R_{r}$ [22], where $n_{p,r}$ are
the effective refractive indices of the pulley waveguide and microring modes
for the sum-frequency wave, and $R_{p}$ and $R_{r}$ denote their radii. For
the SF wave, the waveguide-microring coupling strength is proportional to the
length of the pulley coupler, while inversely proportional to their gap. For
the signal mode, in contrast, the coupling strength varies as
$\mathrm{sinc}(\Delta\Phi)$, where $\Delta\Phi$ is the coupling phase
mismatch. This allows to carefully design the dimensions to create the
desirable over-coupling for both signal and sum-frequency waves.
Figure 2: (a) Optical spectra of the interacting TM00 cavity modes at (i)
1560.15 nm, (ii) 1551.85 nm, and (iii) 778.00 nm, respectively. (b)
Illustration of possible parametric conversion processes in the microring. (c)
Original optical spectrum before any optical filtering when strong pump and
weak signal are applied. In high conversion region, signal starts to be
depleted. Insert gives the zoom-in spectrum around the SF band. After
correcting for the coupling loss, the on-chip power of pump, signal and SF
waves are -13.8 dBm, -44.3 dBm and -44.0 dBm, respectively.
Device characterization: The details of our device fabrication and experiment
setup are presented in Method and Supplementary .2. The fiber-chip-fiber
coupling losses are measured to be 8 $\pm$ 0.15 dB at 1556 nm and 9.5 $\pm$
0.2 dB at 778 nm, respectively. To find phase matching, we first search for
strong SHG across the C-band by sweeping an infrared laser while optimizing
the chip’s temperature. This gives several cavity modes around 1556 nm, based
on which we select a set of cavity modes at 1560.15 nm, 1551.85 nm, and 778.00
nm, considering the over-coupling requirement and limited by our visible band-
pass filter (BPF, bandwidth $\sim$3 nm, 760 to 780 nm). To reduce the Raman
background, we designate the 1560.15 nm mode to the pump. For this set,
$m_{p}=602$, $m_{s}=606$, $m_{f}=1357$, so that $M=149$. The loaded quality
factors $Q_{j,l}$ are measured for each modes, while the coupling $Q_{j,c}$
and intrinsic $Q_{j,0}$ factors are calculated by fitting the resonance
spectra, as shown in Fig. 2 (a). Those Q’s, according to Eq. (2), give the
highest possibl external quantum efficiency of
$\eta_{\mathrm{QE}}^{\mathrm{max}}\approx 65\%$. For an even higher
efficiency, the cavity needs to be further overcoupled.
Low-noise Frequency Conversion: We perform SFG using the setup detailed in
Fig. S1. We first couple a strong pump at 1560 nm and a weak signal at 1552 nm
into the microring, and fine tune the microring temperature and the laser
wavelengths to verify QPM and triple resonance. The resulting spectrum,
measured without any filtering thus manifesting all possible nonlinear
processes, is shown in Fig. 2(c). It exhibits a clean profile with low
baseline, except for a residual second-harmonic peak at $\sim$780 nm by the
strong pump. This peak is significant only in the high conversion regime
(i.e.,$>$50$\%$) and can be conveniently filtered out. In this experiment, we
reject it using a $\sim$3 nm bandpass filter centered at 778 nm. The otherwise
low background over the entire spectral range shows that all other competing
processes are well suppressed, as desirable for quantum applications.
In this experiment, the signal power is fixed to be about 37 nW on chip, while
the pump power is gradually increased. During the measurement, we only need to
slightly optimize the temperature within 45 $\pm$ 0.3 ∘C and tune the
wavelengths of both lasers within $\pm$ 20 pm to compensate for slight phase
mismatch and resonance drift caused by thermo-optical and photorefractive
effects [14]. The quantum efficiency as a function of the on-chip pump power
is shown in Fig. 3. Thanks to the over-coupling and nearly ideal poling,
$(65\pm 3)\%$ quantum efficiency is achieved with at $(104\pm 4)$ $\mu$W pump
power. Taking into account all insertion losses both on and off chip, this
corresponds to about $9\%$ total efficiency. In the low conversion region, the
normalized power transduction efficiency is fitted to be about 4.5$\%$
$\mu$W-1. By fitting the experimental data with the steady-state solution to
Eqs.(S1-S3), as shown in Fig. 3, the photon-photon coupling coefficient $g$ is
determined to be 8.2 MHz.
The above results speaks to the ultrahigh efficiency. For quantum frequency
conversion, it is critical that no significant in-band noise is injected
during the conversion. As illustrated in Fig. 2(b), there are multiple
processes that can produce in-band noise, such as Raman scattering from strong
classical pump to the signal band followed by SFG, Raman scattering by the
pump’s residue SH light, and spontaneous parametric down-conversion (SPDC)
followed by SHG and SFG. To quantify their total contributions, we measure the
noise photon flux generated in the SF band when only the pump is applied. To
ensure total rejection of any out-band noise, the SF photons are passed
through additional free-space filters (see Supplementary Material .2), before
detected by a silicon-based single-photon detector (Si-SPD, quantum
efficiency: 50$\%$, dark count: 250 Hz). The results are plotted in Fig. 3,
where the increase of the noise photon flux is between quadratic and cubic
with the pump power. This result indicates that besides Raman scattering, the
cascaded SPDC and SFG process also presents, similar to what we observed in a
PPLN nanowaveguide [23]. At the $65\%$ peak QE, the on-chip noise photon flux
is 0.3 MHz under continuous-wave pumping. If using $\sim$100 ps pulses that
match the cavity lifetime, the noise photon per pulse is $3\times 10^{-5}$,
which is low especially given the small detuning between the pump and signal
that are both in the telecom C-band. This noise level can be substantially
lowered by, for example, further detuning the pump from the signal [24].
Figure 3: SFG efficiency $\eta_{\mathrm{QE}}$ (left-axis, red square) and the
generated noise photon flux $N_{\mathrm{noise}}$(right-axis, black square)
plotted against the on-chip pump power. $\eta_{\mathrm{QE}}$ = 65$\%$ is
obtained at the pump power around 100 $\mu$W. Solid red curve represents the
prediction of the coupled mode equations (see Supplementary Material.1) using
the actual parameters of the microring with only one fitting parameter: $g$ =
8.2 MHz. Solid black line is fitted to the pump power $P$ as
$N_{\mathrm{noise}}\sim P^{2.4}$. The error bars in left-axis and x-axis are
estimated according to coupling instability. The error bar of noise photon in
right-axis is estimated by uncertainty assuming Poissonian photon counting
statistics. Figure 4: Quantum efficiency $\eta_{\mathrm{QE}}$ versus the
intracavity mean pump photon number. Detector dark counts of 250 Hz have been
subtracted, and the coupling loss of 4.75 dB and detector efficiency of 50$\%$
have been accounted for. Inset is an zoom-in to shown a quantum effciency of
$\eta_{\mathrm{QE}}=10^{-5}$ is achieved at one intracavity pump photon
(corresponding to photon flux $N_{p}=2.8$ GHz), where the intracavity signal
photon number is fixed at 0.25 (corresponding to photon flux $N_{s}=0.7$ GHz).
Solid red line represents the simulated results using the actual parameters of
the microring with only one free parameter $g$ = 9.1 MHz, fitted with the
coupled-mode equations (see Supplementary Material.1). All error bars are
estimated assuming Poissonian photon counting statistics. Figure 5:
Illustration of a TFLN integrated photonic circuit for heralded entanglement
generation.
Interaction Between Photon-level Coherent States: Strong interaction between
single photons is of great values for both fundamental optics studies and
quantum applications such as faithful quantum entanglement swapping [2, 3],
heralded entanglement source [4, 5], and device-independent quantum key
distribution [25]. Here, we characterize the device responses in the single
photon regime by attenuating the pump and signal to weak coherent states at
single photon levels. Their photon fluxes are carefully monitored by two
superconducting single-photon detectors (SNSPDs, see Supplementary Material
.2).
To maximize the detection efficiency for the SF photons, we remove the free-
space filtering system—used in the last section to reject background noise
created by the strong pump—and directly couple the SF photons from the chip to
the Si-SPD via a lensed fiber. To ensure that the pump photons do not generate
background counts (that could contribute to the over-estimation of the single-
photon nonlinearity), we block the signal and verify that even at the highest
on-chip flux ($\sim$ 5 GHz), the photon counts remain at its dark count level
($\sim$ 250 Hz). Here, the total detection efficiency for the sum-frequency
photons is $\eta=\eta_{d}\eta_{c}\approx 17\%$, with detector efficiency
$\eta_{d}\sim 50\%$ and coupling efficiency $\eta_{c}\sim 33.5\%$.
The measurement results are shown in Fig. 4, where we observe linear
dependency of the quantum efficiency on the intracavity pump photon number.
The on-chip quantum efficiency is about $10^{-5}$ when there is one pump
photon on average in the cavity, after correcting for the coupling loss and
finite detector efficiency. By fitting with the coupling-mode equations, we
extract the photon-photon coupling coefficient $g$ to be 9.1 MHz, compared to
8.2 MHz as from the classical measurement. This slightly higher value may come
from better QPM or triple resonance for single-photon light waves, due to the
absence of thermo-optical and photorefrative effects.
The above quantum efficiency gives the probability of a signal photon at the
cavity input being converted to its SF and appear at the cavity output. It has
accounted for the signal coupling loss and the SF extraction loss, thus
constituting a direct measure of the device figure of merit that dictates its
performance in quantum applications. Another measure, of less practical
relevance but nonetheless describing an intrinsic property, is the internal
Rabi rotation angle $\theta=2gQ_{p,i}/\omega_{p}$. That’s, Eq. (1) can be
interpreted as the pump induced Rabi oscillation between signal and SF
photons. $\theta$ then gives how much Rabi rotation can a pump photon induce
during it is lost in the cavity. In our case, $g=9.1\times 10^{6}$ and
$Q_{p,i}=7.1\times 10^{5}$, so that $\theta=0.01$, which is already
appreciable from a fundamental standpoint. Increasing $\theta$ to $\pi/2$
would require improving $Q_{p,i}$ to $10^{8}$, which has been demonstrated in
polished TFLN microdisks [18].
Heralded Entanglement Generation: The demonstrated photon-level nonlinearity
implies unprecedented performance in quantum information science and
technology. In particular, because of lithium niobate’s exceptional optical
properties in multiple aspects, the demonstrated photon conversion and
interaction are ready to be integrated with other passive and active elements
on the same chip, such as PPLN wavegudies [26, 27], electro-optical modulators
[28], frequency comb sources [29, 30, 31, 32], and microring filters [33], to
create functional quantum devices of practical impacts [34]. Compared with the
existing table-top or assembled systems, the detrimental insertion loss will
be eliminated, and the mechanical and optical stabilities are expected to be
exceptional.
As an example, in Fig. 5 we present the design of an integrated chip for
heralded entanglement generation, following the scheme in [2]. It consists of
a PPLN nanowaveguide to create photon pairs by SPDC, a series of thermally-
tuned microrings for optical filtering, directional couplers for photon
combination, and a PPLN microring for photon conversion. The process starts
with passing a visible pump through a PPLN nanowaveguide to generate via SPDC
photon pairs simultaneously over multiple wavelength channels,
$\omega_{s1}+\omega_{i1}$, $\omega_{s2}+\omega_{i2}$,…, and
$\omega_{sn}+\omega_{in}$ [26, 27]. The signal and idler photons will each be
picked and separated into different arms by using coupled add-drop filters for
high extinction while eliminating free spectrum ambiguity. In each arm,
amorphous-silicon over-cladding will be applied to further reject any residue
pump in the visible band. The signal photons will be recombined via cascaded
directional couplers and sent into a PPLN microring, where they are interact
to create a SF photon. The existence of a photon pair in the idler channels
will be heralded upon the detection of the SF photon. To create entanglement
in time bins, the SF photon needs to pass through a Franson interferometer and
be detected in a superposition time-bin state [2]. Unlike schemes using SPDC
and linear optical Bell state measurement, where the success probability is
fundamentally capped at 50% [35], this scheme is deterministic in that upon
heralding, the entangled photon pair exists with nearly certainty. In this
design, with 12 pm net filtering bandwidth and 100 ps pump pulses, one can
drive the SPDC at 1% photon pair production rate per pulse for each channel.
The heralding rate can approach 10 Hz for the demonstrated $10^{-5}$ photon
conversion, which would correspond to orders of magnitude improvement over
previous demonstrations [4, 5]·
Discussion: We have demonstrated photon conversion and interaction with record
high efficiency and low noise in a Z-cut, periodically poled microring on
thin-film lithium niobate. Combining nearly perfect QPM, tight mode
confinement, high cavity quality, and efficient photon extraction, we achieved
65% wavelength transduction with only about 100 microwatt pump power,
advancing the state of the art by large. Despite a small detuning between the
signal and pump, at the peak conversion only $3\times 10^{-5}$ noise photons
are created over the cavity lifetime, thanks to the deep single-mode condition
and suppression of side processes. The same device allowed nonlinear
interaction between two single-photon level coherent states, where the photon-
photon coupling strength reaches 9.1 MHz, and a single photon can produce 0.01
internal Rabi rotation angle. The external quantum efficiency is directly
measured to be about $10^{-5}$, compared with the best reported efficiency of
$10^{-7}$. Our results mark new milestones towards quantum nonlinear optics in
the single photon regime, with broad implications in areas of fundamental
studies and applied quantum information technology. Combining with other
favorable optical proprieties of thin film lithium niobate, a superior
platform of photonic integrated circuits is within sight for scalable quantum
applications.
## Methods
The entire device is fabricated on a magnesium-doped Z-cut LNOI wafer (NANOLN
Inc.), with a 600-nm thick LN thin film bonded on 2 $\mu$m silicon dioxide
layer above a silicon substrate. First, the microring and waveguide structure
are defined using hydrogen silsesquioxane (HSQ, Fox-16) by electron beam
lithography. The top width and the radius of the microring are 1.7 $\mu$m and
80 $\mu$m, respectively. Then, ICP Argon milling is applied to shallowly etch
the structures, where 430-nm thick LN is etched with 170-nm LN remaining, and
the sidewall angle is approximately 64∘. The optimized pulley coupler, shown
in Fig. 1(b) with the pulley top width of $w_{pulley}$ = 400 nm, the gap of
$g_{pulley}$ = 650 nm, and the length of $L_{pulley}$ = 40 $\mu$m, is created
to increase the ring-bus waveguide coupling to attain simultaneously over-
coupling condition for both the visible and IR lightwaves. Then, a concentric
periodically-poled region with period of $\Lambda=3.37$ $\mu$m given by
$\Lambda=2\pi R/M$, M = 149 and near 50$\%$ duty cycle (see Fig. 1(c)) is
created via several 1-ms and 450-V electrical pulses using a similar process
described in [27]. After removing the poling electrodes, the chip is coated
with 1.5 $\mu$m silicon dioxide. Finally the chip is diced and polished.
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## Acknowledgements
The research was supported in part by National Science Foundation (Award
#1641094 & #1842680 & #1806523) and National Aeronautics and Space
Administration (Grant Number 80NSSC19K1618). Device fabrication was performed
in Nanofabrication Facility at Advanced Science Research Center (ASRC), City
University of New York (CUNY).
## Author contributions statement
J. C. and Y. H. conceived the experiments. J. C. and Z. L. fabricated the
device and conducted the experiments. Z. M. and C. T. involved in the
fabrication. H. F. and Y. S. involved in the quantum measurement. J. C.
performed the numerical simulations. Y. H. supervised the project. All authors
contributed to write and review the manuscript.
## Competing interests
The authors declare no competing interests.
## Additional information
Correspondence and requests for materials should be addressed to J.C and Y.H.
Supplemental Materials: Photon conversion and interaction on chip
### .1 Coupled-mode theory
The dynamic of sum-frequency generation (SFG) inside the $\chi^{(2)}$ cavity,
neglecting Rayleigh backscattering and assuming the single-mode condition, is
governed by the following coupled mode equations:
$\displaystyle\frac{da_{p}}{dt}$
$\displaystyle=(i\delta_{p}-\frac{\kappa_{p,t}}{2})a_{p}+ig^{*}a^{*}_{s}a_{f}+i\sqrt{\kappa_{p,c}}F_{p},$
(S1) $\displaystyle\frac{da_{s}}{dt}$
$\displaystyle=(i\delta_{s}-\frac{\kappa_{s,t}}{2})a_{s}+ig^{*}a^{*}_{p}a_{f}+i\sqrt{\kappa_{s,c}}F_{s},$
(S2) $\displaystyle\frac{da_{f}}{dt}$
$\displaystyle=(i(\delta_{p}+\delta_{s}+\Delta\omega)-\frac{\kappa_{f,t}}{2})a_{f}+iga_{p}a_{s},$
(S3)
where $\kappa_{j,o}=\omega_{j}/Q_{j,o}$ are the cavity dissipation rates,
$Q_{j,o}$ are the quality factors of cavity mode, $\omega_{j}$ is the angular
frequency, with $j=p,s,f$ for the pump, signal, sum-frequency modes,
respectively, and $o=0,c,t$ denoting intrinsic, coupling and total dissipation
rates with $\kappa_{t}=\kappa_{0}+\kappa_{c}$,
$\delta_{j}=\omega_{j}-\omega_{j,0}$ is the laser-cavity detuning and
$\omega_{j,0}$ is the cavity resonance, $F_{j}=\sqrt{N_{j}}$ with $N_{j}$ is
the photon number. Because of the energy conservation in parametric conversion
process, $\omega_{f}=\omega_{p}+\omega_{s}$ permits
$\delta_{p}+\omega_{p,0}+\delta_{s}+\omega_{s,0}=\delta_{f}+\omega_{f,0}$ thus
$\delta_{f}=\delta_{p}+\delta_{s}+\Delta\omega$ where
$\Delta\omega=\omega_{p,0}+\omega_{s,0}-\omega_{f,0}\approx 0$, when it is a
triply-resonant cavity. The photon-photon coupling coefficient $g$:
$\displaystyle
g=\sqrt{\frac{\hbar\omega_{p}\omega_{s}\omega_{f}}{2\epsilon_{0}\epsilon_{p}\epsilon_{s}\epsilon_{f}}}\frac{\frac{2}{\pi}d_{eff}\xi}{\sqrt{A_{eff}2\pi
R}}\times\delta(m_{f}-m_{p}-m_{s}-M),$ (S4) $\displaystyle\xi=\frac{\iint
E^{*}_{f}E_{p}E_{s}dxdy}{(\iint|E_{p}|^{2}E_{p}dxdy\iint|E_{s}|^{2}E_{s}dxdy\iint|E_{f}|^{2}E_{f}dxdy)^{1/3}},$
(S5)
where $\epsilon_{0}$ is the vacuum permittivity, $\epsilon_{j}=n^{2}_{j}$ are
the relative permittivity with $n_{j}$ giving the effective refractive
indices, $d_{eff}$ is the effective nonlinear susceptibility, $\xi$ is the
mode overlapping factor and $V_{eff}$ is the effective mode volume, $m_{j}$
are the azimuthal order of the cavity modes, M is the azimuthal poling grating
number. For quasi-phase matched case, $m_{f}-m_{p}-m_{s}-M=0$.
At steady state, the output filed of SF is defined as
$b_{f}=i\sqrt{\kappa_{f,c}}a_{f}$ and the quantum efficiency of photon
conversion is given by $\eta_{\mathrm{QE}}=N_{f}/N_{s}=\lvert
b_{f}\rvert^{2}/N_{s}$. The intracavity pump and signal photon number are
given by $\lvert a_{p}\rvert^{2}$ and $\lvert a_{s}\rvert^{2}$, respectively.
The above quantum efficiency gives the probability of a signal photon at the
cavity input being converted to its sum-frequency and appear at the cavity
output. It has accounted for the signal coupling loss and the SF extraction
loss. It is thus a direct measure of the device figure of merit that dictates
its performance in applications.
Another measure, of less practical relevance but an intrinsic property, is the
internal Rabi rotation angle $\theta=2gQ_{p,i}/\omega_{p}$. The system
$\hat{H}_{\textrm{eff}}=\hbar\mathnormal{g}(\hat{a}_{p}\hat{a}_{s}\hat{a}_{f}^{\dagger}+\hat{a}_{p}^{\dagger}\hat{a}_{s}^{\dagger}\hat{a}_{f})$
can be interpreted as the pump induced Rabi oscillation between signal and SF
photons. $\theta$ then gives how much Rabi rotation can a pump photon induce
during it is lost in the cavity.
We implement time split-step method to numerically solve the coupled-mode
equations. The parameters used in the simulation for Fig.3 and Fig.4 is listed
in Table.S1:
Parameters | Description | Value | Unit
---|---|---|---
$\kappa_{p,t}$ | total dissipation rate of pump mode | $1467.9$ | MHz ($\times 2\pi$)
$\kappa_{p,0}$ | intrinsic dissipation rate of pump mode | $270.8$ | MHz ($\times 2\pi$)
$\kappa_{p,c}$ | coupling dissipation rate of pump mode | $1197.0$ | MHz ($\times 2\pi$)
$\kappa_{s,t}$ | total dissipation rate of signal mode | $1510.3$ | MHz ($\times 2\pi$)
$\kappa_{s,0}$ | intrinsic dissipation rate of signal mode | $304.4$ | MHz ($\times 2\pi$)
$\kappa_{s,c}$ | coupling dissipation rate of signal mode | $1205.9$ | MHz ($\times 2\pi$)
$\kappa_{f,t}$ | total dissipation rate of sum-frequency mode | $1512.2$ | MHz ($\times 2\pi$)
$\kappa_{f,0}$ | intrinsic dissipation rate of sum-frequency mode | $318.7$ | MHz ($\times 2\pi$)
$\kappa_{f,c}$ | coupling dissipation rate of sum-frequency mode | $1193.5$ | MHz ($\times 2\pi$)
$d^{a}_{eff}$ | effective nonlinear susceptibility | 16.4 | pm/V
$d^{b}_{eff}$ | effective nonlinear susceptibility | 18.2 | pm/V
$A_{eff}$ | effective mode area | 1.0 | $\mu$m2
$R$ | radius of the microring | 80 | $\mu$m
$\xi$ | mode overlapping factor | 90% | NA
Table S1: Simulation parameters used in coupled-mode equations models for sum-
frequency generation. a: used in Fig.3, b: used in Fig.4. The fitted $g^{a}$
and $g^{b}$ are 8.2 MHz and 9.1 MHz, respectively.
### .2 Experimental setup
We use the setup shown in Fig.S1 to characterize the device and perform the
photon conversion and interaction experiment. The whole chip is placed on a
thermoelectric cooler and the temperature is set at about 45 ∘C. For linear
characterization, as shown in Fig.S1(a), we use two polarized tunable
continuous-wave (CW) lasers (Santec 550 and Newport TLB-6712) and tapered
fibers (OZ OPTICS) to independently characterize the fiber-chip-fiber
coupling, whose losses are measured to be (8 $\pm$ 0.15) dB around 1556 nm and
(9.5 $\pm$ 0.2) dB around 778 nm, respectively. For its nonlinear optical
properties, we sweep the infrared laser across the whole C-band while
optimizing the chip’s temperature to achieve strong second-harmonic generation
(SHG).
Once identifying the quasi-phase matching resonances, we switch to the setup
in Fig.S1(b) for sum-frequency generation. Additional tunable CW laser
(Coherent, MTP-1000) is used to serve as the signal. We will further optimize
the system around its optimum SHG condition to maximize the SFG.
* •
For classical measurement, optical spectrum analyzer (OSA, Yokogawa AQ6370D)
is used to collect the data.
* •
For quantum noise measurement, a silicon-based single-photon detector
(Excelitas, efficiency 50$\%$, dark count 250 Hz) and a free-space filtering
system are introduced. It consist of two fiber collimators (insertion loss,
IL$\sim$ 2 dB), one short-pass filter ( IL$\sim$0.5 dB, extinction ratio, ER
$\sim$50 dB) and one band-pass filter (10nm, 780 nm, IL$\sim$0.5 dB, ER
$\sim$50 dB) rejecting pump and signal in C-band while passing through visible
light, and three narrow band-pass filters (Alluxa, 3 nm, IL$\sim$1 dB,
ER$>$120 dB) rejecting residue second-harmonic light of 780 nm while passing
through the target light of 778 nm.
* •
For interaction between single-photon coherent states, superconducting
nanowire single-photon detector (ID Quantique, ID281, efficiency 85$\%$, dark
count 100 Hz) are used to monitor the input photon flux. Due to limited
photon-count saturation rate ($\sim$20 MHz) of SNSPDs, each monitor channel
has about 50 dB attenuation. Meanwhile, the free-space filtering system will
be removed to reduce the insertion loss (IL$\sim$4 dB). The SF photons will be
directly measured by silicon-based single-photon detector via a lensed fiber.
Figure S1: Setups for classical (a) and quantum (b) experiments. The blue and
red lines denote the telecome light path and visible path, respectively. FPC:
Fiber Polarization Controller; TEC: thermoelectric cooler; WDM: wavelength-
division multiplexing module; PD: photodetector; DUT: device under test;
VOA:variable optical attenuator; ATT: optical attenuator; OSA: optical
spectrum analyzer; FC: fiber collimator; SP: short-pass filter; BP: band-pass
filter; NBP: narrow band-pass filter; SNSPD: superconducting nanowire single-
photon detector; Si-SPD: silicon single-photon detector.
|
# Effect Size Estimation in Linear Mixed Models
Jürgen Groß Institute for Mathematics and Applied Informatics, University of
Hildesheim, Germany<EMAIL_ADDRESS>and Annette Möller
Faculty of Business Administration and Economics, Bielefeld University,
Germany<EMAIL_ADDRESS>
###### Abstract.
In this note, we reconsider Cohen’s effect size measure $f^{2}$ under linear
mixed models and demonstrate its application by employing an artificially
generated data set. It is shown how $f^{2}$ can be computed with the
statistical software environment R using lme4 without the need for
specification and computation of a coefficient of determination.
###### Key words and phrases:
Hypothesis testing, effect size, Cohen’s f2, linear regression, linear mixed
model, multivariate normal distribution
###### 2010 Mathematics Subject Classification:
62J05, 62J20, 62F03
Support of the second author by the Helmholtz Association’s pilot project
”Uncertainty Quantification” is gratefully acknowledged.
## 1\. Introduction
In studies with a large number of observations, statistical testing procedures
are prone to detect even minor departures from a null hypothesis yielding very
small p-values. However, since significance does not automatically imply
relevance, measures for the size of the effect associated with a possible
rejection of the null hypothesis are often recommended as a useful tool, see
e.g. Wilkinson (1999).
A well known effect size measure in a regression context when a quantitative
variable $Y$ depends on independent regressors (quantitative and/or
qualitative) is Cohen’s $f^{2}$, see Cohen (1988, Chapt. 9). Under
multivariate normality this measure is strongly related to an $F$ test of a
linear hypothesis that a subset $B$ of independent variables does not
substantially contribute to the explanation of $Y$, given a set $A$ of
independent regressors already included in the model. One may argue that even
if the regression coefficients associated with variables $B$ significantly
differ from zero, their contribution may be assessed as relevant when some
meaningful size of their effect is measured.
In this light studies may additionally want to report $f^{2}$ values for each
variable in a regression model given the others, see for example Tables 2 to 5
in Taylor et al. (2020). With regard to the procedure of calculating $f^{2}$,
the authors refer to Selya et al. (2012) who introduced a practical method in
relation with SAS® software. This approach even carries over to linear mixed
models, i.e. the case that some independent variables are associated with
random effects rather than with fixed effects. Hence, the original approach by
Cohen is generalized to a certain extent, involving the additional estimation
of a variance-covariance matrix.
In the following we reconsider and discuss the generalization of $f^{2}$ to
linear mixed models. We also point to an alternative computational procedure
which turns out to be especially useful in context with the statistical
software environment R (R Core Team, 2022) and well known package lme4, see
Bates et al. (2015), for fitting linear mixed models. Our explanations are
illustrated on the basis of an artificially generated data set.
## 2\. Linear Mixed Model
Consider a linear mixed model (LMM) described by
(1) $\bm{y}=\bm{X}\bm{\beta}+\bm{Z}\bm{u}+\bm{e}\;,$
where $\bm{y}$ is an $n\times 1$ observable random vector. It is assumed that
the $n\times p$ model matrix $\bm{X}$ of full column rank $p$ can be
partitioned as
(2) $\bm{X}=(\bm{1}_{n}:\bm{X}_{1}:\bm{X}_{2})\;,$
where $\bm{1}_{n}$ denotes the $n\times 1$ vector of ones, while the $n\times
p_{1}$ and $n\times p_{2}$ matrices $\bm{X}_{1}$ and $\bm{X}_{2}$ contain the
values of $p_{1}+p_{2}=p-1$ regressors. The $p\times 1$ vector $\bm{\beta}$
comprises $p$ unknown parameters $\beta_{0},\beta_{1},\ldots,\beta_{p-1}$
addressed as fixed effects. The $n\times q$ matrix $\bm{Z}$ contains the
values of independent variables associated with an $q\times 1$ vector $\bm{u}$
of unobservable random effects. It is assumed that $\bm{u}$ has expectation
$\bm{0}_{q}$ (the $q\times 1$ vector of zeroes) and variance covariance matrix
$\mathop{\operator@font Cov}\nolimits(\bm{u})=\sigma^{2}\bm{D}$ with unknown
parameter $\sigma^{2}>0$ and $q\times q$ matrix $\bm{D}$, which may depend on
further unknown parameters. For the $n\times 1$ vector $\bm{e}$ of
unobservable random errors it is assumed that ${\operator@font
E}(\bm{e})=\bm{0}_{n}$ and $\mathop{\operator@font
Cov}\nolimits(\bm{e})=\sigma^{2}\bm{T}$, where the $n\times n$ positive
definite matrix $\bm{T}$ may also depend on unknown parameters. Moreover, the
assumption $\mathop{\operator@font Cov}\nolimits(\bm{u},\bm{e})=\bm{0}_{q,n}$
(the $q\times n$ matrix of zeroes) implies
(3) $\mathop{\operator@font
Cov}\nolimits(\bm{y})=\sigma^{2}\bm{V},\quad\bm{V}=\bm{Z}\bm{D}\bm{Z}^{T}+\bm{T}\;.$
Then the above model may also be represented by the triplet
$\\{\bm{y},\bm{X}\bm{\beta},\sigma^{2}\bm{V}\\}$ and can be considered as a
special case of the general Gauss-Markov model, see e.g. Groß (2004). As a
matter of fact, formulas useful under model 1) carry over from a classical
regression model
$\bm{Q}^{-1}\bm{y}=\bm{Q}^{-1}\bm{X}\bm{\beta}+\bm{\varepsilon}$ with
${\operator@font E}(\bm{\varepsilon})=\bm{0}_{n}$ and $\mathop{\operator@font
Cov}\nolimits(\bm{\varepsilon})=\sigma^{2}\bm{I}_{n}$, see Christensen (2020,
Sect 2.7). Here $\bm{Q}$ denotes some nonsingular matrix satisfying
$\bm{Q}\bm{Q}^{T}=\bm{V}$. Let us assume for the moment that $\bm{V}$ is
completely known. Then
(4)
$\widehat{\bm{\beta}}=(\bm{X}^{T}\bm{V}^{-1}\bm{X})^{-1}\bm{X}^{T}\bm{V}^{-1}\bm{y}$
is the best linear unbiased estimator for $\bm{\beta}$, and
(5)
$\widehat{\sigma}^{2}=\frac{1}{\nu}(\bm{y}-\bm{X}\widehat{\bm{\beta}})^{T}\bm{V}^{-1}(\bm{y}-\bm{X}\widehat{\bm{\beta}}),\quad\nu=n-p\;,$
is the usual unbiased estimator for $\sigma^{2}$. Consider the linear
hypothesis $H_{0}:\bm{R}\bm{\beta}=\bm{r}$ versus
$H_{1}:\bm{R}\bm{\beta}\not=\bm{r}$ for a given $r\times p$ matrix $\bm{R}$ of
full row rank and a given $r\times 1$ vector $\bf{r}$. The corresponding $F$
statistic in model (1) is
(6)
$F=\frac{(\bm{R}\widehat{\bm{\beta}}-\bm{r})^{T}(\bm{R}\bm{B}\bm{R}^{T})^{-1}(\bm{R}\widehat{\bm{\beta}}-\bm{r})}{r\widehat{\sigma}^{2}}=\frac{(\bm{R}\widehat{\bm{\beta}}-\bm{r})^{T}(\bm{R}\bm{B}\bm{R}^{T})^{-1}(\bm{R}\widehat{\bm{\beta}}-\bm{r})}{(\bm{y}-\bm{X}\widehat{\bm{\beta}})^{T}\bm{V}^{-1}(\bm{y}-\bm{X}\widehat{\bm{\beta}})}\cdot\frac{\nu}{r}\;,$
where
(7) $\mathop{\operator@font
Cov}\nolimits(\widehat{\bm{\beta}})=\sigma^{2}\bm{B},\quad\bm{B}=(\bm{X}^{T}\bm{V}^{-1}\bm{X})^{-1}\;.$
Under multivariate normality the statistic $F$ follows a $F_{r,f}$
distribution provided $H_{0}$ holds true.
### 2.1. Effect Size
Suppose that we are interested in the effect of the independent variables
represented by the model matrix $\bm{X}_{1}$, given the variables represented
by $\bm{X}_{2}$. The corresponding linear hypothesis reads
$\bm{R}_{1}\bm{\beta}=\bm{0}_{p_{1}}$ with
$\bm{R}_{1}=(\bm{0}_{p_{1}}:\bm{I}_{p_{1}}:\bm{0}_{p_{1},p_{2}})$. Hence, by
reasoning similar to Cohen (1988), an appropriate measure for the size of the
effect based on the above $F$ statistic is provided by
(8)
$f^{2}=\frac{(\bm{R}_{1}\widehat{\bm{\beta}})^{T}(\bm{R}_{1}\bm{B}\bm{R}_{1}^{T})^{-1}(\bm{R}_{1}\widehat{\bm{\beta}})}{(\bm{y}-\bm{X}\widehat{\bm{\beta}})^{T}\bm{V}^{-1}(\bm{y}-\bm{X}\widehat{\bm{\beta}})}\;,$
thereby removing the factor $\nu/r$ from the $F$ statistic. This generalizes
Cohen’s $f^{2}$ in the sense that if $\bm{D}=\bm{0}_{q,q}$, then the
unobservable random vector $\bm{u}$ equals $\bm{0}_{q}$ with probability one,
the LMM reduces to the usual linear model of fixed effects only, and $f^{2}$
becomes identical to the measure provided by formula (9.2.1) in Cohen (1988).
We note that it is also possible to compute $f^{2}$ as
(9) $f^{2}=\frac{R_{A,B}^{2}-R_{A}^{2}}{1-R_{A,B}^{2}}$
for appropriately defined coefficients of determination $R_{A,B}^{2}$ derived
under the full model and $R_{A}^{2}$ derived under a reduced LMM assuming that
variables represented by $\bm{X}_{1}$ are not present at all. Such a formula
is the basis for the widely applied computational procedure suggested by Selya
et al. (2012).
### 2.2. Operational Effect Size
Formula (8) for $f^{2}$ is only operational when $\bm{V}$ is completely known,
a condition not met in practical applications. According to Harville (1977) a
simple LMM is the ordinary mixed and random effects ANOVA model, also referred
to as traditional variance components model, see Christensen (2019, Chapt. 5).
Under this model, the variance-covariance matrix of $\bm{y}$ depends on a
total of $m$ variance components. The $n\times q$ matrix $\bm{Z}$ is
partitioned as $\bm{Z}=(\bm{Z}_{1}:\cdots:\bm{Z}_{m-1})$, where each $n\times
q_{i}$ matrix $\bm{Z}_{i}$ is the design matrix corresponding to a qualitative
variable with a certain number of levels. In such a case one may assume
(10)
$\bm{D}=\text{diag}\left[(\sigma_{1}^{2}/\sigma^{2})\bm{I}_{q_{1}},\ldots,(\sigma_{m-1}^{2}/\sigma^{2})\bm{I}_{q_{m-1}}\right]$
where $\sigma^{2}>0$ and $\sigma_{1}^{2},\ldots\sigma_{m-1}^{2}\geq 0$ are $m$
unknown variance components. Then
(11)
$\bm{V}=\bm{I}_{n}+\sum_{i=1}^{m-1}(\sigma_{i}^{2}/\sigma^{2})\bm{Z}_{i}\bm{Z}_{i}^{T}\;.$
If $\sigma_{i}^{2}=0$ for some $i$, then the corresponding $q_{i}\times 1$
random effect vector $\bm{u}_{i}$ equals $\bm{0}_{q_{i}}$ with probability
one. Such a model, and also more general ones, may be fitted in R with the
package lme4, see Bates et al. (2015). From the fitting procedure its is
possible to obtain an estimate for the variance-covariance matrix
(12) $\widehat{\mathop{\operator@font
Cov}\nolimits}({\widehat{\bm{\beta}}})=\widehat{\sigma}^{2}\widehat{\bm{B}}$
of the estimated fixed effects parameter vector $\bm{\beta}$. Then a
corresponding estimate for $f^{2}$ is given as
(13)
$f^{2}=\frac{1}{\nu}(\bm{R}_{1}\widehat{\bm{\beta}})^{T}(\bm{R}_{1}\widehat{\mathop{\operator@font
Cov}\nolimits}({\widehat{\bm{\beta}}})\bm{R}_{1}^{T})^{-1}(\bm{R}_{1}\widehat{\bm{\beta}})\;.$
where the factor $1/\nu$ is required to compensate for the usage of
$\widehat{\sigma}^{2}$ in the estimated variance-covariance matrix (12). The
application of this formula is illustrated in the following section. The merit
of formula (13) lies in the fact that it can be applied whenever an estimate
(12) is available. But then, of course, the actual outcome also depends on the
estimation procedure, implying that different estimation methods for (12) may
also lead to (slightly) different actual values of (13). This, however, is not
the topic of our discussion. Also, when applying (13) there is no need for
computing coefficients of determination. Nonetheless it is possible to define
an operational version of (9) which corresponds to (13). This is demonstrated
in Sect. 3.4, where the $R^{2}$ measure discussed in Edwards et al. (2008) is
used with an operational version of the $F$ statistic obtained in the same
light as (13) by employing the very same variance-covariance estimate (12).
Again, employing alternative computational methods or alternative measures
$R^{2}$ in linear mixed models, see also Nakagawa and Schielzeth (2013);
Nakagawa et al. (2017), may result in different actual values of $f^{2}$.
Figure 1. Distribution of response variable $Y$ over all $n=1000$ observations
(histogram) and within two groups of sizes $n_{1}=687$ and $n_{2}=313$
indicated by $0$ and $1$ (boxplots)
## 3\. Example
In the following we discuss the performance of measure $f^{2}$ for an
artificially generated data set of $n=1000$ observations intended to further
illustrate some computational aspects. For the sake of simplicity our model
consists of two independent variables $X_{1}$ (categorical/binary) and $X_{2}$
(quantitative) associated with fixed effects and one variable $Z$
(categorical) associated with random effects. However, the same principles
apply when $X_{1}$, $X_{2}$, and $Z$ are extended to possible sets of
variables containing more than one element. Our setting corresponds to the
above mentioned variance components model with $p_{1}=p_{2}=1$ and $m=2$.
### 3.1. Variable of Interest
Figure 1 shows a discernible location difference with respect to the
distribution of the response variable $Y$ in the two groups indicated by the
binary variable $X_{1}$. The Welch two-sample $t$ statistic for the null
hypothesis of no difference in group means reads $|t|=6.0751$ implying a
highly significant result. A corresponding effect size measure is Cohen’s $d$
which may be computed from R package effectsize, see Ben-Shachar et al.
(2020), as $|d|=0.4122$. From Cohen, values $|d|=0.2$, $|d|=0.5$ and $|d|=0.8$
indicate a small, medium and large effect, respectively.
Figure 2. Scatter plot of $n=1000$ pairs $(X_{2},Y)$ with group markings
according to $X_{1}$
### 3.2. Additional Fixed Effects
From Figure 2 one may conclude, however, that the difference in the two groups
may to some extent be explained by the variable $X_{2}$, since there is a
tendency for larger values of $X_{2}$ to come along with larger values of $Y$
and observations from group 1 of variable $X_{1}$. Therefore one might be
interested in the size of the effect of $X_{1}$ when $X_{2}$ is held constant.
This can be achieved by considering a regression model with $X_{1}$ and
$X_{2}$ as independent variables and deriving the measure $f^{2}$ as explained
in (Cohen, 1988, Sect. 9). From package effectsize one gets $f^{2}=0.0017767$.
Here, values $f^{2}=0.02$, $f^{2}=0.15$ and $f^{2}=0.35$ are supposed to
indicate a small, medium and large effect, respectively.
Recently, Groß and Möller (2023) considered a generalized version $d_{\ast}$
of $d$ as an effect size measure for a binary variable $X_{1}$ given further
variables. It may be computed from $f^{2}$ as
(14) $d_{\ast}=\sqrt{f^{2}(n-2-w)\gamma}\;,$
where in our analysis $w=1$ is the number of additional independent variables
incorporated in the model and $\sigma^{2}\gamma$ is the variance of the
regression coefficient for $X_{1}$. For our data $\gamma=0.0065821$, yielding
$d_{\ast}=0.108$ and thus confirming a less than small effect.
Figure 3. Scatterplot of $(\\#\\{X_{1}=1\\},\overline{Y})$ for the 15 groups
of $Z$, where $\\#\\{X_{1}=1\\}$ denotes the number of observations with
$X_{1}=1$
### 3.3. Additional Fixed and Random Effects
As a next step one may take the categorical variable $Z$, admitting 15 groups
in our data set, into account. From Figure 3 it is seen that there is a
tendency for $Z$ groups with larger means of the response variable $Y$ to
contain less observations marked as 1 (referring to variable $X_{1}$) than $Z$
groups with smaller means of $Y$. However, since observations from group 1
were earlier seen to reveal on average larger $Y$ values than observations
from group 0, holding $Z$ constant is expected to contribute in favor of an
effect of $X_{1}$ again.
As noted before, the variable $Z$ is meant to be associated with a random
effects vector $\bm{u}$. The corresponding design matrix $\bm{Z}$ has 15
columns where each row contains 0’s and a single 1, indicating the membership
of the observations to the respective $Z$ variable group. There are two
variance components, the overall $\sigma^{2}>0$ and the random effects
variance denoted by $\sigma_{u}^{2}\geq 0$ here. The model may also be
reparameterized via an unknown $k\geq 0$ by setting
$\sigma_{u}^{2}=k\sigma^{2}$. This gives variance-covariance matrix
(15) $\mathop{\operator@font
Cov}\nolimits(y)=\sigma^{2}\bm{V},\quad\bm{V}=k\bm{Z}\bm{Z}^{T}+\bm{I}_{m}\;,$
and the LMM reduces to the usual fixed effects regression model in case $k=0$.
For $\bm{V}$ from (15) one may compute the effect size $f^{2}$ for $X_{1}$ as
a function of $k$ from formula (8) when all variables $X_{1}$, $X_{2}$ and $Z$
are included in the LMM formulation. Figure 4 shows $f^{2}$ for different
choices of $k$. As expected, the effect size is larger when both, $X_{2}$ and
$Z$ are incorporated into the model compared to the case when only $X_{2}$ is
employed, corresponding to the choice $k=0$.
Figure 4. Values of $f^{2}$ from a LMM fit depending on $k$
The operational version of $f^{2}$ from (13) can easily be computed from
fitting the model by the function lmer from package lme4 as fit <\- lmer(Y ~ 1
+ X1 + X2 + (1|Z)). The estimated variance components are
$\widehat{\sigma}^{2}=393.4455$ and $\sigma_{u}^{2}=180.4234$ giving
$\widehat{k}=\sigma_{u}^{2}/\widehat{\sigma}^{2}=0.4586$ as an estimation for
$k$, see the dashed vertical line in Figure 4. Then the vector
$\widehat{\beta}$ is obtained from fixef(fit) and
$\widehat{\mathop{\operator@font Cov}\nolimits}({\widehat{\bm{\beta}}})$ is
obtained from vcov(fit). By using $\bm{R}_{1}=(0,\,1,\,0)$ and $\nu=n-p=997$,
formula 13 yields $f^{2}=0.0946626$ indicating a small but not medium effect
size of $X_{1}$ when $X_{2}$ and $Z$ are held constant.
### 3.4. Coefficient of Determination
Finally, we confirm that $f^{2}$ may also be obtained from formula (9),
although there is no actual need for this when formula (13) can be used
instead. For this, we define
(16) $R_{AB}^{2}=\frac{(r/\nu)F}{1+(r/\nu)F},\quad r=p-1,\,\nu=n-p\;,$
which is the proposed $R^{2}$ for linear mixed models by Edwards et al. (2008,
Eq. (19)). Here, $F$ is the $F$ statistic from (6) for testing the hypothesis
$H_{0}:\bm{R}\bm{\beta}=\bm{0}_{p-1}$ with
$\bm{R}=(\bm{0}_{p-1}:\bm{I}_{p-1})$. The relationship between $F$ and
$R_{AB}^{2}$ from (16) may be established similar to Example 4.8 from Seber
and Lee (2003). For our data $\bm{R}=(\bm{0}_{2}:\bm{I}_{2})$, $r=2$, and
$\nu=n-p=997$. The measure $R_{A}^{2}$ is defined in the same way, but for a
reduced model with all variables present except for $X_{1}$. For this reduced
model we have $\bm{R}=(0,\,1)$, $\bm{r}=(0)$, $r=1$, $\nu=n-p_{2}-1=998$. This
results in
(17)
$f^{2}=\frac{R_{A,B}^{2}-R_{A}^{2}}{1-R_{A,B}^{2}}=\frac{0.1539263-0.07418754}{1-0.1539263}=0.09424569\;,$
which is nearly the same as the value computed above. The displayed values
were obtained from operational versions of $R_{A,B}^{2}$ and $R_{A}^{2}$
computed in the same light as formula (13) by employing the estimated variance
covariance matrix of the fixed effects resulting from applying vcov() to the
two models in question.
## References
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* Ben-Shachar et al. (2020) M. S. Ben-Shachar, D. Lüdecke, and D. Makowski. effectsize: Estimation of effect size indices and standardized parameters. _Journal of Open Source Software_ , 5:2815, 2020. URL https://doi.org/10.21105/joss.02815.
* Christensen (2019) R. Christensen. _Advanced Linear Modeling: Statistical Learning and Dependent Data_. Springer Nature, 2019.
* Christensen (2020) R. Christensen. _Plane Answers to Complex Questions. Fifth Edition_. Springer, 2020.
* Cohen (1988) J. Cohen. _Statistical Power Analysis for the Behavioral Sciences. Second Edition_. Lawrence Erlbaum Associates, 1988.
* Edwards et al. (2008) L. J. Edwards, K. E. Muller, R. D. Wolfinger, B. F. Qaqish, and O. Schabenberger. An R2 statistic for fixed effects in the linear mixed model. _Statistics in Medicine_ , 27:6137–6157, 2008.
* Groß (2004) J. Groß. The general Gauss-Markov model with possibly singular dispersion matrix. _Statistical Papers_ , 45:311–336, 2004.
* Groß and Möller (2023) J. Groß and A. Möller. A note on Cohen’s d from a partitioned linear regression model. _Journal of Statistical Theory and Practice_ , 17:22, 2023\. URL https://doi.org/10.1007/s42519-023-00323-w.
* Harville (1977) D. A. Harville. Maximum likelihood approaches to variance component estimation and to related problems. _Journal of the American Statistical Association_ , 72:320–338, 1977.
* Nakagawa and Schielzeth (2013) S. Nakagawa and H. Schielzeth. A general and simple method for obtaining R2 from generalized linear mixed-effects models. _Methods in Ecology and Evolution_ , 4:133–142, 2013. URL https://doi.org/10.1111/j.2041-210x.2012.00261.x.
* Nakagawa et al. (2017) S. Nakagawa, P. C. Johnson, and H. Schielzeth. The coefficient of determination R 2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. _Journal of the Royal Society Interface_ , 14:20170213, 2017\. URL https://doi.org/10.1098/rsif.2017.0213.
* R Core Team (2022) R Core Team. _R: A Language and Environment for Statistical Computing_. R Foundation for Statistical Computing, Vienna, Austria, 2022. URL https://www.R-project.org/.
* Seber and Lee (2003) G. A. Seber and A. J. Lee. _Linear Regression Analysis_. John Wiley & Sons, 2003.
* Selya et al. (2012) A. S. Selya, J. S. Rose, L. C. Dierker, D. Hedeker, and R. J. Mermelstein. A practical guide to calculating Cohen’s f2, a measure of local effect size, from PROC MIXED. _Frontiers in Psychology_ , 3:111, 2012. URL https://doi.org/10.3389/fpsyg.2012.00111.
* Taylor et al. (2020) S. Taylor, C. A. Landry, M. M. Paluszek, and G. J. Asmundson. Reactions to COVID-19: Differential predictors of distress, avoidance, and disregard for social distancing. _Journal of Affective Disorders_ , 277:94–98, 2020\. URL https://doi.org/10.1016/j.jad.2020.08.002.
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# Unlearning via Sparse Representations
Vedant Shah
Mila, Université de Montréal
&Frederik Träuble
MPI, Tübingen
&Ashish Malik
Oregon State University
&Hugo Larochelle
Mila, Google DeepMind
&Michael Mozer
Google DeepMind
&Sanjeev Arora
Princeton University
&Yoshua Bengio
Mila, Université de Montréal
&Anirudh Goyal
Google DeepMind
Corresponding Author<EMAIL_ADDRESS>
###### Abstract
Machine _unlearning_ , which involves erasing knowledge about a _forget set_
from a trained model, can prove to be costly and infeasible by existing
techniques. We propose a nearly compute-free zero-shot unlearning technique
based on a discrete representational bottleneck. We show that the proposed
technique efficiently unlearns the forget set and incurs negligible damage to
the model’s performance on the rest of the data set. We evaluate the proposed
technique on the problem of class unlearning using three datasets: CIFAR-10,
CIFAR-100, and LACUNA-100. We compare the proposed technique to SCRUB, a
state-of-the-art approach which uses knowledge distillation for unlearning.
Across all three datasets, the proposed technique performs as well as, if not
better than SCRUB while incurring almost no computational cost.
## 1 Introduction
Machine Unlearning (Cao & Yang, 2015; Nguyen et al., 2022; Zhang et al., 2023;
Xu et al., 2023; Kurmanji et al., 2023) may be defined as the problem of
removing the influence of a subset of the data on which a model has been
trained. Unlearning can be an essential component in addressing several
problems encountered in deploying deep-learning-based solutions in real life.
Neural networks such as Large Language Models (LLMs), which have been trained
on massive amounts of commonly available data, can exhibit harmful behaviors
in the form of generating misinformation, demonstrating harmful biases, or
having other undesirable characteristics. A major culprit behind these
behaviors is the presence of biased or corrupted instances in the training
data of these models. To ensure safe model deployment, it is necessary to
remove these instances. Another reason to remove instances and make a model
behave as if it had not been trained on certain data is concerns about data
privacy and the right of end users to expunge their data (Mantelero, 2013).
For example, an individual might want their data removed from a face
recognition system that was trained on their faces such that it is no longer
able to identify them. All the above problems can be addressed by unlearning a
specific subset of the training data, i.e., the subset of data giving rise to
the harmful behavior of the model in the former cases and an individual’s
private data in the latter cases. Apart from these concerns, unlearning can
also be used for other purposes such as removing outdated data from a model to
free the network capacity for more recent or relevant data. With increasing
concerns about AI safety and the increasing ubiquity of deep learning models
in real-world applications, the problem of unlearning is becoming critical.
The main challenge in unlearning is maintaining the performance of the model
on the data that needs to be retained, called the retain set, while unlearning
the forget set. The naive way to ensure that a model has no information about
the forget set is to train from scratch on the retain set. Unlearning
techniques aim to achieve the same end but at a much lower computational cost
compared to full retraining. Unlearning in a pretrained network is difficult,
especially in densely connected neural networks, since the value of one
parameter may affect the output for all the input examples given to the neural
network. A possible solution is to fine tune the model we wish to unlearn only
on the retain set. While this would ensure that the performance of the model
on the retain set is maintained, it has been shown to be ineffective in
practice in unlearning the forget set (Golatkar et al., 2020a). Other more
effective solutions include retraining the model on the training data with a
negative gradient for the forget set (Golatkar et al., 2020a; Kurmanji et al.,
2023), or using knowledge-distillation-based training objectives to capture
information about the retain set while filtering out information about the
forget set (Kurmanji et al., 2023; Chundawat et al., 2023). However, all of
these approaches require some form of substantial additional compute in order
to facilitate unlearning. Moreover, some of the existing approaches also
require access to the original training data to enable unlearning, which may
not be possible in many practical applications, e.g., for a model in
production which is being trained online on an incoming data stream. The use
of large models is becoming more popular and prevalent with the development of
general purpose transformer models. The requirement for additional compute can
quickly become impractical in the context of these large models, especially in
cases where a model is deployed and needs to be redeployed as quickly as
possible after making the necessary changes.
Figure 1: A summary of the proposed unlearning approach. Left: The structure
of a key-value bottleneck. The encoder is frozen and pre-trained and $R_{1}$
is a random projection matrix. The values corresponding to the selected keys
are retrieved to be used by the decoder. The gradient is backpropagated
through the decoder into the values during training. The figure depicts the
case with 1 codebook in the DKVB. However, in practice we use multiple
codebooks. Center: Examples from the forget set are passed through the trained
model and the key-value pairs selected during the forward pass are recorded.
Right: The recorded key-value pairs are then masked from the bottleneck. As a
result, the key selection is redirected to other non-masked keys, whose
corresponding values are not useful for the example, and just lead to
uninformed prediction.
In this article, we argue that _neural information bottlenecks_ can be means
of highly efficient and specific unlearning. Neural information bottlenecks
have emerged as useful components in neural network architectures, providing
numerous benefits such as improving out-of-distribution (OOD) generalization
capabilities and robustness to noisy data (Goyal et al., 2021; Jaegle et al.,
2021; Liu et al., 2021; 2023), facilitating large scale unsupervised pre-
training and generative modeling (Esser et al., 2021; Oord et al., 2017), and
more recently, helping in continual learning (Träuble et al., 2023). We
particularly focus on using the Discrete Key-Value Bottleneck (DKVB) proposed
in Träuble et al. (2023). DKVB induces sparse representations in the form of
key-value pairs which are trained in a localized and context-dependent manner.
Since these representations are sparse, we hypothesize that it is possible to
remove the information about a subset of the training data without damaging
the information about the rest of the data—the primary desiderata for a useful
unlearning method. Moreover, since the representations are discrete, this may
be achieved zero-shot, i.e., without requiring any additional compute in the
form of retraining or fine tuning, by directly intervening on individual
representations.
In this work, we investigate the above-mentioned idea of zero-shot unlearning
in the Discrete Key-Value Bottleneck. Specifically, we focus on the problem of
class unlearning in multi-class classification tasks, where the aim is to
remove information about a specific output class, called the forget class,
from a trained model. We use the term retain classes to refer to the classes
other than the forget class that are present in the training data. More
specifically, we wish to remove the influence of the forget class (or more
generally speaking the forget set) on the model. We measure this influence
using the performance of the model on held-out test datasets corresponding to
the forget class and the retain classes. We propose two approaches for zero-
shot unlearning on DKVB, one which requires access to training examples from
the forget set—which we call Unlearning via Examples—and one which does not
require access to training examples—called Unlearning via Activations. We show
that the proposed methods achieve unlearning of the forget class while
incurring negligible damage to the model’s performance on the retain classes.
We compare the proposed methods to SCRUB (Kurmanji et al., 2023), a recent
state-of-the-art approach that requires additional compute to unlearn, on
three datasets: CIFAR-10, CIFAR-100, and LACUNA-100, and also further
investigate the effects of retraining the zero-shot unlearned models on the
retain set.
## 2 Related Work
The problem of unlearning has been studied in different forms for over two
decades. Early works such as Tsai et al. (2014), Cauwenberghs & Poggio (2000)
and Duan et al. (2007) study the problem of decremental learning in linear
models, where a small number of samples need to be removed from a model.
Ginart et al. (2019) considers unlearning as a problem of deleting individual
data points from a model. They give a probabilistic definition, formalize the
notion of efficient data deletion, and propose two deletion efficient learning
algorithms. Guo et al. (2019) introduces certified removal \- a theoretical
guarantee of indistinguishability between a model from which data was removed
and a model that never saw the data. Izzo et al. (2021) distinguishes between
exact unlearning and approximate unlearning and proposes a compute-efficient
approximate data deletion method, and a new metric for evaluating data
deletion from these models. Golatkar et al. (2020a) and Kurmanji et al. (2023)
cast unlearning into an information theoretic framework. Golatkar et al.
(2020b) proposes Neural Tangent Kernel (NTK) (Jacot et al., 2018) theory-based
approximation of the weights of the unlearned network. Multiple works also
delve into the more philosophical, ethical, and legal aspects of unlearning
and the “right to be forgotten” (Kwak et al., 2017; Villaronga et al., 2018).
Chundawat et al. (2023); Tarun et al. (2023) learn error minimization and
error maximization-based noise matrices which are used to finetune the trained
model in order to do unlearning. Chundawat et al. (2023) further uses a
generator that generates pseudo data points for unlearning in order to operate
in a data-free regime.
Most relevant to our work, Kurmanji et al. (2023) introduces SCRUB, an
effective knowledge distillation-based unlearning method. SCRUB considers the
original model as a teacher model and trains a student model to obey the
teacher model on the retain set and disobey it on the forget set. This is done
by computing the KL Divergence between the output distributions of the two
models and training the student model to maximize it on the forget set (this
is called a max-step) and minimize it on the retain set (this is called a min-
step). The student model is simultaneously also optimized for minimizing the
task loss on the retain set. The training consists of mstep max-steps. The
max-steps and min-steps are performed in an interleaved fashion. Warnecke et
al. (2021) focus on unlearning in deleting information at the features level
rather than unlearning specific instances or classes. They do so by using an
optimization objective that incentivizes a model to replace the information
about a set of features with a perturbed set of features. The proposed
approaches on the other hand focus on unlearning entire classes of data. Chen
et al. (2023), similarly to us, focuses on class unlearning. Unlearning is
done by destroying the decision boundary of the forget class. The authors
propose two boundary shift methods termed as Boundary Shrink and Boundary
Expanding.
Jia et al. (2023) and Mehta et al. (2022) investigate machine unlearning in
context of model sparsity in context of sparsity. Jia et al. (2023) leverages
the Lottery Ticket Hypothesis Frankle & Carbin (2018) by using parameter
pruning on a trained dense model to identify the token subnetwork. They
observe that applying standard unlearning approaches to a sparsified networks
is better as compared to doing unlearning directly on the dense network. Mehta
et al. (2022) identify the Markovian Blanket of parameters corresponding to
the examples to be unlearnt and updates those parameters. Thus, their approach
can be seen as applying sparse updates to the network for unlearning. Both
these approaches start with dense trained models and leverage sparsity for
unlearning. Whereas the approaches proposed in this paper suggest using
sparsity as an inductive bias in the model during the initial training along
which combined with the architectural prior of the DKVB makes it suitable for
unlearning involving minimal compute requirements. Jia et al. (2023) on the
hand spasifies the model after it has been trained. Mehta et al. (2022)
involves sparse updates to the model parameters as discussed previously.
However, these sparse updates are utilized during unlearning as opposed to
during training of the original model in the proposed approaches.
While most of the above-mentioned approaches improve upon the naive and
intractable baseline of retraining on the retain set, in terms of
computational efficiency, they still require some amount of computation to do
unlearning. This additional compute requirement can quickly become infeasible
where large models are involved. The proposed approach, on the other hand,
requires negligible computation for unlearning. Any computation that may be
required is in the form of running inference on the forget set.
## 3 Background and Notations
Unlearning: Let $\mathcal{D}_{train}=\\{x_{i},y_{i}\\}^{N}_{i=1}$ be a
training dataset and $\mathcal{D}_{test}$ be the corresponding test dataset.
In our experiments, we consider the setting of class unlearning, wherein we
aim to unlearn a class $c$ from a model trained with a multiclass
classification objective on $\mathcal{D}_{train}$. $c$ is called the forget
class or the forget set. Given $c$, we obtain
$\mathcal{D}_{train}^{forget}\subset\mathcal{D}_{train}$ such that
$\mathcal{D}_{train}^{forget}=\\{(x,y)\in\mathcal{D}_{train}|y=c\\}$. The
complement of $\mathcal{D}_{train}^{forget}$ is
$\mathcal{D}_{train}^{retain}$, i.e., subset of $\mathcal{D}_{train}$ that we
wish to retain. Thus
$\mathcal{D}_{train}^{retain}\cup\mathcal{D}_{train}^{forget}=\mathcal{D}_{train}$.
Similarly, from $\mathcal{D}_{test}$, we get
$\mathcal{D}_{test}^{forget}=\\{(x,y)\in\mathcal{D}_{test}|y=c\\}$ and its
complement $\mathcal{D}_{test}^{retain}$. We refer to
$\mathcal{D}_{train}^{retain}$ as the retain set training data,
$\mathcal{D}_{test}^{retain}$ as the retain set test data,
$\mathcal{D}_{train}^{forget}$ as the forget set training data and
$\mathcal{D}_{test}^{forget}$ as the forget set test data.
Discrete Key-Value Bottleneck: A discrete key-value bottleneck (DKVB) (Träuble
et al., 2023) consists of a discrete set of coupled key-value codes.
Specifically, the bottleneck contains $C$ codebooks with each codebook
containing $M$ key-value pairs. Models with DKVB use a pre-trained and frozen
encoder to encode the input into a continuous representation. This input
representation is then projected into $C$ lower dimension heads and each head
is quantized to the $top-k$ nearest keys in the corresponding codebook. The
values corresponding to the selected keys are averaged, and used for the
downstream task. The keys in the codebooks are frozen and initialized to cover
the input data manifold whereas the values are learnable. The mapping between
the keys and values is non-parametric and frozen. Thus, the gradient is not
propagated between the values and keys during training of the model. Since the
values are retrieved and updated sparsely, and all the components except the
value codes and the decoder are frozen, DKVB stores information in the form
input-dependent, sparse and localized representations (i.e., the value codes).
These inductive biases allow the framework to exhibit improved generalization
under distribution shifts during training, as shown empirically in Träuble et
al. (2023). Figure 1 (Left) shows an overview of a model with a DKVB where
$C=1$, $M=5$ and top-$k=1$.
## 4 Unlearning via Sparse Representations
#### Learning a Discrete Key Value Bottleneck.
A Discrete Key Value Bottleneck (DKVB) model is first trained on the given
dataset using the standard negative log-likelihood (cross-entropy loss)
training objective for multi-class classification. We use a non-parametric
average pooling decoder and a CLIP (Radford et al., 2021) pre-trained ViT-B/32
(Dosovitskiy et al., 2020) as the backbone in all our experiments involving
the DKVB. Then we proceed to unlearn a specific subset of data from these
models. Before training with the classification objective, we do a key
initialization for the DKVB models on the same dataset.
Key Initialization in DKVB models. The mapping between keys and values in the
discrete key-value bottleneck is non-parametric and frozen. As a result, there
is no gradient (back)propagation from the values to the keys. Thus, it becomes
essential for the keys to be initialized before learning the values and
decoder, such that they broadly cover the feature space of the encoder. This
ensures that the representations are distributed sparsely enough. As in
Träuble et al. (2023), we use exponential moving average (EMA) updates (Oord
et al., 2017; Razavi et al., 2019) to initialize the keys of the DKVB models.
The key-initialization is done on the same train dataset $\mathcal{D}_{train}$
which we want to train the model on. The key initializations depend solely on
the input encodings of the backbone and hence do not require access to any
labeled data.
#### Inference for Unlearning.
We propose to achieve unlearning in DKVB models by excluding key-value pairs
from the bottleneck such that they cannot be selected again. This masking is
done by setting the quantization distance of the selected keys to ‘infinity’.
Figure 1 (center and right column) shows an overview of the proposed methods.
More specifically, we experiment with two methods, _Unlearning via
Activations_ and _Unlearning via Examples_ , described as follows.
Unlearning via Examples. In this method, we analyze the effect of unlearning a
subset of $N_{e}$ examples belonging to the forget set. $N_{e}$ examples are
randomly sampled from the forget set training data
($\mathcal{D}_{train}^{retain}$) and are input into the model having a DKVB.
All key-value pairs that are selected during forward propagation across the
$N_{e}$ examples are flagged. These key-value pairs are then masked out from
the bottleneck. Technically, this approach requires access to the original
training data corresponding to the forget class. However, it is also possible
to carry out this procedure with a proxy dataset that has been sampled from a
distribution close enough to that of the forget set. This approach is
motivated by the assumption that such a dataset would likely utilize roughly
the same set of selected key-value pairs.
Unlearning via Activations. In this second method, we analyze the effect on
the quality of unlearning by deactivating different numbers of key-value pairs
corresponding to the forget set. We refer to the key-value pairs that have
been selected as inputs to the decoder as activations. The entire forget set
is forward-propagated through the DKVB model and all the key-value pairs
selected across all examples of the forget class are recorded. Next, we mask
the top-$N_{a}$ most frequently selected key-value pairs from the bottleneck.
The requirement of accessing the original training data for this method can be
avoided by caching all the activations corresponding to the forget set during
the last epoch of training. Further, similar to the previous case, unlearning
via activations may also be performed given access to data that has been
sampled from a distribution close enough to the distribution of the forget
set.
Unlearning via Activations and Unlearning via Examples are two different ways
of achieving a common objective: to exclude a subset of activations
corresponding to the forget set. However, using one approach over the other
may be more practical or even necessary, depending on the task at hand. For
selective unlearning where the goal is to forget specific examples, Unlearning
via Examples would be necessary. In both the above approaches, we do not do
any form of retraining or fine-tuning. Hence both these approaches require
negligible additional compute.
## 5 Experiments and Results
The goal of our experiments is three-fold. First, we validate that we can
zero-shot unlearn information via the Unlearning via Activations and
Unlearning via Examples methods in models with a DKVB (Section 5.2). Second,
we show that the proposed method is competitive with SCRUB (Section 5.3).
Third, we show that the proposed method is equally competitive in the less
constrained setting where we allow gradient-based re-training as is commonly
required by previous methods (Section 5.4). Before presenting these results,
we describe our experimental setup.
### 5.1 Experimental Setup
#### Benchmark datasets
We validate the proposed methods using experiments across three base datasets:
CIFAR-10 with 10 distinct classes, CIFAR-100 (Krizhevsky et al., 2009) with
100 distinct classes and LACUNA-100 (Golatkar et al., 2020a) with 100 distinct
classes. LACUNA-100 is derived from VGG-Faces (Cao et al., 2018) by sampling
100 different celebrities and sampling 500 images per celebrity, out of which
400 are used as training data and the rest are used as test images.
#### Models
On the aforementioned three datasets we study the following types of model
architectures:
1. (a)
Backbone + Discrete Key-Value Bottleneck (Ours): Overall, this architecture
consists of three components: 1) the frozen pre-trained backbone 2) the
Discrete Key-Value Bottleneck (DKVB) and 3) a decoder, as shown in Figure 1.
For the DKVB, we use 256 codebooks, with 4096 key-value pairs per codebook
(approximately 1M pairs overall) following Träuble et al. (2023).
2. (b)
Backbone + Linear Layer (Baseline): As a baseline, we replace the Discrete Key
Value bottleneck and the decoder in the above model architecture with a linear
layer. Thus, the two components of this model are 1) a frozen pre-trained
backbone and 2) a linear layer. This model will be used for the SCRUB baseline
method.
In each model, we use a pre-trained frozen CLIP (Radford et al., 2021)
ViT-B/32 as our encoder backbone. We refer to the appendix for additional
implementation details.
#### Training the Base Models
We then train both model architectures on the full training sets of each
dataset. Since the backbone is frozen, for the baseline, only the weights of
the linear layer are tuned during both initial training (and later
unlearning). Since we use only one linear layer, we do not do any sort of pre-
training (beyond the backbone), unlike in previous works (Kurmanji et al.,
2023; Golatkar et al., 2020a; b). Table 1 shows the performance of these
trained models on the train and test splits of the complete datasets. Starting
from these base models trained on the full datasets, we will validate the
ability to unlearn previously learned knowledge.
Table 1: Performance of the models on different sets of data after the initial
training on the three datasets. We use two kinds of models: (a) models having
a Discrete KV Bottleneck which are used for the proposed methods and (b)
models where the DKVB and the decoder are replaced by a Linear Layer. These
are used for the baseline. We wish to reduce the accuracy of these models on
${D}_{test}^{forget}$ to 0% while maintaining the accuracy on
$\mathcal{D}_{test}^{retain}$.
Dataset | $\mathcal{D}_{train}$ | $\mathcal{D}_{train}^{retain}$ | $\mathcal{D}_{train}^{forget}$ | $\mathcal{D}_{test}$ | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$
---|---|---|---|---|---|---
CIFAR-10 | 100% | 100% | 100% | 93.01% | 92.61% | 96.50%
CIFAR-100 | 99.98% | 99.98% | 100% | 78.43% | 78.24% | 96.00%
LACUNA-100 | 98.09% | 98.07% | 100% | 90.38% | 90.28% | 100%
(a) Backbone + DKVB
Dataset | $\mathcal{D}_{train}$ | $\mathcal{D}_{train}^{retain}$ | $\mathcal{D}_{train}^{forget}$ | $\mathcal{D}_{test}$ | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$
---|---|---|---|---|---|---
CIFAR-10 | 93.27% | 92.82% | 97.32% | 93.02% | 92.59% | 96.90%
CIFAR-100 | 86.73% | 86.61% | 99.00% | 78.53% | 78.35% | 96.00%
LACUNA-100 | 95.58% | 95.53% | 100% | 90.68% | 90.59% | 100%
(b) Backbone + Linear Layer
#### Unlearning
We aim to make the problem of unlearning as challenging as possible in order
to fairly evaluate the proposed methods. Therefore, on each dataset we select
the class that is best learned by the respective models with the Discrete Key
Value Bottleneck trained previously, to be the forget class (see Appendix A
for further details). Table 1 further shows the accuracy of the base models on
the train and test splits of the thus defined retain and forget class dataset
splits.
#### Objective & Metrics
We report our results on the test data of retain classes and forget class,
i.e. $\mathcal{D}_{test}^{retain}$ and $\mathcal{D}_{test}^{forget}$. Further,
in our experiments, we aim to achieve complete unlearning \- achieving minimal
accuracy on the forget set while maintaining the performance on the retain
set.111All of our experiments are performed on a RTX8000 GPU with 48GB memory.
While achieving complete unlearning may not always be desirable, such as in
the case of Membership Inference Attacks (MIAs) the proposed methods can be
easily extended to defend against MIAs (We refer to Appendix D for further
discussion on MIAs and the proposed methods).
### 5.2 Zero-Shot Unlearning via Discrete Key-Value Bottleneck
We will now discuss the results of unlearning via activations and examples,
i.e. the two approaches proposed in Section 4 on all three benchmark datasets.
All experiments are performed across 5 random seeds and mean values are
reported.
#### Unlearning via Activations.
As discussed in Section 4, unlearning via activations requires us to set the
hyperparameter $N_{a}$, reflecting the top-$N_{a}$ most frequently activated
key-value pairs which will be masked out after inference on the forget set. We
therefore start by analyzing its role over a wide range of values of $N_{a}$
to probe its choice and effect including $N_{a}=0$ being the limit without any
unlearning. Figures 2(c) \- 2(e) summarize the unlearning and effect of
$N_{a}$ on the retain vs forget test set. In the case of CIFAR-10, the initial
accuracies on the retain and forget test set are 92.61% and 96.50%
respectively. As $N_{a}$ increases, the forget class test accuracy decreases,
slowly for small $N_{a}$ and rapidly for larger $N_{a}$. The model reaches
random accuracy (i.e. 10% for CIFAR-10) on the forget class test data at
$N_{a}=150000$. At this point the retain set test accuracy is 92.97%. The
model unlearns the forget class completely between $N_{a}=170,000$ (0.4%) and
$N_{a}=180,000$ (0%). At this point, the retain set test accuracy is 92.94%,
which is almost identical to the initial accuracy. Further increasing $N_{a}$
up to $N_{a}=200,000$, i.e. about 20% of all key-value pairs, leads to an
additional increase in retain set test accuracy to 93%. As can be seen from
the equivalent analysis on the CIFAR-100 models in Figure 2(d) as well as the
Lacuna-100 models in Figure 2(e), the same trend of maintaining the initial
retain accuracy while minimizing the forget accuracy up to a minimum holds
across all three datasets validating its meaningful unlearning capability.
(c) CIFAR-10
(d) CIFAR-100
(e) LACUNA-100
Figure 2: Performance on the retain set test data vs. Performance on the
forget set test data on (a) CIFAR-10 (b) CIFAR-100 (c) LACUNA-100 in the case
of Unlearning via Activations as the value of $N_{a}$ is increased. The
relative performance on the retain set test data as compared to the original
models increases after unlearning in the case of CIFAR-10 and drops by 0.2%
and 0.17% for CIFAR-100 and LACUNA-100.
#### Unlearning via Examples.
For the second method—unlearning via examples—$N_{e}$ examples are sampled
randomly from the training data of the forget class, and subsequently used for
unlearning by the mechanism described in Section 4. Similar to before, we aim
to assess the effect on the choice of $N_{e}$ over a wide range for each
dataset, including $N_{e}=0$ being the limit without any unlearning. Figures
3(a) \- 3(c) summarize the unlearning and effect of $N_{a}$ on the retain vs.
forget test set. We again begin by focusing on the results with CIFAR-10
(Figure 3(a)). Here, the forget set $\mathcal{D}_{train}^{forget}$ contains
5000 examples. We start off with retain set and forget set test accuracies of
92.61% and 96.50% respectively. Similar to the previous approach – unlearning
via activations – the test accuracy on the forget set decreases with
increasing $N_{e}$. The accuracy on the retain test set, on the other hand,
increases monotonically, although only slightly overall. The model achieves
random accuracy on the forget class around $N_{e}=2500$. The accuracy on
retain set test data is at just under 93% at this transition. Finally, the
accuracy on the forget set drops to 0% (i.e. complete unlearning) between
$N_{e}=3000$ and $N_{e}=3400$ with a retain set test accuracy of just above
93% at $N_{e}=3400$. Further increasing $N_{e}$ does not affect the retain set
test accuracy notably. An equivalent analysis on the CIFAR-100 models in
Figure 3(b) and the Lacuna-100 models in Figure 3(c) exhibits a very similar
trend to CIFAR-10 with the only difference that the initial retain accuracy is
roughly maintained instead of further increasing. Successful minimization of
the forget accuracy up to a minimum is achieved across all three datasets,
validating it as another option for unlearning using discrete key-value
bottlenecks.
(a) CIFAR-10
(b) CIFAR-100
(c) LACUNA-100
Figure 3: Performance on the retain set test data vs. Performance on the
forget set test data on (a) CIFAR-10 (b) CIFAR-100 (c) LACUNA-100 in the case
of Unlearning via Examples as the value of $N_{e}$ is increased. Similar to
Unlearning via Activations, the relative performance on the retain set test
data as compared to the original models increases after unlearning in the case
of CIFAR-10 and drops by 0.36% and 0.09% for CIFAR-100 and LACUNA-100.
#### Summary.
Both methods, Unlearning via Activations and Unlearning via Examples,
successfully demonstrated unlearning of the forget class while having a
negligible effect on the models’ performance on the retain set. Importantly,
this is achieved without any form of training, retraining, or fine-tuning as
is usually required by other methods. The retain set test accuracy remains
more or less constant for all three datasets except for a few minor
fluctuations. This is a result of the fact that due to localized and context-
dependent sparse updates during the initial training of the model, discrete
key-representations corresponding to different classes in the dataset are well
separated from each other, an important prerequisite discussed in (Träuble et
al., 2023). Hence, all the information about a class can be unlearned by
forgetting only a subset of the forget class training data in the case of
Unlearning via Examples, making it very data-efficient.
### 5.3 Comparison with Baseline
We now compare the results of both the proposed methods, which require
Backbone + DKVB models against the SCRUB method, which is optimized for models
without such a bottleneck. For this, we will use the Backbone + Linear Layer
model described in 5.1. On this model, we run SCRUB and compare the
performance drop after unlearning with the two proposed methods.222We refer to
Appendix E.2 for more details on the implementation of this baseline Table 2
shows the comparison between the two previously reported methods and the SCRUB
baseline. We can see that the proposed methods perform at least as well as
SCRUB; in the case of CIFAR-100 and LACUNA-100 favorably, as we see from the
damage incurred on the retain set test performance for CIFAR-100 and
LACUNA-100. Finally, it is important to re-emphasize that the proposed methods
achieve the shown performance without requiring any additional gradient-based
training for unlearning. SCRUB is stopped when the forget set is completely
unlearned without damaging the performance on the retain set. We refer to
Appendix E.3 for further training details.
Table 2: Comparison between the proposed methods and the baseline on CIFAR-10, CIFAR-100 and LACUNA-100. We compare the change in performance on the retain and forget test data relative to the originally trained models. All three methods are able to unlearn the forget sets completely in all three cases and maintain the performance on the retain set of the data in the case of CIFAR-10. For CIFAR-100 and LACUNA-100, the proposed methods are able to maintain the performance on the retain set better than the baseline. | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
Rel. change from base model | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$ | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$ | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$
DKVB via Activations (sec 5.2) | 0.36% | -100% | -0.20% | -100% | -0.17% | -100%
DKVB via Examples (sec 5.2) | 0.45% | -100% | -0.36% | -100% | -0.09% | -100%
Linear Layer + SCRUB | 1.62% | -100% | -0.91% | -100% | -1.10% | -100%
### 5.4 Unlearning beyond the zero-shot setting
Next, we investigate the effect of using additional compute to the proposed
methods. As shown previously, the proposed methods perform competitively to
SCRUB zero-shot. To the best of our knowledge, SCRUB is the most competitive
and relevant unlearning approach. However, it has the inherent drawback of
requiring compute for unlearning. Nevertheless, for a fair comparison, we
additionally explore the implications of this additional compute for the
proposed two methods. Specifically, we retrain the models after zero-shot
unlearning on the training data of the retain set (i.e.,
$\mathcal{D}_{train}^{retain}$) for 10 epochs. For the baseline, we use the
same experimental setting as in Section 5.3 and run it for 10 epochs, instead
of stopping once the forget set has been completely unlearned. Figure 4
highlights the effect of retraining of the proposed methods compared to SCRUB
across multiple epochs, for all three datasets.
Retraining the unlearned models on the retain set does not affect their
performance significantly. The performance of the baseline on the other hand
increases after an initial drop in case of CIFAR-100 and LACUNA-100. The
initial drop may be attributed to the damage to the retain set performance
caused by the initial max-steps. The subsequent increase can be attributed to
the fact that the SCRUB training objective also optimizes the task loss on the
retain set. Thus, once the model unlearns the forget set, SCRUB shifts the
model capacity towards better learning the retain set. For CIFAR-10 this
results in the model performing better than the DKVB models on the retain set
as the retain set test accuracy after unlearning is higher than the original
model. However, the baseline is unable to recover its original performance for
CIFAR-100 and LACUNA-100. We refer to Appendix B for a similar comparison on
the forget set.
(a) CIFAR-10
(b) CIFAR-100
(c) LACUNA-100
Figure 4: Comparison between the performance of proposed methods with added
compute and the baseline on the retain set test data. For the proposed
methods, the plots start from after the initial zero shot unlearning. For the
baseline, the plots start from the original models. Retraining the models
unlearned using the proposed models does not lead to any significant
improvements in performance.
## 6 Limitations and Future Work
The proposed methods inherit the limitations of the DKVB (Träuble et al.,
2023). Two important limitations are 1.) the reliance of DKVB on pre-trained
encoders which can extract meaningful shared representations and 2.) trade-
offs in downstream performance due to the use of an information bottleneck.
Extensions to the model may involve training sparse representations inducing
discrete bottleneck end-to-end. Further, in our experiments, we consider the
setting of class incremental learning where the forget set can be easily
identified and isolated. However, this may not always be true for a given task
and more complicated approaches might be needed to identify the data that
needs to be removed from the model. Thus, future work may involve scaling the
proposed methods to such tasks and developing methods to identify and isolate
the forget set in such cases.
## 7 Conclusion
In this work, we proposed a new approach to unlearning that requires
negligible additional computation in order to unlearn a subset of data. This
approach is based on the use of a discrete architectural bottleneck which
induces sparse representations. These sparse representations facilitate
unlearning a subset of data from the model with minimal to no performance drop
on the rest of the data. We focused on the setting of class unlearning and our
experiments show that the proposed approach, while being compute efficient,
performs competitively with or in some cases better than a state-of-the-art
approach which requires additional compute to perform unlearning. We also
studied a compute-matched condition in which we allowed additional retraining.
We found that the proposed approach did not benefit from such retraining,
indicative of the fact that knowledge is highly localized in the Key-Value
Bottleneck. Consequently, excising the activated key-value pairs from the
model is a highly effective means of unlearning the forget set without
disrupting the retain set.
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## Appendix A Deciding the Forget Class
We assume that this class should be the most difficult one for the model to
forget. Figures 5 \- 5 show the number of mis-classifications per class on the
test data, for all three datasets. For CIFAR-10, class #1 is the best-learned
class with the lowest number of mis-classifications. Thus, we select class #1
as the forget class for the dataset. Similarly, for CIFAR-100 class 58 is the
best-learned class and for LACUNA-100, class 48 is one of the best-learned
classes with zero mis-classifications. Hence, we select classes #58 and #48 as
the forget classes for CIFAR-100 and LACUNA 100 respectively. We use the same
forget classes for experiments on the models with a linear layer in place of
the DKVB (i.e., the baseline) as well.
Figure 5: Number of mis-classifications per class for the test data. The red
bars correspond to the class with the least number of mis-classifications (a)
CIFAR-10: Class 1 has the least number of mis-classifications (b) CIFAR-100:
Class 58 has the least number of mis-classifications (c) LACUNA-100: Classes
34, 48, 65, 76, 82 and 85 have 0 mis-classifications and hence, do not have a
bar
## Appendix B Performance on Forget Class during Re-training
In all three cases, the baseline completely unlearns the forget set quickly.
(a) CIFAR-10
(b) CIFAR-100
(c) LACUNA-100
Figure 6: Comparison between the performance of proposed methods with added
compute and the baseline on the and forget set test data. Note that for the
proposed methods, the plots start from after the initial zero shot unlearning.
For the baseline, the plots start from the original models. The green line
occludes the red line since both of them stay at 0% throughout the training.
## Appendix C Detailed Analysis Zero-Shot Unlearning via Activations
### C.1 Unlearning via Activations
CIFAR-10: For CIFAR-10, we first validate the unlearning with
$N_{a}\in\\{10000n:n\in[0,20]\\}$ and record the performance on
$\mathcal{D}_{test}^{retain}$ and $\mathcal{D}_{test}^{forget}$. Figure 2(c)
shows the scatterplot of forget set test accuracy vs. retain set test accuracy
for different values of $N_{a}$. We start off with $N_{a}=0$, i.e., no
unlearning, to $N_{a}=300000$. The initial forget and retain set test
accuracies for CIFAR-10 as shown in Table 1 are 93.02% and 96.50%
respectively. As $N_{a}$ increases, the forget class test accuracy decreases,
slowly at first and rapidly later. The model reaches random accuracy (i.e. 10%
for CIFAR-10) on the forget class test data at $N_{a}=150000$. At this point
the retain set test accuracy is 92.97%. The model unlearns the forget class
completely between $N_{a}=170000$ (0.4%) and $N_{a}=180000$ (0%). The retain
set test accuracy at $N_{a}=180000$ is 92.94%, which is almost similar to the
initial accuracy of 92.61%. Further increasing $N_{a}$ till 200000 leads to an
increase in retain set test accuracy to 93%.
CIFAR-100 We vary $N_{a}$ for CIFAR-100 as $N_{a}\in\\{5000n:n\in[0,18]\\}$.
Figure 2(d) shows the scatterplot of retain set test accuracy vs. forget set
test accuracy for these values. We start with retain set test accuracy of
78.24% and forget set test accuracy of 96% (see Table 1. On increasing
$N_{a}$, the retain set test accuracy fluctuates in a small range around the
original accuracy, reaching a maximum of 78.32% at $N_{a}=40000$ and $450000$.
The forget set test accuracy drops slowly at first, dropping by 12% till
$N_{a}=20000$. The model reaches random test accuracy (1% for CIFAR-100) on
the forget set at $N_{a}=65000$ where the forget set test accuracy and retain
set test accuracies are 1% and 78.14% respectively. It reaches 0% at
$N_{a}=70000$. At this point, the retain set test accuracy is 78.08%. Thus the
model incurs a 0.2% deterioration in its retain set test accuracy by the time
it has completely unlearned the forget class.
LACUNA-100 We vary $N_{a}$ for LACUNA-100 as
$N_{a}\in\\{10000n:n\in[0,20]\\}$. Figure 2(e) shows the scatterplot of retain
set test accuracy vs forget set test accuracy for LACUNA-100. The initial
retain set test accuracy and forget set test accuracy for LACUNA-100 are
90.28% and 100% respectively (see Table 1. As we increase $N_{a}$, the retain
set test accuracy increases monotonically to 90.43% till $N_{a}=40000$ and
then decreases before starting to fluctuate. Unlike CIFAR-10 and CIFAR-100,
the decrease in forget set test accuracy is rapid from the beginning and slows
down towards the end. The forget set test accuracy reaches a near random
accuracy of 2% at $N_{a}=140000$ where the retain set test accuracy is 90.24%.
The model fully unlearns the forget set (i.e., forget set test accuracy = 0%)
at $N_{a}=150000$ at which point the accuracy of the model on the retain set
test data is 90.13%. Thus there is only a 0.17% drop in performance as
compared to the original model. Moreover as seen in Figure 2(e), the retain
set test accuracy increases slightly as $N_{a}$ is increased further.
### C.2 Unlearning via Examples
CIFAR-10 CIFAR-10 contains 5000 examples in the forget set training data. We
unlearn randomly sampled $N_{e}$ examples out of these 5000 examples. We
experiment with the following values of $N_{e}\in\\{200n:n\in[0,20]\\}$.
Figure 3(a) plots the forget set test performance vs. retain set test
performance of the model for different values of $N_{e}$. We start off with
retain set test and forget set test accuracies of 92.61% and 96.50%
respectively. Similarly to in unlearning via activations, the forget class
test accuracy decreases with an increase in $N_{e}$, and the decrease is slow
at first and then rapid. The retain set test accuracy, on the other hand,
increases monotonically, although marginally. The model achieves random
accuracy on the forget class between $N_{e}=2400$ and $N_{e}=2600$, where the
accuracies are 18.04% and 6.68% respectively. The accuracy on retain set test
data at these points are 92.95% and 92.98% respectively. The forget set test
accuracy drops to 0% (i.e., complete unlearning) between $N_{e}=3000$ (0.08%)
and $N_{e}=3200$ (0%). The retain set test accuracies at these two points are
92.99% and 93.02%. Further increasing $N_{e}$ does not affect the retain set
test accuracy much.
CIFAR-100 CIFAR-100 contains 500 examples of the forget class in the training
data. We vary $N_{e}$ for CIFAR-100 as $N_{e}\in\\{10n:n\in[0,15]\\}$. Figure
3(b) shows the plot of forget set test accuracy vs. retain set test accuracy.
The forget set test accuracy drops rapidly and monotonically from $N_{e}=30$
to $N_{e}=110$ where its value is 5% and the retain set test accuracy is
77.99%. It reaches the near random accuracy on 1.6% in the forget set test
data at $N_{e}=120$ where the retain set test accuracy is 78.01%. The model
completely unlearns the forget set at $N_{e}=150$ where the retain set test
accuracy is 77.96%. The retain set test accuracy first decreases (almost
monotonically) till $N_{e}=90$. It increases thereafter slightly, before
decreasing again towards the end.
LACUNA-100 LACUNA-100 has 400 forget class training examples. We unlearn a
randomly selected subset of size $N_{e}$ from it. $N_{e}$ is varied as
$N_{e}\in\\{20n:n\in[0,15]\\}$. Figure 3(c) shows the plot of retain set test
accuracy to forget set test accuracy for different $N_{e}$. As in the case of
unlearning via examples and unlike CIFAR-10 and CIFAR-100, the forget set test
accuracy decreases rapidly with $N_{e}$ initially till $N_{e}=60$; and rather
slowly after that. The retain set test accuracy increases slightly at first
and then starts fluctuating. The model reaches a random performance of 1% on
the forget set test data at $N_{e}=260$. The retain set test performance at
this point is 90.25%. Finally, the model unlearns the forget class completely
at $N_{e}=300$, where the retain set test accuracy is 90.2%. Thus, the retain
set performance drops by 0.09% by the time the model has completely unlearned
the forget class.
## Appendix D Using the proposed Methods against Membership Inference Attacks
Depending on the application, complete unlearning of the forget set may not
always be the final goal of unlearning. For several use cases such as removing
information about corrupted data from the model or removing harmful biases
exhibited by the model, maximal error on the forget set is desirable. However,
for applications such as Differential Privacy, it is more desirable to achieve
a forget set error which is similar to that of a model trained from scratch
only on the retain set. Otherwise, it makes the unlearned model susceptible to
Membership Inference Attacks (MIA) (Shokri et al., 2017). Although we do not
explore this setting in this work, the proposed method can also be used for
applications where complete unlearning is not desirable. This can be done by
following a procedure similar to SCRUB+R (Kurmanji et al., 2023), wherein
instead of selecting a particular model checkpoint, one can select the model
corresponding to specific values of $N_{a}$ or $N_{e}$ such that the error on
the forget set test data is similar to the reference point as defined in
Kurmanji et al. (2023).
First,it is important to clarify that the proposed approach is not a-priori
suited for selective unlearning, i.e. the setting where we want the model to
forget specific examples or a small subset of examples instead of removing the
information about an entire class. The KV bottleneck induces clusters of
representation, where the members of a particular cluster correspond to the
representations belonging to the same class. When we try to unlearn the
representations corresponding to one particular example belonging to a
particular class, the KV bottleneck routes the selection to other
(key-)representations within the same cluster. Since these representations
also contain information about the same class as the examples we intend to
unlearn, the model would still predict the class to be unlearnt.
Due to the same reason our approach is also not designed for working against
traditional Membership Inference attacks. According to the basic attacks setup
as explained in Kurmanji et al. (2023), the objective is to obtain a model
that has unlearnt a small subset of specific examples (i.e. selective
unlearning) such that the loss of the model on the unlearnt subset of examples
should be indistinguishable from loss on examples that the model never saw
during training.
Nevertheless, we attempt to modify the above setup to that of ”Class
Membership Inference Attacks (CMIA)”. In CMIA, the aim is to defend against an
attacker whose aim is to determine whether a model that has undergone
unlearning ever saw a particular class as a part of its training data. Thus,
we want the model to unlearn a particular class such that the
losses/performance of the model on the unlearnt class is indistinguishable
from a held-out class that the model never saw during its training. We
describe the experimental setup and results below.
Experimental Setup We perform the experiment for CIFAR10. We divide the
dataset into training data ($D_{Train}$), validation data ($D_{Val}$) and test
data ($D_{Test}$). Training Data consists of 4000 examples per class;
validation and test data consist of 1000 examples per class. We first trained
a model on the first 9 classes of CIFAR10. Thus, class number 10 is the held-
out class. Next, we unlearn class 1 from the model using the Unlearning via
Activations approach introduced in the paper. We unlearn the model until the
loss of the model on the validation sets of the forget class and the held-out
class are similar. In our experiments, we find that we reach this point at
approximately $N_{a}=240000$. The loss $l(x,y)$ in our case would be the
cross-entropy loss.
Next, we label the losses corresponding to the validation and test set of the
forget class as 1 and those corresponding to the validation and test set of
the held-out class as 0. We train a binary classifier on the validation losses
of the forget and held-out sets and evaluate it on the test losses. We follow
a similar setting for the baseline model, where we obtain the model suitable
for MIA defense by using SCRUB+R (Kurmanji et al., 2023). For a successful
defense, we would want the accuracy of the classifier to be close to 50%,
indicating that it is unable to distinguish between the unlearnt class and the
held-out class. Same as Kurmanji et al. (2023), we use
sklearn.logistic_regression as our attacker (the binary classifier). We call
the approach described above Partial UvA (Partial Unlearning via Activations).
We run experiments for 3 random seeds, and the mean of the attacker
performance is reported. Note that a similar procedure can also be followed
using Unlearning via Examples.
Observations and Results: We report the results of the experiment described
above in the table given below. We observe that although the baseline performs
better, the proposed approach performs competitively, even though we have not
intended to develop the method for this scenario.
Table 3: Comparison on Class Membership Inference Attacks between the proposed approach and the baseline. A binary classifier is trained on the validation losses of the forget and held-out sets and is evaluated on the test losses. The proposed approach performs competitively to SCRUB + R/ Approach | Attacker Accuracy
---|---
Partial UvA | 53.50%
Linear Layer + SCRUB + R | 51.50%
## Appendix E Training Details and Hyperparameters
We do not use any data augmentation in any experiment. The transforms used are
the same as CLIP (Radford et al., 2021) pretrained ViT/B-32 transforms.
### E.1 Training Details and Hyperparameters for training the original DKVB
Models
Table 4 shows the hyperparameters used for training the base DKVB models
Table 4: Hyperparameters used for training the base DKVB models | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
top-k | 1 | 10 | 10
Key Dimension | 8 | 8 | 8
# of Key Init Epochs | 10 | 10 | 10
Type of Value Init | Gaussian Random | Zeros | Uniform Random
# of Codebooks | 256 | 256 | 256
# of Key-Value Pairs per Codebook | 4096 | 4096 | 4096
Optimizer | Adam | Adam | Adam
LR | 0.1 | 0.3 | 0.3
Batch Size | 256 | 256 | 256
Epochs | 74 | 71 | 7
### E.2 Training Details and Hyperparameters for training the original
Baseline Models
For the baseline models, we deliberately train them to similar test
($\mathcal{D}_{test}$) accuracies as the models with a Discrete Key Value
Bottleneck to ensure a fair comparison for unlearning. Table 5 shows the
hyperparameters used for training the baseline models
Table 5: Hyperparameters used for training the baseline models | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
LR | 0.001 | 0.01 | 0.01
Batch Size | 256 | 256 | 256
Epochs | 2 | 2 | 2
### E.3 Training Details and Hyperparameters for SCRUB
For the baseline, we run SCRUB on the model with linear layer. One epoch
consists of one min step and may or may not contain a max step. One max step
is included in every epoch for the first msteps epochs. We tune the
hyperparameter msteps in our experiments and pick the case where the model is
able to best recover its performance on the retain set test data and consider
this model as the final unlearned model.We mention the hyperparameters used
for running SCRUB for corresponding to the results presented in Section 5.3 in
Table 6. In this case, training of SCRUB is stopped when either (a) the forget
set accuracy has dropped to 0% without damaging the retain set accuracy or (b)
at the end of 10 epochs. In the latter case, the value of the retain set test
accuracy reported is the highest value in the training when under the
constraint that the forget set test accuracy is 0 at that point. Results
presented in Section 5.4 also uses the same set of hyperparameters except min-
step which is always 10 since we train all the methods for 10 epochs. We do
not consider cases where SCRUB is not able to completely unlearn the forget
class within the given computational budget.
Table 6: Hyperparameters for SCRUB + Linear Layer Experiments shown in Section 5.3 | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
Forget Set Batch Size | 256 | 256 | 256
Retain Set Batch Size | 256 | 256 | 256
# of max-steps (msteps) | 3 | 9 | 5
# of min-steps | 3 | 10 | 7
LR | 0.001 | 0.01 | 0.01
Optimizer | Adam | Adam | Adam
Batch Size | 256 | 256 | 256
### E.4 Training Details and Hyperparameters for Retraining Experiments
Once the DKVB models are unlearned using Unlearning via Activations and
Unlearning via Examples, we retraining them in order to make a fair comparison
with the baseline. Thus, during retraining, the initial performance of these
models on the retain set is same as the final performance of the zero-shot
unlearned models. Table 7 show the hyperparameters used for retraining the
unlearned DKVB models.
Table 7: Hyperparameters used for re-training experiments. UvA stands for Unlearning via Activations and UvE stands for Unlearning via Examples | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
| UvA | UvE | UvA | UvE | UvA | UvE
LR | 0.3 | 0.3 | 0.1 | 0.1 | 0.1 | 0.3
Optimizer | Adam | Adam | Adam | Adam | Adam | Adam
Batch Size | 256 | 256 | 256 | 256 | 256 | 256
Gradient Clipping | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1
## Appendix F Comparison of Runtimes
In this section, we present a comparison of runtimes of our approach against
the baseline, i.e. SCRUB (Kurmanji et al., 2023) as a proxy for comparing the
compute requirements of the two approaches. Table 8 compares the runtimes of
the proposed approaches against the baseline. We observe that the runtime for
the proposed approaches is orders smaller than that of the baseline.
Table 8: Comparison of runtimes between the proposed methods and the baseline on CIFAR-10, CIFAR-100 and LACUNA-100. | CIFAR-10 | CIFAR-100 | LACUNA-100
---|---|---|---
Runtimes (in seconds) | | | | | |
DKVB via Activations (sec 5.2) | 5.02 | 1.57 | 2.74
DKVB via Examples (sec 5.2) | 13.98 | 6.65 | 13.03
Linear Layer + SCRUB | 288.80 | 921.56 | 553.31
## Appendix G Experiments with a ResNet Backbone
To demonstrate the the proposed approaches are agnostic to the choice of the
backbone, we run the same set of experiments presented in Section 5.2 and
Section 5.3 on CIFAR-10 with an ImageNet supervised pretrained backbone.
Figure 7 shows the scatter plot of retain set test performance vs. forget set
test performance on CIFAR-10 with a ResNet backbone. Table 9 shows the
comparison against baseline.
Table 9: Comparison between the proposed methods with a pretrained ResNet50 backbone and the baseline on CIFAR-10. We compare the change in performance on the retain and forget test data relative to the originally trained models. For the baseline, we report two cases: Case B where the model unlearns the forget set completely but the retain set performance is not preserved very well, and Case A where the retain set performance is not preserved very well, but the model does not unlearn the forget set completely | CIFAR-10
---|---
Rel. change from base model | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$
DKVB via Activations (sec 5.2) | 0.04% | -100%
DKVB via Examples (sec 5.2) | -0.07% | -100%
Linear Layer + SCRUB (A) | 0.44% | -96.04%
Linear Layer + SCRUB (B) | -11.75% | -100%
(a) CIFAR-10 (via Activations)
(b) CIFAR-10 (via Examples)
Figure 7: Performance on the retain set test data vs. Performance on the
forget set test data for DKVB models with a supervised pretrained ResNet50
backbone on CIFAR-10.
There is a steep but drop in the retain set test accuracy towards the end when
$N_{a}$ and $N_{e}$ are high enough. However, speaking in absolute terms, the
drop seems to be not significant.
## Appendix H Experimental Results on ImageNet-1k
Table 10: Comparison between the proposed methods and the baselines on ImageNet-1k. We compare the change in performance on the retain and forget test data relative to the originally trained models. Rel. change from base model | $\mathcal{D}_{test}^{retain}$ | $\mathcal{D}_{test}^{forget}$
---|---|---
DKVB via Activations (sec 5.2) | 0.15% | -100%
DKVB via Examples (sec 5.2) | -0.03% | -100%
Linear Layer + SCRUB | 7.31% | -100%
(a) ImageNet (via Activations)
(b) ImageNet (via Examples)
Figure 8: Performance on the retain set test data vs. Performance on the
forget set test data for DKVB models on ImageNet-1k
|
Probabilistic Programming with Exact Conditions
Dario Stein
Radboud University Nijmegen
Erasmusplein 1
The Netherlands
Sam Staton
University of Oxford
Parks Road, OX1 3QD
United Kingdom
We spell out the paradigm of exact conditioning as an intuitive and powerful way of conditioning on observations in probabilistic programs. This is contrasted with likelihood-based scoring known from languages such as Stan. We study exact conditioning in the cases of discrete and Gaussian probability, presenting prototypical languages for each case and giving semantics to them. We make use of categorical probability (namely Markov and CD categories) to give a general account of exact conditioning which avoids limits and measure theory, instead focusing on restructuring dataflow and program equations. The correspondence between such categories and a class of programming languages is made precise by defining the internal language of a CD category.
<concept_desc>Theory of computation Denotational semantics</concept_desc>
<concept_desc>Theory of computation Categorical semantics</concept_desc>
<concept_desc>Theory of computation Probabilistic computation</concept_desc>
[500]Theory of computation Denotational semantics
[500]Theory of computation Categorical semantics
[500]Theory of computation Probabilistic computation
§ INTRODUCTION
Probabilistic programming is a programming paradigm that uses code to formulate generative statistical models and perform inference on them [50, 19]. Techniques from programming language theory can be used to understand modelling assumptions such as conditional independence, as well as enable optimizations that improve the efficiency of various inference algorithms (e.g. [45]). We'll also elaborate on this application in <Ref>.
There are two different styles of conditioning on data in a probabilistic program: scoring and exact conditioning. Scoring features constructs to re-weight the current execution trace of the probabilistic program with a given likelihood. By contrast, exact conditioning focuses on a primitive operation $E_1 \eq E_2$ which signifies that expressions $E_1$ and $E_2$ shall be conditioned to be exactly equal. A prototypical exact conditioning program looks as follows, where we infer some underlying value |x| from a noisy measurement |y| (see also <Ref>):
x = normal($\mu$=50, $\sigma$=10) # prior
y = normal($\mu$=x, $\sigma$=5) # noisy measurement
y =:= 40 # make exact observation
Variants of exact conditioning are available in different frameworks: In Hakaru [42], certain exact conditioning queries can be addressed using symbolic disintegration, but $(\eq)$ is not a first-class construct in Hakaru. Infer.NET [33] does allow exact conditioning on variables and employs an approximate inference algorithm to solve the resulting queries (e.g. [23]). The intended formal meaning of exact conditioning is however far from obvious when continuous distributions such as Gaussians are involved (in our example, the observation |y == 40| has probability zero). The goal of this article is to rigorously spell out the exact conditioning paradigm, give semantics to it and analyze its properties.
We note that, even in this simple example, the use of exact conditioning is intuitive and allows the programmer to cleanly decouple the generative model from the data observation stage.
As the example makes clear, exact conditioning lends itself to logical reasoning about programs. For example, after conditioning $s \eq t$, the expressions $s$ and $t$ are known to be equal and can be interchanged.
As we will show, exact conditioning also enjoys good formal properties which allow us to simplify programs compositionally.
Among the desired properties are the following: Program lines can be reordered as long as dataflow is respected. That is, the commutativity equation remains valid for programs with conditioning
\begin{equation}
\begin{array}{l}
\letin x u \\
\letin y v t
\end{array}
\equiv
\begin{array}{l}
\letin y v \\
\letin x u t
\end{array} \label{eqn:initcommutativity}
\end{equation}
where $x$ not free in $v$ and $y$ not free in $u$. We have a substitution law: if $t\eq u$ appears in a program, then later occurrences of $t$ may be replaced by $u$.
\begin{equation}(t\eq u);v[t/x]\quad \equiv
\quad (t\eq u);v[u/x]
\label{eqn:substlong}
\end{equation}
As a special base case, if we condition a normal variable on a constant $\underline c$, then the variable is simply initialized to this value
\begin{equation}\letin x {\normal()} {(x\eq \underline c);t} \quad \equiv
\quad t[\underline c/x]
\label{eqn:initlong}
\end{equation}
We substantiate these claims further and give examples of applications of these laws in our extended introduction (<Ref>).
In order to formally study exact conditioning, we focus on two concrete fragments of probabilistic computation
* finite probability, which deals with finite sets and discrete distributions
* Gaussian probability, which deals with multivariate Gaussian (normal) distributions and affine-linear maps
We give probabilistic languages for each fragment and extend them with an exact conditioning construct. The language for finite probability is well-known, while the Gaussian language is novel. We give its formal description and operational semantics in <Ref>.
Methods: Our goal is to give denotational semantics to these languages, and prove that the desired properties from <ref> hold. We wish to do this in a abstract way, that relies as little as possible on the particular details of finite or Gaussian probability, but instead treats them uniformly and is open to generalization. Categorical probability theory is such an abstract language of mathematical models of probability, which allows us to discuss the relevant notions such as determinism, independence and conditioning in a uniform way. We connect categorical probability theory to probabilistic programming using the following Curry-Howard style correspondence
* probabilistic programs are the internal languages of categorical probability theories; program terms $t$ can be interpreted as morphisms $\sem{t}$ in these categories
* for every probabilistic language, its syntactic category is a categorical model of probability theory. objects are types and morphisms are terms of the language
We prove a particular version of the correspondence in <Ref>: The CD-calculus is the internal language of CD categories, a widespread model of categorical probability theory <Ref>. This language, which resembles a first-order OCaml, serves as a meta-language of which all other languages discussed in this article (finite or Gaussian, with or without conditioning) will be particular instances.
The Curry-Howard correspondence gives us three equivalent formalisms for describing probabilistic models (see <Ref>)
* terms in a probabilistic language
* morphisms in a categorical model
* string diagrams, which are a well-known graphical notation to describe compositions of morphisms in an intuitive way [41]
We will frequently convert back and forth between the different formalisms for convenience, conciseness and to emphasize different mathematical or programming intuitions. We expect some familiarity of the reader for translating string-diagrammatic and algebraic categorical notation, though we will give a brief reminder in the introduction of <Ref>. The correspondence between string diagrams and program terms is a novel technical contribution of this article. It is formally proved in <Ref>, and we will aid the reader by spelling out examples in both programming and categorical terms. A reader not primarily interested in category theory may still view our string diagram manipulations as a concise graphical way of encoding program transformations.
\[ \input{categorical_probability/dag_to_string.tikz} \]
String diagram
\begin{align*}
(f \otimes &((\id_W \otimes g) \circ \cpy_W)) \\ & \circ \cpy_W \\ &\circ \phi
\end{align*}
Categorical composition
\begin{align*}
&\vdash \phi : W \\
w : W &\vdash f : X \\
w : W &\vdash g : Y
\end{align*}
\begin{align*}
\overline{\vdash \letin w \phi (f,(w,g)) : X \ast (W \ast Y)}
\end{align*}
Different formalisms for composition
To connect our development to more traditional probabilistic notions, we consider the formalism of directed graphical models (Bayesian networks). For example, the three expressions of <Ref> can all be understood as encoding the generative structure of the following Bayesian network
\begin{equation} \input{categorical_probability/dag.tikz} \label{eq:graphicalmodel} \end{equation}
This Bayesian network indicates that there are random variables valued in spaces $W$, $X$ and $Y$, with the given conditional independence structure, but this conditional independence means that we can describe the situation by giving the distribution $\phi$ of the random variable valued in $W$, together with the distributions of the random variables valued in $X$ and in $Y$, which both depend on the random choice for $W$, via $f$ and $g$.
A systematic comparison of graphical models and string diagrams is given in [10]. We note that Bayesian networks are a strictly weaker formalism than the other three, i.e. not every string diagram or probabilistic program can be obtained from a Bayesian network.
We build up this categorical machinery to first understand the semantics of probabilistic languages without conditioning. We then use the same framework to extend the language with an exact conditioning operator. On the side of categorical semantics, this amounts to extending a suitable category $\C$ (for conditioning-free computation) with effects $X \to I$ for conditioning on observations. We call the extended category $\cond(\C)$. Constructing this category (<Ref>) and proving its desirable properties is the central contribution of this article.
Importantly, the $\cond$ construction makes no mention of measure theory, densities or limits, which usually feature in discussions of conditional probability, but is defined purely in terms of the categorical structure of $\C$. It is closely tied to reasoning in terms of program transformations: The Cond construction can be understood as giving normal forms for straight-line programs with exact observations, modulo contextual equivalence
\[ x : X \vdash \letin {(y,k) : Y \otimes K} {h} {(k \eqo o); \return y \quad : Y} \]
Our definition of abstract inference problems in <Ref> follows that intuition. The desired properties of exact conditioning can be proved purely abstractly for the $\cond$ construction. Our type-theoretic approach to conditioning will also help addressing counterintuitive behavior such as Borel's paradox (<Ref>).
We give concrete descriptions of the $\cond$ construction applied to our main examples of discrete and Gaussian probability. This means analyzing contextual equivalence for our example languages in detail: For finite probability, we obtain substochastic kernels modulo automatic renormalization (<Ref>). This fully characterizes contextual equivalence for straight-line inference with discrete probability, and refines the semantics using the subdistribution monad. Our discussion reveals interesting connections between the admissibility of automatic normalization and the expressibility of branching in probabilistic programs (<Ref>). For the Gaussian language we give a concrete analysis by proving that the denotational semantics is fully abstract, in <Ref>.
We give an overview over the structure of the article:
* To demonstrate the strengths and intricacies of the exact conditioning approach, we will follow up this introduction with an extended example (<Ref>) elaborating the noisy measurement example, and demonstrate the power of program transformations and compositional reasoning for a Gaussian random walk example.
* In <Ref>, we formally introduce the Gaussian language and its operational semantics.
* In <Ref>, we review string diagrams and categorical probability theory using CD- and Markov categories. We then introduce the CD-calculus (<Ref>) and prove it to be internal language of CD categories, which gives us the central correspondence of probabilistic programs and string diagrams. In <Ref>, we use the categorical notions to develop an abstract theory of inference problems in Markov categories.
* In <Ref>, we present the $\cond$ construction, which is the centerpiece of this article. We prove the well-definedness of its construction (<Ref>) and verify the desired laws for conditioning in full generality (<Ref>).
* In <Ref>, we return to analyze the $\cond$ construction in detail for our two example settings. This is tantamount to studying contextual equivalence for our exact conditioning languages: In <Ref>, we show that the denotational semantics for the Gaussian language is fully abstract. In <Ref>, we conduct a similar analysis for finite probability, arriving at an explicit characterization of $\cond(\mathsf{FinStoch})$. Our discussion reveals interesting connections between the admissibility of automatic normalization and the availability of branching in probabilistic programs (<Ref>).
Note. The starting point for this article is our paper in the Proceedings of LICS 2021 [49], where we introduced the Gaussian language, and used the $\cond$ construction to prove a full abstraction result. For this invited journal submission, we expand upon [49] with greater detail and background throughout Sections <ref>-<ref>, and expose the $\cond$ construction in a self-contained manner with an emphasis on program equations and graphical reasoning in <Ref>. This submission also incorporates otherwise unpublished material that is part of the first author's DPhil thesis [48], such as the formal presentation of the CD calculus. The treatment of finite probability in <Ref> and the recognition of the special role of branching in <Ref> are entirely novel contributions of this article. A Python implementation of the Gaussian Language is available under [47].
§.§ Extended Discussion about Exact Conditioning
We proceed with an extended discussion on the differences between the scoring and exact conditioning paradigms, and the strengths and difficulties related to exact conditioning. This discussion uses an informal Python-like language, and is not technically essential for the rest of the paper. The later sections of the article are fully formal.
Exact Conditioning versus Scoring: In the introduction, we considered an noisy measurement example: Our prior assumption about the distribution of some quantity $X$ is that it is normally distributed with mean $\mu=50$ and standard deviation $\sigma=10$. We only have access to a noisy measurement $Y$, which itself has standard deviation $5$, and observe a value of $Y=40$. Conditioned on that observation, the posterior distribution over $X$ is now $\N(42,\sigma)$ with $\sigma = \sqrt{20} \approx 4.47$.
In probabilistic programming with scoring, the primitive |score(r)| re-weights the current execution trace of the probabilistic program with a score or likelihood $\mlstinline{r} \in \mathbb R_{> 0}$. A derived operation |observe(x,D)| expresses an observation of a value $x$ from some distribution $D$ by scoring with the density $r = \mathsf{pdf}_D(x)$. The scoring implementation of the noisy measurement example therefore looks like this:
x = normal(50, 10) # prior
observe(40, normal(x,5)) # observation
The idea of Monte Carlo simulation is to run the program many times, picking different values for |x| from the normal distribution, but preferring runs with a high likelihood. This makes execution traces more likely whose value of x lies closer to $40$.
Scoring constructs are widely available in popular probabilistic languages such as Stan [5] or WebPPL [18]. Scoring with likelihoods from $\{0,1\}$ is sometimes called a hard constraint, as opposed to more general soft constraints. The prototypical way of performing inference on scoring programs is by likelihood-weighted importance sampling. Hard constraints turn this into mere rejection sampling, because likelihood-zero traces are discarded entirely. Replacing hard constraints by equivalent soft ones can thus be beneficial for inference efficiency.
Exact conditions are strictly more powerful than scoring, because we can express |observe(x,D)| in terms of conditioning on a freshly generated sample as |let y = sample(D) in y =:= x|. On the other hand, not every exact conditioning program can be expressed in terms of scoring:
In the special case of discrete probability, we can express an exact condition $E_1 \eq E_2$ by the hard constraint |score(if $E_1$ == $E_2$ then 1 else 0)| without issue. This causes an execution trace to be discarded whenever the condition is not met. This encoding is no longer viable for continuous distributions such as Gaussians: For example, the program
x = normal(0,1); x =:= 40
should return |x=40| deterministically, because $x$ is conditioned to have that value. On the other hand, the following hard constraint
x = normal(0,1); score(if x == 40 then 1 else 0)
will reject every execution trace, because the probability that |x==40| is true equals zero. It is important to distinguish exact conditioning $(\eq)$ from the boolean equality test $(==)$. This distinction is crucial to making sense of apparent paradoxes such as Borel's paradox (<Ref>).
Compositional Reasoning about Conditions: To elaborate the power of reasoning compositionally about conditioning programs, we consider the example of a simple Gaussian random walk with 100 steps, together with a table |obs| of exact observations (<Ref>). A straightforward implementation would be to first generate the entire random walk, and then condition on the observations
# generative model
for i in range(1,100):
y[i] = y[i-1] + normal(0,1)
# observations
for j in obs:
y[j] =:= obs[j]
The same program is more complicated to express without exact conditioning: Using soft conditions, the observations would need to be known at the time of generation and |observe| commands need to be issued in-place, breaking the decoupling between the model and the data.
On the other hand, rewriting the original model in such a way may improve the efficiency of inference. We can verify such a transformation using compositional reasoning: As we will show, it is consistent to reorder program lines as long as the dataflow is respected (<ref>), so the random walk program is equivalent to the following version with interleaved observations
for i in range(1,100):
y[i] = y[i-1] + normal(0,1)
if i in obs: y[i] =:= obs[i]
In the observation branch, we can now use initialization principle (<ref>) to set |y[i]| to its target value directly as |y[i] = obs[i]|. The remaining condition becomes |(y[i] - y[i-1]) =:= normal(0,1)| so we obtain
for i in range(1,100):
if i in obs:
y[i] = obs[i]
(y[i] - y[i-1]) =:= normal(0,1)
y[i] = y[i-1] + normal(0,1)
In this version of the program, all exact conditions can now be replaced by |observe| statements:
for i in range(1,100):
if i in obs:
y[i] = obs[i]
observe(y[i] - y[i-1] , normal(0,1))
y[i] = y[i-1] + normal(0,1)
We can run this resulting program directly using a Monte Carlo simulation in Stan or WebPPL.
Gaussian random walk (left) and conditioned posterior (right) with four exact observations at $i=20,40,60,80$
In <Ref>, we formalize the operational semantics of our Gaussian language, in which there are two key commands: drawing from a standard normal distribution ($\normal()$) and exact conditioning $(\eq)$. The operational semantics is defined in terms of configurations $(t,\psi)$ where $t$ is a program and $\psi$ is a state, which here is a Gaussian distribution. Each call to $\normal()$ introduces a new dimension into the state $\psi$, and conditioning $(\eq)$ alters the state $\psi$, using a canonical form of conditioning for Gaussian distributions (<Ref>).
In our first version of the random walk example, the operational semantics will first build up the prior distribution shown on the left in <Ref>, and then the second part of the program will condition to yield a distribution as shown on the right. But for the other programs above, the conditioning will be interleaved in the building of the model.
In stateful programming languages, composition of programs is often complicated and local transformations are difficult to reason about. This is what makes programs transformations like the ones we used powerful and nontrivial to verify.
§ A LANGUAGE FOR GAUSSIAN PROBABILITY
In this section, after an overview of the mathematics of Gaussian probability (<Ref>), we formally introduce a typed language (<Ref>) for Gaussian probability and exact conditioning, and provide an operational semantics for it (<Ref>).
Our operational semantics is straightforward, in that it maintains a symbolic description of the distribution over all latent variables as the program runs, expressed as a covariance matrix. Thus the aim of this section is to formally describe language that we are studying in this paper.
Looking beyond this section, the aim of the further sections of this paper is to address that issue that, as usual, the simple operational semantics here is intensional and non-compositional. It is intensional in that if two different programs actually behave in the same way, that might be very unclear from the operational semantics; it is non-compositional in that the role of running subprograms is hidden in the overall run of the operational semantics. The aim of the remainder of the paper, then, is to establish an equational and denotational framework for exact conditioning.
The language in this section is focused on Gaussian probability, for concreteness, but to understand the equational framework it will be helpful in future sections to move to a general setting (<Ref> and <Ref> for the general case without and with exact conditioning respectively). Once this general framework is established, we are able to offer a denotational explanation of exact conditioning, which specializes to this Gaussian language (<Ref>).
§.§ Recap of Gaussian Probability
We briefly recall Gaussian probability, by which we mean the treatment of multivariate Gaussian distributions and affine-linear maps (e.g. [30]). A Gaussian distribution is the law of a random vector $X \in \R^n$ of the form $X=AZ + \mu$ where $A \in \R^{n \times m}$, $\mu \in \R^n$ are not random but the vector $Z$ is a multivariate standard normal random vector. That is, its components $Z_1, \ldots, Z_m \sim \mathcal N(0,1)$ are independent and standard normally distributed, i.e. with the following probability density function:
\[ \varphi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac 1 2 x^2} \]
(Formally, this is a density is regarded with respect to the Lebesgue measure, see Section <ref>, but a high-school knowledge of probability density is sufficient for this section.)
The distribution of $X$ is fully characterized by its mean $\mu$ and the positive semidefinite covariance matrix $\Sigma=AA^T$. Conversely, for any $\mu$ and positive semidefinite matrix $\Sigma$ there is a unique Gaussian distribution of that mean and covariance denoted $\mathcal N(\mu, \Sigma)$. The vector $X$ takes values precisely in the affine subspace $S=\mu + \col(\Sigma)$ where $\col(\Sigma)$ denotes the column space of $\Sigma$. We call $S$ the support of the distribution. We note that while it is common to consider Gaussian distributions with nonsingular (positive definite) covariance matrix, it is convenient to allow the more general positive semidefinite case here, even including vanishing covariance. Natural programming constructs such as the copying of variables results in a singular covariance, and setting $\Sigma = 0$ lets us treat deterministic computation as a special case of probabilistic one. On the flipside, singular covariance requires us to carefully consider supports in our theory of conditioning.
Gaussian probability defines a small convenient fragment of probability theory, with the following properties:
* Affine transformations of Gaussians remain Gaussian. That is, if an affine map $f$ is written as $f(x)=Ax+b$ and $X$ is a random vector with mean $\mu$ and covariance $\Sigma$, then $f(X)$ has mean $A\mu + b$ and covariance $A\Sigma A^T$. This operation is called the pushforward of distributions and is denoted $f_*$, i.e.
\begin{equation} f_*(\mathcal N(\mu,\Sigma)) = \mathcal N(A\mu + b, A\Sigma A^T) \label{def:pushforward} \end{equation}
* Conditional distributions of Gaussians are themselves Gaussian. If we decompose an $(m+n)$-dimensional Gaussian vector $X \sim \mathcal N(\mu, \Sigma)$ into components $X_1, X_2$ with
\begin{equation*}
X = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}, \mu = \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \Sigma = \begin{pmatrix}
\Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22}
\end{pmatrix}\text{ where } \Sigma_{21} = \Sigma_{12}^T
\end{equation*}
there is a well-known explicit formula (e.g. <cit.>) for the conditional distribution $X_1|(X_2 = a)$ of $X_1$ conditional on $X_2=a$ for $a \in \supp(X_2)$. Namely $X_1|(X_2 = a) \sim \mathcal N(\mu',\Sigma')$ where
\begin{equation}
\mu' = \mu_1 + \Sigma_{12}\Sigma_{22}^{-}(a-\mu_2) \quad \Sigma' = \Sigma_{11} - \Sigma_{12}\Sigma_{22}^-\Sigma_{21}
\label{eq:conjugacy_formula}
\end{equation}
and $\Sigma_{22}^-$ is any generalized inverse of $\Sigma_{22}$ .
We elaborate on formula (<ref>) a bit: A generalized inverse of an $(m \times n)$-matrix $M$ is an $(n \times m)$-matrix $M^-$ such that $MM^-M = M$. Such inverses can be shown to always exist, but they need not be unique. If $M$ is invertible, its unique generalized inverse is $M^{-1}$. The posterior covariance matrix $\Sigma' = \Sigma_{11} - \Sigma_{12}\Sigma_{22}^-\Sigma_{21}$ is also known as Schur complement and is independent of the choice of generalized inverse. The matrix $\Sigma_{12}\Sigma_{22}^{-}$ appearing in the calculation of $\mu'$ does depend on the choice of $\Sigma_{22}^-$. However, it takes uniquely defined values on the subspace $\col(\Sigma_{22})$. Therefore, formula (<ref>) is only well-defined if the observation $a$ lies in the support of $X_2$. This caveat is mirrored in our categorical treatment of <Ref>, where conditionals are only unique on supports. A popular choice of generalized inverse is the Moore-Penrose pseudoinverse, which has connections to least squares optimization. For a detailed discussion of these concepts, we refer to <cit.>.
The formula for conditional probability becomes particular simple if we condition on a single real-valued component of a vector: Let $X \sim \mathcal N(\mu, \Sigma)$ and let $Z = uX$ for some $u \in \R^{n \times 1}$, then the covariance of $(X,Z)$, regarded as a random $(n+1)$-vector, decomposes as
\[\begin{pmatrix}
\Sigma & \Sigma u^T \\ u\Sigma^T & \sigma_{22}
\end{pmatrix}
\qquad \text{where $\sigma_{22} = u\Sigma u^T$}\]
and the conditional distribution of $X|(Z=a)$ is $\N(\mu',\Sigma')$ with
\begin{equation}
\mu' = \mu + \frac{a-u\mu}{\sigma_{22}}\Sigma u^T, \quad \Sigma' = \Sigma - \frac 1 {\sigma_{22}} \Sigma u^Tu \Sigma \label{eq:conjugacy_simple}
\end{equation}
whenever $\sigma_{22} > 0$. If $\sigma_{22} = 0$ and $u\mu = a$, the condition is tautologously $0=0$ and we have $\mu' = \mu$, $\Sigma'=\Sigma$. Otherwise, $a \notin \supp(Z)$, and the conditioning problem has no well-defined solution.
Let $X,Y \sim \mathcal N(0,1)$ be independent and $Z=X-Y$. The joint distribution of $(X,Y,Z)$ is $\N(\vec 0,\Sigma)$ with covariance matrix
\[ \Sigma = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & -1 & 2 \end{pmatrix} \]
By (<ref>), the conditional distribution of $(X,Y)$ given $Z=0$ has the following covariance matrix
\[ \Sigma' = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} - \frac 1 2 \begin{pmatrix} 1 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 1 & -1 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \]
The posterior distribution is thus equivalent to the model
\[ X \sim \mathcal N(0,0.5), Y=X \]
with one univariate Normal distribution having mean $0$ and variance $0.5$.
Borel's paradox
Borel's paradox is an important subtlety that occurs when conditioning on the equality of random variables $X=Y$. The original formulation involves conditioning a uniform point on a sphere to lie on a great circle, but we will use Borel's paradox to refer to any situation where conditioning on equivalent equations leads to different outcomes (e.g. [42]). For example, if instead of the condition $X-Y=0$ in <Ref> we had chosen the seemingly equivalent equations $X/Y=1$ or even $[X=Y]=1$ (using Iverson bracket notation), we would have obtained different posteriors:
If $X, Y \sim \mathcal N(0,1)$, then conditioned on $(X/Y=1)$, the variable $X$ can be shown to have density $|x|e^{-x^2}$ [40]. Under the boolean condition $[X=Y]=1$, the inference problem has no solution because the model $X, Y \sim \mathcal N(0,1), Z = [X=Y]$ is measure-theoretically equal to $X, Y \sim \mathcal N(0,1), Z = 0$ (since independent Gaussian random variables are almost surely different), and conditioning on $0=1$ is inconsistent.
We will address Borel's paradox and posit that a careful type-theoretic phrasing (<Ref>) helps alleviate its seemingly paradoxical nature (<Ref>).
§.§ Types and Terms of the Gaussian language
We now describe a language for Gaussian probability and conditioning. The core language resembles first-order OCaml with a construct $\normal()$ to sample from a standard Gaussian, and conditioning denoted as $(\eq)$. Types $\tau$ are generated from a basic type $\rv$ denoting real number or random variable, pair types and unit type $I$.
\[ \tau ::= \rv \s \unit \s \tau \ast \tau \]
Terms of the language are
\begin{align*}
e ::= x &\s e + e \s \alpha \cdot e \s \underline{\beta} \s (e,e) \s () \\
&\s \letin x e e \s \letin {(x,y)} e e \\
&\s \normal() \s e \eq e
\end{align*}
where $\alpha,\beta$ range over real numbers. Typing judgements are
\[ \infer{\Gamma, x : \tau, \Gamma' \vdash x : \tau}{}
\qquad
\infer{\Gamma \vdash () : \unit}{}
\qquad
\infer{\Gamma \vdash (s,t) : \sigma \ast \tau}{\Gamma \vdash s : \sigma \quad \Gamma \vdash t : \tau}
\]
\[ \infer{\Gamma \vdash s + t : \rv}{\Gamma \vdash s : \rv \quad \Gamma \vdash t : \rv}
\qquad
\infer{\Gamma \vdash \alpha \cdot t : \rv}{\Gamma \vdash t : \rv}
\qquad
\infer{\Gamma \vdash \underline{\beta} : \rv}{}
\]
\[ \infer{\Gamma \vdash \normal() : \rv}{}
\qquad
\infer{\Gamma \vdash (s \eq t) : \unit}{\Gamma \vdash s : \rv \quad \Gamma \vdash t : \rv}
\]
\[ \infer{\Gamma \vdash \letin x s t : \tau}{\Gamma \vdash s : \sigma \quad \Gamma, x : \sigma \vdash t : \tau}
\]
\[
\infer{\Gamma \vdash \letin {(x,y)} s t : \tau}{\Gamma \vdash s : \sigma \ast \sigma' \quad \Gamma, x : \sigma, y : \sigma' \vdash t : \tau}
\]
In Section <ref> we will introduce the general CD-calculus, and our Gaussian language is an instance of this[in the CD-calculus, we use projection maps rather than pattern-matching $\mathsf{let}$, but those constructs are interdefinable], with base type $\rv$ and signature
\begin{align}
(+) : \rv \ast \rv \to \rv, \quad \alpha \cdot (-) : \rv \to \rv, \quad \underline \beta : \unit \to \rv, \quad \normal : \unit \to \rv, \quad (\eq) : \rv \ast \rv \to \unit \label{eq:sig_gauss}
\end{align}
This will give us a clear path to denotational semantics: In <Ref>, we will indeed identify our language as the internal language of an appropriate CD category with an exact conditioning morphism.
We use standard syntactic sugar for sequencing $s;t$. We identify the type $\rv^n = \rv \ast (\rv \ast \ldots )$ with vectors $x=(x_1,\ldots,x_n$), and write matrix-vector multiplication $A \cdot x$ in an informal manner. For $\sigma \in \R$ and $e : \rv$, we define $\normal(x,\sigma^2) \equiv x + \sigma \cdot \normal()$. More generally, for a covariance matrix $\Sigma$ and $x : \rv^n$, we write $\normal(x, \Sigma) = x + A\cdot (\normal(), \ldots, \normal())$ where $A$ is any matrix such that $\Sigma = AA^T$. By the simple nature of the typing rules, we can identify any context and type with $\rv^n$ for suitable $n$.
For example, referring to <Ref>, the tuple $(X,Y,Z)$ can be written in our language as
\[\letin {(x,y)} {(\normal(),\normal())} {(x,y,x-y)}
\]
The full example with conditioning can be written
\[\letin {(x,y,z)}{(\letin {(x,y)} {(\normal(),\normal())} {(x,y,x-y)})} {z\eq 0;(x,y)}
\]
This program is contextually equivalent (<Ref>) to
\[\letin {x} {\sqrt{0.5} * \normal()} {\letin y x {(x,y)}}
\]
§.§ Operational Semantics
Informally, our operational semantics works as follows: calling $\normal()$ allocates a latent random variable, and a prior distribution over all latent variables is maintained; calling $(\eq)$ updates this prior by symbolic inference according to the formula (<ref>).
Formally, we define a reduction relation over configurations.
A configuration is either a dedicated failure symbol $\bot$ or a pair
\[(e,\psi)\]
where $\psi$ is a Gaussian distribution on $\R^r$ (i.e. a mean vector and covariance matrix) and $z_1 : \rv, \ldots, z_r : \rv \vdash e$.
Thus a running term $e$ may have free variables; these stand for dimensions in a given multivariate Gaussian distribution $\psi$, reminiscent of a closure in a higher-order language.
To define a reduction relation, we first introduce values, redexes and reduction contexts.
Values $v,w$ and redexes $\rho$ are defined as
\begin{align*}
v,w &::= x \s (v,w) \s v + w \s \alpha \cdot v \s \underline{\beta} \s () \\
\rho &::= \normal() \s v \eq w \s \letin x v e \s \letin {(x,y)} v e
\end{align*}
A reduction context $C$ with hole $[-]$ is of the form
\begin{align*}
C ::= [-] &\s (C,e) \s (v,C) \s C + e \s v + C \s \alpha \cdot C \s C \eq e \s v \eq C \\
&\s \letin x C e \s \letin {(x,y)} C e
\end{align*}
Perhaps the only thing to note is that, in keeping with the call-by-value tradition of most probabilistic programming languages, we do reduce before a let assignment, i.e. $\letin x C e$ is a reduction context.
It is easy to show by induction that every term is either a value or decomposes uniquely as $C[\rho]$. The latent variables $(z_1\dots z_r)$ are taken from a distinct supply of variable names $\{z_i : i \in \mathbb N \}$. We first define reduction on redexes (1–3), and then reduction contexts (4):
* Calling $\normal()$ allocates a fresh latent variable and adds an independent dimension to the prior
\[ (\normal(),\psi) \red (z_{\mathrm{r+1}}, \psi \otimes \mathcal N(0,1))
\]
where $\psi$ is a Gaussian distribution on $\R^r$ with mean $\mu$ and covariance $\Sigma$;
here $(\psi \otimes \mathcal N(0,1))$ is notation for the Gaussian distribution on $\R^{r+1}$
with mean $(\mu,0)$ and covariance matrix $(\begin{smallmatrix}\Sigma & 0 \\ 0 & 1\end{smallmatrix})$.
* To define conditioning, note that every value defines an affine function $\R^r \to \R$. In order to reduce $(v \eq w, \psi)$, we consider an independent random variable $X \sim \psi$ and define the auxiliary real random variable $Z = v(X)-w(X)$. If $0$ lies in the support of $Z$, we denote by $\psi|_{v=w}$ the outcome of conditioning $X$ on $Z=0$, and reduce
\[ (v \eq w, \psi) \red ((), \psi|_{v=w}) \]
Otherwise $(v \eq w, \psi) \red \bot$, indicating that the inference problem has no solution. To be completely precise, since $v$ and $w$ are affine, the function $(v-w)$ is affine too, so we can find $u \in \R^{1 \times r}$ and $b \in \R$ such that $(v(x)-w(x)) = ux + b$ and then we condition on $uX=-b$ using formula (<ref>).
* Let bindings are standard
\begin{align*}
(\letin x v e, \psi) &\red (e[v/x], \psi) \\
(\letin {(x,y)} {(v,w)} e, \psi) &\red (e[v/x,w/y], \psi)
\end{align*}
* Lastly, under reduction contexts, if $(\rho,\psi) \red (e,\psi')$ we define $(C[\rho],\psi) \red (C[e], \psi')$. If $(\rho,\psi) \red \bot$ then $(C[\rho], \psi) \red \bot$.
For every closed typed program $\vdash e : \tau$ either there is a unique value configuration $(v,\psi)$ such that
$(e,())\red^* (v,\psi)$ with $v$ a value, or $(e,())\red^*\bot$. (Here $()$ is the unique $0$-dimensional Gaussian distribution, and $\red^*$ is the reflexive transitive closure of $\red$.)
First, the $\red$ relation is deterministic, and satisfies progress and type preservation lemmas.
These are all shown by induction on typing derivations.
Next, all reduction sequences terminate, because the number of steps is bounded by the number of symbols from $\{\normal, \eq, \letin{}{}\}$ in an expression.
We consider the observable result of this execution either failure, or the pushforward distribution $v_*\psi$ on $\R^n$, as this distribution could be sampled from empirically.
The program
\[ \letin {(x,y)} {(\normal(),\normal())} x \eq y; x+y \]
reduces to $(z_1 + z_2, \psi)$ where
\[ \psi = \mathcal N\left(\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \right) \]
The observable outcome of the run is the pushforward distribution $(1\, 1)_*\psi = \mathcal N(0,2)$ on $\R$.
One goal of this paper is to study properties of this language compositionally, and abstractly, without relying on any specific properties of Gaussians. From the operational semantics, we can define an extensional and compositional contextual equivalence.
We say $\Gamma \vdash e_1, e_2 : \tau$ are contextually equivalent, written $e_1 \approx e_2$, if for all closed contexts $K[-]$ and $i,j \in \{1,2\}$
* when $(K[e_i], !) \red^* (v_i, \psi_i)$ then $(K[e_j], !) \red^* (v_j, \psi_j)$ and $(v_i)_*\psi_i = (v_j)_*\psi_j$
* when $(K[e_i], !) \red^* \bot$ then $(K[e_j],!) \red^* \bot$
Here $(v_i)_*\psi_i$ denotes the pushforward distribution as defined in (<ref>).
We later study contextual equivalence by developing a denotational semantics for the Gaussian language (<Ref>), and proving it fully abstract (<Ref>). We also note nothing conceptually limits the language in this section to only Gaussians. We are running with this example for concreteness, but any family of distributions which can be sampled and conditioned can be used. So we will take care to establish properties of the semantics in a general setting.
§ CATEGORICAL SEMANTICS FOR PROBABILISTIC PROGRAMMING
This section introduces denotational semantics for the probabilistic language in <Ref>, at first without conditioning. We will construct this semantics using a general and reusable strategy, namely
* understanding the structure of mathematical models of probability theory, and
* introducing a metalanguage (the CD calculus) which acts as the internal language of these categorical models
The Gaussian language will be a particular instance of the metalanguage, and its semantics will take place in the Markov category $\gauss$ (defined in <Ref>). The language for finite probability will have semantics in the Markov category $\finstoch$ (<Ref>).
This is a general section on the relationship between probabilistic languages and categorical models. We will recall symmetric monoidal categories and string diagrams, and define CD and Markov categories, which are the relevant categorical models of probability theory. We then introduce the CD calculus as a metalanguage for first-order probabilistic programs, and prove the desired correspondence
* every program term can be interpreted as a morphisms in a CD category
* every string diagram can be encoded as a program term
* every valid manipulation of string diagrams translates to a provable equality of programs, and vice versa
This is formally stated and proven in terms of internal language and syntactic category (<Ref>). This unlocks the equivalent formalisms discussed in the introduction (<Ref>). These formalisms form the basis of our study of conditioning in the later <ref>.
Monoidal Categories and String Diagrams
Recall that a category comprises objects and morphisms between the objects. In this context, the objects are to be thought of as generalized spaces, and the morphisms as stochastic functions. That is, a morphism $X\to Y$ is thought of roughly as something that takes an argument from $X$ and makes some random choices before returning an element of $Y$. (The reader familiar with probability theory can regard them as probability kernels, or parameterized measures). In particular, our categories will be monoidal, which means we have the following constructions (see e.g. [32] for full definitions):
* Monoidal structure: There is a distinguished object $I$ (thought of as the one-point space),
and for any objects $X$ and $Y$ there is an object $X\otimes Y$ (thought of as the product space). The morphisms $I\to X$ are thought of as probability distributions on $X$, and so the morphisms $Y\to X$ can be thought of as distributions on $X$ with parameters from $Y$.
* Categorical composition: for any morphisms $f\colon X\to Y$ and $g\colon Y\to Z$, there is a composite morphism $g \circ f:X\to Z$. This represents running stochastic computation in sequence. Mathematically, in several of the examples (Section <ref>), composition is calculated by a form of integration or summation (integrating over $Y$), and so we can regard this composition as an abstract account of integration. We will often abbreviate the composition $g \circ f$ as $gf$. In the category of sets and functions, morphisms $x : 1 \to X$ can be identified with elements $x \in X$. If $f : X \to Y$ is a function, the composite $f \circ x = fx$ agrees with function application $f(x)$. This notation will be convenient in <Ref> when applied to deterministic states.
* Monoidal composition: for any morphisms $f\colon A\to B$ and $g\colon X\to Y$, there is a composite morphism $(f\otimes g): A\otimes X\to B\otimes Y$. This can be understood as running stochastic computations in parallel: informally, given a pair $(a,x)$ we randomly produce $b$ from $a$ and $y$ from $x$, returning the pair $(b,y)$. Mathematically, in the examples, (Section <ref>), this monoidal composition amounts to product probabilities. In the measure theoretic example (Def. <ref>), the interchange law of the tensor $\otimes$ encodes Fubini's theorem (see <Ref>).
The notation for composing morphisms (with $\circ$ and $\otimes$) can quickly become cryptic, and it is important to find good notations. String diagrams are one such widely used and intuitive notation (e.g. [27, 41]). In the string diagram, the objects of the category become wires and morphisms are boxes; sequential composition ($\circ$) is simply joining wires together, and monoidal composition ($\otimes$) is juxtaposition. States ($I \to X$) and effects ($X \to I$) have special notations, because the unit object $I$ need not be drawn in string diagrams (see <Ref> for an overview).
\[ \input{categorical_probability/string_basics.tikz} \]
Overview of string diagram notation
We read such diagrams from bottom to top. Even without categorical machinery, string diagrams carry an intuitive meaning as dataflow diagrams. We can manipulate string diagrams using intuitive rules (sliding around of boxes, formally: “planar isotopy”), and the axioms of monoidal categories ensure that all such manipulations result in the same overall composite. Even more, if there are different ways of parsing a string diagram into a sequence of composites, then all these ways have provably equal meanings. This result is known as coherence for monoidal categories. We demonstrate this for the so-called interchange law $(g \circ f) \otimes (g' \circ f') = (g \otimes g') \circ (f \otimes f')$ which is derivable from the axioms of monoidal categories. It corresponds to the unambiguous reading of the string diagram
\[ \input{categorical_probability/interchange.tikz} \]
Symmetry, copying and discarding
Monoidal categories are an important general concept, but to discuss probabilities in this axiomatic way, it is appropriate to require further structure, resulting in copy-delete categories and Markov categories as we discuss in Section <ref>. The crucial operations are swapping, copying and discarding of data, which we depict as follows
\[ \input{categorical_probability/comon_structures_simple.tikz} \]
The same building blocks are necessary to interpret terms of a programming language in which variables can be used without linearity restriction. For example, the term-in-context $x,y,z \vdash (y,x,x)$ requires us to swap and copy $x$, as well as discard $z$. The same could be achieved by first copying $x$ and then swapping both copies with $y$. These two diagrams are provably equal from the axioms of CD categories,
\[ \input{categorical_probability/copyswap.tikz} \]
In defining the CD-calculus (<Ref>), we extend the syntax with let-binding, tuples, projections and function calls. The calculus has equivalent expressive power to string diagrams, as showcased in the introduction (<Ref>). In <Ref>, we give the systematic method to associate to every term $t$ a string diagam $\sem{t}$. In particular, manipulations of the string diagrams correspond to valid program transformations. For example, using the definition of the semantics of let-bindings in <Ref>, the validity of the commutativity equation (<ref>) corresponds to the following string diagram manipulation (where $x \notin \fv(t), y \notin \fv(s)$)
\[ \input{categorical_probability/letexample.tikz} \]
In these opening remarks so far, we have started from categorical notions and moved to notations. It is also helpful to follow the opposite route: we can regard notation as primal – be it string diagrams, graphical models, or probabilistic programs. Now to decide whether two composites are equal (two diagrams, two programs) we regard them as morphisms in a category (called the syntactic category), and ask whether they are equal there (<Ref>). In this way, category theory is merely a formalism for compositional theories of equality, which are useful from a foundational perspective as well as for understanding valid program manipulations. We axiomatize this equality from the programming perspective in Section <ref>.
We will now formally introduce CD categories and Markov categories and define the relevant mathematical models for finite and Gaussian probability, before returning to the CD calculus in <Ref>.
§.§ Copy-Delete Categories and Markov Categories
We will recall two closely related notions, namely:
Markov categories to model purely stochastic computation [11] and
CD categories which model potentially unnormalized stochastic computation [6].
Markov categories have been used to formalize various theorems of probability and statistics, such as sufficient statistics (Fisher-Neyman, Basu, Bahadur) [11], stochastic dominance (Blackwell-Sherman-Stein) [12] and zero-one laws [13]. Both types of category admit a convenient graphical language in terms of string diagrams. We will also define the CD-calculus, which is the internal language of CD categories and reminiscent of first-order OCaml (<Ref>). This makes CD categories a natural foundation for probabilistic programming. Denotational semantics will be given to our Gaussian language by recognizing it as the internal language of an appropriate CD category.
A copy-delete category (CD category) is a symmetric monoidal category $(\C, \otimes, I)$ where every object $X$ is equipped with the structure of a commutative comonoid
\[ \cpy_X : X \to X \otimes X \quad \del_X : X \to I \]
graphically depicted as
\[ \input{categorical_probability/comon_structures.tikz} \]
satisfying the axioms
\[ \input{categorical_probability/comonoid1.tikz}
\qquad\quad\input{categorical_probability/comonoid2.tikz} \]
We require that the comonoid structure be compatible with the monoidal structure as follows
\[ \input{categorical_probability/multiplicativity.tikz} \]
It is important that $\cpy$ and $\del$ are not assumed to be natural; explicitly the equations
\begin{equation} \input{categorical_probability/non_copyable.tikz} \label{cd:def_copy_disc} \end{equation}
need not hold in general. We give special names to situations where they do hold.
A morphism $f : X \to Y$ is called copyable if the first equation of (<ref>) holds:
$ \cpy_Y \circ f = (f \otimes f) \circ \cpy_X$.
A morphism $f : X \to Y$ is called discardable if the second equation of (<ref>) holds: $\del_Y \circ f = \del_X$. A morphism is called deterministic if it is copyable and discardable.
A Markov category is a CD category $\C$ in which the following equivalent properties hold
* $\C$ is semicartesian, i.e. the unit $I$ is terminal
* every morphism is discardable
* $\del$ is natural
The definitions of CD- and Markov categories encode a significant amount of properties of stochastic computation, as discussed informally at the beginning of Section <ref>. The discardability condition in Markov categories informally means that probabilities sum to $1$, as is the case in the examples (Section <ref>).
Notation: The presence of explicit copying and discarding maps lets us apply a product-like syntax for CD categories: If $f : A \to X$, $g : A \to Y$, we write a tupling
\[ \langle f,g \rangle \defeq (f \otimes g) \circ \cpy_A \]
and define projection maps
\[ \pi_X : X \otimes Y \to X \quad \pi_Y : X \otimes Y \to Y \]
via discarding. Recall that a state in a symmetric monoidal category is a morphism $\psi : I \to X$. An effect is a morphism $\rho : X \to I$. Note that by terminality of the unit $I$ in a Markov category, all effects $X \to I$ must be trivial; in CD categories, effects will be of interest.
In Markov categories, we will furthermore employ the following probabilistic terminology: We call states $\psi : I \to X$ distributions, and if $f : A \to X \otimes Y$, we define its marginal (of $X$) $f_X : A \to X$ to be $\pi_Xf$. Of course, we generally have $f \neq \langle f_X, f_Y \rangle$ unless $\C$ is cartesian. For every CD category $\C$, the wide subcategory $\C_\det$, which consists of only the deterministic morphisms, is cartesian <cit.>.
§.§ Examples of CD categories
In this subsection, we will briefly introduce the relevant examples of Markov and CD categories we will be working with, in particular finite and Gaussian probability. All examples in this subsection are standard material, and for example covered in [11]. More mathematical detail and the theory of conditioning is given in <Ref>. Readers primarily interested in syntax can proceed with <Ref>.
The Markov category $\finstoch$ has as objects finite sets $X$, and morphisms $X \to Y$ are probability channels, that is stochastic matrices $p \in [0,1]^{Y \times X}$, which are sometimes written in the notation $p(y|x)$. Composition in $\finstoch$ takes the form of the Kolmogorov-Chapman equation
\begin{equation} (pq)(z|x) = \sum_y p(z|y)q(y|x) \label{eq:chapman} \end{equation}
We modify $\finstoch$ as follows to allow for unnormalized (`sub-stochastic') computation:
The CD category $\finsubstoch$ has as objects finite sets $X$, and morphisms $X \to Y$ are subprobability channels $p(y|x)$, that is $p(y|x) \in [0,1]$ and for all $x \in X$,
\[ \sum_y p(y|x) \leq 1 \]
Composition is again given by (<ref>).
A convenient way to understand composition in these categories is using the theory of monads.
The distribution monad $D$ and subdistribution monad $D_{\leq 1}$ on the category of sets are defined as the sets of all (sub)probability distributions on a given set $X$;
\begin{align*}
D(X) &= \{ p : X \to [0,1] \text{ finitely supported } : \sum_x p(x) = 1 \} \\
D_{\leq 1}(X) &= \{ p : X \to [0,1] \text{ finitely supported } : \sum_x p(x) \leq 1 \}
\end{align*}
The unit of both monads is given by taking Dirac distribution $x \mapsto \delta_x$ with
\[ \delta_x(y) = \begin{cases} 1, &\text{if } x = y \\
0 &\text{if } x \neq y
\end{cases} \]
and the monad multiplication $\mu$ is defined as
\[ \mu(\rho)(x) = \sum_{p} \rho(p) \cdot p(x) \]
Kleisli composition for these monads recovers the Kolmogorov-Chapman equation (<ref>). That is, morphisms in $\finstoch(X,Y)$ are Kleisli arrow $X \to D(Y)$ for $D$, and morphisms in $\finsubstoch(X,Y)$ are Kleisli arrows for $X \to D_{\leq 1}(Y)$. The following result shows that Kleisli categories are a general source of CD categories.
Let $\C$ be a category with finite products and $T : \C \to \C$ be a strong, commutative monad. Then the Kleisli category $\kl(T)$ is a CD category, which is furthermore Markov if and only if $T$ is affine, i.e. $T1 \cong 1$.
Again, a morphism in $\finsubstoch(X,Y)$ is a Kleisli arrow $X \to D_{\leq 1}(Y)$ for the subprobability monad on $\set$.
We now define the Markov category which captures the Gaussian probability of Section <ref>.
The Markov category $\gauss$ has objects $n \in \mathbb N$, which represent the affine space $\R^n$, and $m \otimes n = m+n$. Morphisms $m \to n$ are tuples $(A,b,\Sigma)$ where $A \in \R^{n \times m}, b \in \R^n$ and $\Sigma \in \R^{n \times n}$ is a positive semidefinite matrix. The tuple represents a stochastic map $f : \R^m \to \R^n$ that is affine-linear, perturbed with multivariate Gaussian noise of covariance $\Sigma$, informally written
\[ f(x) = Ax + b + \mathcal N(\Sigma) \text{ or } Ax + \mathcal N(b,\Sigma) \]
Such morphisms compose sequentially and in parallel in the expected way, with noise accumulating independently
\begin{align*} (A,b,\Sigma) \circ (C,d,\Xi) &= (AC, Ad + b, A\Xi A^T + \Sigma) \\
(A,b,\Sigma) \otimes (C,d,\Xi) &= \left(
\begin{pmatrix} A & 0 \\ 0 & C \end{pmatrix},
\begin{pmatrix} b \\ d \end{pmatrix},
\begin{pmatrix} \Sigma & 0 \\ 0 & \Xi \end{pmatrix} \right)
\end{align*}
Copy- and discard structure are given using the affine maps
\[ \cpy_n : \R^n \to \R^{n+n}, x \mapsto (x,x) \quad \del_n : \R^n \to \R^0, x \mapsto () \]
Note that we explicitly allow zero covariance in the definition of $\gauss$. This way, the category is able to encode deterministic computation as a special case.
A morphism $(A,b,\Sigma)$ in $\gauss$ is deterministic (<Ref>) iff $\Sigma = 0$, i.e. there is no randomness involved.
Write $f = (A,b,\Sigma)$, then the covariance matrices of $f \circ \cpy$ and $\cpy \circ f$ are
\[ \begin{pmatrix} \Sigma & 0 \\ 0 & \Sigma \end{pmatrix} \text{ and } \begin{pmatrix} \Sigma & \Sigma \\ \Sigma & \Sigma \end{pmatrix} \]
respectively. Thus $f$ is copyable iff $\Sigma = 0$.
It follows that the deterministic subcategory $\gauss_\det$ is the category $\catname{Aff}$ consisting of the spaces $\R^n$ and affine maps between them.
Note that this definition of $\gauss$ involves no measure theory at all; a Gaussian is fully described by its mean and covariance matrix. Measure theory can however be used to build a Markov category that is rather comprehensive in that it includes the previous two examples. We briefly state the definition here and refer to the appendix (<Ref>) for details. (In this paper, we will not use this measure-theoretic Markov category in a crucial way, we will only use it in illustrative examples.)
The Markov category $\borelstoch$ has as objects standard Borel spaces $X$, and morphisms $X \to Y$ are probability kernels $\Sigma_X \times Y \to [0,1]$.
$\borelstoch$ arises as the Kleisli category of the Giry monad $\G$ on standard Borel spaces [16]. Both $\finstoch$ and $\gauss$ are subcategories of $\borelstoch$, i.e. there are faithful inclusion functors which preserve all CD structure.
Lastly, we give an example to show that the formalism of CD categories can not only encompass probabilistic situations but also nondeterminism.
We denote by $\rel$ the CD category of sets $X,Y$ and relations $R \subseteq X \times Y$ between them. We denote by $\rel^+$ the Markov subcategory of sets and left-total relations between them, i.e. $\forall x \in X \exists y \in Y, (x,y) \in R$.
The two categories are obtained as the Kleisli categories of the powerset monad $\mathcal P : \set \to \set$ and the nonempty powerset monad $\mathcal P^+: \set \to \set$ respectively. The category $\rel^+$ is referred to as $\catname{SetMulti}$ in [11].
§.§ Internal Languages and Denotational Semantics
We present the CD calculus, which is the internal language CD categories. It is reminiscent of the first-order fragment of fine-grained call-by-value or the computational $\lambda$-calculus (e.g. [34, 35, 31]), but the commutativity of the tensor allow for some convenient simplifications and a concise equational presentation. To this extent it is a novel calculus.
A CD signature $\mathfrak S = (\tau,\omega)$ consists of sets $\tau$ of base types and function symbols $\omega$. A type is recursively defined by closing the base types under tuple formation
\[ A ::= \tau \s \tunit \s A \ast A \]
Each function symbol $f \in \omega$ is equipped with a unary arity of types, written $f : A \to B$. The terms of the CD-calculus are given by
\[ t ::= x \s () \s (t,t) \s \pi_i\,t \s f\,t \s \letin x t t \quad\quad (i=1,2) \]
subject to the typing rules $x_1 : A_1, \ldots, x_n : A_n \vdash t : B$ given in <Ref>.
\begin{align*}&
\infer{\Gamma, x : A, \Gamma' \vdash x : A}{} \qquad
\infer{\Gamma \vdash () : \tunit}{}
\qquad \infer{\Gamma \vdash (s,t) : A \ast B}{\Gamma \vdash s : A \quad \Gamma \vdash t : B} \qquad
\infer[(f : A \to B)]{\Gamma \vdash f\,t : B}{\Gamma \vdash t : A}
\\[6pt]
&\infer{\Gamma \vdash \pi_1\,t : A_1}{\Gamma \vdash t : A_1 \ast A_2} \qquad \infer{\Gamma \vdash \pi_2\,t : A_2}{\Gamma \vdash t : A_1 \ast A_2} \qquad
\infer{\Gamma \vdash \letin x e t : B}{\Gamma, x : A \vdash t : B \quad \Gamma \vdash e : A}
\end{align*}
Typing rules for the CD calculus
We employ some standard syntactic sugar, for example sequencing
\[ s;t \defeq \letin {x} s t \quad\quad (x \notin \fv(t)) \]
We also define a pattern-matching let as syntactic sugar
\[ (\letin {(x,y)} s t) \defeq (\letin p s \letin x {\pi_1\,p} \letin y {\pi_2\,p} t) \text.\]
Conversely, we can provably recover the projection constructs from this sugar (in a sense made precise by the equational theory in <Ref>):
\[
(\pi_1\,s) = (\letin {(x,y)} s x)\qquad
(\pi_2\,s) = (\letin {(x,y)} s y)
\]
We prefer the projections over pattern-matching when presenting the equational theory, because this means one less binding construct.
§.§ Semantics
We now explain how the CD calculus can be interpreted in CD categories. Types will be interpreted as objects, and terms interpreted as morphisms. Formally, a model of signature $(\tau,\omega)$ is a CD category $\C$ together with an assignment of objects $\sem{A} \in \C$ for each basic type and morphisms $\sem{f} : \sem{A} \to \sem{B}$ for each function symbol $f : A \to B$. Here we extend $\sem{-}$ to arbitrary types and contexts by
\[ \sem{\tunit} = I \quad \sem{A_1 \ast A_2} = \sem{A_1} \otimes \sem{A_2} \quad \sem{A_1, \ldots, A_n} = \sem{A_1} \otimes (\cdots \otimes \sem{A_n}) \]
For any model, the interpretation of a term $\Gamma \vdash t : A$ is defined recursively as
* $\sem{x}$ is the discarding map $\sem{\Gamma,A,\Gamma'} \cong \sem{\Gamma} \otimes \sem{A} \otimes \sem{\Gamma'} \to I \otimes \sem{A} \otimes I \cong \sem{A}$
* $\sem{()}$ is the discarding map $\del_{\sem{\Gamma}} : \sem{\Gamma} \to I$
* $\sem{(s,t)}$ is the map $\sem{\Gamma} \xrightarrow{\cpy_{\sem{\Gamma}}} \sem{\Gamma} \otimes \sem{\Gamma} \xrightarrow{\sem{s} \otimes \sem{t}} \sem{A} \otimes \sem{B} = \sem{A \ast B}$
* $\sem{\pi_i t}$ is marginalization $\sem{\Gamma} \xrightarrow{\sem{t}} \sem{A_1} \otimes \sem{A_2} \to \sem{A_i}$
* $\sem{f\,t}$ is the composite $\sem{\Gamma} \xrightarrow{\sem{t}} \sem{A} \xrightarrow{\sem{f}} \sem{B}$
* $\sem{\letin x e t}$ is given by $\sem{\Gamma} \xrightarrow{\cpy_{\sem{\Gamma}}} \sem{\Gamma} \otimes \sem{\Gamma} \xrightarrow{\id_{\sem{\Gamma}} \otimes \sem{e}} \sem{\Gamma} \otimes \sem{A} \xrightarrow{\sem{t}} \sem{B}$
The semantics can be seen as a procedure for translating every term of the CD calculus into a string diagram, as shown in <Ref>.
\[ \input{categorical_probability/semantics1.tikz} \]
Translating terms into string diagrams (brackets $\sem{-}$ omitted for readability)
The weakening and exchange rules
\[ \infer{\Gamma, x : A\vdash t : B}{\Gamma \vdash t : B} \qquad \infer{\Delta,\Gamma \vdash t : B}{\Gamma, \Delta \vdash t : B} \]
are derivable, and their semantics corresponds to precomposition with the discard and swap morphisms.
Straightforward induction using the comonoid axioms.
As an example, we use the CD calculus to give straightforward denotational semantics to the conditioning-free fragment of the Gaussian language in $\gauss$. We notice that this fragment is precisely the CD calculus for the signature $\mathfrak S$ with base type $\rv$ and function symbols
\[(+) : \rv \ast \rv \to \rv \qquad
\beta \cdot (-) : \rv \to \rv \qquad
\underline \beta : \tunit \to \rv \qquad
\normal : \tunit \to \rv \]
The Markov category $\gauss$ models this signature using $\sem{\rv} = 1$ and the obvious interpretations of the function symbols.
The goal of <Ref> will be to interpret the full Gaussian language in a CD category $\cond(\gauss)$. That category will need to interpret the additional function symbol $(\eq) : \rv \ast \rv \to \rv$.
§.§ Equational Theory
We now give a sound and complete equational theory with respect to CD models.
In call-by-value languages, the substitution
\[ (\letin x e u) \equiv u[e/x] \]
is generally only admissible if $e$ is a value expression, that is it does not produce effects. In the CD calculus, another powerful substitution scheme is valid: We can replace $(\letin x e u) \equiv u[e/x]$ whenever $u$ uses $x$ linearly, i.e. exactly once, even if $e$ is an effectful computation. Using the linear- and value substitution schemes, the theory of the CD calculus can be presented concisely as in <Ref>. Note that we omit the context of equations when unambiguous and identify bound variables up to $\alpha$-equivalence. Whenever we say “use” or “occurrence”, we mean free use and occurrence, and substitution is always capture-avoiding.
Congruence laws:
\begin{gather}
\text{$\equiv$ is reflexive, symmetric and transitive} \label{cd:equiv} \tag{equiv} \\
\infer{(\letin x {e_1} e_2) \equiv (\letin x {e_1'} e_2')}{e_1 \equiv e_1' \quad e_2 \equiv e_2'} \label{cd:let_cong} \tag{let.$\xi$}
\end{gather}
A value expression is a term of the form
\[ V ::= x \s () \s (V,V) \s \pi_i V \s \letin x V V \]
The axioms of the CD calculus are:
\begin{align}
(\letin x e t) &\equiv t[e!x] \label{cd:let_lin} \tag{let.lin} \\
(\letin x V t) &\equiv t[V/x] \label{cd:let_val} \tag{let.val} \\
\pi_i\,(x_1,x_2) &\equiv x_i \label{cd:pair_beta} \tag{$\ast$.$\beta$} \\
(\pi_1\,x, \pi_2\,x) &\equiv x \label{cd:pair_eta} \tag{$\ast$.$\eta$} \\
x &\equiv () \label{cd:unit_eta} \tag{$\tunit$.$\eta$}
\end{align}
where we write $t[e ! x]$ for substituting a unique free occurrence of $x$. For the internal language of Markov categories, extend (<ref>) to all substitutions targeting at most one free occurrence of $x$.
Axioms of the CD-calculus
Every CD model validates the axioms of the CD calculus. That is if $\Gamma \vdash e_1 \equiv e_2 : A$ then $\sem{e_1} = \sem{e_2} : \sem{\Gamma} \to \sem{A}$.
The proofs are straightforward if tedious string diagram manipulations. We showcase the validation of one interesting equation, (<ref>), here and move the remaining derivations to the appendix (<Ref>). Let $\Gamma \vdash e_1 : X_1$, $\Gamma, x_1 : X_1 \vdash e_2 : X_2$ and $\Gamma, x_2 : X_2 \vdash e : Y$. Then showing
\[ \sem{(\letin {x_1} {e_1} \letin {x_2} {e_2} e^{\mathsf w})} \equiv \sem{(\letin {x_2} {(\letin {x_1} {e_1} e_2)} e)} \]
translates to the following manipulation of string diagrams
\begin{equation}
\input{categorical_probability/eq_assoc.tikz} \label{cd:eq_assoc}
\end{equation}
Note that we formally write $e^{\mathsf w}$ to be fully explicit about weakening $e$; its denotation discards the unused $X_1$-wire as per <Ref>.
The equational theory lets us derive many useful program equations, including commutativity.
All axioms of the ground $\lambda_c$-calculus <cit.> and commutativity are derivable.
\begin{align}
(\letin {x_2} {(\letin {x_1} {e_1} e_2)} e) &\equiv (\letin {x_1} {e_1} \letin {x_2} {e_2} e) \quad x_1 \notin \fv(e) \label{cd:assoc} \tag{assoc} \\
(\letin {x_1} {e_1} \letin {x_2} {e_2} e) &\equiv (\letin {x_2} {e_2} \letin {x_1} {e_1} e) \quad x_1 \notin \fv(e_2), x_2 \notin \fv(e_1) \label{cd:comm}~\tag{comm} \\
(\letin x e x) &\equiv e \label{cd:id} \tag{id} \\
(\letin {x_1} {x_2} e) &\equiv e[x_2/x_1] \label{cd:let_beta} \tag{let.$\beta$} \\
f e &\equiv (\letin x e f x) \label{cd:let_f} \tag{let.f}\\
(s,t) &\equiv (\letin x s \letin y t (x,y)) \label{cd:let_ast} \tag{let.$\ast$}
\end{align}
Note that by commutativity (<ref>), the order of evaluation in (<ref>) does not matter.
In the appendix (<Ref>).
We proceed with some syntactic remarks about the CD calculus on the relationship between linear substitution to general nonlinear substitutions: If $t$ is a term with $n$ free occurrences of the variable $x$, let $\hat t$ denote the term $t$ with those occurrences replaced with distinct fresh variables $x_1, \ldots, x_n$ (the order does not matter). By repeated application of (<ref>), we can derive
\begin{equation}
t \equiv \letin {x_1} x \cdots \letin {x_n} x \hat t
\end{equation}
We can now substitute some or all occurrences of $x$ using (<ref>) as follows
\begin{equation}
t[e/x] \equiv \hat t[e!x_1] \cdots [e!x_n] \equiv \letin {x_1} e \cdots \letin {x_n} e \hat t \label{cd:general_subs}
\end{equation}
This means we can reduce questions about substitution to the copying behavior of the term $e$. We adapt the definitions from [14, 29].
A term $e$ is called copyable if
\begin{equation} (\letin x e (x,x)) \equiv (e,e) \label{cd:copyable} \end{equation}
is derivable. A term $e$ is called discardable if
\begin{equation} (\letin x e ()) \equiv () \label{cd:discardable} \end{equation}
is derivable. We call $e$ deterministic if it is both copyable and discardable.
The substitution equation
\[ (\letin x e t) \equiv t[e/x] \]
is derivable in any of the following circumstances:
* $t$ uses $x$ exactly once
* $t$ uses $x$ at least once, and $e$ is copyable
* $t$ uses $x$ at most once, and $e$ is discardable
* $e$ is deterministic (combining the previous two points)
Finally, we remark that the CD calculus is complete with respect to CD models. We employ the usual construction of a syntactic category or free CD category over a given CD signature. Not only can every term be translated into a string diagram, also every string diagram can be parsed into a term, and the theory of $\equiv$ proves all ways of reading a diagram equivalent.
Fix a CD signature $\mathfrak S$. The syntactic category $\catname{Syn}$ has
* objects are types $A$
* morphisms are equivalence classes of terms $x : A \vdash t : B$ modulo $\equiv$
* identities are variables $x : A \vdash x : A$
* composition is let binding; if $x : A \vdash s : B$ and $x : B \vdash t : C$, their composite is
\[ x : A \vdash \letin x e t \]
* Tensor on objects is defined as $A \otimes B = A \ast B$ with unit type $\tunit$. The tensor on morphisms of $x_1 : A_1 \vdash s_1 : B_1$, $x_2 : A_2 \vdash s_1 : B_2$ is
\[ x : A_1 \ast A_2 \vdash \letin {x_1} {\pi_1\,x} \letin {x_2} {\pi_2\,x} (s_1,s_2) \]
* CD structure is given by nonlinear use of variables, that is
\begin{align*}
\cpy_A &= x : A \vdash (x,x) : A \ast A \\
\del_A &= x : A \vdash () : \tunit
\end{align*}
The verification of the CD category axioms is tedious but standard. Note that we can build on existing work [31] because our axioms prove all equations of the ground fragment of $\lambda_c$ (<Ref>).
We expect that the syntactic category is an initial model over a given signature and the definition of the semantics $\sem{-}$ is forced by preserving CD structure, but we won't formalize this here.
§ AN ABSTRACT ACCOUNT OF INFERENCE
In <Ref>, we recalled Markov categories (<Ref>) as abstract formulations of probability theory, equipped with multiple notational formalisms: string diagram as well as programming notations. We now present an abstract theory of inference problems in Markov categories with sufficient structure. We approach the topic from the programming languages side. An informal outline is as follows: Roughly, an inference problem is a closed program of the form
\begin{equation} \letin {(x,k)} \psi (k \eqo o); \return x
\qquad \qquad\text{for some $\psi$ and $o$}
\label{eq:closed_inf} \end{equation}
where $k \eqo o$ is, at this point, simply a notation for recording an exact condition we wish to make — that the second marginal $k$ of $\psi$ is equal to the observation $o$ (see Sec. <ref>).
A solution to this problem is a contextually equivalent program which no longer mentions $k\eqo o$.
Our treatment is now guided by program equations like (<ref>) and (<ref>), and also the symbolic approach of [42]: Finding a conditional for $\psi$ amounts to restructuring the dataflow of (<ref>) in a way that the return value $x$ is expressed in terms of the observation $k$ (where $\psi_K = \pi_2\,\psi$):
\[ \letin k {\psi_K} \letin x {\psi|_K(k)} (k \eqo o); \return x
\qquad
\text{for some $\psi|_K$}\text.\]
By commutativity (<ref>) and (<ref>), this is the same as
\[ \letin k {\psi_K} (k \eqo o); \return \psi|_K(k) \]
We can regard the initialization principle (<ref>) as a fundamental property of exact conditions ($k\eqo o$), and then use this. Informally, if it is possible for $k$ to be $o$ ($k \ll \psi_K$, Def. <ref>), we may simply substitute the observation for the variable $k$,
\[ \letin k o \psi|_K(k)\]
which finally results in the solution $\psi|_K(o)$, which is equivalent to (<ref>). If on the other hand it is impossible for $k$ to be equal to $o$ ($k \not\ll \psi_K$), then the inference problem has no solution and is infeasible.
For the rest of this chapter, we formally express this idea of conditioning via program transformation in terms of Markov categories. We begin by recalling categorical rephrasings of core notations from measure-theoretic probability: conditional probability (<Ref>), and almost-sure equality, absolute continuity and support (<Ref>), mostly due to [6, 11].
We then formulate precisely what an inference problem is, and when it succeeds (<Ref>). For now, we summarize informally: an inference problem is a pair $(\psi,o)$ of a distribution $\psi$ on a compound space $X\otimes K$ and a deterministic observation $o$ about $K$, regarded in the spirit of (<ref>); it succeeds if we are able to infer a conditional distribution, or posterior, about $X$, and fails otherwise. This failure is not sought in practice, but happens for instance if one attempts to record two different exact observations about the same data point, or in general if the observation $o$ is outside the support of $\psi$.
Looking forward, we will develop a notion of open inference problem in <Ref> as part of a compositional framework for collecting conditions, so as to give a compositional semantics to the kind of programming with conditioning demonstrated in <Ref>.
For the rest of this section, we fix a Markov category $\C$ (Def. <ref>).
§.§ Conditionals
In essence, conditioning is a way of recovering a joint distribution only given access to part of its information. Given a joint distribution over $(X,Y)$, we can always form a generative story where the value of $X$ is sampled first, and then $Y$ is computed depending (or conditional) on $X$. The categorical formulations of conditioning trace back to Golubtsov and Cho-Jacobs [17, 6].
A conditional distribution for $\psi : I \to X \otimes Y$ (given $X$) is a morphism $\psi|_X : X \to Y$ such that
\begin{equation} \input{categorical_probability/cond_dist.tikz} \label{eq:cond_dist} \end{equation}
More generally, a (parameterized) conditional for $f : A \to X \otimes Y$ is a morphism $f|_X : X \otimes A \to Y$ such that
\begin{equation} \input{categorical_probability/cond_param.tikz} \end{equation}
It is worth spelling out that (<ref>), expressed in the CD-calculus, corresponds precisely the type of restructuring of dataflow which was discussed in the introduction to this chapter. The equation (<ref>) simply becomes
\[ \psi \quad \equiv \quad (\letin x {\pi_1\,\psi} (x,\psi|_X(x))) \]
Parameterized conditionals can again be specialized to conditional distributions by fixing a parameter
If $f : A \to X \otimes Y$ has conditional $f|_X : X \otimes A \to Y$ and $a : I \to A$ is a deterministic state, then $f|_X(\id_X \otimes a)$ is a conditional distribution for the composite $f \circ a$.
Using determinism of $a$, we check that
\[ \input{categorical_probability/specialization.tikz} \]
$\finstoch$, $\borelstoch$, $\gauss$ and $\rel^+$ have all conditionals.
In $\borelstoch$, the definition of conditionals instantiates to regular conditional distributions which are known to exist under the assumptions of the category (that is on standard Borel spaces) (see <cit.>, <cit.>). As a special case, conditionals in $\finstoch$ are given by the traditional conditional distribution <cit.>
\begin{equation} \psi|_X(y|x) = \frac{\psi(x,y)}{\psi_X(x)} \quad \text{ when } \pi_X(x) > 0 \label{eq:finstoch_conditional} \end{equation}
Conditionals in $\gauss$ exist and can be given using an explicit formula generalizing (<ref>) <cit.>. The property that conditionals of Gaussians are again Gaussian is sometimes called self-conjugacy [24].
In $\rel^+$, the conditional of the state $R \subseteq X \times Y$ with respect to $X$ is given by `slicing' the relation
\[ R|_X(x) = \{ y \in Y : (x,y) \in R \} \]
which is nothing but $R$ itself.
§.§ Almost-sure Equality, Absolute Continuity and Supports
Conditionals in Markov categories are generally not uniquely determined, although there is still a sense in which they are unique. For example, the formula (<ref>) only determines $\psi|_X$ on the set $\{ x : \pi(x) > 0 \}$, which we call the support of $\psi_X$. Outside of the support, the conditional may be modified arbitrarily. Similarly, the formula for conditionals of Gaussian distributions (<ref>) depends on a choice of generalized inverse, which is only unique on the appropriate support.
If the reader is familiar with measure-theoretic probability, they will recall that the essential uniqueness of conditionals, when they exist, is usually stated in terms of `almost-sure equality' and `absolute continuity'. These have established generalizations to Markov categories in general, as we now recall (Definitions <ref> and <ref> respectively). These definitions specialize to the measure-theoretic concepts by fixing a Markov category (Propositions <ref>, <ref>), but for a reader not familiar with measure-theoretic probability, the definitions can be taken as basic. An reference of measure theoretic terminology is given in <Ref>.
Let $\mu : I \to X$ be a distribution. Two morphisms $f, g : X \to Y$ are called $\mu$-almost surely equal (written $f =_\mu g$) if
\[ \langle \id_X, f \rangle\mu = \langle \id_X, g \rangle \mu \]
For our example categories, the abstract definition of almost sure equality recovers the familiar meaning:
* In $\finstoch$, $f, g : X \to D(Y)$ are $\mu$-almost surely equal iff the distributions $f(x) = g(x)$ agree for all $x$ with $\mu(x) > 0$
* In $\borelstoch$, $f, g : X \to \G(Y)$ are $\mu$-almost surely equal iff $f(x) = g(x)$ as measures for $\mu$-almost all $x$, that is the set $\{ x : f(x) \neq g(x) \}$ has $\mu$-measure $0$.
* In $\gauss$, if $\mu : I \to m$ is a distribution with support $S$(in the sense of <Ref>), then $f,g : m \to n$ are $\mu$-almost surely equal iff $f \circ x=g \circ x$ for all elements $x \in S$, seen as deterministic states $x : 0 \to m$.
* In $\rel^+$, if $M \subseteq X$ and $R,S : X \to \mathcal P^+(Y)$ are two left-total relations, then $R =_M S$ iff $R(x) = S(x)$ for all $x \in M$.
The results for $\finstoch$ and $\borelstoch$ are given in <cit.> and <cit.>. The result for $\gauss$ is a strengthening of the result for $\borelstoch$. The morphisms $f, g : m \to n$ can be faithfully considered $\borelstoch$ maps $f, g : \R^m \to \G(\R^n)$, so we have $f(x) = g(x)$ for $\mu$-almost all $x$. Because $f,g$ are furthermore continuous functions and $\mu$ is equivalent to the Lebesgue measure on the support $S$, the equality almost everywhere can be strengthened to equality on all of $S$.
It follows directly from the definitions that conditional distributions are almost surely unique:
If $\psi : I \to X \otimes Y$ is a distribution and $\psi|_X, \psi|_X'$ are two morphisms satisfying (<ref>), then
\begin{equation} \psi|_X =_{\psi_X} \psi|_X' \label{eq:conditionals_as_unique} \end{equation}
That is, conditional distributions are unique almost surely with respect to the marginal $\psi_X$
The important notion of absolute continuity can now be formulated naturally in terms of almost-sure equality:
Given two distributions $\mu,\nu : I \to X$, we say that $\mu$ is absolutely continuous with respect to $\nu$, written $\mu \ll \nu$, if for all $f, g : X \to Y$ we have
\[ f =_\nu g \text{ implies } f =_\mu g \]
Absolute continuity lets us strengthen statements about almost-sure equality to actual equality.
If $\mu : I \to X$, $f =_\mu g$ and $x \ll \mu$ then $fx = gx$
On the other hand, $\ll$ recovers the usual notion of absolute continuity in our example categories.
* In $\finstoch$, $\mu \ll \nu$ iff $\nu(x) = 0$ implies $\mu(x) = 0$
* In $\borelstoch$, $\mu \ll \nu$ iff for all measurable sets $A$, $\nu(A) = 0$ implies $\mu(A) = 0$
* In $\gauss$, $\mu \ll \nu$ iff $\supp(\mu) \subseteq \supp(\nu)$ (in the sense of <Ref>).
* In $\rel^+$, $R \ll S$ iff $R \subseteq S$.
The claim for $\finstoch$ follows immediately from <Ref>, and the result for $\borelstoch$ is given in <cit.>.
<Ref> implies that the support condition for $\gauss$ is sufficient. To see that it is also necessary, let $x \in \supp(\mu) \setminus \supp(\nu)$. Then we can find two affine functions $f,g$ which agree on $\supp(\nu)$ but $f(x) \neq g(x)$. Now $f=_\nu g$ but not $f=_\mu g$, hence $\mu \not \ll \nu$.
In $\finstoch$, we can also rephrase $\mu \ll \nu$ as $\supp(\mu) \subseteq \supp(\nu)$ if we define $\supp(\mu) = \{ x : \mu(x) > 0 \}$. For the purposes of our development, it will suffice to consider the special case of the absolute continuity relation restricted to deterministic states and distributions. We take this as the categorical definition of supports:
If $x : I \to X$ is a deterministic state, we say that $x$ lies in the support of $\mu$ if $x \ll \mu$.
We obtain the following characterization
* In $\finstoch$, $x \ll \mu$ iff $\mu(x) > 0$
* In $\borelstoch$, $x \ll \mu$ iff $\mu(\{x\}) > 0$
* In $\gauss$, $x \ll \mu$ iff $x \in \supp(\mu)$
* In $\rel^+$, $x \ll R$ if $x \in R$
It is crucial that the support of a distribution can change with the surrounding Markov category:
Let $\mu = \mathcal N(0,1)$ be the standard normal distribution. When considered in $\gauss$, its support is $\R$ and in particular for all $x_0 \in \R$ we have $x_0 \ll \mu$. In $\borelstoch$, we have $x_0 \not \ll \mu$ because $\mu(\{x_0\}) = 0$.
This means that smaller Markov categories like $\gauss$ have a stronger notion of support, which in turn allows more interesting conditions to be evaluated. This is reminiscent of the tradeoff between expressiveness and well-behavedness discussed under the notion of “well-behaved disintegrations” in [42].
By combining the notions of conditionals and support, we can now present an abstract theory of inference problems.
§.§ Abstract Inference Problems
Let $\C$ be a Markov category with all conditionals. In order to describe statistical inference categorically, we introduce the following terminology:
* An observation is a constant piece of data, that is a deterministic state $o : I \to K$.
* An inference problem over $X$ is a tuple $(K,\psi,o)$ of an object $K$, a joint distribution $\psi : I \to X \otimes K$ called the model and an observation $o : I \to K$.
The problem is then to infer the posterior distribution over $X$ conditioned on the observation $o$. An inference problem can either succeed, or fail if the observation $o$ is inconsistent with the model.
* We say $(K,\psi,o)$ succeeds if the observation lies in the support of the model, i.e. $o \ll \psi_K$. In that case, a solution to the inference problem is the composite $\psi|_K \circ o : I \to X$ where $\psi|_K : K \to X$ is a conditional to $\psi$ with respect to $K$. The solution is also referred to as a posterior for the problem.
* If $o \not\ll \psi_K$, we say that the inference problem fails or is infeasible.
Solutions to inference problems are unique, i.e. if $(K,\psi,o)$ succeeds and $\psi|_K, \psi|_K'$ are two conditionals then $\psi|_K(o) = \psi|_K'(o)$.
Combine <Ref> and <Ref>.
We call two inference problems observationally equivalent if they either both fail, or they both succeed with equal posteriors.
For the rest of this section, we will rederive <Ref> in terms of the categorical machinery and show that it matches the conditioning procedure from <Ref>.
The <ref> can be written as
\begin{align*}
X &\sim \mathcal N(0,1) \\
Y &\sim \mathcal N(0,1) \\
(X - Y) \,&\eqo\, 0
\end{align*}
which corresponds to the inference problem $(1,\mu,0)$ where $\mu : 0 \to 2 \otimes 1$ has covariance matrix
\[ \Sigma = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & -1 & 2 \end{pmatrix} \]
A conditional with respect to the third coordinate $Z$ is
\[ \mu|_Z(z) = \begin{pmatrix}0.5 \\ 0.5\end{pmatrix}z + \mathcal N\begin{pmatrix}
0.5 & 0.5 \\ 0.5 & 0.5
\end{pmatrix} \]
which can be verified by calculating (<ref>). The marginal $\mu_Z = \mathcal N(2)$ is supported on all of $\R$, hence $0 \ll \mu_Z$ and by <Ref> the composite
\[ \mu|_{Z}(0) = \mathcal N\begin{pmatrix}
0.5 & 0.5 \\ 0.5 & 0.5
\end{pmatrix} \]
is the uniquely defined solution to the inference problem.
The same inference problem would not have a solution when interpreted in $\borelstoch$ instead of $\gauss$. This is because $0 \not\ll \mu_Z$ (<Ref>). In $\borelstoch$, we can only condition on observations of positive probability; this agrees with the classical definition of conditional probability
\[ P(A|B) \defeq \frac{P(A \cap B)}{P(B)} \text{ if } P(B) > 0 \]
In $\gauss$, we can also condition on probability zero observations in a principled way because the notion of support is better behaved.
§ COMPOSITIONAL CONDITIONING – THE COND CONSTRUCTION
In <Ref> we have seen that Markov categories with conditionals allow a general recipe for conditioning. In order to give compositional semantics to a language with conditioning, we need to internalize the conditioning operation as a morphism.
The key step is to move from a closed inference problem $(K,\psi\colon I\to X\otimes K,o\colon I\to K)$, to
open inference problems or conditioning channels, where $\psi$ is replaced with a morphism with more general domain, so that it can be composed.
With some care, we can turn these conditioning channels into a CD-category (<Ref>, a monoidal category where every object has a comonoid structure). This allows us to give a denotational semantics to a CD calculus with conditioning, and in particular a denotational semantics for the Gaussian language with conditioning of <Ref>. Looking forward, in <Ref> we will show that this denotational semantics is fully abstract: it precisely captures the contextual equivalence from the operational semantics.
The construction presented in this chapter is rather involved, but its well-definedness and properties are the central technical contribution of this article. From <Ref> onwards, we can harness the good properties the construction enjoys to return to a higher level picture and use string diagrams for studying a graphical language of conditioning.
Let $\C$ be a Markov category, then a conditioning channel $X \obsto Y$ is given by a morphism $X \to Y \otimes K$ together with an observation (i.e. deterministic state) $o : I \to K$. This represents an intensional open program of the form
\begin{equation}
\label{eq:obs_normalform}
x : X \vdash \letin {(y,k) : Y \otimes K} {f(x)} {(k \eqo o); y}
\end{equation}
We think of $K$ as an additional hidden output wire, to which we attach the observation $o$. Such programs compose in the obvious way, by aggregating observations (<Ref>). Two representations (<ref>) are deemed equivalent if they contextually equivalent, that is roughly they compute the same posteriors in all contexts.
An important caveat is that the primary operation we formalize is that of an exact observation $(\eqo o)$ where $o$ is a deterministic state. Binary exact conditioning $(\eq)$ between two expressions may be encoded in terms of $(\eqo)$, for example as $(x==y)\eqo \true$ for finite sets or as $(x-y) \eqo 0$ for Gaussians. Generally, the choice of encoding does matter (<Ref>), so we consider this choice additional structure and focus on formalizing $(\eqo)$.
For modularity, we present the construction in two stages: In the first stage (<Ref>) we form a category $\obs(\C)$ on the same objects as $\C$ consisting of the data (<ref>) but without any quotienting. This adds, purely formally, for every observation $o : I \to X$ an observation effect $(\eqo o) : X \obsto I$.
In the second stage (<Ref>) – this is the core of the construction – we relate these morphisms to the conditionals present in $\C$, that is we quotient by contextual equivalence. The resulting quotient is called $\cond(\C)$. Under mild assumptions, this will have the good properties of a CD category, showing that conditioning stays commutative. We demonstrate the resulting reasoning methods in <Ref> and <Ref>.
§.§ Obs – Open Programs with Observations
For ease of notation, we will assume $\C$ is a strictly monoidal category, that is all associators and unitors are identities (this poses no restriction by <cit.>). We note that all constructions can instead be performed purely string-diagrammatically.
The following data define a symmetric premonoidal category called $\obs(\C)$:
* the object part of $\obs(\C)$ is the same as $\C$
* morphisms $X \obsto Y$ are tuples $(K,f,o)$ where $K$ is an object of $\C$, $f \in \C(X,Y\otimes K)$ and $o \in \C_{\mathrm{det}}(I,K)$, representing (<ref>)
* The identity on $X$ is $\Id_X = (I,\id_X,!)$ where $!=\id_I$.
* Composition is defined by
\[ (K',f',o') \bullet (K,f,o) = (K' \otimes K, (f' \otimes \id_K)f, o' \otimes o). \]
* if $(K,f,o) : X \obsto Y$ and $(K',f',o') : X' \obsto Y'$, their (premonoidal) tensor product is defined as
\[ (K' \otimes K, (\id_{Y'} \otimes \swap_{K',Y} \otimes \id_K)(f' \otimes f), o' \otimes o) \]
* There is an identity-on-objects functor $J : \C \to \obs(\C)$ that sends $f : X \to Y$ to $(I,f,!) : X \obsto Y$. This functor is strict premonoidal and its image central
* $\obs(\C)$ inherits symmetry and comonoid structure
Recall that a symmetric premonoidal category (due to [39]) is like a symmetric monoidal category where the interchange law $(f_1 \otimes f_2) \circ (g_1 \otimes g_2) = f_1g_1 \otimes f_2g_2$ need not hold. This is the case because $\obs(\C)$ does not yet identify observations arriving in different order. This will be remedied automatically later when passing to the quotient $\cond(\C)$. For an observation $o : I \to K$, the conditioning effect $(\eqo o) : K \obsto I$ is defined by $(I,\id_K,o)$.
A morphism $F : X \obsto Y$ in $\obs(\C)$ consists of a morphism $f : X \to Y \otimes K$ in $\C$ with an extra datum $o : I \to K$. It can be convenient to describe morphisms in $\obs(\C)$ by their underlying morphisms in $\C$, with extra `conditioning wires' into the object $K$ and observations attached to them. Informally, we will highlight these wires dashed and blue, simply to distinguish the codomain object $Y$ from the condition object $K$. Using this notation, composition and tensor in $\obs(\C)$ take the form shown in <Ref>. In <Ref>, we will use actual string diagram in $\cond(\C)$ as opposed to string diagrams in $\C$, which will obviate the need for blue wires.
Composition and tensoring of morphisms in $\obs$
§.§ Cond – Contextual Equivalence of Inference Programs
Let us now assume that $\C$ has all conditionals. We wish to quotient $\obs$-morphisms by contextual equivalence, relating them to the conditionals which can be computed in $\C$. We know how to interpret closed programs, because a state $(K,\psi,o) : I \obsto X$ is precisely an inference problem as in <Ref>: If $o \not \ll \psi_K$, the observation does not lie in the support of the model and conditioning fails. If not, we form the conditional $\psi|_K$ in $\C$ and obtain a well-defined posterior $\mu|_K \circ o$.
The observational equivalence (<Ref>) defines an equivalence relation on states $I \leadsto X$ in $\cond(\C)$. We will extend this relation to a congruence on arbitrary morphisms $X \leadsto Y$ by a general categorical construction.
Given two states $I \leadsto X$ we define $(K,\psi,o) \sim (K',\psi',o')$ if they are observationally equivalent as inference problems, that is either
* $o \ll \psi_K$ and $o' \ll \psi'_{K'}$ and $\psi|_K(o)= \psi'|_{K'}(o')$.
* $o \not \ll \psi_K$ and $o' \not \ll \psi'_{K'}$
<Ref> describes observationally equivalent states $0 \obsto 2$ in $\obs(\gauss)$
We now give a general recipe to extend an equivalence relation on states to a congruence on arbitrary morphisms $f : X \to Y$.
Let $\mathbb X$ be a symmetric premonoidal category. An equivalence relation $\sim$ on states $\mathbb X(I,-)$ is called functorial if $\psi \sim \psi'$ implies $f\psi \sim f\psi'$. We can extend such a relation to a congruence $\approx$ on all morphisms $X \to Y$ via
\[ f \approx g \Leftrightarrow \forall A,\psi : I \to A \otimes X, (\id_A \otimes f)\psi \sim (\id_A \otimes g)\psi. \]
The quotient category $\mathbb X/{\approx}$ is symmetric premonoidal.
We show now that under good assumptions, the quotient by conditioning (<Ref>) on $\mathbb X = \obs(\C)$ is functorial, and induces a quotient category $\cond(\C)$. The technical condition is that supports interact well with dataflow:
A Markov category $\C$ has precise supports if the following are equivalent for all deterministic $x : I \to X$, $y : I \to Y$, and arbitrary $f : X \to Y$ and $\mu : I \to X$.
* $x \otimes y \ll \langle \id_X, f\rangle \mu$
* $x \ll \mu$ and $y \ll fx$
The word `support' here refers to <Ref>.
$\gauss$, $\finstoch$, $\borelstoch$ and $\rel^+$ have precise supports.
This follows from the characterizations of $\ll$ in <Ref>. For $\gauss$, let $\mu$ have support $S$ and $f(x)=Ax+\mathcal N(b,\Sigma)$. Let $T$ be the support of $\mathcal N(b,\Sigma)$. The support of $\langle \id, f \rangle\mu$ is the image space $\{ (x,Ax+c) : x \in S, c \in T \}$. Hence $(x,y) \ll \langle \id, f \rangle\mu$ iff $x \ll \mu$ and $y \ll fx$. Similarly, for $\rel^+$, we readily verify
\[ x \in \{ (x,y) : x \in \mu, (x,y) \in f \} \Leftrightarrow x \in \mu \wedge y \in f(x) \]
For $\finstoch$, an outcome $(x,y)$ has positive probability under $\langle \id, f \rangle\mu$ iff $x$ has positive probability under $\mu$, and $y$ has positive probability under $f(-|x)$.
For $\borelstoch$, the measure $\psi = \langle \id, f \rangle \mu$ is given by
\[ \psi(A \times B) = \int_{x \in A} f(B|x) \mu(\mathrm dx) \]
Hence $\psi(\{(x_0,y_0)\}) = f(\{y_0\}|x)\mu(\{x\})$, which is positive exactly if $\mu(\{x_0\}) > 0$ and $f(\{y_0\}|x)>0$.
Let $\C$ be a Markov category that has conditionals and precise supports. Then $\sim$ is a functorial equivalence relation on $\obs(\C)$.
Let $(K,\psi,o) \sim (K',\psi',o') : I \obsto X$ be equivalent states and $(H,f,v) : X \obsto Y$ be any morphism. We need to show that the composites
\begin{equation}
\label{eq:cond_fun_equiv}
(H \otimes K, (f \otimes \id_K)\psi, v \otimes o) \sim (H \otimes K', (f \otimes \id_{K'})\psi', v \otimes o')
\end{equation}
are equivalent. We analyze different cases.
The states fail If a state $(K,\psi,o)$ fails because $o \not \ll \psi_K$, then any composite must fail too. So both sides of (<ref>) fail and are thus equivalent.
The composite fails Assume from now that the states succeed and thus also have equal posteriors
\begin{equation} \psi|_{K}(o) = \psi'|_{K}(o') \label{eq:assumption_posteriors} \end{equation}
We first show that the success conditions on both sides of (<ref>) are the same, so if the LHS fails so does the RHS. The “precise supports” axiom lets us split the success condition into two statements; that is the following are equivalent (and analogous for $\psi', o'$):
* $v \otimes o \ll (f_H \otimes \id_K)\psi$
* $o \ll \psi_K$ and $v \ll f_H\psi|_K(o)$
To see this, we instantiate <Ref> with the morphisms $\mu = \psi_K$ and $g=f_H \circ \psi|_K$, because the definition of the conditional $\psi|_K$ lets us recover
\[ \langle g, \id_K \rangle \mu = (f_H \otimes \id_K)\psi. \]
It is clear that condition <ref> agrees for both sides of (<ref>). Hence so does <ref>.
The composite succeeds We are left with the case that both sides of (<ref>) succeed, and need to show that the composite posteriors agree
\begin{equation} [(f \otimes \id_K)\psi]|_{H \otimes K}(v \otimes o) = [(f \otimes \id_{K'})\psi']|_{H \otimes K'}(v \otimes o') \label{eq:posterior_conclusion} \end{equation}
We use a variant of the argument from <cit.> that double conditionals can be replaced by iterated conditionals. Consider the parameterized conditional
\[ \beta \defeq (f \circ \psi|_K)|_H : H \otimes K \to Y \]
with univsonersal property
\begin{equation} \input{conditioning/proof_beta_iter.tikz} \label{eq:beta_prop} \end{equation}
Some string diagram manipulation shows that $\beta$ too has the universal property of the double conditional
\[ \beta = [(f \otimes \id_K)\psi]|_{H \otimes K} \]
We check
\begin{equation*} \input{conditioning/proof_beta_double.tikz} \end{equation*}
which further reduces using (<ref>) to the desired
\begin{equation*} \input{conditioning/proof_beta_double_2.tikz} \end{equation*}
By specialization (<Ref>), we can fix one observation $o$ in $\beta$ to obtain a conditional
\begin{equation}
\beta(\id_H \otimes o) = (f \circ \psi|_K(o))|_H \label{eq:beta_special}
\end{equation}
But this conditional agrees with $(f \circ \psi'|_K(o'))|_H$ by assumption (<ref>). Hence we can evaluate the joint posterior successively,
\begin{align*}
[(f \otimes \id_K)\psi]|_{H \otimes K}(v \otimes o) &= \beta(\id_H \otimes o) \circ v \\
&\stackrel{\eqref{eq:beta_special}}= (f \circ \psi|_K(o))|_H \circ v \\
&\stackrel{\eqref{eq:assumption_posteriors}}= (f \circ \psi'|_{K'}(o'))|_H \circ v \\
&\stackrel{\text{symmetric}}= [(f \otimes \id_{K'})\psi']|_{H \otimes K'}(v \otimes o)
\end{align*}
establishing (<ref>).
We can spell out the induced congruence $\approx$ on $\obs(X,Y)$ as follows:
We have $(K,f,o) \approx (K',f',o') : X \obsto Y$ if and only if for all $\psi : I \to A \otimes X$, either
* $o \ll f_K\psi_X$ and $o' \ll f'_{K'}\psi'_X$ and $[(\id_A \otimes f)\psi]|_K(o) = [(\id_A \otimes f')\psi']|_{K'}(o')$
* $o \not \ll f_K\psi_X$ and $o' \not \ll f'_{K'}\psi'_X$
Furthermore, because $\C$ has conditionals, it is sufficient to check these conditions for $A=X$ and $\psi$ of the form $\cpy_X \circ \phi$.
By <Ref>, the quotient $\obs(\C)/{\approx}$ is a well-defined symmetric premonoidal category. We argue now that it is in fact monoidal. Checking the interchange means showing that the order of observations does not matter modulo $\approx$. We can derive this from a general statement about isomorphic conditions.
Let $(K,f,o) : X \obsto Y$ and $\alpha : K \cong K'$ be an isomorphism. Then
\[ (K,f,o) \approx (K',(\id_Y \otimes \alpha)f, \alpha o). \]
In programming terms, the observations $(k \eqo o)$ and $(\alpha k \eqo \alpha o)$ are contextually equivalent.
Let $\psi : I \to A \otimes X$. We first notice that $o \ll \psi_K$ if and only if $\alpha o \ll \alpha \psi_K$, so the success conditions coincide. It is now straightforward to check the universal property
\[ (\id_A \otimes f)\psi|_K = (\id_A \otimes ((\id_X \otimes \alpha)f))\psi|_{K'} \circ \alpha. \]
This requires the fact that isomorphisms are deterministic in a Markov category with conditionals <cit.>. The proof more generally works if $\alpha$ is deterministic and split monic.
We can now give the Cond construction by means of quotienting $\obs(\C)$ modulo contextual equivalence.
Let $\C$ be a Markov category that has conditionals and precise supports. We define $\cond(\C)$ as the quotient category
\[ \cond(\C) = \obs(\C)/{\approx} \]
This quotient is a CD category, and the functor $J : \C \to \cond(\C)$ preserves CD structure.
§.§ Laws for Conditioning
We will now establish convenient properties of $\cond(\C)$ in a purely abstract way. In terms of the internal language, those are the desired program equations for a language with exact conditioning. For example, the fact that $\cond(\C)$ is a well-defined CD category already implies that commutativity equation holds for such programs (<Ref>).
Secondly, we can draw string diagrams in the category $\cond(\C)$. These look like diagrams in $\C$ to which we add effects $(\eqo o) : X \to I$ for every observation $o : I \to X$. For example, <Ref> states diagrammatically that for all isomorphisms $\alpha$ and observations $o$, we have
\[ \input{conditioning/isomorphic_conditions.tikz} \]
We begin by showing that passing to $\cond(\C)$ does generally not collapse morphisms which were distinct in $\C$, that is the functor $J$ is faithful for common Markov categories.
Two morphisms $f, g : X \to Y$ are equated via $J(f) \approx J(g)$ if and only if
\begin{equation} \forall \psi : I \to A \otimes X, (\id_A \otimes f)\psi = (\id_A \otimes g)\psi \label{eq:j_ident} \end{equation}
In particular, $J$ is faithful whenever $I$ is a separator. This is the case for $\gauss$, $\finstoch$, $\borelstoch$ and $\rel^+$.
Directly from the definition of $\approx$.
By construction, the states in $\cond(\C)$ are precisely inference problems up to observational equivalence. Any such problem either fails or computes a well-defined posterior, which gives rise to the following classification:
The states $I \obsto X$ in $\cond(\C)$ are of the following form:
* There exists a unique failure state $\bot_X : I \obsto X$ given by the equivalence class of any $(K,\psi,o)$ with $o \not\ll \psi_K$.[it is a minor extra assumption that there exists a non-instance $o \not\ll \mu$ in $\C$; this should be the case in any Markov category of practical interest]
* Any other state is equal to a conditioning-free posterior, namely $(K,\psi,o) \approx J(\psi|_K \circ o)$. That is diagrammatically
\[ \input{conditioning/cond_states.tikz} \]
* Failure is “strict” in the sense that any composite or tensor with $\bot$ gives $\bot$.
* The only scalars $I \obsto I$ are $\id_I$ and $\bot_I$. Both are copyable, but $\bot_I$ is not discardable.
By definition of $\sim$.
If $o \ll \psi$ then $(\psi \eqo o)$ succeeds without observable effect; in particular, because $o \ll o$, we can always eliminate tautological conditions
\[ \input{conditioning/emptydiag.tikz} \]
The central law of conditioning states that after we enforce a condition, it will hold with exactness. In programming terms, this is the substitution principle (<ref>). Categorically, we are asking how the conditioning effect interacts with copying:
We have
\[ (X,\cpy_X,o) \approx (X,o \otimes \id_X,o) \]
In programming notation, this is
\[ (x \eqo o); x \approx (x \eqo o); o \]
and in string diagrams
\begin{equation} \input{conditioning/exactly.tikz} \label{eq:exact} \end{equation}
Note that the conditioning effect cannot be eliminated; however after the condition takes place, the other wire can be assumed to now contain $o$.
Let $\psi : I \to A \otimes X$; the success condition reads $o \ll \psi_X$ both cases. Now let $o \ll \psi_X$ and let $\psi|_X$ be a conditional distribution for $\psi$. The following maps give the required conditionals
\begin{align*}
[(\id_A \otimes \cpy_X)\psi]|_X = \langle \psi|_X, \id_X \rangle \quad
[(\id_A \otimes o \otimes \id_X)\psi]|_X = \psi|_X \otimes o
\end{align*}
as evidenced by the following string diagrams
\[ \input{conditioning/proof_exact.tikz} \]
Composing with $o$, we obtain the desired equal posteriors
\[ \langle \psi|_X, \id_X \rangle o = \psi|_X(o) \otimes o = (\psi|_X \otimes o)(o) \]
from determinism of $o$.
Conditioning a fresh variable on a feasible observation makes it assume that observation. Formally, if $o \ll \psi$ then
\[ (\letin x \psi (x \eqo o); x) \approx o \]
Combining <Ref> and <Ref>, we have
\[ \input{conditioning/initialization.tikz} \]
Conditioning is idempotent, that is
\[ (x \eqo o); (x \eqo o) \approx (x \eqo o) \]
In other words, the conditioning effect is copyable (but not discardable).
Again by <Ref> and <Ref> we obtain
\[ \input{conditioning/idempotence.tikz} \]
We note that this does not imply that every effect in $\cond(\C)$ is copyable, only that exact observations are.
Conditions can be aggregated
\[ \input{conditioning/aggregation.tikz} \]
By definition of the monoidal structure of $\obs$.
§.§ Example: Graphical Models and Conditioning
We demonstrate the power of our conditioning laws by briefly revisiting graphical models as mentioned in the introduction of <Ref>: Every graphical model can be turned into a string diagram, where the independence structure of the graphical model translates into a factorization of the diagram. For example, in the model (<ref>) of variables $X,Y$ which are conditionally independent on $W$, the joint distribution $\psi$ can be factored as follows
\[ \input{conditioning/cond_ind_state.tikz} \]
Using the conditioning effects in $\cond(\C)$, we can now incorporate observed nodes into this language.
\begin{equation}
\input{conditioning/cond_ind_observed.tikz} \label{eq:cond_ind_observed}
\end{equation}
We want to argue that once the `common cause' $W$ has been observed, $X$ and $Y$ become independent: We can show this purely using graphical reasoning: Applying repeatedly <Ref>, idempotence of scalars and determinism of $w$, we obtain that (<ref>) is the product of its marginals:
\[ \input{conditioning/cond_ind_proof.tikz} \]
§ DENOTATIONAL SEMANTICS AND CONTEXTUAL EQUIVALENCE
In <Ref> we introduced the $\cond$ construction (<Ref>) as a way of building a category that accommodates the abstract inference for Markov categories (<Ref>). As we have seen, we can interpret the CD calculus (<Ref>) in categories built from the $\cond$ construction, and this forms a probabilistic programming language with exact conditioning. In this final section, we will work out in detail what the $\cond$ construction does when applied to our specific example settings of finite and Gaussian probability.
In <Ref>, we show that the Gaussian language (<Ref>) has fully abstract denotational semantics in $\cond(\gauss)$: equality in the category coincides with the operational contextual equivalence from <Ref>.
In <Ref>, we conduct the same analysis for finite probability and show that $\cond(\finstoch)$ consists of substochastic kernels up to automatic normalization. In <Ref>, we spell out the relationship between the admissibility of automatic normalization and the expressibility of branching in the language.
§.§ Full Abstraction for the Gaussian Language
The Gaussian language embeds into the internal language of $\cond(\gauss)$, where $x \eq y$ is translated as $(x - y) \eqo 0$. A term $\vec x : R^m \vdash e : R^n$ denotes a conditioning channel $\sem{e} : m \obsto n$.
If $(e,\psi) \red (e',\psi')$ then $\sem{e}\psi = \sem{e'}\psi'$. If $(e,\psi) \red \bot$ then $\sem{e} = \bot$.
We can faithfully interpret $\psi$ as a state in both $\gauss$ and $\cond(\gauss)$. If $x \vdash e$ and $(e,\psi) \red (e',\psi')$ then $e'$ has potentially allocated some fresh latent variables $x'$. We show that
\begin{equation}
\letin{x}{\psi} (x,\sem{e}) = \letin{(x,x')}{\psi'} (x,\sem{e'}).
\end{equation}
This notion is stable under reduction contexts.
Let $C$ be a reduction context.
\begin{align*}
&\letin x \psi (x,\sem{C[e]}(x)) \\
&= \letin x \psi \letin y {\sem{e}(x)} (x,\sem{C}(x,y)) \\
&= \letin {(x,x')} {\psi'} \letin y {\sem{e'}(x,x')} (x,\sem{C}(x,y)) \\
&= \letin {(x,x')} {\psi'} (x,\sem{C[e']})
\end{align*}
Now for the redexes
* The rules for $\mathrm{let}$ follow from the general axioms of value substitution in the internal language
* For $\normal()$ we have $(\normal(), \psi) \red (x',\psi \otimes \mathcal N(0,1))$ and verify
\begin{align*}
&\letin x \psi (x,\sem{\normal()}) \\
&= \psi \otimes \mathcal N(0,1) \\
&= \letin {(x,x')} {\psi \otimes \mathcal N(0,1)} (x,\sem{x'})
\end{align*}
* For conditioning, we have $(v\eq w,\psi) \red ((), \psi|_{v=w})$. We need to show
\begin{align*}
&\letin x \psi (x, \sem{v \eq w}) = \letin x {\psi|_{v=w}} (x,())
\end{align*}
Let $h=v-w$, then we need to the following morphisms are equivalent in $\cond(\gauss)$:
\[ \input{conditioning/cond_exact.tikz} \]
Applying <Ref> to the left-hand side requires us to compute the conditional $\langle \id, h\rangle\psi|_2 \circ 0$, which is exactly how $\psi|_{h=0}$ is defined.
$\sem{e_1} = \sem{e_2}$ if and only if $e_1 \approx e_2$ (where $\approx$ is contextual equivalence, <Ref>).
For $\Rightarrow$, let $K[-]$ be a closed context. Because $\sem{-}$ is compositional, we obtain $\sem{K[e_1]} = \sem{K[e_2]}$. By <Ref> If both succeed, we have reductions $(K[e_i], !) \red^* (v_i,\psi_i)$ and by correctness $v_1\psi_1 = \sem{K[e_1]} = \sem{K[e_2]} = v_2\psi_2$ as desired. If $\sem{K[e_1]} = \sem{K[e_2]} = \bot$ then both $(K[e_i], !) \red^* \bot$.
For $\Leftarrow$, we note that $\cond$ quotients by contextual equivalence, but all Gaussian contexts are definable in the language.
§.§ Contextual Equivalence for Finite Probability
With programs over a finite domain, we can understand conditioning in terms of rejection sampling.
This means that we run a program $N$ times, with different random choices each time. We reject those runs that violate the conditions, and then we resample from the among the acceptable results. As $N\to \infty$, this random distribution converges to the probability distribution that the program describes.
The following reformulation is semantically equivalent: For a closed program, suppose the program would return some value $x_1$ with probability $p_1$, value $x_2$ with probability $p_2$, and so on.
Then the probability that the program will not fail is $Z=\sum_i p_i$. The result of rejection sampling is a program that actually returns $x_i$ with probability $\frac {p_i} {Z}$, so that we have a normalized probability distribution over $\{x_1\dots x_n\}$, i.e. $\sum \frac {p_i}{Z}=1$. The quantity $Z$ is called the normalization constant of the program, or sometimes model evidence.
For example, the program
\[ \letin {x} {\bernoulli (0.4)} {\letin {y} {\bernoulli (0.4)} {x \eq y; x}}\]
will fail the condition with probability $2\cdot 0.4\cdot 0.6 = 0.48$, return true with probability $0.4^2=0.16$, and return false with probability $0.6^2=0.36$.
Under rejection sampling, once we renormalize, the program is equivalent to $\bernoulli(\frac {0.16}{0.36})\approx\bernoulli(0.44)$.
Rejection sampling makes sense for closed programs. For programs with free variables, we can still understand a program that rejects runs that violate the conditions, but normalization is more subtle. For example, in the program
\begin{equation}\label{eqn:finprobintroB}{\letin {y} {\bernoulli (0.4)} {x \eq y; x}}\end{equation}
the normalizing constant is either $0.4$ or $0.6$ depending on the value of $x$.
If we normalize regardless of the value of $x$, then the meaning of the program must change, because it would simply return $x$, and the context
\[ \letin {x} {\bernoulli (0.4)} {[-]}\]
distinguishes this.
There is nonetheless some normalization that can be done in straight-line programs, since e.g. the meaning is not changed by prefixing a program with a closed program. It is for example safe to regard program (<ref>) as equivalent to
\begin{equation}
\letin {z} {\bernoulli (0.2)} {z\eq \false ; \letin {y} {\bernoulli (0.4)} {x \eq y; x}} \label{eq:autonorm_example}
\end{equation}
because the difference in normalizing constant will be the same for both values of $x$.
Semantically, the interpretation of a program with free variables is a stochastic kernel, and one involving rejection too is a substochastic kernel (<Ref>). As we show, we can accommodate multiplication by a constant if it is uniform across all arguments; this is what we call `projectivized' substochastic kernels, by analogy with the construction of a projective space from a vector space.
The CD-category $\finprojstoch$ of projectivized substochastic kernels is a quotient of the CD-category of $\finsubstoch$ of substochastic maps:
* objects are finite sets $X$
* morphisms $X \to Y$ are equivalence classes $[p]$ of substochastic kernels $p(y|x)$ up to a scalar. That is we identify $p$ and $q$ if there exists a number $\lambda > 0$ such that $p(y|x) = \lambda \cdot q(y|x)$ for all $x\in X, y \in Y$. In this circumstance we write $p\propto q$.
It is routine to verify that the monoidal and CD category structure are preserved by this quotient.
The CD-categories $\cond(\finstoch)$ and $\finprojstoch$ are equivalent.
Sketch. Given a conditioning channel $Q : X \obsto Y$ presented by a finite set of observations $K$, a probability kernel $q(y,k|x)$ and an observation $k_0 \in K$, we associate to it the subprobability kernel $\rho_Q : X \to D_{\leq 1}(Y)$ given by the likelihood function
\[ \rho_Q (y|x) = q(y,k_0|x) \]
Conversely, we associate to every subprobability kernel $\rho : X \to D_{\leq 1}(Y)$ a conditioning channel $Q_\rho = (\{0,1\}, q_\rho, 1)$ with a single boolean observation $b \eqo 1$, defined as
\[ q_\rho(y,b|x) = b \cdot \rho(y|x) + (1-b)\cdot (1-\rho(y|x)). \]
We recover the subprobability kernel $\rho_Q$ from the conditioning channel $Q$ that way: Given any distribution $p(x)$, the posterior in <Ref> is given by
\[ \frac{p(x)q(y,k_0|x)}{\sum_{x,y} p(x)q(y,k_0|x)} \]
We see that $q_\rho$ computes the same posterior, namely
\[ \frac{p(x) q_\rho(y,1|x)}{\sum_{x,y} p(x) q_\rho(y,1|x)} = \frac{p(x) \rho(y|x)}{\sum_{x,y} p(x) \rho(y|x)} = \frac{p(x) q(y,k_0|x)}{\sum_{x,y} p(x) \rho(y,k_0|x)}\]
On the other hand, the subprobability kernel $\rho$ can be recovered from $Q_\rho$ up to a constant. For a uniform prior $p(x) = 1/|X|$, the posterior under $Q_\rho$ in <Ref> becomes
\[ \frac{\rho(y|x)}{\sum_{x,y} \rho(x,y)} \]
from which we can read off $\rho(y|x)$ up to the constant in the denominator.
Identifying scalar multiples is necessary because the Cond construction by definition `normalizes automatically'. That is, it considers two conditioning channels equivalent if they compute the same posterior distributions for all priors. We will explore the relationship with model evidence and branching in <Ref>.
We briefly showcase some of the structure of this category, by characterizing the discardable morphisms, and observing that conditioning gives a commutative monoid structure. The latter gives a characterization for the finite uniform distributions as units for conditioning.
A projectivized subprobability kernel $p : X \to Y$ is discardable (<Ref>) if and only if there exists a constant $\lambda \neq 0$ such that
\[ \forall x, \sum_y p(y|x) = \lambda \]
As an instance of <Ref>, in particular, to give a state in $\finprojstoch(1,Y)$ is to give either a normalizable distribution $p \in \finstoch(1,Y)$ or the failure kernel $\bot_Y = 0$.
In $\cond(\finstoch)$, we define an exact conditioning operation $x \eq y$ by exactly observing $\true$ from the boolean equality test $(x == y)$. We define a morphism $\bullet : X \times X \obsto X$ by
\[ x \bullet y \defeq (x \eq y); x \]
In terms of projectivized subprobability kernels, this is
\[ \bullet(z|x,y) = \begin{cases}
1 & z = x = y \\
0 & \text{otherwise}
\end{cases} \]
We call $\bullet$ the conditioning product.
Concretely for subdistributions $p(x), q(x)$, the subdistribution $(p \bullet q)$ has the product of mass functions
\[ (p \bullet q)(x) = p(x) \cdot q(x) \]
The conditioning product defines a commutative monoid structure on $X$, where the unit is given by the uniform distribution $u_X : 1 \obsto X$.
The operation $\bullet$ is commutative and associative already in $\finsubstoch$, however it does not have a unit. Conditioning with the uniform distribution produces a global factor of $1/|X|$, which is cancelled by the proportionality relation. Therefore, $u_X$ is a unit for $\bullet$ in $\cond(\finstoch)$.
This is intuitive in programming terms: Observing from a uniform distribution gives no new information. Such a conditioning statement can thus be discarded.
Aside on non-determinism
We show the analogous version of <Ref> for nondeterminism. The Cond construction here does nothing more than add the possibility for failure (zero outputs) in a systematic way.
$\cond(\rel^+) \cong \rel$.
Given a conditioning channel $(K,R,k_0)$ with $R \subseteq X \times Y \times K$ left-total in $X$, we define a possibly non-total relation $R' \subseteq X \times Y$ by $R' = \{ (x,y) : (x,y,k_0) \in R \}$. On the other hand, given $R'$, we form the conditioning channel $(2,R'',1)$ with left-total relation $R'' \subseteq X \times Y \times 2$ defined as
\[ (x,y,b) \in R'' \stackrel{\text{def}}\Leftrightarrow ((x,y) \in R' \Leftrightarrow (b=1)) \]
These constructions are easily seen to be inverses. Any relation $R'$ is recovered from $R''$ and we have $(K,R,k_0) \approx (2,R'',1)$ because <Ref> boils down to checking that $(x,y,k_0) \in R \Leftrightarrow (x,y,1) \in R''$.
The conditioning product $\bullet$ in $\rel$ is the relation $\{ (x,x,x) : x \in X \}$, and on states we have $R \bullet S = R \cap S$. The conditioning product has a unit $v_X : 1 \to X$ given by the maximal subset $v_X = X$.
§.§ Automatic Normalization and Straight-line Inference
By automatic normalization we mean that two (open) probabilistic programs which differ by an overall normalization constant $Z$ are considered equivalent, as formalized in <Ref>. As a consequence, the precise value of the normalization constant cannot be extracted operationally from such programs, which is a limitation whenever $Z$ is itself a quantity of interest. On the other hand, auto-normalization is a convenient optimization, as seen in (<ref>) or <Ref>.
In this section, we argue that validity of auto-normalization is tied to the form of branching available in language under consideration. We distinguish straight-line inference programs with a static structure of conditions from programs where we can dynamically choose whether to execute conditions or not. This is sufficient for inference in Bayesian networks, in which the observed nodes are determined statically. For example, the Bayesian network depicted in (<ref>) corresponds to the straight-line program in <Ref>. Auto-normalization is valid for straight-line programs but not for those with more general branching. The quotient from <Ref> arises naturally from studying contextual equivalence of a straight-line inference language. In semantical terms, this means $\cond(\finstoch)$ does not have coproducts.
We consider the CD-calculus (<Ref>) with base types $X$ for all finite sets and function symbols $\ct f$ for all subprobability kernels $f : X \to D_{\leq 1}(Y)$:
\begin{align*}
t ::= x \s () \s (t,t) \s \pi_it \s \ct f(t) \s \letin x {t_1} t_2
\end{align*}
This language can express scoring and exact conditioning because there exist suitable subprobability kernels
\[ \mathsf{score}_p \in D_{\leq 1}(1) \text{ and } (\eq) : X \times X \to D_{\leq 1}(1) \]
We denote the this calculus $\ppisl$, for straight-line inference. Following the development in Section <ref>, the language $\ppisl$ has canonical denotational semantics in $\finsubstoch$. (In fact, $\ppisl$ is precisely the internal language of $\finsubstoch$ as a CD category.)
We also consider a richer language, $\ppi$, which contains the syntax of $\ppisl$ and also if-then-else branching:
\[t ::= \dots \s \ite {t_1}{t_2}{t_3}\]
with typing rule
\[ \infer{\Gamma \vdash \ite{t_1}{t_2}{t_3} :A }{\Gamma \vdash t_1 : 2 \quad \Gamma \vdash t_2 : A \quad \Gamma \vdash t_3 : A} \]
This language can also be interpreted in $\finsubstoch$, via
\[ \sem{\ite{t_1}{t_2}{t_3}}(a|\gamma) = \sem{t_2}(a|\gamma) \cdot \sem{t_1}(\true|\gamma) + \sem{t_3}(a|\gamma) \cdot \sem{t_1}(\false|\gamma) \]
Categorically, this makes use of the distributive coproducts in $\finsubstoch$ [38, 45]. $\ppi$ is a commonly considered probabilistic language, and subprobability kernel semantics are already known to be fully abstract, though we rederive this here.
As explained in the introduction to this section, a program in the language $\ppi$ or $\ppisl$ is typically executed by some sort of inference engine, which tries to sample (usually approximately) from the posterior distribution it defines. In the semantics, we express this top-level normalization for a finite set $X$ using the function $\norm : \subd(X) \to \subd(X)$ which is defined as
\[ \norm(\varphi)(x) = \begin{cases}
\tfrac 1 Z \cdot \varphi(x) & \text{where } Z = \sum_{x\in X} \varphi(x)\neq 0
\\ 0& \text{where }\forall x.\,\varphi(x)=0\end{cases}
\]
The zero distribution is mapped to itself, signaling failure of normalization.
Two closed $\ppi$ programs $s,t:X$ are called observationally equivalent, written $s \approx t$, if the normalized distributions they define are equal, that is $\norm(\sem{s}) = \norm(\sem{t})$.
We say that two open programs $\Gamma \vdash s, t:X$ are contextually equivalent if under every closed context $C[-]$ we have $C[s] \approx C[t]$. The distinguishing power crucially depends on the fragment of the language we are allowed to use in the contexts $C[-]$.
The terms $s,t$ are called straight-line equivalent, written $s \approx_{\ppisl} t$, if for every closed context $C[-]$ in $\ppisl$ (without branching), we have $C[s] \approx C[t]$. The terms $s,t$ are called branching equivalent, written $s \approx_{\ppi} t$, if for every closed context $C[-]$ in $\ppi$ (possibly involving branching), we have $C[s] \approx C[t]$.
The following proposition shows that straight-line equivalence can distinguish subprobability kernels up to a constant.
Two open programs are straight-line equivalent iff their denotations are proportional.
\[ s \approx_{\ppisl} t \Leftrightarrow \sem{s} \propto \sem{t} \]
That is $\finprojstoch$ is fully abstract for the language $\ppisl$.
$\Leftarrow$ Is is easy to show that the semantics of all $\ppisl$ constructs are linear or bilinear and hence respect the relation $\propto$. $\Rightarrow$ Let $s \approx_\ppisl t$ and consider the straight-line context
\[ C[t] \defeq \letin x {u_X} (x,t) \]
where again $u_X$ denotes the uniform distribution on the finite set $X$. Its denotation is
\[ \sem{C[t]}(x,y) = \frac{1}{|X|} \sem{t}(y|x) \]
By assumption $\sem{C[s]} \propto \sem{C[t]}$, so we have $\sem{s} \propto \sem{t}$.
This gives a clear interpretation of <Ref>. In our current terminology, the Cond construction aims to give a canonical semantics for straight-line inference programs: the construction presents a normal form for straight-line programs up to straight-line equivalence. The lack of branching is reflected in the fact that unlike $\finsubstoch$, $\finprojstoch$ does not have coproducts.
Branching inference
Up to straight-line equivalence, programs which differ by a global constant are not distinguishable, that is auto-normalization is valid. This changes if we allow branching in the contexts, because branching can be used to extract the normalization constant. This trick is fundamental to so-called `Bayesian model selection'. If $y$ is a closed program, then the boolean program
\[ \ite{\mathsf{bernoulli(0.5)}} {y;\true} {\false} \]
normalizes to a distribution which returns $\true$ with probability
\[ p=\frac {0.5\cdot Z}{0.5\cdot Z + 0.5}=\frac{Z}{Z+1} \qquad \text{ where } Z = \sum_{x} \sem{y}(x) \]
and $\false$ with probability $\frac 1 {Z+1}$.
Because the assignment $Z \mapsto Z/(Z+1)$ is a bijection $[0,\infty) \to [0,1)$, we can recover $Z$ from the probability $p$. It follows that $\finsubstoch$ is fully abstract for $\ppi$.
Two open programs $s,t$ are branching equivalent if their denotations are equal as subprobability kernels.
\[ s \approx_\ppi t \Leftrightarrow \sem s = \sem t:\sem\Gamma\to D_{\leq 1}(\sem X) \]
That is $\finsubstoch$ is fully abstract for the language $\ppi$.
From straight-line equivalence, we know that $\sem{s} = \lambda \cdot \sem{t}$. We then use the `Bayesian model selection' trick to show $\lambda = 1$.
The tradeoff between straight-line inference and branching inference is an interesting design decision: Branching inference is more general and allows us to extract the normalization constant. On the other hand, restricting ourselves to straight-line inference, we are free to normalize at any point, which leads to an appealing equational theory. For example, an `uninformative' observation form a uniform distribution can be eliminated (<Ref>).
We emphasize that the crucial difference between $\ppisl$ and $\ppi$ lies in putting conditions in branches. Even in $\ppisl$, we can still implement if-then-else when the branches do not involve conditions, because we can make use of the probability kernel
\[
\mathsf{ite} : 2 \times X \times X \to D(X)
\]
with $\mathsf{ite}(1,x,y)=\delta_x$ and $\mathsf{ite}(0,x,y)=\delta_y$. If $t_1,t_2$ are discardable (condition-free) terms, we can define
\[ (\ite c {t_1} {t_2}) \defeq \mathsf{ite}(c,t_1,t_2). \]
In $\ppisl$, the structure of conditions is static, while it is dynamic in $\ppi$.
§ CONTEXT, RELATED WORK AND OUTLOOK
§.§ Symbolic Disintegration, Consistency and Paradoxes
Our line of work can be regarded as a synthetic and axiomatic counterpart of the symbolic disintegration of [42] (see also [15, 36, 37, 51]).
That work provides in particular verified program transformations to convert an arbitrary probabilistic program of type $\rv \otimes \tau$
to an equivalent one that is of the form
\[ \letin x {\mathrm{lebesgue}()} {\letin y M {(x,y)}} \]
Now the exact conditioning $x\eqo o$ can be carried out by substituting $o$ for $x$ in $M$.
We emphasize the similarity to our treatment of inference problems in <Ref>, as well as the role that coordinate transformations play in both our work [49] and [42]. One language novelty in our work is that exact conditioning is a first-class construct in our language, as opposed to a whole-program transformation, which makes the consistency of exact conditioning more apparent.
Consistency is a fundamental concern for exact conditioning. Borel's paradox is an example of an inconsistency that arises if one is careless with exact conditioning (<cit.>, <cit.>): It arises when naively substituting equivalent equations within $(\eq)$. For example, the equation $x - y = 0$ is equivalent to $x/y = 1$ over the (nonzero) real numbers. Yet, in a hypothetical extension of our language which allows division, the following programs would not contextually equivalent, as discussed in <Ref>:
\begin{equation*}
\begin{array}{l}
\mlstinline{x = normal(0,1)} \\
\mlstinline{y = normal(0,1)} \\
\mlstinline{x-y =:= 0} \\
\end{array}
\not \equiv
\begin{array}{l}
\mlstinline{x = normal(0,1)} \\
\mlstinline{y = normal(0,1)} \\
\mlstinline{x/y =:= 1}
\end{array}
\end{equation*}
For that reason, we make it clear in our treatment of inference problems (<Ref>) that conditioning on a deterministic observation $(\eqo)$ is the fundamental notion. Binary conditioning $(\eq)$ is a derived notion which involves further choices, and those choices are not equivalent. Our approach also makes it clear that we should always condition on random variables directly, and not on (boolean) predicates: By presenting conditioning as an algebraic effect, the expressions $(s \eq t) : \unit$ and $(s == t) : \mathrm{bool}$ have a different formal status and can no longer be confused.
§.§ Contextual Equivalence for Exact Conditioning languages
In this article, we have emphasized the role of program equations for manipulating probabilistic programs, and based the Cond construction on an analysis of contextual equivalence of straight-line inference (<Ref>).
While we have fully characterized the case of finite probability (<Ref>), a corresponding explict characterization of contextual equivalence for the Gaussian language is still outstanding. We have given partial results in that direction (see [49]) in the form of a sound equational theory for contextual equivalence. The classification of effects $n \to 0$ in $\cond(\gauss)$ is not straightforward: it is not true that every effect is observing from a unique distribution $0 \to n$, as for example $(\eq) : 2 \to 0$ is not of that form. We believe that by passing to an extension category of Gaussians $\gauss \to \mathsf{GaussEx}$, we can obtain the desired duality and achieve a more explicit characterization.
It is a further challenge to find semantics for exact conditioning with branching. Automatic normalization is no longer valid here (<Ref>) and the subtleties of [25] have to be accounted for. Mathematically, this would be an extension of the Cond construction which produces a distributive Freyd category [38]. An example of a categorical model of the Beta-Bernoulli process with branching (but no first-class conditioning) is in [46].
§.§ Other Directions
Categorical tools
Once a foundation is in algebraic or categorical form, it is easy to make connections to and draw inspiration from a variety of other work: The $\obs$ construction (<Ref>) that we considered here is reminiscent of lenses [7] and the Oles construction [20]. These have recently been applied to probability theory [43], quantum theory [22] and reversible computing [21]. The details and intuitions are different, but a deeper connection or generalization may be profitable in the future.
Probabilistic logic programming
The concept of exact conditioning is reminiscent of unification in Prolog-style logic programming. Our presentation in [49] is partly inspired by the algebraic presentation of predicate logic of [44], which has a similar signature and axioms. Logic programming is also closely related to relational programming, and we note that our laws for conditioning are reminiscent of graphical presentations of categories of linear relations [1, 4, 3].
ProbLog [8] supports both logic variables as well as random variables within a common formalism. We have not considered logic variables in conjunction with the Gaussian language, but a challenge for future work is to bring the ideas of exact conditioning closer to the ideas of unification, both practically and in terms of the semantics. This is again related to the extension $\mathsf{GaussEx}$ by “improper priors”, which are a unit for the conditioning product in the same way uniform distributions are in finite probability (<Ref>). The connections with logic programming are spelled out in more detail in <cit.>.
|
# Girth, words and diameter
Martin W. Liebeck M.W. Liebeck, Department of Mathematics, Imperial College,
London SW7 2BZ, UK<EMAIL_ADDRESS>and Aner Shalev A. Shalev,
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
<EMAIL_ADDRESS>
###### Abstract.
We study the girth of Cayley graphs of finite classical groups $G$ on random
sets of generators. Our main tool is an essentially best possible bound we
obtain on the probability that a given word $w$ takes the value 1 when
evaluated in $G$ in terms of the length of $w$, which has additional
applications. We also study the girth of random directed Cayley graphs of
symmetric groups, and the relation between the girth and the diameter of
random Cayley graphs of finite simple groups.
###### 2010 Mathematics Subject Classification:
Primary 20D06; Secondary 20P05, 05C80, 05C12
We are grateful to Sean Eberhard for some enlightening comments. The second
author acknowledges the support of ISF grant 686/17, BSF grant 2016072 and the
Vinik chair of mathematics which he holds.
## 1\. Introduction
The girth of a graph (resp. directed graph) is the minimal length of a cycle
(resp. directed cycle) in the graph. The girth of finite $k$-regular graphs
has been studied extensively, with a particular focus on graphs of large girth
– see for example [8], [18]. Note the trivial upper bound of
$2\log_{k-1}{v}+1$ for the girth, where $v$ is the number of vertices and
$k>2$.
In [9] the girth of random Cayley graphs of various families of groups was
studied, and large girth results were established. For a finite group $G$ and
a sequence $S$ of elements $g_{1},\ldots,g_{k}$ of $G$, let $\Gamma(G,S)$
(resp. $\Gamma^{*}(G,S)$) denote the associated undirected (resp. directed)
Cayley graph. Corollary 2 of [9] asserts that for finite simple groups $G$,
the girth of $\Gamma(G,S)$ for $k$ random generators tends to $\infty$ almost
surely as $|G|\rightarrow\infty$. Also [9, Thm. 4] shows that for groups $G$
of Lie type of bounded rank, the girth is $\Omega(\log|G|)$, while [9, Thm. 3]
asserts that for $G=S_{n}$, the girth is at least $\Omega((\log|G|)^{1/2})$
almost surely. An error in the proof of the latter result was recently pointed
out in [7], and a slightly weaker bound of the form $\Omega(n^{1/3})$ was
obtained.
The random girth of classical groups $G$ of unbounded rank has apparently
remained unexplored. Denote by $Cl_{n}(q)$ a simple classical group over
$\mathbb{F}_{q}$ with natural module of dimension $n$. Our first result shows
in particular that if the underlying field $\mathbb{F}_{q}$ has bounded size,
then the random girth of such groups is at least $\Omega((\log|G|)^{1/2})$.
###### Theorem 1.
There exists an absolute constant $N$, and for each integer $k\geq 2$ and
prime power $q$ a positive real number $b=b(q,k)$ with the following property.
Let $G=Cl_{n}(q)$ with $n\geq N$, and let $S$ be a sequence of $k$
independently chosen random elements of $G$. Then as $|G|\rightarrow\infty$,
the girth of the Cayley graph $\Gamma(G,S)$ exceeds $b\sqrt{\log|G|}$ almost
surely.
In particular, for bounded $q$ and $k$ the girth is almost surely
$\Omega(\sqrt{\log|G|})$. Also the proof shows that the girth exceeds $Bn$ for
some positive constant $B=B(k)$; in fact we can take
$B=\frac{1}{7(1+2\log_{2}{(2k-1)})}$.
Our next result concerns the girth of directed Cayley graphs of symmetric
groups (namely, the minimal length of a directed cycle in the graph).
###### Theorem 2.
Fix an integer $k\geq 2$, and let $S$ be a sequence of $k$ independently
chosen random elements of $G=S_{n}$. Then the girth of $\Gamma^{*}(G,S)$
almost surely exceeds $c\sqrt{\log_{k}|G|}$, where $c$ is a positive absolute
constant.
Thus the girth of a random directed Cayley graph of $S_{n}$ is at least
$\Omega(\sqrt{n\log n})$.
The diameter of Cayley graphs of finite simple groups (with explicit or with
random generators) has also attracted considerable attention – see for
instance [2], [4], [19], [10], [11], [5], [3] and the references therein.
Clearly if $d$ and $g$ are the diameter and girth respectively, then a trivial
lower bound for $d$ is $\lfloor\frac{g}{2}\rfloor$, and there is interest in
finding families of graphs for which $d$ is bounded in terms of $g$. According
to [1], a family of graphs is $dg$-bounded if the ratio $\frac{d}{g}$ is
bounded. The focus is on graphs of large girth, meaning that the girth is
$\Omega(\log|G|)$, where $G$ ranges over the ambient family of groups. The
main result of [1] is a construction of certain Cayley graphs of $SL_{n}(p)$
with respect to two explicit generators, where $n$ is fixed, which are of
large girth and $dg$-bounded.
It follows from part (i) of the next result that such families of Cayley
graphs exist for all groups of Lie type of bounded rank. Parts (ii) and (iii)
bound the diameter in terms of (nonlinear) functions of the girth almost
surely for other families of finite simple groups.
###### Proposition 3.
Fix $k\in\mathbb{N}$. Let $G$ be a finite simple group and let $S$ be a
sequence of $k$ independently chosen random elements of $G$. Let $d,g$ be the
diameter and girth of $\Gamma(G,S)$ respectively. Suppose
$|G|\rightarrow\infty$.
* (i)
If $G$ is of Lie type of bounded rank, then $\Gamma(G,S)$ has large girth and
is $dg$-bounded (i.e. $d=O(g)$) almost surely.
* (ii)
If $G=Alt_{n}$, then almost surely $d\leq g^{6}(\log g)^{c}$ for some absolute
constant $c$.
* (iii)
If $G=Cl_{n}(q)$ is a classical group of dimension $n$ with $q$ bounded, then
$d\leq C^{g(\log g)^{3}}$ almost surely, for some constant $C=C(k)>1$.
The proof of part (i) is rather short, modulo the deep results in [4, 19]
(which are also used in [1]). Parts (ii) and (iii) require results from [11]
and [3] respectively.
The bound in part (iii) above seems far from best possible, and it would be
nice to obtain a polynomial bound in this case too. Such a bound would follow
from Theorem 1, together with Babai’s conjecture (so far unproved) that the
diameter of any connected Cayley graph of a (nonabelian) finite simple group
$G$ is at most $(\log{|G|})^{c}$, for some absolute constant $c$ (see [2,
1.7]). In fact, a polynomial bound in part (iii) of Proposition 3 would
already follow if the latter bound on the diameter holds almost surely for
random Cayley graphs of classical groups.
The proofs of Theorems 1 and 2 rely on the study of the probability $P_{G}(w)$
that a word $w=w(x_{1},\ldots,x_{k})$ in the free group $F_{k}$ takes the
value 1 when we substitute a sequence $S$ of $k$ independently chosen random
elements of $G$ for $x_{1},\ldots,x_{k}$. It is an elementary observation that
${\textbf{P}}(\hbox{girth}(\Gamma(G,S))\leq L)\leq\sum_{|w|\leq L}P_{G}(w)$
where $|w|$ denotes the length of $w$ (see [9, Sec. 2]). The study of
$P_{G}(w)$ is an important part of the theory of word maps, with a particular
focus on finite simple groups $G$. In [6, Thm. 3] it is shown that if $w\neq
1$ then $P_{G}(w)\rightarrow 0$ as $|G|\rightarrow\infty$; and [12, Thm. 1.1]
shows that for every $w\neq 1$ there exist $\delta=\delta(w)>0$ and $N=N(w)$
such that $P_{G}(w)\leq|G|^{-\delta}$ provided $|G|>N$. The next result gives
an explicit and close to best possible value for the constant $\delta(w)$ in
terms of the length of $w$.
###### Theorem 4.
For any $\epsilon>0$, there exists $c=c(\epsilon)>0$ with the following
property. Let $\ell\in\mathbb{N}$ and let $G=Cl_{n}(q)$ be a classical group
of dimension $n\geq c\ell$. Then, for any reduced word $w\in F_{k}$ of length
$\ell$, we have
$P_{G}(w)\leq|G|^{-\frac{1}{(2+\epsilon)\ell}}.$
In fact the proof gives $c(\epsilon)=4\,(1+\frac{2}{\epsilon})$ for
$G=SL_{n}(q)$, and $c(\epsilon)=7\,(1+\frac{2}{\epsilon})$ for the other
classical groups. For more detailed bounds on $P_{G}(w)$ see Section 2.
Theorem 4 improves a bound of the form
$P_{G}(w)\leq|G|^{-\frac{1}{1800\ell^{2}}+o(1)}$ obtained in the proof of [12,
Thm. 1.1].
As claimed above, Theorem 4 is close to being best possible; indeed, [16, Thm.
1.4] shows that, for a fixed power word $w=x_{1}^{\ell}$ and $G=Cl_{n}(q)$
with $n\rightarrow\infty$ we have $P_{w}(G)=|G|^{-\frac{1}{\ell}+o(1)}$. It
seems an interesting and challenging problem to improve the upper bound in
Theorem 4 to $P_{G}(w)\leq|G|^{-\frac{1}{(1+\epsilon)\ell}}$.
The key to the proof of Theorem 2 is the following result, which is of some
independent interest. Recall that a word $w\in F_{k}$ is said to be positive
(or a semigroup word) if it does not involve inverses of the generators of
$F_{k}$.
###### Proposition 5.
Let $w\in F_{k}$ be a positive word of length $\ell$. Then for all
$n\in\mathbb{N}$, we have
$P_{S_{n}}(w)\leq\left(\frac{2\ell}{n}\right)^{\lfloor\frac{n}{2\ell}\rfloor}\leq(n!)^{-\frac{1}{2\ell}+o_{n}(1)}.$
In fact the inaccurate proof of the above bound for $P_{S_{n}}(w)$ on p.106 of
[9] becomes accurate when $w$ is assumed to be a positive word. Proposition 5
is essentially best possible; indeed for $w=x_{1}^{\ell}$ we have
$P_{S_{n}}(w)=(n!)^{-\frac{1}{\ell}+o_{n}(1)}$ (see for instance [16, 2.17]).
We conclude the introduction with applications of the two results above to
representation varieties and subgroup growth (cf. [12, 1.3, 1.4]).
###### Corollary 6.
Let $\Gamma$ be a non-free group with $k$ generators, and let $\ell$ be the
minimal length of a non-trivial relation (in these generators) which holds in
$\Gamma$. Then for every $\epsilon>0$ there exists $N=N(\ell,\epsilon)$ such
that the following hold for all $n\geq N$ and for any algebraically closed
field $F$:
* (i)
$\dim{\rm Hom}(\Gamma,GL_{n}(F))\leq(k-\frac{1}{(2+\epsilon)\ell})n^{2}$;
* (ii)
$\dim{\rm Hom}(\Gamma,G)\leq(k-\frac{1}{(2+\epsilon)\ell})\dim G$, where $G$
is a simple algebraic group over $F$ of dimension $n$.
Recall that $a_{n}(\Gamma)$ denotes the number of index $n$ subgroups of
$\Gamma$.
###### Corollary 7.
Let $\Gamma$ be a group with $k$ generators which satisfy some non-trivial
positive relation. Let $\ell$ be the minimal length of such a relation. Then
$a_{n}(\Gamma)\leq(n!)^{k-1-\frac{1}{2\ell}+o_{n}(1)}$.
## 2\. Proof of Theorem 4
First we give the proof in the case $G=GL_{n}(q)$. Our method is inspired by
Eberhard’s proof in [7]. Denote by $V=\mathbb{F}_{q}^{n}$ the underlying
vector space. Assume $2\ell<n$.
Let $a_{1},\ldots,a_{k}$ be free generators for $F_{k}$, and let
$w=w_{\ell}\cdots w_{1}$, where each $w_{i}\in\\{a_{1}^{\pm
1},\ldots,a_{k}^{\pm 1}\\}$. Let $g_{1}\ldots,g_{k}$ be a random sequence of
elements of $G$.
Fix $v_{1}\in V\setminus 0$, and define $v_{1}^{0},\ldots v_{1}^{\ell}$ by
$\begin{array}[]{l}v_{1}^{0}=v_{1},\\\
v_{1}^{j}=w_{j}(g_{1},\ldots,g_{k})(v_{1}^{j-1})\;\;(1\leq
j\leq\ell).\end{array}$
Call the sequence $v_{1}^{0},\ldots v_{1}^{\ell}$ the trajectory of $v_{1}$.
Assume $v_{1}^{0},\ldots,v_{1}^{j-1}$ are linearly independent. Then
$v_{1}^{j}\not\in{\rm Sp}\left(\\{v_{1}^{i}\,(i\leq j-1)\,|\,w_{i}=w_{j}\hbox{
or }w_{i+1}=w_{j}^{-1}\\}\right),$ (1)
which excludes at most $q^{j-1}$ possibilities for $v_{1}^{j}$; all other
vectors are equally likely as possibilities for $v_{1}^{j}$, since
$w_{j}(g_{1},\ldots,g_{k})$ is a random element. Hence the conditional
probability
${\textbf{P}}\left(v_{1}^{j}\in{\rm
Sp}(v_{1}^{0},\ldots,v_{1}^{j-1})\,|\,v_{1}^{0},\ldots,v_{1}^{j-1}\right)\leq\frac{q^{j}}{q^{n}-q^{j-1}}.$
It follows that
${\textbf{P}}\left(v_{1}^{0},\ldots,v_{1}^{\ell}\hbox{ lin.
dep.}\right)\leq\sum_{j=1}^{\ell}\frac{q^{j}}{q^{n}-q^{j-1}}\leq\frac{q+q^{2}+\ldots+q^{\ell}}{q^{n}-q^{\ell-1}}<\frac{q}{q-1}\cdot\frac{q^{\ell}}{q^{n}-q^{\ell-1}}.$
Now suppose $v_{1}^{0},\ldots,v_{1}^{\ell}$ are given, and set $V_{1}={\rm
Sp}(v_{1}^{0},\ldots v_{1}^{\ell})$. Pick $v_{2}\not\in V_{1}$, and define the
trajectory of $v_{2}$ to be $v_{2}^{0},\ldots v_{2}^{\ell}$ as above. Assuming
$v_{2}^{0},\ldots v_{2}^{j-1}$ to be linearly independent and also have span
intersecting $V_{1}$ trivially, we have
${\textbf{P}}\left(v_{2}^{j}\in{\rm Sp}(v_{2}^{0},\ldots v_{2}^{j-1}\cup
V_{1})\right)\leq\frac{q^{\ell+j}}{q^{n}-q^{\ell+j-1}},$
and hence, arguing as above, we obtain
${\textbf{P}}\left(v_{2}^{0},\ldots,v_{2}^{\ell}\hbox{ lin.
dep.}\,|\,v_{1}^{0},\ldots,v_{1}^{\ell}\right)\leq\sum_{j=1}^{\ell}\frac{q^{\ell+j}}{q^{n}-q^{\ell+j-1}}<\frac{q}{q-1}\cdot\frac{q^{2\ell}}{q^{n}-q^{2\ell-1}}.$
Repeating this argument $m$ times, where $m\leq\frac{n}{\ell}$ (choosing each
$v_{i}$ ($1\leq i\leq m$) not in the span of the previous trajectories), we
obtain
${\textbf{P}}\left(v_{m}^{0},\ldots,v_{m}^{\ell}\hbox{ lin.
dep.}\,|\,v_{i}^{0},\ldots,v_{i}^{\ell}\hbox{ for
}i<m\right)<\frac{q}{q-1}\cdot\frac{q^{m\ell}}{q^{n}-q^{m\ell-1}}.$
If $w(g_{1},\ldots,g_{k})=1$, then $v_{i}^{\ell}=v_{i}$ for all $i$, and hence
$\begin{array}[]{ll}P_{G}(w)&\leq\prod_{i=1}^{m}{\textbf{P}}\left(v_{i}^{\ell}=v_{i}\,|\,v_{j}^{\ell}=v_{j}\hbox{
for }1\leq j\leq i-1\right)\\\
&<\prod_{i=1}^{m}\frac{q}{q-1}\frac{q^{i\ell}}{q^{n}-q^{i\ell-1}}=(\tfrac{q}{q-1})^{m}\prod_{i=1}^{m}\frac{1}{q^{n-i\ell}-q^{-1}}.\end{array}$
Set $m=\lfloor\frac{n}{\ell}\rfloor$. Define
$a(q)=\prod_{i=1}^{m}\frac{q^{n-i\ell}}{q^{n-i\ell}-q^{-1}}=\prod_{i=1}^{m}\frac{1}{1-q^{-(n-i\ell)-1}}.$
We may and shall assume $\ell\geq 2$, since for $w$ of length $1$ we have
$P_{G}(w)=|G|^{-1}$ for all finite groups $G$. Clearly $a(q)\leq a(2)\leq a$,
where
$a:=\prod_{i=0}^{\infty}\frac{1}{1-2^{-(2i+1)}}=\prod_{i=0}^{\infty}(1+\frac{1}{2^{2i+1}-1})<2.3749,$
where the last inequality is easily verified by computing the sum for $i\leq
6$ and bounding its tail. It follows that
$P_{G}(w)\leq
a\cdot(\tfrac{q}{q-1})^{m}\cdot\prod_{i=1}^{m}q^{-n+i\ell}=a\cdot(\tfrac{q}{q-1})^{m}\cdot
q^{-mn+\frac{1}{2}\ell m(m+1)}.$
Now $\frac{n}{\ell}-1<m\leq\frac{n}{\ell}$. We conclude that
$P_{G}(w)\leq a\cdot(\tfrac{q}{q-1})^{\frac{n}{\ell}}\cdot
q^{-n(\frac{n}{\ell}-1)+\frac{1}{2}n(\frac{n}{\ell}+1)}=a\cdot(\tfrac{q}{q-1})^{\frac{n}{\ell}}\cdot
q^{-\frac{n^{2}}{2\ell}+\frac{3n}{2}}.$
For $\ell\geq 3$, this is less than $q^{-\frac{n^{2}}{(2+\epsilon)\ell}}$,
hence less than $|G|^{-\frac{1}{(2+\epsilon)\ell}}$, provided $n\geq
c(\epsilon)\ell$, where $c(\epsilon)=4\,(1+\frac{2}{\epsilon})$. And for
$\ell=2$, the same assertion holds using the upper bound for $i_{2}(G)$, the
number of involutions in $G$, given by [13, 1.3] (noting that for the word
$w=x^{2}$, $P_{G}(w)=\frac{i_{2}(G)+1}{|G|}$).
This completes the proof of Theorem 4 for $G=GL_{n}(q)$, and the same argument
replacing $G$ by $SL_{n}(q)$ gives the result for $SL_{n}(q)$.
Now let $G=Cl_{n}(q)$ be a classical group with natural module
$V=(\mathbb{F}_{Q})^{n}$, where $Q=q^{2}$ if $G$ is unitary and $Q=q$
otherwise. Let $(\,,\,)$ be the associated bilinear or sesquilinear form on
$V$ preserved by $G$, and when $G$ is orthogonal, let $R$ be the associated
quadratic form. Assume $n>4\ell$.
The proof is rather similar to the previous proof for $GL_{n}$. Let
$a_{1},\ldots,a_{k}$ be free generators for $F_{k}$, and let $w=w_{\ell}\cdots
w_{1}$, where each $w_{i}\in\\{a_{1}^{\pm 1},\ldots,a_{k}^{\pm 1}\\}$. Let
$g_{1}\ldots,g_{k}$ be a random sequence of elements of $G$. Let $v_{1}\in V$
be a nonzero singular vector, and define its trajectory $v_{1}^{0},\ldots
v_{1}^{\ell}$ as before. Assume that $v_{1}^{0},\ldots,v_{1}^{j-1}$ are
linearly independent. Again, (1) holds, excluding at most $q^{j-1}$
possibilities for $v_{1}^{j}$. Moreover the values of $(v_{1}^{j},v_{1}^{i})$
are specified for the vectors $v_{1}^{i}$ for which $w_{i}=w_{j}$ or
$w_{i+1}=w_{j}^{-1}$. Hence there are at least $Q^{n-j}-Q^{j-1}$ possibilities
for $v_{1}^{j}$, and so
${\textbf{P}}\left(v_{1}^{j}\in{\rm
Sp}(v_{1}^{0},\ldots,v_{1}^{j-1})\,|\,v_{1}^{0},\ldots,v_{1}^{j-1}\right)\leq\frac{Q^{j}}{Q^{n-j}-Q^{j-1}}.$
It follows that
${\textbf{P}}\left(v_{1}^{0},\ldots,v_{1}^{\ell}\hbox{ lin.
dep.}\right)\leq\sum_{j=1}^{\ell}\frac{Q^{j}}{Q^{n-j}-Q^{j-1}}\leq\frac{Q}{Q-1}\cdot\frac{Q^{\ell}}{Q^{n-\ell}-Q^{\ell-1}}.$
Now as before define further trajectories $v_{i}^{0},\ldots,v_{i}^{\ell}$ for
$1\leq i\leq m$, where $m<\frac{n}{2\ell}$. Arguing as above we obtain
${\textbf{P}}\left(v_{i}^{0},\ldots,v_{i}^{\ell}\hbox{ lin.
dep.}\,|\,v_{j}^{0},\ldots,v_{j}^{\ell}\hbox{ for
}j<i\right)\leq\frac{Q}{Q-1}\cdot\frac{Q^{i\ell}}{Q^{n-i\ell}-Q^{i\ell-1}}.$
If $w(g_{1},\ldots,g_{k})=1$, then $v_{i}^{\ell}=v_{i}$ for all $i$, and hence
$\begin{array}[]{ll}P_{G}(w)&\leq\prod_{i=1}^{m}{\textbf{P}}\left(v_{i}^{\ell}=v_{i}\,|\,v_{j}^{\ell}=v_{j}\hbox{
for }1\leq j\leq i-1\right)\\\
&\leq(\tfrac{Q}{Q-1})^{m}\prod_{1}^{m}\frac{Q^{i\ell}}{Q^{n-i\ell}-Q^{i\ell-1}}=(\tfrac{Q}{Q-1})^{m}\prod_{1}^{m}\frac{1}{Q^{n-2i\ell}-Q^{-1}}.\end{array}$
Set $m=\lfloor\frac{n}{2\ell}\rfloor$. Arguing as above, this leads to
$P_{G}(w)\leq
a\cdot(\tfrac{Q}{Q-1})^{\frac{n}{2\ell}}Q^{-\frac{n^{2}}{4\ell}+\frac{3n}{2}}.$
As before, this gives $P_{G}(w)\leq|G|^{-\frac{1}{(2+\epsilon)\ell}}$ provided
$n\geq 7\,(1+\frac{2}{\epsilon})\ell$.
This completes the proof of Theorem 4.
## 3\. Deduction of Theorem 1
Let $G$ be a finite group and $S$ a sequence of $k$ random elements of $G$
chosen independently. For $\ell\geq 1$, define $P_{G}(\ell)$ to be the maximum
of $P_{G}(w)$ over all words $w\in F_{k}$ of length $|w|=\ell$. Then as in [9,
Sec. 2] by the well-known union bound, for any positive integer $L$ we have
${\textbf{P}}(\hbox{girth}(\Gamma(G,S))\leq L)\leq\sum_{|w|\leq
L}P_{G}(w)=\sum_{\ell=1}^{L}2k(2k-1)^{\ell-1}P_{G}(\ell).$ (2)
Now let $G=Cl_{n}(q)$ and choose $\epsilon$ with $0<\epsilon\leq\log_{2k-1}q$.
Then Theorem 4 gives $P_{G}(l)\leq|G|^{-\frac{1}{(2+\epsilon)\ell}}$ for
$\ell\leq\frac{n}{c}$, where $c=c(\epsilon)=7\,(1+\frac{2}{\epsilon})$. Hence,
taking $L\leq\frac{n}{c}$, the right hand side in (2) is bounded above by
$E:=\frac{k}{k-1}(2k-1)^{n/c}|G|^{-\frac{c}{(2+\epsilon)n}}.$
Since $\frac{c}{2+\epsilon}=\frac{7}{\epsilon}$, we have
$\log_{2k-1}E\leq 1+\frac{n}{c}-\frac{3.5(n-1)}{\epsilon}\log_{2k-1}q,$
and by the choice of $\epsilon$ this tends to 0 as $|G|\rightarrow\infty$.
Hence the girth is at least $\frac{n}{c}$. Fixing $k$ and $\epsilon$, this is
of the order of $b(q)\sqrt{\log|G|}$, where $b(q)=(\log q)^{-\frac{1}{2}}$.
Theorem 1 follows.
## 4\. Proof of Proposition 3
We first prove part (i). Fix $k\geq 2$. Let $G=G(q)$ be a simple group of Lie
type of fixed rank and let $S$ be a sequence of $k$ independently chosen
random elements of $G$. By [14], $S$ generates $G$ almost surely. By [9, Thm.
4], the girth $g$ of $\Gamma(G,S)$ satisfies
$g\geq c_{1}\log|G|$ (3)
almost surely for some positive absolute constant $c_{1}$. Let $T$ be the
symmetric set consisting of the elements of $S$ and their inverses. Write
$h=\lfloor\frac{g-1}{2}\rfloor$. Since $g$ is the girth, we have
$|T^{h}|\geq(2k-1)^{h}$ almost surely. Let $A=T^{h}$. Then it follows from (3)
that $|A|\geq|G|^{\delta}$ for some positive absolute constant $\delta$ almost
surely.
By the Product Theorem [4, 19], there is a positive absolute constant
$\epsilon$ such that for any symmetric generating subset $B$ of $G$, either
$B^{3}=G$ or $|B^{3}|\geq|B|^{1+\epsilon}$. It follows inductively that if $m$
is chosen minimally such that $\delta(1+\epsilon)^{m}\geq 1$, then we have
$A^{3^{m}}=G.$
Note that $m$ is an absolute constant. It follows that
$d=\hbox{diam}(\Gamma(G,S))\leq 3^{m}\cdot h\leq c_{2}\cdot g$
almost surely, where $c_{2}=3^{m}/2$, as required.
Now we prove part (ii) of Proposition 3. Let $G=Alt_{n}$ and let $S$ be a
sequence of $k$ independently chosen random elements of $G$. It follows from
[7, Thm. 1.1] that $g=\hbox{girth}(\Gamma(G,S))>c_{1}n^{1/3}$ almost surely,
where $c_{1}$ is a positive absolute constant. Also by [11, Thm. 1.1],
$d=\hbox{diam}(\Gamma(G,S))<c_{2}n^{2}(\log n)^{c_{3}}$. The conclusion
follows.
The proof of part (iii) relies on [3, Thm. 1.4], showing that, for
$G=Cl_{n}(q)$, the diameter $d$ of any connected Cayley graph of $G$ satisfies
$d\leq q^{O(n(\log n+\log q)^{3})}.$
Combining this with the fact that the girth $g$ of $\Gamma(G,S)$ satisfies
$g\geq B(k)n$ almost surely (see the remark following Theorem 1) we easily
derive part (iii).
## 5\. Proof of Results 2, 5, 6 and 7
The proof of [9, Thm. 3] given on p.106 was shown to contain an error by
Eberhard [7]. However, the error pointed out in [7, Sec. 3] only pertains if
there is a value of $i$ such that both $a_{i}$ and $a_{i}^{-1}$ occur in the
word $w$. Hence, if we restrict to positive words, the inequality displayed as
(6) on p.106 of [9] holds. Proposition 5 is just this bound. Now Theorem 2
follows, just as in [9, p.106].
To prove Corollaries 6 and 7 we may assume that $\Gamma=\langle
a_{1},\ldots,a_{k}:w(a_{1},\cdots,a_{k})=1\rangle$ where $w$ is the relation
(resp. the positive relation) of minimal length $\ell$ (since our group is a
quotient of the group above).
Note that, for an algebraic group $G$, the variety ${\rm Hom}(\Gamma,G)$ can
be identified with the subvariety of $G^{k}$ defined by the equation
$w(g_{1},\ldots,g_{k})=1$. The proof of Corollary 6 now follows using Theorem
4 and Lang-Weil estimates, as in [16, Sec. 7] and [12, Sec. 4].
Finally, to prove Corollary 7 we combine Proposition 5 with the well-known
inequality
$a_{n}(\Gamma)\leq|{\rm Hom}(\Gamma,S_{n})|/(n-1)!=P_{S_{n}}(w)n!^{k-1}\cdot
n,$
which follows from [17, 1.1].
## References
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|
# Realisations of elliptic operators on compact manifolds with boundary
Lashi Bandara , Magnus Goffeng and Hemanth Saratchandran Lashi Bandara,
Institut für Mathematik, Universität Potsdam, D-14476, Potsdam OT Golm,
Germany http://www.math.uni-potsdam.de/ bandara<EMAIL_ADDRESS>Magnus Goffeng, Centre for Mathematical Sciences, University of Lund, Box 118,
221 00 LUND, Sweden
https://www.lunduniversity.lu.se/lucat/user/e604494124f99d9bb6048e890306f7a4
<EMAIL_ADDRESS>Hemanth Saratchandran The University of Adelaide
Adelaide, South Australia, 5005, Australia
https://researchers.adelaide.edu.au/profile/hemanth.saratchandran
<EMAIL_ADDRESS>
###### Abstract.
This paper investigates realisations of elliptic differential operators of
general order on manifolds with boundary following the approach of Bär-
Ballmann to first order elliptic operators. The space of possible boundary
values of elements in the maximal domain is described as a Hilbert space
densely sandwiched between two mixed order Sobolev spaces. The description
uses Calderón projectors which, in the first order case, is equivalent to
results of Bär-Bandara using spectral projectors of an adapted boundary
operator. Boundary conditions that induce Fredholm as well as regular
realisations, and those that admit higher order regularity, are characterised.
In addition, results concerning spectral theory, homotopy invariance of the
Fredholm index, and well-posedness for higher order elliptic boundary value
problems are proven.
###### Key words and phrases:
Elliptic differential operator, Fredholm boundary conditions, boundary
regularity, Calderón projector
###### 2010 Mathematics Subject Classification:
35J58, 35J56, 58J05, 58J32
###### Contents
1. 1 Introduction
2. 2 Elliptic differential operators and boundary conditions
3. 3 Calderón projectors and the Douglis-Nirenberg calculus
4. 4 The Cauchy data space of an elliptic differential operator
5. 5 Characterisations of regularity via graphical decompositions
6. 6 Higher order boundary regularity
7. 7 Differential operators with positive principal symbols and their Weyl laws
8. 8 Rigidity results for the index of elliptic differential operators
9. 9 Symbol computations with Calderón projectors in the first order case
10. A Lemmas on dimensions and subspaces
11. B Bär-Ballmann’s approach from an abstract perspective
## 1\. Introduction
Elliptic boundary value problems have a long history emerging from classical
problems in physics and engineering. Their mathematical description dates back
to classical works such as [grisvard, horIII, lionsmagenes, grubb68, grubb74,
grubb77, schechter59, vishik]. Elliptic operators – their index theory and
spectral theory – is broadly used in mathematics connecting seemingly disjoint
areas of mathematics, for instance through noncommutative geometry and
geometric analysis. The spectral theory of elliptic boundary value problems
has primarily been focused on Laplace type operators (for an overview see
[ivrii16]) or Dirac type operators (see for instance [bruninglesch99,
bruninglesch01, bosswojc, Grubb03]), but has also been studied in larger
generality (e.g. in [geygru74, gerdsgreenbook, grubb74, grubb77a, grubb84]).
The index theory of elliptic boundary value problems is also a well studied
area. For instance, the index theory of Dirac operators with the spectral APS
boundary condition was computed by Atiyah-Patodi-Singer [APS], and has since
been well studied, see for instance [G99, grubb92, melroseAPS, bruninglesch99,
bosswojc]. The index theory for boundary value problems in the Boutet de
Monvel calculus was computed by [boutetdemonvel] relating to ideas of
Agranovich-Dynin. This formula covers the case of local elliptic boundary
conditions. Boutet de Monvel’s results have been extensively studied, see
[elmaretal, fedosovindex, fedoind, rempelschulze], and were generalised to an
extended Boutet de Monvel calculus by Schulze-Seiler [schulzeseiler].
In this paper we take a new approach to higher order elliptic operators on
manifolds with boundary. The approach is based on work of Bär-Ballmann [BB].
The paper [BB] consists of a coherent overview of first order boundary value
problems (admitting a self-adjoint adapted boundary operator) based on an
analytic description of the Cauchy data space as well as a novel graphical
decomposition of regular realisations of first order operators. The results
were later extended to general first order problems by the first listed author
and Bär in [BBan]. Although this approach was first fully utilised and made
explicit in [BB], the ideas have been around since the ’60s and was implicitly
contained in a paper by Seeley [seeley65, top of page 782]. They were visible
already in Agranovich-Dynin’s classical works on index theory for elliptic
boundary value problems of general order. Similar approaches predating Bär-
Ballmann’s have been studied in the abstract in [bossfury], been applied to
Dirac-Schrödinger operators in [ballbruncarr] as well as having been used in
the study of Maxwell’s equations on Lipschitz domains [HR2019, BuCoSh].
Related ideas can also be seen in for instance [APS, bruninglesch01, grubb68,
grubb77, grubb92, G99]. Since its appearance, the coherence of Bär-Ballmann’s
work [BB] has paved the way for numerous applications [MR4011805, MR4000837,
MR3981455, MR3908762, MR3850258].
The aim of this paper is to present a perspective similar to [BB], providing
an overview of the theory for general order elliptic boundary value problems
from the vantage point of the Cauchy data space. As such, the main novelty
lies in the framework and the methods. Indeed, several of the results in this
paper are, as stand alone results, at best mild generalisations of known
results and are unlikely to surprise experts in the field of boundary value
problems. A key feature in several of the results is that they can treat all
(or at least the regular extensions with the occasional restriction to the
pseudo-local case) closed extensions of an elliptic differential operator on
an equal footing with the special classes of boundary conditions studied
classically. We have attempted to the best of our abilities to indicate where
the results we are generalising can be found in the abundance of literature on
boundary value problems.
The approach we take lies close in spirit to the ideas of noncommutative
geometry [connesbook] whose methods have proven their worth in index theory,
relating to recent work on boundaries in noncommutative geometry [FGMR]. It
takes a more abstract perspective than the semiclassical methods ordinarily
deployed to study boundary value problems, e.g. in [grisvard, horIII,
lionsmagenes, grubb77, schechter59, vishik], and lies closer to methods for
order one elliptic operators seen in [G99, grubb92, bruninglesch99, bosswojc,
bossfury, BB, BBan] rather than the abstract methods of, for instance,
[grubb68, grubb74, malamud, behrndtetal]. The level of abstraction makes the
method feasible for further development of index theory and applications to a
larger class of problems than elliptic differential operators on manifolds
with boundary. We anticipate this includes elliptic operators on singular
manifolds, hypoelliptic operators (cf. the subelliptic boundary conditions in
[epsteinsub]) or geometric operators in Lorentzian geometry (cf. [MR4011805]).
These facts raises optimism for demystifying the general index formulas
appearing in the (extended) Boutet de Monvel calculus relying on abstract
homotopies [boutetdemonvel, rempelschulze], non-explicit inverses to
isomorphisms in $K$-theory [elmaretal] or Chern characters of operator valued
symbols [fedoind, fedosovindex].
The method of Bär-Ballmann for first order operators can be summarised as
describing boundary value problems by means of the Cauchy data space, i.e. the
range of the corresponding trace map onto a mixed Sobolev space on the
boundary. For higher order operators we apply the same ideas to the full trace
map (taking into account all boundary traces up to the order of the operator)
into a mixed Sobolev space on the boundary. The range of the full trace map –
the Cauchy data space – characterises the realisation defining the boundary
value problem when topologising it so that the full trace map is a quotient.
The analytic details involved in this procedure produces precise information
about regularity and Fredholm properties of the realisations in terms of
properties of the boundary condition relative to the range of the full trace
mapping on the maximal domain of the elliptic operator. The general theme in
this framework is to reduce properties of realisations, and their proofs, to
neat functional analysis arguments and Fredholm theory.
The description of the possible realisations of an elliptic operator as
subspaces of a Hilbert space of functions on the boundary relies on a precise
description of the Cauchy data space. Abstractly, the Cauchy data space is
isomorphic to the quotient of the maximal domain by the minimal domain. By
classical results of Lions-Magenes [lionsmagenes63, lionsmagenes] the Cauchy
data space is a concrete subspace of a mixed order Sobolev space on the
boundary. For an elliptic first order differential operator this is described
in detail in [BBan] using an adapted boundary operator. To describe the Cauchy
data space of a general order elliptic differential operator, we employ the
Calderón projection of Seeley [seeley65]. Although our setup is based on the
work of Bär-Ballmann, similar ideas date back much further as discussed above.
We emphasise that this paper, despite allowing for elliptic differential
operators of general order, by no means supersedes [BB, BBan]. A significant
novelty distinguishing [BBan] from this paper is that [BBan] links analysis on
the Cauchy data space to the $\mathrm{H}^{\infty}$-functional calculus, which
in [BBan] forms the key to obtaining the estimates required to topologise the
Cauchy data space. In contrast, in this paper we use Seeley’s work on Calderón
projectors to analyse the Cauchy data space.
### 1.1. Main results and overview of paper
Let $M$ be a compact manifold with boundary $\Sigma:=\partial M$. We fix a
smooth measure $\mu$ on $M$. In our convention, $\Sigma\subset M$ and the
interior of $M$ is denoted by $\mathring{M}=M\setminus\Sigma$. We also choose
a smooth interior pointing vectorfield $\vec{T}$ transversal to
$\Sigma=\partial M$. The smooth measure on $\Sigma$ induced by $\mu$ and
$\vec{T}$ will be denoted by $\nu$.
In this paper, we are mainly concerned with elliptic differential operators
$D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ acting between hermitian
vector bundles $(E,h^{E})\to M$ and $(F,h^{F})\to M$. The hermitian metric
will be implicit in the notations. We let $m$ denote the order of $D$. The
maximal realisation $D_{\rm max}$ of $D$ on ${\rm L}^{2}$ is given by
$\mathrm{dom}(D_{\max}):=\\{f\in{\rm L}^{2}(M;E):Df\in{\rm L}^{2}(M;F)\\}$
where $D$ acts in a distributional sense on ${\rm L}^{2}(M;E)$ as a
consequence of the presence of a unique formal adjoint $D^{\dagger}:{\rm
C}^{\infty}(M;F)\to{\rm C}^{\infty}(M;E)$. The minimal realisation $D_{\rm
min}$ is defined from the domain obtained from closing ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$ in the graph norm of $D$; it is readily seen that
$\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ (cf. Proposition 2.1,
on page 2.1). Many of the general results we prove only rely on abstract
properties of the minimal and maximal realisations using the machinery of
Appendix B (see page B). For $s\in\mathbb{R}$, we use the notation
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m}):=\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;E).$
As described in Subsection 3.1, the space
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$ is easiest thought of as a
graded Sobolev space with respect to a certain grading on
$E\otimes\mathbb{C}^{m}$.
A (closed) realisation of $D$ is a (closed) extension $D_{e}$ of $D_{\rm min}$
such that $D_{e}\subseteq D_{\rm max}$. In other words, a realisation of $D$
is an extension of $D$ acting on ${\rm H}^{\rm m}_{\rm 0}(M;E)$ to a domain on
${\rm L}^{2}(M;E)$ where it acts in the ordinary distributional sense.
The following theorem summarises how the Bär-Ballmann machinery applies in the
setting of higher order elliptic differential operators on manifolds with
boundary.
###### Theorem 1.
Let $D$ be an elliptic differential operator of order $m>0$ as in the
preceding paragraph.
1. (i)
The full trace mapping $\gamma:{\rm H}^{\rm
m}(M;E)\to{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
$u\mapsto(\partial_{x_{n}}^{j}u|_{\Sigma})_{j=0}^{m-1}$ extends to a
continuous trace mapping $\gamma:\mathrm{dom}(D_{\rm
max})\to{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ with
kernel $\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ and range
being the _Cauchy data space_
$\check{\mathrm{H}}(D)=P_{\mathcal{C}}{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\bigoplus(1-P_{\mathcal{C}}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
where $P_{\mathcal{C}}$ is a Calderón projection (for more details on
$P_{\mathcal{C}}$, see Section 3). Equipping $\check{\mathrm{H}}(D)$ with the
Hilbert space topology making the projectors onto
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
respectively, into partial isometries, the canonical isomorphism
$\check{\mathrm{H}}(D)\cong\mathrm{dom}(D_{\rm max})/\mathrm{dom}(D_{\rm
min})$ is a Banach space isomorphism.
2. (ii)
There is a one-to-one correspondence between (closed) realisations of $D$ and
(closed) subspaces $B\subseteq\check{\mathrm{H}}(D)$ as follows. A (closed)
subspace $B\subseteq\check{\mathrm{H}}(D)$ uniquely determines a (closed)
realisation $D_{\rm B}$ of $D$ defined by
$\mathrm{dom}(D_{\rm B})=\\{f\in\mathrm{dom}(D_{\rm max}):\gamma(f)\in B\\},$
and any (closed) realisation $\hat{D}$ uniquely determines a (closed) subspace
$B:=\gamma\mathrm{dom}(\hat{D})$ with $D_{\rm B}=\hat{D}$. Moreover,
$(D_{\rm B})^{*}=D^{\dagger}_{{\rm B}^{*}},$
where $B^{*}$ is the adjoint boundary condition (for more details see
Subsection 4.3, starting on page 4.3, and Appendix B.2, starting on page B.2).
3. (iii)
A realisation $D_{\rm B}$ of $D$ is semi-regular, that is we have the domain
inclusion
$\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E),$
if and only if
$B\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$. A
realisation $D_{\rm B}$ is regular (i.e. $D_{\rm B}$ and $(D_{\rm B})^{*}$ are
semi-regular) if and only if
$B\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and
$B^{*}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
where $B^{*}$ is the adjoint boundary condition.
The reader can find this theorem spread out in the bulk of the paper as
follows. Part i can be found in Theorem 3.18 (on page 3.18). Part ii is proven
in much larger generality in Appendix B, more precisely in Proposition B.5 on
page B.5. Part iii is proven in Proposition 2.5, see page 2.5. The precise
definition of regularity can be found in Definition 2.3 (page 2.3). In [BB,
BBan], regular boundary conditions are called elliptic, we discuss our choice
of terminology further in Remark 2.4 (page 2.4). We remark that Theorem 1
concerns local statements at the boundary and therefore111Using for instance
Lemma 4.1 on page 4.1 or Proposition 4.3 on page 4.3 extends to the case that
$M$ is noncompact with compact boundary when $D$ is complete.
Based on Theorem 1, a boundary condition for $D$ is defined to be a closed
subspace $B\subseteq\check{\mathrm{H}}(D)$ and we write $D_{\rm B}$ for the
associated realisation. We say that a boundary condition $B$ is (semi-)
regular if $D_{\rm B}$ is (semi-) regular.
Theorem 1 has direct consequences to the well-posedness of PDEs. If $D$ is an
elliptic differential operator of order $m>0$ with a semi-regular boundary
condition $B$ such that $\ker(D_{\rm B})=0$, then the partial differential
equation
$Du=f,\qquad\gamma(u)\in B$
is well-posed for $f\in\mathrm{ran}(D_{\rm B})=\ker(D_{{\rm
B}^{*}}^{\dagger})^{\perp}$. For more details, see Proposition 2.20 on page
2.20. In particular, if $D$ is a Dirac type operator (so $m=1$), the partial
differential equation
$Du=f,\qquad\gamma(u)=0$
is well-posed for $f\in\mathrm{ran}(D_{\rm min})$. See more in Corollary 2.22
on page 2.22.
The previous theorem has the following consequence on the spectral theory of
the maximal and minimal realisations of an elliptic operator. We include this
result because we have noticed some confusion surrounding it in the community.
The result is known to experts in the field (cf. the discussion on
[grubbdistop, page 60-61]).
###### Theorem 2.
Let $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;E)$ be an elliptic
differential operator of order $m>0$ acting between sections on a Hermitian
vector bundle $E\to M$ over a compact manifold with boundary with $\dim(M)>1$.
The spectrum of $D_{\rm max}$ on ${\rm L}^{2}(M;E)$ is
$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}.$
That is, the spectrum of $D_{\max}$ is purely discrete and each generalised
eigenspace is of infinite dimension. Moreover, the spectrum of $D_{\rm min}$
on ${\rm L}^{2}(M;E)$ is
$\mathrm{spec}(D_{\min})=\mathbb{C}.$
If $m=1$ and $D$ is a Dirac type operator then
$\mathrm{spec}(D_{\min})=\mathrm{spec}_{\mathrm{res}}(D_{\min})=\mathbb{C}$;
i.e., it consists solely of residual spectrum.
The reader can find the first part of this theorem on general order operators
as Proposition 3.12 (see page 3.12) in the bulk of the text. The statement
concerning $\mathrm{spec}(D_{\min})$ for Dirac-type operators can be found in
Corollary 3.14 (see page 3.14). The property distinguishing Dirac type
operators from general elliptic operators is the so-called _unique
continuation property_ which undermines the existence of eigenvalues for
$D_{\rm min}$. See the discussion in Remark 3.13 on page 3.13.
We now turn to some results on the index theory of realisations of elliptic
differential operators. We say that a boundary condition
$B\subseteq\check{\mathrm{H}}(D)$ is Fredholm if the associated realisation
$D_{\rm B}$ is Fredholm. We also introduce the notation
$\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ for the Hardy space – the
image of $\ker(D_{\rm max})$ under the full trace mapping. Note that
$\mathcal{C}_{D}=P_{\mathcal{C}}\check{\mathrm{H}}(D)=P_{\mathcal{C}}{\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
by Theorem 1.
###### Theorem 3.
Let $D$ be an elliptic differential operator of order $m>0$ as above. A
boundary condition $B\subseteq\check{\mathrm{H}}(D)$ is Fredholm if and only
if $(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$. In
this case, it holds that
$\operatorname{ind}(D_{\rm
B})=\operatorname{ind}(B,\mathcal{C}_{D})+\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger}).$
Moreover, $\operatorname{ind}(D_{\rm B})$ is a homotopy invariant of $(D,B)$
in the sense of Theorem 8.1 on page 8.1 (see also Theorem 8.5 on page 8.5).
The characterisation of Fredholm boundary conditions and the index formula is
proven by abstract principles and can be found in Theorem 2.12 (see page 2.12)
in the body of the text. Results similar to Theorem 3 for operators of order
$m=1$ can be found in [bossfury, bosswojc].
###### Theorem 4.
Let $D$ be an elliptic differential operator of order $m>0$. Assume that $P$
is a projection on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and set
$A:=P_{\mathcal{C}}-(1-P)$. Consider the following statements:
1. (1)
The operator $A$ is Fredholm on
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
2. (2)
The operator $P$ extends by continuity to $\check{\mathrm{H}}(D)$ and
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$, and $A$
defines a Fredholm operator on $\check{\mathrm{H}}(D)$.
3. (3)
The operator $P$ extends by continuity to
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and $A$
defines a Fredholm operator on
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
The following holds:
1. (i)
If 1 and 2 holds, then $P$ is _boundary decomposing_ (see Definition 3.26 on
page 3.26) and in particular
$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$
2. (ii)
If 1 and 3 holds, then the space
$B_{P}:=(1-P){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
defines a regular boundary condition for $D$.
Moreover, if $P$ is a zeroth order pseudodifferential projection in the
Douglis-Nirenberg calculus (see more details in Subsection 3.1), then the
following are equivalent:
1. a)
$P$ is Shapiro-Lopatinskii elliptic with respect to $D$ (in the sense of
Definition 4.16).
2. b)
The operator $P_{\mathcal{C}}-(1-P)\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ is elliptic in the Douglis-Nirenberg
calculus.
3. c)
The space
$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
is a regular boundary condition for $D$, so in particular, the realisation
$D_{\rm B}$ defined from
$\mathrm{dom}(D_{\rm B}):=\\{u\in\mathrm{dom}(D_{\rm max}):P\gamma
u=0\\}=\\{u\in{\rm H}^{\rm m}(M;E):P\gamma u=0\\},$
is regular.
4. d)
The space
$B_{P}:=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
is a Fredholm boundary condition for $D$, so in particular, the realisation
$D_{\rm B}$ defined from
$\mathrm{dom}(D_{\rm B}):=\\{u\in\mathrm{dom}(D_{\rm max}):P\gamma u=0\\},$
is a Fredholm operator.
If any of the equivalent conditions a)-d) holds, and moreover
$[P,P_{\mathcal{C}}]$ is order $-m$ in the Douglis-Nirenberg calculus, then
$P$ is also boundary decomposing.
The reader can find item i) stated as Theorem 3.30 (see page 3.30) and item
ii) stated as Theorem 4.14 (see page 4.14) in the body of the text. The
equivalence of item a) and b) is found in Proposition 4.18 (see page 4.18),
and the equivalence of item b), c) and d) is found in Theorem 4.15 (see page
4.15). The final conclusion of the theorem appears as Corollary 3.32 (see page
3.32).
###### Remark 5.
The reader should be wary of the fact that the structure of the Cauchy data
space $\check{\mathrm{H}}(D)$ is quite different from that of a Sobolev space.
We give three instances of how this manifests:
* •
If $P$ is a zeroth order pseudodifferential projection in the Douglis-
Nirenberg calculus and $P_{\mathcal{C}}-(1-P)$ is elliptic then
$B_{P}=(1-P){\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ is
a regular boundary condition but if $(1-P_{\mathcal{C}})PP_{\mathcal{C}}$ is
not of order $-m$ (in the Douglis-Nirenberg calculus) then $P$ fails to be
boundary decomposing and even fails to act boundedly on
$\check{\mathrm{H}}(D)$. This follows from Lemma 3.31.
* •
The obvious projectors on the Cauchy data space defining Dirichlet or Neumann
conditions for a Laplacian are not bounded operators on the Cauchy data space
by Example 3.34, but a more refined projection (projecting along the Hardy
space) produces a bounded projector as in Subsection 6.3. The
pseudodifferential operators in the Douglis-Nirenberg calculus that act
boundedly on the Cauchy data space are characterised in Lemma 3.31.
* •
We give an example in Section 9 (see page 9) of a formally self-adjoint first
order elliptic operator on the unit disc, with associated adapted boundary
operator $A$ such that the classical pseudodifferential operator
$P_{\mathcal{C}}-\chi^{+}(A)$ on the boundary (i.e. the unit circle) is of
order $-1$ but does not act compactly on $\check{\mathrm{H}}(D)$. Therefore,
the contrast between the approach in this paper to that in [BBan] (where
spectral projectors $\chi^{\pm}(A)$ topologise the Cauchy data space) goes
beyond compact perturbations even in the first order case. The image of
$\chi^{+}(A)$ differs substantially from the image of $P_{\mathcal{C}}$ – the
Hardy space – as seen in Proposition 9.12 containing an example where the two
images’ intersection is a finite-dimensional space of smooth functions.
Much of the work in [BBan] concerning regular boundary conditions relied on
graphical decompositions. The following theorem extends [BBan, Theorem 2.9] to
elliptic differential operators of any order $>0$. The theorem can be found as
Theorem 5.6 (see page 5.6) in the body of the text.
###### Theorem 6.
Let $D$ be an elliptic differential operator of order $m>0$, $\mathcal{P}_{+}$
a boundary decomposing projection (see Definition 3.26 on page 3.26), and $B$
a boundary condition for $D$. The following are equivalent:
1. (i)
$B\subset\check{\mathrm{H}}(D)$ is an regular boundary condition,
2. (ii)
$B$ is $\mathcal{P}_{+}$ graphically decomposable (see Definition 5.3 on page
5.3),
3. (iii)
$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in
${\mathbb{H}}^{{m-\frac{1}{2}}}$ (see Definition 5.4 on page 5.4),
4. (iv)
$B$ is $\mathcal{P}_{+}$ Fredholm decomposable in $\check{\mathrm{H}}$ (see
Definition 5.5 on page 5.5).
The ideas of Theorem 1 can also be applied to study higher boundary
regularity. Inspired by [BBan], we say that a boundary condition
$B\subseteq\check{\mathrm{H}}(D)$ is $s$-semiregular, for $s\geq 0$, if
whenever $\xi\in B$ satisfies that
$(1-P_{\mathcal{C}})\xi\in\bigoplus_{j=0}^{m-1}{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
then
$\xi\in{\mathbb{H}}^{{s+m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$. For
$s=0$, $0$-semiregularity is equivalent to semi-regularity. In [BBan], a
notion of $s$-semiregular boundary conditions were introduced to prove higher
boundary regularity for operators of order $m=1$ that implies our notion of
$s-{\frac{1}{2}}$-semiregularity for $s>{\frac{1}{2}}$. The following theorem
can be found as Theorem 6.6 (see page 6.6) in the body of the text.
###### Theorem 7.
Let $D$ be an elliptic operator of order $m>0$, $s\geq 0$ and $B$ a boundary
condition. Then $B$ is $s$-semiregular if and only if $D_{B}$ is
$s$-semiregular, i.e. that whenever $u\in\mathrm{dom}(D_{\rm B})$ satisfies
$D_{\rm B}u\in{\rm H}^{\rm s}(M;F)$ it holds that $u\in{\rm H}^{\rm
s+m}(M;E)$.
Theorem 7 generalises results from [BB, BBan] from first order to general
order. The method of proof can be localised to the boundary, so in the case
that $M$ is noncompact with compact boundary and $D$ is complete, it holds
that $B$ is $s$-semiregular if and only if whenever $u\in\mathrm{dom}(D_{\rm
B})$ satisfies that $D_{\rm B}u\in{\rm H}^{\rm s}_{\rm\rm loc}(M;F)$ it holds
that $u\in{\rm H}^{\rm s+m}_{\rm\rm loc}(M;E)$ (where we use the notation
${\rm H}^{\rm s}_{\rm\rm loc}$ from [BB, BBan]).
###### Theorem 8.
Let $D$ be a formally self-adjoint elliptic operator of order $m>0$ acting on
a vector bundle $E\to M$. Assume that $D$ has positive interior principal
symbol $\sigma_{D}$ and define
$c_{D}:=\left(\frac{1}{n(2\pi)^{n}}\int_{S^{*}M}\mathrm{Tr}_{E}(\sigma_{D}(x,\xi)^{-\frac{n}{m}})\mathrm{d}x\mathrm{d}\xi\right)^{-\frac{m}{n}},$
where $n$ is the dimension of $M$. Then any lower semi-bounded self-adjoint
realisation $D_{\rm B}$ with $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm
m}(M;E)$ is bounded from below with discrete real spectrum $\lambda_{0}(D_{\rm
B})\leq\lambda_{1}(D_{\rm B})\leq\lambda_{2}(D_{\rm B})\leq\cdots$ satisfying
that
$\lambda_{k}(D_{\rm B})=c_{D}k^{\frac{m}{n}}+o(k^{\frac{m}{n}}),\quad\mbox{as
$k\to+\infty$}.$
Theorem 8 appears as Theorem 7.4 (see page 7.4) in the body of the text. The
result might not be surprising as Weyl laws with error estimates are known
under mild assumptions, for a brief historical overview see Section 7. Our
assumptions are even milder, and we include the result as it showcases yet
another feature of regular realisations that only depend on abstract
principles. The proof of Theorem 8 was suggested to us by Gerd Grubb and
relies on a precise description of the resolvent of $D_{\rm B}$, see Theorem
7.1 on page 7.1, and previous work of Grubb [grubb68, grubb77a]. For
historical note, we mention that for Laplace operators on manifolds with
boundary (see for instance [ivrii16] for a historical overview) and for
elliptic pseudodifferential operators on closed manifolds (see for instance
[horIV, Chapter XXIX] or [shubinsbook, Chapter III]), the remainder terms in
the Weyl law are known more precisely.
### 1.2. Overview of contents
Let us briefly describe the contents of the paper.
In section 2 we set up the theory for elliptic differential operators,
building heavily on an abstract viewpoint presented in Appendix B. The
abstract theory allows us to set up boundary conditions (Subsection 2.2) as in
[BB, BBan] and characterise when they define Fredholm realisations. It allow
us to make some straightforward applications to wellposedness in Subsection
2.4. We also consider a number of examples: in Subsection 2.5 we describe some
classes of boundary conditions, in Subsection 2.6 we consider first order
elliptic operators reconciling with results obtained in [BBan]. In Subsection
2.7, we consider scalar properly elliptic operators reconciling with more
classical theory, e.g. [grubb68, schechter59, lionsmagenes].
In Section 3 we recall Seeley’s work on Calderón projectors and study its
implications for the Cauchy data space and its analytic structure. The
required Douglis-Nirenberg calculus and Hörmander’s description of the
symbolic structure of a Calderón projection is described in Subsection 3.1.
The Calderón projection from [seeley65] and further consequences for boundary
conditions are given in Subsection 3.2. In Subsection 3.3 we introduce the
notion of boundary decomposing projections, the projections that characterise
the analytic structure of the Cauchy data space.
In Section 4 we further analyse the Cauchy data space. The results of
Subsection 4.1 shows that the Cauchy data space can be locally defined and
that the Calderón projection could equally well be replaced with a projection
constructed from a finite number of terms in Hörmander’s construction in
Subsection 3.1. In Subsection 4.2 we give a precise description of the
boundary pairing between the Cauchy data space of an elliptic operator and the
Cauchy data space of its adjoint, allowing us to characterise adjoint boundary
conditions and regularity of pseudolocal boundary conditions in Subsection
4.3.
In Section 5 we characterise regular boundary conditions via graphical
decompositions, in Section 6 we characterise boundary conditions admitting
higher regularity, in Section 7 we prove the Weyl law for elliptic
differential operators with regular boundary conditions, and in Section 8 we
prove a rigidity result for the index of Fredholm realisations of elliptic
differential operators. Following this, Section 9 consists of a computational
exercise comparing a Calderón projection to the spectral projectors used in
[BBan], quantifying their qualitative differences.
### 1.3. Notation
Throughout the paper, we use the analyst’s inequality $a\lesssim b$ to mean
that $a\leq Cb$ for some constant $C$ where the dependence is apparent from
context or explicitly specified. By $a\simeq b$, we mean that $a\lesssim b$
and $b\lesssim a$.
The notation $+$ is used for internal sums of vector subspaces and $\oplus$ is
used for direct sums. For two subspaces $V_{1},V_{2}\subseteq V$ we use the
notation $V_{1}\oplus_{W}V_{2}=V_{1}+V_{2}$ to indicate $W:=V_{1}\cap V_{2}$,
note that $V_{1}\oplus_{W}V_{2}\cong(V_{1}\oplus V_{2})/d(W)$ for the diagonal
embedding $d$. In general, $\oplus$ does not imply that the sum is orthogonal.
We use the notation $\oplus^{\perp}$ for orthogonal direct sums. The term
projection will be used for an idempotent operator on a Hilbert space; in
general, they will not be orthogonal.
All manifolds and their boundaries are assumed to be smooth. For a manifold
$X$ without boundary, we write $\mathcal{D}^{\prime}(X)$ for the Fréchet space
of distributions, i.e. the topological dual of ${\rm C}^{\infty}_{\rm c}(X)$.
For a manifold with boundary $M$ we use the convention that the boundary is
included in $M$. We write $\Sigma:=\partial M$ and
$\mathring{M}:=M\setminus\Sigma$.
For an operator $T:\mathscr{H}\to\mathscr{H}$ over a Hilbert space
$\mathscr{H}$, potentially unbounded, we denote its _domain_ by
$\mathrm{dom}(T)$. The _kernel_ and _range_ are then given by $\ker(T)$ and
$\mathrm{ran}(T)$ respectively. The _graph norm_ of $T$ is
$\|\cdot\|_{T}=\sqrt{\|\cdot\|^{2}+\|T\cdot\|^{2}}$, and the operator $T$ is
said to be _closed_ if $(\mathrm{dom}(T),\|\cdot\|_{T})$ is a Banach space.
Since the graph norm polarises, $(\mathrm{dom}(T),\|\cdot\|_{T})$ is a Hilbert
space if $T$ is closed. If $S,T$ are two operators, then we write $S\subset T$
if $\mathrm{dom}(S)\subset\mathrm{dom}(T)$ and $S=T$ on $\mathrm{dom}(S)$. If
$S\subseteq T$ are two closed operators then
$\mathrm{dom}(S)\subset\mathrm{dom}(T)$ is a closed subspace with respect to
$\|\cdot\|_{T}$.
An operator $S$ is said to be _adjoint_ to $T$ if $\left\langle
Tu,v\right\rangle=\left\langle u,Sv\right\rangle$ for $u\in\mathrm{dom}(T)$
and $v\in\mathrm{dom}(S)$. A densely-defined $T$ has a _unique_ adjoint
$T^{\ast}$ with domain
$\mathrm{dom}(T^{\ast})=\left\\{u\in\mathscr{H}:\exists
C_{u,T}\quad|\left\langle u,Tv\right\rangle|\leq C_{u,T}\|v\|,\,\forall
v\in\mathrm{dom}(T)\right\\}.$
A closed operator $T$ is _Fredholm_ if it has closed range with finite
dimensional $\ker(T)$ and $\ker(T^{\ast})$.
The operator $T$ is said to be _invertible_ if it is injective with dense
range and $T^{-1}:\mathrm{ran}(T)\to\mathscr{H}$ is a bounded map. An
invertible $T$ has a unique extension $T^{-1}:\mathscr{H}\to\mathscr{H}$. The
_spectrum_ of the operator $T$ are the points $\lambda\in\mathbb{C}$ for which
$(\lambda-T)$ is not invertible. The set of all such points are denoted by
$\mathrm{spec}(T)$. There are a number of non-equivalent definitions of
_essential spectrum_ , but for us, this means the points
$\lambda\in\mathrm{spec}(T)$ for which $(\lambda-T)$ fails to be Fredholm. The
_resolvent set_ is the complement of $\mathrm{spec}(T)$ in $\mathbb{C}$ and it
is denoted by $\mathrm{res}(T)$.
A signification notion throughout this paper is that of a _Formally Adjointed
Pair (FAP)_ (cf. Appendix B). This notion was used, without being named, in
[grubb68, Chapter II]. Given Hilbert spaces $\mathscr{H}_{1}$ and
$\mathscr{H}_{2}$, these are a pair of densely-defined and closed operators
$(T_{\min},T_{\min}^{\dagger})$ with
$T_{\min}:\mathscr{H}_{1}\to\mathscr{H}_{2}$ and
$T_{\min}^{\dagger}:\mathscr{H}_{2}\to\mathscr{H}_{1}$ adjoint to each other
(i.e. $T_{\min}\subseteq(T_{\min}^{\dagger})^{*}$ and
$T_{\min}^{\dagger}\subseteq T_{\min}^{*}$). These yield the following
important maximal extensions
$T_{\max}:=(T_{\min}^{\dagger})^{\ast}\quad\mbox{and}\quad
T_{\max}^{\dagger}:=T_{\min}^{\ast}.$
A realisation of a FAP is an extension $T_{\min}\subseteq\hat{T}\subseteq
T_{\max}$. We will see FAPs arising from geometry and elliptic differential
operators.
It is of fundamental importance is to understand extensions, closed or
otherwise, of $T_{\min}$ (and respectively $T_{\min}^{\dagger}$). For this,
the space
$\check{\mathscr{H}}_{T}:=\faktor{\mathrm{dom}(T_{\max})}{\mathrm{dom}(T_{\min})}$
is of fundamental importance. A _Cauchy data space_ is a pair
$(\gamma,\check{\mathrm{H}}(T))$, where
$\gamma:\mathrm{dom}(T_{\max})\to\check{\mathrm{H}}(T)$ is a bounded
surjection with $\ker\gamma=\mathrm{dom}(T_{\min})$. It is clear from the open
mapping theorem that $\gamma$ induces an isomorphism
$\check{\mathscr{H}}(T)\cong\check{\mathrm{H}}(T)$.
### 1.4. Acknowledgements
LB was supported by SPP2026 from the German Research Foundation (DFG). MG was
supported by the Swedish Research Council Grant VR 2018-0350. HS was supported
by the Australian Research Council, through the Australian Laureate Fellowship
FL170100020 held by V. Mathai. The authors would also like to thank Christian
Bär and Andreas Rosén for useful discussions. The authors are most grateful to
Gerd Grubb, who motivated and inspired us through a multitude of comments and
helpful suggestions on earlier versions of the paper and providing us with the
method of proof used in Section 7.
## 2\. Elliptic differential operators and boundary conditions
### 2.1. Setup
Let $M$ be a compact manifold with boundary $\partial M=\Sigma$ carrying a
smooth measure $\mu$. In our convention, $\Sigma\subset M$ and the interior of
$M$ is denoted by $\mathring{M}=M\setminus\Sigma$.
Let $(E,h^{E})\to M$ be a hermitian vector bundle and let ${\rm
C}^{\infty}(M;E)$ denote the smooth sections over $E$. In particular, the
support of such sections are allowed to touch the boundary. The subspace ${\rm
C}^{\infty}_{\rm c}(\mathring{M};E)$ consists of $u\in{\rm C}^{\infty}(M;E)$
such that ${\rm spt}{\text{ }}u\cap\Sigma=\varnothing$. By an abuse of
notation, we write ${\rm C}^{\infty}(\Sigma;E)$ for the space of smooth
sections of $E|_{\Sigma}$, and similarly for other function spaces of
sections. We shall also fix a smooth interior vectorfield $\vec{T}$
transversal to $\partial M=\Sigma$ and let $\nu$ denote the smooth volume
measure on $\Sigma$, induced by $\mu$ and $\vec{T}$. In what is to follow, the
coordinate and derivative, respectively, in this transversal direction will be
denoted $x_{n}$ and $\partial_{x_{n}}$. For $s\in\mathbb{R}$, by ${\rm H}^{\rm
s}(M;E)$ and ${\rm H}^{\rm s}(\Sigma;E)$, we denote the ${\rm L}^{2}$ Sobolev
spaces of $s$ derivatives on $M$ and $\Sigma$, respectively. We let ${\rm
H}^{\rm s}_{\rm 0}(M;E)$ denote the closure of ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$ in ${\rm H}^{\rm s}(\hat{M};E)$ for some closed manifold
$\hat{M}$ containing $M$ as an open subset with smooth boundary. Note that
${\rm H}^{\rm s}(M;E)$ can be identified with the quotient $\faktor{{\rm
H}^{\rm s}(\hat{M};E)}{{\rm H}^{\rm s}_{\rm 0}(\hat{M}\setminus M;E)}$.
For $(F,h^{F})\to M$ another hermitian vector bundle, we will be concerned
with elliptic differential operators $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;F)$ of order $m>0$. Further, by $D^{\dagger}:{\rm
C}^{\infty}(M;F)\to{\rm C}^{\infty}(M;E)$, let us denote the formal adjoint,
obtained via integration by parts (cf. [BB]). By $D_{c}$, we denote $D$ with
domain $\mathrm{dom}(D_{c})={\rm C}^{\infty}_{\rm c}(\mathring{M};E)$, and
analogously $D^{\dagger}_{c}$ is defined from
$\mathrm{dom}(D_{c}^{\dagger})={\rm C}^{\infty}_{\rm c}(\mathring{M};F)$. The
maximal and minimal domains of $D$ are then obtained as follows:
$D_{\max}=(D^{\dagger}_{\mathrm{c}})^{\ast}\quad\text{and}\quad
D_{\min}=\overline{D_{c}}.$
It is clear that both these operators are closed and that $D_{\min}\subset
D_{\max}$. Moreover, it holds that $D_{\rm min}^{*}=D_{\rm max}^{\dagger}$ and
$D_{\rm max}^{*}=D_{\rm min}^{\dagger}$. In the language of Appendix B,
$(D_{\rm min},D_{\rm min}^{\dagger})$ is a formally adjointed pair (FAP), see
Definition B.1. The next result shows that the FAP associated with an elliptic
operator of order $m>0$ is a Fredholm FAP (see Definition B.14).
###### Proposition 2.1.
If $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ is an elliptic
differential operator of order $m>0$ then the following holds:
1. (1)
$\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M,E)$;
2. (2)
$D_{\rm min}$ has “compact resolvent” in the sense that $(1+D_{\rm
min}^{*}D_{\rm min})^{-{\frac{1}{2}}}$ is a compact operator on ${\rm
L}^{2}(M,E)$;
3. (3)
$D_{\rm min}$ has finite-dimensional kernel, closed range and $D_{\rm max}$
has closed range. Moreover,
${\rm
L}^{2}(M;E)=\ker(D_{\min})\oplus^{\perp}\mathrm{ran}(D_{\max}^{\dagger})\quad\mbox{and}\quad{\rm
L}^{2}(M;F)=\ker(D_{\min}^{\dagger})\oplus^{\perp}\mathrm{ran}(D_{\max}).$
###### Proof.
Item 1 follows from the Gårding inequality showing that the graph norm of $D$
is equivalent to the ${\rm H}^{\rm m}(M;E)$-norm on ${\rm C}^{\infty}_{\rm
c}(\mathring{M};E)$. Item 2 follows from the fact that $(1+D_{\rm
min}^{*}D_{\rm min})^{-{\frac{1}{2}}}$ factors over the domain inclusion
$\mathrm{dom}(D_{\rm min})\hookrightarrow{\rm L}^{2}(M,E)$ which is compact by
item 1 as $m>0$. Item 3 follows from item 2 by Proposition B.15. ∎
Using the transversal vectorfield $\vec{T}$ at the boundary, for
$j=0,1,2,\ldots$, we can define a trace mapping
$\gamma_{j}:{\rm C}^{\infty}(M,E)\to{\rm C}^{\infty}(\Sigma,E),\quad
u\mapsto\partial_{x_{n}}^{j}u|_{\Sigma},$
where $x_{n}$ denotes the variable transversal to the boundary defined from
the vector field $\vec{T}$.
###### Theorem 2.2.
Let $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ be an elliptic
differential operator of order $m>0$. The trace mapping
$\gamma:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m}),\quad
u\mapsto(\gamma_{j}u)_{j=0}^{m-1},$
extends to a continuous mapping
$\gamma:\mathrm{dom}(D_{\rm max})\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma;E),$
with dense range and
$\ker\gamma={\rm H}^{\rm m}_{\rm 0}(M;E).$
This result is classical and can be found in [lionsmagenes63, lionsmagenes].
The result is also proven in [seeley65].
Following the notation of Appendix B we use the notation
$\check{\mathscr{H}}_{D}:=\check{\mathscr{H}}_{(D_{\rm min},D_{\rm
min}^{\dagger})}\equiv\faktor{\mathrm{dom}(D_{\rm max})}{\mathrm{dom}(D_{\rm
min})}.$
We note that since $\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm
0}(M;E)=\ker\gamma$, setting
$\check{\mathrm{H}}(D):=\gamma\mathrm{dom}(D_{\max}),$
the map $\gamma$ induces a bounded inclusion
$\check{\mathscr{H}}_{D}\hookrightarrow\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E),$
with range $\check{\mathrm{H}}(D)$. The reader should be aware that this
inclusion is not a Banach space isomorphism onto its image. However, by giving
$\check{\mathrm{H}}(D)$ the induced topology from this inclusion, we obtain
that
$\check{\mathscr{H}}_{D}\cong\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$
Since $\check{\mathrm{H}}(D)$ is isomorphic to the Hilbert space quotient
$\faktor{\mathrm{dom}(D_{\rm max})}{\mathrm{dom}(D_{\rm min})}$, the Banach
space $\check{\mathrm{H}}(D)$ is in fact a Hilbert space (although we shall
not make use of a specific choice of inner product). We also note that since
${\rm H}^{\rm m}(M;E)\subseteq\mathrm{dom}(D_{\max})$ we have continuous
inclusions with dense ranges
$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma,E)\subseteq\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$
Similarly, we obtain $\check{\mathscr{H}}_{D^{\dagger}}$ and
$\check{\mathrm{H}}(D^{\dagger})$ and continuous inclusions with dense ranges
$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma,F)\subseteq\check{\mathrm{H}}(D^{\dagger})\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,F).$
### 2.2. Boundary conditions
Any elliptic operator $D:{\rm C}^{\infty}(M;E)\to{\rm C}^{\infty}(M;F)$ as
above defines the FAP $(D_{\rm min},D_{\rm min}^{\dagger})$ in the terminology
of Appendix B. Therefore, realisations of elliptic operators are readily
described from abstract principles as subspaces of the Cauchy data space
$\check{\mathrm{H}}(D)$. We reiterate that realisations and boundary
conditions of elliptic operators form, to say the least, a well understood
topic. Our ambition is solely to provide a new perspective consistent with the
perspective due to Bär-Ballmann in the first order case.
A _generalised boundary condition_ $B$ for $D$ is a subspace of
$\check{\mathrm{H}}(D)$. It is simply called a _boundary condition_ if $B$ is
further required to be closed. We note that by Proposition B.5, there is a
one-to-one correspondence between realisations of an elliptic operator and its
generalised boundary conditions. This correspondence is set up by associating
the realisation $D_{\rm B}$ defined from
$\mathrm{dom}(D_{\rm B}):=\\{f\in\mathrm{dom}(D_{\rm max}):\gamma(f)\in B\\},$
with $B$, and conversely to associate the subspace
$\gamma(\mathrm{dom}(D_{e}))\subseteq\check{\mathrm{H}}(D)$ with a realisation
$D_{e}$ of $D$. We also note that by Proposition B.7, for any generalised
boundary condition $B$ of $D$, it holds that
$(D_{\rm B})^{*}=D_{{\rm B}^{*}}^{\dagger},$
where $B^{\ast}\subset\check{\mathrm{H}}(D^{\dagger})$ is the _adjoint
boundary condition_ defined by
$B^{*}:=\\{\eta\in\check{\mathrm{H}}(D^{\dagger}):\langle
D^{\dagger}f,g\rangle=\langle f,Dg\rangle\;\quad\forall
f\in\gamma^{-1}(\\{\eta\\}),\,g\in\mathrm{dom}(D_{\rm B})\\}.$
In Subsection 4.3 (see page 4.3) we further describe the adjoint boundary
condition as an annihilator of $B$ with respect to an explicit continuous
sesquilinear form
$\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$,
compare also to Appendix B.2, on page B.2. Note, $B^{\ast}$ is necessarily
closed. For convenience, we shall simply refer to a realisation or boundary
condition for $D$ when speaking of a realisation or boundary condition for
$(D_{\rm min},D_{\rm min}^{\dagger})$. This terminology is consistent with the
terminology for boundary conditions for first order elliptic differential
operators from [BB, BBan].
###### Definition 2.3.
Let $D$ be an elliptic differential operator of order $m>0$ and $D_{\rm B}$ a
closed realisation. We say that that $D_{\rm B}$ is:
* •
_semi-regular_ if $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E)$,
* •
_regular_ if $D_{\rm B}$ and $(D_{\rm B})^{*}$ are semi-regular.
In light of Proposition B.5, we say that $B$ is (semi-) regular if $D_{\rm B}$
is (semi-) regular.
###### Remark 2.4.
The literature on boundary value problems contains a broad range of refined
notions for ellipticity and regularity. Historically, the term ellipticity has
been used for a local condition at the leading symbol level (including
possible boundary symbols) that implies regularity properties. For the
boundary conditions consider in this paper and in [BB, BBan], there is in
general no notion of boundary symbols. In [BB, BBan], the regularity property
$\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm m}(M;E)$ was therefore termed
semi-ellipticity. To better reflect the terminology in bulk of the literature
and to capture the fact that $\mathrm{dom}(D_{\rm B})\subseteq{\rm H}^{\rm
m}(M;E)$ is a regularity property, we use the term semi-regular boundary
condition in this paper.
###### Proposition 2.5.
Let $D$ be an elliptic differential operator of order $m>0$ and $B$ a boundary
condition. Then $D_{\rm B}$ is semi-regular if and only if
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$.
###### Proof.
If $D_{\rm B}$ is semi-regular, then $\gamma(\mathrm{dom}(D_{\rm
B}))\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$
by the trace theorem. We conclude that $B=\gamma(\mathrm{dom}(D_{\rm
B}))\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$.
If $B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$,
then any $x\in\mathrm{dom}(D_{\rm B})$ can be written as $x=x_{0}+x_{\rm B}$
where $x_{0}\in\mathrm{dom}(D_{\rm min})={\rm H}^{\rm m}_{\rm 0}(M;E)$ and
$x_{\rm B}\in\mathrm{dom}(D_{\rm max})$ is a pre-image of $\gamma(x)\in B$
under $\gamma$. By the trace theorem, we can take $x_{\rm B}\in{\rm H}^{\rm
m}(M;E)$ since $\gamma(x)\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;E)$. We conclude that $\mathrm{dom}(D_{\rm
B})\subseteq\mathrm{dom}(D_{\rm min})+{\rm H}^{\rm m}(M;E)={\rm H}^{\rm
m}(M;E)$. ∎
###### Proposition 2.6.
Let $D$ be an elliptic differential operator of order $m>0$ and $B$ a semi-
regular boundary condition. Then, $\ker(D_{\rm B})$ is finite dimensional and
$\mathrm{ran}(D_{\rm B})$ and $\mathrm{ran}(D_{\rm B}^{\ast})$ are closed.
Moreover,
${\rm L}^{2}(M;E)=\ker(D_{{\rm B}})\oplus^{\perp}\mathrm{ran}(D_{{\rm
B}}^{\ast})\quad\mbox{and}\quad{\rm L}^{2}(M;F)=\ker(D_{{\rm
B}}^{\ast})\oplus^{\perp}\mathrm{ran}(D_{{\rm B}}).$
In particular, if $B$ is regular then $D_{\rm B}$ is Fredholm.
###### Proof.
If $B$ is semi-regular, the pair $\mathcal{T}_{\rm B}:=(D_{\rm
B},D^{\dagger}_{\rm min})$ is a FAP (indeed $D_{\rm B}\subseteq D_{\rm
max}=(D^{\dagger}_{\rm min})^{*}$) and $D^{\dagger}_{\rm min}\subseteq D_{{\rm
B}^{*}}^{\dagger}=(D_{\rm B})^{*}$) with compact domain inclusion
$\mathrm{dom}(D_{\rm B})\hookrightarrow{\rm L}^{2}(M;E)$. The claimed result
now follows from Proposition B.15. ∎
###### Remark 2.7.
We return to the study of regular boundary conditions below in Subsection 4.3
and Subsection 5.
For an elliptic operator $D$, we use the notation
$\mathcal{C}_{D}:=\gamma\ker D_{\rm
max}\subseteq\check{\mathrm{H}}(D)\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E).$
The space $\mathcal{C}_{D}$ is called the Hardy space of $D$. We shall later
see that $\mathcal{C}_{D}\subseteq\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-{\frac{1}{2}}-j}(\Sigma,E)$ is, in fact, closed in
$\check{\mathrm{H}}(D)$. In the setting of Appendix B, more precisely
Definition B.11, the space $\mathcal{C}_{D}$ is denoted by
$\mathcal{C}_{(D_{\rm min},D_{\rm min}^{\dagger})}$.
###### Proposition 2.8.
Let $D$ be an elliptic differential operator of order $m>0$ and $B\subseteq
B^{\prime}$ be boundary conditions.
1. (i)
It holds that $D_{{\rm B}}\subseteq D_{{\rm B}^{\prime}}$ and there is a short
exact sequence of Hilbert spaces
$0\to\mathrm{dom}(D_{{\rm B}})\to\mathrm{dom}(D_{{\rm
B}^{\prime}})\to\faktor{B^{\prime}}{B}\to 0.$
In particular,
$\displaystyle\faktor{\mathrm{dom}(D_{\rm B})}{\mathrm{dom}(D_{\rm min})}\cong
B,\quad\text{and}\quad$ $\displaystyle\faktor{\mathrm{dom}(D_{\rm
max})}{\mathrm{dom}(D_{\rm B})}\cong\faktor{\check{\mathrm{H}}(D)}{B}$
hold for a boundary condition $B$.
2. (ii)
The inclusion and restriction mappings fit into a short exact sequence of
Hilbert spaces
$0\to\ker(D_{\rm min})\to\ker(D_{{\rm B}})\to\mathcal{C}_{D}\cap B\to 0.$
In particular, we have the short exact sequence
$0\to\ker(D_{\rm min})\to\ker(D_{\rm
max})\xrightarrow{\gamma}\mathcal{C}_{D}\to 0.$
###### Proof.
The result follows from the Propositions B.10 and B.12. ∎
###### Theorem 2.9.
Let $D$ be an elliptic differential operator of order $m>0$ and $B$ be a
generalised boundary condition. Then the following holds:
1. (i)
$\mathrm{ran}(D_{\rm B})=\mathrm{ran}(D_{{\rm B}+\mathcal{C}_{D}})$ and it is
closed if and only if $B+\mathcal{C}_{D}$ is a boundary condition (i.e. closed
in $\check{\mathrm{H}}(D)$).
2. (ii)
$\ker(D_{\rm B})$ is finite-dimensional if and only if $B\cap\mathcal{C}_{D}$
is finite-dimensional.
3. (iii)
$\mathrm{ran}(D_{\rm B})$ has finite algebraic codimension if and only if
$B+\mathcal{C}_{D}$ has finite algebraic codimension in
$\check{\mathrm{H}}(D)$ if and only if $\mathrm{ran}(D_{\rm B})$ is closed and
$\mathrm{ran}(D_{\rm B})^{\perp}$ is finite-dimensional.
###### Proof.
Item i follows from Theorem B.23, Proposition 2.1, and Lemma B.20. Item ii
follows from Proposition 2.8. Item iii follows from Proposition B.26. ∎
###### Corollary 2.10.
Let $D$ be an elliptic differential operator of order $m>0$ and
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma,E)$ be
closed in $\check{\mathrm{H}}(D)$. Then
$B+\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ is closed.
###### Proof.
The space $B$ defines a semi-regular boundary condition by Proposition 2.5. By
Proposition 2.6, $D_{\rm B}$ has closed range so
$B+\mathcal{C}_{D}\subseteq\check{\mathrm{H}}(D)$ is closed by Theorem 2.9. ∎
A significant consequence of Theorem 2.9 is that it allows us to relate the
closedness of the range of a realisation of a boundary condition to the
closedness of an associated boundary condition. It may seem that this
implicitly asserts that an operator needs to be closed in order to have closed
range. This is not the case, as highlighted by the following quintessential
example.
###### Example 2.11.
Consider the generalised boundary condition
$B_{\rm Sob}=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E),$
for an elliptic operator $D$ of order $m>0$. The space $B_{\rm Sob}$ is not a
closed subspace of $\check{\mathrm{H}}(D)$, so the associated operator
$D_{{\rm B}_{\rm Sob}}$ is not a closed operator. One can in fact show that
$\overline{D_{{\rm B}_{\rm Sob}}}=D_{{\rm B}_{\rm Sob}+\mathcal{C}}=D_{\max}$.
We can prove that $D_{{\rm B}_{\rm Sob}}$ has closed range and as such,
$B_{\rm Sob}+\mathcal{C}_{D}$ is closed in $\check{\mathrm{H}}(D)$. We return
to this example below in in Example 3.21 below and prove, amongst other
things, that $B_{\rm Sob}+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$.
Let us prove that $D_{{\rm B}_{\rm Sob}}$ has closed range. By a similar
argument as in Proposition 2.5, we see that
$\mathrm{dom}(D_{{\rm B}_{\rm Sob}})={\rm H}^{\rm m}(M;E).$
We can assume that $M$ is a domain with smooth boundary in a compact manifold
$\hat{M}$, that $E,F$ extend to Hermitian vector bundles
$\hat{E},\hat{F}\to\hat{M}$ and that there exists an elliptic differential
operator $\hat{D}$ of order $m$ extending $D$ to $\hat{M}$. Let
$\hat{T}\in\Psi^{-m}_{\rm cl}(\hat{M};\hat{F},\hat{E})$ denote a parametrix of
$\hat{D}$ such that $\hat{D}\hat{T}-1$ is a projection onto the finite-
dimensional space $\ker(\hat{D}^{\dagger})$ and $\hat{T}\hat{D}-1$ is a
projection onto the finite-dimensional space $\ker(\hat{D})$. We define the
continuous operator $T:=\hat{T}|:{\rm L}^{2}(M;F)\to{\rm H}^{\rm m}(M;E)$
which by construction inverts $D:{\rm H}^{\rm m}(M;E)\to{\rm L}^{2}(M;F)$
(viewed as a continuous operator) from the right up to smoothing operators.
From the compactness of $(1-DT)$ and Fredholm’s theorem, we conclude that
$\mathrm{ran}(D_{{\rm B}_{\rm Sob}})=\mathrm{ran}(D:{\rm H}^{\rm
m}(M;E)\to{\rm L}^{2}(M;F))\supseteq\mathrm{ran}(DT:{\rm L}^{2}(M;F)\to{\rm
L}^{2}(M;F)),$
has finite algebraic codimension and is closed in ${\rm L}^{2}(M;F)$.
By understanding closedness of the ranges of an operator, we are able to
understand which boundary conditions yield Fredholm operators. In applications
to index theory, it is essential to understand which boundary conditions yield
such realisations. This is described in the following theorem.
###### Theorem 2.12.
Let $D$ be an elliptic operator of order $m>0$ and $B$ be a boundary
condition. Then the following are equivalent:
1. (i)
$(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$;
2. (ii)
$D_{\rm B}$ is a Fredholm operator.
If either of these equivalent conditions hold, we have that
$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$
Moreover,
$\operatorname{ind}(D_{\rm
B})=\operatorname{ind}(B,\mathcal{C}_{D})+\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger}).$
###### Proof.
Follows from Theorem B.28. ∎
###### Remark 2.13.
A direct corollary of Theorem 2.12 is that whenever $B$ and $B^{\prime}$ are
two Fredholm boundary condition that are norm-continuously homotopic via
Fredholm boundary conditions222Norm-continuously homotopic in the sense that
$B$ and $B^{\prime}$ are images of norm-continuously homotopic projectors
where all projections along the homotopy define Fredholm boundary conditions.
then it holds that
$\operatorname{ind}(D_{\rm B})=\operatorname{ind}(D_{{\rm B}^{\prime}}).$
However, Theorem 8.1 below provides a more general result concerning homotopy
invariance so we omit the details of this argument.
###### Corollary 2.14.
Let $D$ be an elliptic operator of order $m>0$. Assume that
$B\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-{\frac{1}{2}}-j}(\Sigma;E)$ is
closed in $\check{\mathrm{H}}(D)$ and satisfies that
$B^{*}\subseteq\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;F)$. Then $(B,\mathcal{C}_{D})$ is a Fredholm pair
in $\check{\mathrm{H}}(D)$ and
$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$
###### Proof.
Follows from Theorem 2.12 by noting that our assumptions are equivalent to $B$
being regular which implies that it is Fredholm by Proposition 2.6. ∎
### 2.3. The boundary pairing
Consider $\check{\mathscr{H}}_{D}$ with the quotient map
$\mathrm{dom}(D_{\max})\to\faktor{\mathrm{dom}(D_{\max})}{\mathrm{dom}(D_{\min})}=\check{\mathscr{H}}_{D}$
as a Cauchy data space. Then, by abstract nonsense, the boundary pairing
$\omega_{D}:\check{\mathscr{H}}_{D^{\dagger}}\times\check{\mathscr{H}}_{D}\to\mathbb{C},$
defined for $f\in\mathrm{dom}(D^{\dagger}_{\rm max})$,
$g\in\mathrm{dom}(D_{\rm max})$ as
$\quad\omega_{D}([f],[g]):=\langle D^{\dagger}f,g\rangle_{{\rm
L}^{2}(M;E)}-\langle f,Dg\rangle_{{\rm L}^{2}(M;F)},$
is well-defined and continuous. By an abuse of notation, we consider the
boundary pairing as a sesquilinear mapping
$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$.
To describe $\omega_{D}$ on $\check{\mathrm{H}}(D)$ and
$\check{\mathrm{H}}(D^{\dagger})$ in greater detail, we introduce the
following $m\times m$-matrix
$\tau=\begin{pmatrix}0&0&\cdots&0&0&1\\\ 0&0&\cdots&0&1&0\\\
0&0&\cdots&1&0&\vdots\\\ \vdots&\vdots&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&0&\vdots&\\\ 0&1&\cdots&0&0&0\\\
1&0&\cdots&0&0&0\end{pmatrix}$ (1)
We introduce the notation $A_{l}=(A_{l}(x_{n}))_{x_{n}\in[0,1]}$ for the
family of differential operators ${\rm C}^{\infty}(\Sigma,E)\to{\rm
C}^{\infty}(\Sigma,F)$ of order $l$ such that
$D=\sum_{l=0}^{m}A_{l}D_{x_{n}}^{m-l},$
near $\Sigma$. The following proposition follows from the computations on
[seeley65, page 794]. See also [gerdsgreenbook, Proposition 1.3.2].
###### Proposition 2.15.
Let $D$ be an elliptic differential operator of order $m>0$. There exists an
$m\times m$ matrix of differential operators
$\widetilde{\scalebox{1.5}{a}}=\begin{pmatrix}A_{0}&0&\cdots&0&0&0\\\
A_{1,m-1}&A_{0}&\cdots&0&0&0\\\ A_{2,m-2}&A_{1,m-2}&\ddots&0&\vdots&\vdots\\\
\vdots&\vdots&\ddots&A_{0}&0&0\\\
A_{m-2,2}&A_{m-3,2}&\cdots&A_{1,2}&A_{0}&0\\\
A_{m-1,1}&A_{m-2,1}&\cdots&A_{2,1}&A_{1,1}&A_{0}\end{pmatrix},$
where $A_{j,k}$ is a differential operator of order $j$ with the same
principal symbol as $A_{j}|_{x_{n}=0}$, such that
$\scalebox{1.5}{a}:=-i\tau\widetilde{\scalebox{1.5}{a}}=-i\begin{pmatrix}A_{m-1,1}&A_{m-2,1}&\cdots&A_{2,1}&A_{1,1}&A_{0}\\\
A_{m-2,2}&A_{m-3,2}&\cdots&A_{1,2}&A_{0}&0\\\ A_{m-3,3}&\mathinner{\mkern
1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise
4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&&A_{0}&0&\vdots\\\
\vdots&&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern
2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern
1.0mu}&0&\vdots&\\\ A_{1,m-2}&A_{0}&\mathinner{\mkern 1.0mu\raise
1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern
2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&&&\\\
A_{0}&0&\cdots&0&0&0\end{pmatrix},$
satisfies
$\langle D^{\dagger}u,v\rangle_{{\rm L}^{2}(M;E)}-\langle u,Dv\rangle_{{\rm
L}^{2}(M,F)}=\langle\gamma u,\scalebox{1.5}{a}\gamma v\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},$
for all $u\in{\rm C}^{\infty}(M;F)$ and $v\in{\rm C}^{\infty}(M;E)$.
The fact that the order of the differential operators appearing in
$\widetilde{\scalebox{1.5}{a}}$ remains constant along diagonals in the matrix
shows that, in fact,
$\widetilde{\scalebox{1.5}{a}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m},F\otimes\mathbb{C}^{m})$ via the Douglis-
Nirenberg calculus we review in Subsection 3.1. The reader should note that we
abuse notation and identify $A_{0}$ with $A_{0}|_{x_{m}=0}$ which is of order
$0$, i.e. $A_{0}\in{\rm C}^{\infty}(\Sigma;\mathrm{Hom}(E,F))$. Since $A_{0}$
is the restriction of the principal symbol to $x_{n}=0$, $\xi^{\prime}=0$ and
$\xi_{n}=1$, ellipticity implies that $A_{0}:E|_{\Sigma}\to F|_{\Sigma}$ is a
vector bundle isomorphism.
###### Lemma 2.16.
Let $D$ be an elliptic differential operator of order $m>0$. Let
$\widetilde{\scalebox{1.5}{a}}$ and $\widetilde{\scalebox{1.5}{a}}_{\dagger}$
denote the matrices of differential operators constructed from $D$ and
$D^{\dagger}$, respectively, as in Proposition 2.15. Recalling that
$\scalebox{1.5}{a}=-i\tau\widetilde{\scalebox{1.5}{a}}$, it holds that
$\scalebox{1.5}{a}^{*}+\scalebox{1.5}{a}_{\dagger}=0.$
###### Proof.
The lemma follows from that a defines the boundary pairing as in Proposition
2.15 and the symmetry condition on the boundary pairing in Proposition B.4. ∎
###### Lemma 2.17.
The operators $\widetilde{\scalebox{1.5}{a}}$ (from Proposition 2.15) and
$\tau$ satisfy the following.
1. (i)
For any $s\in\mathbb{R}$, $\tau$ defines a unitary isomorphism
$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;F)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm m-1+s-j}(\Sigma;F)$
2. (ii)
The matrix of differential operators $\widetilde{\scalebox{1.5}{a}}$ is
invertible as a matrix of differential operators and
$\widetilde{\scalebox{1.5}{a}}^{-1}=\begin{pmatrix}A_{0}^{-1}&0&\cdots&0&0&0\\\
B_{1,m-1}&A_{0}^{-1}&\cdots&0&0&0\\\
B_{2,m-2}&B_{1,m-2}&\ddots&0&\vdots&\vdots\\\
\vdots&\vdots&\ddots&A_{0}^{-1}&0&0\\\
B_{m-2,2}&B_{m-3,2}&\cdots&B_{1,2}&A_{0}^{-1}&0\\\
B_{m-1,1}&B_{m-2,1}&\cdots&B_{2,1}&B_{1,1}&A_{0}^{-1}\end{pmatrix},$
where $B_{j,k}$ is a differential operator of order $j$.
3. (iii)
For any $s\in\mathbb{R}$, $\widetilde{\scalebox{1.5}{a}}$ defines a Banach
space isomorphism
$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s-j}(\Sigma;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;F),$
and $\scalebox{1.5}{a}=-i\tau\widetilde{\scalebox{1.5}{a}}$ defines a Banach
space isomorphism
$\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm m-1+s-j}(\Sigma;F),$
###### Proof.
Item i is immediate. Item ii follows from that
$A_{0}^{-1}\widetilde{\scalebox{1.5}{a}}$ is lower triangular with the
identity operator on the diagonal, and as such, the inverse of
$A_{0}^{-1}\widetilde{\scalebox{1.5}{a}}$ is computed from polynomial
operations on its entries through Gaussian elimination. The fact that the
differential operator $B_{j,k}$ is of order $j$ follows from a short
inspection of the Gaussian elimination. To prove item iii, we note that any
matrix $(C_{j,k})_{j,k=0}^{m-1}$ of differential operators with the order of
$C_{j,k}$ being $j-k$, acts continuously on $\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
s-j}(\Sigma;F)$. Now item iii follows from item ii. ∎
Following Appendix B, we use the notation
$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C},$
for the sesquilinear boundary pairing
$\omega_{D}(\eta,\xi):=\langle D^{\dagger}u,v\rangle_{{\rm
L}^{2}(M;E)}-\langle u,Dv\rangle_{{\rm L}^{2}(M,F)},$
for $u\in\gamma^{-1}(\\{\eta\\})$ and $v\in\gamma^{-1}(\\{\xi\\})$. This
pairing is well defined and non-degenerate by Proposition B.4. Below in
Theorem 4.12, we shall see that it is perfect. For now we settle with the
pairing being perfect on the Sobolev spaces.
###### Proposition 2.18.
The boundary pairing
$\omega_{D}:\check{\mathrm{H}}(D^{\dagger})\times\check{\mathrm{H}}(D)\to\mathbb{C}$
extends by continuity to a perfect pairing
$\omega_{D,s}:\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-
s-j}(\Sigma;F)\times\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-1+s-j}(\Sigma;E)\to\mathbb{C},$
for any $s\in\mathbb{R}$ with
$\omega_{D,s}(\eta,\xi)=\langle\eta,\scalebox{1.5}{a}\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})},$
for $\eta\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-s-j}(\Sigma;F)$ and
$\xi\in\bigoplus_{j=0}^{m-1}{\rm H}^{\rm m-1+s-j}(\Sigma;E)$.
###### Proof.
By Proposition 2.15,
$\omega_{D}([\xi],[\xi^{\prime}])=\langle\xi,\scalebox{1.5}{a}\xi^{\prime}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}$ for $\xi\in{\rm C}^{\infty}(\Sigma;F)$
and $\xi^{\prime}\in{\rm C}^{\infty}(\Sigma;E)$. The statement now follows
from Lemma 2.17. ∎
### 2.4. Well-posedness in the absence of kernels
In the situation of boundary conditions without kernels, we are able to
understand some properties of associated PDEs.
###### Proposition 2.19.
Let $D$ be an elliptic operator of order $m>0$. Suppose that $B$ is a semi-
regular boundary condition such that $\ker(D_{{\rm B}})=0$. Then, the spectrum
of $D_{\rm B}^{\ast}D_{{\rm B}}$ consists solely of discrete spectrum in
$(0,\infty)$ and letting $\lambda_{1}$ denote its smallest nonzero eigenvalue,
we have the Poincaré inequality
$\sqrt{\lambda_{1}}\|u\|_{{\rm L}^{2}(M;E)}\leq\|D_{{\rm B}}u\|_{{\rm
L}^{2}(M;F)}$
for all $u\in\mathrm{dom}(D_{{\rm B}})$.
Under an additional hypothesis on $B$, we give a precise description of the
spectral asymptotics of $D_{\rm B}^{\ast}D_{{\rm B}}$ in Corollary 7.7 below.
###### Proof.
The operator $D_{\rm B}^{\ast}D_{{\rm B}}$ satisfies $\mathrm{dom}(D_{\rm
B}^{\ast}D_{{\rm B}})\subset{\rm H}^{\rm m}(M;E)$, and moreover, it is easy to
see that it is non-negative self-adjoint. By the Rellich embedding theorem, we
obtain that $\mathrm{spec}(D_{\rm B}^{\ast}D_{{\rm B}})$ is discrete, real,
non-negative. Also, $\ker(D_{\rm B}^{\ast}D_{{\rm B}})=\ker(D_{{\rm B}})=0$
and therefore, $\mathrm{spec}(D_{\rm B}^{\ast}D_{{\rm
B}})=\left\\{0<\lambda_{1}\leq\lambda_{2}\leq\dots\right\\}$. Then, via
numerical range considerations,
$\left\langle D_{\rm B}^{\ast}D_{{\rm
B}}u,u\right\rangle\geq\lambda_{1}\|u\|^{2}.$
for $u\in\mathrm{dom}(D_{\rm B}^{\ast}D_{{\rm B}})$. However,
$\mathrm{dom}(D_{\rm B}^{\ast}D_{\rm B})$ is dense in
$\mathrm{dom}(\sqrt{D_{\rm B}^{\ast}D_{\rm B}})=\mathrm{dom}(D_{\rm B})$ and
therefore, we obtain the desired inequality. ∎
###### Proposition 2.20.
Let $D$ be an elliptic operator of order $m>0$. Suppose that $B$ is a semi-
regular boundary condition and that $\ker(D_{\rm B})=0$. Then,
$\begin{cases}D_{\rm B}u=f\\\ \gamma(u)\in B\end{cases}$
for $f\in\mathrm{ran}(D_{{\rm B}})$ is well-posed.
###### Proof.
By Proposition 2.6, we have that $D_{\rm B}$ and $D_{\rm B}^{\ast}$ has closed
range. But ${\rm L}^{2}(M;E)=\ker(D_{\rm B})\oplus\mathrm{ran}(D_{\rm
B}^{\ast})=\mathrm{ran}(D_{\rm B}^{\ast})$, and so on applying [Yosida,
Corollary 1, Chapter VII, Section 5], we obtain that
$D_{{\rm B}}:\mathrm{dom}(D_{\rm B})\subset{\rm
L}^{2}(\Sigma;E)\to\mathrm{ran}(D_{\rm B})$
has a bounded inverse. Therefore,
$\|u\|_{D_{\rm B}}\simeq\|u\|+\|Du\|=\|{D_{{\rm
B}}^{-1}f}\|+\|f\|\lesssim\|f\|.$
This is exactly that the problem in the statement of the corollary is well-
posed. ∎
We apply this to understand the well-posedness of the Dirichlet problem. An
important and well known notion is _weak UCP (unique continuation property)_
for an operator $D$. This means that for smooth $u$, when $Du=0$ and $u=0$ on
a nonempty open subset $\Omega\subset M$, then $u=0$. Similarly, _weak inner
UCP_ is satisfied by $D$ if for smooth $u$, when $Du=0$ and
$u{{\lvert}}_{\Sigma}=0$, this implies $u=0$.
###### Corollary 2.21.
Let $D$ be an elliptic differential operator satisfying weak inner UCP. Then,
$\begin{cases}Du=f\\\ \gamma(u)=0\end{cases}$
is well-posed on $\mathrm{ran}(D_{\mathcal{C}_{D}})=\mathrm{ran}(D_{\min})$.
###### Proof.
We obtain well-posedness on $\mathrm{ran}(D_{\min})$ from Proposition 2.20
with the semi-regular boundary condition $B=0$ upon showing that
$\ker(D_{\min})=0$. By elliptic regularity, $\ker(D_{\min})\subset{\rm
C}^{\infty}(M;E)$ so the weak inner UCP property yields that
$\ker(D_{\min})=0$. Lastly, we have that
$\mathrm{ran}(D_{\min})=\mathrm{ran}(D_{0+\mathcal{C}_{D}})=\mathrm{ran}(D_{\mathcal{C}_{D}})$
from Theorem 2.9 item i. ∎
Now, let us focus on the first order case. In the first order case, $D_{\rm
min}$ corresponds to the Dirichlet problem for $D$. As discussed in [BBL2009,
Section 1.2], weak UCP implies weak inner UCP for elliptic first order
differential operators. However, focusing on Dirac-type operators, the
following holds.
###### Corollary 2.22.
Let $D$ be first order and Dirac-type. Then,
$\begin{cases}Du=f\\\ \gamma(u)=0\end{cases}$
is well-posed on $\mathrm{ran}(D_{\mathcal{C}_{D}})=\mathrm{ran}(D_{\min})$.
###### Proof.
As mentioned in [BBL2009, Remark 2.2], it is well known that all first order
Dirac-type operators satisfy weak UCP. This, in turn, implies weak inner UCP,
and therefore, the conclusion follows. ∎
###### Remark 2.23.
It is a well known fact that well-posedness fails for the Dirichlet problem
for first order Dirac-type operators on ${\rm L}^{2}$. However, the corollary
shows that on a very large space, well-posedness actually holds. In fact, by
Proposition 2.6, it is easily seen that the obstruction to well-posedness is
indeed large. It is precisely $\ker(D^{\dagger}_{max})=\ker(D_{\min}^{*})$. On
a manifold of dimension exceeding 1, we see from Theorem 3.11 that this is,
indeed, an infinite dimensional space. That is, in general, the obstruction to
well-posedness for the Dirichlet problem is infinite dimensional.
### 2.5. Local, pseudo-local and bundle-like boundary conditions
Let us give some examples of boundary conditions.
###### Definition 2.24.
Let $D$ be an elliptic operator of order $m>0$ and $B$ a boundary condition.
* •
We say that $B$ is pseudo-local if there exists an $m\times m$ matrix $P$ of
pseudo-differential operators on $E|_{\Sigma}$ with $P^{2}=P$ and
$B=\check{\mathrm{H}}(D)\cap\ker(P:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})).$
We say that the pseudo-local boundary condition $B$ is defined from $P$, and
to indicate the dependence on $P$ we sometimes use the notation $B_{P}$ for
$B$.
* •
We say that $B$ is local if there exists a differential operator $b$ on
$E|_{\Sigma}\otimes\mathbb{C}^{m}\to E^{\prime}$ (for some vector bundle
$E^{\prime}\to\Sigma$) such that
$B=\check{\mathrm{H}}(D)\cap\ker(b:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E^{\prime})).$
We say that the local boundary condition $B$ is defined from $b$.
* •
If $B$ is a local boundary condition defined from $b$ on
$E|_{\Sigma}\otimes\mathbb{C}^{m}\to E^{\prime}$ where $E^{\prime}\subseteq
E|_{\Sigma}\otimes\mathbb{C}^{m}$ is a subbundle and $b$ is a projection onto
a complement of $E^{\prime}$, we say that the associated boundary condition is
bundle-like and defined from $E^{\prime}$.
In [BBan], the bundle-like boundary conditions were called local. However, in
the higher order case more general local boundary conditions naturally arise
and are well studied, e.g. in [lionsmagenes, grubb68, vishik].
The reader should note that a bundle-like boundary condition is both local and
pseudo-local. If $B$ is a local boundary condition defined from $b$, then
$\mathrm{dom}(D_{\rm B})=\\{u\in\mathrm{dom}(D_{\max}):b\gamma u=0\\}.$
Here $b\gamma u$ is interpreted in a distributional sense. In particular, $B$
is indeed a boundary condition because the topology on $\check{\mathrm{H}}(D)$
induced from the space of distributions is weaker than the norm topology of
$\check{\mathrm{H}}(D)$. A similar argument shows that a pseudo-local boundary
condition indeed is a boundary condition.
###### Example 2.25.
For $0\leq k\leq m$, we consider the order zero operator $b$ that projects
onto the first $k$ coordinates of $E|_{\Sigma}\otimes\mathbb{C}^{m}$. Let
$B_{k}$ be the associated bundle-like boundary condition. We have that
$\displaystyle B_{k}=$
$\displaystyle\\{\xi\in\check{\mathrm{H}}(D):\xi_{0}=\xi_{1}=\ldots=\xi_{k-1}=0\\},\quad\mbox{and}$
$\displaystyle\mathrm{dom}(D_{{\rm B}_{k}})=$
$\displaystyle\\{u\in\mathrm{dom}(D_{\rm
max}):\gamma_{0}u=\gamma_{1}u=\ldots\gamma_{k-1}u=0\\}=\mathrm{dom}(D_{\rm
max})\cap{\rm H}^{\rm k}_{\rm 0}(M).$
###### Proposition 2.26.
Let $D$ be an elliptic operator of order $m>0$ and $B_{P}$ a pseudo-local
boundary condition defined from $P$. Then $(B_{P})^{*}$ is the pseudo-local
boundary condition $B_{P_{\dagger}}$ for $D^{\dagger}$ defined from
$P_{\dagger}:=(\scalebox{1.5}{a}^{*})^{-1}(1-P^{*})\scalebox{1.5}{a}^{*}$
where a is the differential operator from Proposition 2.15.
###### Proof.
We remark that $(\scalebox{1.5}{a}^{*})^{-1}(1-P^{*})\scalebox{1.5}{a}^{*}$ is
again an idempotent matrix of pseudodifferential operators because
$(\scalebox{1.5}{a}^{*})^{-1}$ is a matrix of differential operators by Lemma
2.17. The space $B^{*}$ is characterised by the property that $\eta\in B^{*}$
if and only if $\omega_{D}(\eta,\xi)=0$ for all $\xi\in B$. We need to prove
that
$B^{*}=\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})).$
Since
$B=\check{\mathrm{H}}(D)\cap\ker(P:\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))$,
any $\xi\in B$ satisfies $\xi=(1-P)\xi$. In particular, if
$\eta\in\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))$
and $\xi\in B$ then
$\omega_{D}(\eta,\xi)=\langle\eta,\scalebox{1.5}{a}(1-P)\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=0.$
We conclude that
$\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))\subseteq
B^{*}$.
To prove the converse inclusion, if $\eta\in B^{*}$ then for any
$\xi_{0}\in{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$ we have that
$\omega_{D}(\eta,(1-P)\xi_{0})=\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi_{0}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}.$
Since $\xi:=(1-P)\xi_{0}\in B$, we have that
$\langle(1-P^{*})\scalebox{1.5}{a}^{*}\eta,\xi_{0}\rangle_{{\rm
L}^{2}(\Sigma;F\otimes\mathbb{C}^{m})}=0$ for any $\xi_{0}\in{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$. In particular,
$(1-P^{*})\scalebox{1.5}{a}^{*}\eta=0$ in distributional sense if $\eta\in
B^{*}$. Therefore
$\check{\mathrm{H}}(D^{\dagger})\cap\ker((1-P^{*})\scalebox{1.5}{a}^{*}:\mathcal{D}^{\prime}(\Sigma;F\otimes\mathbb{C}^{m})\to\mathcal{D}^{\prime}(\Sigma;E\otimes\mathbb{C}^{m}))\supseteq
B^{*}$.
∎
###### Example 2.27.
Returning to Example 2.25, assume for simplicity that $E=F$ and that the
elliptic operator $D$ takes the form $D=D_{x_{n}}^{m}+A$ near the boundary for
a differential operator $A$ on $\Sigma$ (with no dependence on $x_{n}$). A
short computation with integration by parts show that
$\scalebox{1.5}{a}=0^{*}\sigma(A)\otimes\tau$ where $0:\Sigma\to T^{*}\Sigma$
denotes the zero section and $\tau$ is as in Equation (1). Using this it is
readily verified via Proposition 2.26 that $B_{k}^{*}=B_{m-k-1}$ for
$k=0,\ldots,m$ with the convention that $B_{-1}=0$. If the order of $D$ is
even, then $B_{m/2}$ should be interpreted as a Dirichlet condition. We return
to this condition below in Subsection 2.7 for scalar properly elliptic
operators.
### 2.6. Elliptic first order operators
The situation on which we have modelled our description of boundary conditions
is that of first order operators described in [BB, BBan]. For simplicity, we
continue to focus on compact manifolds with boundary, while [BB, BBan] allowed
for non-compact manifolds with compact boundary. The theory of boundary value
problems for first order elliptic operators is further described in [G99] and
the special case of Dirac operators is studied in [bosswojc].
We follow the setup of Subsection 2.1 and, throughout this subsection,
consider an elliptic differential operator $D$ order $m=1$. Associated to such
an operator $D$, there are adapted boundary operators $A$ on the bundle
$E|_{\Sigma}$ which are first order elliptic differential operators on
$\Sigma$ whose principal symbols satisfy
$\upsigma(A)(x,\xi^{\prime})=\upsigma(D)(x,\xi^{\prime}=0,\xi_{n}=1)^{-1}\,\circ\,\upsigma(D)(x,\xi^{\prime}).$
Alternatively, an adapted boundary operator $A$ is an operator on $\Sigma$
such that near $\Sigma$ we can write
$D=\sigma\cdot(\partial_{x_{n}}+A+R_{0}),$
for an $x_{n}$-dependent first order differential operator $R_{0}$ on $\Sigma$
such that $R_{0}|_{x_{n}=0}$ is of order zero. Here we have shortened the
notation $\sigma=\upsigma(D)(x,\xi^{\prime}=0,\xi_{n}=1)$. The reader should
note that in the notation of Proposition 2.15,
$\sigma=A_{0}=\scalebox{1.5}{a}\quad\mbox{and}\quad
A_{1}=\sigma\cdot(A+R_{0}).$
In [BBan] by Bär and Bandara, the authors show that there exists an
_admissible cut_ , which is an $r\in\mathbb{R}$ so that $A_{r}:=A-r$ is
invertible $\omega_{r}$-bisectorial. This means that $\mathrm{spec}(A_{r})$ is
contained in a closed bisector
$S_{\omega_{r}}=\left\\{\zeta\in\mathbb{C}:\pm\arg\zeta\leq\omega_{r}\right\\}$
of angle $\omega_{r}<\pi/2$ in the complex plane, and that for
$\mu\in(\omega_{r},\pi/2)$, there exists a constant $C_{\mu}>0$ so that
whenever $\zeta$ is outside of the closed bisector $S_{\mu}$ of angle $\mu$,
we have that $|\zeta|\|(\zeta-A_{r})^{-1}\|\leq C_{\mu}$. The framework in
[BBan] extends the work of Bär and Ballmann in [BB] where they make the
additional assumption that $A$ can be chosen self-adjoint.
Given that the spectrum of $A_{r}$ is contained in the open left and right
half-planes, a consequence of the bisectoriality of $A_{r}$ is that we can
consider spectral projectors $\chi^{\pm}(A_{r})$, where $\chi^{\pm}(\zeta)=1$
for $\pm\mathrm{Re}\ \zeta>0$ and $0$ otherwise. Not only do these spectral
projectors exist, they are pseudo-differential operators of order zero (see
[grubbsec]). This means that they act boundedly across all Sobolev scales and
we define the space
$\check{\mathrm{H}}_{A}(D):=\chi^{-}(A_{r}){\rm
H}^{\rm\frac{1}{2}}(\Sigma;E)\oplus\chi^{+}(A_{r}){\rm
H}^{\rm-\frac{1}{2}}(\Sigma;E).$
The following is an important property of this space that we will use later.
We refer its proof to Appendix A.
###### Lemma 2.28.
Let $A$ be an adapted boundary operator for a first order elliptic operator
acting on a manifold with boundary of dimension $>1$. For any admissible cut
$r\in\mathbb{R}$, the spaces $\chi^{\pm}(A_{r}){\rm L}^{2}(\Sigma;E)$ are
infinite dimensional. Moreover, the space
$\check{\mathrm{H}}_{A}(D)\neq\left\\{0\right\\}$ is an infinite-dimensional
space with
${\rm
H}^{\rm{\frac{1}{2}}}(\Sigma;E)\subseteq\check{\mathrm{H}}_{A}(D)\subseteq{\rm
H}^{\rm-{\frac{1}{2}}}(\Sigma;E),$
and each inclusion is dense.
In [BBan, Theorem 2.3], the trace map $\gamma:u\mapsto u{{\lvert}}_{\Sigma}$,
initially defined on ${\rm C}^{\infty}(\Sigma;E)$, is extended to a bounded
surjection
$u\mapsto
u{{\lvert}}_{\Sigma}:\mathrm{dom}(D_{\max})\to\check{\mathrm{H}}_{A}(D)$
with kernel $\ker(u\mapsto u{{\lvert}}_{\Sigma})=\mathrm{dom}(D_{\min})$. That
is, $\check{\mathrm{H}}(D)=\check{\mathrm{H}}_{A}(D)$ with $\gamma$ is a
Cauchy data space. The topology of $\check{\mathrm{H}}(D)$ is given purely in
terms of an associated differential operator on the boundary, namely, the
adapted boundary operator $A$. We give a direct argument for the equality
$\check{\mathrm{H}}(D)=\check{\mathrm{H}}_{A}(D)$ below in Example 3.33.
A pseudo-differential projection $P$ of order zero, for which
$1-P-\chi^{+}(A_{r})$ is a Fredholm operator induces a regular boundary
condition via $B_{P}=(1-P){\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$. In particular,
for any admissible cut $r\in\mathbb{R}$, $P_{r}=\chi^{+}(A_{r})$ is such a
projection. That is, $B_{r}=\chi^{-}(A_{r}){\rm H}^{\rm\frac{1}{2}}(\Sigma;E)$
is a regular boundary condition. In particular, the (generalised) APS-
realisation $D_{\rm APS}$, defined from
$\mathrm{dom}(D_{\rm APS}):=\\{u\in H^{1}(M;E):\chi^{+}(A_{r})\gamma(u)=0\\},$
is regular. For notational convenience, we assume $r=0$ and drop it from the
notation for the remainder of the paper. The reader can find further examples
of boundary conditions for first order elliptic operators in [BB, BBan,
leschgor] and [G99, Section 4].
### 2.7. Scalar properly elliptic operators
Let us consider an example that historically has played an important role and
has long been well understood, see for instance [lionsmagenes, grubb68,
vishik]. We consider an elliptic differential operator $D$ of order $2m$
acting between sections of $E$ and $F$. We say that the differential operator
is scalar if $E$ and $F$ are line bundles; the terminology is justified by the
fact that $\mathrm{Hom}(E,F)$ is a line bundle (trivialisable at $\Sigma$ due
to the existence of $A_{0}\in C^{\infty}(\Sigma;\mathrm{Aut}(E,F))$), and as
such the symbol calculus is completely scalar. We recall the notion of a
properly elliptic scalar operator, see [schechter59].
###### Definition 2.29.
Let $D$ be a scalar elliptic differential operator of order $2m$. We say that
$D$ is properly elliptic if for each $(x^{\prime},\xi^{\prime})\in
S^{*}\Sigma$, the polynomial equation
$\sigma(D)|_{\Sigma}(x^{\prime},\xi^{\prime},z)=0,$ (2)
has exactly $m$ complex solutions with positive imaginary part.
###### Remark 2.30.
Let us consider some special cases where scalar elliptic differential
operators are automatically properly elliptic. If $\sigma(D)|_{\Sigma}$ is
real-valued, then $z$ is a solution of Equation (2) if and only if $\bar{z}$
is, so real valued principal symbol ensures that a scalar elliptic
differential operator is properly elliptic.
If $\dim(M)>2$, the fibres of the cosphere bundle of $\Sigma$ are connected.
Since $D$ is elliptic, Equation (2) has no real solutions and as such the
number of solutions with positive imaginary part is locally constant. We note
that if $z$ is a solution to Equation (2) at $(x^{\prime},\xi^{\prime})$, then
$-z$ is a solution to Equation (2) at $(x^{\prime},-\xi^{\prime})$ because of
the symmetry condition
$\sigma(D)|_{\Sigma}(x^{\prime},\xi^{\prime},z)=\sigma(D)|_{\Sigma}(x^{\prime},-\xi^{\prime},-z)$.
This proves that all scalar elliptic differential operators are properly
elliptic if $\dim(M)>2$.
For a scalar properly elliptic differential operator $D$ of order $2m$, we
define the bundle-like boundary condition $B_{\rm Dir}$ from the subbundle
$E^{\prime}:=E\otimes\mathbb{C}^{m}$, viewed as a subbundle of
$E\otimes\mathbb{C}^{2m}$ by embedding it in the last $m$ coordinates. Compare
to Example 2.25. We write $D_{\rm Dir}:=D_{{\rm B}_{\rm Dir}}$ and call it the
Dirichlet realisation of $D$. More explicitly, we have that
$\mathrm{dom}(D_{\rm Dir})=\\{u\in\mathrm{dom}(D_{\rm
max}):\gamma_{0}u=\gamma_{1}u=\cdots=\gamma_{m-1}u=0\\}.$
The following theorem reformulates several results from [grubb68] to our
setting, the reader is referred to [grubb68] for proofs.
###### Theorem 2.31.
Let $D$ be a scalar properly elliptic differential operator of order $2m$.
1. (i)
$B_{\rm Dir}$ is a regular boundary condition, and so the realisation $D_{\rm
Dir}$ of $D$ with domain
$\displaystyle\mathrm{dom}(D_{\rm Dir})=$ $\displaystyle{\rm H}^{\rm
2m}(M;E)\cap{\rm H}^{\rm m}_{\rm 0}(M;E)=$ $\displaystyle=$
$\displaystyle\\{u\in{\rm H}^{\rm
2m}(M;E):\gamma_{0}u=\gamma_{1}u=\cdots=\gamma_{m-1}u=0\\},$
is regular.
2. (ii)
Also $D^{\dagger}$ is properly elliptic and
$D_{\rm Dir}^{*}=D^{\dagger}_{\rm Dir},$
is the realisation of $D^{\dagger}$ with domain ${\rm H}^{\rm 2m}(M;F)\cap{\rm
H}^{\rm m}_{\rm 0}(M;F)$.
3. (iii)
The map
$\mathcal{C}_{D}\to\bigoplus_{j=0}^{m-1}{\rm H}^{\rm-j-{\frac{1}{2}}}(M;E),$
induced from the projection $\bigoplus_{j=0}^{2m-1}{\rm
H}^{\rm-j-{\frac{1}{2}}}(M;E)\to\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm-j-{\frac{1}{2}}}(M;E)$, is a Fredholm operator.
4. (iv)
We can write
$\mathrm{dom}(D_{\rm max})={\rm H}^{\rm 2m}(M;E)\cap{\rm H}^{\rm m}_{\rm
0}(M;E)\oplus_{\ker D_{\rm Dir}}\ker(D_{\rm max}),$
and
$\check{\mathrm{H}}(D)=\bigoplus_{j=m}^{2m-1}{\rm H}^{\rm
2m-j-{\frac{1}{2}}}(M;E)\oplus_{\gamma\ker(D_{\rm Dir})}\mathcal{C}_{D}.$
The reader should note that the theorem above holds for more general local
(regular) boundary conditions than Dirichlet conditions, and are studied using
different methods in [grubb68]. We remark that from these type of results,
several Fredholm type results follow readily for properly elliptic boundary
value problems with regular boundary conditions.
## 3\. Calderón projectors and the Douglis-Nirenberg calculus
In this section we will review results of Hörmander and Seeley concerning
Calderón projectors. See more in [horIII, Chapter XX], [seeley65] and also in
[grubbdistop, Chapter 11]. Calderón projections are projections on function
spaces on the boundary $\Sigma$ onto the space of boundary values of
homogeneous solutions $Du=0$ in the interior, i.e. the Hardy space
$\mathcal{C}_{D}$. For higher order operators, it is necessary to keep track
of different order traces, so even though the Calderón projection is a matrix
of pseudodifferential operators, its entries will have different order.
However, the orders remain constant along the diagonals in the Calderón
projection which makes it amenable for study via the calculus of Douglis-
Nirenberg.
### 3.1. Approximate Calderón projectors
First, we will in this subsection recall some constructions found in
Hörmander’s book [horIII]. It will provide a way of constructing the full
symbol of the Calderón projection.
As above, we write $x_{n}$ for the coordinate near the boundary defined from
the transversal vector field $\vec{T}$. The coordinate $x_{n}$ identifies a
neighbourhood of the boundary with a collar $\Sigma\times[0,1]$ with the
boundary defined by the equation $x_{n}=0$. Consider the differential operator
$D_{x_{n}}=-i\partial_{x_{n}}$ defined in the collar neighbourhood of the
boundary. Near the boundary, we can write
$D=\sum_{j=0}^{m}A_{j}D_{x_{n}}^{m-j}.$
For each $j$, $A_{j}=(A_{j}(x_{n}))_{x_{n}\in[0,1]}$ is a family of
differential operators ${\rm C}^{\infty}(\Sigma,E)\to{\rm
C}^{\infty}(\Sigma,F)$ of order $j$. Let $a$ denote the principal symbol of
$D$ and $a_{j}$ the principal symbol of $A_{j}$. In coordinates
$x=(x^{\prime},x_{n})$ on $\Sigma\times[0,1]$ with associated cotangent
coordinates $\xi=(\xi^{\prime},\xi_{n})$ on $T^{*}M|_{\Sigma}$, near $\Sigma$
we can write
$a(x^{\prime},x_{n},\xi^{\prime},\xi_{n})=\sum_{j=0}^{m}a_{j}(x^{\prime},x_{n},\xi^{\prime})\xi_{n}^{m-j}.$
We define the boundary symbol $\sigma_{\partial}(A)$ as the ordinary
differential operator valued function on $T^{*}\Sigma$ given by
$\sigma_{\partial}(D)(x^{\prime},\xi^{\prime}):=\sum_{j=0}^{m}a_{j}(x^{\prime},0,\xi^{\prime})D_{t}^{m-j}.$
(3)
We consider $\sigma_{\partial}(D)$ as an element of ${\rm
C}^{\infty}(T^{*}\Sigma,\mathrm{Hom}(E|_{\Sigma},F|_{\Sigma})\otimes\mathcal{L}({\rm
C}^{\infty}[0,\infty)))$. The conormal symbol of $D$ is the symbol of the
boundary symbol
$\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},\xi_{n}):=\sum_{j=0}^{m}a_{j}(x^{\prime},0,\xi^{\prime})\xi_{n}^{m-j},$
(4)
so that $\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})=\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},D_{t})$ and $\sigma_{\rm
cn}(D)=\sigma(D)|_{\Sigma}$. For more details on the yoga of boundary symbols,
see [gerdsgreenbook, rempelschulze, elmarbdmover] or [melroseAPS] for related
notions.
The following result follows from [horIII, Theorem XX.1.3], which we state
below as Theorem 3.8.
###### Proposition 3.1.
Let $D$ be elliptic of order $m>0$. The following construction produces a
well-defined vector bundle
$E_{+}(D)\to S^{*}\Sigma.$
Define $E_{+}(D)$ as the subset
$E_{+}(D)\subseteq\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}$ whose fibre
over $(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$ consists of
$(v_{0},v_{1},\ldots,v_{m-1})\in E_{x}\otimes\mathbb{C}^{m}$ such that the
solution $v\in{\rm C}^{\infty}([0,\infty),E_{x})$ of the ordinary differential
equation
$\begin{cases}\sigma_{\partial}(D)(x^{\prime},\xi^{\prime})v(t)=0,\;t>0,\\\
D^{j}_{t}v(0)=v_{j},\;j=0,\ldots,m-1,\end{cases}$ (5)
is exponentially decaying as $t\to+\infty$.
The subset $E_{-}(D)\subseteq\pi^{*}(E)|_{S^{*}\Sigma}$ given as the set of
vectors whose solution to (5) is exponentially decaying as $t\to-\infty$ is a
well-defined vector bundle defining a complement to $E_{+}(D)$ in
$\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}$.
###### Definition 3.2.
Let $p_{+}(D)\in{\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Hom}(\pi^{*}(E)|_{S^{*}\Sigma}\otimes\mathbb{C}^{m}))$
denote the projection onto $E_{+}(D)$ along $E_{-}(D)$ and
$p_{-}(D):=1-p_{+}(D)$ its complementary projection.
###### Proposition 3.3.
Let $D$ be elliptic of order $m>0$. It holds that
$p_{+}(D)=\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}^{-1})p_{-}(D^{\dagger})^{*}\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}),$
where a is the matrix of differential operators from Proposition 2.15 and
$\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$ is its principal symbol (in the
Douglis-Nirenberg calculus we review below)
$\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})=-i\begin{pmatrix}\sigma_{m-1}(A_{m-1})&\sigma_{m-2}(A_{m-2})&\cdots&\sigma_{2}(A_{2})&\sigma_{1}(A_{1})&A_{0}\\\
\sigma_{m-2}(A_{m-2})&\sigma_{m-3}(A_{m-3})&\cdots&\sigma_{1}(A_{1})&A_{0}&0\\\
\sigma_{m-3}(A_{m-3})&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&&A_{0}&0&\vdots\\\ \vdots&&\mathinner{\mkern
1.0mu\raise 1.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 2.0mu\raise
4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}&0&\vdots&\\\
\sigma_{1}(A_{1})&A_{0}&\mathinner{\mkern 1.0mu\raise 1.0pt\vbox{\kern
7.0pt\hbox{.}}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}&&&\\\ A_{0}&0&\cdots&0&0&0\end{pmatrix}.$
In particular, $\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$ induces
isomorphisms
$E_{\pm}(D)\cong E_{\mp}(D^{\dagger}).$
We postpone the proof of Proposition 3.3 until after the statement of Theorem
3.8 below.
###### Proposition 3.4.
Let $D$ be an elliptic differential operator of order $m>0$ acting on a
manifold of $\dim(M)>1$. The vector bundles $E_{+}(D),E_{-}(D)\to S^{*}\Sigma$
are non-trivial over each component of $S^{\ast}\Sigma$.
###### Proof.
We note that $\sigma_{\rm
cn}(D)(x^{\prime},\xi^{\prime},z)=(-1)^{m}\sigma_{\rm
cn}(D)(x^{\prime},-\xi^{\prime},-z)$, so $z\in\mathbb{C}$ solves
$\det\sigma_{\rm cn}(D)(x^{\prime},\xi^{\prime},z)=0$ if and only if
$\det\sigma_{\rm cn}(D)(x^{\prime},-\xi^{\prime},-z)=0$. Since
$E\otimes\mathbb{C}^{m}=E_{+}(D)\oplus E_{-}(D)$ by Proposition 3.1, we
conclude that given $(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$, either
$E_{+}(D)_{(x^{\prime},\xi^{\prime})}\neq 0$ or
$E_{-}(D)_{(x^{\prime},\xi^{\prime})}\neq 0$. Therefore, given
$(x^{\prime},\xi^{\prime})\in S^{*}\Sigma$ then at least one of the vector
spaces $E_{+}(D)_{(x^{\prime},\xi^{\prime})}$ and
$E_{+}(D)_{(x^{\prime},-\xi^{\prime})}$ is non-zero and at least one of the
vector spaces $E_{-}(D)_{(x^{\prime},\xi^{\prime})}$ and
$E_{-}(D)_{(x^{\prime},-\xi^{\prime})}$ is non-zero. ∎
###### Remark 3.5.
We remark that if $\dim(M)>2$, then $\pi_{0}(S^{*}\Sigma)=\pi_{0}(\Sigma)$ and
in this case $E_{+}(D),E_{-}(D)\to S^{*}\Sigma$ are non-trivial over each
component of $S^{*}\Sigma$. If $\dim(M)=2$, the “odd-to-even” part of a spinc
Dirac operator twisted by a line bundle provides an example where $E_{+}(D)$
and $E_{-}(D)$ satisfy $\mathrm{rk}(E_{+}(D))+\mathrm{rk}(E_{-}(D))=1$ but are
non-vanishing over each connected component of the boundary; their fibrewise
supports are on complementary components of
$S^{*}\Sigma=S^{1}\times\Sigma\dot{\cup}S^{1}\times\Sigma$.
To construct the Calderón projection, we use the Douglis-Nirenberg calculus.
The reader is referred to [agmon, grubb77], see also [horIII, Chapter XIX.5],
for details. We introduce some notation. If $\boldsymbol{E}\to\Sigma$ is a
Hermitian vector bundle orthogonally graded by $\mathbb{R}$ (viewed as a
discrete group), for $s\in\mathbb{R}$ we define the graded Sobolev space
${\mathbb{H}}^{{s}}(\Sigma,\boldsymbol{E}):=\oplus_{s^{\prime}+t=s}{\rm
H}^{\rm s^{\prime}}(\Sigma,\boldsymbol{E}[t]),$
where $\boldsymbol{E}=\oplus_{t}\boldsymbol{E}[t]$ is the grading of
$\boldsymbol{E}$. If $\boldsymbol{E},\boldsymbol{F}\to\Sigma$ are two
Hermitian vector bundles orthogonally graded by $\mathbb{R}$ (viewed as a
discrete group), we can define the space $\Psi^{\boldsymbol{m}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ of Douglis-Nirenberg operators from
$\boldsymbol{E}$ to $\boldsymbol{F}$ of order $m\in\mathbb{R}$ as the subspace
of classical pseudodifferential operators $\Psi_{\rm
cl}^{*}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ that act homogeneously jointly
in the Sobolev degree and the gradings of the vector bundles. More precisely,
writing $\boldsymbol{E}=\oplus_{k=1}^{n_{E}}E[s_{k}]$ and
$\boldsymbol{F}=\oplus_{j=1}^{n_{F}}F[r_{j}]$ for some
$s_{1},\ldots,s_{n_{E}},r_{1},\ldots,r_{n_{F}}\in\mathbb{R}$, a matrix
$(A_{jk})_{j=1,\ldots,n_{E},k=1,\ldots,n_{F}}\in\Psi_{\rm
cl}^{*}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ belongs to
$\Psi^{\boldsymbol{\alpha}}_{\rm cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ if
and only if the order $\alpha_{j,k}$ of $A_{j,k}$ satisfies
$\alpha_{j,k}=\alpha-r_{k}+s_{j}$ for all $j$ and $k$. By construction,
$\Psi^{-\boldsymbol{\infty}}(\Sigma;\boldsymbol{E},\boldsymbol{F}):=\cap_{s\in\mathbb{R}}\Psi^{-\boldsymbol{s}}(\Sigma;\boldsymbol{E},\boldsymbol{F})=\Psi^{-\infty}(\Sigma;\boldsymbol{E},\boldsymbol{F})$
– the ordinary smoothing operators. We write $\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E}):=\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{E})$.
We define the principal symbol $\sigma_{\boldsymbol{\alpha}}(A)$ to be the
matrix of principal symbols $(\sigma_{\alpha_{j,k}}(A_{j,k}))_{j,k}$ which
defines an element of ${\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Hom}(\pi^{*}\boldsymbol{E},\pi^{*}\boldsymbol{F}))$.
We say that $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ is elliptic if
$\sigma_{\boldsymbol{\alpha}}(A)$ is invertible in all points, i.e. that
$\sigma_{\boldsymbol{\alpha}}(A)\in{\rm
C}^{\infty}(S^{*}\Sigma,\mathrm{Iso}(\pi^{*}\boldsymbol{E},\pi^{*}\boldsymbol{F}))$.
By the standard construction, $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ is elliptic if and only if there
exists a parametrix $B\in\Psi^{-\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{F},\boldsymbol{E})$.
By construction, any $A\in\Psi^{\boldsymbol{\alpha}}_{\rm
cl}(\Sigma;\boldsymbol{E},\boldsymbol{F})$ defines a continuous operator
$A:{\mathbb{H}}^{{s}}(\Sigma;\boldsymbol{E})\to{\mathbb{H}}^{{s^{\prime}}}(\Sigma;\boldsymbol{F}),$
for any $s,s^{\prime}\in\mathbb{R}$ with $s^{\prime}\leq s-\alpha$. If
$\sigma_{\boldsymbol{\alpha}}(A)\neq 0$, this operator is compact if and only
if $s^{\prime}<s-\alpha$. For $\sigma_{\boldsymbol{\alpha}}(A)=0$, this
operator is compact whenever $s^{\prime}\leq s-\alpha$. This operator is
Fredholm if and only if $A$ is elliptic (cf. [horIII, Theorem XIX.5.1 and
XIX.5.2]).
The graded vector bundles we shall be concerned with take the form
$E\otimes\mathbb{C}^{m}$ for a trivially graded Hermitian vector bundle $E$.
We shall tacitly grade this vector bundle as follows. We view $\mathbb{C}^{m}$
as a graded bundle by
$\mathbb{C}^{m}[s]=\begin{cases}\mathbb{C},\;&\mbox{if $s=0,1,\ldots,m-1$},\\\
0,\;&\mbox{otherwise}.\end{cases}.$
We grade $E\otimes\mathbb{C}^{m}$ as a tensor product, i.e.
$(E\otimes\mathbb{C}^{m})[s]=\begin{cases}E,\;&\mbox{if
$s=0,1,\ldots,m-1$},\\\ 0,\;&\mbox{otherwise}.\end{cases}$
In particular,
${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})=\bigoplus_{j=0}^{m-1}{\rm
H}^{\rm s-j}(\Sigma;E).$
We also introduce the graded bundle $\mathbb{C}^{m}_{\rm op}$ which is the
trivial vector bundle of rank $m$, graded by
$\mathbb{C}^{m}_{\rm
op}[s]=\mathbb{C}^{m}[-s]=\begin{cases}\mathbb{C},\;&\mbox{if
$s=0,-1,\ldots,-m+1$},\\\ 0,\;&\mbox{otherwise}.\end{cases}.$
The reader should note that $\mathbb{C}^{m}_{\rm op}$ can be identified with
the graded dual bundle of $\mathbb{C}^{m}$, and the Hermitian structure of $E$
allow us to identify $E\otimes\mathbb{C}^{m}_{\rm op}$ with the graded dual
bundle of $E\otimes\mathbb{C}^{m}$. In particular,
${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm s+j}(\Sigma;E).$
We can conclude the following proposition.
###### Proposition 3.6.
The ${\rm L}^{2}$-pairing induces a perfect pairing
${\mathbb{H}}^{{s}}(\Sigma;E\otimes\mathbb{C}^{m})\times{\mathbb{H}}^{-{s}}(\Sigma;E\otimes\mathbb{C}^{m}_{\rm
op})\to\mathbb{C},$
for any $s\in\mathbb{R}$.
###### Definition 3.7.
Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. An approximate Calderón projection for $D$ is a classical pseudo-
differential operator $Q\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ which is order zero in the Douglis-
Nirenberg sense and satisfies the following
* •
$Q^{2}-Q$ is smoothing.
* •
$Q$ is an approximate projection onto the Hardy space $\mathcal{C}_{D}$ in the
following sense:
1. (1)
If $u\in\ker(D_{\rm max})$ then $v:=\gamma
u\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ satisfies
that $v-Qv\in{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})$.
2. (2)
For any $v\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
there is a $u\in\ker(D_{\rm max})+{\rm C}^{\infty}(M,E)$ (smooth up to the
boundary) such that $Qv=\gamma u$.
###### Theorem 3.8 ([horIII], Theorem XX.1.3).
Let $D$ be an elliptic differential operator of order $m>0$. There exists an
approximate Calderón projection $Q\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ for $D$ with principal symbol given by
$\sigma_{\boldsymbol{0}}(Q)=p_{+}(D),$
(as defined in Defintion 3.2) and the full symbol of
$Q=(Q_{j,k})_{j,k=0}^{m-1}$ computed from the formula [horIII, Equation
XX.1.7]:
$Q_{j,k}f=\sum_{l=0}^{m-1-k}(-i)^{j+l+1}\gamma_{j}\circ
T[A_{m-l-k-1}f\otimes\delta_{t=0}^{(l)}],$ (6)
where $T$ is a pseudodifferential parametrix to $D$ computed in a
neighbourhood of $\Sigma$.
###### Remark 3.9.
We remark that the formula for $Q_{j,k}$ might seem difficult to parse at
first. But the fact that the trace mappings are from inside of $M$ makes it
possible to compute the full symbol by means of residue calculus and the
Paley-Wiener theorem. For instance, [horIII, Equation XX.1.8] computes the
principal symbol of $Q$ by residue calculus as
$\displaystyle\sigma_{\boldsymbol{0}}(Q)_{j,k}(x^{\prime},\xi^{\prime})\equiv$
$\displaystyle\sigma_{k-j}(Q_{j,k})(x^{\prime},\xi^{\prime})=$ (7)
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sum_{l=0}^{m-k-1}\xi_{n}^{j+l}a(x^{\prime},0,\xi,\xi_{n})^{-1}a_{m-k-l-1}(x^{\prime},0,\xi^{\prime})\right]$
where $(x^{\prime},\xi^{\prime})$ are coordinates on $T^{*}\Sigma$ and
$((x^{\prime},t),(\xi^{\prime},\xi_{n}))$ are coordinates on $T^{*}M$ near
$\Sigma$. The proof of [horIII, Theorem XX.1.3] identifies this expression
with $p_{+}(D)$. The lower order terms in $Q$ are, in general, hard to
compute. We shall give some examples of such computations below for order 1
operators. We can condense the formula (7) into
$\displaystyle\sigma_{\boldsymbol{0}}(Q)(x^{\prime},\xi^{\prime})=\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})\right],$
(8)
where $e(\xi_{n}):=v(\xi_{n})\otimes v(\xi_{n})^{*}$ and
$v(\xi_{n})=(\xi_{n}^{j})_{j=0}^{m-1}$. Here, we abuse notation and view
$a^{-1}|_{T^{*}M|_{\Sigma}\setminus\Sigma}$ and $e$ as endomorphisms of
$E|_{\Sigma}\otimes\mathbb{C}^{m}$. Note that
$a^{-1}|_{T^{*}M|_{\Sigma}\setminus\Sigma}$ and $e$ commute because they act
on different factors in the tensor product $E|_{\Sigma}\otimes\mathbb{C}^{m}$.
###### Proof of Proposition 3.3.
In light of Theorem 3.8 and Remark 3.9, the proof consists of an exercise with
boundary symbols and residue calculus. We compute that
$\displaystyle p_{-}(D^{\dagger})^{*}(x^{\prime},\xi^{\prime})=$
$\displaystyle\left(-\sum_{\mathrm{Im}(\xi_{n})<0}\mathrm{Res}_{\xi_{n}}\left[a^{*}(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}_{\dagger})(x^{\prime},\xi^{\prime})\right]\right)^{*}=$
$\displaystyle=$
$\displaystyle-\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\left(a^{*}(x^{\prime},0,\xi^{\prime},\overline{\xi_{n}})^{-1}e(\overline{\xi_{n}})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}_{\dagger})(x^{\prime},\xi^{\prime})\right)^{*}\right]=$
$\displaystyle=$
$\displaystyle-\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a}^{*}_{\dagger})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right]=$
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right].$
In the first equality we used that $p_{-}(D^{\dagger})$ is the complementary
projection to $p_{+}(D^{\dagger})$ which is the sum of all residues in the
lower half plane, with a minus sign due to the orientation. In the second
equality we used that the adjoint is antilinear, and interchanges the upper
and lower half plane. In the last equality we used Lemma 2.16. We arrive at
$\displaystyle\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})$
$\displaystyle(x^{\prime},\xi^{\prime})^{-1}p_{-}(D^{\dagger})^{*}(x^{\prime},\xi^{\prime})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})=$
$\displaystyle=$
$\displaystyle\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})^{-1}\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\right]\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})=$
$\displaystyle=$
$\displaystyle\sum_{\mathrm{Im}(\xi_{n})>0}\mathrm{Res}_{\xi_{n}}\left[a(x^{\prime},0,\xi^{\prime},\xi_{n})^{-1}e(\xi_{n})\sigma_{\boldsymbol{0}}(\scalebox{1.5}{a})(x^{\prime},\xi^{\prime})\right]=p_{+}(D).\qed$
###### Corollary 3.10.
Let $D$ be an elliptic differential operator of order $m>0$ acting on a
manifold of $\dim(M)>1$ and let $Q$ be the approximate Calderón projection
from Theorem 3.8. For any $s\in\mathbb{R}$, the operator
$Q:{\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m}),$
(9)
is continuous but neither $Q$ nor $1-Q$ is compact.
This result should be compared to Lemma A.2 in the appendix.
###### Proof.
Continuity of (9) follows from that $Q$ is order zero in the Douglis-Nirenberg
sense. By standard arguments in pseudodifferential calculus, (9) is compact
for some $s$ if and only if it is compact for all $s$ if and only if the
principal symbol vanishes. The principal symbol of $Q$ is $p_{+}(D)$ which is
non-trivial by Proposition 3.4, so $Q$ is never compact on
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$. By the same argument, the
principal symbol of $1-Q$ is $p_{-}(D)$ which is non-trivial by Proposition
3.4, so $1-Q$ is never compact on
${\mathbb{H}}^{s}(\Sigma;E\otimes\mathbb{C}^{m})$. ∎
###### Theorem 3.11.
Let $D$ be an elliptic differential operator of order $m>0$ on a manifold of
dimension $>1$. Then $\ker(D_{\rm max})$ is infinite-dimensional.
Despite this result being well known and an immediate consequence of results
of Seeley that we recall in the next subsection, let us provide a short proof
that remains at a symbolic level for an approximate Calderón projection.
###### Proof.
By Proposition 2.1, $\ker(D_{\rm max})$ is infinite-dimensional if and only if
the Hardy space $\mathcal{C}_{D}$ is infinite-dimensional. We will argue by
contradiction.
Assume that $\mathcal{C}_{D}$ is finite-dimensional. Since the space
$\mathcal{C}_{D}\subseteq{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
is finite-dimensional it is closed. Consider the operator
$Q:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
as in Equation (9). By Theorem 3.8, for any
$v\in{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ it holds
that
$Qv\in\mathcal{C}_{D}+{\rm C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m}).$
We conclude that
$Q:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
factors over the inclusion
$\mathcal{C}_{D}+{\rm
C}^{\infty}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
which is compact if $\mathcal{C}_{D}$ is finite-dimensional. Therefore, if
$\mathcal{C}_{D}$ is finite-dimensional then $Q$ is compact on
${\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ which
contradicts Corollary 3.10. ∎
###### Corollary 3.12.
Assume that $(F,h^{F})=(E,h^{E})$ and let $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;E)$ be an elliptic differential operator of order $m>0$ acting
on a manifold of $\dim(M)>1$. We then have that
$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}$.
Consequently, $\mathrm{spec}(D_{\min})=\mathbb{C}$.
###### Proof.
Fix $\lambda\in\mathbb{C}$, and note that $D-\lambda$ is again an elliptic
differential operator of order $m>0$. Now, Theorem 3.11 implies that
$\dim\ker(D_{\max}-\lambda)=\infty$. But that is exactly that
$\lambda\in\mathrm{spec}_{\mathrm{pt}}(D_{\max})$ and so we conclude that
$\mathrm{spec}(D_{\max})=\mathrm{spec}_{\mathrm{pt}}(D_{\max})=\mathbb{C}$.
For the statement concerning $D_{\min}$, we note that by mirroring this
argument, we obtain that $\mathrm{spec}(D_{\max}^{\dagger})=\mathbb{C}$ and
since $D_{\min}=(D_{\max}^{\dagger})^{\ast}$, we obtain that
$\mathrm{spec}(D_{\min})=\mathrm{spec}(D_{\max}^{\dagger})^{\mathrm{conj}}=\mathbb{C}$.
∎
###### Remark 3.13.
For $D$ with $\ker(D_{\min})=0$ (i.e., first order Dirac-type operators), we
have that $0\not\in\mathrm{spec}_{\mathrm{pt}}(D_{\min})$. However, from
Corollary 3.12, we know that $0\in\mathrm{spec}(D_{\min})$. In fact,
$0\in\mathrm{spec}_{\mathrm{res}}(D_{\min})$, i.e., it is in the residue
spectrum. This is because we have that ${\rm
L}^{2}(M;E)=\ker(D_{\max}^{\dagger})\oplus\mathrm{ran}(D_{\min})$ from
Proposition 2.6 which shows that $D_{\min}:\mathrm{dom}(D_{\min})\to{\rm
L}^{2}(M;E)$ does not have dense range. In fact, the range fails to be dense
on the infinite dimensional subspace $\ker(D_{\max}^{\dagger})$. However,
despite these shortcomings, and a complete lack of spectral theory,
Proposition 2.20 shows that the Dirichlet problem is indeed well-posed when
restricted to $\mathrm{ran}(D_{\min})=\mathrm{ran}(D_{\mathcal{C}_{D}})$ (see
more in Subsection 2.4).
###### Corollary 3.14.
Assume that $(F,h^{F})=(E,h^{E})$ and that $D:{\rm C}^{\infty}(M;E)\to{\rm
C}^{\infty}(M;E)$ is first order elliptic and Dirac-type. Then,
$\mathrm{spec}(D_{\min})=\mathrm{spec}_{\mathrm{res}}(D_{\min})=\mathbb{C}$.
###### Proof.
For $\lambda\in\mathbb{C}$, the operator $D_{\lambda}=D-\lambda$ is again
first order Dirac-type. By Remark 3.13, we conclude that
$0\in\mathrm{spec}_{\mathrm{res}}((D_{\lambda})_{\min})$. But
$(D_{\lambda})_{\min}=D_{\min}-\lambda$ and so the conclusion holds. ∎
### 3.2. Seeley’s results on Calderón projectors
###### Definition 3.15.
Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. A Calderón projection for $D$ is an approximate Calderón
projection $P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ such that it is a projection onto
$\mathcal{C}_{D}\subseteq{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$
###### Remark 3.16.
Calderón projectors are not uniquely determined. For instance, if
$P_{\mathcal{C}}$ is a Calderón projection, then for any self-adjoint
smoothing operator $R$ which is “small”333How small can be made precise by
pointwise estimates on the kernel, but it is irrelevant for the sake of this
argument. and $R|_{\mathcal{C}_{D}}=0$ we have that
$P_{\mathcal{C}}(1+R)^{-1}$ also satisfies the conditions of the theorem. _We
shall tacitly use only the Calderón projection constructed in the next theorem
of Seeley._
Let us recall the following theorem of Seeley. We remark that Seeley proved
this result in the larger generality of higher regularity (cf. Theorem 6.1
below) on ${\rm L}^{p}$-spaces, $1<p<\infty$. For $p\neq 2$, the spaces on the
boundary are Besov spaces and not Sobolev spaces.
###### Theorem 3.17.
[Seeley [seeley65]] Let $D$ be an elliptic differential operator of order
$m>0$ on a manifold $M$ with boundary $\Sigma$ acting between sections of the
Hermitian vector bundles $E$ and $F$. Then there exists a Calderón projection
$P_{\mathcal{C}}\in\Psi^{\boldsymbol{0}}_{\rm
cl}(\Sigma;E\otimes\mathbb{C}^{m})$ whose full symbol is uniquely determined
as that of the approximate Calderón projection in Theorem 3.8. In particular,
$\sigma_{\boldsymbol{0}}(P_{\mathcal{C}})=p_{+}(D).$
In fact, Seeley obtained even stronger results in [seeley65] that we partly
summarise in the following theorem.
###### Theorem 3.18.
Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. There exists a Calderón projection $P_{\mathcal{C}}$ such that
1. (i)
There exists a continuous Poisson operator
$\mathcal{K}:{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to\ker(D_{\rm
max}),$
with $P_{\mathcal{C}}=\gamma\circ\mathcal{K}$ (and in particular,
$\mathcal{K}=\mathcal{K}\circ P_{\mathcal{C}}$).
2. (ii)
The image of
$\gamma:\mathrm{dom}(D_{\rm
max})\to{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}),$
is precisely the subspace
$P_{\mathcal{C}}{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$
In particular, the identity map induces an isomorphism of Banach spaces
$\check{\mathrm{H}}(D)\cong
P_{\mathcal{C}}{\mathbb{H}}^{-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$
###### Proof.
The first item is proven in [seeley65]. The second item is a reformulation of
a statement on [seeley65, top of page 782] which was stated but not proven
there. We include its proof for the convenience of the reader.
To prove the second item, it suffices to prove that
$(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm
max})\subseteq{\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m}).$
We can assume that $M$ is a domain with smooth boundary in a compact manifold
$\hat{M}$, that $E$ and $F$ extend to Hermitian vector bundles
$\hat{E},\hat{F}\to\hat{M}$ and that there exists an elliptic differential
operator $\hat{D}$ of order $m$ extending $D$ to $\hat{M}$. Write $D^{\prime}$
for the elliptic differential operator of order $m$ defined from restricting
$\hat{D}$ to $M^{c}$. Take a $g\in(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm
max})\subseteq\gamma\mathrm{dom}(D_{\rm max})$ and pick
$f\in\mathrm{dom}(D_{\rm max})$ lifting $g$. Since
$g\in(1-P_{\mathcal{C}})\gamma\mathrm{dom}(D_{\rm max})$, [seeley65, Lemma 5]
implies that there exists an $f^{\prime}\in\ker(D^{\prime})\subseteq{\rm
L}^{2}(M^{c},\hat{E}|_{M^{c}})$ lifting $g$. We define $F\in{\rm
L}^{2}(\hat{M},E)$ as $f$ on $M$ and as $f^{\prime}$ on $M^{c}$. Since
$\gamma(f)=g=\gamma(f^{\prime})$, we have that $F\in\mathrm{dom}(\hat{D}_{\rm
max})$. By elliptic regularity on the closed manifold $\hat{M}$, it holds that
$\mathrm{dom}(\hat{D}_{\rm max})={\rm H}^{\rm m}(\hat{M},E)$. We conclude that
$f\in{\rm H}^{\rm m}(M,E)$ and therefore
$g=\gamma(f)\in{\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
∎
###### Remark 3.19.
It is also proven in [seeley65] that the complementary projection
$(1-P_{\mathcal{C}})$ is (up to a finite rank projection) a projection onto
the Hardy space of the complementary manifold with boundary
$\hat{M}\setminus\mathring{M}$. Note that Seeley’s construction depends on the
choice of the closed manifold $\hat{M}$ and the extension of $D$ to an
elliptic operator $\hat{D}$ on $\hat{M}$. In other words, the Calderon
projection is a projection onto the Hardy space along the Hardy space of the
complement. For this reason, the Calderon projection was denoted by $P^{+}$ in
[seeley65] and $C^{+}$ in [grubbdistop]. Our notation $P_{\mathcal{C}}$ is
chosen to avoid confusion with the notation for spectral projectors from [BB,
BBan].
We remark that a Calderon projection $P_{\mathcal{C}}$ such that
$(1-P_{\mathcal{C}})$ is (up to a finite rank projection) a projection onto
the Hardy space of the complementary manifold is uniquely determined modulo
smoothing operator. We therefore implicitly will use the Calderon projection
$P_{\mathcal{C}}$ from Theorem 3.18 throughout the paper.
###### Corollary 3.20.
Let $D$ be an elliptic differential operator of order $m>0$. The short exact
sequence of Hilbert spaces
$0\to\mathrm{dom}(D_{\rm min})\to\mathrm{dom}(D_{\rm
max})\xrightarrow{\gamma}\check{\mathrm{H}}(D)\to 0,$
is split by the continuous mapping
$E:\check{\mathrm{H}}(D)\to\mathrm{dom}(D_{\rm max}),\quad
E=\mathcal{K}+E_{0}\circ(1-P_{\mathcal{C}}),$
where
$E_{0}:{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\to{\rm
H}^{\rm m}(M;E)$ is any continuous splitting of the trace mapping ${\rm
H}^{\rm
m}(M;E)\to{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
###### Proof.
We have that $E=EP_{\mathcal{C}}+E(1-P_{\mathcal{C}})$. The operators
$EP_{\mathcal{C}}=\mathcal{K}$ and
$E(1-P_{\mathcal{C}})=E_{0}(1-P_{\mathcal{C}})$ are continuous by Theorem
3.17, so $E$ is continuous. Moreover, we have that $\gamma
E=\gamma\mathcal{K}+\gamma
E_{0}(1-P_{\mathcal{C}})=P_{\mathcal{C}}+(1-P_{\mathcal{C}})=1$ so $E$ is a
splitting. ∎
###### Example 3.21.
Let us provide some further context for Example 2.11. We again consider the
generalised boundary condition $B_{\rm Sob}=\bigoplus_{j=0}^{m-1}{\rm H}^{\rm
m-{\frac{1}{2}}-j}(\Sigma;E)$ for an elliptic differential operator $D$ of
order $m>0$. Letting
$\mathrm{P}_{\mathcal{C}^{\mathrm{c}}}=(I-\mathrm{P}_{\mathcal{C}})$, the
complementary projection to $\mathrm{P}_{\mathcal{C}}$,
$B_{\rm
Sob}+\mathcal{C}_{D}=\mathrm{P}_{\mathcal{C}^{\mathrm{c}}}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})\oplus\mathcal{C}_{D}=\mathcal{C}^{\mathrm{c}}\oplus\mathcal{C}=\check{\mathrm{H}}(D),$
by Theorem 3.18. We can now use Theorem B.23 to give an alternate proof of the
fact that $\mathrm{ran}(D_{{\rm B}_{\rm Sob}})$ is closed. We can also use
Lemma B.20 to conclude $\mathrm{ran}(D_{{\rm B}_{\rm
Sob}})=\mathrm{ran}(D_{\max})$ which we formulate in the following corollary.
###### Corollary 3.22.
Let $D$ be an elliptic differential operator of order $m>0$ on a manifold $M$
with boundary $\Sigma$ acting between sections of the Hermitian vector bundles
$E$ and $F$. Then it holds that
$\mathrm{ran}(D_{\max})=D{\rm H}^{\rm m}(M;E),$
as closed subspaces of ${\rm L}^{2}(M;F)$ whose codimension is
$\dim\ker(D^{\dagger}_{\rm min})$.
The next result is another corollary of Theorem 3.18.
###### Corollary 3.23.
Let $D$ be an elliptic differential operator of order $m>0$. The boundary
condition
$\mathcal{C}^{c}:=(1-P_{\mathcal{C}}){\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is regular and satisfies:
* •
$\mathcal{C}^{c}$ is closed in
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and in
$\check{\mathrm{H}}(D)$ and for $x\in\mathcal{C}^{c}$,
$\|x\|_{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})}\simeq\|x\|_{\check{\mathrm{H}}(D)}.$
* •
$\mathcal{C}^{c}\cap\mathcal{C}_{D}=0$ and
$\mathcal{C}^{c}+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$.
* •
The inclusion maps defines a Banach space isomorphism
$\mathcal{C}^{c}\oplus\mathcal{C}_{D}\cong\check{\mathrm{H}}(D),$
where $\mathcal{C}_{D}$ is equipped with either the norm from
$\check{\mathrm{H}}(D)$ or the norm from
${\mathbb{H}}^{{-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$ (the two are
equivalent on $\mathcal{C}_{D}$).
###### Remark 3.24.
It follows from the proof Theorem 2.9 and Corollary 3.23 that an elliptic
differential operator $D$ admits an invertible realisation if and only if
$\ker(D_{\min})=0$ and $\ker(D_{\min}^{\dagger})=0$. Indeed, for any
realisation $D_{\rm B}$ we have the inclusions
$\ker(D_{\min})\subseteq\ker(D_{\rm B})$ and
$\ker(D_{\min}^{\dagger})=\ker(D_{\rm B}^{*})$ so $\ker(D_{\min})=0$ and
$\ker(D_{\min}^{\dagger})=0$ is a necessary condition. The converse follows
from that $\ker(D_{\min})=0$ and $\ker(D_{\min}^{\dagger})=0$ if and only if
$D_{\mathcal{C}^{c}}$ is invertible by the proof Theorem 2.9 and Corollary
3.23.
For general regular boundary conditions, we have a statement similar to
Corollary 3.23:
###### Proposition 3.25.
Let $D$ be an elliptic differential operator of order $m>0$ and $B$ be a
regular boundary condition. Then the following holds:
1. (i)
$(B,\mathcal{C}_{D})$ is a Fredholm pair in $\check{\mathrm{H}}(D)$ and
$B^{\ast}\cap\mathcal{C}_{D^{\dagger}}\cong\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$
2. (ii)
There is a finite dimensional space
$F\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
and a subspace $B_{0}\subseteq B+F$ of finite codimension defining a regular
boundary condition with $\ker D_{\rm B_{0}}=\ker(D_{\min})$ for which we can
write
$\mathrm{dom}(D_{\rm max})=\mathrm{dom}(D_{{\rm B}_{0}})\oplus_{\ker
D_{\min}}\ker D_{\rm max},$
and
$\check{\mathrm{H}}(D)=B_{0}\oplus\mathcal{C}_{D}.$
The contents of item ii) in Theorem 3.25 can be interpreted as the statement
that any regular boundary condition is up to finite dimensional perturbations
a complement to the Hardy space in the Cauchy data space. In light of Theorem
2.12, or rather Corollary B.29, there are constraints on the dimension of $F$
and the codimension of $B_{0}\subseteq B+F$ given by the identity:
$\dim(F)-\dim\left(\faktor{B+F}{B_{0}}\right)=\dim\ker(D_{\min})-\dim\ker(D_{\min}^{\dagger})-\operatorname{ind}(D_{\rm
B}).$
###### Proof.
Item i follows from Theorem 2.12. If $B$ satisfies that $\ker(D_{\rm
B}^{*})=\ker(D_{\rm min}^{\dagger})$ and $\ker D_{\rm B}=\ker(D_{\min})$, then
with $B_{0}=B$ and $F=0$, item ii follows from Corollary B.27 applied to the
FAP with closed range $\mathcal{T}_{\rm B}:=(D_{\rm B},D^{\dagger}_{\rm
min})$.
To prove item ii, we first prove that for any regular boundary condition $B$
there is a finite dimensional space
$F\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
with $(B+F)\cap\mathcal{C}_{D}=B\cap\mathcal{C}_{D}$ and
$B+F+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$. Indeed, $B+F$ is clearly semi-
regular and will be regular since $(B+F)^{*}\subseteq B^{*}$. To construct
such an $F$, we note that item i implies that the inclusion
$\faktor{B}{B\cap\mathcal{C}_{D}}\to\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}$
has finite codimension. Indeed, we have that
$\faktor{(\check{\mathrm{H}}(D)/\mathcal{C}_{D})}{(B/B\cap\mathcal{C}_{D})}=\faktor{\check{\mathrm{H}}(D)}{(B+\mathcal{C}_{D})}.$
As such, there are plenty of finite-dimensional subspaces
$F\subseteq\check{\mathrm{H}}(D)$ with
$(B+F)\cap\mathcal{C}_{D}=B\cap\mathcal{C}_{D}$ and
$B+F+\mathcal{C}_{D}=\check{\mathrm{H}}(D)$. That we can take
$F\subseteq\check{\mathrm{H}}(D)\cap{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
follows from Example 3.21 which implies that the inclusion
${\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\hookrightarrow\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}$
induces an isomorphism
$\faktor{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})}{\left({\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})\cap\mathcal{C}_{D}\right)}\cong\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}.$
To finish the proof, we need to construct $B_{0}$. Upon replacing $B$ with
$B+F$, the argument in the preceding paragraph allow us to assume that the
regular boundary condition $B$ satisfies $\ker(D_{\rm B}^{*})=\ker(D_{\rm
min}^{\dagger})$. We need to construct a regular boundary condition
$B_{0}\subseteq B$ of finite codimension such that
$B_{0}+\mathcal{C}_{D}=B+\mathcal{C}_{D}$ and $B_{0}\cap\mathcal{C}_{D}=0$.
Consider any
$B_{0}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
which forms a complement of $B\cap\mathcal{C}_{D}$ in $B$. Since $B$ is
regular,
$B\cap\mathcal{C}_{D}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
is finite-dimensional so $B_{0}\subseteq B$ has finite codimension. Moreover,
$B_{0}$ is closed in $\check{\mathrm{H}}(D)$ and by construction
$B_{0}+\mathcal{C}_{D}=B+\mathcal{C}_{D}$ and $B_{0}\cap\mathcal{C}_{D}=0$. It
remains to prove that we can take the complement $B_{0}$ to satisfy that
$B_{0}^{*}\subseteq{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m})$.
We have that $B^{*}\subseteq B_{0}^{*}$ has finite codimension and using
Proposition 2.18 we deduce that we can choose $B_{0}$ such that
$\faktor{B_{0}^{*}}{B^{*}}\subseteq\faktor{{\mathbb{H}}^{{m-{\frac{1}{2}}}}(\Sigma;F\otimes\mathbb{C}^{m})}{B^{*}}$
which proves the theorem. ∎
### 3.3. Boundary decomposing projectors
With Theorem 3.17, we are able to prove results of use when studying graphical
decompositions of regular boundary conditions, see more in [BBan] and
Subsection 5 below. We make the following definition that we shall make use of
in the first order case below in Subsection 5
###### Definition 3.26.
Let $D$ be an elliptic differential operator of order $m>0$. We say that a
projection $\mathcal{P}_{+}$ is boundary decomposing for $D$ if the following
conditions are satisfied
1. (P1)
$\mathcal{P}_{+}:{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m})\to{\mathbb{H}}^{\alpha}(\Sigma;E\otimes\mathbb{C}^{m})$
is a bounded projection for
$\alpha\in\left\\{-\frac{1}{2},m-\frac{1}{2}\right\\}$,
2. (P2)
$\mathcal{P}_{+}:\check{\mathrm{H}}(D)\to\check{\mathrm{H}}(D)$, and
$\mathcal{P}_{-}:=(I-\mathcal{P}_{+}):\check{\mathrm{H}}(D)\to{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$,
and
3. (P3)
$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|\mathcal{P}_{-}u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|\mathcal{P}_{+}u\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$
The next proposition follows from the open mapping theorem.
###### Proposition 3.27.
Let $D$ be an elliptic differential operator of order $m>0$ and
$\mathcal{P}_{+}$ a boundary decomposing projection for $D$. Then
$\mathcal{P}_{+}\check{\mathrm{H}}(D)=\mathcal{P}_{+}{\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E)\quad\mbox{and}\quad\mathcal{P}_{-}\check{\mathrm{H}}(D)=\mathcal{P}_{-}{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E),$
with equivalent norms.
Let us state yet another corollary of Theorem 3.17.
###### Corollary 3.28.
The Calderón projection $P_{\mathcal{C}}$ of an elliptic differential operator
$D$ is boundary decomposing for $D$. Moreover, the complementary subspace
$\mathcal{C}^{\mathrm{c}}:=(I-\mathrm{P}_{\mathcal{C}})\check{\mathrm{H}}(D)=(1-P_{\mathcal{C}}){\mathbb{H}}^{m-{{\frac{1}{2}}}}(\Sigma;E\otimes\mathbb{C}^{m})$
defines a regular boundary condition with
$\ker(D_{\mathcal{C}^{\mathrm{c}}})=\ker(D_{\min})$ and characterises
$\mathrm{dom}(D_{\rm max})$ via the following continuously split short exact
sequence of Hilbert spaces
$0\to\mathrm{dom}(D_{\mathcal{C}^{\mathrm{c}}})\to\mathrm{dom}(D_{\rm
max})\xrightarrow{P_{\mathcal{C}}\circ\gamma}\mathcal{C}\to 0.$
###### Proof.
By the fact that $\mathrm{P}_{\mathcal{C}}$ is a pseudo-differential operator
of order zero in the Douglis-Nirenberg calculus, it automatically satisfies 1.
The properties 2 and 3 follow from Theorem 3.18. It follows that
$\mathcal{C}^{\mathrm{c}}$ is a regular boundary condition. Since the
restricted trace map is a surjection
$\ker(D_{\mathcal{C}^{\mathrm{c}}})\to\mathcal{C}^{\mathrm{c}}\cap\mathcal{C}=0$
with kernel $\ker(D_{\min})$, we conclude that
$\ker(D_{\mathcal{C}^{\mathrm{c}}})=\ker(D_{\min})$. ∎
###### Example 3.29.
The Hardy space itself is a boundary condition, so we may consider the closed
operator $D_{\mathcal{C}_{D}}$. The realisation $D_{\mathcal{C}_{D}}$ is
called the soft extension, or the Krein extension, and is self-adjoint if $D$
is formally self-adjoint. Its properties were further studied in [grubbsoft].
It is clear that $\ker(D_{\mathcal{C}_{D}})=\ker(D_{\max})$, which we have
noted is infinite dimensional if $\dim(M)>1$. By Proposition B.10, setting
$B_{2}=\check{\mathrm{H}}(D)$ and $B_{1}=\mathcal{C}_{D}$, we obtain that
$\faktor{\mathrm{dom}(D_{\max})}{\mathrm{dom}(D_{\mathcal{C}_{D}})}\cong\faktor{\check{\mathrm{H}}(D)}{\mathcal{C}_{D}}\cong\mathcal{C}^{\mathrm{c}},$
where $\mathcal{C}^{\mathrm{c}}$ is the kernel of $\mathrm{P}_{\mathcal{C}}$.
By Corollary 3.28, we have that $\mathcal{C}^{\mathrm{c}}$ is infinite
dimensional. That is, $D_{\mathcal{C}}$ is an operator which differs from
$D_{\max}$ on an infinite dimensional subspace but still has infinite
dimensional kernel.
This can be repeated on replacing $\mathcal{C}$ by any infinite dimensional
closed subspace of $\mathcal{C}$. Therefore, there’s an abundance of boundary
conditions with infinite dimensional kernel which is different from $D_{\max}$
on an infinite dimensional subspace.
We now state a sufficient condition for a projection to be boundary
decomposing.
###### Theorem 3.30.
Let $D$ be an elliptic differential operator of order $m>0$ and $P$ a
continuous projection on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$. Assume that
$P$ satisfies the following:
* •
$P$ is continuous also in the $\check{\mathrm{H}}(D)$-norm and the
${\mathbb{H}}^{{-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$-norm.
* •
The operator $A:=P_{\mathcal{C}}-(1-P)$ defines a Fredholm operator on
${\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$ and by
continuity extends to a Fredholm operator on $\check{\mathrm{H}}(D)$.
Then $P$ is boundary decomposing, and in particular
$\|u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}+\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}.$
###### Proof.
By the open mapping theorem and density it suffices to prove that the
assumptions on $P$ ensure that
$\|(1-P)u\|_{\check{\mathrm{H}}(D)}\simeq\|(1-P)u\|_{{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})}\quad\mbox{and}\quad\|Pu\|_{\check{\mathrm{H}}(D)}\simeq\|Pu\|_{{\mathbb{H}}^{-{\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})},$
for $u\in{\mathbb{H}}^{{m-\frac{1}{2}}}(\Sigma;E\otimes\mathbb{C}^{m})$.
|
# High Q Hybrid Mie-Plasmonic Resonances in Van der Waals Nanoantennas on Gold
Substrate
Sam A. Randerson1 Panaiot G. Zotev Department of Physics and Astronomy,
University of Sheffield, Sheffield, S3 7RH, UK Xuerong Hu Department of
Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK
Alexander J. Knight Department of Physics and Astronomy, University of
Sheffield, Sheffield, S3 7RH, UK Yadong Wang Department of Physics and
Astronomy, University of Sheffield, Sheffield, S3 7RH, UK Sharada Nagarkar
Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK Dominic Hensman Department of Physics and Astronomy, University of
Sheffield, Sheffield, S3 7RH, UK Yue Wang Department of Physics, School of
Physics, Engineering and Technology, University of York, York, YO10 5DD, UK
Alexander I. Tartakovskii1
###### Abstract
Dielectric nanoresonators have been shown to circumvent the heavy optical
losses associated with plasmonic devices, however they suffer from less
confined resonances. By constructing a hybrid system of both dielectric and
metallic materials, one can retain the low losses of dielectric resonances,
whilst gaining additional control over the tuning of the modes with the metal,
and achieving stronger mode confinement. In particular, multi-layered van der
Waals materials are emerging as promising candidates for integration with
metals owing to their weak attractive forces, which enable deposition onto
such substrates without the requirement of lattice matching. Here we use
layered, high refractive index WS2 exfoliated on gold, to fabricate and
optically characterise a hybrid nanoantenna-on-gold system. We experimentally
observe a hybridisation of Mie resonances, Fabry–Pérot modes, and surface
plasmon-polaritons launched from the nanoantennas into the substrate. We
achieve experimental quality factors of hybridised Mie-plasmonic modes to be
19 times that of Mie resonances in nanoantennas on silica, and observe
signatures of a supercavity mode with a Q factor of 263 $\pm$ 28, resulting
from strong mode coupling between a higher-order anapole and Fabry–Pérot-
plasmonic mode. We further simulate WS2 nanoantennas on gold with an hBN
spacer, resulting in calculated electric field enhancements exceeding 2600,
and a Purcell factor of 713. Our results demonstrate dramatic changes in the
optical response of dielectric nanophotonic structures placed on gold, opening
new possibilities for nanophotonics and sensing with simple-to-fabricate
devices.
In the last decade, transition metal dichalcogenide (TMD) monolayers have
attracted a large research effort owing to their direct band gap transition
and useful opto-electronic properties [1], rendering the bulk material largely
overlooked. More recently, nanoresonators utilising bulk TMDs to host Mie
resonances [2] have gained interest, however these studies primarily focus on
fabricating such structures on low refractive index dielectric substrates such
as SiO2 [3, 4, 5, 6]. A hybrid TMD-nanoantenna-on-metal system has not been
thoroughly explored, with previous work being dominated by numerical
simulations. One of the first to be numerically characterised was a dielectric
nanowire above a metallic substrate by Oulton et al. [7], which led to the
prediction of hybrid dielectric-plasmonic modes. Since then, a variety of
dielectric-metal nanodevices have been considered, from metallic nanoparticle-
on-TMD monolayer systems [8, 9], to hybrid nanoantennas composed of silicon
and gold coaxial layers [10, 11]. Furthermore, dielectric nanoantennas
situated several nanometers above a silver substrate have been numerically
analysed by Yang et al. [12]. Their work shows hybrid dielectric-plasmonic
modes [12] with quality (Q) factors up to $\sim 10^{3}$ and Purcell
enhancements of $>5000$, along with strong electric field confinement in the
nanoantenna-substrate gap. Shen et al. [13] built on this by simulating an
hBN-WSe2 heterostructure placed within the gap and achieving strong light-
matter coupling. Experimental work has so far been limited to silicon-based
nanoresonators on metallic substrates. In 2018 Xu et al. fabricated silicon
nanodisks on a gold substrate, realising an anapole mode [14]. The anapole was
used to enhance third harmonic generation by two orders of magnitude compared
to the same nanodisk on an insulating substrate. Maimaiti et al. fabricated a
similar device composed of polycrystalline silicon nanodisks on a gold
substrate, but with the introduction of an Al2O3 spacer layer between the
nanodisk and substrate [15]. This led to an electric field enhancement in the
spacer of over 40, and a Purcell factor of 300 for a vertically oriented
dipole in simulation. In experiment, they demonstrated fluorescence
enhancement, and surface-enhanced Raman spectroscopy of coupled molecules with
a more stable signal than solely plasmonic nanodisks. Finally, and most
applicable to this study, Dmitriev et al. fabricated silicon nanorings on gold
with a layer of embedded quantum emitters between them [16]. They observed Mie
resonances from the nanoring in dark field spectroscopy, along with strong
directionality of the coupled emitters normal to the substrate.
TMDs present an attractive alternative to silicon-based nanophotonics for
generating strong mode confinement owing to their higher refractive indices
[17], whilst also having no absorption over large parts of the visible
wavelength range [18]. Additionally, one can achieve high crystalline quality
of thin films (from monolayer up to $\sim 500$ nm) required for fabrication of
nanoantennas, through simple exfoliation from bulk TMDs [19]. Furthermore,
they can be exfoliated onto a range of other materials owing to their inherent
van der Waals attractive forces [1]. This avoids difficulties such as lattice
matching requirements and growth in molecular beam epitaxy chambers, which are
associated with fabricating nanoantennas from other conventional dielectrics
such as GaAs and GaP. Use of TMDs thus opens new possibilities in the design
and fabrication of hybrid dielectric-metallic structures of a variety of
thicknesses, enabling advanced control of photonic and plasmonic resonances on
the nanoscale.
In this study, we fabricate WS2 nanoantennas on a gold substrate and
characterise their optical response with experimental dark field spectroscopy,
which agrees well with finite-difference time-domain (FDTD) simulations. We
carefully examine the Mie and anapole resonances within such devices, and
observe dramatic changes in the mode structure compared to that of
nanoantennas on silica, with improved Q factors. We use both simulation, and
experimental scattering-type scanning near-field optical microscopy (s-SNOM)
to confirm the presence of hybrid Mie-plasmonic modes [20], which can enhance
surface plasmon-polaritons (SPPs) launched by illuminated nanoantennas. We
observe a Fano lineshape for these modes, unlike the Lorentzian lineshape
associated with Mie modes, therefore confirming the identification of hybrid
resonances. We experimentally demonstrate that they can be easily tuned to
different wavelengths by changing nanoantenna geometries. Such hybridised
modes exhibit experimental Q factors up to 94, nearly a factor of 20 higher
than Mie modes in nanoantennas on silica [21], highlighting potential
applications in switching and sensing [22, 23, 24, 25].
We further explore strong mode coupling between a higher-order anapole mode
(HOAM) and a Fabry–Pérot-plasmonic (FPP) mode within a WS2 nanoantenna on a
gold substrate in both experiment and simulation. Such resonances can be tuned
to realise an anti-crossing in the scattering spectra, with an experimental
minimum energy splitting of 48 $\pm$ 5 meV. At the point of anti-crossing, we
observe signatures of a highly confined supercavity mode [26], with a Q factor
of 263 $\pm$ 28, resulting from the destructive interference of the HOAM and
FPP mode outside of the hybrid nanoantenna structure. This offers one of the
first practical solutions to realising a supercavity mode in finite-sized
nanophotonic devices.
Finally, we model novel structures that include WS2 nanoantennas on top of 5
nm thick layers of hBN attached to a gold substrate. With this device, we
numerically achieve strong electric field enhancement up to 2647 in the hBN
spacer between the nanoantenna and gold. From this, we calculate Purcell
enhancements up to 713 for a dipole polarised normal to the substrate within
the hBN. This configuration offers the potential for enhancing the emission of
coupled single photon emitters (SPEs), or interlayer and moiré excitons [27,
28] in TMD heterostructures placed within the gap.
Results
Sample Fabrication We realised WS2 nanoantennas on a gold substrate using
well-established nanofabrication techniques (see Methods). We began by
fabricating gold film substrates using two methods: electron-beam evaporation
of gold onto a silicon substrate with either a titanium or nickel adhesion
layer, and template-stripping [29]. We measured RMS roughness values down to
1.2 and 0.7 nm respectively, before mechanically exfoliating bulk WS2 [19]
directly onto the gold. Flakes of varying thicknesses were found in each
exfoliation run, measured using atomic force microscopy (AFM), before
selecting those with thicknesses matching our simulations for nanoantenna
fabrication. A positive resist was then spun onto our sample, before using
electron beam lithography (EBL) to pattern arrays of circles of increasing
radii from 100 to 400 nm. We then carried out reactive ion etching (RIE) to
remove excess WS2 leaving behind nanoantennas. We used an isotropic etching
recipe with SF6 gas (pressure and DC bias detailed in Methods) to achieve
nanoantennas with hexagonal geometries. Fluorine radicals in the plasma etch
the armchair axis of the WS2 crystal much faster than the zig-zag axis,
leaving the adjacent etched nanostructure facets at 120° to each other [30, 6,
17]. The true radii of the nanoantennas, as measured by scanning electron
microscopy (SEM), were smaller than those in the resist pattern due to the
isotropic etching process, which slightly reduces their lateral size. The gold
substrate is chemically inert to SF6. This means that it acts as a natural
etch stop unlike SiO2, which would be etched along with the WS2, leaving the
nanoantennas on a small pedestal of substrate material. Here, the use of gold
eliminates this problem. Figures 1(a)-(d) illustrate the fabrication
procedure, and Figure 1(e) shows both optical and SEM images of the finalised
nanoantennas.
Figure 1: WS2 nanoantennas on gold fabrication and imaging. (a)-(d) Standard
nanofabrication technique of WS2 nanoantennas on gold. Material thicknesses
not to scale. (e) Optical image of nanoantennas arranged into arrays of 30
with increasing radii. Smaller nanoantennas not visible. Inset shows SEM image
illustrating the hexagonal shape of the nanoantennas.
Dark Field Spectroscopy of WS2 Nanoantennas on Gold In order to optically
characterise our fabricated samples, we carried out dark field spectroscopy on
individual WS2 nanoantennas (monomers) on gold. We measured three different
heights of nanoantennas (41, 78, 180 nm as measured by AFM) with a range of
radii for each, and plot the normalised scattering intensity in Figures
2(a)-(c). Panels (d)-(f) show the simulated scattering cross sections for the
same range of heights and radii using the FDTD method, exhibiting good
agreement. Note that panels (c) and (f) correspond to double nanoantennas
(dimers) with a gap between them on the order of 500 nm. We do not expect a
significant change in the scattering intensity between monomers and dimers,
especially for such large gap sizes [6].
Figure 2: Optical characterisation of WS2 nanoantennas on gold. (a)-(c)
Normalised scattering cross section from experimental dark field spectroscopy
of nanoantennas on gold of heights 41, 78, and 180 nm respectively. (d)-(f)
Simulated normalised scattering cross section of WS2 nanoantennas for the same
geometries as in experiment. Left and central columns show data corresponding
to single pillars (monomers), whilst right column corresponds to double
pillars (dimers) with a separation of 475 nm. ED corresponds to the electric
dipole mode, and AM and HOAM correspond to the anapole and higher-order
anapole modes respectively. MP, FP, and FPP correspond to Mie-plasmonic,
Fabry–Pérot, and Fabry–Pérot-plasmonic modes respectively. X0 represents the
WS2 exciton.
The mode structure becomes increasingly complex as we increase the nanoantenna
height. In the simplest case of monomers with height 41 nm (Figures 2(a) and
(d)), we observe a variety of maxima and minima in the scattering spectra that
red-shift with increasing radius. These are known as Mie resonances [2], which
are caused by bound charge oscillations within the WS2 crystal of the
nanoantennas. An example is the electric dipole mode, which leads to increased
scattering in the far-field as seen by the peaks labelled ED in Figure 2. The
dark band can be identified as an anapole mode (AM), which is a destructive
interference of the electric dipole mode and a magnetic toroidal mode [31],
causing suppression of the scattering in the far-field [32]. The dip in
scattering at a wavelength of 625 nm is due to WS2 excitonic absorption, where
we observe avoided crossings with the Mie modes as previously reported [3]. We
also label one of the modes MP, or Mie-plasmonic, which will be discussed in
greater detail in the next section.
We extracted the quality factor of the ED and MP mode in experiment for WS2
nanoantennas on gold of height 41 nm throughout the range of radii in Figure
2(a). A Lorentzian function was fitted to the ED mode, and a Fano curve for
the MP mode to calculate each respective Q factor. We extracted maximum Q
factors of 24 $\pm$ 0.6 for the ED mode of a nanoantenna with radius 48 nm,
and 94 $\pm$ 5 for the MP mode of a nanoantenna with radius 125 nm.
Furthermore, we compared these Q factors to those previously measured in
experiment for the ED mode in WS2 nanoantennas on a SiO2 substrate, which
reach a maximum Q factor of 5 $\pm$ 0.3 [21]. We observe a remarkable 19-fold
increase in Q factor when comparing the MP mode in nanoantennas on a gold
substrate, with the ED mode for nanoantennas on a SiO2 substrate.
This behaviour is further supported by simulation. We calculated maximum Q
factors from Figure 2(d) of 61 $\pm$ 0.6 for the ED mode of a nanoantenna of
radius 40 nm on gold, and 137 $\pm$ 1.6 for the MP mode of a nanoantenna of
radius 100 nm on gold. By comparing these values to that of the ED mode in
nanoantennas on a SiO2 substrate (Q = 4 $\pm$ 0.1 [21]), we observe that the
MP mode has a factor 34 higher Q factor than the ED mode for an all-dielectric
system in simulation.
For h = 78 nm, we observe a narrowing of the electric dipole mode compared to
h = 41 nm, along with a stronger red-shift with increasing radius. Additional
modes such as the HOAM and two MP modes are visible in both simulation and
experiment. When the height is increased to 180 nm, the mode structure becomes
much more complex. Not only do we see anapole and higher-order anapole modes,
but similar to previous simulations [33, 34, 35, 26, 36] and experiments [37,
38, 39, 40] of various dielectric nanostructures, we observe a Fabry–Pérot
(FP) mode trapped between the TMD-gold interface at the bottom of the
nanoantenna, and the TMD-air interface at the top. This mode shifts very
little with changing radius, as would be expected from a vertically
propagating FP mode. A more rigorous definition with comparison to FP mode
theory is provided in Supplementary Note 1. We also observe an anti-crossing
of a HOAM and FPP mode [40] in both the experimental and simulated scattering
spectra of the dimer nanoantennas plotted in Figure 2(c) and (f). This occurs
for nanoantennas with radii between 270 and 280 nm at a wavelength of around
800 nm, and is investigated in more detail later in this study.
Mie-Plasmonic Mode characterisation The mode structure of WS2 nanoantennas in
vacuum and on low-index substrates is very different compared to the case with
a gold substrate (see Supplementary Note 2 for a comparison of substrates). In
order to gain further insight into the origin of the modes observed in WS2-on-
gold nanoantennas, we simulated a monomer of a fixed geometry positioned at a
varied distance above the gold substrate. We moved the nanoantenna towards the
gold in 1 nm increments, starting at 20 nm, and calculated the scattering
cross section as shown in Figures 3(a) and (b). This was done for monomer
heights of 41 nm and 78 nm, and radii of 200 nm and 290 nm respectively, as
studied in experiment and reported in Figure 2. In order to characterise the
Mie resonances within the nanoantennas, we performed rigorous multipole
expansions of the scattered light using the open source software MENP [41].
From this analysis, partial scattering cross sections attributed to the
individual Mie modes can be extracted in order to assign them to their
respective peaks and dips in the spectra. This was carried out for a monomer
simulated in vacuum, as a homogeneous environment is necessary for the
expansion.
Figure 3: Characterisation of resonant modes in WS2 nanoantennas on gold. (a),
(b) Simulated normalised scattering cross section of WS2 nanoantennas with
increasing distance from a gold substrate, S. Geometries are h = 41 nm, r =
200 nm, and h = 78 nm, r = 290 nm respectively. X0 corresponds to the WS2
exciton, ED corresponds to the electric dipole mode, MP corresponds to the
Mie-plasmonic mode, AM and HOAM correspond to the anapole, and higher-order
anapole modes respectively. (c) and (d) show the Q factors of each simulated
resonance as a function of S. AM, MP and HOAM are fitted with a Fano curve,
and ED mode fitted with a Lorentzian. Error bars correspond to error in the
fitting. (e)-(j) Electric field distributions for different resonant modes
within a nanoantenna of height 41 nm and radius 200 nm for S = 0 nm. Both the
xy (top panel) and xz (bottom panel) views show a slice through the middle of
the nanoantennas in the corresponding planes. White dashed lines represent the
edges of the nanoantennas. Bottom right value corresponds to the incident
wavelength. (k) Experimental s-SNOM scattering amplitude data for three WS2
nanoantennas on a gold substrate for varying incident wavelengths as labelled
above each image. All data normalised to the scattering amplitude from an area
of gold free from SPP effects. Nanoantenna heights are all 41 nm with radii 76
nm (top), 163 nm (middle), and 266 nm (bottom) for comparison.
The green dashed line in Figures 3(a) and (b) follows the ED mode as S is
decreased from 20 to 0 nm. The peak red-shifts by 140 nm and narrows for the
nanoantennas in Figure 3(a) (h = 41 nm, r = 200 nm), therefore increasing the
quality factor from 6 to 8 as shown in Figure 3(c). We observe the electric
field distribution in the xz plane of the ED mode in Figure 3(h), where the
central lobe protrudes upwards and out of the top of the nanoantenna with the
introduction of the gold substrate, making the mode volume larger compared to
in vacuum (see Supplementary Note 3). We therefore attribute the red-shift of
the resonance to this increased mode volume. Conversely, the AM blue-shifts.
Again, by considering the mode volume of the anapole resonance, as shown in
Figure 3(i), we see a confinement of the mode owing to the gold substrate. The
central field maxima extends slightly less outside of the nanoantenna and into
the gold than compared to the case of a nanoantenna in vacuum (see
Supplementary Note 3), hence we attribute the small blue-shift to this
confinement. Furthermore, a resonance peak not seen in purely dielectric
systems emerges in the spectra when S is reduced to the order of 10 nm, which
we name Mie-plasmonic (MP) [42]. This peak is much sharper than that of the ED
mode, and red-shifts more strongly by 159 nm from S = 4 nm to S = 0 nm for the
h = 41 nm nanoantenna (Figure 3(a)). This behaviour is also seen for the MP
mode of the h = 78 nm nanoantenna. An enlarged plot of the MP mode evolution
is shown in Supplementary Note 4 for clarity, where an avoided crossing with
the AM and exciton is also visible. The electric field distribution of this
mode in the xz plane is very strongly localised to the gold-TMD boundary shown
in Figure 3(j), which suggests that there is a contribution from plasmons,
hence the naming Mie-plasmonic. This can also be observed for the ED and AM
cases. However, the relative electric field enhancement at the boundary is
much weaker. Interestingly, the MP mode takes a Fano lineshape in the spectra,
which suggests an interference between a discrete state and a continuum of
states [43, 24, 25]. Both the AM and HOAM also exhibit Fano lineshapes,
whereas they can be described by a Lorentzian function for the case of
nanoantennas on a dielectric substrate or surrounded by vacuum [44].
Supplementary Note 2 shows scattering cross sections for identical geometries
of WS2 nanoantennas on gold, SiO2 and in vacuum for further confirmation of
the different lineshapes with different substrates, suggesting that the gold
substrate introduces an interference between a continuum (i.e. plasmons) and a
discrete state (a Mie mode within the nanoantennas). Furthermore, the MP mode
is not observed in the cases with nanoantennas on SiO2 or in vacuum, and
neither does it appear in previous experimental or numerical dark field
studies [3, 4, 6]. This evidence further supports that there is a
hybridisation of both Mie and plasmonic modes present in our TMD nanoantenna-
on-gold system.
In addition to the previously measured Q factors from Figure 2, we note that
the AM and HOAM can reach even higher Q factors than the MP resonance when the
distance between the nanoantenna and gold, S, is varied in simulation. This is
shown in Figures 3(c) and (d). For a nanoantenna of height 78 nm and radius
290 nm, the MP mode reaches a maximum Q factor of 63 $\pm$ 0.1 at S = 0 nm,
whilst the AM has a maximum Q of 85 $\pm$ 0.4 likely owing to the strong
lateral confinement of this type of mode. The HOAM has an even larger maximum
Q factor of 145 $\pm$ 1.7 at S = 7 nm, suggesting an even stronger mode
confinement. The dips in Q factor correspond to values of S where different
modes cross each other in the scattering spectra, and so could be experiencing
an interference effect. Whilst the Q factor of the ED mode in Figure 3(c) only
increases very slightly with the introduction of a gold substrate, we see a
localisation of the mode towards the substrate in the electric field
distribution in Figure 3(h). This suggests that the ED mode also hybridises
with plasmons in the gold along with the MP mode, but not as strongly.
To further characterise the nanoantennas and verify the mode hybridisation
with plasmons, we performed s-SNOM imaging on arrays of WS2 nanoantennas of
height 41 nm on gold, for a range of wavelengths from 700 - 1000 nm. s-SNOM
involves probing the near-field response of a sample with an illuminated AFM
tip, and using interferometric techniques to resolve both the amplitude and
phase of the light scattered from the tip-sample interaction region. The tip
can then be scanned across the sample, as in Figure 3(k), which shows the
near-field scattering amplitude from the tip-sample interaction at each point,
exhibiting the formation of ripples around the nanoantennas. Such ripples are
the result of the tip illumination source interfering with SPPs on the gold,
or with scattered light from features on the sample [45, 46, 47, 12, 48, 49,
50, 51, 52]. As the tip is moved across the sample, the contributions to the
s-SNOM signal either interfere constructively or destructively. This produces
a pattern of bright and dark fringes in the amplitude of the scattered light,
which is observed as ripples. There are several methods by which this can
happen, and each produce different interference patterns [52, 47, 45, 46].
However, there are two prominent methods in the case of nanoantennas on gold,
the first being tip-launched SPPs. As the incident light reaches the tip, it
becomes strongly localised at the tip’s apex. This strong near-field
enhancement, and matching of the photon and plasmon wavevectors, causes tip-
launched plasmons that emanate radially into the gold [53]. Such SPPs can then
reflect from nearby structures and interfere with the incident light back at
the tip. The second mechanism involves SPPs launched from the nanoantennas
when illuminated [53], which travel to the tip and interfere with the incoming
light. A combination of these two effects is observed in the ripple patterns
in Figure 3(k).
As the incident wavelength is varied, the wavelength of the SPPs produced
changes according to their dispersion relation, resulting in a change of the
distance between the maxima of the ripple intensities. From Figure 3(k), we
note that the distance between the ripple maxima increases with the wavelength
of the excitation laser, but is constant across nanoantenna radius. This
behaviour can also be seen in the s-SNOM images of the full array of
nanoantennas in Supplementary Note 5. In contrast, the amplitude of the
ripples corresponds strongly to the nanoantenna size and incident wavelength.
By comparing the s-SNOM data in Figure 3(k) to the dark field spectra of the
same nanoantennas in Figure 2(a) (see Supplementary Note 5 for direct
comparison), we note a correlation between the amplitude of the SPP ripples
and peaks in the corresponding dark field spectra. At shorter wavelengths,
such as 700 nm, we observe the highest amplitude ripples from the smallest
nanoantenna (r = 76 nm), followed by the r = 163 nm nanoantenna. This agrees
with our experimental dark field data, where we see a strong resonant ED mode
around 700 nm for the r = 76 nm nanoantenna, and the MP mode for the r = 163
nm nanoantenna (see Supplementary Note 5). As the incident wavelength is
increased to 800 nm, the r = 76 nm nanoantenna begins to resonate less in the
s-SNOM images, and the r = 163 nm nanoantenna shows a stronger SPP
interference pattern, which we attribute to the excitation of the MP mode. At
900 nm illumination, both the r = 163 and 266 nm nanoantennas exhibit strong
SPP ripples, following the red-shift of the ED mode in the dark field data
(see Supplementary Note 5). Finally, at 1000 nm illumination wavelength, the
amplitude of the SPP interference pattern around all of the nanoantennas is
much lower, corresponding to the dip in overall scattering in the dark field
spectra (Figure 2(a)). This observation demonstrates that van der Waals
nanoantennas on gold can both scatter light to the far-field, and couple light
to SPPs detectable in the near-field. Such SPPs can be further enhanced via
the excitation of various Mie resonances, such as the ED and MP modes within
the nanoantennas, hence providing further evidence for the coupling between
Mie and plasmonic modes.
We do not observe SPP ripples on areas of gold far away from the nanoantennas
(see Supplementary Note 6), as in this case, there are no features on the
substrate to launch SPPs from. Only tip-launched SPPs are present, which
travel away from the tip radially and have no edges to reflect back from.
Therefore, no SPPs interfere with incident light at the tip, and so the s-SNOM
image appears uniform. In Supplementary Note 6, we also show an area of the
sample with pillars of resist between 25-35 nm in height on the gold, left
over from the nanoantenna fabrication process. We observe negligible optical
response from the resist pillars, which can be attributed to their low
refractive index of 1.49 [54]. We therefore expect tip-launched plasmons to be
mostly transmitted through such structures with little reflection back to the
tip. In addition, light incident on the resist pillars is not expected to lead
to well confined photonic resonances unlike the WS2 nanoantennas. This
observation further supports our previous statement, suggesting that only the
high refractive index WS2 nanoantennas can launch SPPs through strongly
confined, hybrid Mie-plasmonic resonances.
Supercavity Mode in WS2 Nanoantennas on Gold In addition to the plasmon
hybridisation discussed previously, we observe signatures of a highly-
confined, non-radiating supercavity mode in both experiment and simulation by
tuning the radii of WS2 nanoantennas on gold. This occurs as a result of the
destructive interference of two different photonic modes outside of the
nanoantenna, thus forming an extremely confined mode with a Q factor that, in
theory, increases to infinity [26]. This is analogous to a Friedrich-Wintgen
bound state in the continuum (BIC) [55], but for a finite-sized structure such
as a nanoantenna. We observe signs of this in the form of an anti-crossing
between the HOAM and FPP mode from Figure 2(c). In Figure 4(a), we investigate
such behaviour in greater depth, by fitting the two modes to a coupled
oscillator model.
Figure 4: Supercavity mode characterisation for a WS2 dimer nanoantenna of
height 180 nm and separation 475 nm, on a gold substrate. (a) Experimental
peak center positions of the anti-crossed HOAM and FPP mode fitted to a
coupled oscillator model, yielding a minimum energy splitting of 48 $\pm$ 5
meV. Error bars represent uncertainty in the measured radii of the
nanoantennas. (b) Quality factor of the two modes with respect to radius.
Error bars represent both uncertainty in the measured radii, and error in the
fits. Q factor calculated as the central wavelength of each peak divided by
its respective full-width-at-half-maximum. (c) Simulated scattering spectra
corresponding to WS2 dimers of height 200 nm over a range of radii increasing
in steps of 1 nm on a gold substrate. Pink and blue circles correspond to the
FPP mode and HOAM respectively. White central circle denotes the position of
the high Q factor supercavity mode (SM), and the purple circle corresponds to
the low Q factor lossy mode. Remaining insets show electric field
distributions through the center of a single nanoantenna in the xz plane at
the corresponding circles in the scattering spectra. White boxes highlight the
edges of the nanoantennas, and yellow lines correspond to the position of the
gold substrate. (d) Waterfall of normalised simulated scattering spectra from
(c) for different radii showing suppression of the high Q factor mode for the
nanoantenna radius corresponding to the minimum wavelength splitting between
the two modes (r = 302 nm).
We first fitted the HOAM and FPP peaks with a double Fano formula to account
for the hybridisation of the individual modes with plasmons, as detailed in
Supplementary Note 7. We then extracted the peak center positions and plotted
them in terms of energy against nanoantenna radius as shown in Figure 4(a).
The error bars represent the uncertainty in the measured radii of each
nanoantenna, with one data point at r = 293 nm showing a notably large error.
We attribute this uncertainty to fabrication imperfections of this particular
nanoantenna, which had a more irregular hexagonal cross-section than others
when imaged with SEM, thus making the determination of its radius less
reliable. The error in the fitted peak energy is negligible. The peak center
positions were then fitted to a coupled oscillator model, yielding the upper
and lower energy branches, shown as red and blue lines respectively. We refer
to these as the high and low Q factor modes respectively. The green dotted
lines represent the uncoupled HOAM and FPP mode for reference. From this
fitting, we extract a minimum energy splitting of 48 $\pm$ 5 meV. This is
greater than the sum of the half linewidths of the uncoupled modes (34 $\pm$ 1
meV), and hence we confirm strong mode coupling [36].
Furthermore, we calculate the Q factor of each peak and plot against
nanoantenna radius in Figure 4(b). Whilst the low Q factor mode remains mostly
constant, the high Q factor mode increases significantly at a radius of 273
nm. This radius corresponds to the closest point to the anti-crossing in
Figure 4(a). We observe a maximum Q factor of 263 $\pm$ 28; an order of
magnitude larger than for the ED mode reported earlier in this study. This
sharp increase in Q factor along with the observation of strong mode coupling
are both signatures of a supercavity mode. We expect to observe much greater Q
factors by fabricating more optimised structures with changes in radii down to
1 nm, in line with our simulations detailed in Supplementary Note 7.
In order to better understand the mode behaviour around the anti-crossing, we
performed FDTD simulations of WS2 nanoantennas of height 200 nm on gold for a
range of radii of 250 to 350 nm, with a much finer step in radius of 1 nm as
shown in Figure 4(c). This greater height was chosen in order to red-shift the
anti-crossing away from other modes in the scattering spectra to aid with
fitting. The anti-crossing is clearly reproduced in the simulated spectra, and
we extract an energy splitting of 37.5 $\pm$ 0.3 meV (see Supplementary Note
7), agreeing with our experimental observations. We also simulate the electric
field distribution within the nanoantennas at the points marked by the
coloured circles in Figure 4(c). Away from the point of anti-crossing, the FPP
mode displays clear maxima and minima in the xz plane as would be expected
from a Fabry–Pérot mode confined vertically within the nanoantenna [36].
However, we also see evidence of hybridisation with plasmonic modes at the
gold-WS2 boundary, similar to the MP modes described previously in this study.
In addition, we do not observe this mode in WS2 nanoantennas on a SiO2
substrate in either simulation or experiment [21]. We therefore attribute this
resonance to a hybrid Fabry–Pérot-plasmonic mode as a result of reflections
from, and SPPs at, the WS2-gold interface. For smaller radii the FPP mode
appears mostly plasmonic, with a field maxima near the bottom of the
nanoantenna. In contrast, the HOAM field distribution is strongly localised at
the center of the nanoantenna, exhibiting little hybridisation with plasmons.
As the radius is increased, the electric field profile of the FPP mode
hybridises with that of the HOAM to form a supercavity mode labelled SM. Upon
increasing the nanoantenna radius further, the HOAM returns to a similar field
distribution as before the anti-crossing, with the central field maxima
localised closer to the gold interface. However, the FPP mode field maxima is
pushed up towards the top of the nanoantenna, whilst retaining the
characteristic plasmon field distribution at the bottom.
We further investigate the suppression of the high Q factor mode as the
nanoantenna radius is tuned. Figure 4(d) shows individual simulated scattering
spectra from Figure 4(c) for a range of radii close to the anti-crossing.
Whilst the low Q factor mode at higher wavelength remains mostly the same, the
high Q factor mode at lower wavelength becomes almost invisible for a radius
of 302 nm. This suppression of scattering corresponds to the point where the
HOAM and FPP mode destructively interfere near perfectly, forming a highly
confined resonance within the nanoantenna. Our simulations thus provide
additional evidence to support our observation of a supercavity mode in hybrid
WS2-on-gold nanoantennas in experiment.
Electric Field Confinement Between WS2 Nanoantennas and Gold The strong
localisation of the electric field at the TMD-metal boundary, depicted in
Figure 3(j), prompted further study into Purcell enhancement of emitters at
this position. We simulated the electric field distribution within an hBN
layer of 5 nm thickness between a WS2 nanoantenna and the gold substrate as
shown in Figure 5(a). hBN was chosen owing to its transparency throughout the
visible wavelength range [56], low refractive index of 2.2 [57], and the
presence of single photon-emitting defects, radiating at a variety of
wavelengths from around 550 - 850 nm [58, 59, 60, 61]. The results are shown
in Figures 5(b) and (c) for a nanoantenna of height 60 nm and radius 210 nm.
The geometry was optimised for the maximum possible Purcell factor within the
hBN layer over the wavelengths previously reported for hBN SPEs. We calculated
a maximum electric field enhancement of 2647 within the hBN spacer at 773 nm
wavelength (the MP mode); two orders of magnitude higher than the maximum
field within the nanoantennas for the ED mode (Figure 3(h)), and one order of
magnitude higher than that of the MP mode inside a nanoantenna directly on a
gold substrate (Figure 3(j)). Based on these calculations, we spatially mapped
the Purcell factor of a dipole emitter placed within the hBN (see Figure
5(a)), mimicking an SPE. This mapping is shown in Figures 5(e) and (f), where
the dipole was oriented perpendicular (along z), and parallel (along x) to the
substrate respectively.
Figure 5: Simulated electric field and Purcell factor throughout an hBN layer
between a WS2 nanoantenna and a gold substrate. (a) Schematic of the hBN
spacer showing the z position at which the dipole was placed to simulate the
Purcell factor. Crossed arrows represent the two polarisations considered.
Nanoantenna of height 60 nm and radius 210 nm. (b), (c) correspond to electric
field distributions within the nanoantenna-substrate gap when illuminated by a
773 nm plane wave in the xy plane, and the xz plane through the center of the
nanoantenna respectively. Dashed white lines represent the edges of the
nanoantenna, yellow denotes the gold-hBN boundary, and grey corresponds to the
top surface of the hBN. (d) Polarisation dependency of the total emitted
intensity of a dipole over a range of 550 - 850 nm at the position marked by
the green cross in (e). Dipole was rotated 360° in the xz plane. (e), (f) Maps
of the maximum Purcell factor for a dipole oriented perpendicular (polarised
along z), and parallel (polarised along x) to the substrate respectively in
the same plane as (b). Dipole wavelength set to 773 nm and 731 nm
respectively, corresponding to the MP and AM modes.
From Figure 5(e), we calculated a maximum Purcell factor of 713 for a dipole
polarised perpendicular to the substrate at an emission wavelength of 773 nm
(the MP mode), over two times greater than previously reported for silicon
nanoantennas above gold for the same gap size [15]. Comparing to the dipole
polarised parallel to the substrate in Figure 5(f), we observed a much lower
maximum Purcell factor of 22 at 731 nm (the AM). We integrated the total
emitted intensity of the dipole in all directions over a wavelength range of
550 - 850 nm for varying polarisations as shown in Figure 5(d). The dipole was
located at the position of the Purcell factor maxima from Figure 5(e), marked
by a green cross, with the polarisation being rotated in the xz plane. We
observed a strong sensitivity to dipole orientation within the nanoantenna-
substrate gap which suggests that particular enhancement is expected in
structures where the dipole is oriented vertically. An example is hetero- and
homobilayer TMDs, where interlayer excitons can be observed having the
electron and hole in adjacent layers [62]. Therefore, we predict that by
placing a TMD heterostructure within the nanoantenna-gold gap, one could
achieve excitonic emission enhancement up to 32 times greater than for
intralayer excitons in a similar structure, where the exciton lies in plane
with the substrate.
Conclusion
In this study, we fabricated and characterised a hybrid dielectric-metallic
nanophotonic system composed of WS2 nanoantennas on a gold substrate. Such
TMD-based nanoantennas are easy to fabricate on metals using standard
nanofabrication techniques owing to their van der Waals forces acting between
the TMD thin film and the substrate, with the added benefit of gold providing
a natural etch stop during RIE. We then investigated resonant Mie modes within
the structures through simulation and observed excellent agreement in
experimental dark field spectroscopy. We demonstrated that all the resonant
modes identified can be tuned to different wavelengths simply by changing the
nanoantenna radii, and that additional, higher-order modes can be introduced
by increasing the nanoantenna height. Fano resonances were observed in WS2
nanoantennas on a gold substrate, not present in the same structures on SiO2,
which we identify as a hybridisation of Mie modes and SPPs through both
simulation and experiment. These modes couple to the far-field, as measured
with dark field spectroscopy, and produce SPPs detectable in the near-field
with s-SNOM imaging. The SPP intensities exhibited resonant behaviour,
following the red-shift of the modes upon increasing nanoantenna radii,
further supporting our claims of hybridised Mie-plasmonic modes. Such hybrid
Fano resonances also have high Q factors, almost 20 times higher than Mie
modes in nanoantennas placed on a low-index SiO2 substrate in experiment [21],
hence enabling applications in switching and sensing [22, 23, 24, 25].
We further demonstrated strong mode coupling of Mie and Fabry–Pérot-plasmonic
modes within WS2 nanoantennas on a gold substrate in experiment and
simulation. We calculated a minimum energy splitting of 48 $\pm$ 5 meV, and
with careful tuning of the nanoantenna geometry we discovered signatures of a
quasi-BIC supercavity mode at the point of anti-crossing, including a
significantly increased experimental Q factor of over 260, and near complete
suppression of scattering in simulation. To the best of our knowledge, the use
of a gold substrate reveals one of the first realistic methods of achieving a
supercavity mode in van der Waals nanoantennas in experiment.
Finally, we observed in simulations that very strong electric field
enhancement of over 2600 occurs in a nanometer scale gap between the studied
WS2 nanoantennas and gold substrate. For a gap filled with 5 nm of hBN, we
calculated a Purcell factor of 713 for an emitter within the hBN polarised
perpendicular to the substrate; 32 times higher than for parallel
polarisation. This introduces opportunities for enhancing emitters placed in
this nanoscale gap, such as SPEs in TMDs [63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74] and hBN [58, 59, 60, 61], as well as interlayer excitons in TMD
bilayers [62] and van der Waals heterostructures [75, 76, 77, 78, 79, 80].
We have shown experimentally that merits from both plasmonic and dielectric
regimes can be achieved, and that hybrid Mie-plasmonic resonances can be tuned
easily for any desired application by changing nanoantenna geometries,
resulting in high Q factors not previously observed in solely dielectric
nanoantennas. We believe that our hybrid nanoantenna system will open up
additional pathways for future nanophotonic structures and resonators, with
immediate applications for single photon emitters and photoluminescence
enhancement in van der Waals heterostructures. Coupling of this system to
other photonic devices such as waveguides, photonic crystals, and gratings,
offers near limitless combinations for using TMDs and metals together to
fabricate nano-optical circuits with strong field confinement and low losses.
Methods
FDTD Simulations In order to predict the behaviour of light within and around
our nanoantennas, the software package Lumerical from Ansys was used to
perform FDTD simulations.
Scattering Simulations The scattering cross-sections in Figure 2 were
calculated by simulating WS2 nanoantennas of varying geometries on a semi-
infinite gold substrate. To emulate dark field experiments as closely as
possible, a total-field scattered-field (TFSF) plane wave source was used
which subtracts the incident wave outside of its area of effect. This way,
only the scattered light in the far-field was measured by a power monitor
placed above the nanoantenna. The incident wave was set to propagate normal to
the substrate and was polarised along the x axis. Anti-symmetric and symmetric
boundary conditions were used along the x = 0 and y = 0 planes respectively,
to reduce simulation time and memory requirements.
Field Distributions To visualise the electric and magnetic field distributions
within and around the nanoantennas, frequency-domain field and power monitors
which perform discrete Fourier transforms (DFTs) at chosen frequencies were
used. The monitors were set as 2D surfaces through the middle of the
nanoantennas used in the scattering simulations along various planes, and
returned the electric and magnetic field intensities normalised to the
incident, vacuum wave.
Purcell Factor Calculations We considered a dipole emitter placed in an hBN
spacer between our WS2 nanoantennas and a gold substrate to emulate an SPE.
The wavelength was set to a range of 550 - 850 nm and the orientation of the
dipole rotated in the xz plane to consider different polarisations. The
Purcell factor was then calculated as the total integrated power of the system
divided by the total integrated power of the same dipole in vacuum.
Substrate Preparation The gold substrates were fabricated using either
template stripping (used in the structures measured in Figure 2(a)) or
electron beam evaporation of roughly 150 nm of 99.99% pure gold onto a silicon
wafer with a 10 nm titanium (used in the structures measured in Figures 2(b)),
or nickel layer (used in the structures measured in Figure 2(c)) to promote
adhesion to the gold. These had rms roughness values down to 0.7 nm, 1.2 nm,
and 2.5 nm respectively.
TMD Exfoliation WS2 bulk crystal from HQ-graphene was mechanically exfoliated
onto the gold substrates by hand. A temperature of 105 °C was used to ensure
good flake adhesion. Uniform thickness flakes of sizes 50 $\mu$m and upwards
were recorded for patterning.
Electron Beam Lithography A positive resist (ARP-9 AllResist GmbH) was first
spin-coated onto the sample at 3500 rpm for 60s before heating for 2 minutes
at 180°C. EBL was then performed using a Raith GmbH Voyager system at 50 kV
accelerating voltage and 560 pA beam current. The pattern formed an array of
circles of varying radii across the resist to cover several WS2 flakes.
Reactive Ion Etching A chemical etching recipe was used to achieve hexagonal
nanoantenna geometries. Plasma etching was performed for 40 s with SF6 gas at
0.13 mbar pressure with a DC bias of 50V. The armchair crystal axis of the
bulk WS2 was preferentially etched faster than the zigzag axis leading to 120°
angles between them, forming hexagonal pillars [30]. The gold substrate was
etched much slower than the WS2, and so acted as a natural etch stop, leaving
nanoantennas on a flat gold surface, rather than on a pedestal of substrate
material. The leftover resist was then removed using warm 1165 resist remover,
before bathing in acetone, followed by IPA for 5 min respectively. A final UV-
ozone treatment of 20 min removed any residual organic debris.
Dark Field Spectroscopy Spectroscopy involving illuminating a sample whilst
rejecting the reflected light and collecting only the scattered light was
achieved using a Nikon LV150N microscope with a fitted circular beam block
between the illumination source (tungsten halogen lamp) and the dark field
objective lens (50x with 0.8 NA). The beam block used was slightly smaller
than the diameter of the beam, so that the central part was discarded and only
the outer ring of light entered the objective via redirection from an annular
mirror. The sample was illuminated at a high oblique angle causing light to be
scattered from the sample. The vertically scattered light was then collected
by the objective and passed back through the hole in the annular mirror
towards a 50 $\mu$m pinhole before a fiber coupler. The pinhole ensured that
only light scattered at a low angle to the normal was allowed to propagate
into the 100 $\mu$m diameter core of the multi-mode fiber. Another fiber
coupler then sent the beam into a free space path, where two achromatic lenses
were used to minimise beam diversion along the path to the spectrometer.
Finally, a single achromatic lens was used to focus the beam onto the slit of
a Princeton Instruments spectrometer, where the wavelength components were
separated and detected by a charge coupled device.
s-SNOM Probing of the near-field scattering from our samples at the nanoscale
was done using a commercial neaspec modular s-SNOM system in conjunction with
a Coherent Chameleon Compact OPO-Vis pulsed laser. This technique combined a
sharp AFM tip with incident radiation to strongly confine near-fields at the
tip-sample interface, and measure the phase and amplitude of the scattered
light. The laser was aligned onto a Platinum-Iridium (PtIr) coated cantilever
tip (NanoWorld Arrow NCPt), with radius of curvature less than 25 nm, using a
parabolic mirror within the s-SNOM system. The beam made a 60° angle with the
tip, and was polarised parallel to the plane of incidence (p-polarised), to
maximise the component along the tip axis. A strongly-confined near-field was
generated at the tip, which then interacted with the sample as it was scanned
below. Background scattering signal owing to the large spot size (few microns)
compared to the tip size, was suppressed using neaspec’s patented pseudo-
heterodyne interferometry system. A reference beam with a phase modulation
induced via an oscillating mirror was interfered with the scattered signal at
the detector. This formed sidebands of frequency $n\Omega+m\Delta$, where
$\Omega$ is the tapping frequency of the tip, and $\Delta$ is the modulation
frequency of the reference mirror. The detector then locked in at harmonics of
the tapping and sideband frequencies in order to eliminate background signal.
Through using pseudo-heterodyne detection, both the amplitude and phase of the
scattered light were measured simultaneously.
The s-SNOM measurements in this report were demodulated at either the 3rd
(Figure 3(k) and Supplementary Note 5) or 4th (Supplementary Note 6) harmonic
of $\Omega$ and the first sideband in order to reduce the background as much
as possible, whilst still keeping a good signal to noise ratio.
Acknowledgements
S.A.R., P.G.Z., X.H., A.J.K., S.N., D.H. and A.I.T. acknowledge support from
the European Graphene Flagship Project under grant agreement number 881603 and
EPSRC grants EP/S030751/1, EP/V006975/1, EP/V006975/1 and EP/V026496/1. Yadong
Wang and A.I.T. acknowledge support from UKRI fellowship TWIST-NANOSPEC
EP/X02153X/1. Yue Wang acknowledges a Research Fellowship (TOAST) awarded by
the Royal Academy of Engineering. S.A.R., P.G.Z., S.N. and D.H. acknowledge IT
Services at The University of Sheffield for the provision of services for High
Performance Computing.
Author Contributions
Yue Wang fabricated the gold film substrates. S.A.R. exfoliated bulk WS2 onto
the gold substrates and conducted AFM to ascertain flake thicknesses. Yue Wang
and X.H. patterned and etched the WS2 flakes into hexagonal nanoantennas using
EBL and RIE. S.A.R. and Yadong Wang characterised nanoantenna radii through
SEM. S.A.R. and P.G.Z. measured dark field spectra for all nanoantennas and
analysed the results. S.A.R. and A.J.K. characterised the nanoantennas using
s-SNOM imaging at different wavelengths. S.A.R., P.G.Z., S.N., and D.H. all
contributed to FDTD simulations of the scattering spectra, electric field
distributions, and Purcell factors. S.A.R., P.G.Z. and A.I.T. wrote the
manuscript. A.I.T. managed the whole project.
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|
11institutetext: University of Mannheim, Chair of Data Science, 68159
Mannheim, Germany22institutetext: University of Mannheim, Chair of Public Law,
Regulatory Law and Tax Law, 68159 Mannheim, Germany
# Winning at Any Cost - Infringing the Cartel Prohibition With Reinforcement
Learning††thanks: The work presented in this paper has been conducted in the
_KarekoKI_ project, which is funded by the _Baden-Württemberg Stiftung_ in the
_Responsible Artificial Intelligence_ program.
Michael Schlechtinger 11 0000-0002-4181-3900 Damaris Kosack 22
0000-0002-7599-3233 Heiko Paulheim 11 0000-0003-4386-8195 Thomas Fetzer 22
0000-0002-5148-4610
###### Abstract
Pricing decisions are increasingly made by AI. Thanks to their ability to
train with live market data while making decisions on the fly, deep
reinforcement learning algorithms are especially effective in taking such
pricing decisions. In e-commerce scenarios, multiple reinforcement learning
agents can set prices based on their competitor’s prices. Therefore, research
states that agents might end up in a state of collusion in the long run. To
further analyze this issue, we build a scenario that is based on a modified
version of a prisoner’s dilemma where three agents play the game of rock paper
scissors. Our results indicate that the action selection can be dissected into
specific stages, establishing the possibility to develop collusion prevention
systems that are able to recognize situations which might lead to a collusion
between competitors. We furthermore provide evidence for a situation where
agents are capable of performing a tacit cooperation strategy without being
explicitly trained to do so.
###### Keywords:
Multi Agent Reinforcement Learning Pricing Agents
Algorithmic Collusion.
## 1 Introduction
Dynamic reinforcement learning based pricing strategies supersede static ones
in terms of average daily profits [Kropp.2019]. As 27 percent of the
respondents of a 2017 study by KPMG identified price or promotion as the
factors that are most likely to influence their decision regarding which
product or brand to buy online [KPMG.], it is to be expected that successful
companies (such as Amazon [Chen.2016]) base their decisions on these
algorithms to learn from and react to their competitor’s pricing policies as
well as to adjust to external factors, such as a transformation of demand or
product innovations [Ezrachi.2017]. Monitoring these AIs is getting
increasingly complex as the market is distributed worldwide, the barriers of
entry are minimal, and the amount of created pricing data grows quicker by the
day.
Primarily legal scholars have commented on the possibility of self-learning
algorithms to quickly learn to achieve a price-setting collaboration
especially within oligopolies (e. g., [Ezrachi.2016, Ezrachi.2015,
Ezrachi.2017]). With the power of modern hardware, AIs would be able to
monitor the market in which they act, resulting in a rapidly arising tacit
collusion. Researchers investigated the issue by creating game theory like
scenarios with the intention of pushing the agents towards a Nash equilibrium
(e. g., [Ezrachi.2017, Waltman.2008]). In essence, it seems to be “incredibly
easy, if not inevitable” to achieve “such a tacitly collusive, profit-
maximizing equilibrium” [Schwalbe.2018]. While collusion has been presumed to
appear in enclosed multi agent reinforcement learning scenarios, scholars have
neither studied how to spot the origin of collusion nor if competitors can
apply tacit collusion by displacing the others.
In an effort to simplify the dynamic pricing data analysis, we aim to train a
competitive multi agent reinforcement learning (MARL) game simulation. In this
game, the agents play a three-player version of rock paper scissors (RPS). We
aim to analyze the effect of the competitive RPS scenario on the agents’
learning performances and potential collaboration strategies. In specific, we
aspire to analyze whether RL agents are capable of performing a tacit
cooperation or communication strategy without being explicitly trained to do
so.
## 2 Related Work
### 2.1 Infringing the Cartel Prohibition
In its most recent proposal for an Artificial Intelligence Act, the European
Commission emphasises the importance of the safety and lawfulness of AI
systems, of legal certainty with regard to AI, the governance and effective
enforcement of existing law on fundamental rights and the installation of
safety requirements [EuropeanCommission.20210421]. In line with these goals,
AI price policies must oblige to competition law just as prices that are set
by humans. Both European and German competition law distinguish three possible
conducts of infringing the cartel prohibition, see Article 101 (1) Treaty on
the Functioning of the European Union (”TFEU”)111Corresponding provision under
German law: § 1 Act against Restraint of Competition; corresponding provision
under US law Section 1 Sherman Antitrust Act of 1890.: (a) _agreements between
undertakings_ , (b) _decisions by associations of undertakings_ , and (c)
_concerted practices_. Independent undertakings shall independently decide
over their market behavior and must not coordinate it with their competitors
(“requirement of independence”). This requirement does strictly preclude any
direct or indirect contact by which an undertaking may influence the conduct
on the market of its actual or potential competitors or disclose to them its
decisions or intentions concerning its own conduct on the market where the
object or effect of such contact is to create conditions of competition which
do not correspond to the normal conditions of the market
[EuropeanCourtofJustice.].
The independently chosen intelligent adaption of an undertaking’s market
behavior to the observed market behavior of its competitors (generally) is
permitted. Drawing a clear line between the adaption of an observed market
behavior and a conduct through which competition knowingly is replaced by a
practical cooperation and therefore constitutes a concerted practice within
the meaning of Article 101 (1) TFEU222For US law see [DOJandFTC.2123June2017,
Gulati.] is often difficult and sometimes even impossible. Especially on
transparent markets with few market participants, the market outcome of
collusion can often hardly be traced back to be (or not to be) the product of
a concerted practice (cf. petrol station market). Although collusion as a
market outcome can be detrimental to consumers, innovation and economic growth
and is therefore undesirable from a welfare economic point of view, the
difficulty from a dogmatic perspective is that legal responsibility cannot be
attached to a market outcome as such [Weche.2020].
Our goal is to disclose whether a certain sequence of actions or a specific
pattern can be identified as a situation in which the uncertainty about the
competitor’s next moves is replaced by a practical cooperation. It is
conceivable that such accurate determination might not be possible due to the
increased market transparency achieved by the self-learning algorithms: their
ability to quickly process large amounts of competition-relevant data and to
react to price movements in an almost unlimited frequency might lead to such a
high degree of transparency on a market that makes it impossible to determine
from its outcome whether or not the result of collusion is due to intelligent
market observation and parallel behavior or a concerted practice.
### 2.2 Multi Agent Reinforcement Learning
A tacit collaboration between some reinforcement learning agents can only
occur in certain situations. The agents have to interact within a multi agent
reinforcement learning (MARL) environment, where competing agents and prices
are recognized as a part of such [Charpentier.2020]. Due to that, the
environment is usually subjective for every agent, resulting in a differing
learning performance and a diverse landscape of achieved competencies. It is
unclear whether one of these competencies might arise in the skill to
communicate with specific other agents to adjust their pricing policies
accordingly; resulting in a higher producer’s pension and a displacement of a
competitor.
Researchers have investigated circumstances which can be juxtaposed with
collusion between pricing agents, such as bidding processes [Schwind.2007,
Dutting.12.06.2017] or economy simulations [Zheng.28.04.2020]. However, the
authors did not control for or induce communication or collaboration. To
combat this shortcoming, scholars within the economics realm created
oligopolistic models (particularly Cournot oligopolies) to show collusion
between agents. A Cournot oligopoly is characterized by an imperfect
competition, where firms individually have some price-setting ability but are
constrained by rivals [AugustinA.Cournot.1836]. Izquierdo and Izquierdo
[IZQUIERDO.2015] show that simple iterative procedures, such as the win-
continue, lose-reverse (WCLR) rule are able to achieve collusive outcomes.
However, the results are not robust in terms of minor, independent
perturbations in the firms’ cost or profit functions. Similar results were
achieved with basic Q-learning [Waltman.2008]. As a case in point, using a
price-setting duopoly model with fixed production, in which two firms follow a
Q-learning algorithm, Tesauro and Kephart [Tesauro.2002] observed convergence
to prices higher than the competitive level. Major takeaways from these
studies are, that cooperation is more likely to occur in simplified, static
environments with a homogeneous good and that communication is vital to
achieve collusive outcomes, particularly when more than two firms operate in a
market. Such results suggest that the ability to communicate could also be
pivotal for algorithmic collusion to occur [Schwalbe.2018].
## 3 Methodology
### 3.1 Problem Definition
Oroojlooy and Hajinezhad [OroojlooyJadid.11.08.2019] recommend to model a MARL
problem based on (i) centralized or decentralized control, (ii) fully or
partially observable environment and (iii) cooperative or competitive
environment. Our case demands for a decentralized control, with a partially to
fully observable environment, so that every agent is able to make its own
decisions based on the information given by the environment. Lastly, we apply
a cooperative inside of a competitive environment, so that agents are able to
team up against other agents.
### 3.2 Approach
Table 1: Three player RPS combinatorics. Agent 1 | Agent 2 | Agent 3 | $r_{1}$ | $r_{2}$ | $r_{3}$
---|---|---|---|---|---
Rock | Paper | Scissors | 0 | 0 | 0
Rock | Rock | Rock | 0 | 0 | 0
Scissors | Scissors | Scissors | 0 | 0 | 0
Paper | Paper | Paper | 0 | 0 | 0
Scissors | Rock | Rock | -1 | 0.5 | 0.5
Rock | Paper | Paper | -1 | 0.5 | 0.5
Paper | Scissors | Scissors | -1 | 0.5 | 0.5
Paper | Rock | Rock | 2 | -1 | -1
Rock | Scissors | Scissors | 2 | -1 | -1
Scissors | Paper | Paper | 2 | -1 | -1
… | … | … | … | … | …
Expected Reward $r$ | 0 | 0 | 0
With the intention of simplifying a realistic economy simulation, we choose to
build a MARL-game based on a three player version of RPS. Every agent
$i=\\{1,\,...,\,3\\}$ represents a player with a set of legal game actions
$A=\\{1,\,...,\,3\\}$ comprising the moves of rock, paper and scissors. The
agents interact with a stochastic environment $E$ which solely contains the
chosen actions of every agent of the current time step $t$. Hence, a state at
$t$ can be described as $s_{t}=\\{a_{1}^{\prime},\,...,\,a_{i}^{\prime}\\}$.
Following a collective action, every agent receives a reward out of
$R=\\{\text{-}1,\,0,\,0.5,\,2\\}$ mapped to the possible game outcomes
presented in table 1, resulting in a direct association between input and
output. This formalism gives rise to a finite Markov decision process (MDP) in
which every $t$ relates to a distinct state, encouraging an application of
standard reinforcement learning methods for MDPs. The goal of the agent is to
interact with $E$ by selecting actions in a way that maximises future rewards.
As the agents receive a reward at the end of every timestep, we will not apply
any discount to future rewards. We define the optimal action-value function
$Q^{\ast}(s,a)$ as the maximum expected return achievable by following any
strategy, after seeing some sequence $s$ and then taking some action $a$,
$Q^{\ast}(s,a)=\max_{\pi}\mathbb{E}[R_{t}|s_{t}=s,a_{t}=a,\pi]$, where $\pi$
is a policy that maps sequences to actions (or distributions over actions). In
an attempt to induce strategic behavior, resulting in a tacit communication
within this competitive MARL environment, we utilize a Deep Q-Network (DQN)
[Mnih.19.12.2013] with an experience replay and a target network [L.Lin.1992].
After performing experience replay, the agent selects and executes an action
according to an $\epsilon$-greedy policy. The agents select the action $a^{t}$
that maximizes the expected value of $r+Q^{\ast}(s^{\prime},a^{\prime})$,
updating the Q-values by:
$Q^{\ast}(s,a)=\mathbb{E}_{s^{\prime}\sim\varepsilon}[r+\underset{a^{\prime}}{\max}\,Q^{\ast}(s^{\prime},a^{\prime})|s,a]$
(1)
Our main argument for the selection of this specific scenario is the
controlled, unambiguous reward allocation in combination with the restricted
moveset of the agents. Thus, every step $t$ develops into a zero-sum game (as
shown in Table 1). On the one hand, we create an environment, where no agent
can earn a profit, if it does not communicate with another agent. On the other
hand, we counteract the poor learning performance of MARL [Allen.2009] (due to
the defined equilibrium/ local optimum) as well as increase the
comprehensibility of the neural network’s predictions. We expect the agents to
converge to a collusive state after several episodes, as described by
economics and law scholars (e. g., [Ezrachi.2017, Waltman.2008]).
We also attempt to induce a displacement of one agent due to the actions
selected by the other two agents. In our use case, they need to learn a
specific policy which would force two colluding agents to not repeat their
allied agents’ actions. While this would not necessarily result in a better
short term step reward for these agents, it would however eliminate the
ability to achieve a ”big win” (e. g., playing Paper if the opponents play
Rock and Rock) for the third, competing agent. Generally speaking, if two
agents avoid choosing the same action, the expected reward for the third
player is negative. We aim to simulate this circumstance in diverging
settings.
In mode 1, collusion is induced by explicit communication as suggested by
Schwalbe [Schwalbe.2018]. More specifically, we designate two ’cheating’
agents $i_{c}\subset i$ and a ’fair’ agent $i_{f}\in i,\,i_{f}\not\in i_{c}$
ahead of a training session. Before its turn, one of the cheating agents
transmits his picked action to the other cheating agent. The message will be
enclosed to input to the receiver’s DQN.333It is important that in the eyes of
the receiving agent, this is just a variable with the values of $A$ which does
_not_ have the specific semantics of _this is the other agent’s next move_. In
mode 2, instead of making the players communicate explicitly, we provoke tacit
communication by adjusting the reward of the cheating agents $r^{t}_{i_{c}}$
to $r^{t}_{i_{c}}=-r^{t}_{f}$. In other words, they will try to maximize their
_joint_ instead of their _individual_ reward, which is equivalent to
minimizing $i_{f}$’s reward. We additionally denoise the rewards; hence,
$i_{c}$ will receive 1 for a loss or a tie with $i_{f}$ and -1 for a win of
$i_{f}$. To further stress this issue, we perform control-runs, where $i_{f}$
is replaced with an agent that plays random actions (which is the best
strategy in a competitive 3-player version of RPS).
### 3.3 Implementation
Figure 1: DQN Architecture
The main weakness of RPS in a real world scenario is the unpredictability of
an opponent’s move. The best player would just play random, however since
playing this game is psychologically based on personal human short-term memory
behavior, there is a tendency to follow specific patterns, like not repeating
moves or trying to play unpredictably [Ali.2000]. In an artificial MARL-
problem, we can model that by not only reacting to an opponent’s last move,
but learning from a history of its last moves. After testing, we chose to
apply a history size of 100 games to accommodate for a stable learning
process. Regarding the experience replay, we chose to use the last 3 timesteps
as an input for the neural net. The network is made up of four dense layers
(input, two hidden layers, output), whose main task is to compress the given
information and provide the chosen action. For that matter, we design a DQN
with an input layer comprising 8100 neurons (300 steps * 3 one-hot encoded
actions * 3 players * 3 time steps), two hidden layers with 2700 and 9 neurons
and a dense output with 3 neurons to choose either rock, paper or scissors
(cf. figure 1). The neurons required for the number of players will increase
by 1 for the cheating player to accommodate for the action received by the
messaging agent. We use TensorFlow 2 to build and train the DQNs; the code can
be found on
Github444https://gitfront.io/r/user-7017325/1eb2ef3332343def1c7f67d5fce5953f1e003681/
AiCollusionDQN/.
## 4 Results
To counter inconsistent MARL outcomes, we chose to train the agents for 10
runs with 100 episodes each (300 steps per episode), comprising three
different learning rates (0.001, 0.005, 0.01), resulting in 30 runs with
900.000 games of RPS per scenario. We picked learning rates that are fairly
small to counteract quickly developing local optima, causing repetitions of
the same action, due to the straightforward connection between action and
reward. For every scenario with $i_{c}$ involved, we also performed another 15
runs (5 per learning rate) where $i_{f}$ is replaced with an agent that
randomly picks actions in order to further stress the issue by simulating a
perfect RPS policy.
### 4.1 Collusion between all agents
Figure 2: Episode reward distribution within the different learning rate
scenarios. Table 2: Action samples from two different runs, divided in stages
1, 2, 3a and 3b
[head to column names, tabular=*4—c—, table head=step A0 A1 A2
, late after line=
, filter expr= test
|
# Incorporating Background Knowledge in Symbolic Regression using a Computer
Algebra System
Charles Fox Department of Computer Science and Electrical Engineering,
University of Maryland Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250 Neil Tran Department of Chemical,
Biochemical, and Environmental Engineering, University of Maryland Baltimore
County,
1000 Hilltop Circle, Baltimore, MD 21250 Nikki Nacion Department of
Chemical, Biochemical, and Environmental Engineering, University of Maryland
Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250 Samiha Sharlin Department of
Chemical, Biochemical, and Environmental Engineering, University of Maryland
Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250 Tyler R. Josephson Department of
Chemical, Biochemical, and Environmental Engineering, University of Maryland
Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250 Department of Computer Science and
Electrical Engineering, University of Maryland Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250
###### Abstract
Symbolic Regression (SR) can generate interpretable, concise expressions that
fit a given dataset, allowing for more human understanding of the structure
than black-box approaches. The addition of background knowledge (in the form
of symbolic mathematical constraints) allows for the generation of expressions
that are meaningful with respect to theory while also being consistent with
data. We specifically examine the addition of constraints to traditional
genetic algorithm (GA) based SR (PySR) as well as a Markov-chain Monte Carlo
(MCMC) based Bayesian SR architecture (Bayesian Machine Scientist), and apply
these to rediscovering adsorption equations from experimental, historical
datasets. We find that, while hard constraints prevent GA and MCMC SR from
searching, soft constraints can lead to improved performance both in terms of
search effectiveness and model meaningfulness, with computational costs
increasing by about an order-of-magnitude. If the constraints do not correlate
well with the dataset or expected models, they can hinder the search of
expressions. We find incorporating these constraints in Bayesian SR (as the
Bayesian prior) is better than by modifying the fitness function in the GA.
## 1 Introduction
### 1.1 Symbolic Regression for Scientific Discovery
Symbolic Regression (SR) generates mathematical expressions that are optimized
for complexity and accuracy to a given dataset. Since John Koza pioneered the
paradigm of programming by means of natural selection, many applications for
SR in scientific discovery have emerged [1]. Unlike other applications of
machine learning techniques, scientific research demands explanation and
verification, both of which are made more feasible by the generation of human-
interpretable mathematical models (as opposed to fitting a model with
thousands of parameters) [2, 3, 4]. Furthermore, SR can be effective even with
very small datasets ($\sim$10 items) such as those produced by difficult or
expensive experiments which are not easily repeated. The mathematical
expressions produced by SR can easily be extrapolated to untested or otherwise
unreachable domains within a dataset (such as extreme pressures or
temperatures).
For decades, SR has discovered interesting models from data in many
applications including inferring process models at the Dow Chemical Company
[5], rainfall-runoff modeling [6], and rediscovering equations for double-
pendulum motion [7]. Symbolic regression has been applied across all scales of
scientific investigation, including the atomistic (interatomic potentials
[8]), macroscopic (computational fluid dynamics [9]), and cosmological (dark
matter overdensity [10]) scales. Some techniques facilitate search through
billions of candidate expressions, such as the space of nonlinear descriptors
of material properties [11]. While most applications of SR in science focus on
identifying empirical patterns in data, such "data-only" approaches do not
account for potential insights from background theory. In fact, some SR works
emphasize their capabilities of discovery “without any prior knowledge about
physics, kinematics, or geometry” [7]. Nonetheless, we posit that prior
knowledge need not be discarded, and in this work, we explore how theory may
be incorporated into symbolic regression to demonstrate machine learning in
the context of background knowledge.
### 1.2 Incorporating Background Knowledge into Symbolic Regression
One particularly important step towards effective use of SR in specific
domains is the addition of prior knowledge. This step has the potential to
take a general purpose SR algorithm and use it to find novel models with
physical meaning. For example, AI-DARWIN is uses prior knowledge of chemical
reaction mechanisms in the form of predefined functions that a genetic
algorithm may use in its search of equation space, ensuring that each
generated model is mechanistically meaningful [12]. This approach specifically
encodes the prior knowledge in the form of functions available instead of
limitations on functions generated. In another recent example, Engle and
Sahinidis use a deterministic symbolic regression algorithm that constrains
the space of possible equations, not to those constructed from a library of
meaningful function components, but to those functions that obey derivative
constraints from theory. This improves the quality of generated expressions
for thermodynamic equations of state [13]. Another approach to incorporating
background knowledge in symbolic regression is the Bayesian Machine Scientist
(BMS) [14]. BMS rigorously incorporates background knowledge in the form of a
Bayesian prior on symbolic expressions; expressions are _a priori_ more likely
if their distribution of mathematical operators aligns with the distribution
of operators in a corpus of prominent equations. However, their approach to
the Bayesian prior does not incorporate meaning from particular scientific
domains.
Checking consistency of equations _after_ the search is complete is also
possible. Previously, we showed that generated expressions can be compared to
rich background knowledge (expressed as axioms for the environment under
study), by posing generated expressions as conjectures to an automated theorem
prover (ATP) [15]. However, state-of-the-art ATPs are too slow to incorporate
this logical check as symbolic expressions are generated, and therefore cannot
be easily used to bias the search for equations in light of that background
knowledge. Moreover, translating scientific theories into a computer-
interpretable form is not straightforward.
We address these specific drawbacks by combining symbolic regression systems
(both genetic algorithm and Bayesian approaches) with a computer algebra
system (CAS) that checks constraints as an equation search is conducted. This
is similar to Logic Guided Genetic Algorithms (LGGA), which uses “auxiliary
truths” (ATs) corresponding to datasets in order to weigh items in a dataset
as well as augment it with more information [16]. LGGA follows an iterative
approach of training an arbitrary genetic algorithm with some dataset,
augmenting that dataset with ATs, and training that algorithm again with more
informative data. An important distinction between our work and LGGA is that
the dataset is not altered in any way and the addition of extra information is
performed during the execution of the GA. Another related approach is shape-
constrained symbolic regression [17, 18], in which constraints on function
shape (e.g. partial derivatives and monotonicity) are incorporated into
symbolic regression using an efficient application of integer arithmetic. Our
approach considers a broader range of constraints (any that can be defined and
checked by the CAS), and considers both a genetic programming and Bayesian
symbolic regression approach.
### 1.3 Adsorption
Adsorption, the phenomenon in which molecules bind to a surface, enables
chemical processes including carbon capture, humidity control, removal of
harmful pollutants from water, and hydrogen production [19] [20] [21] [22].
Models of adsorption enable prediction and design of engineered adsorption
processes, and many have been proposed over the years (selected equations are
shown in Table 1) [23, 24, 25, 26]. These models relate the amount adsorbed at
equilibrium as a function of pressure or concentration and are commonly
expressed as equations that are either empirical or derived from theory. For
example, the Freundlich isotherm [27], is an empirical function designed to
fit observed data, the Langmuir [28] and BET [29] isotherms are derived from
physical models, and the Sips [30] isotherm is Langmuir-inspired with
empirical terms added for fitting flexibility. We wonder, “What kinds of
models could be generated by a machine learning system, and what role can
background knowledge play in the search for accurate and meaningful
expressions?”
Table 1: Some well-known isotherms written as SR might find them, and their complexities. Isotherm | Literature Expression | Symbolic Regression Form | SR Complexity
---|---|---|---
Langmuir [28] | $\frac{q_{max}K_{eq}p}{1+K_{eq}p}$ | $\frac{c_{1}p}{c_{2}+p}$ | 7
Dual-Site Langmuir [28] | $\frac{q^{a}_{max}K^{a}_{eq}p}{1+K^{a}_{eq}p}+\frac{q^{b}_{max}K^{b}_{eq}p}{1+K^{b}_{eq}p}$ | $\frac{c_{1}p}{c_{2}+p}+\frac{c_{3}p}{c_{4}+p}$ | 15
BET [29] | $\frac{v_{m}*c*(p/p_{0})}{(1-p/p_{0})*(1+(c-1)p/p_{0})}$ | $\frac{c_{1}p}{p^{2}+c2p+c3}$ | 13
Freundlich [27] | $c_{1}p^{\frac{1}{n}}$ | $c_{1}p^{c_{2}}$ | 5
Sips [30] | $\frac{c_{1}p^{\frac{1}{n}}}{1+c_{1}p^{\frac{1}{n}}}$ | $\frac{p^{c_{2}}}{c_{1}+p^{c_{2}}}$ | 9
### 1.4 Thermodynamic Constraints
We consider models to be more _meaningful_ when they satisfy thermodynamic
constraints on the functional forms appropriate for modeling these phenomena.
That is, a random equation that fits data, but does not approach zero loading
correctly, is less trustworthy outside the training data than an equation
constrained to follow thermodynamics. We have identified three constraints
relevant for single-component adsorption [15]:
$\displaystyle\lim_{p\to 0}f(p)=0$ (1) $\displaystyle\lim_{p\to
0}f^{\prime}(p)<\infty$ (2) $\displaystyle\forall p>0\qquad f^{\prime}(p)\geq
0$ (3)
Constraint 1 ensures that, in the limit of zero pressure, all molecules must
desorb, and loading cannot be negative. Constraint 2 requires that in the
limit of zero pressure, the slope of the isotherm must be a positive finite
constant. Talu and Myers show that, as pressure approaches zero, the slope of
the adsorption isotherm equals the adsorption second virial coefficient
$B_{1S}$, which characterizes the interaction between one molecule and the
surface, and must be a finite positive number [31] [32]:
$\displaystyle\lim_{p\to 0}\frac{df}{dp}=\frac{B_{1S}}{RT}=c$ (4)
Constraint 3 requires that loading does not decrease with increasing pressure
(the isotherm is monotonically non-decreasing) for all ($\forall$) positive
values of pressure. Note that this does not hold for mixture adsorption (in
which competition plays a role), nor in BET adsorption, which exhibits a
discontinuity at the saturation pressure, instead of a monotonic increase.
### 1.5 PySR: Symbolic Regression using Genetic Algorithms
PySR, Python for Symbolic Regression, is a Python library that uses a genetic
algorithm for symbolic regression [33]. PySR is a Python wrapper that calls a
Julia library by the same author, SymbolicRegression.jl (SR.jl), for numerical
performance. Due to the nature of the modifications needed to the algorithm
for this work, the base Julia library was used, but all added functionality
should be inherited by the Python wrapper library as well.
The basic premise is that one or more populations of models move towards more
optimal solutions via random mutations. At each generation, some members of a
population are removed based on their fitness, age, or some other criteria
(PySR replaces the oldest members). Beneficial solutions are encouraged by
having more optimal members of a population mutate and reproduce.
Figure 1: All mutations (except for random tree generation and simplification)
in PySR in succession (read from left to right, top to bottom). Changes from
each previous expression tree are shown in orange.
Changes include mutating a single constant or operator, simplifying the
expression, or performing crossover between two expressions (Fig. 1 and Fig.
2). PySR uses multiple populations in a method similar to the island
methodology [34]. This aims to allow for specialization by separately evolving
unique populations, occasionally allowing some members to move between them to
share that specialization. Specifically, PySR implements the so-called Hall of
Fame (HOF), which is a Pareto front built from the best members across each
population. After a number of generations, each population submits its top 10
best members (based on score) which are then compared and pared down via
Pareto front. Expressions that remain in the HOF are used for future mutations
in each of the populations.
Figure 2: An example of the crossover mutation between two expression trees.
### 1.6 Bayesian Symbolic Regression
The Bayesian Machine Scientist (BMS) by Guimera et. al. [14] approaches
symbolic regression from a Bayesian perspective. Bayesian Symbolic Regression
(BSR) frames the search for accurate, concise and informed models as sampling
the marginal posterior distribution of symbolic models with respect to a prior
and fit to a dataset. Markov chain Monte Carlo (MC) is used to generate new
expression trees (Fig. 3), which are accepted or rejected based on their
likelihood. The authors define three MC moves: node replacement, root
addition/removal, and elementary tree replacement, which together enable
construction of expression trees while maintaining detailed balance, ensuring
proper sampling of the posterior.
Figure 3: Illustrating the moves available to the BMS algorithm, as applied to
adsorption equations. In contrast to the mutations available in PySR, these
transformations satisfy detailed balance [14].
Specifically, the probability of some model given some data is defined as:
$\displaystyle
p(f_{i}|D)=\frac{1}{Z}\int_{\Theta_{i}}d\theta_{i}p(D|f_{i},\theta_{i})p(\theta_{i}|f_{i})p(f_{i})=\frac{\exp[-\mathcal{L}(f_{i})]}{Z}$
(5)
where $Z$ is the probability of the dataset $p(D)$, $\Theta_{i}$ is the space
of possible values for parameters $\theta_{i}$ and $\mathcal{L}$ is the
description length of the model.
A central idea in BSR is the inclusion of a prior to emphasize expressions
that are _a priori_ more likely than others, regardless of the data. Guimera,
et al. based their prior off of a corpus of 4080 mathematical expressions
collected from Wikipedia (from the “list of scientific equations named after
people"), and assigned the prior likelihood of a function $p(f_{i})$ (and its
"energy" EP) using the counts of each unique operator ($n_{o}$) in the corpus,
by fitting parameters $\alpha$ and $\beta$ like so:
$\displaystyle\text{EP}=-\log(p(f_{i}))=\sum_{o\in
O}\big{[}\alpha_{o}n_{o}(f_{i})+\beta_{o}n_{o}^{2}(f_{i})\big{]}$ (6)
While this method leads to a distribution of expressions that resembles the
corpus when run with no data, $p(f_{i})$ can also be set to a constant value
so that there is no bias based on operators present in the search process. For
our problem, we crafted a prior especially for adsorption thermodynamics (see
details in Methods).
## 2 Methods
### 2.1 Checking Thermodynamic Constraints
Three constraint checking functions for the thermodynamic constraints
described in Section 1.4) were developed using the Python library SymPy, an
open-source computer algebra system[35]. Each function returns either true or
false, depending on if its constraint is met or not (if a time limit is
exceeded, the constraint is returned as false). For both PySR and BSR, we
found that hard constraints (rejecting every expression that fails any
constraint) severely hinder the search process, cutting off intermediate
expressions between better expressions that may also pass the constraints.
Consequently, we impose these as “soft” constraints, penalizing expressions
for constraint violation, without outright rejecting them. [17] and [18] also
found soft constraints to be more effective than hard constraints. This
approach (as implemented in PySR) is detailed in algorithm 1.
Constraints 1 and 2 could be checked using SymPy’s limit and derivative
functionality, but Constraint 3 was more challenging. Though SymPy can check
if an expression is strictly increasing in a given range, the check for
monotonicity returns false if any change in curvature (critical point) exists
for the expression – thus preventing functions such as $x^{3}$ from being
considered monotonically non-decreasing. To allow for zero slope, we
implemented a custom monotonic non-decreasing check function (see alg. 2).
Instead of just checking the slope in one range, it checks the ranges between
all critical points (as well as to the start and end of the original range in
question).
We hypothesize that the “equation space” explored by SR includes accurate, but
not thermodynamically consistent expressions that can be rejected through the
incorporation of background knowledge, guiding the search to more theory-
informed expressions.
### 2.2 PySR Modifications
In PySR, each member in a population has a score to be minimized, which
combines the loss and complexity (defined by total nodes in the expression
tree). When a thermodynamic constraint is violated, we multiply the loss
function by a penalty, raising the score and making the expression less fit.
This allows any number of constraints to be checked in any order (as
multiplication is commutative), and confers larger penalties to expressions
that violate multiple constraints.
$\displaystyle\text{Loss:
}L=\ell_{2}^{R}*\prod_{i=1,2,3}c_{i}^{\delta_{i}}\text{ where
}\delta_{i}=\left\\{\begin{array}[]{lr}1&\text{if constraint $i$ passed}\\\
0&\text{if constraint $i$ failed}\end{array}\right\\}$ (9)
$\displaystyle\text{Member Score: }S=L+n_{nodes}*c_{l}$ (10)
The above equations detail how the loss and score are calculated in PySR.
$\ell_{2}^{R}$ is the L2 norm, $c_{i}$ is the penalty for constraint $i$,
$\delta_{i}$ indicates if constraint $i$ is passed and $c_{l}$ is the penalty
for the length / complexity of an expression.
PySR also has the option to take any operators defined in Julia or Python,
including custom user-defined operators. For this work only the operators $+$,
$-$, $*$ and $\div$ were used to manage the size of the search space.
Expressions written in their canonical form may use other operators such as
exponents but these are only due to simplification of generated expressions.
### 2.3 BMS Modifications
The prior used in the Bayesian Machine Scientist code by Guimera et al. [14]
incorporates “background knowledge” in its equation search by considering
mathematical operation frequency among named equations in Wikipedia. The
authors found this to be helpful for searching for general scientific
equations, but we aim to color this background knowledge according to our
domain of inquiry. We consider the thermodynamic constraints described above
to be our “prior knowledge,” and construct the following expression:
$\displaystyle\text{EP}=\sum_{o\in
O}\big{[}c_{ops}n_{o}(f_{i})\big{]}+\sum_{i=1,2}c_{i}*\delta_{i}\text{ where
}\delta_{i}=\left\\{\begin{array}[]{lr}1&\text{if constraint $i$ passed}\\\
0&\text{if constraint $i$ failed}\end{array}\right\\}$ (13)
where $c_{ops}$ is the constraint penalty for operators (analogous to the
parsimony parameter in PySR), and $n_{o}$ is the count of each operator in
expression $f_{i}$. This expression directly replaces Eq. 6, changing the
prior distribution. Note that we checked all three constraints with PySR, and
only the first two constraints with BSR (omitting the monotonic non-decreasing
check).
## 3 Results
### 3.1 Datasets
To examine the effects of adding constraints to SR during a search, four
experimental adsorption datasets were identified: adsorption of nitrogen and
methane on mica [28], adsorption of isobutane in silicalite, [36] and
adsorption of nitrogen on Fe-Al203 catalyst [29]. The first and second
datasets come from the landmark paper introducing the Langmuir isotherm model
[28]. This model assumes there are discrete loading "sites" that do not
interact with each other, and that each site can either be occupied or not.
The isobutane dataset is well-described by a dual-site Langmuir model which
has two unique types of sites. The fourth dataset (referred to as the BET
dataset) was used by the authors of BET theory to support their model for
multilayer adsorption. These data and the respective ground truth model fits
are shown in Fig. 4.
Figure 4: Each dataset and corresponding ground truth model with constants fit
using SciPy. Isobutane is shown with log scaled pressure so that the two
separate curves are visible. BET is shown with pressure increasing to 1 so the
asymptote is visible.
### 3.2 Langmuir Datasets
The main results of this work are shown in two plot types. The left column
contains Pareto fronts which show the best expressions based on complexity and
accuracy. In these, the horizontal axis shows increasing complexity (defined
as the total number of nodes in an expression tree), and the vertical axis
shows loss, which is logarithmically scaled so the trend of the Pareto front
is more apparent. The best expression at each complexity is taken from each of
8 runs (gray curves), with the overall Pareto front shown in orange. The
“ground truth" expression for each dataset is also shown in the form it would
likely be expressed by SR, along with loss found using fit constants. The
right column of each figure shows the dataset and select expressions from the
overall Pareto front for that test. Only some expressions are shown so plots
remain readable and because expressions longer than the ground truth are
usually overfit and overlay the ground truth expression too closely for
distinction. The ground truth is plotted with a dotted line so that
expressions with similar accuracy can still be seen. Plotting the generated
expressions on the data helps to illustrate how they may or may not follow the
thermodynamic constraints and how similar they are to the ground truth.
Figure 5 shows the results from both SR algorithms with constraints on and off
on the Langmuir nitrogen dataset. The first and second rows show BSR and PySR
respectively with constraints off and clearly show that BSR finds the ground
truth while PySR does not. The expression that defines the corner at
complexity 7 in the BSR Pareto front plot (Fig. 5(a)) is indistinguishable
from the ground truth (both written mathematically and drawn on the data) when
viewed in the isotherm plot (Fig. 5(b)). The BSR plot (Fig. 5(a)) has a much
larger variance in terms of best Pareto fronts across 8 runs (as shown by the
grey lines) than PySR, but this may indicate longer time needed for the
algorithm to converge. The corresponding isotherm plots (the right column)
show how expressions fit the data better as they become more complex,
following the general trend of the Pareto fronts. These plots also show how
some expressions can fit the data reasonably well while violating the
constraints from theory, as is the case in the plot for PySR (Fig. 5(d)). In
fact, only 2.2% of expressions generated by PySR (without enforcing
constraints) pass the first constraint and only 33% pass the second constraint
(Table 2). Without constraints enforced, BSR finds more consistent expressions
than PySR, with 37% of its expressions passing the first constraint and 67%
passing the second.
When the thermodynamic constraints are enabled, the effect is clearly shown in
the Pareto fronts (bottom two rows). Both SR methods find the ground truth and
achieve the same or similar accuracy (accuracy is less for the same expression
when the constants were not optimized as thoroughly in the search). Datasets
that are well represented by the Langmuir isotherm show the effects of the
constraints well because it is typically very accurate as well as being
concise. The isotherm plots show, as before, how the expressions fit the data
better as they become more complex but showing anything beyond a complexity of
7 is redundant as the ground truth is discovered and matches the pre-fit
ground truth almost perfectly. The trend of slightly more variation across BSR
runs also continues here to some extent and the variation across PySR runs
appears roughly similar to with constraints disabled. Importantly, PySR sees a
5x increase in expressions passing the first constraint (though still only
10%) and a marginal improvement across the other two constraints (8% and 13%).
The change is more stark in BSR where twice as many expressions now pass the
first constraint (up to 72%) and a significant portion pass the third
constraint (up to 19% from 0.46%) even though it was not included in the
Bayesian prior.
While the results are mostly similar for the methane dataset, there are some
important differences. Like with the nitrogen dataset, BSR finds the ground
truth without constraints enabled while PySR does not. This is apparent in the
Pareto fronts (Fig. 6(a) and Fig. 6(c)). In this case, PySR finds an
expression with complexity 9 with more accuracy than the ground truth, though
with an extra constant in the numerator, it violates the thermodynamic
constraints (Fig. 6(d)). Imposing the constraints penalized the loss for this
expression relative to the ground truth, but not enough to overcome the
increased accuracy (Fig. 6(h)). As with nitrogen, BSR does a better job of
finding expressions that pass the constraints, even when they are not enabled,
as it finds 33% passing the first and 51% passing the second (where PySR finds
4.1% and 48% respectively).
### 3.3 Isobutane Dataset
Unlike the methane and nitrogen datasets (Fig. 5 and 6) which are best modeled
by the Langmuir isotherm, the isobutane dataset (Fig. 7) is best modeled by
the dual-site Langmuir isotherm, which has twice the complexity. Despite this
significant complexity, the dual-site Langmuir isotherm is not significantly
more accurate than many expressions shorter than it. This is best seen in Fig.
7(c) and 7(g) which show the Pareto fronts for PySR with constraints off and
on respectively. In both plots, expressions with half the complexity reach
almost the same accuracy, creating a plateau from complexity 7 onward. This is
also shown well in the corresponding isotherm plots which show that the
expressions found at complexity 7 match the data as well as the ground truth.
Importantly, these expressions do not satisfy the thermodynamic constraints.
Unlike PySR, BSR does not find expressions with accuracy close to the ground
truth until the same complexity.
For BSR, including constraints shifts the whole Pareto front down (Fig. 7(a)
to Fig. 7(e)), indicating that more accurate expressions were found at many
complexity levels. While PySR did not find accurate expressions consistent
with the constraints, BSR did. In this case, BSR finds the ground truth
expression while PySR does not. This is not apparent on either the Pareto
fronts or isotherm plots however, because the accuracy of the expression found
is about 10x worse than the fit ground truth and the best expressions found at
that complexity. This is likely because, while the ground truth is found, the
form it was originally produced in (before being simplified) is much more
complex.
In PySR, penalizing expressions that violate constraints actually led to
populations of equations that violated constraints two and three more often,
with a decrease of about 10% in each case (see Table 2). This was surprising –
we anticipated that imposing penalties would lead to fewer violating
expressions, but the opposite occurred. For BSR as well, including constraints
in the prior actually led to a decrease in expressions satisfying the second
constraint (from 46% to 36%), and a slight increase in the first and third
constraints.
### 3.4 BET Dataset
The BET dataset is unique because the ground truth expression diverges to
infinity as the pressure approaches 1 (pressure in this case is relative vapor
pressure, $p/p^{\mathrm{sat}}$; the vapor being adsorbed becomes a liquid as
$p/p^{\mathrm{sat}}\rightarrow 1$. So in this case, the third constraint (that
it is monotonically non-decreasing) no longer holds for all pressure (seen in
Fig. 8). Nonetheless, we found that whether or not constraints were enabled,
many of the most accurate expressions generated by PySR for this dataset pass
the third constraint (78.65% without constraints and 81.29% with), contrary to
the ground truth theory. Furthermore, PySR satisfies the first two constraints
less frequently with constraints on compared to with constraints off. One
possible explanation for this behavior is that the dataset itself is more
easily fit by expressions with expressions that are monotonically non-
decreasing, at least from the perspective of the PySR algorithm. Overall,
while PySR can find accurate expressions for the BET dataset, it fails to find
expressions that also follow the constraints, even when they are enabled.
In contrast, BSR did not generate many expressions that were monotonically
nondecreasing, and the incorporation of constraints had a substantial effect
on the search. Specifically, the second constraint is passed about 92% of the
time both with it enabled and disabled and the portion passing the first
constraint increases dramatically from 16% to 85% once it is enabled. This
leads to a large number of models which agree with the requisite constraints
for BET, but none of these are the ground truth rediscovered. Instead, many
expressions with close to (or better than) the accuracy of the ground truth
are found by both algorithms in both cases, none of the isotherms plotted
appear similar. The asymptote at a partial pressure of 1 is not replicated by
any similarly accurate expressions and the slight curve of the ground truth in
the middle of the dataset is also absent. These results together seem to
indicate that the constraints, while thermodynamically correct, do not provide
enough information (or even provide contradictory information) for
rediscovering the BET ground truth expression.
Dataset | Constraints Active | BSR C1 | BSR C2 | BSR C3 | PySR C1 | PySR C2 | PySR C3
---|---|---|---|---|---|---|---
Nitrogen | False | 37% | 67% | 0.46% | 2.2% | 33% | 46%
Nitrogen | True | 72% | 73% | 19% | 10% | 41% | 59%
Methane | False | 33% | 51% | 0.51% | 4.1% | 48% | 61%
Methane | True | 59% | 59% | 2.5% | 5.7% | 54% | 62%
Isobutane | False | 24% | 46% | 0.45% | 3.4% | 65% | 68%
Isobutane | True | 36% | 36% | 1.3% | 5.5% | 56% | 58%
BET | False | 16% | 92% | 0.09% | 6.2% | 35% | 79%
BET | True | 85% | 92% | 2.4% | 4.7% | 30% | 81%
Table 2: Percentage of expressions generated passing each of the three
constraints. Results are shown across both SR methods, all datasets and with
constraints active and disabled.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 5: BSR and PySR on the Nitrogen dataset. The left column shows combined
Pareto fronts across 8 runs and the right column shows interesting isotherms
found at the defining corners of those Pareto fronts. The constraints are
disabled in the top four subplots and enabled in the bottom four. The rows
alternate between BSR and PySR.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 6: BSR and PySR on the Methane dataset. The left column shows combined
Pareto fronts across 8 runs and the right column shows interesting isotherms
found at the defining corners of those Pareto fronts. The constraints are
disabled in the top four subplots and enabled in the bottom four. The rows
alternate between BSR and PySR.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 7: BSR and PySR on the Isobutane dataset. The left column shows
combined Pareto fronts across 8 runs and the right column shows interesting
isotherms found at the defining corners of those Pareto fronts. The
constraints are disabled in the top four subplots and enabled in the bottom
four. The rows alternate between BSR and PySR.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 8: BSR and PySR on the BET dataset. The left column shows combined
Pareto fronts across 8 runs and the right column shows interesting isotherms
found at the defining corners of those Pareto fronts. The constraints are
disabled in the top four subplots and enabled in the bottom four. The rows
alternate between BSR and PySR.
## 4 Discussion
### 4.1 Effectiveness
This work highlights that sometimes, a stochastic search through equation
space finds equations that are superior to ground truth expressions – in our
case achieving comparable accuracy to the ground truth expressions while also
being less complex (shorter expression length). This is particularly observed
in the case of isobutane (Fig. 7); both with constraints on and off, PySR
finds the expression $\frac{c_{1}p+c_{2}}{c_{3}+p}$, which fits the data well
but diverges from the ground truth as it approaches 0. But many of these
expressions, while consistent with the data, are inconsistent with
thermodynamics, as they violate our tested constraints. We have demonstrated
that accounting for these constraints in the search process can guide the
population or distribution of expressions toward thermodynamically consistent
expressions, and aid in the identification of the ground truth expression.
Sometimes this leads to more consistent equations, and sometimes it doesn’t
improve the search at all.
### 4.2 Exceptions To Constraints
While the three constraints presented in this work do follow from a broader
thermodynamic theory, not all isotherm models in this work satisfy all the
constraints. Specifically, the BET isotherm does not satisfy the third
constraint because it reaches an asymptote as $p/p_{0}$ approaches 1. While
this does break the monotonicity constraint, it also seems reasonable when
considering what $p_{0}$ represents (the pressure at which the adsorbate
becomes a liquid and the interaction fundamentally changes). This raises the
question: what constraints to include and why? The decision to test if
expressions pass the third constraint necessarily excludes the BET model
because, while it is theoretically grounded, it only applies from in the range
$(0,p_{0})$. Furthermore, the data used does not extend past that range so
expressions that do pass the third constraint may not appear much different in
terms of what is relevant. We recommend carefully examining what constraints
one may want to test and in what places they should actually be checked.
Introducing incorrect constraints may hinder the search with our biases, and
prevent algorithms from discovering phenomena outside our assumptions.
### 4.3 Computational Complexity / Runtime
As seen in Fig. 10), consideration of constraints increases runtime by an
order of magnitude; this is even after we carefully integrated the computer
algebra system into the SR algorithms to reduce overhead, and leveraged memory
to avoid redundant checks on expressions previously visited. On average,
numerically checking models is much faster than manipulating them symbolically
(especially for larger expressions) – checking _every_ new expression is quite
expensive. This is unfortunately necessary if the constraints are to be
considered as an integral part of the search. If a cheaper solution is needed,
the search can be performed without constraints, and constraints checked after
the fact. In fact, this approach enables even more elaborate methods of
considering background knowledge, such as comparing against complex, multi-
premise background theories using an automated theorem prover [15].
### 4.4 Challenges around complexity, simplification, and canonical form
In this work, simplification is necessary in order to identify whether a
generated expression matches the ground truth and to assign generated
expressions an appropriate complexity. We augmented SymPy’s “simplify”
function, to shorten the numerous rational expressions we generated into a
"canonical form" (details in the Supporting Information). While some methods
such as BSR attempt simplification during runtime, PySR does not because of
the added computation needed per expression. Generating a "canonical form" for
expressions generated by PySR sometimes increases, and sometimes decreases,
the complexity. Some expressions are generated as complex expression trees
that are much more complex than their canonical forms (Fig. 9).
Simplification is a crucial challenge of this work because complexity plays a
significant role in SR. After all, models are compared via accuracy and
complexity to make decisions during the search. A single model may very well
take on different scores / likelihoods because of how it is written,
influencing not just its standing, but subsequent steps in the search.
Ideally, every model would always be written in the simplest form, but this is
computationally intractable in some circumstances [37]. Because of this,
comparing functions based on behavior (symbolic constraints) may be more
appropriate, because limiting behavior is invariant to the numerous ways an
expression can be written.
Figure 9: The effect of the canonical form checking function on a Pareto front
showing the results from PySR on the Nitrogen dataset. Points on this plot are
marked as “Passing" only if they pass all constraints checked.
### 4.5 Reducing Underspecification Through Inductive Biases
Machine learning researchers at Google recently highlighted the role of
underspecification in machine learning pipelines [38]. They suggested that one
way to combat underspecification is to use credible inductive biases to allow
for selection from otherwise similarly effective models, and that these
constraints should have little negative effect on accuracy if selected
correctly. In this work, we find expressions that are roughly equivalent in
terms of accuracy and complexity but have different functional forms, leading
to different behavior outside the range of the data – signatures similar to
those discussed in [38]. We find that adding thermodynamic constraints can
help improve the search for good expressions, but this doesn’t necessarily
restrict the hypothesis space in the same way that inductive biases do; we
were unable to effectively search with hard constraints, and so our hypothesis
space still included expressions that are inconsistent with constraints.
Instead, we can reduce the hypothesis space after the search is complete; by
rejecting accurate-but-inconsistent expressions using our background
knowledge, we improve on the issues of underspecification. Nonetheless, for
datasets with reasonably complex behavior, there still exist multiple distinct
thermodynamically-consistent expressions of similar accuracy and complexity.
The space of equations defined by the limited number of operators considered
here, even for one dimensional datasets, is just that vast!
## 5 Conclusions
In this work, we couple a computer algebra system to two symbolic regression
algorithms in order to check the consistency of generated expressions with
background knowledge. We find that including appropriate mathematical
constraints can improve search effectiveness or break the search entirely,
depending on the dataset and implementation details. Although computational
costs increase by an order of magnitude, tightly integrating SR with a
computer algebra system is a practical way to check for constraints on each
expression generated during the search.
We have shown that consideration of constraints helps in rediscovering ground-
truth isotherm models from experimental data, including the Langmuir and the
dual-site Langmuir isotherms (though the dual-site Langmuir isotherm was not
identified on the Pareto front, it was present in the generated models). In
contrast, the BET isotherm was not rediscovered; more accurate and concise
models were generated instead, and the most meaningful model (BET) was
consequently missed. We found that Bayesian Symbolic Regression is a more
effective and intuitive platform for incorporating symbolic constraints in a
Bayesian prior, rather than by modifying the fitness function in traditional
genetic algorithms; the resulting populations of expressions were more attuned
to the constraints with BSR. Finally, though background knowledge can screen
out accurate yet inconsistent solutions, symbolic regression pipelines remain
underspecified in our context, capable of generating multiple distinct
solutions with similar performance and adherence to constraints.
## 6 Acknowledgements
We thank Marta Sales-Pardo and Roger Guimerà for discussions about the
Bayesian Machine Scientist, and Miles Cranmer for assistance with PySR. This
material is based upon work supported by the National Science Foundation under
Grant No. #2138938, as well as startup funds from the University of Maryland,
Baltimore County.
## 7 Supporting Information
The modified version of PySR and the code used to run it are both available on
GitHub. The PySR code was forked from the original repository on June 6th,
2020 and is available at https://github.com/CharFox1/SymbolicRegression.jl.
The code for running PySR, parsing its output, and plotting the results, and
is available at https://github.com/ATOMSLab/pySR_adsorption. The modified
version of BMS code used in this paper is available at
https://github.com/ATOMSLab/BayesianSymbolicRegression.
The Supporting Information includes 1) further description of the adsorption
models considered here, 2) further discussion of the changes we implemented in
PySR and BMS codes to implement thermodynamic constraint checking, including
pseudocode for new algorithms, 3) description of our pipeline for collecting
and analyzing generated expressions, 4) further discussion of the nuances
around identifying of “interesting” expressions in automated pipelines and
algorithms for simplification and pattern-matching, 5) details of constant
fitting, 6) experiments comparing runtime of algorithms on different datasets,
and 7) details of the testing environment on the UMBC supercomputer.
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## 8 Supporting Information
### 8.1 Langmuir
The Langmuir isotherm model was originally presented in 1918 and remains in
common use today [28].
$\displaystyle q=\frac{q_{m}K_{L}p}{1+K_{L}p}\quad\text{ or
}\quad\theta=\frac{K_{L}p}{1+K_{L}p}$ (14)
The original Langmuir isotherm relates the volume adsorbed onto the surface
($q_{e}$), the adsorption strength ($K_{L}$), the gas pressure $p$ and the
maximum adsorption capacity ($q_{m}$) [26] [28]. The isotherm is often more
simply written with only the fractional adsorption $\theta_{A}$, for the
fraction of the surface that is occupied by adsorbed molecules. In practice,
$q_{m}$ is not known _a priori_ , so we aim to rediscover the expression for
$q_{e}$. However, this expression is _never_ found by SR, because an
equivalent expression with shorter length (7 vs. 11) can be written as
$\displaystyle q=\frac{q_{m}K_{L}P}{1+K_{L}P}\rightarrow
q=\frac{c1*p}{(c2+p)}$ (15)
The basic premise of the Langmuir model is to represent the adsorbent (the
surface atoms or molecules are adsorbing onto) as a simple surface with some
number of free and full spots where the adsorbate could stick. The model is
not concerned with rough or porous surfaces, molecules that may take up
multiple “sites" or otherwise interact with each other after being adsorbed
or, most importantly, any stacking of molecules. Wang et. al. classify the
Langmuir model as a chemical adsorption model because it is only concerned
with mono-layer adsorption in which a chemical bond is formed between the
adsorbent and adsorbate [26].
### 8.2 Dual-Site
The Langmuir model can be extended to describe more complex materials.
Specifically, if a surface has two types of adsorption sites (with different
properties), another term can be added to obtain the “dual-site Langmuir"
model:
$\displaystyle
q=\frac{q_{a}K_{A}p}{(1+K_{A}p)}+\frac{q_{b}K_{B}p}{(1+K_{B}p)}\rightarrow
q=\frac{c1*p}{(c2+p)}+\frac{c3*p}{(c4+p)}$ (16)
In this case, because there are two unique Langmuir terms, there are also two
unique terms for both the maximum volume that can be adsorbed ($q_{a}$ and
$q_{b}$) as well as the “Langmuir Constants" ($K_{A}$ and $K_{B}$). This model
is considered to be the ground truth for the isobutane dataset because the
adsorbent material, the MFI zeolite, has two distinct adsorption sites for
isobutane [39].
### 8.3 BET
The BET model extends the Langmuir model to consider multi-layer adsorption
[29]. Beyond the first layer, van der Waals forces attract adsorbates toward
the surface as well as to adsorbed molecules. Because of this, Wang et. al.
classifies the BET isotherm as a physical model as opposed to a chemical one
[26]. The main BET model assumes infinite possible layers; alternative forms
can be derived for a finite maximum number of layers [29]. With a max of one
layer, it simplifies to the Langmuir model (which only models one layer), with
$n$ layers, it becomes significantly more complex (and unlikely to be found
via SR). Fortunately the form from $n=\infty$ is concise and can be simplified
to:
$\displaystyle q=\frac{v_{m}*c*(p/p_{0})}{(1-p/p_{0})*(1+(c-1)p/p_{0})}$ (17)
with $p_{0}$ being the saturated vapor pressure, and $v_{m}$ and $c$ being
constants describing monolayer adsorption and interaction energies in the
system. Data can be provided to SR as $p$ or as $p/p_{0}$; using the second
choice is a form of _a priori_ feature selection, which assumes we know that
pressure should be normalized to $p_{0}$. Because we don’t assume this,
$p_{0}$ becomes a third constant fit by SR. SR generally fits this to an
incorrect value, since it is unconstrained to match $p_{0}$ to anything
physical, and prioritizes fit to the given data.
### 8.4 Thermodynamic Constraint Functions
Three constraint checking functions (which follow from the three thermodynamic
constraints introduced previously in section 1.4) were developed using the
Python library SymPy [35]. Each function returns a Boolean TRUE or FALSE,
depending on if its constraint is met or not. While these functions are useful
for examining the expressions generated after a run, simply discarding an
expression for failing one or more constraint during a run can severely hinder
search potential by cutting off intermediary steps between better expressions
that may also pass the constraints. Because of this, each function has a
corresponding weight that allows it to act as a “soft constraint".
Specifically, a constraint will not affect the score of an expression if it is
passed but will multiply (worsen) that score if it is not. This approach (as
implemented in PySR) is detailed in Algorithm 1. Algorithm 2 describes our
test whether our function is monotonically non-decreasing, using SymPy.
Algorithm 1 Modified genetic algorithm
$tree,dataset,options$
$score,loss$
function scoreFunc($tree,dataset,options$)
$loss\leftarrow\text{evalLoss}(tree,dataset,options)$ $\triangleright$
Traditional error calculation
$penalties\leftarrow options.penalties$
if $penalties\text{ not empty }$ then $\triangleright$ Skip slow calculation
if possible
$expr,var\leftarrow\text{parseTree}(tree)$
for $p\in penalties\text{ and }tf\in thermoFunctions$ do
if $tf(expr,var)=\text{False }$ then
$loss\leftarrow loss*p$
end if
end for
end if
$score\leftarrow\text{scoreFunc}(loss,tree,options)$ $\triangleright$ Factor
in complexity
return $score,loss$
end function
Algorithm 2 Monotonic Non-decreasing Check
$expr,var,start,stop$
$passing$ (If the expression is monotonically non-decreasing)
function montonicNondecreasing($expr,var,start,stop$)
if $expr\text{ is constant}$ then
return True
end if
$turningPoints\leftarrow\text{inflection points (zeros) of }expr$
$\triangleright$ Calculated using SymPy
if $turningPoints\text{is empty}$ then
if $expr(stop)-expr(start)\geq 0$ then $\triangleright$ Measure slope
return True
else
return False
end if
end if
$turningPoints\leftarrow[start,turningPoints,stop]$ $\triangleright$ Include
bounds
for $\text{each sequential pair of points }p,q\in turningPoints$ do
if $expr(q)-expr(p)\leq 0$ then $\triangleright$ Measure slope between zeros
return False
end if
end for
return True
end function
### 8.5 Implementation of Thermodynamic Constraints
In order to maintain parity between PySR and BMS, the Julia library PyCall was
used. This allowed the same Python code that checked thermodynamic constraints
in BMS to be used within the Julia code of the SymbolicRegression.jl library
(the back-end of PySR). Beyond the concern of a fair comparison across
platforms, PyCall showed itself to be somewhat necessary for this work. As of
writing, there is no Julia library close to as full-featured as SymPy (and
SymPy can have some serious limitations/difficulties). One method explored was
using the Julia library MATLAB.jl to call MATLAB code from within Julia in a
similar way to how PyCall works with Python code. While this eventually worked
(MATLAB does have a robust symbolic math toolbox), this proved very difficult
to set up due to intricacies in how the two languages store data. MATLAB was
not significantly faster than Python in our tests.
One downside to the current structure of our version of SR.jl and the
languages it relies on is that it must be run in distributed mode instead of
threaded mode (meaning multiple processes must be created, requiring more
overhead). This is due to the fact that PyCall expects only one Julia instance
to attempt to use Python at once. With multiple processes, a new instance of
the PyCall Julia package is created for each process, thus avoiding memory
access conflicts.
### 8.6 PySR Data Collection
Though PySR generates and handles many expressions, accessing complete
information about all members and all populations over time proved difficult.
PySR’s goal each iteration is to produce a Pareto front showing what it has
found to be the most accurate expression at each complexity level. This means
the vast majority of expressions in a population are not shown (although some
may be duplicates). Indeed, if an expression never makes it to the Pareto
front at any point during the run, by the definition of the search, it is not
as important. However, this prevented us from investigating whether
expressions congruent with the ground truth were generated and discarded,
because they did not land on the Pareto front (due to insufficient parameter
optimization or simplification).
Nonetheless, to track the expressions we could easily access (generally
hundreds of Pareto fronts per run), the full output text is parsed using
Python and stored in a Pandas DataFrame [40]. The values collected from the
raw text are the expression itself, score, loss, complexity, runtime,
iteration and run number. While this is a significant amount of data already,
there are a few interesting attributes yet to be calculated – specifically
those relating to the simplified form of the expressions and those related to
the thermodynamic constraints. To make further parsing manageable, once the
simplified form of each expression is generated, only the most accurate
expression with that form is retained. This step often reduces the number of
rows in the DataFrame by about 1000x. The simplified form of each expression
is calculated both with original (optimized/fit) and substituted constants.
The new complexity of the simplified form may or may not be smaller than the
original complexity so both attributes are kept. Finally, the thermodynamic
constraints that may have been used to guide the search are checked for each
expression.
### 8.7 Identifying “interesting" expressions
An important component of this work is identification of interesting or
meaningful expressions generated by the SR algorithms. Expressions that
satisfy constraints are more interesting than those that do not, but the most
meangingful expressions are those which are _derivable_ [15]. In our case,
this is indicated by being able to be simplified to the canonical form of an
already known expression such as the Langmuir or BET adsorption isotherms.
Unfortunately, determining if one expression is equivalent to another is a
very difficult, and sometimes undecidable problem. In fact, while not
applicable to this work due to our limits on operators and constants used,
Richardson’s theorem shows that, including certain operators and the
transcendental numbers $\pi$ and $e$, it may be impossible to show that one
expression is equivalent to another [37]. Consequently, we identify exact
expressions by either automatically finding a direct match to the canonical
form, or by manually inspecting expressions that satisfy all relevant
constraints and given low errors, but this does not guarantee that we find all
instances of rediscovered ground truth expressions.
### 8.8 Simplification Function
We were first inclined to leave the constants as symbolic and treat them as
variables (e.g. c1, c2…), but we found that SymPy did not reliably simplify
fully symbolic expressions of even modest size. We consequently substituted
either the fitted parameters from the search, or by replacing symbolic
constants with a sequence of prime numbers. We simplify the expression first
using SymPy’s default tool, but this doesn’t simplify some expressions to
their shortest form. For example, if the resulting expression is rational (has
only integer exponents and no division by zero), it can be written as a
fraction and possibly simplified further. Specifically, if there is a common
factor between the numerator and denominator because they are both degree 1 or
larger, it may be possible to remove a leading constant, providing a simpler
expression than that from SymPy’s default tool. For example,
$\displaystyle\frac{2x}{3x^{2}+4x+5}\rightarrow\frac{x}{(3/2)x^{2}+2x+5/2}\quad\text{or}\quad\frac{(2/3)x}{x^{2}+(4/3)x+5/3}$
(18)
$\displaystyle\frac{c_{1}x}{c_{2}x^{2}+c_{3}x+c_{4}}\rightarrow\frac{x}{c_{1}x^{2}+c_{2}x+c_{3}}\quad\text{or}\quad\frac{c_{1}x}{x^{2}+c_{2}x+c_{3}}$
(19)
In this case, an expression of 4 constants is reduced to an expression of 3
constants.
Some expressions have the same constant appearing in multiple places. For
example, the BET expression from literature, as well as its simplified form,
have constants $p_{0}$ or $c_{2}$ appearing multiple times:
$\displaystyle\frac{v_{m}*c*(p/p_{0})}{(1-p/p_{0})*(1+(c-1)p/p_{0})}\rightarrow\frac{c_{1}*p}{(p^{2}+c_{2}*p+c_{2})}$
(20)
In our case, we accept solutions with the same form as the ground truth, but
with non-unique constants. The expression $c1*p/(p^{2}+c_{2}*p+c_{3})$ is
equal in complexity (using our metrics) while have more ability to fit the
data; if we required $c_{2}=c_{3}$, we would never obtain the ground truth.
Algorithm 3 Simplification Function
$expr,vars,pars$
$can$ (Canonical form of expression)
function simplify($expr,vars,pars$)
$expr\leftarrow\text{expression}$
$vars\leftarrow\text{variables}$
$pars\leftarrow\text{parameters}$
for $p\textbf{ in }pars$ do
$p\leftarrow\text{prime number}$ $\triangleright$ Substitute constants with
unique primes
end for
Simplify using SymPy
if $expr\text{ is rational}$ then
$num,denom\leftarrow expr\text{ as fraction}$ $\triangleright$ Calculated
using SymPy
if $\text{degree}(num)>\text{degree}(denom)$ then
$factor\leftarrow\text{ leading term of }num$
else
$factor\leftarrow\text{ leading term of }denom$
end if
$expr\leftarrow\frac{num/factor}{denom/factor}$ $\triangleright$ Remove common
factor
end if
return $expr$
end function
### 8.9 Fitting Constants in PySR
By default, PySR uses Nelder-Mead optimization to fit constants for
expressions as they are generated [33] [41]. Nelder-Mead is well suited to
optimizing parameters for arbitrary generated expressions because it does not
require any derivative or gradient information. The algorithm evaluates the
function at $n+1$ points where $n$ is the number of parameters. The next point
is selected by finding the point with the highest function evaluation and
looking to the opposite side of the remaining points (a simplex). This
iteratively moves the worst point to a likely better location, eventually
moving towards a local optimum regardless of how the function is evaluated.
One downside to this method is that it is not guaranteed (or even expected) to
find global minima. This issue is usually rectified by allowing for multiple
starts from random locations, and selecting the best result. In PySR, each
expression gets 8 attempts by default, randomizing the parameters between
each. This is typically enough but there are occasionally cases where an
expression should be much more accurate than it is due to poorly fitted
constants.
To fit ground-truth expressions and check constants in post-processing, we
also applied Nelder-Mead optimization using SciPy [42]. We allowed many more
iterations in post-processing to ensure ground truths and interesting
functional forms were most optimized.
### 8.10 Runtime
An important consideration when examining the effectiveness of the
thermodynamic constraints in guiding SR is the impact on computation time.
While not extreme, symbolic math (in SymPy) can be slow, especially compared
to the otherwise efficient and optimized Julia code running behind the PySR
front-end. The following plot shows the difference in iteration time (the time
between one Pareto front and the next being printed by PySR) across different
datasets, and more importantly, across different constraint penalties. It
shows that runtime without SymPy is fast and not dependent on the dataset, and
that it is increased by about an order of magnitude when checking constraints.
Figure 10: Average runtimes across all datasets and combinations of
thermodynamic constraint penalties. Runs with all penalties set to 1.0 are
highlighted in orange. Standard deviation is shown by error bars at the top of
each bar.
### 8.11 Environment
Testing was done on both the batch and cpu2021 partitions of the UMBC High
Performance Computing Facility (https://hpcf.umbc.edu/). This allowed for the
longer runtimes and larger parameter exploration necessitated by adding
thermodynamic constraints with variable penalties.
While BMS is entirely based in Python, SymbolicRegression.jl (the backend for
PySR) is entirely in Julia. Because of the need to use SymPy, the Julia
package PyCall.jl was used to allow Python code and libraries to be run by
Julia (at the time this work was completed, symbolic math libraries in Julia
could not evaluate the constraints considered in this work). To allow for
parallel use of Python / SymPy from within Julia, PySR was run in distributed
mode, necessitating more overhead than threaded mode.
|
# Decoupling of dipolar and hydrophobic motions in biological membranes
Hanne S. Antila Department of Theory and Bio-Systems, Max Planck Institute of
Colloids and Interfaces, 14424 Potsdam, Germany Anika Wurl Institute for
Physics, Martin-Luther University Halle-Wittenberg, 06120 Halle (Saale),
Germany O. H. Samuli Ollila Institute of Biotechnology, University of
Helsinki, 00014 Helsinki, Finland Markus S. Miettinen Department of Theory
and Bio-Systems, Max Planck Institute of Colloids and Interfaces, 14424
Potsdam, Germany Tiago M. Ferreira<EMAIL_ADDRESS>Institute for Physics, Martin-Luther University Halle-Wittenberg
###### Abstract
Cells use homeostatic mechanisms to maintain an optimal composition of
distinct types of phospholipids in cellular membranes. The hydrophilic dipolar
layer at the membrane interface, composed of phospholipid headgroups,
regulates the interactions between cell membranes and incoming molecules,
nanoparticles, and viruses. On the other hand, the membrane hydrophobic core
determines membrane thickness and forms an environment for membrane-bound
molecules such as transmembrane proteins. A fundamental open question is to
what extent the motions of these regions are coupled and, consequently, how
strongly the interactions of lipid headgroups with other molecules depend on
the properties and composition of the membrane hydrophobic core. We combine
advanced solid-state nuclear magnetic resonance spectroscopy methodology with
high-fidelity molecular dynamics simulations to demonstrate how the rotational
dynamics of choline headgroups remain nearly unchanged (slightly faster) with
incorporation of cholesterol into a phospholipid membrane, contrasting the
well known extreme slowdown of the other phospholipid segments. Notably, our
results suggest a new paradigm where phospholipid headgroups interact as
quasi-freely rotating flexible dipoles at the interface, independent of the
properties in the hydrophobic region.
## I Introduction
Out of the plethora of lipids found in nature, the most ubiquitous are
glycerophospholipids which consist of a glycerol backbone attached to two
hydrophobic fatty acid chains and a phosphodiester bridge connecting to a
hydrophilic headgroup Marsh (2013). Cells use vast amounts of energy, and rely
on complex synthetic pathways, to adjust and maintain the specific composition
of different types of phospholipid headgroups across cellular organelles
Harayama and Riezman (2018). This chemical homeostasis implies that the
headgroups play a key role in fundamental biological processes, and evidence
for phospholipid-specific functionality concerning compartmentalization,
signaling, transport, ion binding, peptide insertion, and regulation of
membrane protein function has been found Lemmon (2008); van Meer et al.
(2008); Harayama and Riezman (2018); Corradi et al. (2019). However, the
molecular details on how lipid composition of a cellular membrane connects to
its overall properties and to specific biological processes remain poorly
understood.
A key open question is to what degree the behavior of the water-facing
headgroups in biological membranes correlate with the properties of the acyl
chains in the membrane hydrophobic core Rajan et al. (1981); Huang and
Feigenson (1999); Roberts and Redfield (2004); Ali et al. (2007); Klauda et
al. (2008); Sivanandam et al. (2009); Alwarawrah et al. (2010); Leeb and
Maibaum (2018). Two limiting cases can be considered: 1) the conformational
ensemble and dynamics of the headgroup, though positionally connected through
the glycerol backbone, are uncoupled from the acyl chain region (freely
rotating/weak coupling limit), or 2) the orientation and dynamics of the
headgroup and hydrophobic acyl chains are strongly interdependent (strong
coupling limit). These two limiting scenarios will give rise to very distinct
biophysical behaviour. In the case of strong coupling, the headgroups in lipid
domains with ordered acyl chains and hindered dynamics, such as cholesterol-
induced lipid rafts Simons and Ikonen (1997, 2000), would exhibit slower
dynamics and possibly a different conformational ensemble. The acyl chain
behavior would then indirectly affect the interactions between lipid
headgroups and molecules in the aqueous media or within the membrane, such as
proteins or drugs. Such scenario is implicit, for example, in the popular
umbrella model for lipid–cholesterol interactions van Meer et al. (2008);
Huang and Feigenson (1999); Ali et al. (2007). In the weak coupling limit, the
behavior of lipid headgroups is similar irrespective of the acyl chain
structure, order, and dynamics, and consequently any cellular processes that
depend on the conformation and dynamics of the headgroups are unaffected by
the properties of the hydrophobic region.
We address the question of headgroup-tail decoupling using a
phosphatidylcholine (PC)–cholesterol bilayer system (the most abundant
phospholipid and sterol in eukaryotic cells)—a model cellular membrane from
which a wealth of both experimental and simulation data can be obtained.
Cholesterol is known to drive lateral heterogeneity and make membranes more
ordered (the so-called cholesterol condensing effect), which manifests as a
substantial increase in hydrophobic acyl chain C–H bond order parameters
($S_{\rm{CH}}$) in nuclear magnetic resonance (NMR) experiments Vist and Davis
(1990); Hofsäss et al. (2003); Tiburu et al. (2004); Warschawski and Devaux
(2005); Vermeer et al. (2007); Davis et al. (2009); Leftin and Brown (2011);
Ferreira et al. (2013a); Andersson et al. (2017). In contrast, the headgroup
and glycerol backbone order parameters are essentially unaffected up to the
highest cholesterol concentrations possible to incorporate in PC membranes
Brown and Seelig (1978); Ferreira et al. (2013b). The $\alpha$-carbon order
parameter of the choline headgroup, in particular, remains unchanged also upon
other bilayer perturbations that significantly affect acyl chain order
parameters, such as temperature, acyl chain composition, or membrane phase
(Table S1 in the supplementary information and references therein). On the
other hand, the choline headgroup orientation (and consequently the headgroup
order parameters) is highly sensitive to hydration level Ulrich and Watts
(1994), hydrostatic pressure Bonev and Morrow (1995), and the inclusion of
charges Altenbach and Seelig (1984); Scherer and Seelig (1989); Macdonald et
al. (1991a) or molecular dipoles in the membrane Bechinger and Seelig (1991a).
Therefore, a picture of a rotationaly decoupled headgroup, whose orientation
is independent of the hydrophobic region but can be affected by the
environment, emerges.
Although the C–H bond order parameters contain accurate information on
conformational ensembles, they do not convey how fast that ensemble is sampled
(conformational dynamics). The motional time-scales have been dominantly
assessed through spin-lattice relaxation ($R_{1}$) and spin-lattice relaxation
in the rotating frame ($R_{1\rho}$) NMR measurements Brown et al. (1979,
1983); Morrison and Bloom (1994); Klauda et al. (2008); Ferreira et al.
(2015); Sivanandam et al. (2009); Roberts M F (2009); Le Guernevé and Auger
(1995) which are sensitive to different (limited) time-scales depending on
experimental conditions and from which a physically meaningful change in the
dynamics (speedup vs. slowdown) can be challenging to interpret without
multiple measurements under different magnetic fields. Such measurements
demonstrate that the cholesterol-induced order in acyl chains is accompanied
by slower rotational dynamics not only of the acyl chains but also of the
glycerol backbone segments for which there is only a marginal conformational
change Sivanandam et al. (2009); Roberts M F (2009).
However, in a crucial contrast, the impact of cholesterol on the headgroup
motional time-scales, potentially occurring either via direct interaction or
through the observed glycerol backbone slowdown, has remained unclear. An
observation of increase of headgroup 13C $R_{1\rho}$ rates of DMPC upon
incorporation of cholesterol Le Guernevé and Auger (1995) suggests that the
cholesterol-induced slowdown of tail dynamics propagates to the phospholipid
headgroups although neither the statistical significance nor the quantitative
interpretation of the $R_{1\rho}$ increase in terms of physically meaningful
correlation times was provided. In stark contrast, comparison of 13C cross
polarization (CP) and refocused insensitive nuclear enhanced polarization
transfer (rINEPT) intensities suggest that the headgroup motional time-scales
remain unchanged even by the addition of 50% cholesterol Ferreira et al.
(2013a).
Here, we show that the dynamics of the PC headgroup are unaffected by
cholesterol, and consequently, that the motion of phospholipid headgroups is
decoupled from the hydrophobic region (the freely rotating limit). To this
end, we employ our novel NMR methodology Ferreira et al. (2015) where
segmental effective correlation times ($\tau_{\rm{e}}$) are determined from
solid-state NMR measurements of $R_{1}$, $R_{1\rho}$ and $S_{\rm{CH}}$. The
analysis of $\tau_{\rm{e}}$ values enables us to interpret the relaxation
rates in terms of a single, physically meaningful average time-scale for each
carbon where lower $\tau_{\rm{e}}$) value denotes slowdown and vise-versa.
Additionally, we decipher the origin of the decoupled motion by analysing
distinct all-atom (CHARMM36 and Slipids) and united-atom (Berger) MD
simulations which provide either realistic uncoupled (CHARMM36 and Slipids) or
non-realistic coupled (Berger) headgroup motions. The MD simulation models
indicate that the decoupled motion originates from dihedral rotations that are
present in all other glycerophospholipid types in addition to PCs. This
suggests that biological membranes have independent rotational dynamics of the
headgroups from the hydrophobic region, a feature that may be relevant in the
machinery of biological cell membrane processes.
## II Materials and Methods
### II.1 Sample Preparation
1-palmitoyl,2-oleoyl-$sn$-glycero-3-phosphocholine (POPC), cholesterol and
chloroform were purchased from Sigma-Aldrich. The samples were prepared by
mixing the lipids with chloroform and rapidly evaporating the organic solvent
under a nitrogen gas flow to obtain a homogeneous lipid film. Subsequently,
the lipid film was dried under vacuum overnight. The film was then hydrated in
a 0.5 ml tube by adding 50 %wt of water and manually mixing with a thin metal
rod multiple times alternated by sample centrifugation until a homogeneous
mixture was attained. The resulting mixture was then centrifuged into a KEL-F
Bruker insert with a sample volume of approximately 25 $\mu$l specifically
designed for solid-state NMR 4 mm rotors and left to equilibrate for at least
24 hours at room temperature before measurements.
### II.2 NMR Experiments
The solid-state NMR experiments to measure $R_{1}$ and $R_{1\rho}$ were
performed on a Bruker Avance II-500 NMR spectrometer operating at a 13C Larmor
frequency of 125.78 MHz equipped with an E-free CP-MAS 4 mm (13C/31P/1H). The
R-PDLF measurements were performed on a Bruker Avance III 400 spectrometer
operating at a 1H Larmor frequency of 400.03 MHz equipped with a standard 4 mm
CP-MAS HXY probe. All experiments were performed under magic-angle spinning
(MAS) conditions at a rate of 5 kHz. The R-PDLF, $R_{1}$, and $R_{1\rho}$
experiments were performed as previously described in references Löser et al.
(2018); Ferreira et al. (2015). More details on experimental set up are given
in the supplementary information (SI).
### II.3 MD Simulations
We performed MD simulations for two systems: a pure POPC bilayer and a bilayer
containing additional 50% cholesterol, both accompanied by enough water
molecules per lipid to result in fully hydrated bilayers. We used three lipid
MD models (force fields): CHARMM36 Klauda et al. (2010), Slipids Jämbeck and
Lyubartsev (2012), and Berger Ollila et al. (2007)/Höltje Höltje et al.
(2001); Ferreira et al. (2013c) together with either TIP3P Jorgensen et al.
(1983); MacKerell et al. (1998) or SPC Berendsen et al. (1981) water. The
choice of force fields was based on previous works where their ability to
capture structure Botan et al. (2015) and dynamics Antila et al. (2021) of
lipid headgroup and glycerol backbone was assessed against NMR measurables.
The simulations were performed using the GPU-version of Gromacs2020 Abraham et
al. (2015) MD engine, with sampling rate of 10 ps and maintaining 303 K
temperature. The list of all simulated systems, along with the trajectory
lengths and links to the freely available simulation data, is given in Table
1. Further details of the simulations are presented in the SI.
Table 1: Summary of simulated systems: the force fields used, numbers of POPC,
cholesterol, and water molecules, trajectory lengths, and access links to the
simulation files.
Force-field POPC/water+cholesterol | POPC/chol | water | length (ns) | files
---|---|---|---|---
CHARMM36 Klauda et al. (2010)/TIP3P MacKerell et al. (1998) | 122/0 | 4480 | 840 | [51]
Slipids Jämbeck and Lyubartsev (2012)/TIP3P Jorgensen et al. (1983) | 122/0 | 4480 | 1200 | [52]
Berger-POPC-07Ollila et al. (2007)/SPC Berendsen et al. (1981) | 256/0 | 10342 | 1200 | [53]
CHARMM36 Klauda et al. (2010)/TIP3P MacKerell et al. (1998)+CHARMM36 Lim et al. (2012) | 122/122 | 9760 | 1240 | [51]
Slipids Jämbeck and Lyubartsev (2012)/TIP3P Jorgensen et al. (1983)+Slipids Jämbeck and Lyubartsev (2013) | 122/122 | 9760 | 1200 | [56]
Berger-POPC-07 Ollila et al. (2007)/SPC Berendsen et al. (1981)+Höltje-CHOL-13 Höltje et al. (2001); Ferreira et al. (2013c) | 256/256 | 20480 | 1200 | [53]
Figure 1: Effect of cholesterol on the dynamics (panel B) and structure (panel
C) of headgroup ($\alpha$, $\beta$ and $\gamma$) and glycerol backbone
($g_{1}$, $g_{2}$, and $g_{3}$) carbons in POPC lipid membranes. (A) Chemical
structure of POPC with carbon labels and refocused-INEPT spectra from POPC
membranes with (red) and without (black) cholesterol. (B) 13C spin-lattice
relaxation rates, $R_{1}$, and spin-lattice relaxation in the rotating frame
rates, $R_{1\rho}$, showing the independent motion of the headgroup and
slowdown of the glycerol backbone upon cholesterol incorporation. A Larmor
frequency of 500 MHz for 1H nuclei and a spin lock field equal to 50 kHz were
used. The corresponding experimental decays for each data value are shown in
Figure S1. (C) Dipolar recoupling profiles acquired with R-PDLF spectroscopy
from POPC membranes with (red) and without (black) cholesterol. Similar
magnitudes of the splittings indicate structural independence of both the
headgroup and glycerol backbone on cholesterol incorporation. Note that the
line shapes are highly sensitive to the experimental setup and that the
relevant information on conformations is in the splittings, which are
proportional to $|S_{\rm{CH}}|$. (D) Overlayed lipid conformations in MD
simulations with (right) and without (left) cholesterol (yellow) illustrating
the experimental observations.
## III Results
## IV Experimental demonstration of the uncoupled motion
Figure 1 shows the effect of cholesterol on both the dynamics and structure of
POPC headgroup and glycerol backbone. Chemical shift resolution for all the
distinct carbons in the refocused-INEPT spectra displayed in Figure 1A is
enabled by simultaneous magic-angle spinning and heteronuclear decoupling. The
effect of cholesterol on the phospholipid dynamics is assessed by measuring
the $R_{1}$ and $R_{1\rho}$ values from POPC and POPC/cholesterol (1:1) multi-
lamellar vesicles and the effect on phospholipid structure by R-PDLF
spectroscopy. The relaxation rates for the headgroup and glycerol backbone are
shown in Figure 1B. The complete set of decays and relaxation rates measured
for all the phospholipid segments resolved in the 13C spectrum is given in
supplementary information Figures S1 (headgroup and glycerol backbone) and S2
(acyl chains). The dipolar splittings used to calculate the $S_{\rm{CH}}$
order parameters of headgroup and glycerol backbone are presented in Figure
1C. The resulting order parameters confirm the previously reported values
Ferreira et al. (2013a).
$R_{1}$ rates remain constant for both the glycerol backbone and the
headgroup, showing that the C–H bond motions with time-scales close to ns are
not affected by cholesterol for these carbons. On the other hand, the
$R_{1\rho}$ rates in the glycerol backbone (carbons $g_{1}$, $g_{2}$, and
$g_{3}$) increase by approximately a factor of two upon cholesterol addition.
In sharp contrast, the $R_{1\rho}$ values for the choline headgroup ($\alpha$,
$\beta$, and $\gamma$ segments) are unaffected by cholesterol and
significantly lower than in the glycerol backbone. The invariance of both the
headgroup carbon dipolar couplings (Figure 1C) and the relaxation rates
(Figure 1B) upon cholesterol incorporation implies that the conformational
ensemble and the time required to span all the available conformations is the
same irrespective of the glycerol backbone slowdown induced by cholesterol.
For acyl chains both structural ($S_{\rm{CH}}$) and dynamic observables
($R_{1}$ and $R_{1\rho}$) vary with incorporation of cholesterol (SI Figures
S2 and S3), signaling the expected ordering and slowdown in line previously
reported results Ferreira et al. (2013a); Sivanandam et al. (2009).
Figure 2: Impact of cholesterol (hollow bars: pure POPC, filled bars: 50%
POPC+50% cholesterol) on the effective correlation times, $\tau_{\rm{e}}$, of
different carbons in the headgroup and glycerol backbone of POPC quantified
experimentally and from lipid bilayer MD simulations with the CHARMM36,
Slipids and Berger force-fields. Note the different $y$-scales used on the
left and right plots to appreciate the significant difference of effective
correlation times for the choline headgroup (0.1-0.5 ns) and glycerol backbone
segments (2-5 ns).
To give an intuitive measure, sensitive to all time-scales, for the headgroup
and glycerol backbone dynamics, we quantified the effective correlation times
$\tau_{e}$. (Figure 2). To this end, we used the experimental data presented
in panels B and C of Figure 1 to calculate Ferreira et al. (2015)
$\tau_{e}=\frac{5R_{1\rho}-3.82R_{1}}{4\pi^{2}d_{\rm{CH}}^{2}N(1-S^{2}_{\rm{CH}})},$
(1)
where $N$ denotes the number of protons covalently bound to the carbon and the
coupling constant $d_{\rm{CH}}$ is approximately -22 kHz. The $\tau_{e}$
values of the choline headgroup are within 0.1–0.5 ns and remain constant
within the experimental accuracy upon addition of 50% cholesterol, while the
$\tau_{\rm{e}}$ values for the glycerol backbone slowdown with a factor of
approximately two, and are one order of magnitude slower than in the
headgroup.
## V Comparison of experiments with MD simulations
Figure 3: Effect of cholesterol (hollow symbols: pure POPC, filled symbols:
50% POPC+50% cholesterol) on the POPC headgroup and glycerol backbone C–H bond
order parameters. The experimental values were determined by R-PDLF
spectroscopy. The grey areas show the range of $S_{\rm{CH}}$ values for PC
bilayer systems with and without cholesterol reported until date (see e.g.
References Leftin and Brown (2011); Ferreira et al. (2013a)). For comparing
with the effect on the acyl chains see Figure S3. Figure 4: Effect of
cholesterol on internal structure and dynamics of POPC. (A) Dihedral angle
distributions for pure POPC membranes (darker thick lines) and
POPC/cholesterol membranes (lighter thin lines) from MD simulations using the
CHARMM36 (red), Slipids (blue) and Berger (green) force-fields. (B) Reduced
and normalized dihedral torsion autocorrelation functions (see Eq. 3) showing
their corresponding dihedral effective correlation times ($\tau_{c}^{\phi}$).
Note that the extremely short correlation times for the outermost right column
are due to the very narrow angle range accessible for this torsion dihedral.
Also included in Figure 2 are the $\tau_{\rm{e}}$ values calculated from three
sets of MD simulations using the CHARMM36, Slipids, and Berger force-fields.
Notably, CHARMM36 simulations reproduce the experimental $\tau_{\rm{e}}$
values of the choline headgroup, both with and without cholesterol, flawlessly
within experimental uncertainty. CHARMM36 also captures well the experimental
choline C-H bond order parameters which remain the same with the presence of
the sterol (Figure 3). For the glycerol backbone, CHARMM36 simulations
slightly overestimate the slowdown of the effective correlation times, but
give the best structural model among the three force-fields used. Slipids
simulations show the best agreement with experiments for the effect of
cholesterol on the glycerol backbone $\tau_{\rm{e}}$ values, although they
fail to capture the glycerol backbone structure, and overestimate the
effective correlation time for the $\gamma$ carbon.
The Berger force-field clearly produces the least realistic dynamics, giving a
significant overestimation of $\tau_{\rm{e}}$ for the choline headgroup
segments (both with and without cholesterol) and predicts an erroneous, large
(approximately 5-fold) cholesterol-induced slowdown of the choline headgroup
dynamics (see Figure S4 for $\tau_{e}$s from the Berger simulations drawn to
scale). Both the structural and dynamical force field properties observed here
are in line with the previously reported Botan et al. (2015); Antila et al.
(2021).
In contrast to the Berger model, both the CHARMM36 and Slipids models,
although not perfect, capture the key experimental observation in this work:
The structure and dynamics of the choline headgroup are not affected by
incorporation of cholesterol despite the increased acyl chain order and
hindered dynamics in the acyl chains and glycerol backbone, i.e, the headgroup
is uncoupled from the glycerol backbone in these models.
## VI Effect of cholesterol on the time-scale of internal motions of the
headgroup and glycerol backbone
Knowing their ability to reproduce the NMR measurables, we proceed to exploit
the temporal and spatial resolution in the distinct MD models to gain insight
on the rotational dynamics of specific sites on the molecules as well as the
origin and degree of (un)coupling between the headgroup and the rest of the
phospholipid. To this end, Figure 4 shows the distributions of selected
headgroup and glycerol dihedral angles, $\phi$, and the corresponding dihedral
effective correlation times, $\tau_{c}^{\phi}$, which are extracted from
autocorrelation functions
$G^{\phi}(\tau)=\langle\cos\phi(t)\cos\phi(t+\tau)\rangle,$ (2)
where the angular brackets denote an average over time and over the number of
molecules in the system. We define the dihedral effective correlation time
($\tau_{c}^{\phi}$) as simply the area under the reduced normalised
autocorrelation function
$g^{\phi}(\tau)=\frac{G^{\phi}(\tau)/G^{\phi}(0)-\langle\cos\phi\rangle^{2}}{1-\langle\cos\phi\rangle^{2}},$
(3)
where $\langle\cos\phi\rangle^{2}$ is the value of $G^{\phi}(\tau)$ at
infinitely long $\tau$. $\tau_{c}^{\phi}$ provides a measure of how much time
is needed for dihedral motion to sample its angle distribution.
For the more realistic force-fields (CHARMM36 and Slipids) cholesterol does
not affect the dihedral distributions of POPC as expected from the
$S_{\rm{CH}}$ data alone. More interestingly, for both of these force-fields,
the dihedral angles over the phosphate linkage between the glycerol and the
alpha carbon, ($g_{3}$)O–P(O) and (O)P–O($\alpha$), and the glycerol backbone
dihedral $g_{2}$–$g_{3}$, have no angles which are unavailable, in contrast to
the other dihedrals analysed. Furthermore, the ($g_{3}$)O–P(O) and
(O)P–O($\alpha$) exhibit fast correlation times ($\tau_{c}^{\phi}\leq$0.5 ns)
which are very close to the effective correlation times in Figure 2.
The most notable differences between the more realistic simulations and Berger
simulations are the slower $\tau_{c}^{\phi}$ in Berger, and how these are
affected by cholesterol. While in CHARMM36 and Slipids only minor changes are
observed, with a slight speedup of the sampling by cholesterol, in the Berger
model these internal dynamics slowdown considerably.
## VII Polar and azimuthal motion of the choline dipole and of the glycerol
backbone
To investigate the correlation between motions of headgroup and other parts of
lipid molecules, we quantified the autocorrelation functions of the polar and
azimuthal angles, $\theta$ and $\varphi$ (coordinate system where the
z-direction coincides with membrane normal), for a number of selected vectors
between intramolecular atomic pairs. The definition of the autocorrelation
functions $G^{\theta}(\tau)$ and $G^{\varphi}(\tau)$ are the same as in Eq. 2
but using $\theta$ and $\varphi$ as angles, respectively. Note that for
$G^{\theta}(\tau)$ a non-zero plateau at the long $\tau$ is expected since the
different $\theta$ angles are not equally likely to occur. On the other hand,
$G^{\varphi}(\tau)$ is always zero at long $\tau$ due to the lipid uniaxial
motion.
In Figure 5 we show these correlation functions for the choline dipole
orientation, P$\rightarrow$N, and for the interatomic vector connecting the
carbonyl carbons in the sn-1 and sn-2 positions, calculated from the most
realistic force-field (CHARMM36). The $\varphi$ autocorrelation functions
clearly show the contrasting effects of cholesterol on these vectors. While
for the choline dipole a speedup of reorientational motion is observed, the
$\varphi$ dynamics of the vector connecting the carbonyl carbons become slower
by almost an order of magnitude. To extract effective correlation times
$\tau_{c}^{\theta}$ and $\tau_{c}^{\varphi}$, we again integrate the reduced
and normalised autocorrelation functions. While cholesterol induces more than
a 4-fold slowdown for the $\varphi$ dynamics of the vector connecting carbonyl
carbons, the correlation times of the P$\rightarrow$N orientation remain
essentially the same. The complete set of reduced autocorrelations functions
analysed is given in SI Figures S7-S13 together with $\theta$ distributions,
and $\tau_{c}^{\theta}$ and $\tau_{c}^{\varphi}$ values.
Figure 5: The effect of cholesterol on the time-scales for the reorientations
of P$\rightarrow$N and (sn-1)O=C$\rightarrow$C=O(sn-2) vectors over the
spherical angles $\varphi$ and $\theta$. The autocorrelation functions shown
here are described in Equation 2.
## VIII Discussion
Our experimental and MD simulation results show that the phospholipid
headgroup conformational ensemble and dynamics remain unaffected by addition
of 50% cholesterol to the lipid membrane, despite the significant acyl chain
ordering and reduction in both the acyl chain and glycerol backbone dynamics.
Therefore, our results do not support models that contain interdependence
between the structure or dynamics of the hydrophobic and hydrophilic regions
of cellular or model phospholipid membranes.
The observed slowdown of the glycerol backbone upon addition of cholesterol
(Figures 2 and S5) arises from the longer timescales to which $R_{1\rho}$ is
sensitive to. The internal motions of the glycerol backbone are not affected
by cholesterol in the most realistic MD simulations used (CHARMM36 and
Slipids, see Figure 4) in line with the invariance of the glycerol backbone
$R_{1}$ values (Figure 1). Therefore, the slower glycerol backbone dynamics
induced by cholesterol most likely arises from a slowdown of the rotational
diffusion of the whole phospholipid body as previously suggested by Roberts et
al. Sivanandam et al. (2009); Roberts M F (2009), rather than restrictions in
internal dynamics.
The independence of headgroup and hydrophobic chain motions must result from a
set of fast internal rotations around phospholipid bonds with specific
orientations that decouple these motions. The MD simulations in best agreement
with the NMR experiments show a high flexibility for dihedral angles in the
headgroup region with a wide range of accessible conformations (Figures 4, S6
as well as Ref. Bacle et al. (2021)). From the set of dihedral distribution
functions, one clearly observes the highly flexible nature of the
($g_{3}$)O–P, P–O($\alpha$) and $g_{2}$-$g_{3}$ dihedral angles with all
angles over a complete dihedral rotation having non-zero probability in
contrast to the remaining torsions. This applies for all three force-fields,
though Berger has a less even distribution than CHARMM36 and Slipids. For
CHARMM36 and Slipids, the correlation times for the rotations around the
($g_{3}$)O–P and P–O($\alpha$) bonds are lower than 0.5 ns (Figure 4) and very
close to the effective correlation times measured for the C–H bonds from the
$\alpha$ and $\beta$ carbons. These $\tau_{c}^{\phi}$ values slightly decrease
with the addition of cholesterol, i.e. cholesterol induces a slight speed-up
of the torsion dynamics for these particular dihedrals, most likely because
fewer steric hindrances are present due to an increased average distance
between headgroups.
The highly flexible dihedral rotations around the phosphate P–O bonds, as well
as the g2–g3 torsion, are much faster than the transverse and longitudinal
rotational diffusion of the molecular frame whose correlation times have been
approximated to be 10–20 ns and 100 ns, respectively Klauda et al. (2008). The
most probable $\theta$ angle of the the (g3–)O–P and g2–g3 bonds is less than
10∘ (Figures S8 and S11) which is very close to be parallel with the bilayer
normal axis. A flexible, fast-rotating dihedral aligned with the membrane
normal leads to decoupling of headgroup from the longitudinal rotational
diffusion of the molecular frame. Decoupling from the transverse rotational
diffusion is mostly enabled by the fast motion over the P–O(–Cα).
The torsions around the phosphate group and g2–g3 dihedrals act analogously to
a frictionless spherical-joint which decouples the choline headgroup structure
and dynamics from the glycerol backbone. This is in line with the previous
analysis Klauda et al. (2008) where a partial decoupling of the headgroup from
the main phospholipid body due to the rotation of a phosphate dihedral was
suggested based on a comparison of CHARMM C27r MD simulation of pure
dipalmitoylphosphatidylcholine bilayers to 31P-NMR R1 data under several
magnetic fields. Here, we demonstrate that such decoupling is strong enough to
prevent the propagation of the slowdown effect of cholesterol from the acyl
chains and glycerol backbone to the headgroup.
We base our molecular interpretation of the decoupled motion on the CHARMM36
force field, which gives the best overall description for the headgroup and
glycerol backbone structure and dynamics among the available models (Figures 2
and 3, and Refs. Botan et al. (2015); Antila et al. (2021)). However, not all
the NMR observables calculated are within experimental errors even in CHARMM36
simulations and we cannot exclude the possibility that improved future models
correctly capturing the decoupling effect, effective correlation times, and
structural order parameters may give an alternative molecular interpretation.
The headgroup decoupling is not observed in Berger force-field, although it
also provides fast dynamics over the phosphate group dihedrals (Figure 4).
This is most likely due to an overestimation of the attractive interaction
between cholesterol and the choline group. Such interaction has been
interpreted previously as a consequence of the so-called umbrella effect where
a reorientation of the headgroup due to presence of cholesterol is often
assumed Alwarawrah et al. (2012). However, the combination of MD simulations
and experiments presented here indicates that such interaction is artefactual
and that both the orientation and dynamics of the headgroup are unaffected by
the sterol presence. The implicit assumption in the umbrella model of a
cholesterol effect on headgroup reorientation, either through a change of the
conformational ensemble or a change of dynamics, is not supported by our
experimental results or the more realistic MD models.
Correlation functions of P$\rightarrow$N and other intramolecular vectors,
calculated from CHARMM36 simulations, further support the idea of decoupled
motions between headgroup and the acyl chains. Cholesterol induces a
significant slowdown of the reorientation of the interatomic vectors between
atoms belonging to the glycerol backbone and acyl chain segments (Figures
S7-S13), while the effect on the P$\rightarrow$N vector, representing the
choline dipole, is negligible with only a slight speed up of the dynamics most
likely due to the increase of the distance between phospholipids headgroups.
The molecular description suggested here has rather strong implications for
membrane biophysics and should motivate a number of additional experiments and
simulations. It implies that the dipolar surface of glycerophospholipid
bilayers consists of freely rotating dipoles with timescales faster than 2 ns
that do not depend on the dynamics of the acyl chains or glycerol backbone.
The time-scale of reorientation of the dipoles is expected to influence the
interaction of the headgroups with charged molecules, e.g. proteins, that
approach the lipid biomembrane. For instance, it is known that the tilt of the
headgroup dipole is highly sensitive to membrane surface charge Scherer and
Seelig (1989). Under a positive surface charge, the headgroups tilt to a more
upright orientation (increase of the alpha and beta $S_{\mathrm{CH}}$ values)
and vice-versa for a negative surface charge due to the charge-dipole
electrostatic interactions. The results presented here suggest that the
phosphate and the g2–g3 dihedrals enable an unconstrained response of
headgroups to the electrostatic field and effectively uncouple the
interactions occurring in the membrane surface from the hydrophobic region.
Although we only investigate here PC headgroups, it is foreseable that the
decoupling applies to all other glycerophospholipids since the same molecular
bearings are present irrespectively of the substituent headgroup Marsh (2013).
In summary, our results suggest that for describing the dipolar interactions
at the surface of membranes, the hydrophobic structure may be neglected to a
good approximation and that the relevant headgroup physics lie on the
electrostatic interactions—which is remarkably useful considering the complex
molecular arrangement in the hydrophobic region of biological membranes.
## IX Acknowledgments
O.H.S.O acknowledges CSC – IT Center for Science for computational resources
and Academy of Finland (grants 315596 and 319902) for financial support.
H.S.A. gratefully acknowledges financial support from the Osk. Huttunen
Foundation, Finnish Academy of Science and Letters (Foundations’ Post Doc
Pool), Instrumentarium Science Foundation, and the Alexander von Humboldt
Foundation. T.M.F. was supported by the Ministry of Economics, Science and
Digitalisation of the State of Saxony-Anhalt, Germany.
T.M.F. greatly acknowledges Kay Saalwächter and Alexey Krushelnitsky for
invaluable support and discussions.
## X Competing interests
The authors declare no competing interests.
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## XI Supplementary Information
## XII Solid-state NMR experiments
### XII.1 R-PDLF experiments
A total of 32 points in the indirect dimension with increments equal to two
R18 blocks; SPINAL64 was used for proton decoupling during 13C acquisition,
with a nutation frequency of approximately 50 kHz, a total acquisition time of
0.07 s and a spectral width of 200 ppm; the rINEPT pulses were set at a
nutation frequency of 78.12 kHz.
### XII.2 $R_{1}$ and $R_{1\rho}$ experiments
RF $\pi/2$ and $\pi$ pulses were set to a nutation frequency of 63.45 kHz.
TPPM was used for proton decoupling during 13C acquisition, with a nutation
frequency of approximately 50 kHz, a total acquisition time of 0.1 s, recycle
delay of 10 s and a spectral width of 140 ppm. The spin-lock frequency for
$R_{1\rho}$ was 50 kHz.
For quantifying $R_{1}$ and $R_{1\rho}$ for a given carbon segment, we
determined the decay over the indirect dimension by fitting gaussian
lineshapes in the direct dimension and using the analytic areas of the fitted
functions. The decay was then fitted with a single exponential decay and the
error bounds for both the $R_{1}$ and $R_{1\rho}$ values presented are the 95
% confidence bounds from these fits.
## XIII MD simulations
The force field parameters were acquired from CHARMM-GUI (CHARMM36), the
Slipids web page (http://www.fos.su.se/~sasha/SLipids/, the pre 2020 version),
and from existing simulations in zenodo repository Ollila (2014); Bacle and
Fuchs (2018) (Berger). Similarly, the initial configurations, where the
bilayers spanned the x–y plane of a rectangular simulation box with periodic
boundary condition, were obtained either from CHARMM-GUI or constructed from
existing trajectories Ollila (2014); Bacle and Fuchs (2018). The initial
configurations were then relaxed using the steepest decent algorithm (5000
steps), followed by 2.5 ns + 41 ns of NPT simulation with semi-isotropic
pressure coupling maintaining 1 bar pressure. In the first 2.5 ns the pressure
set using the Berendsen barostat Berendsen et al. (1984) with coupling
constant 1 ps while the rest of the 41 ns pre-equilibration was done by
utilizing the Parrinello–Rahman barostat Parrinello and Rahman (1981) with 1
ps coupling constant. The latter barostat settings where then used for the
following 0.84 $\mu$s-1.24 $\mu$s of NPT production runs used for analysis
(See Table 1 in main text). The temperature in the simulations was set to 303
K using the Nose-Hoover thermostat Nose (1984); Hoover (1985) with coupling
constant 1 ps. The electrostatic interactions were modelled utilizing the
particle-mesh-Ewald algorithm Essman et al. (1995). For CHARMM36 and Slipids
simulations a cutoff of 1.2 nm was used to separate the real and reciprocal
space contributions to electrostatics whereas the Lennard-Jones potentials
were smoothly fixed to zero between cutoffs 1 nm and 1.2 nm. In Berger
simulations we utilized verlet list scheme with cutoff 1 nm for both
electrostatic and Lennard-Jones interactions. The leap-frog algorithm with
time step 0.002 ps was used to integrate the movement of the particles. To
avoid the need for a shorter time-step, the fast vibrations of the C–H bonds
of the lipids where removed using the LINCS algorithm Hess (2008) and SETTLE
Miyamoto and Kollman (1992) constrains were used for water. The resulting
simulation trajectories were analysed using in-house scripts. The exact
methodology utilized for extracting the NMR measurables, and their error
estimates, from the simulation is detailed in Ref. 49.
System | method | conditions | $T$ / ∘C | Phase | $S^{\alpha}_{\rm{CH}}$ | $-S^{\beta}_{\rm{CH}}$ | ref.
---|---|---|---|---|---|---|---
DMPC | 2H NMR | 150 mM NaCl, 10 mM HEPES | 30 | Lα | 0.048 | 0.046 | Macdonald et al. (1991b)
DMPC | 2H NMR | 150 mM NaCl, 10 mM HEPES | 15 | Lβ | 0.051 | 0.062 | Macdonald et al. (1991b)
DOPC | 2H NMR | water only | 30 | Lα | 0.049 | 0.029 | Ulrich and Watts (1994)
POPC | 2H NMR | NaCl saturated solution | 23 | Lα | 0.048 | 0.044 | Bechinger and Seelig (1991b)
DPPC | 2H NMR | water only | 45-90 | Lα | 0.048 | 0.046-0.024 | Gally et al. (1975)
DPPC/chol. (1:1) | 2H NMR | 0.2 M sodium acetate/acetic acid | 10-70 | Lα | 0.048/0.048-0.040 | 0.03-0.024 | Brown and Seelig (1978)
POPC/chol. (1:1) | R-PDLF | water only | 30 | Lα | 0.052 | 0.034 | here
POPC | R-PDLF | water only | 30 | Lα | 0.052 | 0.040 | here
Table S1: Previously published $\alpha$ and $\beta$ C–H bond order parameters
from 2H NMR spectroscopy for a number of phosphatidylcholine lamellar systems
together with values reported here using 1H-13C dipolar recoupling on POPC and
POPC/cholesterol (1:1). Figure S1: The $R_{1}$ and $R_{1\rho}$ decays
measured for the headgroup and glycerol backbone carbons in the POPC (blue)
and POPC/cholesterol (red) systems. Each point corresponds to the integral
determined from a Gaussian fit of the corresponding 13C peak in the high
resolution chemical shift spectrum acquired under MAS of 5 kHz. The spin lock
field for the $R_{1\rho}$ measurement was 50 kHz and the 13C Larmor frequency
was 125 MHz. Figure S2: The $R_{1}$ and $R_{1\rho}$ decays for the acyl chain
carbons measured in the POPC (blue) and POPC/cholesterol (red) systems. Each
point corresponds to the integral determined from a gaussian fit of the
corresponding 13C peak in the high resolution chemical shift spectrum acquired
under MAS of 5 kHz. The spin lock field for the $R_{1\rho}$ measurement was 50
kHz and the 13C Larmor frequency was 125 MHz. Figure S3: Effect of cholesterol
on the C–H bond order parameter magnitudes, $|S_{\rm{CH}}|$, of different
segments in the acyl chains of POPC measured experimentally and calculated
from lipid bilayer MD simulations with the CHARMM36 force-field. Figure S4:
Same as Figure 2 but with proper scale for visualizing the Berger data.
Figure S5: Effect of cholesterol on the effective correlation times,
$\tau_{\rm{e}}$, of different segments in the acyl chains of POPC calculated
experimentally and from lipid bilayer MD simulations with the CHARMM36 force
field. Figure S6: Dihedral angle distributions from the POPC (thick black
lines) and POPC/cholesterol (thin colored lines) MD simulations using the
CHARMM36 (top), Slipids (middle) and Berger (bottom) force-fields. Figure S7:
Orientation distribution, $p(\theta)$ (outermost left), and auto-correlation
functions for the polar, $\theta$ (middle row), and azimuthal, $\varphi$
(outermost right), angles of vector P$\rightarrow$N from POPC (top) and
POPC/cholesterol (bottom) CHARMM36 MD simulations. The grey lines represent
data for each individual phospholipid molecule in the simulation. $\theta$ and
$\varphi$ are the spherical angles in the laboratory coordinate frame defined
by the simulation box axes. Figure S8: Orientation distribution, $p(\theta)$
(outermost left), and auto-correlation functions for the polar, $\theta$
(middle row), and azimuthal, $\varphi$ (outermost right), angles of vector
($g_{3}$-)O$\rightarrow$P from POPC (top) and POPC/cholesterol (bottom)
CHARMM36 MD simulations. The grey lines represent data for each individual
phospholipid molecule in the simulation. $\theta$ and $\varphi$ are the
spherical angles in the laboratory coordinate frame defined by the simulation
box axes. Figure S9: Orientation distribution, $p(\theta)$ (outermost left),
and auto-correlation functions for the polar, $\theta$ (middle row), and
azimuthal, $\varphi$ (outermost right), angles of vector
($g_{1}$-)O$\rightarrow$O(-$g_{2})$ from POPC (top) and POPC/cholesterol
(bottom) CHARMM36 MD simulations. The grey lines represent data for each
individual phospholipid molecule in the simulation. $\theta$ and $\varphi$ are
the spherical angles in the laboratory coordinate frame defined by the
simulation box axes. Figure S10: Orientation distribution, $p(\theta)$
(outermost left), and auto-correlation functions for the polar, $\theta$
(middle row), and azimuthal, $\varphi$ (outermost right), angles of vector
$g_{1}\rightarrow g_{3}$ from POPC (top) and POPC/cholesterol (bottom)
CHARMM36 MD simulations. The grey lines represent data for each individual
phospholipid molecule in the simulation. $\theta$ and $\varphi$ are the
spherical angles in the laboratory coordinate frame defined by the simulation
box axes. Figure S11: Orientation distribution, $p(\theta)$ (outermost left),
and auto-correlation functions for the polar, $\theta$ (middle row), and
azimuthal, $\varphi$ (outermost right), angles of vector $g_{2}\rightarrow
g_{3}$ from POPC (top) and POPC/cholesterol (bottom) CHARMM36 MD simulations.
The grey lines represent data for each individual phospholipid molecule in the
simulation. $\theta$ and $\varphi$ are the spherical angles in the laboratory
coordinate frame defined by the simulation box axes. Figure S12: Orientation
distribution, $p(\theta)$ (outermost left), and auto-correlation functions for
the polar, $\theta$ (middle row), and azimuthal, $\varphi$ (outermost right),
angles of vector $g_{2}$ $\rightarrow$P from POPC (top) and POPC/cholesterol
(bottom) CHARMM36 MD simulations. The grey lines represent data for each
individual phospholipid molecule in the simulation. $\theta$ and $\varphi$ are
the spherical angles in the laboratory coordinate frame defined by the
simulation box axes. Figure S13: Orientation distribution, $p(\theta)$
(outermost left), and auto-correlation functions for the polar, $\theta$
(middle row), and azimuthal, $\varphi$ (outermost right), angles of vector
C1(sn-1)$\rightarrow$C1(sn-2) from POPC (top) and POPC/cholesterol (bottom)
CHARMM36 MD simulations. The grey lines represent data for each individual
phospholipid molecule in the simulation. $\theta$ and $\varphi$ are the
spherical angles in the laboratory coordinate frame defined by the simulation
box axes.
|
11institutetext: Anatoli Ivanov 22institutetext: Pennsylvania State University
at Wilkes-Barre, 44 University Drive, Dallas, PA 18612, USA, 22email:
<EMAIL_ADDRESS>33institutetext: Sergiy Shelyag 44institutetext: Flinders
University, Tonsley Innovation District, 1284 South Rd, Tonsley, SA, 5042,
Australia, 44email<EMAIL_ADDRESS>
# Explicit Periodic Solutions in a Delay Differential Equation
Anatoli Ivanov and Sergiy Shelyag
###### Abstract
We construct stable periodic solutions for a simple form nonlinear delay
differential equation (DDE) with a periodic coefficient. The equation involves
one underlying nonlinearity with the multiplicative periodic coefficient. The
well-known idea of reduction to interval maps is used in the case under
consideration, when both the defining nonlinearity and the periodic
coefficient are piece-wise constant functions. The stable periodic dynamics
persist under a smoothing procedure in a small neighborhood of the
discontinuity set. This work continues the research in recent paper IvaShe23
on stable periodic solutions of differential delay equations with periodic
coefficients.
## 1 Introduction
Scalar delay differential equations of the form
$x^{\prime}(t)=-\mu x(t)+f(x(t-\tau))$ (1)
are used as mathematical models in a broad variety of applications. Some of
the most well-known ones are the Mackey-Glass physiological models MacGla77 ,
Nicholson’s blowflies models GurBlyNis80 ; PerMalCou78 , and many others
GlaMac88 ; Kua93 . In spite of its formal simplicity equation (1) can exhibit
quite complex dynamical behaviors. Unlike scalar ordinary differential
equations the DDE (1) can have complicated dynamics of solutions including
oscillations about equilibrium, stable and unstable periodicity, and
complex/chaotic behaviors DieSvGSVLWal95 ; HalSVL93 .
More accurate mathematical models of real world phenomena take into account
some periodicity factors such as seasonal changes or circadian rhythm
parameter fluctuations and several others. Such improved models with respect
to equation (1) lead to similar type DDEs with periodic coefficients:
$x^{\prime}(t)=-\mu(t)x(t)+a(t)f(x(t-\tau)),$ (2)
where $\mu(t)$ and $a(t)$ are periodic functions with the same period. One of
the natural questions that appears in this context is whether DDE (2) has
periodic solutions with the same period as the periodic coefficients. This
work is an attempt to answer such question in a particular case.
Note that the periodicity problem we consider is different from similar ones
in many other publications, such as in e.g. Far17 (also see additional
references therein). We assume the negative feedback conditions on $f,x\cdot
f(x)<0,\;\forall x\neq 0,$ implying that DDE (2) admits the trivial solutions
$x\equiv 0$ (for any functions $\mu$ and $a$). This situation is similar to
the case of autonomous equation (1), when it admits a constant solution (with
the negative feedback in many cases). In Far17 and elsewhere, when such
constant positive equilibrium is perturbed by a multiplicative periodic
coefficient, the new problem does not admit constant solutions (positive
equilibrium) any longer.
A particular case of equation (2) is considered when $\mu(t)\equiv 0$ (see
equation (3) below). The initial choice of the nonlinearity $f$ and the
periodic coefficient $a(t)$ is as piecewise constant functions. Such simple
form allows for an explicit calculation of solutions for all forward times as
piecewise affine functions (made up continuously of line segments). The
problem of existence of slowly oscillating solutions and their stability is
reduced to the existence of fixed points of a simple interval map and their
attractivity. The dynamics of both is shown to persist when the two
discontinuous functions are replaced by close to them continuous functions (or
even of $C^{\infty}$ class).
The results of this paper are a continuation and an expansion of recently
obtained similar results by the same authors IvaShe23 .
## 2 Preliminaries
As in IvaShe23 we consider the special case of DDE (2) when $\mu(t)\equiv 0$:
$x^{\prime}(t)=a(t)f(x(t-1))$ (3)
Here initially the nonlinear function $f$ is given by
$f(x)=f_{0}(x)=-\mathrm{sign}(x)$ and the periodic coefficient
$a=a(t,a_{1},a_{2},p_{1},p_{2})$ is defined as
$a(t)=a_{0}(t)=\begin{cases}a_{1},\;\text{if}\;0\leq t<p_{1}\\\
a_{2},\;\text{if}\;p_{1}\leq t<p_{1}+p_{2}\\\ \text{periodic extension
outside}\;[0,p_{1}+p_{2})\;\text{for all}\;t\in\mathbf{R},\end{cases}$ (4)
with all the values $a_{1},a_{2},p_{1},p_{2}$ being positive and
$p_{1}+p_{2}:=T>1$.
For arbitrary initial function $\varphi(s)\in C=C([-1,0],\mathbf{R})$ such
that $\varphi>0\;\forall s\in[-1,0]$ the corresponding unique solution
$x=x(t,\varphi)$ exists for all $t>0$ (it is calculated by the consecutive
integration, the ”step method”). The solution is a continuous piece-wise
affine function made up of straight line segments. It is differentiable
everywhere except a countable set of points where two line segments with
different slopes match.
## 3 Main Results
### 3.1 Explicit Periodic Solutions
Given $f=f_{0}$ and $a=a_{0}$ as above for arbitrary initial function $\phi\in
C$ the corresponding solution $x=x(t,\phi)$ is constructed for all forward
times $t\geq 0$. It is a continuous piecewise differentiable function composed
of consecutive segments of affine (linear) functions.
We assume that periodic solutions have the form as depicted in Fig. 1 (this
will be shown to be the case for particular values of the parameters
$a_{1},a_{2},p_{1},p_{2}$). For arbitrary initial function $\phi\in C$ such
that $\phi(s)>0\;\forall s\in[-1,0],$ the corresponding forward solution
$x(t),t\geq 0,$ depends only on the value $\phi(0):=h>0$ and does not depend
on the other values $\phi(s),s\in[-1,0)$ of the initial interval. There exist
consecutive zeros $t_{1}>0$ and $t_{2}=t_{1}+2$ of the solution such that
$x(t)>0,t\in[0,t_{1})$, $x(t)<0,t\in(t_{1},t_{1}+2)$, and
$x(t)>0,t\in[t_{1}+2,p_{1}]$. The necessary condition for such first two zeros
to exist is that $p_{1}>2$. We also assume that $p_{1}<t_{1}+3<p_{1}+p_{2}$
and that the solution remains positive on the interval $[p_{1},p_{1}+p_{2}]$.
For the latter to be true one only has to require that
$x_{3}:=x(p_{1}+p_{2})>0$.
Figure 1: Slowly oscillating piece-wise affine solution
We next calculate the explicit values of
$t_{1},x_{1}=x(p_{1}),x_{2}=x(t_{1}+3),x_{3}=x(p_{1}+p_{2})$ in terms of the
parameters $a_{1},a_{2},p_{1},p_{2}$ and given initial value $h>0$. It is
straightforward to find that:
$t_{1}=\frac{h}{a_{1}},\quad x_{1}=x(p_{1})=-h+a_{1}p_{1}-2a_{1},\;$
$x_{2}=x(t_{1}+3)=\left(\frac{a_{2}}{a_{1}}-1\right)h+a_{1}p_{1}-2a_{1}+3a_{2}-a_{2}p_{1},$
$x_{3}=x(p_{1}+p_{2})=\left(2\frac{a_{2}}{a_{1}}-1\right)h+a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]:=mh+b$
In addition to the previous assumptions the condition $b>0$, i.e.
$b=a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]>0,$ (5)
guaranties that $x_{3}>0$, and thus that the solution
$x(t),t\in[0,p_{1}+p_{2}],$ is of desired shape (as shown in Fig. 1).
A fixed point $h_{*}>0$ of the affine map $F(x)=mx+b$ gives rise to a slowly
oscillating periodic solution $x=x(t,h_{*})$ of equation (1). Moreover, by the
linearity of $F$ and the construction, such periodic solution is
asymptotically stable if $|m|<1$. The latter is equivalent to $0<a_{2}<a_{1}$.
The unique fixed point $x=h_{*}$ is easily found as
$h_{*}=b/(1-m)=\frac{a_{1}[a_{1}(p_{1}-2)+a_{2}[6-(2p_{1}+p_{2})]]}{2(a_{1}-a_{2})}.$
(6)
Therefore, we arrive at the following statement
###### Theorem 3.1
Suppose that the parameters $a_{1},a_{2},p_{1},p_{2}$ are such that the
inequality (5) is satisfied and $a_{1}>a_{2}$. Then DDE (3) has an
asymptotically stable slowly oscillating periodic solution. The periodic
solution is generated by the initial function $\phi(s)\equiv
h_{*},s\in[-1,0],$ where $h_{*}$ is given by (6).
Note that due to the symmetry property of the nonlinearity $f$ (oddness) and
the procedure of construction of the periodic solution $x_{h_{*}}$, under the
assumption of Theorem 3.1 there also exists the symmetric to $x_{h_{*}}$
periodic solution generated by the initial function $\phi(s)\equiv-h_{*}$. The
two periodic solutions are related by $x_{-h_{*}}(t)\equiv-x_{h_{*}}(t)$.
### 3.2 Smoothing of Nonlinearities
In this subsection we demonstrate that the stable periodic solution derived in
the previous subsection persists under the standard smoothing procedure for
functions $f$ and $a$. The procedure is a well known one which has been used
in many publications (see e.g. IvaSha91 ; Pet80 and further references
therein). It consists in replacing each jump discontinuity by a continuous
affine function in a small neighborhood of a discontinuity point. We repeat
here the exact constructions as in IvaShe23 .
Consider the functions $f$ and $a$ as defined in subsection 3.1 :
$f(x)=f_{0}(x)$ and $a(t)=a_{0}(t)$. Let $\delta_{0}>0$ be small, and for
every $\delta\in(0,\delta_{0}]$ introduce the continuous functions
$f_{\delta}(x)$ and $a_{0}^{\delta}(t)$ by:
$f(x)=f_{\delta}(x)=\begin{cases}+1\;\text{if}\;x\leq-\delta\\\
-1\;\text{if}\;x\geq\delta\\\
-({1}/{\delta})x\;\text{if}\;x\in[-\delta,\delta],\end{cases}$ (7)
and
$a(t)=a_{\delta}(t)=\begin{cases}a_{2}+\frac{a_{1}-a_{2}}{2\delta}(t+\delta)\;\text{if}\;t\in[-\delta,\delta]\\\
a_{1}\;\text{if}\;t\in[\delta,p_{1}-\delta)\\\
a_{1}+\frac{a_{2}-a_{1}}{2\delta}[t-(p_{1}-\delta)]\;\text{if}\;t\in[p_{1}-\delta,p_{1}+\delta]\\\
a_{2}\;\text{if}\;t\in[p_{1}+\delta,p_{1}+p_{2}-\delta)\\\
a_{2}+\frac{a_{1}-a_{2}}{2\delta}[t-(p_{2}-\delta)]\;\text{if}\;t\in[p_{1}+p_{2}-\delta,p_{1}+p_{2}+\delta]\\\
\text{periodic extension on}\;\mathbf{R}\;\text{outside
interval}\;[0,p_{1}+p_{2}).\end{cases}$ (8)
The following statement is an extension of the main Theorem 3.1 to the case of
continuous functions $f_{\delta}$ and $a_{\delta}$.
###### Theorem 3.2
Suppose the assumptions of Theorem 3.1 are satisfied. There exists
$\delta_{0}>0$ such that for every $\delta\in[0,\delta_{0}]$ DDE (3) with
$f=f_{\delta}$ and $a=a_{\delta}$ has an asymptotically stable slowly
oscillating periodic solution $x_{\delta}(t).$ The solution $x_{\delta}(t)$ is
close in the uniform metric to the periodic solution $x_{h_{*}}$ of Theorem
3.1 for small $\delta\geq 0$, moreover $\lim_{\delta\to
0}\\{\sup_{t\in\mathbf{R}}|x_{\delta}(t)-x_{h_{*}}(t)|\\}=0$.
The proof is based on the explicit calculation of the similar value $x_{3}$ of
the solution $x(t,h)$ of the corresponding DDE (3). Its shape is similar to
that shown in Fig. 1, except that the corner points (where the derivative
$x^{\prime}$ is not defined) are replaced by a smooth $C^{1}$ curve, based on
the affine representation of both $f$ and $a$ around their discontinuity
points. The exact calculations repeat those in IvaShe23 . The final result is
a representation of $x_{3}(h)$ in the form $x_{3}(h)=\tilde{F}(h)$, where
$\tilde{F}(h)$ is $C^{1}$-close to the $F$ defined in subsection 3.1 (see
pages 74-75 of IvaShe23 for more details). Therefore, the existence,
stability, and closeness follow.
### 3.3 Numerical Demonstration
Periodic solutions described in subsections 3.1 and 3.2 can be easily verified
numerically. It is clear that sets of parameters leading to the existence of a
stable periodic solution, as described by Theorem 3.1, are not exceptional.
Any sufficiently small perturbation of a particular quadruple of the
parameters will produce an asymptotically stable periodic solution. They are
open in the set of all parameters. The exemplary parametric values for which
numerical periodic solutions have been obtained and confirmed to be of the
described form are given in Table 1.
$~{}~{}~{}~{}~{}a_{1}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}a_{2}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}p_{1}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}p_{2}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}h_{*}~{}~{}~{}~{}~{}$ | $~{}~{}~{}~{}~{}T~{}~{}~{}~{}~{}$
---|---|---|---|---|---
1 | 0.25 | 2.5 | 1.5 | 0.25 | 4
2 | 0.5 | 2.5 | 2 | 1/3 | 4.5
2 | 0.25 | 2.5 | 1 | 4/7 | 3.5
1 | 0.5 | 3 | 1 | 0.5 | 4
2 | 1 | 3 | 1.5 | 0.5 | 4.5
2.5 | 0.5 | 3 | 4 | 0.31 | 7
3 | 0.5 | 3 | 4.5 | 0.45 | 7.5
5 | 0.5 | 3 | 3 | 1.94 | 6
5 | 1 | 3 | 2 | 1.88 | 5
Table 1: Parametric values for which stable periodic solutions of equation (3)
with the coefficient defined in equation (4) have been demonstrated
numerically.
## 4 Discussion
This paper demonstrates the existence of another type of stable periodic
solutions in DDE (3). They are complimentary to those which existence was
shown in our recent work IvaShe23 . There is a possibility of existence of
additional types of stable periodic solutions in (3) with piece-wise constant
defining functions $f$ and $a$, the question we intend to tackle in our next
research considerations. It is of interest to show the existence of stable
periodic solutions in a general DDE of the form (3), where $f$ and $a$ are
smooth functions of arbitrary type. An interesting and seemingly very
nontrivial problem is the existence of (stable) periodic solutions in DDE (3)
with smooth functions, when $f$ is arbitrary and fixed nonlinearity with the
negative feedback property and the periodic coefficient $a$ is parameter
dependent, e.g., $a=a(t,\varepsilon)$, such that $a(t,0)=a_{0}>0$ and
$a(t+T,\varepsilon)=a(t,\varepsilon)$, for some $T>0$. As the value
$\varepsilon=0$ gives periodic solutions in the well-known autonomous case of
equation (1), and some larger positive values of $\varepsilon_{0}$ result in
$T$-periodic solutions with the same period as $a$, the intermediate values of
$\varepsilon\in(0,\varepsilon_{0})$ may yield varied dynamics and complex
transition patterns. This is an open problem worthy further investigation.
###### Acknowledgements.
The authors thank the mathematical research institute MATRIX in Australia
where part of this research was performed. Its final version resulted from
collaborative activities of the authors during the workshop ”Delay
Differential Equations and Their Applications” (https://www.matrix-
inst.org.au/events/delay-differential-equations-and-their-applications/) held
in December 2023. The authors are also grateful for the financial support
provided for these research activities by Simons Foundation (USA), Flinders
University (Australia), and the Pennsylvania State University (USA). A.I.’s
research was also supported in part by the Alexander von Humboldt Stiftung
(Germany) during his visit to Justus-Liebig-Universität, Giessen, in June-
August 2023.
## References
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* (2) Faria, T. (2017). Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems. J. Differential Equations, 263, no. 1, 509–533.
* (3) Glass, L., Mackey, M.C.: From Clocks to Chaos. The Rhythms of Life. Prinston University Press, (1988).
* (4) Gurney, W.S.C., Blythe, S.P., Nisbet, R.M. (1980). Nicholson’s blowflies revisited. Nature, 287, 17–21.
* (5) Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer Applied Mathematical Sciences, vol. 99 (1993).
* (6) Ivanov, A.F., Sharkovsky, A.N. (1991). Oscillations in singularly perturbed delay equations. Dynamics Reported (New Series), Springer Verlag, vol. 1, 165–224.
* (7) Ivanov, A., Shelyag, S. (2023). Stable periodic solutions in scalar periodic differential delay equations. Archivum Mathemticum (Brno), 59 (1), 69–76.
* (8) Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc. Series: Mathematics in Science and Engineering, Vol. 191, 398 pp. (1993).
* (9) Mackey, M.C., Glass, L. (1977). Oscillation and chaos in physiological control systems. Science, 197, 287–289.
* (10) Perez, J.F., Malta, C.P., Coutinho, F.A.B. (1978). Qualitative analysis of oscillations in isolated populations of flies. J. Theoret. Biol., 71, 505–514.
* (11) Peters, H. (1980). Comportement chaotique d’une equation differentielle retardee. C. R. Acad. Sci. Paris Sér. A-B, 290, no. 24, A1119–-A1122. (in French)
|
# FEVA: Fast Event Video Annotation Tool
Snehesh Shrestha, William Sentosatio, Huiashu Peng, Cornelia Fermüller,
Yiannis Aloimonos
The University of Maryland College Park
College Park, MD 20740
<EMAIL_ADDRESS>
###### Abstract
Video Annotation is a crucial process in computer science and social science
alike. Many video annotation tools (VATs) offer a wide range of features for
making annotation possible. We conducted an extensive survey of over 59 VATs
and interviewed interdisciplinary researchers to evaluate the usability of
VATs. Our findings suggest that most current VATs have overwhelming user
interfaces, poor interaction techniques, and difficult-to-understand features.
These often lead to longer annotation time, label inconsistencies, and user
fatigue. We introduce FEVA, a video annotation tool with streamlined
interaction techniques and a dynamic interface that makes labeling tasks easy
and fast. FEVA focuses on speed, accuracy, and simplicity to make annotation
quick, consistent, and straightforward. For example, annotators can control
the speed and direction of the video and mark the onset and the offset of a
label in real time with single key presses. In our user study, FEVA users, on
average, require 36% less interaction than the most popular annotation tools
(Advene, ANVIL, ELAN, VIA, and VIAN). The participants (N=32) rated FEVA as
more intuitive and required less mental demand. The code and demo are
available at http://www.snehesh.com/feva.
Figure 1: “Speed Label” enables you to create annotations (red rectangle) in
real-time by marking start ($st_{p}$) and end time ($et_{p}$) with a single
key press respectively. FEVA automatically adjusts these times of the event
annotation based on your reaction-time ($\Delta r$) in order to give you the
most precise intended times ($st_{i}$ and $et_{i}$).
_K_ eywords Video Annotation Tools $\cdot$ User Interface Design $\cdot$ User
Interaction
## 1 Introduction
Social scientists need to code conversations and behaviors of videotaped
interviews and experiments that quickly add up to hours of footage [1, 2, 3].
Computer scientists need datasets with appropriately labeled ground truth for
machine learning that contains video clips spanning hundreds of hours [4, 5,
6], or even thousands of hours [7]. Annotating videos is thus time-consuming
and tedious.
There are existing VATs [8, 9, 10, 11, 12, 13] that offer a range of different
features for video annotation activities. However, they often fail to meet the
need of the researchers due to the steep learning curves with complicated
features, overwhelming interfaces, and poor interaction techniques leading to
longer annotation time, inconsistencies, and user fatigue.
For example, to analyze soccer games, researchers annotate player ball
possessions, kicks, and assists. Annotating with tools that pause for the user
to name the annotation every time, needing to annotate with mouse context menu
options, and not allowing overlapping annotations make it an extremely tedious
and time-consuming task. On the contrary, coding by hand is more
straightforward. An annotator can play the video at a slower speed, use a
clicker count or a stopwatch [14] to lap the timestamps in real-time, then
enter them into a spreadsheet when completed. As a result, many annotators
still code by hand and rely on spreadsheets [15]. At scale, when you have
hours of such footage, computer scientists often outsource or crowdsource such
annotation tasks [16, 17, 18, 19]. This can raise concerns about privacy and
reliability [20]. To get around this, researchers blur faces to obfuscate
participant identity. However, this could compromise the quality of the
emotional judgment, causing inconsistencies in the results.
In this paper, we aim to design a video annotation tool that makes labeling
tasks easy and fast. We interviewed researchers from different disciplines to
understand standard practices, workflows, and tools used for video annotation.
Specifically, we interviewed 13 researchers from 5 fields (neuroscience,
behavioral psychology, film studies, and computer science). Researchers
expressed reservations with existing VATs leading to avoiding them. We further
surveyed 59 VATs (the list is available in the appendix ). We categorized
their main features and interface design choices from firsthand experience and
analyzed video tutorials for tools that are not accessible. According to the
interviews and survey results, we propose five design criteria that would
benefit video annotation activities, which are detailed in section 4:
* •
D1. Organize space based on operational workflow
* •
D2. Streamline high-frequency actions
* •
D3. Use algorithmic support when possible
* •
D4. Adopt what works and redesign what doesn’t
* •
D5. Allow flexibility
Figure 2: The shift keys activate the fine-tuning feature. The left and right
shift keys correspond to the start and end time of the label, respectively.
Pressing the arrow key while holding down the shift key adjusts the label’s
start or end time by a single frame.
These criteria inform the design of Fast Event Video Annotation (FEVA), a
video annotation tool with streamlined interaction techniques and a dynamic
interface that makes labeling tasks easy and fast. Simplified UI and features
such as real-time labeling with reaction time adjustments (figure 1), and
precise fine-tuning mechanisms (figure 2), help annotators create a large
number of accurate labels faster than any VAT.
To evaluate FEVA, we conducted two comparative studies. In the first study, we
compared the number of inputs users need to complete a task with FEVA versus
five other event-based VATs [8, 9, 10, 12, 13]. As seen in table 3, on
average, FEVA required 36.0% fewer inputs than competing VATs to do the same
task. In the second study, we asked 34 participants to perform annotations
with one of the VATs in random order. We found that regardless of the
background, users thought FEVA was more intuitive at 88% and easier to use at
91% of the time. Users also rated FEVA as requiring lower mental demand by 46%
(p < 0.00003), caused fewer frustrations by 62% (p <0.00147), had less
physical demand by 41% (p < 0.00187), and required less effort by 34% (p <
0.00324).
In Summary, in this paper, our contributions are as follows:
* •
a comprehensive understanding of existing VATs with interviews, tool surveys,
and pilot tests.
* •
a list of criteria for VAT tool design
* •
FEVA, an event-based video annotation tool that lets you annotate faster and
more accurately.
* •
a comparative study and a user study that demonstrates FEVA as more intuitive
and requiring less effort while creating more consistent and accurate
annotations.
## 2 Related Work
In this section, we introduce event-based annotation and tools we build upon,
the workflow used in the annotation process, and finally, the user interface
and the related interaction techniques. To understand the annotation workflow
and goals, we interviewed researchers and surveyed the literature on the steps
required to complete the desired annotations. We found two primary groups of
annotators: ones that annotate with the assistance of VAT and ones that rely
on more heuristic methods.
### 2.1 Event Based Annotation
Video annotation is the process of marking regions of interest (ROI) either a)
spacial (object annotation, OA) or b) temporal (event annotation, EA). OA is
marked on a single-time instance, referred to as frame-based annotation. EA is
the period when the event occurs. OA is prevalent in the CV community for
object detection and recognition, and bounding boxes [21, 22, 23, 24], dots
[25, 12], or polygon a.k.a segmentation [26, 27, 28] or masks [25, 29, 23] are
used to mark the annotation. While some VATs also track objects [30, 31] over
time [12, 25, 22, 24], it is different from EA. EA focuses on what is
happening in the scene, what the objects are doing, or what is done to the
objects [32, 33] rather than the objects themselves. So the start and the end
time marks are used. EA includes behaviors, interactions, emotional response,
speech, and movements [34, 35, 36]. Their application range across disciplines
from computer science for activity recognition [37, 12, 38, 39, 40, 41],
psychology for behavior analysis [11, 42, 35], to journalism to track and
present stories over timelines [43]. However, events annotation can be
challenging due to the complex nature of temporal navigation, the ability to
mark at the exact desired time of sliding events, use of available features of
the tools to facilitate the annotation. Therefore, the focus of developing
FEVA was to simplify the workflow and optimize the steps for the annotation of
events to make annotation easier and faster.
### 2.2 Heuristic Workflows
Some researchers prefer not to use specialized video annotation tools. For
example, some researchers thought it was easier to use a clicker to count the
time certain events occurred. This is error-prone, less reliable, and makes it
difficult to review the records in the future. Even though some VATs can
provide a similar functionality [12] using keypresses, researchers showed
hesitation using VATs with the fear of the initial setup time and the repeated
learning curve for new coders. In another study [14], research assistants (RA)
used a stopwatch to obtain the time taken "from when the OK button was pressed
to when the device beeped to signal completion of the RR count." These clicker
counts or stopwatch intervals are recorded either by hand with pen and paper
or entered in a spreadsheet. In another workflow, an interpersonal
relationship researcher described how, from a video of two people interacting,
they asked RAs to respond to research questions set up by the researchers
while watching a video. For example, multiple RAs focus on the entire
interaction or a specific individual in the video and respond to a Likert
scale on how attentive one of the partners was when the other spoke. These are
either recorded on paper or online using Qualtrics or Google forms.
### 2.3 Video Annotation Tools Workflows
The primary workflow using a VAT does not differ from the heuristic methods
for simple coding. The main difference is the higher learning curve and a
longer setup process. However, as the coding gets more complex, heuristic
methods have a diminishing return on speed as you have more annotators. The
annotation process is much slower, needing to do a lot of steps typically
tools might facilitate manually. However, not all VAT are created equal. The
workflow design, screen layout of features, interaction steps, and techniques
differ significantly across VATs.
Computer scientists and engineers design most VATs for computer vision,
machine learning, and robotics research and applications [12, 35, 4, 16, 7].
However, there are a handful of tools specifically designed for and used by
other domains such as film studies [13, 44] where VAT allows for character
analysis and scene analysis. Journalists use VAT to annotate and synchronize
events from multiple sources to present a cohesive story [43]. In sports,
teams annotate games and significant events and move to study for their teams
[45, 46]. While most tools have different focuses, the primary annotation is
marking a binary exists or not and a temporal range of events of interest. To
this end, events annotation is the foundation that makes further work easier
or, in some cases, possible. So it is essential to have accurate annotations.
Support for synchronized multi-camera views, micro and macro visualization of
events in a timeline, and convenient searching and reviewing video features
make FEVA a favorable choice in these areas.
#### 2.3.1 Layout design: Balance of features and space usage
Some tools have a lot of features with complicated workflows and many
permutations of features that can be customized and used, so the UI is packed
with small windows, buttons, and layers of menu items to get to them. These
tools take much longer to learn and get used to; however, they can be powerful
once you learn them. Due to many features, such as media player controls,
annotation controls, and visualizations, the available screen real estate can
be quite challenging. So some tools group them into smaller windows [13, 9,
8], organize them through multiple levels of menus [9, 10], or through
different operational modes [12, 10, 13]. While this helps organize the
functionality, it isn’t easy to access, and one needs a lot of practice to
remember the various steps required to use the software. While there are other
much simpler tools that serve a very niche purpose [46, 47, 44, 48, 49], and
the layout is simple, and they make excellent use of the screen real estate
for the functionality they offer. These tools are easy to learn and start to
use. However, the features are limited. Our interviews found that most
researchers’ and annotators’ needs are in between. Most features are never
used in the complex VAT, while simple VAT limits them from doing more than the
niche they offer, and they need to pair with other tools to augment the gap to
complete their needs. We designed FEVA with the motivation of creating a tool
that is simple to navigate and use but comes with features that more complex
tools offer without it being overwhelming that you have to take a course to
use a tool.
### 2.4 Adoption
The keyboard shortcuts for VIA [12] are intuitive and practical, especially
the hints that pop up based on the context is something most tools lack that
FEVA adopts as well. FEVA uses popular shortcuts that have become the standard
for media players, such as the spacebar to play or pause, ctrl+Z, and ctrl+Y
to undo and redo, etc. VIAN [13] is the only tool you don’t have to select
before dragging the label with the mouse, which is the same in FEVA. Advene
[9], ELAN [10] and VIA [12] provide alternative ways to visualize or execute
the playback option, such as continuous mode. While useful in some instances,
most annotators used the default way without changing, which could be
confusing. While some tools [12] allows one to create a label when the movie
is playing, it only expects the start time with a fixed length. Unfortunately,
most users need to return to the label and readjust them. FEVA improves upon
this interaction by allowing a second key press to mark the endpoint.
## 3 Understanding the State-Of-The-Art VATs and Their Comparisons
### 3.1 Target Users
To understand if researchers from different disciplines annotate their data,
what that entails, and what the workflow looks like, we interviewed 3
neuroscience researchers, 3 behavior psychology researchers, 2 film study
instructors, and 5 computer scientists. The interview was semi-structured to
answer the following 3 questions:
* •
IQ1: The nature of their research involving human studies, the kinds of data
collected, and if it entails video.
* •
IQ2: The workflow in the data collection process, post-processing of these
data, and code generation.
* •
IQ3: The structure and workflow for annotating the videos, and how the
annotations are used after.
Post-interview, the responses were tallied and coded for technical challenges.
The interview insights (II) are as follows:
* •
II1. There were three primary temporal annotations. 1) A binary label to mark
the presence or absence of certain events. For example, researchers were
interested in counting "how many times a person touched their face as one
indicator of how nervous the participants were." 2) A range label marks the
beginning and end of an event of interest. "Participants annotate videos for
specific moments. For instance, a couple might interact, then the researchers
have each couple member watch the video and annotate whenever a specific
thought or feeling occurred to them. They do this with both participants to
see convergence and dissonance of thoughts, feelings, goals, etc., etc." 3)
Certain kinds of labels, such as mood or scenario, lasted longer than an
action label. "For instance, we might want to compute a rating of how
responsive or caring one individual is to their relationship partner. So a
Research Assistant might watch an entire interaction, focusing on a specific
individual in the video, and then answer a question about how attentive they
are when their partner speaks." Most VATs do a poor job of supporting these
labels, so researchers had workarounds such as creating multiple label tracks,
each dedicated to a specific response.
* •
II2: The time, effort, and cost for data annotation are exponentially high. So
any system that made some improvements to make the annotation faster and more
reliable was always a huge win. "We spend a lot of our RA hours doing these
annotations. If there were a tool that could cut the time by even an hour, I
definitely would be using that tool."
* •
II3: Even with very explicit codes, annotated data often had low temporal
precision, so the agreement rules were relaxed. "Many RAs review and annotate
the videos. There will be slight variations in when each RA thinks a certain
event happened."
* •
II4: Collaboration and sharing of the annotation during the annotation process
and analysis step was cumbersome and required multiple steps to be in sync
between the teams. "…the students need to refresh the page if they are working
on annotating the same clip simultaneously to see what their classmates are
writing. That is also a problem for me since I like to add my comments while
they are working (to encourage them to elaborate on points or explore a new
point)."
* •
II5: Current VAT provided a poor interface for researchers to explore,
analyze, and search the annotated data.
* •
II6: Crowd-sourced or AI-generated annotation often needed so much review that
it was easier and faster for researchers’ RAs to do the annotations. "So I
used the [online automatic speech recognition tool]. With this, with the
premium, I still have to go in and edit everything. The labels for the actions
and the labels for the word-for-word speech need to match up in terms of start
and end time. So creating the labels by hand is actually easier for me."
This paper focuses on II2, II3, and II5, which help shape the design decisions
made in section 4.
### 3.2 Comparing Video Annotation Tools specifications
We created an extensive list of 59 VAT from the literature as listed in
appendix B. We were able to find their websites, download links or shared open
source codes, or at the minimum online videos of either talk by the authors or
how-to videos. We cataloged typical features most software supported and
unique features and techniques distinctive to each tool. In this extensive
survey, we share the table for the narrowed-down selected five tools. We
detail the selection method in section 6.1. Two types of tables were created:
* •
based on high-level taxonomy as shown in table 1 and
* •
based on features as shown in table 2
SN | Features | Description | Advene | ANVIL | ELAN | VIA | VIAN | FEVA (Ours)
---|---|---|---|---|---|---|---|---
0 | Last Updated | Month Year | Jun 2020 | Mar 2019 | Mar 2020 | Jul 2020 | May 2020 | Sep 2020
1 | SW Platform | Cloud vs Edge | Edge | Edge | Edge | Edge | Edge | Cloud or Edge
| | Native vs Web Based | Native | Native | Native | Web Based | Native | Web Based
| | Modular vs Static | Static | Static | Static | Static | Static | Modular
2 | License | Open Source vs Proprietary | Open Source | Proprietary | Open Source | Open Source | Open Source | Open Source
| | Commercial vs Open Access | Open Access | Open Access | Open Access | Open Access | Open Access | Open Access
| | Maintained vs Outdated | Maintained | Maintained | Maintained | Maintained | Maintained | Maintained
3 | Cost | Free, low cost, vs Expensive | Free | Free | Free | Free | Free/Low Cost | Free
| | One time vs Subscription | N/A | N/A | N/A | N/A | N/A | N/A
4 | Collaboration | Single User | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Multi-User (Simultaneous) | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
| | Crowd | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
5 | Target Users | Technical vs Non-Technical | Technical | Technical | Technical | Technical | Technical | Both
| | Academic vs Commercial | Academic | Academic | Academic | Academic | Academic | Academic
6 | Input Type | Image | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
| | Video | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Audio | ✗ | ✗ | ✓ | ✓ | ✗ | ✓
7 | Annotation Type | Object | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
| | Action | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Events | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Hybrid | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
8 | Annotation Approach | Manual | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Automatic | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
| | Hybrid | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
9 | Annotation Format | JSON | ✗ | ✓ | ✗ | ✓ | ✗ | ✓
| | XML | ✓ | ✗ | ✓ | ✗ | ✗ | ✗
| | SQL | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
| | Others? | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
Table 1: VAT taxonomy comparison table. SN | Features | Description | Advene | ANVIL | ELAN | VIA | VIAN | FEVA
---|---|---|---|---|---|---|---|---
1 | Annotation types | Object Bounding Box | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
| | Object Mask | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
| | Object Dot | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
| | Temporal Events | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
2 | Playback controls | Play Pause FF RR | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Speed +/- | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Timeline Jump | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
3 | Preview | Thumbnail Previews | ✓ | ✗ | ✗ | ✗ | ✗ | ✓
4 | Label | Multi-track | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Group tracks | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
| | User-defined Label Types | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Show/Hide/Collapse/Expand | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
5 | Speed Label | Sudo-Pedal | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
| | Transcribing Pedal Support | ✗ | ✗ | ✗ | ✗ | ✗ | ✓
6 | Resize | | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
7 | Move | | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
8 | Add | | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
9 | Delete | | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
10 | Edit | | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
11 | Import | | ✓ | ✓ | ✓ | ✗ | ✓ | ✓
12 | Import other formats | | ✓ | ✓ | ✓ | ✗ | ✓ | ✓
13 | Video Support | MP4 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
| | Others | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
14 | Cameras | Multi-Cam | ✓ | ✗ | ✗ | ✗ | ✗ | ✓
| | Switch View | ✓ | ✗ | ✗ | ✗ | ✗ | ✓
| | Instant Switch View | ✓ | ✗ | ✗ | ✗ | ✗ | ✓
15 | History | Undo/ Redo | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
16 | Search | Keyword | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
| | Filter by label type | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
17 | User Config | Remember/ Save | ✗ | ✗ | ✓ | ✗ | ✗ | ✓
18 | Modular/ API | Add-In Support | ✓ | ✗ | ✓ | ✗ | ✓ | ✗
| | Full Open Source Support | ✓ | ✗ | ✓ | ✓ | ✗ | ✓
| | Custom Layers | ✗ | ✗ | ✗ | ✓ | ✓ | ✗
| | Custom Tracks | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
19 | Layers | Show/Hide Layers | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
| | Human Joints Keypoints Support | ✗ | ✗ | ✗ | ✗ | ✗ | ✓
| | Human Bounding Box Support | ✗ | ✗ | ✗ | ✓ | ✗ | ✓
| | Human Mask Support | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
20 | Export Support | Video Clips | ✓ | ✗ | ✓ | ✗ | ✗ | ✓
| | Image Frames | ✗ | ✗ | ✓ | ✗ | ✗ | ✓
| | Closed Caption | ✓ | ✗ | ✓ | ✗ | ✗ | ✓
Table 2: VAT features comparison table. Figure 3: Example of steps needed to
annotate in 3 different VATs. The Upper left screenshot is ANVIL, the upper
right is VIAN, and the bottom screenshot is of VIA.
As a contrasting difference, here is an example of the steps required to
create a new annotation using 3 different VATs, as shown in figure 3. We make
a more thorough step-by-step comparison in the evaluation section in 6.2.
* •
In the upper left corner is ANVIL. It requires you to double-click the
starting point and click on the endpoint time mark to select the region. Then
right-click the shaded area to select from the context menu to create and edit
the label.
* •
On the upper right corner is VIAN. In VIAN, you right-click in the space
between the tracks and the ruler. Then select create an annotation, then
select text as the type for text annotation.
* •
In VIA, you create the track for the particular annotation and name it ("Bird"
in this case), then play the video till the movie’s starting point and hit the
letter ’A’ to create a fixed-size annotation. You can go back and adjust the
annotation.
We surveyed the layout and the functionalities available in a number of events
annotation software by using them to annotate videos or watch videos posted by
the authors. We noticed the following:
* •
Most VATs except for VIA [12] screen was filled with buttons, menus, and
windows. While essential features were buried in multiple levels of menu
items, there needed to be more screen real estate utilized. The new VAT
requires to be simple to understand and easy to navigate.
* •
Observations through video instructions and pilot testing, we noticed that
only limited features and controls were needed, primarily a) project and data
management, b) annotation management, c) video navigation, d) annotations
space, and e) the tool configurations. The new VAT needs a layout such that
these features are upfront and eliminate unnecessary features or allocate them
into rarely used spaces.
* •
The annotation visualization was limiting. An entire track was dedicated to a
single label [13], and no overlapping time labels were possible [8, 10]. Only
some [12, 9] allow overlapping time labels. However, it is difficult to
distinguish and manipulate the labels. The new VAT needs to organize the
annotations automatically and better use annotation space.
* •
While some tools [12, 11] have good annotation UX controls to create and edit
temporal placements, the use of mouse and keyboard had to be used
interchangeably between annotation steps. Some tools [8, 9, 10] have a very
cumbersome way to move or resize annotations. Some tools provided a history
option to undo/ redo [9, 10], while others provided no way to backtrack user
mistakes or perform experimental steps. The new VAT needed simple mouse
control and default keyboard shortcuts while allowing users to redefine them.
* •
All tools required you to stop and annotate except VIA [12], which provided
real-time annotation during video playback. However, VIA only allowed fixed-
length annotation flags requiring adjustment of the endpoint later.
Furthermore, VIA has poor visualization and lacks control for overlapping
labels. The new VAT needs a fast real-time way to continue annotating without
stopping.
## 4 Design Considerations
According to the interviews, literature survey, and the use of the tools, we
propose five design criteria that would benefit video annotation activities:
* •
D1. Organize space based on operational workflow: Features should be laid out
in a logical flow of the workflow while not veering too far away from standard
software conventions. Features not needed in that context should not be
visible or active to free up valuable screen real estate. High-frequency
controls and features should be in the middle of the screen.
* •
D2. Streamline high-frequency actions: The highly repeated actions, such as
creating annotations and fine-tuning them, should be optimized for easier and
faster execution. Try to accomplish them with a single key press when
possible. Stick to a single device (i.e., do not require some steps to use the
keyboard and the other steps mouse movements, using up precious time in
transitioning between the devices.) Leverage D1 when possible to minimize user
interaction required to accomplish the task.
* •
D3. Use algorithmic support when possible: Whenever possible, offload the
user and rely on the algorithm to take on the burden. For example, when the
timing is concerned, consider user reaction time and adjust for the lag in the
user input from the intended time. Additionally, allowing external modules
such as movement detection, human detection, speech detection, and recognition
can offload users’ need to find and annotate the events manually. However, we
should be cautious that no matter how good machine learning modules are, no
algorithm is 100% accurate. These should be considered as only additional
assistance and not to be completely relied on for ground truth generation.
* •
D4. Adopt what works and reinvent what doesn’t: Instead of reinventing the
wheel, adopt from other tools what works well based on user tests or pilot
tests and not instincts and personal preferences.
* •
D5. Allow flexibility: While having D4, tested default input methods is
great, adding flexibility by providing redundancies with mouse context menus
and keyboard shortcuts might be more intuitive for some users. Additionally,
let users redefine the key mapping as people have personal preferences.
## 5 FEVA
Figure 4: A screenshot of FEVA, where a participant interacting with a robot,
is being annotated.
The design considerations inform the minimalist design of FEVA, with most of
the screen real estate allocated for the video and the annotation. Inspired by
the clicker/ stopwatch methods and transcribing pedal, we created the speed
label. This feature lets you create labels during video play, where a user can
press a key to mark the starting and stopping points and continue to watch,
creating multiple labels with desired lengths. The system additionally
accounts for the user’s reaction time and adjusts the marked times as seen in
figure 1.
The speed label enables users to annotate in real-time, the fine-tuning
control lets the user make frame level adjustments, and the label organizer
arranges the label without ever overlapping them for direct access. The flat
UI is responsive and aesthetically pleasing. Using F-shape design [50], the
initial workflow items are on the top left, while the most viewed items are in
the middle of the screen, and the most interacted components are at the bottom
of the screen. Keeping most used elements in the hot zones while less used
elements such as camera selection and configuration buttons are on the right-
hand side in the less noticed area, making them easily tuned out unless
needed. The UI has redundant user input for flexible and fast interaction to
cater to novice and expert users. And underlying context and configuration-
dependent UI model comes to life only when you need them. Expert users can
annotate videos faster than in real-time by using media speed control and
speed labeler without touching the mouse. Researchers can also use the zoom
feature to visualize and analyze labels at a micro or macro level. FEVA UI can
display the most number of labels on one screen without losing their meaning.
### 5.1 Workflow
To create any project, upon selecting a video, FEVA imports it and creates an
empty annotation file, also referred to as a label or the dataset file (D1,
D2). Default tracks will be loaded, which can be customized from the
configuration (D3). Users can add or remove tracks. To play or pause the
movie, users can use the media player overlay on the main video or the
keyboard shortcut ’spacebar.’ You can use your mouse scroll button to navigate
the video in the timeline. You can also use the arrow keys with or without the
’ctrl’ key to move the video by different amounts. You can click and drag the
filmstrip, roll over the global timeline and click at the desired time, or
double-click the filmstrip or the local timeline window (D1, D2, D5). You can
also change the movie playback speed. You can zoom in and out at different
time intervals using the + and - icons around the white box on the global
timeline or roll over the timeline and ctrl and scroll.
To annotate, users can right-click on the tracks and select the label type
they want to create. Users can also hit the letter ’A’ key on the keyboard to
mark the start and a second time to mark the stop. See figure 1. This is
called the speed label, as you can keep annotating without stopping the movie.
Speed annotation requires a two-pass, but in our pilot tests, the speed label
is at least 1.5x faster than the traditional methods.
Following left-to-right and top-to-bottom conventions, workflow such as
loading the project, dataset, and manipulating the video and labels are
organized in that order. The main video is at the center of the screen
occupying the most space. Below the video are the annotation tracks that
follow conventions and eyes and the hand layout of users’ gaze and action
areas. See figure 5 and figure 4.
### 5.2 View/ Layout
As seen in figure 5, ’b’ shows the project selector from which you create a
new project and import videos. The ’d’ shows your label file selector to
create, load, save, merge, import, and export labels. You can search using the
’e’ and filter labels by type. You can double-click the label from the ’2’
label list to find the corresponding label in the timeline ’j.’ You can see
the thumbnail preview in ’h’ along with the current time window ’g’ and the
global timeline ’f.’
Figure 5: FEVA Screen Default Layout black boxed areas are 1) Toolbar, 2)
Label list, main video player, and multi-camera selector, 3) Video navigation
timeline, and 4) Label tracks. Key components are a) Logo, b) Project
selection, c) toolbar icons, d) Label data file selection, e) Search bar, f)
Global progress bar timeline, g) Local timeline ruler, h) Filmstrips, i) Label
type, and j) tracks and labels.
### 5.3 Control: User input system
Every media control controlled by the mouse also has its associated keyboard
shortcuts. For flexibility and efficiency, GUI for novice users that are based
on principles of recognition rather than recall [51], and shortcuts for expert
users for faster control. Users can press the play button overlayed on the
video or use the spacebar key at any point to play or pause the video. You can
also choose to speed up or slow down. To create an annotation while the video
is in play, you can press the letter ’A’ twice to indicate the start of the
label and end the label. This can be customized from the configuration. A
blank label is created after adjusting for your reaction time. You can also
double-click an empty space on a track or right-click the tracks and select a
label type you wish to create. While these shortcuts were selected to be
consistent with existing standards from other VATs, all shortcuts can be user-
defined easily in the user configuration.
To navigate, you can use your mouse to scroll or click and drag the
filmstrips, double click the desired point in the filmstrip or the global
progress bar. You can also jump to a specific label from the label list by
double-clicking it from the label list. Users can choose what they prefer by
providing multiple redundancies with both keyboard and mouse and keyboard
shortcuts that can be re-customized by the user.
### 5.4 Model/ State: Underlying UI support system
The UI makes use of the limited screen space (x-y plain) by using the z-axis
to layer components displayed and state changes based on contexts such as if
the video is playing, if labels are selected, if labels are being edited, or
if your mouse is hovering over a component or a specific feature is enabled in
the configuration. Every area is compacted with features that feel intuitive
based on affordance users naturally would assign those components. An example
is the video player area. When the video is playing, one would only see the
video. When a mouse pointer hovers over, media controls, current time and
frame number, and layer control are displayed. From the configuration, you can
also enable showing or hiding the video, human body keypoints [52], the
bounding box of humans or objects, segmentation masks, etc., that are
extensible for researchers to customize.
## 6 Evaluation
To evaluate FEVA, we compared FEVA with existing state-of-the-art (SOTA) VAT
with two studies.
* •
Interaction Benchmark: To evaluate the theoretical limits of how fast users
could annotate with each VAT, we counted the number of user inputs required to
perform various tasks.
* •
User Study: To evaluate user experience based on the user’s perceived workload
with each VAT, we conducted user studies where the users provided feedback
based on their experience.
### 6.1 The State-of-the-Art VAT Selection Method
We first created a master list of highly cited VAT that we could download and
use. In this list, we only included software that supported temporal
annotation that could be downloaded, installed, and run without taking extreme
measures for practical reasons. Therefore, VCode [11], SVAT [53], and VACA
[35] could not be included. Tools that were too specific such as ToolScape
[47], HistoryTracker [46], and CASAM [54] due to missing functionalities such
as start and end time, were removed from the list. Tools focused on crowd-
sourcing such as Glance [55] and CoAT [42], were excluded as we conducted a
single-user study. Using these criteria, we could narrow down the tools to be
compared in the study. EagleView [36] was extremely unstable to run, so we
could not test them. We narrowed down to Advene [9], ANVIL [8], Elan [10], VIA
[12], and VIAN [13].
### 6.2 Interaction Benchmark
To compare the steps required to do a particular task with each VAT, we
counted the number of clicks, double-clicks, mouse movements, and keyboard key
presses and took a cumulative sum as seen in table 3. If there were multiple
ways of completing a task, we included the fastest method for that tool. For
example, if you can press Ctrl+N to create a new project (keypresses count =
2) or can move your mouse to the main file menu, click the file, move the
mouse to a new project, and click on the menu item (mouse move = 2 and mouse
clicks = 2, total = 4), then we took the lesser of the two.
Table 3 shows the 15 tasks considered for the evaluation. These included basic
setting, label creation, and manipulation tasks. We selected tasks that the
majority of the VATs could do. If a VAT missed a feature, we assigned the
worst count received by competing with the VATs. For example, [12] does not
support "undo" or "redo" and received a count of 2.
The number of inputs required in FEVA is significantly less than the SOTA, as
shown in table 3. On average, FEVA requires 36% less input than the SOTA.
Based on the T-test, FEVA required significantly less input than all tools
except VIA, which was not statistically significant.
SN | Tasks | Advene | ANVIL | ELAN | VIA | VIAN | FEVA
---|---|---|---|---|---|---|---
1 | Create a project + Import a video | 7 | 8 | 8 | 7 | 14 | 3
2 | Create a single label | 4 | 6 | 7 | 2 | 5 | 2
3 | Create multiple labels | 8 | 12 | 14 | 3 | 9 | 3
4 | Create and name label | 4 | 9 | 7 | 7 | 9 | 7
5 | Edit labels | 4 | 7 | 3 | 3 | 3 | 5
6 | Resize labels | 6 | 5 | 6 | 4 | 4 | 4
7 | Move labels | 6 | 12 | 6 | 5 | 4 | 4
8 | Change label type | 6 | 6 | 6 | 6 | 6 | 4
9 | Delete labels | 3 | 5 | 4 | 3 | 3 | 3
10 | Find labels | 3 | 3 | 5 | 5 | 3 | 2
11 | Save labels | 2 | 2 | 2 | 2 | 2 | 2
12 | Load labels | 4 | 4 | 4 | 4 | 8 | 4
13 | Navigate video | 2 | 6 | 2 | 1 | 2 | 1
14 | Play/ Pause video | 2 | 1 | 2 | 1 | 1 | 1
| Play only label video | 6 | 5 | 4 | 3 | 6 | 2
15 | Undo/ Redo | 2 | 2 | 2 | 2 | 2 | 2
| TOTAL SCORE | 69 | 93 | 82 | 58 | 81 | 49
| FEVA Faster by | 29% | 47% | 40% | 16% | 40% |
| p-value | 0.0255 | 0.0013 | 0.0149 | 0.1321 | 0.0223 |
Table 3: The list of tasks done using the fastest possible methods in each
software (shortcuts where applicable). Each number reflects a cumulative sum
of mouse clicks, double clicks, movement, and key presses. The last row shows
how much FEVA is faster than the SOTA in percent (%) and the T-test p-values.
### 6.3 User Study
To evaluate user experience, we conducted a user study where participants used
two VAT, FEVA, and one another selected SOTA VATs (Advene, ANVIL, ELAN, VIA,
and VIAN) in a round-robin fashion. We counterbalanced the order of the two
tools by alternating the order with the next participant. Due to the COVID-19
regulations, we conducted our study via Zoom remote shared screen and control
feature. This introduced some lag in the user experience. However, since both
tools were remote, we assumed that the effects of the lag on the outcome were
not significantly discriminant. Participants were given an approximate time
range where an event occurs with clear descriptions of the events to annotate,
for instance, as seen in figure 6, "between 4 minutes and 20 seconds and 4
minutes 40 seconds, please annotate bunny jump roping." Participants completed
approximately 24 tasks until they gave up on the tool. After completing all
the tasks in the first software, they filled out the NASA Task Load Index [56]
questionnaire with a 5-point Likert scale. And this was repeated with the
second tool.
#### 6.3.1 Participants
We recruited 34 participants in the University community via email and social
media forums. In our study of N=32, the participants were 53% male and 47%
female, had a mean age of 30.4 +/- 5.9, with 84% not having video annotation
experience, and 66% had no experience with video editing. Two participants had
to be dropped due to zoom connection issues and are not counted in N=32.
#### 6.3.2 Procedure
For this study, we trained annotators for 3 minutes by watching a short
training video that taught the basics of media controls, video timeline
navigation, and how to create and edit annotations, followed by practice
trials for each item with the research coordinator answering any questions.
They then spent another 2 minutes exploring the tool on their own. The
participants spent the next 5 minutes practicing the tasks assigned
individually by the researcher where they were allowed to ask questions. Once
they got comfortable, they were randomly given four categories of tasks shown
in the list below, with each type of task repeated at least three times. The
tasks chosen were the most fundamental and repeated tasks annotators must do
during video annotation. The tasks ranged in complexity, with some tasks
requiring combinations of the fundamental steps. For instance, some tasks
simply asked the participant to navigate the video to 2 minutes and 20
seconds. In contrast, others asked participants to annotate three consecutive
events between a specific time and name them appropriately. They were no time
limits to perform the tasks. They worked on the standard freely available "Big
Buck Bunny" video which is approximately 10 minutes long at 720p resolution.
Figure 6 demonstrates one example task. After completing all the tasks with
the first tool, the participants filled out the NASA TLX workload
questionnaire and repeated the tasks with the second VAT.
Figure 6: An example of a task where a participant was asked to annotate an
event of the bunny jump roping between 04:28 minutes mark and 04:31 minutes
mark in FEVA.
We conducted the following four categories of basic tasks during the user
study:
* •
Navigate the video a) play/ pause the video, b) jump to a specific point in
the timeline, and c) jump to a precise point where a particular label is.
* •
Label Creation a) Create a new label at a specific time with a specific
length, b) Create a label when a participant shows a specific behavior (e.g.,
a character yawns, eats an apple, etc.), and c) create multiple labels in a
row.
* •
Label Content Manipulation a) Write text annotation for a label created and b)
Modify annotation text.
* •
Label Temporal Manipulation a) Move the label by a specific number of seconds
and b) Resize the label to change its starting time or ending time to match a
specific behavior by the person in the video
#### 6.3.3 Results
In this study, on average, the users felt less metal demand by 46% (p <
0.00003) with FEVA than the SOTA, less physical demand by 41% (p < 0.00187),
less effort was required by 34% (p < 0.00324), and felt less frustration by
62% (p <0.00147). The difference in the temporal demand and the performance
level indexes was not significant. We attributed this to there not being a
time limit enforced during the study, and except for one user on VIAN, where
the user gave up, all other users completed all the tasks.
FEVA | Mental Demand | Physical Demand | Temporal Demand | Performance Level | Effort | Frustration Level
---|---|---|---|---|---|---
MEAN (n=32) | 1.8 | 1.5 | 1.8 | 4.3 | 2.3 | 1.7
Table 4: Shows FEVA’s average score on a NASA Task Load Index with a 5-point
Likert scale.
FEVA vs. (%) | Mental Demand | Physical Demand | Temporal Demand | Performance Level | Effort | Frustration Level
---|---|---|---|---|---|---
Advene (n=5) | 75% | 57% | 0% | 14% | 67% | 183%
ANVIL (n=8) | 25% | 36% | -8% | 3% | 17% | 6%
ELAN (n=8) | 29% | 42% | 13% | 12% | 38% | 83%
VAI (n=8) | 44% | 36% | 6% | 11% | 22% | 36%
VIAN (n=3) | 120% | 40% | 20% | 17% | 83% | 140%
MEAN (n=32) | 46% | 41% | 5% | 10% | 34% | 62%
Table 5: Shows how FEVA compared to the other VAT. A positive number indicates
how much people perceived FEVA to be better than other VATs in the NASA TLX
respective six dimensions, and a negative number indicates how much worse.
### 6.4 User Feedback
#### 6.4.1 The Good
On average, users expressed FEVA was more intuitive 88% of the time and that
FEVA was easier to use 91% of the time. The features users liked the most were
the speed label, fine adjustments, "cooler feel," locating label and label
playback. A few users wished they could go back and change their feedback for
the first tool once they used the second tool. This was typical when they felt
they gave the first tool too high scores after using FEVA. In contrast, this
did not happen when it was the other way around. One user said the user wanted
to start a fan club and wanted to volunteer to annotate because it was "so
fun."
#### 6.4.2 The bad
One user mentioned that the user preferred traditional windowed UI for serious
work. So the user thought FEVA looked too mainstream tablet app-like. Another
user stated, "while I think it was fine for me, I don’t think my mom will be
able to use either of the tools. So I gave low scores to both of them." A few
users complained that they did not like pressing the enter key to confirm the
label after editing them. Clicking elsewhere, causing the loss of what was
just typed as a cancel feature, was not popular.
#### 6.4.3 The ugly
The majority of the confusion, however, was about the global and the local
timeline due to needing a clear separation and sharing the same preview
component. Participants remotely controlling the UI of the VATs over a Zoom
call on the research coordinator’s computer noticed a lag in the effect of
their actions. Some users complained that the UI did not update fast enough
due to the Zoom lags. "Maybe because I am controlling your computer through
Zoom, but a huge delay made it harder for me to resize the labels."
## 7 Implementation
### 7.1 Framework and Dependencies
We used ReactJS [57], an efficient component-based JavaScript library, and
wrote the architecture to be lightweight and responsive, so it works on most
people’s computers. The installation has only two dependencies of Python and
Flask. The front end relies on standard HTML, Javascript, ReactJS, and CSS. We
designed all the controls to optimize for performance and flexibility to
customize. We detail the UI layout breakdown in figure 5 and section 5.
### 7.2 Architecture
FEVA uses a simple server-client architecture typical for many web-based
applications. The server side runs on Python 3.5x or newer with Flask as the
webserver. The server side primarily handles servicing data (web content,
annotation data, and video streaming) when requested by the client-side
application. For FEVA, most of the modules and the design are on the client
side. We show more details of all the modules and their interaction with other
modules in the block diagram in appendix 8.
Figure 7: FEVA Client-Server Architecture Diagram. We include a more detailed
block diagram of the different modules on the client side in the appendix 8.
## 8 Discussion
This is the first version release of our tool FEVA, where we focused on
building the fundamental tool while streamlining the user interface and
interactions to make annotating events faster, intuitive, robust, and more
accurate. While these early results look promising, there are more research
questions that need to be further explored.
### 8.1 User Study
In our pilot and user study, we conducted a short-duration study. In our user
study, we assumed that 15 minutes was sufficient time for participants to
learn and practice annotating, which is how we designed the first evaluation
of comparing multiple VATs. However, we need to further our research by
conducting a longitudinal user study to understand the impact of our design on
users as they get more comfortable with the software.
### 8.2 User Input
In the input sector, we considered the keyboard and mouse/ trackpad as the
interaction devices at this stage. Still, we need to expand this to other
kinds of inputs, such as touch, speech, and gesture, to explore the potential
benefits of multi-modal methods.
### 8.3 Target Users
As more general public gets involved in the coding process, we focused this
study so anyone can participate in the coding process. Future studies to
gather feedback from seasoned coders will be valuable in understanding how
they use VAT.
### 8.4 VAT benchmarking study
In section 3 study, we counted the number of inputs as the pilot study showed
the correlation between the number of clicks and the time taken to complete a
task. A more comprehensive study should be considered, including the mouse
movements and time taken that can reflect user confusion and a more accurate
performance metric for completing the task.
### 8.5 Implicit
In section 6.3, we focused on the user’s perception to reflect their
experience. Future studies should expand to more quantitative measures
implicit in evaluating user confusion, performance, and success. We would also
like to conduct more studies to understand labeling consistency.
### 8.6 The layout
Based on user feedback, lessons learned from our observations, and the process
of comparing with existing tools, we could have done better. The multi-camera
selector layout takes up a lot of screen real estate. Many users found the
global timeline being so close to the local timeline and the thumbnail view
without any separation confusing and more challenging to get used to. We have
planned to redesign those experiences in the subsequent versions. We will
focus on several optimization opportunities in the next release to make FEVA
even faster.
### 8.7 Future Work
Beyond the incremental improvements, there were key features that we have
planned for the future:
* •
Adaptive: All the input controls were linear. We plan to explore dynamic and
adaptive control systems to various new interaction techniques for faster
annotation and a more intuitive experience.
* •
Extensible: An easier workflow for the open-source community to extend the
features.
* •
AI assisted: While this had some algorithmic support, better integration with
machine learning and deep neural network models is needed. We will further
research how AI can augment the annotation process while exploring ways to
inform the users of these models’ inaccuracies, uncertainties, and inherited
bias.
* •
Remote videos: While our internal prototypes support YouTube, there are
optimization opportunities that we need to explore before they can be used
seamlessly as an alternative to the local MPEG videos. We will also explore
other online streaming platforms.
* •
Case study: While we are working with some research labs in evaluating the
FEVA for their video annotation [58], we want to invite other interested
research labs to try out FEVA, collaborate with us, and grow as a community to
address needs that may not have been realized by our research so far.
## 9 Conclusion
We present a new event video annotation tool with streamlined interaction
techniques and a dynamic UI, contextually visible, and active features
organized based on the workflow and usage frequency. With features like speed
labeling, users can accurately annotate videos in real-time. With simplified
onboarding and workflow, researchers can set up and start annotating videos
using minimal time. We release FEVA’s source code in GitHub for everyone to
try and further extend its features. The community can also find project
samples, tutorial videos, GitHub issues for support, and future updates on the
GitHub page. As we expand our case studies, we invite more researchers to use
FEVA or contact us if you wish to collaborate.
## Acknowledgments
We thank Chethan Parameshwara, Levi Burner, Lindsay Little, and peers from UMD
and the Perception and Robotics Group for their valuable feedback and
discussions. We extend special thanks to all our project contributors Johnny
Chiu, Rachelle Sims, John Gao, Leya Abraham, Vikram Sehgal, Swagata
Chakroborty, Lucas Stuart, and Lin Chen. The support of NSF under grant OISE
2020624 is greatly acknowledged.
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## Appendix A Online Resources
Please visit FEVA website http://www.snehesh.com/feva for more information,
code, instruction videos, samples, and any updates.
## Appendix B List of Video Annotation Tools surveyed
* •
A Formative Study for Record-time Manual Annotation of First-person Videos
* •
A multi-level video annotation tool based on XML-dictionaries
* •
A Semi-Automatic Video Annotation Tool to Generate Ground Truth for
Intelligent Video Surveillance Systems
* •
Advene
* •
AIBU
* •
An innovative web-based collaborative platform for video annotation
* •
An Ontology Web Application-based Annotation Tool for Intangible Culture
Heritage Dance Videos
* •
Anvil
* •
atlas.ti
* •
Augmented Studio: Projection Mapping on Moving Body for Physiotherapy
Education
* •
Automatic tagging of video based on voice and localization
* •
Automatically Freezing Live Video for Annotation during Remote Collaboration
* •
AVISA: An annotation tool for video understanding
* •
BeaverDam: Video Annotation Tool for Computer Vision Training Labels
* •
BEDA: Visual Analysis of Relationships between Behavioral and Physiological
Sensor Data
* •
CASAM: collaborative human-machine annotation of multimedia
* •
CLIPPER: Audiovisual Annotation in the Study of Physics
* •
CoAT: A Web-based, Collaborative Annotation Tool
* •
CoVidA: Pen-based collaborative video annotation
* •
Crowd-Guided Ensembles: How Can We Choreograph Crowd Workers for Video
Segmentation?
* •
CrowdSport: Crowd-based Semantic Event Detection and Video Annotation for
Sports Videos
* •
DarkLabel
* •
Demo: Semantic Human Activity Annotation Tool - Using Skeletonized
Surveillance Videos
* •
EagleView: A Video Analysis Tool for Visualising and Querying Spatial
Interactions of People and Devices
* •
Elan
* •
Generating annotations for how-to videos using crowdsourcing
* •
Glance: rapidly coding behavioral video with the crowd
* •
HistoryTracker: Minimizing Human Interactions in Baseball Game Annotation
* •
iSeg: Semi-automatic ground truth annotation in videos: An interactive tool
for polygon-based object annotation and segmentation
* •
iVAT: An interactive tool for manual, semi-automatic and automatic video
annotation
* •
LabelMe
* •
Marquee: a tool for real-time video logging
* •
MediaDiver: viewing and annotating multi-view video
* •
MoViA: a mobile video annotation tool
* •
MRAS: Annotations for streaming video on the web
* •
Multimodal Video Annotation for Contemporary Dance Creation
* •
MuLVAT: A Video Annotation Tool Based on XML-Dictionaries and Shot Clustering
* •
NVivo
* •
Oudjat is dedicated to the manual annotation facial expressions of
emotion(FEE)
* •
Redesigning video analysis: an interactive ink annotation tool
* •
Rethinking Engagement with Online News through Social and Visual Co-Annotation
* •
Sirio, orione and pan: an integrated web system for ontology-based video
search and annotation
* •
Stabilized Annotations for Mobile Remote Assistance
* •
SVAT
* •
The MESH mobile video annotation tool
* •
Timelinely
* •
Tool Eval: Rapid Model-Driven Annotation and Evaluation for Object Detection
in Videos
* •
ToolScape: Enhancing the Learning Experience of How-to Videos
* •
VACA: a tool for qualitative video analysis
* •
VAnnotator: Annotations as multiple perspectives of video content
* •
VATIC
* •
VCode and VData: Illustrating a new Framework for Supporting the Video
Annotation Workflow
* •
VIA: The VIA Annotation Software for Images, Audio and Video
* •
ViBAT
* •
VideoAnt
* •
VideoJot: A Multifunctional Video Annotation Tool
* •
VidOR: Annotating Objects and Relations in User-Generated Videos
* •
ViTBAT: Video Tracking and Behavior Annotation Tool
* •
VoTT
## Appendix C FEVA Client Architecture
Figure 8: FEVA Client side block diagram of the various modules
|
$\displaystyle\Big{\|}\|v_{1}(t,x_{1},x^{\prime})+v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\Big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}$
$\displaystyle\leq\Big{(}\big{\|}\|v_{1}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}(I)}+\big{\|}\|v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}(I)}\Big{)}^{\frac{\mathfrak{a}}{\mathfrak{b}}}$
$\displaystyle\leq\big{\|}\|v_{1}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}+\big{\|}\|v_{2}(t,x_{1},x^{\prime})\|_{L^{\mathfrak{c}}_{x^{\prime}}(\mathbb{R}^{d-1})}\big{\|}^{\mathfrak{a}}_{L^{\mathfrak{b}}_{t}(I)}.$
Integrating in $x_{1}$ yields
$\|v_{1}+v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}\leq\|v_{1}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}+\|v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{\mathfrak{a}}\,.$
After raising both sides to the power $\frac{r}{\mathfrak{a}}$ and using again
that $(a+b)^{s}\leq a^{s}+b^{s}$, for $s=\frac{r}{\mathfrak{a}}\leq 1$, it
follows
$\|v_{1}+v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r}\leq\|v_{1}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r}+\|v_{2}\|_{L^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}_{e_{1}}}^{r},$
as desired.
Second, we assume that $\mathfrak{a}\geq\mathfrak{b}$, and in particular that
$r\leq\mathfrak{b}$. Then, (3) and the triangle inequality yield
$\|v_{1}+v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r}\begin{aligned}
&\leq\Big{(}\int_{\mathbb{R}}\big{(}\big{\|}\|v_{1}(x_{1})\|_{L^{\mathfrak{c}}_{x^{\prime}}}\|^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}+\big{\|}\|v_{2}(x_{1})\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\big{)}^{\frac{\mathfrak{a}}{\mathfrak{b}}}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x_{1}\Big{)}^{\frac{r}{\mathfrak{a}}}\\\
&=\Big{\|}\big{\|}\|v_{1}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}+\big{\|}\|v_{2}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}^{\frac{r}{\mathfrak{b}}}\\\
&\leq\Big{(}\Big{\|}\big{\|}\|v_{1}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}+\Big{\|}\big{\|}\|v_{2}\|_{L^{\mathfrak{c}}_{x^{\prime}}}\big{\|}^{\mathfrak{b}}_{L^{\mathfrak{b}}_{t}}\Big{\|}_{L^{\frac{\mathfrak{a}}{\mathfrak{b}}}_{x_{1}}}\Big{)}^{\frac{r}{\mathfrak{b}}}\\\
&=\Big{(}\|v_{1}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{\mathfrak{b}}+\|v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{\mathfrak{b}}\Big{)}^{\frac{r}{\mathfrak{b}}}\\\
&\leq\|v_{1}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r}+\|v_{2}\|_{L_{e_{1}}^{(\mathfrak{a},\mathfrak{b},\mathfrak{c})}(I\times\mathbb{R}^{d})}^{r},\end{aligned}$
as desired.
Having shown that (3) holds, the desired bound (3) follows for
$r=\frac{2}{1-\varepsilon_{0}}$, which indeed satisfies
$r\leq\min\\{\mathfrak{a},\mathfrak{b}\\}$ for any directional norm in the
definition of $X_{N}(I)$ or $Y_{N}(I)$. The proof of the desired results is
finished. ∎
## 4\. The multilinear estimates
We start this section by proving a bilinear estimates in terms of directional
space-time norms. Then, we derive crucial trilinear estimates that allow us to
control the cubic nonlinearity in (1).
###### Lemma 4.1.
Fix any $0<\varepsilon\lesssim 1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon}1$. For any two functions
$h_{+},h_{-}\colon\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{C}$ and any
$N_{+},N_{-}\in 2^{\mathbb{N}}$ with $N_{+}\geq N_{-}$ it holds that
$\displaystyle\|h_{+}h_{-}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim_{\varepsilon_{0}}N_{+}^{\varepsilon}\Big{(}\frac{N_{+}}{N_{-}}\Big{)}^{-\frac{1}{2}}\times\begin{dcases}N_{-}^{\mathfrak{s}_{c}}\|h_{+}\|_{X_{N_{+}}(\mathbb{R})}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{Y_{N_{+}}(\mathbb{R})}\|h_{-}\|_{Y_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{X_{N_{+}}(\mathbb{R})}\|h_{-}\|_{Y_{N_{-}}(\mathbb{R})},\\\
\|h_{+}\|_{Y_{N_{+}}(\mathbb{R})}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})}\,,\end{dcases}$
where $\mathfrak{s}_{c}\coloneqq\frac{d-2}{2}$.
###### Proof.
Let $\mathop{\kern 0.0pt\mathrm{U}}\mathopen{}_{e_{l}}$ be as in Lemma 2.5,
and recall that $\sum_{l}\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}=\mathop{\kern
0.0pt\mathrm{Id}}\mathopen{}$. The triangle inequality implies
$\|h_{+}\,h_{-}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\leq\sum_{l=1}^{d}\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\,,$
and therefore it suffices to prove (4.1) for $h_{+}$ replaced by
$\mathop{\kern 0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}$.
By Fubini’s Theorem and Hölder’s inequality we get that
$\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\begin{aligned}
&=\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L_{e_{l}}^{(2,2,2)}(\mathbb{R})}\\\
&\lesssim\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}})}(\mathbb{R})}\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{2}{\varepsilon_{0}})}(\mathbb{R})}\,.\end{aligned}$
From the definitions (3) and (3) of the norms $X_{N}$ and $Y_{N}$ we directly
obtain the first three bounds in (4.1).
To prove the last estimate in (4.1), we use Hölder’s inequality to obtain the
bound
$\big{\|}(\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+})\,h_{-}\big{\|}_{L_{e_{l}}^{(2,2,2)}(\mathbb{R})}\lesssim\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{1}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})})}(\mathbb{R})}\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{1-\varepsilon_{0}})}(\mathbb{R})}\,.$
Then from the definition (3) of the norm $X_{N}$ it follows that
$\|h_{-}\|_{L_{e_{l}}^{(\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{1-\varepsilon_{0}})}(\mathbb{R})}\lesssim
N_{-}^{\frac{1}{2}}\|h_{-}\|_{X_{N_{-}}(\mathbb{R})}\,.$
To control thus we can use interpolation to control $\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}$ we use interpolation. Indeed, since
$\mathfrak{c}_{0}=2\frac{d-1}{d-2}\geq 2$, it holds that
$\frac{2}{1-\varepsilon_{0}}\leq\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})}\leq\frac{2}{\varepsilon_{0}}$
for any $0<\varepsilon_{0}<2^{-100}$. Thus, interpolation with an
appropriately chosen parameter $\theta\in[0,1]$, followed by referencing the
definition (3) of the norm $Y_{N}$, gives that
$\displaystyle\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2\mathfrak{c}_{0}}{\mathfrak{c}_{0}-2(1-\varepsilon_{0})})}(\mathbb{R})}$
$\displaystyle\leq\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}})}(\mathbb{R})}^{1-\theta}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{L_{e_{l}}^{(\frac{2}{\varepsilon_{0}},\frac{2}{1-\varepsilon_{0}},\frac{2}{\varepsilon_{0}})}(\mathbb{R})}^{\theta}$
$\displaystyle\lesssim N_{+}^{\frac{1-\theta}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}^{1-\theta}N_{+}^{\frac{\theta}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}^{\theta}$
$\displaystyle=N_{+}^{-\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}h_{+}\|_{Y_{N_{+}}(\mathbb{R})}$
and the proof is finished. ∎
The above lemma is the crucial ingredient for showing that product of three
functions taken from the spaces $Y^{S}$ or $X^{\mathfrak{s}}$ lies in the
space $X^{*,\sigma_{*}}$, which we use to control the non-homogeneity $h$ in
(3). The specific regularity of the product depends on whether the functions
in play belong to $Y^{S}$ or to $X^{\mathfrak{s}}$ and on the respective
regularity exponents $S$ and $\mathfrak{s}$. Since the three-fold product of
functions is estimated in the norm $X^{*,\sigma_{*}}$, defined by duality in
(3), the proof of Proposition 4.2 naturally reduces to showing bounds on a
$4$-linear integral form, formulated in Lemma 4.3. The $4$-linear estimate
(4.3) of Lemma 4.3 in turn can be reduced using the Cauchy-Schwarz inequality
to the bilinear $L^{2}$ estimate of Lemma 4.1.
###### Proposition 4.2.
Fix any
$0<S_{1}\leq S_{2}\leq S_{3}<\mathfrak{s}_{c}<\mathfrak{s}\,,$
and choose any $0<\varepsilon\lesssim_{S_{j},\mathfrak{s}}\\!1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S_{j},\mathfrak{s}}\\!1$. Then for any
$z_{j}$ and $v_{j}$, $j\in\\{1,2,3\\}$, the following estimates hold:
$\displaystyle\big{\|}z_{1}z_{2}z_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|z_{2}\|_{Y^{S_{2}+\varepsilon}(\mathbb{R})}\|z_{3}\|_{Y^{S_{3}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with }\sigma_{*}\coloneqq
S_{1}+\min\Big{(}S_{2},\frac{1}{2}\Big{)}+\min\Big{(}S_{3},\frac{1}{2}\Big{)}\,.$
$\displaystyle\big{\|}z_{1}z_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|z_{2}\|_{Y^{S_{2}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with }\sigma_{*}\coloneqq
S_{1}+\min\Big{(}S_{2},\frac{1}{2}\Big{)}+\frac{1}{2}\,.$
$\displaystyle\big{\|}z_{1}v_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|z_{1}\|_{Y^{S_{1}+\varepsilon}(\mathbb{R})}\|v_{2}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with
}\sigma_{*}\coloneqq\min\big{(}S_{1}+\mathfrak{s},\,S_{1}+1\big{)}\,.$
$\displaystyle\big{\|}v_{1}v_{2}v_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\lesssim\|v_{1}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{2}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}\|v_{3}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}$
$\displaystyle\qquad\text{with
}\sigma_{*}\coloneqq\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})\,.$
The implicit constants are allowed to depend on $\varepsilon$,
$\varepsilon_{0}$, $\mathfrak{s}$, and $S_{j}$.
We deduce Proposition 4.2 by duality from Lemma 4.3 below, which is proved at
the end of the section.
Before proceeding, we introduce some notation. For $\sigma_{*}$ as in each
case of Proposition 4.2 (detailed below) we choose $v_{*}\in
X^{-\sigma_{*}}(\mathbb{R})$. Next, for any function
$h\in\\{v_{*},z_{j},v_{j}\\}$, $j=1,2,3$, and $N\in 2^{\mathbb{N}}$ we define
quantities
$Z_{N}(h)\coloneqq\begin{cases}\|v_{*}\|_{X_{N}(\mathbb{R})}&\text{if
}h=v_{*}\,,\\\ \|v_{j}\|_{X_{N}(\mathbb{R})}&\text{if }h=v_{j}\,,\\\
\|z_{j}\|_{Y_{N}(\mathbb{R})}&\text{if
}h=z_{j}\,,\end{cases}\qquad\alpha(h)\coloneqq\begin{cases}-\sigma_{*}&\text{if
}h=v_{*}\,,\\\ \mathfrak{s}&\text{if }h=v_{j}\,,\\\ S_{j}&\text{if
}h=z_{j}\,,\end{cases}$
and we denote by $\mathscr{S}(h_{1},\cdots,h_{4})$ the set of permutations of
a quadruple $(h_{1},\cdots,h_{4})$.
###### Lemma 4.3.
Fix exponents as in (4.2), and choose any $0<\varepsilon\lesssim 1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S_{j},\mathfrak{s}}1$. Then for any
$N_{j}\in 2^{\mathbb{N}}$, $j\in\\{1,2,3,4\\}$, we have that
$\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern 0.0pt\mathrm{d}}\mathopen{}x=0$
unless $N_{j}\lesssim\sum_{j^{\prime}\neq j}N_{j^{\prime}}$ for all
$j\in\\{1,2,3,4\\}$. Furthermore,
$\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\lesssim(N_{1}N_{2}N_{3}N_{4})^{\varepsilon}\prod_{j=1}^{4}N_{j}^{\alpha(h_{j})}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}$
provided that one of the following conditions is satisfied:
Case $zzz$:
$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},z_{2},z_{3})$ and $\sigma_{*}$
as in (4.2).
Case $zzv$:
$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},z_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).
Case $zvv$:
$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},z_{1},v_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).
Case $vvv$:
$(h_{1},\cdots,h_{4})\in\mathscr{S}(v_{*},v_{1},v_{2},v_{3})$ and $\sigma_{*}$
as in (4.2).
###### Proof of Proposition 4.2.
Choose any $h_{j}\in\\{z_{j},v_{j}\\}$, $j=\\{1,2,3\\}$. The definition of
$X^{*,\sigma_{*}}$ and $X_{N}$ in (3), $\mathop{\kern
0.0pt\mathrm{Id}}\mathopen{}=\sum_{N\in 2^{\mathbb{N}}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}$, the triangle inequality, and
$\Big{(}\sum|a_{j}|^{2}\Big{)}^{\nicefrac{{1}}{{2}}}\leq\sum|a_{j}|$ imply
that
$\big{\|}h_{1}h_{2}h_{3}\big{\|}_{X^{*,\sigma_{*}}(\mathbb{R})}\begin{aligned}
&=\Big{(}\sum_{N\in 2^{\mathbb{N}}}N^{2\sigma_{*}}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\big{(}h_{1}h_{2}h_{3}\big{)}\big{\|}_{X^{*}_{N}(\mathbb{R})}^{2}\Big{)}^{\nicefrac{{1}}{{2}}}\\\
&\leq\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}N^{\sigma_{*}}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\big{(}P_{N_{1}}h_{1}P_{N_{2}}h_{2}P_{N_{3}}h_{3}\big{)}\big{\|}_{X^{*}_{N}(\mathbb{R})}\\\
&\leq\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}N^{\sigma_{*}}\sup_{v_{*}}\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\,,\end{aligned}$
where the upper bound in the last expression is taken over all $v_{*}\in
C^{\infty}_{c}(\mathbb{R}\times\mathbb{R}^{d})$ with $\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\|_{X_{N}}\leq 1$. Note that the
assumptions of Lemma 4.3 on $S_{j}$, $\mathfrak{s}$, $\varepsilon$,
$\varepsilon_{0}$, and $\sigma_{*}$ coincide with those of Proposition 4.2,
and therefore (4.3) with $\varepsilon$ replaced by $\frac{\varepsilon}{8}$
yields
$\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}v_{*}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\begin{aligned}
&\lesssim(N_{1}N_{2}N_{3}N)^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\\\
&=(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\,.\end{aligned}$
We may assume that $N\lesssim\max_{j}N_{j}$ since otherwise, by (4.3), the
left-hand side vanishes. We have that
$(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\lesssim
N^{-\frac{\varepsilon}{4}}$, and consequently
$\displaystyle\sum_{N,N_{1},N_{2},N_{3}\in
2^{\mathbb{N}}}(N_{1}N_{2}N_{3})^{-\frac{3\varepsilon}{8}}N^{\frac{\varepsilon}{8}}\prod_{j=1}^{3}N_{j}^{\alpha(h_{j})+\varepsilon}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}$ $\displaystyle\hskip
100.00015pt\lesssim\sum_{N\in
2^{\mathbb{N}}}N^{-\frac{\varepsilon}{4}}\prod_{j=1}^{3}\Big{(}\sum_{N_{j}\in
2^{\mathbb{N}}}N_{j}^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}\Big{)}\,.$
Using the Cauchy-Schwarz inequality and the definitions of $Z_{N_{j}}(h_{j})$,
$\alpha(h_{j})$, $X^{\mathfrak{s}}$, and $Y^{S}$ we have
$\displaystyle\sum_{N_{j}\in
2^{N}}N^{\alpha(h_{j})+\varepsilon/2}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})$
$\displaystyle\lesssim\Big{(}\sum_{N\in
2^{\mathbb{N}}}N_{j}^{-\varepsilon}\Big{)}^{1/2}\Big{(}\sum_{N\in
2^{N}}N_{j}^{2\alpha(h)+2\varepsilon}Z_{N_{j}}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j}\big{)}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim_{\varepsilon}\begin{cases}\|z_{j}\|_{Y^{S_{j}+\varepsilon}(\mathbb{R})}&\text{if
}h=z_{j}\,,\\\ \|v_{j}\|_{X^{\mathfrak{s}+\varepsilon}(\mathbb{R})}&\text{if
}h=v_{j}\,.\end{cases}$
∎
###### Proof of Lemma 4.3.
Note that the right-hand sides of (4.3) and (4.3), as well as the subsequent
conditions are the same if we exchange $P_{N_{j}}h_{j}$ with
$P_{N_{j^{\prime}}}h_{j^{\prime}}$ $j,j^{\prime}\in\\{1,2,3,4\\}$, and
therefore we assume without loss of generality, that $N_{1}\geq N_{2}\geq
N_{3}\geq N_{4}$.
First, we prove an orthogonality observation (4.3). For a contradiction
suppose that $N_{1}\lesssim N_{2}+N_{3}+N_{4}$ does not hold, and in
particular, suppose that $N_{1}\geq 2^{5}N_{2}$. By (2) we have $\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1})\big{)}\subset B_{N_{1}}\setminus
B_{2^{-2}N_{1}}$, since $N_{1}\geq 2^{5}N_{2}\geq 2^{5}>1$. On the other hand,
since $\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(h_{2}h_{3}h_{4})=\widehat{h_{2}}*\widehat{h_{3}}*\widehat{h_{4}}$,
it holds that
$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(h_{2}h_{3}h_{4})\big{)}\subset\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{2}}\big{)}+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{3}}\big{)}+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{4}}\big{)},$
and since $N_{2}\geq N_{3}\geq N_{4}$, $N_{1}\geq 2^{5}N_{2}$, and
$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}h\big{)}\Big{)}\subset B_{N}$ by (2), we have
$\mathop{\kern 0.0pt\mathrm{spt}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\big{)}\Big{)}\subset
B_{N_{2}}+B_{N_{3}}+B_{N_{4}}\subset B_{2^{3}N_{2}}\subset B_{2^{-2}N_{1}}\,,$
Thus (4.3) holds by Plancherel’s identity.
We henceforth suppose that $N_{1}\approx N_{2}$ and we proceed to prove bound
(4.3). To estimate the left-hand side of (4.3) we apply the Cauchy-Schwarz
inequality pairing one function with high frequency ($N_{1}$ or $N_{2}$) with
a function with a lower frequency ($N_{3}$ or $N_{4}$). This gives the bound
$\displaystyle\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern 0.0pt\mathrm{d}}\mathopen{}x\Big{|}$
$\displaystyle\hskip 50.00008pt\leq\min\Big{(}\begin{aligned} &\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})},\\\
&\qquad\|\mathop{\kern 0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{4}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\Big{)}.\end{aligned}$
By definitions in (4) and Lemma 4.1 for any $l<l^{\prime}\in\\{1,2,3,4\\}$ it
holds that
$\displaystyle\|\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{l}}h_{l}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{l^{\prime}}}h_{l^{\prime}}\|_{L^{2}_{t}L^{2}_{x}(\mathbb{R}\times\mathbb{R}^{d})}$
$\displaystyle\lesssim
N_{l}^{\varepsilon-\nicefrac{{1}}{{2}}-\alpha(h_{l})}N_{l^{\prime}}^{\nicefrac{{1}}{{2}}-\alpha(h_{l^{\prime}})+\beta(h_{l},h_{l^{\prime}})}$
$\displaystyle\hskip
50.00008pt\times\prod_{j\in\\{l,l^{\prime}\\}}N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})\,,$
where
$\beta(h_{l},h_{l^{\prime}})\coloneqq\begin{cases}\mathfrak{s}_{c}&\text{if
}h_{l},h_{l^{\prime}}\in\\{v_{*},v_{1},v_{2},v_{3}\\}\,,\\\
0&\text{otherwise}.\end{cases}$
Hence, since $N_{1}\approx N_{2}$, we obtain
$\displaystyle\Big{|}\int_{\mathbb{R}\times\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{1}}h_{1}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{2}}h_{2}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{3}}h_{3}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{4}}h_{4}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}t\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\Big{|}\leq
N_{1}^{2\varepsilon-1-\alpha(h_{1})-\alpha(h_{2})}N_{3}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})}N_{4}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})}$
$\displaystyle\hskip
50.00008pt\times\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}\prod_{j=1}^{4}N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})\,.$
By homogeneity of the required bound (4.3) we can assume, without loss of
generality, that $N_{j}^{\alpha(h_{j})}Z_{N_{j}}(\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N_{j}}h_{j})=1$, $j\in\\{1,2,3,4\\}$. Thus,
recalling that $N_{1}\approx N_{2}$, we reduce (4.3) to proving
(4.11)
$N_{3}^{\nicefrac{{1}}{{2}}-\alpha(h_{3})-\varepsilon}N_{4}^{\nicefrac{{1}}{{2}}-\alpha(h_{4})-\varepsilon}\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}\\\
\lesssim N_{1}^{1+\alpha(h_{1})+\alpha(h_{2})}$
We first claim that $\alpha(h_{1})+\alpha(h_{2})+1\geq 0$. Indeed,
$\alpha(h)<0$ only if $h=v_{*}$, in which case $\alpha(h)=-\sigma_{*}$.
Assuming, without loss of generality, that $h_{1}=v_{*}$, in each of the cases
zzz, zzv, or zvv, we replace the minimum in (4.2), in (4.2), or in (4.2) by
$\frac{1}{2}$ or by $S+1$ to obtain that
$\sigma_{*}\leq S_{1}+1\leq S_{j}+1\leq\mathfrak{s}+1,\qquad j\in\\{1,2,3\\};$
the claim follows. Finally,in case vvv we have $h_{2}=v$, and therefore
$\alpha(h_{2})=\mathfrak{s}$. Consequently,
$\sigma_{*}=\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})\leq\mathfrak{s}+1,$
and the claim follows.
Thus, since $N_{1}\geq N_{3}$ it is sufficient to show (4.11) for
$N_{1}=N_{3}$, that is, to show
$\displaystyle\min\Big{(}N_{3}^{\beta(h_{2},h_{3})}N_{4}^{\beta(h_{1},h_{4})},N_{3}^{\beta(h_{1},h_{3})}N_{4}^{\beta(h_{2},h_{4})}\Big{)}$
$\displaystyle\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})}N_{4}^{-\nicefrac{{1}}{{2}}+\alpha(h_{4})}$
After standard algebraic manipulations, (4) follows if we show that for any
$N_{3}\geq N_{4}$ either
$N_{4}^{\beta(h_{1},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})}\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{2},h_{3})}$
or
$N_{4}^{\beta(h_{2},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})}\lesssim
N_{3}^{\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{1},h_{3})}\,.$
Since $N_{3}\geq N_{4}$ are arbitrary, it suffices to show that either
$\max\\{0,\beta(h_{1},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})\\}\leq\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{2},h_{3})$
or
$\max\\{0,\beta(h_{2},h_{4})+\nicefrac{{1}}{{2}}-\alpha(h_{4})\\}\leq\nicefrac{{1}}{{2}}+\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})-\beta(h_{1},h_{3})$
holds. The expressions above can be rewritten in more symmetric form as
$\beta(h_{2},h_{3})+\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})$
or
$\beta(h_{1},h_{3})+\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{2},h_{4})\\}\leq\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})$
Next, we discuss each case separately.
Case $zzz$. Then,
$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+S_{2}+S_{3}$
and $\beta(h_{j},h_{k})=0$ for any $j\neq k$, and therefore (4) is equivalent
to
$\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},0\\}\leq-\sigma_{*}+S_{1}+S_{2}+S_{3}\,.$
Clearly the left hand side is largest if $\alpha(h_{4})=S_{3}$, and
consequently (4) holds if
$\sigma_{*}\leq
S_{1}+S_{2}+S_{3}-\max\\{S_{3}-\nicefrac{{1}}{{2}},0\\}=S_{1}+S_{2}+\min\\{S_{3},\nicefrac{{1}}{{2}}\\}\,,$
which holds by (4.2).
Case $zzv$. Then,
$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}$
and either $\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$. Suppose
$\beta(h_{2},h_{3})=0$, the case $\beta(h_{1},h_{3})=0$ follows analogously by
proving (4) instead of (4). Then, (4) is equivalent to
$\max\\{\alpha(h_{4})-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}\,.$
Since $\alpha(h_{4})\leq\mathfrak{s}$ and
$\beta(h_{1},h_{4})\leq\mathfrak{s}_{c}$, then (4) follows if we show that
$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+S_{1}+S_{2}+\mathfrak{s}\,,$
which is equivalent to
$\sigma_{*}\leq
S_{1}+S_{2}+\mathfrak{s}+\min\\{\nicefrac{{1}}{{2}},\mathfrak{s}-\mathfrak{s}_{c}\\}\,,$
that follows from (4.2).
Case $zvv$. Then,
$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+S_{1}+\mathfrak{s}+\mathfrak{s}\,.$
We discuss three cases. If $h_{4}=v$, then $z_{1}\in\\{h_{1},h_{2},h_{3}\\}$
and consequently either $\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$.
Suppose $\beta(h_{2},h_{3})=0$, the case $\beta(h_{1},h_{3})=0$ follows
analogously by proving (4) instead of (4). Then, (4) is equivalent to
$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\beta(h_{1},h_{4})\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,.$
Since $\beta(h_{1},h_{4})\leq\mathfrak{s}_{c}$, then (4) follows if we show
that
$\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,,$
which is equivalent to
$\sigma_{*}\leq
S_{1}+\mathfrak{s}+\min\\{\nicefrac{{1}}{{2}},\mathfrak{s}-\mathfrak{s}_{c}\\}\,,$
that follows from (4.2).
If $h_{4}=z_{1}$, then $\beta(h_{1},h_{4})=0$,
$\beta(h_{2},h_{3})\leq\mathfrak{s}_{c}$, and $\alpha(h_{4})=S_{1}$ implies
that (4) holds if
$\mathfrak{s}_{c}+\max\\{S_{1}-\nicefrac{{1}}{{2}},0\\}\leq-\sigma_{*}+S_{1}+2\mathfrak{s}\,.$
After standard algebraic manipulations, the latter is equivalent to
$\sigma_{*}\leq
2\mathfrak{s}-\mathfrak{s}_{c}+\min\\{\nicefrac{{1}}{{2}},S_{1}\\}\,,$
which follows from (4.2).
If $h_{4}=v_{*}$, then $\alpha(h_{4})-\nicefrac{{1}}{{2}}<0$, and either
$\beta(h_{2},h_{3})=0$ or $\beta(h_{1},h_{3})=0$. As above, we suppose
$\beta(h_{2},h_{3})=0$ and we note that (4) follows if we show
$\mathfrak{s}_{c}\leq-\sigma_{*}+S_{1}+2\mathfrak{s},$
or equivalently
$\sigma_{*}\leq S_{1}+2\mathfrak{s}-\mathfrak{s}_{c},$
which follows from (4.2).
Case $vvv$. Then,
$\alpha(h_{1})+\alpha(h_{2})+\alpha(h_{3})+\alpha(h_{4})=-\sigma_{*}+3\mathfrak{s}$
and $\beta(h_{j},h_{k})\leq\mathfrak{s}_{c}$ for each $j\neq k$. Then, (4)
follows if we show
$\mathfrak{s}_{c}+\max\\{\mathfrak{s}-\nicefrac{{1}}{{2}},\mathfrak{s}_{c}\\}\leq-\sigma_{*}+3\mathfrak{s}\,.$
The latter is equivalent to
$\sigma_{*}\leq
2(\mathfrak{s}-\mathfrak{s}_{c})+\min\\{\nicefrac{{1}}{{2}}+\mathfrak{s}_{c},\mathfrak{s}\\}\,,$
and (4) follows from (4.2). ∎
We conclude this section by a crucial estimate that justifies the regularity
computation encoded by the defintion (1.3) of $\mu(k,S)$
###### Lemma 4.4 (Inductive property of $\mu(k,S)$).
For any $S>0$ and any $k_{1},k_{2},k_{3}\in\mathbb{N}\setminus\\{0\\}$, with
$k_{1}\leq k_{2}\leq k_{3}$, it holds that
$\mu(k_{1}+k_{2}+k_{3},S)\leq\mu(k_{1},S)+\min\big{(}\mu(k_{2},S),\nicefrac{{1}}{{2}}\big{)}+\min\big{(}\mu(k_{3},S),\nicefrac{{1}}{{2}}\big{)}.$
Furthermore, setting $k\coloneqq k_{1}+k_{2}+k_{3}$
$0<\varepsilon\lesssim_{k}1$, and any
$0<\varepsilon_{0}\lesssim_{k,S,\varepsilon}1$. Then for any functions
$z_{j}$, $j\in\\{1,2,3\\}$, it holds that
$\|z_{1}z_{2}z_{3}\|_{X^{*,\mu(k,S)}}\lesssim\prod_{j=1}^{3}\|z_{j}\|_{Y^{\mu(k_{j},S)+\varepsilon}}\,,$
where the implicit constant depends on $k$, $\varepsilon$, $\varepsilon_{0}$,
and $S$.
###### Proof of Lemma 4.4.
Without loss of generality, suppose $k_{1}\leq k_{2}\leq k_{3}$. Let
$J\coloneqq\mu(k_{1},S)+\min\big{(}\mu(k_{2},S),\nicefrac{{1}}{{2}}\big{)}+\min\big{(}\mu(k_{3},S),\nicefrac{{1}}{{2}}\big{)}.$
Trivially, we have $\mu(|k_{j}|,S)\geq S$, $j\in\\{1,2,3\\}$. If
$\mu(k_{3},S)\geq\mu(k_{2},S)\geq\frac{1}{2}$ then
$J\geq S+\frac{1}{2}+\frac{1}{2}=S+1\geq\mu(k_{1}+k_{2}+k_{3},S)\,,$
and (4.4) follows. If $\mu(k_{3},S)\geq\frac{1}{2}\geq\mu(k_{2},S)$, then
$J\geq S+S+\frac{1}{2}=2S+\frac{1}{2}\geq\mu(k_{1}+k_{2}+k_{3},S)\,,$
and (4.4) follows. Finally, if $\frac{1}{2}\geq\mu(k_{3},S)\geq\mu(k_{2},S)$
then
$J=\mu(k_{1},S)+\mu(k_{2},S)+\mu(k_{3},S)$
and in addition $\mu(k_{j},S)=k_{j}S$, $j\in\\{1,2,3\\}$ since
$\min(S+1,2S+\frac{1}{2})>\frac{1}{2}$ and $k_{1}\leq k_{2}$. Then we deduce
that
$J=\big{(}k_{1}+k_{2}+k_{3}\big{)}S\geq\mu(k_{1}+k_{2}+k_{3},S),$
and (4.4) follows.
Finally, bound (4.4) follows from bound (4.2) with $S$ replaced by
$S+\varepsilon$. This completes the proof. ∎
## 5\. The deterministic fixed point
This section we prove Theorem 1.5.
Recall that solution $u$ to (1) is a fixed point of the map
$u\mapsto\mathbb{I}_{f}(|u|^{2}u)$ with $\mathbb{I}_{f}$ given by (1). Having
assumed that $u$ is a function of the form given by (1.5) for some
$M\in\mathbb{N}$, it follows by direct substitution that $u$ is a solution to
(1) if and only if the remainder term $u^{\\#}_{M}\coloneqq u-z_{\leq M}$,
with $z_{\leq M}\coloneqq\sum_{k\leq M}z_{k}$, is fixed point of the iteration
map
$u^{\\#}_{M}\mapsto\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$, given by (1.3).
The boundedness of the map $\mathbb{I}_{0}$ from
$X^{*,\mathfrak{s}_{*}}(\mathbb{R})$ to $X^{*,\mathfrak{s}_{*}}(\mathbb{R})$,
for some $\mathfrak{s}_{*}>\mathfrak{s}$, has already been established in
Proposition 3.1. Thus, the proof of Theorem 1.5 relies on estimates on
$\Phi_{z_{\leq M}}[v]$ and $\big{[}z,z,z\big{]}_{>M}$ in the
$X^{*,\mathfrak{s}_{*}}([0,T])$ norm for some $\mathfrak{s}_{*}>\mathfrak{s}$
and for small enough intervals $T>0$.
In Lemma 5.1 we establish bounds on the Lipschitz constant of the nonlinear
map $X^{\mathfrak{s}}(I)\ni v\mapsto\Phi_{z}[v]\in
X^{*,\mathfrak{s}_{*}}\big{(}I\big{)}$ for an interval $I\subset\mathbb{R}$.
This will allow us to prove Theorem 1.5 using the uniqueness of a fixed point
of $v\mapsto\mathcal{I}(v)$ by using the Banach fixed point theorem.
###### Lemma 5.1.
Fix $0<S<\mathfrak{s}_{c}<\mathfrak{s}<\mathfrak{s}_{c}+\nicefrac{{1}}{{2}}$.
Choose $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. For any interval
$I\subseteq\mathbb{R}$, any $z\in Y^{S}(I)$ and $v_{1},v_{2}\in
X^{\mathfrak{s}}(I)$ and any
$\mathfrak{s}_{*}<\min\big{(}\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\nicefrac{{1}}{{2}},\,S+1,S+\mathfrak{s}\big{)}-\varepsilon\,,$
it holds that
$\big{\|}\Phi_{z}[v_{1}]\big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}\|_{X^{\mathfrak{s}}(I)},$
$\Big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{2}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}-v_{2}\|_{X^{\mathfrak{s}}(I)}\,,$
and
$\Big{\|}\Phi_{z}[v]-\Phi_{\widetilde{z}}[v]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(I)}\lesssim\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}+\|\widetilde{z}\|_{Y^{S}(I)}\big{)}^{2}\|z-\widetilde{z}\|_{Y^{S}(I)}.$
In particular, if
$\mathfrak{s}<\min\big{(}2S+\nicefrac{{1}}{{2}},\,S+1\big{)}$, then for any
sufficiently small $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$, (5.1) holds with
$\mathfrak{s}_{*}=\mathfrak{s}+\varepsilon$.
We remark that the implicit constants may depend on $S$, $\mathfrak{s}$,
$\mathfrak{s}_{*}$, $\varepsilon$, and $\varepsilon_{0}$ but not on $I$, $z$,
$v_{1}$ or $v_{2}$.
###### Proof of Lemma 5.1.
Without loss of generality, we assume that $I=\mathbb{R}$. The general
statement follows by replacing $z$ with $z\mathbbm{1}_{I}$ and using that
$\|z\|_{Y^{S}(I)}=\|z\mathbbm{1}_{I}\|_{Y^{S}(\mathbb{R})}$.
By (1.3) we have $\Phi_{z}[0]=0$, and therefore (5.1) follows from (5.1) after
setting $v_{2}=0$. Thus we concentrate on proving (5.1). Using the expression
(1.3) for $\Phi_{z}[v]$ it follows that
$\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]=G_{1}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}(v_{1}-v_{2})+G_{2}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}(\overline{v_{1}-v_{2}})\,,$
where $G_{j}[z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}]$,
$j\in\\{1,2\\}$, are homogeneous polynomials of degree 2 in the variables $z$,
$\overline{z}$, $v_{1}$, $\overline{v_{1}}$, $v_{2}$, $\overline{v_{2}}$.
Assume $\varepsilon<\min\big{(}S,\mathfrak{s}-\mathfrak{s}_{c}\big{)}$ and
observe that
$\displaystyle\mathfrak{s}_{*}<\min\big{(}2S+\nicefrac{{1}}{{2}},\,S+1\big{)}-\varepsilon,$
$\displaystyle\mathfrak{s}_{*}<\min\big{(}S+\mathfrak{s},\,S+1\big{)}-\varepsilon,$
$\displaystyle\mathfrak{s}_{*}<\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c})-\varepsilon.$
Then, (4.2), (4.2), and (4.2) with $\mathfrak{s}-\varepsilon/3$ and
$S_{j}-\varepsilon/3$ replacing, respectively, $\mathfrak{s}$ and $S_{j}$,
$j=1,2,3$ imply
$\displaystyle\Big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}$
$\displaystyle\leq\begin{aligned}
&\Big{\|}G_{1}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}\big{(}v_{1}-v_{2}\big{)}\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}\\\
&\qquad+\Big{\|}G_{2}\big{[}z,\overline{z},v_{1},\overline{v_{1}},v_{2},\overline{v_{2}}\big{]}\big{(}\overline{v_{1}-v_{2}}\big{)}\Big{\|}_{X^{*,\mathfrak{s}_{*}}(\mathbb{R})}.\end{aligned}$
$\displaystyle\lesssim\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}-v_{2}\|_{X^{\mathfrak{s}}(I)}\,,$
as desired.
Bound (5.1) follows analogously by representing
$\Phi_{z}[v]-\Phi_{\widetilde{z}}[v]=\Psi_{1}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}(z-\widetilde{z})+\Psi_{2}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}\overline{(z-\widetilde{z})}$
with
$\Psi_{1}\big{[}z,\overline{z},\widetilde{z},\overline{\widetilde{z}},v,\overline{v}\big{]}$,
$j\in\\{1,2\\}$, are homogeneous polynomials of degree 2 in the input
variables. ∎
Next, we define the time-delayed version of the iteration map
$v\mapsto\mathbb{I}_{0}\big{(}\Phi_{z}[v]+h\big{)}$ with non-trivial intial
data. Define for any $t_{0}>0$ the map
$\mathcal{K}(v):=e^{i(t-t_{0})\Delta}v_{0}\mp
i\int_{t_{0}}^{t}e^{i(t-s)\Delta}(\Phi_{z}[v]+h)\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}s$
In the following lemma we show that $v\mapsto\mathcal{K}(v)$ is a contraction
on $X^{\mathfrak{s}}([t_{0},t_{1}))$ if the time interval $[t_{0},t_{1})$ is
short enough, or if $v_{0}$,$z$, and $h$ are small in appropriate norms.
###### Lemma 5.2.
Let $I$ with $I\ni t_{0}$ be a time interval. Fix
$0<S<\mathfrak{s}_{c}<\mathfrak{s}<\mathfrak{s}_{c}+\nicefrac{{1}}{{2}}$.
Choose $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. Then there exist
constants $C=C(\varepsilon,\varepsilon_{0},S,\mathfrak{s})>0$ and
$c=c(\varepsilon,\varepsilon_{0},S,\mathfrak{s})>0$ such that for any $z\in
Y^{S}(I)$, $h\in X^{\mathfrak{s}+\varepsilon}(I)$, and $v_{1},v_{2}\in
X^{\mathfrak{s}}(I)$ the following bounds hold:
$\Big{\|}\mathcal{K}(v_{1})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\begin{aligned}
\Big{(}&\|v_{0}\|_{H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\\\
&\quad+\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\|v_{1}\|_{X^{\mathfrak{s}}(I)}\Big{)}\end{aligned}$
and
$\displaystyle\Big{\|}\mathcal{K}(v_{1})-\mathcal{K}(v_{2})\Big{\|}_{X^{\mathfrak{s}}(I)}$
$\displaystyle\qquad\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\big{(}\|v_{1}\|_{X^{\mathfrak{s}}(I)}+\|v_{2}\|_{X^{\mathfrak{s}}(I)}+\|z\|_{Y^{S}(I)}\big{)}^{2}\big{\|}v_{1}-v_{2}\big{\|}_{X^{\mathfrak{s}}(I)}\,.$
As a consequence:
* •
For any $v_{0}\in H^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})$, $z\in
Y^{S}(I)$, $h\in X^{*,\mathfrak{s}+\varepsilon}(I)$, and $\delta_{0}>0$ we
define
$\displaystyle
T_{\delta_{0}}\big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})},\|z\|_{Y^{S}(I)},\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\big{)}$
$\displaystyle\quad\coloneqq\frac{1}{2^{100}(C\delta_{0})^{\nicefrac{{1}}{{c}}}}\big{\langle}\delta_{0}+\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(\mathbb{R})}+\|z\|_{Y^{S}(\mathbb{R})}\big{\rangle}^{-\nicefrac{{3}}{{c}}}\,.$
Then for any interval $J=[t_{0},t_{1}]\subseteq I$ with
$|J|<T_{\delta_{0}}\big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})},\|z\|_{Y^{S}(I)},\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}\big{)}$
the map $v\mapsto\mathcal{K}(v)$ is a contraction on the set
$\overline{\mathcal{B}}_{\delta_{0}}^{J}\coloneqq\Big{\\{}v\in
X^{\mathfrak{s}}(J)\colon\|v\|_{X^{\mathfrak{s}}(J)}\leq\delta_{0}\Big{\\}}\,,$
and thus it admits a unique fixed point in
$\overline{\mathcal{B}}_{\delta_{0}}^{J}$.
* •
If $I=\mathbb{R}$, there exists $\delta_{0}>0$ such that if
$\|z\|_{Y^{S}(\mathbb{R})}\leq\frac{\delta_{0}}{2C},\qquad\|v_{0}\|_{H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}\leq\frac{\delta_{0}}{2C},\qquad\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(\mathbb{R})}\leq\frac{\delta_{0}}{2C},$
then $v\mapsto\mathcal{K}(v)$ is a contraction on the set
$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}\coloneqq\Big{\\{}v\in
X^{\mathfrak{s}}(\mathbb{R})\colon\|v\|_{X^{\mathfrak{s}}(\mathbb{R})}\leq\delta_{0}\Big{\\}}\,,$
and thus admits a unique fixed point in
$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}$.
###### Proof of Lemma 5.2.
Assume that $\varepsilon>0$ is chosen small, and in particular that it
satisfies
$\varepsilon<\frac{1}{2}\min\big{(}S,2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\nicefrac{{1}}{{2}}-\mathfrak{s},S+1-\mathfrak{s}\big{)}.$
Fix $z\in Y^{S}(I)$, $h\in X^{\mathfrak{s}+\varepsilon}(I)$, and
$v_{1},v_{2}\in X^{\mathfrak{s}}(I)$. Then, by (3.1)
$\displaystyle\Big{\|}\mathcal{K}(v_{1})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\Big{(}\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})}+\|h\|_{X^{*,\mathfrak{s}+\varepsilon}(I)}+\big{\|}\Phi_{z}[v_{1}]\big{\|}_{X^{*,\mathfrak{s}+\varepsilon}(I)}\Big{)}\,.$
Since
$\mathfrak{s}+\varepsilon<\min\big{(}\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),\,2S+\nicefrac{{1}}{{2}},S+1,S+\mathfrak{s}\big{)}-\varepsilon$
for $\varepsilon>0$ small enough, (5.2) follows from (5.1) with
$s_{*}=\mathfrak{s}+\varepsilon$. Next, (3.1) implies
$\Big{\|}\mathcal{K}(v_{1})-\mathcal{K}(v_{2})\Big{\|}_{X^{\mathfrak{s}}(I)}\leq
C\big{\langle}|I|^{-1}\big{\rangle}^{-c}\big{\|}\Phi_{z}[v_{1}]-\Phi_{z}[v_{2}]\big{\|}_{X^{*,\mathfrak{s}+\varepsilon}(I)}.$
and (5.1) with $s_{*}=\mathfrak{s}+\varepsilon$ yields (5.2).
Assuming, without loss of generality, that $C>1>c>0$, direct computation
allows us to deduce from (5.2) that $\mathcal{K}$ maps the sets
$\overline{\mathcal{B}}_{\delta_{0}}^{J}$ and
$\overline{\mathcal{B}}_{\delta_{0}}^{\mathbb{R}}$ to themselves under the
corresponding assumptions on $\delta_{0}$, $z$, and $h$. Similarly, bound
(5.2) shows that $\mathcal{K}$ is a contraction on these sets. The existence
and uniqueness of the fixed point follows from the Banach contraction mapping
principle. ∎
Finally, let us record the bound on $[z,z,z]_{>M}$ appearing in the defintion
of the iteration map
$\mathcal{J}_{z,M}(v)\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$.
###### Lemma 5.3.
Fix $0<S<\mathfrak{s}_{c}$, $0<\varepsilon\lesssim_{S,\mathfrak{s}}1$, and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,\mathfrak{s}}1$. Then it holds that
$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}$
and the the map
$\vec{z}_{M}=(z_{k})_{k\leq M}\mapsto[z,z,z]_{>M}$
is Lipschitz on bounded sets with a Lipschitz constant bounded by
$C\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{2}$
for some $C>0$ with respect to distance introduced in (1.5).
###### Proof of Lemma 5.3.
To prove (5.3) we note that, according to (1.3) it holds that
$[z,z,z]_{>M}=\sum_{\mathclap{\qquad\begin{subarray}{c}\hskip
3.5ptk_{1}+k_{2}+k_{3}\geq M+1\\\ k_{j}\leq
M\end{subarray}}}z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\,,$
Since the sum above is finite, we estimate each summand separately in the
space $X^{*,\mu(M+1,S)}(\mathbb{R})$. Without loss of generality, assume that
$k_{1}\leq k_{2}\leq k_{3}$ and use (4.4) to obtain the desired claim.
To prove Lipschitz regularity it is sufficient to note that
$[z,z,z]_{>M}-[\widetilde{z},\widetilde{z},\widetilde{z}]_{>M}=\sum_{\mathclap{\qquad\begin{subarray}{c}\hskip
3.5ptk_{1}+k_{2}+k_{3}\geq M+1\\\ k_{j}\leq
M\end{subarray}}}\Big{(}(z_{k_{1}}-\widetilde{z}_{k_{1}})\overline{z_{k_{2}}}z_{k_{3}}+\begin{aligned}
&\widetilde{z}_{k_{1}}\overline{(z_{k_{2}}-\widetilde{z}_{k_{2}})}z_{k_{3}}\\\
&+\widetilde{z}_{k_{1}}\overline{\widetilde{z}_{k_{2}}}(z_{k_{3}}-\widetilde{z}_{k_{3}})\Big{)}\,,\end{aligned}$
and apply the same reasoning as above. ∎
We are now ready to prove Theorem 1.5.
###### Proof of Theorem 1.5.
We suppose that $\varepsilon>0$ is small enough, and in particular it
satisfies
$\varepsilon<\frac{1}{3}\min\Big{(}S,2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\frac{1}{2}-\mathfrak{s},S+1-\mathfrak{s},\mu(M+1,S)-\mathfrak{s}\Big{)}.$
Let $C,c$ be constants from Lemma 5.2.
According to the discussion at the beginning of this section, $u$ is a
solution to (1) on $[0,T)$ if and only if $u^{\\#}_{M}$ is a fixed point of
the iteration map
$u^{\\#}_{M}\mapsto\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$ given by (1.3). Henceforth, we use $v$ as a
shorthand for the remainder term $u^{\\#}_{M}$.
Local existence of solutions. Let $T=T_{\delta_{0}=1}$, where $T_{\delta_{0}}$
is as in (• ‣ 5.2). Note that
$\mathcal{J}_{z,M}(v)\coloneqq\mathcal{K}(v)$
with $v_{0}=0$, $z=z_{\leq M}$, $h=[z,z,z]_{>M}$, and $t_{0}=0$. Thus
existence of fixed point in $X^{\mathfrak{s}}([0,T))$ follows from Lemma 5.2.
The bound from below on the time of existence follows from Lemma 5.2, from the
bounds (5.3) from Lemma 5.3, which gives
$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}\,,$
and from the fact that
$\|z_{\leq M}\|_{Y^{S}(I)}\leq
M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)}(I)}.$
The bound on
$\|v\|_{C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}$
follows from the next part of the proof.
Global existence of solutions. Analogously to the proof above, note that
$\mathcal{J}_{z,M}(v)\coloneqq\mathcal{K}(v)$
with $v_{0}=0$, $z=z_{\leq M}$, $h=[z,z,z]_{>M}$, and $t_{0}=0$. Thus
existence of fixed point in $X^{\mathfrak{s}}([0,+\infty))$ again follows from
Lemma 5.2. Indeed, the bounds above show that $\|z_{\leq M}\|_{Y^{S}(I)}$ and
$\big{\|}[z,z,z]_{>M}\big{\|}_{X^{*,\mu(M+1,S)}(\mathbb{R})}$ can be made
arbitrarily small as long as $\delta_{0}$ in (Global existence of solutions: )
is chosen small enough. Note that the value of $\delta_{0}$ here is chosen
differently, smaller, than the value of $\delta_{0}$ appearing in Lemma 5.2.
Time-continuity of the solution We show that if $v\in X^{\mathfrak{s}}(I)$ for
any interval $I\subset[0,T_{0})$, then $\mathcal{J}_{z,M}(v)\in
C^{0}\big{(}I,H^{\mathfrak{s}+\varepsilon}_{x}(\mathbb{R}^{d})\big{)}$.
Indeed, from Proposition 3.3, we have
$\displaystyle\|\mathcal{J}_{z,M}(v)\|_{C^{0}(I,H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d}))}\lesssim$
$\displaystyle\begin{aligned} &\big{\|}\Phi_{z_{\leq
M}}[v]\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\\\
&+\big{\|}[z,z,z]_{>M}\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\,.\end{aligned}$
$\displaystyle\lesssim$ $\displaystyle\begin{aligned}
&\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(I)}\big{)}^{3},\end{aligned}$
where we used the bounds
$\displaystyle\big{\|}[z,z,z]_{>M}\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\lesssim\Big{(}\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}\,.$
$\displaystyle\big{\|}\Phi_{z_{\leq
M}}[v]\big{\|}_{X^{\ast,\mathfrak{s}+2\varepsilon}(I)}\lesssim\big{(}\|v\|_{X^{\mathfrak{s}}(I)}+M\max_{k\in\\{1,\ldots,M\\}}\|z_{k}\|_{Y^{\mu(k,S)}(I)}\big{)}^{2}\|v\|_{X^{\mathfrak{s}}(I)}$
from (5.3) of Lemma 5.3 and from (5.1) of Lemma 5.1, respectively. The
assumptions of the latter are satisfied since because (5) implies that
$\mathfrak{s}+2\varepsilon<\min\big{(}S+\mathfrak{s},\mathfrak{s}+2(\mathfrak{s}-\mathfrak{s}_{c}),2S+\frac{1}{2},\,S+1\big{)}-\varepsilon\,.$
This concludes the proof of our claim.
Uniqueness of the solution and Blow-up criterion. Next, we show the uniqueness
of the fixed point without any a-priori assumption on the smallness of its
$X^{\mathfrak{s}}$ norm (cf. Lemma 5.2). For any two fixed points $v_{j}\in
X^{\mathfrak{s}}([0,T_{j}])$, $j\in\\{1,2\\}$ of the map
$v\mapsto\mathcal{J}_{z,M}(v)$, set $T_{\cap}\coloneqq\min(T_{1},T_{2})$ and
$t_{0}\coloneqq\sup\big{(}t\in[0,T_{\cap})\colon v_{1}(s)=v_{2}(s)\text{ for
all }s\in[0,t]\big{)}$
and assume, by way of contradiction, that $t_{0}<T_{\cap}$. Note that $t_{0}$
is well defined since $v_{1}(0)=v_{2}(0)$, and therefore the supremum is taken
over non-empty set. Since $v_{1},v_{2}\in
C_{t}^{0}\big{(}[0,T_{\cap}),H_{x}^{\mathfrak{s}+\varepsilon}(\mathbb{R}^{d})\big{)}$,
thanks to (Time-continuity of solutions: ), it holds that
$v_{1}(t_{0})=v_{1}(t_{0})$. Thereby, from the group properties of the linear
Schrödinger evolution $t\mapsto e^{it\Delta}$ for all $t\in[t_{0},T_{\cap}]$
we have that
$v_{j}(t)=e^{i(t-t_{0})\Delta}v_{0}\mp
i\int_{t_{0}}^{t}e^{i(t-s)\Delta}(\Phi_{z_{\leq
M}}[v_{j}]+[z,z,z]_{>M})\,ds\qquad j=1,2\,.$
with $v_{0}=v_{1}(t_{0})=v_{1}(t_{0})$, that is $v_{j}$ are both fixed points
of the map $v\mapsto\mathcal{K}(v)$ with $z_{\leq M}$ in place of $z$, and
with $[z,z,z]_{>M}$ in place of $h$.
Thanks to (5), we can fix $R>0$ for which
$\|z_{\leq M}\|_{Y^{S}([0,T_{\cap}))}\leq
R,\quad\|[z,z,z]_{>M}\|_{X^{\mathfrak{s}}([0,T_{\cap}))}\leq
R,\quad\|v_{0}\|_{H^{\mathfrak{s}+\varepsilon}}\leq R.$
Estimate (5.2) shows that there exists $T$ with $t_{0}<t_{0}+T<T_{\cap}$ for
which
$\|v_{1}\|_{X^{\mathfrak{s}}([t_{0},t_{0}+T])}\leq 1\qquad\textrm{and
}\quad\|v_{2}\|_{X^{\mathfrak{s}}([t_{0},T])}\leq 1$
and the map $v\mapsto\mathcal{K}(v)$ has a unique fixed point on
$\overline{\mathcal{B}}_{1}^{[t_{0},t_{0}+T]}\big{\\{}v\colon\|v\|_{X^{\mathfrak{s}}([t_{0},t_{0}+T])}\leq
1\big{\\}}$. This contradicts our assumption that $t_{*}<T_{\cap}$ is the
upper bound of times $s$ for which $v_{1}(s)=v_{2}(s)$ and proves the desired
assertion.
The uniqueness above allows us to define $T_{\mathrm{max}}$ as the upper bound
of for the existence and uniqueness time of solutions. To prove the blow-up
criterion, let us assume that $T_{\mathrm{max}}<T_{0}$ and by way of
contradiction let us assume that we can fix $R>0$ and find
$t_{0}<T_{\mathrm{max}}$, arbitrarily close to $T_{\mathrm{max}}$ for which
(5) holds with $V_{0}=u^{\\#}_{M}(t_{0})$. Then Lemma 5.2 guarantees that we
can extend our solution to at least $\big{[}0,\min(t_{0}+T,T_{0})\big{)}$ by
finding a fixed point to the map $v\mapsto\mathcal{K}(v)$ on
$\big{[}t_{0},\min(t_{0}+T,T_{0})\big{]}$ and extending the remainder
$u^{\\#}_{M}$ past $t_{0}$. By subadditivity of norms, the $u^{\\#}_{M}$
extended this way satisfies
$\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}\big{(}[0,\min(t_{0}+T,T_{0}))\big{)}}+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,\min(t_{0}+T,T_{0}));H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}<\infty.$
This contradicts our assumption that $T_{\mathrm{max}}<T_{0}$ as long as
$t_{0}$ is chosen close enough to $T_{\mathrm{max}}$.
Time-continuity and sattering of the multilinear data Clearly
$z_{1}=e^{it\Delta}f$ so the scattering for $z_{1}$ is immediate. It also
holds that
$z_{1}=\mathbb{I}_{f}(0)$
so (3.3) from Proposition 3.3 we obtain
$\|z_{1}\|_{C^{0}([0,T_{0});H^{S}_{x}(\mathbb{R}^{d}))}\leq\|f\|_{H^{S}_{x}(\mathbb{R}^{d})}.$
Next we focus on $z_{k}$ with $2\leq k\leq M$.
According to (1) it holds that that
$z_{k}=\mp i\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k+1\\\
k_{1},k_{2},k_{3}\leq
k\end{subarray}}}\mathbb{I}_{0}(z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}})\,,$
Using (3.3) from Proposition 3.3 we obtain that
$\|z_{k}\|_{C^{0}\big{(}[0,T_{0});H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}\leq\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k+1\\\
k_{1},k_{2},k_{3}\leq
k\end{subarray}}}\Big{\|}z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}h\Big{\|}_{X^{*,\mu(k,S)+\varepsilon}([0,T_{0}))}.$
Using (4.4) applied with $S+\varepsilon$ in place of $S$ we obtain
$\displaystyle\|z_{k}\|_{C^{0}\big{(}[0,T_{0});H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}$
$\displaystyle\lesssim\|z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\|_{Y^{\mu(k,S)+\varepsilon}([0,T_{0}))}$
$\displaystyle\leq\prod_{j=1}^{3}\|z_{k_{j}}\|_{Y^{\mu(k_{j},S)+(M+1)\varepsilon}([0,T_{0}))}.$
This implies the desired claim as long as we replace $\varepsilon$ with
$\nicefrac{{\varepsilon}}{{(M+1)}}$ above.
To show that $z_{k}$, $2\leq k\leq M$ scatters set
$\mathfrak{g}_{k}\coloneqq\lim_{t\to+\infty}e^{-it\Delta}z_{k}=\mp
i\sum_{\mathclap{\begin{subarray}{c}k_{1}+k_{2}+k_{3}=k\\\
k_{1},k_{2},k_{3}<k\end{subarray}}}\lim_{t\to+\infty}e^{-it\Delta}\mathbb{I}_{0}(z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}})\,,$
where we used the inductive definition (1) of $z_{k}$. As before, using (4.4)
applied with $S+\varepsilon$ in place of $S$
$\|z_{k_{1}}\overline{z_{k_{2}}}z_{k_{3}}\|_{Y^{\mu(k,S)+\varepsilon}([0,+\infty))}\lesssim\prod_{j=1}^{3}\|z_{k_{j}}\|_{Y^{\mu(k_{j},S)+(M+1)\varepsilon}([0,+\infty))}\,,$
while from (3.4) and (3.4) it follows that
$\lim_{t\to+\infty}\Big{\|}z_{k}(t)-e^{it\Delta}\mathfrak{g}_{k}\Big{\|}_{H^{\mu(k,S)}_{x}(\mathbb{R}^{d})}=\lim_{t\to+\infty}\Big{\|}e^{-it\Delta}z_{k}(t)-\mathfrak{g}_{k}\Big{\|}_{H^{\mu(k,S)}_{x}(\mathbb{R}^{d})}=0.$
This is the desired claim as long as we replace $\varepsilon$ with
$\nicefrac{{\varepsilon}}{{(M+1)}}$ above.
Scattering of global solutions. Now let us show that the remainder
$u^{\\#}_{M}$ scatters in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ under the
assumptions (Global existence of solutions: ), Since $u^{\\#}_{M}$ satisfies
$u^{\\#}_{M}=\mathbb{I}_{0}\big{(}\Phi_{\mathfrak{z}_{\leq
M}}[u^{\\#}_{M}]+[\mathfrak{z},\mathfrak{z},\mathfrak{z}]_{>M}\big{)},\,$
it is sufficient to set
$\mathfrak{g}^{\\#}_{M}=\lim_{t\to+\infty}e^{-it\Delta}\mathbb{I}_{0}\big{(}\Phi_{\mathfrak{z}_{\leq
M}}[u^{\\#}_{M}]+[\mathfrak{z},\mathfrak{z},\mathfrak{z}]_{>M}\big{)}\,,$
to show that $\mathfrak{g}^{\\#}_{M}\in H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$,
and to show that that (Scattering of global solutions: ) holds. Using (5.1)
and (5.3) we obtain that
$\Big{\|}\Phi_{z_{\leq
M}}[u^{\\#}_{M}]+[z,z,z]_{>M}\Big{\|}_{X^{*,\mathfrak{s}+\varepsilon}([0,+\infty))}\\\
\lesssim\Big{(}\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,+\infty))}+\max_{k\leq
M}\|z_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\Big{)}^{3}<\infty.$
The existence of the limit in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ defining
$\mathfrak{g}^{\\#}_{M}$ and (Scattering of global solutions: ) follow, again,
from (3.4) and (3.4).
Continuous dependence on the multilinear data. We have already argued that for
any $f\in H^{S+\varepsilon}_{x}(\mathbb{R}^{d})$ with $d(0,f)\leq R$ there
exists a unique solution $u\in
C^{0}\big{(}[0,T);H^{S+\varepsilon}_{x}(\mathbb{R}^{d})\big{)}$ of the form
(1.5), which satisfies $\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,T))}<\infty$ and
$u^{\\#}_{M}=\mathcal{J}_{z,M}(u^{\\#}_{M})\coloneqq\mathbb{I}_{0}\big{(}\Phi_{z_{\leq
M}}[v]+[z,z,z]_{>M}\big{)}$. Proposition 3.1 and bound (3.1) shows that
$\mathbb{I}_{0}$ is a bounded linear map from
$X^{*,\mathfrak{s}+\varepsilon}([0,T))$ to $X^{\mathfrak{s}}([0,T))$. In Lemma
5.3 we have shown that the map
$\vec{z}_{M}\mapsto[z,z,z]_{>M}$
is Lipschitz contious with respect to the distance we introduced . Finally,
the map
$(\vec{z}_{M},v)\mapsto\Phi_{z_{\leq M}}[v]$
is Lipshitz continuous thanks to estimates (5.1) and (5.1). ∎
## 6\. Probabilistic estimates
In this section, we focus on establishing probabilistic estimates for the
multilinear correction terms $\mathfrak{z}_{k}$, defined in (1). We thereby
prove Theorem 1.6.
To prove these estimates we have to keep track of how the Wiener randomization
of $f$, encated to obtain $\mathfrak{f}$, reflects on the terms
$\mathfrak{z}_{k}$. We do this by introducing efficient bookkeeping notation
by indexing all terms that explicitly depend on the initial datum
$\mathfrak{f}$ using ternary trees, defined below. We stress that trees serve
merely as a notation, and we do not use any graph theory or subtle properties
of trees.
More precisely, we define the set $\mathbb{T}$ of ternary trees as
$\mathbb{T}:=\bigcup_{n\geq 1}\mathbb{T}_{n}$, where for each
$n\in\mathbb{N}\setminus{0}$ the set of trees $\mathbb{T}_{n}$ is given by
induction as follows:
* •
We say $\tau\in\mathbb{T}_{1}$ if $\tau=[\bullet]$.
* •
We say $\tau\in\mathbb{T}_{n}$ for $n\geq 1$ if
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$ for some $\tau_{j}\in\mathbb{T}_{n_{j}}$
and $n=n_{1}+n_{2}+n_{3}$ with $1\leq n_{j}<n$.
The index $n$ can be viewed as the number of leaves of of a tree.
Pictorially, our inductive construction of $\tau=[\tau_{1},\tau_{2},\tau_{3}]$
can be represented as follows
$\bullet$$\tau_{1}$$\tau_{2}$$\tau_{3}$
where $\tau_{j}$, $j=1,2,3$ are ternary trees. If $\tau\in\mathbb{T}_{n}$, we
set $|\tau|=n$ In particular, if $\tau=[\tau_{1},\tau_{2},\tau_{3}]$ then
$|\tau|=|\tau_{1}|+|\tau_{2}|+|\tau_{3}|$. Under this convention, $|\tau|$
corresponds to the number of nodes with no descendants, that we refer to as
“leaves”. We omit the proof, as this fact does not explicitly enter our
discussion.
In the literature, one usually allows a node of a general ternary trees to
have zero, one, two, or three descendants. However, in the present manuscript,
we only consider trees in which each node has either three descendants
(children) or no descendants at all (a leaf). By induction, one can show there
are no ternary trees with an even number of leaves, that is,
$\mathbb{T}_{n}=\emptyset$ when $n\in 2\mathbb{N}$. For example, we have
$\Big{|}\,\big{[}[\bullet],[\bullet],[\bullet]\big{]}\,\Big{|}=3\,,\qquad\Big{|}\,\Big{[}[\bullet],\big{[}[\bullet],[\bullet],[\bullet]\big{]},[\bullet]\Big{]}\,\Big{|}=5.$
Graphically, these trees can be represented as
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$
To each tree $\tau\in\mathbb{T}_{n}$ with $n\in 2\mathbb{N}+1$ we inductively
assign an $n$-(real) linear tree operator $R_{\tau}$ mapping $n$-tuples of
functions $(f_{1},\ldots,f_{n})\in L^{2}(\mathbb{R}^{d})$ to functions on
$\mathbb{R}\times\mathbb{R}^{d}$.
* •
We set $R_{[\bullet]}[f](t,x)\coloneqq e^{it\Delta}f(x).$
* •
Inductively, we set
$\displaystyle
R_{[\tau_{1},\tau_{2},\tau_{3}]}[\mathbf{f}_{1}\oplus\mathbf{f}_{2}\oplus\mathbf{f}_{3}](t,\cdot)$
$\displaystyle\coloneqq\mp
i\int_{0}^{t}e^{i(t-s)\Delta}\Big{(}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{)}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}s\,,$
where the choice of the sign is exactly the opposite to the sign on the right
hand side of (1). Here
$\mathbf{f}_{j}=(f_{j,1},\ldots,f_{j,|\tau_{j}|})\in\big{(}L^{2}(\mathbb{R}^{d})\big{)}^{|\tau_{j}|}$
are $|\tau_{j}|$-tuples of $L^{2}(\mathbb{R}^{d})$ functions and
$\mathbf{f}_{1}\oplus\mathbf{f}_{2}\oplus\mathbf{f}_{3}=\big{(}f_{1,1},\ldots,f_{1,|\tau_{1}|},f_{2,1},\ldots,f_{2,|\tau_{2}|},f_{3,1},\ldots,f_{3,|\tau_{3}|}^{3}\big{)}\,.$
To simplify notation, given $\tau\in\mathbb{T}$ and one function $f\in
L^{2}(\mathbb{R}^{d})$, we write
$R_{\tau}[f]:=R_{\tau}\big{[}f,\ldots,f\big{]}\,,$
where the right-hand side contains $|\tau|$ copies of $f$.
The main result of this section shows that the functions
$R_{\tau}[\mathfrak{f}]$ lie, almost surely, in the space $Y^{\mu(|\tau|,S)}$
with an appropriate regularity $\mu(|\tau|,S)$ depending on $S$, the
regularity of $f\in H^{S}(\mathbb{R}^{d})$, and on $|\tau|$, the order of
multilinearity of the operator $R_{\tau}$.
###### Proposition 6.1.
For any $S\in\mathbb{R}$, $n\geq 1$, let $\mu(k,S)$ be as defined in (1.3).
Fix $\tau\in\mathbb{T}$, $S>0$, and let $0<\varepsilon\lesssim_{|\tau|}1$ be
small. Then for every $0<\varepsilon_{0}\lesssim_{\varepsilon,|\tau|,S}1$
there exist constants $C=C(\varepsilon_{0},\varepsilon,|\tau|,S)>0$
independent of $f\in L^{2}(\mathbb{R}^{d})$, such that for any $\lambda>0$ it
holds that
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{|\tau|}}}{C\|f\|_{H^{S+\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,,$
where $\mathfrak{f}$ is the unit scale randomization of $f$ given by (1.1). In
particular,
$\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}<\infty\qquad\text{almost
surely}$
for any $f\in H^{S+\varepsilon}_{x}(\mathbb{R}^{d})$.
Before proceeding with the proof we use the above result to deduce the
probability distribution for $\|\mathfrak{z}_{k}\|_{Y^{S}(\mathbb{R})}$,
$\|\mathfrak{z}_{\leq M}\|_{Y^{S}(\mathbb{R})}$, defined in (1.3). We show
that Theorem 1.6 holds and thereby we have for any $\lambda>1$ that
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega\colon\big{\|}\mathfrak{z}_{\leq
M}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-\frac{\lambda^{\frac{2}{M}}}{C\|f\|_{H_{x}^{S+\varepsilon}(\mathbb{R}^{d})}^{2}}\bigg{)}\,.$
###### Proof of Theorem 1.6.
The set $\bigcup_{k\leq M}\mathbb{T}_{k}$ is finite, and therefore by (1)
there is a constant $\widetilde{C}=\widetilde{C}(M)$ such that
$\displaystyle\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}\mathfrak{z}_{k}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\bigcup_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}$
$\displaystyle\leq\sum_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}\,.$
$\displaystyle\leq\sum_{\begin{subarray}{c}\tau\in\mathbb{T}_{k}\end{subarray}}\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(k,S)}(\mathbb{R})}>\widetilde{C}^{-1}\lambda\big{\\}}\Big{)}\,,$
From (6.1) of Proposition 6.1 we obtain that
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{\|}\mathfrak{z}_{k}\big{\|}_{Y^{S}(\mathbb{R})}>\lambda\big{\\}}\Big{)}\lesssim\widetilde{C}CC\exp\Bigg{(}-\frac{\lambda^{\frac{2}{k}}}{\widetilde{C}^{2}C\|f\|_{H^{S+\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)},$
as required.
To obtain (6) it is sufficient to note that $S\leq\mu(k,S)$ and apply the
triangle inequality. ∎
Next, we proceed to the proof of Proposition 6.1. We rely on the deterministic
estimate of Lemma 6.2, which establishes regularity for the operators
$R_{\tau}$ when evaluated on functions with bounded frequency support. Then,
using Lemma 6.5 below, we show that the probabilistic bounds of Proposition
6.1 follow from estimates on large enough moments of the random variable
$\big{\|}R_{\tau}[\mathfrak{f}]\|_{Y^{\mu(|\tau|,S)}(\mathbb{R})}$. Such
moment bounds are established with help of multi-parameter Wiener chaos
estimates (Lemma 6.3).
First, we state and prove, or provide references for the required lemmata.
Finally, we prove Proposition 6.1.
###### Lemma 6.2.
Let $\mu(n,S)$ be as in (1.3) and fix $\tau\in\mathbb{T}$ and $S,R_{0}>0$.
Choose any $0<\varepsilon\lesssim_{|\tau|}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,S,|\tau|}1$. For any tuple
$\mathbf{f}=(f_{1},\ldots
f_{|\tau|})\in\big{(}L^{2}(\mathbb{R}^{d})\big{)}^{|\tau|}$ of functions with
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{j}))\leq 2R_{0}$,
$j\in\\{1,\ldots,|\tau|\\}$, it holds that
$\Big{\|}R_{\tau}\big{[}\mathbf{f}\big{]}\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim\prod_{j=1}^{|\tau|}\|f_{j}\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}.$
The implicit constant may depend on $|\tau|$, $\varepsilon$,
$\varepsilon_{0}$, and $R_{0}$.
The following lemma establishes the relation between the regularity parameter
$\mu(|\tau|,S)$ and the trilinear bound (4.2).
Our probabilistic estimates depend crucially on the Wiener chaos estimates. If
$n=1$ and $g_{j}$ having standard normal distribution, the claim of Lemma 6.3
reduces to the classical Khintchine inequality for Gaussian sums. Here, we
provide a version of the general statement.
###### Lemma 6.3.
[TT10, Proposition 2.4] Fix integers $M,n\geq 1$ and let $(g_{j})_{1\leq j\leq
M}\in\mathcal{N}_{\mathbb{C}}(0,1)$ be a sequence of complex $L^{2}$
normalized independent Gaussian random variables. Given
$c\colon\mathbb{N}^{n}\to\mathbb{C}$ define the random variable
$\Xi_{n,M}\coloneqq\sum_{\mathrlap{k_{1},\ldots,k_{n}\in\\{1,\cdots,M\\}}}\hskip
10.00002ptc(k_{1},\cdots,k_{n})g_{k_{1}}\cdots g_{k_{n}}.$
Then, for all $\gamma\geq 2$ it holds that
$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\Xi_{n,M}\big{|}^{\gamma}\big{)}^{\frac{1}{\gamma}}\lesssim_{n}(\gamma-1)^{\frac{n}{2}}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\Xi_{n,M}\big{|}^{2}\big{)}^{\frac{1}{2}}$
with the implicit constant independent of $M$ and $\gamma$.
###### Remark 6.4.
The assertion of Lemma 6.3 is slightly different compared to [TT10,
Proposition 2.4], where the summation is taken over the simplex
$A(n,M)=\big{\\{}(k_{1},\ldots,k_{n})\colon k_{1}\leq k_{2}\leq\ldots\leq
k_{n}\leq M\big{\\}}\,.$
However, the proof of Lemma 6.3 follows the proof of [TT10, Proposition 2.4]
line by line, with $A(n,M)$ replaced by $\\{1,\cdots,M\\}^{n}$. Otherwise,
Lemma 6.3 can be deduced from [TT10, Proposition 2.4] simply by successive
applications of the triangle inequality.
Finally, we formulate a slight modification of [Tzv09, Lemma 4.5]. We omit the
proof: it coincides with the one in the reference and follows from the
Chebyshev inequality and an appropriate optimization.
###### Lemma 6.5.
Let $F$ be a random variable and suppose that there exist $K>0$,
$\gamma_{0}\geq 1$, and $k\geq 1$ such that for any $\gamma\geq\gamma_{0}$ we
have
$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|F|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\leq\gamma^{\frac{k}{2}}K.$
Then, there exist $c>0$ and $C>0$ depending on $\gamma_{0}$ and $k$, but
independent of $K$ and $\gamma$, such that for every $\lambda>0$,
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:|F|>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{\frac{2}{k}}}{K^{\frac{2}{k}}}\bigg{)}\,.$
In particular, we have
$\mathbb{P}(\\{\omega\in\Omega:|F|<\infty\\})=1.$
From the above we can deduce an estimate on the probability distribution of
the Sobolev norms of the randomized initial data.
###### Corollary 6.6.
For some $C,c>0$ it holds that
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\|\mathfrak{f}\|_{H^{S}_{x}(\mathbb{R}^{d})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{2}}{\|f\|^{2}_{H^{S}_{x}(\mathbb{R}^{d})}}\bigg{)}\,.$
###### Proof.
For any $\gamma\geq 2$, the Minkowski inequality implies
$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\leq\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\big{\|}_{L^{2}_{x}(\mathbb{R}^{d})}$
According to the definition of the unit-scale Wiener randomization (1.1), the
pointwise value $\mathfrak{f}(x)$ has the form required by Lemma 6.3, and
therefore
$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\lesssim(\gamma-1)^{\nicefrac{{1}}{{2}}}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{2}\big{)}^{\nicefrac{{1}}{{2}}}\,.$
Finally, using independence of the random variables $g_{k}$ and Plancherel’s
identity, we obtain that
$\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|\mathfrak{f}|^{2}\big{)}^{\nicefrac{{1}}{{2}}}\big{\|}_{L^{2}_{x}(\mathbb{R}^{d})}^{2}\begin{aligned}
&=\sum_{k,k^{\prime}\in\mathbb{Z}^{d}}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}(g_{k}\overline{g_{k^{\prime}}})\int_{\mathbb{R}^{d}}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f(x)\overline{\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k^{\prime}}f(x)}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\\\
&\lesssim\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\big{|}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f(x)\big{|}^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}|\chi_{k}(\xi)|^{2}\big{|}\widehat{f}(\xi)\big{|}^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}x\\\
&\lesssim\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}\,,\end{aligned}$
where in the last inequality we used that
$\sum_{k\in\mathbb{Z}^{d}}|\chi_{k}(\xi)|^{2}\lesssim 1$, since
$|\chi_{k}(\xi)|\leq 1$ and for any $\xi\in\mathbb{R}^{d}$ there are finitely
many $k\in\mathbb{Z}^{d}$ for which $\xi\in\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\chi_{k}$. Thus,
$\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\lesssim(\gamma-1)^{\nicefrac{{1}}{{2}}}\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}$
and from Lemma 6.5 follows that
$\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\|\mathfrak{f}\|_{L^{2}_{x}(\mathbb{R}^{d})}>\lambda\big{\\}}\Big{)}\leq
C\exp\bigg{(}-c\frac{\lambda^{2}}{\|f\|_{L^{2}_{x}(\mathbb{R}^{d})}^{2}}\bigg{)}\,.$
Since the unit-scale Wiener randomization commutes with the operator
$\langle\Delta\rangle^{S}$, we can replace $f$ with
$\langle\Delta\rangle^{S}f$ and the claim follows. ∎
Next, we prove Lemma 6.2, and finally we show Proposition 6.1.
###### Proof of Lemma 6.2 .
We prove the claim by induction on $|\tau|$. First, we claim that for each
$t\in\mathbb{R}$ and $\tau\in\mathbb{T}$ we have
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau}[\mathbf{f}](t))\big{)}\Big{)}\leq
2R_{0}|\tau|.$
For the base step, if $|\tau|=1$, that is, $\tau=[\bullet]$, then
$R_{[\bullet]}[f]=e^{it\Delta}f$. Thus, for every $t\in\mathbb{R}$, we have
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{[\bullet]}[f](t))\big{)}\big{)}\leq 2R_{0}$,
since $\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(e^{it\Delta}f)(\xi)=e^{-4\pi^{2}i|\xi|^{2}}\widehat{f}(\xi)$
and $\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\hat{f}\big{)}\big{)}\leq 2R_{0}$.
Next, fix $n>1$ and assume that (6) holds for any $\tau\in\mathbb{T}$ with
$|\tau|<n$. Choose any $\tau\in\mathbb{T}$ with $|\tau|=n$ and let
$\tau_{j}\in\mathbb{T}$, $j\in\\{1,2,3\\}$, be such that
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$. For any functions $h_{j}\in
L^{2}(\mathbb{R}^{d})$, $j\in\\{1,2,3\\}$ it holds that
$\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{1}\overline{h_{2}}h_{3}}\big{)}\subset\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{1})-\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{2})+\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{h}_{3})\,,$
and thus
$\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\widehat{h_{1}\overline{h_{2}}h_{3}}\big{)}\Big{)}\leq\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{1})\big{)}+\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{2})\big{)}+\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}(\widehat{f}_{3})\big{)}.$
Since $n=|\tau|=|\tau_{1}|+|\tau_{2}|+|\tau_{3}|$ and $|\tau_{j}|\geq 1$, we
obtain that $|\tau_{j}|<n$, $j\in\\{1,2,3\\}$, and we can use the induction
hypothesis to deduce for any $s\in\mathbb{R}$ that
$\displaystyle\mathop{\kern 0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot))\big{)}\Big{)}$
$\displaystyle\leq\sum_{j=1}^{3}\mathop{\kern
0.0pt\mathrm{diam}}\mathopen{}\Big{(}\mathop{\kern
0.0pt\mathrm{spt}}\mathopen{}\big{(}\mathop{\kern
0.0pt\mathcal{F}}\mathopen{}(R_{\tau_{j}}[\mathbf{f}_{j}](s,\cdot))\big{)}\Big{)}\leq\sum_{j=1}^{3}2R_{0}|\tau_{j}|=2R_{0}|\tau|.$
The induction step follows since the multiplication by
$e^{4\pi^{2}i(t-s)|\xi|^{2}}$, or integration in time does not change the
support in Fourier space.
Next, let us inductively prove the bound
$\Big{\|}R_{\tau}\big{[}\mathbf{f}\big{]}\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim_{R_{0},|\tau|,S}\prod_{j=1}^{|\tau|}\|f_{j}\|_{H^{S+2|\tau|\varepsilon}(\mathbb{R}^{d})}\,,$
from which (6.2) follows, since $\varepsilon\lesssim_{|\tau|}1$ is small. If
$|\tau|=1$, that is, $\tau=[\bullet]$, then $\mu(|\tau|,S)=S$. Also,
$R_{[\bullet]}[f]=e^{it\Delta}\mathop{\kern 0.0pt\mathrm{Q}}\mathopen{}_{k}f$
and (6) follows from (3.1) with $h=0$.
Fix $n\in 2\mathbb{N}+1$, $n>1$ and assume that (6) holds for all
$\tau\in\mathbb{T}$ with $|\tau|<n$. As above, let
$\tau_{j}\in\mathbb{T}_{n_{j}}$ be such that
$\tau=[\tau_{1},\tau_{2},\tau_{3}]$ with $|\tau_{j}|<n$ for each
$j\in\\{1,2,3\\}$. For the rest of the proof, we allow all our constants to
depend on $|\tau|,S$, and $R_{0}$.
Since $R_{\tau}[\mathbf{f}]$ has bounded support as shown in (6), then from
(3.1) with $v_{0}=0$ and
$\mu(|\tau|,S+\varepsilon)\geq\mu(|\tau|,S)+\varepsilon$ we obtain that
$\big{\|}R_{\tau}[\mathbf{f}]\big{\|}_{Y^{\mu(|\tau|,S)+\varepsilon}(\mathbb{R})}\begin{aligned}
&\lesssim\big{\|}R_{\tau}[\mathbf{f}]\big{\|}_{Y^{\mu(|\tau|,S+\varepsilon)}(\mathbb{R})}\\\
&\lesssim\Big{\|}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{\|}_{X^{*,\mu(|\tau|,S+2\varepsilon)}(\mathbb{R})}\,.\end{aligned}$
Consequently, (4.4) yields
$\displaystyle\Big{\|}R_{\tau_{1}}[\mathbf{f}_{1}](s,\cdot)\,\overline{R_{\tau_{2}}[\mathbf{f}_{2}](s,\cdot)}\,R_{\tau_{3}}[\mathbf{f}_{3}](s,\cdot)\Big{\|}_{X^{*,\mu(|\tau|,S+2\varepsilon)}(\mathbb{R})}$
$\displaystyle\lesssim\prod_{j=1}^{3}\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+2\varepsilon)+\varepsilon}(\mathbb{R})}\lesssim\prod_{j=1}^{3}\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+3\varepsilon)}(\mathbb{R})}.$
Using the inductive hypothesis with $S+3\varepsilon$ in place of $S$ we obtain
$\Big{\|}R_{\tau_{j}}[\mathbf{f}_{j}]\Big{\|}_{Y^{\mu(|\tau_{j}|,S+3\varepsilon)}(\mathbb{R})}\lesssim\prod_{k=1}^{|\tau_{j}|}\|f_{j}\|_{H^{S+(2|\tau_{j}|+3)\varepsilon}}.$
Since $|\tau_{j}|\leq|\tau|-2$, then $2|\tau_{j}|+3\leq 2|\tau|$, and
therefore
$\Big{\|}R_{\tau}[\mathbf{f}]\Big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\lesssim\prod_{k=1}^{|\tau|}\|f_{k}\|_{H^{S+2|\tau|\varepsilon}}\,,$
as desired ∎
###### Proof of Proposition 6.1.
Fix $\tau\in\mathbb{T}$. Since $\mathbb{T}_{n}=\emptyset$ if $n\in
2\mathbb{N}$, we only consider $|\tau|\in 2\mathbb{N}+1$. According to Lemma
6.5, it suffices to prove that
$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}\gamma^{\frac{|\tau|}{2}}\|f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{|\tau|}$
for all $\gamma$ large enough. Fix $\gamma>\frac{2}{\varepsilon_{0}}$ and
recall that $\varepsilon_{0}$ is a small, fixed constant appearing in the
definition (3) of the space $Y$ depending on $S$, $|\tau|$, $\varepsilon$.
Since $\gamma>\frac{2}{\varepsilon_{0}}>2$ and $\varepsilon_{0}<2^{-100}$,
from the definition of $Y^{S}$ and Minkowski’s inequality it follows that
$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\begin{aligned}
&=\Big{(}\mathop{\kern 0.0pt\mathbb{E}}\mathopen{}\big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{\|}_{Y_{N}(\mathbb{R})}^{2}\big{)}^{\frac{\gamma}{2}}\Big{)}^{\frac{1}{\gamma}}\\\
&\leq\Big{(}\sum_{N\in 2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{\|}^{\gamma}_{Y_{N}(\mathbb{R})}\big{)}^{\frac{2}{\gamma}}\Big{)}^{\frac{1}{2}}.\end{aligned}$
Minkowski’s inequality also gives that $(\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\|f\|_{L^{p}}^{\gamma})^{\nicefrac{{1}}{{\gamma}}}\leq\big{\|}\big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}|f|^{\gamma}\big{)}^{\nicefrac{{1}}{{\gamma}}}\big{\|}_{L^{p}}$
whenever $\gamma\geq p$, where the $L^{p}$ norm is over $t,x_{1}$, or
$x^{\prime}$. Having assumed that $\gamma$ is larger than any $L^{p}$
integrability exponent of appearing in the definition of the norms
$Y_{N}(\mathbb{R})$ (see (3)), we obtain that
$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}\Big{)}^{\frac{1}{2}}.$
Henceforth, we fix $N\in 2^{\mathbb{N}}$ and we focus on bounding each term of
the sum on the right-hand side of (6) individually.
Recall that the function $\mathfrak{f}$ is obtained via the randomization
procedure $\mathfrak{f}\coloneqq\sum_{k\in\mathbb{Z}^{d}}g_{k}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f$. We then use that the map
$\mathbf{f}=(f_{1},\ldots,f_{|\tau|})\mapsto R_{\tau}[\mathbf{f}]$ is linear
in the odd entries and anti-linear in even entries $f_{j}$ to obtain that
$\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\begin{aligned}
&=\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\Big{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}g_{\mathbf{k}}R_{\tau}\big{[}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{1}}f,\ldots,\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{|\tau|}}f\big{]}(t,x)\Big{)}\\\
&=\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}\Big{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}g_{\mathbf{k}}R_{\tau}^{\mathbf{k}}\big{[}f\big{]}(t,x)\Big{)}\,,\end{aligned}$
where $\mathbf{k}=(k_{1},\ldots,k_{|\tau|})$, $g_{\mathbf{k}}\coloneqq
g_{k_{1}}\overline{g_{k_{2}}}\ldots
g_{k_{|\tau|-2}}\overline{g_{k_{|\tau|-1}}}g_{k_{|\tau|}}$, and we use the
notation $R_{\tau}^{\mathbf{k}}[f]\coloneqq R_{\tau}\big{[}\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{1}}f,\ldots,\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{|\tau|}}f\big{]}(t,x)$. We claim that the
following crucial estimate
$\Big{\|}\Big{(}\mathop{\kern 0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}\lesssim_{|\tau|}\begin{aligned}
&(\gamma-1)^{|\tau|}\\\
&\times\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\end{aligned}$
holds.
Assuming (6) holds, let us prove (6.1) first. Applying bound (6) to (6) we
obtain that
$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}$
$\displaystyle\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}$
$\displaystyle\times\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\bigg{)}^{\frac{1}{2}}\,.$
Using the Cauchy-Schwarz inequality in $N$ and the definition (3) of then norm
$Y^{\sigma}(\mathbb{R})$, it follows from the bound (6.2) of Lemma 6.2 that
$\sum_{N\in 2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}(\mathbb{R})}\\\
\begin{aligned} &\leq\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\Big{\|}^{2}_{Y_{N}(\mathbb{R})}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{N\in
2^{\mathbb{N}}}N^{2\mu(|\tau|,S)}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}^{2}_{Y_{N}(\mathbb{R})}\Big{)}^{\frac{1}{2}}\\\
&=\big{\|}R_{\tau}^{\mathbf{k}}[f]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\big{\|}R_{\tau}^{\mathbf{l}}[f]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}\\\
&\lesssim\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{l_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\,,\end{aligned}$
and hence,
$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}$
$\displaystyle\qquad\times\bigg{(}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{(}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\Big{)}\Big{(}\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{l_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}\Big{)}\bigg{)}^{\frac{1}{2}}\,.$
Since $(g_{k})_{k\in\mathbb{Z}^{d}}$ satisfy $\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\lesssim_{|\tau|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}|g_{0}|^{2|\tau|}\big{)}\lesssim_{|\tau|}1$
and $\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}=0$
if any fixed $m\in\mathbb{Z}^{d}$ appears an odd number of times in sequence
$(k_{1},\cdots,k_{|\tau|},l_{1},\cdots,l_{|\tau|})$ of elements in
$\mathbb{Z}^{d}$. Then (6) becomes
$\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{\|}R_{\tau}[\mathfrak{f}]\big{\|}_{Y^{\mu(|\tau|,S)}(\mathbb{R})}^{\gamma}\Big{)}^{\frac{1}{\gamma}}\begin{aligned}
&\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{\mathrlap{\mathbf{k}\in(\mathbb{Z}^{d})^{|\tau|}}}\hskip
20.00003pt\prod_{j=1}^{|\tau|}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k_{j}}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\bigg{)}^{\frac{1}{2}}\\\
&\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\bigg{)}^{\frac{|\tau|}{2}}\,.\end{aligned}$
We claim that
$\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\leq\|f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\,.$
Indeed, since $0\leq\psi(\xi)\leq 1$, then
$\sum_{k\in\mathbb{Z}^{d}}|\psi(\xi-k)|^{2}\leq\sum_{k\in\mathbb{Z}^{d}}|\psi(\xi-k)|=1$,
and therefore
$\sum_{k\in\mathbb{Z}^{d}}\|\mathop{\kern
0.0pt\mathrm{Q}}\mathopen{}_{k}f\|_{H^{S+\varepsilon}(\mathbb{R}^{d})}^{2}\begin{aligned}
&=\sum_{k\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}(1+|\xi|)^{2S+2\varepsilon}|\psi(\xi-k)|^{2}|\hat{f}(\xi)|^{2}\mathop{\kern
0.0pt\mathrm{d}}\mathopen{}\xi\\\
&\leq\int_{\mathbb{R}^{d}}(1+|\xi|)^{2S+2\varepsilon}|\hat{f}(\xi)|^{2}d\xi\end{aligned}$
and the required bound (6) follows.
We conclude the proof by showing that (6) holds. By (6) and Lemma 6.3 applied
point-wise in $(t,x)$, we have
$\displaystyle\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}$
$\displaystyle\lesssim_{|\tau|}(\gamma-1)^{\frac{|\tau|}{2}}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}](t,x)\big{|}^{2}\Big{)}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=(\gamma-1)^{\frac{|\tau|}{2}}\bigg{(}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\Big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\Big{)}\;R_{\tau}^{\mathbf{k}}\big{[}f\big{]}(t,x)\overline{R_{\tau}^{\mathbf{l}}\big{[}f\big{]}(t,x)}\bigg{)}^{\nicefrac{{1}}{{2}}}\,.$
Next, for any $v\colon\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{C}$ we define
the norm $Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})$ by halving all
integrability exponents in the expression (3) for the norm
$Y_{N}(\mathbb{R})$:
$\|v\|_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}\coloneqq\|v\|_{L_{t}^{\frac{1}{\varepsilon_{0}}}L_{x}^{\frac{1}{1-\varepsilon_{0}}}(\mathbb{R}\times\mathbb{R}^{d})}+\|v\|_{L_{t}^{\frac{1}{1-\varepsilon_{0}}}L_{x}^{\frac{d}{d-2}\frac{1}{1-\varepsilon_{0}}}(\mathbb{R}\times\mathbb{R}^{d})}\\\
\begin{aligned}
&+\sum_{l=1}^{d}\Big{(}N^{-\frac{1}{2}}\|v\|_{L_{e_{l}}^{(\frac{1}{1-\varepsilon_{0}},\frac{1}{\varepsilon_{0}},\frac{1}{\varepsilon_{0}})}(\mathbb{R})}+N^{-\frac{1}{2}}\|v\|_{L_{e_{l}}^{(\frac{1}{1-\varepsilon_{0}},\frac{1}{\varepsilon_{0}},\frac{\mathfrak{c}_{0}}{2(1-\varepsilon_{0})})}(\mathbb{R})}\Big{)}\\\
&+\sum_{l=1}^{d}\Big{(}N^{\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}v\|_{L_{e_{l}}^{(\frac{1}{\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}})}(\mathbb{R})}+N^{\frac{1}{2}}\|\mathop{\kern
0.0pt\mathrm{U}}\mathopen{}_{e_{l}}v\|_{L_{e_{l}}^{(\frac{1}{\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}},\frac{1}{1-\varepsilon_{0}})}(\mathbb{R})}\Big{)}.\end{aligned}$
Note that
$\big{\|}|v|^{\nicefrac{{1}}{{2}}}\big{\|}_{Y_{N}(\mathbb{R})}^{2}\approx_{\varepsilon_{0}}\|v\|_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}$,
and consequently after taking the $Y_{N}(\mathbb{R})$ norm on both sides of
(6) and using the triangle inequality for
$Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})$ we have that
$\displaystyle\Big{\|}\Big{(}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}[\mathfrak{f}]\big{|}^{\gamma}\Big{)}^{\nicefrac{{1}}{{\gamma}}}\Big{\|}_{Y_{N}(\mathbb{R})}^{2}$
$\displaystyle\hskip
50.00008pt\lesssim_{|\tau|,\varepsilon_{0}}(\gamma-1)^{|\tau|}\sum_{\mathrlap{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{d})^{|\tau|}}}\;\big{|}\mathop{\kern
0.0pt\mathbb{E}}\mathopen{}\big{(}g_{\mathbf{k}}\overline{g_{\mathbf{l}}}\big{)}\big{|}\Big{\|}\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{k}}[f]\mathop{\kern
0.0pt\mathrm{P}}\mathopen{}_{N}R_{\tau}^{\mathbf{l}}[f]\Big{\|}_{Y_{N}^{\nicefrac{{1}}{{2}}}(\mathbb{R})}.$
Finally, by Hölder’s inequality one has
$\big{\|}v_{1}v_{2}\big{\|}_{Y_{N}^{1/2}(\mathbb{R})}\lesssim\big{\|}v_{1}\big{\|}_{Y_{N}(\mathbb{R})}\big{\|}v_{2}\big{\|}_{Y_{N}(\mathbb{R})}$
and (6) follows. ∎
## 7\. Conclusion - Proof of main theorems
In this section, we use Theorem 1.5 and Theorem 1.6 proved respectively in
Section 5 and Section 6 to deduce the remaining theorems stated in Section 1.
###### Proof of Theorem 1.3 .
By Theorem 1.6 the assumptions of Theorem 1.5 on $z_{k}$ are satisfied almost
surely. Then, the first assertion follows from (Time-continuity and scattering
of multilinear data: ) and Theorem 1.6 and the second one follows from the
“Time-continuity and scattering of the multilinear data” claim of Theorem 1.5.
∎
###### Proof of Theorems 1.1, 1.2, and 1.4.
Let us relabel $S$ as $S^{\prime}$ in the theorems above. Since
$\lim_{M\to\infty}\mu(M,S_{\mathrm{min}})\geq\mathfrak{s}_{c}$, we fix $M$
such that $\mu(M+1,S^{\prime})>\mathfrak{s}_{c}$ and an arbitrary
$\mathfrak{s}$ with $\mathfrak{s}_{c}<\mathfrak{s}<\mu(M+1,S^{\prime})$, for
Theorem 1.4.
Next, we fix $S>S_{\mathrm{min}}$ such that $S<S^{\prime}<\min(2S,S+1)$ and
$\mu(M+1,S^{\prime})>\mathfrak{s}$. Finally we choose
$0<\varepsilon\lesssim_{M,S,S^{\prime}}1$ and
$0<\varepsilon_{0}\lesssim_{\varepsilon,M,S,S^{\prime}}1$ such that such that
$\varepsilon<\frac{\min(S,1,S^{\prime}-S)}{3M}$ and the conditions of Theorem
1.5 and of Theorem 1.6 are fulfilled.
According to Theorem 1.6 with $S$ replaced by $S+\varepsilon$, and because
$\mu(k,S+\varepsilon)\geq\mu(k,S)+\varepsilon$, we obtain for all $k\leq M$
that
$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}>\lambda\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$
This allows us to restrict our attention from now on to the probability set
$\Omega_{0}$ with $\mathbb{P}(\Omega_{0})=1$ for which
$\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}<+\infty$
Define the random time
$T=\begin{dcases}\frac{1}{C}\big{(}1+\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\big{)}^{-\nicefrac{{3}}{{c}}}&\text{if
}R>\delta_{0}\,,\\\ +\infty&\text{if }\begin{aligned}
&\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0}\\\
&\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\quad
k\in\\{1,\ldots,M\\}\end{aligned}\end{dcases}$
with $\delta_{0}$, $c$, and $C$ as in Theorem 1.5. The local existence of a
random solution $u=\mathfrak{z}_{\leq M}+u^{\\#}_{M}$ on $[0,T)$ with
$\|u^{\\#}_{M}\|_{X^{\mathfrak{s}}([0,T))}+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$
is guaranteed by the local and global existence claims Theorem 1.5. Such a
solution is unique among those satisfying $u^{\\#}_{M}\in
C^{0}\big{(}[0,T);H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ as claimed by the
uniqueness claim of Theorem 1.5. Furthermore, since
$\mathfrak{s}<\mu(M+1,S^{\prime})$ is arbitrary, the uniqueness claim of
Theorem 1.5 shows that
$\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,T);H^{\widetilde{\mathfrak{s}}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$
for any $\widetilde{\mathfrak{s}}<\mu(M+1,S^{\prime})$, as required by Theorem
1.4.
The measurability of $u$ follows from the fact that $u^{\\#}_{M}$ depends
continuously on the multilinear data $(\mathfrak{f},\vec{\mathfrak{z}}_{M})$.
To establish almost-certain continuity of the full solution $u$ we recall that
Theorem 1.3 applied with $S+\varepsilon$ in place of $\varepsilon$ shows that
almost surely
$\|\mathfrak{z}_{k}\|_{C^{0}\big{(}[0,\infty);H^{\mu(k,S)}_{x}(\mathbb{R}^{d})\big{)}}<\infty\,,$
while for $k=1$ it holds that
$\|\mathfrak{z}_{1}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}<\infty$
since, using Corollary 6.6, we get
$\|\mathfrak{z}_{1}(t)\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}=\|e^{it\Delta}\mathfrak{f}\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}=\|\mathfrak{f}\|_{H^{S^{\prime}}_{x}(\mathbb{R}^{d})}<\infty\,.$
Since we chose $S$ such $\mu(k,S)>S^{\prime}$ for $k\geq 2$, the triangle
inequality implies
$\|u\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}\leq\begin{aligned}
\|\mathfrak{z}_{1}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}&+\sum_{k=2}^{M}\|\mathfrak{z}_{k}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}\\\
&+\|u^{\\#}_{M}\|_{C^{0}\big{(}[0,\infty);H^{S^{\prime}}_{x}(\mathbb{R}^{d})\big{)}}<\infty\,,\end{aligned}$
as required.
Next, since $T\geq\frac{1}{C}\big{(}1+\max_{k\leq
M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\big{)}^{-\nicefrac{{3}}{{c}}}$,
accroding to the defintion (7). The probability estimates (7) imply for any
$\lambda\geq\delta_{0}$ and $k\leq M$ that
$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}>\lambda\Big{)}\leq
C\exp\Bigg{(}-\frac{\lambda^{\frac{2}{M}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$
Then (1.1) follows after standard algebraic manipulations.
To prove Theorem 1.2, choose $\delta_{0}$ as in Theorem 1.5 and note that by
choosing appropriate $\varepsilon,\varepsilon_{0},\mathfrak{s}_{c}$, and $M$
in Theorem 1.5, then $\delta_{0}$ depends only on $S$ and $\mathfrak{s}_{c}$.
Set
$\Omega_{\mathrm{glob}}=\\{\omega:\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0},\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\quad
k\in\\{1,\ldots,M\\}\\}$
and note that by (7), $T=\infty$ on $\Omega_{\mathrm{glob}}$. Then (7) implies
(7.3)
$\mathbb{P}\Big{(}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\Big{)}\geq
1-C\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,,$
and therefore
(7.4)
$\mathbb{P}\Big{(}\max_{k=1,\ldots,M}\|\mathfrak{z}_{k}\|_{Y^{\mu(k,S)+\varepsilon}(\mathbb{R})}\leq\delta_{0}\Big{)}\geq
1-CM\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}\,.$
Moreover, by (6.6) we have
(7.5)
$\mathbb{P}\Big{(}\|\mathfrak{f}\|_{H^{S+\varepsilon}_{x}}\leq\delta_{0}\Big{)}\geq
1-C\exp\Bigg{(}-\frac{\delta_{0}^{\frac{2}{k}}}{C\|f\|_{H^{S+2\varepsilon}_{x}(\mathbb{R}^{d})}^{2}}\Bigg{)}$
and (1.2) follows.
Scattering for the random mulitilinear terms $\mathfrak{z}_{k}$ follows from
Theorem 1.3 while scattering in $H^{\mathfrak{s}}_{x}(\mathbb{R}^{d})$ of
$u^{\\#}_{M}$ follows from (Scattering of global solutions: ) since
$T=+\infty$ on $\Omega_{\mathrm{glob}}$. ∎
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|
# Prompting4Debugging: Red-Teaming Text-to-Image Diffusion Models
by Finding Problematic Prompts
Zhi-Yi Chin1, Chieh-Ming Jiang1, Ching-Chun Huang1, Pin-Yu Chen,2 Wei-Chen
Chiu1
###### Abstract
Text-to-image diffusion models, e.g. Stable Diffusion (SD), lately have shown
remarkable ability in high-quality content generation, and become one of the
representatives for the recent wave of transformative AI. Nevertheless, such
advance comes with an intensifying concern about the misuse of this generative
technology, especially for producing copyrighted or NSFW (i.e. not safe for
work) images. Although efforts have been made to filter inappropriate
images/prompts or remove undesirable concepts/styles via model fine-tuning,
the reliability of these safety mechanisms against diversified problematic
prompts remains largely unexplored. In this work, we propose
Prompting4Debugging (P4D) as a debugging and red-teaming tool that
automatically finds problematic prompts for diffusion models to test the
reliability of a deployed safety mechanism. We demonstrate the efficacy of our
P4D tool in uncovering new vulnerabilities of SD models with safety
mechanisms. Particularly, our result shows that around half of prompts in
existing safe prompting benchmarks which were originally considered “safe” can
actually be manipulated to bypass many deployed safety mechanisms, including
concept removal, negative prompt, and safety guidance. Our findings suggest
that, without comprehensive testing, the evaluations on limited safe prompting
benchmarks can lead to a false sense of safety for text-to-image models.
WARNING: This paper contains model outputs that may be offensive or upsetting
in nature.
## Introduction
In recent years, generative models have been making remarkable advancements
across multiple domains, such as text, images, and even code generation,
blurring the distinction between the works created by AI systems and those
crafted by human experts. One prominent area of focus upon generative AI is
text-to-image (T2I) generation (Li et al. 2019a; Ramesh et al. 2021; Rombach
et al. 2022; Ramesh et al. 2022; Saharia et al. 2022), where most of the
state-of-the-art T2I methods are built upon the diffusion models, in which
these T2I diffusion models enable the transformation of textual information
into images. They not only bridge the gap between natural language processing
and visual content creation, but also enhance the interaction and
understanding across these two modalities. One of the main factors leading to
the exceptional performance of T2I diffusion models nowadays stems from the
vast amount of training data available on the internet, allowing the models to
generate a wide range of content, including natural animals, sketches, cartoon
images, and even artistic images. However, such large-scale training data
collected from the Internet can be a double-edged sword, as it can lead the
models to unconsciously generate inappropriate content such as copyright
infringement and NSFW materials.
Figure 1: Given an existing text-to-image (T2I) diffusion model
${\mathcal{G}}^{\prime}$ with safety mechanism which ideally can remove the
target concept (e.g. nudity) from the generated image (while the same input
prompt would lead to inappropriate image content for the typical T2I diffusion
model ${\mathcal{G}}$), our proposed Prompting4Debugging (P4D) red-teams
${\mathcal{G}}^{\prime}$ to automatically uncover the safety-evasive prompts.
To this end, there are several recent research works proposing the diffusion
models equipped with safety mechanisms, e.g. Stable Diffusion with negative
prompts (Rombach et al. 2022), SLD (Schramowski et al. 2023), and ESD
(Gandikota et al. 2023), which either restrict the text embedding space during
inference or finetune the model for attempting to prevent the model from
generating copyrighted or inappropriate images. Although these safety
mechanisms are shown to be partially effective according to their evaluation
schemes, there are already studies that demonstrate their potential flaws. For
example, (Rando et al. 2022) has found that the state-of-the-art Stable
Diffusion model equipped with NSFW safety filter (Rombach et al. 2022) will
still generate sexual content if users give the text prompt ”A photo of a
billboard above a street showing a naked man in an explicit position”.
However, these problematic prompts are discovered manually and thus are hard
to scale. Here hence comes an urgent need for developing an automated and
scalable red-teaming tool for developers to systematically inspect the model
safety and reliability before deployment.
On the other hand, as the rapid increase of size (e.g. even growing up to
billions of parameters) for recent T2I diffusion models (Ramesh et al. 2022;
Rombach et al. 2022; Ramesh et al. 2021; Saharia et al. 2022), model
finetuning becomes extremely expensive and infeasible upon limited computation
resources while building the red-teaming tool. As a result, in this work, we
utilize prompt engineering (Brown et al. 2020; Li et al. 2019b; Cui et al.
2021; Petroni et al. 2019; Jiang et al. 2020; Lester, Al-Rfou, and Constant
2021; Shin et al. 2021; Schick and Schütze 2020b) as our basis for developing
the red-teaming technique, which achieves comparable performance to
traditional approaches of finetuning heavy models but with the advantage of
learning only a minimal number of prompts.
Overall, we propose a Prompting4Debugging (P4D) framework to help
debugging/red-teaming the T2I diffusion models equipped with safety mechanisms
via utilizing prompt engineering techniques as well as leveraging an
unconstrained diffusion model to automatically and efficiently find the
problematic prompts that would lead to inappropriate content. Moreover, the
problematic prompts discovered by our P4D testing tool can be used for
understanding model misbehavior and as important references for follow-up
works to construct stronger safe mechanisms. The illustration of our proposed
P4D is provided in Figure 1. Our main contributions of this work are
summarized as follows.
* •
Our proposed Prompting4Debugging (P4D) serves as a debugging tool to red-team
T2I diffusion models with safety mechanisms for finding problematic prompts
resulting in safety-evasive outputs.
* •
Our extensive experiments based on the Inappropriate Image Prompts (I2P)
dataset reveal the fact that around half of the prompts which originally can
be tackled by the existing safety mechanisms are actually manipulable by our
P4D to become problematic ones.
* •
We also observe that some of the existing safety mechanisms in T2I diffusion
models could lead to a false sense of safety by “information obfuscation” for
red-teaming: when turning off the safety mechanism during the debugging
process, it even becomes easier for our P4D to find the problematic prompts
which are still effective to pass the safety mechanism and produce
inappropriate image content during the inference time.
## Related work
AI red-teaming tools. Red-teaming is an active cybersecurity assessment method
that exhaustively searches for vulnerabilities and weaknesses in information
security, where the issues found by red-teaming can further help companies or
organizations improve their defense mechanisms and strengthen overall
cybersecurity protection. Recently, with the popularity and increasing demand
for generative AI, red teaming is also being applied to AI models (especially
language models (Perez et al. 2022; Shi et al. 2023; Lee et al. 2023)) to
enhance model security and stability. (Perez et al. 2022) proposes to prompt
language models with a variety of methods, such as few-shot generation and
reinforcement learning, to generate test cases that are able to find
vulnerabilities in models. (Shi et al. 2023) fools the model for detecting
machine-generated text by revising output, e.g. replacing synonyms words or
altering writing style in generated sentences. On the other hand, (Lee et al.
2023) constructs a pool of user inputs and employs Bayesian optimization to
iteratively modify diverse positive test cases which eventually lead to model
failures. However, these methods are only applicable to red-team language
models, while our P4D focuses on the text-to-image models, which is a field
that has been rarely explored in AI red-teaming.
Prompt engineering. Prompt engineering originates from the field of natural
language processing and aims to adapt a pretrained language model to various
downstream tasks by modifying input text with prompts. Prompt engineering can
be categorized into two groups: hard prompts and soft prompts. Hard prompts,
also known as discrete tokens, usually consist of interpretable words that are
hand-crafted by users. For instance, (Brown et al. 2020) first demonstrates
the remarkable generalizability of pretrained language models via adopting
manually crafted hard prompts to a wide range of downstream tasks in few-shot
learning. Then (Schick and Schütze 2020a; Jiang et al. 2020; Gao, Fisch, and
Chen 2020) reformulate input texts into specific cloze-style phrases, thus
maintaining the form of hard prompts, to prompt the language models. On the
other hand, soft prompts consist of appended continuous-valued text vectors or
embeddings, providing a larger search space compared to hard prompts. For
instance, prompt-tuning (Lester, Al-Rfou, and Constant 2021) and prefix-tuning
(Shin et al. 2020) automate the soft prompts in continuous space. However,
soft prompts are often uninterpretable or non-transferrable (i.e. cannot be
shared by different language models). As a consequence, some discrete
optimization methods are proposed to strike a balance between hard prompts and
soft prompts, e.g. AutoPrompt (Shin et al. 2020), FluentPrompt (Shi et al.
2022), and PEZ (Wen et al. 2023) that learns hard prompts through continuous
gradient-based optimization. Additionally, PEZ extends its capabilities to
discover prompts that can be matched with given images, achieved by measuring
the CLIP Score (Hessel et al. 2021) using the same optimization method. These
studies demonstrate the potential of prompt engineering across various tasks
and domains, motivating us to integrate prompt engineering into the field of
red-teaming T2I diffusion models.
Diffusion models with safety mechanisms. In response to the emerging issues of
generating inappropriate images from diffusion models, several works have
devoted to address the concern. These works fall into two categories:
guidance-based and finetuning-based methods. For guidance-based methods like
Stable Diffusion with negative prompts (Rombach et al. 2022) and SLD
(Schramowski et al. 2023), they block the text embedding of certain words or
concepts (e.g. nudity, hate, or violence), in order to prevent the generation
of the corresponding image content during the inference process. Rather than
using guidance-based techniques, ESD (Gandikota et al. 2023) takes a different
approach by finetuning the partial model weights (e.g. the U-Net to perform
denoising in Stable Diffusion) to remove unwanted contents from the image
output. Nonetheless, certain corner cases still bypass the safety mechanisms
of these diffusion models (Rando et al. 2022). To enable profound testing, our
P4D serves as a debugging tool, allowing developers to identify problematic
prompts at scale by employing red-teaming strategies on T2I diffusion models.
Meanwhile, the models can enhance their robustness by attempting to tackle the
more challenging prompts uncovered through our P4D.
## Background
In this section, we first briefly introduce how diffusion models learn to
generate unconditional images. Moreover, as all the state-of-the-art T2I
diffusion models used in this work are based on latent diffusion models, we
also describe how latent diffusion models improve the efficiency of diffusion
processes and extend to support conditional generation.
Diffusion Models (Sohl-Dickstein et al. 2015; Ho, Jain, and Abbeel 2020) are
powerful generative models that learn to simulate the data generation process
by progressively denoising the (intermediate) noisy states of data, where such
denoising steps stand for the backward process to the opposite forward one
composed of diffusion steps which gradually add random noise to data. Given an
input image $x$, Denoising Diffusion Probabilistic Models (DDPM) (Ho, Jain,
and Abbeel 2020) first generates intermediate noisy image $x_{t}$ at time step
$t$ via the forward diffusion steps, where $x_{t}$ can be written as a close
form depending on $x$, $t$, and noise $\epsilon$ sampled from Gaussian
distribution $\mathcal{N}(0,I)$. Then the diffusion model training is based on
the backward process for learning a model parameterized by $\theta$ to predict
$\epsilon$, where such model takes both $x_{t}$ and the corresponding time
step $t$ as input. The objective is defined as:
$\mathcal{L}_{DM}=\mathbb{E}_{x,\epsilon\sim\mathcal{N}(0,1),t}\left[\|\epsilon-\epsilon_{\theta}(x_{t},t)\|^{2}_{2}\right]$
(1)
where $t$ ranges from $1$ to the maximum time step $T$.
Latent Diffusion Models (Rombach et al. 2022) proposes to model both forward
and backward processes in the latent space, for alleviating the efficiency
issue of DDPM which stems from having the model operate directly in the pixel
space, where the transformation between latent and pixel spaces is based on a
variational autoencoder (composed of an encoder $\mathcal{E}$ and a decoder
$\mathcal{D}$). Furthermore, they extend DDPM to enable conditional image
generation, via incorporating diverse conditions such as text prompts.
Specifically, given the latent representation $z=\mathcal{E}(x)$ of input
image $x$ as well as the intermediate noisy latent vector $z_{t}$ at time step
$t$ (analogously, depending on $z$, $t$, and $\epsilon\sim\mathcal{N}(0,I)$),
a model parameterized by $\theta$ is trained to make prediction for the noise
$\epsilon_{\theta}(z_{t},c,t)$ that is conditioned on $z_{t}$, time step $t$,
and a text condition $c$. The objective for learning such conditional
generation process (based on image–condition training pairs $\\{(x,c)\\}$) is:
$\mathcal{L}_{LDM}=\mathbb{E}_{\mathcal{E}(x),c,\epsilon\sim\mathcal{N}(0,1),t}\left[\|\epsilon-\epsilon_{\theta}(z_{t},c,t)\|^{2}_{2}\right].$
(2)
## Methdology
Figure 2: An overview of our proposed Prompting4Debugging (P4D) framework,
which employs prompt engineering techniques to red-team the text-to-image
(T2I) diffusion model ${\mathcal{G}}^{\prime}$ with safety mechanism (e.g.
Stable Diffusion with negative prompts (Rombach et al. 2022), SLD (Schramowski
et al. 2023), and ESD (Gandikota et al. 2023)). With the help of an
unconstrained T2I diffusion model $\mathcal{G}$, our P4D optimize to find the
safety-evasive prompts (i.e. $P^{\ast}_{\text{cont}}$) which can bypass the
safety mechanism in ${\mathcal{G}}^{\prime}$ and still lead to generation of
inappropriate image concept/objects (e.g. nudity). Such optimization procedure
is composed of three sequential steps, please refer to the section of proposed
method for more detailed description.
In this paper, we aim to develop a red-teaming tool named Prompting4Debugging
(P4D) for Text-to-image (T2I) diffusion models to test the reliability of
deployed safety mechanisms. In particular, three models, including Stable
Diffusion (SD) with negative prompts (Rombach et al. 2022), SLD (Schramowski
et al. 2023), and ESD (Gandikota et al. 2023), are considered as our targets
of study. The overview of our proposed P4D is visualized in Figure 2, in which
we detail its designs in the following.
Given an input text prompt $P$ which is able to lead an unconstrained/standard
T2I diffusion model $\mathcal{G}$ for generating the output image with an
inappropriate concept/object $\mathcal{C}$ (i.e. $\mathcal{G}$ does not have
the safety mechanism, and $P$ is a problematic prompt), when taking such
prompt $P$ as the input for another T2I diffusion model
${\mathcal{G}}^{\prime}$ equipped with the safety mechanism specific for
$\mathcal{C}$, ideally the resultant output image should be free from
$\mathcal{C}$ (i.e. ${\mathcal{G}}^{\prime}$ successfully defends the
generated image against the problematic prompt $P$). Our red-teaming tool P4D
now attempts to counteract the safety mechanism of ${\mathcal{G}}^{\prime}$
such that the inappropriate concept/object $\mathcal{C}$ now again appears in
the generated image (i.e. the safety mechanism of ${\mathcal{G}}^{\prime}$ is
bypassed).
Specifically, our red-teaming tool P4D adopts the technique of prompt
engineering to circumvent the safety mechanism in ${\mathcal{G}}^{\prime}$,
where a new or modified prompt $P^{\ast}$ is optimized for making
${\mathcal{G}}^{\prime}$ conditioned on $P^{\ast}$ to produce the
inappropriate content as what would be obtained by having $\mathcal{G}$
conditioned on $P$. As the state-of-the-art T2I diffusion model, i.e. Stable
Diffusion (SD), as well as the choices for the T2I diffusion models with
safety mechanism ${\mathcal{G}}^{\prime}$ in this work (e.g. SD with negative
prompts (Rombach et al. 2022), SLD (Schramowski et al. 2023), and ESD
(Gandikota et al. 2023)) are all based on the latent diffusion models, the
optimization for $P^{\ast}$ in our P4D is actually realized in the latent
space, following the procedure below (cf. Figure 2):
1. 1.
With an unconstrained T2I diffusion model $\mathcal{G}$ (e.g. Stable Diffusion
in our experiments), an original text prompt $P$ is first used to generate an
image $x$ having the inappropriate concept/object $\mathcal{C}$. Note that the
noise predictor in the backward process of $\mathcal{G}$ is parameterized by
$\theta$.
2. 2.
We then obtain the latent representation $z=\mathcal{E}(x)$ of $x$ via the
encoder $\mathcal{E}$ of $\mathcal{G}$ (noting that $\mathcal{G}$ is based on
latent diffusion models thus has the corresponding variational autoencoder),
followed by computing the intermediate noisy latent vector $z_{t}$ at an
arbitrary time step $t$ according to the diffusion process of $\mathcal{G}$.
3. 3.
Given a T2I diffusion model with safety mechanism ${\mathcal{G}}^{\prime}$ in
which its noise predictor in the backward process is parameterized by
${\theta}^{\prime}$, we now aim to find a prompt $P^{\ast}$ such that
${\mathcal{G}}^{\prime}$ conditioned on $P^{\ast}$ can produce the output
$x^{\ast}$ similar to $x$, thereby also having the similar inappropriate
concept/object $\mathcal{C}$. The optimization for $P^{\ast}$ happens on the
latent space directly to encourage similarity between the noise predictions
$\epsilon_{\theta}(z_{t},P,t)$ and
$\epsilon_{{\theta}^{\prime}}(z_{t},P^{\ast},t)$. The basic idea is that,
starting from the same noisy latent vector $z_{t}$ at an arbitrary time step
$t$, if both the noise predictors of $\mathcal{G}$ and
${\mathcal{G}}^{\prime}$ which respectively take $P$ and $P^{\ast}$ as text
prompt are able to reach the same noise prediction, then our goal of assuring
the similarity between $x^{\ast}$ and $x$ in pixel space ideally can be also
achieved.
Notably, the text prompt is typically fed into the noise predictor in the form
of embeddings (according to the common practice for our $\mathcal{G}$ and
${\mathcal{G}}^{\prime}$). To this end, the noise prediction happens in
$\mathcal{G}$ is actually operated as
$\epsilon_{\theta}(z_{t},\mathcal{W}(P),t)$, where $\mathcal{W}$ is a pre-
trained and fixed text encoder (e.g. CLIP) for extracting the embedding
$\mathcal{W}(P)$ of text prompt $P$. While for the noise prediction in
${\mathcal{G}}^{\prime}$ that involves our optimization target $P^{\ast}$, we
adopt the similar design of prompt engineering as PEZ (Wen et al. 2023) to
automate the optimization (a benefit of soft prompt) while making the
resultant prompt more transferable (a benefit of hard prompt): We start from a
continuous/soft embedding $P^{\ast}_{\text{cont}}=[e_{1},\dots,e_{N}]$
composed of $N$ tokens $e_{i}\in\mathbb{R}^{d}$, followed by projecting
$P^{\ast}_{\text{cont}}$ into the corresponding discrete/hard embedding
$P^{\ast}_{\text{disc}}=\mathcal{F}(P^{\ast}_{\text{cont}})$ via a projection
function $\mathcal{F}$ (where each token in $P^{\ast}_{\text{cont}}$ is mapped
to its nearest vocabulary embedding). As a result, noise prediction in
${\mathcal{G}}^{\prime}$ is now operated as
$\epsilon_{{\theta}^{\prime}}(z_{t},P^{\ast}_{\text{disc}},t)$, and the
objective $\mathcal{L}$ for our debugging process is defined as
$\mathcal{L}=\left\|\epsilon_{\theta}(z_{t},\mathcal{W}(P),t)-\epsilon_{{\theta}^{\prime}}(z_{t},P^{\ast}_{\text{disc}},t)\right\|^{2}_{2}$
(3)
with noting that both noise predictors in $\mathcal{G}$ and
${\mathcal{G}}^{\prime}$ are kept fixed in such optimization.
It is also worth noting that, as projection function $\mathcal{F}$ acts as a
vector quantization operation and is non-differentiable, during the
optimization procedure we follow the practice of PEZ (Wen et al. 2023) to
directly update $P^{\ast}_{\text{cont}}$ by the gradient of $\mathcal{L}$ with
respect to $P^{\ast}_{\text{disc}}$, where
$P^{\ast}_{\text{cont}}=P^{\ast}_{\text{cont}}-\gamma\nabla_{P^{\ast}_{\text{disc}}}\mathcal{L}$.
Last but not least, the resultant $P^{\ast}_{\text{disc}}$ can be transformed
into legible texts $P^{\ast}$ by the off-the-shelf text decoder/tokenizer.
We experiment two variants for $P^{\ast}_{\text{cont}}$: P4D-$N$ and P4D-$K$,
where the former initializes $N$ tokens in $P^{\ast}_{\text{cont}}$ from
scratch via randomly drawing $N$ vocabulary embeddings, while the latter
inserts learnable tokens after every $K$ tokens of $\mathcal{W}(P)$ (i.e. the
embedding of the original text prompt $P$). Basically,
$P^{\ast}_{\text{cont}}$ in P4D-$N$ has fixed length of $N$ which is
independent from the length of $\mathcal{W}(P)$, it would potentially be
insufficient for debugging the images with complex content as the original
prompt length are not taken into consideration. In comparison, the length of
$P^{\ast}_{\text{cont}}$ in P4D-$K$ (and the number of trainable tokens being
inserted) varies with the length of $\mathcal{W}(P)$ thus alleviating the
aforementioned concern in P4D-$N$. Later in experiments, we observe that both
P4D-$N$ and P4D-$K$ have the comparable debugging performance but the hard
prompt found by P4D-$K$ demonstrates better interpretability as the original
prompt $P$ is taken as its part.
## Experiments
Dataset. We evaluate the performance of our P4D on concept-related and object-
related datasets. For concept-related dataset, we focus on Inappropriate Image
Prompts (I2P) dataset (Schramowski et al. 2023), which encompasses various
uncomfortable and inappropriate prompts (including hate, harassment, violence,
self-harm, nudity contents, shocking images, and illegal activity).
Specifically, nudity contents are most prohibitive due to privacy and respect
considerations, we hence specifically set this concept aside for separate
evaluation. On the other hand for the object-related datasets, we utilize the
“car” and “French-horn” classes from ESD (Gandikota et al. 2023) for our
evaluation (as ESD only offers finetuned weights for these two classes).
Notably, the original French-horn dataset comprises merely 10 identical
prompts with different evaluation seeds. We hence extend the size of French-
horn prompts from 10 to 305 by experimenting with a wider array of evaluation
seeds.
In order to enhance the assessment of P4D’s capabilities, we additionally
filter the aforementioned datasets. We generate 3 images per prompt from the
original dataset via diffusion models, where a prompt (or an image) is
considered “unsafe” if any of the resultant images (or itself, respectively)
contains the target inappropriate concept/objects. For the purpose of
debugging and validating the reliability of safe prompts, our objective is to
select ideal prompts that yield safe images (i.e. having no inappropriate
content) through T2I diffusion models with safety mechanism while producing
unsafe images through unconstrained T2I ones. The reasons are that: 1) if the
T2I diffusion model with safety mechanism generates unsafe images through a
given prompt, such prompt has already been considered as a problematic one; 2)
if the unconstrained T2I diffusion model generates a safe image with a given
prompt, such prompt is less useful to our evaluation as we need the unsafe
prompts for our inspection on the safety mechanisms. Table 1 provides the size
of the filtered dataset. For simplicity purposes, we abbreviate “unconstrained
T2I diffusion models” and “T2I diffusion models with safety mechanism” to
“standard T2I models” and “safe T2I models” respectively.
Standard T2I and safe T2I models. In our experiments, we adopt the typical
Stable Diffusion (Rombach et al. 2022) (denoted as standard SD) for our
standard T2I model, while using ESD (Gandikota et al. 2023), SLD (Schramowski
et al. 2023) (where we adopt two superior variants of SLD, i.e. SLD-MAX and
SLD-STRONG, provided in their release code), and SD with negative prompts
(Rombach et al. 2022) (denoted as SD-NEGP) for our safe T2I models. For
standard SD, ESD, and SLD, we apply the Stable Diffusion v1-4 model backbone,
while for SD-NEGP, we use the Stable Diffusion v2-0 model backbone. When
generating an image from any of the aforementioned T2I models, the number of
inference steps is set to 25 and the setting of random seed aligns with the
used dataset, where guidance scale is set to 7.5 if not specified in the
dataset.
Implementation details. We set $N=16$ and $K=3$ respectively for our P4D-$N$
and P4D-$K$. Please note that in $P^{\ast}_{\text{cont}}$ of P4D-$K$ only the
inserted tokens are trainable while the other tokens from $\mathcal{W}(P)$ are
kept untouched. We set the batch size to 1, learning rate to 0.1, weight decay
to 0.1, and use AdamW (Loshchilov and Hutter 2017) as the optimizer. All the
prompts $P^{\ast}_{\text{cont}}$ are optimized with 3000 gradient update
steps. We measure the optimized prompts every 50 steps and update the optimal
prompts based on the cosine similarity between the generated $x^{\ast}$ and
original $x$ images.
| Category | Total | Safe T2I models | Ideal
---|---|---|---|---
_Concept_ | Nudity | 854 | ESD | 361
SLD-MAX | 204
SLD-STRONG | 112
SD-NEGP | 209
All in I2P | 4703 | SLD-MAX | 1667
_Object_ | Car | 731 | ESD | 91
French-horn | 305 | 200
Table 1: The statics upon the size of the datasets and their filtered
counterparts used in our experiments. “Total” denotes the number of prompts in
the original dataset, while “Ideal” denotes the number of ideal prompts after
applying our dataset filtering operation. The ideal prompts are the ones which
can yield safe images through safe T2I models while leading to unsafe images
by standard T2I models.
Baselines. To the best of our knowledge, there are currently no automated
tools available for red-teaming T2I diffusion models. As a result, we propose
three distinct baselines, namely Random-$N$, Random-$K$, and Shuffling.
Random-$N$ is analogous to P4D-$N$ where $N$ vocabulary embeddings are
randomly drawn to be the input prompt for safe T2I models, but without any
further optimization being performed. Similarly, Random-$K$ is analogous to
P4D-$K$ (i.e. inserting random vocabulary embeddings after every $K$ tokens in
$\mathcal{W}(P)$) but excludes further optimization. Furthermore, as some
research works in the natural language field have discovered that shuffling
the word order in a sentence can make ChatGPT (Ouyang et al. 2022) generate
inappropriate responses, we therefore introduce similar approach to construct
our Shuffling baseline, which involves randomly permuting the words in the
prompt $P$.
Method | Nudity | All in I2P | Car | French-horn
---|---|---|---|---
ESD | SLD-MAX | SLD-STRONG | SD-NEGP | SLD-MAX | ESD | ESD
Random-$N$ | 1.39% | 11.27% | 12.50% | 4.31% | 17.10% | 6.60% | 22.00%
Random-$K$ | 16.62% | 28.43% | 25.89% | 16.27% | 23.46% | 25.47% | 23.50%
Shuffling | 13.85% | 32.35% | 23.21% | 13.88% | 25.61% | 22.64% | 23.50%
OURS (P4D-$N$) | 54.29% | 27.94% | 34.82% | 27.75% | 24.00% | 42.86% | 70.50%
OURS (P4D-$K$) | 49.58% | 42.16% | 38.39% | 21.53% | 27.83% | 36.26% | 33.50%
OURS (P4D-UNION) | 70.36% | 57.35% | 56.25% | 44.02% | 44.57% | 59.34% | 82.00%
Table 2: Quantitative evaluation among various approaches for debugging
performance, where the failure rate (FR) indicating the proportion of
problematic prompts with respect to the overall amount of data is adopted as
the evaluation metric. Figure 3: Visualization of images generated by
different prompts and T2I models. The images are generated using the displayed
prompts (i.e. the sentence below the image) with the specified T2I models
(i.e. indicated by the model name on top of the image). Problematic prompts
found by our P4D are colored in dark red. Notably, P4D demonstrates the
capability to jailbreak safe T2I models and create images containing specific
target concepts or objects that should have been restricted by safe T2I
models.
Evaluation. During the evaluation process, we generate 3 images for every
prompt (either the original prompts in the dataset or the ones produced by the
baselines/P4D) by safe T2I models, where a prompt is considered problematic if
any of the resulting images is identified as “unsafe” by the classifiers or
detectors described below. For categories in the concept-related I2P dataset,
we employ the NudeNet image classifier (Bedapudi 2019) with an unsafe
percentage threshold of 0.45 to detect nudity content, while adopting the Q16
classifier (Schramowski, Tauchmann, and Kersting 2022) for identifying the
other categories in I2P. For object-related categories (i.e. car and French-
horn), we utilize YOLO v5 vehicle detector (Boneh 2023) with the confidence
threshold at 0.7 for identifying cars, while employing a pre-trained ResNet-18
classifier (Ma 2021) from the Imagenette dataset (Howard 2019) for detecting
French-horns. Although these classifiers and detectors are not error-free,
they act as a low-cost third-party auditor, capable of evaluating our P4D and
the baselines in a scalable and fair manner.
Metric. We report failure rate (FR) in experimental results, showing how many
problematic prompts are identified from the entire dataset. The higher FR
indicates better debugging performance for red-teaming.
### Experimental Results
Main Results. Quantitative results and some qualitative examples are reported
in Table 2 and Figure 3 respectively. Please refer to our appendix for more
qualitative results. Regarding concept-related I2P dataset, we inspect all
safe T2I models for the nudity category; while we only examine SLD-MAX for all
the other categories, as SLD-MAX is the sole model capable of resisting
additional concepts such as shocking, self-harm, and illegal content.
Regarding object-related categories, we inspect ESD for cars and French-horns.
From Table 2, we observe that P4D-$N$ and P4D-$K$ demonstrate promising and
comparable results across a range of safe T2I models and categories,
indicating P4D-$K$ preserves its prompt interpretability without compromising
the debugging performance. Furthermore, we unify problematic prompts from
P4D-$N$ and P4D-$K$ and obtain P4D-UNION, which significantly increases the
failure rate across various safe T2I models and categories (either concept-
related or object-related ones), indicating most problematic prompts found by
P4D-$N$ and P4D-$K$ are not repeated. Notably, for the nudity category, as our
P4D achieves the highest failure rate in ESD, in which it indicates that ESD
originally (before our red-teaming) provides the most effective safety
protection against nudity content among all safe T2I models. However, the
finetuning-based concept-removal safety mechanism of ESD may only learn to
disassociate certain concept-related words with the unsafe image content, but
it may not be resistant to optimized prompts. On the other hand, guidance-
based safe T2I models such as SLD and SD-NEGP, restrict the textual embedding
space for P4D to explore as well as prevent the generation of particular
concepts/objects with their text filters, resulting in a lower failure rate
compared to ESD with P4D. We observe that deactivating these text filters
during training encourages P4D to investigate a broader range of problematic
prompts (i.e. larger explorable textual embedding space). We refer to this
phenomenon as ”information obfuscation”, which will be elaborated in the
subsequent section.
### Ablation Studies and Extended Discussion
For the experiments used in the following studies, we focus on the nudity
category unless otherwise specified.
“Information Obfuscation” of Text Filters. We delve into the phenomenon of a
misleading sense of security caused by “information obfuscation” while
applying P4D to red-teaming the guidance-based safe T2I models (i.e. SLD and
SD-NEGP). The detailed computation procedure for such safe T2I models is as
follows: our trainable discrete prompt is firstly concatenated with the safety
concept for SLD (or the negative prompt for SD-NEGP) before feeding it into
the denoising model (i.e. the UNet for noise prediction); After denoising, the
safety-oriented guidance for SLD (or the classifier-free guidance for SD-NEGP)
is applied on the predicted noise prior to the loss calculation. This safety
process functions as a meticulously controlled text filter, ensuring the
protection of these safe T2I models. For the purpose of debugging, we have the
option to selectively deactivate some components of the inspected model. We
experiment with deactivating this safety filter during the P4D training phase
while keeping it operational during inference (noting that the deactivation is
done by excluding the concatenation with the safety concept and skipping the
safety-oriented guidance for SLD, whole similar deactivation holds for SD-
NEGP). The results are outlined in Table 3. Notably, when the safety filter is
disabled during the debugging process, P4D becomes capable of identifying more
problematic prompts. We hypothesize that the text filter actually obscures the
search for optimized textual prompts (i.e. constraining the explorable textual
embedding space), thereby leading to the failure of uncovering certain
problematic prompts. However, the removal of the text filter eliminates such
constraint on the embedding search space, thereby facilitating the
identification of problematic prompts. This phenomenon draws parallels with
the concept of “obfuscated gradients” of AI security as discussed in (Athalye,
Carlini, and Wagner 2018), where “obfuscated gradients“ foster a false sense
of security in defenses against adversarial examples. Similarly, in our study,
the text filter induces a false sense of safety through “information
obfuscation”, as evidenced by the fact that removing this filter allows P4D to
find more problematic prompts. Please also note that, due to such information
obfuscation properties of SLD and SD-NEGP, in the following studies, we remove
the text filter when optimizing the prompts for SLD and SD-NEGP, for more
efficient computation.
Safe T2I P4D-$N$ P4D-$K$ w/ TF w/o TF w/ TF w/o TF SLD-MAX 27.94% 44.12%
42.16% 42.16% SLD-STRONG 34.82% 50.89% 38.39% 42.86% SD-NEGP 27.75% 30.14%
21.53% 33.97%
Table 3: Percentage of problematic prompts (i.e. failure rate) found for SLD
and SD-NEGP with and without the safety text filter (w/ or w/o TF) in nudity
category of I2P dataset.
Prompt Length. We further conduct investigation upon the number of tokens in a
prompt (i.e. prompt length) for optimization. For P4D-$N$, we test three
different values of $N$: 8, 16 (default), and 32. For P4D-$K$, we also test
inserting a learnable token every 1, 3 (default), and 5 tokens in the
embedding $\mathcal{W}(P)$ of the original input prompt $P$. From Table 4, we
can observe that there is no optimal prompt length in either P4D-$N$ or
P4D-$K$. We argue that a complex scenario requires a longer prompt for
description, whereas simpler scenarios can be adequately described with
shorter prompts. Consequently, we recommend aggregating/unioning the
problematic prompts found by using various settings of length for more
efficient red-teaming.
P4D-$N$ $N$ Safe T2I 8 16 32 _Union_ ESD 54.85% 54.02% 59.00% 78.67% SLD-MAX
35.29% 44.12% 38.24% 67.16% SLD-STRONG 47.32% 50.89% 45.54% 78.57% SD-NEGP
36.84% 30.14% 34.45% 61.72% P4D-$K$ $K$ Safe T2I 1 3 5 _Union_ ESD 52.63%
49.58% 49.31% 74.52% SLD-MAX 38.73% 42.16% 40.69% 69.12% SLD-STRONG 40.18%
42.86% 50.00% 72.32% SD-NEGP 32.06% 33.97% 32.06% 61.24%
Table 4: Ablation study for prompt length.
Text and Image Similarity. We calculate cosine similarities for both original
and optimized prompts, as well as their generated images. In a nutshell, we
suggest T2I safety research should jointly safeguard the text and image
domains. Please refer to appendix for details.
Prompt Generalizability. We accumulate all non-repeated problematic prompts
(while selecting the prompt with the highest toxicity score if repeated) found
by P4D across all safe T2I models (e.g. ESD, SLD, and SD-NEGP) as another
dataset/collection to test the generalizability of these problematic prompts
across different safe T2I models. As shown in Table 5, over 50% prompts found
by P4D are able to red-team multiple safe T2I models at the same time.
Moreover, we report the failure rate of universal problematic prompts that are
able to red-team all the safe T2I models simultaneously, which we term the
“intersection”. We can observe that over 30% problematic prompts found in both
P4D-$N$ and P4D-$K$ are robust and general enough to red-team across all safe
T2I models simultaneously.
| | P4D-$N$ | P4D-$K$
---|---|---|---
| Data size | 405 | 380
Failure rate (FR,%) | ESD | 61.23% | 64.64%
SLD-MAX | 89.14% | 83.37%
SLD-STRONG | 90.37% | 91.02%
SD-NEGP | 54.81% | 54.35%
| Intersection | 37.28% | 31.93%
Table 5: Evaluation upon prompt generalizability. We create a collection of
the problematic prompts discovered by P4D across all safe T2I models, and
assess such collection using each safe T2I model. Intersection refers to the
percentage of universal problematic prompts that are able to red-team all safe
T2I models simultaneously.
## Conclusion
In this paper, we propose an automated red-teaming debugging tool called P4D
to unveil unprecedented weaknesses of several safety mechanisms used in T2I
diffusion models. P4D proactively finds problematic prompts that may lead to
inappropriate (e.g. nudity or violent) images that bypass deployed safety
mechanisms. Our extensive experimental results demonstrate the effectiveness
of P4D for debugging, providing developers with a red-teaming tool to
safeguard and test the reliability of safe T2I diffusion models.
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|
# Multi-channel Hybrid Access Femtocells:
A Stochastic Geometric Analysis
Yi Zhong and Wenyi Zhang, _Senior Member, IEEE_ The authors are with
Department of Electronic Engineering and Information Science, University of
Science and Technology of China, Hefei 230027, China (email:
<EMAIL_ADDRESS>[email protected]). The research has been
supported by the National Basic Research Program of China (973 Program)
through grant 2012CB316004, National Natural Science Foundation of China
through grant 61071095, Research Fund for the Doctoral Program of Higher
Education of China through grant 20103402120023, and by MIIT of China through
grants 2010ZX03003-002 and 2011ZX03001-006-01.
###### Abstract
For two-tier networks consisting of macrocells and femtocells, the channel
access mechanism can be configured to be open access, closed access, or hybrid
access. Hybrid access arises as a compromise between open and closed access
mechanisms, in which a fraction of available spectrum resource is shared to
nonsubscribers while the remaining reserved for subscribers. This paper
focuses on a hybrid access mechanism for multi-channel femtocells which employ
orthogonal spectrum access schemes. Considering a randomized channel
assignment strategy, we analyze the performance in the downlink. Using
stochastic geometry as technical tools, we model the distribution of
femtocells as Poisson point process or Neyman-Scott cluster process and derive
the distributions of signal-to-interference-plus-noise ratios, and mean
achievable rates, of both nonsubscribers and subscribers. The established
expressions are amenable to numerical evaluation, and shed key insights into
the performance tradeoff between subscribers and nonsubscribers. The
analytical results are corroborated by numerical simulations.
###### Index Terms:
Channel management, femtocell, hybrid access, Neyman-Scott cluster process,
spatial Poisson process, two-scale approximation, two-tier network
## I Introduction
In current cellular network services, about 50% of phone calls and 70% of data
services take place indoors [1]. For such indoor use cases, network coverage
is a critical issue. One way to improve the indoor performance is to deploy
the so-called femtocell access points (FAPs) besides macrocell base stations
(MBSs). Femtocells are small cellular base stations, typically designed for
use in home or small business [2][3]. The use of femtocells not only benefits
the users, but also the operators. As the distance between transmitter and
receiver is reduced, users will enjoy high quality links and power savings.
Furthermore, the reduced transmission range also creates more spatial reuse
and reduces electromagnetic interference.
Among the many challenges faced by femtocells, and more generally, two-tier
networks, is the issue of interference; see Figure 1. The two-tier
interference problem differs from that in traditional single-tier networks in
several important aspects: First, due to the limitations of access mechanism,
a user equipment (UE) may not be able to connect to the access point which
offers the best service. Second, since femtocells connect to operator’s core
network via subscribers’ private ISP, coordination between macrocells and
femtocells and among femtocells is limited. Finally, compared to planned
macrocell deployments, femtocells are usually deployed in an ad hoc manner,
and the randomly placed femtocells make it difficult to manage the
interference. In two-tier networks, interference can be categorized into two
types: (a) cross-tier, referring to the interference from one tier to the
other; (b) co-tier, referring to the interference within a tier.
Figure 1: Downlink two-tier network model for hybrid access femtocell.
In this paper, we consider two-tier networks based on multicarrier techniques,
for example those deploying LTE or WiMAX standards, which use orthogonal
frequency-division multiple access (OFDMA) techniques. In multicarrier
systems, the available spectrum is divided into orthogonal subcarriers, which
are then grouped into multiple subchannels, assigned to different users. Due
to the flexibility in channel assignment, the interference may be alleviated.
The access mechanism of femtocells (see, e.g., [4]) is a key factor that
affects the performance of two-tier networks, and generally can be classified
as follows, where we call the UEs registered to a femtocell as subscribers,
and those not registered to any femtocell as nonsubscribers.
* •
Closed access: An FAP only allows its subscribers to connect.
* •
Open access: An FAP allows all its covered UEs, no matter registered or not,
to connect.
* •
Hybrid access: An FAP allows its covered nonsubscribers to connect via a
subset of its available subchannels, and reserves the remaining subchannels
for its subscribers.
Hybrid access [5] is an intermediate approach, in which a fraction of resource
is allocated to nonsubscribers. By doing so, nonsubscribers near an FAP may
handover into the femtocell to avoid high interference; meanwhile, with
certain amount of resource reserved for subscribers, the performance of
subscribers may be well assured even in the presence of nonsubscribers.
In hybrid access, a central issue is how to allocate the resource between
subscribers and nonsubscribers. Previous studies [6] [7] indicate that hybrid
access improves the network performance at the cost of reduced performance for
subscribers, therefore suggesting a tradeoff between the performance of
nonsubscribers and subscribers. In this paper, we consider a hybrid access
mechanism that uses a randomized channel assignment strategy, and analyze the
performance in the downlink of both macrocells and femtocells. We employ
stochastic geometry to characterize the spatial distributions of users as well
as access points; see, e.g., [8] and references therein for its recent
applications in wireless networks. In order to make the work integral, we will
carry out the analysis in two different cases. As a general assumption, we
first assume that the FAPs are distributed as a Poisson point process (PPP).
Then, we switch to the case when the FAPs are distributed as a Neyman-Scott
cluster process. The cluster process is likely to be more realistic because
the FAPs are typically deployed in populous locations, like commercial or
residential area. Accordingly, we derive the key performance indicators
including mean achievable rates and distributions of the signal-to-
interference-plus-noise ratios (SINRs) of both nonsubscribers and subscribers.
In our study, we establish general integral expressions for the performance
indicators, and closed form expressions under specific model parameters. With
the obtained results, we reveal how the performance of subscribers and
nonsubscribers trades off each other.
The introduction of stochastic geometry in the analysis of wireless network is
not our original, and an overview of related works is as follows. In [9], the
authors proposed to study key performance indicators for cellular networks,
such as coverage probabilities and mean achievable rates. In [10], the
considered scheme divides the spectrum resource into two orthogonal parts
which are assigned to macrocells and femtocells, respectively, with femtocells
being closed access. In [11], the authors considered two-tier femtocell
networks using time-hopped CDMA, examining the uplink outage probability and
the interference avoidance capability. In [12], the success probabilities
under Rayleigh fading for both macrocells and femtocells are derived in uplink
and downlink respectively. In [13] and [14], stochastic geometry tools are
applied in the coexistence analysis of cognitive radio networks. In [15] and
[16], the authors studied the performance of various femtocell access
mechanisms, under substantially different system models from ours. More
explicitly, the work in [15], which does not make use of stochastic geometry,
focused on the uplink with only one MBS and one FAP in the model. Though the
work in [16] also applies the Laplace transform of interference in the
derivation, the substantially difference of our work lies in that we take the
load (measured by the number of UEs in a cell) into consideration, utilizing
the size distribution of Voronoi cells to derive the distribution of the load.
Moreover, we model the mechanism for sharing sub-channels in multi-channel
systems, and propose to use a two-scale approximation which substantially
simplifies the analysis. All the works mentioned above are based on the PPP
assumption. The works applying the clustered model can be found in [17] which
derived the success probability for transmission in clustered ad-hoc networks
and in [18] which discussed the property of interference with clustered
interferers.
The main contribution of our work is detailed as follows. The existing works
mostly ignore the network load which is a key factor that affects the
distribution of interfering access points (APs). For example, when the load is
uniformly distributed in the plane, the APs with larger coverage may
experience more load, thus leading to more interference to the network.
Moreover, the optimal proportion of shared resources of a femtocell also
depends on the distribution of network load. In our work, we focus on the
performance analysis in the context of multi-channel systems, in which case
not all sub-channels are occupied and not all APs cause interference to a
given subchannel. We evaluate the two-tier interference when the FAPs are
distributed as the PPP and the Neyman-Scott cluster respectively. In addition,
we propose two-scale approximation to simplify the analysis and verify the
effectiveness of the approximation by simulation.
The remaining part of the paper is organized as follows. Section II describes
the two-tier network model, the channel assignment strategy, and the hybrid
access mechanism. Based on a two-scale approximation for the spatial
distributions of FAPs, Section III analyzes the statistical behavior of UEs,
deriving the distributions of the number of UEs connecting to either an FAP or
an MBS, as well as the probabilities of a subchannel being used by either an
FAP or an MBS. Built upon those statistics, Section IV and V establish
expressions for the distributions of SINRs, and mean achievable rates in the
cases when the FAPs are distributed as PPP and Neyman-Scott cluster
respectively. Section VII illustrates the aforementioned analysis by numerical
results, which are also corroborated by simulations. Finally, Section VIII
concludes the paper.
## II Network Model
### II-A Hybrid Access Femtocells
In the two-tier network, we consider two types of access points, MBSs and
FAPs. The MBSs constitute the macrocell tier, and they induce a Voronoi
tessellation of the plane (see Figure 2). When a UE attempts to access the
macrocell network, it chooses to connect to the MBS in the Voronoi cell in
which the UE is situated. An FAP provides network access to UEs in its
vicinity, and we assume that all FAPs have a covering radius of $R_{f}$.
Within the covered circular area of each FAP are two types of UEs, called
subscribers and inside nonsubscribers. Inside nonsubscribers are those UEs who
gather around an FAP without subscribing to its service; for example,
transient customers in a shop or a restaurant. Besides those two types of UEs,
we also consider a third type of UEs, outside nonsubscribers, who are
uniformly scattered over the whole plane, corresponding to those regular
macrocell network users.
(a) FAPs are distributed as PPP.
(b) FAPs are distributed as Neyman-Scott cluster.
Figure 2: The Voronoi macrocell topology, in which each Voronoi cell is the
coverage area of a macrocell and each small circle represents a femtocell.
The available spectrum is evenly divided into $M$ subchannels, which are to be
shared by both macrocell tier and femtocell tier. Each FAP is configured to
allocate a fixed number, $M_{s}$, of subchannels for its covered inside
nonsubscribers. These $M_{s}$ subchannels are called shared subchannels, and
the remaining $M_{r}=M-M_{s}$ subchannels are called reserved subchannels as
they are reserved for the subscribers. In the considered hybrid access
mechanism, each FAP selects its shared subchannels randomly, and independently
of other FAPs. We assume that each UE, whether subscriber or nonsubscriber,
needs one subchannel for transmitting. When a UE accesses an MBS or an FAP,
the serving subchannel is selected randomly (see Figure 3).
Figure 3: Spectrum allocation in each hybrid access femtocell. All the $M_{s}$
shared subchannels are randomly selected by each FAP.
The hybrid access mechanism operates as follows.
* •
A subscriber accesses to one of the $M_{r}$ reserved subchannels of its
corresponding FAP. When there are more than $M_{r}$ subscribers in an FAP,
they are served by time-sharing with equal time proportion.
* •
An inside nonsubscriber accesses to one of the $M_{s}$ shared subchannels of
its covering FAP. When there are more than $M_{s}$ inside nonsubscribers in an
FAP, they are served by time-sharing with equal time proportion.
* •
An outside nonsubscriber accesses the MBS located in the Voronoi cell in which
the outside nonsubscriber is situated. When there are more than $M$ outside
nonsubscribers in the Voronoi cell, they are served by time-sharing with equal
proportion.
### II-B Mathematical Model and Two-scale Approximation
To formulate the aforementioned hybrid access scenario mathematically, we
model the spatial distributions of the nodes using spatial point processes as
follows. The MBSs constitute a homogeneous Poisson point process (PPP)
$\Phi_{m}$ of intensity $\lambda_{m}$ on the plane. The distribution of FAPs
will be divided into two cases for discussion:
* •
Case 1: the FAPs constitute another homogeneous PPP $\Phi_{f}$ of intensity
$\lambda_{f}$.
* •
Case 2: the FAPs are distributed as a Neyman-Scott cluster process $\Phi_{f}$
[19]. The center of the clusters are assumed to be distributed according to a
stationary PPP $\Phi_{p}$ of intensity $\lambda_{p}$, which is called the
parent process. For each cluster center $x\in\Phi_{p}$, the FAPs are
distributed according to an independent PPP $\Phi^{x}$ of intensity
$\lambda_{c}$ in the circular covered area of radius $R_{c}$ around the center
$x$. The complete distribution of all FAPs is given as
$\Phi_{f}=\bigcup_{x\in\Phi_{p}}\Phi^{x}.$ (1)
In this case, the number of FAPs in a typical cluster is a Poisson random with
parameter $\pi R_{c}^{2}\lambda_{c}$ and the intensity of all FAPs is
$\lambda_{f}=\pi R_{c}^{2}\lambda_{c}\lambda_{p}$.
In the circular covered area of radius $R_{f}$ of each FAP, the subscribers
are distributed according to a homogeneous PPP of intensity $\lambda_{s}$, and
the inside nonsubscribers are distributed according to another homogeneous PPP
of intensity $\lambda_{\mathrm{in}}$. Outside nonsubscribers constitute on the
whole plane a homogeneous PPP of intensity $\lambda_{\mathrm{out}}$. All the
PPPs are mutually independent.
In this paper, we focus on the downlink performance. The transmit power is set
to a constant value $P_{m}$ for an MBS, and $P_{f}$ for an FAP. For the sake
of convenience, we adopt a standard path loss propagation model with path loss
exponent $\alpha>2$. Regarding fading, we assume that the link between the
serving access point (either an MBS or an FAP) and the served UE experiences
Rayleigh fading with parameter $\mu$. The received signal power of a UE at a
distance $r$ from its serving access point therefore is $P_{m}hr^{-\alpha}$
(MBS) or $P_{f}hr^{-\alpha}$ (FAP) where $h\sim\mathrm{Exp}(\mu)$. The fading
of interference links may follow an arbitrary probability distribution, and is
denoted by $g$. Furthermore, considering the typical scenario of indoor
femtocell deployment, we introduce a wall isolation at the boundary of each
FAP coverage area, which incurs a wall penetration loss factor $W<1$. For all
receivers, the noise power is $\sigma^{2}$.
The different point processes corresponding to different entities in the
network interact in a complicated way, thus making a rigorous statistical
analysis extremely difficult. For example, an inside nonsubscriber may be
covered by more than one FAPs, thus leading to the delicate issue of FAP
selection, and furthermore rendering the subchannel usage distributions among
FAPs and MBSs intrinsically correlated. To overcome the technical difficulties
due to spatial interactions, in the subsequent analysis we propose a two-scale
approximation for the network model, motivated by the fact that the covered
area of an FAP is significantly smaller than that of an MBS. The two-scale
approximation consists of two views, the macro-scale view and the micro-scale
view. The macro-scale view concerns an observer outside the coverage area of
an FAP, and in that view the whole coverage area of the FAP shrinks to a
single point, marked by the numbers of subscribers and inside nonsubscribers
therein. The micro-scale view concerns an observer inside the coverage area of
an FAP, and in that view the coverage area is still circular with radius
$R_{f}$ in which the subscribers and inside nonsubscribers are spatially
distributed. By such a two-scale approximation, an inside nonsubscriber can
only be covered by a unique FAP, and the coverage area of an FAP can only be
within a unique Voronoi cell of an MBS. Meanwhile, at the cell edge the
outside nonsubscribers are clearly divided by the boundary and the subscribers
and inside nonsubscribers are all attached to the corresponding FAPs which are
also clearly divided. These consequences substantially simplify the
performance analysis. In Section VII, we validate the two-scale approximation
method through comparing analytical results and simulation results, for
network parameters of practical interest.
## III Statistics of UEs and Subchannels
In this section, we characterize the distributions of UEs connecting to
different types of access points, and the distributions of used subchannels in
MBSs and FAPs. The analysis is based on a snapshot of the network model, and
the obtained results will then be applied for characterizing the distributions
of SINRs and achievable rates in Section IV.
### III-A Distributions of UEs
Let $U_{s}$ be the number of subscribers accessing a given FAP and from our
model we have $U_{s}\sim\mathrm{Poisson}(\lambda_{s}\pi R_{f}^{2})$.
Similarly, let $U_{\mathrm{in}}$ be the number of inside nonsubscribers
accessing a given FAP, and we have
$U_{\mathrm{in}}\sim\mathrm{Poisson}(\lambda_{\mathrm{in}}\pi R_{f}^{2})$.
The number of outside nonsubscribers who access a given MBS, denoted by
$U_{\mathrm{out}}$, is characterized as follows. We note that the macrocell
coverage area is a Voronoi cell, and denote by $S$ the area of the Voronoi
cell. There is no known closed form expression of the probability density
function (pdf) of $S$, whereas a simple approximation [20] has proven
sufficiently accurate for practical purposes. Considering scaling, the
approximate pdf of the size of a macrocell coverage area is given by
$f(S)=\frac{343}{15}\sqrt{\frac{7}{2\pi}}(S\lambda_{m})^{\frac{5}{2}}\exp(-\frac{7}{2}S\lambda_{m})\lambda_{m}.$
(2)
Conditioning upon $S$, the number of outside nonsubscribers is a Poisson
random variable with mean $\lambda_{\mathrm{out}}S$. The probability
generating function of the unconditioned $U_{\mathrm{out}}$ is thus given by
$G(z)=\int_{0}^{\infty}\exp\Big{(}\lambda_{\mathrm{out}}(z-1)S\Big{)}f(S)dS.$
(3)
Plugging in the approximate pdf of $S$ and simplifying the integral, we get
$G(z)=\frac{343}{8}\sqrt{\frac{7}{2}}\Big{(}\frac{7}{2}-\frac{\lambda_{\mathrm{out}}}{\lambda_{m}}(z-1)\Big{)}^{-\frac{7}{2}}.$
(4)
The distribution of $U_{\mathrm{out}}$ is therefore given by the derivatives
of $G(z)$,
$\mathbb{P}\\{U_{\mathrm{out}}=i\\}=\frac{G^{(i)}(0)}{i!},\quad i=0,1,\ldots.$
(5)
### III-B Distributions of Subchannel Usage
Since the subchannels are uniformly and independently selected by each FAP, it
suffices to analyze an arbitrary one of them. Let us examine the probability
that a given subchannel is used by an MBS or an FAP. First, we evaluate the
average number of subchannels used by an MBS or an FAP, and then we normalize
the average number by the total number of subchannels, $M$.
The probability that a subchannel is used by an FAP is
$\displaystyle
P_{\mathrm{busy},f}=\frac{1}{M}\Big{(}\sum_{i=0}^{\infty}\min\\{i,M_{r}\\}\mathbb{P}\\{U_{s}=i\\}$
$\displaystyle+\sum_{j=0}^{\infty}\min\\{j,M_{s}\\}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\Big{)}.$
(6)
For a Poisson random variable $N\sim\mathrm{Poisson}(\lambda)$, its cumulative
distribution function (cdf) is
$\sum_{i=0}^{n}\mathbb{P}\\{N=i\\}=\sum_{i=0}^{n}\frac{\lambda^{i}}{i!}e^{-\lambda}=\frac{\Gamma(n+1,\lambda)}{n!},$
(7)
where $\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\mathrm{d}t$ is the
incomplete gamma function. Using (7) to simplify $P_{\mathrm{busy},f}$, we get
$\displaystyle
P_{\mathrm{busy},f}=1+\frac{M_{r}}{M}\frac{1}{M_{r}!}\Big{(}\lambda_{s}\pi
R_{f}^{2}\Gamma(M_{r},\lambda_{s}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{r}+1,\lambda_{s}\pi R_{f}^{2})\Big{)}$
$\displaystyle+\frac{M_{s}}{M}\frac{1}{M_{s}!}\Big{(}\lambda_{\mathrm{in}}\pi
R_{f}^{2}\Gamma(M_{s},\lambda_{\mathrm{in}}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{s}+1,\lambda_{\mathrm{in}}\pi R_{f}^{2})\Big{)}.$ (8)
The probability that a subchannel is used by an MBS is
$\displaystyle
P_{\mathrm{busy},m}=\frac{1}{M}\sum_{i=0}^{\infty}\min\\{i,M\\}\mathbb{P}\\{U_{\mathrm{out}}=i\\},$
(9)
where $\mathbb{P}\\{U_{\mathrm{out}}=i\\}$ is given by (5).
The spatial point process of FAPs that use a given subchannel is the
independent thinning of the original process of FAPs $\Phi_{f}$ by the
probability $P_{\mathrm{busy},f}$, denoted by $\Phi_{f}^{\prime}$. The term
“independent thinning” means that $\Phi_{f}^{\prime}$ can be viewed as
obtained from $\Phi_{f}$ by independently removing points with probability
$1-P_{\mathrm{busy},f}$. When the FAPs are distributed as a homogeneous PPP,
the resulting point process is still a homogeneous PPP with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. For the case when the
FAPs are distributed as Neyman-Scott cluster process, the resulting point
process is a Neyman-Scott cluster process with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. Moreover, the intensity
of the parent process is still $\lambda_{p}$ and the intensity of the FAPs in
each cluster is reduced to
$\lambda_{c}^{\prime}=\lambda_{c}P_{\mathrm{busy},f}$. As for the MBSs, the
correlations between the sizes of neighboring cells may lead to the dependence
in the thinning of the original PPP of MBSs. However, in order to facilitate
the analysis, we assume that the independent thinning assumption still holds.
Therefore, the spatial process of MBSs that use a given subchannel is the
independent thinning of the original PPP of MBSs $\Phi_{m}$ by the probability
$P_{\mathrm{busy},m}$, denoted by $\Phi_{m}^{\prime}$ with intensity
$\lambda_{m}^{\prime}=\lambda_{m}P_{\mathrm{busy},m}$. Meanwhile, the
computation of $P_{\mathrm{busy},m}$ and the SINR distribution can also be
decoupled. The simulation results validate that our independent thinning
assumption is rather reliable. These two independently thinned point processes
will prove useful in the subsequent analysis.
## IV Performance With Poisson Distribution Of FAPs
In this section, we derive the distributions of the SINRs and the mean
achievable rates of UEs served by MBS and FAP respectively when the FAPs are
distributed as the PPP. For each type of UEs, we begin with general settings,
and then simplify the general results under specific parameters to gain
insights. The mean achievable rates are the averaged instantaneous achievable
rates over both channel fading and spatial distributions of UEs and access
points.
### IV-A Macrocell UEs
#### IV-A1 General Case
For an active UE served by an MBS, it must be occupying one subchannel of the
MBS. The following theorem gives the cdf of SINR and the mean achievable rate
of each active macrocell UE in general case.
###### Theorem 1
The cdf of the SINR of a macrocell UE, denoted by
$Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is given by
$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(T,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
Tv^{\frac{\alpha}{2}}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$ (10)
The mean achievable rate of a macrocell UE is given by
$\displaystyle\tau_{m}=\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(e^{t}-1,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
v^{\frac{\alpha}{2}}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}t\mathrm{d}v.$
(11)
In (10) and (11), $\beta(T,\alpha)$ is given by
$\displaystyle\beta(T,\alpha)=\frac{2(\mu
T)^{\frac{2}{\alpha}}}{\alpha}\mathbb{E}_{g}\Big{(}g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},\mu
Tg)$
$\displaystyle-(1+\frac{\lambda_{f}^{\prime}(WP_{f})^{\frac{2}{\alpha}}}{\lambda_{m}^{\prime}P_{m}^{\frac{2}{\alpha}}})\Gamma(-\frac{2}{\alpha})\Big{)}\Big{)}.$
(12)
The proof of Theorem 1 is in Appendix A.
In Theorem 1, $Z_{m}(T)$ in (10) gives the probability that the SINR is below
a given target level $T$, and $\tau_{m}$ in (11) gives the mean achievable
rate of a macrocell UE. The integrals in (10) and (11) can be evaluated by
numerical methods, and furthermore, they can be simplified to concise forms in
the special case when the interference experiencing Rayleigh fading and the
path loss exponent being $\alpha=4$ with no noise, $\sigma^{2}=0$.
#### IV-A2 Special Case When Interference Experiences Rayleigh Fading
Here we consider the case where the interference experiences Rayleigh
distribution with mean $\mu$, i.e. $g\sim\mathrm{Exp}(\mu)$. In this case, the
results are as follows.
###### Corollary 1
When the interference follows Rayleigh fading, the cdf of the SINR is
$\displaystyle\\!\\!\\!Z_{m}(T)$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$ $\displaystyle
1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(T,\alpha)$ (13)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}WT}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$
The mean achievable rate is
$\displaystyle\\!\\!\\!\tau_{m}$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)$ (14)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}W(e^{t}-1)}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}v\mathrm{d}t,$
where
$\varphi(T,\alpha)=T^{2/\alpha}\int_{T^{-2/\alpha}}^{\infty}\frac{1}{1+u^{\alpha/2}}\mathrm{d}u.$
(15)
The results in Corollary 1 are further simplified in the following special
cases.
Specifically, when $\alpha=4$ we obtain
$\displaystyle Z_{m}(T)$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!1-\frac{1}{1+\sqrt{T}\Big{(}\arctan\sqrt{T}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}^{\prime}}\sqrt{\frac{WP_{f}}{P_{m}}}\Big{)}P_{\mathrm{busy},m}}.$
The mean achievable rate of a macrocell UE is simplified into
$\tau_{m}=\int_{0}^{\frac{\pi}{2}}\frac{2}{\tan
y+(\frac{\pi}{2}-y)P_{\mathrm{busy},m}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}}\mathrm{d}y.$
(17)
From a practical perspective, it is desirable to shape $Z_{m}(T)$ to make it
small for small values of $T$. From (LABEL:equ:SINR_Simple), we see that there
are two approaches to shape $Z_{m}(T)$. First, $Z_{m}(T)$ decreases as
$P_{\mathrm{busy},m}$, the probability that a subchannel is used by an MBS,
decreases. This may be interpreted as an effect of frequency reuse. Second,
$Z_{m}(T)$ decreases as the whole term,
$\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}$,
decreases, which corresponds to a number of network parameters, representing
the effect due to the deployment of the femtocell tier.
### IV-B Femtocell UEs
#### IV-B1 General Case
A UE served by an FAP occupies one subchannel of the FAP. The following
theorem gives the cdf of the SINR and the mean achievable rate of each active
femtocell UE in general case.
###### Theorem 2
The cdf of the SINR of a femtocell UE in general case is given by
$\displaystyle
Z_{f}(T)=1-\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\exp\Big{(}-\rho(\alpha)T^{2/\alpha}v$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{f}}\Big{)}\mathrm{d}v.$
(18)
The mean achievable rate of a femtocell UE is given by
$\displaystyle\tau_{f}=\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\exp\Big{(}-\rho(\alpha)(e^{t}-1)^{2/\alpha}v$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{f}}\Big{)}\mathrm{d}t\mathrm{d}v.$ (19)
in (18) and (2), $\rho(\alpha)$ is given by
$\displaystyle\rho(\alpha)=-\frac{2\pi\mu^{2/\alpha}}{\alpha}\Gamma\Big{(}-\frac{2}{\alpha}\Big{)}\Big{(}\lambda_{m}^{\prime}\Big{(}\frac{WP_{m}}{P_{f}}\Big{)}^{2/\alpha}$
$\displaystyle+\lambda_{f}^{\prime}W^{4/\alpha}\Big{)}\mathbb{E}_{g}(g^{2/\alpha}).$
(20)
The proof of Theorem 2 is in Appendix B.
#### IV-B2 Special Case When Interference Experiences Rayleigh Fading
For Rayleigh fading, the results are essentially the same as that in the
general fading case. The only additional simplification is the evaluation of
$\rho(\alpha)$. We just give the result in the very special case when
$\sigma^{2}=0$ and $\alpha=4$.
When the interference experiences Rayleigh fading, we have
$\mathbb{E}_{g}(g^{\frac{1}{2}})=\mu\int_{0}^{\infty}g^{\frac{1}{2}}e^{-\mu
g}\mathrm{d}g=\frac{1}{2}\sqrt{\frac{\pi}{\mu}}$, and consequently,
$\rho(4)=\frac{\pi^{2}}{2}\Big{(}\lambda_{m}^{\prime}\sqrt{\frac{WP_{m}}{P_{f}}}+\lambda_{f}^{\prime}W\Big{)}.$
(21)
The SINR distribution is then given by
$Z_{f}(T)=1-\frac{1-e^{-\rho(4)\sqrt{T}R_{f}^{2}}}{\rho(4)\sqrt{T}R_{f}^{2}}.$
(22)
Let $y=\rho(4)R_{f}^{2}\sqrt{e^{t}-1}$, the mean achievable rate is simplified
into
$\tau_{f}=2\int_{0}^{\infty}\frac{1-e^{-y}}{y^{2}+\rho^{2}(4)R_{f}^{4}}\mathrm{d}y.$
(23)
These expressions are convenient for numerical evaluation.
## V Performance With Clustered FAPs
In this section, we derive the distributions of SINRs and the mean achievable
rates in the case when the FAPs are distributed as the Neyman-Scott cluster
process. To facilitate the analysis, we only focus on the case when the
interference links experience exponential fading.
### V-A Macrocell UEs
The following theorem gives the cdf of SINR and the mean achievable rate of
each active macrocell UE.
###### Theorem 3
In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
macrocell UE, denoted by $Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by
$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{Tv^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(T,\alpha)\bigg{)}\mathrm{d}v.$ (24)
The mean achievable rate of a macrocell UE is
$\displaystyle\tau_{m}=\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{(e^{t}-1)v^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu(e^{t}-1)v^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)\bigg{)}\mathrm{d}v\mathrm{d}t,$
(25)
where $\varphi(T,\alpha)$ is given by (15). Let $C(o,R_{c})$ be the circle
centered at the origin with radius $R_{c}$ and $\eta(s,x)$ is given as
$\eta(s,x)=\exp\Big{(}\int_{C(o,R_{c})}\frac{-\lambda_{c}^{\prime}}{1+\frac{1}{s}|x+y|^{\alpha}}\mathrm{d}y\Big{)}.$
(26)
The proof of Theorem 3 is in Appendix C.
### V-B Femtocell UEs
The following theorem gives the cdf of SINR and the mean achievable rate of
each active femtocell UE.
###### Theorem 4
In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
femtocell UE, denoted by $Z_{f}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by
$\displaystyle Z_{f}(T)=1-\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}$ $\displaystyle-\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{WTP_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta(Tr^{\alpha}W^{2},x)\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta(Tr^{\alpha}W^{2},y-z)\mathrm{d}y\bigg{)}r\mathrm{d}r.$
(27)
The mean achievable rate of a femtocell UE is given by
$\displaystyle\tau_{f}=\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{\infty}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu(e^{t}-1)r^{\alpha}\sigma^{2}}{P_{f}}-$
$\displaystyle\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{W(e^{t}-1)P_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\big{(}(e^{t}-1)r^{\alpha}W^{2},x\big{)}\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta\Big{(}(e^{t}-1)r^{\alpha}W^{2},y-z\Big{)}\mathrm{d}y\bigg{)}r\mathrm{d}r\mathrm{d}t,$
(28)
where $z=(r,0)$ and $\eta(s,x)$ is given by (26).
The proof of Theorem 4 is in Appendix D.
## VI Mean Achievable Rates of Nonsubscribers and Subscribers
There are two types of nonsubscribers, outside nonsubscribers who access MBSs
and inside nonsubscribers who access FAPs. When the number of outside
nonsubscribers in a macrocell is no greater than the total number of
subchannels (i.e., $U_{\mathrm{out}}\leq M$), each nonsubscriber UE
exclusively occupies a subchannel, and its mean achievable rate is $\tau_{m}$.
However, when $U_{\mathrm{out}}>M$, those $U_{\mathrm{out}}$ UEs share the $M$
subchannels with mean achievable rate $\frac{M}{U_{\mathrm{out}}}\tau_{m}$.
Since the evaluation is conditioned upon the existence of at least one UE, the
mean achievable rate of an outside nonsubscriber UE is given by
$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{out}}=\frac{\sum_{i=1}^{M}\mathbb{P}\\{U_{\mathrm{out}}=i\\}+\sum_{i=M+1}^{\infty}\mathbb{P}\\{U_{\mathrm{out}}=i\\}\frac{M}{i}}{1-\mathbb{P}\\{U_{\mathrm{out}}=0\\}}\tau_{m}.$
(29)
Similarly, the mean achievable rate of an inside nonsubscriber is given by
$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{in}}=\frac{\sum_{j=1}^{M_{s}}\mathbb{P}\\{U_{\mathrm{in}}=j\\}+\sum_{j=M_{s}+1}^{\infty}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\frac{M_{s}}{j}}{1-\mathbb{P}\\{U_{\mathrm{in}}=0\\}}\tau_{f}.$
(30)
When averaged over both inside and outside nonsubscribers, the overall mean
achievable rate of nonsubscriber is obtained as
$\\!\\!\\!\\!\tau_{n}=\frac{\lambda_{out}\tau_{\mathrm{out}}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}\tau_{\mathrm{in}}}{\lambda_{out}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}}.$ (31)
Regarding subscribers, note that they are exclusively served by FAPs. Similar
to the analysis of nonsubscribers, the mean achievable rate of a subscriber is
given by
$\tau_{s}=\frac{\sum_{i=1}^{M_{r}}\mathbb{P}\\{U_{s}=i\\}+\sum_{i=M_{r}+1}^{\infty}\mathbb{P}\\{U_{s}=i\\}\frac{M_{r}}{i}}{1-\mathbb{P}\\{U_{s}=0\\}}\tau_{f}.$
(32)
## VII Numerical Results
The numerical results are obtained according to both the analytical results we
have derived and Monte Carlo simulation. The default configurations of system
model are as follows (also see Table I). The total number of subchannels is
$M=20$ and the coverage radius of each femtocell is $R_{f}=10$m. The transmit
power of FAP is $P_{f}=13$dBm, and that of MBS is $P_{m}=39$dBm. We set the
path loss exponent as $\alpha=4$, with all links experiencing Rayleigh fading
of normalized $\mu=1$. The wall penetration loss is set as $W=-6$dB. We focus
on the interference-limited regime, and for simplicity we ignore the noise
power (i.e., $\sigma^{2}=0$). The intensity of MBSs is set as
$\lambda_{m}=0.00001$ and of FAPs $\lambda_{f}=0.0001$. In the clustered case,
the intensity of parent process is set as $\lambda_{p}=0.00001$ and of FAPs in
each cluster $\lambda_{c}=0.00127$. By setting the radius of each cluster as
$R_{c}=50$m, we get the intensity of FAPs as $\lambda_{f}=0.0001$. So the
average coverage area of an MBS is roughly equal to a circle with a radius of
$180$m, and on average there are ten FAPs within the coverage area of an MBS.
Unless otherwise specified, the subscribers and inside nonsubscribers are
distributed within an FAP coverage are with intensities
$\lambda_{s}=\lambda_{\mathrm{in}}=0.015$. The intensity of outside
nonsubscribers is set as $\lambda_{\mathrm{out}}=0.0001$.
TABLE I: SYSTEM PARAMETERS Symbol | Description | Value
---|---|---
$\Phi_{m},\Phi_{f}$ | Point processes defining the MBSs and FAPs | N/A
$\lambda_{m}$ | Density of MBSs | 0.00001 MBS/m2
$\lambda_{f}$ | Density of FAPs | 0.0001 FAP/m2
$\lambda_{p}$ | Density of parent process for the clustered FAPs | 0.00001 center/m2
$\lambda_{c}$ | Density of FAPs in each cluster | 0.00127 FAPs/m2
$R_{c}$ | Radius of each cluster | $50$m
$R_{f}$ | Radius of femtocell | $10$m
$\lambda_{\mathrm{out}}$ | Densities of outside nonsubscribers | 0.0001 user/m2
$\lambda_{s},\lambda_{\mathrm{in}}$ | Densities of subscribers and inside nonsubscribers | 0.015 user/m2
$P_{m},P_{f}$ | Transmit power at MBS and FAP | $39$dBm,$13$dBm
$M$ | Number of subchannels at each access point | $20$
$M_{s},M_{r}$ | Number of subchannels shared and reserved by each femtocell | Not fixed
$\alpha$ | Path loss exponent | $4$
$\mu$ | Rayleigh fading parameter | $1$ (normalized)
$W$ | Wall penetration loss | $-6$dB
$\sigma^{2}$ | Noise power | $0$ (interference-limited regime)
$P_{\mathrm{busy,m}},P_{\mathrm{busy,f}}$ | Probabilities that a given subchannel is used by an MBS and an FAP | Not fixed
$\Phi_{m}^{\prime},\Phi_{f}^{\prime}$ | Point processes defining MBSs and FAPs that interfere a given subchannel | N/A
$\lambda_{m}^{\prime},\lambda_{f}^{\prime}$ | Densities of MBSs and FAPs that interfere a given subchannel | Not fixed
$U_{\mathrm{out}},U_{\mathrm{in}}$ | Numbers of nonsubscribers that access a given MBS and a given FAP | Not fixed
$U_{s}$ | Numbers of subscribers that access a given FAP | Not fixed
$\tau_{f},\tau_{m}$ | Mean achievable rates of femtocell UEs and macrocell UEs | Not fixed
$\tau_{s},\tau_{n}$ | Mean achievable rates of subscribers and nonsubscribers | Not fixed
Figure 4 displays the SINR distributions of macrocell UEs and femtocell UEs,
when the number of shared subchannels is set as $M_{s}=10$. We plot in dotted
curves the analytical results, and we also plot the empirical cdfs obtained
from Monte Carlo simulation. The curves reveal that the simulation results
match the analytical results well, thus corroborating the accuracy of our
theoretical analysis. From the SINR distributions, we observe that while the
macrocell UEs experience a fair amount of interference, due to the shrinking
cell size, the interference for femtocell UEs is substantially alleviated. We
also observe that the performance of femtocell UEs is worse in the clustered
case than in the Poisson case when the intensity of the FAPs is set as the
same value; however, the performance of the macrocell UEs is just the reverse.
This result reveals that the gathering of FAPs leads to more interference to
femtocell UEs and at the same time reduces the chance that a macrocell UE
being interfered by the nearby FAPs.
Figure 4: Cdfs of SINRs for macrocell UEs and femtocell UEs, when $M_{s}=10$.
Figure 5 displays the mean achievable rates of macrocell UEs and femtocell UEs
as the outside nonsubscribers intensity $\lambda_{\mathrm{out}}$ increases. We
observe that the mean achievable rates of macrocell and femtocell UEs drop
initially and then tend to be stable. To interpret this behavior, we note that
as the intensity of outside nonsubscribers begins to increase, more
subchannels become occupied by MBSs, incurring more macrocell tier
interference; however, when the intensity of outside nonsubscribers is
sufficiently large, almost all the subchannels are persistently occupied by
MBSs with the UEs served by time-sharing, and then the interference saturates
thus leading to stable performance for UEs. In the case when the FAPs are
clustered, we observe that the mean achievable rate of femtocell UEs only
mildly decreases with the increasing of $\lambda_{\mathrm{out}}$, suggesting
that the performance of femtocell UEs is mainly limited by the interference
from the nearby FAPs and has little correlation with the intensity of
interfering MBSs.
Figure 5: Mean achievable rates of macrocell UEs and femtocell UEs as
functions of the intensity of outside nonsubscribers $\lambda_{\mathrm{out}}$.
Figure 6 displays the mean achievable rates of nonsubscribers and subscribers
as functions of the number of shared subchannels, in which the rate of
nonsubscribers is averaged over both outside and inside nonsubscribers. When
few subchannels are to be shared by each FAP (i.e., small $M_{s}$), the mean
achievable rate of nonsubscribers is small while subscribers enjoy a good
spectral efficiency. On the contrary, when most of the subchannels are shared
to nonsubscribers, the mean achievable rate of subscribers deteriorates
seriously. Nevertheless, we observe from the figure that there exists a stable
compromise at which the rates of both subscribers and nonsubscribers do not
drop much from their maxima. For the default configuration in our numerical
study, the stable compromise in the PPP case as well as in the clustered case
occurs when the value of $M_{s}$ lies in the range of $[7,12]$, corresponding
to a reasonably wide tuning range for system designer when provisioning the
resource.
Figure 6: Performance of nonsubscribers and subscribers as a function of the
number of shared subchannals $M_{s}$.
Figure 7 displays the mean achievable rates of nonsubscribers and subscribers
with different proportions of inside nonsubscribers and subscribers in
femtocells. For a fair comparison, we fix the sum intensity of the two types
of femtocell UEs, as $\lambda_{s}+\lambda_{\mathrm{in}}=0.03$. Figure 7(a)
gives the performance in the case when most of the femtocell UEs are
nonsubscribers, while Figure 7(b) corresponds to the opponent case. From the
curves, we observe that in order to achieve a good performance for both types
of UEs, the number of shared subchannels $M_{s}$ should be adjusted based on
the intensity of inside nonsubscribers. Moreover, we also find that the tuning
range of $M_{s}$ in the PPP case is almost the same as that in the clustered
case; thus illustrating that it is the load of the network rather than the
spatial distribution of FAPs that contributes the major impact on the choice
of $M_{s}$.
(a) $\lambda_{s}=0.003$ and $\lambda_{\mathrm{in}}=0.027$.
(b) $\lambda_{s}=0.027$ and $\lambda_{\mathrm{in}}=0.003$.
Figure 7: Performance of nonsubscribers and subscribers with changing
proportions of inside nonsubscribers and subscribers.
## VIII Conclusions
In this paper, we explored the application of stochastic geometry in the
analysis of hybrid access mechanisms for multi-channel two-tier networks,
focusing on the evaluation of the tradeoff between nonsubscribers and
subscribers. We characterized several key statistics of UEs and subchannels,
and established SINR distribution and mean achievable rate for each type of
UEs.
Our analysis revealed the interaction among the various parameters in the
network model, and thus shed useful insights into the choice of network
parameters and the provisioning of resource in system design. From our
numerical study, we observe that although there is an apparent conflict
between the interests of nonsubscribers and subscribers, there usually exists
a reasonably wide tuning range over which nonsubscribers and subscribers
attain a stable compromise at which the rates of both subscribers and
nonsubscribers do not drop much from their maxima. We also found that although
the spatial distributions of FAPs are different, the tuning range is almost
the same when the intensities of different types of UEs are fixed.
## Appendix A
Assume that the typical marcocell UE is located at the origin $o$, and let $r$
be the distance between it and its serving MBS. Since an outside nonsubscriber
always chooses its nearest MBS to access, the cdf of $r$ is obtained by
$\displaystyle\mathbb{P}\\{r\leq R\\}$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\\{\mathrm{no\ MBS\ closer\ than\ }R\\}$ (33) $\displaystyle=$
$\displaystyle 1-e^{-\lambda_{m}\pi R^{2}}.$
Then, the pdf of $r$ is $f(r)=e^{-\lambda_{m}\pi r^{2}}2\pi\lambda_{m}r$.
Assuming that the considered UE is at distance $r$ from its serving MBS,
$g_{\cdot}$ is the fading of an interference link, and $R_{\cdot}$ is the
distance between the UE and an interfering access point, the SINR experienced
by the UE is $\mathrm{SINR}=\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}$,
where
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
is the interference from the macrocell tier (excluding the serving MBS itself
which is denoted by $b_{0}$) and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$ is the
interference from the femtocell tier with the wall penetration loss taken into
account.
Thus, the cdf of the SINR is given by
$\displaystyle Z_{m}(T)=1-\mathbb{P}\\{\mathrm{SINR}>T\\}$ (34)
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}>T\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}h>\frac{Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{E}\Big{\\{}e^{-\frac{\mu
Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})}\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{m}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{m}}\Big{)}\mathrm{d}r.$
Now, we evaluate the Laplace transform for the interference conditioning on
the fact that the typical UE is served by the nearest MBS $b_{0}$ which is at
the distance $r$. First, we derive the Laplace transform for the interference
of MBSs, denoted as $I_{m}$.
$\displaystyle\mathcal{L}_{I_{m}}(s)$ $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{\\{}\exp(-s\\!\\!\\!\\!\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!P_{m}g_{i}R_{i}^{-\alpha})\Big{\\}}$
(35) $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!E_{g_{i}}\\{\exp(-sg_{i}R_{i}^{-\alpha}P_{m})\\}\Big{)}$
$\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!\mathcal{L}_{g}(sR_{i}^{-\alpha}P_{m})\Big{)},$
The probability generating functional of PPP $\Phi$ in the region $D$ with
intensity $\lambda$, denoted by
$G_{p}(v)=\mathbb{E}\\{\prod_{x\in\Phi}v(x)\\}$, is given by [19] as follows
$G_{p}(v)=\exp\Big{(}-\lambda\int_{D}(1-v(x))\mathrm{d}x\Big{)}.$ (36)
Let $C(o,r)$ be the circle centered at the origin $o$ with radius $r$. As
there is no MBS in the circle $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. This implies that the interfering MBSs
are distributed as PPP on the space $R^{2}$ exclusive of the region $C(o,r)$.
Let $D=R^{2}\setminus C(o,r)$ and $v(x)=\mathcal{L}_{g}(s|x|^{-\alpha}P_{m})$,
by applying the probability generating functional of PPP we get
$\displaystyle\mathcal{L}_{I_{m}}(s)=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\int_{r}^{\infty}(1-\mathcal{L}_{g}(sx^{-\alpha}P_{m}))x\mathrm{d}x\Big{)}$
$\displaystyle=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\underbrace{\int_{0}^{\infty}\\!\\!\int_{r}^{\infty}\\!\\!(1-e^{-sx^{-\alpha}P_{m}g})f(g)x\mathrm{d}x\mathrm{d}g}_{(A)}\Big{)}.$
(37)
where the last equation follows from the exchange of the integral order.
Let $y=sP_{m}gx^{-\alpha}$ and integrate by parts, from the properties of
Gamma function we obtain
$\displaystyle(A)=-\frac{r^{2}}{2}+\frac{1}{\alpha}(sP_{m})^{\frac{2}{\alpha}}\mathbb{E}_{g}\Big{\\{}$
$\displaystyle
g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},sP_{m}gr^{-\alpha})-\Gamma(-\frac{2}{\alpha})\Big{)}\Big{\\}}.$
(38)
The evaluation of the Laplace transform for $I_{f}$ is almost the same except
that the interfering FAPs are distributed on the whole space $R^{2}$. Similar
to (37), we get
$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\exp\Big{(}-2\pi\lambda_{f}^{\prime}\int_{0}^{\infty}\\!\\!(1-\mathcal{L}_{g}(sx^{-\alpha}WP_{f}))x\mathrm{d}x\Big{)}$
(39) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}(sWP_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}\Big{\\{}g^{\frac{2}{\alpha}}\Big{\\}}.$
Substituting $\mathcal{L}_{I_{m}}(s)$ and $\mathcal{L}_{I_{f}}(s)$ into
$Z_{m}(T)$ hence leads to (10).
Now we evaluate the mean achievable rate. Since for a positive random variable
$X$, $\mathbb{E}\\{X\\}=\int_{t>0}\mathbb{P}\\{X>t\\}\mathrm{d}t$, we have
$\displaystyle\tau_{m}$ $\displaystyle=$
$\displaystyle\mathbb{E}\\{\ln(1+\mathrm{SINR})\\}$ (40) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\ln(1+\mathrm{SINR})>t\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\mathrm{SINR}>e^{t}-1\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}(1-Z_{m}(e^{t}-1))\mathrm{d}t.$
Plugging $Z_{m}(T)$ into (40), we arrive at (11) and thus establish Theorem 1.
## Appendix B
Assume that the typical UE is located at the origin $o$. Let $r$ be the
distance between a femtocell UE and its serving FAP. Because the femtocell UEs
are uniformly distributed in the circular coverage area of radius $R_{f}$ of
each FAP, the pdf of $r$ is given by $f(r)={2r}/{R_{f}^{2}}$.
Denote by $I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$
the interference strengths from MBSs and FAPs respectively. Similar to the
derivation of (34), we have
$Z_{f}(T)=1-\int_{0}^{R_{f}}\frac{2r}{R_{f}^{2}}e^{-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{f}}\Big{)}\mathrm{d}r.$ (41)
Since $\Phi_{m}^{\prime}$ is a homogeneous PPP with intensity
$\lambda_{m}^{\prime}$, we obtain the Laplace transform for $I_{m}$ similar as
the derivation of (39)
$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}(sWP_{m})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(42)
The FAPs are distributed as a homogeneous PPP; however, the serving FAP is not
included when calculating the interference. By the Slivnyak-Mecke Theorem, the
reduced Palm distribution of the Poisson p.p. is equal to its original
distribution. Thus, the Laplace transform for $I_{f}$ can still be obtained
similar as the derivation of (39)
$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{f}}\big{\\}}=\exp\Big{(}-\pi\lambda_{f}^{\prime}(sW^{2}P_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(43)
In the above, it is noteworthy that the interference from an FAP penetrates
two walls thus the loss becoming $W^{2}$ instead of $W$. Substituting the
Laplace transform for $I_{m}$ and $I_{f}$ into (41) with $v=r^{2}$, we get the
SINR distribution, and similar to (40), we get the mean achievable rate.
## Appendix C
The derivation is exactly the same as Appendix A till the equation (34). The
essential distinction of the derivation lies in that the Laplace transform for
the interference is different from that in the PPP case. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
and $I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$. Since
there is no MBS in the disk $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. Referring to the previous derivation in
the PPP case, we obtain the Laplace transform for $I_{m}$ as follows
$\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}r^{2}\varphi\Big{(}\frac{sP_{m}}{\mu
r^{\alpha}},\alpha\Big{)}\Big{)}$ (44)
The FAPs are distributed as a Neyman-Scott cluster process and the generating
functional $G(v)=\mathbb{E}(\prod_{x\in\Phi}v(x))$ is given by [19, Page 157]
$\displaystyle G(v)=\exp\Big{(}-\lambda_{p}\int_{R^{2}}\Big{(}1-$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x+y))\mathrm{d}y\Big{)}\Big{)}\mathrm{d}x\Big{)}$
(45)
Similar to the derivation of (35), we get the Laplace transform for $I_{f}$
$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\mathcal{L}_{g}(sR_{j}^{-\alpha}WP_{f})\Big{\\}}$
(46) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\frac{\mu}{\mu+sR_{j}^{-\alpha}WP_{f}}\Big{\\}}$
Let $v(x)=\frac{\mu}{\mu+sP_{f}W|x|^{-\alpha}}$ and plugging into the
generating functional of Neyman-Scott cluster process (45), we get the Laplace
transform for $I_{f}$. Having derived the Laplace transform for $I_{m}$ and
$I_{f}$, similar to the derivations in Appendix A, we obtain the results in
Theorem 3.
## Appendix D
Different from the above proofs, we assume that the serving FAP rather than
the typical UE is located at the origin. The typical UE is distributed in the
circle centered at the origin with radius $R_{f}$. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$.
First, we evaluate the Laplace transform for the interference conditioned on
the fact that the typical UE is located at distance $r$ from the serving FAP
located at the origin. Without loss of generality, we assume that the typical
UE is located at $z=(r,0)$. Since $\Phi_{m}^{\prime}$ is a homogeneous PPP
with intensity $\lambda_{m}^{\prime}$, we obtain the Laplace transform for
$I_{m}$ similar as the derivation of (39)
$\\!\\!\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}\Big{(}\frac{sWP_{m}}{\mu}\Big{)}^{\frac{2}{\alpha}}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})\Big{)}$
(47)
The FAPs are distributed as a Neyman-Scott cluster process. Since the serving
FAP, located at the origin, is not included when calculating the interference,
we should consider the reduced Palm distribution of the cluster process when
evaluating the Laplace transform for $I_{f}$. Let
$G_{o}^{!}(v)=\mathbb{E}_{o}^{!}(\prod_{x\in\Phi}v(x))$ denotes the generating
functional of the reduced Palm distribution of the cluster process. The
notation $\mathbb{E}_{o}^{!}(\cdot)$ denotes the conditional expectation for
the point process given that there is a point of the process at the origin but
without including the point. The conditional generating functional
$G_{o}^{!}(v)$ is given by the Lemma 1 in [17] as follows
$\displaystyle G_{o}^{!}(v)=\frac{1}{\pi R_{c}^{2}}G(v)\int_{C(o,R_{c})}$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x-y))\mathrm{d}x\Big{)}\mathrm{d}y$
(48)
where $G(v)$ is the generating functional of Neyman-Scott cluster process and
is given by (45).
Referring to (46), let $v(x)=\frac{\mu}{\mu+sP_{f}W|x-z|^{-\alpha}}$. Plugging
$v(x)$ into the conditional generating functional of Neyman-Scott cluster
process (48), we get the Laplace transform for $I_{f}$. Having derived the
Laplace transform for $I_{m}$ and $I_{f}$, similar to the derivation in
Appendix B, we obtain the results in Theorem 4.
## Acknowledgement
The authors wish to thank the anonymous reviewers for their valuable comments
on this work.
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|
# Some results on space-like self-shrinkers
Huaqiao Liu and Y. L. Xin School of Mathematics and Information Sciences,
Henan University, Kaifeng 475004, China<EMAIL_ADDRESS>Institute of
Mathematics, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
###### Abstract.
We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space
$\mathbb{R}^{m+n}_{m}$with index $m$. We derive drift Laplacian of the basic
geometric quantities and obtain their volume estimates in pseudo-distance
function. Finally, we prove rigidity results under minor growth conditions in
terms of the mean curvature or the image of Gauss maps .
###### 1991 Mathematics Subject Classification:
58E20,53A10.
The authors are supported partially by NSFC
## 1\. Introduction
Let $\mathbb{R}^{m+n}_{m}$ be an $(m+n)$-dimensional pseudo-Euclidean space
with the index $m$. The indefinite flat metric on $\mathbb{R}^{m+n}_{m}$ is
defined by
$ds^{2}=\sum_{i=1}^{n}(dx^{i})^{2}-\sum_{\alpha=n+1}^{m+n}(dx^{\alpha})^{2}.$
In what follows we agree with the following range of indices
$A,\,B,\,C,\cdots=1,\cdots,m+n;\,i,\,j,\,k\cdots=1,\cdots,n;$
$s,t=1,\,\cdots,m;\,\alpha,\,\beta,\cdots=n+1,\cdots,m+n.$
Let $F:M\to\mathbb{R}^{m+n}_{m}$ be a space-like $n$-dimensional submanifold
in $\mathbb{R}^{m+n}_{m}$ with the second fundamental form $B$ defined by
$B_{XY}\mathop{=}\limits^{def.}\left(\bar{\nabla}_{X}Y\right)^{N}$ for
$X,Y\in\Gamma(TM)$. We denote $(\cdots)^{T}$ and $(\cdots)^{N}$ for the
orthogonal projections into the tangent bundle $TM$ and the normal bundle
$NM$, respectively. For $\nu\in\Gamma(NM)$ we define the shape operator
$A^{\nu}:TM\to TM$ by $A^{\nu}(V)=-(\bar{\nabla}_{V}\nu)^{T}.$ Taking the
trace of $B$ gives the mean curvature vector $H$ of $M$ in
$\mathbb{R}^{m+n}_{m}$ and
$H\mathop{=}\limits^{def.}\text{trace}(B)=B_{e_{i}e_{i}},$ where $\\{e_{i}\\}$
is a local orthonormal frame field of $M.$ Here and in the sequel we use the
summation convention. The mean curvature vector is time-like, and a cross-
section of the normal bundle.
We now consider a one-parameter family $F_{t}=F(\cdot,t)$ of immersions
$F_{t}:M\to\mathbb{R}^{m+n}_{m}$ with the corresponding images
$M_{t}=F_{t}(M)$ such that
(1.1) $\begin{split}\frac{d\,}{d\,t}F(x,t)&=H(x,t),\qquad x\in M\\\
F(x,0)&=F(x)\end{split}$
are satisfied, where $H(x,t)$ is the mean curvature vector of $M_{t}$ at
$F(x,t).$ There are many interesting results on mean curvature flow on space-
like hypersurfaces in certain Lorentzian manifolds [10, 11, 12, 13]. For
higher codimension we refer to the previous work of the second author [19].
A special but important class of solutions to (1.1) are self-similar shrinking
solutions, whose profiles, space-like self-shrinkers, satisfy a system of
quasi-linear elliptic PDE of the second order
(1.2) $H=-\frac{X^{N}}{2}.$
Besides the Lagrangian space-like self-shrinkers [2, 14, 9], there is an
interesting paper on curves in the Minkowski plane [15]. The present paper is
devoted to general situation on space-like self-shrinker.
For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ we have the
Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$. The target manifold is a pseudo-
Grassmann manifold, dual space of the Grassmann manifold $\mathbb{G}_{n,m}$.
In the next section we will describe its geometric properties, which will be
used in the paper.
Choose a Lorentzian frame field $\\{e_{i},e_{\alpha}\\}$ in
$\mathbb{R}^{m+n}_{m}$ with space-like $\\{e_{i}\\}\in TM$ and time-like
$\\{e_{\alpha}\\}\in NM$ along the space-like submanifold
$F:M\to\mathbb{R}^{m+n}_{m}$. Define coordinate functions
$x^{i}=\left<F,e_{i}\right>,\;y^{\alpha}=-\left<F,e_{\alpha}\right>.$
We then have
$|F|^{2}=X^{2}-Y^{2},$
where $X=\sqrt{\sum_{i=1}^{n}(x^{i})^{2}},\quad
Y=\sqrt{\sum_{\alpha=n+1}^{m+n}(y^{\alpha})^{2}}.$ We call $|F|^{2}$ the
pseudo-distance function from the origin $0\in M$.
We always put the origin on $M$ in the paper. We see that $|F|^{2}$ is
invariant under the Lorentzian action up to the choice of the origin in
$\mathbb{R}^{m+n}_{m}$. Set $z=|F|^{2}$. It has been proved that $z$ is proper
provided $M$ is closed with the Euclidean topology (see [4] for $m=1$ and [16]
for any codimension $m$).
Following Colding and Minicozzi [6] we can also introduce the drift Laplacian,
(1.3) $\displaystyle\mathcal{L}=\Delta-\frac{1}{2}\langle
F,\nabla(\cdot)\rangle=e^{\frac{z}{4}}div(e^{-\frac{z}{4}}\nabla(\cdot)).$
It can be showed that $\mathcal{L}$ is self-adjoint with respect to the
weighted volume element $e^{-\frac{z}{4}}d\mu,$ where $d\mu$ is the volume
element of $M$ with respect to the induced metric from the ambient space
$\mathbb{R}^{m+n}_{m}$. In the present paper we carry out integrations with
respect to this measure. We denote $\rho=e^{-\frac{z}{4}}$ and the volume form
$d\mu$ might be omitted in the integrations for notational simplicity.
For a space-like submanifold in $\mathbb{R}^{m+n}_{m}$ there are several
geometric quantities. The squared norm of the second fundamental form
$|B|^{2}$, the squared norm of the mean curvature $|H|^{2}$ and the
$w-$function, which is related to the image of the Gauss map. In §3 we will
calculate drift Laplacian $\mathcal{L}$ of those quantities, see Proposition
3.1.
Corresponding to the weighted measure and drift Laplacian there is so-called
the Baker-Emery Ricci tensor. It is noted that in [3]
$\text{Ric}_{f}\geq\frac{z}{4}$ with $f=\frac{z}{4}$. Using the comparison
technique the weighted volume of the geodesic ball can be estimated from above
in terms of the distance function [18].
For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ there are 3
kind global conditions: Closed one with Euclidean topology; entire graph;
complete with induced Riemannian metric. A complete space-like one has to be
entire graph, but the converse claim is not always the case. Closed one with
Euclidean topologg is complete under the parallel mean curvature assumption
(see [4] for codimension one and [16] for higher codimension).
In our case of closed one with Euclidean topology, the pseudo-distance
function $z$ is always proper. It is natural to consider the volume growth in
$z$. For the proper self-shrinkers in Euclidean space Ding-Xin [8] gave the
volume estimates. It has been generalized in [5] for more general situation.
But, the present case does not satisfy the conditions in Theorem 1.1 in [5].
However, the idea in [8] is still applicable for space-like self-shrinkers. In
§4 we will give volume estimates for space-like self-shrinkers, in a similar
manner as in [8], see Theorem 4.1.
Finally, using integral method we can obtain rigidity results as follows.
###### Theorem 1.1.
Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.
###### Theorem 1.2.
Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.
###### Remark 1.1.
In the special situation, for the Lagrangian space-like self-shrinkers, the
rigidity results hold without the growth condition (see [9]). Let ${\tenmsb
R}^{2n}_{n}$ be Euclidean space with null coordinates
$(x,y)=(x_{1},\cdots,x_{n};\,y_{1},\cdots,y_{n})$, which means that the
indefinite metric is defined by $ds^{2}=\sum_{i}dx_{i}dy_{i}.$ If
$M=\\{(x,Du(x))\big{|}\ x\in{\tenmsb R}^{n}\\}$ is a space-like submanifold in
${\tenmsb R}^{2n}_{n}$, then $u$ is convex and the induced metric on $M$ is
given by $ds^{2}=\sum_{i,j}u_{ij}dx_{i}dx_{j}$. $M$ is a space-like Lagrangian
submanifold in $\mathbb{R}^{2n}_{n}$. It is worthy to point out that if $M$ is
entire gradient graph the potential function $u$ is proper, as the following
consideration. On $\mathbb{R}^{n}$ set $\rho=|x|=\sqrt{\sum x_{i}^{2}}$. At
any direction $\theta\in S^{n-1}$
$u_{i}=u_{\rho}\frac{\partial\rho}{\partial x_{i}}=\frac{x_{i}}{\rho}u_{\rho}$
and the pseudo-distance
$z=x_{i}u_{i}=\rho u_{\rho},$
which is positive when the origin is on $M$, since it is space-like. It
implies that $u$ is increasing in $\rho$. Moreover,
$z_{\rho}=u_{\rho}+\rho u_{\rho\rho}>0,$
which means that $z$ is also increasing in $\rho$. Hence,
$u(\rho)-u(\epsilon)=\int_{\epsilon}^{\rho}u_{\rho}d\rho=\int_{\epsilon}^{\rho}\frac{z}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\to\infty$
as $\rho\to\infty$.
###### Remark 1.2.
Rigidity problem for space-like extremal submanifolds was raised by E. Calabi
[1], and solved by Cheng-Yau [4] for codimension $1$. Later, Jost-Xin
generalized the results to higher codimension [16]. The rigidity problem for
space-like submanifolds with parallel mean curvature was studied in [20][22]
and [16] (see also in Chap. 8 of [21]).
## 2\. Geometry of $G_{n,m}^{m}$
In $\mathbb{R}^{n+m}_{m}$ all space-like $n-$subspaces form the pseudo-
Grassmannian $G^{m}_{n,m}.$ It is a specific Cartan-Hadamard manifold which is
the noncompact dual space of the Grassmann manifold $G_{n,m}.$
Let $P$ and $A\in G_{n,m}^{m}$ be two space-like $n-$plane in $R_{m}^{m+n}.$
The angles between $P$ and $A$ are defined by the critical values of angel
$\theta$ between a nonzero vector $x$ in P and its orthogonal projection
$x^{*}$ in $A$ as $x$ runs through $P$.
Assume that $e_{1},\cdots,e_{n}$ are orthonormal vectors which span the space-
like $P$ and $a_{1},\cdots,a_{n}$ for space-like $A.$ For a nonzero vector in
$P$
$x=\sum_{i}x_{i}e_{i},$
its orthonormal projections in $A$ is
$x^{*}=\sum_{i}x_{i}^{*}a_{i}.$
Thus, for any $y\in A,$ we have
$\langle x-x^{*},y\rangle=0.$
Set
$W_{i\,j}=\langle e_{i},a_{j}\rangle,$
We then have
$x_{j}^{*}=\sum_{i}W_{i\,j}x_{i}.$
Since $x$ is a vector in a space-like $n-$plane and its projection $x^{*}$ in
$A$ is also a space-like vector. We then have a Minkowski plane $R_{1}^{2}$
spanned by $x$ and $x^{*}.$ Then angle $\theta$ between $x$ and $x^{*}$ is
defined by
$\cosh\theta=\frac{\langle x,x^{*}\rangle}{|x||x^{*}|}.$
Let
$W=(W_{i\,j})=\left(\begin{array}[]{lll}\langle
e_{1},a_{1}\rangle&\cdots&\langle e_{n},a_{1}\rangle\\\ \quad\vdots&\
\vdots&\quad\vdots\\\ \langle e_{n},a_{1}\rangle&\cdots&\langle
e_{n},a_{n}\rangle\end{array}\right)$
Now define the $w-$function as
$w=\langle e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle=\det W.$
$W^{T}W$ is symmetric, its eigenvalues are $\mu_{1}^{2},\cdots,\mu_{n}^{2},$
then there exist $e_{1},\cdots,e_{n}$ in $P$, such that
$W^{T}W=\left(\begin{array}[]{lll}\mu_{1}^{2}&&0\\\ &\ddots&\\\
0&&\mu_{n}^{2}\end{array}\right),$
in which $\mu_{i}\geq 1$ and $\mu_{i}=\cosh\theta_{i}.$ Then
(2.1)
$w=\prod_{i}\cosh\theta_{i}=\prod_{i}\frac{1}{\sqrt{1-\lambda_{i}^{2}}},\;\lambda_{i}=\tanh\theta_{i}.$
The distance between $P$ and $A$ in the canonical Riemannian metric on
$\mathbb{G}_{n,m}^{m}$ is (see [17] for example)
$d(P,A)=\sqrt{\sum_{i}\theta_{i}^{2}}.$
For the fixed $A\in G_{n,m}^{m},$ which is spanned by $\\{a_{i}\\}$, choose
time-like $\\{a_{n+s}\\}$ such that $\\{a_{i},a_{n+s}\\}$ form an orthonormal
Lorentzian bases of $R^{n+m}_{m}.$
Set
$\displaystyle e_{i}$
$\displaystyle=\cosh\theta_{i}a_{i}+\sinh\theta_{i}a_{n+i}$ $\displaystyle
e_{n+i}$
$\displaystyle=\sinh\theta_{i}a_{i}+\cosh\theta_{i}a_{n+i}\,(\;\text{and}\;e_{n+\alpha}=a_{n+\alpha}\;\text{if}\;m>n).$
Then $e_{i}\in T_{p}M,e_{n+i}\in N_{p}M.$ In this case
(2.2) $\displaystyle w_{i\,\alpha}$ $\displaystyle=$ $\displaystyle\langle
e_{1}\wedge\cdots\wedge e_{i-1}\wedge e_{\alpha}\wedge e_{i+1}\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ (2.3) $\displaystyle=$
$\displaystyle\cosh\theta_{1}\cosh\theta_{i-1}\sinh\theta_{i}\cosh\theta_{i+1}\cosh\theta_{n}=\lambda_{i}w\delta_{n+i\
\alpha},$
which is obtained by replacing $e_{i}$ by $e_{\alpha}$ in $w$. We also have
$w_{i\alpha j\beta}$ by replacing $e_{j}$ by $e_{\beta}$ in $w_{i\,\alpha}.$
We obtain
(2.7) $\displaystyle w_{i\alpha
j\beta}=\left\\{\begin{array}[]{lll}\lambda_{i}\lambda_{j}w&&\alpha=n+i,\beta=n+j\\\
-\lambda_{i}\lambda_{j}w&&\alpha=n+j,\beta=n+i\\\
0&&otherwise.\end{array}\right.$
## 3\. Drift Laplacian of some geometric quantities
The second fundamental form $B$ can be viewed as a cross-section of the vector
bundle Hom($\odot^{2}TM,NM$) over $M.$ A connection on Hom($\odot^{2}TM,NM$)
can be induced from those of $TM$ and $NM$ naturally. There is a natural fiber
metric on Hom($\odot^{2}TM,NM$) induced from the ambient space and it becomes
a Riemannian vector bundle. There is the trace-Laplace operator $\nabla^{2}$
acting on any Riemannian vector bundle.
In [19] we already calculate $\nabla^{2}B$ for general space-like
$n-$submanifolds in $\mathbb{R}^{m+n}_{m}$.
Set
$B_{i\,j}=B_{e_{i}\,e_{j}}=h_{i\,j}^{\alpha}e_{\alpha},\;S_{\alpha\,\beta}=h_{i\,j}^{\alpha}h_{i\,j}^{\beta}.$
From Proposition 2.1 in [19] we have
(3.1)
$\displaystyle\langle\nabla^{2}B,B\rangle=\langle\nabla_{i}\nabla_{j}H,B_{i\,j}\rangle+\langle
B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S_{\alpha\,\beta}^{2},$
where $R^{\perp}$ denotes the curvature of the normal bundle and
$|R^{\perp}|^{2}=-\langle
R_{e_{i}\,e_{j}}\nu_{\alpha},R_{e_{i}\,e_{j}}\nu_{\alpha}\rangle.$
Then from the self-shrinker equation (1.2) we obtain
$\displaystyle\nabla_{i}F^{N}$ $\displaystyle=$
$\displaystyle[\bar{\nabla}_{i}(F-\langle F,e_{j}\rangle e_{j})]^{N}$
$\displaystyle=$ $\displaystyle[e_{i}-\bar{\nabla}_{i}\langle F,e_{j}\rangle
e_{j}-\langle F,e_{j}\rangle\bar{\nabla}_{e_{i}}e_{j}]^{N}$ $\displaystyle=$
$\displaystyle-\langle F,e_{j}\rangle B_{i\,j},$
and
$\displaystyle\nabla_{i}\nabla_{j}F^{N}$ $\displaystyle=$
$\displaystyle-\nabla_{i}[\langle F,e_{k}\rangle B_{k\,j}]$ $\displaystyle=$
$\displaystyle-\delta_{i}^{k}B_{k\,j}-\langle F^{N},B_{k\,i}\rangle
B_{k\,j}-\langle F,e_{k}\rangle\nabla_{i}B_{k\,j}$ $\displaystyle=$
$\displaystyle-B_{i\,j}-\langle F^{N},B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}$ $\displaystyle=$ $\displaystyle-
B_{i\,j}+\langle 2H,B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$
Set $P_{i\,j}=\langle B_{i\,j},H\rangle,$ then
(3.2)
$\displaystyle\nabla_{i}\nabla_{j}H=\frac{1}{2}B_{i\,j}-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$
Substituting (3.2) into (3.1)we obtain
$\displaystyle\langle\nabla^{2}B,B\rangle$ $\displaystyle=$
$\displaystyle\langle\frac{1}{2}B_{i\,j},B_{i\,j}\rangle-\langle
H,B_{k\,i}\rangle\langle B_{k\,j},B_{i\,j}\rangle+\frac{1}{2}\langle
F,e_{k}\rangle\langle\nabla_{k}B_{i\,j},B_{i\,j}\rangle$
$\displaystyle+\langle B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$
This also means that
(3.3) $\displaystyle\langle\nabla^{2}B,B\rangle=\frac{1}{2}\langle
B,B\rangle+\frac{1}{4}\langle F^{T},\nabla\langle
B,B\rangle\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$
Note that $\Delta\langle
B,B\rangle=2\langle\nabla^{2}B,B\rangle+2\langle\nabla B,\nabla B\rangle,$ so
(3.4) $\displaystyle\Delta\langle B,B\rangle$ $\displaystyle=\langle
B,B\rangle+\frac{1}{2}\langle F^{T},\nabla\langle
B,B\rangle\rangle-2|R^{\perp}|^{2}-2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}$
$\displaystyle+2\langle\nabla B,\nabla B\rangle.$
We denote
$|B|^{2}=-\langle B,B\rangle=\sum_{i,j,\alpha}h_{\alpha ij}^{2},\;|\nabla
B|^{2}=-\langle\nabla B,\nabla B\rangle.$ $|H|^{2}=-\langle
H,H\rangle,\;|\nabla H|^{2}=-\langle\nabla H,\nabla H\rangle$
then
(3.5) $\displaystyle\Delta|B|^{2}=|B|^{2}+\frac{1}{2}\langle
F^{T},\nabla|B|^{2}\rangle+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2}$
From (3.2) we also obtain
$\displaystyle\nabla^{2}H=\frac{1}{2}H-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H.$
Since
$\displaystyle\Delta|H|^{2}=-\Delta\langle
H,H\rangle=-2\langle\nabla^{2}H,H\rangle-2\langle\nabla H,\nabla H\rangle,$
we obtain
(3.6) $\displaystyle\Delta|H|^{2}$
$\displaystyle=-2\langle\frac{1}{2}H-P_{k\,i}B_{k\,i}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H,H\rangle-2\langle\nabla H,\nabla H\rangle$
$\displaystyle=|H|^{2}+2|P|^{2}+\frac{1}{2}\langle
F^{T},\nabla|H|^{2}\rangle+2|\nabla H|^{2},$
where $|P|^{2}=\sum_{i,j}P_{ij}^{2}$.
In the pseudo-Grassmann manifold $\mathbb{G}_{n,m}^{m}$ there are
$w-$functions with respect to a fixed point $A\in\mathbb{G}_{n,m}^{m}$, as
shown in §2. For the space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$
we define the Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$, which is obtained
by parallel translation of $T_{p}M$ for any $p\in M$ to the origin in
$\mathbb{R}^{m+n}_{m}$. Then, we have functions $w\circ\gamma$ on $M$, which
is still denoted by $w$ for notational simplicity.
For any point $p\in M$ around $p$ there is a local tangent frame field
$\\{e_{i}\\}$, and which is normal at $p$. We also have a local orthonormal
normal frame field $\\{e_{\alpha}\\}$, and which is normal at $p$. Define a
$w-$function by
$w=\left<e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\right>,$
where $\\{a_{i}\\}$ is a fixed orthonormal vectors which span a fixed space-
like $n-$plane $A$. Denote
$e_{i\,\alpha}=e_{1}\wedge\cdots\wedge e_{\alpha}\wedge\cdots\wedge e_{n},$
which is got by substituting $e_{\alpha}$ for $e_{i}$ in
$e_{1}\wedge\cdots\wedge e_{n}$ and $e_{i\alpha j\beta}$ is obtained by
substituting $e_{\beta}$ for $e_{j}$ in $e_{i\,\alpha}.$ Then
(3.7) $\displaystyle\nabla_{e_{j}}w$ $\displaystyle=\sum_{i=1}^{n}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{j}}e_{i}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}\langle e_{1}\wedge\cdots\wedge B_{i\,j}\cdots
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle e_{1}\cdots\wedge
e_{\alpha}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle
e_{i\,\alpha},a_{1}\wedge\cdots\wedge a_{n}\rangle.$
Furthermore,
(3.10) $\displaystyle\nabla_{e_{i}}\nabla_{e_{j}}w$ $\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}(e_{1}\wedge\cdots\wedge
e_{n}),a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle
e_{1}\wedge\cdots\wedge\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge\bar{\nabla}_{e_{i}}e_{l}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle e_{1}\wedge\cdots\wedge
B_{j\,k}\wedge\cdots\wedge B_{il}\wedge\cdots e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{T}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
Note that
$\displaystyle(\ref{3.13})$ $\displaystyle=$ $\displaystyle\sum_{k\neq
l}h_{j\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle(\ref{3.14})$
$\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k},e_{k}\rangle
w=-\langle\bar{\nabla}_{j}e_{k},\bar{\nabla}_{i}e_{k}\rangle w=-\langle
B_{j\,k},B_{i\,k}\rangle w=h_{j\,k}^{\alpha}h_{i\,k}^{\alpha}w$
$\displaystyle(\ref{3.15})$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}(B_{j\,k}+\nabla_{e_{j}}e_{k}))^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle\nabla_{i}B_{j\,k},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge
a_{n}\rangle=-\langle\nabla_{k}B_{i\,j},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$
where we use the Codazzi equation in the last step. Thus, we obtain
$\Delta w=\sum_{i,k\neq l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{k\beta
l}^{\alpha},a_{1}\wedge\cdots\wedge
a_{n}\rangle+|B|^{2}w-\langle\nabla_{k}H,e_{\alpha}\rangle\langle e_{\alpha
k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$
Since
$\nabla_{i}F^{N}=-\langle F,e_{j}\rangle B_{i\,j},$
from (1.2), we obtain
(3.11) $\displaystyle\nabla_{i}H$ $\displaystyle=\frac{1}{2}\langle
F,e_{j}\rangle B_{i\,j}$ $\displaystyle\langle\nabla_{i}H,e_{\alpha}\rangle$
$\displaystyle=-\frac{1}{2}\langle F,e_{j}\rangle h_{i\,j}^{\alpha},$
so,
(3.12) $\displaystyle\Delta w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,e_{i}\rangle
h_{k\,i}^{\alpha}\langle e_{\alpha k},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,\nabla w\rangle,$
where (3.7) has been used in the last equality.
###### Proposition 3.1.
For a space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
we have
(3.13)
$\mathcal{L}|B|^{2}=|B|^{2}+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2},$ (3.14) $\mathcal{L}|H|^{2}=|H|^{2}+2|P|^{2}+2|\nabla H|^{2},$ (3.15)
$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}.$
###### Proof.
From (1.3,3.5,3.6), we can obtain (3.13) and (3.14) easily.
From (1.3,3.12) we have
(3.16) $\displaystyle\mathcal{L}w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}w_{\alpha k\beta l}$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})w,$
Furthermore, since
$\mathcal{L}(\ln w)=\frac{1}{w}\mathcal{L}w-\frac{|\nabla w|^{2}}{w^{2}},$
we obtain
$\mathcal{L}(\ln w)=|B|^{2}+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})-\frac{|\nabla
w|^{2}}{w^{2}}.$
From (3.7),we obtain
$\displaystyle|\nabla w|^{2}$
$\displaystyle=\sum_{j=1}^{n}|\nabla_{e_{j}}w|^{2}=\sum_{j=1}^{n}(\sum_{i=1}^{n}\sum_{\alpha}h_{ij}^{\alpha}w_{i\,\alpha})^{2}$
$\displaystyle=\sum_{j=1}^{n}(\sum_{i=1}^{n}h_{ij}^{n+i}\lambda_{i}w)^{2}=\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}w^{2}h_{i\,j}^{n+i}h_{k\,j}^{n+k}.$
in the case of $m\geq n$ we rewrite (otherwise, we treat the situation
similarly)
$|B|^{2}=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}.$
So, we obtain
(3.17) $\displaystyle\mathcal{L}(\ln w)$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}(h_{i\,j}^{n+i}h_{j\,k}^{n+k}-h_{i\,j}^{n+k}h_{j\,k}^{n+i})-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+i}h_{j\,k}^{n+k}-\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+k}h_{j\,k}^{n+i}-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{i\,j}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
144.54pt-\sum_{i,j}\lambda_{i}^{2}(h_{ij}^{n+i})^{2}-\sum_{ij,k\neq
i}\lambda_{i}\lambda_{k}h_{ij}^{n+k}h_{jk}^{n+i}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
180.67499pt-2\sum_{j}\sum_{k<i}\lambda_{k}\lambda_{i}h_{jk}^{n+i}h_{ij}^{n+k}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
231.26378pt+\sum_{j}\sum_{i<k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j,k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\prod_{i}(1-\lambda_{i}^{2})\sum_{i,j,k}(h_{ij}^{n+k})^{2}.$
Noting (2.1) the inequality (3.15) has been proved.
###### Remark 3.1.
For a space-like graph $M=(x,f(x))$ with $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$
its induced metric is
$ds^{2}=(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})dx^{i}dx^{j}.$ Set
$g=\det(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})$ then
$w=\frac{1}{\sqrt{g}}$.
###### Remark 3.2.
(3.15) is a generalization of a formula (5.8) for space-like graphical self-
shrinkers in [7] to more general situation.
∎
## 4\. Volume growth
To draw our results we intend to integrate those differential inequalities
obtained in the last section. We need to know the volume growth in the pseudo-
distance function $z$ on the space-like submanifolds. In [16] the following
property has been proved.
###### Proposition 4.1.
(Proposition 3.1 in [16]) Let $M$ be a space-like $n-$submanifold in
$\mathbb{R}^{m+n}_{m}$. If $M$ is closed with respect to the Euclidean
topology, then when $0\in M$, $z=\left<F,F\right>$ is a proper function on
$M$.
we also need a lemma from [8]:
###### Lemma 4.1.
If $f(r)$ is a monotonic increasing nonnegative function on $[0,+\infty)$
satisfying $f(r)\leq C_{1}r^{n}f(\frac{r}{2})$ on $[C_{2},+\infty)$ for some
positive constant $n,C_{1},C_{2}$, here $C_{2}>1$, then $f(r)\leq
C_{3}e^{2n(\log r)^{2}}$ on $[C_{2},+\infty)$ for some positive constant
$C_{3}$ depending only on $n,C_{1},C_{2},f(C_{2})$.
Using the similar method as in [8] we obtain the following volume growth
estimates.
###### Theorem 4.1.
Let $z=\langle F,F\rangle$ be the pseudo-distance of $R_{m}^{n+m},$ where
$F\in R_{m}^{n+m}$ is the position vector with respect to the origin $0\in M$.
Let $M$ be an $n-$dimensional space-like self-shrinker of $R^{n+m}_{m}$.
Assume that $M$ is closed with respect to the Euclidean topology, then for any
$\alpha>0$, $\int_{M}e^{-\alpha z}d\mu$ is finite, in particular $M$ has
finite weighted volume.
###### Proof.
We have
$z_{i}\mathop{=}\limits^{def.}e_{i}(z)=2\left<F,e_{i}\right>,$
$z_{ij}\mathop{=}\limits^{def.}Hess(z)(e_{i},e_{j})=2\left(\delta_{ij}-y^{\alpha}h_{ij}^{\alpha}\right),$
(4.1) $\Delta z=2n-2y^{\alpha}H^{\alpha}=2n+Y^{2},$
where the self-shrinker equation (1.2) has been used in third equality. For
our self-shrinker $M^{n}$ in ${\tenmsb R}^{n+m}_{m}$, we define a functional
$F_{t}$ on any set $\Omega\subset M$ by
$F_{t}(\Omega)=\frac{1}{(4\pi
t)^{n/2}}\int_{\Omega}e^{-\frac{z}{4t}}d\mu,\quad\mathrm{for}\quad t>0.$
Set $B_{r}=\\{p\in\mathbb{R}^{m+n}_{m},z(p)<r^{2}\\}$ and $D_{r}=B_{r}\bigcap
M$. We differential $F_{t}(D_{r})$ with respect to $t$,
$F_{t}^{\prime}(D_{r})=(4\pi)^{-\frac{n}{2}}t^{-(\frac{n}{2}+1)}\int_{D_{r}}(-\frac{n}{2}+\frac{z}{4t})e^{-\frac{z}{4t}}d\mu.$
Noting (4.1)
(4.2) $\displaystyle-e^{\frac{z}{4t}}\mathrm{div}(e^{-\frac{z}{4t}}\nabla z)$
$\displaystyle=-\Delta z+\frac{1}{4t}\nabla z\cdot\nabla z$
$\displaystyle=-2n-Y^{2}+\frac{X^{2}}{t}$
$\displaystyle\geq\frac{z}{t}-2n\quad(\ when\ 0<t\leq 1\ ).$
Since
$\nabla z=2F^{T}$
and the unit normal vector to $\partial D_{r}$ is $\frac{F^{T}}{X}$, then
(4.3) $\displaystyle F_{t}^{\prime}(D_{r})\leq$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{D_{r}}-\mathrm{div}(e^{-\frac{z}{4t}}\nabla
z)d\mu$ $\displaystyle=$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{\partial
D_{r}}-2Xe^{-\frac{z}{4t}}\leq 0.$
We integrate $F_{t}^{\prime}(D_{r})$ over $t$ from $\frac{1}{r}$ to $1,\,r\geq
1$, and get
$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\leq
r^{\frac{n}{2}}\int_{D_{r}}e^{-\frac{zr}{4}}d\mu,$
Since
$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\geq e^{-\frac{r^{2}}{4}}\int_{D_{r}}1d\mu$
and
$\displaystyle\int_{D_{r}}e^{-\frac{zr}{4}}d\mu$ $\displaystyle=$
$\displaystyle\int_{D_{r}\backslash
D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu+\int_{D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu$
$\displaystyle\leq$ $\displaystyle
e^{-\frac{r^{3}}{16}}\int_{D_{r}}1d\mu+\int_{D_{\frac{r}{2}}}1d\mu.$
Set $V(r)=\int_{D_{r}}1d\mu$. Then,
$(e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}})V(r)\leq
r^{\frac{n}{2}}V(\frac{r}{2}).$
Let $g(r)=e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}}$. $g(r)>0$
when $r$ sufficiently large (say $r\geq 8n$) Since
$\displaystyle g^{\prime}(r)$
$\displaystyle=-\frac{r}{2}e^{-\frac{r^{2}}{4}}-\frac{n}{2}r^{\frac{n}{2}-1}e^{-\frac{r^{3}}{16}}+\frac{3r^{2}}{16}r^{\frac{n}{2}}e^{-\frac{r^{3}}{16}}$
$\displaystyle>(-\frac{r}{2}-\frac{n}{2}r^{\frac{n}{2}-1}+\frac{3}{16}r^{\frac{n}{2}+2})e^{-\frac{r^{3}}{16}}>0,$
$g(r)$ is increasing in $r$ and $g^{-1}(r)$ is decreasing in $r$. Therefore,
$g^{-1}(r)\leq\frac{1}{e^{-16n^{2}}-e^{-32n^{3}}(8n)^{\frac{n}{2}}}=C_{1}$
We then have
$V(r)\leq C_{1}r^{n}V(\frac{r}{2})\quad\quad\text{for $r$ sufficiently large
(say, $r\geq 8n$)}.$
By Lemma 4.1, we have
$V(r)\leq C_{4}e^{2n(\log r)^{2}}\quad for\quad r\geq 8n,$
here $C_{4}$ is a constant depending only on $n,V(8n)$. Hence, for any
$\alpha>0$
$\displaystyle\int_{M}e^{-\alpha z}d\mu=$
$\displaystyle\sum_{j=0}^{\infty}\int_{D_{8n(j+1)}\setminus D_{8nj}}e^{-\alpha
z}d\mu\leq\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}V(8n(j+1))$
$\displaystyle\leq$ $\displaystyle
C_{4}\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}e^{2n(\log(8n)+\log(j+1))^{2}}\leq
C_{5},$
where $C_{5}$ is a constant depending only on $n,V(8n)$. So we obtain our
estimates. Certainly, $M$ has weighted finite volume. ∎
###### Corollary 4.1.
Any space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
with closed Euclidean topology has finite fundamental group.
From the Gauss equation we have
$\text{Ric}(e_{i},e_{i})=\left<H,B_{ii}\right>-\sum_{j}\left<B_{ij},B_{ij}\right>,$
and
$\text{Hess}(f)(e_{i},e_{i})=\frac{1}{4}\text{Hess}(z)(e_{i},e_{i})=\frac{1}{2}\delta_{ij}+\frac{1}{2}\left<F,B_{ij}\right>=\frac{1}{2}\delta_{ij}-\left<H,B_{ij}\right>,$
It follows that
$\text{Ric}_{f}(e_{i},e_{i})=\text{Ric}(e_{i},e_{i})+\text{Hess}(f)(e_{i},e_{i})\geq\frac{1}{2}.$
Set $B_{R}(p)\subset M$, a geodesic ball of radius $R$ and centered at $p\in
M$. From Theorem 3.1 in [18] we know that for any $r$ there are constant
$A,\;B$ and $C$ such that
(4.4) $\int_{B_{R}(p)}\rho\leq A+B\int_{r}^{R}e^{-\frac{1}{2}t^{2}+Ct}dt$
## 5\. Rigidity results
Now, we are in a position to prove rigidity results mentioned in the
introduction.
###### Theorem 5.1.
Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.
###### Proof.
Let $\eta$ be a smooth function with compact support in $M,$ then by (3.14) we
obtain
(5.1) $\displaystyle\int_{M}(\frac{1}{2}|H|^{2}+|P|^{2}+|\nabla
H|^{2})\eta^{2}\rho$
$\displaystyle=\frac{1}{2}\int_{M}(\mathcal{L}|H|^{2})\eta^{2}\rho=\frac{1}{2}\int_{M}\text{div}(\rho\nabla|H|^{2})\eta^{2}$
$\displaystyle=-\int_{M}\eta\rho\nabla|H|^{2}\cdot\nabla\eta$
$\displaystyle=2\int_{M}\eta\rho\langle\nabla_{i}H,H\rangle\cdot\nabla_{i}\eta$
$\displaystyle\leq\int_{M}|\nabla
H|^{2}\eta^{2}\rho+\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$
We then have
(5.2)
$\int_{M}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\eta^{2}\rho\leq\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$
Let $\eta=\phi(\frac{|F|}{r})$ for any $r>0,$ where $\phi$ is a nonnegative
function on $[0,+\infty)$ satisfying
$\displaystyle\phi(x)=\left\\{\begin{array}[]{lllll}1&&&if&x\in[0,1)\\\
0&&&if&x\in[2,+\infty),\end{array}\right.$
and $|\phi^{\prime}|\leq C$ for some absolute constant. Since $\nabla
z=2F^{T}$,
$\nabla\eta=\frac{1}{r}\phi^{\prime}\nabla\sqrt{z}=\frac{1}{r}\phi^{\prime}\frac{F^{T}}{\sqrt{z}}.$
By (1.2) we have
$|\nabla\eta|^{2}\leq\frac{C^{2}}{r^{2}}\frac{|F^{T}|^{2}}{z}=\frac{1}{r^{2}z}C^{2}(z+4|H|^{2}).$
It follows that (5.2) becomes
(5.3)
$\int_{D_{r}}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\rho\leq\frac{C^{2}}{r^{2}}\int_{D_{2r}\setminus
D_{r}}|H|^{2}(1+\frac{4|H|^{2}}{z})\rho.$
By Theorem 4.1 then under the condition on $|H|$, we obtain that the right
hand side of (5.3) approaches to zero as $r\rightarrow+\infty.$ This implies
that $H\equiv 0.$
According to Theorem 3.3 in [16] we see that $M$ is complete with respect to
the induced metric from $\mathbb{R}^{m+n}_{m}$. In a geodesic ball $B_{a}(x)$
of radius $a$ and centered at $x\in M$ we can make gradient estimates of
$|B|^{2}$ in terms of the mean curvature. From (2.9) in [16] we have
$|B|^{2}\leq k\frac{2m(n-4)a^{2}}{(a^{2}-r^{2})^{2}}.$
Since $M$ is complete we can fix $x$ and let $a$ go to infinity. Hence,
$|B|^{2}=0$ at any $x\in M$ and $M$ is an $n-$plane. ∎
###### Theorem 5.2.
Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.
###### Proof.
(3.15) tells us
$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}\geq 0.$
As an application to (4.4) (Theorem 3.1 in [18]) the Corollary 4.2 in [18]
tells us that $\ln w$ is constant. This forces $|B|^{2}\equiv 0$. ∎
## References
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* [13] K. Ecker and G. Huisken: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys. 135(1991), 595-613.
* [14] Rongli Huang and Zhizhang Wang, On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations 41 (2011), 321-339.
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|
# Measuring the Variance of the Macquart Relation in z-DM Modeling
Jay Baptista Department of Astronomy and Astrophysics, Yale University, New
Haven, CT 06520, USA Kavli Institute for Particle Astrophysics & Cosmology,
Stanford University, P.O. Box 2450, Stanford, CA 94305, USA J. Xavier
Prochaska Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA 95064, USA Kavli Institute for the Physics and Mathematics of
the Universe (Kavli IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan Alexandra G. Mannings Department of Astronomy
and Astrophysics, University of California, Santa Cruz, CA 95064, USA C.W.
James International Centre for Radio Astronomy Research, Curtin University,
Bentley, WA 6102, Australia R. M. Shannon Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122,
Australia Stuart D. Ryder Department of Physics & Astronomy, Macquarie
University, NSW 2109, Australia A. T. Deller Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122,
Australia Danica R. Scott International Centre for Radio Astronomy Research,
Curtin University, Bentley, WA 6102, Australia Marcin Glowacki International
Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102,
Australia Nicolas Tejos Instituto de Física, Pontificia Universidad Católica
de Valparaíso, Casilla 4059, Valparaíso, Chile
###### Abstract
The Macquart relation describes the correlation between the dispersion measure
(DM) of fast radio bursts (FRBs) and the redshift $z$ of their host galaxies.
The scatter of the Macquart relation is sensitive to the distribution of
baryons in the intergalactic medium (IGM) including those ejected from
galactic halos through feedback processes. The width of the distribution in
DMs from the cosmic web (${\rm DM}_{\rm cosmic}$) is parameterized by a
fluctuation parameter $F$, which is related to the cosmic DM variance by
$\sigma_{\rm DM}=Fz^{-0.5}$. In this work, we present a new measurement of $F$
using 78 FRBs of which 21 have been localized to host galaxies. Our analysis
simultaneously fits for the Hubble constant $H_{0}$ and the DM distribution
due to the FRB host galaxy. We find that the fluctuation parameter is
degenerate with these parameters, most notably $H_{0}$, and use a uniform
prior on $H_{0}$ to measure $\log_{10}F>-0.89$ at the $3\sigma$ confidence
interval and a new constraint on the Hubble constant
$H_{0}=85.3_{-8.1}^{+9.4}\,{\rm km\,s^{-1}\,Mpc^{-1}}$. Using a synthetic
sample of 100 localized FRBs, the constraint on the fluctuation parameter is
improved by a factor of $\sim 2$. Comparing our $F$ measurement to simulated
predictions from cosmological simulation (IllustrisTNG), we find agreement
between $0.4<z<2$. However, at $z<0.4$, the simulations underpredict $F$ which
we attribute to the rapidly changing extragalactic DM excess distribution at
low redshift.
Radio transient sources (2008), Radio bursts (1339), Cosmological parameters
(339), Intergalactic medium (813), Hubble constant (758)
## 1 Introduction
In galaxy formation models, AGN and stellar feedback have provided mechanisms
for regulating star formation and evacuating gas out of low-mass halo (Cen &
Ostriker, 2006; Davé et al., 2011). In simulations without baryonic outflows,
galaxies simply produce too many stars and have higher than observed star
formation rates (Davé et al., 2011). These outflow processes are also critical
in understanding how the IGM becomes enriched and how the galaxies and the IGM
co-evolve.
Not only is understanding the nature of feedback crucial in reproducing
realistic galaxy properties in cosmological-baryonic simulations but also in
understanding the channels where these “missing” baryons may have left halos
and are prevented from re-accretion. Gas accretion onto galaxies from cold gas
filaments is exceptionally efficient. Baryonic feedback is a preventative
process that not only enriches the IGM but also removes baryons and restricts
accretion from the IGM (Kereš et al., 2005). For example, in the Simba suite
of cosmological hydrodynamic simulations, feedback from AGN jets can cause 80%
of baryons in halos to be evacuated by $z=0$ (Davé et al., 2019; Appleby et
al., 2021; Sorini et al., 2022).
A comparison of simulation suites shows that different feedback prescriptions
can eject baryons at various distances beyond the halo boundary—feeding gas
into the reservoir of diffuse baryons (e.g. Ayromlou et al., 2023). Thus, to
constrain the strength of AGN and stellar feedback processes, one must be able
to constrain the distribution of baryons in the IGM to discriminate between
these feedback models.
Determining the distribution of these ejected baryons is difficult. Emission
and absorption lines from baryons in the IGM are extremely difficult to detect
due to their high temperatures and low densities (Fukugita et al., 1998; Cen &
Ostriker, 2006; Shull et al., 2012; McQuinn, 2014). However, the advent of
fast radio bursts (FRBs) presents a new opportunity to probe the intergalactic
distribution of baryons and provide a novel approach to measuring feedback
strength (McQuinn, 2014; Muñoz & Loeb, 2018).
FRBs are sensitive to the line-of-sight free electron density where the
integrated free electron density yields the dispersion measure (DM) of the
signal (Lorimer et al., 2007). A redshift can be estimated if the spatial
localization of the FRB overlaps with a galaxy with a known redshift (assuming
the FRB progenitor indeed lies within that galaxy; Aggarwal et al., 2021).
This FRB redshift-extragalactic dispersion measure (z-${\rm DM}_{\rm EG}$)
correlation known as the Macquart relation (Macquart et al., 2020) is
sensitive to cosmological properties of the universe (e.g. James et al.,
2022a). By using a sophisticated FRB observational model that can account for
observational biases, intergalactic gas distribution, burst width, and DM, it
is possible to use FRB surveys to infer the distribution of baryons in the
universe (McQuinn, 2014; James et al., 2022a; Lee et al., 2022).
Halos with weaker feedback retain their baryons more effectively leading to
halos with higher ${\rm DM}_{\rm EG}$ contributions and voids with lower ${\rm
DM}_{\rm EG}$ contributions (McQuinn, 2014). An FRB can travel either through
extremely low DM voids or pass through extremely high DM halos suddenly,
leading to enhanced scatter in the $z$-DMEG distribution. On the other hand,
halos with stronger feedback will cause the $z$-DMEG distribution to show less
scatter. Feedback processes are able to more effectively relocate halo baryons
into the IGM, causing inter-halo voids to have higher DM. This leads to a more
homogeneous universe. The goal of this work is to forecast and measure a
precise galactic feedback prescription as determined by a sample of FRBs.
This paper is organized as follows: Section 2 outlines the modeling of the
$z-{\rm DM}$ distribution and how the fluctuation parameter $F$ influences
observations of the FRB distribution; Section 3.1 presents our measurements on
the fluctuation parameter $F$ and the fits on other parameters used in the
$z$-DMEG model; Section 3.2 details our forecast on $F$ by sampling 100
synthetic FRBs and finding the probability distribution functions of our model
parameters based on the synthetic survey; Section 4 discusses our results in
the context of constraining cosmic feedback strength, parameter degeneracies,
and compares our measurements to simulations.
## 2 Methods
### 2.1 Basic Formalism
This work makes use of the FRB code zdm developed by James et al. (2022a) to
model observables of FRB populations. The model assumes that the DM
measurement of an FRB can be decomposed as ${\rm DM}_{\rm FRB}={\rm DM}_{\rm
ISM}+{\rm DM}_{\rm halo}+{\rm DM}_{\rm EG}$, where ${\rm DM}_{\rm ISM}$ and
${\rm DM}_{\rm halo}$ are the DM contributions due to the Milky Way’s
interstellar medium (modeled using NE2001; Cordes & Lazio, 2002a) and diffuse
ionized gas in the Galaxy’s halo (Prochaska & Zheng, 2019; Cook et al., 2023;
Ravi et al., 2023). The ${\rm DM}_{\rm EG}$ term is the extragalactic DM
contribution and is decomposed into ${\rm DM}_{\rm EG}={\rm DM}_{\rm
cosmic}+{\rm DM}_{\rm host}$ where ${\rm DM}_{\rm cosmic}={\rm DM}_{\rm
IGM}+{\rm DM}_{\rm halo,EG}$ is the contribution due to baryons in the IGM and
intersecting halos in the line-of-sight, and ${\rm DM}_{\rm host}$ is the
contribution due to the host galaxy of the FRB signal. The contribution from
the host (${\rm DM}_{\rm host}$) is modeled as a log-normal distribution with
a width of $\exp(\mu)$ and a logarithmic scatter of ${\sigma}_{\rm host}$
where $\mu$ and ${\sigma}_{\rm host}$ are free parameters of the model.
The width of the $z$-DMEG distribution at a fixed redshift is characterized in
part by the probability distribution of measuring the ${\rm DM}_{\rm cosmic}$
of an FRB above or below $\langle{\rm DM}_{\rm cosmic}\rangle$. This
distribution is described by $p_{\rm cosmic}(\Delta)$, where $\Delta\equiv{\rm
DM}_{\rm cosmic}/\langle{\rm DM}_{\rm cosmic}\rangle$:
$p_{\rm
cosmic}(\Delta)=A\Delta^{-\beta}\exp(-\frac{(\Delta^{-\alpha}-C_{0})^{2}}{2\alpha^{2}\sigma_{\rm
DM}^{2}}),$ (1)
where $\alpha\simeq 3$ and $\beta\simeq 3$ are the inner and outer slopes of
the gas profile density of intervening halos (based on numerical simulations
from Macquart et al., 2020), $C_{0}$ shifts the distribution such that
$\langle\Delta\rangle=1$, and $\sigma_{\rm DM}$ represents the spread of the
distribution (Macquart et al., 2020). This non-Gaussian probability
distribution function is motivated by theoretical treatments of the IGM and
galaxy halos (Macquart et al., 2020). For example, in the limit where
$\sigma_{\rm DM}$ is small, the distribution becomes Gaussian to capture the
Gaussianity of large-scale structures. This is physically motivated as halo
gas is more diffuse in this limit and thus contributions to the variance due
to halo gas are insignificant. In the limit where $\sigma_{\rm DM}$ is large,
the halo gas contribution becomes significant (Macquart et al., 2020).
Owing to the approximately Poisson nature of intersecting halos, one expects
$\sigma_{\rm DM}\propto z^{-1/2}$ (Macquart et al., 2020) and one is motivated
to introduce a fluctuation parameter $F$
$\sigma_{\rm DM}(\Delta)=Fz^{-0.5}\;\;\;,$ (2)
As the fluctuation parameter increases, i.e. $F\sim 1$, the spread of ${\rm
DM}_{\rm cosmic}$ increases. Figure 1 shows $p({\rm DM}|z)$ for two extreme
values of $F$ and the resultant, substantial changes to the width of the ${\rm
DM}_{\rm EG}$ distribution at any given redshift.
Figure 1: Upper panel: The $p({\rm DM_{\rm EG}}|z)$ distribution which admits
a high fluctuation parameter (low galactic feedback efficiency). Lower panel:
The $p({\rm DM_{\rm EG}}|z)$ distribution which admits a low fluctuation
parameter (high galactic feedback efficiency). The white dashed-line indicates
the 95 percentile contour. Note that the distribution primarily falls below
the mean due to the rare population of high DM FRBs that result from
intersections with the host galaxy and/or very massive galaxy halos along the
LOS.
The variance in ${\rm DM}_{\rm EG}$, however, is influenced by both
${\sigma}_{\rm host}$ and $F$. However, at high redshift, the contribution to
the variance of ${\rm DM}_{\rm EG}$ due to ${\sigma}_{\rm host}$ may decrease
relative to the contributions by ${\rm DM}_{\rm cosmic}$ and, inherently, $F$.
In James et al. (2022b), their work assumes that uncertainties attributed to
the fixed value of $F$ can be aggregated into uncertainties in ${\sigma}_{\rm
host}$; however, at high redshift, the assumption breaks down as the
uncertainty in $F$ becomes larger than the true constraint in ${\sigma}_{\rm
host}$.
Although Macquart et al. (2020) restrict their fitting of the fluctuation
parameter to $F\in[0.09,0.32]$ based on semi-analytic models, we sample a wide
range of $F\in[0,1]$. We opt for a logarithmic sampling of the fluctuation
parameter to efficiently sample this domain: $\log_{10}F\in[-2,0]$.
The additional parameters used in the model include $H_{0}$ (acceleration of
the Universe’s expansion), the ${\rm DM}_{\rm EG}$ contribution due to the FRB
host galaxy, and other parameters that govern the FRB luminosity function and
redshift distribution. The model assumes that the ${\rm DM}_{\rm EG}$
contribution from the FRB host galaxy can be modeled as a log-normal
distribution with a mean of $\mu_{\rm host}$ (or ${\rm DM}_{\rm host}$) and a
spread of $\sigma_{\rm host}$.
In terms of the luminosity function, the maximum burst energy is given as
$E_{\rm max}$, and the integral slope of the FRB luminosity function is
controlled by $\gamma$. The volumetric burst rate ($\Phi$) is controlled by
the parameter $n_{\rm sfr}$ assuming a star-formation rate: $\Phi\propto{\rm
SFR(z)}^{n_{\rm sfr}}$. Additionally, $\alpha$ is the spectral index that sets
a frequency-dependent FRB rate as $\Phi(z,\nu)=\Phi(z)\nu^{\alpha}$ (James et
al., 2022b).
### 2.2 Measuring $F$ Using FRB Survey Data
To measure the fluctuation parameter, we perform a simultaneous fit of the
parameters in the zdm model implemented by James et al. (2022b). We obtain the
probability distributions of each parameter by a brute-force grid search based
on the ranges specified in Table 2, and calculating the likelihoods for each
permutation of parameter values.
We fit these parameters using both the FRB sample used in James et al. (2022b)
and newly detected or analyzed FRBs (see Table 1) which were collected from
the Parkes and ASKAP telescopes. Of this sample of 78 measured FRBs, 57 FRBs
do not have measured redshifts. Constraining the redshift of an FRB greatly
increases the statistical power as a single FRB with a redshift can have the
same constraining power as roughly 20 FRBs without redshifts (James et al.,
2022b). Thus our inclusion of seven new CRAFT/ICS FRBs detected over three
frequency ranges (four with host redshifts), and our identification of the
host galaxy of FRB20211203C at $z=0.344$, provides a significant increase in
statistical precision.
There is a slight bias in this sample, as we include FRB20220610A, which has
an energy exceeding the previously estimated turnover $E_{\rm max}$ by a
factor of 3.5–10, depending on the assumed spectral behavior (Ryder et al.,
2022). FRB20210912A has a lower DM of 1234.5 ${\rm pc\,cm^{-3}}$, but it does
not have an identified redshift, perhaps due to the distance to its host
galaxy (Marnoch et al., in prep). Therefore, the inclusion of some data is
redshift-dependent. Given that our sample is statistically limited, we assume
the resulting bias to be small compared to the gain in precision.
In contrast to the James et al. (2022b) analysis, we hold model parameters
that are not degenerate with $F$ to their fiducial values. These parameters
were determined to be non-degenerate with $F$ running the model using a low-
resolution grid search on synthetic data to determine if $F$ correlated with
any of the other model parameters. From this preliminary analysis, we fix the
following parameters that were found to be non-degenerate with $F$: $\alpha$,
$\gamma$, $E_{\rm max}$, and $n_{\rm sfr}$. On the other hand, we expect the
fluctuation parameter to be degenerate with the other model parameters. In
particular, we expect the Hubble constant $H_{0}$–—the cosmological parameter
that quantifies the expansion of the universe–—to be degenerate with the
fluctuation parameter $F$.
We examine this degeneracy further in Figure 2 which shows the 95 percentile
contours for $p({\rm DM}|z)$ for two models with very different $F$ and
$H_{0}$ values. One notes that the lower contours (${\rm DM}_{\rm EG}\lesssim
720,\,z\lesssim 1.3$) of both realizations look nearly identical. Although the
contours differ above the mean, the bulk of the constraining power on $F$ is
in the lower contour or “DM cliff”. Therefore, we anticipate $F$ and $H_{0}$
to be highly correlated.
The distribution at the low DM end of $p({\rm DM_{\rm EG}}|z)$ exhibits a
sharp cut-off and provides strong constraints on $H_{0}$ since there is a
minimum imparted ${\rm DM}_{\rm cosmic}$ from voids and is not impeded by the
${\rm DM}_{\rm EG}$ contributions from large-scale structures like filaments
or halos. And while the contours do have modest differences at high $z$, high
${\rm DM}_{\rm EG}$, these can be difficult to distinguish from host galaxy
contributions to ${\rm DM}_{\rm EG}$.
Figure 2: $95^{\rm th}$ percentile contours of two $p({\rm DM_{\rm EG}}|z)$ distributions with different prescriptions on the Hubble constant $H_{0}$ and the fluctuation parameter $F$. Note that the lower contours (“the DM cliff”) of both models are nearly identical to each other. Since the DM cliff places a stronger constraint on $H_{0}$ and $F$ than the upper contour, we expect a high degree of degeneracy between $H_{0}$ and $F$. The units of ${\rm DM}_{\rm EG}$ are in pc cm-3 and the units of $H_{0}$ are $\rm km\,s^{-1}\,Mpc^{-1}$. Table 1: New FRB detections detected in 2022 used in addition to the FRB surveys used in James et al. (2022b). The FRB name, SNR-maximizing DM, ${\rm DM}_{\rm ISM}$ estimated using the NE2001 model of Cordes & Lazio (2002b), central frequency of observation $\nu$, measured signal-to-noise ratio SNR, redshift $z$, and original reference. Where redshifts are not given, this is because (a): no voltage data were dumped, preventing radio localization; (b) optical follow-up observations are not yet complete; (c) Substantial Galactic extinction has challenged follow-up optical observations; (d) the host galaxy appears too distant to accurately measure a redshift. Name | DM | ${\rm DM}_{\rm ISM}$ | $\nu$ | SNR | $z$ | Ref.
---|---|---|---|---|---|---
| (${\rm pc\,cm^{-3}}$) | (${\rm pc\,cm^{-3}}$) | (MHz) | | |
CRAFT/ICS $900\,{\rm MHz}$
20211203C | 636.2 | 63.4 | 920.5 | 14.2 | 0.344 | Shannon et al. (in prep.)
20220501C | 449.5 | 30.6 | 863.5 | 16.1 | 0.381
20220725A | 290.4 | 30.7 | 920.5 | 12.7 | 0.1926
CRAFT/ICS $1.3\,{\rm GHz}$
20220531A | 727.0 | 70.0 | 1271.5 | 9.7 | – | Shannon et al. (in prep.)
20220610A | 1458.1 | 31.0 | 29.8 | 1.016 | Ryder et al. (2022)
20220918A | 656.8 | 40.7 | 26.4 | – | Shannon et al. (in prep.)
CRAFT/ICS $1.6\,{\rm GHz}$
20220105A | 583.0 | 22.0 | 1632.5 | 9.8 | 0.2785 | Shannon et al. (in prep.)
20221106A | 344.0 | 34.8 | 1631.5 | 35.1 | –
### 2.3 Forecasting the fluctuation parameter $F$ using Synthetic FRBs
Table 2: z-DM grid parameters
Parameter | Unit | Fiducial | Min | Max | N
---|---|---|---|---|---
$H_{0}$ | km s-1 Mpc-1 | 67.4 | 60.0 | 80.0 | 21
$\log F$ | - | 0.32 | -1.7 | 0 | 30
${\mu}_{\rm host}$ | $\log{\rm pc\,cm^{-3}}$ | 2.16 | 1.7 | 2.5 | 10
${\sigma}_{\rm host}$ | $\log{\rm pc\,cm^{-3}}$ | 0.51 | 0.2 | 0.9 | 10
$\log_{10}E_{\rm max}$ | $\log{\rm erg}$ | 41.84 | – | – | –
$n_{\rm sfr}$ | - | 1.77 | – | – | –
$\alpha$ | - | 1.54 | – | – | –
$\gamma$ | - | -1.16 | – | – | –
Note. — This table indicates the parameters of the high-resolution grid run.
Non-degenerate parameters are held to the fiducial values. $N$ is the number
of cells between the minimum and maximum parameter values.
Future radio surveys are expected to widely increase the number of sub-
arcsecond localized FRBs. Thus, the constraining power on $F$ will greatly
increase. To explore this scenario we generate a forecast on the fluctuation
parameter by replicating our analysis using a synthetic FRB survey. A sample
of 100 localized synthetic FRBs was drawn assuming the distribution of FRBs
followed the fiducial $z-{\rm DM}$ distribution (Table 2). With this synthetic
survey, we calculate the associated 4D likelihood matrix and make a forecast
on the fluctuation parameter by adopting different priors on $H_{0}$.
## 3 Results
### 3.1 Parameter Likelihoods from FRB Surveys
In Figure 3, we present the 1D PDFs of each parameter determined from the 78
FRBs collected from the ASKAP and Parkes Radio Telescopes. In comparison to
James et al. (2022b), there is a significant loss of constraining power on
$H_{0}$ by including $F$ as a free parameter. We measure a Hubble constant to
be $85.3_{-8.1}^{+9.4}$ which is 1.5 times more uncertain than the $H_{0}$
measurement in James et al. (2022b). We attribute the uncertainty to the
degeneracy between $H_{0}$ and $F$ as indicated by the strong anti-correlation
in Figures 2 and 5.
To understand how our survey data at different frequencies contribute to the
constraining power on $H_{0}$ and $F$, we replicate the survey contribution
determination from James et al. (2022b) which provides the 1D parameter
likelihood across different FRB surveys with the Murriyang (Parkes) and
Australian Square Kilometre Array (ASKAP). Figure 4 shows the 1D PDFs of
$H_{0}$ and $\log_{10}F$ across the different surveys used in this analysis.
We observe that the CRAFT 1.3 GHz and 900 MHz surveys tend to have stronger
constraining power as they contain more FRBs with measured distances (10 and 7
redshifts respectively) than the rest of the surveys.
Figure 3: The calculated 1D likelihood functions using 78 FRBs (21 FRBs with
redshifts). $F$ is measured to be $\log_{10}F=-0.75_{-0.25}^{+0.33}$ with no
priors on $H_{0}$. There is a loss of constraining power on $H_{0}$ compared
to the measurement by James et al. (2022b) when allowing the $F$ parameter to
vary.
Figure 4: Upper panel: 1D likelihood functions of $H_{0}$ based on different
FRB surveys used in this work. Compared to the survey contribution constraints
(see Figure 7 in James et al., 2022b), the constraining power of each survey
is diminished. Lower panel: Same as upper panel for likelihood functions of
$\log_{10}F$. The CRAFT/ICS 900 MHz and 1.3 GHz surveys provide the most
constraining power on $F$ and $H_{0}$ as they contain more redshifts than the
other surveys.
In Figure 5, we present the 2D likelihoods of each parameter against $F$. We
observe correlations between $F$ and the FRB host galaxy parameters
(${\mu}_{\rm host}$, ${\sigma}_{\rm host}$), which we expect to be degenerate
given that they both influence the variance of $\langle{\rm DM}_{\rm
cosmic}\rangle$. As expected from Figure 2, the degeneracy in the lower bound
(or cliff) of the $z$-DMEG distribution results in the strong anti-correlation
of $H_{0}$ and $F$, resulting in a loss of constraining power on $H_{0}$ when
allowing $F$ to vary.
Our initial simultaneous fit does not implement any priors on the model
parameters. As motivated by the $H_{0}$-$F$ degeneracy in Figure 2, we
determine the 1D likelihoods of $\log_{10}F$ by limiting our grid to different
values of $H_{0}$. We consider a uniform prior on $H_{0}$ between 67.4 and
73.04 ${\rm km\,s^{-1}\,Mpc^{-1}}$—the lower bound is motivated by the $H_{0}$
constraint from Planck Collaboration et al. (2020), and the upper bound is
motivated by cosmological constraints using type-1a supernovae (SNe) from
Riess et al. (2022).
In Figure 6, we present the 1D likelihood of the fluctuation parameter
assuming different priors on $H_{0}$. Assuming a uniform prior between the CMB
and SNe-derived values of $H_{0}$, we measure the fluctuation parameter to be
$\log_{10}F=-0.48^{+0.26}_{-0.18}$ within $1\sigma$
($\log_{10}F=\log_{10}F>-0.89$ with 99.7% confidence). We present all
measurements of the $F$ parameter with different priors on $H_{0}$ in Table 3.
Figure 5: The 2D likelihood functions for each parameter compared against
$\log_{10}F$ derived from 78 FRBs (21 FRBs with redshifts). There is a strong
anti-correlation between $F$ and $H_{0}$. Additionally, we observe strong
correlations between $F$ and the host galaxy ${\rm DM}_{\rm EG}$ contribution
(${\mu}_{\rm host},{\sigma}_{\rm host}$). Figure 6: 1D survey likelihoods of
$\log_{10}F$ assuming different priors on $H_{0}$. The dotted gray line is the
original 1D likelihood without any priors on $H_{0}$. The blue line is the
likelihood which adopts a uniform prior on $H_{0}\in[67.4,73.04]$. The dashed
gray line is the likelihood adopting $H_{0}=67.0\,{\rm kms^{-1}Mpc^{-1}}$. The
dash-dotted gray line is the likelihood adopting $H_{0}=73.0\,{\rm
kms^{-1}Mpc^{-1}}$. Adopting a prior on $H_{0}$ or fixing the values of
$H_{0}$ greatly improves the constraint on $F$. Table 3: Measurements of the
$F$ Parameter
Survey | No Prior | Uniform $H_{0}$ Prior | CMB $H_{0}$ | SNe $H_{0}$
---|---|---|---|---
Observed | $-0.75_{-0.25}^{+0.33}$ | $-0.48^{+0.26}_{-0.18}$ | $-0.35_{-0.15}^{+0.23}$ | $-0.52_{-0.17}^{+0.26}$
Synthetic | $-0.58_{-0.15}^{+0.15}$ | $-0.60_{-0.10}^{+0.09}$ | $-0.52_{-0.06}^{+0.07}$ | $-0.67_{-0.06}^{+0.06}$
Note. — This table lists the measurements of the $F$ parameter from the
observational FRB survey (76 localized FRBs with 16 redshifts) and the
synthetic CRACO survey (100 localized FRBs; all with redshifts). The
measurements are presented without a prior, a uniform prior between the CMB
($H_{0}=67.0\,{\rm km\,s^{-1}\,Mpc^{-1}}$) and SNe ($H_{0}=73.0\,{\rm
km\,s^{-1}\,Mpc^{-1}}$) with their respective Gaussian errors on each side,
and fixing $H_{0}$ to the CMB or SNe estimates.
### 3.2 Parameter Likelihoods from Synthetic Surveys
Figure 7: The 1D likelihood functions for each parameter using 100 synthetic
FRBs. The constraint on $F$ is enhanced as the uncertainty due to sample size
is reduced. Similarly, the constraint on $H_{0}$ is improved compared to the
observational fit with only 21 redshifts.
We use a synthetic sample of 100 localized CRACO FRBs to investigate the
improvement in constraining power on both $F$ and $H_{0}$. In Figure 7, we
present the PDFs of each parameter in the grid. We observe that the constraint
on $H_{0}$ has significantly improved by a factor of 1.7 and is more Gaussian
than the previous run with $69.2_{-4.9}^{+5.5}$. Assuming a survey of 100
localized FRBs, the best measurement we can make on $H_{0}$ if we adopt a
Gaussian prior on $\log_{10}F$ (assuming $1\sigma$ corresponds to a 20% error
in the measurement) is $67.6_{-3.4}^{+3.5}\,{\rm km\,s^{-1}\,Mpc^{-1}}$ (see
Table 4).
In Figure 8 we show posterior estimates for $\log_{10}F$ using different
priors (see Table 3). Using the uniform prior, we obtain a forecast on the
fluctuation parameter of $\log_{10}F=-0.60_{-0.18}^{+0.19}$ within $2\sigma$.
We note that when compared to Figure 3, there is a definitive upper limit on
the fluctuation parameter rather than only a lower limit. Incorporating the
uniform prior enhances the constraint on $F$ by a factor of $\sim 1.5$, and
fixing the value of $H_{0}$ can increase the constraint by a factor of $2.5$.
Figure 8: Same as Figure 6 but based on fits to the synthetic FRB sample.
Assuming a uniform prior between CMB and SNe values of $H_{0}$ equipped with
their associated Gaussian errors on both sides, we find
$\log_{10}F=-0.60_{-0.10}^{+0.09}$. Fixing the value of $H_{0}$ also greatly
enhances the constraint on $F$ by a factor of $>1.5$. Table 4: Measurements of
$H_{0}$
Survey | No Prior | Gaussian Prior
---|---|---
Observed | $85.3_{-8.1}^{+9.4}$ | -
Synthetic | $69.2_{-4.9}^{+5.5}$ | $67.6_{-3.4}^{+3.5}$
Note. — This table lists the measurements of $H_{0}$ from the observational
FRB survey (76 localized FRBs with 16 redshifts) and the synthetic CRACO
survey (100 localized FRBs; all with redshifts). The measurements are
presented without a prior on $F$ and a Gaussian prior on $F$ centered at
$\log_{10}F\simeq-0.49$ with $\sigma\simeq 0.1$ (20% error on $F$).
## 4 Discussion
### 4.1 Measurement of the fluctuation parameter
Our principle result from the population analysis of 78 FRBs (21 with
redshifts) is a lower limit on $F$ which is $\log_{10}F=-0.48^{+0.26}_{-0.18}$
($\log_{10}F>-0.89$ at 99.7% confidence). This measurement is motivated by
James et al. (2022b), where they noted that for future localization of FRBs
beyond $z\gtrsim 1$, $F$ may need to be fitted explicitly. We note that this
observation is only made when adopting a prior between the CMB and SNe values
of $H_{0}$.
### 4.2 Fluctuation Parameter Degeneracies
Our findings indicate a strong degeneracy between the Hubble constant $H_{0}$
and the fluctuation parameter $F$ when simultaneously fitting both within the
z-DM modeling framework adopted by James et al. (2022a) which uses $F=0.32$
which falls within the accepted range of our measurement.
Aside from the degeneracy between $F$ and $H_{0}$, we would like to call
attention to the possible degeneracy between $F$ and $\sigma_{8}$–the RMS
amplitude of the matter density field when smoothed with an $8h^{-1}$ Mpc
filter. In the case of w feedback ($F\rightarrow 1$), more mass would be
concentrated within cosmic filaments, increasing the variance of a fixed-mass
filter (i.e., $\sigma_{8}$). We expect these two parameters to be inversely
coupled. A preliminary analysis varying $\sigma_{8}$ in the CAMELS
IllustrisTNG cosmological simulations does show a positive correlation between
$F$ and $\sigma_{8}$ (Medlock et al. in prep.).
### 4.3 Forecasting enhanced constraints on $F$
Using a sample of 100 synthetic FRBs (see Figure 8), we are able to constrain
both upper and lower limits on the fluctuation parameter out to $3\sigma$.
Since we are only able to effectively constrain a lower limit on $F$, we
compare the lower-sided half-maximum widths. We find the left-sided half-
maximum width of the synthetic distribution is half the width of the current
measured distribution. We expect this constraint to only improve with more
localizations, which will be easily facilitated with next-generation all-sky
radio observatories.
Additionally, it is of interest to see how this method compares to other ways
of measuring the baryon distribution in the IGM. For example, an alternative
method to constrain AGN and stellar feedback focuses on small-scale deviations
in the matter power spectrum (van Daalen et al., 2020). As baryonic feedback
significantly influences the mass distribution at smaller scales (higher $k$),
probes of the gas density at those scales (thermal Sunyaev-Zel’dovich effect)
can measure the intergalactic baryon distribution (Pandey et al., 2023).
### 4.4 Comparing with Fluctuation Parameter in IllustrisTNG
In a work by Zhang et al. (2021) to highlight the utility of FRBs in probing
the IGM, they generated thousands of FRB sightlines in IllustrisTNG and fitted
the observed extragalactic DM excess $p_{\rm cosmic}(\Delta)$. They provide
the fitted parameters as well as the dispersion in the $z$-DMEG distribution
$\sigma_{\rm DM}$. We convert these values into the fluctuation parameter $F$
by assuming $\sigma_{\rm DM}=Fz^{-0.5}$.
In Figure 9, we present these derived $\log_{10}F$ values as a function of
redshift compared to our measured values. Between $0.4<z<2$, our measurements
are in fine agreement. However, we observe that the fluctuation parameter in
Illustris appears to be higher at $z\lesssim 0.4$ and lower when $z>2$.
From the redshift-dependent ${\rm DM}_{\rm IGM}$ distributions derived from
IllustrisTNG (figure 2 from Zhang et al. (2021)), distributions between
$0.1<z<0.4$ are wider and the modes of each distribution are spread further
apart. This may explain why the IllustrisTNG fluctuation parameter is higher
than our measurement as the ${\rm DM}_{\rm IGM}$ distribution functions have
larger variance at those redshifts.
To make a proper comparison between our work and Zhang et al. (2021), it may
be necessary to introduce a free parameter for the redshift evolution of
$\sigma_{\rm DM}$ instead of simply fixing the redshift exponent to $-1/2$
(Equation 2).
Figure 9: Fluctuation parameters derived from this work, the fiducial value
from James et al. (2022b), and the IllustrisTNG values from Zhang et al.
(2021). Our measurement on $F$ agrees with the simulated $F$ parameter between
$0.4<z<2$.
## 5 Conclusions
In this work, we have implemented variance in ${\rm DM}_{\rm cosmic}$ as a
free parameter in a forward model of the $z$-DMEG distribution of FRBs. With
this adapted model and a survey of 78 ASKAP and Parkes FRBs, we constrain a
value for the fluctuation parameter, explore degeneracies within the model,
and generate a forecast of the constraint on the fluctuation parameter with a
synthetic survey of 100 localized FRBs. The conclusions we draw from this
analysis are:
* •
Incorporating survey data of 78 (21 with redshifts) FRBs yields a firm lower
limit on $F$. We place the lower limit on $F$ as measured by the survey sample
to be $\log_{10}F>-0.89$ at 99.7% confidence. The 900 MHz and 1.3 GHz surveys
dominate this constraint due to their higher number of localizations to host
galaxies and their associated redshifts.
* •
Forward modeling the FRB data from Parkes and ASKAP, the fluctuation parameter
is degenerate with the Hubble constant $H_{0}$.
* •
We forecast that 100 localized FRBs are sufficient to constrain both an upper
and lower limit on the fluctuation parameter. With the greater count of
localizations, the half-maximum width of the distribution decreases by
$\approx 50\%$.
* •
Extrapolation of the fluctuation parameter from IllustrisTNG shows agreement
between $0.4<z<2.0$. Zhang et al. (2021) measure a higher fluctuation
parameter at low redshift ($z<0.4$) and a lower fluctuation parameter beyond
$z>2$. The former result is likely to be an effect of the rapidly evolving
${\rm DM}_{\rm IGM}$ distribution at low redshift.
Next-generation radio observatories will significantly improve the constraint
on the fluctuation parameter. For example, the Deep Synoptic Array 2000
(DSA-2000) is expected to localize on the order of 10,000 FRBs each year —
enough FRBs to sufficiently characterize the baryonic contents of the IGM
(Hallinan et al., 2019; Ravi et al., 2019).
Additionally, the FRB Line-of-sight Ionization Measurement From Lightcone
AAOmega Mapping (FLIMFLAM) survey is an upcoming spectroscopic survey that
seeks to map the intervening cosmic structures and diffuse cosmic baryons in
front of localized FRBs (Lee et al., 2022). These FRB foreground data taken in
the Southern hemisphere will be used in conjunction with ASKAP FRB
measurements to improve the constraints on the intergalactic baryon
distribution (Lee et al., 2022).
These expansions in FRB surveys with localizations are expected to greatly
improve the constraints on the fluctuation parameter. With these improved
constraints on $F$, one may leverage this novel observable for investigating
feedback and cosmological prescriptions in simulations.
Combining the $F$ parameter with other observables like the thermal SZ effect
that trace the intergalactic baryon distribution, there is ample opportunity
to better inform subgrid feedback models (Muñoz & Loeb, 2018; Pandey et al.,
2023).
## Acknowledgements
J.B. acknowledges support from the University of California Santa Cruz under
the Lamat REU program, funded by NSF grant AST-1852393, and the Yale Science
Technology and Research Scholars Fellowship funded by the Yale College Dean’s
Office. Authors A.G.M. and J.X.P., as members of the Fast and Fortunate for
FRB Follow-up team, acknowledge support from NSF grants AST-1911140,
AST-1910471 and AST-2206490. The authors acknowledge the use of the Nautilus
cloud computing system which is supported by the following US National Science
Foundation (NSF) awards: CNS-1456638, CNS-1730158, CNS-2100237, CNS-2120019,
ACI-1540112, ACI-1541349, OAC-1826967, OAC-2112167.
CWJ and MG acknowledge support by the Australian Government through the
Australian Research Council’s Discovery Projects funding scheme (project
DP210102103).
RMS and ATD acknowledge support through Australian Research Council Future
Fellowship FT190100155 and Discovery Project DP220102305.
The Australian SKA Pathfinder is part of the Australia Telescope National
Facility (https://ror.org/05qajvd42) which is managed by CSIRO. Operation of
ASKAP is funded by the Australian Government with support from the National
Collaborative Research Infrastructure Strategy. ASKAP uses the resources of
the Pawsey Supercomputing Centre. Establishment of ASKAP, the Murchison Radio-
astronomy Observatory and the Pawsey Supercomputing Centre are initiatives of
the Australian Government, with support from the Government of Western
Australia and the Science and Industry Endowment Fund. We acknowledge the
Wajarri Yamatji people as the traditional owners of the Observatory site.
This research is based on observations collected at the European Southern
Observatory under ESO programmes 0102.A-0450(A), 0103.A-0101(A),
0103.A-0101(B), 105.204W.001, 105.204W.002, 105.204W.003, 105.204W.004,
108.21ZF.001, 108.21ZF.002, 108.21ZF.005, 108.21ZF.006, and 108.21ZF.009.
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|
11institutetext: Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The
Netherlands 22institutetext: TNO, Oude Waalsdorperweg 63, 2597 AK The Hague,
The Netherlands
22email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Spot What Matters: Learning Context
Using Graph Convolutional Networks
for Weakly-Supervised Action Detection
Michail Tsiaousis 1Corresponding author1Corresponding authorThis work was
carried out during an internship at TNOThis work was carried out during an
internship at TNO 0000-0002-9355-5026 Gertjan Burghouts 22 0000-0001-6265-7276
Fieke Hillerström 22 0000-0003-1301-3073 Peter van der Putten 11
0000-0002-6507-6896
###### Abstract
The dominant paradigm in spatiotemporal action detection is to classify
actions using spatiotemporal features learned by 2D or 3D Convolutional
Networks. We argue that several actions are characterized by their context,
such as relevant objects and actors present in the video. To this end, we
introduce an architecture based on self-attention and Graph Convolutional
Networks in order to model contextual cues, such as actor-actor and actor-
object interactions, to improve human action detection in video. We are
interested in achieving this in a weakly-supervised setting, i.e. using as
less annotations as possible in terms of action bounding boxes. Our model aids
explainability by visualizing the learned context as an attention map, even
for actions and objects unseen during training. We evaluate how well our model
highlights the relevant context by introducing a quantitative metric based on
recall of objects retrieved by attention maps. Our model relies on a 3D
convolutional RGB stream, and does not require expensive optical flow
computation. We evaluate our models on the DALY dataset, which consists of
human-object interaction actions. Experimental results show that our
contextualized approach outperforms a baseline action detection approach by
more than 2 points in Video-mAP. Code is available at
https://github.com/micts/acgcn.
###### Keywords:
Weakly-Supervised action detection graph convolutional networks relational
reasoning actor-context relations
## 1 Introduction
††Paper presented at the International Workshop on Deep Learning for Human-
Centric Activity Understanding (DL-HAU2020), January 11, 2021
Human action recognition is an important part of video understanding, with
potential applications in robotics, autonomous driving, surveillance, video
retrieval and healthcare. Given a video, spatiotemporal action detection aims
to localize all human actions in space and time, and classify the actions
being performed. The dominant paradigm in action detection is to extend CNN-
based object detectors [13, 18] to learn appearance and motion representations
in order to jointly localize and classify actions in video. The desired output
are action tubes [9]: sequences of action bounding boxes connected in time
throughout the video.
In contrast to object detection, action detection requires learning of both
appearance and motion features. Although spatiotemporal features are essential
for action recognition and detection, they might prove insufficient for
actions that share similar characteristics in terms of appearance and motion.
For example, spatiotemporal features might not be sufficient to differentiate
the action ”Taking Photos” in Fig. 1 from a similar one, such as ”Phoning”,
since both share similar characteristics in space and time (i.e. similar
posture, motion around the head). As humans, we make use of context to put
actions and objects in perspective, which can be an important cue to improve
action recognition. Such contextual cues can refer to actor-object and actor-
actor interactions. For instance, a person holding a camera is more likely to
perform the action ”Taking Photos” than ”Phoning”, and vice versa. CNNs are
able to capture such abstract or distant visual interactions only implicitly
by stacking several convolutional layers, which increases the overall
complexity and number of parameters. Hence, an approach to explicitly model
contextual cues would be beneficial.
We introduce an approach to explicitly learn contextual cues, such as actor-
actor and actor-object interactions, to aid action classification for the task
of action detection. Our model, inspired by recent work on graph neural
networks [12, 27, 32], learns context by performing relational reasoning on a
graph structure using Graph Convolutional Networks (GCN) [12]. A high-level
overview is illustrated in Fig. 1. Given a detected actor in a short video
clip, we construct a graph with an actor node encoding actor features, and
context nodes encoding context features, such as objects and other actors in
the scene. The graph’s adjacency matrix consists of relation values encoding
the importance of context nodes to the actor node, and is learned during
training via gradient descent. Graph convolutions accumulate the learned
context to the actor to obtain contextualized/updated actor features for
action classification. Our model aids explainability by visualizing the
learned adjacency matrix as an attention map that highlight the relevant
context for recognizing the action.
We are interested in an approach to learn these contextual cues using as few
annotated data as possible. Recent works [7, 23, 25, 33] that model contextual
cues for action detection rely on full supervision in terms of actor bounding
box annotations. However, extensive video annotation is time consuming and
expensive [15]. In this work, we are interested in learning context for the
task of weakly-supervised action detection, i.e. action detection when only a
handful of annotated frames are available throughout the action instance.
Following the setting of sparse spatial supervision [31], we train our
contextual model by using up to five actor bounding box annotations throughout
the action instance.
We evaluate our models on the challenging Daily Action Localization in YouTube
(DALY) dataset [31], which consists of 10 action classes of human-object
interactions (e.g. Drinking, Phoning, Brushing Teeth), and is annotated based
on sparse spatial supervision. Therefore, DALY is a suitable test bed to model
context for the task of weakly-supervised action detection.
Our contributions are as follows: 1) We introduce an architecture employing
Graph Convolutional Networks [12] in order to model contextual cues to improve
classification of human actions in videos; 2) Our model aids explainability by
visualizing the graph’s adjacency matrix in the form of attention maps that
highlight the learned context, even in a zero-shot setting, i.e. for actions
and objects unseen during training; 3) We achieve 1) and 2) in a weakly-
supervised setting, i.e. when annotated data are sparse throughout the action
instance; 4) We introduce an intuitive metric based on recall of retrieved
objects in attention maps, in order to quantitatively evaluate how well the
model highlights the important context. As attention maps are often used for
qualitative inspection only, this metric may be of general use beyond our use
case.
In Section 3, we present the baseline model and our approach on learning
context by performing reasoning on a graph structure using GCN [12].
Additionally, we discuss training of these models using sparse spatial
supervision [31]. We conduct experiments and report results in Section 4. In
Section 5, we perform a qualitative and quantitative analysis of attention
maps.
Figure 1: Given spatiotemporal features extracted from an input clip, we
construct a graph with an actor (grey) node and context (numbered) nodes in
order to model relations, such as actor-actor and actor-object interactions.
Graph convolutions accumulate the learned context to the actor to obtain
updated actor features for classification.
## 2 Related Work
### 2.1 Action Recognition and Detection
Action recognition aims to classify the action taking place in a video. Early
approaches relied on two-stream 2D CNNs [21] operating on RGB and optical flow
inputs, or on detectors to classify bounding boxes of components in keyframes
leaving action classification at the video level to traditional machine
learning techniques [3]. Recent works focus on (two-stream) 3D CNNs [4, 17,
24], which perform spatiotemporal (3D) convolutions. While action recognition
considers the classification task, action detection carries out both
classification and detection of actions. Although action detection is usually
addressed using full supervision [7, 9, 23, 25, 30, 33], we are interested in
weak supervision, which allows us to reduce annotation cost by training models
using very few action bounding box annotations per action instance.
Sivan and Xiang [22] approach weakly-supervised action detection using
Multiple Instance Learning (MIL). Their approach requires binary labels at the
video level, indicating the presence of an action. Mettes et al. [15] propose
action annotation using points, instead of boxes. Chéron et al. [5] present a
unified framework for action detection by incorporating varying levels of
supervision in the form of labels at the video level, a few bounding boxes,
etc. Weinzaepfel et al. [31] introduce DALY and the setting of sparse spatial
supervision, in which, up to five bounding boxes are available per action
instance. Chesneau et al. [6] produce full-body actor tubes inferred from
detected body parts, even when the actor is occluded or part of the actor is
not included in the frame.
### 2.2 Visual Relational Reasoning
There has been recent research on augmenting deep learning models with the
ability to perform visual relational reasoning.
Santoro et al. [20] propose the relation network, which models relations
between pairs of feature map pixels for visual question answering. This idea
has been extended for action detection [23, 25] to model actor-context
relations. Similar to [23, 25], we treat every 1$\times$1$\times$1 location of
the feature map as context. In these works, learned actor-context relations
are used either directly [23] or to highlight features of the feature map [25]
for action classification. In contrast, we encode actor-context relations as
edges in a graph, and graph convolutions output updated actor features for
action classification.
With regard to visual attention, non-local neural networks [28] compute the
output of a feature map pixel as a weighted sum of all input pixels. For
action detection, Girdhar et al. [7] extend the Transformer architecture [26]
for action detection. Although a Transformer can be represented as a graph
neural network and vice versa, we argue that a graph representation is simpler
and more intuitive compared to the Transformer representation of Queries, Keys
and Values. Whilst [7] use two transformations of a feature map to represent
context as Keys and Values, this representation does not have a direct
interpretation in a graph. In contrast, we use a single transformation to
obtain context features, which correspond to context nodes in the graph.
Furthermore, our model does not require residual connections [11] nor Layer
Normalization [2], two essential components of the Transformer.
In this work, we apply Graph Convolutional Networks (GCN) [12], which provide
a structured and intuitive way to model relations between nodes in a graph.
Recently, GCN have been used for visual relational reasoning for the tasks of
action recognition [29] and group activity recognition [32]. Zhang et al. [33]
employ GCN [12] for action detection, where nodes represent detected actors
and objects. As we do not require an external object detector, our approach is
suitable to reason with respect to arbitrary context and objects which cannot
be detected, e.g. because the object detector has not been trained to do so.
Whilst aforementioned approaches [7, 23, 25, 33] rely on full actor
supervision during training, our work focuses on learning contextual cues
using weak supervision in terms of actor bounding box annotations. Although
similar attention maps are also presented in [7, 23, 25], our work is the
first to generate them in a zero-shot setting, and introduce a metric to
quantitatively evaluate them.
## 3 Learning Context with GCN
We propose an approach to learn contextual cues, such as actor-actor and
actor-object interactions, by performing relational reasoning on a graph
structure using Graph Convolutional Networks (GCN) [12]. We expect such
contextual cues to improve action classification, as they can be
discriminative for the actions performed, and provide more insight into what
the model has learnt, which benefits interpretability. Moreover, we are
interested in learning context using weak supervision in terms of actor
bounding box annotations. An overview of our proposed GCN model is illustrated
in the second branch of Fig. 2. The input is a short video clip with at least
one actor performing an action. A 3D convolutional network extracts
spatiotemporal features for the input clip, up to a convolutional layer. We
treat every $1\times 1\times 1$ spatiotemporal location of the output feature
map as context, while actor features are extracted by RoI Pooling [8] on the
detected actor’s bounding box and subsequent 3D convolutional layers. We
construct a graph consisting of context nodes and an actor node, with
connections drawn from every context node to the actor node. Relation values
between nodes are encoded in the adjacency matrix, and learned using a dot-
product self-attention mechanism [26, 27]. Graph convolutions accumulate the
learned context to the actor in order to obtain updated/contextualized actor
features for action classification. We compare the GCN model to a baseline
model that uses no context, and classifies the action using the feature
representation corresponding to the actor bounding box.
In this section, we first present the 3D convolutional backbone network to
learn spatiotemporal features using weak supervision. Next, we present our
approach on learning contextual cues for action detection using GCN and we
provide implementation details.
Figure 2: The lowest part shows the graph representation of the GCN model for
a single graph, while its implementation using matrix operations is shown in
the middle part. The top part illustrates the construction of multiple graphs
(multi-head attention) in order to learn different types of actor-context
relations.
### 3.1 Feature Extraction with Weak Supervision
#### 3.1.1 Backbone Network
Spatiotemporal features for the whole input clip are extracted using a 3D
convolutional backbone network. There are several 3D architectures in
literature [4, 17, 24]. We opt for I3D [4], which is widely used and has
demonstrated very positive results in action recognition. The input is a
sequence of frames of size $C\times T\times H\times W$, where $C$ denotes the
number of channels, $T$ is the number of input frames, and $H$ and $W$
represent the height and width of the input sequence. Features are extracted
up to Mixed_4f layer, which has an output feature map of size
$D^{\prime}\times T^{\prime}\times H^{\prime}\times W^{\prime}$, where
$D^{\prime}$ denotes the number of feature channels, $T^{\prime}=\frac{T}{8}$,
$H^{\prime}=\frac{H}{16}$, and $W^{\prime}=\frac{W}{16}$.
#### 3.1.2 Actor Feature Extraction with Weak Supervision
We are interested in learning spatiotemporal features using only a handful of
annotated frames per action instance. For an annotated frame, also called
keyframe, annotation is in the form of an action bounding box and
corresponding class label. Due to the limited number of available annotated
frames, training a Region Proposal Network (RPN) [18] to produce actor box
proposals would be sub-optimal. To this end, we train our models using sparse
spatial supervision as introduced in [31]. In detail, a Faster R-CNN [18]
detects all actors in each frame, and detections are tracked throughout the
action instance using a tracking-by-detection approach [30], which produces
class-agnostic action tubes. In practice, we use tubes provided by [31]. Tubes
are labeled based on spatiotemporal Intersection over Union (IoU) with sparse
annotations, i.e. ground truth tubes comprised of up to 5 bounding boxes
throughout the action instance. Tubes with spatiotemporal IoU greater than 0.5
are assigned to the action class of the ground truth tube with the highest
IoU. If no such ground truth tube exists, the action tube is labeled as
background. The backbone is augmented with a RoI pooling layer [8] to extract
features for each actor for action classification. Boxes of each tube are
appropriately scaled and mapped to the output feature map of Mixed_4f layer,
with a temporal stride of four frames. For each action tube, RoI pooling
extracts actor features of size $D^{\prime}\times T^{\prime}\times 7\times 7$.
Actor features are then passed through I3D tail consisted of 3D convolutional
layers Mixed_5b and Mixed_5c. Finally, a spatiotemporal (3D) average pooling
layer reduces the size to $D^{\prime\prime}\times 1\times 1\times 1$.
### 3.2 Graph Convolutional Networks
#### 3.2.1 Learning Relations
Our graph consists of two types of nodes: context nodes and actor nodes.
Context node features, $f_{j}^{\prime}\in\mathbb{R}^{D^{\prime}\times 1\times
1\times 1}$, $j=1,2,\dots,M$, $M=T^{\prime}H^{\prime}W^{\prime}$, correspond
to every $1\times 1\times 1$ spatiotemporal location of the output feature map
of Mixed_4f layer. Actor node features,
$a_{i}^{\prime}\in\mathbb{R}^{D^{\prime\prime}\times 1}$, $i=1,2,\dots,N$,
where $N$ is the number of detected actors in the input clip, are extracted as
described in Section 3.1. Relations between actor features and context
features, shown in orange arrows in Fig. 2, are learned using a dot-product
self-attention operation [26, 27], after projecting the features in a lower
dimensional space using a linear transformation. Formally,
$\displaystyle e_{ij}=\theta(a_{i}^{\prime})^{T}\cdot\phi(f_{j}^{\prime})$ (1)
where
$\displaystyle
a_{i}=\theta(a_{i}^{\prime})=\mathbf{W}_{\theta}a_{i}^{\prime}+\mathbf{b}_{\theta}$
(2) $\displaystyle
f_{j}=\phi(f_{j}^{\prime})=\mathbf{W}_{\phi}f_{j}^{\prime}+\mathbf{b}_{\phi}$
(3)
Eq. 2–3 are transformations for actor features and context features,
respectively, with $\mathbf{W}_{\theta}\in\mathbb{R}^{D^{\prime\prime}\times
D},\mathbf{W}_{\phi}\in\mathbb{R}^{D^{\prime}\times D}$;
$\mathbf{b}_{\theta},\mathbf{b}_{\phi}\in\mathbb{R}^{D\times 1}$;
$D<D^{\prime},D^{\prime\prime}$. In matrix form,
$\mathbf{A}\in\mathbb{R}^{N\times D}$ for transformed actor features and
$\mathbf{F}\in\mathbb{R}^{M\times D}$ for transformed context features. The
graph is represented by an adjacency matrix, $\mathbf{G}\in\mathbb{R}^{N\times
M}$, where $g_{ij}\in\mathbf{G}$ denotes the relation or attention value,
indicating the importance of context feature, $f_{j}$, to actor feature,
$a_{i}$. Consequently, $\mathbf{G}$ is a directed graph connecting every
context node to every actor node. Relation or attention values, $g_{ij}$, are
obtained by applying softmax normalization on $e_{ij}$ (output of dot-product)
across context features
$\displaystyle g_{ij}=\frac{\exp(e_{ij})}{\sum_{k}\exp(e_{ik})}$ (4)
#### 3.2.2 Graph Convolutions
Having defined the graph and a mechanism for learning actor-context relations,
we perform reasoning on the graph in order to obtain updated actor features.
This is achieved by accumulating information from context nodes to the actor
node using graph convolutions. Updated actor features,
$\mathbf{Z}\in\mathbb{R}^{N\times D}$, are obtained by
$\displaystyle\mathbf{Z}=\sigma\Big{(}\Big{(}\mathbf{G}\mathbf{F}+\mathbf{A}\Big{)}\mathbf{W}\Big{)}$
(5)
The operation is shown in blue arrows in Fig. 2. The weighted average of
$\mathbf{F}$ with the relation values $\mathbf{G}$ produces weighted context
features. Adding actor features $\mathbf{A}$ to the resulting representation
imposes identity links for all actor nodes in the graph. The output is passed
through a learnable linear transformation $\mathbf{W}\in\mathbb{R}^{D\times
D}$ and a non-linear activation function $\sigma(\cdot)$ implemented as ReLU
[10].
In order to capture multiple types of relations between the actor and the
context, we perform multi-head attention [26] by constructing multiple graphs
at a given layer and merging their outputs using concatenation or summation.
Weight matrices $\mathbf{W}_{\theta},\mathbf{W}_{\phi}$, $\mathbf{W}$ are
independent across graphs. Finally, in order to encode updated actor features
on a higher level, we stack multiple GCN layers by providing the output of
multiple graphs as input to the next GCN layer.
#### 3.2.3 Location Embedding
Location information, such as the position of an actor with respect to other
actors and objects, is important for modeling contextual cues. However, such
information, encoded indirectly by regular convolutions, is lost when applying
convolutions on a graph structure.
We incorporate location information in both context features and actor
features. For context features, we concatenate coordinates $(x,y)$ along the
channel dimension before applying $\mathbf{W}_{\phi}$, indicating the location
of the feature on the output feature map. For actor features, we concatenate
coordinates $(cx,cy,w,h)$ before applying $\mathbf{W}_{\theta}$, corresponding
to the average center, width and height of the actor tube across the input
clip. Coordinates are normalized in $[-1,1]$.
### 3.3 Implementation Details
We implement our models in PyTorch [16]. I3D is pre-trained on ImageNet [19]
and then on the Kinetics [4] action recognition dataset, while the external
detector is pre-trained on the MPII Human Pose dataset [1]. The input is a
clip of 32 RGB frames with spatial resolution of $224\times 224$. The output
feature map of Mixed_4f layer has $D^{\prime}=832$ channels, while actor
features have $D^{\prime\prime}=1024$. Transformations $\mathbf{W}_{\theta}$,
$\mathbf{W}$ are implemented as fully connected layers and $\mathbf{W}_{\phi}$
as a 3D convolutional layer with kernel size $1\times 1\times 1$. We set
$D=256$. We apply 3-dimensional dropout to context features before
$\mathbf{W}_{\phi}$. Additionally, 1-dimensional dropout is applied to actor
features before $\mathbf{W}_{\theta}$ in the first GCN layer, before
$\mathbf{W}$ in all GCN layers and prior to the final classification layer (in
both GCN and baseline model). Dropout probability is 0.5 in all cases. All
fully connected layers are initialized using a Normal distribution according
to [10]. We set the gain parameter to 1 for $\mathbf{W}_{\theta}$ and to
$\sqrt{2}$ for the rest of the fully connected layers. $\mathbf{W}_{\phi}$ is
initialized using a Uniform distribution according to [10] in the range
$(-b+0.01,b-0.01)$ for the first GCN layer, and in the range $(-b,b)$ for
subsequent layers, using a gain of $\frac{1}{\sqrt{3}}$. Biases of all layers
are initialized to zero.
Models are optimized using SGD and cosine learning rate annealing, with
learning rate $2.5\cdot 10^{-4}$ over 150 epochs, and $4.7\cdot 10^{-5}$ over
450 epochs, for the baseline and GCN model, respectively. We use a batch size
of 3 clips, where each clip is randomly sampled from a video in the training
set. Tubes of each clip are scored using the softmax scores produced by the
model. During inference, we sample 10 32-frame clips from each video, and
tubes are scored by averaging the softmax scores across the clips. The same
clips are sampled in order to facilitate fair comparison between different
models. Training time is approximately one day for GCN and less than half a
day for the baseline on a GTX 1080 Ti GPU.
## 4 Experiments
In this section, we first describe the DALY dataset and the evaluation metric
used throughout the experiments. Next, we conduct experiments to evaluate the
performance of the GCN model, and we compare it with the baseline and the
state of the art on DALY. Finally, we evaluate the GCN model using minimal
spatial supervision i.e. one bounding box per action instance.
### 4.1 Dataset and Evaluation Metric
We develop and evaluate our models on the Daily Action Localization in Youtube
(DALY) [31] dataset. It consists of 510 videos of 10 human actions, such as
”Drinking”, ”Phoning” and ”Brushing Teeth”. In this paper, we do not perform
temporal localization, and we assume that the temporal boundaries of each
action instance within a video are known. An action instance has an average
duration of 8 seconds and may contain more than one person performing an
action. Each of the 10 classes contains an interaction between a person and an
object that define the action taking place. There are 31 training videos and
20 test videos per class. We fine-tune our models by holding out a subset of
the training set as a validation set, consisted of 10 videos from each class.
We evaluate models using Video-mAP at 0.5 IoU threshold<EMAIL_ADDRESS>[9].
### 4.2 Evaluation of Architecture Choices
In this section, we experimentally evaluate the GCN model with respect to
several architecture choices. Specifically, we experiment with up to two GCN
layers and up to three graphs per layer. Additionally, we compare
concatenation and summation as merging functions to combine the output of
multiple graphs. Finally, we measure the impact of including the location
embedding and the I3D tail (convolutional layers Mixed_5b and Mixed_5c) to
extract actor features.
Results with respect to different number of layers and graphs are shown in
Table 1, along with the number of parameters for every configuration (I3D
parameters are not included). Note that for two GCN layers, the first layer
always employs concatenation as a merging function. Building multiple graphs
is beneficial for model performance, for both functions. It is interesting
that mAP increases for a 2-layer GCN model with concatenation, but not with
summation. Concatenation outperforms summation in nearly all configurations.
For the rest of the experiments, we choose a 2-layer, 2-graph GCN model with
concatenation, which provides a good trade-off between performance and number
of parameters.
In order to measure the impact on model performance obtained by the location
embedding and I3D tail, we remove them from the architecture and examine the
difference in model performance. By removing the location embedding, the model
has no information of the actor’s location relatively to other actors and
objects, and relations are calculated based solely on visual features. This
results in a decrease of 1.1 points in mAP (50.7), which indicates that
modeling spatial actor-context relations improves performance. By removing the
I3D tail, actor features are extracted from the output feature map of Mixed_4f
layer. This results in a significant decrease of more than four points in mAP
(47.42), highlighting the importance of using the I3D tail to encode actor
features.
Table 1: Validation mAP with respect to different number of layers, number of graphs per layer and merging functions to combine the output of multiple graphs. The number of model parameters are provided for every configuration. # Layers | # Graphs | Merging Function | # Parameters | Val. mAP
---|---|---|---|---
1 | 1 | - | 543K | 49.39
1 | 2 | Sum | 1.084M | 51.39
Concat | 1.086M | 51.3
1 | 3 | Sum | 1.624M | 50.7
Concat | 1.630M | 50.98
2 | 1 | - | 887K | 47.28
2 | 2 | Sum | 1.903M | 50.1
Concat | 1.906M | 51.82
2 | 3 | Sum | 3.050M | 49.98
Concat | 3.055M | 52.09
### 4.3 Comparison with Baseline and State of the Art
We compare the GCN model with the baseline model and the state-of-the-art [6,
31] on the DALY test set in Table 2.
The baseline model classifies actor features obtained from I3D (see Section
3.1) using a linear layer that outputs classification scores for $C$ action
classes and a background class. Results are shown in Table 2. Across five
repetitions, the GCN model outperforms the baseline model by 2.24 (3.7%)
points in mean mAP, and by 2.94 (4.9%) points in maximum mAP. The left-hand
side of Fig. 3 illustrates per-class average precision for the baseline and
GCN model. GCN performs comparably or better than the baseline model in all
classes except ”TakingPhotosOrVideos”. On the right-hand side of Fig. 3, we
visualize t-sne [14] actor feature embeddings, colored by the respective
action, for the GCN (top) and baseline model (bottom). The GCN model produces
tighter and more distinct clusters compared to the baseline model.
Comparing the GCN model with the state-of-the-art [6, 31], we obtain slightly
improved performance in comparison to [31], while Chesneau et al. [6] achieve
better performance by 1.69 points in mean mAP and 0.78 points in maximum mAP.
We argue that this is due to the following reasons. Firstly, our models are
trained using fewer videos, since we hold out a part of the training set as a
validation set for fine-tuning. Secondly, [6, 31] train their models on the
region proposals produced by the detector (see Section 3.1), while we train
our models on the data provided by [31], which are only the final detections
of the detector. Consequently, we train our models using fewer videos and
boxes compared to [6, 31]. It is worth noting that, in contrast to [6, 31],
our model does not employ expensive optical flow computation.
Table 2: Comparison of GCN model with the baseline model and state-of-the-art
on the test set. We report model architecture and input modalities (RGB,
Optical Flow).
Model | Architecture | Input | Test mAP
---|---|---|---
Weinzaepfel et al. [31] | Fast R-CNN (VGG-16) | RGB, OF | 61.12
Chesneau et al. [6] | Fast R-CNN (VGG-16) | RGB, OF | 63.51
Baseline (Ours) | I3D | RGB | 59.58 ($\pm$ 0.22) (59.79)
GCN (Ours) | I3D | RGB | 61.82 ($\pm$ 0.51) (62.73)
.
Figure 3: Per-class Video-AP on the test set across five repetitions of GCN
and baseline model, and t-SNE actor feature embeddings of GCN (top) and
baseline model (bottom).
### 4.4 Reducing Annotation to One Bounding Box
We examine model performance when minimal spatial supervision is used, i.e.
one bounding box per action instance. We label tubes based on spatial IoU with
the ground truth box of a randomly selected keyframe for each action instance.
A GCN model is then trained using the newly labeled tubes. Using only a single
keyframe to label tubes, we obtain a small decrease in mAP, from 61.82 to
61.07. On the other hand, when one keyframe is used during evaluation too,
performance decreases by 3.15 points in mAP. The reason for such a decrease is
that mAP is not adequately estimated using only one evaluation keyframe.
## 5 Analysis of Attention
Our GCN model aids explainability by visualizing the adjacency matrix in the
form of attention maps that highlight the learned context, even in a zero-shot
setting, i.e. for actions and objects unseen during training. Although similar
attention maps are presented in previous works [7, 23, 25] (albeit not in a
zero-shot setting), in this paper, we go one step further to quantitatively
evaluate the ability of the attention to highlight the relevant context. To
this end, we propose a metric based on recall of objects retrieved by
attention maps. In this Section, we present qualitative and quantitative
results of attention maps.
### 5.1 Evaluation of Attention Maps
The adjacency matrix contains the relation or attention values, indicating the
importance of every context node (spatiotemporal location of feature map) to
the actor node. By visualizing the adjacency matrix, we obtain an attention
map that highlights, for a given actor, the important context regions the
model pays most attention to. The map is interpolated to the original input
size and overlaid on the input clip. Our model is even able to generalize its
attention in a zero-shot setting i.e. for actions that the model has not been
trained to recognize. To achieve this, we train a GCN model by excluding two
action classes, and then visualize the attention maps for the excluded
classes.
#### 5.1.1 Qualitative Evaluation
Fig. 4 illustrates examples of attention maps, where each one is the
combination of four adjacency matrices (2 GCN layers; 2 graphs per layer) by
summing their values along the spatial dimensions. Each example contains four
attention maps, representing time progression along the input clip. The last
row of Fig. 4 contains zero-shot attention maps for classes ”Ironing” and
”TakingPhotosOrVideos”. The attention maps show that our GCN model highlights
relevant context, such as objects, hands and faces, and is also able to track
objects along time. Finally, our model is able to highlight relevant objects
(e.g. Iron, Camera) for actions unseen during training (last row of Fig. 4).
Figure 4: Per-class object recall curves, along with visualizations of
attention maps (last row illustrates zero-shot cases) for actions “Drinking”,
“CleaningFloor”, “CleaningWindows”, “BrushingTeeth”, “FoldingTextile”,
“Ironing”, “TakingPhotosOrVideos”.
#### 5.1.2 Quantitative Evaluation
We evaluate how well the attention maps highlight relevant objects by
introducing a metric based on recall of objects retrieved by the attention.
DALY provides object bounding box annotations on annotated frames. Given the
attention map produced for a detected actor, we sum the attention values
inside the object’s bounding box. An instance is a true positive if the sum of
values is larger than a threshold, and a false negative, otherwise.
A per-class quantitative evaluation of attention maps is shown in Fig. 4, with
recall on the $y$-axis and the attention threshold on the $x$-axis. Dashed
curves correspond to zero-shot cases. Our metric suggests that objects are
retrieved by the attention with relatively high recall, even for large
attention thresholds, which shows the effectiveness of our model to highlight
the relevant context.
## 6 Conclusion
We propose an approach using Graph Convolutional Networks [12] to model
contextual cues, such as actor-actor and actor-object interactions, to improve
action detection in video. On the challenging DALY dataset [31], our model
outperforms a baseline, which uses no context, by more than 2 points in Video-
mAP, performing on par or better in all action classes but one. The learned
adjacency matrix, visualized as an attention map, aids explainability by
highlighting the learned context, such as objects relevant for recognizing the
action, even in a zero-shot setting, i.e. for actions unseen during training.
We quantitatively evaluate the attention maps using our proposed metric based
on recall of objects retrieved by the attention. Results show the
effectiveness of our model to highlight the relevant objects with high recall.
All the above are achieved in a weakly-supervised setting using only up to
five or even one actor box annotation per action instance. Future work
includes end-to-end model training using weak supervision and modeling
relations between consecutive clips and videos of the same action.
#### Acknowledgements
We would like to thank Philippe Weinzaepfel for providing us with the
predicted action tubes of their tracking-by-detection model.
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|
# A phenomenological note on the missing $\rho_{2}$ meson
Shahriyar Jafarzade
###### Abstract
The $\rho_{2}$ meson is the missing isovector member of the meson nonet with
the quantum numbers $J^{PC}=2^{--}$. It belongs to the class of $\rho$-mesons
such as the vector meson $\rho(770)$, the excited vector $\rho(1700)$ and the
tensor $\rho_{3}(1690)$. Yet, despite the rich experimental and theoretical
studies for other $\rho$-meson states, no resonance that could be assigned to
the $\rho_{2}$ meson has been measured. In this note, we present the results
for the mass and dominant decay channels of the $\rho_{2}$ meson within the
extended Linear Sigma Model.
## 1 Introduction
PDG contains various mesons denoted with the letter $\rho$ [1] . These are the
isovector resonances with quantum number of isospin ($I=1$), of parity
$(P=-1)$, and of charge conjugation ($C=-1$). For instance, the vector mesons
$\rho(770)$ with quantum number $J^{PC}=1^{--}$ [2], the excited vector mesons
$\rho(1450)$, $\rho(1700)$ [5, 4, 3], and the tensor meson $\rho_{3}(1690)$
with quantum number $J^{PC}=3^{--}$ [6]. Despite the prediction of the
$\rho_{2}$ in the Relativistic Quark model [7], it is still missing
experimentally. We only have the following data which were observed from
different experimental groups and listed as “further states” in PDG [1]:
$\rho_{2}(1940)$ and $\rho_{2}(2225)$ with the total decay widths
$\Gamma^{\text{tot}}_{\rho_{2}(1940)}\simeq 155\pm 40$ MeV and
$\Gamma^{\text{tot}}_{\rho_{2}(2225)}\simeq 335^{+100}_{-50}$ MeV accordingly.
Axial tensor mesons are studied in recent LQCD simulations [8], where the
authors consider the mass of $\rho_{2}$ is about $1.7$ GeV as the
$\rho_{3}(1690)$. We present the results about the missing $\rho_{2}$ [9]
within a chiral effective model which is so-called the extended Linear Sigma
Model (eLSM) [2].
## 2 Effective Model and Results
The physical resonances such as the pseudoscalar mesons
{$\pi,K,\eta,\eta^{\prime}(958)$}, the vector mesons {$\rho(770)$,
$\overline{K}^{\star}(892)$, $\omega(782)$, $\phi(1020)$}, and the tensor
mesons {$a_{2}(1320)$, $\overline{K}^{\star}_{2}(1430)$, $f_{2}(1270)$,
$f_{2}^{\prime}(1525)$} together with their chiral partners construct chiral
nonets. Table 1 describes the transformation of the chiral fields under
different symmetries for (pseudo) scalars (P) S, (axial-) vector
($A_{1}^{\mu}$) $V^{\mu}$ and (axial-) tensor ($A_{2}^{\mu\nu}$) $T^{\mu\nu}$
nonets.
Nonet | Parity $(P)$ | Charge conjugation $(C)$ | $U_{R}(3)\times U_{L}(3)$
---|---|---|---
$\Phi(t,\vec{x}):=S(t,\vec{x})+iP(t,\vec{x})$ | $\Phi^{\dagger}(t,-\vec{x})$ | $\Phi^{t}(t,\vec{x})$ | $U_{L}\Phi U^{\dagger}_{R}$
$R^{\mu}(t,\vec{x}):=V^{\mu}(t,\vec{x})-A_{1}^{\mu}(t,\vec{x})$ | $L_{\mu}(t,-\vec{x})$ | $-(L^{\mu}(t,\vec{x}))^{t}$ | $U_{R}R^{\mu}U_{R}^{\dagger}$
$L^{\mu}(t,\vec{x}):=V^{\mu}(t,\vec{x})+A_{1}^{\mu}(t,\vec{x})$ | $R_{\mu}(t,-\vec{x})$ | $-(R^{\mu}(t,\vec{x}))^{t}$ | $U_{L}L^{\mu}U_{L}^{\dagger}$
$\mathbf{R}^{\mu\nu}(t,\vec{x}):=T^{\mu\nu}(t,\vec{x})-A_{2}^{\mu\nu}(t,\vec{x})$ | $\mathbf{L}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{L}^{\mu\nu}(t,\vec{x}))^{t}$ | $U_{R}\mathbf{R}^{\mu\nu}U^{\dagger}_{R}$
$\mathbf{L}^{\mu\nu}(t,\vec{x}):=T^{\mu\nu}(t,\vec{x})+A_{2}^{\mu\nu}(t,\vec{x})$ | $\mathbf{R}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{R}^{\mu\nu}(t,\vec{x}))^{t}$ | $U_{L}\mathbf{L}^{\mu\nu}U^{\dagger}_{L}$
Table 1: Transformations of the chiral multiplets under P, C, and
$U_{R}(3)\times U_{L}(3)$.
The chiral invariant Lagrangian that generates the masses of the spin-2 mesons
reads
$\displaystyle\mathcal{L}_{\text{mass}}=\text{Tr}\Big{[}\Big{(}\frac{m_{\text{ten}}^{2}}{2}+\Delta^{\text{ten}}\Big{)}\Big{(}\mathbf{L}_{\mu\nu}^{2}+\mathbf{R}_{\mu\nu}^{2}\Big{)}+\mathbf{R}^{\mu\nu}\mathbf{R}_{\mu\nu}\Big{]}+2h_{3}^{\text{ten}}\text{Tr}\Big{[}\Phi\mathbf{R}^{\mu\nu}\Phi^{\dagger}\mathbf{L}_{\mu\nu}\Big{]}\text{,}$
(1)
where
$\Delta^{\text{ten}}=\text{diag}\\{0\,,0\,,\delta_{S}^{\text{ten}}=m_{K_{2}}^{2}-m_{\textbf{a}_{2}}^{2}\\}$.
The following three equations are coming from the extended version of the
above lagrangian and relate the masses of spin-2 chiral partners:
$\displaystyle
m_{\rho_{2}}^{2}=m_{a_{2}}^{2}-h_{3}^{\text{ten}}\phi_{N}^{2},\quad
m_{K_{2A}}^{2}=m_{K_{2}}^{2}-\sqrt{2}h_{3}^{\text{ten}}\phi_{N}\phi_{S},\quad
m_{\omega_{2,S}}^{2}$
$\displaystyle=m_{f_{2,S}}^{2}-2h_{3}^{\text{ten}}\phi_{S}^{2}\text{ ,}$ (2)
which leads to $m_{\rho_{2}}=1663$ MeV where we have assumed
$m_{K_{2A}}=m_{K_{2}(1820)}$. The same assumption in the last term of Eq (1)
implies $\Gamma(\rho_{2}\rightarrow a_{2}(1320))\pi\approx 88$ MeV which is
200 MeV in [10]. Our prediction for the mass of $\rho_{2}$ from the
spontaneous breaking of the chiral symmetry is near to the prediction in [7].
The simplest Lagrangian which describes tree level decays has the following
form
$\displaystyle\mathcal{L}=\frac{g_{2}^{\text{ten}}}{2}\Big{(}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}\\{L^{\mu},L^{\nu}\\}\Big{]}+\text{Tr}\Big{[}\mathbf{R}_{\mu\nu}\\{R^{\mu},R^{\nu}\\}\Big{]}\Big{)}\text{
.}$ (3)
We firstly present the results for $a_{2}(1320)$ with the quantum number
$J^{PC}=2^{++}$ based on the Lagrangian (3).
Decay process (in model) | eLSM [9] | PDG [1]
---|---|---
$\,\;a_{2}(1320)\longrightarrow\bar{K}\,K$ | $4.06\pm 0.14$ | $7.0^{+2.0}_{-1.5}$
$\,\;a_{2}(1320)\longrightarrow\pi\,\eta$ | $25.37\pm 0.87$ | $18.5\pm 3.0$
$\,\;a_{2}(1320)\longrightarrow\pi\,\eta^{\prime}(958)$ | $1.01\pm 0.03$ | $0.58\pm 0.10$
Table 2: Decay rates of the $a_{2}(1320)$ into the pseudoscalar mesons in MeV.
Secondly, we present the results in Table 3 for the missing $\rho_{2}$. Note
that, we have used the PDG data in Table 2 to obtain the coupling
$g_{2}^{\text{ten}}$ for presenting the results in the second row of the Table
3. We expect the dominant $\rho_{2}$ decay widths in the interval between the
second and the third rows of the following table which implies it is being
broad despite some uncertainties.
Decay process (in model) | eLSM | eLSM (LQCD) | LQCD [8]
---|---|---|---
$\,\;\rho_{2}(?)\longrightarrow\rho(770)\,\eta$ | $87$ | $30$ | $-$
$\,\;\rho_{2}(?)\longrightarrow\bar{K}^{\ast}(892)\,K+\textbf{c}.\mathrm{c}.$ | $77$ | $27$ | $36$
$\,\;\rho_{2}(?)\longrightarrow\omega(782)\,\pi$ | $376$ | $122$ | $125$
$\,\;\rho_{2}(?)\longrightarrow\phi(1020)\,\pi$ | $0.8$ | $0.3$ | $-$
Table 3: Decay rates of $\rho_{2}$ into the vector and the pseudoscalar mesons
in MeV.
We finally present the result for the well-established tensor meson
$\rho_{3}(1690)$ in Table 4. The decay channel of
$\Gamma(\rho_{3}(1690)\rightarrow\omega(782)\pi)$ which is measured
experimentally too, is 5-6 times smaller than
$\Gamma(\rho_{2}\rightarrow\omega(782)\pi)$ within LQCD simulations in spite
of having the same mass.
Decay process (in model) | PDG [1] | eLSM [6] | LQCD [8]
---|---|---|---
$\,\;\rho_{3}(1690)\longrightarrow\rho(770)\,\eta$ | $-$ | $3.8\pm 0.8$ | $-$
$\,\;\rho_{3}(1690)\longrightarrow\bar{K}^{\ast}(892)\,K+\textbf{c}.\mathrm{c}.$ | $-$ | $3.4\pm 0.7$ | $2$
$\,\;\rho_{3}(1690)\longrightarrow\omega(782)\,\pi$ | $25.8\pm 9.8$ | $35.8\pm 7.4$ | $22$
$\,\;\rho_{3}(1690)\longrightarrow\phi(1020)\,\pi$ | $-$ | $0.036\pm 0.007$ | $-$
Table 4: Decay rates of $\rho_{3}(1690)$ into the vector and the pseudoscalar
mesons in MeV.
## 3 Conclusion
We have studied $\rho_{2}$ axial-tensor meson, chiral partner of the tensor
meson $a_{2}(1320)$ in the framework of a chiral model for low-energy QCD. We
predict its mass to be around $1.663$ GeV similar to the Relativistic Quark
model prediction. Because of the chiral symmetry, the parameter determined in
the tensor sector allows to make predictions for unknown ground-state axial-
tensor resonance. The effective model fitting to the LQCD results is also
presented.
## Acknowledgement
I am thankful to Adrian Königstein for collaboration in [6] , Milena
Piotrowska, Arthur Verijeken in [9] and Francesco Giacosa for his reading the
manuscript as well as collaboration in [6, 9]. I acknowledge financial support
through the project “Development Accelerator of the Jan Kochanowski University
of Kielce”, co-financed by the European Union under the European Social Fund,
with no. POWR.03.05. 00-00-Z212 / 18 and support through the NCN OPUS no.
2018/29/B/ST2/02576.
## References
* [1] R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01 doi:10.1093/ptep/ptac097
* [2] D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev. D 87 (2013) no.1, 014011 doi:10.1103/PhysRevD.87.014011 [arXiv:1208.0585 [hep-ph]].
* [3] M. Piotrowska and F. Giacosa, PoS Hadron2017 (2018), 237 doi:10.22323/1.310.0237 [arXiv:1712.05617 [hep-ph]].
* [4] M. Piotrowska and F. Giacosa, EPJ Web Conf. 182 (2018), 02097 doi:10.1051/epjconf/201818202097 [arXiv:1712.01087 [hep-ph]].
* [5] M. Piotrowska, C. Reisinger and F. Giacosa, Phys. Rev. D 96 (2017) no.5, 054033 doi:10.1103/PhysRevD.96.054033 [arXiv:1708.02593 [hep-ph]].
* [6] S. Jafarzade, A. Koenigstein and F. Giacosa, Phys. Rev. D 103 (2021) no.9, 096027 doi:10.1103/PhysRevD.103.096027 [arXiv:2101.03195 [hep-ph]].
* [7] S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985), 189-231 doi:10.1103/PhysRevD.32.189
* [8] C. T. Johnson et al. [Hadron Spectrum], Phys. Rev. D 103 (2021) no.7, 074502 doi:10.1103/PhysRevD.103.074502 [arXiv:2012.00518 [hep-lat]].
* [9] S. Jafarzade, A. Vereijken, M. Piotrowska and F. Giacosa, Phys. Rev. D 106 (2022) no.3, 036008 doi:10.1103/PhysRevD.106.036008 [arXiv:2203.16585 [hep-ph]].
* [10] D. Guo, C. Q. Pang, Z. W. Liu and X. Liu, Phys. Rev. D 99 (2019) no.5, 056001 doi:10.1103/PhysRevD.99.056001 [arXiv:1901.03518 [hep-ph]].
|
# Remarks on pseudo-continuity
John Cotrina Universidad del Pacífico, Lima, Perú. Email<EMAIL_ADDRESS>
###### Abstract
In this work we extend a maximum theorem proposed by Morgan and Scalzo. We
also show some results of minimax inequalities which are equivalent to the
famous Ky Fan minimax inequality. Additionally, we prove that the existence
result of Nash equilibria proposed by Morgan and Scalzo is actually equivalent
to a classical result in the literature.
Keywords: Maximum theorem; Minimax inequality; Nash games; Pseudo-continuity
MSC (2010): 47H10; 54C60; 91A10
## 1 Introduction
Historically, the concept of pseudo-continuity was introduced by Morgan and
Scalzo [18], where they used first the name _sequential pseudo-continuity_.
They, in [17, 21], showed that pseudo-continuity for functions is equivalent
to continuity for preference relations, among other properties. They also
studied, in [20], the asymptotical behaviour of finite economies. Moreover,
they generalized the Tikhonov well-posed result for optimization problems in
[19].
The notion of pseudo-continuity has gained more attention, for instance Anh,
Khanh and Van [3] studied the well-posedness for quasi-equilibrium problems
and quasi-optimization problems. Wangkeeree, Bantaojai and Yimmuang [29] dealt
with continuity properties of the set-valued solution map for a special kind
of quasi-equilibrium problem. Al-Homidan, Hadjisavvas and Shaalan [1]
implicitly used the concept of pseudo-continuity in order to characterize
quasi-convex functions as a composition of two functions where one of them has
the property that every local minimum is a global minimum. Recently, in [5]
the authors used the pseudo-continuity assumption in order to establish the
existence of solution for a particular generalized Nash game proposed by Rosen
[26], which generalizes many existence results in the literature.
The Berge maximum theorem [2] is an important tool in the area of general
equilibrium theory and in mathematical economics. It establishes that the
argmax correspondence is upper hemicontinuous and its value function is
continuous, under certain continuity assumptions. In [17, 21], Morgan and
Scalzo presented a maximum theorem under a pseudo-continuity assumption.
However, they do not say anything about the value function. In that sense, we
extend the result proposed by Morgan and Scalzo showing that the value
function is pseudo-continuous.
Relaxing the continuity assumption allows us to deal with a large class of
problems in optimization and game theory. One of the first existence result of
Nash equilibria for discontinuous games is due Dasgupta and Maskin [9], which
generalizes the one given by Debreu [10] for continuous games. In a similar
way, Morgan and Scalzo [21] presented a generalization of Debreu’s result
using pseudo-continuity, despite their result being a consequence of the one
proposed by Reny [25]. However, we will show that their result can be obtain
from Debreu’s theorem thanks to a Scalzo’ result in [27]. On the other hand,
Qiu and Peng [23] showed a result of minimax inequality under the pseudo-
continuity assumption. We will show that this result is actually equivalent to
Ky Fan’s minimax inequality.
The remainder of the paper is organized as follows. In Section 2, we introduce
the concept of pseudo-continuity for functions and continuity correspondences
used in our study. Section 3 is devoted to the Berge maximum theorem. In
Section 4 we present some results about pseudo-continuity, which are
equivalent to the famous Ky Fan’s minimax inequality. Finally, in Section 5 we
show that the results on the existence of Nash equilibria proposed by Morgan
and Scalzo [17, 21], Debreu [10] and, Arrow and Debreu [4] are equivalent.
## 2 Definitions, Notations and Preliminary results
Given a topological space $X$, an extended real-valued function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$ is said to be:
* •
_upper semi-continuous_ if, for any $x\in X$ and any $\lambda\in\mathbb{R}$
such that $f(x)<\lambda$, there exists a neighbourhood $\mathscr{V}_{x}$ of
$x$ satisfying
$f(x^{\prime})<\lambda,\mbox{ for all }x^{\prime}\in\mathscr{V}_{x};$
* •
_upper pseudo-continuous_ if, for any $x,y\in X$ such that $f(x)<f(y)$, there
exists a neighbourhood $\mathscr{V}_{x}$ of $x$ satisfying
$f(x^{\prime})<f(y),\mbox{ for all }x^{\prime}\in\mathscr{V}_{x};$
* •
_transfer upper continuous_ if, for any $x,y\in X$ such that $f(x)<f(y)$,
there exist a neighbourhood $\mathscr{V}_{x}$ of $x$, and $y^{\prime}\in X$
satisfying
$f(x^{\prime})<f(y^{\prime}),\mbox{ for all }x^{\prime}\in\mathscr{V}_{x}.$
###### Remark 2.1.
Clearly, upper semi-continuity implies upper pseudo-continuity, which in turn
implies transfer upper continuity. The converse implications are not true in
general. The concept of pseudo-continuity was introduced by Morgan and Scalzo
in [21, 17], while the transfer continuity was studied by Tian and Zhou in
[28].
On the other hand, it is well known that the sum of two upper semi-continuous
function is upper semi-continuous. However, in [8] the authors showed that the
same does not hold for transfer upper continuity. In a similar way, it does
not hold for upper pseudo-continuity, a contra-example can be found in [29].
Associated to the function $f$ we consider the following set
$U_{f}(\lambda)=\\{x\in X:~{}f(x)\geq\lambda\\},$
where $\lambda\in\mathbb{R}$. It is clear that a function $f$ is upper semi-
continuous if, and only if, the set $U_{f}(\lambda)$ is closed, for all
$\lambda\in\mathbb{R}$. In a similar way, $f$ is upper pseudo-continuous if,
and only if, the set $U_{f}(\lambda)$ is closed, for all $\lambda\in f(X)$,
see [21, 17]. Moreover, in [28] the authors showed that $f$ is transfer upper
continuous if, and only if,
$\bigcap_{x\in X}U_{f}(x)=\bigcap_{x\in X}\overline{U_{f}(x)},$
where we use $U_{f}(x)$ instead $U_{f}(f(x))$.
The function $f$ is said to be lower (pseudo) semi-continuous, if $-f$ is
upper (pseudo) semi-continuous. Also, a real-valued function is (pseudo)
continuous if, and only if, it is both upper and lower (pseudo) semi-
continuous.
The following result is Theorem 3.2 in [27].
###### Proposition 2.2 (Scalzo).
Let $X$ be a connected topological space and $f:X\to\mathbb{R}$ be a function.
Then $f$ is pseudo-continuous if, and only if, there exist a continuous
function $u:X\to\mathbb{R}$ and a increasing function $h:u(X)\to\mathbb{R}$
such that
$f=h\circ u.$
###### Remark 2.3.
A similar result to the previous one is Proposition 4 in [11].
Thanks to Proposition 2.2 in [20], we can state Rader’s utility representation
[24] as follow.
###### Proposition 2.4 (Rader).
Let $X$ be a topological space with a countable base and $f:X\to\mathbb{R}$ be
a function. Then $f$ is upper pseudo-continuous if, and only if, there exist a
upper semi-continuous function $u:X\to\mathbb{R}$ and an increasing function
$h:u(X)\to\mathbb{R}$ such that
$f=h\circ u.$
Given a convex set $X$ of a vector space, a function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$ is said to be _quasi-concave_ if, for
all $\lambda\in\mathbb{R}$, $U_{f}(\lambda)$ is convex. Also, $f$ is called
_quasi-convex_ if, $-f$ is quasi-concave.
As a consequence of Proposition 2.4, we give an answer to a question proposed
by Al-Homidan, Hadjisavvas and Shaalan in [1]. Before that we need to
introduce the following definition: A function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$ is said to be _neatly quasi-
concave_ [1] if it is quasi-concave and for every $x$ with $f(x)<\sup f$, the
sets $U_{f}(x)$ and $U_{f}^{<}(x)=\\{y\in\mathbb{R}^{n}:~{}f(y)>f(x)\\}$ have
the same closure.
The following result is Theorem 4.1 in [1], but we state it in terms of quasi-
concavity instead of quasi-convexity.
###### Theorem 2.5.
For every quasi-concave and upper pseudo-continuous function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$, there exist a neatly quasi-
concave and upper pseudo-continuous function $g:\mathbb{R}^{n}\to\mathbb{R}$,
and a nonincreasing function
$h:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$ such that $f=h\circ g$.
The question proposed by Al-Homidan, Hadjisavvas and Shaalan in [1] is the
following: _Is it possible to choose an upper semicontinuous function $g$ in
the previous theorem when $f$ is upper semicontinuous_? The following
corollary gives a positive answer to this question.
###### Corollary 2.6.
For every quasi-concave and upper pseudo-continuous function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$, there exist a neatly quasi-
concave and upper semicontinuous function $g:\mathbb{R}^{n}\to\mathbb{R}$, and
a nonincreasing function $h:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$
such that $f=h\circ g$.
###### Proof.
From Theorem 2.5, there exist a neatly quasi-concave function
$g_{1}:\mathbb{R}^{n}\to\mathbb{R}$, and a nonincreasing function
$h_{1}:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$ such that
$f=h_{1}\circ g_{1}$. Moreover, the function $g_{1}$ is upper pseudo-
continuous. Thus, Proposition 2.4 guarantees the existence of an upper semi-
continuous function $g:\mathbb{R}^{n}\to\mathbb{R}$ and an increasing function
$v:g(\mathbb{R}^{n})\to\mathbb{R}$ such that $g_{1}=v\circ g$. It is not
difficult to see that $g$ is neatly quasi-concave. Since $h_{1}\circ v$ is a
nonincreasing function, the result follows. ∎
Given a convex set $X$ of a vector space and a function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$, the _quasi-concave regularization_
$f_{q}$ of $f$ is defined as
$f_{q}(x)=\sup\\{\lambda\in\mathbb{R}:~{}x\in\operatorname*{conv}(U_{f}(\lambda))\\},$
where $\operatorname*{conv}(U_{f}(\lambda))$ means the convex hull of
$U_{f}(\lambda)$. The function $f_{q}$ is the smallest quasi-concave function
which is lower bounded by $f$.
###### Lemma 2.7.
Let $X$ be a convex set of a vector space and $f,g:X\to\mathbb{R}$ be two
functions such that $f\leq g$. If $g$ is quasi-concave, then $f_{q}$ is a
real-valued function. Moreover, $f\leq f_{q}\leq g$.
###### Proof.
It follows from definition of quasi-concave regularization. ∎
We finish this section recalling continuity notions for correspondences.
Before that, we introduce the concept of correspondence.
Let $U,V$ be non-empty sets. A _correspondence_ or _set-valued map_
$T:U\rightrightarrows V$ is an application $T:U\to\mathcal{P}(V)$, that is,
for $u\in U$, $T(u)\subset V$. The _graph_ of $T$ is defined as
$\operatorname{gra}(T)=\big{\\{}(u,v)\in U\times V\>:\>v\in T(u)\big{\\}}.$
Let $T:X\rightrightarrows Y$ be a correspondence with $X$ and $Y$ two
topological spaces. The map $T$ [2] is said to be:
* •
_closed_ , when $\operatorname{gra}(T)$ is a closed subset of $X\times Y$;
* •
_lower hemicontinuous_ when for all $x\in X$ and any open set $V\subset Y$,
with $T(x)\cap V\neq\emptyset$, there exists $\mathscr{V}_{x}$ neighbourhood
of $x$ such that $T(x^{\prime})\cap V\neq\emptyset$ for all
$x^{\prime}\in\mathscr{V}_{x}$;
* •
_upper hemicontinuous_ when for all $x\in X$ and any open set $V$, with
$T(x)\subset V$, there exists $\mathscr{V}_{x}$ neighbourhood of $x$ such that
$T(\mathscr{V}_{x})\subset V$;
* •
_continuous_ when it is upper and lower hemicontinuous.
We state below the well known Kakutani’s fixed point theorem, see [14, 13].
###### Theorem 2.8 (Kakutani-Fan-Glicksberg).
Let $X$ be a non-empty convex and compact subset of a Hausdorff locally convex
topological vector space $Y$ and let $T:X\rightrightarrows X$ be a
correspondence. If $T$ is upper hemicontinuous with convex, closed and non-
empty values, then there exists $x_{0}\in X$ such that $x_{0}\in T(x_{0})$.
The following result is an extension to the previous one, see [16] for more
details.
###### Theorem 2.9 (Fan-Browder).
Let $X$ be a compact, convex and nonempty subset of a topological vector space
and $T:X\rightrightarrows X$ be a correspondence with convex values. If $T$
has open fibres, i.e. $\\{x\in X:y\in T(x)\\}$ is open for all $y\in X$, then
there exists $x_{0}\in X$ such that $T(x_{0})=\emptyset$ or $x_{0}\in
T(x_{0})$.
## 3 On the Morgan-Scalzo-Berge maximum theorem
The Berge maximum theorem can be stated as follows (see [2]).
###### Theorem 3.1.
Let $X$, $Y$ be two topological spaces, $S:X\rightrightarrows Y$ be a
continuous correspondence with non-empty and compact values, and $f:X\times
Y\to\mathbb{R}$ be a continuous function. Then the “argmax” correspondence
$M:X\rightrightarrows Y$, defined as
$\displaystyle M(x)=\\{y\in S(x):f(x,y)=m(x)\\}$ (1)
is upper hemicontinuous and has non-empty compact values. Moreover, the “value
function” $m:X\to\mathbb{R}$ defined as
$\displaystyle m(x)=\max_{y\in S(x)}f(x,y)$ (2)
is continuous.
Let $X$ and $Y$ be topological spaces and $T:X\rightrightarrows Y$ be a set-
valued map. A function $f:X\times Y\to\mathbb{R}$ is called _quasi-transfer
upper continuous_ [28] on $T$ if, for all
$(x,y),~{}(x,z)\in\operatorname{gra}(T)$ with $f(x,y)<f(x,z)$, there exists a
neighbourhood $\mathscr{V}_{(x,y)}$ such that for any
$(x^{\prime},y^{\prime})\in\mathscr{V}_{(x,y)}\cap\operatorname{gra}(T)$ there
is $z^{\prime}\in T(x^{\prime})$ satisfying
$f(x^{\prime},y^{\prime})<f(x^{\prime},z^{\prime}).$
Tian and Zhou [28] noticed that if $f$ is continuous and $T$ is lower
hemicontinuous, then $f$ is quasi-transfer upper continuous on $T$. In a
similar way, we have the same result with pseudo-continuity instead continuity
of $f$.
###### Proposition 3.2.
Let $X$ and $Y$ be topological spaces, $T:X\rightrightarrows Y$ be a set-
valued map and $f:X\times Y\to\mathbb{R}$ be a function. If $f$ is pseudo-
continuous and $T$ is lower hemicontinuous, then $f$ is quasi-transfer upper
continuous on $T$.
###### Proof.
Let $(x,y),~{}(x,z)\in\operatorname{gra}(T)$ such that $f(x,y)<f(x,z)$. If
there exists $(a,b)\in\operatorname{gra}(T)$ such that
$\displaystyle f(x,y)<f(a,b)<f(x,z)$ (3)
then by pseudo-continuity of $f$ there are neighbourhoods
$\mathscr{V}_{x},\mathscr{V}_{y}$ and $\mathscr{V}_{z}$ respectively of $x,y$
and $z$, such that
$\displaystyle f(x^{\prime},y^{\prime})<f(a,b)<f(x^{\prime},z^{\prime})\mbox{
for all
}(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z}.$
(4)
Since $\mathscr{V}_{y}\cap T(x)\neq\emptyset$ and $\mathscr{V}_{z}\cap
T(x)\neq\emptyset$, we deduce that there is a neighbourhood
$\hat{\mathscr{V}}_{x}$ of $x$ such that
$\mathscr{V}_{y}\cap T(x^{\prime})\neq\emptyset\mbox{ and }\mathscr{V}_{z}\cap
T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x},$
due to the lower hemicontinuity of $T$. We set
$\mathscr{U}_{x}=\mathscr{V}_{x}\cap\hat{\mathscr{V}}_{x}$, and we can see
that for all $(x^{\prime},y^{\prime})\in\mathscr{U}_{x}\times\mathscr{V}_{y}$,
there exists $z^{\prime}\in\mathscr{V}_{z}\cap T(x^{\prime})$ such that (4)
holds.
If there is not $(a,b)\in\operatorname{gra}(T)$ such that (3) holds, then
there exist neighbourhoods $\mathscr{V}_{x},\mathscr{V}_{y}$ and
$\mathscr{V}_{z}$ respectively of $x,y$ and $z$, such that
$\displaystyle f(x^{\prime},y^{\prime})<f(x,z)\mbox{ and
}f(x,y)<f(x^{\prime},z^{\prime})\mbox{ for all
}(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z},$
because $f$ is pseudo-continuous. This implies
$\displaystyle f(x^{\prime},y^{\prime})\leq f(x,y)\mbox{ and }f(x,z)\leq
f(x^{\prime},z^{\prime}),$ (5)
for all
$(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z}$
such that
$(x^{\prime},y^{\prime}),(x^{\prime},z^{\prime})\in\operatorname{gra}(T)$.
Now, following the same steps in the previous case, thanks the lower semi-
continuity of $T$, there is a neighbourhood $\hat{\mathscr{V}}_{x}$ of $x$
such that
$\mathscr{V}_{y}\cap T(x^{\prime})\neq\emptyset\mbox{ and }\mathscr{V}_{z}\cap
T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x}.$
Take $\mathscr{U}_{x}=\mathscr{V}_{x}\cap\hat{\mathscr{V}}_{x}$, thus for all
$(x^{\prime},y^{\prime})\in\mathscr{U}_{x}\times\mathscr{V}_{y}\cap\operatorname{gra}(T)$,
there exists $z^{\prime}\in\mathscr{V}_{z}\cap T(x^{\prime})$ such that (5)
holds. Hence $f(x^{\prime},y^{\prime})<f(x^{\prime},z^{\prime})$, and this
proves that $f$ is quasi-transfer upper continuous on $T$. ∎
We state below a generalization of Berge’s maximum theorem due to Tian and
Zhou [28].
###### Theorem 3.3.
Let $X$ and $Y$ be two topological spaces, $T:X\rightrightarrows Y$ be a non-
empty compact-valued closed correspondence and $f:X\times Y\to\mathbb{R}$ be a
function. Then the best response correspondence $M$, defined as in (1), is
non-empty, compact-valued and closed if, and only if, the function
$f(x,\cdot)$ is transfer upper continuous on $T(x)$, for every $x\in X$; and
$f$ is quasi-transfer upper continuous in $(x,y)$ with respect to $T$. If, in
addition $T$ is upper hemicontinuous, then so is $M$.
The following result is an extension of Theorem 3.1 in [17] and a
generalization of Theorem 3.1 in [21].
###### Theorem 3.4.
Let $X$ and $Y$ be two topological spaces, $T:X\rightrightarrows Y$ be a
correspondence and $f:X\times Y\to\mathbb{R}$ be a function. If $f$ is pseudo-
continuous and $T$ is continuous with non-empty and compact values, then the
argmax correspondence $M$, defined as in (1), is upper hemicontinuous and the
value function $m$, defined as in (2), is real-valued and pseudo-continuous.
###### Proof.
From Proposition 3.2 and Theorem 3.3, we deduce that the argmax correspondence
$M$ is upper hemicontinuous.
Since for each $x$, the set $T(x)$ is non-empty and compact, the function
$f(x,\cdot)$ attains it maximum. Thus $m$ is real-valued. Now, we will show
that $m$ is pseudo-continuous. First, we will prove that $m$ is lower pseudo-
continuous. Let $x_{1}$ and $x_{2}$ be two elements of $X$ such that
$m(x_{1})>m(x_{2})$. There exist $y_{1}\in T(x_{1})$ and $y_{2}\in T(x_{2})$
such that
$f(x_{1},y_{1})=m(x_{1})>m(x_{2})=f(x_{2},y_{2}).$
Since $f$ is lower pseudo-continuous, there are neighbourhoods
$\mathscr{V}_{x_{1}}$ and $\mathscr{V}_{y_{1}}$, respectively of $x_{1}$ and
$y_{1}$, such that
$f(x^{\prime},y^{\prime})>f(x_{2},y_{2}),\mbox{ for all
}(x^{\prime},y^{\prime})\in\mathscr{V}_{x_{1}}\times\mathscr{V}_{y_{1}}.$
On the other hand, as $\mathscr{V}_{y_{1}}\cap T(x_{1})\neq\emptyset$ there is
a neighbourhood $\hat{\mathscr{V}}_{x_{1}}$ of $x_{1}$ satisfying
$\mathscr{V}_{y_{1}}\cap T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x_{1}},$
this is a consequence of the lower semi-continuity of $T$. Therefore, we
deduce that for all
$(x^{\prime},y^{\prime})\in(\mathscr{U}_{x_{1}}\times\mathscr{V}_{y_{1}})\cap\operatorname{gra}(T)$,
where $\mathscr{U}_{x_{1}}=\hat{\mathscr{V}}_{x_{1}}\cap\mathscr{V}_{x_{1}}$,
the following holds
$m(x^{\prime})\geq f(x^{\prime},y^{\prime})>f(x_{2},y_{2})=m(x_{2}).$
Finally, we will prove that $m$ is upper pseudo-continuous. Let $x_{1}$ and
$x_{2}$ be two elements of $X$ such that $m(x_{1})<m(x_{2})$. There exist
$y_{1}\in T(x_{1})$ and $y_{2}\in T(x_{2})$ such that
$f(x_{1},y_{1})=m(x_{1})<m(x_{2})=f(x_{2},y_{2}).$
We distinguish the following two cases.
First, if there exists $(x_{0},y_{0})\in X\times Y$ such that
$m(x_{1})<f(x_{0},y_{0})<m(x_{2})$. We have that $f(x_{1},y)<f(x_{0},y_{0})$,
for all $y\in K(x_{1})$. Thus, there are open neighbourhoods $\mathscr{V}_{y}$
and $\mathscr{V}_{x_{1}}^{y}$, respectively of $y$ and $x_{1}$ such that
$\displaystyle f(x^{\prime},y^{\prime})<f(x_{0},y_{0}),\mbox{ for all
}(x^{\prime},y^{\prime})\in\mathscr{V}_{x_{1}}^{y}\times\mathscr{V}_{y}.$ (6)
The family of set $\\{\mathscr{V}_{y}\\}_{y\in K(x_{1})}$ is an open covering
of $K(x_{1})$, which is compact, thus it can be covered by $n$ neighbourhoods
$\mathscr{V}_{y_{i}}$. That means
$K(x_{1})\subset\bigcup_{i=1}^{n}\mathscr{V}_{y_{i}}$. By upper semicontinuity
of $T$, there exists open neighbourhood $\mathscr{V}_{x_{1}}^{0}$ such that
$\displaystyle K(x)\subset\bigcup_{i=1}^{n}\mathscr{V}_{y_{i}},\mbox{ for all
}x\in\mathscr{V}_{x_{1}}^{0}.$ (7)
For each $x\in\mathscr{V}_{x_{1}}=\bigcap_{i=0}^{n}\mathscr{V}_{x_{1}}^{i}$
and each $y\in K(x)$, we have from (6) and (7) the following
$f(x,y)<f(x_{0},y_{0}).$
Consequently, $m(x)\leq f(x_{0},y_{0})<m(x_{2})$.
Second, assume that there is not any $(x_{0},y_{0})\in X\times Y$ such that
$m(x_{1})<f(x_{0},y_{0})<m(x_{2})$. We have that $f(x_{1},y)<f(x_{2},y_{2})$,
for all $y\in K(x_{1})$. By the same steps given in the previous part, there
exists a neighbourhood $\mathscr{V}_{x_{1}}$ of $x_{1}$ such that
$f(x,y)<f(x_{2},y_{2}),\mbox{ for all }x\in\mathscr{V}_{x_{1}}\mbox{ and all
}y\in K(x).$
Since $f(x,\cdot)$ attains its maximum on $K(x)$, we deduce that
$m(x)<m(x_{2})$. ∎
## 4 Equivalent results of Ky Fan’s minimax inequality
The following result is the famous minimax inequality due to Ky Fan [15].
###### Theorem 4.1 (Ky Fan).
Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ be a real-valued function defined on $X\times X$ such that
1. (i)
for each $y\in X$, $f(\cdot,y)$ is lower semicontinuous;
2. (ii)
for each $x\in X$, $f(x,\cdot)$ is quasi-concave.
Then, the minimax inequality
$\min_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}f(x,x)$
holds.
The Ky Fan minimax inequality is equivalent to the Fan-Browder theorem, we
suggest [16] in order to see this equivalence.
Now, we state a similar result where we use pseudo-continuity instead lower
semicontinuity.
###### Proposition 4.2.
Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ be a real-valued function defined on $X\times X$ such that
1. (i)
$f$ is pseudo-continuous;
2. (ii)
for each $x\in X$, $f(x,\cdot)$ is quasi-concave.
Then, the minimax inequality
$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{x\in X}f(x,x)$
holds.
###### Proof.
By Theorem 3.4, the argmax correspondence $M:X\rightrightarrows X$ defined as
$M(x)=\left\\{y\in X:~{}f(x,y)=\max_{z\in X}f(x,z)\right\\}$
is upper hemicontinuous with compact, convex and nonempty values. Thus, there
exists $x_{0}\in X$ such that $x_{0}\in M(x_{0})$, due to Kakutani’s fixed
point theorem. That means $f(x_{0},x_{0})\geq f(x_{0},y)$, for all $y\in X$.
Therefore,
$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{y\in
X}f(x_{0},y)=f(x_{0},x_{0})\leq\max_{x\in X}f(x,x).$
∎
As a direct consequence of the previous proposition we recover the following
result concerning the existence of fixed points.
###### Corollary 4.3 (Browder).
Let $X$ be a compact, convex and nonempty subset of $\mathbb{R}^{n}$ and
$h:X\to X$ be a continuous function. Then there exists $x_{0}\in X$ such that
$x_{0}=h(x_{0})$.
###### Remark 4.4.
Since Corollary 4.3 implies Browder-Fan’s theorem, so this last one is a
consequence of Proposition 4.2.
At first glance it seems that Theorem 4.1 and Proposition 4.2 are independent,
because pseudo-continuity does not imply lower semicontinuity, and conversely
lower semicontinuity in the second variable does not imply pseudo-continuity
in both variables. However, we will show that these are equivalent.
###### Proposition 4.5.
Theoren 4.1 and Proposition 4.2 are equivalent.
###### Proof.
First, we will prove that Theorem 4.1 implies Proposition 4.2. Since $X\times
X$ is a convex set, in particular it is connected. By Proposition 2.2, there
exists a continuous function $u:X\times X\to\mathbb{R}$ and an increasing
function $h:u(X\times X)\to\mathbb{R}$ such that $f=h\circ u$. Moreover, for
each $x$ we have $U_{f(x,\cdot)}(y)=U_{u(x,\cdot)}(y)$, for all $y\in X$. In
other words, $u(x,\cdot)$ is quasi-concave, for every $x\in X$. Thus, by
Theorem 4.1 we have
$\min_{x\in X}\max_{y\in X}u(x,y)\leq\max_{x\in X}u(x,x).$
Since $u$ is continuous, there exist $x_{0}$ and $x_{1}$ both in $X$ such that
$u(x_{0},y)\leq u(x_{1},x_{1}),$
for all $y\in X$. Thus, $f(x_{0},y)=h(u(x_{0},y))\leq h(u(x_{1},x_{1}))\leq
f(x_{1},x_{1})$ and consequently
$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{x\in X}f(x,x).$
Conversely, since Proposition 4.2 implies Theorem 2.9 and this is equivalent
to Theorem 4.1, the result follows.
∎
Another consequence of Ky Fan’s minimax inequality is stated below.
###### Theorem 4.6.
Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ and $g$ be two real-valued function defined on $X\times X$ such that
1. (i)
for all $x,y\in X$, $f(x,y)\leq g(x,y)$;
2. (ii)
for each $y\in X$, the function $f(\cdot,y)$ is lower semicontinuous;
3. (iii)
for each $x\in X$, the function $g(x,\cdot)$ is quasi-concave;
4. (iv)
for all $x\in X$, $g(x,x)\leq 0$.
Then, there exists $x_{0}\in X$ such that $f(x_{0},y)\leq 0$, for all $y\in
X$.
###### Proof.
For each $x$, we denote by $f_{q}(x,\cdot)$ the quasi-concave regularization
of $f(x,\cdot)$. Since $g$ is quasi-concave in its second argument,
$f_{q}(x,\cdot)$ is real-valued, due to Lemma 2.7. Moreover, $f(x,y)\leq
f_{q}(x,y)\leq g(x,y)$, for all $x,y\in X$. By Proposition 3.10 in [7],
$f_{q}$ is lower semicontinuous in its first argument. Thus, there exists
$x_{0}\in X$ such that $f_{q}(x_{0},y)\leq 0$, for all $y\in X$, due to
Theorem 4.1. Therefore, the result follows. ∎
###### Remark 4.7.
The previous result is actually equivalent to Theorem 4.1. For that, it is
enough to show that Theorem 4.1 is a consequence of Theorem 4.6. In that
sense, we denote $\alpha=\sup_{x\in X}f(x,x)$. If $\alpha=+\infty$ there is
nothing to prove. Otherwise, we define the function $h:X\times X\to\mathbb{R}$
by $h(x,y)=f(x,y)-\alpha$, which satisfies all assumptions of Theorem 4.6.
Other similar result to the previous one was considered by Qiu and Peng [23],
where they used lower pseudo-continuity instead of lower semicontinuity, but
they need both functions to vanish on the diagonal. We will give a simple
proof in order to show that it is a consequence of Fan-Browder’s theorem which
is equivalent to Theorem 4.1.
###### Theorem 4.8 (Qiu and Peng).
Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ and $g$ be two real-valued function defined on $X\times X$ such that
1. (i)
for all $x,y\in X$, $f(x,y)\leq g(x,y)$;
2. (ii)
for each $y\in X$, the function $f(\cdot,y)$ is lower pseudo-continuous;
3. (iii)
for each $x\in X$, the function $g(x,\cdot)$ is quasi-concave;
4. (iv)
for all $x\in X$, $f(x,x)=g(x,x)=0$.
Then, there exists $x_{0}\in X$ such that $f(x_{0},y)\leq 0$, for all $y\in
X$.
###### Proof.
We consider two correspondences $F,G:X\rightrightarrows X$ defined by
$F(x)=\\{y\in X:~{}f(x,y)>0\\}\mbox{ and }G(x)=\\{y\in X:~{}g(x,y)>0\\}$
Clearly $G$ has convex values and does not have fixed points. Moreover,
$F(x)\subset G(x)$, for all $x\in X$. Also, $F$ has open fibres, due to $f$
being lower pseudo-continuous in its first argument and the fact that $f$
vanishes on the diagonal of $X\times X$. By Lemma 5.1 in [30] the
correspondence $\operatorname*{conv}(F):X\rightrightarrows X$ defined as
$\operatorname*{conv}(F)(x)=\operatorname*{conv}(F(x))$ has open fibres.
Furthermore, $\operatorname*{conv}(F)(x)\subset G(x)$, for all $x\in X$.
Hence, by Fan-Browder’s theorem there exists $x_{0}\in X$ such that
$\operatorname*{conv}(F(x_{0}))=\emptyset$. Consequently, $F(x_{0})=\emptyset$
and this means $f(x_{0},y)\leq 0$, for all $y\in X$. ∎
###### Remark 4.9.
Theorem 4.8 is equivalent to Theorem 4.1. In order to see that, it is enough
to show that Theorem 4.8 implies Theorem 4.2. Indeed, if $f$ is pseudo-
continuous then there exists a continuous function $u$ and an increasing
function $h$ such that $f=h\circ u$, due to Proposition 2.2. Moreover, $u$ is
quasi-concave in its second argument. Now, we consider the function $u_{0}$
defined by $u_{0}(x,y)=u(x,y)-u(x,x)$, which is continuous and it vanishes on
the diagonal. We now apply Theorem 4.8 to $u_{0}$ and recover Theorem 4.1.
## 5 Equivalence results in Nash games
A _Nash game_ , [22, Nash 1951], consists of $p$ players, each player $i$
controls the decision variable $x_{i}\in C_{i}$ where $C_{i}$ is a subset of a
Hausdorff locally convex topological vector space $E_{i}$. The “total strategy
vector” is $x$ which will be often denoted by
$x=(x_{1},x_{2},\dots,x_{i},\dots,x_{p}).$
Sometimes we write $(x_{i},x_{-i})$ instead of $x$ in order to emphasize the
$i$-th player’s variables within $x$, where $x_{-i}$ is the strategy vector of
the other players. Player $i$ has a payoff function
$\theta_{i}:C\to\mathbb{R}$ that depends on all player’s strategies, where
$C=\prod_{i=1}^{p}C_{i}$. Given the strategies $x_{-i}$ of the other players,
the aim of player $i$ is to choose a strategy $x_{i}$ solving the problem
$P_{i}(x_{-i})$:
$\displaystyle\max_{x_{i}}\theta_{i}(x_{i},x_{-i})~{}\mbox{ subject to
}~{}x_{i}\in C_{i}.$
A vector $\hat{x}\in C$ is a _Nash equilibrium_ if for all $i$, $\hat{x}_{i}$
solves $P_{i}(x_{-i})$.
Thanks to Debreu [10], Glicksberg [14] and Fan [13], we have the following
existence result of Nash equilibria.
###### Theorem 5.1.
For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
payoff function $\theta_{i}$ is continuous and quasi-concave in $x_{i}$, then
there exists at least one Nash equilibrium.
We now present an existence result of Nash equilibria for discontinuous games,
due to Morgan and Scalzo [21].
###### Theorem 5.2.
For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
payoff function $\theta_{i}$ is pseudo-continuous and quasi-concave in
$x_{i}$, then there exists at least one Nash equilibrium.
In a generalized Nash game, each player’s strategy must belong to a set
identified by the correspondence $K_{i}:C^{-i}\rightrightarrows C_{i}$ in the
sense that the strategy space of player $i$ is $K_{i}(x_{-i})$, which depends
on the rival player’s strategies $x_{-i}$. Given the strategy $x_{-i}$, player
$i$ chooses a strategy $x_{i}$ such that it solves the following problem
$\displaystyle\max_{x_{i}}\theta_{i}(x_{i},x_{-i})~{}\mbox{ subject to
}~{}x_{i}\in K_{i}(x_{-i}).$ ($GP_{i}(x_{-i})$)
Thus, a _generalized Nash equilibrium_ (GNEP) is a vector $\hat{x}\in C$ such
that the strategy $\hat{x}_{i}$ is a solution of the problem
($GP_{i}(x_{-i})$) associated to $\hat{x}_{-i}$, for any $i$.
The following result is due to Arrow and Debreu [4], but we state it as in
[12].
###### Theorem 5.3.
For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
following hold:
1. 1.
the payoff function $\theta_{i}$ is continuous and quasi-concave in $x_{i}$,
2. 2.
the correspondence $K_{i}$ is lower and upper hemicontinuous with convex,
closed and non-empty values;
then, there exists at least one generalized Nash equilibrium.
Below, an existence result of generalized Nash equilibria for discontinuous
generalized Nash games, due to Morgan and Scalzo [17].
###### Theorem 5.4.
For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
following hold:
1. 1.
the payoff function $\theta_{i}$ is pseudo-continuous and quasi-concave in
$x_{i}$,
2. 2.
the correspondence $K_{i}$ is lower and upper hemicontinuous with convex,
closed and non-empty values;
then, there exists at least a generalized Nash equilibrium.
Clearly, Theorem 5.4 implies Theorem 5.3. However, the next result says that
they are actually equivalent.
###### Theorem 5.5.
Theorems 5.4 and 5.3 are equivalent.
###### Proof.
Since any convex set is connected, by Proposition 2.2, we have that for each
$i$ there exists a continuous function $u_{i}:C\to\mathbb{R}$ and an
increasing function $h_{i}:u_{i}(C)\to\mathbb{R}$ such that
$\theta_{i}=h_{i}\circ u_{i}.$
As $\theta_{i}$ is quasi-concave in $x_{i}$, it is not difficult to show that
so is $u_{i}$. Now, the generalized Nash game defined by the functions $u_{i}$
and the correspondences $K_{i}$, admits a generalized Nash equilibrium, due to
Theorem 5.3, say $\hat{x}$. Since each function $h_{i}$ is increasing we have
that
$\theta_{i}(\hat{x})=h_{i}\circ u_{i}(\hat{x})\geq h_{i}\circ
u_{i}(x_{i},\hat{x}_{-i})=\theta_{i}(x_{i},\hat{x}_{-i}),\mbox{ for all
}x_{i}\in K_{i}(\hat{x}_{i}).$
The result follows. ∎
It is clear that any Nash game is a generalized Nash game. Moreover, Theorem
5.3 implies Theorem 5.1, and Theorem 5.4 implies Theorem 5.2. Thus, as a
direct consequence of the previous result we have that Theorem 5.1 is
equivalent to Theorem 5.2. Furthermore, the following result establishes that
Theorem 5.3 is equivalent to Theorem 5.1 on Banach spaces.
###### Theorem 5.6.
Theorem 5.1 implies Theorem 5.3 on Banach spaces.
In order to prove the previous result, we need the following lemma, which is
inspired by Theorem 4.1 in [6].
###### Lemma 5.7.
Theorem 5.1 implies Proposition 4.2 on Banach spaces.
###### Proof.
By Proposition 2.2, we assume without loss of generality that $f$ is
continuous. Now, consider the two-player game defined by the strategy sets and
the payoff functions as follows
$C_{1}=C_{2}=X,~{}\theta_{1}(x_{1},x_{2})=f(x_{2},x_{1})-f(x_{2},x_{2})\mbox{
and }\theta_{2}(x_{1},x_{2})=-\|x_{2}-x_{1}\|,$
where $\|\cdot\|$ is the norm. It is clear that this game satisfies all
assumption of Theorem 5.1, thus there exists $(\hat{x}_{1},\hat{x}_{2})\in
X\times X$ such that
$\theta_{1}(\hat{x}_{2},\hat{x}_{1})\geq\theta_{1}(\hat{x}_{2},x_{1})\mbox{
and }\|\hat{x}_{2}-\hat{x}_{1}\|\leq\|x_{2}-\hat{x}_{1}\|,~{}\mbox{for all
}x_{1},x_{2}\in X.$
From the second part we deduce that $\hat{x}_{1}=\hat{x}_{2}=\hat{x}$. Thus,
using this in the first part we obtain $0\geq\theta_{1}(\hat{x},x_{1})$ for
all $x_{1}\in X$. Therefore,
$\max_{z\in X}f(z,z)\geq f(\hat{x},\hat{x})\geq f(\hat{x},y),\mbox{ for all
}y\in X.$
∎
###### Proof of Theorem 5.6.
Since Theorem 5.3 is a consequence of Theorem 2.8 and this is consequence of
Fan-Browder’s theorem. The result follows from Lemma 5.7 and Remark 4.4. ∎
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# T4DT: Tensorizing Time for Learning Temporal 3D Visual Data
Mikhail Usvyatsov ETH Zurich, Switzerland, {mikhailu<EMAIL_ADDRESS>Rafael Ballester-Ripoll IE University, Madrid, Spain<EMAIL_ADDRESS>Lina Bashaeva Skolkovo Institute of Science and Technology, Moscow, Russia,
{lina.bashaeva, g.ferrer<EMAIL_ADDRESS>Konrad Schindler ETH
Zurich, Switzerland, {mikhailu<EMAIL_ADDRESS>Gonzalo Ferrer Skolkovo
Institute of Science and Technology, Moscow, Russia, {lina.bashaeva, g.ferrer,
<EMAIL_ADDRESS>Ivan Oseledets Skolkovo Institute of Science and
Technology, Moscow, Russia, {lina.bashaeva, g.ferrer<EMAIL_ADDRESS>
###### Abstract
Unlike 2D raster images, there is no single dominant representation for 3D
visual data processing. Different formats like point clouds, meshes, or
implicit functions each have their strengths and weaknesses. Still, grid
representations such as signed distance functions have attractive properties
also in 3D. In particular, they offer constant-time random access and are
eminently suitable for modern machine learning. Unfortunately, the storage
size of a grid grows exponentially with its dimension. Hence they often exceed
memory limits even at moderate resolution. This work proposes using low-rank
tensor formats, including the Tucker, tensor train, and quantics tensor train
decompositions, to compress time-varying 3D data. Our method iteratively
computes, voxelizes, and compresses each frame’s truncated signed distance
function and applies tensor rank truncation to condense all frames into a
single, compressed tensor that represents the entire 4D scene. We show that
low-rank tensor compression is extremely compact to store and query time-
varying signed distance functions. It significantly reduces the memory
footprint of 4D scenes while remarkably preserving their geometric quality.
Unlike existing, iterative learning-based approaches like DeepSDF and NeRF,
our method uses a closed-form algorithm with theoretical guarantees.
(a) Original, $284\text{\,}\mathrm{GB}$
(b) 1:12345, $24\text{\,}\mathrm{MB}$
(c) 1:793, $368\text{\,}\mathrm{MB}$
(d) 1:227, $1.24\text{\,}\mathrm{GB}$
Figure 1: T4DT with different compression levels in OQTT format for a
longshort-flying-eagle scene of resolution $512^{3}$ with 284 time frames.
Only frames 1, 142, and 284 are depicted. The compression ratio is different
from the actual memory consumption due to the padding of the time dimension to
512. High compression is achieved with $r_{\max}=400$, $\text{MSDM2}=0.45$ in
Fig. 1(b), medium compression with $r_{\max}=1800$, $\text{MSDM2}=0.32$ in
Fig. 1(c), and low compression / high quality with $r_{\max}=4000$,
$\text{MSDM2}=0.29$ in Fig. 1(d).
## 1 Introduction
Recent advances in hardware development have made it possible to collect rich
depth datasets with commodity devices. For example, modern AR/VR hardware can
record videos of 3D data, thus capturing their temporal evolution to yield 4D
datasets. Nevertheless, working with 4D data presents computational challenges
due to the curse of dimensionality. Furthermore, while 2D data mostly come in
the form of raster images, there is an entire zoo of popular representations
for 3D data, ranging from point clouds and meshes to voxel grids and implicit
(neural) functions.
In this work, we address the case of representations based on regular grids.
Such representations are highly structured and allow for efficient
manipulation, e.g., they offer random access, slicing, and other operations in
constant time. However, their storage requirements become a bottleneck: a grid
of resolution $I$ along each dimension $D$ has $I^{D}$ elements. We propose
applying tensor decompositions to leverage the high temporal and spatial
correlation in grids that arise from time-evolving 3D data. Besides exploring
the general idea of compression via low-rank constraints, we compare different
tensor decomposition schemes and their potential for 4D video. For example,
although the Tucker format Tucker, (1963, 1966) is known to yield good results
for the 3D case, it still suffers from the curse of dimensionality since it
requires $O(r^{D})$ elements w.r.t. the Tucker rank $r$, while $r$ must
typically be chosen $\approx\frac{I}{\text{const.}}$ to achieve reasonable
accuracy. Therefore, we explore hybrid low-rank formats based on the tensor
train (TT) decomposition Oseledets, (2011).
The Signed Distance Function (SDF, sometimes also called Signed Distance
Field) and its truncated version, known as Truncated SDF or TSDF, is a
convenient format for data fusion: several methods convert partial 3D views,
even if initially in mesh format, to TSDFs to fuse them into complete models
Izadi2011KinectFusionR3. This pipeline is useful for multi-view stereo
acquisition, laser scanners, as well as resource-limited devices (e.g., mobile
augmented reality). TSDFs offer 1-to-1 correspondence across time, which is
not easy to achieve with meshes if the objects undergo non-rigid deformation;
while every mesh can be converted to a TSDF on the same voxel grid (with
appropriately chosen resolution to avoid loss of detail). We experimentally
analyze the effects of the low-rank constraint, using TSDF sequences of 3D
scenes from the CAPE dataset Ma et al., (2020); Pons-Moll et al., (2017) and
problem-specific quality metrics. Our method can compress time-varying grids
to a usable size, while the same scenes in uncompressed form would each
require hundreds of GigaBytes of memory.
In the following, Section 2 reviews applicable tensor methods and techniques
for 3D and 4D data compression. Section 4 gives a detailed outline of our
proposed approach, Section 5 demonstrates its the performance in an
experimental study, followed by a discussion of limitations in Section 6, and
concluding remarks in Section 7. Our contributions are:
1. 1.
The first tensor decomposition framework for compressing temporal sequences in
3D voxel space;
2. 2.
A benchmark and analysis of different decomposition schemes for both the
spatial and temporal dimensions;
3. 3.
An open source implementation of our method based on the tntorch framework
Usvyatsov et al., (2022), available at https://github.com/prs-eth/T4DT.
## 2 Background and Related Work
In our work, we apply tensor methods to temporal sequences of 3D scenes. We
consider grid representations of the 3D data, i.e., a tensor
$\displaystyle{\bm{\mathsfit{A}}}\in\mathbb{R}^{I_{1}\times\dots\times I_{D}}$
constitutes a discrete sampling of a $D$-dimensional space on a grid
$\mathbb{I}=I_{1}\times\dots\times I_{D}$, with $I_{d}$ samples along
dimension $d$.
### 2.1 SDF and Truncated SDF
The SDF at position $p\in\mathbb{R}^{3}$ is defined as:
$\text{SDF}(p)=\begin{cases}\text{dist}(p,\partial\Omega)&\text{if
}p\in\Omega,\\\
\hfill-\text{dist}(p,\partial\Omega)&\text{otherwise},\end{cases}$ (1)
while the truncated SDF clamps the SDF as follows:
$\text{TSDF}(p)=\begin{cases}\hfill-\tau&\text{if }\text{SDF}(p)\leq-\tau,\\\
\hfill\tau&\text{if }\text{SDF}(p)\geq\tau,\\\
\hfill\text{SDF}(p)&\text{otherwise}.\end{cases}$ (2)
Depending on the available input data and the desired application, different
formats may be more or less suitable. For example, colored images can be
converted to NERF or NELF Mildenhall et al., (2020); Sitzmann et al., (2021),
which optimizes the reconstruction error via differentiable rendering. Depth
images can be fused into TSDFs as described in Curless and Levoy, (1996);
meshes can be converted into TSDFs, too, and also the inverse transformation
is possible with variants of marching cubes Lorensen and Cline, (1987); point
clouds can be turned into meshes or directly into TSDFs; etc.
### 2.2 Compression
Storing signed distance fields constitutes a computational and memory
bottleneck, especially for time-varying data. In Tang et al., (2018), a real-
time compression method was proposed that uses a learnable, implicit temporal
TSDF representation, combining independent color and texture encodings with
learnable geometry compression. Multi-dimensional visual data compression
based on the Tucker decomposition, combined with bit-plane coding, was applied
to volumetric data in Ballester-Ripoll et al., (2019). Later, Boyko et al.,
(2020) proposed compressing single-frame TSDFs with a block-based neural
network. They showed that, although TSDF tensors usually cannot be written
exactly as a low-rank expansion, in practice, one can store them in the low-
rank TT representation with low compression error. Here, we build on the idea
of tensor decompositions as a tool for compressing high-dimensional visual
data. We address the task of compressing dynamic 3D scenes, whose 4D grid
representations quickly exceed practical memory limits when stored in
uncompressed form.
Compressing TSDF with tensor methods not only allows one to work with data
that otherwise would not fit the memory, but it can also be useful for:
1. 1.
Fusion, as demonstrated in Boyko et al., (2020);
2. 2.
Constant-time TSDF random access,111The complexity of accessing a single
element depends on the choice of tensor format and rank but is independent of
the location of the query point within the tensor which may be useful to test
if a given point is inside or outside of a mesh, and potentially for ray-
casting;
3. 3.
Constant-time computation of the TSDF gradient, used to compute surface
normals Sommer et al., (2022);
4. 4.
Unlike neural network-based learning approaches, tensor methods rely on SVD-
based schemes, which provide theoretical guarantees Oseledets, (2011) and make
the framework more interpretable.
Throughout this work, we use the term compression ratio (or just compression,
if clear from the context) to denote the ratio between the number of
coefficients used for the compressed format and the number of coefficients
required to store the uncompressed tensor.
## 3 Tensor Formats
### 3.1 Notation
##### Tensor diagram notation.
To visualize tensor networks, we adopt Penrose’s graphical notation Penrose,
(1971). See Fig. 2 for visualization of a vector, a matrix and a matrix-vector
product.
(a) vector
(b) matrix
(c) matrix-vector product
Figure 2: Each tensor is denoted as a node whose edges represent its
dimensions. Whenever an edge is shared, tensor contraction is assumed: e.g.,
Fig. 2(c) depicts the matrix-vector product
$\sum_{j}{\bm{M}}_{ij}{\bm{v}}_{j}$.
### 3.2 Tucker
The Tucker decomposition Tucker, (1963) factors a tensor of dimension $D$ into
a $D$-dimensional _core_ tensor
$\displaystyle{\bm{\mathsfit{G}}}\in\mathbb{R}^{r_{1}\times\dots\times r_{D}}$
and $D$ matrices $\displaystyle\\{{\bm{A}}_{d}\\}_{d=1}^{D}$,
$\displaystyle{\bm{A}}_{d}\in\mathbb{R}^{r_{d}\times I_{d}}$. The format is
defined as
$\displaystyle{\bm{\mathsfit{A}}}[i_{1},\dots,i_{D}]={\bm{\mathsfit{G}}}{\bm{\mathsfit{A}}}_{1}[:,i_{1}]\dots\bm{\mathsfit{A}}_{D}[:,i_{D}],$
(3)
where $r_{d}$ are the Tucker-ranks. The Tucker decomposition has
$\mathcal{O}\big{(}(\max_{d}[r_{d}])^{D}+\max_{d}[r_{d}]\cdot\max_{d}[I_{d}]\big{)}$
storage cost. See Fig. 3(a) for visualization of Tucker format in graphical
notation.
(a) Tucker
(b) TT
Figure 3: Graphical representation of the Tucker Fig. 3(a) and TT Fig. 3(b)
decompositions.
### 3.3 Tensor Train
The TT decomposition Oseledets, (2011) factors a tensor of dimension $D$ into
a sequence of $D$ 3-dimensional tensors. The format is defined as
$\displaystyle{\bm{\mathsfit{A}}}[i_{1},\dots,i_{D}]={\bm{\mathsfit{Q}}}_{1}[0,i_{1},:]{\bm{\mathsfit{Q}}}_{2}[:,i_{2},:]\dots\bm{\mathsfit{Q}}_{D}[:,i_{D},0],$
(4)
where the tensors
$\displaystyle\\{{\bm{\mathsfit{Q}}}_{d}\\}_{d=1}^{D},{\bm{\mathsfit{Q}}}_{d}\in\mathbb{R}^{r_{d-1}\times
I_{d}\times r_{d}}$, are called TT-cores; and $r_{d}$ are the _TT-ranks_
($r_{0}\\!=\\!r_{D}\\!=\\!1$). The TT decomposition has
$\mathcal{O}\big{(}D\cdot(\max_{d}[r_{d}])^{2}\cdot\max_{d}[I_{d}]\big{)}$
storage cost, and leads to a linear tensor network; see Fig. 3(b) for a
graphical example.
### 3.4 QTT
Quantics TT (QTT) is an extension of TT that includes reshaping222With zero-
padding where needed the input into a tensor of shape $2\times 2\dots\times
2=2^{\sum_{d=1}^{D}\log_{2}(I_{d})}$, with $I_{j}$ the size of the $j$-th mode
Kazeev et al., (2017). The sub-dimensions $x,y,z$ are then grouped side-by-
side for each octet, which imitates the traversal of a $z$-space filling curve
and makes QTT similar to a tensorized octree. Last, the resulting tensor is
then subject to standard TT decomposition. This scheme is also connected to
the wavelet transform Kazeev and Oseledets, (2013).
##### Octet QTT.
We also propose to reuse a variant of QTT with a base dimension of size eight
instead of two by merging the three sub-dimensions of each octet into one.
This format was originally proposed by qttnf2022. In this way, the resulting
tensor has shape $8\times 2\times...\times 8\times 2$. We refer to this format
as OQTT for octet QTT and found it to reduce discontinuity artifacts (Section
5).
## 4 Proposed Method
We introduce T4DT, a method to compress high-resolution temporal TSDF fields
with tensor decompositions discussed in Sections 3.3 \- 3.4. We exploit that
individual TSDF frames can be compressed into a low-rank TT decomposition with
reasonable error Boyko et al., (2020). Under the low-rank constraint, a zero-
level set can be reconstructed with sufficiently good quality. However, this
becomes more challenging when considering time-evolving data since
uncompressed 4D grid scenes at fine spatial resolutions can range in the
hundreds of GBs. Algorithm 1 gives a high-level overview of our pipeline.
${\mathbb{I}}=I_{1}\times I_{2}\times I_{3}\times I_{4}$ – temporal 3D grid
${\bm{\mathsfit{X}}}\in\mathbb{R}^{{\mathbb{I}}}$ – input temporal TSDF
$R_{s}$ – maximal rank along spatial dimensions
$R_{t}$ – maximal rank along time dimension
$\displaystyle{\bm{\mathsfit{X}}}^{\prime}$ – Collection of compressed TSDF
frames
$\displaystyle{\bm{\mathsfit{Y}}}$ – Compressed temporal TSDF
for $\text{i}=1,2,\cdots,I_{4}-1,I_{4}$ do
$\displaystyle{\bm{\mathsfit{X}}}^{\prime}_{\text{i}}\leftarrow$
truncate($\displaystyle{\bm{\mathsfit{X}}}[\cdots,\text{i}],R_{s}$)
end for
while $\displaystyle{\bm{\mathsfit{X}}}^{\prime}\neq\varnothing$ and
$|{\bm{\mathsfit{Y}}}|\neq 1$ do
$\displaystyle{\bm{\mathsfit{Y}}}\leftarrow\text{truncate}({\bm{\mathsfit{X}}}_{0}^{\prime}\cup{\bm{\mathsfit{Y}}},R_{t})$
$\text{remove}({\bm{\mathsfit{X}}}_{0}^{\prime})$
end while
Algorithm 1 T4DT pipeline. Since the whole 4D scene is not expected to fit
into memory, we first compress each frame individually. Next, the collection
of individually compressed frames is assembled into a single compressed scene
using a tree-like merge procedure
${\bm{\mathsfit{X}}}_{0}^{\prime}\cup{\bm{\mathsfit{Y}}}$. See Section 4.2 for
the merge procedure details. Figure 4: We compress and merge individual frames
progressively into a compressed scene (for simplicity, all tensors are shown
in uncompressed form). Two of the previous level tensors are merged at each
tree level using the concatenation procedure (which increases tensor rank),
followed by rank truncation. On the first level, an additional core for the
temporal dimension is inserted into each frame.
### 4.1 Framewise Decomposition
The first technical choice is the base decomposition at the frame level. We
considered three variants: Tucker, TT, and QTT. In 3D, Tucker is an attractive
choice since it is equivalent to TT plus an additional compression step along
the second dimension. For the 4D case, we propose to combine the best of both
worlds by using a TT-Tucker blend, where spatial dimensions are stored in
Tucker formatted cores, and the temporal dimension is represented with a TT
core. Each core or factor is responsible for a single dimension in both the TT
and Tucker formats. However, the Tucker model shares the core tensor between
all dimensions.
Many real-world datasets are sparse in the sense that most of the occupied
volume is empty. An octree is an excellent choice to pack such sparse data
into a compact representation. We argue that QTT is a tensor analog of an
octree data structure. Indeed, it is easy to see that, padded and reshaped to
$2^{\sum_{d=1}^{D}\log_{2}(I_{d}))}$, the scene becomes a piece-wise
separated, reshaped set of octets after permutation of the corresponding sub-
dimensions. Rank truncation is a way to reduce the number of coefficients to
encode each octet sub-dimension.
### 4.2 Frame Concatenation
Since the original scene can by far exceed the available memory, we developed
a progressive procedure to merge compressed frames into a single compressed
scene. The algorithm is akin to the _pairwise summation_ method to reduce
round-off errors in numerical summation Higham, (2002): we stack the frames by
pairs along the time dimension, recompress each pair via tensor rank
truncation Oseledets, (2011), and repeat the procedure recursively in a
binary-tree fashion to ultimately yield a single 4D compressed tensor. See
Fig. 4 for an illustration.
(a) Chamfer distance
(b) Hausdorff distance
(c) IoU
(d) MSDM2
Figure 5: Ranks vs. performance for TT-Tucker compressed longshort-flying-
eagle scene of resolution $512^{3}$ with 284 time frames. Metrics are averaged
between frames 1, 142, and 284. Each data point is annotated with the
corresponding compression ratio scaled by $10^{6}$.
Note that this procedure is compatible with any format that supports rank
truncation, which includes the TT and Tucker formats and mixtures thereof that
allow for concatenation in compressed format. They are implemented via
concatenation of the cores, increasing the rank of the resultant cores to a
sum of the ranks of the operands. Afterward, rank truncation is applied to
reduce the rank back to the desired maximal value with an SVD-like algorithm
Oseledets, (2011).
##### QTT scene.
The scene stored in QTT format has the shape $x_{1}\times y_{1}\times
z_{1}\times t_{1}\times x_{2}\times y_{2}\times z_{2}\times
t_{2}\times\cdots\times x_{k}\times y_{k}\times z_{k}\times t_{k}$, where
$x_{i}=y_{i}=z_{i}=t_{i}=2$ and
$\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}=W$,
$\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}=D$,
$\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}=H$,
$\prod_{i=1}^{\lceil\log_{2}I_{4}\rceil}t_{i}=T$, with uncompressed scene
shape $W\times D\times H\times T$. In order to reconstruct the $i$-th frame
from a scene compressed in this way, one must first compute the binary
representation of $i$ and then decompress the sub-tensor at
$[:,:,:,i_{\text{bit}_{k}},:,:,:,i_{\text{bit}_{k-1}},\cdots,:,:,:,i_{\text{bit}_{0}}]$.
##### OQTT scene.
The scene stored in OQTT format has the shape $x_{1}y_{1}z_{1}\times
t_{1}\times x_{2}y_{2}z_{2}\times t_{2}\times\cdots\times
x_{k}y_{k}z_{k}\times t_{k}$, where $x_{i}=y_{i}=z_{i}=t_{i}=2$ and
$\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}=W$,
$\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}=D$,
$\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}=H$,
$\prod_{i=1}^{\lceil\log_{2}I_{4}\rceil}t_{i}=T$, with uncompressed scene
shape $W\times D\times H\times T$. In order to reconstruct the $i$-th frame,
one must again first compute the binary representation of $i$, then decompress
the sub-tensor at
$[:,i_{\text{bit}_{k}},:,i_{\text{bit}_{k-1}},\cdots,:,i_{\text{bit}_{0}}]$.
${\mathbb{I}}=I_{1}\times I_{2}\times I_{3}$ – 3D grid
${\bm{\mathsfit{X}}}\in\mathbb{R}^{{\mathbb{I}}}$ – input 3D volumetric frame
$R$ – maximal rank
$\displaystyle{\bm{\mathsfit{Y}}}$ – Compressed 3D volumetric frame
for $\text{d}=1,2,3$ do
${\bm{\mathsfit{X}}}\leftarrow\text{pad}(X,2^{\lceil\log_{2}I_{d}\rceil})$
end for
${\bm{\mathsfit{X}}}\leftarrow\text{reshape}(X,2^{\sum_{d=1}^{3}\lceil\log_{2}(I_{d})\rceil)})$
${\bm{\mathsfit{X}}}\leftarrow\text{permute}(X,(x_{1},y_{1},z_{1},\cdots,x_{\lceil\log_{2}I_{1}\rceil},y_{\lceil\log_{2}I_{2}\rceil},z_{\lceil\log_{2}I_{3}\rceil}))$,
$I_{1}=\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}$,
$I_{2}=\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}$ ,
$I_{3}=\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}$
${\bm{\mathsfit{Y}}}\leftarrow\text{TT}({\bm{\mathsfit{X}}},R)$
Algorithm 2 Conversion of a single 3D volumetric frame into OQTT format.
## 5 Results
### 5.1 Data
For our experiments, we use selected scenes from the CAPE dataset Ma et al.,
(2020); Pons-Moll et al., (2017). The dataset consists of 3D meshes of 15 (10
male, 5 female) clothed people in motion. For each scene frame, we compute its
TSDF and discretize it at resolution $512^{3}$, using the PySDF library. We
use $\tau=0.05$ in Eq. 2 to allow $\approx 10$ voxels with distinct levels
near the TSDF zero level set. Since all scene frames must share a common
coordinates frame, we recover the scene’s bounding box and compute the SDF of
each frame within that box.
We further demonstrate the performance of T4DT on selected scenes from the
Articulated Mesh Animation dataset Vlasic et al., (2008). That real-world
dataset contains 10 mesh sequences depicting 3 different humans performing
various actions. For each scene frame, we compute its TSDF and discretize it
at resolution $512^{3}$, using the PySDF library. We use $\tau=0.05$ in Eq. 2
to allow $\approx 10$ voxels with distinct levels near the TSDF zero level
set.
### 5.2 Influence of Tensor Ranks
We provide several error metrics computed for the TT-Tucker format, averaged
across the first, middle, and last frames of the longshort-flying-eagle scene,
for different spatial and temporal ranks in Fig. 5. See Section A.1 for the
definition of the error metrics. Note that the time dimension is far from
full-rank in the TT-Tucker format since the error metrics saturate well below
the total sequence length of 284 frames.
The best compression/performance was obtained for the OQTT base format.
Selected visualizations are provided in Section A.2: Fig. 9 shows error
metrics for the TT format, as a function of the rank. Qualitative results are
shown in Fig. 6 for the TT-Tucker format, in Fig. 7 for the TT format, and in
Fig. 8 for QTT. For visual results of OQTT please refer to Fig. 1. In the
cases of TT and TT-Tucker, severe rank truncation smoothes the reconstructed
surface and erodes smaller features like fingers or facial details. In
contrast, QTT preserves more details at the cost of discontinuities due to the
separation and independent compression of octet sub-dimensions. OQTT reduces
these discontinuity artifacts by encoding the octets as a single dimension
without compressing the ranks between sub-dimensions of a single octet. See
Fig. 1 for qualitative results for OQTT. We also present quantitative OQTT
compression results in Table 1, at a rank that offers a good compromise
between visual quality and compression rate.
| Crane | Swing | Handstand | Samba
---|---|---|---|---
Resolution | $512^{3}\times 174$ | $512^{3}\times 174$ | $512^{3}\times 174$ | $512^{3}\times 149$
Metric | | | |
L2 $\downarrow$ | 2.67 | 2.07 | 2.45 | 1.71
Chamfer distance $\downarrow$ | $5e^{-5}$ | $4e^{-5}$ | $5e^{-5}$ | $4e^{-5}$
Hausdorff distance $\downarrow$ | 0.190 | 0.015 | 0.012 | 0.013
MSDM2 $\downarrow$ | 0.36 | 0.38 | 0.35 | 0.36
IoU $\uparrow$ | 0.41 | 0.40 | 0.64 | 0.54
Compression | 1:954 | 1:1356 | 1:938 | 1:935
Size | $549\text{\,}\mathrm{MB}$ | $386\text{\,}\mathrm{MB}$ | $558\text{\,}\mathrm{MB}$ | $560\text{\,}\mathrm{MB}$
Table 1: Quantitative performance of OQTT for selected scenes from the
Articulated Mesh Animation dataset Vlasic et al., (2008). The metrics are
averaged across the first, middle, and last frames. For all scenes,
$r_{\max}=4000$, chosen to provide a good trade-off between visual quality and
compression rate.
## 6 Failure Cases and Limitations
We did not observe significant failures while working with the CAPE and AMA
datasets. However, how T4DT fares for 3D data that are not structurally sparse
(i.e., not mostly free space) remains to be tested. Unfortunately, there are
no such scenes in the two datasets.
We do note that TT decomposition is not invariant against the rotation of the
input. Hence, one could attack the scheme by rotating the scene w.r.t. the
grid axes in a way that maximizes the reconstruction error. A straightforward
measure to mitigate the influence of the grid orientation is to pre-rotate the
3D scene to its principal axes before grid sampling and OQTT compression.
## 7 Conclusions
We have presented T4DT, a scalable and interpretable compression pipeline for
3D time sequence data based on tensor decomposition. Our scheme improves over
the related TT-TSDF method in two ways: (i) We are able to process temporally
varying data. We can do so even though the TSDF field takes hundreds of GBs,
and our method works fully in-memory; (ii) We tested various tensorization
schemes, such as a TT-Tucker hybrid, the QTT format, and the new OQTT variant,
and found OQTT to outperform previous decompositions. OQTT is tailored
specifically to 3D volumetric scene representations, and our experiments
quantitatively and qualitatively support its special reshaping of the 3D grid.
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## Appendix A Appendix
### A.1 Metrics
Throughout this work, we convert temporal 3D data from mesh format to tensor
format for compression using sampled truncated SDF and from tensor format back
to temporal 3D data in the form of meshes for quantitative tests using the
marching cubes algorithm.
We selected a tensor-based l2 reconstruction metric for the comparison in
tensorial form. We use the metrics below to evaluate the quality of the
obtained mesh.
#### A.1.1 Intersection over Union (IoU)
IoU is a commonly used metric for comparing 3D shapes Rezatofighi et al.,
(2019). For two given 3D volumes ${\bm{\mathsfit{A}}}$ and
${\bm{\mathsfit{B}}}$:
$\text{IoU}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\frac{\|{\bm{\mathsfit{A}}}\cap{\bm{\mathsfit{B}}}\|}{\|{\bm{\mathsfit{A}}}\cup{\bm{\mathsfit{B}}}\|}.$
(5)
3D mesh is converted to an occupancy grid, and the operation is easily
performed with implementation from Fuji Tsang et al., (2022).
#### A.1.2 Hausdorff distance
One-sided Hausdorff distance is computed as:
$d_{\text{h}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\max\\{\min\\{||a-b||:a\in{\bm{\mathsfit{A}}}\\}:b\in{\bm{\mathsfit{B}}}\\}.$
(6)
Note, that $d_{\text{h}}$ is not symmetric. The symmetric version is defined
as :
$d_{\text{H}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\max\\{d_{\text{h}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}}),d_{\text{h}}({\bm{\mathsfit{B}}},{\bm{\mathsfit{A}}})\\}.$
(7)
We use the implementation from Jacobson et al., (2018).
#### A.1.3 Chamfer distance
Symmetric Chamfer distance is defined as:
$d_{\text{CD}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\sum_{a\in{\bm{\mathsfit{A}}}}\min_{b\in{\bm{\mathsfit{B}}}}||x-y||^{2}+\sum_{b\in{\bm{\mathsfit{B}}}}\min_{a\in{\bm{\mathsfit{A}}}}||x-y||^{2}.$
(8)
Following Boyko et al., (2020), we sample 30’000 points from each mesh and
compute sampled symmetric Chamfer distance. Finally, we use the implementation
from Fuji Tsang et al., (2022).
#### A.1.4 Mesh Structural Distortion Measure (MSDM2)
Introduced in Lavoué, (2011) MSDM2 metric is used to estimate the correlation
with human visual perception. The metric assumes one mesh to be original and
the second distorted. See Algorithm 3 for an algorithmic view of MSDM2
computation.
Algorithm 3 MSDM2 high-level computation scheme
$M_{o}$ – original mesh, $M_{d}$ – distorted mesh, $\\{v_{i}\in M\\}$ – set of
mesh vertices
for $v_{i}\in M_{o}$ do
Find $v_{j}=\min_{v_{k}\in M_{d}}\text{dist}(v_{i},v_{k})$
end for
for $v_{i}\in M_{o}$ do
calculate curvature of $v_{i}$
end for
for $v_{i}\in M_{d}$ do
calculate curvature of $v_{i}$
end for
interpolate curvature per face for $M_{o}$ and $M_{d}$
for $v_{i}\in M_{d}$ do
calculate MSDM2 for $v_{i}$ based on curvature features and nearest neighbours
$v_{j}$ from $M_{o}$
end for
Integrate global asymmetric MSDM2
Refer to Lavoué et al., (2006); Lavoué, (2011) for the more details on metric
formulation. We use original implementation from Lavoué, (2011) and self
written pybind11 Jakob et al., (2017) interface to Python.
### A.2 Selected Visualizations
(a) Original, $284\text{\,}\mathrm{GB}$
(b) Compression 1:769230, $0.2\text{\,}\mathrm{MB}$
(c) Compression 1:1333, $33\text{\,}\mathrm{MB}$
(d) Compression 1:116, $385\text{\,}\mathrm{MB}$
Figure 6: T4DT with different compression levels in TT-Tucker format for a
longshort-flying-eagle scene of resolution $512^{3}$ with 284 time frames.
Only frames 1, 142, and 284 are depicted.
(a) Original, $284\text{\,}\mathrm{GB}$
(b) compression 1:476190, $0.61\text{\,}\mathrm{MB}$
(c) compression 1:6250, $16\text{\,}\mathrm{MB}$
(d) compression 1:2000, $48.5\text{\,}\mathrm{MB}$
Figure 7: T4DT with different compression levels in TT format for a longshort-
flying-eagle scene of resolution $512^{3}$ with 284 time frames. Only frames
1, 142, and 284 are depicted. The compression ratio is different from the
actual memory consumption due to the padding of the time dimension to 512.
(a) Original, $284\text{\,}\mathrm{GB}$
(b) compression 1:6666, $0.08\text{\,}\mathrm{MB}$
(c) compression 1:1587, $332\text{\,}\mathrm{MB}$
(d) compression 1:244, $1.4\text{\,}\mathrm{GB}$
Figure 8: Different levels of compression in QTT format for a longshort-
flying-eagle scene of resolution $512^{3}$ with 284 time frames. Only frames
1, 142, and 284 are depicted. The compression ratio is different from the
actual memory consumption due to the padding of the time dimension to 512.
(a) Chamfer distance
(b) Hausdorff distance
(c) IoU
(d) MSDM2
Figure 9: Ranks vs. performance for TT compressed longshort-flying-eagle scene
of resolution $512^{3}$ with 284 time frames. Metrics are averaged between
frames 1, 142, and 284. Each data point is annotated with the corresponding
compression ratio scaled with 1e6.
(a) TT
(b) TT Tucker
Figure 10: Ranks vs. $L_{2}$ for longshort-flying-eagle scene of resolution
$512^{3}$ with 284 time frames. $L_{2}$ is averaged between frames 1, 142, and
284. Each data point is annotated with the corresponding compression ratio
scaled with 1e6.
|
# Electron power absorption in micro atmospheric pressure plasma jets driven
by tailored voltage waveforms in He/N2
Máté Vass1,2∗, Sebastian Wilczek1, Julian Schulze1,3, Zoltán Donkó2,
1 Department of Electrical Engineering and Information Science, Ruhr-
University Bochum, D-44780, Bochum, Germany
2 Institute for Solid State Physics and Optics, Wigner Research Centre for
Physics, H-1121 Budapest, Konkoly-Thege Miklós str. 29-33, Hungary
3 Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams
(Ministry of Education), School of Physics, Dalian University of Technology,
Dalian 116024, People’s Republic of China
E-mail<EMAIL_ADDRESS>
> In atmospheric pressure capacitively coupled microplasma jets, Voltage
> Waveform Tailoring (VWT) was demonstrated to provide ultimate control of the
> Electron Energy Distribution Function (EEDF), which allows to enhance and
> adjust the generation of selected neutral species by controlling the
> electron power absorption dynamics. However, at the fundamental level, the
> physical origin of these effects of VWT remained unclear. Therefore, in this
> work, the electron power absorption dynamics is investigated in a He/N2 jet
> with a nitrogen concentration of 0.05% driven by a valleys voltage waveform
> at a base frequency of 13.56 MHz for different numbers of harmonics using a
> self-consistent Particle in Cell simulation coupled with a spatio-temporally
> resolved analysis of the electron power absorption based on moments of the
> Boltzmann equation. Due to the local nature of the transport at atmospheric
> pressure, ohmic power absorption is dominant. Increasing the number of
> harmonics, due to the peculiar shape of the excitation waveform the sheath
> collapse at the grounded electrode is shortened relative to the one at the
> powered electrode. As a consequence, and in order to ensure flux
> compensation of electrons and positive ions at this electrode, a high
> current is driven through the discharge at the time of this short sheath
> collapse. This current is primarily driven by a high ohmic electric field.
> Close to the grounded electrode, where the electron density is low and the
> electric field is, thus, high, electrons are accelerated to high energies
> and strong ionization as well as the formation of a local electron density
> maximum are observed. This electron density maximum leads to a local
> ambipolar electric field that acts as an electric field reversal and
> accelerates electrons to even higher energies. These effects are understood
> in detail to fundamentally explain the unique potential of VWT for EEDF
> control in such plasmas.
## 1 Introduction
Radio frequency (RF-) driven atmospheric pressure microplasma jets
($\mu$APPJs) have gained considerable scientific attention in the last few
years due to their effectiveness in generating reactive species [1, 2, 3, 4,
5, 6]. This ability makes microplasma jets a versatile tool having
applications in a variety of areas, such as plasma medicine (sterilization,
wound healing, cancer treatment) [7, 8, 9, 10, 11, 12], and materials
processing [13, 14, 15]. For such applications, optimizing and controlling the
generation of radicals is of utmost importance. This can be achieved by
controlling the electron energy distribution function (EEDF), as radicals are
generated by electron impact excitation/dissociation of the background gas.
This control might be realized if a thorough understanding of electron power
absorption, i.e. the way electrons gain and lose their energy within the
plasma, is achieved. One might be tempted to conclude that due to the high
collisionality of the system, kinetic effects play a less prominent role as
compared to plasmas at low pressure. However, previous investigations have
shown that kinetic simulations are still preferable [16, 17]. Previous
investigations on electron dynamics in atmospheric pressure plasmas were
conducted using fluid models [18, 19, 20], hybrid models, where collisions
were incorporated in a Monte Carlo Collision (MCC) approach [21] and fully
kinetic Particle in Cell (PIC)/MCC simulations [22, 23, 24].
Previous studies of electron power absorption in atmospheric pressure plasma
jets ($\mu$APPJ) uncovered different power absorption mechanisms, viz. the
$\Omega$\- and Penning-mode [25, 26, 27, 28, 29] and mode transitions between
them [21]. In the $\Omega$-mode, electrons are mainly accelerated at the times
of sheath expansion and collapse within the RF period. Per electron, maximum
power absorption occurs at positions close to the electrodes, where the
electron density is low and, thus, the resistivity is high. Under such
discharge conditions, the ohmic electric field is high due to the atmospheric
pressure and it can effectively increase the electrons’ energy. The spatio-
temporal ionization and excitation dynamics observed in this power absorption
mode are similar compared to those found in the $\alpha$-mode in low pressure
capacitively coupled plasmas (CCP) [30], but the underlying mechanism of
electron power absorption is entirely different. The Penning-mode will be
present, if ionization maxima are found within the sheath regions at the times
of maximum sheath voltage within the RF period. In contrast to the
$\gamma$-mode in low pressure CCPs [31], where these maxima are a result of
energetic secondary electrons accelerated by the sheath electric field thereby
acquiring enough energy to ionize, in the atmospheric pressure case the
ionization is largely due to Penning-ionization, where energetic electrons
excite the helium atoms creating metastable species in the vicinity of the
electrodes that can then lead to the ionization (see sec. 3).
Typically, $\mu$APPJs are driven by a single driving frequency [32]. This
leads to an inherent spatio-temporal symmetry of the electron power absorption
dynamics within each RF period and strongly limits its control [22]. Such
control is, however, needed to tailor the EEDF to enhance the generation of
specific neutral species by electron impact driven reactions with particular
electron energy thresholds. Recently Voltage Waveform Tailoring (VWT) was
demonstrated to provide ultimate control of the EEDF and to allow enhancing
and controlling the generation of selected neutral species such as helium
metastables and atomic oxygen compared to single frequency operation [33, 24,
23, 34]. The energy efficiency of such plasma processes was demonstrated to be
higher compared to classical single frequency operation [35], which is
strongly beneficial for applications. Korolov et al. demonstrated that this is
caused by a breaking and control of the spatio-temporal symmetry of the
electron power absorption dynamics induced by tailored voltage waveforms [33,
24, 23, 34]. For instance, driving a $\mu$APPJ by a valleys-/peaks-waveform
was found to cause strong electron impact excitation and high neutral particle
densities adjacent to only one electrode.
These unique effects, however, have not been understood fundamentally up to
now, i.e. the question why valleys-/peaks- driving voltage waveforms induce
such a symmetry breaking in $\mu$APPJs is not answered. In low pressure CCPs,
using an analysis based on the first velocity moment equation of the Boltzmann
equation aided by PIC/MCC simulations, originally applied by Surendra et al.
[36], provides a fully self-consistent description of electron power
absorption. In recent years, this method, also called the Boltzmann term
analysis, has been applied to various low pressure CCPs including
electropositive [37, 38, 39, 40], electronegative gases [41, 42, 43] and
magnetized discharges [44, 45, 46], by which a considerable amount of new
insight into the physics of electron power absorption has been gained.
Generally and to our knowledge, no fully self-consistent description of
electron power absorption has been provided for $\mu$APPJs.
In this paper, we use the Boltzmann term analysis and apply it to a He/N2
$\mu$APPJ excited by a valleys voltage waveform synthesized from different
numbers of harmonics to investigate the characteristics of electron power
absorption at atmospheric pressure and to provide a fundamental explanation
for the more energy efficient and controllable generation of reactive and
excited neutrals by VWT in such microplasmas. Based on the Boltzmann term
analysis, we show that contrary to the low pressure scenario, ohmic power
absorption is dominant. By increasing the number of driving harmonics, due to
the peculiar nature of the excitation waveform, the duration of the sheath
collapse at both electrodes become different. In order to ensure flux
compensation of positive ions and electrons at the electrode, where the sheath
collapse is short, within each period of the fundamental driving frequency
[47], a large electron current must be drawn to this electrode during the
local sheath collapse. This is not the case at the other electrode, where the
sheath collapse is much longer and electrons have more time to get to this
electrode. To drive such a high current, a strong ohmic electric field is
generated at the time of the short sheath collapse throughout the plasma. Due
to the decrease of the electron density and, thus, the conductivity towards
the electrodes, this ohmic field is space dependent and maximum close to the
electrode, where the sheath collapse is short. Consequently, electrons are
accelerated to high energies at this position and time within the fundamental
RF period and cause strong ionization. Jointly with an enhanced Penning
ionization at the grounded electrode due to locally enhanced helium metastable
densities, this ionization, in turn, leads to the formation of a local
electron density maximum close to this electrode. The related electron density
gradient causes an ambipolar electric field that enhances the acceleration
towards this electrode even further. Ultimately, this mechanism, which happens
only at one electrode for a valleys driving voltage waveform, results in a
strong electric field reversal during the short local sheath collapse, which
leads to the generation of highly energetic electrons locally. This explains,
at a fundamental level, the superior performance of VWT for generating and
controlling radical and excited species densities in $\mu$APPJs.
The paper is structured as follows: in section 2, the theoretical background
of the Boltzmann term analysis is introduced. Section 3 gives an overview of
the specifications of the PIC/MCC simulation method from which all physical
parameters necessary for the analysis can be obtained. In section 4, results
are presented and discussed. Finally, in section 5 conclusions are drawn.
## 2 Theoretical background
The basis of the Boltzmann term analysis is the first velocity moment equation
of the 1D Boltzmann equation (i.e. the momentum balance equation). Rearranging
this equation, the total electric field can be divided into three distinct
terms as $E_{\rm tot}=E_{\rm in}+E_{\nabla p}+E_{\rm Ohm}$ [40], where the
terms are given by
$\displaystyle E_{\rm in}$
$\displaystyle=-\frac{m}{ne}\left[\frac{\partial}{\partial
t}(nu)+\frac{\partial}{\partial x}(nu^{2})\right],$ $\displaystyle E_{\nabla
p}$ $\displaystyle=-\frac{1}{ne}\frac{\partial}{\partial x}p_{\parallel},$
$\displaystyle E_{\rm Ohm}$ $\displaystyle=-\frac{\Pi_{\rm c}}{ne}.$
Here $n$, $u$, $m$ and $e$ are the density, mean velocity, mass and charge of
the electrons, respectively. $p_{\parallel}$ stands for the diagonal element
of the electron pressure tensor, where ‘parallel’ refers to the direction of
the electric field, which is perpendicular to the electrode surfaces.
$\Pi_{\rm c}$ is the electron momentum loss due to collisions.
Each term in equation (2) represents a distinct physical mechanism. The
inertia term, $E_{\rm in}$, is the electric field needed to balance the
changes in the electron momentum. $E_{\rm Ohm}$, the ohmic electric field, is
a result of collisions between electrons and the particles of the background
gas. $E_{\nabla p}$, which is the electric field originating from the electron
pressure gradient, can be split into two separate terms ($E_{\nabla
p}=E_{\nabla n}+E_{\nabla T}$), given by
$\displaystyle E_{\nabla n}$
$\displaystyle=-\frac{T_{\parallel}}{ne}\frac{\partial n}{\partial x},$
$\displaystyle E_{\nabla T}$ $\displaystyle=-\frac{1}{e}\frac{\partial
T_{\parallel}}{\partial x}.$ (2)
Here $T_{\parallel}$ denotes the parallel electron temperature (in units of
eV), which is defined by, $T_{\parallel}=p_{\parallel}/n$. $E_{\nabla n}$ is,
in quasineutral regions, identical to the classical ambipolar electric field
[48], and $E_{\nabla T}$ originates from the gradient of the parallel electron
temperature.
Based on the above, the total electric field and the electron power absorption
can be determined using input parameters taken from simulations. To obtain the
power dissipated to electrons, the electric field must be multiplied by the
electron conduction current density, $j_{\rm e}$. This Boltzmann term analysis
allows to understand the fundamental mechanisms of electron power absorption
and electric field generation by splitting the corresponding quantities up
into the different terms mentioned above and by analyzing their respective
spatio-temporal dynamics. For a more detailed description of the Boltzmann
term analysis, we refer to [38, 39].
## 3 Computational method
The simulations are based on a 1d3v Particle-in-Cell Monte Carlo Collisions
(PIC/MCC) code [49, 50, 51]. The code has previously been applied to
atmospheric pressure plasma jets [33, 22, 23] and atmospheric-pressure
nanosecond pulsed discharges [52]. As the implementation details of the
discharge model have already been discussed in the references above, here we
only give a brief overview of its main features.
Due to the complex nature of the plasma chemistry of the reactive He-N2 plasma
in the jet, the model is not able to account for all possible reactions.
However, a simplified version of the plasma chemistry model with a limited
number of species and reactions was found to reproduce the main features of
the electron dynamics and the corresponding ionization and excitation profiles
in the plasma [33, 22, 23].
The species traced in the simulation are electrons, He+, He${}_{2}^{+}$, and
N${}_{2}^{+}$ ions. For electrons, the possible reactions are electron impact
collisions (cross sections are taken from [53] for e-\+ He collisions and from
[54] for e-\+ N2 collisions, where the latter is based on the Siglo cross
section set, accessible via LxCat [55]). These collisions are assumed to be
isotropic, therefore, the elastic momentum cross sections are used. In case of
electron impact excitation of He, we assume that 50% of these events lead to
the formation of either a singlet (21S) or a triplet (23S) metastable state by
direct excitation to these levels or via cascades from higher states [52, 24].
Unless otherwise indicated, in the rest of the paper we do not distinguish
between these two types and by ‘metastable state’ we generally mean both the
singlet and triplet state. The model distinguishes between two types of
electron impact ionization:
$\displaystyle{\rm e}^{-}+{\rm He}$ $\displaystyle\rightarrow{\rm e}^{-}+{\rm
e}^{-}+{\rm He}^{+},$ (3) $\displaystyle{\rm e}^{-}+{\rm N}_{2}$
$\displaystyle\rightarrow{\rm e}^{-}+{\rm e}^{-}+{\rm N}_{2}^{+}.$ (4)
Since our studies are limited to small concentrations of nitrogen in the
background gas ($c_{\rm N_{2}}=0.05\%$), elastic collisions are only
considered between positive ions and He atoms as targets. For He+ \+ He
collisions, an isotropic and a backward scattering channel are taken into
account [59]. For the collisions of He${}_{2}^{+}$ and N${}_{2}^{+}$ ions with
He atoms, the corresponding Langevin cross sections are used. Ions are created
either by electron impact ionization (processes 3-4) or via ion conversion:
${\rm He}^{+}+{\rm He}+{\rm He}\rightarrow{\rm He}_{2}^{+}+{\rm He},$ (5)
and through Penning ionization:
${\rm He^{*}}+{\rm N}_{2}\rightarrow{\rm He}(1^{1}{\rm S})+{\rm
N}_{2}^{+}+{\rm e}^{-}.$ (6)
The treatment of the latter two processes is based on a Monte Carlo approach:
Whenever He+ and He∗ particles are created, random lifetimes are assigned to
them based on the reaction rates [60, 61] of these processes. After these
lifetimes elapsed, the process is executed.
For the various species, different time steps are used based on their
collision frequencies. As the collisionality of electrons is extremely high,
these particles require the smallest time steps. An upper bound for this time
step can be found by requiring $P=1-\exp(-\nu\Delta t)\leq 0.1$ to hold for
the collision probability. This results in $\Delta t_{\rm e}=4.5\times
10^{-14}$ s. For the ions, longer time steps are allowed based on their
collisionalities. In our case, these values are $\Delta t_{\rm
He^{+}}=10\,\Delta t_{\rm e}$ and $\Delta t_{\rm He_{2}^{+}}=\Delta t_{\rm
N_{2}^{+}}=100\,\Delta t_{\rm e}$.
The excitation waveform used in this study is the “valleys” waveform, given by
$\phi_{N}(t)=\phi_{\rm
pp}\sum\limits_{k=1}^{N}\frac{2(N+k-1)}{(N+1)^{2}}\cos(2\pi kf_{\rm
b}t+(k+1)\pi),$ (7)
where $N$ denotes the number of harmonics, $\phi_{\rm pp}$ the peak-to-peak
voltage and $f_{\rm b}$ denotes the base frequency. In this study, we use
$\phi_{\rm pp}=500$ V and $f_{\rm b}=13.56$ MHz.
The gas pressure is kept constant at $p=10^{5}$ Pa along with a gas
temperature of $T_{\rm g}=300$ K. The electrode gap is $L=1$ mm. For the
electrodes, an electron reflection probability of $e_{\rm refl}=0.5$ is
assumed, together with ion-induced secondary electron emission coefficients
for the respective ion species given by $\gamma_{\rm He^{+}}=$ 0.2,
$\gamma_{\rm He_{2}^{+}}=$ 0.12, and $\gamma_{\rm N_{2}^{+}}=$ 0.07. These
specifications follow [23]. Electron emission due to metastable atoms is
neglected, since only a few of these atoms are expected to reach the surfaces
due to the decreased diffusion speed at high pressure [62] and due the high
probability of the Penning ionization. Our simulations resemble the COST
reference plasma jet, a parallel plate low temperature jet that is broadly
used for fundamental laboratory studies and applications [32]. In order to get
a high spatial resolution for the power absorption terms, we use $N_{\rm g}$ =
800 grid points to resolve the gap between the electrodes. The number of
superparticles in the simulations is $\sim 5\cdot 10^{4}$ and data for our
analysis are collected over $100-200$ RF cycles following the convergence of
the simulations. Due to the huge number of time steps per RF cycle the above
data collection period provides high quality data.
## 4 Results
In the following, results are presented for the “valleys” waveform (eq. 7)
with $\phi_{\rm pp}=500$ V and $f_{\rm b}=13.56$ MHz for different number of
harmonics ranging from $N=1-4$.
Figure 1: “Valleys” driving voltage waveform for different numbers of
harmonics.
Figure 1 shows the “valleys” driving voltage waveform given by eq. 7 for
different numbers of harmonics over two RF-cycles (where $T_{\rm
RF}=\frac{1}{f_{\rm b}}$). The $N=1$ case is a simple cosine-shaped function
with a frequency of $f_{\rm b}$. Increasing the number of harmonics has the
effect of introducing a “valley” at $t=0.5T_{\rm RF}$, whose width decreases
and depth is enlarged as the number of harmonics is increased. At the same
time, the voltage is in the vicinity of its maximum for an increasing fraction
of time. Thus, while in the single harmonic case the waveform monotonically
decreases from its maximum towards its minimum, in case of $N=4$ harmonics,
the waveform oscillates near its maximum and then sharply decreases towards
its minimum. This will have very important consequences for the electron
dynamics and the overall behaviour of the plasma.
To understand the properties of electron power absorption at atmospheric
pressure, fig. 2 shows the spatio-temporally averaged electron power
absorption terms as well as the total electron power absorption as a function
of the number of harmonics for a valleys driving voltage waveform. The first
obvious observation is that ohmic heating is the dominant power absorption
mechanism due to the high pressure and the corresponding high collisionality.
Figure 2: Spatio-temporally averaged power absorption terms as a function of
the number of harmonics as well as the proportion of ambipolar power
absorption compared to the total power absorption in percentage for valleys
driving voltage waveforms. Discharge conditions: $\phi_{\rm pp}=500$ V,
$f_{\rm b}=13.56$ MHz, $L=1$ mm. Figure 3: Spatial distribution of the
temporal average of the total electron power absorption as well as that of the
different power absorption terms for valleys waveforms with 1 harmonic (a) and
4 harmonics (b). The dashed blue lines indicate the maximum of the sheath
width, $s_{\rm max}$, calculated according to [64]. Discharge conditions:
$\phi_{\rm pp}=500$ V, $f_{\rm b}=13.56$ MHz, $L=1$ mm. The powered and
grounded electrodes are situated at $x=0$ and $x=1$ mm, respectively.
The other power absorption terms (inertial power absorption and pressure
heating) are negligible on space- and time average. Even the ambipolar power
absorption, which is found to be the dominant power absorption term at low
pressure [38, 39] is less than one percent of the total power absorption. By
increasing the number of harmonics, the ohmic power absorption, and
correspondingly, the total power absorption monotonically increase. In order
to explain this behaviour, the details of electron power absorption need to be
investigated. In the following, we will only do this in the two limiting
cases, i.e. the single harmonic ($N=1$) and the four harmonics case ($N=4$).
Figure 3 shows the temporal average of the different power absorption terms
according to eq. 2 along the discharge gap for a single harmonic (a) and four
harmonics (b). According to panel (a), due to the atmospheric pressure, the
dominant (and, essentially, the only relevant) power absorption term on time
average is ohmic heating, which is a direct consequence of the atmospheric
pressure and the corresponding high collisionality of electrons. This is in
strong contrast to the low pressure scenario, where the dominant power
absorption mechanism is usually Pressure heating [38, 39]. The spatial profile
of ohmic heating shows a symmetrical shape with a maximum in the middle of the
discharge. As the number of harmonics is increased, the amplitude of the
different power absorption terms increases, as noted above. In this case,
ohmic heating is still the dominant power absorption term, but there is a
region, in the vicinity of the grounded electrode ($x=1$ mm), where $P_{\nabla
n}$, the ambipolar power absorption, and $P_{\nabla T}$, the power absorption
due to the temperature gradient is non-negligible. The inertial power
absorption, $P_{\rm in}$, just as in the single harmonic case, is completely
negligible on time average as well. Due to the difference of the excitation
waveform, the shape of the temporal average of the power absorption is
different compared to the single harmonic case: instead of a spatially
symmetric profile with a maximum in the middle of the discharge, ohmic heating
has two, unequal maxima around the maximum of the corresponding sheath widths,
$s_{\rm max}$, with a plateau in the bulk region. The maximum near the
grounded electrode is greater than that near the powered electrode. The maxima
of $P_{\nabla n}$ and $P_{\nabla T}$ are also situated in the vicinity of the
grounded electrode. At the immediate vicinity of the grounded electrode, the
ohmic power absorption shows a strong increase, which is compensated by a
negative ambipolar heating. The maximal sheath widths are smaller in the four
harmonics case, which is an indication for the increased electron number
density. To understand the physical origin of these observations regarding
electron power absorption, one needs to investigate the spatio-temporal
distribution of the conduction current density, $j_{\rm c}$, and given its
dominance due to the high collisionality and a correspondingly local transport
of the plasma, the ohmic electric field, $E_{\rm Ohm}$.
Figure 4 shows the electron conduction current density, $j_{\rm c}$, (a,c) and
the ohmic electric field, $E_{\rm Ohm}$, (b,d) for one and four harmonics,
respectively. The current density in case of a single harmonic (a) is
temporally symmetric, with the maximum during the middle of sheath expansion.
Due to the local transport, it is the ohmic field that needs to drive this
current through the whole discharge. Thus, in panel (b), we see a similar
spatio-temporal structure for the ohmic electric field as for the current. The
relatively large maximum sheath width ($\sim 0.45$ mm) explains the observed
maximum in the ohmic power absorption in the discharge center in fig. 3(a), as
around this position the conduction current density is nonzero in the whole RF
period, and given that ohmic heating is always positive, its temporal average
at this position will be maximal. Changing the number of harmonics to $N=4$, a
temporal asymmetry between sheath expansion and collapse is clearly visible in
the conduction current density in panel (c). Due to the specific shape of the
valleys driving voltage waveform, the sheath at the grounded electrode is
collapsed for a short fraction of the fundamental RF period, i.e. for about 5
ns, while the sheath at the powered electrode is collapsed for a much longer
time, i.e. for about 40 ns.
Figure 4: Spatio-temporal distribution of the electron conduction current
density, $j_{\rm c}$, and ohmic electric field, $E_{\rm Ohm}$ for valleys
driving voltage waveforms with 1 harmonic (a, b) and 4 harmonics (c, d).
Discharge conditions: $\phi_{\rm pp}=500$ V, $f_{\rm b}=13.56$ MHz, $L=1$ mm.
The black lines indicate the sheath edges.
As the fluxes of electrons and positive ions must balance at each electrode on
time average [47] and ions flow to both electrodes continuously, a much higher
electron current must flow to the grounded electrode during the short local
sheath collapse compared to the situation at the powered electrode. Therefore,
the electron conduction current density is higher during the sheath collapse
at the grounded compared to that at the powered electrode (-1800 Am-2 compared
to 1000 Am-2, see fig. 4(c)) and a higher ohmic electric field is needed to
drive this higher current to the grounded electrode during the local sheath
collapse (-600 kVm-1 compared to 400 kVm-1, see fig. 4(d)). As the plasma
density is lower in the vicinity of the electrodes compared to the bulk
center, the ohmic electric field will increase in this region. This is the
reason for the unequal maxima of the temporally averaged ohmic heating in fig.
3(b). In the vicinity of the grounded electrode, two distinct local maxima can
be observed in the ohmic electric field during the local sheath collapse: one
in the immediate vicinity of the electrode and one on the bulk side of the
sheath edge. The maximum close to the electrode surface is due to the
decreased electron density at this position, and is the reason for the sharp
increase in the temporally averaged ohmic power absorption around $x\approx 1$
mm in fig. 3(b). The other local maximum in panel (d) is also due to a local
minimum of the electron density. The reason for this is as follows: Due to the
low electron density in the vicinity of the grounded electrode during the
local sheath collapse, there is a high instantaneous local ohmic electric
field that causes strong ionization at this position. This, in turn,
contributes to the formation of a local maximum of the electron density.
Additionally, enhanced local Penning ionization due to a locally enlarged
helium metastable density contributes to the formation of this electron
density peak. The presence of this electron density maximum and the
corresponding density gradient leads to the formation of an ambipolar electric
field on the bulk side of this maximum that accelerates electrons towards the
grounded electrode even more strongly, as $E_{\nabla n}\propto-\frac{1}{n_{\rm
e}}\frac{\partial n_{\rm e}}{\partial x}$. At the position, where the electron
density is maximum close to the grounded electrode, this ambipolar field as
well as the ohmic field are low due to the low local density gradient and the
high local electron density. Overall, this mechanism leads to the formation of
a strong local maximum of $n_{e}$. Due to the different durations of sheath
collapse at both electrodes induced by the tailored driving voltage waveform,
this happens only at one and not at the other electrode. Correspondingly,
electrons are only accelerated to high energies at the electrode, where the
sheath collapse is short, and ionization as well as dissociation and
excitation of neutrals occur predominantly at this position.
Figure 5: Spatio-temporal distribution of the ionization source function for
nitrogen, $S_{\rm ion,N_{2}^{+}}$ and He metastable density, $n_{\rm
He^{\ast}}$ for valleys driving voltage waveforms with 1 harmonic (a,b) and 4
harmonics (c,d). Discharge conditions: $\phi_{\rm pp}=500$ V, $f_{\rm
b}=13.56$ MHz, $L=1$ mm. The black lines indicate the sheath edges. The white
arrows in panels (a) and (c) indicate the ionization peaks due to electron
impact.
Ultimately, this explains in detail why such tailored voltage waveforms allow
to break the spatio-temporal symmetry of the electron dynamics in such
discharges, which is inherently present in single frequency jets, and why the
EEDF can be controlled in space and time to optimize the generation of
selected radicals [22, 24, 23, 34, 35].
To see the local maximum of the ionization caused by the increased ohmic
electric field during sheath collapse at the grounded electrode in case of
$N=4$ harmonics, one needs to look at the possible electron impact ionization
channels. As given by eqs. 3, 4 and 6, the three possibilities are electron
impact ionization of either helium atoms or nitrogen molecules, or Penning
ionization. As the ionization threshold for helium is higher than that of
nitrogen (24.59 eV vs. 15.6 eV), the electron impact ionization of helium can
be discarded. Figure 5 shows the spatio-temporal profiles of the ionization
source function for nitrogen, $S_{\rm ion,N_{2}^{+}}$ (a,c), and the He∗
metastable density, $n_{\rm He^{\ast}}$ (b,d) for one and four harmonics,
respectively. In panel (a) the contributions from the two channels, i.e.
electron impact ionization of $N_{2}$ and Penning ionization, can be
distinguished: the local maxima during sheath expansion at either electrode
are due to electron impact ionization (indicated by the white arrows), whereas
the “background”, which resembles the maxima of the helium metastable density
shown in panel (b) is due to Penning ionization, as according to eq. 6 Penning
ionization is caused by helium metastables whose density is approximately
constant in time.
Increasing the number of harmonics has the effect of introducing a spatial
asymmetry to the ionization as well as the metastable density profile: the
reason for this is the peculiar shape of the excitation waveform: at the
grounded electrode the sheath voltage is high for most of the time within the
fundamental RF-cycle. Therefore, secondary electrons can gain a high enough
energy to excite the background atoms, i.e. helium, which correspondingly
increases the metastable density and also the contribution of Penning
ionization at this electrode. As noted above, there is a local increase of
$S_{\rm ion,N_{2}^{+}}$ in panel (c), during sheath collapse at the grounded
electrode, which is a consequence of the increased ohmic electric field in
this region. This field is needed to drive a high current through the
discharge to ensure flux conservation at the grounded electrode. This has
important consequences for the spatio-temporal distribution of the electron
density and, thus, for the ambipolar electric field, which fundamentally
influences the electron dynamics.
Figure 6: Spatio-temporal distribution of the electron number density, $n_{\rm
e}$, and its temporal snapshots as a function of space for valleys driving
voltage waveforms with 1 harmonic (a,b) and 4 harmonics (c,d). Discharge
conditions: $\phi_{\rm pp}=500$ V, $f_{\rm b}=13.56$ MHz, $L=1$ mm. The black
lines in panels (a,c) indicate the sheath edges.
Figure 6 shows the spatio-temporal profile of the electron density, $n_{\rm
e}$ (a,c), and its temporal snapshots at given specific time instances in the
fundamental RF-cycle as a function of position (b,d) for one and four
harmonics, respectively. Panels (a) and (b) show the spatially symmetric
nature of the electron density in the single frequency case: this is in
accordance with the spatio-temporal profile of $j_{\rm c}$ in fig. 4 (a). The
position of the maximum of the density is temporally modulated, which is due
to the relatively low electron density, which also leads to the large sheath
widths. On the other hand, the electron density profile in case of four
harmonics shows a completely different behaviour. The electron density is
increased compared to the one harmonic case; the reason for this being the
peculiar nature of the excitation waveform: when the number of harmonics is
increased, the electrons get a stronger “kick” at $t=0.5T_{\rm RF}$. This
increases the energy of the electrons, which can then contribute more to
ionization. The waveform shape is also responsible for the slightly shifted
shape of $n_{\rm e}$: the density is quite low in the vicinity of the powered
electrode up to the maximum sheath width, as the sheath is relatively small
during most of the time in the RF-cycle, where little ionization happens, but
near half of the RF-cycle the sheath width is sharply increased resulting in
higher ionization. In the vicinity of the grounded electrode during the sheath
expansion/collapse phase of the “valley” a local electron density increase is
present; this is also shown in panel (d) at the time of $t=0.5T_{\rm RF}$,
near $\approx 0.9$ mm the electron density first decreases and then increases
as a function of position, which is a consequence of the local ionization
maximum in fig. 5(c) that is due to the local maximum of the ohmic electric
field in fig. 4(d) and a local enhancement of the Penning ionization due to
the high local metastable density.
Figure 7: Spatio-temporal distribution of the parallel electron temperature,
$T_{\parallel}$ and normalized electron density gradient, $\nabla n_{\rm
e}/n_{\rm e}$ for valleys driving voltage waveforms with 1 harmonic (a, b) and
4 harmonics (c, d). Discharge conditions: $\phi_{\rm pp}=500$ V, $f_{\rm
b}=13.56$ MHz, $L=1$ mm. The black lines indicate the sheath edges. The dashed
black lines in panels (b,d) indicate the points where $\nabla n_{\rm e}$ is
zero.
The presence of this local density increase will result in a positive density
gradient between the position of the local minimum near the maximum sheath
width at the grounded electrode and the position of the local maximum,
resulting in an electric field reversal due to the ambipolar field.
In order to investigate the ambipolar electric field, given by $E_{\nabla
n}=-\frac{T_{\parallel}}{n_{\rm e}}\frac{\partial n_{\rm e}}{\partial x}$, one
needs to take a look at fig. 7, which shows the parallel electron temperature,
$T_{\parallel}$ (a,c) and the normalized electron density gradient,
$\frac{1}{n_{\rm e}}\frac{\partial n_{\rm e}}{\partial x}$ (b,d) for one and
four harmonics, respectively.
The electron temperature in the single harmonics case (panel (a)) is increased
during sheath expansion due to the fact that electrons are accelerated by the
expanding sheath. At this time, within the RF period, the temperature
decreases as a function of position from the sheath edge towards the bulk and
then, around the position of maximum sheath edge at the opposite electrode,
begins to increase again. The reason for this behaviour is the local character
of the transport: as shown previously, due to the atmospheric pressure, ohmic
heating is the dominant power absorption, thus, the profile of the temperature
is primarily determined by the spatio-temporal distribution of the ohmic
electric field. As in the bulk region the electron density, $n_{\rm e}$, is
high as shown in fig. 6(a), thus the resistivity is low, the ohmic electric
field is smaller, and as a consequence, the temperature will also decrease in
this region. The spatially periodic patterns present inside the sheath are due
to the small energy relaxation length of the secondary electrons, emitted from
the electrodes, and are reminiscent of the Franck-Hertz experiment [23]. Panel
(b) shows the normalized electron density gradient, $\frac{1}{n_{\rm
e}}\frac{\partial n_{\rm e}}{\partial x}$, for the single harmonic case. As
one moves away from the powered electrode, this quantity is positive, i.e. the
electron density increases, then reaches its maximum (whose axial position is,
under these conditions, time-dependent), and then decreases towards the
grounded electrode in accordance with fig. 6(a).
The spatio-temporal distribution of the temperature in case of four harmonics
is shown in panel (c). The temporal asymmetry between sheath expansion and
collapse is present in this quantity as well, which is due to the increased
current during sheath collapse at the grounded electrode to ensure flux
conservation at this boundary surface. Thus, during sheath collapse at the
grounded electrode, the electron temperature is higher along the whole
discharge than during the collapse phase. There is a local maximum of the
electron temperature at the position of the local ohmic electric field maximum
in fig. 4(d), i.e. in the vicinity of the grounded electrode during its sheath
collapse. As seen in the normalized electron density gradient, panel (d), this
is related to the local maximum of the electron density, which results in a
negative electron density gradient, and, thus, a negative ambipolar field at
the grounded electrode, i.e. a local electric field reversal. Thus, this
electric field will increase the energy of incoming electrons accelerated by
the high ohmic field in this region, resulting in more ionization and a higher
local maximum of the electron density. Therefore, even though on time average
the ambipolar power absorption is very small, it has an important effect on
the details of the electron dynamics.
The spatio-temporal distributions of the electron power absorption terms
investigated in this paper are shown in fig. 8 for one harmonic (first column)
and four harmonics (second column). As the inertial power absorption proved to
be negligible and hence unimportant, here we only show the other power
absorption terms.
Figure 8: Spatio-temporal distribution of electron power absorption terms in
units of MWm-3 for valleys driving voltage waveforms with 1 harmonic (first
column) and 4 harmonics (second column). Discharge conditions: $\phi_{\rm
pp}=500$ V, $f_{\rm b}=13.56$ MHz, $L=1$ mm. The black lines indicate the
sheath edges.
Panel (a) of figure 8 shows the spatio-temporal distribution of the ambipolar
power absorption, $P_{\nabla n}$, for a single harmonic. It is similar to that
found in the case of low pressure: there is a temporally symmetric shape
(although shifted in time due to the high pressure and the corresponding
resistivity), where during sheath expansion, there is positive ambipolar
heating in the vicinity of the powered electrode. Electrons are accelerated in
this region by the expanding sheath, whereas during its collapse, incoming
electrons are decelerated which results in a negative power absorption. In
panel (c) of figure 8, the power absorption due to the temperature gradient is
negative during sheath expansion, i.e. as the conduction current density,
$j_{\rm c}$, is negative (because during sheath expansion electrons move
towards the grounded electrode), $E_{\nabla T}=-\frac{1}{e}\frac{\partial
T_{\parallel}}{\partial x}$ is positive, which indicates a negative
temperature gradient in the bulk region during the first half of the RF-cycle,
which then near the maximum sheath width at the grounded electrode changes
sign and becomes positive. This means, that the parallel electron temperature
first decreases as the electrons move away from the powered electrode and then
increases as they reach the maximum sheath width of the grounded electrode, as
shown in fig. 7(a). Ohmic heating (panel (e) of figure 8) is positive (meaning
that electrons lose momentum during collisions as they move towards the
opposite electrode) in the entire discharge region. Note, that in accordance
with fig 3 (a), the magnitude of ohmic heating is much higher than that of the
other two power absorption terms.
As the number of harmonics is increased, the spatio-temporal distributions of
the power absorption terms show notable differences as previously discussed.
Panels (b) and (d) of figure 8, i.e. $P_{\nabla n}$ and $P_{\nabla T}$ show
the presence of the electric field reversal in the vicinity of the grounded
electrode. Similarly, there is a temporal asymmetry in case of ohmic heating
(panel (f)), which is the result of the strong “kick” of the rapid sheath
expansion at the grounded electrode. Due to this a high current has to flow
through the discharge in order to ensure flux compensation at the grounded
electrode. At positions where the electron density is low in the spatial
region near the grounded electrode during the local sheath collapse, the ohmic
power absorption shows a local maximum, as due to the low density the
resistivity of the plasma increases. At the position of the local electron
density maximum close to the grounded electrode, the ohmic power absorption is
depleted. The ambipolar power absorption follows the gradient of the electron
density caused by its local maximum. The behaviour of $P_{\nabla T}$ is a
direct consequence of the spatio-temporal dynamics of the electron temperature
discussed before.
## 5 Conclusions
The electron power absorption dynamics in a micro atmospheric pressure RF
plasma jet driven by a “valleys” waveform with 500 V peak-to-peak voltage and
13.56 MHz base frequency operated in a He-N2 gas mixture with a nitrogen
concentration of 0.05% and an electrode gap of 1 mm has been investigated for
one and four harmonics using the Boltzmann term analysis, which offers a self-
consistent, spatio-temporally resolved description of electron power
absorption [38, 39, 40]. The input parameters needed for the analysis were
obtained by 1d3v PIC/MCC simulations.
Due to the atmospheric pressure and the corresponding local nature of the
transport, the dominant, and essentially only relevant power absorption term
on space and time average is found to be ohmic heating. In the single
frequency case, the spatial profile of the time-averaged ohmic heating has a
symmetrical shape with a maximum in the center of the discharge. The reason
for this is the relatively low electron density and the correspondingly large
sheath widths, due to which the maximum of the electron density is time
modulated. The spatio-temporal profiles of $P_{\nabla n}$ and $P_{\nabla T}$,
i.e. the ambipolar power absorption and the power absorption corresponding to
the gradient of the parallel electron temperature, are found to be similar to
that found in low pressure CCPs ([38, 39]), with the important difference that
in this case these power absorption terms are found to be negligible on time
average. $P_{\rm in}$, i.e. inertial power absorption is found to be
completely negligible in the cases considered.
Increasing the number of harmonics from 1 to 4 causes notable differences.
Although, on space- and time average still ohmic heating is dominant,
ambipolar power absorption is found to have a non-negligible contribution to
the time-averaged electron power absorption in a spatial region near the
grounded electrode. Furthermore, the spatial profile of the time-averaged
electron power absorption becomes asymmetric, which is the consequence of the
peculiar shape of the excitation waveform, as during sheath collapse at the
grounded electrode electrons are strongly accelerated, which increases their
energy contributing to an increased electron power absorption on time average.
This is the reason for an increase of the electron density compared to the
single frequency case at otherwise identical discharge conditions. An
additional consequence of the “valleys” waveform is a temporal asymmetry in
the electron conduction current, $j_{\rm c}$, as the amplitude of the current
density is found to be higher during sheath collapse at the grounded electrode
than during sheath expansion at this electrode. The reason for this temporal
asymmetry is the different durations of sheath collapse at the electrodes: due
to the valley driving voltage waveform the sheath collapse at the grounded
electrode is shorter than at the powered electrode. Thus, in order to ensure
flux compensation of positive ions and electrons at this electrode, the
electron conduction current density during sheath collapse at the grounded
electrode has to be higher compared to the sheath collapse at the other
electrode. Due to the local nature of the transport at atmospheric pressure,
it is primarily the ohmic electric field that needs to drive this current
through the discharge and is, thus, temporally asymmetric as well. During the
sheath collapse at the grounded electrode, electrons accelerated by this high
ohmic field as well as a locally enhanced Penning ionization due to locally
enhanced helium metastable densities lead to the generation of a local
electron density maximum. In the spatial region between the maximum sheath
width at the grounded electrode and this maximum, i.e. where the electron
density gradient is positive, a negative ambipolar electric field is formed,
leading to an electric field reversal that accelerates electrons towards the
grounded electrode and enhances the local generation of energetic electrons in
the vicinity of the grounded electrode.
These findings explain at the fundamental level how the spatio-temporal
electron power absorption dynamics work in $\mu$APPJs and why its spatio-
temporal symmetry can be broken as well as controlled by Voltage Waveform
Tailoring. This is the physical origin of the enhanced EEDF control and the
enhanced generation of radicals as well as excited neutrals by Waveform
Tailoring in such discharges. Thus, these insights might provide the basis for
the optimization of applications of such microplasmas that rely on
high/controlled fluxes of neutral particle generated by electron impact driven
reactions with specific electron energy thresholds and dependencies.
## Acknowledgements
This work was funded by the German Research Foundation in the frame of the
project, “Electron heating in capacitive RF plasmas based on moments of the
Boltzmann equation: from fundamental understanding to knowledge based process
control” (No. 428942393). Support by the DFG via SFB TR 87, project C1, SFB
1316, project A4 and by the Hungarian Office for Research, Development, and
Innovation (NKFIH 134462) is gratefully acknowledged.
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|
# Relational quantum computing using only maximally mixed initial qubit states
Terry Rudolph Dept. of Physics, Imperial College London, London SW7 2AZ,
<EMAIL_ADDRESS>Shashank Soyuz Virmani Dept. of Mathematics, Brunel
University London, Kingston Ln, London, Uxbridge UB8 3PH,
<EMAIL_ADDRESS>
In [1] we showed that the (non-destructive) 2-qubit singlet/triplet (s/t)
measurement is universal for quantum computing, given only an initial Haar-
twirled qubit ensemble of the form
$\int\\!\\!d\Omega\,\,U(\Omega)^{\otimes 3N}\left(\rho_{a}^{\otimes
N}\otimes\rho_{b}^{\otimes N}\otimes\rho_{c}^{\otimes
N}\right)U^{\dagger}(\Omega)^{\otimes 3N}.$ (1)
Here $N$ grows polynomially in the computation size. The Bloch vectors of
$\rho_{a}$, $\rho_{b}$, $\rho_{c}$ needed only to satisfy the constraints that
they are non-zero and linearly independent. Universality of s/t measurements
under other assumptions was considered recently in [2]. Here we will show that
in fact replacing (1) with even a resource of only maximally mixed single-
qubits suffices to make the s/t measurement universal.111Almost any resource
of qubits can be turned into such by repeatedly performing the s/t
measurement, keeping singlet outcomes, and discarding a member of each pair.
There are two steps to the procedure. The first is to note that we can
efficiently prepare a state exponentially close to the maximally mixed state
over the symmetric subspace of $N$ qubits, $\rho^{sym}_{N}$. The second will
be to convert several copies of $\rho^{sym}_{N}$ to (1) using a procedure
known as measurement-induced relative localisation [3].
Creating the maximally mixed state in the symmetric subspace: Suppose that we
start with $K$ qubits prepared in $\rho^{sym}_{K}$. We bring in a new qubit in
the maximally mixed state and randomly pick pairs of qubits222More efficient
choices than random pairings exist - for example interpreting the switches of
the networks in [4] as s/t measurement locations. to undergo a polynomial
number of s/t measurements. If we only ever find triplet outcomes, we perform
an exponentially good approximate projection [2] into the state
$\rho^{sym}_{K+1}$. This occurs with probability $P(K)=(K+2)/(2K+2)>1/2$. If
we ever find a singlet outcome we discard those two qubits and the remaining
$K-1$ qubits are left in $\rho^{sym}_{K-1}$.
We can interpret the protocol as a 1-d random walk process where we begin at
$K=1$, and have a probability $P(K)$ of stepping to the right, and $1-P(K)$ of
stepping to the left. The boundary at $K=0$ is absorbing (fail, restart) and
let us consider our target is to create $\rho_{N}^{sym}$ for some fixed $N$.
The solution to this problem can be found in [5]. Note that the particle must
eventually be absorbed at one or the other boundaries. Eq. (2.7) of [5] for
our case yields that the probability it is absorbed at the right hand boundary
is $(N+1)/2N$, i.e. slightly higher than 1/2. Thus we have finite probability
of eventual success. To ensure the resources consumed (qubits/time steps) are
polynomial we need to compute the conditional mean for the number of steps
before stopping (absorption at a boundary). This can be found by solving the
recurrence relations (3.1)-(3.3) in [5]. For starting at $K=1$ we find the
expected number of steps before absorption is $(N^{2}+3N-4)/6$, which grows
polynomially with $N$. (The method we have outlined may possibly be improved
using a variant, with an appropriate input state, of the splitting protocol of
[2].)
Relative Localization: The second step is to see how two sufficiently large
maximally mixed symmetric states can be converted into an ensemble equivalent
to a Haar-twirled product state over pure states with fixed overlap, i.e. a
state of the form:
$\int\\!\\!d\Omega\,\,U(\Omega)^{\otimes
2N}\left(\left|a\right\rangle\left\langle a\right|^{\otimes
N}\otimes\left|b\right\rangle\left\langle b\right|^{\otimes
N}\right)U^{\dagger}(\Omega)^{\otimes 2N}$
for some arbitrary (but known) $\left|a\right\rangle$, $\left|b\right\rangle$
with $0<|\langle a|b\rangle|^{2}<1$.
This can be done by using a “measurement induced localization” of the relative
angle between initial (mixtures of) spin coherent states, in full analogy with
the cases for optical phase, Bose-Einstein condensate phase, and particle
position studied in detail in [3]. (If we can localize two such ensembles then
we can use the same procedure to relationally localize further ensembles to
the first two).
The basic intuition is simple: we start with two sources
$\rho^{sym}_{N+M}\otimes\rho^{sym}_{N+M}$, as created in the first step, and
interpret each as an ensemble of $N+M$ copies of a randomly selected pure
state. We pair up $M$ of the spins from each source and perform the
singlet/triplet measurement on each pair, enabling us to get a good estimate
of the overlap between the two (random) pure states. We then use the remaining
$2N$ qubits for computation, under the assumption that the overlap is the
estimated one [6]. As we now demonstrate, a fixed overall error across the
$2N$ qubits requires $M$ to grow only polynomially in $N$.
Because of the collective unitary freedom, we are free to decide that the
first symmetric state is actually a source of $\left|0\right\rangle^{\otimes
N+M}$, and the second state $\left|\theta\right\rangle^{\otimes N+M}$ is
specified by the relative angle $\theta\in[0,\pi)$ its Bloch vector makes with
the first source state, where $\theta$ has p.d.f $\sin(\theta)/2$. An s/t
measurement on $\left|0\right\rangle\otimes\left|\theta\right\rangle$ gives a
triplet outcome with probability
$q=(1+\cos^{2}(\theta/2))/2=(3+\cos(\theta))/4.$ The total probability over
the $M$ measurements of obtaining $n_{1}$ triplet outcomes is
$\displaystyle
P(n_{1})=\int^{\pi}_{0}\\!\\!\\!d\theta\,\binom{M}{n_{1}}q^{n_{1}}(1-q)^{M-n_{1}}{\sin(\theta)\over
2}$
$\displaystyle=2\binom{M}{n_{1}}\int^{1}_{1/2}\\!\\!\\!dq\,q^{n_{1}}(1-q)^{M-n_{1}}$
This has the convenient interpretation that the probability of seeing a given
number of triplets is described by a Bernoulli trial with a uniformly chosen
$q$ in the interval $[1/2,1]$. Estimating $\theta$ corresponds to estimating
$q$ given the observed $M$, $n_{1}$, so we will also write
$\left|q\right\rangle:=\left|\theta\right\rangle$.
Considering the function
$\displaystyle T(a,b)$ $\displaystyle=$
$\displaystyle\int^{1}_{1/2}\\!\\!\\!dq\,q^{a}(1-q)^{b}$ $\displaystyle=$
$\displaystyle{a!b!\over(a+b+1)!}{1\over
2^{a+b+1}}\sum^{a}_{j=0}\binom{a+b+1}{j}$
we can use standard identities (e.g here) for partial sums of binomial
coefficients to see that $T(a,b)$ is exponentially close (in $M=(a+b)$) to
$\left(\binom{a+b}{a}(a+b+1)\right)^{-1}$ when $a>(a+b)/2$. Applying this to
$P(n_{1})$ we find that it is exponentially close to $2/(M+1)$, which means
that with high probability on any given run of the procedure we will observe
$n_{1}>M/2$ triplet outcomes, and from now on, we consider only situations
where this has occurred.
The probability density of $q$ given $n_{1}$ triplet outcomes is (over the
domain $q\in[1/2,1]$):
${\rm Pr}(q|n_{1},M)={q^{n_{1}}(1-q)^{M-n_{1}}\over T(n_{1},M-n_{1})},$
from which we wish to bound the goodness of our estimated value of $q$ (and
hence $\theta)$. The mean and variance for this inference problem are given by
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle={T(n_{1}+1,M-n_{1})\over
T(n_{1},M-n_{1})}\approx{n_{1}+1\over M+2}$ $\displaystyle\sigma^{2}$
$\displaystyle=$ $\displaystyle={T(n_{1}+2,M-n_{1})\over
T(n_{1},M-n_{1})}\approx{(n_{1}+1)(M+1-n_{1})\over(M+2)^{2}(M+3)},$
where $\approx$ denotes exponential closeness. A simple upper bound on the
variance is then $\sigma^{2}<1/M$.
Now, we are roughly in the following situation: we will operate as if $q=\mu$,
i.e. the state of the second $N$ qubits is $\left|\mu\right\rangle^{\otimes
N}$ (by collective rotational freedom a state in the right semicircle of the
XZ plane in the Bloch sphere), but with a low probability ($\leq 1/h^{2}$ by
the Chebyshev inequality) the actual value of $q$ could be further than
$h\sigma$ from this. In later calculations we will pick $h=M^{1/6}$. The error
we want to understand will ultimately arise from the trace distance between
the estimated state and the actual one, and so we wish to bound this.
To make things simpler we first ask, for any pair of $q_{1},q_{2}$ with a
fixed value of $|q_{1}-q_{2}|$, what is the largest possible trace distance
between the corresponding quantum states $\left|q_{1}\right\rangle$,
$\left|q_{2}\right\rangle$? Elementary considerations [7] yield:
$\|\left|q_{1}\right\rangle\left\langle
q_{1}\right|-\left|q_{2}\right\rangle\left\langle q_{2}\right|\|\leq
2\sqrt{8|q_{1}-q_{2}|}$ (2)
We can use this to bound the overall error via:
$\displaystyle\|\left|\mu\right\rangle\left\langle\mu\right|^{\otimes
N}-\int\\!\\!dq\,{\rm Pr}(q|n_{1},M)\left|q\right\rangle\left\langle
q\right|^{\otimes N}\|$ $\displaystyle\leq\int\\!\\!dq\,{\rm
Pr}(q|n_{1},M)\|\left|\mu\right\rangle\left\langle\mu\right|^{\otimes
N}-\left|q\right\rangle\left\langle q\right|^{\otimes N}\|$
$\displaystyle\leq\sqrt{N-1}\int\\!\\!dq\,{\rm
Pr}(q|n_{1},M)\|\left|\mu\right\rangle\left\langle\mu\right|-\left|q\right\rangle\left\langle
q\right|\|$ $\displaystyle\leq 2\sqrt{8(N-1)}\int\\!\\!dq\,{\rm
Pr}(q|n_{1},M)\sqrt{|q-\mu|}$ $\displaystyle\leq
2\sqrt{8(N-1)}\sqrt{\int\\!\\!dq\,{\rm Pr}(q|n_{1},M)|q-\mu|}$
$\displaystyle\leq 2\sqrt{8(N-1)}\sqrt{{1\over h^{2}}+h\sigma}\leq
2\sqrt{8(N-1)}\sqrt{{2\over M^{1/3}}}$
where the first inequality is the triangle inequality, the second is because
for pure states it holds that $\|\psi^{\otimes N}-\phi^{\otimes
N}\|\leq\sqrt{N-1}\|\psi-\phi\|$, the third is from the bound (2), the fourth
is concavity of the square root, the fifth is from the largest probabilities
(and $|q-\mu|$ values) consistent with the Chebyshev inequality, and the last
is obtained by using $\sigma\leq 1/\sqrt{M}$ and picking $h=M^{1/6}$.
We deduce that given target overall error of $\epsilon$, we can choose
$M\sim(N/\epsilon^{2})^{3}$, which is a polynomial cost.
Acknowledgements: SSV acknowledges thought provoking discussions with Nihaal
Virmani. TR acknowledges interaction with one of two indistinguishable
Virmanicles.
## References
* Rudolph and Virmani [2005] T. Rudolph and S. S. Virmani, New Journal of Physics 7, 228 (2005).
* Freedman _et al._ [2021] M. H. Freedman, M. B. Hastings, and M. S. Zini, (2021), arXiv:2105.04649 [quant-ph] .
* Cable _et al._ [2005] H. Cable, P. L. Knight, and T. Rudolph, Phys. Rev. A 71, 042107 (2005).
* Czumaj [2015] A. Czumaj, in _Proceedings of STOC ’15_, STOC ’15 (2015) p. 703–712.
* El-Shehawey [2000] M. A. El-Shehawey, J. Phys A 33, 9005 (2000).
* [6] We remark that the relative localisation will happen more efficiently using approximate total angular measurement protocols of [2] together with some form of global angular momentum inference scheme (see e.g. [8]).
* [7] In brief: the state $\left|q\right\rangle$ has $z$ component of its Bloch vector given by $z=4q-3$, so a given value of $|q_{1}-q_{2}|$ constrains the Bloch vectors of $\left|q_{1}\right\rangle$, $\left|q_{2}\right\rangle$ to have a projection on the z-axis to a fixed interval $4|q_{1}-q_{2}|$. Positioning one end of this interval at either the north or south poles of the Bloch sphere yields the largest possible trace distance consistent with this projected value.
* Fanizza _et al._ [2020] M. Fanizza, M. Rosati, M. Skotiniotis, J. Calsamiglia, and V. Giovannetti, Phys. Rev. Lett. 124, 060503 (2020).
|
# Statistical tests of young radio pulsars with/without supernova remnants:
implying two origins of neutron stars
Xiang-Han Cui1,2, Cheng-Min Zhang1,2,3, Di Li1,2,4, Jian-Wei Zhang1, Bo
Peng1,2,5, Wei-Wei Zhu1,2, Qing-Dong Wu6, Shuang-Qiang Wang6, Na Wang6, De-Hua
Wang7, Yi-Yan Yang8, Zhen-Qi Diao7, Chang-Qing Ye7, and Hsiang-Kuang Chang9
1National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China
2School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China
3School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
4NAOC-UKZN Computational Astrophysics Centre, University of KwaZulu-Natal,
Durban 4000, South Africa
5Guizhou Radio Astronomy Observatory, Chinese Academy of Sciences, Guiyang
550025, China
6Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi,
Xinjiang 830011, China
7School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550001, China
8School of Physics and Electronic Science, Guizhou Education University,
Guiyang 550018, China
9Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan,
China [email protected](CMZ)
(Accepted XXX. Received YYY; in original form 2021)
###### Abstract
The properties of the young pulsars and their relations to the supernova
remnants (SNRs) have been the interesting topics. At present, 383 SNRs in the
Milky Way galaxy have been published, which are associated with 64 radio
pulsars and 46 pulsars with high energy emissions. However, we noticed that
630 young radio pulsars with spin periods of less than half a second have been
not yet observed the SNRs surrounding or nearby them, which arises a question
of that could the two types of young radio pulsars with/without SNRs hold
distinctive characteristics? Here, we employ the statistical tests on the two
groups of young radio pulsars with (52) and without (630) SNRs to reveal if
they share different origins. Kolmogorov-Smirnov (K-S) and Mann-Whitney-
Wilcoxon (M-W-W) tests indicate that the two samples have the different
distributions with parameters of spin period ($P$), derivative of spin period
($\dot{P}$), surface magnetic field strength ($B$), and energy loss rate
($\dot{E}$). Meanwhile, the cumulative number ratio between the pulsars with
and without SNRs at the different spindown ages decreases significantly after
$\rm 10-20\,Kyr$. So we propose that the existence of the two types of
supernovae (SNe), corresponding to their SNR lifetimes, which can be roughly
ascribed to the low-energy and high-energy SNe. Furthermore, the low-energy
SNe may be formed from the $\rm 8-12\,M_{\odot}$ progenitor, e.g., possibly
experiencing the electron capture, while the main sequence stars of $\rm
12-25\,M_{\odot}$ may produce the high-energy SNe probably by the iron core
collapse.
###### keywords:
pulsars: general - stars: neutron - supernovae: general - methods: statistical
††pubyear: 2021††pagerange: Statistical tests of young radio pulsars
with/without supernova remnants: implying two origins of neutron
stars–Appendix C: Tests of spin period and its derivative with the different
ages
## 1 Introduction
The associations between the pulsars and supernova remnants (SNRs) have been
of considerable interest topics in neutron stars (NSs) astrophysics, since the
radio pulses were firstly observed in the Crab Nebula in 1968 (Staelin &
Reifenstein 1968), as well as the discovery of the Vela pulsar (Large,
Vaughan, & Mills 1968). The story is continuing with identifying the potential
NS in SN 1987A (Page et al. 2020; Greco et al. 2021; Soker 2021). Thanks the
efforts and developments of astronomical facilities in recent years, the
number of pulsars and SNRs has increased significantly. Up to now, there are
more than 3000 pulsars
(ATNF111https://www.atnf.csiro.au/research/pulsar/psrcat/: 2781,
GPPS222http://zmtt.bao.ac.cn/GPPS/: 201 (Han et al. 2021),
CRAFT333https://crafts.bao.ac.cn/: 125) observed in the radio band and 383
SNRs (SNRcat444http://snrcat.physics.umanitoba.ca/SNRtable.php) in the Milky
Way galaxy had been published (Ferrand & Safi-Harb 2012). Among these, there
are 110 pulsars that have been identified in SNRs, including 6 anomalous X-ray
pulsars (AXPs), 5 soft gamma-ray repeaters (SGRs), a total of 13 magnetar
candidates, and 15 central compact objects (CCOs) or CCO candidates. By
comparing with ANTF Pulsar Catalogue, there are also 18 pulsars without radio
emissions (NRAD) in SNRcat, only observed at the infrared or higher
frequencies (Manchester et al. 2005). In short, 64 radio pulsars with SNRs
have been published, as seen in Table 1. Interestingly, the number ratio
between the radio pulsars (64) and SNRs (337) is about 1/5, which is
consistent with the estimation by the beaming fraction of radio pulsars
(Taylor & Manchester 1977; Lorimer et al. 1993; Lorimer & Kramer 2012).
Table 1: List of various types of pulsars with SNRs
Source | Number | Ref.
---|---|---
SNR | 383 | [1]
Pulsara | 110 | [1]
Radio pulsar | 64b | [2]
Magnetar or candidate | 13c | [1, 3, 4]
CCOd or candidate | 15 | [1]
NRADe | 18 | [2]
Note:
a Various pulsars with SNRs.
b The total number of radio pulsars with SNRs is 64, but only 52 of them are
analyzed in this article (selection details in Section 2.1).
c AXP (anomalous X-ray pulsar): 6, SGR (soft gamma-ray repeater): 5.
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html
d CCO: central compact object.
e Pulsars without radio emissions and do not contain the magnetars, CCOs, and
their candidates.
Ref.: [1] Ferrand & Safi-Harb (2012); [2] Manchester et al. (2005); [3]
Olausen & Kaspi (2014); [4] Esposito, Rea, & Israel (2021).
From Table 1, the question of why only a fraction of SNRs have been found with
pulsars can be solved by their beaming cone angles across the earth.
Meanwhile, some other explanations are worthy of mentioning as pointed out as
below. First of all, not every SNe could generate a NS or some NSs might not
be detectable as pulsars (Radhakrishnan & Srinivasan 1980; Srinivasan,
Bhattacharya, & Dwarakanath 1984; Manchester 1987; Narayan & Schaudt 1988).
Next, the flux density or luminosity of young radio pulsars may be
overestimated, implying that some young radio pulsars are too faint to be
observed in some SNRs (Stollman 1987; Lorimer et al. 1993). Finally, the kick
velocity of pulsar may be quite high after birth, which could result in the
pulsars to escape from their SNRs (Frail, Goss, & Whiteoak 1994).
Although some pulsars can be found in the pulsar wind nebulae (Gaensler &
Slane 2006), it is still an open question that so many young radio pulsars
have not seen their SNRs. Nowadays, there exist 630 young radio pulsars (spin
period less than 0.5 s, and details as described in Section 2) without SNRs,
which provides us a new aspect to further study the association between
pulsars and SNRs, as well as the NS origins. In the former studies of X-ray
pulsars (Knigge, Coe, & Podsiadlowski 2011) and double neutron star (DNS)
(Yang et al. 2019) population, researchers suggested that the NSs may birth
from the different origins, i.e., the existence of the electron capture and
iron core collapse SNe (Janka 2012). In the recent work about pulsars and
SNRs, Malov (2021) noticed that the mean values of radio luminosity of pulsars
observed inside and outside SNRs are significantly different with one order of
magnitude.
Inspired by these researches, we conjecture that there may exist two origins
for radio pulsars. Therefore, we attempt to use the statistical methods to
study the properties of radio pulsars and their relations with SNRs. We apply
some physical criteria to select two samples of pulsars, that is, the radio
pulsars with SNR (SNR-PSRs, 52) and without SNR (non SNR-PSRs, 630) (see
Section 2.1 for details) By drawing the cumulative distribution function (CDF)
and utilizing the Kolmogorov-Smirnov (K-S) and Mann-Whitney-Wilcoxon (M-W-W)
tests for these two samples, we find that the obtained results support the
different distributions for two samples. Based on spin period evolution model,
we estimate the spindown age (not characteristic age) of radio pulsars in
these two samples. After a further discussion, it is inferred that the two
samples of pulsars may origin from two types of progenitors, such as the low-
energy and high-energy SNe (e.g., electron capture and iron core collapse),
respectively. However, the energy boundary is still unclear, or there may be
an overlapping part between them, because some researches in SNRs indicated
that SNRs may have a continuous energy distribution like lognormal (Leahy
2017; Leahy, Ranasinghe, & Gelowitz 2020). Meanwhile, we notice that the
cumulative number ratios of SNR-PSRs to non SNR-PSRs are decreased quickly
after $\rm 10-20\,Kyr$. Finally, according to the initial mass function (IMF)
or Salpeter function (Salpeter 1955), it is possible to statistically
distinguish the two types of their progenitor stars at the mass boundary of
$\rm\sim 12\,M_{\odot}$.
The structure of our paper is presented as follows. In Section 2, we describe
the data selection of pulsars and introduce the spin period evolution model.
In Section 3, we apply the statistical tests on the two samples and analyze
the results. Finally, in Section 4, we discuss a possible physical
significance for the two distributions of the pulsars, and main conclusions
are summarized also.
## 2 Data and Model
### 2.1 Data Selection
Our data are taken from ATNF Pulsar Catalogue (Manchester et al. 2005) and
SNRcat (Ferrand & Safi-Harb 2012). Data selections are made to construct the
two samples of the young radio pulsars with and without SNRs, and the
selection criteria are described below.
1) We choose the pulsar samples in the two data bases with the spin periods
($P$) less than 0.5 s. Due to the short duration of SNRs, the pulsars with
SNRs are taken as the young pulsars (Gaensler & Johnston 1995; Malov 2021).
Considering that the characteristic age of pulsar usually has a significant
error compared to the real age (Lai 1996; Kaspi et al. 2001; Tian & Leahy
2006; Lyne & Graham-Smith 2012), we apply the spin period to represent the age
under the spin period evolution model (details in Section 2.2). Generally
speaking, the lifetime of SNRs usually less than 300 Kyr, so the spin period
of pulsars may be around 0.5 s based on the magnetic dipole model (see Section
2.2 and Appendix A for calculation details). Meanwhile, for all radio pulsars,
the median of period is about 0.5 s, which can be known from ATNF database
(Manchester et al. 2005). Additionally, because the periods for most rotating
radio transients (RRATs) and intermittent pulsars are greater than 0.5 s
(McLaughlin et al. 2006; Kramer et al. 2006), we eliminate these special radio
pulsars from our samples. Therefore, we regard the radio pulsars with spin
periods of less than 0.5 s to be the young pulsar, and they are roughly
consistent with the arguments based on that their magnetic fields are
comparable with those measured by cyclotron absorption lines of X-rays (Ye et
al. 2019).
2) The data with the surface magnetic field strength ($B$) ranged from $\rm
10^{11}\,G$ to $\rm 10^{14}\,G$ are used. If the pulsar’s B-field is lower
than $\rm 10^{11}\,G$, it may be a millisecond pulsar (MSP) (Bhattacharya &
van den Heuvel 1991; Lorimer 2008) or a CCO (Halpern & Gotthelf 2010;
Gotthelf, Halpern, & Alford 2013). If the pulsar’s B-field is higher than $\rm
10^{14}\,G$, it may be a magnetar (Duncan & Thompson 1992; Ferrario &
Wickramasinghe 2008; Kaspi & Beloborodov 2017; Esposito, Rea, & Israel 2021).
3) We select the data to satisfy that their spin periods and magnetic fields
are distributed above the spin-up line in B-P diagram (Bhattacharya & van den
Heuvel 1991). While, the pulsars below the spin-up line may have a more
complicated evolutionary tracks, such as experiencing the accretion spin-up in
binary systems (Zhang & Kojima 2006).
4) Some types of the special pulsars are removed from our samples, including
the MSPs, magnetars, recycled pulsars (Zhang & Kojima 2006), CCOs, RRATs,
intermittent pulsars and NRADs. The physical characteristics between the young
radio pulsars and these special pulsars are significantly different, so they
may follow the different birth conditions and evolutionary paths.
After setting these selections ($P<0.5s$, $10^{11}G<B<10^{14}G$, above the
spin-up line, and removing the special pulsars), we obtain the two groups of
samples of 52 SNR-PSRs and 630 non SNR-PSRs, respectively, as illustrated in
Figure 1.
Figure 1: Data distribution graph in the magnetic field versus spin period
(B-P) diagram after the selection criteria. The blue stars and orange dots
stand for the SNR-PSRs and non SNR-PSRs, respectively. The green dashed lines
represent the constraint lines of surface B-field strength (B) upper
($B_{upper}=\rm 10^{14}\,G$) and lower ($B_{lower}=\rm 10^{11}\,G$) limit and
spin period ($P<0.5\,\rm s$). The red solid line represents the spin-up line
of accretion pulsars in binaries (Bhattacharya and van den Heuvel 1991). The
red, green and blue area stand for the approximate ranges of the magnetars,
recycled pulsars and millisecond pulsars (MSPs).
### 2.2 The Model
Because of employing the spin period to represent the pulsar’s age (spindown
age), we briefly introduce the model(Shapiro & Teukolsky 1983; Camilo,
Thorsett, & Kulkarni 1994; Lorimer & Kramer 2012; Lyne & Graham-Smith 2012).
The energy loss rate of a spin-powered pulsar can be expressed as,
$\dot{E}={dE}/{dt}={d(I\Omega^{2}/2)}/{dt}=-I\Omega\dot{\Omega}$, where $E$ is
total energy of pulsar, $\dot{E}$ is energy loss rate, $\Omega$ and
$\dot{\Omega}$ are the spin angular velocity and its derivative, and $I$ is
moment of inertia (Shapiro & Teukolsky 1983). The pulsar kinetic energy and
the radiation energy loss rate are equal, then we have,
$I\Omega\dot{\Omega}=k\Omega^{\rm n+1}$, with $k$ a coefficient, where the
braking index n=3 for the magnetic dipole model. The spin period evolution
equation can be written as (calculation details in Appendix A)
$\begin{split}P(t)=P_{0}\left(\frac{t}{\tau}_{0}+1\right)^{1/(\rm
n-1)},\end{split}$ (1)
where $P_{0}$, $\dot{P}_{0}$, and $\tau_{0}$ are the spin period, the
derivative of spin period and the spindown age at present, respectively. The
spindown age of a pulsar is defined as $\tau={P}/{[({\rm n-1})\dot{P}]}$
(Camilo, Thorsett, & Kulkarni 1994). When $t\gg\tau_{0}$ ($P\gg P_{0}$), the
spin period evolution equation can be expressed as
$\begin{split}P(t)\approx P_{0}\left(\frac{t}{\tau}_{0}\right)^{1/(\rm
n-1)}.\end{split}$ (2)
With n=3 and parameter $k=-{B_{p}^{2}R^{6}}/{(6c^{3})}$ is obtained by the
magnetic dipole model, where $B_{p}$ is polar magnetic, $R$ is NS radius, and
$c$ is speed of light (Shapiro & Teukolsky 1983). Then the Eq.(2) can be
written as $P(t)\approx P_{0}(t/\tau_{0})^{1/2}$. Because n=3 is a theoretical
value and the measured values of many pulsars are generally less than it
(Johnston & Galloway 1999; Kou & Tong 2015), we take n=2.7 to estimate the
ages of pulsars, and the reasons are interpreted as follows. Firstly, Lyne,
Manchester, & Taylor (1985) and Lorimer (2004) estimated that the number of
observable pulsars in Milky Way galaxy is $\sim 2-7\times 10^{4}$, which
corresponds to the relevant life time of radio pulsars to be $\rm\sim
20\,Myr$. If the braking index n is too small, like n=2.1, the corresponding
life time of pulsars will decrease to one or two million years to reach the
observational limit spin period of about 10 s. This will result in too few
observable pulsars, which is inconsistent with the observational facts at
present. Therefore, it is the reason that we do not apply the Crab pulsar’s
breaking index as an average value for the whole pulsar samples, which is from
2.1 to 2.5 or 2.6 (Lyne et al. 2015; Čadež et al. 2016). Secondly, almost all
radio pulsar periods are less than 10s (only two are longer than 10s, e.g.,
J2251-3711 with 12.1s (Morello et al. 2020) and J0250+5854 with 23.5s (Tan et
al. 2018)). By considering the above two reasons, we take the braking index
n=2.7 as a statistical value to estimate the evolutionary time, which can give
the spin period to be about 10s after evolving about 20 Myrs. Then we select
the initial conditions of the Crab-like pulsars to perform the calculations,
e.g., $P_{0}=0.033$ s and $\dot{P}_{0}=4.21\times 10^{-13}\,\rm ss^{-1}$
(Goldreich & Julian 1969; Lyne et al. 2015), implying $\tau_{02.7}=1462$ yr
(n=2.7) and $\tau_{03}=1243$ yr (n=3, characteristic age). The spin period
evolutionary curves are plotted in Figure 2 based on Eq. (2). Here, we point
out that, although initial spin periods of neutron stars may be not like that
of the Crab pulsar’s (Popov & Turolla 2012; Igoshev & Popov 2013), we still
insist to employ the Crab pulsar as a reference sample since it is the sole
pulsar with the known real age.
Figure 2: The diagram of spin period evolution with various braking index.
Different colors label the evolution curves with the breaking index n=2.1
(green), n=2.7 (red), and n=3 (blue), respectively. The orange dashed
horizontal line shows the required time for a pulsar period evolving to 10s.
## 3 Statistics and Results
In the following analysis, we employ the two statistical tests, K-S and M-W-W
(Yang et al. 2019; Cui et al. 2021), to check whether two samples hold the
same distribution. The test results are represented by the parameters,
p-values, and the procedure is described in the following. If p-value is less
than 0.05, it indicates that this test rejects the null hypothesis (the two
samples have the same distribution) at 5% significance level.
Figure 3: The Cumulative distribution function (CDF) of spin period for SNR-
PSRs and non SNR-PSRs. The dashed (solid) line stands for SNR-PSRs (non SNR-
PSRs).
In order to see whether the two groups of samples of SNR-PSRs and non SNR-PSRs
hold the same distribution, we draw the CDF curves in Figure 3, where the two
curves are conspicuously separated to each other. To show the difference of
the two samples more quantitatively, we apply the K-S test and M-W-W test, and
the p-values of these two tests are as low as $1.98\times 10^{-12}$ and
$5.85\times 10^{-13}$. The very low p-values indicate that the distributions
of the two samples should have the different origins. Therefore, with the
results of two CDF curves and p-values, we believe that the two samples may
come from the different statistical distributions. The further test results
for the other parameters are shown in Appendix B.
Figure 4: The cumulative number distribution of SNR-PSRs and non SNR-PSRs in
different ages. The blue (orange) solid lines stand for SNR-PSRs (non SNR-
PSRs) under n=2.7, and dashed lines stand for the cases of n=3.
Next, we draw a cumulative number distribution (Figure 4) for the different
ages that are estimated by Eq.(2). Interestingly, for the age less than $\sim$
10 Kyr the two curves of SNR-PSRs and non SNR-PSRs coincide together, however,
after $\rm 10-20\,Kyr$, the two curves are drifted away. The cumulative number
ratio, expressed as $N_{snr}/N_{nsnr}$, for the two samples with the different
ages probably can infer a fact that the two samples hold the different
origins.
In order to see the variation of this ratio between two samples more
intuitively, we plot the different ratio values from the age of 5 Kyr to 100
Kyr in Figure 5. We find that, before the age of 25 Kyr, we obtain 5 ratio
points for n=2.7 and n=3. While, from 25 Kyr to 100 Kyr, only 2 ratio points
are obtained because of the inadequate data. From the data point of these
ratio values, we obtain that the number ratio between SNR-PSR and non SNR-PSR
is close to unity at the age of $\sim$ 10 Kyr. However, after $\rm
10-20\,Kyr$, there exists a sharp decline in the ratio values. Meanwhile, for
the ratio value after $\sim$ 10 Kyr, with n=2.7 (n=3), we obtain a relation
between the ratio and the age $\mathcal{\phi}_{2.7}=1.1(t/10kyr)^{-1.01}$
($\mathcal{\phi}_{3}=1.2(t/10kyr)^{-0.91}$) and goodness of fit
$\mathcal{R}_{2.7}^{2}=0.998$ ($\mathcal{R}_{3}^{2}=0.996$), as described in
figure 5.
Figure 5: The evolution diagram of cumulative number ratio between SNR-PSRs
($N_{snr}$) and non SNR-PSRs ($N_{nsnr}$). With n=2.7 (n=3), the orange stars
(green triangles) stand for the ratio in different ages, and the solid blue
(red) line is the fitting curve of these ratio points.
## 4 Discussions and Conclusions
On the reasons for a sharp drop of the cumulative number ratio between the
SNR-PSRs and non-SNR-PSRs, after $\rm 10-20\,Kyr$, together with the the
different distributions, we think that there may exist two ways of birth for
radio pulsars. The long aged SNR-PSRs may be involved in the high-energy SNe,
whereas the non SNR-PSRs may be related to the low energy cases with short
duration. For high-mass stars, their SNRs may survive longer time.
Correspondingly, the low-energy SNe may be involved in the explosions of the
low-mass progenitor stars, in which the SNRs may last a shorter duration.
However, can the effect of decline in cumulative number ratio is caused by the
age difference? For example, Leahy, Ranasinghe, & Gelowitz (2020) analysed a
15-Galactic-SNR sample with the best-fit mean energy of $\rm 2.7\times
10^{50}\,erg$, and they concluded that SNRs become incomplete and hard to
identify after 30 Kyr. In order to discuss this question more clearly, we need
to test for samples with the different spindown ages. According to Figures 4
and 5, the two curves are separated after about $\rm 10-20\,Kyr$ for various
braking index. So, we can take 10 Kyr as the critical boundary and divide SNR-
PSRs and non SNR-PSRs into two groups by the ages of less and greater than 10
Kyr. For the pulsar ages less than 10 Kyr, the $P$ distributions of SNR-PSRs
and non SNR-PSRs are same, while their $\dot{P}$ distributions are different.
However, for the pulsar ages more than 10 Kyr, both distributions of $P$ and
$\dot{P}$ are different (details in Appendix C). These results may infer that
their initial periods are independent on the types of pulsars, but their
braking mechanisms are different. The birth of the NSs spin are due to the
transfer of angular momentum of progenitors to NSs (Lyne & Graham-Smith 2012),
then the energy of SN explosions might little effect on their spin periods.
Meanwhile, the difference in $\dot{P}$ will affect the evolution of $P$,
resulting in a different distribution of $P$ after 10 Kyr. The above evidence
support that their intrinsic differences may lead to different distribution
among SNR-PSRs and non SNR-PSRs. Specifically, when radio pulsars are just
born, the differences of progenitor mass or explosion energy may create
differences in the two groups. The effect of age is to amplify the differences
during the evolution, which origin from different initial $\dot{P}$. Thus,
this indicates that the differences between SNR-PSRs and non SNR-PSRs in
Figures 3, 4, and 5 are more likely to be caused by multiple reasons, rather
than only an age difference. The reasons may include different neutron star
generation mechanisms and evolution over time.
We need to emphasize that the duration of $\rm 10-20\,Kyr$ is not a strict
time, but a statistical value. The result based on SNR evolutionary models
(Leahy, Ranasinghe, & Gelowitz 2020) may be slightly different from our
result. Although our results ($\rm 10-20\,Kyr$) are not completely the same
with their (30 Kyr), at least it shows a life time boundary of two type SNRs,
no matter from the perspectives of pulsars in radio and SNRs in X-ray. It can
be inferred from this boundary that two types of pulsars generated by two
types of SNRs can be roughly distinguished. Therefore, the cumulative number
ratio of $\sim 1$ at $\sim$ 10 Kyr may represent the ratio of these two kinds
of pulsar production, that is
$\begin{split}\psi=\frac{N_{snr}}{N_{nsnr}}\sim\frac{N_{high-energy}}{N_{low-
energy}}\sim\frac{N_{high-mass}}{N_{low-mass}}\sim 1,\end{split}$ (3)
where $N_{snr}$ ($N_{nsnr}$), $N_{high-energy}$ ($N_{low-energy}$) and
$N_{high-mass}$ ($N_{low-mass}$) represent the numbers of SNR-PSRs (non SNR-
PSRs), high (low)energy SNe, and high (low) mass stars. Specifically for the
Crab pulsar, it may born from a low-energy SN ($\sim\rm 10^{50}\,erg$) and
low-mass progenitor star (Yang & Chevalier 2015). However, the earlier view
believed that the Crab Nebula has a more extended remnant (Chevalier 1977),
which indicates a higher energy ($\sim\rm 10^{51}\,erg$). Although no medium
have been detected in the surrounding area at radio or X-ray band (Frail et
al. 1995; Seward, Gorenstein, & Smith 2006), it also reminds us that total
energy of the Crab Nebula is still an open question.
The mass range of the progenitor stars for NS formations approximately lies in
$\rm 8-25\,M_{\odot}$ (Arnett & Schramm 1973; Miyaji et al. 1980; Heger et al.
2003). If we consider the initial mass function (IMF) by Salpeter, described
as $\rm dN/dm=\xi_{0}m^{-2.35}$ (Salpeter 1955), where m is star mass in $\rm
M_{\odot}$ units and $\rm\xi_{0}$ is normalization coefficient, to calculate
the boundary mass value for the high and low stellar masses, corresponding to
the SNR-PSRs and non SNR-PSRs, we obtain this critical mass to be be at
$\rm\sim 12\,M_{\odot}$, as shown in Figure 6. So, the SNRs by the low-mass
($\rm 8-12\,M_{\odot}$) progenitor stars may be survived with a shorter time
around $\rm 10-20\,Kyr$ (Braun, Goss, & Lyne 1989), which may be a reason for
the lots of young pulsars without SNRs.
Figure 6: The mass distribution of stars for SNR-PSRs and non SNR-PSRs
according to initial mass function (IMF) (Salpeter 1955). The blue solid line
is the curve of IMF. The left and right orange solid lines are lower ($\rm
8\,M_{\odot}$) and upper ($\rm 25\,M_{\odot}$) mass limit for NS production,
respectively. The area of $\rm 8-12\,M_{\odot}$ ($\rm 12-25\,M_{\odot}$)
stands for the progenitor stars for non-SNR-PSRs (SNR-PSRs).
The physical process of these two types of SNe can be described as the iron
core collapse and electron capture, and the latter is driven by the neutrino
(Janka 2012, 2017). The iron core collapse is the dominant process of high-
energy SNe that is generated from high-mass main sequence stars (Heger et al.
2003). The electron capture may be an explanation for the low-energy SNe,
while the degenerate oxygen-neon core is collapsed to form the NS(Barkat,
Reiss, & Rakavy 1974; Nomoto 1984), and the mass range of these progenitor
stars is about $\rm\sim 8-10\,M_{\odot}$ (Nomoto & Leung 2017; Leung, Nomoto,
& Suzuki 2020).
Moreover, the mass boundary values given by some researchers are similar to
ours, as $\rm\sim 12\,M_{\odot}$ (Sugimoto & Nomoto 1980; Miyaji et al. 1980).
However Nomoto (1984) and Heger et al. (2003) obtained the boundary mass as
$\rm\sim 10\,M_{\odot}$. It is remarked that our result is based on a
statistics, but not a numerical calculation result from a stellar theoretical
model. In addition, because of less pulsar samples at the young age ($<$10
Kyr), the ratio in Eq.(3) may be biased, which will directly affect the mass
boundary.
Finally, the main conclusions are summarized below: The 52 SNR-PSRs and 630
non SNR-PSRs have been tested (K-S and M-W-W) and analyzed (cumulative number
ratio), implying different $\dot{P}$ and other properties for the two sets,
perhaps associated with the different mass ranges of their progenitor masses
for SN explosions. The critical mass of different progenitor stars is
estimated by the Salpeter initial mass function, obtained as $\rm
12\,M_{\odot}$. The low-mass stars (high-mass) with $\rm\sim 8-12\,M_{\odot}$
($\rm\sim 12-25\,M_{\odot}$) will generate the low-energy (high-energy) SNe in
the shorter (longer) SNR duration of about $\rm<10-20\,Kyr$
($\rm>10-20\,Kyr$). These conjectures can explain why many young radio pulsars
are not seen inside SNRs. In the future, with the observations by FAST (Li et
al. 2018) and launch of James Webb Space Telescope (JWST) (Gardner et al.
2006), more fainter and weaker pulsars and SNRs would be discovered, which
will present the better constraints on our conclusions.
## Acknowledgments
This work is supported by the National Natural Science Foundation of China
(Grant No. 11988101, No. U1938117, No. U1731238, No. 11703003 and No.
11725313), the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008, the National Key R&D Program of China
No. 2016YFA0400702, and the Guizhou Provincial Science and Technology
Foundation (Grant No. [2020]1Y019).
## Data Availability
The data underlying this article are available in the references below: (1)
pulsars data are taken from ATNF Pulsar Catalogue, available at
https://www.atnf.csiro.au/research/pulsar/psrcat/; (2) SNR data are taken from
SNRcat, available at http://snrcat.physics.umanitoba.ca/SNRtable.php.
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## Appendix A: Derivation of period evolution equation
Energy loss rate of a spin-powered pulsar can be expressed as
$\begin{split}\dot{E}=\frac{dE}{dt}=\frac{d(I\Omega^{2}/2)}{dt}=-I\Omega\dot{\Omega},\end{split}$
(4)
where $E$ is total energy of pulsar, $\dot{E}$ is energy loss rate, $\Omega$
is spin angular velocity, $\dot{\Omega}$ is rate of spin angular velocity, and
$I$ is moment of inertia. When the pulsar kinetic energy and the radiation
energy loss rate are equal, then there is
$\begin{split}I\Omega\dot{\Omega}=K\Omega^{\rm n+1},\end{split}$ (5)
where $K$ is coefficient, We combined with $\Omega=2\pi/P$ and integrate both
sides of above equation,
$\begin{split}\int_{P_{0}}^{P(t)}P^{\rm
n-2}d\dot{P}=\int_{0}^{t}-\frac{K}{I}(2\pi)^{\rm n-1}dt.\end{split}$ (6)
With assuming that the rate has been constant since pulsar birth, then spin
period evolution equation can be written as
$\begin{split}P(t)=\left[({\rm n-1})P_{0}^{\rm n-2}\dot{P}_{0}t+P_{0}^{\rm
n-1}\right]^{1/(\rm n-1)}.\end{split}$ (7)
In the above equation, $P_{0}$ and $\dot{P}_{0}$ is spin period and rate at
present, respectively. The spindown age of pulsar is $\tau=P/[({\rm
n-1})\dot{P}]$, which is different with characteristic age $\tau_{c}=P/(\rm
2\dot{P})$. Then Eq.(7) can be rewritten as
$\begin{split}P(t)=P_{0}\left(\frac{t}{\tau}_{0}+1\right)^{1/(\rm
n-1)}.\end{split}$ (8)
When $t\gg\tau_{0}$, the equation can be simplified to
$\begin{split}P(t)\approx P_{0}\left(\frac{t}{\tau}_{0}\right)^{1/(\rm
n-1)}.\end{split}$ (9)
## Appendix B: Further tests of derivative of spin period, B-field and energy
loss rate
If the origin of pulsars that with or without SNRs are indeed different, then
the distribution of other physical parameters should also be different. Here
we discuss the derivative of spin period ($\dot{P}$), B-field strength ($B$
with n=3) and energy loss rate ($\dot{E}$) by applying K-S and M-W-W tests,
and the results are shown in Table 2, and the CDFs are shown in Figure 7, 8
and 9, where the distributions for two samples of SNR-PSRs and non SNR-PSRs
are different respect to these three parameters. For $\dot{P}$, the two groups
with the different ages share the significantly different distributions, which
implies that the braking mechanisms of them perhaps are different. The
physical parameter distributions of two samples are quite different, which
possibly could be ascribed to the different origins of radio pulsars.
Figure 7: The Cumulative distribution function (CDF) of derivative of spin
period ($\dot{P}$) of SNR-PSRs and non SNR-PSRs. The dashed line is for SNR-
PSRs, and the solid line is for non SNR-PSRs. Figure 8: The Cumulative
distribution function (CDF) of surface magnetic field strength ($B$ with n=3)
of SNR-PSRs and non SNR-PSRs. The dashed line is for SNR-PSRs, and the solid
line is for non SNR-PSRs. Figure 9: The Cumulative distribution function (CDF)
of spin down energy loss rate ($\dot{E}$) of SNR-PSRs and non SNR-PSRs. The
dashed line is for SNR-PSRs, and the solid line is for non SNR-PSRs. Table 2:
P-values of K-S and M-W-W test for different parameters
Physical parametersa | K-S test | M-W-W test
---|---|---
$P$ | $1.98\times 10^{-12}$ | $5.85\times 10^{-13}$
$\dot{P}$ | $1.55\times 10^{-15}$ | $6.41\times 10^{-21}$
$B$ | $3.24\times 10^{-11}$ | $2.12\times 10^{-14}$
$\dot{E}$ | $1.55\times 10^{-15}$ | $4.33\times 10^{-24}$
a $P$ is spin period, $\dot{P}$ is derivative of spin period, $B$ is surface
magnetic field strength, and $\dot{E}$ is energy loss rate of radio pulsars.
## Appendix C: Tests of spin period and its derivative with the different ages
Figure 10: The Cumulative distribution function (CDF) of spin period ($P$) and
its derivative ($\dot{P}$) of SNR-PSRs and non SNR-PSRs with the spindown ages
of less and over than 10 Kyr. The sub-figures of a and b are CDF of $P$, and
the sub-figures of c and d are CDF of $\dot{P}$. For all sub-figures, the
dashed line represents SNR-PSRs, and the solid line is for non SNR-PSRs. The
text in each sub-figure shows their age ranges. Table 3: P-values of K-S and
M-W-W test for $P$ and $\dot{P}$ with different ages
Physical parametersa | Spindown age | K-S test | M-W-W test
---|---|---|---
$P$ | ¡10 Kyr | 0.99 | 0.85
¿10 Kyr | $2.75\times 10^{-3}$ | $4.83\times 10^{-4}$
$\dot{P}$ | ¡10 Kyr | $3.41\times 10^{-2}$ | $5.34\times 10^{-2}$b
¿10 Kyr | $4.38\times 10^{-13}$ | $1.64\times 10^{-13}$
a $P$ is spin period, $\dot{P}$ is derivative of spin period.
b Although this value is slightly larger than 0.05, it may also imply that the
two samples have different statistical distributions at a 90% probability
(e.g. if p-value is less than 0.1, it indicates that this test rejects the
null hypothesis that the two samples have the same distribution at 10%
significance level).
In this part, we plot distributions of $P$ with different age ranges of less
and over than 10 Kyr in Figure 10 (subplots a & b). After K-S and M-W-W tests
(Table 3), we find that when the spindown age is less than 10 Kyr the $P$
distributons of SNR-PSRs and non SNR-PSRs are the same. But the distributions
of $P$ after 10 Kyr are different. Meanwhile, distributions of $\dot{P}$ are
also tested with the same age ranges as that of $P$ (less and more than 10
Kyr) in Figure 10 (subplots c & d). Interestingly, regardless of the age
ranges, the $\dot{P}$ distributions are different under K-S tests in Table 3.
The possible physical explanation is that although the initial $P$
distribution is the same, the initial $\dot{P}$ is different, which makes
pulsars no longer have the same $P$ distribution after evolution over 10 Kyr.
Thus, the above evidence supports that the differences between the SNR-PSRs
and non SNR-PSRs are possibly the result of a combined effect of different
mechanisms and evolution, but not just caused by age difference.
|
# Two high capacity text steganography schemes based on color coding
Juvet K. Sadié1,2,3,4 Leonel Moyou Metcheka1,2,3,4 René
Ndoundam1,2,3,4,111Corresponding author<EMAIL_ADDRESS>
1Team GRIMCAPE
2Sorbonne Unversity, IRD, UMMISCO, F-93143, Bondy, France
3CETIC, Yaounde, Cameroon
4Department of Computer Science, University of Yaounde I, P.o. Box 812
Yaounde, Cameroon
E.mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Text steganography is a mechanism of hiding secret text message inside another
text as a covering message. In this paper, we propose a text steganographic
scheme based on color coding. This include two different methods: the first
based on permutation, and the second based on numeration systems. Given a
secret message and a cover text, the proposed schemes embed the secret message
in the cover text by making it colored. The stego-text is then sent to the
receiver by mail. After experiments, the results obtained show that our models
performs a better hiding process in terms of hiding capacity as compared to
the scheme of Aruna Malik et al. in which our idea is based.
Keywords : Steganography, text steganography, covert medium, stego-object,
permutation, embedding capacity, numeration systems.
## 1 Introduction
The word steganography is of Greek origin and means covered writing. It is the
hiding of a message within another (cover medium) such as web pages, images or
text, so that the presence of the hidden message is indiscernible. When a
message is hidden in the cover medium, the resulting medium is called a stego-
object. The key concept behind steganography is that the message to be
transmitted should not be detectable with bare eyes. From the definition,
Steganography is used to ensure data confidentiality, like encryption.
However, the main difference between the two method is that with encryption,
anybody can see that both parties are communicating in secret. Steganography
hides the existence of a secret message and in the best case nobody can detect
the presence of the message. When combined, steganography and encryption can
provide more security. Steganography dates back to ancient Greece, where
common practices consisted of etching messages in wooden tablets and covering
them with wax. A number of steganographic methods have been introduced on
different cover media such as images [1, 2, 3], video files [4, 5] and audio
files [6]. In text based steganographic methods, text is used as a cover media
for hiding the secret data. Due to the lack of large scale redundancy of
information in a text file, the human eye is very susceptible to any change
between the original and the modified texts. Therefore, text steganography
seems to be the most difficult kind of steganography [7], as compare to
others.
In this paper, we propose a text steganographic scheme based on color coding,
permutation and numeration systems. Given a secret message and a cover text,
the proposed scheme embed the secret message in the cover text by making it
colored, using a permutation algorithm for the first method and numeration
systems for the second one. After the first section devoted to the
introduction, section 2 presents some preliminaries and related works. Section
3 concerns the presentation of the first method of our scheme. Section 4
labels the second approach of our scheme, and finally conclusion is stated in
section 5.
## 2 Preliminaries and related works
In this section, the focus is to present some preliminaries that lead us to
the comprehension of our scheme. Also, we present related works in the field
of text steganography.
### 2.1 Text Steganography
There are many techniques in text steganography. In Syntactical steganography,
punctuation marks such as full stop (.), comma (,) etc, are used to hide bits
in cover text. The problem with this method is that it requires identification
of correct places to insert punctuation [8, 9]. In lexical steganography,
words are used to hide secret bits. A word could be replaced by its synonyms
and the choice of word to be chosen from the list of synonyms would depend on
secret bits. Sms texting is a combination of abbreviated words used in sms
[10]. This technique proposes to hide binary data by using full form or its
abbreviated form. For instance, to hide 0, full form of the word is used and
to hide 1, abbreviated form of word is used [10]. The CSS technique encrypts a
message using RSA public key cryptosystem and cipher text is then embedded in
a cascading style Sheet (CSS) by using End of Line on each CSS style
properties, exactly after a semi-colon. A space after a semi-colon embeds bit
0 and a tab after a semicolon embeds bit 1 [11]. Anandaprova Majumder and al
[12] proposed an approach for text steganography through a technique that uses
reflection symmetry of the English alphabet. Ekodeck and Ndoundam [13]
proposed different approaches of PDF file based steganography, essentially
based on the Chinese Remainder Theorem. Here, after a cover PDF document has
been released from unnecessary characters of ASCII code A0, a secret message
is hidden in it using one of the proposed approaches, making it invisible to
common PDF readers, and the file is then transmitted through a non-secure
communication channel. Aruna Malik and al [14], proposed a high capacity text
steganography scheme based on LZW compression and color coding. Their scheme
uses the forward mail platform to hide secret data. The algorithm first
compresses secret data and then hide the compressed data into the email
addresses and also, in the cover message of email. The secret data is embedded
in the message by making it colored using a color table. Here below, some
limits of that scheme will be presented.
### 2.2 Critic and limits
LZW is a lossless compression technique that performs high compression ratio
when the source contains repetition pattern. In the LZW based steganographic
scheme propose by Aruna Malik [14], they apply this lossless compression on
the secret message to increase the embedding capacity. But in the example
proposed, there is no compression. In other words, the size of the compressed
text is much greater than the size of the secret. To show this, we will give
three different implementations of LZW algorithm applied to the secret
message.
#### 2.2.1 The LZW Algorithm with initial dictionary fixed and known
This algorithm [15] starts by initializing the dictionary with the 256
characters of the ASCII code from 0 to 255. The output codes start at a
minimum bit size equal to 9 and in general, as long as the indexes considered
are strictly inferior to n = $2^{k}$ \- 1, we can represent them on k bits.
When the first integer greater than or equal to $2^{k}$ \- 1 is met, the
sequence 1. . . 1 (k times bit 1) and continue with coding the integers on k +
1 bits. Applying this method to the following secret message: "underlying
physiological mechanisms", we obtain the outputs presented in Table 1. The
binary compressed text is obtained by converting the indexes of the output
column of the array to 9 bits :
001110101 001101110 001100100 001100101 001110010 001101100 001111001
001101001 001101110 001100111 000100000 001110000 001101000 001111001
001110011 001101001 001101111 001101100 001101111 001100111 001101001
001100011 001100001 001101100 000100000 001101101 001100101 001100011
001101000 001100001 001101110 001101001 001110011 001101101 001110011. Hence,
the size of the output is 35*9 = 315 bits.
Buffer | input-char | Output | New Item | Buffer | input-char | Output | New Item
---|---|---|---|---|---|---|---
u | n | 117 | 256=un | l | o | 108 | 273=lo
n | d | 110 | 257=nd | o | g | 111 | 274=og
d | e | 100 | 258=de | g | i | 103 | 275=gi
e | r | 101 | 259=er | i | c | 105 | 276=ic
r | l | 114 | 260=rl | c | a | 99 | 277=ca
l | y | 108 | 261=ly | a | l | 97 | 278=al
y | i | 121 | 262=yi | l | | 108 | 279=l
i | n | 105 | 263=in | | m | 32 | 280= m
n | g | 110 | 264=ng | m | e | 109 | 281=me
g | | 103 | 265=g | e | c | 101 | 282=ec
| p | 32 | 266= p | c | h | 99 | 283=ch
p | h | 112 | 267=ph | h | a | 104 | 284=ha
h | y | 104 | 268=hy | a | n | 97 | 285=an
y | s | 121 | 269=ys | n | i | 110 | 286=ni
s | i | 115 | 270=si | i | s | 105 | 287=is
i | o | 105 | 271=io | s | m | 115 | 288=sm
o | l | 111 | 272=ol | m | s | 109 | 289=ms
| | | | s | | 115 |
Table 1: LZW Algorithm output with initial dictionary fixed and known
#### 2.2.2 The LZW algorithm with sharing of the initial dictionary
In this version [15], initial dictionary contains only the character of the
secret message. The output code is represented on height bits. The
particularity of this implementation is from the initial dictionary which must
be shared between the two parties in order to be able to decompress the binary
code. Table 2 presents the initial dictionary for the same secret message:
Here are the output code : 1 2 3 4 5 6 7 8 2 9 10 11 12 7 13 8 14 6 14 9 8 15
16 6 10 17 4 15 12 16 2 8 13 17 13 and in binary we have :
00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000
00000010 00001001 00001010 00001011 00001100 00000111 00001101 00001000
00001110 00000110 00001110 00001001 00001000 00001111 00010000 00000110
00001010 00010001 00000100 00001111 00001100 00010000 00000010 00001000
00001101 00010001 00001101. Hence, the size of the output is the sum of the
size of initial dictionary and the output code: 17+35 = 52 bytes = 416 bits.
Index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
char | u | n | d | e | r | l | y | i | g | space | p | h | s | o | c | a | m
Table 2: Initial Dictionnary
#### 2.2.3 The Unix compress command
The ncompress package [17] is a compression utility available on Linux which
contains the compress command for fast compression and decompression using LZW
algorithm. The algorithm behind this command is explained at page 153 of the
data compression book [15]. The initial dictionary size is 512 and the minimum
output code size is 9 bits. This package can be installed by using the command
"sudo apt-get install ncompress". Based on the Ubuntu 16.04 platform, this
command produces a file with .Z extension as compress file. By Applying this
command "compress -v source.txt" to the secret message contained in text
source file with the -v option, the given output indicates that there is no
compression and the .Z file size is 44 bytes = 352 bits.
Finally the table 3 shows the comparison in bit between the original text size
and the output size after the compression using the three different approaches
of LZW implementation: The LZW Algorithm with initial dictionary fixed and
known, The LZW algorithm with sharing of the initial dictionary and The Unix
compress command.
Secret message size | Output size 1 | Output size 2 | Output size 3
---|---|---|---
280 | 315 | 416 | 352
Table 3: Secret message size comparison
From Aruna and al. paper [14], the size obtained was 264 bits, but we have
proven above that there is no compression for this example. This is the
principal limit of this steganographic scheme, where for some messages the
reduction of the message size will not be possible. Our paper uses:
* •
The idea of color coding contained in the paper of Aruna Malik and al [14];
* •
The permutation generation method of W. Myrvold and F. Ruskey [19];
* •
The numeration systems;
to present a new scheme where the secret message embedding capacity is better
than the scheme of Aruna Malik and al [14].
### 2.3 Permutation Generation Methods
Permutation is one of the most important combinatorial object in computing,
and can be applied in various applications, for example, the scheduling
problems. Permutation generation can form the basis of a backtracking program
to solve any problem involving reordering a set of items. It is well-known
that, for n distinct items, the total number of permutations is n!.
Permutation generation has a long history. Surveys in the field have been
published in 1960 by D.H. Lehmer [20]. Several authors [19, 21, 22, 23] have
since developed many methods to generate all the possible permutations of n
elements. Also, several works [13, 24, 25, 26, 27] in steganography taking
advantage of permutations have been done. In particular, H. Hioki [26] in
2013, proposed a permutation steganography, which is an effective method for
hiding messages provided where the contents of cover objects are not affected
by the rearrangement of their elements. In their paper, W. Myrvold and F.
Ruskey [19] proposed a ranking function for the permutations on n symbols
which assigns a unique integer in the range [0, $n$! - 1] to each of the $n$!
permutations. Also, they proposed an unranking function for which, given an
integer $r$ between 0 and $n$! - 1, the value of the function is the
permutation of rank r. Their algorithms is presented below [19].
#### 2.3.1 Unranking function
First of all, recall that a permutation of order $n$ is an arrangement of $n$
symbols. An array $\pi[0\cdots n-1]$ is initialized to the identity
permutation $\pi[i]=i$, for $i=0,1,\cdots n-1$.
Procedure $unrank(n,r,\pi)$[19]
begin
if $n>0$ then
$\phantom{salutsal}swap(\pi[n-1],\pi[r$ mod $n])$;
$\phantom{salutsal}unrank(n-1,\left\lfloor r/n\right\rfloor,\pi)$;
end;
end;
Note: $swap(a,b)$ exchanges the values of variables $a$ and $b$.
#### 2.3.2 Ranking function
To rank, first compute $\pi^{-1}$. This can be done by iterating
$\pi^{-1}[\pi[i]]=i$, for $i=0,1,\cdots,n-1$.
In the algorithm below, both $\pi$ and $\pi^{-1}$ are modified.
function $rank(n,\pi,\pi^{-1})$:integer[19]
begin
if $n=1$ then return(0) end;
$\phantom{salutsal}s:=\pi[n-1]$ ;
$\phantom{salutsal}swap(\pi[n-1],\pi[\pi^{-1}[n-1]])$ ;
$\phantom{salutsal}swap(\pi^{-1}[s],\pi^{-1}[n-1])$ ;
return$(s+n.rank(n-1,\pi,\pi^{-1}))$ ;
end;
## 3 Scheme design based on permutation
In this section, we present the first method of our scheme.
### 3.1 Embedding Algorithm
Input:
$\phantom{saluts}C$: the cover text;
$\phantom{saluts}M$: the secret message to embed;
The key $\pi$: the initial permutation of $n$ colors;
$\phantom{saluts}e$: the e-mail address of the receiver
Output:
$\phantom{saluts}C^{\prime}$: the stego-message;
begin:
1\. Compute $m$, the binary representation of $M$;
2\. Compute $t=\left\lfloor log_{2}(n!)\right\rfloor$
3\. Divide $m$ into $p$ blocks of $t$ bits each, $b_{1},b_{2},\cdots,b_{p}$;
4\. Divide $C$ into $k$ blocks of $n$ characters each
$c_{1},c_{2},\cdots,c_{k}$;
5\. For each block $b_{i},1\leq i\leq p$:
a. compute $Nperm=(b_{i})_{10}$, the decimal representation of $b_{i}$;
b. compute $\pi^{\prime}=unrank(n,Nperm,\pi)$, the permutation corresponding
to the number $Nperm$. $\pi^{\prime}$ can be considered as
$\pi^{\prime}(1),\pi^{\prime}(2),\cdots,\pi^{\prime}(n)$;
c. color each character of $c_{i}$ by the corresponding color given by the
permutation $\pi^{\prime}$ and obtain the string $c^{\prime}_{i}$;
d. compute $C^{\prime}\leftarrow C^{\prime}||c^{\prime}_{i}$; where a$||$b is
the concatenation of a and b.
6\. If the next character is EOF (End of File) then
begin
a. Use e to send C’ by mail to the receiver;
end
Else
begin
a. Colour the next character with a color different of permutation colors.
This color is shared by the sender and the receiver. However, this color will
not be very distant from the others.
b. Randomly color the rest of characters of C by the colors of colors table,
until the EOF character is obtained;
c. Use e to send C’ by mail to the receiver;
end;
end;
### 3.2 Retrieval Algorithm
Input:
$\phantom{saluts}C^{\prime}$: the stego-text;
The key $\pi$: the initial permutation of $n$ colors;
Output:
$\phantom{saluts}M$: the secret message;
begin:
1\. Retrieve all characters coloured by the permutation colors, until a color
different from the colors in the colors table, or the EOF character is
obtained. Lets call them $C^{\prime\prime}$;
2\. Divide $C^{\prime\prime}$ into $p$ blocks of $n$ characters each
$c_{1},c_{2},\cdots,c_{p}$;
3\. For each block $c_{k},1\leq k\leq p$:
a. use the color order of characters to compute the relative permutation, that
we call $\pi^{\prime}$. $\pi^{\prime}$ can be considered as
$\pi^{\prime}(1),\pi^{\prime}(2),\cdots,\pi^{\prime}(n)$;
b. compute the number $Nperm=rank(n,\pi^{\prime},\pi^{\prime-1})$.
c. compute $m^{\prime}=(Nperm)_{2}$, the binary representation of $Nperm$;
d. compute $M\leftarrow M||m^{\prime}$;
end;
### 3.3 Experimentation
In this subsection, we present some experimentations of this method. First, we
propose a theoretical estimation of our embedding capacity for $n$ colors.
Secondly, we present a practical experimentation in the case of 10, 16, 32 and
64 colors, based on example 1 and figure 5 of [14]. These colors are given in
figure 1, figure 2, figure 3 and figure 4.
Figure 1: The table of 10 colors
.
Figure 2: The table of 16 colors
.
Figure 3: The table of 32 colors
.
Figure 4: The table of 64 colors
.
In order to present our embedding capacity, we use as cover text and secret
message those of example 1 and figure 5 of Aruna Malik and al [14].
#### 3.3.1 Theoretical estimations
The table 4 presents the embedding capacity of our scheme for some different
values of n: 10, 16, 20 32,60, 64. This theoretical estimation is based on our
embedding algorithm.
More generally, in a set of n colors, the number of permutation of n distinct
colors is n !. According to the stirling formula [28] we have:
$n!\sim\left(\frac{n}{e}\right)^{n}\times\sqrt{2\pi n}$ (1)
Where $\pi=3.14$ is the area of the circle with unit radius, $e=2.718$ is the
base of the natural logarithm, and $\sim$ means approximate equality.
we know that:
$n=2^{log_{2}(n)}$
By replacing the value of n in equation 1 we have:
$\displaystyle n!$
$\displaystyle\sim\left(\frac{2^{log_{2}(n)}}{2^{1.442695}}\right)^{n}\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{log_{2}(n)-1.442695}\right)^{n}\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times\sqrt{2\pi
n}$ $\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times
2^{log_{2}(\sqrt{2\pi n})}$
$\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n}\right)\times
2^{\frac{1}{2}log_{2}(2\pi n)}$
$\displaystyle\sim\left(2^{nlog_{2}(n)-1.442695n+{\frac{1}{2}log_{2}(2\pi
n)}}\right)$
Proposition : the embedding capacity (E) using n colors to hide a secret is :
$E=\frac{M\times 100}{n\times 8}$
where $M={n(log_{2}(n)-1.442695)+{\frac{1}{2}log_{2}(2\pi n)}}$, and $n$ the
number of colors.
$\blacksquare$
n | M= $\left\lfloor log(n!)\right\rfloor$ | P=M/8 | 100*(P/n),(embedding capacity)
---|---|---|---
10 | 21 | 2.6 | 26.25%
16 | 44 | 5.5 | 34.37%
20 | 61 | 7.6 | 38%
32 | 117 | 14.6 | 45.63%
60 | 272 | 34 | 56.67%
64 | 295 | 36.9 | 57.66%
Table 4: Theoretical estimations of the proposed scheme
Remark: As far as the space characters of the stego-text are not coloured, the
embedding capacity can decrease in the experimentations.
#### 3.3.2 Experimentation 1
Here, the secret message is : underlying physiological mechanisms
and the cover text is:
Only boats catch connotes of the islands sober wines only ships wrap the slips
on the cleats of twining lines only flags flap in tags with color that assigns
only passage on vessels
Here we present the embedding process.
1. 1.
We compute the binary representation of the secret and obtain the following
result:
01110101 01101110 01100100 01100101 01110010 01101100 01111001 01101001
01101110 01100111 00100000 01110000 01101000 01111001 01110011 01101001
01101111 01101100 01101111 01100111 01101001 01100011 01100001 01101100
00100000 01101101 01100101 01100011 01101000 01100001 01101110 01101001
01110011 01101101 01110011
2. 2.
We compute $t=\left\lfloor log_{2}(10!)\right\rfloor=21$ ;
3. 3.
The binary secret is then divided into blocks of 21 bits each. For instance,
the first block $b_{1}$ = 011101010110111001100 and the second block $b_{2}$ =
100011001010111001001, …
4. 4.
We divide the cover text into blocks of 10 characters. For instance, the first
block $c_{1}=$ Only boats c, the second block $c_{2}=$ atch connot, …;
5. 5.
We color the cover text:
* •
For the block $b_{1}$ = 011101010110111001100, Nperm = 961996, its decimal
representation.
* •
The permutation relative to 961996 is given by $\pi^{\prime}$ = unrank(10,
961996, $\pi)$ = 3 8 5 2 1 4 9 0 7 6, $\pi$ = 0 1 2 3 4 5 6 7 8 9, with the
corresponding colors given by figure 1.
* •
The block $c_{1}=$ Only boats c is coloured relatively to the permutation
$\pi^{\prime}$. We then obtain the color text given in figure 5
Figure 5: The Stego-Text
.
* •
For the block $b_{2}$ = 100011001010111001001, Nperm = 1152457, its decimal
representation.
* •
The permutation relative to 1152457 is given by $\pi^{\prime}$ = unrank(10,
1152457, $\pi)$ = 2 9 1 6 3 8 4 5 0 7, $\pi$ = 0 1 2 3 4 5 6 7 8 9, with the
corresponding colors given by figure 1.
* •
The block $c_{2}=$ atch connot is coloured relatively to the permutation
$\pi^{\prime}$. We then obtain the color text given in figure 5
6. 6.
The process is the same, and finally we obtain the stego-text given by the
figure 6. That stego-text is then send by mail to the receiver.
Figure 6: The Stego-Text
.
With this example :
* •
in the case of 10 colors, the embedding capacity is 20.58 %;
* •
with 16 colors, the embedding capacity is 25.5 %;
* •
With 32 colors, the embedding capacity is 29.5 %;
* •
with 64 colors, the embedding capacity is 45.45 %.
#### 3.3.3 Experimentation 2
In the example of figure 5 [14], the secret message is : behind using a cover
text is to hide the presence of secret messages the presence of embedded
messages in the resulting stego-text cannot be easily discovered by anyone
except the intended recipient.
and the cover-text is:
in the research area of text steganography, algorithms based on font format
have advantages of great capacity, good imperceptibility and wide application
range. However, little work on steganalysis for such algorithms has been
reported in the literature. based on the fact that the statistic features of
font format will be changed after using font-format-based steganographic
algorithms, we present a novel support vector machine-based steganalysis
algorithm to detect whether hidden information exists or not. this algorithm
can not only effectively detect the existence of hidden information, but also
estimate the hidden information length according to variations of font
attribute value. as shown by experimental results, the detection accuracy of
our algorithm reaches as high as 99.3 % when the hidden information length is
at least 16 bits. Our scheme present experimentation based on different colors
number.
In the case of 10 colors, We apply our embedding algorithm and obtain the
following stego-text, given by figure 7.
Figure 7: The Stego-Text
.
With this example:
* •
In the case of 10 colors, the embedding capacity is 22.32 %;
* •
With 16 colors, the embedding capacity is 29.64 %;
* •
With 32 colors, the embedding capacity is 38 %;
* •
With 64 colors, the embedding capacity is 44 %.
## 4 Scheme design based on numeration systems
In this new approach, we improve the method of the first scheme with the
assertion that each color can be repeated as many times on some positions of a
given group of characters. Unlike the previous scheme in which each color
could only appear once in a group of precise characters.
### 4.1 The Scheme description
we give a brief description of how this new scheme works by following these
steps:
1. 1.
Choose a base $B$ such that $2\leq B\leq 2^{24}$. where $2^{24}$ is the number
of existing colors ;
2. 2.
choose B colors from the set of $2^{24}$ colors number from 0 to $B-1$;
3. 3.
convert the secret m to base B such that : $m=(m_{q-1}...m_{1}m_{0})_{B}$,
where $0\leq m_{i}\leq B-1$;
4. 4.
We assume that the number of characters of the covert text is : $n$ and $q\leq
n$;
5. 5.
For $i=0$ to $q-1$ do
The character $c_{i}$ is coloured with the color relative to $m_{i}$
6. 6.
The text coloured is then send to the receiver.
The reverse procedure consists to extract the secret conceal in the colors
distribution. These steps must be performed by the receiver of the stego-text
:
1. 1.
Take the text with the first $q$ characters which has been coloured;
2. 2.
For $i=0$ to $q-1$ do
Find the color number $z_{i}$ associated to the character $c_{i}$ by using the
reference color table shared between the sender and the receiver;
3. 3.
Convert $z=(z_{q-1}...z_{1}z_{0})_{B}$ to binary and get the secret message.
### 4.2 Embedding Algorithm
Input
C: the cover text;
M: the secret message to embed;
$\beta$ : The base;
T : a table of $\beta$ color;
e : the e-mail address of the receiver;
Output
C’: the stego-message;
Begin
1. 1.
Convert the secret M to base B such that : $m=(m_{n-1}...m_{1}m_{0})_{B}$,
where $0\leq m_{i}\leq B-1$;
2. 2.
For $i=n-1$ to $0$ do
1. (a)
Find in the color table, the color $a_{i}$ associated to the value $m_{i}$;
2. (b)
Coloured the character $c_{i}$ of $C$ with the color $a_{i}$ and obtain
$c^{\prime}_{i}$ ;
3. (c)
Compute $C^{\prime}\longleftarrow C^{\prime}\mid\mid c^{\prime}_{i}$; where
$a\mid\mid b$ is the concatenation of a and b.
3. 3.
If the next character is not EOF (End of File) then
1. (a)
Colour the next character with a color different from the colors table T. This
color is shared by the sender and the receiver. However, this color will not
be very distant from the others;
2. (b)
Randomly color the rest of characters of C by the colors from the colors
table, until obtain the EOF character;
3. (c)
Compute $C^{\prime}\longleftarrow C^{\prime}\mid\mid c^{\prime}_{j}$ : $n\leq
j\leq m$, where $m$ is the position of the last character of $C$;
4. 4.
Use e to send $C^{\prime}$ by mail to the receiver;
End
### 4.3 Retrieving Algorithm
Input
C’: the stego-text;
T : a table of $\beta$ color;
$\beta$ : The base;
Output
M: the secret message;
Begin
1. 1.
Retrieve all characters coloured with the table colors, until obtain a color
different from those of the colors table, or obtain the EOF character. Lets
call them $C^{\prime\prime}$ and $\mid C^{\prime\prime}\mid=n$;
($C^{\prime\prime}=c_{n-1}c_{n-2}...c_{1}c_{0}$);
2. 2.
For $i=n-1$ to $0$ do
1. (a)
get the color $a_{i}$ associated to the color of the character $c_{i}$ of
$C^{\prime\prime}$;
2. (b)
Find in the color table, the value $m_{i}$ associated to the color $a_{i}$ ;
3. (c)
compute $M\longleftarrow M\mid\mid m_{i}$;
4. (d)
Compute $M_{2}$, the binary representation of the secret $M$;
End
### 4.4 Experimentation
This subsection presents some experimentations for this method. We first
propose a theoretical estimation of our embedding capacity for $B$ colors.
Secondly, we present a practical experimentation in the case of 10, 16 and 32
colors, based on example 1 and figure 5 of [14]. These colors are given in
figure 1, figure 2 and figure 3.
#### 4.4.1 Theoretical Estimation
We want to color a block of text with $\eta$ characters. Each character is
coloured with a single color. The number of colors used is $B$. Knowing that a
color can appear as many times on some positions, the total number of
colouring possibilities for each character is : $B$. For the $\eta$
characters, the total number of colouring possibilities is : $B^{\eta}$.
The number of bits used to color the $\eta$ characters is :
$log_{2}(B^{\eta})$
The embedding capacity [14, 18] is define as the ratio of the secret bits
message by the stego cover bits :
$\displaystyle Capacity$
$\displaystyle=\frac{Bits\;of\;secret\;message}{Bits\;of\;stego\;cover}$ (2)
$\displaystyle Capacity$ $\displaystyle=\frac{log_{2}(B^{\eta})}{\eta\times
8}$ (3) $\displaystyle Capacity$ $\displaystyle=\frac{log_{2}(B)}{8}$ (4)
The following table gives a theoretical estimate of the capacity as a function
of the base $B$ used:
Table 5: Embedding capacity estimation as a function of $B$ $B$ | Capacity $\times 100$
---|---
2 | 12.5%
4 | 25%
8 | 37.5%
10 | 41.5%
16 | 50%
32 | 62.5%
64 | 75%
#### 4.4.2 Experimentation 1
This experimentation is based on example 1 of [14], where the number of color
$B$ is equal to 10. The figure 8 presents the results of the embedding process
based on this second method for 10 colors.
Figure 8: The Stego-Text for a table of 10 colors
.
With this example :
* •
in the case of 10 colors, the embedding capacity is 34.31 %;
* •
with 16 colors, the embedding capacity is 41.17 %;
* •
With 32 colors, the embedding capacity is 52.23 %;
#### 4.4.3 Experimentation 2
The figure 9 presents the results of our embedding process for 10 colors,
based on the example gives by figure 5 of [14].
Figure 9: The Stego-Text for a table of 10 colors
.
With this example :
* •
in the case of 10 colors, the embedding capacity is 35.29 %;
* •
with 16 colors, the embedding capacity is 42.85 %;
* •
With 32 colors, the embedding capacity is 53.22 %;
The table 6 recapitulates the embedding capacity of our schemes in comparison
with the scheme of Aruna and al [14], in the case of 10 colors.
| First Method | Second Method | The scheme of Aruna and al [14]
---|---|---|---
example 1 [14] | 20.58 % | 34.31 % | 6.03 %
example of figure 5 [14] | 22.32 % | 35.29 % | 13.43%
Table 6: Comparison between our scheme and the scheme of Aruna [14], in terms
of embedding capacity, for 10 colors
## 5 Conclusion
In this paper, two text steganographic schemes based on color coding have been
proposed. The first based on permutation and the second based on numeration
systems. Given a secret message and a cover text, the proposed schemes embed
the secret message in the cover text by making it coloured. Using 32 colors,
the first scheme achieves a theoretical and practical embedding capacity of
45.63 % and 38 % respectively. While with the second scheme the theoretical
and practical embedding capacity are 62.5% and 53.22% respectively with the
same number of colors. These two high capacity text steganographic scheme
significantly improve the existing work of Aruna and al.
## 6 Acknowledgments
This work was supported by UMMISCO, CETIC and the University of Yaounde 1.
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|
$h_{[a_{1},b_{1}],[a_{2},b_{2}]}$ are the R-symmetry blocks, solving the
$SO(5)$ quadratic Casimir equation and $\beta_{i}$ are unknown constants to be
determined up to an overall factor. The Ward Identities constrain the
coefficients up to an overall constant which is normalised to
$\beta_{5}=\frac{-1}{5}$ for the lowest conformal weight $g_{2,2}$ to match
the normalisation conventions.
$\displaystyle\beta_{1}$ $\displaystyle=\frac{3}{70}\beta_{5}$
$\displaystyle\beta_{2}$ $\displaystyle=-\frac{3}{70}\beta_{5}$
$\displaystyle\beta_{3}$ $\displaystyle=-\frac{1}{4}\beta_{5}$
$\displaystyle\beta_{4}$ $\displaystyle=\frac{1}{4}\beta_{5}$
$\displaystyle\beta_{6}$ $\displaystyle=\frac{3}{40}\beta_{5}$ (4.31)
This gives a fully fixed superconformal block which depends on the two
spacetime and 5 R-symmetry cross-ratios. In turn, this block can be decomposed
like the correlator in terms of $F_{i}$ and $f_{i}$ functions. In terms of the
Ward identity channels, this gives
$\displaystyle f^{0}_{\mathcal{D}_{2},\mathcal{D}_{2}}$ $\displaystyle=1$
(4.32) $\displaystyle f^{1}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=\sum_{k,l}\frac{(k+1)(l+1)(2(k+1)l+3k+2)}{\nu_{2}^{2}\chi_{1}^{2}(k+l+1)}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.33) $\displaystyle f^{2}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=-\sum_{k,l}\frac{(k+1)(l+1)(2k(l+1)+3l+2)}{\nu_{2}^{2}\chi_{1}^{2}(k+l+1)}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.34) $\displaystyle f^{3}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=-\sum_{k,l}\frac{(k+1)(k+2)(l+1)(l+2)}{2\nu_{2}^{2}\chi_{1}^{2}}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}$
(4.35)
One can define the bloc
$\displaystyle
G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}=\frac{3(2)_{k+l}}{(5)_{k}(5)_{l}}\chi_{1}^{k+2}\nu_{2}^{l+2}$
(4.36)
$\displaystyle\sum_{k,l}G_{\mathcal{D}_{2},\mathcal{D}_{2}}^{k,l}=3\chi_{1}^{2}\nu_{2}^{2}F_{2}\left(2,1,1;5,5\,|\,\chi_{1},\nu_{2}\right)$
(4.37)
and the function $f^{3}_{\mathcal{D}_{2},\mathcal{D}_{2}}$ is then the
derivative of this Appell function. What is remarkable in these explicit forms
of the blocks above (4.32) and listed in the Appendix C.2 is that the
superconformal blocks are simple functions of the cross-ratios (essentially of
the same form as the Appell functions of the scalar conformal blocks), even in
the case where the functions came from the sum of 50 terms.444This being said,
one must be careful of which variable to use in the OPE expansion as an
awkward choice can lead to quite cumbersome expressions.
#### Superconformal Blocks in the Asymmetric Channel
In the other channel, where we have the OPE expansion
$\displaystyle\mathcal{D}_{1}\times\mathcal{D}_{2}\sim\mathcal{D}_{1}+\mathcal{D}_{3}+\sum_{h}\mathcal{L}^{h}_{0,[0,1]}$
(4.38)
in addition to the one above. This can be seen in the diagram
$\mathcal{D}_{2}(X_{5})$$\mathcal{D}_{1}(X_{1})$$\mathcal{D}_{1}(X_{4})$$\mathcal{D}_{1}(X_{2})$$\mathcal{D}_{1}(X_{3})$$\mathcal{I}+\mathcal{D}_{2}+\mathcal{L}^{h_{1}}_{0,[0,0]}$$\mathcal{D}_{1}+\mathcal{D}_{3}+\mathcal{L}^{h_{2}}_{0,[0,1]}$
Figure 4.2: OPE diagram for the five-point function
$\langle\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}\rangle$
for the Asymmetric OPE channel, where the same operators are exchanged from
(12) and (45).
The OPE limits in this diagram can be seen at the vertices, and we can choose
variables that vanish in these limits; for example, it will be useful to write
the results in terms of
$\displaystyle\nu_{1}$ $\displaystyle=\frac{\chi_{1}-\chi_{2}}{1-\chi_{2}}$
$\displaystyle\nu_{2}$ $\displaystyle=1-\chi_{2}$ (4.39)
In this expansion, the non-vanishing blocks are given by
$\displaystyle\mathcal{A}=$ $\displaystyle
c_{110}^{2}c_{112}\mathcal{G}_{\mathcal{I},\mathcal{D}_{1}}+c_{112}^{3}\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{1}}+c_{112}c_{123}^{2}\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{3}}+\sum_{h_{1}}c_{112}c_{11h_{1}}^{2}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{D}_{1}}$
(4.40)
$\displaystyle+\sum_{h_{1}}c_{11h_{1}}c_{13h_{1}}c_{123}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{D}_{3}}+\sum_{h_{2}}c_{112}c_{12h_{2}}^{2}\mathcal{G}_{\mathcal{D}_{2},\mathcal{L}^{h_{2}}_{0,[0,1]}}+\sum_{h_{1},h_{2}}c_{11h_{1}}c_{12h_{2}}c_{1h_{1}h_{2}}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{L}^{h_{2}}_{0,[0,1]}}$
In particular, the topological limit is given by
$\displaystyle
g^{0}=c_{\mathcal{D}_{1},\mathcal{I}}+c_{\mathcal{D}_{2},\mathcal{D}_{1}}+c_{\mathcal{D}_{2},\mathcal{D}_{3}}$
(4.41)
### 4.4 Free Theory Data
We can find the free theory OPE coefficients in both channels from the block
expansion above. The Free result is obtained by Wick contracting the
elementary fields and gives555To do this, we compute the six-point GFF result
and pinch points 5 and 6.
$\displaystyle\mathcal{A}=\sqrt{2}\left(\frac{r_{1}}{\chi_{1}^{2}}+\frac{r_{2}}{\chi_{2}^{2}}+\frac{s_{1}}{(\chi_{1}-1)^{2}}+\frac{s_{2}}{(\chi_{2}-1)^{2}}+\frac{t}{(\chi_{1}-\chi_{2})^{2}}+1\right)$
(4.42)
which gives the $F_{i}=\sqrt{2}$ and
$\displaystyle f_{0}^{(0)}$ $\displaystyle=6\sqrt{2}$ (4.43) $\displaystyle
f_{1}^{(0)}$
$\displaystyle=\frac{\sqrt{2}(1-2\chi_{1})}{(\chi_{1}-1)\chi_{1}}$ (4.44)
$\displaystyle f_{2}^{(0)}$
$\displaystyle=\frac{\sqrt{2}(1-2\chi_{2})}{(\chi_{2}-1)\chi_{2}}$ (4.45)
$\displaystyle f_{3}^{(0)}$
$\displaystyle=-\frac{\sqrt{2}}{(\chi_{1}-\chi_{2})^{2}}$ (4.46)
For the symmetric channel, this gives us the following:
$\displaystyle
c_{\mathcal{I},\mathcal{D}_{2}}=c_{\mathcal{D}_{2},\mathcal{I}}=c_{112}=\sqrt{2}$
(4.47) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{2}}=c_{112}^{2}c_{222}=4\sqrt{2}$ (4.48)
$\displaystyle
c_{\mathcal{L}^{\Delta}_{0,[0,0]},\mathcal{D}_{2}}=c_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,0]}}=c_{112}c_{11\Delta}c_{22\Delta}=\left(\frac{1+(-1)^{\Delta}}{2}\right)\frac{\sqrt{\pi}2^{-2\Delta+\frac{1}{2}}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.49) $\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,0]}}=c_{11\Delta_{1}}c_{11\Delta_{2}}c_{2\Delta_{1}\Delta_{2}}=-\frac{\left(\pi\,2^{-2\Delta_{1}-2\Delta_{2}-\frac{3}{2}}(\Delta_{1})_{3}(\Delta_{2})_{3}\Gamma(\Delta_{1}+\Delta_{2})\right)}{\Gamma\left(\Delta_{1}+\frac{3}{2}\right)\Gamma\left(\Delta_{2}+\frac{3}{2}\right)}$
(4.50)
where the last coefficient is vanishing for $\Delta_{1}$ and $\Delta_{2}$ odd.
For the asymmetric channel, this gives:
$\displaystyle c_{\mathcal{I},\mathcal{D}_{1}}=c_{110}^{2}c_{112}=\sqrt{2}$
(4.51) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{1}}=c_{112}^{3}=2\sqrt{2}$ (4.52)
$\displaystyle
c_{\mathcal{D}_{2},\mathcal{D}_{3}}=c_{112}c_{123}^{2}=3\sqrt{2}$ (4.53)
$\displaystyle
c_{\mathcal{L}^{\Delta}_{0,[0,0]},\mathcal{D}_{1}}=c_{112}c_{11\Delta}^{2}=\left(\frac{1+(-1)^{\Delta}}{2}\right)\frac{\sqrt{\pi}2^{-2\Delta-\frac{1}{2}}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.54) $\displaystyle
c_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,1]}}=c_{112}c_{12\Delta}^{2}=-\frac{\sqrt{\pi}2^{-2\Delta-\frac{7}{2}}(\Delta-2)\left((-1)^{\Delta}(\Delta+1)(\Delta+2)-12\right)\Gamma(\Delta+4)}{3(\Delta+1)\Gamma\left(\Delta+\frac{3}{2}\right)}$
(4.55) $\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,1]}}=c_{11\Delta_{1}}c_{12\Delta_{2}}c_{1\Delta_{1}\Delta_{2}}$
(4.56)
$\displaystyle=\frac{1+(-1)^{\Delta_{1}}}{2}\frac{2\sqrt{2}(\Delta_{1}-1)\Delta_{1}(\Delta_{1}+1)(\Delta_{2}-\Delta_{1})\Gamma(\Delta_{1}+3)\Gamma(\Delta_{2}-1)\Gamma(\Delta_{1}+\Delta_{2}+4)}{\Gamma(2\Delta_{1}+3)\Gamma(2\Delta_{2}+3)}$
$\displaystyle\text{for}\quad\Delta_{2}>\Delta_{1}\quad\text{otherwise}$
$\displaystyle
c_{\mathcal{L}^{\Delta_{1}}_{0,[0,0]},\mathcal{L}^{\Delta_{2}}_{0,[0,1]}}=0$
(4.57)
This gives the OPE coefficients for the protected operators
$\displaystyle c_{112}$ $\displaystyle=\sqrt{2}$ (4.58) $\displaystyle
c_{222}$ $\displaystyle=2\sqrt{2}$ (4.59) $\displaystyle c_{123}$
$\displaystyle=\sqrt{3}$ (4.60)
which agree with [37] Those of the uncharged long operators
$\displaystyle c_{11\Delta}$
$\displaystyle=\left(\frac{1+(-1)^{\Delta}}{2}\right)\sqrt{\frac{\sqrt{\pi}2^{-2\Delta-1}(\Delta-1)\Gamma(\Delta+3)}{\Gamma\left(\Delta+\frac{3}{2}\right)}}$
(4.61) $\displaystyle c_{22\Delta}$ $\displaystyle=2c_{11\Delta}$ (4.62)
$\displaystyle c_{2\Delta_{1},\Delta_{2}}$
$\displaystyle=\sqrt{\frac{2\Delta_{1}^{2}(\Delta_{1}+1)(\Delta_{1}+2)\Delta_{2}^{2}(\Delta_{2}+1)(\Delta_{2}+2)\Gamma(\Delta_{1}+\Delta_{2})^{2}}{(\Delta_{1}-1)(\Delta_{2}-1)\Gamma(2\Delta_{1}+2)\Gamma(2\Delta_{2}+2)}}\qquad\\{\Delta_{1},\Delta_{2}\\}\,\text{even}$
(4.63)
where the last expression is only valid when both $\Delta_{1}$ and
$\Delta_{2}$ are even given the form of the OPE expansion.
For the charged channel, just as for the protected operators, the Ward
Identities only constrain the blocks up to a constant. Unlike the protected
operators, no topological sector fixes this constant. We thus impose a minimal
condition of reality and choose a normalisation factor of
$-\frac{(-1)^{\Delta+1}\left((-1)^{\Delta}(\Delta+1)(\Delta+2)-12\right)}{\Delta+1}$
for the $\mathcal{G}_{\mathcal{D}_{2},\mathcal{L}^{\Delta}_{0,[0,1]}}$ block
which gives the OPE coefficient for the charged long and mixed
uncharged/charged long, respectively:
$\displaystyle c_{12\Delta}$
$\displaystyle=\sqrt{\frac{\sqrt{\pi}2^{-2\Delta-4}(\Delta-2)\Gamma(\Delta+4)}{3\Gamma\left(\Delta+\frac{3}{2}\right)}}$
(4.64) $\displaystyle c_{1\Delta_{1}\Delta_{2}}$
$\displaystyle=\sqrt{\frac{12\Delta_{1}\Delta_{2}(\Delta_{2}-1)(\Delta_{1}-\Delta_{2})^{2}\left((\Delta_{2}-1)_{4}\right){}^{2}\Gamma(\Delta_{1}+\Delta_{2}+4)^{2}}{\Gamma(2\Delta_{1}+2)\Gamma(2\Delta_{2}+2)\left((\Delta_{1}-1)_{3}\right){}^{2}(\Delta_{1}-1)_{4}(\Delta_{2}-2)_{6}}}\qquad\Delta_{1}\,\,\text{even
and }\,\Delta_{2}>\Delta_{1}$ (4.65)
One can already recognise patterns observed in four-point functions, such as
the absence of uncharged long operators of odd dimensions. This is, however,
not the case for the charged long operators
$\mathcal{L}^{\Delta_{2}}_{0,[0,1]}$. For the final OPE coefficient, we
observe a bound which seems to come from the OPE
$\displaystyle\mathcal{D}_{1}\times\mathcal{L}^{\Delta_{1}}_{0,[0,0]}\sim\sum_{\Delta_{2}>\Delta_{1}}\mathcal{L}^{\Delta_{2}}_{0,[0,1]}+...$
(4.66)
### 4.5 Conformal Bootstrap
We will focus on bootstrapping the function$f_{3}$ since it has both braiding
and crossing symmetry and deducing the rest of the functions from there. The
crossing is
$\displaystyle f_{3}(\chi_{1},\chi_{2})$
$\displaystyle=f_{3}(1-\chi_{2},1-\chi_{1})$ (4.67)
and the braiding is
$\displaystyle f_{3}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
f_{3}(\chi_{2},\chi_{1}).$ (4.68)
For the first order functions, we start with the Ansatz which satisfies this
crossing and braiding
$\displaystyle f_{3}^{(1)}(\chi_{1},\chi_{2})$
$\displaystyle=p(\chi_{1},\chi_{2})+r_{1}(\chi_{1},\chi_{2})\log(\chi_{1})+r_{1}(\chi_{2},\chi_{1})\log(\chi_{2})+r_{1}(1-\chi_{1},1-\chi_{2})\log(1-\chi_{1})$
$\displaystyle+r_{1}(1-\chi_{2},1-\chi_{1})\log(1-\chi_{2})+r_{5}(\chi_{1},\chi_{2})\log(\chi_{2}-\chi_{1})$
(4.69)
where
$\displaystyle p(\chi_{1},\chi_{2})$ $\displaystyle=p(\chi_{2},\chi_{1})$
(4.70) $\displaystyle p(\chi_{1},\chi_{2})$
$\displaystyle=p(1-\chi_{2},1-\chi_{1})$ (4.71) $\displaystyle
r_{5}(\chi_{1},\chi_{2})$ $\displaystyle=r_{5}(\chi_{2},\chi_{1})$ (4.72)
$\displaystyle r_{5}(\chi_{1},\chi_{2})$
$\displaystyle=r_{5}(1-\chi_{2},1-\chi_{1})$ (4.73)
The easiest way to see the effect of crossing and braiding is for the
individual R-symmetry channels where the relevant transformations are:
$\displaystyle F_{0}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}\left(\frac{\chi_{2}-1}{\chi_{2}-\chi_{1}},\frac{\chi_{2}}{\chi_{2}-\chi_{1}}\right)$
(4.74) $\displaystyle F_{1}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{2}-\chi_{1}}{\chi_{2}-1},\frac{\chi_{2}}{\chi_{2}-1})$
(4.75) $\displaystyle F_{2}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{1}}{\chi_{2}},\frac{1}{\chi_{2}})$ (4.76) $\displaystyle
F_{3}(\chi_{1},\chi_{3})$ $\displaystyle\simeq
F_{5}(\frac{\chi_{1}}{\chi_{1}-1},\frac{\chi_{2}-\chi_{1}}{1-\chi_{1}})$
(4.77) $\displaystyle F_{4}(\chi_{1},\chi_{2})$ $\displaystyle\simeq
F_{5}(\frac{1}{\chi_{1}},\frac{\chi_{2}}{\chi_{1}})$ (4.78)
So there are only three rational functions to fix. The rational functions can
only have poles when operators collide; that is
$\displaystyle\\{\chi_{1},\chi_{2},1-\chi_{1},1-\chi_{2},\chi_{2}-\chi_{1},\frac{1}{\chi_{1}},\frac{1}{\chi_{2}}\\}\rightarrow
0$ (4.79)
Additionally, the OPE expansion dictates the behaviour of the overall function
as well as the logarithms in the OPE limits; this gives the following boundary
conditions
$\displaystyle r_{1}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O(\chi_{1}^{1},(1-\chi_{2})^{0})$ (4.80) $\displaystyle
r_{5}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O((1-\chi_{2})^{-2},(\chi_{1}-\chi_{2})^{0})$ (4.81) $\displaystyle
f_{3}(\chi_{1},\chi_{2})$ $\displaystyle\sim
a_{\mathcal{D}_{1},\mathcal{I}}(\chi_{1}-\chi_{2})^{-2}+O(\chi_{1}^{0},(1-\chi_{2})^{0},(\chi_{1}-\chi_{2})^{0})$
(4.82)
Additionally, the anomalous dimension’s growth constrains the poles’ maximal
order in the other variables. For example, $r_{1}(\chi_{1},\chi_{2})$ cannot
have poles in $\\{\chi_{1},1-\chi_{2}\\}$ so one can parametrise it as
$\displaystyle r_{1}(\chi_{1})$
$\displaystyle=\frac{\chi_{1}\sum_{\\{k,l\\}=0}r_{1,kl}\chi_{1}^{k}\chi_{2}^{l}}{\chi_{2}^{N_{r_{1}}}(\chi_{2}-\chi_{1})^{M_{r_{1}}}}$
(4.83)
where the growth of the anomalous dimension fixes
$\displaystyle N_{r_{1}}$ $\displaystyle=l+2$ $\displaystyle M_{r_{1}}$
$\displaystyle=l$ (4.84)
Likewise, one can use the crossing symmetry to best parametrise
$\displaystyle r_{5}(\chi_{1},\chi_{2})$
$\displaystyle=\left(\frac{\tilde{r_{5}}(\chi_{1},\chi_{2})}{\chi_{1}^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(\chi_{2},\chi_{1})}{\chi_{2}^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(1-\chi_{1},1-\chi_{2})}{(1-\chi_{1})^{N_{r_{5}}}}+\frac{\tilde{r_{5}}(1-\chi_{2},1-\chi_{1})}{(1-\chi_{2})^{N_{r_{5}}}}\right)$
(4.85)
where the growth of the anomalous dimension sets
$\displaystyle N_{r_{5}}=l+1$ (4.86)
Possible cancellations between the rational functions and the logarithms in
the OPE limits mean that the boundary conditions for $p_{1}$ are relaxed from
those of $f_{3}$ to
$\displaystyle p_{1}(\chi_{1},\chi_{2})$ $\displaystyle\sim
O(\chi_{1}^{-1},(1-\chi_{2})^{-1},(\chi_{1}-\chi_{2})^{-2})$ (4.87)
so we can parameterise $p_{1}$ accordingly as
$\displaystyle p_{1}$
$\displaystyle=\frac{p_{1,1}}{(\chi_{2}-\chi_{1})^{2}}+\frac{\sum_{k,l}p_{k,l}\chi_{1}^{k},\chi_{2}^{l}}{\chi_{1}\chi_{2}(1-\chi_{1})(1-\chi_{2})}$
(4.88)
The crossing symmetry (which still relates some coefficients of $p_{1}$), the
boundary conditions of $f_{1},f_{2},f_{3}$ from the OPE expansion, as well as
the constraint that $f_{0}$ is a constant are all extremely constraining along
with the Regge behaviour fixes all the coefficients up to an overall constant
and $p_{1,1}$. At this point, two physical inputs are needed, known from
integrability: The OPE coefficient of the protected operator.
$\displaystyle
c_{\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}}=2-\frac{3}{2\sqrt{2\lambda}}-\frac{9}{16\sqrt{2}\lambda}+O(\lambda^{-3/2})$
(4.89)
and the value of the topological sector at large N
$\displaystyle
f_{0}=\frac{6\mathbb{I}_{2}^{2}}{\lambda\mathbb{I}_{1}^{2}}\,\frac{2(\mathbb{I}_{1}-2)(\mathbb{I}_{1}+28)+\lambda(2\mathbb{I}_{1}-19)}{\sqrt{3\lambda-(\mathbb{I}_{1}-2)(\mathbb{I}_{1}+10)}}$
(4.90)
where
$\displaystyle\mathbb{I}_{i}=\frac{\sqrt{\lambda}I_{0}(\sqrt{\lambda})}{I_{a}(\sqrt{\lambda})}$
(4.91)
and $I_{a}$ are modified Bessel functions of the first kind. The explicit
expansion at strong coupling is
$\displaystyle f_{0}$
$\displaystyle=6\sqrt{2}-\frac{33}{\sqrt{2\lambda}}+\frac{189}{8\sqrt{2}\lambda}+O(\lambda^{-3/2})$
(4.92)
This fixes the free rational functions to
$\displaystyle r_{1}(\chi_{1})$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\lambda}}\frac{\chi_{1}\left(\chi_{1}^{2}-3\chi_{1}\chi_{2}+4\chi_{2}^{2}\right)}{\chi_{2}^{2}(\chi_{2}-\chi_{1})^{3}}$
(4.93) $\displaystyle r_{5}(\chi_{1}-\chi_{2})$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\lambda}}\left(\frac{1}{\chi_{1}^{2}}+\frac{1}{(\chi_{1}-1)^{2}}+\frac{1}{\chi_{2}^{2}}+\frac{1}{(\chi_{2}-1)^{2}}\right)$
(4.94) $\displaystyle p(\chi_{1},\chi_{2})$
$\displaystyle=\frac{1}{2\sqrt{2}\sqrt{\lambda}}\left(\frac{4}{(\chi_{1}-1)(\chi_{2}-1)}+\frac{4}{\chi_{1}\chi_{2}}+\frac{19}{(\chi_{1}-\chi_{2})^{2}}\right)$
(4.95)
This gives a degenerate anomalous dimension in all the channels, independent
of multiplying OPE coefficients. This solves the mixing between all these
operators
$\displaystyle\gamma^{(1)}_{\Delta}=\frac{-1}{2\sqrt{\lambda}}\Delta(\Delta+3)$
(4.96)
### 4.6 Comments
The low dimensionality of this system, along with the high symmetry of the
insertions, allows for a simple analysis of a presumably complex quantity: the
five-point correlator of operator insertions on the Wilson line. The system is
set up to allow the recursive analytic conformal bootstrap to be implemented,
and the first-order bootstrap solution is presented where the higher-order
solutions will be found in [8]. The structure of the superconformal blocks is
a non-trivial result in itself, which enables the bootstrap process for the
defect correlators but could potentially also inform more general setups such
as the bulk-defect blocks presented in section 3.4 of [34]. The bootstrap of
the five-point quantity means that the CFT one can access is much greater and
includes the OPE coefficients with 2 long operators. The analysis of the six-
point function in [8] will also shed light on the full long OPE coefficient
$c_{\Delta_{1}\Delta_{2}\Delta_{3}}$.
## Chapter 5 Effective Theories in AdS2
“Deep in the human unconscious is a pervasive need for a logical universe that
makes sense, But the real universe is always one step beyond logic.” -
Princess Irulan
– Frank Herbert, Dune
A common trait of the theories above, as well as other defect theories (such
as the 1/2-BPS minimal surface string solution in
$\text{AdS}_{3}\times\text{S}^{3}\times\text{S}^{3}\times\text{S}^{1}$ [61]
and in $\text{AdS}_{3}\times\text{S}^{3}\times T^{4}$), is that they have an
effective theory description in $\text{AdS}_{2}$. For this reason, it is
beneficial to understand features of QFT in $\text{AdS}_{2}$ in general; such
as generic diagrams, properties from different interaction vertices, and tools
to compute and interpret quantities within this framework effectively.
As introduced in subsection 2.3.1, boundary (and bulk) correlators in
$\text{AdS}_{2}$ are computed using Witten diagrams. These have the same
structure as Feynman diagrams. However, the difference in metric does bring
complications when it comes to propagators but simplifications when it comes
to the symmetries of the correlators. The conformal symmetry can help solve
some of these diagrams, particularly involving exchange diagrams. Another
method is to use the embedding space formalism to map the problem to flat-
space coordinates, though the bulk propagators remain difficult. Yet another
method uses the “dimension-independent representation of Conformal
Theories”[64] provided by the Mellin transform. However, this description is
redundant for one-dimensional defects and does not allow for the Mellin
transforming of analytically-obtained results. This Chapter will present two
methods specific to one-dimensional defects addressing these issues. The first
is a tool for Witten diagrams in $\text{AdS}_{2}$ that uses the one-
dimensional boundary to simplify direct spacetime computations of contact
diagrams. The second is a Mellin formalism tailored to one-dimensional systems
that provides a useful tool for studying effective theories in
$\text{AdS}_{2}$.
### 5.1 Explicit Computations in AdS2
Perturbative computations in Anti-de-Sitter space are done through Witten
diagrams, whose structure has been studied extensively [23, 118, 56, 63, 117,
102, 94, 95, 96, 109, 65]. Yet, the complexity relative to their flat-space
counterpart is still a roadblock to perturbative analysis. There is still a
search for the full equivalent of Feynman rules [108]. As such, the most
efficient methods for perturbative correlators are through the conformal
bootstrap [47, 53]. However, explicit computations remain a reliable way to
make progress in perturbation theory and can provide some insight into
assumptions that may simplify the bootstrap process. First-order four-point
correlators with quartic interactions in the strong coupling limit can be
written in terms of $D$-functions [56], which are four-point Witten contact
diagrams. At higher order, with loops and exchanges corresponding to
additional integrated bulk points, some diagrams can be related to contact
integrals through differential equations [63, 120]. As such, the $n$-point
$D$-functions are used beyond the first order and can be seen as a starting
point to build ‘master integrals’ for Witten diagrams. AdS2 is a perfect place
to look at these integrals as it provides a simple framework with relevance in
its own right (e.g. in defect theories) and corresponds to a diagonal limit
($\chi=\bar{\chi}$ for four points) of its higher-dimensional counterparts.
Boundary correlators in AdS2 enjoy a one-dimensional conformal symmetry. In
higher dimensions, the conformal symmetry simplifies perturbative
computations. However, the structure of AdS2 provides a framework in which
another elementary method can be used to compute perturbative quantities: the
residue theorem. Using contour integration for one of the AdS2 integrals, the
contact diagram in $\lambda_{n}\phi_{\Delta}^{n}$ theory for $n$ scalars of
low conformal weights is remarkably simple, leading to the results for the
integral 2.33
$\displaystyle I_{\Delta=1,n}(x_{i})$
$\displaystyle=\frac{\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\log\left(\frac{x_{ij}}{2i}\right),$ (5.1) $\displaystyle
I_{\Delta=2,n}(x_{i})=\sum_{i}\sum_{j\neq
i}\frac{-\pi}{2(2i)^{2n-4}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-5}}{\prod_{k\neq
j,k\neq i}x_{ik}^{2}x_{jk}^{2}}\log\frac{x_{ji}}{2i}\right)$
$\displaystyle+\sum_{i}\sum_{j\neq
i}\partial_{x_{i}}\frac{-\pi}{(2i)^{2n-2}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-4}}{\prod_{k\neq
j,k\neq i}x_{ik}^{2}x_{jk}^{2}}\log\frac{x_{ji}}{2i}\right),$ (5.2)
Above, $I_{\Delta,n}(x_{i})$ is the integral corresponding to the Witten
contact diagram of $n$ fields $\phi_{\Delta}$ of conformal dimension $\Delta$
inserted at positions $x_{i}$ defined in 2.33. The normalisation $C_{\Delta}$
in (2.31) and the vertex coupling have been factored out. These expressions
are in terms of the operators’ positions and combine naturally into the cross-
ratios obtained with the usual conformal transformations (see discussion in
section 1.1 around equation (1.17) and Appendix D.1).
#### 5.1.1 Contour Integration for Witten Diagrammatics
In the case of massless scalar fields, the integral corresponding to the
contact Witten diagram with $n$ external points is
$\displaystyle
I_{\Delta=1}(x_{1},...,x_{n})=\int_{0}^{\infty}dzz^{n-2}\int_{-\infty}^{\infty}dx\frac{1}{\Pi_{i=1}^{n}(z^{2}+(x-x_{i})^{2})}.\vspace{.4cm}$
(5.3)
The advantage of working in AdS2 is that since the boundary only has one
dimension, the integrated boundary coordinate $x$ can be analytically
continued to the complex plane and the integral can be evaluated with the
residue theorem. The contribution from the contour around infinity
($\mathcal{C}_{\infty}$ in Figure 5.1) vanishes since the integrand is
appropriately bounded at large $|x|$.
$x_{i}+iz$$x_{i}-iz$$x\in\mathbb{C}$$\mathcal{C}_{\mathbb{R}}$$\mathcal{C}_{\infty}$
Figure 5.1: Contour used for the integral over the variable $x$ parametrising
the AdS2 boundary. The contour can be chosen to close in the upper or lower
complex half-plane since the integrand is appropriately bounded at large
$|x|$.
The integrand in (5.3) has $2n$ poles at
$\displaystyle x=x_{j}\pm iz$ (5.4)
where $1\leq j\leq n$, with residues
$\displaystyle\pm\frac{1}{2iz\Pi_{i\neq j}(x_{ij}^{2}+2izx_{ij})},$ (5.5)
which are depicted in Figure 5.1. Since, for $z>0$, the poles in the upper
half-plane (UHP) come with a positive sign and those in the lower half-plane
(LHP) come with a minus sign, the result is independent of the choice of
closing the contour. However, when $z$ is real, these poles will have an
additional factor $\textrm{sgn}(z)$. This is because the poles cross the
$x\in\mathbb{R}$ axis when $z$ crosses $0$.
We are thus left with the integral
$\displaystyle
I(x_{i})=\pi\int_{0}^{\infty}dzz^{n-3}\sum_{j=1}^{n}\frac{1}{\Pi_{1=1,i\neq
j}^{n}(x_{ij}^{2}+2izx_{ij})}.$ (5.6)
The integrand of (5.6) has a leading large $z$ behaviour
$\displaystyle\pi\sum_{j=1}^{n-1}\frac{z^{-1}}{(2i)^{n-2}}\frac{1}{\Pi_{i=1,i\neq
j}^{n}x_{ij}}+O(z^{-2}),$ (5.7)
which vanishes thanks to the identity
$\displaystyle\sum_{j\in J}\frac{1}{\Pi_{i\in J,i\neq j}x_{ij}}=0,$ (5.8)
so the integral is convergent for $n\geq 3$ as expected. Notice that the
integrand of (5.6) has the same parity as the number of external fields. This
leads to a simplification in computing the odd $n$-point functions since the
$z$-integral can be extended to the range $\\{-\infty,\infty\\}$ and evaluated
using contour integration once again (see Appendix D.1). The symmetry of the
integrand dictates that all contact diagrams where $\sum\Delta$ is odd will
lead to a rational function and not a logarithm.
For generic $n$, equation (5.6) can be evaluated explicitly with the pole-
matched, partial fraction decomposition of the integrand
$\displaystyle\sum_{j}\frac{z^{n-3}}{\Pi_{k\neq
j}\left(2ix_{kj}(z-i\frac{x_{kj}}{2})\right)}$
$\displaystyle=\frac{1}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq j}x_{ik}x_{kj}}\frac{-1}{(z+a_{ij})},$
(5.9)
where
$\displaystyle a_{ij}=\frac{x_{ij}}{2i}.$ (5.10)
Using this decomposition, we obtain logarithm functions whose branch cut is
chosen to be on the negative real axis. The choice of the branch of the
logarithm is arbitrary since we do not cross any branch cut in the definite
integration.111The author thanks Luke Corcoran for a discussion on this point.
The convergent commuting of the sum and the integral is ensured by only taking
the upper bound $\Lambda$ to infinity at the end of computations. This gives
the result
$\displaystyle I(x_{i})$
$\displaystyle=\lim_{\Lambda\rightarrow\infty}\frac{-\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\left(\ln(a_{ij}+\Lambda)-\ln(a_{ij})\right),$ (5.11)
which can be simplified by averaging over the permutation of the two indices
$i$ and $j$. The first consequence is that the divergent term cancels in both
cases. In the even-$n$ case
$\displaystyle\log(\Lambda)\sum_{i\neq j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}=0.$ (5.12)
In the odd-$n$ case, we have a vanishing leading term since
$\displaystyle\ln(\Lambda-i\frac{x_{ij}}{2})-\ln(\Lambda+i\frac{x_{ij}}{2})\xrightarrow{\Lambda\rightarrow\infty}0.$
(5.13)
Thus, we can write the result as
$\displaystyle I(x_{i})$ $\displaystyle=\frac{\pi}{(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{kj}}\ln\left(\frac{x_{ij}}{2i}\right),$ (5.14)
which is a real quantity for both the even case
$\displaystyle I_{even}(x_{i})$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\ln\left(x_{ij}^{2}\right),$ (5.15)
and the odd-$n$ case
$\displaystyle I_{odd}(x_{i})$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\sum_{i\neq
j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\left(\ln(a_{ij})-\ln(-a_{ij})\right)$
$\displaystyle=\frac{\pi}{2(2i)^{n-2}}\left(i\pi\sum_{i>j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq
i\neq j}x_{ik}x_{jk}}-i\pi\sum_{i<j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq i\neq
j}x_{ik}x_{jk}}\right)$
$\displaystyle=\frac{\pi^{2}}{2(2i)^{n-3}}\sum_{i>j}\frac{x_{ij}^{n-4}}{\Pi_{k\neq
i\neq j}x_{ik}x_{jk}}.$ (5.16)
The correlator (2.38) follows from (5.1.1) and (5.15). This matches known
literature for the case of the four-point functions, for example;
$\displaystyle
I_{\Delta=1,n=4}=-\frac{\pi}{2}\left(\frac{\log\left(\chi_{1}\right)}{1-\chi_{1}}+\frac{\log\left(1-\chi_{1}\right)}{\chi_{1}}\right).$
(5.17)
More cases are listed in Appendix D.1.
#### 5.1.2 Massive Scalar Fields
The method used in subsection 5.1.1 is compelling in the generic $n$ case but
quickly increases in complexity when $\Delta>1$. For $\Delta=2$, the result
can still be computed with this method relatively efficiently.222Another
method can be used to obtain the massive $n$-point functions from the massless
cases, as seen in 2.3.1 . The integral
$\displaystyle I_{\Delta=2}(x_{i})=\int dzz^{2n-2}\int
dx\frac{1}{\Pi_{i=1}^{n}(z^{2}+(x-x_{i})^{2})^{2}}.$ (5.18)
is evaluated by contour integration for the $x-$integral and partial fraction
decomposition for the $z-$integral. Double poles lead to the less compact
formula
$\displaystyle I_{\Delta=2,n}=\sum_{i}\sum_{j\neq
i}\frac{-\pi}{2(2i)^{2n-4}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-5}}{\prod_{k\neq
j,k\neq i}x_{kj}^{2}x_{ki}^{2}}\ln\frac{x_{ji}}{2i}\right)$
$\displaystyle+\sum_{i}\sum_{j\neq
i}\partial_{x_{i}}\frac{-\pi}{(2i)^{2n-2}x_{ij}^{2}}\partial_{x_{j}}\left(\frac{x_{ji}^{2n-4}}{\prod_{k\neq
j,k\neq i}x_{kj}^{2}x_{ik}^{2}}\ln\frac{x_{ji}}{2i}\right),$ (5.19)
which is derived in Appendix D.1. One expects a similar structure at higher
$\Delta$, with a double sum over the external coordinates $x_{i,j}$ and
$\partial^{2\Delta}$ derivatives and $\Delta$ terms. Some evidence of this is
the pinching presented in subsection 2.3.1 though subtleties in the order of
limits prevent a general analysis. As such, the residue method loses its
efficiency as we increase the dimension of the external operators.
#### 5.1.3 An Application: Topological Correlators
We now consider non-Abelian gauge theories in AdS2 and an alternative
construction to the Witten diagram computation in Appendix A.2 of [212]. For
consistency with the notation in [212], we denote the boundary coordinate by
$t$ instead of $x$. We review the setting of [212] where the strong coupling
action is that of Yang-Mills theory in AdS2, completed with a regulating
boundary term
$\displaystyle S_{YM}$
$\displaystyle=\frac{1}{2g_{YM}^{2}}\int_{\text{AdS}_{2}}dx^{2}\sqrt{-g}\textrm{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right)$
(5.20) $\displaystyle S_{b^{y}}$
$\displaystyle=\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}-2A^{i}F_{\mu
i}n^{\mu}\right),$ (5.21)
where $\mu,\nu$ are the indices in the bulk coordinates of AdS2, $i$ those of
the boundary coordinates, and $n^{\mu}$ is a unit vector normal to the
boundary of AdS2.
In radial coordinates, the equation of motion is solved by
$\displaystyle F_{r\varphi}$ $\displaystyle=Q\sinh r$ $\displaystyle
A_{\varphi}$ $\displaystyle=Q(\cosh r-1)$ $\displaystyle A_{r}$
$\displaystyle=0,$ (5.22)
where $Q=Q_{a}T^{a}$ is an element of the Lie algebra of the theory, and in
the following, indices $a,b,a_{i}$ are those of the gauge algebra. This gives
the on-shell action
$\displaystyle\left(S_{tot}\right)_{\textrm{on-
shell}}=-2\pi\frac{\textrm{Tr}(Q^{2})}{g_{YM}^{2}}.$ (5.23)
To relate the boundary fields to the bulk fields, the variation of the bulk
action needs to be written in terms of the variation of the boundary field
$\displaystyle\delta S_{tot}$
$\displaystyle=\frac{2}{g_{YM}^{2}}\int_{\partial
B}dx\sqrt{-\gamma}\textrm{Tr}\left(A^{i}\delta a\right)$ $\displaystyle a$
$\displaystyle=\lim_{x^{\mu}\rightarrow\partial B}\left(A_{i}-F_{\mu
i}n^{\mu}\right),$ (5.24)
where $a$ is thus the corresponding boundary field and $i$ is the index
corresponding to the boundary coordinate ($t$). The on-shell action (5.23) can
be written in terms of the boundary fields $a$ through the equation
$\displaystyle a(\varphi)$
$\displaystyle=-uQu^{-1}+iu\partial_{\varphi}u^{-1}$ (5.25) $\displaystyle
u_{0}Qu_{0}^{-1}$
$\displaystyle=\frac{i}{2\pi}\log\left(P\exp\left(i\int_{0}^{2\pi}d\varphi
a(\varphi)\right)\right),$ (5.26)
where the $\varphi-$dependant large gauge transformations at the boundary are
parametrised by $u$ and $P\exp$ denotes the usual path ordered exponential.
The expression for the on-shell action is then proportional to the trace of
(5.26) squared,
$\displaystyle\textup{Tr}(Q^{2})=\textup{Tr}((u_{0}Qu_{0}^{-1})^{2})=-\frac{1}{4\pi^{2}}\Omega(a).$
(5.27)
The expression for $\Omega(a)$ is a standard result in quantum mechanics and
is solved by the Magnus expansion [213, 214]
$\displaystyle\exp\left(\Omega\right)=P\exp\left(i\int d\varphi
a(\varphi))\right).$ (5.28)
This can be used to find the dual correlators through the holographic
dictionary
$\displaystyle<j^{a}(\varphi_{1})j^{b}(\varphi_{2})>$
$\displaystyle=\frac{\delta^{ab}}{4\pi g_{YM}^{2}}$ (5.29)
$\displaystyle<j^{a}(\varphi_{1})j^{b}(\varphi_{2})j^{c}(\varphi_{3})>$
$\displaystyle=-\frac{f^{abc}\textrm{sgn}{\varphi_{12}\varphi_{23}\varphi_{31}}}{4\pi
g^{2}_{YM}}$ (5.30)
$\displaystyle<j^{a_{1}}(\varphi_{1})j^{a_{2}}(\varphi_{2})j^{a_{3}}(\varphi_{3})j^{a_{4}}(\varphi_{4})>$
$\displaystyle=-\frac{f^{aa_{1}a_{2}}f^{aa_{3}a_{4}}}{4\pi
g^{2}_{YM}}\left(\textrm{sgn}{\varphi_{12}\varphi_{24}\varphi_{43}\varphi_{31}}-\textrm{sgn}{\varphi_{21}\varphi_{14}\varphi_{43}\varphi_{32}}\right)$
$\displaystyle\quad+(2\leftrightarrow 3)+(2\leftrightarrow 4),$ (5.31)
where the indices $a,b,c,a_{i}$ are those of the gauge algebra. Through Witten
diagrams, these correlators of boundary terms can be computed explicitly using
the contour integral method detailed above. The bulk-to-boundary propagators
in Poincaré coordinates for the gauge field $A_{\mu}$ [118, 215] are
$\displaystyle G_{\mu}(z,t;t_{i})$
$\displaystyle=\frac{z^{2}+(t-t_{i})^{2}}{2\pi
z}\partial_{\mu}\left(\frac{t-t_{i}}{z^{2}+(t-t_{i})^{2}}\right),$ (5.32)
or explicitly
$\displaystyle G_{z}(z,t;t_{i})$
$\displaystyle=\frac{t_{i}-t}{\pi\left((t-t_{i})^{2}+z^{2}\right)}$
$\displaystyle G_{t}(z,t,t_{i})$
$\displaystyle=\frac{z^{2}-(t-t_{i})^{2}}{2\pi
z(t-t_{i})^{2}+z^{2}}.\vspace{3mm}$ (5.33)
The on-shell action is a pure boundary term
$\displaystyle S_{on-shell}$
$\displaystyle=\frac{1}{2g_{YM}^{2}}\int_{\text{AdS}_{2}}dx^{2}\sqrt{-g}\textrm{Tr}\left(D_{\mu}A_{\nu}F^{\mu\nu}\right)+\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}-2A^{i}F_{\mu
i}n^{\mu}\right)$
$\displaystyle=\frac{1}{g_{YM}^{2}}\int_{\partial\text{AdS}_{2}}dx\sqrt{-\gamma}\textrm{Tr}\left(A_{i}A^{i}+A^{i}F_{i\mu}n^{\mu}\right).\vspace{3mm}$
(5.34)
Explicitly, in the $(z,t)$ Poincaré coordinates, this gives333Note that the
vector pointing out of the boundary goes in the $-z$ direction.
$\displaystyle S_{on-shell}$ $\displaystyle=-\frac{1}{g_{YM}^{2}}\int
dtz\textrm{Tr}\left(A_{t}A_{t}-zA_{t}F_{tz}\right)|_{z=0}.$ (5.35)
The two-point correlators are given by Wick contractions acting on this term
$\displaystyle<a^{a}(t_{1})a^{b}(t_{2})>$ $\displaystyle=\lim_{z\rightarrow
0}-\frac{1}{2g_{YM}^{2}}\int
dt\delta^{ab}zG_{t}(z,t,t_{2})\left(G_{t}(z,t;t_{1})+z\partial_{[z}G_{t]}(z,t;t_{1})\right)$
(5.36) $\displaystyle=\lim_{z\rightarrow
0}\frac{(t_{1}-t_{2})^{2}\delta^{ab}}{4\pi
g_{YM}^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}$ (5.37)
$\displaystyle=\frac{\delta^{ab}}{4\pi g^{2}_{YM}}.$ (5.38)
The three-point vertex is
$\displaystyle S_{3}=-\frac{1}{g_{YM}^{2}}\int
dtdzz^{2}f_{abc}A^{a}_{z}A^{b}_{t}\partial_{[z}A^{c}_{t]},$ (5.39)
which gives a correlator
$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{2})a^{c}(t_{3})\rangle=\frac{1}{g_{YM}^{2}}\text{Perm}\left(f^{abc}I(t_{1},t_{2},t_{3})\right),$
(5.40)
where we define the single-Wick-contracted integral
$\displaystyle I(t_{1},t_{2},t_{3})=\int
dtdzz^{2}G_{z}(z,t;t_{1})G_{t}(z,t;t_{2})\partial_{[z}G_{t]}(z,t;t_{3}).$
(5.41)
The anti-symmetrised derivative removes the $t_{3}$ dependence, and the parity
of this integrand under $z\rightarrow-z$ is the same as that of the odd $n$
massless scalar case (see subsection 5.1.1), so we can evaluate both the $z$
and $t$ integrals with a complex contour.444There is a convergence issue in
$I({t_{1},t_{2},t_{3}})$, which is solved when considering the sum of the Wick
contractions. The leading term ($t^{-1}$) is always cancelled by the odd
permutation of the indices $(1,2,3)$ and the next-to-leading term is
convergent. Extending the $z$ variable to the entire real line, we have
$\displaystyle I({t_{1},t_{2},t_{3}})$
$\displaystyle=\int_{-\infty}^{\infty}dt\int_{0}^{\infty}dz\frac{(t-t_{1})\left((t-t_{2})^{2}-z^{2}\right)}{4\pi^{3}z\left((t-t_{1})^{2}+z^{2}\right)\left((t-t_{2})^{2}+z^{2}\right)}$
(5.42)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dz\,\,\textrm{sgn}(z)\frac{(t-t_{1})\left((t-t_{2})^{2}-z^{2}\right)}{4\pi^{3}z\left((t-t_{1})^{2}+z^{2}\right)\left((t-t_{2})^{2}+z^{2}\right)}.$
(5.43)
This integral has a $t_{i}$-independent contribution from the behaviour at
$t\rightarrow\infty$. However, this is cancelled by the permutation and the
antisymmetry of the structure constants. We will therefore ignore this
contribution and evaluate the integral by contour integration. The
$t-$integral evaluates to
$\displaystyle I({t_{1},t_{2},t_{3}})$
$\displaystyle=\int_{-\infty}^{\infty}\frac{dz}{z}\frac{2z(t_{1}-t_{2})+i(t_{1}-t_{2})^{2}+4iz^{2}}{8\pi^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}\textrm{sgn}(z)^{2}$
(5.44)
$\displaystyle=\int_{-\infty}^{\infty}\frac{dz}{z}\frac{2z(t_{1}-t_{2})+i(t_{1}-t_{2})^{2}+4iz^{2}}{8\pi^{2}\left((t_{1}-t_{2})^{2}+4z^{2}\right)}.$
(5.45)
The factors of $\textrm{sgn}(z)$ cancel and leave an analytic function in $z$.
This integral also has a pole at 0 and at $\infty$, which can also be
cancelled using the antisymmetry of the structure constants of the algebra by
considering a combination of Wick contractions; for example,
$f^{a_{1},a_{2},a_{3}}\left(I_{t_{1},t_{2},t_{3}}-I_{t_{2},t_{1},t_{3}}\right)$.
With this in mind, the integral can be evaluated using contour integration.
The only remaining pole is at $z=\pm i\frac{t_{1}-t_{2}}{2}$ and therefore the
integral will have a factor of $\textrm{sgn}(t_{1}-t_{2})$, multiplying the
residue at that point (see Figure 5.2).
$i(t_{2}-t_{1})$$-i(t_{2}-t_{1})$$z\in\mathbb{C}$$\mathcal{C}_{\mathbb{R}}$$\mathcal{C}_{\infty}$
Figure 5.2: Contour used for the $z-$integral in equation (5.45). The origin
of the topological factor sgn$(t_{1}-t_{2})$ is clear in this setup. The
contour is closed in the UHP (the same analysis holds for the LHP closing).
The pole contained within this contour depends on the sign of $(t_{1}-t_{2})$
where, in this example, we have shown the case $t_{1}<t_{2}$.
The single-Wick-contracted integral is then
$\displaystyle I(t_{1},t_{2},t_{3})=\frac{1}{2\pi}\textrm{sgn}(t_{1}-t_{2}),$
(5.46)
and therefore
$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{2})a^{c}(t_{3})\rangle=\frac{1}{8\pi
g_{YM}^{2}}\left(\sum_{\sigma(\\{1,2,3\\})}\left(f^{a_{1}a_{2}a_{3}}\textrm{sgn}(t_{1}-t_{2})\right)\right)|_{\\{a_{1},a_{2},a_{3}\\}\rightarrow\\{a,b,c\\}}.$
(5.47)
Using the total antisymmetry of the structure constants, we have
$\displaystyle\sum_{\sigma(\\{1,2,3\\})}\left(f^{a_{1}a_{2}a_{3}}\textrm{sgn}(t_{1}-t_{2})\right)|_{\\{a_{1},a_{2},a_{3}\\}\rightarrow\\{a,b,c\\}}=-2f^{abc}\,\textrm{sgn}(t_{12}t_{23}t_{31}),$
(5.48)
which gives the final result
$\displaystyle\langle
a^{a}(t_{1})a^{b}(t_{1})a^{c}(t_{1})\rangle=-\frac{1}{4\pi
g_{YM}^{2}}f^{abc}\,\textrm{sgn}(t_{12}t_{23}t_{31}).$ (5.49)
This agrees with equation (5.30) through a simple change of coordinates.555For
higher-point functions, there is the subtlety that the boundary field $a$ is
not the boundary limit of the gauge field $A_{\mu}$, but rather has a
dependence on both $A_{\mu}$ and $F_{\mu\nu}$. This implies that the bulk-to-
boundary propagator receives corrections from multi-source terms. These
questions are addressed in [212].
#### 5.1.4 A Basis of Interaction Terms
This subsection considers deformations from generalized free field theory
produced by effective interactions in a bulk AdS2 field theory. In this
holographic AdS2/CFT1 setup, the background AdS2 metric is not dynamical,
corresponding to the absence of a stress tensor in the boundary CFT1.
According to the usual dictionary, a massive free scalar field $\Phi$ in AdS2
is dual to a boundary 1d generalised free field $\phi$. We deform this theory
with quartic self-interactions with an arbitrary number $L$ of derivatives
$\displaystyle S=\int
dxdz\,\sqrt{g}\,\big{[}\,g^{\mu\nu}\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi+m^{2}_{\Delta_{\phi}}\Phi^{2}+g_{L}\,(\partial^{L}\Phi)^{4}\,\big{]}\,,\qquad
L=0,1,\dots$ (5.50)
where we use the AdS2 metric in Poincaré coordinates
$ds^{2}=\frac{1}{z^{2}}(dx^{2}+dz^{2})$. The mass
$m^{2}_{\Delta_{\phi}}=\Delta_{\phi}(\Delta_{\phi}-1)$ is fixed in units of
the AdS radius so that $\Delta_{\phi}$ is the dimension (independent of
$g_{L}$) of the field $\Phi$ evaluated at the boundary, $\phi(x)$.666When we
introduce an interaction, such as (5.50), there will be Witten diagrams
contributing to the mass renormalization of $\Phi$. We can always choose the
bare mass so that the dictionary is preserved and $\Delta_{\phi}$ is not
modified. We will limit our analysis to tree-level correlators and thus
consider only contact diagrams, whose building blocks are the $D$-functions
[194, 56, 97] reviewed in Appendix D.1. The writing $(\partial^{L}\Phi)^{4}$
above is symbolic, denoting a complete and independent set of quartic vertices
with four fields and up to $4L$ derivatives.777The fact that a complete and
independent basis of vertices is labelled by 1/4 the number of derivatives can
be seen using integration by parts and the equations of motion, or by noticing
that the counting of physically distinct four-point interactions is equivalent
to the counting of crossing-symmetric polynomial $S$-matrices in 2D Minkowski
space (see discussion in [55, 49]). In the following, we will present a
particularly convenient basis for these interactions, allowing us to derive a
closed-form expression for the tree-level correlator in Mellin space. Consider
the interaction Lagrangian
$\mathcal{L}_{L}=g_{L}\left[\prod_{k=0}^{L-1}\left(\tfrac{1}{2}\partial_{\mu}\partial^{\mu}-(\Delta_{\phi}+k)(2(\Delta_{\phi}+k)-1)\right)\Phi^{2}\right]^{2}\,.$
(5.51)
This looks like a very complicated term, but it contains four fields $\Phi$
and $4L$ derivatives, so by the argument above, it must be effectively a
linear combination of operators like $(\partial^{\ell}\Phi)^{4}$ for $\ell\leq
L$. The advantage of this interaction is that the corresponding correlator
computed via Witten diagrams reads
$\displaystyle
f^{(1)}_{L}(z)=\frac{4^{L-1}\pi^{-\frac{3}{2}}\Gamma(2\Delta_{\phi}-\frac{1}{2}+2L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})^{4}}z^{2\Delta_{\phi}}\,(1+z^{2L}+(1-z)^{2L})\bar{D}_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(z)$
(5.52)
where the $\bar{D}$-functions are listed in Appendix D.1. If one starts with
some specific $4L$-derivative interaction, such as $(\partial^{L}\Phi)^{4}$,
the explicit computation through Witten diagrams shows several other
combinations of $D$-functions with different weights. Nevertheless, by the
argument above, these results cannot be independent of those obtained using
$\mathcal{L}_{L}$ and therefore, the result must be expressible as a linear
combination $\sum_{\ell}a_{\ell}f^{(1)}_{\ell}(z)$. This requires a series of
non-trivial identities among $\bar{D}$ functions, some of which are derived in
[2].
Here we consider the interaction Lagrangian (5.51) and show that it leads to
the correlator (5.52) using Witten diagrams. The result of the Wick
contractions is
$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}=\\\
\sum_{\text{perms}}g_{L}\int\frac{dxdz}{z^{2}}\mathcal{D}\left(K_{\Delta_{\phi}}(x,z;x_{1})K_{\Delta_{\phi}}(x,z;x_{2})\right)\mathcal{D}\left(K_{\Delta_{\phi}}(x,z;x_{3})K_{\Delta_{\phi}}(x,z;x_{4})\right)\,,$
(5.53)
where we defined
$\displaystyle\mathcal{D}=\prod_{k=0}^{L-1}\left(\frac{1}{2}\partial_{\mu}\partial^{\mu}-(\Delta_{\phi}+k)(2(\Delta_{\phi}+k)-1)\right)$
(5.54)
acting on the bulk point, and we used the bulk-to-boundary propagator with the
conventions of (2.31). Using the identity recursively
$\displaystyle-2x_{ij}^{2}\Delta^{2}\tilde{K}_{\Delta+1}(x,z;x_{i})\tilde{K}_{\Delta+1}(x,z;x_{j})=(\tfrac{1}{2}\partial_{\mu}\partial^{\mu}-\Delta(2\Delta-1))(\tilde{K}_{\Delta}(x,z;x_{i})\tilde{K}_{\Delta}(x,z;x_{j}))\,,$
(5.55)
which can be derived from (D.34), we obtain
$\displaystyle\mathcal{D}\left(\tilde{K}_{\Delta_{\phi}}(x,z;x_{1})\tilde{K}_{\Delta_{\phi}}(x,z;x_{2})\right)=(-2x_{12}^{2})^{L}\left(\frac{\Gamma(\Delta_{\phi}+L)}{\Gamma(\Delta_{\phi})}\right)^{2}\tilde{K}_{\Delta_{\phi}+L}(x,z;x_{1})\tilde{K}_{\Delta_{\phi}+L}(x,z;x_{2})\,.$
(5.56)
Inserting this into equation (5.53), summing over the permutations and
remembering the definition of $\mathcal{C}_{\Delta}$ in (2.31), we get
$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}_{L}=g_{L}[(x^{2}_{13}x^{2}_{24})^{L}+(x^{2}_{12}x^{2}_{34})^{L}+(x^{2}_{14}x^{2}_{23})^{L}]\times\\\
2^{2L-1}\pi^{-2}\left(\frac{\Gamma(\Delta_{\phi}+L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})}\right)^{4}D_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(x_{1},x_{2},x_{3},x_{4})\,.$
(5.57)
Using (D.35) we immediately get
$\braket{\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})}^{(1)}_{L}=g_{L}(1+\chi^{2L}+(1-\chi)^{2L})\times\\\
\frac{4^{L-1}\pi^{-\frac{3}{2}}\,\Gamma(2\Delta_{\phi}-\frac{1}{2}+2L)}{\Gamma(\Delta_{\phi}+\frac{1}{2})^{4}\,(x^{2}_{13}\,x^{2}_{24})^{\Delta_{\phi}}}\bar{D}_{\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L,\Delta_{\phi}+L}(\chi)$
(5.58)
in perfect agreement with (5.52).
### 5.2 Mellin Amplitudes
In the higher-dimensional case, the Mellin representation of conformal
correlators [64, 109] has proven to be an excellent tool, especially for the
study of holographic CFTs [65, 108, 120]. The counting of independent cross-
ratios for an $n$-point correlation function of local operators in a
$d$-dimensional CFT is identical to that of independent variables scattering
in $d+1$ dimensions. The Mellin representation, or Mellin amplitude, makes
this correspondence manifest, expressing the correlators in a form that is the
natural AdS counterpart of flat-space scattering amplitudes. This construction
has several nice features. First, the Mellin amplitude has simple poles
located at the values of the twist of exchanged operators (there are, however,
infinitely many accumulation points of such poles). Secondly, the crossing
symmetry of the correlator maps to the amplitude crossing symmetry. Finally,
the language of Mellin amplitudes is particularly suitable for large $N$ gauge
theories, where perturbation theory is described in terms of Witten diagrams.
An extensive introduction to higher dimensional Mellin amplitudes is presented
in Appendix E.1. In the following, we will use these properties as guiding
principles for the definition of Mellin amplitudes for 1d CFTs.
#### 5.2.1 Nonperturbative Mellin Amplitude
Let $f(t)$ be a function describing a four-point correlator in a 1d CFT where
the cross ratio $t$ is defined as
$\displaystyle t=\frac{\chi}{1-\chi}.$ (5.59)
We define the one-parameter Mellin transform $\mathcal{M}_{a}[f(t)]$ as
$\displaystyle\mathcal{M}_{a}[f(t)]==\int_{0}^{\infty}dt\,\left(\frac{t}{1+t}\right)^{a}f(t)\,t^{-1-s}\,,$
(5.60)
and Mellin amplitude $M_{a=0}(s)$ as
$\displaystyle
M(s)=\frac{1}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\int_{0}^{\infty}dt\,f(t)\,t^{-1-s}\,$
(5.61)
which we multiply by an overall factor for future convenience. In this case,
the crossing relation becomes
$\displaystyle f(t)$ $\displaystyle=t^{2\Delta_{\phi}}f(\frac{1}{t}),$
$\displaystyle M(s)$ $\displaystyle=M(2\Delta_{\phi}-s)\,.$ (5.62)
The goal of our discussion is to infer the analytic properties of the Mellin
amplitude $M(s)$ from the physical requirements on the correlator $f(t)$.
First, following [216], we recall a general result for the one-dimensional
Mellin transform (5.61).
##### A Theorem
Consider a function $F(t)$ in the vector space $\mathcal{F}_{H}^{\Theta}$ of
complex valued functions that are holomorphic for $\textrm{arg}(t)\in\Theta$
and obey
$|F(t)|\leq\frac{C(h)}{|t|^{h}}\qquad h\in H\,,$ (5.63)
where H is a subset of $\mathbb{R}$, typically of the form
$H=(h_{min},h_{max})$. Consider also the function $\hat{M}(s)$ in the vector
space $\mathcal{M}_{H}^{\Theta}$ of complex valued functions that are
holomorphic for $\text{Re}(s)\in H$ and exponentially suppressed in the limit
$|\text{Im}(s)|\rightarrow\infty$
$\displaystyle|\hat{M}(s)|\leq
K(\textrm{Re}(s))e^{-|\text{Im}(s)\sup_{\Theta}\textrm{arg}(t)|}\qquad|\text{Im}(s)|\to\infty\,.$
(5.64)
These two vector spaces exist independently but the following theorem holds
Theorem:
Given a function $F(t)\in\mathcal{F}_{H}^{\Theta}$, its Mellin transform
$\mathcal{M}[F](s)$ exists and $\mathcal{M}[F](s)\in\mathcal{M}_{H}^{\Theta}$.
Furthermore, $\mathcal{M}^{-1}\mathcal{M}[F](t)=F(t)$ for any
$\arg(t)\in\Theta$. Conversely, given $\hat{M}(s)\in\mathcal{M}_{H}^{\Theta}$
its inverse Mellin transform exists and
$\mathcal{M}^{-1}[\hat{M}](s)\in\mathcal{F}_{H}^{\Theta}$.
Furthermore, $\mathcal{M}\mathcal{M}^{-1}[\hat{M}](s)=\hat{M}(s)$ for any
$s\in H+i\mathbb{R}$.
This is a classical result for the one-dimensional Mellin transform, so we
will not prove it here. Instead, we will discuss how the physical 1d
correlator violates the hypothesis of the theorem and how we can overcome this
issue. The convergence of the $s$-channel OPE for $|\arg(t)|<\pi$ ensures that
the function $f(t)$ is indeed analytic in a sectorial domain $\Theta$.
Nevertheless, the condition (5.63) is violated in two ways :
* •
When light operators ($\Delta<\Delta_{\phi}$) are exchanged in the OPE, the
region $H$ is not well defined and the Mellin transform does not exist. This
issue is analogous to the higher dimensional case of [216] and we will solve
it by implementing a finite number of subtractions in subsection 5.2.2.
* •
The correlator $f(t)$ is not bounded for $t\to e^{i\pi}$ where it has a
singularity controlled by the Regge limit (5.86). This issue does not spoil
the existence of the Mellin transform, but it gives a result that is not
bounded by (5.64).
To understand this second point, let us present a simple example which will be
useful to explain the issue. Consider the function
$F(t)=(\frac{t}{1+t})^{2\Delta_{\phi}}$. This function is analytic for
$|\arg(t)|<\pi$ and gives a convergent integral (5.61) for
$0<\text{Re}(s)<2\Delta_{\phi}$. However, although the bound (5.63) holds
along the real axis, it is violated for $t\to e^{i\pi}$. The Mellin transform
of this function reads
$\int_{0}^{\infty}dt\left(\frac{t}{1+t}\right)^{2\Delta_{\phi}}t^{-1-s}=\frac{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}{\Gamma(2\Delta_{\phi})}\,.$
(5.65)
From this explicit expression we see immediately that for
$|\text{Im}(s)|\to\infty$ the r.h.s. is not bounded by
$e^{-\pi|\text{Im}(s)|}$. It is bounded by
$\frac{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}{\Gamma(2\Delta_{\phi})}\leq
K(\textrm{Re}(s))|\text{Im}(s)|^{2\Delta_{\phi}-1}e^{-\pi|\text{Im}(s)|}\qquad|\text{Im}(s)|\to\infty\,.$
(5.66)
The exponential decay is correctly predicted by the theorem, while the
additional polynomial divergence can be related to the behaviour of the
function $F(t)$ for $t\to e^{i\pi}$. In section 5.2.3, we will show that this
is a specific instance of a general relation between the large $s$ asymptotics
of $M(s)$ to the Regge limit of $f(t)$.
#### 5.2.2 Convergence and Subtractions
Let us now discuss the convergence of the integral (5.61). Let $f(t)$ be well-
behaved for $t\in\mathbb{R}^{+}$; we do not want divergences in $t$ other than
at $t=0$ and $t\rightarrow\infty$. This behaviour coincides with the CFT1
correlators we are interested in. Consider the behaviour of $f(t)$ close to
$t=0$. Using the conformal block expansion (1.22), we find that the leading
power is $f(t)\sim t^{\Delta_{0}}$ where $\Delta_{0}$ is the dimension of the
lightest exchanged operator. Analogously, using the crossing symmetry relation
(5.62), we find that the large $t$ behaviour of $f(t)$ is $f(t)\sim
t^{2\Delta_{\phi}-\Delta_{0}}$. Therefore the integral converges in the strip
$\displaystyle 2\Delta_{\phi}-\Delta_{0}<\text{Re}(s)<\Delta_{0}\,,$ (5.67)
which is a well-defined interval _only for $\Delta_{0}>\Delta_{\phi}$_. To
give a nonperturbative definition of the Mellin transform, which allows for
lighter operators to be exchanged, we need to perform some subtractions along
the lines of [216].888See in particular Appendix B in [216] for the one-
dimensional case. One obvious example is GFF, where the identity operator is
exchanged. We will consider this case explicitly in Appendix E.2. For the
moment, we consider the Mellin transform of the connected part of the
correlator. Let us consider the following subtractions
$\displaystyle f_{0}(t)$
$\displaystyle=f_{\text{conn}}(t)-\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}t^{\Delta+k}\,,$
(5.68) $\displaystyle f_{\infty}(t)$
$\displaystyle=f_{\text{conn}}(t)-\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}t^{2\Delta_{\phi}-\Delta-k}\,,$
(5.69)
where, for convenience, we write
$c_{\Delta_{\phi}\Delta_{\phi}\Delta}^{2}\equiv c_{\Delta}$. For the function
$f_{0}(t)$ we subtracted the $s$-channel contribution of all the operators
(primaries and descendants) with scaling dimension below the threshold
$\Delta=\Delta_{\phi}$, making use of the series expansion of the
hypergeometric function in the conformal block. This improves the behaviour of
the function at $t=0$. On the other hand, for $f_{\infty}(t)$ we subtracted
all the $t$-channel operators below the threshold, thus improving the
behaviour at $t=\infty$. The idea is to split the integral (5.61) into two
parts, which are defined on (possibly non-overlapping) semi-infinite regions
of the complex $s$ plane
$\displaystyle\psi_{0}(s)$
$\displaystyle=\int_{0}^{1}dt\,f_{\text{conn}}(t)\,t^{-1-s}$
$\displaystyle\text{Re}(s)$ $\displaystyle<\Delta_{0}\,,$ (5.70)
$\displaystyle\psi_{\infty}(s)$
$\displaystyle=\int_{1}^{\infty}dt\,f_{\text{conn}}(t)\,t^{-1-s}$
$\displaystyle\text{Re}(s)$ $\displaystyle>2\Delta_{\phi}-\Delta_{0}.$ (5.71)
When the two regions do not overlap, we analytically continue $\psi_{0}(s)$
and $\psi_{\infty}(s)$ by considering the integrals of the functions (5.68)
and (5.69) and adding a finite number of poles
$\displaystyle\\!\\!\\!\\!\psi_{0}(s)$
$\displaystyle=\int_{0}^{1}\\!\\!\\!dt\,f_{0}(t)\,t^{-1-s}+\\!\\!\\!\\!\\!\\!\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\\!\\!\\!\\!\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\tfrac{(-1)^{k}}{k!}\tfrac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\tfrac{1}{s-\Delta-k}\,,\qquad\small{\text{Re}(s)<\tilde{\Delta}_{0}}\,,$
(5.72) $\displaystyle\\!\\!\\!\\!\psi_{\infty}(s)$
$\displaystyle=\int_{1}^{\infty}\\!\\!\\!\\!\\!\\!dt\,f_{\infty}(t)\,t^{-1-s}+\\!\\!\\!\\!\\!\\!\sum_{\Delta_{0}\leq\Delta\leq\Delta_{\phi}}\\!\\!\\!\\!\sum_{k=0}^{[\Delta_{\phi}-\Delta]}c_{\Delta}\tfrac{(-1)^{k}}{k!}\tfrac{\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\tfrac{1}{s-2\Delta_{\phi}+\Delta+k}\,,\small{\text{Re}(s)>2\Delta_{\phi}-\tilde{\Delta}_{0}}\,,$
(5.73)
where $\tilde{\Delta}_{0}>\Delta_{\phi}$ is the lightest exchanged operator
above the threshold (notice that this operator could be either a primary or a
descendant). Both these functions are now well-defined on the non-vanishing
strip $2\Delta_{\phi}-\tilde{\Delta}_{0}<\text{Re(s)}<\tilde{\Delta}_{0}$ and
therefore their sum yields a well-defined Mellin transform
$\displaystyle M(s)$
$\displaystyle=\frac{\psi_{0}(s)+\psi_{\infty}(s)}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\,,$
$\displaystyle 2\Delta_{\phi}-\tilde{\Delta}_{0}$
$\displaystyle<\text{Re(s)}<\tilde{\Delta}_{0}\,.$ (5.74)
The price to pay is a deformation of the integration contour in the inverse
Mellin transform, which reads
$\displaystyle f(t)=\int_{\mathcal{C}}\frac{ds}{2\pi
i}\,\Gamma(s)\Gamma(2\Delta_{\phi}-s)\,M(s)\,t^{s}\,.$ (5.75)
To understand the form of the contour $\mathcal{C}$, we need to discuss the
analytic structure of $M(s)$. One can follow the above-mentioned strategy to
extend the definition (5.74) to the whole complex $s$ plane. To analytically
continue $\psi_{0}(s)$ from the region $\text{Re}(s)<\Delta_{0}$ to the region
$\text{Re}(s)<\tilde{\Delta}_{0}$ we subtracted a few exchanged operators in
$f(t)$ and added a finite number of poles in (5.72). By adding more and more
poles, we can further extend the area of analyticity. We then conclude that
the Mellin block expansion defined by
$M(s)=\frac{\psi_{0}(s)+\psi_{\infty}(s)}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}$
(5.76)
with
$\displaystyle\psi_{0}(s)$
$\displaystyle=\sum_{\Delta}\sum_{k=0}^{\infty}c_{\Delta}\frac{(-1)^{k+1}\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{k!\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\frac{1}{s-\Delta-k}\,,$
(5.77) $\displaystyle\psi_{\infty}(s)$
$\displaystyle=\sum_{\Delta}\sum_{k=0}^{\infty}c_{\Delta}\frac{(-1)^{k}\Gamma(\Delta+k)^{2}\Gamma(2\Delta)}{k!\Gamma(\Delta)^{2}\Gamma(2\Delta+k)}\frac{1}{s-2\Delta_{\phi}+\Delta+k}$
(5.78)
provides a representation of $M(s)$, which is valid on the whole complex $s$
plane (excluding the point at infinity, which will be discussed in detail in
subsection 5.2.3). In particular, the expression (5.76) immediately allows us
to read off the position of the poles of $M(s)$.999In principle, there could
be an additional singularity at $\infty$, but we postpone this discussion to
section 5.2.4. For any exchanged primary operator of dimension $\Delta$, two
infinite sequences of poles run to the right of $s=\Delta$ and to the left of
$s=2\Delta_{\phi}-\Delta$. Following the common nomenclature, we denote them
as
$\displaystyle\text{\emph{right} poles}:s_{R}$
$\displaystyle=\Delta+k\,,~{}~{}\qquad\,\,\qquad k=0,1,2,\dots$ (5.79)
$\displaystyle\text{\emph{left} poles}:s_{L}$
$\displaystyle=2\Delta_{\phi}-\Delta-k\,,\qquad k=0,1,2,\dots$ (5.80)
$\displaystyle\text{Res}[M(s)]|_{s_{L}}$
$\displaystyle\equiv-\text{Res}[M(s)]|_{s_{R}}=\frac{(-1)^{k}\Gamma(2\Delta)\Gamma(\Delta+k)}{k!\,\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)}\,.$
(5.81)
Notice that the precise identification of the sum over $k$ in (5.77) with the
sum over descendants in the block expansion is a consequence of the choice
$a=0$ in (5.60). The different choices of $a$ in (5.60) would lead to a less
transparent correspondence between poles and conformal descendants.
Given this structure of poles, we can now precisely define the contour
$\mathcal{C}$ in (5.75). The contour $\mathcal{C}$ is chosen in such a way as
to leave all the _right_ poles of $M(s)$ on its right and all the _left_ poles
on its left. Suppose the lightest exchanged operator has dimension
$\Delta_{0}>\Delta_{\phi}$. In that case, no analytic continuation is required
in (5.70) and (5.71) (in other words, the set of left and right poles do not
overlap) and any contour within the interval (5.67) will suffice, see the
straight one on the left in Figure 5.3.
When lighter operators are exchanged, the contour needs to be deformed because
the set of right poles intersects with the set of left poles. Figure 5.3 shows
an example with a single operator below the threshold. It is clear from the
picture that a more complicated situation arises when a left and a right pole
coincide. For instance, this happens in the GFF case, which we address in
Appendix E.2. More generally, this happens whenever there is an exchanged
operator with dimension $\Delta=\Delta_{\phi}+\frac{\mathbb{Z}}{2}$. We do not
expect this to be the case in a generic spectrum.
$s$$\mathcal{C}$$\Delta_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\Delta_{0}$
$s$$\mathcal{C}$$\Delta_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\Delta_{0}$$\tilde{\Delta}_{0}$$2\Delta_{\phi}\\!\\!-\\!\\!\tilde{\Delta}_{0}$
Figure 5.3: Left: The contour for the inverse Mellin transform when
$\Delta_{0}>\Delta_{\phi}$. Left poles are marked in red and right poles are
green. Right: When $\Delta_{0}<\Delta_{\phi}$ left and right poles intersect,
the contour must be deformed.
We conclude this section by noticing that we can perform the sum over $k$ in
(5.76), resumming all the conformal descendants in a crossing symmetric Mellin
block expansion
$\displaystyle M(s)$
$\displaystyle=\frac{1}{\Gamma(s)\Gamma(2\Delta_{\phi}-s)}\sum_{\Delta}\,c_{\Delta}[\mathcal{F}_{\Delta}(s)+\mathcal{F}_{\Delta}(2\Delta_{\phi}-s)]\,,$
(5.82) $\displaystyle\mathcal{F}_{\Delta}(s)$
$\displaystyle=\frac{{}_{3}F_{2}({\Delta,\Delta,\Delta-s};{2\Delta,1+\Delta-s};-1)}{\Delta-s}\,.$
(5.83)
It is worth noting the importance of these finite subtractions in the context
of generalized free field theory. This simple yet subtle example is shown in
Appendix E.2.
#### 5.2.3 Regge Limit and Mellin Boundedness
In this subsection, a bound on the large $s$ behaviour of the Mellin amplitude
$M(s)$ will be derived using the Regge behaviour of the function
$f(t)$.101010In terms of the cross-ratio $t$, this corresponds to the limit
$t\to e^{i\pi}$ described in (5.86). The Regge limit is 111111This limit can
be taken along any direction excluding the real line to avoid the branch cuts,
but for definiteness we take it along the imaginary axis.
$\displaystyle z\to\frac{1}{2}+i\infty.$ (5.84)
This limit can be understood in terms of the higher-dimensional correlator in
the diagonal limit, where it corresponds to the $u$-channel Regge
limit.121212In 1d there is no $u$-channel OPE expansion as it is impossible to
bring $x_{1}$ close to $x_{3}$ without $x_{2}$ in-between. However, one can
resort on the higher dimensional picture to understand that while the
$u$-channel OPE would correspond to $z\to i\infty$ and $\bar{z}\to-i\infty$,
the $u$-channel Regge limit is $z,\bar{z}\to i\infty$. In particular, four-
point functions of a unitary CFT are bounded in the Regge limit [217, 218],
and we have [55]
$\displaystyle\left|\tilde{g}\left(\textstyle{\frac{1}{2}}+iT\right)\right|\,\text{is
bounded as }T\rightarrow\infty.$ (5.85)
Translating into the $t$ cross-ratio (5.59), the line parametrized by
$z=\frac{1}{2}+i\,\xi$ is mapped into the unit circle $t=e^{i\theta}$ for
$\theta\in(-\pi,\pi)$ and the Regge limit occurs when $\theta\to\pi$. The
Regge boundedness condition (5.85) for the function $f(t)$ then reads
$f(e^{i\theta})=\mathcal{O}\left((\pi-\theta)^{-2\Delta_{\phi}}\right)\qquad\theta\to\pi.$
(5.86)
Looking at the direct definition of the Mellin transform (5.61), it may seem
surprising that the large $s$ behaviour is controlled by a region ($t\sim-1$)
which is far away from the integration contour. We must start by considering
the inverse Mellin transform (5.75) where the contour $\mathcal{C}$ is a
straight line parametrized by $s=c+i\eta$ for some constant
$2\Delta-\tilde{\Delta}_{0}<c<\tilde{\Delta}_{0}$ (the additional poles that
are included in (5.72) and (5.73) for the analytic continuation will not
affect this argument) and $\eta\in\mathbb{R}$. We take $t=e^{i\theta}$ and we
integrate over $\eta$
$f(e^{i\theta})=e^{ic\theta}\int_{-\infty}^{\infty}d\eta\,\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\,M(c+i\eta)\,e^{-\theta\eta}\,.$
(5.87)
We are interested in the behaviour of the integrand for $|\eta|\to\infty$. In
this limit
$\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\sim
e^{-\pi|\eta|}\,\eta^{2\Delta_{\phi}-1}\qquad|\eta|\to\infty\,.$ (5.88)
This means that the Gamma function prefactor accounts for the exponential
behaviour (5.64) of $\hat{M}(s)$ for $|\text{Im}(s)|\to\infty$ predicted by
the theorem in subsection 5.2.1. This essentially motivates our choice of
prefactor in (5.61). In particular, the exponential in (5.88) combined with
that in (5.87) shows that the regime $\theta\to\pm\pi$ is controlled by the
region $\eta\sim\mp\infty$. Let us make this more precise by defining
$\displaystyle H(\eta)$
$\displaystyle\equiv\Gamma(c+i\eta)\Gamma(2\Delta_{\phi}-c-i\eta)\,M(c+i\eta)\,e^{\pi|\eta|}$
(5.89)
so that the integral (5.87) can be rewritten as
$f(e^{i\theta})=e^{ic\theta}\int_{0}^{\infty}d\eta\,H(-\eta)\,e^{-\eta(\pi-\theta)}+e^{ic\theta}\int_{0}^{\infty}d\eta\,H(\eta)\,e^{-\eta(\pi+\theta)}\,,$
(5.90)
where we recognize two Laplace transforms of the functions $H(\pm\eta)$. A
singular behaviour for $\theta=\pi$ originates from the first term in the sum
(5.90), while the singularity at $\theta=-\pi$ arises from the second term.
More specifically, Tauberian theorems for the Laplace transform imply that for
a function $H(\eta)\sim k\,\eta^{\alpha}$ as $\eta\to\infty$ then
$\int_{0}^{\infty}d\eta\,H(\eta)\,e^{-\eta(\pi+\theta)}\sim
k\,\Gamma(\alpha+1)(\theta+\pi)^{-\alpha-1}\qquad\theta\to-\pi$ (5.91)
and similarly for the case $\theta\to\pi$. We are then led to the conclusion
that the Regge behaviour (5.86) is reproduced by asking that
$H(\eta)\sim|\eta|^{2\Delta_{\phi}-1}\qquad|\eta|\to\infty\,.$ (5.92)
Combining this with (5.89) and (5.88) we conclude that
$M(c+i\eta)=\mathcal{O}(|\eta|^{0})\qquad|\eta|\to\infty\,.$ (5.93)
Assuming that no Stokes phenomenon occurs for physical correlators, we can
extend this behaviour for any $\text{arg}(s)$ such that
$M(s)=\mathcal{O}(|s|^{0})\qquad|s|\to\infty\,.$ (5.94)
The absence of Stokes phenomenon is an assumption for which we do not have
proof. This assumption, however, is verified in all our examples and was also
made in the higher-dimensional case [216].
We conclude this subsection with an important remark about the perturbative
regime, which we will consider in subsection 5.2.5. The result (5.94) is valid
for the full non-perturbative Mellin amplitude. If the correlator contains a
small parameter, it is often the case that, order-by-order in the perturbative
expansion, the Regge behaviour is worse than in the full non-perturbative
correlator.131313A typical example of this phenomenon is the function
$\frac{1}{1-gz}$, which is regular for $z\to\infty$ but its expansion at small
$g$ is more and more divergent. In Appendix E.3, we illustrate this in detail
in the context of the analytic sum rules discussed in the next section. Given
this aspect, it is thereby useful to formulate our result in a more general
form. Let us consider a correlator $\tilde{f}(z)$ with a Regge behaviour
$\tilde{f}(z)=\mathcal{O}(z^{2\Delta_{\phi}+n})\qquad z\to\frac{1}{2}+i\infty$
(5.95)
for some positive integer $n$, then the associated Mellin amplitude will have
a large $s$ asymptotics
$M(s)=\mathcal{O}(|s|^{n})\qquad|s|\to\infty\,.$ (5.96)
#### 5.2.4 Sum Rules
A common way to express the well-known fact that an arbitrary set of CFT data
does not necessarily lead to a consistent CFT is through a set of sum rules
for the CFT data. In the following, we will start with our Mellin amplitude
definition and derive an infinite set of sum rules. As we mentioned in the
Introduction, these sum rules are not dispersive, according to the definition
of [122]. This is related to the behaviour at infinity obtained using our one-
dimensional definition. In subsection 5.2.3, we described how the product of
Gamma functions in our definition (5.61) leads to a nice behaviour for the
Mellin amplitude $M(s)$ at $s=\infty$. However, introducing that prefactor
also leads to the appearance of spurious poles in the integral (5.75). In a
generic CFT, it is not expected that operators with the exact dimension
$s=2\Delta_{\phi}+n$ are present in the spectrum. Thus, the poles of the Gamma
functions must be compensated by zeros in the Mellin amplitude. This strategy
was used in [216, 219] to derive dispersive sum rules for the higher
dimensional case, where the Mellin amplitude needs to have double zeros. Here,
we will use the same idea to derive a new set of sum rules characterised by
single zeros of the Mellin amplitude. This makes these sum rules different and
less powerful than the dispersive ones. Still, we believe that their
derivation and the check of their validity on a set of known examples provide
an important consistency check of our results.
One may be concerned because the presence of single or double zeroes for the
Mellin amplitude seems to be related to the choice of the prefactor in (5.61).
This is not the case. The choice to factor out a prefactor in (5.61) is
related to having a polynomial behaviour for the function $M(s)$ as
$s\to\infty$. If we were to pick a different prefactor (for instance, using
Gamma function squared, leading to double poles for the Mellin amplitude), the
Mellin amplitude would contain an essential singularity at $s=\infty$, and
this divergence would have to be compensated by the function $F_{p}(s)$, which
we will use in (5.100) to derive our sum rules. We can safely conclude that
choosing a prefactor is a convenient trick, but it does not affect the
resulting sum rules.
Finally, let us emphasize some important differences compared to the higher-
dimensional strategy of the Mellin Polyakov bootstrap [54, 52, 53]. The
derivation of the non-perturbative Polyakov consistency conditions used in
[216, 219] is quite subtle due to accumulation points in the twist spectrum of
higher-dimensional CFTs. In our case, however, the situation is simpler. The
twist accumulation points are related to the presence of a spin or,
equivalently, to the need of introducing two Mandelstam variables. For us
there is no spin and the only quantum number is the scaling dimension of the
operators. Therefore, we do not expect any accumulation point in the spectrum
and we will be able to impose the conditions (5.98) without recurring to any
analytic continuations.
We start by summarizing the main properties of the Mellin amplitude $M(s)$ in
(5.61):
* •
$M$ is crossing symmetric
$M(s)=M(2\Delta_{\phi}-s)\,.$ (5.97)
* •
$M$ has poles at the location of the physical exchanged operators in the two
channels, i.e. $s=\Delta+k$ and $s=2\Delta_{\phi}-\Delta-k$ for
$k\in\mathbb{N}\,.$
* •
Generically, $M$ has single zeros compensating the poles of the prefactor
$\displaystyle M(2\Delta_{\phi}+k)=0\quad\text{and}\quad
M(-k)=0\quad\text{for}\quad k\in\mathbb{N}\,.$ (5.98)
Some of these zeros might be absent if the spectrum contains protected
operators.
* •
$M$ is bounded for $|s|\to\infty$, see (5.94) .
* •
$M$ admits a crossing-symmetric Mellin block expansion
$\displaystyle M(s)=\sum_{\Delta}\,c_{\Delta}\,M_{\Delta}(s)$ (5.99)
with $M_{\Delta}(s)$ given by the comparison with (5.82).
The properties above will allow us to define a set of sum rules along the
lines of [216, 219]. Let $\omega_{p}$ be the functional
$\displaystyle\omega_{p_{i}}=\oint_{\mathbb{C}|_{\infty}}\frac{ds}{2\pi
i}M(s)F_{p_{i}}(s)\,,$ (5.100)
where the contour here is a very large circle around infinity. When
$F_{p_{i}}(s)$ is a sufficiently suppressed function at $s\to\infty$, we can
take the limit of infinite radius for the circle, and we get
$\displaystyle\omega_{p_{i}}[M]=0\,.$ (5.101)
For a nonperturbative Mellin amplitude characterised by the asymptotic
behaviour (5.94) it is sufficient to ask that $F_{p_{i}}(s)\sim
s^{-1-\epsilon}$ for $\epsilon>0$ as $|s|\to\infty$. As we mentioned at the
end of subsection 5.2.3, when considering a perturbative expansion around GFF,
the Regge behaviour may worsen and a sufficiently suppressed function $F$
would be required as detailed in Appendix E.3.
The strategy to derive the sum rules consists of deforming the integration
contour in (5.100) to include all the poles of the integrand such that
$\displaystyle\omega_{p_{i}}=\sum_{s^{*}}\text{Res}_{s=s^{*}}\left[M(s)\right]F_{p_{i}}(s^{*})+\sum_{s^{**}}M(s^{**})\text{Res}_{s=s^{**}}\left[F_{p_{i}}(s)\right]=0\,.$
(5.102)
This equation already looks like a sum rule, but it depends on the value
$M(s^{**})$ of the Mellin amplitude at the poles of $F_{p_{i}}(s)$. To avoid
this issue, one can choose $F_{p_{i}}(s)$ to have simple poles at the position
of the zeros of $M(s)$. Therefore, we need a function $F_{p_{i}}(s)$ with
poles at $s=-k$ or at $s=2\Delta_{\phi}+k$. Furthermore, the function
$F_{p_{i}}(s)$ must not be crossing symmetric. Indeed, using the position of
the poles in (5.79) and (5.80) and crossing symmetry for the residues (5.81)
we get
$\displaystyle\omega_{p_{i}}=\sum_{s_{R}}\text{Res}_{s=s_{R}}(M(s))(F_{p_{i}}(s_{R})-F_{p_{i}}(2\Delta_{\phi}-s_{R}))\,,$
(5.103)
so that any crossing symmetric function $F$ would lead to a trivial vanishing
of $\omega_{p_{i}}$. Using the explicit expression for the residues (5.81), we
find the set of sum rules
$\displaystyle\sum_{\Delta,k}c_{\Delta}\frac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)}(F_{p_{i}}(\Delta+k)-F_{p_{i}}(2\Delta_{\phi}-\Delta-k))=0\,.$
(5.104)
A natural choice for the function $F$ is
$\displaystyle F_{p_{1},p_{2}}(s)=\frac{1}{(s+p_{1})(s+p_{2})}\,,\qquad
p_{1},p_{2}\in\mathbb{N}\,.$ (5.105)
Notice that, despite the function $F_{p_{1},p_{2}}(s)\sim\frac{1}{s^{2}}$ for
$s\to\infty$, thanks to (5.103) only the crossing antisymmetric part of it
matters, i.e. $F_{p_{1},p_{2}}(s)-F_{p_{1},p_{2}}(2\Delta-s)$, and one can
easily check that this combination decays as $\frac{1}{s^{3}}$ for
$s\to\infty$. Using this function, we can derive the nonperturbative sum rules
$\displaystyle\sum_{\Delta,k}c_{\Delta}\tfrac{(-1)^{k+1}\Gamma(2\Delta)\Gamma(\Delta+k)}{\Gamma(\Delta)^{2}\Gamma(2\Delta+k)\Gamma(2\Delta_{\phi}-\Delta-k)\Gamma(k+1)}\tfrac{2(\Delta+k-\Delta_{\phi})(p_{1}+p_{2}+2\Delta_{\phi})}{(\Delta+k+p_{1})(\Delta+k+p_{2})(2\Delta_{\phi}-\Delta-k+p_{1})(2\Delta_{\phi}-\Delta-k+p_{2})}=0\,.$
(5.106)
Performing the sum over $k$, one obtains sum rules of the form
$\displaystyle\sum_{\Delta}c_{\Delta}\alpha_{\Delta}=0$ (5.107)
with
$\displaystyle\alpha_{\Delta}$
$\displaystyle=\frac{\Gamma(\Delta)}{\Gamma(2\Delta)\Gamma(2\Delta_{\phi}-\Delta)}\left(\mathcal{F}_{p_{1},p_{2}}(\Delta)+\mathcal{F}_{-2\Delta_{\phi}-p_{1},-2\Delta-
p_{2}}(\Delta)\right)\,,$ (5.108)
$\displaystyle\mathcal{F}_{p_{1},p_{2}}(\Delta)$
$\displaystyle=\tfrac{1}{p_{1}-p_{2}}\left(\tfrac{{}_{3}F_{2}(\Delta,p_{1}+\Delta,1+\Delta-2\Delta_{\phi};2\Delta,1+p_{1}+\Delta;1)}{(p_{1}+\Delta)}-\tfrac{{}_{3}F_{2}(\Delta,p_{2}+\Delta,1+\Delta-2\Delta_{\phi};2\Delta,1+p_{2}+\Delta;1)}{(p_{2}+\Delta)}\right)\,.$
(5.109)
As already mentioned in the Introduction, these sum rules differ from those
found in [55, 49, 216, 219], which are dispersive sum rules having double
zeros at the dimension of double twist operators (twist in higher-$d$). Our
functionals $\alpha_{\Delta}$ have single zeros at $\Delta=2\Delta_{\phi}+k$
for $k\in\mathbb{N}$ and $k\neq p_{1},p_{2}$, implying that the functional
changes sign at any of these zeros. The absence of any positivity property
makes these sum rules less powerful and harder to use with the standard method
of the modern conformal bootstrap. Recent developments in this direction have
these positivity conditions [220] and have improved upon this method,
confirming some of the results presented in [2].
#### 5.2.5 Perturbative Results
In this subsection, we consider the Lagrangian presented in (5.51) and use the
Mellin transform to find the corresponding four-point correlators and
anomalous dimension for generic $L$. Using (5.52) as a basis for
$4L$-derivative results, we can take its Mellin transform. The first step is
to compute the Mellin transform of the function
$\bar{D}_{\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}}(t)$. In this
section we consider the reduced Mellin amplitude $\hat{M}(s)\equiv
M(s)\Gamma(s)\Gamma(2\Delta_{\phi}-s)$ and we need to compute
$\displaystyle\hat{M}_{\Delta_{\phi}}(s)=\int_{0}^{\infty}dt\,\bar{D}_{\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}\Delta_{\phi}}(t)\,\Big{(}\frac{t}{1+t}\Big{)}^{2\Delta_{\phi}}\,t^{-1-s}\,.$
(5.110)
A closed-form expression for the $\bar{D}$ functions is unavailable, and
dealing with integral representations is quite hard. Therefore, we considered
the case of integer $\Delta_{\phi}$, where simple explicit expressions for the
$\bar{D}$ functions are known (see (D.39)-(D.41)) and we inferred the general
form
$\displaystyle\hat{M}_{\Delta_{\phi}}(s)$ $\displaystyle=\pi\csc(\pi
s)\,\Big{(}\pi\cot(\pi
s)P_{\Delta_{\phi}}(s)-\sum_{k=1}^{2\Delta_{\phi}-1}\frac{P_{\Delta_{\phi}}(k)}{s-k}\Big{)}\,,$
(5.111) $\displaystyle P_{\Delta_{\phi}}(s)$
$\displaystyle=2\sum_{n=0}^{\Delta_{\phi}-1}(-1)^{n}\frac{\Gamma(2n+1)\Gamma^{4}(\Delta_{\phi})\Gamma(\Delta_{\phi}+n)}{\Gamma^{4}(n+1)\Gamma(\Delta_{\phi}-n)\Gamma(2(\Delta_{\phi}+n))}(2\Delta_{\phi}-s)_{n}(s)_{n}\,,$
(5.112)
The functions $P_{\Delta_{\phi}}(s)$ are effectively just polynomials of order
$2\Delta_{\phi}-2$. Defining
$\displaystyle Q_{\Delta_{\phi}}(s(s-2\Delta_{\phi}))\equiv
P_{\Delta_{\phi}}(s)\,,$ (5.113)
we have, for the first few cases
(5.120)
The functions $P_{\Delta_{\phi}}(s)$ can also be rewritten as
$\displaystyle P_{\Delta_{\phi}}(s)$
$\displaystyle=2\frac{\Gamma(\Delta_{\phi})^{4}}{\Gamma(2\Delta_{\phi})}{}_{4}{F}_{3}(\\{\textstyle\frac{1}{2},s,1-\Delta_{\phi},2\Delta_{\phi}-s\\};\\{1,1,\Delta_{\phi}+\frac{1}{2}\\};1)\,,$
(5.121)
Notice that in cross-ratio space a closed-form expression for the $\bar{D}$
functions is not known, while in Mellin space it looks reasonably simple, at
least for integer $\Delta_{\phi}$. This is similar to what happens in the
higher dimensional case, where this occurrence is even more striking as the
reduced Mellin transform of the $\bar{D}$ functions is simply a product of
Gamma functions.141414It is often said that the Mellin transform of contact
interactions is one, but this assumes that the correct product of Gamma
function has been factored out [64, 109]. In the one-dimensional case, we
could not find such a simple representation for the contact interactions, but
the fact we could write the result in a closed form is already a notable
improvement compared to cross-ratio space, and as we will see, it will allow
us to extract new CFT data successfully.
Knowing the Mellin transform for the $\bar{D}$ functions, it is simple to
compute the Mellin transform of (5.52)
$\displaystyle\hat{M}^{(1)}_{L}(s)$
$\displaystyle=\int_{0}^{\infty}dt\,f^{(1)}_{L}(t)\,\Big{(}\frac{t}{1+t}\Big{)}^{2\Delta_{\phi}}\,t^{-1-s}=\,\sum_{k=0}^{2L}c_{k,L}\,\hat{M}_{\Delta_{\phi}+L}(s+k)\,,$
(5.122) $\displaystyle 2c_{k,L}$
$\displaystyle=\frac{\Gamma(2L+1)}{\Gamma(k+1)\Gamma(2L-k+1)}+\delta_{k,0}+\delta_{k,2L}\,.$
(5.123)
Notice that the presence of double poles for integer values of $s$ in (6.1) is
not in contradiction with the general single-pole structure of the
nonperturbative Mellin amplitude (5.76)-(5.78), but just a consequence of the
perturbative expansion of those single poles at this first order of
perturbation theory as detailed in Appendix E.3. Moreover, the structure of
(6.1) is such that both single and double poles cancel (as they should) within
the region of convergence (5.67) of the integral (5.110), which in this case
($\Delta_{0}=2\Delta_{\phi}$) is $0<\text{Re}(s)<2\Delta_{\phi}$. The
cancellation of the double poles is evident, given the poles of $\cot(\pi s)$
and the explicit poles in the sum. The cancelling of the single poles stems
from a property of the polynomial $P_{\Delta_{\phi}}(s)$, which ensures the
cancellation of the finite part in the expression multiplying $\csc(\pi s)$
when expanded around integer values of $s$, $0<s<2\Delta_{\phi}$. As we will
see, this structure is consistent with the OPE expansion.
We stress that equation (5.122) is a closed-form expression for the first-
order perturbation around GFF generated by a quartic interaction with any
number of derivatives. Under the assumption that the deformation from GFF
described by these interactions only modifies two-particle data, see
(5.128)-(5.129) below, one can extract these data. In particular, in section
5.2.6, we will show that the anomalous dimension of two-particle operators
receives the following correction
$\Delta=2\Delta_{\phi}+2n+g_{L}\hat{\gamma}_{L,n}(\Delta_{\phi})$ (5.124)
with
$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\frac{\Gamma(L+\Delta_{\phi})^{4}}{\Gamma(2L+2\Delta_{\phi})}\sum_{p=0}^{2n}\sum_{k=2L-p}^{2L}\sum_{l=0}^{k+p-2L}(-1)^{k}c_{k,L}\times$
(5.125)
$\displaystyle\frac{(4\Delta_{\phi}+2n-1)_{p}(-2n)_{p}(2L-k-p)_{l}(1-\Delta_{\phi}-L)_{l}(2\Delta_{\phi}+k+p)_{l}(\tfrac{1}{2})_{l}}{(l!)^{3}(2\Delta_{\phi})_{p}(\tfrac{1}{2}+\Delta+L)_{l}}\,.$
To compare these results with those computed with bootstrap methods in [55,
49] for $L=0,1,2,3$, we have to change the basis in the space of couplings.
Since the bootstrap approach is blind to the specific values of the couplings
$g_{L}$ in (5.124), one needs to establish a criterium to organize the set of
independent data. The criterium that is used in [55, 49] consists of setting
$\gamma_{L,n}(\Delta_{\phi})=0\qquad n<L\,.$ (5.126)
In our approach, this is implemented by taking a linear combination
$\gamma_{L,n}=\sum_{\ell=0}^{L}a_{\ell}\hat{\gamma}_{\ell,n}$ (5.127)
and fixing the $L+1$ $a_{\ell}$ coefficients in (5.127), using the $L$
conditions (5.126) and the normalization $\gamma_{L,L}(\Delta_{\phi})=1$.
Following this strategy in section 5.2.6, we will reproduce the known results
for $L\leq 3$ and present new results for $L\leq 8$ at any $\Delta$ and $n$.
We stress, however, that equation (5.125) is valid for any $L$, so, up to the
algorithmic procedure of fixing the $a_{\ell}$ coefficients, one can easily
extract the result for any given $L$.
Below we will show how the interaction (5.51) leads to the correlator (5.52)
through explicit Witten diagrammatics. We will also see, for the cases
$L=0,1,2$, how other interaction terms lead to results that can be rearranged
as linear combinations of the eigenfunctions (5.52). We will then proceed to
extract CFT data in the bootstrap normalization.
#### 5.2.6 CFT Data
Given the closed-form expression (5.122) for the perturbative Mellin
amplitude, we can use it to extract the CFT data
$\displaystyle\Delta$
$\displaystyle\equiv\Delta_{n,L}=2\Delta_{\phi}+2n+g_{L}\,\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})+\dots\,,$
(5.128) $\displaystyle c_{\Delta}$ $\displaystyle\equiv
c_{n,L}=c_{n}^{(0)}(\Delta_{\phi})+g_{L}\,c_{L,n}^{(1)}(\Delta_{\phi})+\dots\,,$
(5.129)
where $n\in\mathbb{N}$ and $c_{n}^{(0)}$ is given in (E.81). In
(5.128)-(5.129), we assume that the AdS interaction only modifies the CFT data
of the two-particle operators exchanged in GFF. If we insert (5.128)-(5.129)
in the general Mellin OPE expansion (5.76) at first order in $g_{L}$, one
obtains double and single poles at $s=2\Delta_{\phi}+p$, $p\in\mathbb{N}$. One
can then compare the corresponding residues with the ones in the tree-level
Mellin amplitude (5.122), which amounts to solving the equations formally
written as
$\displaystyle\sum_{n}c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\,{\textstyle{\frac{(-1)^{p}\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\lim_{s\rightarrow
2\Delta_{\phi}+p}(s-2\Delta_{\phi}-p)^{2}\hat{M}^{(1)}_{L}(s)\,,$ (5.130)
$\displaystyle\sum_{n}(c_{L,n}^{(1)}(\Delta_{\phi})\\!+c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\partial_{n})\,{\textstyle{\frac{(-1)^{p}\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\text{Res}_{s=2\Delta_{\phi}+p}\hat{M}^{(1)}_{L}(s)$
(5.131)
for the leading-order corrections $\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
and $c_{L,n}^{(1)}(\Delta_{\phi})$. Since the function $\hat{M}^{(1)}_{L}(s)$
is known explicitly, it is possible to write down a linear system for the
anomalous dimensions $\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$. To do this,
let us use the form (6.1) for $\hat{M}^{(1)}_{\Delta_{\phi}}(s)$ to rewrite
(5.130) as
$\displaystyle\sum_{n}c_{n}^{(0)}(\Delta_{\phi})\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})\,{\textstyle{\frac{\Gamma(4\Delta_{\phi}+4n)\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(2\Delta_{\phi}+2n)^{2}\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}}}\,=\sum_{k=0}^{2L}(-1)^{k}c_{k,L}P_{\Delta_{\phi}+L}(2\Delta_{\phi}+p+k)\,.$
(5.132)
Notice that the sum on the left hand side of (5.132) is truncated by the
$\Gamma(p-2n+1)$ in the denominator. This means that at a fixed value of $p$,
equation (5.132) provides an invertible linear system which can be solved for
$\gamma$. The system can also be inverted explicitly using the identity
$\displaystyle\sum_{p=0}^{2n}\frac{(4m+4\Delta_{\phi}-1)\,\Gamma(p+4\Delta_{\phi}+2m-1)(-1)^{p}}{\Gamma(2m-p+1)\,\Gamma(2\Delta+p)^{2}}\frac{\Gamma(2\Delta_{\phi}+p)^{2}}{\Gamma(4\Delta_{\phi}+2n+p)\Gamma(p-2n+1)}=\delta_{m,n}$
(5.133)
yielding (5.125):
$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\frac{\Gamma(L+\Delta_{\phi})^{4}}{\Gamma(2L+2\Delta_{\phi})}\sum_{p=0}^{2n}\sum_{k=2L-p}^{2L}\sum_{l=0}^{k+p-2L}(-1)^{k}c_{k,L}\times$
(5.134)
$\displaystyle\frac{(4\Delta_{\phi}+2n-1)_{p}(-2n)_{p}(2L-k-p)_{l}(1-\Delta_{\phi}-L)_{l}(2\Delta_{\phi}+k+p)_{l}(\tfrac{1}{2})_{l}}{(l!)^{3}(2\Delta_{\phi})_{p}(\tfrac{1}{2}+\Delta+L)_{l}}\,.$
Equation (5.134) has been derived under the assumption that $\Delta_{\phi}$
takes integer values. However, one can argue that it holds for any
$\Delta_{\phi}$ by noticing that the result for $L=0$ agrees with that of
[55], which has been obtained without assuming integer $\Delta_{\phi}$.
Furthermore, equation (5.122) implies that the CFT data at $L=0$ fully
determines that at higher $L$ or, in other words, equation (5.134) could be
rewritten as a combination of anomalous dimensions for $L=0$. This shows that
(5.134) holds for any $\Delta_{\phi}$.
Notice that all the sums in (5.134) are finite, so for a given value of $L$
and $n$, it is straightforward to extract the value of the anomalous dimension
$\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$. It turns out expression (5.134) can
be rewritten as
$\displaystyle\hat{\gamma}_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})\,,$
(5.135)
where
$\displaystyle\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})=\tfrac{\sqrt{\pi}4^{-2\Delta-L+1}\Gamma(2\Delta)^{2}\Gamma(L+\frac{1}{2})\Gamma(L+\Delta)^{4}\Gamma(L+2\Delta-\frac{1}{2})\Gamma(n+\Delta+\frac{1}{2})\Gamma(L-n+\Delta)}{\Gamma(L+1)\Gamma(L+\Delta+\frac{1}{2})^{2}\Gamma(L+2\Delta)\Gamma(n+\Delta)^{3}\Gamma(2n+2\Delta-\frac{1}{2})\Gamma(L+n+\Delta+\frac{1}{2})}\,,$
(5.136)
while $\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})$ is a polynomial in $n$ and in
$\Delta_{\phi}$ of degree $6L$. It is easy to extract these polynomials from
(5.125), a short Mathematica notebook was attached to the [2] arXiv submission
where the function FindPolynomial[L,$\Delta$,n] allows to extract
$\hat{\mathcal{P}}_{L,n}$ for many values of $L$ (the function gets slower and
slower at higher $L$, but in principle it works for any $L$).
To make contact with the bootstrap results, the coefficients $a_{\ell}$ in the
definition (5.122) of $M_{L}(s)$ should be chosen to have the bootstrap
normalization, namely $\gamma^{(1)}_{\ell,L}=0$ for $0\leq\ell<L$, and
$\gamma^{(1)}_{L,L}=1$. In this case, we have
$\displaystyle\gamma_{L,n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\mathcal{G}_{L,n}(\Delta_{\phi})\mathcal{P}_{L,n}(\Delta_{\phi})\,,$
(5.137)
where
$\displaystyle\mathcal{G}_{L,n}(\Delta_{\phi})=\frac{4^{-L}\left(L+\frac{1}{2}\right)_{\Delta_{\phi}}\left(L+\Delta_{\phi}\right)_{\Delta_{\phi}}(-L+n+1)_{\Delta_{\phi}-1}\left(L+n+\Delta_{\phi}+\frac{1}{2}\right)_{\Delta_{\phi}-1}}{\Gamma\left(\Delta_{\phi}\right)\left(\Delta_{\phi}\right)_{3L}\left(2L+\Delta_{\phi}+\frac{1}{2}\right)_{\Delta_{\phi}-1}\left(L+2\Delta_{\phi}-\frac{1}{2}\right)_{2L}\left(n+\frac{1}{2}\right)_{\Delta_{\phi}}\left(n+\Delta_{\phi}\right)_{\Delta_{\phi}}}\,,$
(5.138)
while $\mathcal{P}_{L,n}(\Delta_{\phi})$ is a polynomial of degree $4L$ in $n$
and $5L$ in $\Delta_{\phi}$. The explicit polynomials for the first few values
of $L$ are detailed in Appendix E.4 and, up to $L=3$, they perfectly agree
with the result of [49]. The attached Mathematica notebook has values of $L$
ranging from $L=0$ to $L=8$ as well as a function
FindBootstrapPolynomial[L,$\Delta$,n] to compute
$\mathcal{P}_{L,n}(\Delta_{\phi})$ for arbitrary $L$.
#### 5.2.7 Bootstrapping the Anomalous Dimension
Since the interaction terms in (5.51) form a basis, we can use the results for
the anomalous dimensions to directly bootstrap the result and find the
resulting effective action in AdS2. Consider an effective theory of massless
scalar in $\text{AdS}_{2}$ with the Lagrangian based on the one in (5.50)
$\displaystyle S=\int
dxdz\,\sqrt{g}\,\big{[}\,g^{\mu\nu}\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi+m^{2}_{\Delta_{\phi}}\Phi^{2}+\sum_{L}g_{L}\,(\partial^{L}\Phi)^{4}\,\big{]}\,,\qquad
L=0,1,\dots$ (5.139)
From the Mellin space analysis, the resulting anomalous dimension will be
$\displaystyle\hat{\gamma}_{n}^{(1)}(\Delta_{\phi})$
$\displaystyle=\sum_{L}g_{L}\hat{\mathcal{G}}_{L,n}(\Delta_{\phi})\hat{\mathcal{P}}_{L,n}(\Delta_{\phi})\,.$
(5.140)
Explicitly for massless fields, we have
$\displaystyle\gamma^{(1)}_{h}=\frac{2g_{0}}{(h-1)h}+\frac{3g_{1}\left((h-1)^{2}h^{2}-4\right)}{35(h-1)h}+\sum_{L\geq
2}g_{L}\hat{\gamma}_{L}^{(1)}.$ (5.141)
For a given Regge bound $\gamma^{1}\sim_{\Delta>>1}\Delta^{l}$, this series
truncates. Additionally, the requirement that the anomalous dimension vanishes
for a certain protected operator fixes the ratio of the two coefficients, for
example:
$\displaystyle\gamma^{(1)}_{h=2}$ $\displaystyle=0$
$\displaystyle\qquad\Rightarrow g_{0}$ $\displaystyle=0$ (5.142)
$\displaystyle\gamma^{(1)}_{h=4}$ $\displaystyle=0$
$\displaystyle\qquad\Rightarrow g_{0}$ $\displaystyle=-6g_{1}$ (5.143)
A particular choice stands out, which is for $\gamma^{(1)}_{h=\Delta=1}=0$.
This gives a polynomial anomalous dimension proportional to the quadratic
Casimir:
$\displaystyle\gamma^{(1)}_{h,(\gamma_{h=1}=0)}$
$\displaystyle=\frac{g_{0}}{2}h(h-1)$ (5.144)
where $g_{1}=\frac{35}{6}g_{0}$.
This analysis also points to one of the limits of the analytic conformal
bootstrap. When the constraint from the growth of the anomalous dimension
depends on the perturbative order, each order has additional contact terms
which can potentially be added and the number of additional constraints to fix
the full correlator increases. Up to this point, in the bootstrap of 1d dCFTs
this has not yet been an issue (at third-order in ABJM and fourth-order in
$\mathcal{N}=4$ SYM). However, this may indicate an upper limit for the order
of perturbation one can bootstrap.151515The author thanks Miguel Paulos for
useful comments about this fact. The next and final Chapter will reflect on
this and other areas of the analytic bootstrap that can be improved.
## Chapter 6 Optimisation of the Bootstrap Procedure
End? No, the journey doesn’t end here.
– J.R.R. Tolkien, The Lord of the Rings
As illustrated in the first Chapters of this thesis, the analytic conformal
bootstrap is an extremely powerful tool to compute correlators in the context
of conformal line defects in holographic theories. However, many improvements
can still be made in a number of the steps essential to the bootstrap process.
The first concerns the Ansatz. As this is the starting point of the analytic
bootstrap process, its accuracy and reliability are crucial. Additionally,
there are limits to the bootstrap procedure; these take the form of the
increased number of constants to fix at each order of the bootstrap and the
resolving the mixing problem. Finally, improvements can still be made for the
physical input. All-order results may not be available depending on the setup,
especially in non-supersymmetric cases. However, there is hope that QFT
methods for non-perturbative computations might pave the way for these all-
order results through lattice computations.
### 6.1 Ansatz and Mellin Bootstrap
The first obvious choice of improvement is through the Ansatz: a rigorous
treatment along the lines of the master integral’s formalism for flat-space
amplitudes is yet to be carried out. There has been work done in this
direction in Mellin space [108] and for contact integrals as seen above [4],
and the diagrams’ structure is relatively well understood. However, in
physical examples, the results are much simpler than the individual diagrams
(usually at least one order of transcendentality less). This has already been
observed for supergravity amplitudes in $\text{AdS}_{5}\times\text{S}^{5}$ and
could be linked to the exchange diagram being expressed as a finite truncation
of the originally infinite sum of $D-$functions[63]. There is some evidence
that this could be linked to Yangian symmetry[101] given both ABJM and
$\mathcal{N}=4$ are integrable theories. Such an explicit realisation of
Yangian symmetry would be a powerful additional tool for the bootstrap [221,
222].
The requirements of having well-defined functions under crossing and braiding
are already highly constraining. However, additional factors linked to the
reasons above seem to favour powers of logarithms to polylogarithms in the
results obtained in Chapters 3 and 4. Additionally, the Ansatz will change
drastically depending on the point around which one is expanding. This
difference is visible between the free theories at weak and strong coupling,
where the weak coupling sector seems to have a quicker increase in
transcendentality [79, 119]. Finally, there might be simpler frameworks for
the Ansatz and the results; for example in [102], the Mellin amplitude was
essential for an all-order result. In the perturbative results above, there is
a drastic simplification between the generic $D-$functions and the result for
correlators. For example, the $D_{1111}$ function, corresponding to a four-
point contact diagram with external massless scalars, has the Mellin transform
(as defined in section 5.2)
$\displaystyle\hat{M}_{\Delta_{\phi}=1}(s)$ $\displaystyle=\pi\csc(\pi
s)\,\Big{(}2\pi\cot(\pi s)-\frac{2}{s-1}\Big{)}$ (6.1)
Comparing this to the ABJM four-point functions, one immediately sees the
simpler forms. The first-order solution $f^{(1)}(\chi)$ in equation 3.166 has
the Mellin transform
$\displaystyle\hat{M}[f^{(1)}(\chi)](s)$
$\displaystyle=6\epsilon\Gamma(-s-2)\Gamma(s-1),$ (6.2)
and the second order is simple enough to be written as
$\displaystyle\hat{M}[f^{(2)}(\chi)](s)$
$\displaystyle=6(\epsilon^{2}p_{0}(s)+1)\Gamma(-s+\epsilon^{2}p_{1}(s)-2)\Gamma(s+\epsilon^{2}p_{2}(s)-1)|_{\epsilon^{2}}$
(6.3) $\displaystyle p_{0}(s)$
$\displaystyle=\frac{30\gamma(s-2)((s-1)s+2)-2s(2s(s((s-5)s+17)-21)+69)+298}{3(s-3)(s-2)^{2}}$
(6.4) $\displaystyle p_{1}(s)$
$\displaystyle=\frac{(s+2)(s(s(s(2s-9)+8)+34)+3)}{6(s-3)(s-2)}$ (6.5)
$\displaystyle p_{2}(s)$
$\displaystyle=-\frac{s\left(s\left(2s^{2}+s-7\right)-31\right)+38}{6(s-2)}.$
(6.6)
Additionally, the differential operators relating $f(\chi)$ to the other
correlators of elements of the displacement multiplet are linear in the cross-
ratio. Therefore, the Mellin transform will correspond to a linear combination
of shifted Mellin amplitudes and will therefore keep the ‘simple form’. For
example, the massless bosonic correlator is expressed in 3.114 in terms of two
functions $G_{1}(\chi)$ and $G_{2}(\chi)$. These can be expressed in terms of
a differential equation acting on $f(\chi)$ in equation (3.116) and can be
written in terms of the Mellin amplitude
$\displaystyle G_{1}(\chi)$ $\displaystyle=f(\chi)-\chi
f^{\prime}(\chi)+\chi^{2}f^{\prime\prime}(\chi)$ (6.7) $\displaystyle
G_{2}(\chi)$
$\displaystyle=-\chi^{2}f^{\prime}(\chi)-\chi^{3}f^{\prime\prime}(\chi)$ (6.8)
$\displaystyle M[G_{1}(\chi)](s)$ $\displaystyle=(s^{2}+1)M[f(\chi)](s)$ (6.9)
$\displaystyle 6(s^{2}+1)\Gamma(-s-2)\Gamma(s-1)$ (6.10) $\displaystyle
M[G_{2}(\chi)](s)$ $\displaystyle=-(s-3)(s-1)M[f(\chi)](s-1)$ (6.11)
$\displaystyle=-6(s-3)(s-1)\Gamma(-s-1)\Gamma(s-2).$ (6.12)
This can also be done for the correlator of the mixed fluctuations detailed in
B.49, for example:
$\displaystyle M[h(\chi)]$
$\displaystyle=(s-2)(s-1)((s-2)s+3)M[f(\chi)](s)-(-3+s)(-2+s)(-1+s)^{2}M[f(\chi)](s-1)$
(6.13) $\displaystyle=12(1-3s+s^{2})\Gamma(s+1)\Gamma(-s-2).$ (6.14)
The Mellin transform from [2] seems promising at tree-level given the
simplifications which occur when compared the generic $D-$functions. Even
beyond this, there seems to be a clear perturbative structure which might be
exploited (for example 6.3). Moreover, the different aspects of the bootstrap
up to the normalisation can be implemented directly in Mellin space.
### 6.2 Physical Non-Perturbative Input
One of the dangers of the ever-increasing bound on the large-$\Delta$
behaviour of the anomalous dimension is that one has an ever-increasing number
of terms to fix with the bootstrap.111This is discussed in subsection 5.2.7
This implies that there might be a hard upper limit beyond which one cannot
use the same algorithm to constrain the correlators via the analytic
bootstrap. Even in the lucky cases of the 1/2-BPS defect theories in ABJM and
$\mathcal{N}$=4 SYM, there is still a constant to fix at each order requiring
input from other computations. In the latter, the topological sector is
amenable to a localisation computation which gives an all-order result. In the
former, the integrated correlator conditions [195] relate to another quantity,
the Bremsstrahlung function, which can be computed non-perturbatively [180,
181]. In theory, another such integrated correlator condition should exist for
the four-point function [79], and one could also use the analogous quantities
for higher-point functions. This is one of the difficulties when extending
these methods to systems with fewer symmetries; they often lack non-
perturbative results.
One way to counter this is to use the non-perturbative framework developed for
the standard model: Lattice Field Theory. A well-defined discretised theory
would provide non-perturbative data for analytic results and give insight into
the whole range of the AdS/CFT correspondence. However, such a discretisation
which preserves the symmetries, matches with continuum data and is
renormalisable is hard to come by. This section presents the analysis carried
out in [3] of the cusped Wilson line in $\mathcal{N}=4$ SYM studied in [223,
208, 209, 205], uncovering a discretisation which preserves more symmetry.
Despite this, the lattice model requires fine-tuning to have finite
quantities.
#### 6.2.1 Green-Schwarz AdS${}_{5}\times\text{S}^{5}$
The Green-Schwarz AdS${}_{5}\times\text{S}^{5}$ string is expected to be
defined at the non-perturbative level. A valid question is whether the non-
perturbative regime of the $\sigma$-model, which describes the
AdS${}_{5}\times~{}\text{S}^{5}$ string at tree-level in string perturbation
theory, is accessible through a lattice discretization of the worldsheet
(while target space remains continuous). This question is motivated by the
success of the lattice as a UV non-perturbative regulator of Quantum
Chromodynamics. This approach has been pioneered in [224, 223, 209, 225],
where a lattice-discretized version of $S_{\text{cusp}}$ has been introduced
and also used to perform Monte Carlo simulations.222Other lattice approaches
to AdS/CFT include [226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236,
237, 238, 239, 240, 241, 242], see also [243] and references therein. The goal
of this system is to have a benchmark for a case where analytic results are
still known to extend to systems where they are not.
#### 6.2.2 $U(1)\times SU(4)$ Invariant Discretization
In the framework of the AdS/CFT correspondence [140, 141], the expectation
value of a light-like cusped Wilson loop in $\mathcal{N}=4$ super Yang-Mills
is equal to the partition function of an open string propagating in
AdS${}_{5}\times\text{S}^{5}$ space and ending on the loop at the AdS
boundary. In practice, one writes
$\left\langle\mathcal{W}_{\text{cusp
}}\right\rangle=\int\mathcal{D}Y\mathcal{D}\Psi\,e^{-S_{\text{cusp}}(X_{\text{cl}}+Y,\Psi)}\equiv
e^{-\frac{f(g)}{8}V_{2}}\,.$ (6.15)
This gauge-fixed configuration in AdS${}_{5}\times$S5 has a superstring action
describing quantum fluctuations around this null-cusp background [244]
$\displaystyle S^{\text{cont}}_{\rm cusp}=g\int dtds$
$\displaystyle\Bigg{\\{}\left|\partial_{t}x+\tfrac{m}{2}x\right|^{2}+\tfrac{1}{z^{4}}\left|\partial_{s}x-\tfrac{m}{2}x\right|^{2}+\left(\partial_{t}z^{M}+\tfrac{m}{2}z^{M}+\tfrac{i}{z^{2}}z_{N}\eta_{i}\left(\rho^{MN}\right)_{\phantom{i}j}^{i}\eta^{j}\right)^{2}$
(6.16)
$\displaystyle+\tfrac{1}{z^{4}}\left(\partial_{s}z^{M}-\tfrac{m}{2}z^{M}\right)^{2}+i\,\left(\theta^{i}\partial_{t}\theta_{i}+\eta^{i}\partial_{t}\eta_{i}+\theta_{i}\partial_{t}\theta^{i}+\eta_{i}\partial_{t}\eta^{i}\right)-\tfrac{1}{z^{2}}\left(\eta^{i}\eta_{i}\right)^{2}$
$\displaystyle+2i\,\Big{[}\tfrac{1}{z^{3}}z^{M}\eta^{i}\left(\rho^{M}\right)_{ij}\left(\partial_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}-\tfrac{i}{z}\eta^{j}\left(\partial_{s}x-\tfrac{m}{2}x\right)\right)$
$\displaystyle\qquad+\tfrac{1}{z^{3}}z^{M}\eta_{i}\big{(}{\rho^{M}}^{\dagger}\big{)}^{ij}\left(\partial_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}+\tfrac{i}{z}\eta_{j}\left(\partial_{s}x-\tfrac{m}{2}x\right)^{*}\right)\Big{]}\,\Bigg{\\}}\,,$
where
* •
$x$ is a complex bosonic field whose real and imaginary parts parametrize the
fluctuations of the string (in light-cone gauge) at the boundary of AdS5
* •
$z^{M}$ are six real bosonic fields333$z=\sqrt{z^{M}z^{M}}$ is the radial
coordinate of the AdS5 space, while $u^{M}=z^{M}/z$ identifies points on
$S_{5}$.
* •
the Graßmann-odd fields
$\theta^{i}=(\theta_{i})^{\dagger},~{}\eta^{i}=(\eta_{i})^{\dagger},\,i=1,2,3,4$
are complex anticommuting variables (no Lorentz spinor indices appear);
* •
the matrices $(\rho^{MN})_{i}^{\hphantom{i}j}=(\rho^{[M}\rho^{\dagger
N]})_{i}^{\hphantom{i}j}$ are the $SO(6)$ generators. $\rho^{M}_{ij}$444By
convention, we will write the indices of $\rho$ as down and those of
$\rho^{\dagger}$ as up. are the (traceless) off-diagonal blocks of $SO(6)$
Dirac matrices $\gamma^{M}$ in chiral representation.
The massive parameter $m$ keeps track of the dimensionful light-cone momentum
$P_{+}$, set to 1 in [245]. The action (6.16) is invariant under a $U(1)\times
SU(4)$ global symmetry defined by
$\displaystyle z^{M}\to\text{Ad}(U)^{MN}z^{N}\ ,\quad\theta^{i}\to
U^{i}_{\phantom{i}j}\theta^{j}\ ,\quad\eta^{i}\to
U^{i}_{\phantom{i}j}\eta^{j}\ ,$ (6.17) $\displaystyle x\to e^{i\alpha}x\
,\quad\theta^{i}\to e^{i\alpha/2}\theta^{i}\ ,\quad\eta^{i}\to
e^{-i\alpha/2}\eta^{j}\ ,$ (6.18)
where $U$ is an element of $SU(4)$ and its representative in the adjoint,
$\text{Ad}(U)$, is an element of $SO(6)$. While the original Green-Schwarz
AdS${}_{5}\times\text{S}^{5}$ string action is invariant under diffeomorphisms
and $\kappa$-symmetry, these local symmetries have been fixed by the choice of
light-cone gauge in equation (6.16). Notice that the action is not invariant
under worldsheet rotations, parity ($s\rightarrow-s$), or time reversal
($t\rightarrow-t$).
To define the lattice-discretised theory, we need to provide a discretised
action and an explicit expression for the measure. We choose to use a flat
measure for the fields but we keep in mind that this choice is quite arbitrary
as it is not invariant under reparametrisation of the target AdS${}_{5}\times
S_{5}$ target space. Given a generic observable $A$, expectation values in the
lattice discretised theory are defined by
$\langle A\rangle=\frac{1}{Z_{\text{cusp}}}\int dxdx^{*}d^{6}zd^{4}\theta
d^{4}\theta^{\dagger}d^{4}\eta d^{4}\eta^{\dagger}\,e^{-S_{\text{cusp}}}A\,,$
(6.19)
where $df\equiv\prod_{s,t}df(s,t)$ is the discretised measure. The partition
function $Z_{\text{cusp}}$ is fixed by the requirement $\langle 1\rangle=1$,
and $S_{\text{cusp}}$ refers now to the discretised action that we choose to
be
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\rm cusp}=g\sum_{s,t}a^{2}$
$\displaystyle\\!\\!\\!\\!\\!\Bigg{\\{}\\!\\!\left|b_{+}\hat{\partial}_{t}x+\tfrac{m}{2}x\right|^{2}\\!\\!+\tfrac{1}{z^{4}}\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}\\!\\!+\big{(}b_{+}\hat{\partial}_{t}z^{M}+\tfrac{m}{2}z^{M}+\tfrac{i}{z^{2}}z^{N}\eta_{i}(\rho^{MN})_{\phantom{i}j}^{i}\eta^{j}\big{)}^{2}$
(6.20)
$\displaystyle+\tfrac{1}{z^{4}}\big{(}\hat{\partial}_{s}z^{M}\hat{\partial}_{s}z^{M}+\tfrac{m^{2}}{4}z^{2}\big{)}+2i\,\big{(}\theta^{i}\hat{\partial}_{t}\theta_{i}+\eta^{i}\hat{\partial}_{t}\eta_{i}\big{)}-\tfrac{1}{z^{2}}\left(\eta^{i}\eta_{i}\right)^{2}$
$\displaystyle+2i\,\Big{[}\tfrac{1}{z^{3}}z^{M}\eta^{i}\left(\rho^{M}\right)_{ij}\big{(}b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}-\tfrac{i}{z}\eta^{j}\big{(}b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\big{)}\big{)}$
$\displaystyle\qquad+\tfrac{1}{z^{3}}z^{M}\eta_{i}\big{(}{\rho^{M}}^{\dagger}\big{)}^{ij}\big{(}b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}+\tfrac{i}{z}\eta_{j}\big{(}b_{-}\hat{\partial}_{s}x^{*}-\tfrac{m}{2}x^{*}\big{)}\\!\big{)}\\!\Big{]}\Bigg{\\}}\,.$
The action is written in terms of the forward and backward discrete
derivatives
$\hat{\partial}_{\mu}f(\sigma)\equiv\frac{f\left(\sigma+ae_{\mu}\right)-f\left(\sigma\right)}{a}\,,\qquad\bar{\partial}_{\mu}f(\sigma)\equiv\frac{f\left(\sigma\right)-f\left(\sigma-
ae_{\mu}\right)}{a}\,$ (6.21)
where $e_{\mu}$ is the unit vector in the direction $\mu=0,1$, and $\sigma$ is
a shorthand notation for $(s,t)$.
Notice that the proposed discretized action (6.20) depends on four parameters:
$g$, $m$, and the auxiliary parameters $b_{\pm}$. The discretized action
$S_{\rm cusp}$ reduces to the desired continuum action $S^{\text{cont}}_{\rm
cusp}$ in the naïve $a\to 0$ limit if $b_{\pm}\to 1$. However, as we will
discuss in detail, the naïve choice $b_{\pm}=1$ produces undesired UV
divergences at one loop. The values of $b_{\pm}$ need to be tuned so that
these UV divergences cancel. This is a sign that the lattice regulator does
not manage to reproduce the cancellation of UV divergences that occurs in
dimensional regularisation.
An important feature of the proposed discretised action and measure is that
they are invariant under the full $U(1)\times SU(4)$ internal symmetry group.
This is in contrast to the discretisation previously presented in [209]. The
key ingredient is the use of forward and backward discrete derivatives for
both the bosonic and fermionic parts of the action. This is normally avoided
for fields that satisfy first-order equations of motion (usually fermions)
since it breaks parity and time reversal. This is not an issue in our case
because these symmetries are already broken in the continuum action. In [209],
instead, the symmetric derivative was used and, as in lattice QCD, a Wilson-
like term was included to solve the resulting doubling problem while breaking
either the $U(1)$ or the $SU(4)$ symmetry. The perturbative series is obtained
on the lattice as in the continuum by expanding the action around its minima.
The $SU(4)$ symmetric point is a singularity for the action because of the
terms proportional to inverse powers of the radial coordinate $z$.555All
fields vanish in this point. Consequently, the minimum of the action
spontaneously breaks the internal symmetry. In the continuum, an absolute
minimum of the action is given by $x=x^{*}=0$ and $z^{M}=\delta^{M6}$, and any
other absolute minimum is obtained by acting with the $SU(4)$ symmetry. One
can easily check that these minima are relative minima for the discretised
action. We parametrise the fluctuations around the chosen minimum in the same
way as it is done in the continuum [244]
$z=e^{\phi}\,,\qquad z^{a}=e^{\phi}\frac{y^{a}}{1+\frac{1}{4}y^{2}}\,,\qquad
z^{6}=e^{\phi}\frac{1-\frac{1}{4}y^{2}}{1+\frac{1}{4}y^{2}}\,,\qquad
y^{2}=\sum_{a=1}^{5}(y^{a})^{2},\,$ (6.22)
where $a=1,\dots,5$. In terms of the new variables $\phi$ and $y^{a}$, the
path-integral measure over the $z^{M}$ fields reads
$\displaystyle\prod_{M=1}^{6}dz^{M}=e^{\sum_{s,t}\left\\{6\phi+5\log\left(1+\frac{y^{2}}{4}\right)\right\\}}\,d\phi\prod_{a=1}^{5}dy^{a}\,.$
(6.23)
The contribution of the Jacobian determinant above can be conveniently
included in the effective action
$\displaystyle
S_{\text{eff}}=S_{\text{cusp}}-\sum_{s,t}\left\\{6\phi+5\log\left(1+\frac{y^{2}}{4}\right)\right\\}\,,$
(6.24)
in terms of which the expectation values of observables are
$\langle A\rangle=\frac{1}{Z_{\text{eff}}}\int dxdx^{*}d\phi d^{5}yd^{4}\theta
d^{4}\theta^{\dagger}d^{4}\eta d^{4}\eta^{\dagger}\,e^{-S_{\text{eff}}}A\,.$
(6.25)
Notice that the sum in (6.23) does not come with the corresponding $a^{2}$
factor, which means that in the naïve continuum limit it diverges like
$a^{-2}$. This should not be surprising: in the continuum, this term would be
proportional to $\delta^{2}(0)$ which yields a quadratic divergence in a hard-
cutoff regularization (but it is set to zero in dimensional regularization).
The perturbative expansion, i.e. the expansion in powers of $g$, is obtained
by splitting the action $S_{\text{eff}}=S_{0}+S_{\text{int}}$, where $S_{0}$
contains all quadratic terms in the fields with a coefficient proportional to
$g^{-1}$, and $S_{\text{int}}$ contains all other terms. Notice that
$S_{\text{int}}$ also contains $g$-independent quadratic terms, which come
from expanding the Jacobian determinant. We focus here on the leading-order
quadratic action
$\displaystyle S_{0}=g\,a^{2}\sum_{s,t}\Bigg{\\{}$
$\displaystyle\left|b_{+}\hat{\partial}_{t}x+\tfrac{m}{2}x\right|^{2}+\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}$
(6.26)
$\displaystyle+b_{+}^{2}(\hat{\partial}_{t}y^{a})^{2}+mb_{+}y^{a}\hat{\partial}_{t}y^{a}+(\hat{\partial}_{s}y^{a})^{2}$
$\displaystyle+b_{+}^{2}(\hat{\partial}_{t}\phi)^{2}+mb_{+}\phi\hat{\partial}_{t}\phi+(\hat{\partial}_{s}\phi)^{2}+m^{2}\phi^{2}+2i\left(\theta^{i}\hat{\partial}_{t}\theta_{i}+\eta^{i}\hat{\partial}_{t}\eta_{i}\right)$
$\displaystyle+2i\eta^{i}(\rho^{6})_{ij}\left(b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}\right)+2i\eta_{i}({\rho^{6}}^{\dagger})^{ij}\left(b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}\right)\Bigg{\\}}\,.$
#### 6.2.3 One-Point Functions
Let us turn to the one-point functions of the perturbative fields. Notice that
$\langle x\rangle=0$ because of the $U(1)$ symmetry, and $\langle
y^{a}\rangle=0$ because of the $SO(5)\subset SO(6)\simeq SU(4)$ which leaves
the perturbative vacuum invariant. $\phi$ is the only field with a non-
vanishing one-point function, which has been calculated in dimensional
regularisation [245, 244, 246]. This one-point function and any $n$-point
function of bare fields are not expected to be UV finite. It is known that
$\langle\phi\rangle$ is UV divergent in dimensional regularisation, and we
will see that it turns out to be UV divergent also in the lattice
regularisation. The interest in this one-point function lies in the fact that
it appears as a sub-diagram in any other $n$-point function, and ultimately
its UV divergence contributes to any physical observable. We will give an
example of this mechanism in the next subsection.
There are two classes of vertices contributing to the one-point function of
$\phi$: single-field vertices coming from the measure
$\displaystyle S_{\phi}=-6\sum_{s,t}\phi\ ,$ (6.27)
and three-field vertices coming from the action
$\displaystyle S_{\phi\bullet\bullet}$ $\displaystyle=$ $\displaystyle
g\sum_{s,t}a^{2}\bigg{\\{}-4\phi\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}+c_{+}\hat{\partial}_{t}\phi\hat{\partial}_{t}(\phi^{2})+\hat{\partial}_{s}\phi\hat{\partial}_{s}\phi^{2}-4\phi(\hat{\partial}_{s}\phi)^{2}$
(6.28) $\displaystyle\hskip
45.5244pt+2c_{+}\hat{\partial}_{t}y^{a}\hat{\partial}_{t}(\phi
y^{a})-c_{+}\hat{\partial}_{t}\phi\hat{\partial}_{t}(y^{2})+2\hat{\partial}_{s}y^{a}\hat{\partial}_{s}(\phi
y^{a})-\hat{\partial}_{s}\phi\hat{\partial}_{s}(y^{2})-4\phi(\hat{\partial}_{s}y^{a})^{2}$
$\displaystyle\hskip
45.5244pt-4i\phi\left[\eta^{i}(\rho^{6})_{ij}\left(b_{+}\bar{\partial}_{s}\theta^{j}-\tfrac{m}{2}\theta^{j}\right)+\eta_{i}({\rho^{6}}^{\dagger})^{ij}\left(b_{+}\bar{\partial}_{s}\theta_{j}-\tfrac{m}{2}\theta_{j}\right)\right]\bigg{\\}}\
.$
Notice that the insertion of $S_{\phi}$ produces a tree-level diagram, while
the insertion of $S_{\phi\bullet\bullet}$ produces a one-loop diagram.
However, because of the mismatch in the power of $g$ in $S_{\phi}$ and
$S_{\phi\bullet\bullet}$, all these diagrams contribute to the same order in
$g$, yielding
$\displaystyle\langle\phi\rangle$ $\displaystyle=$
$\displaystyle\frac{3}{gm^{2}a^{2}}+\frac{2}{gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{-}|\hat{q}_{1}|^{2}+\tfrac{m^{2}}{4}}{c_{+}|\hat{q}_{0}|^{2}+c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{2}}$
(6.29)
$\displaystyle-\frac{1}{2gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{0}|^{2}-|\hat{q}_{1}|^{2}}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}+m^{2}}-\frac{5}{2gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{0}|^{2}-|\hat{q}_{1}|^{2}}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}}$
$\displaystyle-\frac{8}{gm^{2}}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{|\hat{q}_{0}|^{2}+c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}+O(g^{-2})\
.$
With the special choice $b_{\pm}=\bar{b}_{\pm}$, i.e. $c_{\pm}=1$, one can use
the symmetry of the integrals under $p_{0}\leftrightarrow p_{1}$ exchange to
simplify
$\displaystyle\langle\phi\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{g}\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+\frac{m^{2}}{4}}+O(g^{-2})$
(6.30) $\displaystyle=$
$\displaystyle\frac{1}{g}\left\\{\frac{1}{4\pi}\log\frac{(am)^{2}}{4}+\frac{1}{4\pi}-I_{0}^{(0,0)}+O(a\log
a)\right\\}+O(g^{-2})\ ,$
which is logarithmically divergent. The definition of the numerical constant
$I_{0}^{(0,0)}\simeq 0.355$ is given in the appendix of [3]. Notice that the
measure, fermion-loop and $x$-loop contributions are separately quadratically
divergent, and the cancellation of these divergences is highly non-trivial.
In the general case $c_{\pm}=1+(am)\delta c_{\pm}$ where $\delta
c_{\pm}=O(a^{0})$ and, after a lengthy calculation, one gets
$\displaystyle\langle\phi\rangle=\frac{1}{g}\bigg{\\{}\frac{-8\delta
c_{+}+\delta c_{-}}{\pi
a}+\frac{1}{4\pi}\log\frac{(am)^{2}}{4}+\frac{1}{4\pi}-I_{0}^{(0,0)}+\frac{8\delta
c_{+}^{2}-\delta c_{-}^{2}}{2\pi}+O(a\log a)\bigg{\\}}+O(g^{-2})\ .$ (6.31)
Notice that the naïve choice $b_{\pm}=1$ corresponds to the choice $\delta
c_{\pm}=\mp 1/2$ which yields indeed a linear divergence for
$\langle\phi\rangle$:
$\displaystyle\langle\phi\rangle=\frac{1}{g}\left\\{\frac{9}{2\pi a}+O(\log
a)\right\\}+O(g^{-2})\ .$ (6.32)
#### 6.2.4 Two-Point Function
We turn now to the two-point function of the field $x$, which we calculate at
one loop. We will use the two-point function to extract the dispersion
relation of the $x$ particle propagating on the worldsheet. In dimensional
regularisation and at one loop [244], both the two-point function and the
dispersion relation are UV finite without the need for renormalisation. We
will also see that this is true at one loop in lattice perturbation theory,
provided that one has chosen $c_{\pm}=1$. The naïve choice $b_{\pm}=1$
generates UV divergences in the dispersion relation. A valid question is
whether these divergences can be eliminated with a renormalisation procedure.
There are two classes of vertices contributing to the two-point function of
$x$ at one loop: three-field vertices
$\displaystyle S_{xx^{*}\bullet}$ $\displaystyle=$ $\displaystyle
g\sum_{s,t}a^{2}\bigg{\\{}-4\phi\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}$
(6.33) $\displaystyle\hskip
51.21495pt+2\eta^{i}\rho^{6}_{ij}\eta^{j}\left(b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right)-2\eta_{i}({\rho^{6}}^{\dagger})^{ij}\eta_{j}\left(b_{-}\hat{\partial}_{s}x^{*}-\tfrac{m}{2}x^{*}\right)\bigg{\\}}\
,$
and four-field vertices
$\displaystyle
S_{xx^{*}\bullet\bullet}=8g\sum_{s,t}a^{2}\phi^{2}\left|b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right|^{2}\
,$ (6.34)
combined to give Feynman diagrams with the three different topologies
illustrated in Figure 6.1. Notice that the tadpole contribution will be
proportional to $\langle\phi\rangle$.
Figure 6.1: Topologies of diagrams contributing to the two-point function
$\langle xx^{*}\rangle$ at one-loop in the discretized model in (6.20) .
On general grounds, one sees that the two-point function has the following
form
$\\!\\!\\!\\!\\!\langle\tilde{x}(p)x^{*}(0)\rangle=\frac{1}{g}\left\\{c_{+}|\hat{p}_{0}|^{2}+c_{-}|\hat{p}_{1}|^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(c_{-}|\hat{p}_{1}|^{2}+\frac{m^{2}}{4}\right)\Pi_{a}(p)+O(g^{-2})\right\\}^{-1}\
.$ (6.35)
The factor $\left(c_{-}|\hat{p}_{1}|^{2}+\tfrac{m^{2}}{4}\right)$ comes from
the fact that, in all interaction vertices, $x$ always appears in the
combination $\left(b_{-}\hat{\partial}_{s}x-\tfrac{m}{2}x\right)$ or its
complex conjugate. The function $\Pi_{a}(p)$ has a representation in terms of
amputated Feynman diagrams, and it is explicitly given by
$\displaystyle\Pi_{a}(p)$ $\displaystyle=$
$\displaystyle-4g\langle\phi\rangle+4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{c_{+}|\hat{q}_{0}|^{2}+|\hat{q}_{1}|^{2}+m^{2}}$
(6.36)
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{c_{+}|\hat{q}_{0}|^{2}+c_{-}|\hat{q}_{1}|^{2}+\frac{m^{2}}{2}}\frac{1}{c_{+}|\widehat{p+q}_{0}|^{2}+|\widehat{p+q}_{1}|^{2}+m^{2}}$
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{\hat{q}_{0}}{|\hat{q}_{0}|^{2}+c_{+}|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}\frac{\widehat{p+q}_{0}^{*}}{|\widehat{p+q}_{0}|^{2}+c_{+}|\widehat{p+q}_{1}|^{2}+\frac{m^{2}}{4}}\
.$
All integrals in the above formula are logarithmically divergent, while the
term proportional to $\langle\phi\rangle$ generally contains a linear
divergence. Up to terms that vanish in the $a\to 0$ limit, one can replace
$c_{\pm}=1$ in the above integrals, obtaining the simpler expression
$\displaystyle\Pi_{a}(p)$ $\displaystyle=$
$\displaystyle-4g\langle\phi\rangle+4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+m^{2}}-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{|\hat{q}_{1}|^{2}+\frac{m^{2}}{4}}{|\hat{q}|^{2}+\frac{m^{2}}{2}}\frac{1}{|\widehat{p+q}|^{2}+m^{2}}$
(6.37)
$\displaystyle-8\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{\hat{q}_{0}}{|\hat{q}|^{2}+\frac{m^{2}}{4}}\frac{\widehat{p+q}_{0}^{*}}{|\widehat{p+q}|^{2}+\frac{m^{2}}{4}}+O(a\log
a)\ .$
As in the continuum, the leading divergence of the above integrals does not
depend on the external momentum, therefore the subtracted quantity
$\Delta\Pi_{a}(p)=\Pi_{a}(p)-\Pi_{a}(0)$ has a finite $a\to 0$ limit given by
the corresponding continuum integrals, i.e.
$\displaystyle\Delta\Pi_{0}(p)$ $\displaystyle=$
$\displaystyle-8\int_{-\infty}^{\infty}\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{1}^{2}+\frac{m^{2}}{4}}{q^{2}+\frac{m^{2}}{2}}\left\\{\frac{1}{(p+q)^{2}+m^{2}}-\frac{1}{q^{2}+m^{2}}\right\\}$
(6.38)
$\displaystyle-8\int_{-\infty}^{\infty}\frac{d^{2}q}{(2\pi)^{2}}\frac{q_{0}}{|\hat{q}|^{2}+\frac{m^{2}}{4}}\left\\{\frac{p_{0}+q_{0}}{(p+q)^{2}+\frac{m^{2}}{4}}-\frac{q_{0}}{q^{2}+\frac{m^{2}}{4}}\right\\}+O(a\log
a)\ ,$
while all the divergences are contained in
$\displaystyle\Pi_{a}(0)=-4g\langle\phi\rangle-4\int_{-\pi/a}^{\pi/a}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{|\hat{q}|^{2}+\frac{m^{2}}{4}}+\frac{1}{\pi}+O(a\log
a)\ ,$ (6.39)
where we have used the symmetry of the integrals under $p_{0}\leftrightarrow
p_{1}$ exchange to simplify them.
With the choice $c_{\pm}=1$, using equation (6.30) one immediately sees that
all divergences cancel and $\Pi_{0}(0)=1/\pi$. The two-point function is
finite in the continuum limit and
$\displaystyle\lim_{a\to
0}\langle\tilde{x}(p)x^{*}(0)\rangle=\frac{1}{g}\left\\{p^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\Pi_{0}(p)+O(g^{-2})\right\\}^{-1}.$
(6.40)
The two-point function has poles at $p_{0}=\pm iE(p_{1})$ for every value of
$p_{1}$, where $E(p_{1})$ is the energy of a single excitation with the
quantum numbers of the field $x$, propagating on the worldsheet with momentum
$p_{1}$. In the continuum limit, this is found to be
$\displaystyle E(p_{1})^{2}$ $\displaystyle=$ $\displaystyle
p_{1}^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\Pi_{0}\left(\sqrt{p_{1}^{2}+\frac{m^{2}}{2}},p_{1}\right)+O(g^{-2})$
(6.41) $\displaystyle=$ $\displaystyle
p_{1}^{2}+\frac{m^{2}}{2}-\frac{1}{gm^{2}}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)^{2}+O(g^{-2})\
,$
where we have used the on-shell value of $\Pi_{0}$. The obtained dispersion
relation coincides with the result in [244].666When comparing to [244], notice
that one has to redefine the worldsheet coordinates, resulting in square
masses of the fluctuations rescaled by a factor of four.
However in the general case $c_{\pm}=1+(am)\delta c_{\pm}$ where $\delta
c_{\pm}=O(a^{0})$, $\Pi_{a}(0)$ and $E(p_{1})$ inherit the linear divergence
from $\langle\phi\rangle$. Using equation (6.31) one obtains
$\displaystyle\Pi_{a}(0)=\frac{32\delta c_{+}-4\delta c_{-}}{\pi
a}+\frac{1-16\delta c_{+}^{2}+2\delta c_{-}^{2}}{\pi}+O(a\log a)\ .$ (6.42)
For instance, for the naïve choice $b_{\pm}=1$, which corresponds to $\delta
c_{\pm}=\mp 1/2$, one obtains for the dispersion relation
$\displaystyle
E(p_{1})^{2}=p_{1}^{2}+\frac{m^{2}}{2}+\frac{1}{g}\left(p_{1}^{2}+\frac{m^{2}}{4}\right)\left[-\frac{18}{\pi
a}+O(\log a)\right]+O(g^{-2})\ .$ (6.43)
It is interesting to notice that once we have set $b_{\pm}=1$, the divergence
in the dispersion relation cannot be eliminated by renormalising the remaining
available parameters, i.e. $g$ and $m$. In other words, the choice $b_{\pm}=1$
is not stable under renormalisation. On the other hand, if one allows the
coefficients $b_{\pm}$ to be renormalised along with $m$ and $g$, then the
divergences in the dispersion relation are eliminated, e.g. by choosing
$\displaystyle
b_{+}=1+\frac{1}{g_{R}}\frac{\frac{a\,m_{R}}{8}}{2+\frac{a\,m_{R}}{2}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\
,$ (6.44) $\displaystyle
b_{-}=1-\frac{1}{g_{R}}\frac{1+\frac{5a\,m_{R}}{8}}{2+\frac{a\,m_{R}}{2}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\
,$ (6.45) $\displaystyle
m^{2}=m_{R}^{2}\left[1+\frac{1}{2g_{R}}\left(\Pi_{a}(0)-\frac{1}{\pi}\right)\right]\
,$ (6.46) $\displaystyle g=g_{R}\left[1+O(g^{-1})\right]\ .$ (6.47)
This choice yields a dispersion relation in the continuum limit of the same
form as equation (6.41), except that the mass $m$ needs to be replaced by its
renormalised counterpart $m_{R}$. One could also see that the one-loop
renormalisation of the coupling constant can be chosen so that the cusp
anomaly is finite. With this discussion, we do not want to imply that the
chosen lattice theory is renormalisable (we do not know this). However, we
conclude that if the lattice theory is renormalisable, then it is not
sufficient to renormalize $m$ and $g$; one also needs to introduce additional
coefficients in the action and either fine-tune their tree-level value or
renormalise them.
## Chapter 7 Conclusions and Outlook
In the first Chapters (2, 3, 4), the analytic conformal bootstrap was
introduced and applied to two examples: the four-point correlator of operator
insertions on the 1/2-BPS Wilson line in ABJM and the five-point correlator of
operator insertions on the 1/2-BPS Wilson line in $\mathcal{N}=4$ SYM. In the
first example, the superspace structure was developed in order to relate all
the correlators of fields in the displacement multiplet to a single function
$f(z)$. This function was then bootstrapped entirely up to third order in a
strong coupling expansion and to all orders in the double-scaling limit. The
mixing problem for the CFT data was solved at first order to obtain these
results. The CFT data (generically mixed) was then extracted up to third
order.
In the second example, multipoint Ward identities and selection rules were
used to compute the superblock expansion of the five-point correlator of
1/2-BPS operators inserted on the 1/2-BPS Wilson line in $\mathcal{N}=4$ SYM.
This allows for the computation of the first-order correlator through the
analytic conformal bootstrap. These higher-point correlators access a wider
range of CFT data, which were then extracted. The analysis of the different
channels confirms that the anomalous dimension does not contribute to mixing
at this order.
Extensions to this work fall into two parallel categories: bootstrapping
correlators and improving the constraints from the bootstrap. One can easily
find more results to bootstrap; by increasing the order of perturbation, the
number of insertions, or by considering different setups. Higher-order
contributions increase in complexity due to the mixing of operators and
somewhat because of the type of functions appearing in the ansatz. The higher-
point correlators increase in complexity because of the wealth of possible
operators exchanged in the many OPEs. Systems with lower supersymmetry will
have fewer constraints from the corresponding selection rules. As such, these
three extensions will have upper limits as to what can be fixed by the
symmetry.
When looking at improving the bootstrap, many constraints have not yet been
explored. The first category are additional constraints from the symmetries of
the theory. For example, integrated correlator constraints relate the
$n$-point results to functions known through integrability and should exist
for most superconformal defects and for increasing number of insertions [196].
Integrability has also been extremely powerful in determining the spectrum of
the 1/2-BPS setup in $\mathcal{N}=4$ SYM[149]. This symmetry should also
constraint the perturbative correlators through Yangian symmetry. The second
category covers more general defects setups and can be explored in toy models.
Three examples are the constraints from single-valuedness, from the flat-space
limit and from non-perturbative sum rules.
Looking forwards, an immediate extension is the unmixing of the CFT data at
higher orders, which would enable the computation of the correlators at the
next orders in perturbation theory. One way to do this is to look at more
general correlators, such as was done to solve the first-order mixing in the
1/2-BPS Wilson line in ABJM. A particular family of interest in this case are
the 1/3-BPS operators, which are charged under R-symmetry. These form an
infinite tower of protected operators which may have a topological limit.
Additionally, these operators preserve the same symmetries as the 1/2-BPS
operators on the 1/3-BPS Wilson line, whose parallel computation would be
enlightening. A direct extension of the 5-point bootstrap which is underway is
|
# Boosting Graph Search with Attention Network for Solving the General
Orienteering Problem
Jing Xu State Grid Corporation of China<EMAIL_ADDRESS>Jintao Su Zhejiang
University<EMAIL_ADDRESS>Tao Xiao State Grid Corporation of China
<EMAIL_ADDRESS>Zongtao Liu1 Jing Xu2 Jintao Su1 Tao Xiao2&Yang Yang1
1Zhejiang University
2State Grid Corporation of China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Recently, several studies have explored the use of neural network to solve
different routing problems, which is an auspicious direction. These studies
usually design an encoder-decoder based framework that uses encoder embeddings
of nodes and the problem-specific context to produce node sequence(path), and
further optimize the produced result on top by beam search. However, existing
models can only support node coordinates as input, ignore the self-referential
property of the studied routing problems, and lack the consideration about the
low reliability in the initial stage of node selection, thus are hard to be
applied in real-world.
In this paper, we take the orienteering problem as an example to tackle these
limitations. We propose a novel combination of a variant beam search algorithm
and a learned heuristic for solving the general orienteering problem. We
acquire the heuristic with an attention network that takes the distances among
nodes as input, and learn it via a reinforcement learning framework. The
empirical studies show that our method can surpass a wide range of baselines
and achieve results close to the optimal or highly specialized approach. Also,
our proposed framework can be easily applied to other routing problems. Our
code is publicly
available111https://anonymous.4open.science/repository/7cb20ede-b50e-4a9a-99a5-e3c3626bf1a1/.
## 1 Introduction
The orienteering problem(OP) is an important routing problem that originates
from the sports game of orienteering. In this game, each competitor starts at
a specific control point, visits control points in the tour, and will arrive
at a destination point (usually the same as the start point). Each control
point has an associated prize, and the travel between control points involves
a certain cost. The goal is to select a sequence of points such that the total
prize is maximized within the cost constraint. This problem relates to several
practical applications, such as resource delivery, tourist trip guide and
single-ring design in telecommunication networks(Golden et al. (1987);
Souffriau et al. (2008); Thomadsen and Stidsen (2003)). Golden et al. (1987)
have shown the OP to be an NP-hard problem, which motivates vigorous research
into the design of heuristic solvers and approximation algorithms.
A recent trend in the machine learning community for solving routing problems
is using deep neural networks to learn heuristic algorithms. With the help of
other meta-algorithm such as beam search, some of these methods can achieve
near start-of-the-art performance on several tasks (Li et al. (2018); Kool et
al. (2018); Vinyals et al. (2015); Khalil et al. (2017); Nazari et al.
(2018)). These tasks include the traveling salesman problem(TSP) and the
orienteering problem(OP), and the vehicle routing problem(VRP). These studies
deem solving such problems as a sequence generation process from the
coordinates of nodes, and use recurrent neural networks(RNNs) or graph neural
networks(GNNs) to build a bridge between node patterns and the policy. The OP
belongs to the routing problem family and thus is fit for this methodology.
However, current learning-based algorithms have several limitations. Firstly,
existing methods do not support direct input of travel cost information. They
implicitly assume the travel cost, saying travel distance here, between each
pair of points is their Euclidean distance, and take the coordinate of each
position as network input to indirectly acquire the distance information.
However, the distance metric used in real-world is not limited to the
Euclidean distance, thus such kind of methods has limited practicability. For
the term convenience, we here define the OP which only supports coordinates as
input as the coordinate orienteering problem, and define the OP that supports
direct input of costs among locations as the general orienteering problem.
Secondly, the routing problems considered in the previous works have self-
referential property, but these works lack insight into it. For instance, in
the orienteering problem, each step of point selection can be viewed as the
initial step in another distinct orienteering problem, in which the competitor
starts from the last selected points and will choose from the rest points.
Lacking consideration about such property in the existing studies might
degrade the utilization of samples when training.
Besides, most existing studies usually rely on a step-by-step beam search to
acquire a better solution (Kaempfer and Wolf (2018); Vinyals et al. (2015);
Nowak et al. (2017)); however, it might not work well on the routing problem
with relative large problem size. The step-by-step beam search at each step
prunes non-promising partial solutions based on a heuristic function and
stores a fix-sized set (also called beam) of best alternatives. However, as
the difficulty in the early stage of a routing problem is much more than that
in the later stage, the estimated heuristic score (probability or action
value) to perform beam search in the initial step might also be less credible.
To overcome this shortcoming, the stored partial solutions in the early stage
of a search procedure should be more than those in the later stage (since the
less credible heuristic scores might mislead the pruning). How to design such
a search algorithm is an important issue.
In this paper, we therefore propose a novel combination of a variant beam
search algorithm and the learned heuristic for solving the general
orienteering problem. We acquire the heuristic with an attention network that
takes both node attribute (prize) and edge attribute (cost) as input, and
learn it via a reinforcement learning framework. Rather than formulating such
routing tasks as sequence generation, we instead consider each step of point
selection separately; that is,we view each state in the decision process as
another different OP. This insight helps provide more direct reinforcement in
each state and improves the utilization of training instances. The
experimental result shows that our method can surpass a wide range of
baselines and achieve results close to the optimal or highly specialized
approach.
## 2 Related Work
Orienteering problem. Over the years, many challenging applications in
logistics, tourism, and other fields were modeled as OP. Meanwhile, quite a
few studies for orienteering problems have been conducted. Golden et al.
(1987) prove that the OP is NP-hard, i.e., no polynomial-time algorithm has
been designed or is expected to be developed to solve this problem to
optimality. Thus the exact solution algorithms are very time consuming, and
for practical applications, heuristics will be necessary. Exact solution
methods (using branch-and-bound and branch-and-cut ideas) have been presented
by Laporte and Martello (1990); Ramesh et al. (1992). Heuristic approaches,
which use traditional operation research techniques, have been developed by
Tsiligirides (1984); Gendreau et al. (1998); Tang and Miller-Hooks (2005).
Applications of neural networks in routing problems. Using neural networks
(NNs) for optimizing decisions in routing problems is a long-standing
direction, which can date back to the early work (Hopfield and Tank (1985);
Wang et al. (1995)). These studies usually design an NN and learn a solution
for each instance in an online fashion. Recent studies focus on using (D)NNs
to learn about an entire class of problem instances. The critical point of
these studies is to model permutation-invariant and variable-sized input
appropriately.
Earlier researches leverage sequence-to-sequence models and the attention
mechanism to produce path, and update network parameters via supervised or
reinforcement learning (Vinyals et al. (2015); Bello et al. (2016); Nazari et
al. (2018)). More recently, several studies leverage the power of GNNs that
can handle variable-sized and order-independent input to tackle the routing
problems (Khalil et al. (2017); Nowak et al. (2017); Kool et al. (2018)).
However, these studies only focus on coordinate routing problem; thus the
practicability is limited in real-word. Besides, current methods usually use a
set-to-sequence/node-to-sequence model to perform routing, which ignores the
self-referential property of many routing problems (including the OP).
## 3 Problem Definition
Formally, we define each orienteering problem instance $s$ as a tuple with six
elements $\langle V,C,R,T,v_{start},v_{end}\rangle$. More specifically,
$V=\\{v_{1},...,v_{M}\\}$ refers to the set of nodes (control points)
containing the start node $v_{start}$, the prized nodes, and the end node
$v_{end}$, where $M$ denotes the number of nodes. $C$ refers to the cost map
where $C_{v_{i},v_{j}}$ represents the travel cost from $v_{i}$ to $v_{j}$.
$R$ is the prize map where $R_{v}$ indicates the prize collected when $v$ is
visited. Therefore, the problem is to find a path from $v_{start}\in V$ to
$v_{end}\in V$ such that the total cost of the path is less than the
prescribed cost limit $T$, and the overall prize collected from the nodes
visited is maximized. The orienteering problem can also be formalized as a
mixture integer problem, which can be referenced in Gunawan et al. (2016).
For notation convenience, here we introduce some operators or functions among
these elements. $cost(P)$, $prize(P)$, and $last(P)$ represent the total cost,
the total prize, and the last selected point of the (partial) selected path
$P$, respectively. $(P+v_{i})$ represents a selected path obtained by adding a
new node $v_{i}$ to the original path $P$. $(V-P)$ represents the node set
that excludes the nodes in $P$ from $V$.
## 4 Method
### 4.1 Cost-Level Beam Search with the Learned Heuristic
Before introducing the search method proposed in this study, we first present
an exact search method. We divide the maximal cost $T$ into $\lceil
T/\tau\rceil$ intervals with the length of $\tau$. For each range, we maintain
a queue to save all partial paths(solutions) with the total cost in that
interval . Starting from interval $t=0$, we iteratively retrieve the
incomplete solution $P$ from the front of the queue and scan all the nodes
that are unselected. For each node $v$ scanned, if $cost(P+v+v_{end})$ is no
greater than the cost limit $T$, we then add the updated partial path $(P+v)$
to the queue in interval $(t+\lceil C_{last(P),v}/\tau\rceil)$. Finally, from
all stored paths with the end node, the path with the highest total prize is
the optimal path.
This method cannot finish computation in polynomial time. As the value of $t$
increases, the number of stored partial paths per window increases
exponentially. To reduce the space and time occupied by the search, at each
step of the iteration, it is practical to prune some partial paths based on a
predefined heuristic score $f(P,s)$ with the input of the current partial
solution $P$ and the problem instance $s$. To follow up this idea, we apply
the idea of beam search and replace the queue maintained in each window $t$ by
a priority queue sized $K$, which is used to save the paths with the K highest
heuristic scores in each window. We introduce a data-driven heuristic score
function $f$ to estimate the total prize that can be reached along the partial
path $P$:
$f(P,s)=prize(P)+e(s^{\prime};\theta)$ (1)
where $f$ consists of two terms, in which the term $prize(P)$ computes the
total prize of the given partial path, and the term $e(s^{\prime};\theta)$ is
an estimation of the subsequent prize along the current path under $\pi$ till
the end of the problem. $e(\cdot~{};\theta)$ is an estimation function
parameterized $\theta$ that computes the total potential prize given by an OP
under the policy $\pi_{\theta}$ and $s^{\prime}$ represents the subproblem
$\langle V-P,C,S,T-cost(P),last(P),v_{end}\rangle$ of the original OP.
A question that naturally arises here is how to obtain a reliable function
$e(\cdot~{};\theta)$ to estimate the total prize of an orienteering problem.
In the next section, we will present our solution with the attention-based
neural networks learned by Q-learning.
Algorithm 1 Cost-Level Beam Search
1: initialize a list of priority queues PQ[$\lceil T/\tau\rceil$]
2: initialize the empty path $P_{g}$ and insert it to PQ[0]
3: for t=0 to $\lceil T/\tau\rceil$ do
4: while PQ[t] is not empty do
5: pop partial path $P_{c}$ from PQ[t]
6: for v in $(V-P_{c})$ do
7: pq = PQ[$\lceil cost(P_{c}+v)/\tau\rceil$]
8: if $cost(P_{c}+v+v_{end})>=T$ then continue
9: insert $(P_{c}+v)$ to pq
10: if $prize(P_{c}+v)>prize(P_{g})$ then $P_{g}=(P_{c}+p)$
11: if the size of pq is larger than K then
12: pop $argmin_{P}f(P,s)$ from pq
13: end if
14: end for
15: end while
16: end for
17: return path $P_{g}$
### 4.2 Prize Estimation via Attention Network
Kool et al. (2018) propose a network architecture based on self-attention to
model several coordinate routing problems, including TSP, OP, and several VRP
variants. With the benefit of reinforcement learning, the learned heuristic
achieves better performance over other learning-based heuristic methods.
Following the idea of using the attention mechanism to model the node
interactions with permutation-invariant property, we propose an attention-
based network, named Attention Network (AN). Our proposed structure is not
limited in handling coordinate OP, but can directly take all the node
attributes (prize of each node), edge attributes (costs among nodes) and
global attribute (the remaining cost of the agent) as input.
Figure 1: An illustration of our Attention Network.
Edge-Aware Graph Attention Networks Layer. The graph attention
networks(GAT)Veličković et al. (2017) utilize the attention mechanism to
aggregate the attributes of node neighbors adaptively, and extract high-level
representation feature vector of each node. We modify the original structure
of GAT; thus it can aggregate information from the connected edges, as well as
the node neighbors and global information:
$\displaystyle\alpha_{vk}=\frac{\exp\left(\text{LeakyReLU}\left(\mathbf{a}^{T}\left[{\mathbf{W}}[\mathbf{x}_{v}\|\mathbf{x}_{k}\|\mathbf{u}_{vk}\|\mathbf{g}]\right]\right)\right)}{\sum_{j\in\mathcal{N}_{v}}\exp\left(\text{LeakyReLU}\left(\mathbf{a}^{T}\left[{\mathbf{W}}[\mathbf{x}_{v}\|\mathbf{x}_{j}\|\mathbf{u}_{vj}\|\mathbf{g}]\right]\right)\right)}$
(2)
$\displaystyle\mathbf{h}_{\mathcal{N}_{v}}=\sigma\left(\sum_{k\in\mathcal{N}_{v}}\alpha_{vk}\left[{\mathbf{W}}[\mathbf{x}_{v}\|\mathbf{x}_{k}\|\mathbf{u}_{vk}\|\mathbf{g}]\right]\right)$
(3) $\displaystyle
MHA:\mathbf{h}_{\mathcal{N}_{v}}=\sigma\left(\frac{1}{M}\sum_{m=1}^{M}\sum_{k\in\mathcal{N}_{v}}\alpha_{vk}^{m}\left[{\mathbf{W}^{m}}[\mathbf{x}_{v}\|\mathbf{x}_{k}\|\mathbf{u}_{vk}\|\mathbf{g}]\right]\right)$
(4)
where $x_{v}$ denotes $v$’s node attribute (prize $R_{v}$), $u_{vw}$ denotes
the edge attribute (travel cost $C_{v,w}$) between node $v$ and $w$, and $g$
indicates the problem-level feature (the maximal cost $T$). After aggregating
information from connected edges and node neighbors, we obtain the
intermediate representation of each node $H=\\{h_{v_{1}},h_{v_{i}},...,\\\
h_{v_{M}}\\}$, where $h_{v}$ is the representation of $v$.
Transformer Encoding Layer(TEL). Recently, Kool et al. (2018) also
demonstrates the effectiveness of the Transformer to model many routing
problems dealing with coordinates of points. Following this idea, we use the
encoding layer of the Tranformer in this study to extract the node features
for the intermediate representation set
$H=\\{h_{v_{1}},h_{v_{2}},...,h_{v_{M}}\\}$. Since the resulting node
representations are invariant to the input order, we do not use a positional
encoding. Feeding the immediate embeddings $H$ into $N$ stacked Tranformer
encoding layers, we obtain the updated embeddings
$H^{\prime}=\\{h^{\prime}_{v_{1}},h^{\prime}_{v_{2}},...,h^{\prime}_{v_{M}}\\}$.
Linear Projection Layer. Finally, we use the linear combination of $v_{i}$’s
embedding $h^{\prime}_{i}$ to estimate the potential total prize $r_{i}$ when
selecting $v_{i}$ in the next step. where $U\in\mathbf{R}^{|H|}$ is a
learnable weight vector. Thus, we obtain the prize estimation
$e(s;\theta)=max_{v\in V}q(s,v;\theta)$. Although it would be more convenient
here to estimate the total prize of the given OP directly, we adopt this way
for the sake of facilitating Q-learning to train the networks.
### 4.3 Training via Q-learning
We use a fitted double Q-learning algorithm (Van Hasselt et al. (2016)) to
obtain the potential future prize of each selection. We formulate the states,
actions, rewards, transitions, and policy in the reinforcement learning
framework as follows:
* •
States: We define the state $s$ as the current subproblem of the original OP,
i.e., $\langle V-P,C,S,T-cost(P),last(P),v_{end}\rangle$.
* •
Actions: The action $v$ is a node selection of $V$ that is in the part of the
current state $s$. Both the definition of states and actions is applicable
across node set with various size.
* •
Rewards: We define the reward $r$ of each action $v$ as the prize collected
when the node is visited.
* •
Transitions: A transition is a tuple $\langle
s_{t},v_{t},r_{t+1},s_{t+1}\rangle$ which represents the agent takes action
$v_{t}$ in state $s_{t}$ and then observes the immediate reward $r_{t+1}$ and
resulting state $s_{t+1}$.
* •
Policy: Given the current state $s$, we can compute the q-value
$q(s,v;\theta)$ of each unselected node $v$ and apply a deterministic greedy
policy $\pi(v|s)=\operatorname*{\arg\\!\max}_{v\in|V|}q(s,v;\theta)$.
We use double Q-learning (Van Hasselt et al. (2016)) to learn a greedy policy
$\pi_{\theta}$ parametrized by the attention network and obtain the expected
total prizes $q(s,v;\theta)$ of each action $v$ simultaneously. The network
parameters can be updated by minimizing the following loss function of the
transitions picked up from the replayed memory. More details can be referenced
in supplementary part.
### 4.4 Discussions
Comparison with the step-by-step beam search. The learned heuristic score
function $f$ of (1) can also be applied to the step-by-step beam search.
However, as the routing in the early stage is much complicated than that in
the later stage, the estimated heuristic score in the early stage might also
be less reliable. In the initial stage of step-by-step beam search, the
proportion of reliable selection could be very low and thus degrade the
routing performance. Compared with the step-by-step beam search, the cost-
level beam search store the early partial paths as many as possible.
Running times. As the computation of the estimated action values in each time
window is parallelizable, the presented beam search algorithm can be
significantly accelerated. The basic idea is to compute the action value on
the GPU or TPU.
Extensibility to other routing problems. By simple modification, our
framework can be easy to be extended to other routing problems. Here we take
the classical TSP as an example. Similar to OP, we set the learned heuristic
score function as $f(P,s)=-cost(P)-e(s^{\prime},\theta)$, which represents the
negative of the expected total cost of the whole tour, while the edge
attribute is placed the same as those in the OP (the node and global
attributes are not needed). The reward in each selection is defined as the
negative of the increased cost. The cost constraint $T$ can be set as the
total travel cost given by another simple heuristic method.
## 5 Experimental Results
Table 1: Performance comparison. The gap % is w.r.t. the best value across all methods. | | $n=20,T=2$ | $n=50,T=3$ | $n=100,T=4$
---|---|---|---|---
Prize | Method | Obj. | Gap | Time | Obj. | Gap | Time | Obj. | Gap | Time
Uniform | Gurobi | 5.85 | $0.00\%$ | (1s) | - | -
Compass | $5.84$ | $0.17\%$ | (0s) | 16.46 | $0.00\%$ | (2s) | 33.30 | $0.00\%$ | (6s)
Random | $2.13$ | $63.59\%$ | (0s) | $3.33$ | $79.77\%$ | (0s) | $4.37$ | $86.88\%$ | (0s)
Tsili (greedy) | $4.85$ | $17.09\%$ | (0s) | $12.46$ | $22.94\%$ | (0s) | $25.48$ | $23.48\%$ | (0s)
AN-dqn (greedy) | ${5.15}$ | ${11.97\%}$ | (0s) | ${13.91}$ | ${15.49\%}$ | (0s) | ${26.51}$ | ${20.39\%}$ | (1s)
AN-a2c (greedy) | ${4.97}$ | ${15.04\%}$ | (0s) | ${14.13}$ | ${14.16\%}$ | (0s) | ${24.31}$ | ${27.00\%}$ | (1s)
AN-at (greedy) | ${3.98}$ | ${31.97\%}$ | (0s) | ${8.22}$ | ${50.06\%}$ | (0s) | ${9.75}$ | ${70.72\%}$ | (1s)
AN-pn (greedy) | ${4.46}$ | ${23.76\%}$ | (0s) | ${11.10}$ | ${32.56\%}$ | (1s) | ${10.06}$ | ${69.79\%}$ | (2s)
Tsili (beam) | $5.54$ | $5.30\%$ | (0s) | $13.54$ | $15.63\%$ | (2s) | $25.91$ | $22.19\%$ | (9s)
AN-dqn (beam) | ${5.74}$ | ${18.80\%}$ | (1s) | ${15.76}$ | ${4.25\%}$ | (10s) | ${30.61}$ | ${8.07\%}$ | (40s)
AN-a2c (beam) | ${5.57}$ | ${4.79\%}$ | (1s) | ${15.13}$ | ${8.08\%}$ | (5s) | ${27.40}$ | ${17.72\%}$ | (23s)
AN-at (beam) | ${4.45}$ | ${23.93\%}$ | (1s) | ${9.45}$ | ${16.46\%}$ | (4s) | ${10.55}$ | ${68.32\%}$ | (20s)
AN-pn (beam) | ${4.84}$ | ${17.26\%}$ | (1s) | ${13.75}$ | ${42.59\%}$ | (4s) | ${12.08}$ | ${63.72\%}$ | (24s)
CS (20, 0.05) | ${5.82}$ | ${0.51\%}$ | (1s) | ${16.13}$ | ${2.00\%}$ | (7s) | ${30.93}$ | ${7.12\%}$ | (40s)
Distance | Gurobi | 5.39 | $0.00\%$ | (4s) | - | -
Compass | $5.37$ | $0.37\%$ | (0s) | 16.17 | $0.00\%$ | (3s) | 33.19 | $0.00\%$ | (8s)
Random | $1.89$ | $64.94\%$ | (0s) | $2.90$ | $82.07\%$ | (0s) | $3.81$ | $88.52\%$ | (0s)
Tsili (greedy) | $4.08$ | $24.30\%$ | (0s) | $12.46$ | $22.94\%$ | (0s) | $25.69$ | $22.60\%$ | (0s)
AN-dqn (greedy) | ${4.77}$ | ${11.50\%}$ | (0s) | ${13.70}$ | ${15.28\%}$ | (0s) | ${26.60}$ | ${19.86\%}$ | (1s)
AN-a2c (greedy) | ${4.57}$ | ${15.21\%}$ | (0s) | ${13.70}$ | ${15.28\%}$ | (0s) | ${22.32}$ | ${32.75\%}$ | (1s)
AN-at (greedy) | ${3.62}$ | ${32.84\%}$ | (0s) | ${9.29}$ | ${42.55\%}$ | (1s) | ${10.52}$ | ${68.30\%}$ | (2s)
AN-pn (greedy) | ${3.75}$ | ${30.43\%}$ | (0s) | ${11.37}$ | ${29.68\%}$ | (1s) | ${10.77}$ | ${67.55\%}$ | (2s)
Tsili (beam) | $5.26$ | $1.68\%$ | (0s) | $15.50$ | $14.50\%$ | (2s) | $30.53$ | $8.04\%$ | (10s)
AN-dqn (beam) | ${5.27}$ | ${11.5\%}$ | (1s) | ${15.82}$ | ${4.14\%}$ | (9s) | ${30.28}$ | ${8.77\%}$ | (35s)
AN-a2c (beam) | ${5.02}$ | ${6.86\%}$ | (1s) | ${15.13}$ | ${6.43\%}$ | (6s) | ${27.83}$ | ${16.15\%}$ | (24s)
AN-at (beam) | ${4.42}$ | ${18.00\%}$ | (1s) | ${10.68}$ | ${33.95\%}$ | (4s) | ${11.10}$ | ${66.56\%}$ | (21s)
AN-pn (beam) | ${4.24}$ | ${21.34\%}$ | (1s) | ${12.15}$ | ${24.86\%}$ | (4s) | ${12.48}$ | ${62.40\%}$ | (17s)
CS (20, 0.05) | ${5.35}$ | ${0.74\%}$ | (1s) | ${15.99}$ | ${1.11\%}$ | (6s) | ${31.16}$ | ${6.12\%}$ | (42s)
### 5.1 Dataset Creation
To evaluate our proposed method against other algorithms, we generate problem
instances with different settings. We sample each node $v$’s coordinates
$x_{v}$ uniformly at random in the unit square and then compute the travel
cost from node $v$ to node $w$ by their Euclidean distance $dis(v,w)$, where
$dis(v,w)$ denotes the Euclidean distance between node $v$ and $w$. We set the
distance metrics as Euclidean distance because although many OP algorithms can
handle the OP with different distance metrics, most released implementations
can only support planar coordinates as input for the sake of convenience. One
more to mention is that selecting the Euclidean distance metric does not mean
that we will compare the performance of our approach with those specialized
methods designed for coordinate OP like Khalil et al. (2017); Kool et al.
(2018), because such comparison is unfair.
For the prize distribution, we consider three different settings described by
Fischetti et al. (1998); Kool et al. (2018). Besides, we normalize the prizes
to make the normalized value fall between 0 and 1. Uniform.
$r^{\prime}_{v}\thicksim DiscreteUniform(1,100);r_{v}=r^{\prime}_{v}/100$. The
prize of each node is discretized uniformed.
Distance. $r_{v}=(1+\lfloor 99\cdot\frac{dis(x_{v_{start}},x_{v})}{max_{v\in
V}dis(x_{v_{start}},x_{v})}\rfloor)/100$. Each node has a (discretized) prize
that is proportional to the distance to the start node, which is a challenging
task as the node with the largest prize are the furthest away from the start
nodeFischetti et al. (1998).
We choose the maximal travel cost $T$ as a value close to half of the average
optimal length of the corresponding TSP tour. Setting approximately the half
of the nodes can be visited results in the most challenging problem instances
(Vansteenwegen et al. (2011)). Finally, we set the value of $T$ for the OP
sized 20, 50, 100 as 2, 3, and 4, respectively.
### 5.2 Performance Comparison
Comparable Methods. We evaluate the performance of our method under the above-
mentioned instance generation settings . Firstly, we compare our proposed
method with an exact method Gurobi (Optimization (2014)) and a state-of-the-
art heuristic method Compass (Kobeaga et al. (2018)).We further compare with
several policy-based methods, which compute the node selection probability
$p(v|s)$ or action value $q(s,v)$ in state $s$.These methods include (1)Random
selection (Random), (2)Tsili (Tsiligirides (1984)), (3)policy learned by deep
Q-learning algorithm (AN-dqn), (4)policy learned by advantage actor-critic
algorithm (AN-a2c). We report their results based on two meta-algorithm:
(1)greedy: the node with the highest probability or action value is selected
in each state. (2) beam: we apply a step-by-step beam search for some of the
below methods to obtain the target path. The beam size is set as 100.
(a) With/without MHA in GAT. (b) Varing #layer of TLE. (c) Varing hidden size.
Figure 2: Model parameter/structure analysis. The x-axis is the number of
episodes. The y-axis denotes the average total prize on 200 test instances
with greedy policy using the estimated action values from AN. This experiment
is conducted five times, and we show the average performance and the
corresponding standard error.
In addition to the above methods as benchmark approaches, the comparison
between $AN-a2c$, $AN-at$, and $AN-pn$ also serves to demonstrate the
effectiveness of our network architecture compared with the encoder-decoder
schemes used in the previous works. This is because $AN-a2c$ can be view as a
policy gradient version of our proposed DQN framework proposed in Section 4.3,
which considers each step of node selection as a distinct OP. In contract,
$AN-at$ and $AN-pn$ view each selection is related to the previous ones (via
RNN or self-attention mechanism). Lastly, we report the performance of our
cost-level beam search algorithm CS(K, $\tau$). $K$ and $\tau$ represent the
beam size and cost interval, respectively.
Comparison Results. Table 1 reports the performance and average running time
of each approach on 10K test instances. Presenting running time is not for the
comparison between the efficiency of different methods. The results of
different approaches might not be comparable due to the differences in their
hardware usages (e.g., CPU or GPU), details of implementation (e.g.,
C++/Python with/without some optimizations), the settings of hyperparameters
(e.g., beam size). We report the time spent mainly to show that in general,
our method and implementation are time-affordable with the help of GPU
acceleration.
We note that our search method with the learned heuristic ($CS$) surpasses all
comparative methods except the exact approach Gurobi and the specialized
genetic algorithm Compass (the gap is relatively small compared with those of
other benchmarks), which demonstrates the effectiveness of our proposed
methods. AN-a2c is a competitive method concerning other baselines with
attention network as a shared bottom. However, since the step-by-step beam
search retains relative fewer partial paths at the start of the selection
process, our search method significantly outperforms this variant.
The comparison between $AN-dqn(beam)$ and $CS$ demonstrates the effectiveness
of the cost-level beam search.
By comparison of the result of AN-a2c, AN-at, and AN-pn in a greedy fashion or
a beam search fashion, we can find which scheme of network design works better
for the OP. We find that An-a2c significantly outperforms the rest two
approaches. This shows it is necessary to take into account the self-
referential property of the OP and use a non-sequential network to model this
problem.
### 5.3 Parameter/Structure Analysis
We conduct sensitive analysis in this section. We present the performance of
AN-dqn (greedy policy using DQN) on 200 instances in the learning process.
Each instance has 50 prized nodes, and the prize of each node is proportional
to its distance to the start node (distance). We note that by introducing
multi-head, the performance increases, which shows that the multi-head helps
the GAT better aggregate edge information. Following the GAT, the Transformer
encoding layer(TEL) further extracts more high-level representation for nodes.
From Figure 2(b), we observe that the performance improves as the number of
layer increases. Lastly, we explore the relationship between the hidden size
$\mathbb{R}^{H}$ and the quality of the learned model.
### 5.4 Visualization
Finally, we visualize the output paths generated by $AN-dqn$, $CS-p$, and
$CS-l$ with the problem size of 50 and the prize type of distance in Figure 3.
We observe that compared with the results of the greedy selection with action
values ($AN-dqn$) or the cost-level beam search with a simple heuristic
function ($CS-p$), the combination of our cost-level beam search and the
learned heuristic ($CS-l$) can conduct more look-forwarding selection, thus
have better performance.
(a) result of AN-dqn on Instance 1 (b) result of AN-dqn on Instance 2
Figure 3: Visualization of the output paths by AN-dqn, and CS. The star marker
denotes the start/end node, and the circle marker represents the prized node,
the area of which is proportional to its prize.
## 6 Conclusion and Future Work
In this paper, we solve the NP-hard problem of the general orienteering
problem by conducting a novel combination of a cost-level beam search method
and the learned heuristic score function. To estimate the potential prize of
the current partial solution, we design an attention-based network. It first
uses a variant GAT to aggregate both the edge attribute(travel cost) and the
node attribute(prize), then applies stacked Transformer encoding layers to
produce high-level node representations. Rather than modeling the OP in an
encoder-decoder structure, we use the extracted node representations to
directly estimate the target action values, which is a better scheme for the
OP with the self-referential property. We further conduct sufficient
experiments to show that our method surpasses most of the benchmark methods
and get results close to those of the exact method and the highly optimized
and specialized method. We also note that the performance of our method has
been improved by introducing the deep neural network. Besides, with a simple
modification, our proposed framework can also be straightforward to apply to
other routing problems.
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|
# Analysis of Many-body Localization Landscapes and Fock Space Morphology via
Persistent Homology
Gregory A. Hamilton<EMAIL_ADDRESS>Bryan K. Clark Institute for Condensed
Matter Theory and IQUIST and Department of Physics, University of Illinois
Urbana-Champaign, Urbana, IL 61801, USA
###### Abstract
We analyze functionals that characterize the distribution of eigenstates in
Fock space through a tool derived from algebraic topology: persistent
homology. Drawing on recent generalizations of the localization landscape
applicable to mid-spectrum eigenstates, we introduce several novel persistent
homology observables in the context of many-body localization that exhibit
transitional behavior near the critical point. We demonstrate that the
persistent homology approach to localization landscapes and, in general,
functionals on the Fock space lattice offer insights into the structure of
eigenstates unobtainable by traditional means.
## I Introduction
Many-body localization (MBL) is a paradigmatic example of a dynamical phase
transition whereby eigenstates fail to thermalize [1; 2; 3]. MBL is
characterized by a multitude of observables including vanishing conductivity
[4; 5], logarithmic entanglement growth [6], area law entanglement [7], level
statistics [8], quasilocal integrals of motion [9; 10; 2], and a litany of
others [11; 1]. A variety of models feature in the MBL literature, including
the disordered model considered here, quasiperiodic potentials [12], and
quadratic Stark potentials [13]. While microscopic mechanisms for the phase
transition have been proposed and elucidated upon [14; 15; 16; 17], much
controversy still surrounds the stability of the MBL phase in the
thermodynamic limit [18; 19] as well as the structure of eigenstates near the
transition [20].
In response to this challenge a large body of recent work probes the
correlational structure of eigenstates in Fock space (here taken to be the
configurational $\sigma_{z}$ basis). The guiding intuition for this approach
is, as the many-body Hamiltonian maps to a tight binding model (TBM) on a Fock
space lattice (or graph), eigenstate spatial correlations on the lattice
dictate the behavior of autocorrelation functions that diagnose the phase
transition [21].
Fock space eigenstate structure has been a fundamental question since the
early days of Anderson localization (1BL), as the inverse participation ratio
(IPR) and participation entropies $S_{i}^{P}$ serve as indicators for the
Anderson transition [11; 22; 23; 3]. From the participation entropy stems a
notion of multifractality with multifractal exponents identifying the
universality class; an analogous approach to the participation entropy exists
in the many-body case.
In pursuit of a more nuanced understanding of the transition and MBL
stability, recent approaches look beyond the participation entropies to
quantify the “anatomy” of Fock space correlations [24; 21; 25; 17]. From this
TBM picture several phenomenological models for the MBL transition have been
proposed, including so-called “avalanche” mechanisms [26; 27; 28], and,
pertinent to this work, percolation models [24; 29; 30; 21; 25; 31; 32; 33].
We approach this Fock space viewpoint from a morphological lens by leveraging
a new functional: the localization landscape.
New insights into the 1BL problem hinge on a functional called the
localization landscape (LL) [34], which bounds remarkably well the spatial
extent of low-energy eigenstates [34; 35; 36]. The LL is the solution to a
simple differential equation:
$\displaystyle H\bm{u}=\bm{1},$ (1)
for Hamiltonian $H$ that satisfies mild constraints ($\bm{1}$ is the vector of
all ones). The LL (or rather, its inverse $V_{u}=1/\bm{u}$) operates as an
“effective potential”, and can be used in place of the physical potential to
well approximate both the integrated density of states via the classical Weyl
law [36; 37; 38] and localization lengths for low-energy eigenstates [38]. The
effective potential, being numerically tractable, has already been used
extensively in the context of quantum transport in disordered semiconductors
[39; 37; 36].
Connections between percolation theory and transport extend back several
decades [40], and the effective potential $V_{u}$ lends itself to an
interpretation wherein a transition from localized to extended eigenstates is
percolative, as has been recently hypothesized [39; 41]. This core idea, that
the morphological shift in an effective potential (or more generally, a
potential in a configuration or phase space) might serve as an indicator of a
phase transition, has been well-developed in the context of classical phase
transitions and serves as the impetus for our work here [42; 43; 44; 45].
As we describe in Sec. II, the LL is perfectly well-defined for the many-body
case by the TBM mapping [41], and variants of the LL equally apply to interior
eigenstates [46; 47]. The localization landscape variant we use here
(discussed in detail in Sec. II) is termed the $L_{2}$ landscape, and takes
the form
$\displaystyle(u^{(2)})^{2}=\text{diag}(M^{-1}),\,M:=H^{\dagger}H,$ (2)
where $H$ is the Hamiltonian [46]. As we discuss in Sec. II, with a proper
choice of energy $E_{0}$ the $L_{2}$ landscape bounds mid-spectrum eigenstates
just as the LL bounds low-energy eigenstates. In what follows we analyze the
$L_{2}$ landscape of a 1D spinless fermion model with open boundary
conditions, given by
$\displaystyle H=H_{t}+H_{W}+H_{V}$ (3) $\displaystyle H_{t}=t\sum_{\langle
ij\rangle}\left(c_{i}^{\dagger}c_{j}+\text{ H.c.}\right),$ $\displaystyle
H_{W}=\sum_{i}h_{i}\left(n_{i}-\frac{1}{2}\right),$ $\displaystyle
H_{V}=V\sum_{\langle
ij\rangle}\left(n_{i}-\frac{1}{2}\right)\left(n_{j}-\frac{1}{2}\right),$
with $P(h_{i})\in[-W,W]$ uniform and $\langle ij\rangle$ denoting nearest
neighbor (NN) pairs. The dimensionless parameter $V/t$ sets the strength of NN
interactions; for a choice of $(t,V)=(1/2,1)$ this model reduces to the
standard XXZ model. The $L_{2}$ landscape, while simple to define, is a
complex functional on the Fock space lattice. To ascertain what the $L_{2}$
landscape can tell us about the MBL transition, we leverage a new tool from
computational geometry and topological data analysis: persistent homology.
Persistent homology offers a computable handle on the small and large-scale
structure of a point cloud or underlying functional by building a filtration
of topological objects, usually simplicial complexes. Persistent homology is
largely parameter-free and yields novel observables fundamentally tied to
discrete Morse theory and algebraic topology [48]. While persistent homology
has recently been used to identify classical and quantum phase transitions
[49; 50; 51], we leverage it here to gain a deeper understanding of whether
the $L_{2}$ landscape exhibits signs of a percolation transition.
To this end we generate a large set of observables that quantify the
morphology structure of the LL with respect to the high-dimensional Fock
space. The observables and topological summaries we garner from this approach
paint a complex picture. While some results are easy to interpret in the
context of standard observables, still others defy simple interpretations.
Part of this difficulty lies in the incoincidence of persistent homology with
standard statistical techniques, an impediment we confront several times in
this work.
Nonetheless, we do find indications that persistent homology accesses the
fundamental structure of mid-spectrum eigenstates and coincides with critical
points found in previous works. We provide novel ways to probe the “anatomy”
of Fock space at the morphological level. In Sec. IV we describe many more
ways future research can extend and refine our approach.
To briefly state the novel contributions of our work:
* •
We describe a new tool, persistent homology, to probe the morphology of Fock
space. In contrast to recent investigations of Fock space correlations [33],
this technique is parameter-free and probes clustering at all scales.
* •
We identify the proliferation of extremal points on the $L_{2}$ as an
indicator for the MBL transition.
* •
We demonstrate several topological observable that probe the clustering
properties of the $L_{2}$ landscape scale with the Hilbert space dimension.
The scaling exponents exhibit cross-over behavior near the MBL transition.
* •
We give a new, persistent homology-based definition for fractal dimensions
applicable to both many-body eigenstates and functionals on Fock space.
The outline of this exploratory work is as follows. In Sec. II we describe the
localization landscape, its extension to the many-body case, and the variant
on the LL utilized in this work. We also give an introduction to persistent
homology and its primary output, the persistence diagram. In Sec. III we apply
our persistent homology pipeline to the Hamiltonian in Eq. 3. We give a
phenomenological understanding of several observables extracted from the
persistent homology pipeline, and assess the implications for percolative MBL
phase transition hypothesis. Finally, in Sec. IV we summarize our work and
suggest future directions of study.
Figure 1: Pictorial representation of a superlevel set filtration for a one-
dimensional function. The dashed line denote different level sets, while the
black lines underneath each box denote the intervals in the superlevel set. As
the level decreases, intervals emerge and merge until a final connected
components is formed. The bottom plot shows the corresponding persistence
diagram. The rightmost point $(a,-\infty)$ is displayed but is ignored with
respect to topological summaries of the persistence diagram. The green
dendrogram denotes the merge tree derived from the subset filtration. External
vertices correspond to the maxima, while internal vertices correspond to the
minima.
## II Theory
In this section we overview the LL and $L_{2}$ landscape. We motivate and
describe how to apply persistent homology to Fock space functionals before
describing topological summaries of Fock space morphology extracted from
persistence diagrams and hierarchical clustering.
### II.1 Localization landscapes
The original LL construction takes as input a single-body Hamiltonian $H$ with
eigenspectrum $\\{(E^{\beta},\ket{\phi^{\beta}})\\}$, expressed in orthonormal
basis $\\{\ket{I}\\}$ such that $H$ is positive-definite and $H_{IJ}<0$ for
all $I\neq J$. For Eq. 3 these conditions are easily achieved by adding an
offset to $H\mapsto-H+\Gamma I$ to ensure positivity (though care should be
taken regarding the choice of $\Gamma$ [52]). Then the solution $u$ to the
equation
$\displaystyle Hu=1$ (4)
is component-wise positive. Alternatively, we can express $u$ as [46]
$\displaystyle
u_{J}=\sum_{\beta}\frac{\phi_{J}^{\beta}}{E^{\beta}}\sum_{I}\phi_{I}^{\beta},$
(5)
where $\phi_{J}^{\beta}=\braket{J}{\phi^{\beta}}$.
Remarkably, $u$ bounds eigenstate amplitudes, as we have
$\displaystyle|\phi_{J}^{\beta}|\leq|E^{\beta}|||\phi^{\beta}||_{\infty}u_{J},$
(6)
where $||\phi||^{p}:=\left(\sum_{I}|\phi_{I}|^{p}\right)^{1/p}$ denotes the
$\ell_{p}$ norm. The inverse $V_{u}:=1/u$ serves as an effective potential,
the valleys of which indicate where low-energy eigenstates tend to localize.
The peak ranges, or “valley network” of $V_{u}$ (the unstable manifold of $u$
in the context of discrete Morse theory) demarcate the domains of eigenstate
support [38; 46].
To apply the LL to the many-body case, we map the Hamiltonian in Eq. 3 to a
TBM on the Fock space lattice/graph $G_{\mathcal{F}}=(V,E)$. Here we closely
follow the notation and setup given in Ref. [31], in which several properties
of the Fock space lattice and on-site energies are described.
The physical lattice has $N$ sites with $N_{e}=\lceil\nu N\rceil$ fermions at
filling fraction $\nu$, set to $\lfloor N/2\rfloor$ in this work. The
dimension of the Fock space graph is then
$|V|=\,^{N}C_{N_{e}}=\binom{N}{N_{e}}$, with nodes $V$ indexed by
$\\{\ket{I}\\}$, the eigenstates of $H_{0}=H_{W}+H_{V}$. This is in contrast
to the Anderson basis which diagonalizes $H_{t}+H_{W}$. Each $I$ can be
represented by a tuple of $N$ occupation numbers $n_{i}^{(I)}=0$ or $1$ for
each physical site $i$ such that $\sum_{i}n_{i}^{(I)}=N_{e}$. We note that
$N_{\mathcal{H}}\propto\exp(s_{\infty}N)$ where
$\displaystyle s_{\infty}=-\left(\nu\ln\nu+(1-\nu)\ln(1-\nu)\right)$ (7)
is the per-site configurational entropy [31].
The edges of $G_{\mathcal{F}}$ correspond to states (nodes) $I,J$ connected by
$H_{t}$, i.e.,
$\displaystyle H=\sum_{I}\mathcal{E}_{I}\ket{I}\bra{I}+\sum_{\langle
IJ\rangle}t\ket{I}\bra{J}.$ (8)
We depict in the bottom right of Fig. 2 $G_{\mathcal{F}}$ for $N=8=2N_{e}$.
We denote by $N(I)$ the neighborhood of $I$, such that the coordination number
$Z_{I}=|N(I)|$. The average coordination number over $G_{\mathcal{F}}$ is
given by [31]
$\displaystyle\overline{Z}=2(1-\nu)N_{e}.$ (9)
The competition of the extensive coordination number $\overline{Z}$ (favoring
delocalization), with the strong correlations of neighboring on-site energies
(discussed in Sec. III) is fundamental to the enigma of whether MBL survives
in the thermodynamic limit [31].
Turning back to the LL, the TBM in Eq. 8 $u_{2}$ is defined so long as $H\geq
0$, which is always achievable by adding a positive offset. In fact, any
stoquastic Hamiltonian yields a positive $u$ [52]. From $V_{u}$ a derived,
degenerate Agmon metric on Fock space bounds the extent of low-energy
eigenstate spread into classically disallowed regions [52].
However, one significant shortcoming of the LL is its ineffectiveness in
bounding eigenstates other those close to the ground state. This follows from
the definition of $u$ in Eq. 5: low-energy eigenstates contribute more to the
LL. An analogue dual formulation of the LL can be made for eigenstates at the
top of the spectrum [53], but both parts of the spectrum localize at zero
disorder strength in the thermodynamic limit for the Hamiltonian in Eq. 8.
Thus, the traditional LL is not well-equipped to probe mid-spectrum eigenstate
Fock space anatomy.
Directly in response to this limitation, a new localization landscape was
introduced in Ref. [46], termed the $L_{2}$ landscape. The function $u$ is
modified to take the form
$\displaystyle u_{I}^{(2)}:=\sqrt{M^{-1}_{II}},\,M=H^{\dagger}H.$ (10)
Similar to Eq. 6, this new landscape satisfies
$\displaystyle|\phi_{I}^{\beta}|\leq|E^{\beta}|||\phi^{\beta}||_{2}\sqrt{M^{-1}_{II}},$
(11)
where by orthonormality the $\ell_{2}$ norm equals unity (hence the name
“$L_{2}$ landscape”). Since $M>0$ for any $H$, we shift $H\to H-(E_{0}+i\eta)$
and probe the eigenspectrum near $E_{0}$, where $\eta\in\mathbb{R}^{+}$ is an
infinitesimal offset. From this shift the $L_{2}$ landscape can be written as
a function of the resolvent $G^{\pm}(E_{0}):=(E_{0}\pm i\eta-H)^{-1}$:
$\displaystyle(u^{(2)}_{J}(E_{0}))^{2}=\left(G^{+}(E_{0})G^{-}(E_{0})\right)_{JJ},$
(12)
which for finite systems $u^{(2)}$ is exactly given by
$\displaystyle(u_{J}^{(2)}(E_{0}))^{2}=\sum_{\beta}\frac{|\phi_{J}^{\beta}|^{2}}{|E_{0}+i\eta-
E_{\beta}|^{2}}.$ (13)
Closely related is the local density of states
$\displaystyle\rho_{J}(E_{0}):=\lim_{\eta\to 0^{+}}\frac{1}{2\pi
i}\left(G^{+}_{JJ}(E_{0})-G^{-}_{JJ}(E_{0})\right).$ (14)
It therefore follows [46]
$\displaystyle\rho_{J}(E_{0})=\lim_{\eta\to 0^{+}}\eta(u_{J}^{(2)})^{2}.$ (15)
The behavior of the resolvent $G^{\pm}$ is the central consideration of
several recent works [15], which we discuss further in Sec. III.
The $L_{2}$ landscape incorporates information from the full eigenspectrum,
and has a nonlinear dependence on the offset $\eta$, the consequences of which
we address in Sec. III. Due to the connectivity of $G_{\mathcal{F}}$, it is
not immediately obvious what observables can be extracted from the $L_{2}$
landscape. However, the authors in Ref. [46] where the $L_{2}$ landscape was
first defined posed an open question of how fractal dimensions like those
extracted from the participation entropies can be generalized to the $L_{2}$
landscape. An analogue of the fractal dimension for $\rho(E_{0})$ was recently
proposed [54], but we explore in Sec. III a novel construction by leveraging a
tool originating in Morse theory and rooted in algebraic topology: persistent
homology.
### II.2 Persistent Homology: motivation and theory
Persistent homology characterizes the small and large-scale topological
structure of an underlying dataset by computing how the homology of a
filtration of topological spaces changes as a function of scale. This
fundamental idea underlies the many generalizations of persistent homology;
for the interested reader there exist many reviews such as Ref. [55].
Rather than focus on the technical aspects of persistent homology, here we
give an illustrative example quite similar in spirit to the construction we
later present in Sec. III. Consider a function $f:[0,1]\to\mathbb{R}$ depicted
in Fig. 1. Our objective is to characterize the structure of $f$; i.e., how
the extrema relate to one another. To quantify this structure, we consider the
superlevel set $L_{c}^{+}(f)=\\{x\in[0,1]|f(x)\geq c\\}$. For any value of $c$
(the dashed lines in Fig. 1, known as the filtration value) the superlevel set
is a set of intervals in $[0,1]$, indicated by the black bars at the bottom of
each panel. The number of intervals is the number of connected components
(clusters); in topological terms, the Betti number $\beta_{0}$. As we decrease
$c$, the number of connected clusters fluctuates until, when
$c=c_{min}:=\min_{x}f(x)$, we have $L_{c_{max}}^{+}=[0,1]$, the domain. Along
the filtration we can monitor which clusters merge into one another.
The end result of capturing the progression of homology groups is a
persistence diagram showing the filtration value at which a connected
component appears in the filtration (known as the birth time) on the ordinate,
and when a cluster merges with another cluster (the death time on the
abscissa. Each persistence point has a distance from the diagonal: the farther
from the diagonal, the longer-lived the cluster was in the filtration. Fig. 1
depicts the persistence diagram with notation indicating the minima and maxima
of the functional that identify each cluster. Note that the point
$(a,-\infty)$ indicates the largest component, the domain, which technically
has infinite lifetime. Next to the persistence diagram in Fig. 1 we depict the
dendrogram, or merge tree corresponding to the superlevel set filtration:
external (internal) nodes denote the maxima (minima).
One remarkable property of persistent homology is its stability with respect
to perturbations of the underlying functional. In particular, a series of
algebraic stability results imply the Wasserstein metric of two persistence
diagrams is continuous with respect to $\ell_{p}$ norms between the underlying
filtration functionals [56; 57]. Put in simpler terms, small perturbations to
the superlevel sets result in small perturbations to the persistence diagrams.
Figure 2: Example of the Fock space $G_{\mathcal{F}}$; colors of nodes denote
the value of a random potential. For each value of
$\epsilon\in\\{0.7,0.5,0.2,0.0\\}$ the superlevel set (subgraph)
$G[V_{\epsilon}]$ is displayed.
The unique idea of persistent homology is that the aforementioned pipeline is
readily generalizable to higher dimensions. The filtration of intervals
generalizes to filtrations of simplicial complexes, and the notion of clusters
generalizes to homology groups of a sequence of simplicial complexes. A
filtration need not only be superlevel set filtrations of an underlying
functional: any filtered sequence of topological spaces (given a total
ordering) has a unique representation via a persistence diagram [58].
The toy example described above informs our study of the morphology of the
$L_{2}$ landscape. Instead of a functional on a continuous domain, we perform
a superlevel set filtration on $G_{\mathcal{F}}$ given the function
$u^{(2)}(E_{0})$ on $V(G)$. For each filtration value $c$, the analogue to the
superlevel set is the induced subgraph $G[V_{c}]$, where $V_{c}:=\\{I\in
V(G_{\mathcal{F}})|u_{I}^{(2)}\geq c\\}$. Note that the induced subgraph
$G[V_{c}]$ includes both the vertex set $V_{c}$ as well as all of the edges
$E$ that have both endpoints in $V_{c}$.
While perhaps a less intuitive than a sublevel set filtration, we chose to
perform a superlevel filtration for one major reason. The peaks of the $L_{2}$
landscape indicate where eigenstates localize, while the minima demarcate and
separate the regions eigenstate near $E_{0}$ localize. If we performed a
sublevel filtration, in the extended regime a large fraction of Fock space
sites $I$ would be connected into one cluster well before the maxima
indicating the localization centers would appear. Upon appearing these maxima
would immediately merge into a connected component and therefore not be
recorded in the persistence diagram. By performing a superlevel filtration we
ensure the localization center feature in the diagrams.
In Fig. 2 we depict for visual reference a superlevel set filtration on
$G_{\mathcal{F}}$ for $N=8=2N_{e}$ in 1D and with open boundary conditions.
Due to the absence of odd-length cycles (the graph is bipartite), our
filtration fails to produce any interesting homology in dimensions greater
than zero, as any loops generated persistent throughout the rest of the
filtration, and no higher order simplices are generated. However, a different
interaction term $H_{t}$ will generate higher dimensional homology, as we
discuss in Sec. IV. As noted above, our pipeline for 0D persistent homology
corresponds to single-linkage hierarchical clustering (depicted via the merge
tree in Fig. 1).
There are a multitude of other persistent homology construction that we could
have attempted in this work. Of particular interest would be filtrations based
on the distance function derived from the Agmon metric [52; 41], which has
deep theoretical connections to localization bounds. While an Agmon metric for
the $L_{2}$ does not currently exist, it is an exciting prospect and the
subject of future work.
## III Results
With the theoretical underpinnings of the many-body localization landscape and
the persistent homology pipeline established, we turn our methodology to the
prototypical MBL: the disordered Heisenberg Hamiltonian given in Eq. 3. Our
objective again is to assess whether the $L_{2}$ landscape shows indications
of a morphological transition upon approaching the MBL phase. To build these
indicators we leverage novel topological summaries and observables generated
from persistence diagrams and merge trees. Broadly speaking, we find these
observables best discriminate the transition when compared across system
sizes.
For the Hamiltonian in Eq. 3 we assume open boundary conditions and performed
$1000$ disorder realizations for the $N=[10,14]$ system sizes and over $500$
for the $N=15$ system size. We sampled $W\in[0.1,8.0]$ in intervals of $\delta
W=.2$ as well as sampled $W\in\\{10,12,14,16,18,20\\}$ deep in the MBL phase
for comparison. We use exact diagonalization to compute the eigenspectrum for
system sizes $N\in[10,15]$.
The critical disorder strength for the phase transition in Eq. 3 is typically
given in the range $W_{c}\in[3.5,4]$, though some works place $W_{c}$ at much
higher values [59; 28; 60]. Whether or not the phase transition survives the
thermodynamic limit remains an open question, with recent work posited a
strong distinction between the MBL phase and the finite-size MBL regime that
can be probed with numerics [28]. Given our limited system sizes we take no
position on the stability issue or the absence of mobility edges in the
thermodynamic limit in this work, choosing instead to see if morphological
indicators coincide with critical values claimed in the literature.
Before moving to the persistent homology topological summaries, we briefly
digress to discuss the statistics of eigenenergies in Fock space. Given the
$L_{2}$ landscape’s close correspondence to the local density of states and
$G^{\pm}$, the topological summaries explored below probe similar statistics,
greatly expounded upon in recent works we now refer to [31].
Note that both the on-site energies and eigenenergies $\mathcal{E},E$ share
the same $\overline{E}$, as $H_{t}+H_{V}$ is disorder independent. Due to
parity and our choice of $V=1$ we have $\overline{E}=-1/4$ for even system
sizes. Meanwhile the variance $\mu_{E}^{2}$ goes as $\mathcal{O}(N)$.
We depict in Fig. 3 the eigenenergy probability distribution (normalized
histogram) for $W\in[1.0,5.0,12.0]$ at $N=15$, as well as the Gaussian
approximation to the density of states
$\displaystyle
D(\omega)=(2\pi\mu_{E}^{2})^{-1}\exp\left(-(\omega-\overline{E})^{2}/2\mu_{E}^{2}\right),$
(16)
shown as the dashed line for comparison.
Figure 3: Normalized histogram for $L=15$ of the eigenenergies for
$W=(1,5,12)$, the dashed lines show the Gaussian approximation given in Eq. 16
To evaluate $u^{(2)}$, we must select $E_{0}$ and a finite value for $\eta$
which should be smaller than the level spacing near $E_{0}$, but not so small
as to impact numerical stability [46]. The choice of $\eta$ dictates the
nonlinear extent to which eigenstates far from $E_{0}$ contributes to the
$L_{2}$ landscape. To first order Eq. 13 shifts under $\eta\to\eta+d\eta$ as
$\displaystyle(u_{J}^{(2)})^{2}\to(u_{J}^{(2)})^{2}-2\eta
d\eta\sum_{\beta}\frac{|\phi_{J}^{\beta}|^{4}}{|E_{0}+i\eta-E_{\beta}|^{4}}.$
(17)
One natural choice for $\eta$ is the eigenvalue spacing near $E_{0}$, given by
$(N_{\mathcal{H}}D(E_{0}))^{-1}$. Here $D(E_{0})$ is the density of states
given in Eq. 16, such that $D(E_{0}=\overline{E})=(\sqrt{2\pi}\mu_{E})$. This
then implies $\eta=\sqrt{2\pi}\mu_{E}/N_{\mathcal{H}}$.
However, as noted above, $\mu_{E}$ depends both on $N$ and $W$. Moreover, the
level statistics of the model in Eq. 3 has long been known to shift under the
phase transition from GOE (Wegner-Dyson) to Poissonian statistics [61]. The
precise nature of the level statistics near the transition is still very much
controversial [62], and so it is unclear if a more canonical choice for $\eta$
can be made. We make the choice $\eta=\sqrt{2\pi}\mu_{E}/N_{\mathcal{H}}$ for
two main reasons.
First, the spectral width $\mu_{E}$ is non-parametric in the sense that we do
not need to average over some number $k$ level spacing near $\overline{E}$.
The number of level spacing necessary for estimation could very well depend
upon the disorder strength, rendering this technique possibly biased.
Second, the spectral width $\mu_{E}$ is fundamental to the analysis given in
Refs. [31], on which we have based our notation and exposition for Fock space
spectral statistics. There the authors analyze the stability of the MBL phase
in the thermodynamic limit by considering functionals of the resolvent
$G_{IJ}^{\pm}$ (particularly, the Feenburg self-energy) under a rescaling
$\tilde{E}_{\beta}\to(E_{\beta}-\overline{E})/\mu_{E}$; i.e., the
eigenspectrum is normalized to have vanishing mean and a standard deviation of
one. Under this rescaling the offset becomes
$\tilde{\eta}=\eta/\mu_{E}=\sqrt{2\pi}/N_{\mathcal{H}}$. The rescaling proves
central to a mean-field theory approach to the self-consistent determination
of the typical self-energy [31]. Note that $u^{(2)}(\overline{E})$ would be
rescaled to $\tilde{u}^{(2)}(0)$ with offset $\tilde{\eta}$, and so we have
$\tilde{u}^{(2)}(0)=\mu_{E}u^{(2)}(\overline{E})$. Choosing $\eta$ is a
nuanced issue and should be the study of future work.
Figure 4: Representative persistence diagrams for disorders strengths
$W=\\{1.0,3.0,5.0,8.0\\}$, plot on a log-log scale. Note that $d<b$ due to the
superlevel set filtration. Inset shows the same persistence diagram on a
linear scale.
Persistence Diagram – Turning to the output of the persistent homology
pipeline, Fig. 4 depicts representative persistence diagrams for the disorder
strengths $W=\\{2,8,14\\}$. Recall that the abscissa and ordinate are the
birth and death times, respectively; due to the superlevel set filtration, the
death times are smaller than the birth times. We have plotted the persistence
diagram in both log-log and linear scale (inset) for comparison. The
persistence pairs for the extended regime $W=2$ cluster at high filtration
values with a broad spread of death times. In contrast, in the localized
regime the birth and death times are much smaller (corresponding to a small
value of $u^{(2)}$).
A perhaps naive interpretation of this behavior is that
$|\phi_{J}^{\beta}|^{2}\sim(N_{\mathcal{H}})^{-1}$ on
$\mathcal{O}(N_{\mathcal{H}})$ many sites in the extended regime. In contrast,
in the localized regime $|\phi_{J}^{\beta}|^{2}\sim N_{\mathcal{H}}^{-\alpha}$
for $\alpha<1$ on $\mathcal{O}(N_{\mathcal{H}}^{\alpha})$ sites. We note that
the number of persistence pairs for each diagram corresponds to the total
number of maxima and is therefore not constant, a fact that proves
consequential when we consider a generalization of the fractal dimension.
Figure 5: Statistics on the birth, death, and lifetime distributions on the
persistence diagrams. (a-b) depict the mean (solid), median (dashed) and
standard deviation of the birth values as a function of disorder strength $W$.
(c-d) and (e-f) show the corresponding statistics for the death and lifetimes
values, respectively.
$(b,d,l)$ statistics – Many topological summaries center on properties of the
distributions of the set birth, death, and lifetime values, which we write as
vectors $b,\,d,\,l$, respectively. Note that we exclude the persistence pair
corresponding to the infinitely long-lived component ($(a,-\infty)$ in Fig.
1). As we explore in Sec. III.0.1, norms of these vectors correspond to
observables relevant to percolation theory.
Fig. 5(a) and (b) depicts the arithmetic mean $\overline{b}$, geometric mean
$l_{typ}$ (dashed lines), and standard deviation $\mu_{b}$ of the birth
distribution against $W$. Fig. 5(c-f) show equivalent data for the death and
lifetime distributions. All statistics are calculated within for each disorder
realization before taking averaging over realizations.
Plots (a) and (c) show the mean birth and death times decay monotonically with
$W$, visually reflecting the change in spread of points in Fig. 4. The
geometric mean (or typical value) of the birth/death times deviate strongly
from the arithmetic mean towards the localized regime, with the deviation more
pronounced with larger system sizes and particularly for the birth times. This
implies large outliers in the birth times, coincident with the onset of
localization centers in the localized regime. In contrast, the difference
between arithmetic and geometric mean death times is less pronounced, as the
death times (minima of $u^{(2)}$) are lower bounded by zero.
While the arithmetic mean of $l$ is the difference in the mean birth and death
times, the same cannot be said for the typical lifetime $l_{typ}$, shown in
(e). Here the distinction between arithmetic and geometric mean is even more
pronounced, and a peak is observed at $W\approx 1.5$ for $l_{typ}$ and
$W\approx 2$ for $\overline{l}$ for the $N=15$ system size.
Plot (b), (d), and (f) depict $\mu_{X}$ for $X=(b,d,l)$, respectively, wherein
we see peaks generally obey $\mu_{l}^{max}\geq\mu_{b}^{max}>\mu_{d}^{max}$.
This follows from the fact that $\mu_{l}$ should add in quadrature, i.e.,
$\mu_{l}^{2}\approx\mu_{b}^{2}+\mu_{d}^{2}$. The asymmetry in the peak
$\mu_{b},\,\mu_{d}$ values again stems from the fact that both $u^{(2)}\geq 0$
and $d_{i}<b_{i}$ for any persistence pair $(b_{i},d_{i})$.
Figure 6: Fractal scaling $\nu_{p}$ for $p=\\{0,0.5,1.0,1.5\\}$, corresponding
to plots (a-d), respectively. Figure 7: (a) Slope $\tilde{\beta}_{p}$ of
$\log||l||_{p}^{p}$ against $\log N_{\mathcal{H}}$ for
$p=(0.0,0.5,1.0,1.5,2.0)$. (b) Fractal dimension $\dim_{p}(PD)$ for the case
$p\to 0$. (c) The fractal dimension plotted on a semilog scale.
#### III.0.1 $\ell_{p}$-norms of the persistence diagrams
We now assess to what extent $\ell_{p}$ norms of the $(b,d,l)$ vectors are
sensitive to the phase transition. As we outline below, several of these norms
correspond to meaningful quantities in the context of percolation theory, and
thus should help shed light on the morphological structure of Fock space under
the $L_{2}$ landscape.
Fractal dimensions – The first observable we consider is how the $\ell_{p}$
norms of $l$ scale with $(N,W)$. We define the function
$\displaystyle\nu_{p}(N,W)=\frac{\log\langle||l||_{p}^{p}\rangle}{\log
N_{\mathcal{H}}},$ (18)
where $\langle\cdot\rangle$ denotes averaging over disorder realizations
holding $W$ fixed. For $p<1$ $\nu_{p}$ is a semi-norm that emphasizes small-
scale homological features. While persistent homology is often used to
identify large-scale topological structure, the small-scale structure is
relevant in the context of fractality and estimating curvature [63; 64].
Indeed, $\nu_{p}$ was first defined to give a notion of “persistent fractal
dimension” derived from persistent homology and comparable to more traditional
measures of fractality like the correlation and box-counting dimensions [63].
In its original formulation the persistent fractal dimension is applicable to
filtrations built with respect to a finite metric space, such a Vietoris-Rips
or Cěch filtration [63]. In our case we apply the same methodology to the
$u^{(2)}$ landscape. We interpret $\nu_{p}$ as quantifying the relative weight
of small-scale homological features (short-lived clusters) as a function of
$(N,W)$. Note that $||l||_{0}$ counts the number of $u^{(2)}$ maxima.
We define a persistent fractal dimension as [63]
$\displaystyle\text{dim}_{p}(PD)=\frac{p}{1-\tilde{\beta}_{p}},$ (19)
$\displaystyle\tilde{\beta}_{p}=\lim_{n\to\infty}\sup\nu_{p}.$ (20)
Here $\tilde{\beta}_{p}$ amounts to the slope of $\log||l||_{p}^{p}$ against
$\log N_{\mathcal{H}}$. Note that the tilde on $\beta$ is to distinguish
$\tilde{\beta}_{0}$ from the Betti number $\beta_{0}$, and does not refer to
any rescaling such as $u^{(2)}\to\tilde{u}^{(2)}$.
Fig. 6(a-d) depicts $\nu_{p}$ for $p\in\\{0,0.5,1.0,1.5\\}$. For $p<1$ we
observe a noticeable “kink” in $\nu_{p}$ as $W$ increases, close to
$W_{c}\approx 3.5$. For $p>0$ $\nu_{p}$ decreases monotonically with $W$,
which upon considering Fig. 5(e) seems contradictory. However, given that
$\overline{l}=||l||_{1}/||l||_{0}$, the peak in Fig. 5(e) stems from the
relative decays rates of $\nu_{1}$ versus $\nu_{0}$.
For each $p\in\\{0,.5,1.0,1.5,2.0\\}$ we calculated $\tilde{\beta}_{p}$,
depicted in Fig. 7(a). While $\tilde{\beta}_{p}$ visually appears to scale as
$\log W$ in the extended regime, there is a small sub-logarithmic factor. Note
that for $p\geq 1/2$ the fractal dimension $\text{dim}_{p}(PD)$ is either
divergent at some value of $W$ or negative; however, $\tilde{\beta}_{p}<0$ for
$p=0$. Fig. 7(b) and (c) depict $\text{dim}_{p}(PD)/p$ for $p=0^{+}$. The
inset (c) shows the same plot as (b) but on a log scale. For $W\geq W_{c}$
$\text{dim}_{0+}(PD)/0^{+}\approx\log W^{a}+b$ with $(a,b)=(.79,3.0)$.
There are several interesting remarks to make concerning the case of $p=0$.
First, note that $||l||_{0}$, which we also write as $|PD|$, equates to the
total number of persistence pairs in a persistence diagram (hence $|PD|$
denotes cardinality). This quantity is invariant under any monotonically
increasing transformation of the underlying filtration; in particular, $|PD|$
is the same whether we use $u^{(2)}$ or $\tilde{u}^{(2)}$. Second, $|PD|$
equates to the number of maxima in the $L_{2}$ landscape. These peaks
qualitatively correspond with Fock space sites where eigenstates near
$\overline{E}$ also peak. Third, we can compare the scaling behavior of
$\nu_{0}$ against the scaling of the Hilbert space dimension $N_{\mathcal{H}}$
against $N$. As noted above, $N_{\mathcal{H}}\propto\exp^{s_{\infty}N}$, where
$s_{\infty}$ is the configurational entropy; therefore, $\log
N_{\mathcal{H}}\propto s_{\infty}N$. Thus, we could equivalently have computed
$\tilde{\beta}_{0}$ by fitting $\log||l||_{0}$ against $N$, the system size.
Then $\tilde{\beta}_{0}\to\tilde{\beta}_{0}/s_{\infty}>1$ which implies the
fractal dimension would become negative.
To try and understand this behavior a bit more analytically, we return to
$\nu_{0}$ and consider the behavior of the local density of states
$D_{I}(\overline{E}_{0})$. Deep in the ergodic phase the eigenstates can be
well-approximated by Gaussian random vectors with vanishing mean and a
standard deviation that scales as $N_{\mathcal{H}}^{-1/2}$ [20]. By time-
reversal symmetry the eigenstates are real, and thus the contribution of each
eigenstate to the $L_{2}$ landscape is effectively random apart from the
denominator, which can also be well-approximated by a Gaussian, see Eq. 16.
The lack of correlations for $u^{(2)}$ serves as a proxy for a proliferation
of local maxima, as neighboring Fock space sites are effectively uncorrelated.
As the disorder becomes non-vanishing and the model moves from integrability
the eigenstates drift from uncorrelated Gaussians, thus lowering the number of
maxima. A more simple picture is that, as the eigenstates begin to localize in
regions (clusters of Fock space), the appearance of large amplitude maxima
necessitates a decrease in the total number of maxima.
In the other extreme, deep in the MBL phase, we expect eigenstates near
$\overline{E}_{0}$ to be tightly clustered around localization centers. Away
from these centers, the $L_{2}$ landscape is very small (see Fig. 4) and again
effectively random, which drives up the number of local maxima.
This qualitative description holds as well for the non-interacting $(V=0)$
case, as deep in the MBL phase the model is perturbatively connected to the
non-interacting limit [20]. We can leverage the perturbative connection to the
non-interacting limit to partly explain the $\log W$ behavior in $\nu_{p}$. As
the non-interacting eigenstates are Slater determinants of the Anderson
orbitals, the localization length for an eigenstate scales as $(\log W)^{-1}$
[3]. Thus the proliferation of uncorrelated regions of the $L_{2}$ landscape
should rise in proportion to the decrease in the localization length, which by
proxy implies the number of local maxima scale with $\log W$ as observed.
Because the number of maxima decrease in the ergodic regime and increase in
the MBL regime, there is a natural point between where the number of maxima is
minimal, near $W=3.0$.
Connectivity threshold – In the context of percolation theory, one figure of
merit is the probability of obtaining a giant connected component on Erdős-
Rényi random graphs, wherein a probability is assigned to each edge [65]. For
these systems of random graphs a large body of literature focuses on analogues
of the MBL localization/delocalization transition with respect to the
adjacency matrix [66; 67]. In the context of our study of the $L_{2}$
landscape morphology, an analogous figure of merit is played by the
connectivity threshold $d_{f}$, which we define as the filtration value at
which only one connected component remains. By construction, this corresponds
to the smallest value of $u^{(2)}$ (global minimum), or equivalently, as
$||d||_{-\infty}$. Fig. 8 depicts the connectivity threshold $d_{f}$ against
the $W\in[2.0,3.0]$ wherein we observe the different system sizes cross at
various points. The inset shows $d_{f}$ on a log scale across the entire set
of $W$ probed. In the localized regime the power law decay goes as
$W^{-\alpha}$ with $\alpha\approx 1.15$, which we depict with the dashed black
line. The crossing points in the main plot of Fig. 8 are again substantially
lower than $W_{c}$; it is unclear if the reduced finite size effects at larger
values of $N$ would resolve these crossing points.
Figure 8: Log of the connectivity threshold $d_{f}$, the last superlevel set
wherein at most two connected clusters remain, against $W$. The main plot
depict the connectivity threshold near where different system size curves
coincide, the inset depicts the curves from $W\in[0,20]$. the dashed black
line depicts an the approximate power law scaling $d_{f}\propto W^{-\alpha}$
for $\alpha=1.15$.
Maximum spanning tree – Also related to percolation theory is the notion of
maximum (MaxST) and minimum spanning trees (MinST). On a graph $G=(V,E)$ with
edge weight function $w:E\to\mathbb{R}$, the MaxST (MinST) (if it exists, else
a spanning forest) is an acyclic connected subgraph such that the sum of the
edge weights is maximal (minimal). The weight of the MaxST corresponds to the
sum of the death times $||d||_{1}$ of the persistent diagram and is therefore
equivalent to the integral of the reduced Betti number with respect to the
filtration [68]. The reduced Betti number at filtration value $c$ is
specifically the rank of the reduced homology group $H_{0}(G[V_{c}],I_{max})$,
where $I_{max}:=\text{argmax}_{V(G)}u_{I}^{(2)}$. Put more plainly, the
reduced Betti number is generally one less than the regular Betti number.
Fig. 9 (a) depicts $w(\text{MaxST})/N_{\mathcal{H}}$ against $W$ on a log-log
scale, whereby the power law scaling in the localized regime is manifest. The
number of terms in $w(\text{MaxST})$ scales with $|PD|\propto\log W$, which
implies the average death time scales as $W^{\gamma}/\log W$ for some scaling
exponent $\gamma$ that varies slightly for different system sizes.
Figure 9: (a) the weight of the maximal spanning tree $w(\text{MaxST})$
against $W$, normalized by $N_{\mathcal{H}}$. (b) The persistent entropy
$S(PD)$ against $W$. Figure 10: The exponent $\alpha_{S(PD)}$ from fitting
$S(PD)\propto N^{\alpha}$. Inset shows the range $W\in[2,7]$.
Persistent entropy – We turn now to an observable that closely mirrors the
participation entropy in accounting both for cardinality and magnitude of
persistence pairs: the persistent entropy. By normalizing $l$ to a probability
vector $\tilde{l}:=l/||l||_{1}$, we define the persistent entropy as
$\displaystyle S(PD):=-S(\tilde{l}),$ (21)
where $S(\cdot)$ as before is the entropy function [69]. We can equate $S(PD)$
to a functional of the $\ell_{p}$ norms:
$\displaystyle S(PD)=\frac{S(l)}{||l||_{1}}-\sum_{i}\ln l_{i}.$ (22)
The persistent entropy has been extensively used in the context of detecting
both classical and quantum phase transitions [70]. Qualitatively, for a fixed
value $|PD|$ a large $S(PD)$ implies the lifetimes are largely constant. In
contrast, a small $S(PD)$ implies the prevalence of vanishing few clusters
with lifetime disproportionately large. Clearly $S(PD)\leq\log|PD|$, just as
the Schmidt rank bounds the entanglement entropy and $\log N_{\mathcal{H}}$
bounds the participation entropies.
Fig. 9(b) depicts the persistent entropy $S(PD)$ as a function of $W$. In
contrast to the observables examined above, $S(PD)$ flattens out to nearly
constant, with a very slow, sublogarithmic scaling in $W$. Qualitatively, near
constant $S(PD)$ is analogous to the area law entanglement entropy exhibited
by localized states. Heuristically, the lifetimes of the Fock site clusters
indicating eigenstates near $\overline{E}$ roughly correspond to the large
Schmidt coefficients of a localized eigenstate under a bipartition. We expect
a relatively large contribution to the $L^{2}$ landscape from these
localization centers while the rest of the landscape is vanishingly small.
There is a competition between the number of local maxima (increasing as $\log
W$) versus the relative persistence of these maxima. Fig. 9 indicates that the
maxima proliferation outweighs the disproportional lifetime of these maxima in
slowly driving up the persistent entropy.
Just as $S(\rho_{A})$ scales with $N$ in the extended regime, so too does
$S(PD)$; in Fig. 10 we depict the coefficient $\alpha$ from a fit
$S(PD)\propto N^{\alpha}$, with inset zoomed into the region $W\in[2.0,7.0]$.
The coefficient $\alpha$ plateaus in the region $W\in[4,6]$ before slowly
increasing with $W$. This behavior is very similar to Fig. 7 (b) which
depicted the fractal dimension, though the minimum value for $\alpha$ seems be
at a higher $W$ and the scaling is not quite logarithmic.
Figure 11: The peak value of $\beta_{0}$ during the filtration (equivalently,
the maximum number of independent clusters at any point during the
filtration), normalized by $N_{\mathcal{H}}$. Note that $\beta_{0}^{max}$ goes
as $\log W$ in the localized regime. Figure 12: Scaling exponents from
fitting $\beta_{0}^{max}\propto N_{\mathcal{H}}^{a}\left(\log
N_{\mathcal{H}}\right)^{b}$. The horizontal dashed lines at $0$ and $.5$
roughly correspond to the maximum (minimum) value of $b$ ($a$), which occurs
at $W\approx 3.0$. Inset depicts representative fitting of $\beta_{0}^{max}$
against $N_{\mathcal{H}}$ at $W=3.0$.
Maximum Betti number – The final observable we consider is the maximum value
of the Betti number $\beta_{0}^{max}$ encountered during the superlevel set
filtration. This observable has been leveraged in recent works as an indicator
for quantum phase transitions [51]. Fig. 11 depicts
$\beta_{0}^{max}/N_{\mathcal{H}}$ against $W$ wherein we see that, quite
similar to $|PD|$, a logarithmic growth in the extended regime. Appealing to
the heuristical argument given above, in the extended regime $\beta_{0}^{max}$
occurs very late in the filtration, growing in proportional to the extent of
decorrelated Fock space. Fig. 12 depicts the scaling of
$\beta_{0}^{max}\propto N_{\mathcal{H}}^{a}\left(\log
N_{\mathcal{H}}\right)^{b}$ with exponents $(a,b)$ disorder-dependent.
Intriguingly, at $W\approx 3.0$ $(a,b)\approx(.5,0)$, which implies
$\beta_{0}^{max}\propto\sqrt{N_{\mathcal{H}}}$. This is roughly the same
disorder value for which the minimum in $\text{dim}_{0+}(PD)$ occurred. Though
it is at present unclear if the extremal behavior of the scaling exponents
$(a,b)$ would shift for larger values of $N$, the simplicity of the scaling at
$W\approx 3.0$ indicates deeper behavior that merits exploration in future
work.
## IV Conclusion
This work takes an exploratory approach to leveraging persistent homology as a
descriptor for the complex correlational structure of eigenstates in Fock
space. We utilize a new construction, the $L_{2}$ landscape, to procure
topological summaries of the mid-spectrum Fock space structure.
Overall, we found that several observables, including the cardinality $|PD|$
of the persistence diagrams (Fig. 7), the persistent entropy (Fig. 10) and the
peak Betti number (Fig. 11) all exhibit scaling in either $N$ or
$N_{\mathcal{H}}$ near $W_{c}$ indicative of a transition. This is especially
evident for $|PD|$ which is perhaps the most easily interpreted observable
presented here, as it corresponds to the number of maxima of an approximate
local density of states.
Along with these observables we proposed a novel notion of a fractal dimension
readily applicable to other functionals defined on Fock space, including but
not limited to self-energies and the local density of states. While a recent
study expanded the participation entropy definition to a normalized version of
the local density of states [54], our approach is fundamentally different. The
notion of fractal dimension presented here incorporates not only the Fock
space lattice but also the complex morphology of level sets.
Still other observables painted a less intuitive picture of the MBL phase
transition, and it should be the aim of future work to discriminate
observables that inaccurately pinpoint the phase transition due to finite-size
effects from observables that do not probe the transition at all. Two possible
indicators for the transition, the connectivity threshold $d_{f}$ (Fig. 8) and
the $(b,d,l)$ statistics (Fig. 5), show substantial drift in the crossing of
different system size curves, indicative of the finite-size effects at the
relatively small system sizes probed here.
Throughout this work we have posed a heuristical understanding that is largely
based upon the local density of states which, by Eq. 15, the $L_{2}$ is
connected to. However, lacking a deeper, more analytic understanding of how
$u^{(2)}$ evolves with $W$ limits the scope to which a topological data
analysis lens can shed light on the transition. Much of this limitation stems
from the difficulty in interpreting persistent homology in terms of standard
statistical observables. While persistent homology is strongly connected to
discrete Morse theory [71], to our knowledge there exists a limited amount of
work in the topological data analysis literature to bridge homological and
statistical observables. We conclude with several proposed future directions
to take this exploratory analysis, as the use of persistent homology in the
context of quantum information and many-body systems is still nascent.
Persistent Homology for functionals – The PH pipeline explored here can
equally be applied to individual eigenstate as performed in Ref. [49] for the
single-body Aubry-André model. In fact, the persistent homology pipeline could
be readily applied to any functional represented on $G_{\mathcal{F}}$,
including the local density of states and the self-energies. The edges
$E(G_{\mathcal{F}})$ can have filtration values that differ from their
boundary nodes.
As noted in Sec. II, the original $LL$ gives rise to a degenerate Agmon metric
on $G_{\mathcal{F}}$. From this metric a series of localization theorems limit
the extent eigenstate support extended into “classically disallowed regions”
[52]. This Agmon metric determines a filtration on the complete graph $G$ with
vertex set $V(G)$ the Fock space sites. However, a complete graph would become
computationally prohibitive with large $N$, at which point approximate methods
such as witness complexes could be used [72].
Higher-dimensional Homology – The analysis given in this work is restricted
to 0D homology due to the interaction graph $G_{\mathcal{F}}$: the
bipartiteness of the graph precludes any $2$-simplices from forming. This
obstruction can, of course, be overcome by simply choosing a different
Hamiltonian; e.g., including a $\sigma_{x}$ term that breaks total
magnetization conservation. Under this inclusion persistent homology could be
used to study the occurrence of higher-dimensional homology features like
cycles and voids.
Approximations for larger system sizes – For larger system sizes exact
diagonalization is computationally expensive. Thus, approximate forms of the
$L_{2}$ landscape could be computed via recursive approximations to the
diagonal of the Green function [46]. The original LL equation in Eq. 4 can be
written as a Rayleigh quotient and is therefore amenable to DMRG and tensor
network approaches. Recent tensor network representations of Green functions
could be leveraged to scale the $u^{(2)}$ calculation [73].
###### Acknowledgements.
GH acknowledges useful conversations with Felix Leditzky, Di Luo, Yuliy
Baryshnikov, Shmuel Weinberger, Giuseppe De Tomasi, and Henry Adams. This work
made use of the Illinois Campus Cluster, a computing resource that is operated
by the Illinois Campus Cluster Program (ICCP) in conjunction with the National
Center for Supercomputing Applications (NCSA) and which is supported by funds
from the University of Illinois at Urbana-Champaign. We acknowledge support
from the Department of Energy grant DOE DESC0020165.
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|
# WaveCatBoost for Probabilistic Forecasting of Regional Air Quality Data
††thanks: This work was supported in part by ASEAN-India Science and
Technology Collaboration (AISTIC) Funding from the Department of Science and
Technology, India (CRD/2020/000320) for providing incentives. ††thanks: J.
Borah (email<EMAIL_ADDRESS>& S. Majumdar (email<EMAIL_ADDRESS>is
with the Dept. of ECE, NIT Meghalaya, India ††thanks: T. Chakraborty (email:
<EMAIL_ADDRESS>is with the Sorbonne University, Abu Dhabi, UAE and
Sorbonne Center for Artificial Intelligence, Paris, France ††thanks: M.S.M.
Nadzir is with Dept. of Earth Sciences and Environment, Universiti Kebangsaan,
Selangor, Malaysia. (e-mail: [email protected]). ††thanks: M.G.
Cayetano is with the Institute of Environmental Science and Meteorology,
University of Philippines, Manila, Philippines (e-mail:
[email protected]). ††thanks: J. Borah and T. Chakraborty (Joint first
authors) have equal contributions.
Jintu Borah, Tanujit Chakraborty, Md. Shahrul Md. Nadzir, Mylene G. Cayetano,
and Shubhankar Majumdar
###### Abstract
Accurate and reliable air quality forecasting is essential for protecting
public health, sustainable development, pollution control, and enhanced urban
planning. This letter presents a novel WaveCatBoost architecture designed to
forecast the real-time concentrations of air pollutants by combining the
maximal overlapping discrete wavelet transform (MODWT) with the CatBoost
model. This hybrid approach efficiently transforms time series into high-
frequency and low-frequency components, thereby extracting signal from noise
and improving prediction accuracy and robustness. Evaluation of two distinct
regional datasets, from the Central Air Pollution Control Board (CPCB) sensor
network and a low-cost air quality sensor system (LAQS), underscores the
superior performance of our proposed methodology in real-time forecasting
compared to the state-of-the-art statistical and deep learning architectures.
Moreover, we employ a conformal prediction strategy to provide probabilistic
bands with our forecasts.
###### Index Terms:
Air quality, CatBoost, Wavelet analysis, Conformal prediction, Real-time
forecasting.
## I Introduction
Air pollution is a critical global concern, adversely affecting public health
and the environment. The rapid expansion of industrialization and urbanization
have surged the emission of air pollutants, resulting in significant
challenges to sustainable living. In response to these concerns, the World
Health Organization has issued guidelines for six major air pollutants and set
National Ambient Air Quality Standards. These pollutants, including nitrogen
dioxide ($NO_{2}$), ozone ($O_{3}$), carbon monoxide ($CO$), sulfur dioxide
($SO_{2}$), particulate matter with diameters of 2.5 mm or less ($PM_{2.5}$),
and 10 mm or less ($PM_{10}$), originate from various sources such as
transportation, industry, and natural processes. Despite considerable efforts
to reduce emissions and lower ambient concentrations of these pollutants, air
pollution remains a significant cause of mortality worldwide.
To mitigate these concerns related to polluted air, several countries have
implemented real-time air quality forecasting systems [1]. Accurate prediction
of air pollutant concentration levels is paramount for public awareness
campaigns, environmental management, and healthcare interventions, among many
others. By leveraging these forecasts, government regulations and public
policies can be designed to address pollution-related health concerns and
promote sustainable development initiatives. In recent years, machine learning
algorithms have emerged as powerful tools for enhancing the accuracy of air
quality predictions [2]. These computational techniques, driven by their
ability to decipher complex patterns and relationships within vast datasets,
offer a promising avenue for advancing pollutant forecasting models [3, 4].
However, existing methods encounter challenges adapting to diverse monitoring
environments and accurately predicting pollutant concentrations in real-time
[5]. Traditional models often struggle with the dynamic nature of air quality,
hindered by limitations in handling non-linear relationships and non-
stationary variations in pollutant dynamics. To address these challenges in
the forecasting methodologies, wavelet-based transformation is widely applied
in other applied domains, including epidemiology [6], economics [7], and
others. These wavelet-based forecasting approaches transform non-stationary
seasonal data into a series of uncorrelated distinct components. Subsequently,
the component series are modeled with various statistical and deep learning
[8] architectures to solve diverse problems in time series forecasting. While
the accuracies of these wavelet-based deep learning architectures surpass
traditional approaches for ultra-long time series, their training and
prediction times obstruct the generation of real-time forecasts. To mitigate
this challenge, decision tree-based boosting architectures are favored for
their faster convergence time in the presence of large datasets [9]. Several
boosting algorithms, including random forest, extreme gradient boosting, and
light gradient boosting machines, offer higher accuracy compared to deep
learning methods, but they overlook the sequential nature of the time series,
resulting in target leakage [10]. The recently developed CatBoost algorithm
employs an ordered boosting methodology to overcome this issue [11].
Motivated by this background, in this letter, we develop a novel wavelet-based
CatBoost (WaveCatBoost) model for generating accurate real-time forecasts of
air pollutant concentration levels to mitigate the overarching challenges of
reliable forecasting of air quality data. This integration aims to leverage
the strengths of both techniques, providing a robust and versatile model
capable of delivering reliable point and probabilistic forecasts across a
range of air pollutants via deploying a conformal prediction approach. By
conducting an extensive experimental evaluation and comparing existing models,
we demonstrate the superiority of our proposed approach in advancing air
quality forecasting and addressing the critical gaps present in the current
literature.
Figure 1: Pipeline of the proposed air quality forecasting system Figure 2:
Depiction of the forecast horizons considered in this study
## II Material and Methodology
### II-A Data Collection and Preprocessing
Real-time data on air pollutant concentration levels are gathered through two
wireless sensors, the Central Air Pollution Control Board (CPCB) sensor
network and a low-cost air quality sensor system named ID1, strategically
positioned in Meghalaya, India. These air quality sensors continuously monitor
the concentrations of NO2 (in ppb), O3 (in ppb), CO (in ppb), SO2 (in ppb),
PM2.5 (in $\mu$g/m$3$), and PM10 (in $\mu$g/m$3$) and transmit the data to the
server at one-minute intervals. The transmitted information is sequentially
stored in a tabular data format within the database for further analysis. We
extract the raw data from the server database for a year, from July 1, 2022,
to July 31, 2023. The preprocessing phase of our experiment involves adjusting
the missing data entries and computing hourly averages to generate near-real-
time data on the concentration of various air pollutants. Additionally, we
normalize the dataset using the min-max normalization strategy, which is
crucial for optimizing the convergence of data-driven forecasting algorithms
used in our analysis. A visual representation outlining the processes of data
collection, pre-processing, and analysis employed in this study is depicted in
Fig. 1.
### II-B Proposed WaveCatboost
The key concept behind the wavelet-based CatBoost framework, which we named
WaveCatBoost, involves employing the scale-invariant maximal overlapping
discrete wavelet transformation (MODWT) [12] to denoise the air pollutant
concentrations series and modeling them using an ensemble of CatBoost models.
The MODWT decomposition is applied to filter the normalized air pollutant data
$(Y_{t})_{t=1}^{N}$ using a sequence of wavelet
$\left\\{p_{m};m=0,1,\ldots,\mathcal{M}-1\right\\}$ and scaling
$\left\\{q_{m};m=0,1,\ldots,\mathcal{M}-1\right\\}$ filters. These filters
adhere to the even-length scaling assumption,
$\sum_{m=0}^{\mathcal{M}-1}p_{m}p_{m+2n}=\sum_{m=0}^{M-1}q_{m}q_{m+2n}=0$ for
any non-zero integers $n$. Consequently, applying the pyramid algorithm [12]
generates wavelet and scaling coefficients for the transformed time series,
allowing the original series $Y_{t}$ to be represented as a set of equal-sized
uncorrelated time series:
$Y_{t}=\sum_{k=1}^{K}D_{k,t}+S_{K,t}\quad t=1,2,\ldots,N,$
where $K=\log_{e}N$ is the number of decomposition levels applied,
$D_{k,t}(k=1,2,\ldots,K)$ is the $k^{th}$ details series of $Y_{t}$, captures
the local fluctuations of the series, and the smooth series $S_{K,t}$
preserves the low-frequency trend of $Y_{t}$. This iterative decomposition
essentially helps filter the true signal from the noisy data and aids in
modeling long-term dependent, non-stationary, and seasonal data by breaking it
into several lower-resolution components. Subsequently, the task of generating
$h$-step ahead forecasts of $Y_{t}$ based on its historical data can be solved
by forecasting these low-resolution components based on their corresponding
prior observation. In our proposed WaveCatBoost architecture, we generate the
forecasts for each of the wavelet and scaling coefficients of $Y_{t}$ using
the gradient boosting CatBoost [11] algorithm and ensemble them using an
inverse MODWT (IMODWT) [12] approach as:
$\hat{Y}_{N+h}=\operatorname{IMODWT}\left(\hat{D}_{1,N+h},\ldots,\hat{D}_{K,N+h},\hat{S}_{K,N+h}\right),$
(1)
$\text{with
}\hat{D}_{k,N+h}=f\left(D_{k,1},D_{k,2},\ldots,D_{k,N}\right);k=1,2,\ldots,K$
and $\hat{S}_{K,N+h}=f\left(S_{K,1},S_{K,2},\ldots,S_{K,N}\right),$ where the
function $f$ indicates the CatBoost model applied on each of the $p$ lagged
values of details and smooth series. The CatBoost framework operates by
iteratively combining weak symmetric decision trees into a strong regressor.
Unlike traditional gradient boosting methods, it is prone to prediction shifts
from target leakage. CatBoost architecture employs an ordered boosting
mechanism. During the learning process, the training data undergoes a series
of time-dependent permutations, denoted as
$\left(\sigma_{0},\sigma_{1},\ldots,\sigma_{u}\right)$. The initial
permutation $\sigma_{0}$ determines the leaf values, while the subsequent $u$
permutations decide the structure of the weak learners. At each iteration $v$,
the algorithm considers a randomly chosen permutation $\sigma_{r}$. The
supporting models $M_{r,j}$ are then fitted to the preceding $j$ observations
of $\sigma_{r}$, generating the output $M_{r,j}(i)$ for the $i^{th}$ instance
in $\sigma_{r}$. To construct the tree $T_{v}$ at the current iteration,
gradients for the $i^{th}$ time-index are computed by considering previous
time steps in the $\sigma_{r}$ permutation, i.e.,
$\operatorname{grad}_{r,\sigma(i)-1}(i)$ where
$\operatorname{grad}_{r,j}(i)=\left.\frac{\partial
L\left(y_{i},A\right)}{\partial A}\right|_{A=M_{r,j}(i)}.$ (2)
The leaf value for the $i^{th}$ instance, used for evaluating candidate
splits, is obtained by averaging the corresponding gradients
$\operatorname{grad}_{r,\sigma_{r}(i)-1}$ of preceding examples belonging to
the same leaf node $\operatorname{Leaf}_{r\left(i\right)}$. Once the tree
$T_{v}$ is constructed, it is used for boosting all the supporting models
$M_{r^{\prime},j}$ with varied sets of leaf values depending on the first $j$
instances in the $\sigma_{r^{\prime}}$ permutation. When all the weak learners
are constructed, the leaf value of the CatBoost model is calculated by the
standard gradient boosting procedure. For generating the forecasts
$\hat{D}_{k,N+h};k=1,2,\ldots,K$ and $\hat{S}_{K,N+h}$, the leaf values are
computed based on target statistics learned from the CatBoost architecture
using lagged inputs. The algorithmic structure of the proposed WaveCatBoost
model is given in Algorithm 1.
### II-C Probabilistic Forecasting using Conformal Pediction
Along with the point forecasts generated by the WaveCatBoost model, we focus
on quantifying the uncertainty associated with these forecasts using the
conformal prediction approach [13]. This non-parametric procedure generates a
probabilistic band around the point forecasts based on a conformal score
($CS_{t}$). To calculate $CS_{t}$ corresponding to $Y_{t}$ [14], the
architecture considers $p$ lagged values of the target series ($Y_{t-p}$) and
applies the WaveCatBoost and an uncertainty model $\mathbf{\hat{U}}$ such
that,
$CS_{t}=\frac{\left|Y_{t}-\operatorname{WaveCatBoost}\left(Y_{t-p}\right)\right|}{\mathbf{\hat{U}}\left(Y_{t-p}\right)}.$
Following the sequential nature of $Y_{t}$ and the $CS_{t}$, the conformal
quantile can be computed with a weighted aggregation method having a fixed
window $w_{t}=\mathbb{1}\left(t^{\prime}\geq
t-\kappa\right),\forall\;t^{\prime}<t$ of length $\kappa$ as
$CQ_{t}=\operatorname{inf}\left\\{q:\frac{1}{\operatorname{min\left(\kappa,t^{\prime}-1\right)}}\sum_{t^{\prime}=1}^{t-1}CS_{t^{\prime}}w_{t}\geq
1-\alpha\right\\}.$
The conformal prediction intervals based on these weighted conformal quantiles
can be estimated as
$\left[\operatorname{WaveCatBoost}\left(Y_{t-p}\right)\pm
CQ_{t}\mathbf{\hat{U}}\left(Y_{t-p}\right)\right].$ (3)
Algorithm 1 WaveCatBoost Model
0: Air pollutant concentration data $\left\\{Y_{t}\right\\}_{t=1}^{N}$
0: $h$-step ahead forecast
$\left\\{\hat{Y}_{N+1},\ldots,\hat{Y}_{N+h}\right\\}$
1: Initialize the MODWT algorithm with Haar filter and $K=\log_{e}N$ levels.
2: Transform the training data $\left\\{Y_{t}\right\\}$ using the MODWT
approach into $K$ uncorrelated details series
$\left\\{D_{k,t}\right\\}_{k=1}^{K}$ and a smooth series
$\left\\{S_{K,t}\right\\}$
3: for $k=1,2,\ldots,K$ do
4: Apply CatBoost model to the $k^{th}$ detail series $D_{k,t}$.
5: $\sigma_{0},\sigma_{1},\ldots,\sigma_{u}\leftarrow$ Time-dependent
permutations of $D_{k,t}$.
6: $\sigma_{0}$ decides the leaf nodes and the next $u$ permutations determine
the structure of the symmetric weak learners.
7: for $v=1,2,\ldots,V$ do
8: $\sigma_{r}\longleftarrow\;\text{random permutations
of}\;\left[\sigma_{0},\sigma_{1},\ldots,\sigma_{u}\right]$.
9: Fit $M_{r,j}$ to $j$ preceeding observations of $\sigma_{r}$ and generate
output $M_{r,j}(i)$ for $i^{th}$ instance of $\sigma_{r}$.
10: Calculate leaf values of the week learner $T_{v}$ by averaging the
gradients $\operatorname{grad}_{r,\sigma_{r}(i)-1}$ (Eq. 2) of previous
examples belonging to the same node $\operatorname{Leaf}_{r\left(i\right)}$.
11: Boost the models $M_{r^{\prime},j}$ of
$\sigma_{r^{\prime}};\;r^{\prime}\neq r$ using $T_{v}$.
12: end for
13: Calculate leaf value for the CatBoost model by applying gradient boosting
approach to $V$ week learners’ leaf nodes and generate the one-step ahead
forecast for $D_{k,t}$.
14: Iteratively generate multi-step ahead forecast.
15: end for
16: Similarly, utilize a CatBoost architecture on $S_{K,t}$ to generate the
corresponding one-step ahead and multi-step ahead forecast iteratively.
17: Apply Inverse MODWT transform on the component forecasts from the ensemble
of CatBoost models and generate the final forecasts of the desired horizon as
in Eq. 1.
## III Experimental Results
In this section, we present experimental results showcasing the efficacy of
our proposed WaveCatBoost framework compared to benchmark forecasting methods.
We forecast hourly air pollution levels across different horizons through
subsequent experiments. It helps identify each model’s temporal sensitivity,
guiding the selection of the most suitable model for specific prediction
needs. Additionally, it ensures the models apply to a range of real-world
scenarios, from immediate health advisories to long-term urban planning. This
analysis informs resource allocation, enabling tailored responses for short-
term and long-term forecasts, and supports the construction of ensemble models
for enhanced accuracy. Furthermore, it deepens our scientific understanding of
air quality dynamics and aids in crafting effective policies and mitigation
strategies, addressing different timeframes, from rapid interventions to long-
term sustainability planning. Our analysis utilizes a rolling window
forecasting technique, assessing model performance across four distinct train-
test scenarios: with forecast horizons set at 1 day (24 test points), 7 days,
14 days, and 31 days. Fig. 2 presents a graphical representation of these
train-test configurations.
In our analysis we compare the performance of the proposed WaveCatBoost
architecture with state-of-the-art forecasting methodologies including
NLinear, DLinear, light gradient-boosting machines (LGBM), extreme gradient-
boosting (XGB), NBeats, NHiTS, Transformer, temporal convolution network
(TCN), recurrent neural network (RNN), gated recurrent unit (GRU), long-short-
term memory (LSTM), temporal fusion transformer (TFT), and CatBoost model [5].
We measure the forecast accuracy of various frameworks using the mean absolute
scaled error (MASE) metric, one of the most recommended performance measures
in forecasting literature, on the test data as follows:
$\text{MASE}=\frac{N-S}{h}\frac{\sum_{t=N+1}^{N+h}\left|\hat{Y}_{t}-Y_{t}\right|}{\sum_{t=S+1}^{N}\left|Y_{t}-Y_{t-S}\right|},$
where $h$ is the forecast horizon, $N$ is the length of training data,
$\hat{Y}_{t}$ is the forecast of $Y_{t}$ at time $t$ and $S$ is the
seasonality of the data.
(a)
(b)
Figure 3: MCB plots for MASE metric of the various forecasting models across
all forecast horizons utilizing two different sensors (a) ID1, (b) CPCB
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Actual vs Forecasts of the targeted pollutants for ID1 Sensor for 1
day forecast horizon (a) $NO_{2}$, (b) $O_{3}$, (c) $CO$, (d) $SO_{2}$, (e)
$PM_{2.5}$, and (f) $PM_{10}$
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: Actual vs Forecasts of the targeted pollutants for ID1 Sensor for 7
day forecast horizon (a) $NO_{2}$, (b) $O_{3}$, (c) $CO$, (d) $SO_{2}$, (e)
$PM_{2.5}$, and (f) $PM_{10}$ TABLE I: Performance of the proposed
WaveCatBoost model and selected baseline forecasters for CPCB and ID1 sensors,
evaluated based on the MASE metric for different forecast horizons.
Pollutant | Horizon | CPCB Sensor | ID1 Sensor
---|---|---|---
RNN | LGBM | TCN | TFT | Transformer | XGB | CatBoost | WaveCatBoost | RNN | LGBM | TCN | TFT | Transformer | XGB | CatBoost | WaveCatBoost
$NO_{2}$ | 1d | 2.45 | 0.79 | 1.45 | 1.41 | 3.28 | 0.91 | 0.82 | 0.57 | 2.05 | 1.27 | 2.15 | 1.62 | 2.07 | 1.40 | 1.20 | 0.91
7d | 1.35 | 0.66 | 0.67 | 2.65 | 0.93 | 0.66 | 1.04 | 0.84 | 1.66 | 0.82 | 1.64 | 1.76 | 1.76 | 0.95 | 0.84 | 0.52
14d | 0.81 | 0.71 | 1.79 | 4.27 | 2.05 | 0.72 | 0.73 | 0.72 | 1.57 | 0.85 | 1.56 | 2.06 | 1.63 | 0.92 | 1.69 | 0.42
31d | 1.98 | 0.75 | 1.87 | 1.18 | 1.35 | 0.76 | 1.17 | 0.47 | 1.47 | 1.15 | 1.48 | 1.95 | 1.86 | 1.14 | 1.36 | 0.40
$O_{3}$ | 1d | 5.19 | 1.87 | 1.85 | 2.43 | 11.47 | 1.52 | 2.69 | 0.98 | 3.65 | 1.61 | 3.32 | 2.79 | 3.75 | 1.72 | 1.49 | 1.42
7d | 5.46 | 1.68 | 4.66 | 6.16 | 5.82 | 1.51 | 4.84 | 0.76 | 3.57 | 1.64 | 2.55 | 2.84 | 3.18 | 1.96 | 1.79 | 0.72
14d | 6.24 | 1.23 | 4.38 | 14.68 | 9.54 | 1.27 | 1.77 | 1.11 | 3.85 | 1.80 | 3.67 | 3.74 | 3.12 | 2.06 | 2.32 | 0.64
31d | 3.56 | 2.46 | 4.26 | 2.71 | 3.56 | 2.44 | 5.37 | 0.25 | 3.59 | 2.72 | 3.63 | 3.56 | 3.61 | 2.87 | 2.66 | 1.11
$CO$ | 1d | 2.36 | 3.36 | 2.90 | 2.45 | 0.97 | 3.21 | 2.87 | 2.69 | 4.47 | 2.48 | 3.82 | 2.95 | 4.47 | 2.54 | 2.32 | 1.53
7d | 3.74 | 3.62 | 3.94 | 5.29 | 3.74 | 3.59 | 3.57 | 2.61 | 2.85 | 1.52 | 2.64 | 2.95 | 2.84 | 1.46 | 1.36 | 0.79
14d | 2.76 | 3.06 | 2.79 | 4.34 | 2.77 | 2.88 | 3.15 | 1.63 | 3.08 | 1.93 | 3.08 | 3.71 | 3.06 | 1.83 | 3.36 | 0.83
31d | 1.92 | 2.05 | 1.96 | 3.29 | 1.91 | 1.98 | 2.03 | 0.61 | 2.79 | 2.58 | 2.79 | 3.48 | 3.57 | 2.22 | 3.26 | 0.71
$SO_{2}$ | 1d | 34.35 | 11.46 | 28.11 | 7.76 | 40.15 | 10.62 | 9.35 | 23.93 | 1.81 | 0.45 | 0.85 | 0.49 | 1.79 | 0.43 | 0.44 | 1.31
7d | 16.83 | 15.43 | 14.62 | 20.72 | 11.59 | 14.39 | 5.84 | 7.30 | 2.36 | 0.34 | 2.23 | 0.61 | 2.21 | 0.33 | 0.30 | 0.61
14d | 10.70 | 7.35 | 9.37 | 24.75 | 9.57 | 8.14 | 11.31 | 6.17 | 2.15 | 0.22 | 2.82 | 1.66 | 3.18 | 0.26 | 0.24 | 0.58
31d | 9.06 | 11.22 | 10.34 | 11.44 | 10.79 | 9.13 | 9.81 | 5.33 | 2.35 | 0.41 | 2.95 | 0.81 | 1.88 | 0.26 | 0.31 | 1.31
$PM_{2.5}$ | 1d | 2.88 | 3.81 | 2.50 | 3.52 | 6.57 | 2.95 | 3.02 | 5.06 | 7.31 | 4.67 | 7.03 | 4.68 | 7.35 | 5.53 | 4.45 | 0.85
7d | 6.94 | 6.86 | 7.05 | 7.38 | 6.69 | 5.92 | 6.57 | 5.07 | 7.09 | 3.01 | 5.22 | 4.22 | 5.56 | 2.65 | 2.43 | 0.58
14d | 4.40 | 6.98 | 4.40 | 7.07 | 4.41 | 6.20 | 6.30 | 2.34 | 8.20 | 4.43 | 8.71 | 7.10 | 7.39 | 3.67 | 5.58 | 0.69
31d | 2.73 | 2.85 | 2.83 | 4.41 | 2.73 | 2.83 | 2.83 | 1.57 | 7.21 | 5.26 | 7.01 | 5.97 | 5.05 | 4.49 | 5.29 | 0.42
$PM_{10}$ | 1d | 1.01 | 6.69 | 1.79 | 6.37 | 4.16 | 3.81 | 5.77 | 3.12 | 7.31 | 4.34 | 7.06 | 4.24 | 7.35 | 5.59 | 4.22 | 0.86
7d | 5.20 | 5.30 | 5.32 | 6.19 | 5.11 | 4.99 | 5.24 | 3.70 | 7.37 | 2.83 | 5.42 | 4.34 | 5.81 | 2.46 | 2.27 | 0.57
14d | 3.53 | 4.59 | 3.55 | 5.90 | 3.54 | 4.36 | 4.89 | 1.48 | 8.48 | 4.29 | 9.08 | 7.08 | 7.65 | 3.29 | 5.44 | 0.67
31d | 2.39 | 2.40 | 2.51 | 3.64 | 2.37 | 2.43 | 2.40 | 0.98 | 7.57 | 5.39 | 7.35 | 6.09 | 5.36 | 4.50 | 5.47 | 0.41
Table I displays the performance of selected forecasting architectures,
incorporating only the best those that perform in at least one scenario. For
the CPCB sensor, the MASE metric values in the table indicate that the
WaveCatBoost model outperforms all the baseline architectures for long-range
forecasting periods of 14 days and 31 days for most air pollutants. This
improved performance is attributed to the hybrid training approach, where
MODWT aids in extracting signals from noise, and CatBoost accurately
extrapolates these transformed signals. In the 7-day forecasting scenario,
WaveCatBoost performs comparably to other boosting methods like LGBM, XGB, and
CatBoost across various pollutants. Furthermore, for short-term forecasts with
a 1-day horizon, WaveCatBoost demonstrates competitive performance with
transformer-based architectures. Additionally, the MASE metric for ID1 sensors
in Table I indicates that our proposed model outperforms all baseline models
across different horizons for all air pollutants except for SO2, where XGB and
CatBoost models offer competitive results.
Furthermore, we study the statistical significance of the improvement in model
performance using multiple comparisons with the best (MCB) test procedure
[15]. This distribution-free test ranks each model based on its MASE metric
across different air pollutants and computes their average rank. The model
with the lowest MASE metric is identified as the “best” method, determined by
achieving the minimum rank. The MCB plots depicted in Fig. 3 illustrate that
WaveCatBoost models exhibit superior performance for both the ID1 (Fig. 3a)
and CPCB (Fig. 3b) sensors. Additionally, the boundary of the critical
distance (blue line) of the best-performing model serves as the reference
value (shaded region) for the test. Given that most of the baseline
forecasters have non-overlapping critical distance (blue line) with the
reference value of the test, we can conclude that their performance is
significantly inferior to that of the proposed WaveCatBoost architecture. The
overall experimental analysis underscores the effectiveness of our proposed
model in outperforming a diverse range of benchmark models, emphasizing its
potential as a reliable tool for real-time air quality forecasting. To
quantify the uncertainty in the WaveCatBoost forecasts, we utilize the
conformal prediction approach as in Eq. 3. Figure 4 and 5 showcase the point
forecast generated by the WaveCatBoost model, ground truth, and the conformal
prediction intervals at $\alpha=0.05$ for the 1-day and 7-day horizon of ID1
sensors, respectively (as it gives a probabilistic band, therefore sometimes
the band consists of negative values which for this case should be considered
as zero). From the plot, it is evident that the proposed framework is
generalizable and robust, offering valuable insights for environmental
monitoring and public health interventions.
## IV Conslusion
In this letter, we designed a WaveCatBoost model to forecast real-time air
pollutant concentration levels. These air pollutants, including gaseous and
particulate matter, lead to several carcinogenic and non-carcinogenic health
hazards. Our architecture integrates wavelet decomposition with the CatBoost
approach to effectively capture the non-stationary and long-term dependencies
inherent in pollutant time series data. Experimental evaluation using two
real-world datasets from Meghalaya, India, demonstrates that our proposal
outperforms baseline forecasting methods. Furthermore, we assess the
robustness of the model through statistical significance tests and provide
probabilistic bands for our forecasts based on a conformal prediction
approach. These findings are crucial for advancing the field of air quality
forecasts and guiding future research endeavors to enhance predictive models
for environmental sustainability. As a future direction, further analysis of
the WaveCatBoost architecture in forecasting air pollutant levels by
considering spatial dependencies emerges as a promising avenue in air quality
research.
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# A SPA-based Manifold Learning Framework for Motor Imagery EEG Data
Classification11footnotemark: 1
Xiangyun Lia Peng Chenb<EMAIL_ADDRESS>Zhanpeng Baob West China
Biomedical Big Data Center, West China Hospital, Sichuan University,Chengdu
610041, PR China School of Mechanical Engineering, Southwest Jiaotong
University, Chengdu 610031, PR China
###### Abstract
The electroencephalography (EEG) signal is a non-stationary, stochastic, and
highly non-linear bioelectric signal for which achieving high classification
accuracy is challenging, especially when the number of subjects is limited. As
frequently used solution, classifiers based on multilayer neural networks has
to be implemented without large training data sets and careful tuning. This
paper proposes a manifold learning framework to classify two types of EEG data
from motor imagery (MI) tasks by discovering lower dimensional geometric
structures. For feature extraction, it is implemented by Common Spatial
Pattern (CSP) from the preprocessed EEG signals. In the neighborhoods of the
features for classification, the local approximation to the support of the
data is obtained, and then the features are assigned to the classes with the
closest support. A spherical approximation (SPA) classifier is created using
spherelets for local approximation, and the extracted features are classified
with this manifold-based method. The SPA classifier achieves high accuracy in
the 2008 BCI competition data, and the analysis shows that this method can
significantly improve the decoding accuracy of MI tasks and exhibit strong
robustness for small sample datasets. It would be simple and efficient to tune
the two-parameters classifier for the online brain-computer
interface(BCI)system.
###### keywords:
Motor imagery electroencephalography , Brain-computer interface , Manifold
learning , Local manifold approximation , Classification algorithm
## 1 Introduction
Electroencephalography (EEG)-based brain-computer interface (BCI) technology
is a promising tool that enables users to use their brain activity to directly
control or interact with an external environment or device [1], especially for
persons with severe motor-related diseases [2]. Despite its low spatial
resolution, EEG is the most widely used technique in the field of BCI due to
its good temporal resolution [3], low equipment cost, and high compatibility
[4]. The BCI technique has been studied via EEG signals arising in different
physiological mechanisms, such as motor imagery (MI) [5], steady-state visual
evoked potentials (SSVEP) [6], and P300 [7]. Compared with BCI based on SSVEP
or P300, MI methods may have higher potential because they are independent of
external stimuli, thus allowing asynchronous control and communication, and
facilitating the application of BCI system [8]. Although there are slight
differences of EEG patterns between, recent advances in this field show that
patients can produce a cortical activation similar to healthy individuals
during the imagined movements [9]. Therefore, the MI-based BCI system is an
effective way to promote recovery during rehabilitation interventions, which
is particularly suitable for tetraplegic individuals to restore neuromuscular
connectivity and motor function [10]. The goal of MI-based system is to
provide a neurophysical communication channel between people and external
devices.For safety purpose of related technologies, noninvasive EEG-based BCIs
are widely used, in such applications as word spellers [11], wheelchair
control [12], and video games [13]. In addition, non-invasive BCI may also be
used in evaluating the brain activity of severely paralyzed patients to
predict the efficiency of invasive brain-computer interfaces [14]. For healthy
persons, BCI can also greatly enhance the experience of multimedia and video
games and thus have have vast commercial value [15].
The major challenges in a typical MI task are the effective extraction of EEG
features and and their proper classification. EEG signals is usually
represented as high dimensional data whose classification is prone to
overfitting and bias, especially for small training datasets [16].
Dimensionality reduction is an effective method to solve these problems for
obtaining a more compact representation of the high-dimensional space.
Manifold learning [17] is one of the most important nonlinear dimensionality
reduction techniques, and the goal of most manifold learning algorithms is to
obtain low-dimensional embedding of high-dimensional data so that the proximal
data points from the high-dimensional space remain close to each other as in
the high-dimensional space, so are the distant data points [18].
With the advantages promoting the classification performance with the limited
data settings due to its simple and direct mathematical representations, many
studies have recently proposed manifold learning algorithms for BCI
applications,and EEG classification on high-dimensional Riemannian manifold
has recently received increasing attention [19]. Barachant et al. [20]
proposed two classification algorithms, one of which introduces the concept of
Riemannian geodesic distance to compare the minimum Riemannian distance
between unlabeled data points and the Riemannian mean of labeled data points.
The other algorithm maps all data points in the Riemannian manifold into its
tangent space (TS),called the optimal hyperplane for classification [21],and
then applies the linear classification method on the tangent space. It is
different from the method of global sampling manifold information in reference
[20]. Xie et al used a simple but effective bilinear sub-meridian learning
(BSML) algorithm [18] in which the intrinsic sub-meridians are learned by
identifying a bilinear mapping that maximally preserves the local geometry and
overall structure of the original manifold for dimensionality reduction of the
SPD matrix space in the motor imagery BCIs. Li et al. [22] proposed a local
manifold approximation classification, which is mainly based on obtaining a
local approximation to the support of the data in each class within the
neighborhood of the feature to be classified, and assigning the feature to the
class with the closest support. The classifier performs a local approximation
using easily fitted spheres to result in a spherical approximation (SPA)
classifier.
Unlike general machine learning tasks, classification tasks for BCI often lack
sufficient labeled training data because of the limited number of subjects. In
practice, It is usually that case there is an insufficient number or even no
training samples at all, known as the small training data set problem. If the
amount of available data is small, the model is difficult to generalize, and
the classification accuracy tends to be lowered [16]. There is significant
variation in EEG signals between different individuals, characterized by the
inconsistent distribution of data in feature space, which prevents prevent the
model transferred from one person to other [23]. Therefore, designing an
algorithm with high classification accuracy constrained by small training data
sets is a challenging task in the current BCI field.
The innovations and contributions of this paper include the following three
aspects:
1. 1.
The SPA classifier based on manifold learning is modified and applied to the
BCI system for decoding the features of MI-based EEG signals. In the case of
no geodesic information of the unknown manifold, SPA uses sphere for local
approximation, which is different from other manifold learning methods that
approximate geodesic distance from the global point of view. The sphere is a
simple geometric object that is easy to fit, and it also provides the
improvements of the hyperplane, so that the accuracy can be promoted by
approximating the local curvature. It is the first application of SPA in the
classification of EEG signals, especially for MI data.
2. 2.
Compared with the usual algorithms for a large number of training samples,
This algorithm is suitable for the system such as BCI which lacks enough
training data, and can still present effective feature decoding performance of
EEG signals for a small number of samples with the SPA classifier, as
demonstrated by the experimental results. In clinical rehabilitation training,
such a classifier could work with the limited number of subjects.
3. 3.
The design of the SPA classifier is simple, and there are only two parameters
needed to be tuned. The main parameters need to be adjusted are the size of
the local neighborhoods K and the dimension of the manifold approximating P.
After these two parameters are determined, the algorithm execution will become
very fast, and become suitable for online BCI systems requiring high real-time
performance.
The rest of this article is divided into the following sections. Section 2
describes the MI task experimental example and basic information on the EEG
data set for the 2008 BCI competition, as well as the preprocessing methods
for EEG signals, and presents the Common Space Pattern Feature Extraction
algorithm for MI-based classification problems. Section 3 describes the SPA
classifier based on manifold learning in detail. In Section 4, the
experimental results of this algorithm and other commonly used algorithms are
compared for evaluation and discussion.In the end, Section 5 provides
conclusions and outlines the future applications of SPA classifiers.
## 2 Motor imagery EEG signals classification preparation
### 2.1 Introduction to the BCI competition dataset
The dataset used in this paper is from the BCI Competition IV Dataset 2a,
which consisted of four MI tasks: motor imagery of the left hand, right hand,
both feet, and tongue for nine subjects. EEG signals from 22 Ag/AgCl
electrodes and 3 monopolar electrooculography (EOG) channels (with the left
mastoid as reference) were recorded at a sampling frequency of 250 Hz. The EEG
signal is band-pass filtered between 0.5 and 100 Hz, and the power line
interference is filtered by a 50 Hz notch filter. The timing scheme of the
experimental paradigm is shown in Figure 1. More detailed information about
the EEG experiment can be found in [24]. Taking into account the non-
stationary of the EEG data, the EEG data consists of two sessions, recorded on
different dates. Each session contains a total of 288 trials, and each type of
task is executed 72 times. It indicates that the dataset includes contaminated
data and such data contains some artifacts. However, these contaminated
samples are usually not removed to examine the robustness of the proposed
algorithm. In the experiment, only the data related to the motor imagery of
the left hand and right hand were extracted from the dataset, while the motor
imagery data of the tongue and feet were excluded.
Figure 1: BCI competition IV dataset 2a experimental paradigm
### 2.2 EEG data preprocessing process
#### 2.2.1 Time Interception
The MI task requires subjects to repeatedly imagine limb movements for a
certain amount of time to generate stable and effective brain activity [25].
For the continuously recorded EEG signals, it only focuses on the motor
imagery part and then segments the band-pass filtered continuous EEG signals
through the time window. The segmented time length corresponds to a single
motor imagery trial. Besides, this study used signals between 500 ms and 2500
ms after the stimulus occurred and processed only the intercepted segments.
#### 2.2.2 Band-pass filter
During motor imagery, event-related synchronization and desynchronization
(ERS/ERD) phenomena occur mainly in the 8-30 Hz band, covering both mu (8-13
Hz) and beta (14-28 Hz) rhythms [26]. Typically, a bandpass filter is used to
highlight the frequency components of interest while reducing some artifacts.
In this study, a fifth-order Butterworth bandpass filter was designed, which
eliminates most of the muscle activity, such as EMG (above 35 Hz) caused by
swallowing or facial movements. It also removes some ocular artifacts that
mainly affect the low-frequency components of the EEG signal, such as the EOG
(below 8 Hz) generated by blinking and motion.
### 2.3 Common Spatial Pattern (CSP) for Feature Extraction
At present, the spatial domain-based feature extraction method has become the
main feature extraction method for MI EEG signals, which is represented by the
common spatial pattern (CSP). CSP was first proposed by Fukunaga in 1970 and
then introduced into the analysis and processing of EEG signals by Koles [27],
and it has made a very large number of applications on MI EEG signals [28].
CSP is a supervised learning method, which requires the training data to be
labeled. CSP is mainly a feature extraction algorithm for two classification
tasks, capable of extracting the spatial distribution components of each class
from multi-channel brain-computer interface data.
Suppose $X_{1}$ and $X_{2}$ are the multi-channel evoked response time-space
signal matrices under the binary classification MI tasks, respectively. The
dimensions of $X_{1}$ and $X_{2}$ are all N*T, where N is the number of EEG
channels and T is the number of samples collected per channel.
To calculate its covariance matrix, assume that N<T. The normalized covariance
matrices $R_{1}$ and $R_{2}$ of $X_{1}$ and $X_{2}$ are:
$R_{1}=(X_{1}X_{1}^{T})/(trace(X_{1}X_{1}^{T}))R_{2}=(X_{2}X_{2}^{T})/(trace(X_{2}X_{2}^{T}))$
(1)
Then, the mixed space covariance matrix R is calculated by:
$R=\overline{R_{1}}+\overline{R_{2}}$ (2)
Perform the eigenvalue decomposition on the mixed space covariance matrix R:
$R=U\lambda U^{T}$ (3)
Arrange the eigenvalues in descending order, and calculate the whitening value
matrix P:
$P=\sqrt{\lambda^{-1}}U^{T}$ (4)
Then, the normalized covariance matrices $R_{1}$ and $R_{2}$ are transformed
as follows:
$S_{1}=PR_{1}P^{T}\quad S_{2}=PR_{2}P^{T}$ (5)
Next, do the principal component decomposition for $S_{1}$ and $S_{2}$:
$S_{1}=B_{1}\lambda_{1}B_{1}^{T}\quad S_{2}=B_{2}\lambda_{2}B_{2}^{T}$ (6)
The above formula proves that the eigenvector matrices of matrices $S_{1}$ and
$S_{2}$ are equal, i.e., $B_{1}$=$B_{2}$. Sort the eigenvalues in
$\lambda_{1}$ in descending order, and the corresponding eigenvalues in
$\lambda_{2}$ in ascending order. The transformation of the whitened EEG
signal with the eigenvectors corresponding to the largest eigenvalues in
$\lambda_{1}$ and $\lambda_{2}$ is optimal for separating the variances in two
signal matrices. The spatial filter corresponding to the projection matrix W
is:
$W=B^{T}P$ (7)
For the test data $X_{i}$, the feature vector f is extracted as follows:
$\left\\{\begin{array}[]{c}Z_{i}=W*X_{i}\\\
f_{i}=\log\left(\frac{{VAR}\left(Z_{i}\right)}{{sum}\left({VAR}\left(z_{i}\right)\right)}\right)\end{array}\right.$
(8)
The work flow of the system is shown as Figure 2, and the BCI competition IV
Dataset 2a with relatively few dimensions is utilized for classification.
Figure 2: Workflow of the system for EEG data classification
## 3 Principle and Implementation of SPA Classifier
The basic idea of manifold learning is to assume that the dataset under study
is uniformly sampled on a manifold structure in the high-dimensional data
space, then identify the low-dimensional manifold structure hidden in the
high-dimensional observation data space while keeping the neighborhood
relationship between the data sets unchanged, and the nonlinear mapping
relationship from the high-dimensional observation space to the low-
dimensional embedding space is constructed for dimension reduction or
visualization.
Manifold learning has the description [29]: given a dataset
$X=\left\\{x_{i},i=1,\ldots,N\right\\}\subset R^{m}$, assume that the samples
in X are generated from a low-dimensional dataset Y through an unknown
nonlinear transformation f, i.e.:
$x_{i}=f\left(y_{i}\right)+\varepsilon_{i}$ (9)
Among them, $\varepsilon_{i}$ denotes the noise, $y_{i}\in Y\subset R^{d}$,
$d\gg m$, and $f:R^{d}\rightarrow R^{m}$ is the embedding map of $C^{\infty}$.
Thus, the purpose of manifold learning is to get low-dimensional expressions
based on the given observed dataset X:
$Y=\left\\{y_{i},i=1,\ldots,N\right\\}\subset R^{d}$ (10)
Although many global and local methods have been proposed to identify low-
dimensional embeddings, few of them sample the manifold information from the
original data. Most global methods perform low-dimensional embedding by
approximating the geodesic distance without the geodesic information of the
unknown manifold, which can lead to bias [17]. The SPA method introduced in
this paper mainly uses local manifold approximation and easy-to-fit spheres
for local manifold approximation to classify the MI data. The following
describes the details of the SPA classifier.
For the binary classification problem of MI, suppose there are two types of MI
data with different types of motions, labeled as 1 and 2, respectively. The
features of the MI data for the two different types of motions tend to
approach two different manifolds $M_{1}$ and $M_{2}$, and both of which are
embedded in $R^{D}$ with intrinsic dimension $p<D$. The two manifolds of the
EEG signals might be highly nonlinear and complex, even have different
curvatures or even gaps.
For a given test sample x, the distances between the sample and the two
manifolds, denoted by $d_{1}$ and $d_{2}$, are first calculated, and then the
sample is assigned to the group with a shorter distance. Actually, the two
manifolds $M_{1}$ and $M_{2}$ are unknown, but the training data contains n
different sample data and the corresponding labels. With this known
information, it is possible to obtain an exact local approximation of the
manifolds $M_{1}$ and $M_{2}$ in a high-order domain of a feature x to be
processed. In this way, a wide variety of local approximations can be
considered, and the limited training data can be used completely and
efficiently.
Algorithm 1: SPCA(Spherical Principal Components Analysis)algorithm
---
1 Input: $X=\left\\{x_{i}\right\\}_{i=1}^{n}$, Dimension of manifold p
2 Output: Sphere $S_{P}(V,c,r)$
3 $V=\left(v_{1},\ldots,v_{p+1}\right)$, Eigenvalues are in decreasing order,
where $v_{j}$ is the j-th eigenvector of the covariance matrix
4 $\xi_{i}=\bar{X}+V\bar{V}\left(X_{i}-\bar{X}\right)$
5
$c=-\frac{1}{2}\left\\{\sum_{\mathrm{i}=1}^{n}\left(\xi_{i}-\bar{\xi}\right)\left(\xi_{i}-\bar{\xi}\right)^{\top}\right\\}^{-1}\left\\{\sum_{\mathrm{i}=1}^{n}\left(\xi_{i}^{\top}\xi_{i}-\frac{1}{n}\sum_{j=1}^{n}\xi_{j}^{\top}\xi_{j}\right)\left(\xi_{i}-\bar{\xi}\right)\right\\}$
6 $r=\frac{1}{n}\sum_{\mathrm{i}=1}^{n}\left\|\xi_{i}-c\right\|$
The algorithm regards the local spherical approximation as the sphere which is
a simple geometric object easy to fit, and the sphere provides a hyperplane
generalization that can significantly improve accuracy by approximating local
curvature. First, the center, radius and size of each sphere are optimized to
provide the best local approximation. Then, the distances $d_{1}$ and $d_{2}$
between the test sample x and the two manifolds can be easily calculated based
on the local spherical approximation. The sample to be classified by the
distance to different manifolds embed in 3D space is shown as Figure 3.
Figure 3: Sample to be classified by the distance to different manifolds
Let $X_{[k]}^{l}$ be the k-nearest neighbors between x samples with label $l$.
The points are fitted to a sphere using the spherical principal component
analysis (SPCA) algorithm to obtain a local manifold approximation
$\widehat{M_{l}}$ around x in class l. Then, the value of $\mathrm{d}_{l}$ is
approximated by $\widehat{\mathrm{d}_{l}}:=d\left(x,\widehat{M_{l}}\right)$,
and the label y is selected as the value of $l$. The SPCA algorithm produces
an estimated p-dimensional sphere $S_{P}(V,c,r)$ with center c and radius r in
the subspace V when applied to data $X=\left\\{x_{i}\right\\}_{i=1}^{n}$.
Algorithm 2: SPA(SPherical Approximation)algorithm
---
1 Input: training dataset$\left\\{x_{i},y_{i}\right\\}_{i=1}^{n}$, parameters
k and p,test data x
2 Output: Prediction y of test data x
3 Let L be the number of array
4 for l=1:L
5 Find the K-nearest neighbors of x under label l, denoted as
$X_{[k]}^{l}\subset\left\\{x_{i}\mid y_{i}=l\right\\}$
6 Compute the p-dimensional spherical approximation using Algorithm 1
7 Compute the projection of x onto the sphere field S by :
$({\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\frown$}}\over{x}}^{l}})=Pro{j^{l}}(x)={c_{l}}+\frac{{{r_{l}}}}{{\left\|{{V_{l}}{V_{l}}^{\top}(x-{c_{l}})}\right\|}}{V_{l}}{V_{l}}^{\top}(x-{c_{l}})$
8 Calculate the distance $\widehat{\mathrm{d}}_{l}$ between x and the sphere
$S_{p}^{l}\left(V_{l},c_{l},r_{l}\right):\widehat{\mathrm{d}}_{l}=\left\|x-\widehat{x}^{\imath}\right\|$
9 End
10 Assign x to the group with the smallest distance:
$y=\mathop{argmin}\limits_{l=1,\ldots,L}\widehat{{{\rm{d}}_{l}}}$
The overall process of the SPA classifier is to first obtain the parameters V,
c and r of the spherical classifier by Algorithm 1 (SPCA algorithm), and then
implement the classification by Algorithm 2 (SPA algorithm).
The SPA classifier is simple to implement and efficient to perform, which
could detect non-linear support differences between groups. The parameters to
be tuned for the SPA classifier are the size of the local neighborhood k, and
the dimension p that approximates the data denoising support. When these two
parameters are fixed, the algorithm becomes fast.
For the selection of parameters k and p, k and p are brought in from a certain
range, and the highest accuracy is the accuracy of the test. For the selection
of p, the interval is 1,2,3,4; For the selection of k, the minimum value is
set to 8, increasing in the order of 1, and the maximum value is set to the
number of the small number of the two types of samples. If the number is too
large, the maximum value is set to 46. If p is greater than the number of a
small number of samples, the algorithm will fail.
The SPA classifier theoretically guarantees the impact of the growth of the
number of training samples n on the classification performance. Theorem 1,
which corresponds to the data in the presence of noise, is given below.
Theorem 1: Let $M_{1}$ and $M_{2}$ correspond to two Compact Riemannian
Manifolds. Assume that
$\left\\{z_{i}\right\\}_{i=1}^{n}{}_{\sim}{}^{iid}\rho$, $supp(\rho)=M_{1}\cup
M_{2}$, $z_{i}\in M_{yi}$ and $x_{i}=z_{i}+\epsilon_{i}$, where
$\epsilon_{i}\sim N\left(0,\sigma^{2}I_{D}\right)$. Given a test sample x with
label y, let the SPA classifier obtain the predicated label $\widehat{y_{n}}$.
Let $B_{\delta}(M):=\\{x\mid\mathrm{d}(x-n)<\delta\\}$ and $\delta>0$, then:
$\lim_{n\rightarrow\infty}{P}\left(y\neq\widehat{y_{n}}\right)\leq\rho\left(B_{\delta}\left(M_{1}\right)\cap
B_{\delta}\left(M_{2}\right)\right)+\exp\left\\{-\frac{\delta^{2}}{8\sigma^{2}}+\frac{D}{2}\log\left(\frac{\delta^{2}}{4\sigma^{2}}\right)-\frac{D}{2}(\log(D)-1)\right\\}$
(11)
In theorem 1, the data in class $l$ are distributed around $M_{l}$ as Gaussian
noise. This theorem shows that as the size of training sample n increases, the
probability of the algorithm producing an error class label is gradually
bounded by:
$\rho\left(B_{\delta}\left(M_{1}\right)\cap
B_{\delta}\left(M_{2}\right)\right)+\exp\left\\{-\frac{\delta^{2}}{8\sigma^{2}}+\frac{D}{2}\log\left(\frac{\delta^{2}}{4\sigma^{2}}\right)-\frac{D}{2}(\log(D)-1)\right\\}$
(12)
When the noise level attenuates to 0, i.e., $\sigma\rightarrow 0$. Let
$\delta=\sqrt{\sigma}\rightarrow 0$, so the first term converges to the
$\rho\left(M_{1}\cap M_{2}\right)$, which is the intersection region between
$M_{1}$ and $M_{2}$ are assigned by the probability p. Since
$\frac{\delta^{2}}{\sigma^{2}}\rightarrow\infty$, the second term converges to
0, which is the case that the noise is zero. The probability of correct
classification by the algorithm is ultimately greater than
$1-\rho\left(M_{1}\cap M_{2}\right)$. As the deduction, the limit is 1 when
$\rho\left(M_{1}\cap M_{2}\right)=0$, which means the SPA classifier has
perfect classification performance when it has enough training samples, as
long as the classes are geometrically separable. Theorem 1 takes into account
the noise, which is more practical in most applications.
$\frac{\delta^{2}}{\sigma^{2}}$ can be considered as the signal-noise ratio.
The larger the $\frac{\delta^{2}}{\sigma^{2}}$ , the better the performance of
the SPA classifier.
## 4 Results Evaluation and Discussion
The Accuracy [30] denotes the percentage of the number of correctly classified
samples to the total number of samples, and is calculated as:
$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}\times 100\%$ (13)
In the above formula, True Positive (TP) indicates the number of samples with
positive category and positive classification prediction. True Negative (TN)
indicates the number of samples with negative category and negative
prediction. False Negative (FN) indicates the number of samples with positive
category and negative prediction. And False Positive (FP) indicates the number
of samples with negative category and positive prediction.
Algorithm evaluation: The algorithm proposed in this paper is evaluated by
comparing the CSP+SPA algorithm with the following five algorithms.
1. 1.
CSP+LDA: Motion imagery classification using CSP and LDA algorithm [31].
2. 2.
CSP+SVM: Motion imagery classification using CSP and SVM algorithm [32].
3. 3.
MDRM: The minimum distance to the Riemannian mean is used for classification
on high-dimensional Riemannian manifolds [20].
4. 4.
TS+LDA: LDA was applied to the high-dimensional tangent space for
classification [20].
5. 5.
TS+SVM: SVM was applied to the high-dimensional tangent space for
classification [33].
Parameter Settings: In the TS+LDA algorithm, since the tangent space is
$m=n\times(n+1)/2$-dimensional space, there may be a problem that the sample
dimension exceeds the number of trials per class, and the regularized
classification algorithm is usually used to solve this problem. In the TS+SVM
algorithm, the radial basis function kernel (RFB) is selected as the SVM
kernel[32].
The following two experiments are designed for the BCI competition data,
Experiment 1 for the normal number of samples and Experiment 2 for the small
training dataset.
Since cross-validation allows testing the model in the training phase, in this
paper, the performance of the proposed algorithm is first tested on the BCI IV
Dataset 2a using a 10-fold cross-validation procedure. The BCI IV Dataset 2a
is randomly divided into 10 subsets of equal size. In each run, 9 subsets are
elected for training and 1 subset is left for testing.
Table 1: 10-fold cross-validation results (%) Subject | Method
---|---
TS+LDA | TS+SVM | MDRM | CSP+SPA | CSP+SVM | CSP+LDA
S01 | 84.29 | 86.07 | 80.71 | 88.93 | 87.86 | 82.14
S02 | 59.64 | 60.36 | 54.29 | 70.36 | 58.57 | 59.29
S03 | 96.43 | 96.07 | 91.07 | 97.50 | 93.93 | 94.29
S04 | 81.07 | 82.50 | 76.07 | 83.93 | 76.79 | 75.71
S05 | 59.64 | 60.00 | 57.50 | 71.43 | 61.07 | 60.71
S06 | 69.29 | 68.93 | 68.21 | 73.93 | 68.93 | 68.21
S07 | 80.00 | 81.07 | 75.36 | 90.36 | 83.57 | 82.14
S08 | 95.00 | 94.64 | 92.14 | 98.57 | 95.36 | 94.64
S09 | 86.07 | 85.36 | 77.14 | 86.43 | 82.50 | 81.79
Mean | 79.05 | 79.44 | 74.72 | 84.60 | 78.73 | 77.66
Std | ($\pm$)12.85 | ($\pm$)12.70 | ($\pm$)12.34 | ($\pm$)10.04 | ($\pm$)12.68 | ($\pm$)12.20
BCI IV Dataset 2a is a four-category dataset, and this paper focuses on
extracting the motion imagery of the left hand and right hand in the dataset.
The classification accuracies of all algorithms are given in Table 1, and the
classification accuracy of CSP+SPA is the highest on each subject. CSP+SPA,
TS+LDA and TS+SVM not only have higher average classification accuracy but
also have a lower standard deviation of classification accuracy. Among them,
CSP+SPA has the lowest standard deviation, which indicates that the proposed
SPA classifier is more robust to the variance of subjects than the other
algorithms. The above results show that the SPA classifier works well for
classifying the data of BCI IV Dataset 2a, which reflects the effectiveness of
the algorithm.
For many reasons, the training set available in BCI applications is usually
quite small. Reducing the number of training trials required for a specific
task is an important goal for BCI, so the experiment is set up to evaluate the
performance of the SPA classifier on small datasets. In the experiment, the
training and testing samples of each subject are aggregated into a total
sample set for that subject, from which different proportions of samples are
selected for the experiment. 1/2, 1/3, 1/6, 1/12 of the BCI IV Dataset 2a
(i.e., 144, 96, 48, and 24 trials randomly selected) were used as training
samples in the evaluation, and 20 replicate experiments were conducted each
time. Table 2 records the average classification accuracy with 1/2 and 1/6 of
the total data as the training dataset. As the training sample size decreases
from 1/2 to 1/6, the performance of all algorithms decreases. However,
compared to other methods, the proposed method demonstrates less performance
degradation and has the highest classification accuracy in most subjects.
Table 2: Accuracy in 1/2 and 1/6 training sets results (%) Sample Number | Method | Mean | subject
---|---|---|---
S01 | S02 | S03 | S04 | S05 | S06 | S07 | S08 | S09
1/2 Training Sample (144 trials) | TS+LDA | 75.65 | 82.08 | 56.84 | 92.74 | 76.11 | 57.08 | 64.72 | 76.49 | 92.15 | 82.60
TS+SVM | 74.57 | 80.20 | 55.83 | 92.66 | 73.84 | 56.36 | 63.64 | 74.88 | 91.67 | 82.02
MDRM | 71.32 | 81.25 | 47.78 | 90.38 | 66.18 | 52.22 | 60.07 | 75.94 | 91.32 | 76.74
CSP+SPA | 77.94 | 85.69 | 60.59 | 94.83 | 75.73 | 61.28 | 65.49 | 81.77 | 95.76 | 80.35
CSP+SVM | 75.25 | 84.39 | 54.86 | 93.96 | 72.36 | 53.91 | 63.47 | 79.67 | 93.91 | 80.73
CSP+LDA | 74.15 | 78.92 | 55.52 | 92.99 | 72.19 | 55.03 | 63.54 | 78.33 | 91.46 | 79.38
1/6 Training Sample (48 trials) | TS+LDA | 70.26 | 77.21 | 53.21 | 89.40 | 66.10 | 53.00 | 60.13 | 66.02 | 88.19 | 79.10
TS+SVM | 67.90 | 72.62 | 55.27 | 85.60 | 60.51 | 54.46 | 58.31 | 60.01 | 85.33 | 78.94
MDRM | 68.64 | 79.15 | 49.40 | 89.90 | 58.54 | 49.67 | 57.56 | 67.77 | 88.50 | 77.31
CSP+SPA | 72.79 | 80.25 | 56.96 | 91.50 | 68.52 | 56.42 | 59.23 | 74.27 | 91.94 | 76.00
CSP+SVM | 71.35 | 79.16 | 54.75 | 90.77 | 64.07 | 51.61 | 58.80 | 71.35 | 91.34 | 80.28
CSP+LDA | 65.62 | 67.71 | 51.67 | 84.04 | 61.52 | 51.48 | 55.48 | 66.23 | 83.79 | 68.67
To further analyze the effect of the number of training samples on the
performance of different subjects, the average classification accuracy of
different subjects in the BCI IV Dataset 2a with the training set percentage
of 1/2, 1/3, 1/6 and 1/12 is given. The line charts of the well-performing
subject S08 and the poor-performing subject S06 are given below.
Figure 4: Classification accuracy of S08 Figure 5: Classification accuracy of
S05
As can be seen in Figure 2, the classification accuracy of the SPA classifier
and the other algorithms on the well-performing subject S08 increases as the
proportion of the training set increases. It is also obvious from the figure
that the SPA classifier has the best performance in different proportions of
the datasets. In Figure 3, there are cases where the classification accuracy
increases despite a small proportion, which is due to the randomness of the
poor performance of the subjects.
## 5 Conclusion and Future Research
MI does not rely on any external visual or auditory stimuli, which facilitates
the implementation of portable BCI systems. It has broad application prospects
in the fields of neural engineering, robotics, and rehabilitation engineering.
Our work proposed a new classifier based on the spherical local manifold
approximation, and optimized it for the best size of the local neighborhoods
and the dimension of manifold facing the application of BCI systems.
To evaluate the performance of the proposed classification method, it runs on
the 2008 BCI competition dataset. The experimental results demonstrate the
effectiveness of the method in decoding and classifying two types of MI tasks,
and the proposed method has higher decoding accuracy and shows strong
robustness for small training datasets compared to other algorithms. The
future work will focus on applying and improving the SPA classifier on the
online rehabilitation devices for clinical experiments.
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# VisualCheXbert: Addressing the Discrepancy Between Radiology Report Labels
and Image Labels
Saahil Jain<EMAIL_ADDRESS>Stanford UniversityUSA , Akshay Smit
<EMAIL_ADDRESS>Stanford UniversityUSA , Steven QH Truong
VinBrainVietnam , Chanh DT Nguyen , Minh-Thanh Huynh VinBrainVietnam ,
Mudit Jain USA , Victoria A. Young Stanford UniversityUSA , Andrew Y. Ng
<EMAIL_ADDRESS>Stanford UniversityUSA , Matthew P. Lungren
<EMAIL_ADDRESS>Stanford UniversityUSA and Pranav Rajpurkar
<EMAIL_ADDRESS>Stanford UniversityUSA
(2021)
###### Abstract.
Automatic extraction of medical conditions from free-text radiology reports is
critical for supervising computer vision models to interpret medical images.
In this work, we show that radiologists labeling reports significantly
disagree with radiologists labeling corresponding chest X-ray images, which
reduces the quality of report labels as proxies for image labels. We develop
and evaluate methods to produce labels from radiology reports that have better
agreement with radiologists labeling images. Our best performing method,
called VisualCheXbert, uses a biomedically-pretrained BERT model to directly
map from a radiology report to the image labels, with a supervisory signal
determined by a computer vision model trained to detect medical conditions
from chest X-ray images. We find that VisualCheXbert outperforms an approach
using an existing radiology report labeler by an average F1 score of 0.14 (95%
CI 0.12, 0.17). We also find that VisualCheXbert better agrees with
radiologists labeling chest X-ray images than do radiologists labeling the
corresponding radiology reports by an average F1 score across several medical
conditions of between 0.12 (95% CI 0.09, 0.15) and 0.21 (95% CI 0.18, 0.24).
natural language processing, BERT, medical report labeling, chest X-ray
diagnosis
††journalyear: 2021††copyright: rightsretained††conference: ACM Conference on
Health, Inference, and Learning; April 8–10, 2021; Virtual Event,
USA††booktitle: ACM Conference on Health, Inference, and Learning (ACM CHIL
’21), April 8–10, 2021, Virtual Event, USA††doi:
10.1145/3450439.3451862††isbn: 978-1-4503-8359-2/21/04††ccs: Computing
methodologies Natural language processing††ccs: Computing methodologies
Information extraction
Figure 1. The VisualCheXbert training procedure. VisualCheXbert uses a
biomedically-pretrained BERT model to directly map from a radiology report to
the labels obtained by a radiologist interpreting the associated X-ray image.
The training procedure for VisualCheXbert is supervised by a computer vision
model trained to detect medical conditions from chest X-ray images.
VisualCheXbert architecture
## 1\. Introduction
Because manually annotating a large number of medical images is costly (Nguyen
et al., 2021; Bustos et al., 2020; Abràmoff et al., 2016; Gulshan et al.,
2016; Shih et al., 2019; Wang et al., 2020; Demner-Fushman et al., 2016), an
appealing solution is the use of automatic labelers to extract labels from
medical text reports that accompany the images. On the task of chest X-ray
interpretation, high-performing vision models have been successfully trained
(Rajpurkar et al., 2020a; Pham et al., 2020; Ye et al., 2020; Rajpurkar et
al., 2020b; Tang et al., 2020b; Rajpurkar et al., 2020c) on large, publicly
available chest X-ray datasets (Irvin et al., 2019; Johnson et al., 2019; Wang
et al., 2017; Phillips et al., 2020) labeled by automated radiology report
labelers (Irvin et al., 2019; Peng et al., 2017; McDermott et al., 2020; Smit
et al., 2020). However, training these vision models on labels obtained from
reports assumes that the report labels are good proxies for image labels.
Prior work has found that report labels may not accurately reflect the visual
content of medical images (Oakden-Rayner, 2019; Olatunji et al., 2019; Tang et
al., 2020a).
We investigate this assumption in the setting of automated chest X-ray
labeling and develop methods to produce labels from radiology reports that
better agree with radiologists labeling the corresponding X-ray images. Our
primary contributions are:
1. (1)
We quantify the agreement between radiologists labeling reports and
radiologists labeling images across several medical conditions. We find that
there is significant disagreement between board-certified radiologists when
labeling a chest X-ray image and when labeling the corresponding radiology
report.
2. (2)
Upon board-certified radiologist review of examples of disagreements between
radiologists labeling reports and radiologists labeling images, we find
various reasons for disagreement related to (a) label hierarchy relationships,
(b) access to clinical history, (c) the use of the Impression and Findings
section of radiology reports, and (d) the inherent noise of the labeling task.
3. (3)
We find many significant relationships between presence of conditions labeled
using reports and presence of conditions labeled using images. We report and
clinically interpret various radiology report labels that increase (or
decrease) the odds of particular conditions in an image with statistical
significance.
4. (4)
We learn to map textual radiology reports directly to the X-ray image labels.
Our best performing method, called VisualCheXbert, uses a biomedically-
pretrained BERT model to directly map from a radiology report to the image
labels. We find that VisualCheXbert better agrees with radiologists labeling
chest X-ray images than do radiologists labeling the corresponding radiology
reports by an average F1 score across several medical conditions of between
0.12 (95% CI 0.09, 0.15) and 0.21 (95% CI 0.18, 0.24). We also find that
VisualCheXbert outperforms an approach using the CheXpert radiology report
labeler (Irvin et al., 2019) by an average F1 score of 0.14 (95% CI 0.12,
0.17).
We expect that our methods of addressing the discrepancy between medical
report labels and image labels are broadly useful across the medical domain
and may facilitate the development of improved medical imaging models.
## 2\. Data
We made use of two large publicly available datasets of chest X-rays: CheXpert
(Irvin et al., 2019) and MIMIC-CXR (Johnson et al., 2019). For both datasets,
we use the Impression section of the radiology reports, which summarizes the
key findings in the radiographic study. Each of the X-rays in these datasets
was labeled for 14 commonly occurring medical conditions. CheXpert consists of
224,316 chest radiographs, with labels generated from the corresponding
radiology report impression by the automatic, rules-based CheXpert labeler.
Given a radiology report impression as input, the CheXpert labeler labels each
medical condition (except “No Finding”) as “positive”, “negative”, “uncertain”
or “blank”. A “blank” label is produced by the CheXpert labeler if the
condition was not mentioned at all in the report impression. If the condition
was mentioned but its presence was negated, a “negative” label is produced. If
the condition was mentioned but its presence was uncertain, an “uncertain”
label is produced. For “No Finding”, the CheXpert labeler only produces
“positive” or “blank” labels. “No Finding” is only labeled as “positive” if no
medical abnormality whatsoever was mentioned in the report impression. The
MIMIC-CXR dataset consists of 377,110 chest X-rays and their corresponding
radiology reports, and it has also been labeled by the CheXpert labeler.
The CheXpert dataset contains a separate set of 200 chest X-ray studies called
the “CheXpert validation set” and another set of 500 chest X-ray studies
called the “CheXpert test set”. The CheXpert validation set is labeled by the
majority vote of 3 board-certified radiologists examining the X-ray images and
labeling each of the 14 conditions as “positive” or “negative”, similar to the
image ground truth on the CheXpert test set, which is described below. No
radiologist report labels are obtained for the validation set.
The CheXpert test set, which was collected by Irvin et al. (2019), is labeled
by radiologists in two distinct ways:
##### Image ground truth
5 board-certified radiologists looked at each X-ray image and labeled each of
the 14 conditions as “positive” or “negative”. The final label is their
majority vote. These radiologists only observed the X-ray images and did not
have access to the radiology report or patients’ historical records at the
time of image labeling.
##### Radiologist report labels
A board-certified radiologist looked at each radiology report impression
corresponding to the X-rays and labeled each of the 14 conditions as being
“positive”, “negative”, “uncertain”, or “blank”. This radiologist did not
observe any X-ray images. A condition was labeled as “blank” if it was not at
all mentioned in the report impression. If the condition was mentioned but its
presence in the chest X-ray was negated, then the condition was labeled as
“negative”. If the condition was mentioned but its presence was uncertain, it
was labeled as “uncertain”.
Table 1. Agreement between radiologists looking at reports and radiologists
looking at the corresponding X-ray images. The high and low scores are
obtained by mapping uncertain labels in the radiologist report labels to the
image ground truth labels and the opposite of the image ground truth labels
respectively.
| Condition
---
(n = # positive)
Low F1 | High F1 | Low Kappa | High Kappa
Atelectasis (n=153) | 0.230 | 0.595 | -0.014 | 0.457
Cardiomegaly (n=151) | 0.422 | 0.463 | 0.290 | 0.344
Edema (n=78) | 0.453 | 0.581 | 0.335 | 0.492
Pleural Effusion (n=104) | 0.638 | 0.710 | 0.511 | 0.613
Enlarged Cardiom. (n=253) | 0.089 | 0.208 | -0.053 | 0.097
Lung Opacity (n=264) | 0.683 | 0.686 | 0.401 | 0.405
Support Devices (n=261) | 0.863 | 0.863 | 0.737 | 0.737
No Finding (n=62) | 0.381 | 0.381 | 0.292 | 0.292
Average | 0.470 | 0.561 | 0.312 | 0.430
Weighted Average | 0.492 | 0.575 | 0.320 | 0.427
## 3\. Evaluation
We only evaluate our models on medical conditions for which at least 50 out of
the 500 chest X-ray studies in the CheXpert test set were marked positive by
the radiologists labeling the X-ray images (image ground truth). These
conditions, which we refer to as the evaluation conditions, are: Atelectasis,
Cardiomegaly, Edema, Pleural Effusion, Enlarged Cardiomediastinum, Lung
Opacity, Support Devices, and No Finding. We evaluate models using the average
and weighted average of the F1 score across conditions on the CheXpert test
set with the image ground truth. To compute the weighted average, each
condition is weighted by the portion of positive labels for that condition in
the CheXpert test set.
Table 2. Clinical explanations of disagreements between radiologists looking
at reports and radiologists looking at images on the CheXpert test set. Given
access to the X-ray image, the full radiology report, the radiology report
impression, the radiology report labels, and the image ground truth, a board-
certified radiologist explained disagreements between radiologist report
labels and the image ground truth. We show select examples with explanations
in this table.
Report Impression and Labels | Clinical Explanation
---|---
| 1\. single ap upright view of the chest showing a mildly increased
---
opacity at the left lung base that could represent atelectasis
versus consolidation.
Cardiomegaly Radiologist Report Label: Negative Image Ground Truth: Positive
| The radiologist looking at the report marks Cardiomegaly as
---
negative as it is not mentioned in the report. Since the image is a
Intensive Care Unit (ICU) film and cardiomegaly is not a clinically
relevant condition for the population selected for in ICU films, the
presence of cardiomegaly was never mentioned in the report,
resulting in the discrepancy between radiologists looking at the
report and radiologists looking at the image.
| 1\. pulmonary vascular congestion. left lower lobe opacity comp-
---
-atible with atelectasis and/or consolidation.
Cardiomegaly Radiologist Report Label: Negative Image Ground Truth: Positive
| Although cardiomegaly was mentioned in the radiology report
---
”Findings” section, cardiomegaly was not mentioned in the report
”Impression”. Since the radiologist looking at the report only had
access to the ”Impression” section, they labeled Cardiomegaly as
negative when it was actually present in the image.
| 1\. decreased pulmonary edema. stable bilateral pleural effusions
---
and bibasilar atelectasis.
Edema Radiologist Report Label: Positive Image Ground Truth: Negative
| The phrase ”decreased pulmonary edema” shows that the radiologist
---
writing the report had relevant clinical context, as the edema has
”decreased” compared to a previous report or image. However, the
radiologist looking at the image does not have this clinical context,
resulting in a discrepancy.
| 1\. single frontal radiograph of the chest is limited secondary to
---
poor inspiration and rotation.
2\. cardiac silhouette is partially obscured secondary to rotation.
lungs demonstrate bibasilar opacities, likely reflecting
atelectasis. possible small right pleural effusion. no pneumotho-
-rax.
3\. visualized osseous structures and soft tissues unremarkable.
Pleural Effusion Radiologist Report Label: Positive Image Ground Truth:
Negative
| The phrase ”possible small right pleural effusion” indicates the
---
uncertainty regarding the presence of pleural effusion. This natural
uncertainty may explain the disagreement between radiologists
looking at the image and radiologists looking at the report. On
review, it was noted that pleural effusion was borderline in this
example.
| 1\. crowding of the pulmonary vasculature. cannot exclude mild
---
interstitial pulmonary edema.
2\. no focal air space consolidation. the cardiomediastinal silhou-
-ette appears grossly within normal limits.
Pleural Effusion Radiologist Report Label: Negative Image Ground Truth:
Positive
| Upon review by a board-certified radiologist, there was an error in
---
the radiology report, which did not mention the presence of pleural
effusion. The error in the report itself may explain the disagreement
between the image and report labels.
## 4\. Experiments
### 4.1. Do radiologists labeling reports agree with radiologists labeling
X-ray images?
We first investigate the extent of the disagreement between board-certified
radiologists when labeling a chest X-ray image and when labeling the
corresponding radiology report.
#### 4.1.1. Method
We compute the level of agreement between radiologists labeling X-ray images
and radiologists labeling the corresponding radiology reports on the CheXpert
test set. The CheXpert test set contains a set of labels from radiologists
labeling X-ray images as well as another set of labels from radiologists
labeling the corresponding radiology reports. Using the labels from X-ray
images as the ground truth, we compute Cohen’s Kappa (Cohen, 1960) as well as
the F1 score to measure the agreement between these two sets of labels. To
compare the radiologist report labels to the image ground truth labels, we
convert the radiologist report labels to binary labels as follows. We map the
blank labels produced for the radiology report to negative labels. We map
uncertain labels to either the image ground truth label or the opposite of the
image ground truth label, and we record the results for both these strategies
to obtain “Low F1”, “High F1”, “Low Kappa”, and “High Kappa” scores. The low
and high scores represent the most pessimistic and optimistic mapping of the
uncertainty labels.
#### 4.1.2. Results
We find that there is significant disagreement, which is indicated by low
Kappa and F1 scores for almost all conditions evaluated. For example, Enlarged
Cardiomediastinum and No Finding have a relatively small “High Kappa” score of
0.097 and 0.292 and a “High F1” score of 0.208 and 0.381, indicating high
levels of disagreement even when assuming the most optimistic mapping of the
uncertainty labels. Atelectasis, Cardiomegaly, Edema, Pleural Effusion, and
Lung Opacity also have a low “High Kappa” score of 0.457, 0.344, 0.492, 0.613,
and 0.405 respectively and a “High F1” score of 0.595, 0.463, 0.581, 0.710,
and 0.686 respectively. Support Devices has the highest Kappa score, with a
“High Kappa” of 0.737, and the highest F1 score, with a “High F1” of 0.863.
The average Kappa score is between 0.312 and 0.430, and the average F1 score
is between 0.470 and 0.561. The low and high F1 / Kappa scores for the
evaluation conditions are shown in Table 1.
### 4.2. Why do radiologists labeling reports disagree with radiologists
labeling X-ray images?
We investigate why there is disagreement between board-certified radiologists
when labeling a chest X-ray image and when labeling the corresponding
radiology report.
#### 4.2.1. Method
A board-certified radiologist was given access to the chest X-ray image, the
full radiology report, the radiology report impression section, the image
ground truth across all conditions, and the radiologist report labels across
all conditions for each of the 500 examples in the CheXpert test set. The
radiologist then explained examples where radiologists labeling reports
disagree with radiologists labeling X-ray images. We also calculated the
counts of disagreements between radiologists labeling reports and radiologists
labeling X-ray images for each condition on the CheXpert test set. A board-
certified radiologist explained why there were large numbers of disagreements
on certain conditions.
#### 4.2.2. Results
We find various reasons why radiologists labeling reports might disagree with
radiologists labeling images. First, there is a difference between the setup
of the report labeling and image labeling tasks related to the label
hierarchy. On the report labeling task on the CheXpert test set, radiologists
were instructed to label only the most specific condition as positive and
leave parent conditions blank. For example, although Lung Opacity is a parent
condition of Edema, a radiologist marking a report as positive for Edema would
leave Lung Opacity blank. Blank report labels are typically mapped to negative
image labels. However, radiologists labeling images label each condition as
positive or negative independent of the presence of other conditions. Second,
radiologists labeling reports have access to clinical report history, which
biases radiologists towards reporting certain conditions in reports while a
radiologist labeling the image may not observe the condition on the image.
Busby et al. (2018) explain biases from clinical history in terms of framing
bias, where the presentation of the clinical history can lead to different
diagnostic conclusions, and attribution bias, where information in the
clinical history can lead to different diagnostic conclusions. Third,
radiologists labeling reports were only given access to the report impression
section when labeling the CheXpert test set. Sometimes, conditions are
mentioned in the Findings section of the report but not mentioned in the
Impression section. This results in more negative labels when radiologists
looked at reports. For chest CT scan reports, Gershanik et al. (2011) also
find that a condition mentioned in the Findings section is not always
mentioned in the Impression section of the report. Fourth, labeling images and
reports is inherently noisy to a certain extent, resulting in disagreement.
Drivers of noise include mistakes on the part of radiologists labeling reports
and radiologists labeling images, uncertainty regarding the presence of a
condition based on an image or report, and different thresholds for diagnosing
conditions as positive among radiologists. Brady et al. (2012) describe
additional factors that contribute to discrepancies in radiologist
interpretations, including radiologist specific causes of error like under
reading as well as system issues like excess workload. Wood et al. (2020), in
their analysis on MRI neuroradiology reports, also note factors, such as a
difference in observations within reports depending on the referrer, that
likewise result in discrepancies.
Next, we explain the counts of the largest disagreements between radiologists
labeling reports and radiologists labeling images. Out of the 500 examples on
the CheXpert test set, there were 223 examples where the image was labeled
positive while the report was labeled negative for Enlarged Cardiomediastinum.
We hypothesize that this results from the difference in the task setup related
to the label hierarchy. Since Enlarged Cardiomediastinum is a parent condition
of Cardiomegaly, radiologists labeling reports were instructed to leave
Enlarged Cardiomediastinum blank if they labeled Cardiomegaly positive. There
were 101 examples where the image was labeled positive while the report was
labeled negative for Cardiomegaly. Diagnosis of cardiomegaly on chest
radiographs can depend on patient positioning and clinical history. Further,
particularly in the ICU setting in which multiple consecutive radiographs are
taken, cardiomegaly is not consistently described in the report even when
present unless a clinically significant change is observed (i.e. pericardial
effusion). There were 100 examples where the image was labeled positive while
the report was labeled negative for Lung Opacity. We hypothesize that this
results from the difference in task setup related to label hierarchy, as Lung
Opacity is a parent condition. Further, particularly in the setting of
atelectasis, lung opacity may not have risen to clinical relevance for the
reporting radiologist despite being seen on the isolated imaging task. There
were 65 examples where the image was labeled negative while the report was
labeled positive for Pleural Effusion. We hypothesize that this partially
results from both the variant thresholds for diagnosis of pleural effusion
among radiologists and the clinical setting in which the reporting radiologist
has access to prior films. It was common to see the report state ”decreased”
or ”trace residual” effusion due to the context of prior imaging on that
patient. However, in the isolated image labeling task, the perceived
likelihood of the condition fell below the threshold of a board-certified
radiologist. There were 49 examples where the image was labeled negative,
while the report was labeled positive for Edema. Similar to the effusion
example, clinical context and prior imaging played a role in these
discrepancies as, again, diagnoses were carried forward from prior studies and
language such as ”some residual” or ”nearly resolved” in the report were used
to indicate the presence of edema based on the clinical context. However, when
labeling the corresponding image in isolation, the presence of edema fell
below the threshold of a board-certified radiologist. Table 2 contains
specific examples of these disagreements with clinical explanations. Table 3
shows the counts of disagreements between radiologists labeling reports and
radiologists labeling images by condition.
Table 3. Counts of disagreements by condition between radiologists labeling
reports and radiologists labeling the corresponding X-ray images on the
CheXpert test set. The first column reports the number of times the image
ground truth was positive, while the radiologist report label was negative.
The second column reports the number of times the image ground truth was
negative, while the radiologist report label was positive.
Condition | | Positive on Image
---
Negative on Report
| Negative on Image
---
Positive on Report
No Finding | 38 | 40
Enlarged Cardiom. | 223 | 5
Cardiomegaly | 101 | 15
Lung Opacity | 100 | 50
Lung Lesion | 2 | 12
Edema | 26 | 49
Consolidation | 16 | 17
Pneumonia | 6 | 5
Atelectasis | 75 | 31
Pneumothorax | 1 | 13
Pleural Effusion | 11 | 65
Pleural Other | 3 | 15
Fracture | 3 | 21
Support Devices | 53 | 13
### 4.3. Are there significant relationships between conditions labeled from
reports and conditions labeled from images?
To determine whether there are significant relationships between conditions
labeled from reports and conditions labeled from images, we learn a mapping
from the output of radiologists labeling reports to the output of radiologists
labeling images. We then analyze the significant relationships implied by this
mapping from a clinical perspective.
#### 4.3.1. Method
We train logistic regression models to map the radiologist report labels for
all conditions to the image ground truth for each of the evaluation
conditions. We quantitatively measure the relationship between the radiologist
report labels and the image ground truth by obtaining odds ratios from the
coefficients of these logistic regression models. We review the odds ratios
from these models with a board-certified radiologist to understand how
particular radiologist report labels might clinically change the odds of image
labels.
#### 4.3.2. Training details
We one-hot encode the radiologist report labels and provide these binary
variables as inputs to a logistic regression model. For example, the
”Atelectasis Positive” variable is 1 if the radiologist labels Atelectasis as
positive on the report and 0 otherwise. Similarly, the ”Atelectasis Negative”
variable is 1 if the radiologist labels Atelectasis as negative on the report
and 0 otherwise. The same logic applies to the ”Atelectasis Uncertain”
variable as well as the other variables for each condition. We then train the
logistic regression model with L1 regularization ($\alpha=0.5$) on the
CheXpert test set using the one-hot encoded radiologist report labels (for all
conditions) as input and the image ground truth for a condition as output. In
total, we train different logistic regression models to map the radiologist
report labels to binary image labels for each of the 8 evaluation conditions.
We compute odds ratios by exponentiating the coefficients of the logistic
regression models.
#### 4.3.3. Results
After training the logistic regression models, we find that particular
radiology report labels increased (or decreased) the odds of particular
conditions in an image with statistical significance (P ¡ 0.05). As expected,
we find that radiology report labels associated with a condition increase the
odds of that same condition in the image; for example, a Cardiomegaly positive
report label increases the odds of Cardiomegaly in the image. We also find
that the regression model corrects for label hierarchy. A Cardiomegaly
positive report label increases the odds of Enlarged Cardiomediastinum (the
parent of Cardiomegaly) on the image by 9.6 times. We similarly observe the
model correcting for the label hierarchy of Lung Opacity. Radiology report
labels of Edema positive, Consolidation positive, and Atelectasis positive,
which all correspond to child conditions of Lung Opacity, increase the odds of
Lung Opacity. We also find that the model maps particular uncertainties in
report labels to the presence of a condition in the image. For example,
Atelectasis uncertain report labels and Edema uncertain report labels increase
the odds of Lung Opacity by 2.9 and 7.9 times respectively.
Next, we find that the model maps positive report labels to the presence of
other conditions in the image. A Pleural Effusion positive report label
increases the odds of Lung Opacity by 4.4 times. We hypothesize that this
results from co-occurrence between Pleural Effusion and child conditions of
Lung Opacity such as Atelectasis and Edema. Pleural effusion physiologically
often leads to adjacent lung collapse, atelectasis, and is often seen in
physiologic fluid overload conditions, edema. We find that an Atelectasis
positive report label decreases the odds of Support Devices in the image by
0.28 times. On the patient population who have support devices, many of whom
are in an Intensive Care Unit (ICU) setting, it is not clinically useful for
radiologists to comment on the presence of atelectasis on reports, as they
would rather focus on more clinically relevant changes. This may explain the
mechanism by which the presence of atelectasis in a report signals that there
are no support devices in the image. We find that a Fracture positive report
label decreases the odds of Support Devices by 0.17 times. We hypothesize that
this results from a negative co-occurrence between Fractures and Support
Devices, as the two observations select for different patient populations:
X-rays for fractures are often done in the Emergency Department (ED) or other
outpatient settings rather than the ICU setting. We find that an Edema
positive report label increases the odds of Enlarged Cardiomediastinum on the
image by 2.1 times. This may be explained by the fact that Edema and Enlarged
Cardiomediastinum often co-occur in a clinical setting, as they can both be
caused by congestive heart failure. Lastly, we find that a Support Devices
positive report label decreases the odds of No Finding in the image by 0.03
times. This may be explained by the fact that patients with support devices
are usually in the ICU setting and sick with other pathologies. We visualize
these statistically significant odds ratios for each type of radiologist
report label (such as ”Atelectasis Negative”) as a factor for the presence of
an evaluation condition in the X-ray image in Figure 2.
Figure 2. Odds ratios for radiologist report labels as factors for the
presence of a condition in the X-ray image. We map the radiologist report
labels across all conditions to the image ground truth using a logistic
regression model. We obtain odds ratios for the input variables, which are the
one-hot encoded radiologist report labels, and only display odds ratios for
which the corresponding P value (two-sided t test) is less than 0.05. Table 4.
F1 scores obtained by the Zero-One and LogReg baselines, evaluated on the
CheXpert test set. The weighted average is weighted by prevalence (n = #
positive).
| Condition
---
(n = # positive)
Zero-One Baseline | LogReg Baseline
Atelectasis (n=153) | 0.52 | 0.63
Cardiomegaly (n=151) | 0.46 | 0.56
Edema (n=78) | 0.53 | 0.47
Pleural Effusion (n=104) | 0.65 | 0.65
| Enlarged Cardiom. (n=253)
---
0.20 | 0.67
Lung Opacity (n=264) | 0.69 | 0.81
Support Devices (n=261) | 0.85 | 0.84
No Finding (n=62) | 0.39 | 0.55
Average | 0.54 | 0.65
Weighted Average | 0.56 | 0.70
Table 5. F1 scores for BERT+Thresholding and BERT+LogReg trained on the
MIMIC-CXR and CheXpert datasets. We refer to the BERT+Thresholding method on
the MIMIC-CXR dataset as VisualCheXbert. The models here are evaluated on the
CheXpert test set.
| Condition
---
(n = # positive)
| BERT+Thresholding on
---
MIMIC-CXR, DenseNet
Labels
| BERT+LogReg on
---
MIMIC-CXR, DenseNet
Labels
(VisualCheXbert)
| BERT+Thresholding on
---
CheXpert, DenseNet
Labels
| BERT+LogReg on
---
CheXpert, DenseNet
Labels
Atelectasis (n=153) | 0.65 | 0.64 | 0.67 | 0.66
Cardiomegaly (n=151) | 0.53 | 0.62 | 0.61 | 0.61
Edema (n=78) | 0.55 | 0.54 | 0.49 | 0.53
Pleural Effusion (n=104) | 0.64 | 0.65 | 0.57 | 0.67
| Enlarged Cardiom. (n=253)
---
0.44 | 0.73 | 0.60 | 0.70
Lung Opacity (n=264) | 0.81 | 0.83 | 0.70 | 0.83
Support Devices (n=261) | 0.85 | 0.87 | 0.80 | 0.84
No Finding (n=62) | 0.44 | 0.54 | 0.46 | 0.52
Average | 0.61 | 0.68 | 0.61 | 0.67
Weighted Average | 0.65 | 0.73 | 0.65 | 0.72
### 4.4. Can we naively map labels obtained from reports to X-ray image
labels?
We map the output of an automated radiology report labeler to X-ray image
labels using simple uncertainty handling strategies.
#### 4.4.1. Method
For a baseline approach, we naively map labels obtained from running the
CheXpert labeler on the radiology report impressions to X-ray image labels.
The CheXpert labeler is an automatic, rules-based radiology report labeler
(Irvin et al., 2019). The labels produced by the CheXpert labeler include 4
classes per medical condition (positive, negative, uncertain, and blank).
Since the image ground truth only has positive or negative labels per
condition, we must map the labels produced by the CheXpert labeler to binary
labels. We map the blank labels produced by the CheXpert labeler to negative
labels. We do not change the positive and negative labels produced by the
CheXpert labeler. To handle the uncertain labels, we use the two common
uncertainty handling strategies in Irvin et al. (2019): we map the uncertain
labels to either all negative labels (zeros-uncertainty handling strategy) or
all positive labels (ones-uncertainty handling strategy). We record the F1
score from the better performing strategy on the CheXpert test set, using as
ground truth the labels provided by radiologists labeling X-ray images (image
ground truth). We refer to this method as the Zero-One Baseline. Since we only
report the maximum of the zeros-uncertainty handling strategy and the ones-
uncertainty handling strategy, the F1 scores for the Zero-One Baseline
represent the most optimistic global mapping of the uncertainty labels for
this method.
#### 4.4.2. Results
We find that the average and weighted average F1 scores across the evaluation
conditions for the Zero-One Baseline are 0.54 and 0.56 respectively, which are
in between the average / weighted average “Low F1” and “High F1” scores for
radiologists labeling reports (see Table 1). This indicates that the Zero-One
Baseline is not strictly better or worse than radiologists labeling reports,
who we previously show to have poor agreement with radiologists labeling
images. The Zero-One Baseline F1 scores for Atelectasis, Cardiomegaly, Edema,
Pleural Effusion, and Enlarged Cardiomediastinum are 0.52, 0.46, 0.53, 0.65,
and 0.20 respectively, which are all between the respective “Low F1” and “High
F1” scores for radiologists labeling reports. The Zero-One Baseline F1 scores
for Lung Opacity and No Finding are 0.69 and 0.39 respectively, which are
slightly higher ($\sim 0.01$ difference) than the respective “High F1” scores
for radiologists labeling reports. Similarly, the Zero-One Baseline F1 score
for Support Devices is 0.39, which is slightly lower ($\sim 0.01$ difference)
than the “Low F1” Support Devices score for radiologists labeling reports. The
F1 scores for the Zero-One Baseline across the evaluation conditions are shown
in Table 4.
### 4.5. Can we learn to map labels obtained from reports to X-ray image
labels?
We map the output of an automated radiology report labeler to X-ray image
labels, similarly to how we previously map the output of radiologists labeling
reports to the output of radiologists labeling images. Previous work by
Dunnmon et al. (2019) showed that labels obtained from noisy labeling
functions on radiology reports can be mapped to labels that are of similar
quality to image labels produced by radiologists for the simpler task of
classifying X-rays as normal or abnormal.
#### 4.5.1. Method
This approach, motivated by a prior experiment in which we map radiologist
report labels to image labels, improves upon the naive uncertainty mapping
strategy used in the Zero-One Baseline. As before, we obtain report labels by
running the CheXpert labeler on radiology report impressions. For each of the
evaluation conditions, we train a logistic regression model that maps the
CheXpert labeler’s output on a radiology report impression to a positive or
negative label for the target condition. This approach makes use of the
automated report labels for all 14 conditions to predict the label for each
target condition. We refer to this approach as the LogReg Baseline.
#### 4.5.2. Training details
We one-hot encode the report labels outputted by the CheXpert labeler and
provide these binary variables as inputs to a logistic regression model. We
train a logistic regression model with $L2$ regularization ($C=1.0$) and a max
iteration of $500$ using the one-hot encoded report labels (for all
conditions) as input and the image ground truth for a condition as output. The
class weights are the inverse prevalence of the respective class in the
training set. We use a leave-one-out cross-validation strategy to train and
validate the logistic regression model on the CheXpert test dataset. For each
of the 8 evaluation conditions, we train different logistic regression models
to map the labels produced by the CheXpert labeler to binary image labels.
#### 4.5.3. Results
We find that the LogReg Baseline approach improves upon the Zero-One Baseline
for most conditions. Compared to the Zero-One Baseline, the LogReg Baseline
increases the average F1 score from 0.54 to 0.65 and the weighted average F1
score from 0.56 to 0.70. The LogReg Baseline increases the F1 score compared
to the Zero-One Baseline from 0.52 to 0.63 for Atelectasis, 0.46 to 0.56 for
Cardiomegaly, 0.20 to 0.67 for Enlarged Cardiomediastinum, 0.69 to 0.81 for
Lung Opacity, and 0.39 to 0.55 for No Finding. However, the LogReg Baseline
decreases the F1 scores compared to the Zero-One Baseline from 0.53 to 0.47
for Edema and 0.85 to 0.84 for Support Devices. For Pleural Effusion, both the
LogReg Baseline and the Zero-One Baseline have an F1 score of 0.65. Although
the LogReg Baseline is not better than the Zero-One Baseline for all
conditions, these results suggest that a learned mapping from radiologist
report labels to X-ray image labels can outperform naively mapping all
uncertain labels to positive or negative for most conditions. The F1 scores
obtained by the LogReg Baseline, along with head-to-head comparisons to the
Zero-One Baseline, are shown in Table 4.
Table 6. Improvement in F1 score obtained by VisualCheXbert, evaluated on the
CheXpert test set and reported with 95% confidence intervals. The left-most
column shows the improvement over the Zero-One Baseline. The middle column
shows the improvement over the radiologist report labels with uncertains
mapped to the image ground truth label. The right-most column shows the
improvement over the radiologist report labels with uncertains mapped to the
opposite of image ground truth label.
| Condition
---
(n = # positive)
| Improvement over
---
Zero-One Baseline
| Improvement over
---
Higher Radiologist
Score
| Improvement over
---
Lower Radiologist
Score
Atelectasis (n=153) | 0.12 (0.04, 0.20) | 0.04 (-0.04, 0.12) | 0.41 (0.32, 0.49)
Cardiomegaly (n=151) | 0.16 (0.07, 0.25) | 0.15 (0.07, 0.25) | 0.20 (0.11, 0.28)
Edema (n=78) | 0.01 (-0.05, 0.07) | -0.04 (-0.09, 0.02) | 0.09 (0.02, 0.17)
Pleural Effusion (n=104) | -0.01 (-0.04, 0.03) | -0.06 (-0.10, -0.02) | 0.01 (-0.03, 0.05)
Enlarged Cardiom. (n=253) | 0.53 (0.46, 0.60) | 0.52 (0.44, 0.60) | 0.64 (0.57, 0.71)
Lung Opacity (n=264) | 0.14 (0.09, 0.20) | 0.15 (0.09, 0.20) | 0.15 (0.10, 0.20)
Support Devices (n=261) | 0.02 (-0.01, 0.06) | 0.01 (-0.02, 0.04) | 0.01 (-0.02, 0.04)
No Finding (n=62) | 0.15 (0.05, 0.26) | 0.16 (0.05, 0.28) | 0.16 (0.05, 0.28)
Average | 0.14 (0.12, 0.17) | 0.12 (0.09, 0.15) | 0.21 (0.18, 0.24)
Weighted Average | 0.17 (0.15, 0.20) | 0.15 (0.13, 0.18) | 0.24 (0.21, 0.26)
### 4.6. Can we learn to map the text reports directly to the X-ray image
labels?
Previously, we mapped the output of an existing automated report labeler,
which takes text reports as input, to X-ray image labels. We now map the
textual radiology report directly to the X-ray image labels.
#### 4.6.1. Method
We develop a deep learning model that maps a radiology report directly to the
corresponding X-ray image labels.
Since it is too expensive to obtain labels from radiologists for hundreds of
thousands of X-ray images to supervise our model, we instead train a single
DenseNet model (Huang et al., 2018) to detect medical conditions from chest
X-ray images, as is described by Irvin et al. (2019), and we use this computer
vision model as a proxy for a radiologist labeling an X-ray image. We use the
DenseNet model to output probabilities for each of the 14 conditions for all
X-rays in the MIMIC-CXR dataset and the CheXpert training dataset. To obtain
the output of the vision model on the MIMIC-CXR dataset, we train the DenseNet
on the CheXpert training dataset. Similarly, to obtain the output of the
vision model on the CheXpert training dataset, we train the DenseNet on the
MIMIC-CXR dataset. We find that the DenseNet trained on the CheXpert training
set has an AUROC of 0.875 on the CheXpert test set across all conditions, and
the DenseNet trained on the MIMIC-CXR dataset has an AUROC of 0.883 on the
CheXpert test set across all conditions.
We then use the probabilities outputted from these computer vision models as
ground truth to fine-tune a BERT-base model. We train one BERT model using the
MIMIC-CXR dataset and one using the CheXpert training dataset. The BERT model
takes a tokenized radiology report impression from the MIMIC-CXR or CheXpert
dataset as input and is trained to output the labels produced by the DenseNet
model. We feed the BERT model’s output corresponding to the [CLS] token into
linear heads (one head for each medical condition) to produce scores for each
medical condition. We use the cross-entropy loss to fine-tune BERT. The BERT
model is initialized with biomedically pretrained weights produced by Peng et
al. (2019). This model training process is shown in Figure 1.
After training the BERT model, we map the outputs of BERT, which are
probabilities, to positive or negative labels for each condition. To do so, we
try two different methods. Our first method uses optimal probability
thresholds to convert the BERT outputs to binary labels. We calculate optimal
thresholds by finding the threshold for each condition that maximizes Youden’s
index (Youden, 1950) (the sum of sensitivity and specificity minus one) on the
CheXpert validation dataset. We refer to this approach as BERT+Thresholding.
Our second method trains a logistic regression model to map the output of BERT
across all 14 conditions to a positive or negative label for the target
condition. We refer to this approach as BERT+LogReg. Ultimately, we develop
four different models by using both methods on outputs from a BERT model
trained on the MIMIC-CXR dataset and a BERT model trained on the CheXpert
training dataset. The four resulting models are called BERT+Thresholding on
MIMIC-CXR, BERT+LogReg on MIMIC-CXR, BERT+Thresholding on CheXpert, and
BERT+LogReg on CheXpert. We refer to the BERT+LogReg model trained on the
MIMIC-CXR dataset with labels provided by the DenseNet model, which is our
best performing approach, as VisualCheXbert.
#### 4.6.2. Training details
We train the BERT model on 3 TITAN-XP GPUs using the Adam optimizer (Kingma
and Ba, 2017) with a learning rate of $2\times 10^{-5}$, following Devlin et
al. (2019) for fine-tuning tasks. We use a random 85%-15% training-validation
split, as in Smit et al. (2020). The BERT model is trained until convergence.
We use a batch size of 18 radiology report impressions. For the BERT+LogReg
approach, the logistic regression model uses $L2$ regularization ($C=1.0$) and
a max iteration of 500. Similar to the LogReg Baseline, the class weights are
the inverse prevalence of the respective class in the training set, and we use
a leave-one-out cross-validation strategy to train and test the logistic
regression model on the CheXpert test dataset. We train different logistic
regression models to map the probabilities outputted by the BERT model to the
binary image labels for each of the 8 evaluation conditions.
#### 4.6.3. Results
We compare the performance of the different BERT approaches on the CheXpert
test set. First, we find that on most conditions, BERT+LogReg outperforms
BERT+Thresholding. This finding holds true on both the CheXpert and MIMIC-CXR
datasets. Second, we find that despite being trained on datasets from
different institutions, the models trained on MIMIC-CXR and CheXpert datasets
perform similarly. This indicates that the BERT model trained on radiology
report impressions from the MIMIC-CXR distribution (Beth Israel Deaconess
Medical Center Emergency Department between 2011–2016) (Johnson et al., 2019)
can perform as well as a model trained on radiology report impressions from
the CheXpert distribution (Stanford Hospital between 2002-2017) (Irvin et al.,
2019), even when both models are evaluated on a test set from the CheXpert
distribution. Since we obtain a slightly higher average and weighted average
F1 using the MIMIC-CXR dataset, we use BERT trained on MIMIC-CXR in our final
approach called VisualCheXbert. The performance of the BERT approaches is
shown in Table 5.
Next, we compare VisualCheXbert to the Zero-One Baseline. When comparing
VisualCheXbert to the Zero-One Baseline as well as the higher and lower scores
of radiologists labeling reports described below, we report the improvements
by computing the paired differences in F1 scores on 1000 bootstrap replicates
and providing the mean difference along with a 95% two-sided confidence
interval (Efron and Tibshirani, 1986). Overall, VisualCheXbert improves the
average F1 and weighted average F1 over the Zero-One Baseline with statistical
significance, increasing the average F1 score by 0.14 (95% CI 0.12, 0.17) and
the weighted average F1 score by 0.17 (95% CI 0.15, 0.20). We find that
VisualCheXbert obtains a statistically significant improvement over the Zero-
One Baseline on most conditions. VisualCheXbert increases the F1 score on
Enlarged Cardiomediastinum, Cardiomegaly, No Finding, Lung Opacity, and
Atelectasis compared to the Zero-One Baseline by 0.53 (95% CI 0.46, 0.60),
0.16 (95% CI 0.07, 0.25), 0.15 (95% CI 0.05, 0.26), 0.14 (95% CI 0.09, 0.20),
and 0.12 (95% CI 0.04, 0.20), respectively. VisualCheXbert obtains similar
performance (no statistically significant difference) to the Zero-One Baseline
on the rest of the conditions, which are Edema, Pleural Effusion, and Support
Devices, with improvements of 0.01 (95% CI -0.05, 0.07), -0.01 (95% CI -0.04,
0.03), and 0.02 (95% CI -0.01, 0.06), respectively.
Lastly, we compare the F1 scores for VisualCheXbert to the higher and lower
scores of radiologists labeling reports. The higher scores for radiologists
labeling reports are obtained by mapping the uncertain radiologist report
labels to the image ground truth label, while the lower scores for
radiologists labeling reports are obtained by mapping the uncertain
radiologist report labels to the opposite of the ground truth. Overall,
VisualCheXbert obtains a statistically significant improvement over both the
higher and lower radiologist scores, increasing the average F1 score by 0.12
(95% CI 0.09, 0.15) over the higher radiologist score and 0.21 (95% CI 0.18,
0.24) over the lower radiologist score and increasing the weighted average F1
score by 0.15 (95% CI 0.13, 0.18) over the higher radiologist score and 0.24
(95% CI 0.21, 0.26) over the lower radiologist score. Statistically
significant improvements over the higher radiologist score are observed for
Cardiomegaly (0.15 [95% CI 0.07, 0.25]), Enlarged Cardiomediastinum (0.52 [95%
CI 0.44, 0.60]), Lung Opacity (0.15 [95% CI 0.09, 0.20]), and No Finding (0.16
[95% CI 0.05, 0.28]). VisualCheXbert performs similarly (no statistically
significant difference) to the higher radiologist score on Atelectasis (0.04
[95% CI -0.04, 0.12]), Edema (-0.04 [95% CI -0.09, 0.02]), and Support Devices
(0.01 [95% CI -0.02, 0.04]). VisualCheXbert performs slightly worse than the
higher radiologist score on one condition, which is Pleural Effusion (-0.06
[95% CI -0.10, -0.02]). VisualCheXbert observes considerable, statistically
significant improvements compared to the lower radiologist score on all but
two conditions. There is no statistically significant difference between
VisualCheXbert and the lower radiologist score on these two conditions, which
are Pleural Effusion (0.01 [95% CI -0.03, 0.05]) and Support Devices (0.01
[95% CI -0.02, 0.04]). We show the improvements obtained by VisualCheXbert
over the Zero-One Baseline and the improvements over radiologists labeling
reports in Table 6.
## 5\. Limitations
Our work has the following limitations. First, our study only made use of the
Impression section of the radiology reports, which is a summary of the
radiology report. Prior work regarding automated chest X-ray labeling has also
extensively used the impression section in radiology reports (Irvin et al.,
2019; Johnson et al., 2019; Wang et al., 2017). However, conditions are
sometimes mentioned in the Findings section of the report but not in the
Impression section. As a result, negative and blank labels are more frequent
when using the Impression section, and this could increase the disparity
between labels extracted from the impression and the corresponding chest X-ray
image labels. Second, the VisualCheXbert model has a maximum input size of 512
tokens. In practice, only 3 of the report impressions in the entire CheXpert
dataset were longer than this limit. Third, the CheXpert test set, on which we
evaluated our models, consists of 500 radiology studies and is therefore
limited in size. As a result, some of the medical conditions contained very
few positive examples; we only evaluated our models on conditions for which at
least 10% of the examples in the CheXpert test set were positive. Using a
larger test set would allow evaluation on rarer conditions. Fourth, our models
are evaluated on chest X-rays from a single institution. Further evaluation on
data from other institutions could be used to evaluate the generalizability of
our models.
## 6\. Conclusion
We investigate the discrepancy between labels extracted from radiology reports
and the X-ray image ground truth labels. We then develop and evaluate methods
to address this discrepancy. In our work, we aim to answer the following
questions.
Do radiologists labeling reports agree with radiologists labeling X-ray
images? We find that there is significant disagreement between radiologists
labeling reports and radiologists labeling images. On the CheXpert test set,
we observe low Kappa scores for almost all conditions evaluated. The average
Kappa across the evaluation conditions is between 0.312 and 0.430. These
bounds are based on the most pessimistic mapping and most optimistic mapping
of uncertain radiology report labels.
Why do radiologists labeling reports disagree with radiologists labeling X-ray
images? Upon a board-certified radiologist review of examples of disagreements
between radiologists labeling reports and radiologists labeling images, we
find four main reasons for disagreement. First, on the CheXpert test set,
radiologists labeling reports typically do not mark a parent condition as
positive if a child condition is positive. An example of a parent and child
condition would be Lung Opacity and Edema, respectively. Second, radiologists
labeling reports have access to clinical report history, which biases their
diagnoses compared to radiologists labeling images who do not have access to
this information. Third, conditions are sometimes reported in the Findings
section of radiology reports but not the Impression section of radiology
reports. However, the Impression section of radiology reports is commonly used
to label reports. This discrepancy can cause radiologists labeling reports to
miss pathologies present on the X-ray image. Fourth, labeling images and
reports is noisy to a certain extent due to factors such as human mistakes,
uncertainty in both reports and images, and different thresholds for
diagnosing conditions as positive among radiologists.
Are there significant relationships between conditions labeled from reports
and conditions labeled from images? We find many significant relationships
between conditions labeled from reports and conditions labeled from images. We
report and clinically interpret various radiology report labels that increase
(or decrease) the odds of particular conditions in an image with statistical
significance (P¡ 0.05). As expected, we find that positive report labels for a
condition increase the odds of that condition in an image. We find that
positive report labels for children of a condition increase the odds of the
parent condition in an image, a phenomenon that is correcting for the label
hierarchy. We find that particular uncertain report labels for a condition
increase the odds of the condition (and/or its parent condition). We also find
that positive report labels for certain conditions increase (or decrease) the
odds of other conditions in the image. One example is that a positive
Atelectasis report label decreases the odds of Support Devices in the X-ray
image by 0.28 times. We explain potential mechanisms by which the presence of
a condition in a report signals the presence (or absence) of another condition
in the image.
Can we learn to map radiology reports directly to the X-ray image labels? We
learn to map a textual radiology report directly to the X-ray image labels. We
use a computer vision model trained to detect diseases from chest X-rays as a
proxy for a radiologist labeling an X-ray image. Our final model,
VisualCheXbert, uses a biomedically-pretrained BERT model that is supervised
by the computer vision model. When evaluated on radiologist image labels on
the CheXpert test set, VisualCheXbert increases the average F1 score across
the evaluation conditions by between 0.12 (95% CI 0.09, 0.15) and 0.21 (95% CI
0.18, 0.24) compared to radiologists labeling reports. VisualCheXbert also
increases the average F1 score by 0.14 (95% CI 0.12, 0.17) compared to a
common approach that uses a previous rules-based radiology report labeler.
Given the considerable, statistically significant improvement obtained by
VisualCheXbert over the approach using an existing radiology report labeler
(Irvin et al., 2019) when evaluated on the image ground truth, we hypothesize
that VisualCheXbert’s labels could be used to train better computer vision
models for automated chest X-ray diagnosis.
## 7\. Code Repository
The code to run our model is available in a public code repository:
https://github.com/stanfordmlgroup/VisualCheXbert.
###### Acknowledgements.
We would like to acknowledge the Stanford Machine Learning Group
(stanfordmlgroup.github.io) and the Stanford Center for Artificial
Intelligence in Medicine and Imaging (AIMI.stanford.edu) for infrastructure
support.
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# Parametrized topological complexity of sphere bundles
M. Farber School of Mathematical Sciences, Queen Mary University of London
and S. Weinberger Department of Mathematics, University of Chicago
###### Abstract.
Parametrized motion planning algorithms [1] have high degree of flexibility
and universality, they can work under a variety of external conditions, which
are viewed as parameters and form part of the input of the algorithm. In this
paper we analyse the parameterized motion planning problem in the case of
sphere bundles. Our main results provide upper and lower bounds for the
parametrized topological complexity; the upper bounds typically involve
sectional categories of the associated fibrations and the lower bounds are
given in terms of characteristic classes and their properties. We explicitly
compute the parametrized topological complexity in many examples and show that
it may assume arbitrarily large values.
###### 1991 Mathematics Subject Classification:
55M30
M. Farber was partially supported by a grant from the EPSRC
S. Weinberger was partially supported by a grant from the NSF
## 1\. Introduction
The motion planning problem of robotics is one of the central themes which
makes possible autonomous robot motion, see [9]. A motion planning algorithm
takes as input the initial and the desired states of the system and produces
as output a motion of the system starting at the initial and ending at the
desired states. A robot is “told” where it needs to go and the execution of
this task, including selection of a specific route of motion, is made by the
robot itself. In this approach it is understood that the external conditions
(such as the positions of the obstacles and the geometry of the enclosing
domain) are known.
In recent papers [1], [2], motion planning algorithms of a new type were
considered. These are parametrized motion planning algorithms, which, besides
the initial and desired states, take as input the parameters characterising
the external conditions. The output of a parametrized motion planning
algorithm is a continuous motion of the system from the initial to the desired
state respecting the given external conditions. The papers [1, 2] laid out the
new formalism and analysed in full detail the problem of moving a number $n$
of robots in the domain with $m$ unknown obstacles. The authors used
techniques of algebraic topology and were able to find the answer by using a
combination of upper and lower bounds. The lower bounds use the structure of
the cohomology algebras. A brief introduction into the concept of parametrized
topological complexity is given below in §3.
The purpose of this article is to analyse the parametrized topological
complexity of sphere bundles. The Stiefel - Whitney and Euler characteristic
classes play an important role in these estimates. Our main results give upper
and lower bounds for the parametrized topological complexity and we compute
the parametrized topological complexity of a number of examples.
It would be interesting to adopt the theory of weights of cohomology classes
of E. Fadell and S. Husseini [4] with the purpose of strengthening the
cohomological lower bounds in application to the parametrized topological
complexity.
The authors thank the referee for very helpful comments.
## 2\. Sectional category of sphere bundle
In this section we recall some well-known results, see [11], which will be
useful later in this paper.
Let $\xi:E\to B$ be a rank $q$ vector bundle. We shall denote
$q={\rm{rk}}(\xi)$ and shall write $E(\xi)$ instead of $E$ when dealing with
several bundles at once. In this article we shall always assume that vector
bundles are equipped with metric structures, i.e. with continuous scalar
product on fibres.
We shall denote by $\dot{\xi}:\dot{E}\to B$ the associated bundle of
$(q-1)$-dimensional spheres, i.e. $\dot{\xi}=\xi|_{\dot{E}}$. Here
$\dot{E}\subset E$ is the set of vectors of length 1. If $\xi$ is oriented,
its Euler class ${\mathfrak{e}}(\xi)\in H^{q}(B)$ is defined, see [10]. Here
the cohomology is taken with integral coefficients. We shall adopt the
convention of skipping ${\mathbb{Z}}$ from the notations while indicating
explicitly all other coefficient groups.
For a cohomology class $\alpha\not=0$ we shall denote by
${\mathfrak{h}}(\alpha)$ its height, i.e. the largest integer $k$ such that
its $k$-th power is nonzero, $\alpha^{k}\not=0$. We shall also set
${\mathfrak{h}}(\alpha)=0$ for $\alpha=0$.
Recall that the sectional category (or Schwarz genus [11]) of a fibration with
base $B$ is defined as the minimal integer $k\geq 0$ such that there exists an
open cover
$B=U_{0}\cup U_{1}\cup\dots\cup U_{k}$
with the property that over each set $U_{i}$ the fibration admits a continuous
section.
###### Lemma 1.
Let $\xi:E\to B$ be a vector bundle, $q={\rm{rk}}(\xi)$. Then
1. (A)
The sectional category of the sphere fibration $\dot{\xi}:\dot{E}\to B$
satisfies
${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\rm w}_{q}(\xi))$
where ${\rm w}_{q}(\xi)\in H^{q}(B;{\mathbb{Z}}_{2})$ is the top Stiefel-
Whitney class of $\xi$.
2. (B)
If the bundle $\xi$ is orientable then
${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\mathfrak{e}}(\xi)).$
3. (C)
Moreover, if $\xi$ is orientable and the base $B$ is a CW-complex whose
dimension satisfies
$\dim B\leq q\cdot{\mathfrak{h}}({\mathfrak{e}}(\xi))+q,$
then
$\displaystyle{\sf{secat}}(\dot{\xi})={\mathfrak{h}}({\mathfrak{e}}(\xi)).$
###### Proof.
(A) First we observe that $\dot{\xi}^{\ast}({\rm w}_{q}(\xi))=0\in
H^{q}(\dot{E};{\mathbb{Z}}_{2})$. Indeed, using functoriality of the Stiefel-
Whitney classes, we see that $\dot{\xi}^{\ast}({\rm w}_{q}(\xi))$ is the top
Stiefel - Whitney class of the induced fibration $\dot{\xi}^{\ast}(\xi)$ over
$\dot{E}(\xi)$. However this fibration admits a nonzero section $s(x)=x$ where
$x\in\dot{E}(\xi)$. The top Stiefel - Whitney class of a vector bundle having
a section vanishes, hence $\xi^{\ast}({\rm w}_{q}(\xi))={\rm
w}_{q}(\dot{\xi}^{\ast}(\xi))=0$.
Finally we apply the general cohomological lower bound for the sectional
category, see [11], Theorem 4; this gives
${\sf{secat}}(\dot{\xi})\geq{\mathfrak{h}}({\rm w}_{q}(\xi))$.
(B) follows similarly.
To prove (C) we apply Theorem 3 of Schwarz [11] which identifies the sectional
category of $\dot{\xi}$ with the smallest number $k$ such that the
$(k+1)$-fold fiberwise join $(\dot{\xi})^{\ast(k+1)}$ admits a continuous
section. Note that $(\dot{\xi})^{\ast(k+1)}$ is fiberwise homeomorphic to the
unit sphere bundle of the vector bundle
$(k+1)\xi=\xi\oplus\xi\oplus\dots\oplus\xi$, the Whitney sum of $k+1$ copies
of $\xi$. The obstructions for a section of $(\dot{\xi})^{\ast(k+1)}$ lie in
the groups
$H^{i}(B;\pi_{i-1}(S^{q(k+1)-1})),\quad i=1,2,\dots.$
The first obstruction (with $i=q(k+1)$) equals
${\mathfrak{e}}(\xi)^{k+1}\,=\,{\mathfrak{e}}((k+1)\xi)\,\,\in H^{q(k+1)}(B).$
Taking $k={\mathfrak{h}}({\mathfrak{e}}(\xi))$ we obtain
${\mathfrak{e}}(\xi)^{k+1}=0$, i.e. the first obstruction vanishes. The
further obstructions (with $i=q(k+1)+j$ where $j=1,\,2,\dots$) also vanish
because of our assumption $\dim B\leq
q\cdot({\mathfrak{h}}({\mathfrak{e}}(\xi))+1)$. This completes the proof. ∎
###### Example 2.
Let $\eta:E\to{\mathbb{CP}}^{n}$ denote the canonical complex line bundle over
${\mathbb{CP}}^{n}$. We shall view $\eta$ as a rank 2 real vector bundle. Its
Euler class ${\mathfrak{e}}(\eta)$ is the generator of
$H^{2}({\mathbb{CP}}^{n})$ and ${\mathfrak{h}}({\mathfrak{e}}(\eta))=n$. Since
$\dim{\mathbb{CP}}^{n}=2n\leq 2\cdot(n+1)$, Lemma 1 (C) applies and gives
${\sf{secat}}(\eta)=n$.
###### Example 3.
Let $\eta:E\to{\mathbb{CP}}^{n}$ be as in the previous Example. For $k\leq n$
consider $\xi_{k}=k\eta=\eta\oplus\eta\oplus\dots\oplus\eta$, the Whitney sum
of $k$ copies of $\eta$. We have ${\rm{rk}}(\xi_{k})=2k$, and
${\mathfrak{e}}(\xi_{k})={\mathfrak{e}}(\eta)^{k}$ implying
${\mathfrak{h}}({\mathfrak{e}}(\xi_{k}))=\lfloor n/k\rfloor$. The inequality
$\dim B=2n\leq 2k\cdot(\lfloor n/k\rfloor+1)$ is satisfied and using Lemma 1
we obtain
(1) $\displaystyle{\sf{secat}}(\xi_{k})\,=\lfloor n/k\rfloor$
for any $k=1,2,\dots,n$. Formula (1) is also true for $k>n$ as then the bundle
$\xi_{k}$ admits a section and hence ${\sf{secat}}(\xi_{k})=0$.
###### Remark 4.
There is a version of statement (C) of Lemma 1 for non-orientable bundles; in
this case the Euler class lies in the cohomology $e(\xi)\in
H^{q}(B;\tilde{\mathbb{Z}})$ with twisted coefficients and its powers
$e(\xi)^{k}$ lie in the groups $H^{kq}(B,(\tilde{\mathbb{Z}})^{\otimes k})$.
It is easy to see that $(\tilde{\mathbb{Z}})^{\otimes k}={\mathbb{Z}}$ for $k$
even and $(\tilde{\mathbb{Z}})^{\otimes k}=\tilde{\mathbb{Z}}$ for $k$ odd.
## 3\. Parametrized topological complexity
In this section we briefly recall the notion of parametrized topological
complexity which was recently introduced in [1], [2]. It is a generalization
of the concept of topological complexity of robot motion planning problem
introduced in [5]; see also [6].
Let $X$ be a path-connected topological space viewed as the space of states of
a mechanical system. The motion planning problem of robotics asks for an
algorithm which takes as input an initial state and a desired state of the
system, and produces as output a continuous motion of the system from the
initial state to the desired state, see [9]. That is, given $(x_{0},x_{1})\in
X\times X$, the algorithm will produce a continuous path $\gamma:I\to X$ with
$\gamma(0)=x_{0}$ and $\gamma(1)=x_{1}$, where $I=[0,1]$ denotes the unit
interval.
Let $X^{I}$ denote the space of all continuous paths in $X$, equipped with the
compact-open topology. The map $\pi:X^{I}\to X\times X$, where
$\pi(\gamma)=(\gamma(0),\gamma(1))$, is a fibration in the sense of Hurewicz.
A solution of the motion planning problem, a motion planning algorithm, is a
section of this fibration, i.e. a map $s:X\times X\to X^{I}$ with $\pi\circ
s={\rm id}_{X\times X}$. If $X$ is not contractible, no section can be
continuous, see [5]. The topological complexity of $X$ is defined to be the
sectional category, or Schwarz genus, of the fibration $\pi:X^{I}\to X\times
X$; notation: ${\sf TC}(X)={\sf{secat}}(\pi)$. In other words, ${\sf TC}(X)$
is the smallest integer $k$ for which there exists an open cover $X\times
X=U_{0}\cup U_{1}\cup\dots\cup U_{k}$ such that the fibration $\pi$ admits a
continuous section $s_{j}:U_{j}\to X^{I}$ for each $j=0,1,\dots,k$.
In the parametrized setting developed in [1], one assumes that the motion of
the system is constrained by external conditions, such as obstacles or
variable geometry of the containing domain. The initial and terminal states of
the system, as well as the motion between them, must live under the same
external conditions.
This situation is modelled by a fibration $p:E\to B$, with path-connected
fibers, where the base $B$ is a topological space encoding the variety of
external conditions. For $b\in B$, the fiber $X_{b}=p^{-1}(b)$ is viewed as
the space of achievable configurations of the system given the constraints
imposed by $b$. A parametrized motion planning algorithm takes as input
initial and terminal states of the system (consistent with external conditions
$b$), and produces a continuous path between them, achievable under external
conditions $b$. The initial and terminal points, as well as the path between
them, all lie within the same fiber $X_{b}$.
To define the parametrized topological complexity of the fibration $p:E\to B$
one needs to introduce the associated fibration $\Pi:E^{I}_{B}\to
E\times_{B}E,$ where $E\times_{B}E$ is the space of all pairs of
configurations lying in the same fiber of $p$, while $E_{B}^{I}$ stands for
the space of continuous paths in $E$ lying in a single fiber of $p$; the map
$\Pi$ sends a path to its endpoints.
###### Definition 5.
The parametrized topological complexity of the fibration $p:E\to B$ is defined
as the sectional category of the fibration
(2) $\displaystyle\Pi:E_{B}^{I}\to
E\times_{B}E,\quad\Pi(\gamma)=(\gamma(0),\gamma(1)).$
In more detail,
${\sf TC}[p:E\to B]:={\sf{secat}}(\Pi:E_{B}^{I}\to E\times_{B}E)$
is the minimal integer $k$ such that $E\times_{B}E$ admits an open cover
$E\times_{B}E=U_{0}\cup U_{1}\cup\dots\cup U_{k}$ with the property that each
set $U_{i}$ admits a continuous section of $\Pi$, where $i=0,1,\dots,k$.
Note that $\Pi:E_{B}^{I}\to E\times_{B}E$ is a Hurewicz fibration assuming
that $p:E\to B$ is a Hurewicz fibration, see [2], Proposition 2.1.
If $B^{\prime}\subset B$ is a subset and $E^{\prime}=p^{-1}(B^{\prime})\subset
E$, then the topological complexity ${\sf TC}[p^{\prime}:E^{\prime}\to
B^{\prime}]$ of the restricted fibration (where $p^{\prime}=p|_{E^{\prime}}$)
clearly satisfies
${\sf TC}[p^{\prime}:E^{\prime}\to B^{\prime}]\leq{\sf TC}[p:E\to B].$
In particular, we obtain the inequality
(3) $\displaystyle{\sf TC}(X)\leq{\sf TC}[p:E\to B]$
where $X$ is the fibre of $p:E\to B$.
###### Lemma 6.
Let $p:E\to B$ be a locally trivial fibration with fibre $X$. (A) If ${\sf
TC}[p:E\to B]=0$ then $X$ is contractible. (B) Conversely, if the the fibre
$X$ is contractible and the base $B$ is paracompact then there exists a
globally defined continuous parametrized motion planning algorithm
$s:E\times_{B}E\to E^{I}_{B}$ and therefore ${\sf TC}[p:E\to B]=0$.
###### Proof.
If ${\sf TC}[p:E\to B]=0$ then ${\sf TC}(X)=0$ because of (3). By Theorem 1
from [5] this implies that $X$ is contractible. To prove (B) we shall apply
Corollary 3.2 from the paper of A. Dold [3]. It implies that a locally trivial
fibre bundle $p:E\to B$ with paracompact base and contractible fibre is
shrinkable; this means that there exists a continuous section $\sigma:B\to E$
and a homotopy $H:E\times I\to E$ such that for any $e\in E$ one has
$H(e,0)=e$, $H(e,1)=\sigma p(e)$ and $p(H(e,t))=p(e)$. We may define the
section $s:E\times_{B}E\to E^{I}_{B}$ by the formula
(7) $\displaystyle
s(e,e^{\prime})(t)=\left\\{\begin{array}[]{lll}H(e,2t),&\mbox{for}&0\leq t\leq
1/2,\\\ \\\ H(e^{\prime},2-2t),&\mbox{for}&1/2\leq t\leq 1,\end{array}\right.$
where $(e,e^{\prime})\in E\times_{B}E$ and $t\in I$. Since $H(e,1)=\sigma
p(e)=\sigma p(e^{\prime})=H(e^{\prime},1)$, we see that both parts of the
formula (7) match and hence $s$ is continuous. We clearly have
$s(e,e^{\prime})(0)=e$ and $s(e,e^{\prime})(1)=e^{\prime}$. Besides,
$p(s(e,e^{\prime})(t))=p(e)=p(e^{\prime})$, i.e. $s$ is a continuous
parametrized motion planning algorithm. ∎
Next we mention the upper and lower bounds for the parametrized topological
complexity established in [1].
###### Proposition 7 (Proposition 7.2 from [1]).
Let $p:E\rightarrow B$ be a locally trivial fibration with fiber $X$, where
the spaces $E,B,X$ are CW-complexes. Assume that the fiber $X$ is r-connected.
Then
(8) $\displaystyle{\sf TC}[p:E\to B]<\frac{2\dim X+\dim B+1}{r+1}.$
###### Proposition 8 (Proposition 7.3 from [1]).
Let $p:E\rightarrow B$ be a fibration with path-connected fiber. Consider the
diagonal map $\Delta:E\rightarrow E\times_{B}E$, where $\Delta(e)=(e,e)$. Then
the parametrized topological complexity ${\sf TC}[p:E\rightarrow B]$ is
greater than or equal to the cup-length of the kernel
$\ker[\Delta^{\ast}:H^{\ast}(E\times_{B}E;R)\rightarrow H^{\ast}(E;R)]$, where
$R$ is an arbitrary coefficient ring.
###### Proposition 9.
[Proposition 4.7 from [1]] If $p:E\rightarrow B$ is a locally trivial
fibration, and the spaces $E$ and $B$ are metrizable separable ANRs, then in
Definition 5, instead of open covers one may use arbitrary covers of
$E\times_{B}E$ or, equivalently, arbitrary partitions
$E\times_{B}E=F_{0}\sqcup F_{1}\sqcup\cdot\cdot\cdot\sqcup F_{k},\quad
F_{i}\cap F_{j}=\emptyset,\quad i\not=j$
admitting continuous sections $s_{i}:F_{i}\rightarrow E_{B}^{I}$, where
$i=0,1,...,k.$
###### Example 10.
As an illustration consider the canonical complex line bundle
$\eta:E\to{\mathbb{CP}}^{n}=B$ viewed as a real rank 2 vector bundle. The unit
sphere bundle $\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}$ is a principal
$S^{1}$-bundle, its total space $\dot{E}(\eta)$ is the sphere $S^{2n-1}$, the
set of unit vectors $z\in{\mathbb{C}}^{n}$. The unit circle
$S^{1}\subset{\mathbb{C}}$ acts by multiplication, this action is free and the
quotient is ${\mathbb{CP}}^{n}$. We claim that
(9) $\displaystyle{\sf TC}[\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}]=1.$
Indeed, using (3) we get ${\sf
TC}[\dot{\eta}:\dot{E}(\eta)\to{\mathbb{CP}}^{n}]\geq{\sf TC}(S^{1})=1$. To
obtain the inverse inequality we consider the following partition
$\dot{E}(\eta)\times_{B}\dot{E}(\eta)=F_{0}\sqcup F_{1}$
where $F_{0}$ is the set of all pairs $(z_{1},z_{2})$ of unit vectors
$z_{1},z_{2}\in S^{2n-1}$ lying in the same fibre but not antipodal, i.e.
$z_{1}\not=-z_{2}$; the set $F_{1}$ is the set of antipodal pairs $(z,-z)$.
For $(z_{1},z_{2})\in F_{0}$ we can write $z_{2}/z_{1}=e^{i\phi}$ where
$\phi\in(-\pi,\pi)$ and a continuous section $s_{0}$ of the fibration (2) over
$F_{0}$ can be defined as follows:
$\displaystyle s_{0}(z_{1},z_{2})(t)\,=\,e^{i\phi t}z_{1},\quad t\in[0,1].$
On the other hand, over $F_{1}$ we can define a continuous section $s_{1}$
where
(10) $\displaystyle s_{1}(z,-z)(t)\,=\,e^{i\pi t}\cdot z,\quad t\in[0,1].$
This proves (9).
This example is a special case of a more general statement that the
parametrized topological complexity of any principal bundle equals the
Lusternik - Schnirelmann category of the fibre, see Proposition 4.3 in [1].
The main result of [1] is the computation of the parametrized topological
complexity of the Fadell - Neuwirth fibration which, in term of robotics, can
be understood as the complexity of controlling multiple robots in the presence
of multiple movable obstacles.
## 4\. The cup-length associated with a section
Material of this section will play an auxiliary role in the sequel. We shall
use notations introduced in the beginning of §2.
Let $\xi:E\to B$ be an oriented vector bundle of rank $q\geq 2$ equipped with
scalar product structure $\langle\,,\,\rangle$. As above, let
$\dot{\xi}:\dot{E}\to B$ denote the unit sphere bundle; its fibre is
homeomorphic to $S^{q-1}$. In this section we shall assume that the fibration
$\dot{\xi}:\dot{E}\to B$ has a continuous section $s:B\to\dot{E}$ and our goal
will be to identify the kernel
(11) $\ker[s^{\ast}:H^{\ast}(\dot{E})\to H^{\ast}(B)]$
and its cup-length, i.e. the length of the longest nontrivial products of
elements of this kernel. This result will be used in the following sections to
estimate the parametrized topological complexity from below.
We shall use the following remark. The oriented sphere fibration
$\dot{\xi}:\dot{E}\to B$ admits a cohomological extension of the fibre (see
[12], chapter 5, §7) if and only if its Euler class vanishes,
${\mathfrak{e}}(\xi)=0$. In particular, any oriented sphere fibration with
trivial Euler class ${\mathfrak{e}}(\xi)=0$ satisfies the conclusion of the
Leray - Hirsch theorem, see [12].
Let $\dot{F}\subset\dot{E}$ denote the set $\\{e\in\dot{E};\,e\perp
s(\xi(e))\\}$; it is the set of unit vectors perpendicular to the section. The
projection
$\eta:\,\dot{F}\to B,\quad\eta=\dot{\xi}|_{\dot{F}}$
is an oriented bundle of $(q-2)$-dimensional spheres. Let
${\mathfrak{e}}(\eta)\,\in\,H^{q-1}(B)$ denote the Euler class of $\eta$. The
mod-2 reduction of the class ${\mathfrak{e}}(\eta)$ equals the Stiefel -
Whitney class ${\rm w}_{q-1}(\xi)\in H^{q-1}(B;{\mathbb{Z}}_{2})$ of $\xi$.
###### Theorem 11.
The cup-length of the kernel (11) equals
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$ where
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ denotes the height of the Euler class
${\mathfrak{e}}(\eta)\in H^{q-1}(B)$.
###### Proof.
Let $U\in H^{q-1}(\dot{E})$ denote a fundamental class: for any $b\in B$ the
restriction $U|_{\dot{E}_{b}}$ is the fundamental class of the fibre
$\dot{E}_{b}$. By the Leray - Hirsch theorem every cohomology class in
$H^{\ast}(\dot{E})$ has a unique representation in the form
$\xi^{\ast}(u)+\xi^{\ast}(v)\smile U,$
where $u,v\in H^{\ast}(B)$.
Let $W,W^{\prime}\subset\dot{E}$ denote the following subsets:
$W=\\{e\in\dot{E};\langle e,s(\xi(e))\rangle\geq 0\\}\quad\mbox{and}\quad
W^{\prime}=\\{e\in\dot{E};\langle e,s(\xi(e))\rangle\leq 0\\}.$
Clearly $W\cup W^{\prime}=\dot{E}$ and $W\cap W^{\prime}=\dot{F}$. One can
identify $W$ with the unit disc bundle of the sphere bundle $\dot{F}$.
Therefore the quotient $\dot{E}/W^{\prime}=W/\dot{F}$ can be naturally
identified with the Thom space of the fibration $\eta$.
Next we observe that the fundamental class $U\in H^{q-1}(\dot{E})$ can be
chosen such that $U|_{W^{\prime}}=0$. Indeed, starting with an arbitrary
choice $U^{\prime}$ we can replace it by $U=U^{\prime}-\xi^{\ast}(x)$ where
$x\in H^{q-1}(B)$ is such that
$(\xi|_{W^{\prime}})^{\ast}(x)=U^{\prime}|_{W^{\prime}}$. Here we use the
observation that $\xi|_{W^{\prime}}:W^{\prime}\to B$ is a homotopy equivalence
and hence the class $x$ mentioned above exists and is unique. With this choice
clearly $U|_{W^{\prime}}=0$.
Once the fundamental class $U$ satisfies $U|_{W^{\prime}}=0$ we have the
following formulae which fully describe the multiplicative structure of
$H^{\ast}(\dot{E})$:
(12) $\displaystyle s^{\ast}(U)={\mathfrak{e}}(\eta)\,\in\,H^{q-1}(B)$
and
(13) $\displaystyle U\smile U=\xi^{\ast}({\mathfrak{e}}(\eta)))\smile
U\,\in\,H^{2(q-1)}(\dot{E}).$
To prove (13) we note that $U|_{W}=0$ implies that the class $U$ can be
refined to a relative class $\tilde{U}\in
H^{q-1}(\dot{E},W^{\prime})=H^{q-1}(\dot{E}/W^{\prime})=H^{q-1}(W/\dot{F})$.
We already mentioned that the quotient $\dot{E}/W^{\prime}$ can be identified
with the Thom space of the vector bundle having $\eta$ as its unit sphere
bundle. Examining the long exact sequence in cohomology
$\dots\to
H^{q-2}(\dot{E})\stackrel{{\scriptstyle\simeq}}{{\to}}H^{q-2}(W^{\prime})\to
H^{q-1}(\dot{E},W^{\prime})\to H^{q-1}(\dot{E})$
we see that the refinement $\tilde{U}$ is unique and coincides with the Thom
class. Now, by the definition (see [10], §9), we have
(14)
$\displaystyle{\mathfrak{e}}(\eta)\,=\,s^{\ast}(\tilde{U}|_{W})\,=\,s^{\ast}(U),$
which proves (12). From (14) we also obtain
$\tilde{U}\smile\tilde{U}=(\tilde{U}|_{W})\smile\tilde{U}=\xi^{\ast}({\mathfrak{e}}(\eta))\smile\tilde{U}\,\,\in\,\,H^{2(q-1)}(\dot{E},W^{\prime}).$
Applying the restriction homomorphism $H^{2(q-1)}(\dot{E},W^{\prime})\to
H^{2(q-1)}(\dot{E})$ to both sides of this equality gives (13).
Note that the order of the factors in the RHS of formula (13) is irrelevant:
if $q$ is odd then the classes commute and for $q$ is even the Euler class
${\mathfrak{e}}(\eta)$ has order two.
Consider now an arbitrary class $x\in H^{\ast}(\dot{E})$ satisfying
$s^{\ast}(x)=0$. We can write
$x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile
U\quad\mbox{with}\quad\alpha,\,\beta\in H^{\ast}(B).$
Applying $s^{\ast}$ and using (12) we get
(15) $\displaystyle
s^{\ast}(x)\,=\,0\,=\,\alpha+\beta\smile{\mathfrak{e}}(\eta).$
Conversely, any two classes $\alpha,\,\beta$ satisfying (15) produce a class
$x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile U$ lying in the kernel of
$s^{\ast}$. A particular choice $\alpha=-{\mathfrak{e}}(\eta)$ and $\beta=1$
gives the class
$x_{0}=U-\xi^{\ast}({\mathfrak{e}}(\eta))\,\in H^{q-1}(\dot{E}).$
Using (13) we have
$x_{0}^{2}=U^{2}-2\xi^{\ast}({\mathfrak{e}}(\eta))\smile
U+\xi^{\ast}({\mathfrak{e}}(\eta))^{2}=-\xi^{\ast}({\mathfrak{e}}(\eta))\smile
x_{0}$
and we obtain by induction
$\displaystyle x_{0}^{n}$ $\displaystyle=$
$\displaystyle(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta)^{n-1})\smile x_{0}$
$\displaystyle=$
$\displaystyle(-1)^{n}\xi^{\ast}({\mathfrak{e}}(\eta)^{n})+(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta))^{n-1}\smile
U.$
For $n=h({\mathfrak{e}}(\eta))+1$ the class $x_{0}^{n}$ equals
$(-1)^{n-1}\xi^{\ast}({\mathfrak{e}}(\eta))^{n-1}\smile U$ and is obviously
nonzero (as follows from the Leray - Hirsch theorem). This implies that the
cup-length of the kernel $\ker s^{\ast}$ is at least
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$.
If $x=\xi^{\ast}(\alpha)+\xi^{\ast}(\beta)\smile U\,\in\,H^{\ast}(\dot{E})$ is
an arbitrary class with $s^{\ast}(x)=0$ then
$\alpha=-\beta\smile{\mathfrak{e}}(\eta)$ (see above) and thus
$x=\xi^{\ast}(\beta)\smile x_{0}$. In other words, the kernel $\ker
s^{\ast}\subset H^{\ast}(\dot{E})$ is the principal ideal generated by the
class $x_{0}$. We see that the cup-length of the kernel equals the highest
nonzero power of $x_{0}$ which, as we have shown above, is
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$. ∎
The following Corollary is an analogue of Theorem 11 where we use
$\mathbb{Z}_{2}$ coefficients. The role of the Euler class plays the top
Stiefel - Whitney class of the bundle $\eta$ of vectors orthogonal to the
section. The advantage of this statement is that the answer is given in terms
of the original bundle $\xi$ and its characteristic class ${\rm
w}_{q-1}(\xi)\in H^{q-1}(B;\mathbb{Z}_{2})$.
###### Corollary 12.
Let $\xi:E\to B$ be a rank $q\geq 2$ vector bundle (not necessarily
orientable). Let $s:B\to\dot{E}$ be a continuous section of the unit sphere
bundle. Then the cup-length of the kernel
$\ker[s^{\ast}:H^{\ast}(\dot{E};{\mathbb{Z}}_{2})\to
H^{\ast}(B;{\mathbb{Z}}_{2})]$ equals ${\mathfrak{h}}({\rm w}_{q-1}(\xi))+1$.
###### Proof.
One repeats the arguments of the proof of Theorem 11 replacing the integer
coefficients by ${\mathbb{Z}}_{2}$. The bundle $\eta:\,F\to
B,\quad\eta=\xi|_{F}$ is the bundle of vectors orthogonal to the section, i.e.
$F=\\{e\in E;\,e\perp s(\xi(e))\\}$ and the arguments of the proof of Theorem
11 show that the the kernel
$\ker[s^{\ast}:H^{\ast}(\dot{E};{\mathbb{Z}}_{2})\to
H^{\ast}(B;{\mathbb{Z}}_{2})]$ is the principal ideal generated by the class
$x_{0}=U-\xi^{\ast}({\rm w}_{q-1}(\eta))$. The height of $x_{0}$ equals one
plus the height of the class ${\rm w}_{q-1}(\eta)$. However,
$\xi=\eta\oplus\epsilon$ where $\epsilon$ is the trivial line bundle
determined by the section and therefore ${\rm w}_{q-1}(\eta)={\rm
w}_{q-1}(\xi)$ and the result follows. ∎
## 5\. Parametrized topological complexity of sphere bundles
Let $\xi:E\to B$ be an oriented rank $q\geq 2$ vector bundle equipped with
fibrewise scalar product. Let $\dot{\xi}:\dot{E}\to B$ denote the unit sphere
bundle; its fibre is the sphere of dimension $q-1$. Our goal is to estimate
the parametrized topological complexity ${\sf TC}[\dot{\xi}:\dot{E}\to B]$. By
Proposition 7 we have an upper bound
(17) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to B]<2+\frac{\dim B+1}{q-1}.$
To state our result, consider the bundle
(18) $\displaystyle\ddot{\xi}:\ddot{E}\to\dot{E},$
where $\ddot{E}\subset\dot{E}\times_{B}\dot{E}$ is the space of pairs of
mutually orthogonal unit vectors $(x,y)\in\dot{E}\times_{B}\dot{E}$, $x\perp
y$. The projection $\ddot{\xi}$ acts as $\ddot{\xi}(x,y)=x.$ The map (18) is
an oriented locally trivial fibration with fibre sphere of dimension $q-2$.
Consider its Euler class
(19) $\displaystyle{\mathfrak{e}}(\ddot{\xi})\in H^{q-1}(\dot{E}).$
An obvious property of the class ${\mathfrak{e}}(\ddot{\xi})$ is that for any
point $b\in B$ the restriction ${\mathfrak{e}}(\ddot{\xi})|_{\dot{E}_{b}}$ is
the Euler class of the tangent bundle of the sphere $\dot{E}_{b}$.
###### Remark 13.
A section $s$ of bundle (18) associates with a unit vector $e\in\dot{E}(\xi)$
a unit vector $s(e)$ which is perpendicular to $e$ and the integer
${\sf{secat}}(\ddot{\xi})$ is a measure of complexity of construction such a
section $s$ globally, i.e. over all $\dot{E}$. In particular,
${\sf{secat}}(\ddot{\xi})=0$ if the vector bundle $\xi$ admits a complex
structure: in this case one can define the section by $s(e)=\sqrt{-1}\cdot e$.
###### Theorem 14.
One has the estimates
(20) $\displaystyle{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,\leq\,{\sf
TC}[\dot{\xi}:\dot{E}\to B]\leq{\sf{secat}}(\ddot{\xi})+1.$
Moreover, if $B$ is a CW-complex satisfying $\dim
B\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ then
(21) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to
B]\,=\,{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,=\,{\sf{secat}}(\ddot{\xi})+1.$
###### Proof.
Consider the diagonal map $\Delta:\dot{E}\to\dot{E}\times_{B}\dot{E}$ and
apply Proposition 8; we obtain that the parametrized topological complexity
${\sf TC}[\dot{\xi}:\dot{E}\to B]$ is greater than or equal to the cup-length
of the kernel $\ker[\Delta^{\ast}:H^{\ast}(\dot{E}\times_{B}\dot{E})\to
H^{\ast}(\dot{E})]$. However, $\Delta$ is a section of the sphere fibration
$\dot{E}\times_{B}\dot{E}\to\dot{E}$ given by projection on the first vector;
hence we may apply Theorem 11 which describes the cup-length of the kernel of
the induced map. The bundle of vectors perpendicular to the section is exactly
the bundle $\ddot{\xi}$. By Theorem 11 the cup-length of the kernel
$\ker[\Delta^{\ast}:H^{\ast}(\dot{E}\times_{B}\dot{E})\to H^{\ast}(\dot{E})]$
equals ${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1$. This gives the lower
bound in (20).
To prove the right inequality in (20) consider the set
$U\subset\dot{E}\times_{B}\dot{E}$ consisting of pairs
$(e,e^{\prime})\in\dot{E}\times_{B}\dot{E}$ with $e\not=-e^{\prime}$. Over
$U$, we can define a continuous motion planning algorithm
$s:U\to\dot{E}_{B}^{I}$ by setting
(22) $\displaystyle
s(e,e^{\prime})(t)=\frac{(1-t)e+te^{\prime}}{||(1-t)e+te^{\prime}||},\quad
t\in[0,1].$
In view of Proposition 9 it remains to construct a motion planning algorithm
over the complementary set
$V=\\{(e,-e);\,e\in\dot{E}\\}\subset\dot{E}\times_{B}\dot{E}.$ Denote by
$p_{1},p_{2}:V\to\dot{E}$ the projections $p_{1}(e,-e)=e$ and
$p_{2}(e,-e)=-e.$
Consider again the bundle $\ddot{\xi}:\ddot{E}\to\dot{E}$ and suppose that
$A\subset\dot{E}$ is a subset such that the bundle $\ddot{\xi}$ admits a
continuous section $s_{A}:A\to\ddot{E}$ over $A$. Using this section we may
construct a section $s^{\prime}_{A}$ of the fibration
$\Pi:\dot{E}_{B}^{I}\to\dot{E}\times_{B}\dot{E}$
over the set $p_{1}^{-1}(A)$ as follows:
(23) $\displaystyle s^{\prime}_{A}(e,-e)(t)\,=\,\cos\left(t\pi\right)\cdot
e\,+\,\sin\left(t\pi\right)\cdot s_{A}(e),\quad t\in[0,1].$
Let $\dot{E}=A_{0}\cup A_{1}\cup\dots\cup A_{k}$ be an open covering, where
$k={\sf{secat}}(\ddot{\xi})$, with the property that
$\ddot{\xi}:\ddot{E}\to\dot{E}$ admits a continuous section over each $A_{i}$.
Then the sets $p_{1}^{-1}(A_{i})$ cover $V$ and over each of these sets the
fibration $\Pi$ admits a continuous section. Thus we get an inequality ${\sf
TC}[\dot{\xi}:\dot{E}\to B]\leq{\sf{secat}}(\ddot{\xi})+1$.
Finally we apply Lemma 1 which claims that the sectional category of
$\ddot{\xi}$ equals ${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ under an
additional assumption that
$\dim\dot{E}\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+(q-1)$
which is equivalent to $\dim
B\leq(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$. Hence under this
assumption we obtain ${\sf TC}[\dot{\xi}:\dot{E}\to
B]\,=\,{\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))+1\,=\,{\sf{secat}}(\ddot{\xi})+1$.
This completes the proof. ∎
###### Corollary 15.
For a vector bundle $\xi:E\to B$ satisfying ${\sf{secat}}(\ddot{\xi})=0$ one
has ${\sf TC}[\dot{\xi}:\dot{E}\to B]=1.$
###### Proof.
The inequality (20) gives ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq 1.$ On the
other hand, ${\sf TC}[\dot{\xi}:\dot{E}\to B]\geq{\sf TC}(S^{2r-1})=1$ by (3).
∎
###### Example 16.
Consider the canonical rank 2 bundle $\xi$ over ${\mathbb{CP}}^{n}$ as in
Example 10. In this case $\dot{E}(\xi)=S^{2n-1}$ and the bundle
$\ddot{\xi}:\ddot{E}\to\dot{E}$ is the trivial bundle with fibre $S^{0}$, i.e.
${\sf{secat}}(\ddot{\xi})=0={\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi})).$
We obtain from (20) that ${\sf TC}[\dot{\xi}:S^{2n-1}\to{\mathbb{CP}}^{n}]=1$
confirming the result of Example 10.
Generalising Example 16 we may state:
###### Corollary 17.
For any vector bundle $\xi:E\to B$ of even rank ${\rm{rk}}(\xi)=2r$ admitting
a complex structure, one has
${\sf TC}[\dot{\xi}:\dot{E}\to B]=1={\sf TC}(S^{2r-1}).$
###### Proof.
In this case ${\sf{secat}}(\ddot{\xi})=0$ (by Remark 13) and the result
follows from Corollary 15. ∎
###### Remark 18.
Introducing the bundle $\ddot{\xi}$ over $\dot{E}$ and using its sectional
category to estimate the parametrized topological complexity we made an
approximation of the space of paths on the sphere connecting a pair of
antipodal points by the sphere of one dimension below. This sphere is however
only the first term in the James’ construction $JS^{q-2}$, see [8], which
gives a CW complex having the homotopy type of this space of paths.
Note that for $q$ even the Euler class ${\mathfrak{e}}(\ddot{\xi})\in
H^{q-1}(\dot{E})$ has order 2, i.e. $2\cdot{\mathfrak{e}}(\ddot{\xi})=0$. We
shall focus below on the case when $q$ odd. Compared with Theorem 14,
Corollary 19 stated below has the advantage of dealing with cohomology of the
base $B$.
###### Corollary 19.
For $q\geq 3$ odd, let $\eta:E(\eta)\to B$ be an oriented vector bundle of
rank $q-1$. Let $\xi=\eta\oplus\epsilon$ be the sum where $\epsilon$ is the
trivial line bundle over $B$. Then one has
(24) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}(\xi)\to
B]\geq{\mathfrak{h}}({\mathfrak{e}}(\eta))+1.$
Moreover, if the height ${\mathfrak{h}}({\mathfrak{e}}(\eta))$ is even and the
integral cohomology of the base $B$ in dimension
$(q-1)\cdot{\mathfrak{h}}({\mathfrak{e}}(\eta))$ has no 2-torsion then
(25) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}(\xi)\to
B]\geq{\mathfrak{h}}({\mathfrak{e}}(\eta))+2;$
###### Proof.
Consider the Euler class ${\mathfrak{e}}(\ddot{\xi})\in
H^{q-1}(\dot{E}(\xi))$. Applying the Leray - Hirsch theorem we see that any
class in $H^{q-1}(\dot{E}(\xi))$ has a unique representation as
$\dot{\xi}^{\ast}(a)+bU$ where $U\in H^{q-1}(\dot{E})$ is a fundamental class,
$a\in H^{q-1}(B)$ and $b\in{\mathbb{Z}}$. Let $s:B\to\dot{E}(\xi)$ be the
section determined by the trivial summand $\epsilon$. We showed in the proof
of Theorem 11 that the fundamental class $U$ can be chosen so that
(26) $\displaystyle s^{\ast}(U)={\mathfrak{e}}(\eta),$
see formula (12). Note that
(27) $\displaystyle s^{\ast}(\ddot{\xi})=\eta.$
Besides,
(28)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})=\dot{\xi}^{\ast}(a)+2U,\quad\mbox{for
some class}\quad a\in H^{q-1}(B).$
Indeed, the class ${\mathfrak{e}}(\ddot{\xi})$ restricted to each fibre
$\dot{E}_{b}(\xi)$ equals twice the fundamental class of the sphere
$\dot{E}_{b}(\xi)\simeq S^{q-1}$ (here we use our assumption that $q$ is odd,
and hence the Euler characteristic of $S^{q-1}$ equals 2). Applying $s^{\ast}$
to both sides of equation (28) we find
$s^{\ast}({\mathfrak{e}}(\ddot{\xi}))={\mathfrak{e}}(s^{\ast}(\ddot{\xi}))={\mathfrak{e}}(\eta)$,
and $s^{\ast}(\dot{\xi}^{\ast}(a))=a$ which together with (26) give
$a=-{\mathfrak{e}}(\eta).$ Therefore we have
(29)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})=-\dot{\xi}^{\ast}({\mathfrak{e}}(\eta))+2U.$
Using $U^{2}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta))\smile U$ (see (13)) we
find
${\mathfrak{e}}(\ddot{\xi})^{2}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2})$
and therefore the even and odd powers of the class
${\mathfrak{e}}(\ddot{\xi})$ are as follows
(30)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})^{2n}=\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n})$
and
(31)
$\displaystyle{\mathfrak{e}}(\ddot{\xi})^{2n+1}=-\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n+1})+2\dot{\xi}^{\ast}({\mathfrak{e}}(\eta)^{2n})\smile
U.$
From formulae (30) and (31) we see that the height
${\mathfrak{h}}({\mathfrak{e}}(\ddot{\xi}))$ either equals to
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ or it equals
${\mathfrak{h}}({\mathfrak{e}}(\eta))+1$; the second possibility happens iff
${\mathfrak{h}}({\mathfrak{e}}(\eta))$ is even and the group
$H^{(q-1){\mathfrak{h}}({\mathfrak{e}}(\eta))}(B)$ has no 2-torsion.
Applying Theorem 14 completes the proof. ∎
###### Example 20.
Consider the situation of Corollary 19 in the case when
$\eta:E(\eta)\to{\mathbb{CP}}^{n}$ is the canonical bundle over the complex
projective space as in Example 10. Taking $\xi=\eta\oplus\epsilon$ we have
${\rm{rk}}(\xi)=q=3$ is odd and ${\mathfrak{h}}({\mathfrak{e}}(\eta))=n$. By
Corollary 19 we get ${\sf TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\geq
n+1$ and moreover for $n$ even ${\sf
TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\geq n+2$. On the other hand,
the upper bound (17) gives ${\sf
TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]\leq n+2$. Thus, we see that
${\sf TC}[\dot{\xi}:\dot{E}(\xi)\to{\mathbb{CP}}^{n}]=n+2$
for all even $n$. In particular, we see that the parametrized topological
complexity of sphere bundles can be arbitrarily large. This contrasts the
situation with the usual (i.e. unparametrized) topological complexity which
takes the values $1$ and $2$ only for spheres.
Finally we describe an explicit parametrized motion planning algorithm having
complexity $n+2$ for the unit sphere bundle associated with the vector bundle
$\xi=\eta\oplus\epsilon$ over $B={\mathbb{CP}}^{n}$ as considered in Example
20. We shall describe a partition
(32) $\displaystyle\dot{E}(\xi)\times_{B}\dot{E}(\xi)\,=\,F_{0}\sqcup
F_{1}\sqcup\dots\sqcup F_{n+2}$
and a continuous section $s_{i}$ of the fibration
$\Pi:\dot{E}(\xi)^{I}_{B}\,\to\,\dot{E}(\xi)\times_{B}\dot{E}(\xi)$
over each of the sets $F_{i}$ where $i=0,1,\dots,n+2$.
The set $F_{0}\subset\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ will be defined as
the set of pairs $(x,y)\in\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ with $x\not=-y$.
The section $s_{0}$ over $F_{0}$ can be defined by formula (22).
The unit sphere bundle of the trivial summand $\epsilon$ gives the sections
$\pm\sigma:B\to\dot{E}(\epsilon)\subset\dot{E}(\xi)$. Let
$\dot{E}(\xi)^{\ast}$ denote the complement $\dot{E}(\xi)-\dot{E}(\epsilon)$.
We define the set $F_{1}\subset\dot{E}(\xi)\times_{B}\dot{E}(\xi)$ to be the
set of all pairs $(x,-x)$ with $x\in\dot{E}(\xi)^{\ast}$. Let ${\rm
pr}:\dot{E}(\xi)^{\ast}\to\dot{E}(\eta)\subset\dot{E}(\xi)$ denote the
retraction given by the formula
(33) $\displaystyle{\rm pr}(x)\,=\,\frac{x-\langle
x,\sigma(b)\rangle\cdot\sigma(b)}{||x-\langle
x,\sigma(b)\rangle\cdot\sigma(b)||}\quad\mbox{where}\quad b=\xi(x).$
Here the symbol $\langle\,,\,\rangle$ denotes scalar product in the fibre. The
deformation
$\alpha_{t}(x)\,=\,\frac{x-t\cdot\langle
x,\sigma(b)\rangle\cdot\sigma(b)}{||x-t\cdot\langle
x,\sigma(b)\rangle\cdot\sigma(b)||}\quad\mbox{where}\quad
b=\xi(x),\,t\in[0,1],$
satisfies $\alpha_{0}(x)=x$ and $\alpha_{1}(x)={\rm pr}(x)$. The homotopy
$t\mapsto(\alpha_{t}(x),\alpha_{t}(-x))$ deforms the initial pair $(x,-x)$ to
a pair of antipodal points lying in the equatorial sphere
$\dot{E}(\eta)\subset\dot{E}(\xi)$. Note that the circle $S^{1}$ acts freely
on $\dot{E}(\eta)$, see Example 10. We may define a continuous section $s_{1}$
over $F_{1}$ by setting $s_{1}(x,-x)(t)$ to be the concatenation of the
following three paths: (a) the deformation $\alpha_{t}(x)$, (b) the section
(10) of Example 10, and (c) the reverse of the deformation $\alpha_{t}(-x)$.
In more detail,
$s_{1}(x,-x)(t)=\left\\{\begin{array}[]{lll}\alpha_{3t}(x),&\mbox{for}&t\in[0,1/3],\\\
\\\ e^{i\pi(3t-1)}\cdot{\rm pr}(x),&\mbox{for}&t\in[1/3,2/3],\\\ \\\
\alpha_{3(1-t)}(-x),&\mbox{for}&t\in[2/3,1].\end{array}\right.$
Finally we define the sections $s_{i}:F_{i}\to\dot{E}(\xi)^{I}_{B}$ for
$i=2,3,\dots,n+2$ as follows. The base $B={\mathbb{CP}}^{n}$ has the well-
known cell decomposition ${\mathbb{CP}}^{n}=e^{0}\sqcup e^{2}\sqcup\dots\sqcup
e^{2n}$ with a single cell $e^{2i}$ in each even dimension $2i\leq 2n$. For
$i=2,3,\dots,n+2$, let $F_{i}$ denote the set of pairs $(x,-x)$ with
$x=\pm\sigma(b)$ for $b=\xi(x)$ lying in the cell $e^{2(i-2)}$. Since the cell
$e^{2(i-2)}$ is contractible, the bundle $\eta$ admits a continuous section
$\phi_{i}$ over $e^{2(i-2)}$. Hence we may define the section
$s_{i}:F_{i}\to\dot{E}(\xi)^{I}_{B}$ by the formula
$s_{i}(x,-x)(t)=\cos(\pi t)\cdot x+\sin(\pi t)\cdot\phi_{i}(\xi(x)),$
similarly to (23). Here we assume that the Euclidean structure on the vector
bundle $\xi$ is the orthogonal sum of the Euclidean structures of $\eta$ and
$\epsilon$.
We conclude the paper with the following observations.
Below we always assume that the base $B$ is an ANR.
###### Lemma 21.
Let $\xi:E\to B$ be a vector bundle such that $\xi=\eta\oplus\tau$ and
${\rm{rk}}(\tau)\geq 2$. Then
(34) $\displaystyle\hskip
28.45274pt{\sf{secat}}(\ddot{\xi}:\ddot{E}(\xi)\to\dot{E}(\xi))\,\leq\,{\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))+{\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to
B)+1$
and consequently
(35) $\displaystyle{\sf TC}[\dot{\xi}:\dot{E}\to
B]\leq{\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))+{\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to
B)+2.$
###### Proof.
It is enough to prove the inequality (34) as (35) follows from (34) and
Theorem 14.
If $\xi=\eta\oplus\tau$ then for any point of the base $b\in B$ we have
$E(\xi)_{b}=E(\eta)_{b}\oplus E(\tau)_{b}$. The scalar product can be chosen
so that the spaces $E(\eta)_{b},\,E(\tau)_{b}\subset E(\xi)_{b}$ are mutually
orthogonal. We shall denote by $P^{\eta}_{b}$ and $P^{\tau}_{b}$ the
orthogonal projections of $E(\xi)_{b}$ onto $E(\eta)_{b}$ and $E(\tau)_{b}$
correspondingly.
Let $k$ denote ${\sf{secat}}(\ddot{\tau}:\ddot{E}(\tau)\to\dot{E}(\tau))$ and
let $\ell$ denote ${\sf{secat}}(\dot{\tau}:\dot{E}(\tau)\to B)$. Let
$\dot{E}(\tau)=G_{0}\sqcup G_{1}\sqcup\dots\sqcup G_{k}$ be a partition such
that for each $j=0,\dots,k$ there exists a continuous map
$\sigma_{j}:G_{j}\to\dot{E}(\tau)$ with the property that for every $e\in
G_{j}$ the vectors $e$ and $\sigma_{j}(e)$ lie in the same fibre and are
perpendicular to each other. Besides, let $B_{0}\sqcup B_{1}\sqcup\dots\sqcup
B_{\ell}=B$ be a partition of the base $B$ with continuous sections
$\nu_{i}:B_{i}\to\dot{E}(\tau)$, where $i=0,\dots,\ell$.
We want to show that $\dot{E}(\xi)$ can be partitioned as
$\dot{E}(\xi)=F_{0}\sqcup F_{1}\sqcup\dots\sqcup F_{k}\sqcup
F_{k+1}\sqcup\dots\sqcup F_{k+\ell+1}$
such that there exist continuous sections $s_{i}:F_{i}\to E(\ddot{\xi})$ of
the fibration $\ddot{\xi}$. In view of the result of J. M. Garcia-Calcines
[7], this is equivalent to ${\sf{secat}}(\ddot{\xi})\leq k+\ell+1$.
For $i=0,1,\dots,k$ we set
$F_{i}=\\{e\in\dot{E}(\xi);P_{\xi(e)}^{\tau}(e)\not=0,\,\mbox{and}\,||P_{\xi(e)}^{\tau}(e)||^{-1}\cdot
P_{\xi(e)}^{\tau}(e)\in G_{i}\\}.$
and for $i=k+1,\dots,k+\ell+1$, we set
$F_{i}=\\{e\in\dot{E}(\xi);\,P_{\xi(e)}^{\tau}(e)=0\,\,\mbox{and}\,\,\xi(e)\in
B_{i-k-1}\\}.$
Next we construct the continuous sections $s_{i}:F_{i}\to E(\ddot{\xi})$.
For $i=0,1,\dots,k$, given a unit vector $e\in F_{i}$, consider
$e^{\prime}=P_{\xi(e)}^{\tau}(e)$ which is a nonzero vector of $E(\tau)_{b}$,
where $b=\xi(e)$. Then $e^{\prime\prime}=||e^{\prime}||^{-1}\cdot e^{\prime}$
is a unit vector and $\sigma_{i}(e^{\prime\prime})\in\dot{E}(\tau)_{b}$ is a
unit vector satisfying $\sigma_{i}(e^{\prime\prime})\perp e^{\prime}$. In
particular we see that the unit vectors $e,\,\sigma_{i}(e^{\prime\prime})\in
E(\xi)_{b}$ are linearly independent. Hence the unit vector
$e^{\prime\prime\prime}=||\sigma_{i}(e^{\prime\prime})-\langle\sigma_{i}(e^{\prime\prime}),e\rangle\cdot
e||^{-1}\cdot(\sigma_{i}(e^{\prime\prime})-\langle\sigma_{i}(e^{\prime\prime}),e\rangle\cdot
e)$
is perpendicular to $e$ and depends continuously on $e$. Thus, we may define
the section $s_{i}$ by setting $s_{i}(e)=(e,e^{\prime\prime\prime})$.
Next we describe the sections $s_{i}$ over the sets $F_{i}$ where
$i=k+1,\dots,k+\ell+1$. If $e\in F_{i}$ then $P_{b}^{\tau}(e)=0$ and
$b=\xi(e)$ lies in $B_{i-k-1}$. In other words, $e\in\dot{E}(\eta)$ and
$b=\xi(e)\in B_{i-k-1}$. The section $\nu_{i-k-1}:B_{i-k-1}\to\dot{E}(\tau)$
defines a unit vector $\nu_{i-k-1}(b)\in\dot{E}(\tau)\subset\dot{E}(\xi)$
which is perpendicular to $e$. Therefore we may define the section $s_{i}$ of
$\ddot{\xi}$ over $F_{i}$ by the formula $s_{i}(e)=(e,\nu_{i-k-1}(e)).$
∎
###### Corollary 22.
Let $\xi:E\to B$ be a vector bundle such that $\xi=\eta\oplus\tau$ where
$\tau:E(\tau)\to B$ admits a complex structure and has a nowhere zero
continuous section. Then ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq 2$.
###### Proof.
This follows from Remark 13 and Lemma 21. ∎
###### Corollary 23.
Let $\xi:E\to B$ be a vector bundle admitting two continuous linearly
independent nowhere zero sections. Then ${\sf TC}[\dot{\xi}:\dot{E}\to B]\leq
2$.
###### Proof.
This reduces to the previous Corollary with $\tau$ the trivial bundle of rank
2. ∎
## References
* [1] D.C. Cohen, M. Farber, S. Weinberger, Topology of Parametrized Motion Planning Algorithms, SIAM J. of Applied Algebra and Geometry, 5(2021), pp. 229–249.
* [2] D.C. Cohen, M. Farber, S. Weinberger, Parametrized topological complexity of collision-free motion planning in the plane, arXiv:2010.09809. To appear in ”Annals of Mathematics and Artificial Intelligence”.
* [3] A. Dold, Partitions of unity in the theory of fibrations. Ann. of Math. (2) 78 (1963), 223–255.
* [4] E. Fadell and S. Husseini, Category weight and Steenrod operations, Bol. Soc. Mat. Mexicana (2) 37(1992), no. 1-2, 151–161.
* [5] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221.
* [6] M. Farber, Invitation to topological robotics, Zurich Lectures in Advanced Mathematics, EMS, 2008.
* [7] J. M. García-Calcines, A note on covers defining relative and sectional categories, Topology Appl., 265 (2019), 106810.
* [8] I.M. James, Reduced product spaces, Ann. Math. 62(1955), pp. 170 – 197.
* [9] S. M. LaValle, Planning algorithms, Cambridge University Press, 2006.
* [10] J. W. Milnor, J. D. Stasheff, Characteristic classes, Princeton University Press, 1974.
* [11] A.S. Schwarz, The genus of a fibre space. Trudy Moscow Math Society 11(1962), 99 – 126.
* [12] E. Spanier, Algebraic Topology, 1966.
|
# Two fields quintessential Higgs Inflation
Mehdi Es-haghi eshaghi249(AT)gmail.com Ahmad Sheykhi
asheykhi(AT)shirazu.ac.ir Physics Department and Biruni Observatory, Shiraz
University, Shiraz 71454, Iran
###### Abstract
We study a two-field quintessential inflation model with non-minimal coupling
term where an exponential quintessence potential coupled to Higgs potential
from inflationary era. This non-minimal two-field model is shown to provide
very good consistency to the recent cosmological observation. Although, the
quintessence field is sub-dominant during inflation, it can remove Higgs
instability in the early universe which is a serious problem. Also,
considering quintessence from early time helps overcoming its initial
condition problem.
## I Introduction
Cosmologists consider two accelerating eras for cosmic expansion history. The
first is inflationary era in the early universe Guth which was driven by a
scalar field, called inflaton, with a nearly flat potential and negative
pressure. Based on the latest Planck data Aghanim:2018eyx ; Akrami:2018odb ,
the complicated inflationary models with multiple fields and non-canonical
ones have been eliminated. In exchange, single field inflationary potentials
with plateau-like shapes are the most favored models by Cosmic Microwave
Background (CMB) data.
The second is the late time accelerating era, discovered during 1998
Riess:1998cb . The most straightforward justification for this accelerating
expansion is an infinitesimal positive cosmological constant $\Lambda$ which
maybe introduced via the vacuum state of a scalar field with non-zero constant
energy $V_{0}$ called dark energy. Dark energy which is thought to be
homogeneous and has very low density and negative pressure, makes up about
70$\%$ of the universe compounds.
Following the strange nature of dark energy and considering the similarity of
this component with inflaton field, an alternative to cosmological constant
for explanation of the late time accelerated expansion is a dynamical scalar
field with infinitesimal energy density, called quintessence, which its slow-
rolling evolution at the late time can accelerate the universe Copeland:1997et
. This scalar field avoids the extreme fine-tuning of the cosmological
constant Tsujikawa:2013fta . Although, $\Lambda$CDM model with dark energy
equation of state $\omega_{\Lambda}=-1$ satisfies the cosmological data very
well and we may not need any other explanation for dark energy, but if future
cosmological experiments show a deviation from $\omega_{\Lambda}=-1$ for dark
energy equation of state, then quintessence may help us to explain this
deviation Akrami:2017cir .
In both inflation and quintessence models, potential energy gives the dominant
contribution to the total energy density of a scalar field which slowly rolls
down its almost flat potential. As, quintessence has its own tuning problems,
namely its initial conditions, explaining both inflation and dark energy in a
unified scenario with a single dynamical scalar field, named single field
quintessential inflation, may overcome this difficulty Peebles:1998qn ;
Dimopoulos:2017zvq . In this scenario, the single scalar field needs two
plateau shoulders with an extremely large difference in their heights. So, it
needs that the scalar field rolls down from its inflationary plateau to its
quintessential ones very very slowly without oscillating around the potential
minimum. It means that the reheating of the universe must have happened at the
end of inflation via a mechanism, called gravitational reheating Ford ;
Peebles:1998qn , instead of inflaton field decay. Indeed, through a mechanism,
called kination, almost all of inflaton potential energy becomes converted to
its kinetic ones to prevent the field decays completely and by rolling the
field towards its large negative values, it freezes at some $\phi_{F}$. After
the densities of radiation and cold dark matter (CDM) become sufficiently
small, the field starts rolling down again and acts as quintessence field
Akrami:2017cir ; Dimopoulos:2017zvq .
Another scenario, called two-field quintessential inflation, assumes that from
early times, there is a two-dimensional potential energy in which one field is
responsible for dark energy, while inflation is driven by another field
Akrami:2017cir . During inflation, the inflaton field has main role in the
evolution of the universe, while the role of quintessence field is
insignificant. When the inflation ends, the inflaton field falls down to the
quintessence valley. In this case, the reheating occurs due to oscillations of
the inflaton field near the minimum point of its potential. Because of the
tiny slope of the potential along the quintessence field axis, the
quintessence field does not start rolling until the density of products of
reheating (radiation and matter) decrease significantly.
A general theory of inflation in which a single inflaton field couples non-
minimally to gravity has been extensively studied during past years
Futamase:1987ua . One of the favored inflationary models which satisfys recent
CMB data well Akrami:2018odb is Higgs inflation in which Standard Model Higgs
boson with non-minimal coupling to gravity has the role of inflaton scalar
field Salopek ; Bezrukov:2007ep . A characteristic feature of this model is
that the only scalar field which is discovered yet is Higgs boson
Chatrchyan:2012ufa . However, quantum corrections during inflation make self-
interacting coupling constant of Higgs boson negative at an energy scale
around $10^{11}$GeV which may leads to serious problem, called Higgs
instability Bezrukov:2007ep ; Han:2018yrk . One proposal for overcoming this
problem is a model of a specific coupling between Higgs boson and quintessence
field Denef:2018etk ; Han:2018yrk . The interesting feature of this proposal
is that the constraints come from the stability of electroweak vacuum in
inflationary era and the observations for dark energy are in agreement with
the proposed Swampland Conjecture Brennan:2017rbf . In this paper, we aim to
describe a new model of two-field quintessential inflation including Higgs and
quintessence fields which have a trilinear coupling with each other during
inflation era and in later times and Higgs field has also a non-minimal
coupling to gravity throughout inflation.
In this work, we consider a two-field quintessential inflation model where a
quintessence potential coupled to the Higgs potential from inflationary era.
These Higgs and quintessence fields have non-minimally and minimally coupling
to gravity, respectively. In the following, we study the behaviors of their
coupled potential in the Einstein frame and during the inflationary era.
Finally, we derive the cosmological parameters of the model and compare them
with the recent cosmological data.
## II The model
In Jordan frame, the generalized action with two fields, non-minimally coupled
to gravity, is as follow
$\displaystyle S_{J}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}M_{P}^{2}~{}\Omega^{2}(\phi_{1},\phi_{2})R-\frac{1}{2}\partial_{\mu}\phi_{1}\,\partial^{\mu}\phi_{1}-\frac{1}{2}\partial_{\mu}\phi_{2}\,\partial^{\mu}\phi_{2}-V(\phi_{1},\phi_{2})\right]~{},$
(1)
where $R$ is the Ricci scalar and $\Omega^{2}(\phi_{1},\phi_{2})$ is the
conformal transformation factor defined as
$\displaystyle\Omega^{2}(\phi_{1},\phi_{2})=1+f(\phi_{1},\phi_{2})~{},$ (2)
where $f(\phi_{1},\phi_{2})$ is usually a polynomial function of $\phi_{1}$
and $\phi_{2}$ and also includes non-minimal coupling constants $\xi_{1}$ and
$\xi_{2}$. This metric has signature $(-,+,+,+)$.
In this paper, we consider $V(\phi_{1},\phi_{2})$ in the form of a proposed
two-field potential in Denef:2018etk , in which there is a trilinear coupling
between the Higgs boson and the dilaton particle as a quintessence field. This
potential is a special combination of the well-known Higgs boson potential
$V_{H}(h)$ and a familiar exponential quintessence potential. Of course in
this case, the quintessence field $\phi$ is decoupled from all other Standard
Model fields.
The Higgs boson potential is
$V_{H}(h)=\frac{\lambda}{4}(h^{2}-\nu^{2})^{2},$ (3)
where $\lambda$ is the self-interacting coupling constant of the Higgs field
and $\nu\approx 246$ GeV is the Higgs vacuum expectation value.
The above-mentioned exponential quintessence potential is as
$V_{Q}(\phi)=\Lambda\>e^{-\delta\phi/M_{P}}~{}~{}~{}(\delta>0),$ (4)
where $\Lambda$ is the present value of dark energy density and
$\delta\equiv\lvert M_{P}\nabla_{\phi}V_{Q}/V_{Q}\rvert$. For present
accelerating universe, $\lvert\nabla V_{Q}\rvert_{today}=\Lambda\sim
10^{-120}$ in Planck units. The scale factor of the universe in this model
grows as $t^{2/\delta^{2}}$ and therefore for having an accelerating universe,
it needs $\delta<\sqrt{2}$ Kallosh:2003mt . On the other hand, in recent
years, some efforts in string theory to find suitable cosmological models led
to obtain two so-called Swampland criteria which gave testable predictions
about dark energy. During these efforts, Vafa et al. found that specific
quintessence models can satisfy these bounds Agrawal:2018own . Based on
Swampland criterias and current observational constrains on dark energy,
marginalized upper bounds on $\delta$ gives $\delta<0.6~{}$, $\delta<1.35$
with 68% CL (and 95% CL) and 99.7% CL, respectively Agrawal:2018own ;
Heisenberg:2018yae . Then, by comparing the above potential to the data,
Akrami et al. Showed that models with $\delta\gtrsim 1.02$ were ruled out by
the data at the 99.7% CL Akrami:2018ylq .
It should be mentioned that the above exponential quintessence model cannot
alone act as a single field quintessential inflationary model, i.e. it cannot
describe inflation and quintessence, simultaneously. Because, it supports
inflation for $\delta\ll 1$, but then inflation never ends Akrami:2017cir .
Therefore, one should consider another adequate field act as the inflaton.
Labeling the fields $\phi_{1}=h$ and $\phi_{2}=\phi$ as the Higgs and the
quintessence fields, respectively, our selected two-field potential
$V(\phi_{1},\phi_{2})$ in this paper is the combined form of (3) and (4)
introduced in Denef:2018etk as
$V(\phi_{1},\phi_{2})=V(\phi,h)=e^{-\delta(\phi-\phi_{0})/M_{P}}(V_{H}(h)+\Lambda)~{},$
(5)
where $\phi_{0}=\phi_{today}$.
For obtaining a canonical form for the action (1) to satisfy recent CMB data,
it needs moving to the Einstein frame. For this purpose, one may consider a
conformal transformation, defined as
$\hat{g}_{\mu\nu}=\Omega^{2}\,g_{\mu\nu}~{}.$ (6)
Supposing that the quintessence has minimal coupling to gravity, we define
$\xi_{1}=\xi$ and $\xi_{2}=0$. So for a choice of quadratic scalar field, we
simplify (2) as
$\displaystyle\Omega^{2}=1+f(h)=1+\xi\frac{h^{2}}{M_{P}^{2}}~{},$ (7)
which leads to a non-minimal kinetic term for Higgs field in the Einstein
frame action. Therefore, it is helpful to make a change to the new scalar
field $\chi$ with
$\dfrac{d\chi}{dh}=\sqrt{\dfrac{1}{\Omega^{2}}+6(\dfrac{\Omega^{\prime}}{\Omega})^{2}}~{},$
(8)
where $\chi$ is the canonically normalized form of h. This change gives the
following well-known form for the action (1) in the Einstein frame
Starobinsky:2001xq
$S_{E}=\int
d^{4}x\sqrt{-\hat{g}}\left[\frac{1}{2}M_{P}^{2}\hat{R}-\frac{1}{2}\partial_{\mu}\chi\,\partial^{\mu}\chi-\frac{1}{2}e^{2b}\partial_{\mu}\phi\,\partial^{\mu}\phi-\hat{V}(\chi,\phi)\right]~{},$
(9)
with
$\displaystyle\hat{V}(\chi,\phi)=\frac{V(h(\chi),\phi)}{\Omega^{4}}~{},$ (10)
$\displaystyle b=-\frac{1}{2}\ln(\Omega^{2})~{}.$ (11)
The conformal factor $\Omega^{4}$, gives a plateau-like shape to the potential
(5) for large field values and therefore the cosmological parameters of (10)
are consistent with CMB data well.
To study the behavior of the potential (10) for $\xi\gg 1$, we consider two
general regions in the following; super-Planckian region $h\gg
M_{P}/\sqrt{\xi}$ and sub-Planckian one $h\ll M_{P}/\sqrt{\xi}$.
### II.1 Inflationary era
For $h\gg M_{P}/\sqrt{\xi}$, i.e. during inflation, integrating both sides of
(8) gives
$\displaystyle
h=\frac{M_{P}}{\sqrt{\xi}}\exp\left(\frac{\chi}{\sqrt{6}M_{P}}\right)~{}.$
(12)
Putting (5), (7) and (12) in (10) and ignoring the infinitesimal constants
$\nu$ and $\Lambda$ during inflation, one finds a two dimensional potential,
we call quintessential Higgs inflation
$\hat{V}(\phi,\chi)=\frac{\lambda}{4}\frac{M_{P}^{4}}{\xi^{2}}e^{-\delta(\phi-\phi_{0})/M_{P}}\left(1+e^{-\sqrt{\frac{2}{3}}\chi/M_{P}}\right)^{-2}=V(\phi)\hat{V}(\chi)~{},$
(13)
where
$\displaystyle V(\phi)=e^{-\delta(\phi-\phi_{0})/M_{P}}~{},\>\>\>\>\>\
\hat{V}(\chi)=\frac{\lambda}{4}\frac{M_{P}^{4}}{\xi^{2}}\left(1+e^{-\sqrt{\frac{2}{3}}\chi/M_{P}}\right)^{-2}~{}.$
(14)
The potential (13) is shown in Fig. 1. Without coefficient
$e^{-\delta(\phi-\phi_{0})/M_{P}}$, (13) is reduced to ordinary Higgs
inflation potential in Bezrukov:2007ep which suffers from Higgs instability,
since the quantum fluctuations of the Higgs field may have exceeded the
instability scale. If one defines a new self coupling for this potential as
$\lambda^{\prime}=\lambda e^{-\delta(\phi-\phi_{0})/M_{P}}$, the coefficient
$e^{-\delta(\phi-\phi_{0})/M_{P}}\gtrsim 1.08$ gives stability to the Higgs
potential during inflation Han:2018yrk .
Figure 1: Two-field quentessential Higgs potential.
The dynamics of the two scalar fields are described by the Klein-Gordon and
Friedmann equations respectively
$\displaystyle\ddot{\chi}+3H\dot{\chi}+\hat{V}_{,\chi}=b_{,\chi}e^{2b}\dot{\phi}^{2}~{},$
(15)
$\displaystyle\ddot{\phi}+(3H+2b_{,\chi}\dot{\chi})\dot{\phi}+e^{-2b}\hat{V}_{,\phi}=0~{},$
(16) $\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\chi}^{2}+\frac{1}{2}e^{2b}\dot{\phi}^{2}+\hat{V}\right)~{},$
(17)
$\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}(\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2})~{}.$
(18)
where the dot and the subscript comma denote partial derivative with respect
to time and to scalar fields, respectively.
For having an epoch of accelerating expansion, we need a nearly constant
Hubble parameter, H with a slow-roll evolution, i.e. $\epsilon_{H}\ll 1$ with
$\displaystyle\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}~{},$ (19)
called Hubble slow-roll parameter. On the other hand, the potential slow-roll
parameters of the two fields are defined as
$\displaystyle\epsilon_{\chi}\equiv\frac{M_{P}^{2}}{2}\left(\frac{\hat{V}_{,\chi}}{\hat{V}}\right)^{2},\>\>\>\>\>\eta_{\chi}\equiv
M_{P}^{2}\frac{\hat{V}_{,\chi\chi}}{\hat{V}}~{},$ (20)
$\displaystyle\epsilon_{\phi}\equiv\frac{M_{P}^{2}}{2}\left(\frac{\hat{V}_{,\phi}}{\hat{V}}\right)^{2}e^{-2b},\>\>\>\>\>\eta_{\phi}\equiv
M_{P}^{2}\frac{\hat{V}_{,\phi\phi}}{\hat{V}}e^{-2b}~{}.$ (21)
Although, models include single canonical scalar fields need a nearly flat
potential in order to give accelerated expansion and also satisfy CMB data,
multiple field models can also give acceleration, even in the presence of
steep potential. In other words, in the slow-roll regime for multi-field case,
$\epsilon_{H}\ll 1$ but not necessarily $\epsilon_{V}\ll 1$.
Under the assumption of slow-roll conditions, (15)-(18) simplify as
$\displaystyle 3H\dot{\chi}+\hat{V}_{,\chi}=0~{},$ (22) $\displaystyle
3H\dot{\phi}+e^{-2b}\hat{V}_{,\phi}=0~{},$ (23) $\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\hat{V}~{},$ (24)
$\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}(\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2})~{}.$
(25)
To identify the inflaton field direction in field space during inflation, it
is conventional to define a new field, called adiabatic field $\sigma$, as
$\displaystyle\dot{\sigma}^{2}=\mathcal{G}_{ij}\dot{\varphi}^{i}\dot{\varphi}^{j}~{},$
(26)
where $\mathcal{G}_{ij}$ is the field-space metric. This field represents the
path length along the clasical background trajectory, while fluctuations
orthogonal to the background trajectory, showing non-adiabatic perturbations,
are represented by another field, called entropy field s Gordon:2000hv . In
another word, one can decompose an arbitrary perturbation in multi-field
models into an adiabatic $\delta\sigma$ and an entropy $\delta s$ component.
Following Gordon:2000hv , the adiabatic and entropy fields of our model, are
defined as
$\displaystyle\delta\sigma=\cos\theta~{}\delta\chi+\sin\theta~{}e^{b}\delta\phi~{},$
(27) $\displaystyle\delta
s=\cos\theta~{}e^{b}\delta\chi-\sin\theta~{}\delta\phi,$ (28)
where $\theta$ is the angle of the tangent to the background trajectory and
$\displaystyle\cos\theta=\frac{\dot{\chi}}{\sqrt{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}}~{},$
(29)
$\displaystyle\sin\theta=\frac{e^{b}\dot{\phi}}{\sqrt{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}}~{}.$
(30)
Therefore, the Klein-Gordon and Friedmann equations respectively are re-
defined as
$\displaystyle\ddot{\sigma}+3H\dot{\sigma}+\hat{V}_{,\sigma}=0~{},$ (31)
$\displaystyle
H^{2}=\frac{1}{3M_{P}^{2}}\left(\frac{1}{2}\dot{\sigma}^{2}+\hat{V}\right)~{},$
(32) $\displaystyle\dot{H}=-\frac{1}{2M_{P}^{2}}\dot{\sigma}^{2}~{},$ (33)
where
$\displaystyle\hat{V}_{,\sigma}=\cos\theta~{}\hat{V}_{,\chi}+\sin\theta~{}e^{-b}\hat{V}_{,\phi}~{},$
(34) $\displaystyle\dot{\sigma}^{2}=\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}~{}.$
(35)
On the other hand, substitution of (32) and (33) into (19) and using slow-roll
conditions gives
$\displaystyle\epsilon_{H}=\frac{3}{2}\frac{\dot{\sigma}^{2}}{\hat{V}}=\frac{3}{2}\frac{\dot{\chi}^{2}+e^{2b}\dot{\phi}^{2}}{\hat{V}}~{}.$
(36)
Then, substituting $\dot{\chi}$ and $\dot{\phi}$ from (22) and (23) into (36)
and using (20) and (21), one obtains the relation between the Hubble and
potential slow-roll parameters of this work as
$\displaystyle\epsilon_{H}=\epsilon_{\chi}+e^{-2b}\epsilon_{\phi}~{}.$ (37)
Putting (13) in (20), the inflationary slow-roll parameters of the model can
be calculated as
$\displaystyle\epsilon_{\chi}=\frac{4}{3}(1-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})^{2}~{},$
(38)
$\displaystyle\eta_{\chi}=\frac{4}{3}(1-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})^{2}(2-e^{\sqrt{\frac{2}{3}}\frac{\chi}{M_{P}}})~{},$
(39)
The number of e-folds to the end of inflation, $N_{k}$ is related to the
Hubble parameter as $dN_{k}=Hdt$. Therefore, in the case of the separable
potential (13) and by the help of (22)-(24), one can find $N_{k}$ as a
function of $\chi_{k}$ or $\phi_{k}$ as
$N_{k}=\frac{1}{M_{P}^{2}}\int^{\chi_{k}}_{\chi_{end}}\frac{\hat{V}(\chi)}{\hat{V}(\chi)_{,\chi}}d\chi=\frac{1}{M_{P}^{2}}\int^{\phi_{k}}_{\phi_{end}}\frac{V(\phi)}{V(\phi)_{,\phi}}e^{2b}d\phi~{},$
(40)
Substituting (38) in (40) and integrating, one obtains $N(\chi_{k})$ as
$N(\chi_{k})=\frac{\sqrt{6}}{4M_{P}}(\chi_{k}-\chi_{\textrm{end}})+\frac{3}{4}\left(e^{\sqrt{\frac{2}{3}}\frac{\chi_{k}}{M_{P}}}-e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right)~{}.$
(41)
This equation can be inverted by the use of the lower Lambert function to
obtain $\chi_{k}(N)$ as
$\chi_{k}(N)\simeq\sqrt{\frac{3}{2}}M_{P}\ln\left[\frac{4}{3}N-\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}+e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right]~{}.$
(42)
On the other hand, $\phi_{k}(N)$ can be also determined by the equation of
motion, (23) as
$\phi_{k}(N)\simeq-
M_{P}^{2}\int^{N}_{0}e^{-2b}\frac{V(\phi)_{,\phi}}{V(\phi)}dN\simeq
M_{P}~{}\delta\int^{N}_{0}(1+e^{\sqrt{\frac{2}{3}}\frac{\chi_{k}(N)}{M_{P}}})dN~{}.$
(43)
By substituting (42) in (43), the above integration gives
$\phi_{k}(N)\simeq
M_{P}~{}\delta\left[\left(1-\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}+e^{\sqrt{\frac{2}{3}}\frac{\chi_{end}}{M_{P}}}\right)N+\frac{2}{3}N^{2}\right]+\phi_{end}~{}.$
(44)
For calculating the value of $N_{k}$ from (41), one needs $\chi_{end}$ and
$\chi_{k}$.
As inflation ends by the breakdown of slow-roll condition, i.e.
$\epsilon_{\chi}(\chi_{end})\simeq 1$, one can find
$\displaystyle\chi_{end}\simeq 0.94\>M_{P}~{}.$ (45)
In order to determine $\chi_{k}$, one need to consider the connection between
the times of horizon crossing and re-entering to horizon of the cosmological
observable scales Liddle:2003as
$\displaystyle\frac{k}{a_{0}H_{0}}=\frac{a_{k}H_{k}}{a_{0}H_{0}}=e^{-N_{k}}\frac{a_{end}}{a_{re}}\frac{a_{re}}{a_{eq}}\frac{H_{k}}{H_{\textrm{eq}}}\frac{a_{\textrm{eq}}H_{\textrm{eq}}}{a_{0}H_{0}}~{},$
(46)
where the comoving wave number $k$ equals the Hubble scale $a_{k}H_{k}$ and
the subscripts refer to different eras, including the horizon exit ($k$),
reheating (re), radiation-matter equality (eq) and the present time (0).
Accepting the assumption of entropy conservation between the end of reheating
and today and using the slow-roll approximation in which $H^{2}_{k}\simeq
V_{k}/3M_{\textrm{Pl}}^{2}$, one can obtain (Martin:2010, ; Akrami:2018odb, )
$\displaystyle
N_{k}=67-\ln\left(\frac{k}{a_{0}H_{0}}\right)+\frac{1}{4}\ln\left(\frac{V_{k}^{2}}{M_{\textrm{Pl}}^{4}}\rho_{\textrm{end}}\right)+\frac{1-3\omega_{\textrm{int}}}{12(1+\omega_{\textrm{int}})}\ln\left(\frac{\rho_{\textrm{re}}}{\rho_{\textrm{end}}}\right)-\frac{1}{12}\ln
g_{\textrm{re}}~{},$ (47)
where $V_{k}$ is the potential energy when $k$ leaves the Hubble horizon
during inflation, $\rho_{\textrm{end}}$ and $\rho_{\textrm{re}}$ are the
energy densities at the end of inflation and reheating, respectively,
$\omega_{\textrm{int}}$ is the e-fold average of the equation of state between
the end of inflation and the end of reheating and $g_{\textrm{re}}$ is the
number of effective bosonic degrees of freedom at the end of reheating. To
find $\chi_{k}$, one should first determine the right hand side terms of (47),
one by one.
In the third term of (47), $V_{k}$ is dependent on the value of
$\lambda/\xi^{2}$ and $e^{-\delta(\phi-\phi_{0})/M_{P}}$. The value of
$\lambda/\xi^{2}$ can be calculated through the normalization of the power
spectrum Baumann:2009ds
$\displaystyle\Delta_{\zeta}^{2}=\frac{k^{3}}{2\pi^{2}}P_{\zeta}(k)=\frac{1}{24\pi^{2}M_{P}^{4}}\frac{\hat{V}_{k}}{\epsilon_{H}}~{}.$
(48)
Putting (13) in (48) and doing some calculations, we find
$\frac{\lambda}{\xi^{2}}=96\>\pi^{2}\Delta_{\zeta}^{2}\>\epsilon_{H}\>e^{\delta(\phi-\phi_{0})/M_{P}}\>\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\chi_{k}}{M_{P}}}\right)^{-2}.$
(49)
On the other hand, if one studies the evolution of the quintessence field
between the end of reheating and the EW symmetry breaking by combining the
Friedmann equation and the equation of motion as in $\phi$ Han:2018yrk , for
the permitted range $0.35<\delta<1.02$, one finds
$\displaystyle 1.08<e^{-\delta(\phi-\phi_{0})/M_{P}}<2.25~{}.$ (50)
The energy density of reheating $\rho_{re}$ is related to the reheating
temperature, $T_{re}$ through
$\displaystyle\rho_{\textrm{re}}=\frac{\pi^{2}}{30}g_{\textrm{re}}T_{\textrm{re}}^{4}~{},$
(51)
and $\rho_{end}$ depends on the potential energy at the end of inflation
$V_{end}$ via
$\displaystyle\rho_{\textrm{end}}=\frac{3}{2}V_{\textrm{end}}~{},$ (52)
obtained by setting $\omega_{end}=-1/3$. It is well known that $\omega_{int}$
of the reheating phase for large field models is given by Turner
$\omega_{int}=\dfrac{p-2}{p+2}~{},$ (53)
where $p$ is the power of the inflaton field in the corresponding potential
when it oscillates around its minimum. In the case of the potential (58) which
is proportional to $h^{4}$, (53) gives $\omega_{int}=1/3$ and as a result, the
fourth term on right hand side of (47) vanishes at $\omega_{int}=1/3$.
Therefore, deriving $\chi_{k}$ is independent of reheating temperature.
Considering $k=0.002$ Mp$c^{-1}$, $g_{re}=1000$, $\Delta_{\zeta}^{2}=2\times
10^{-9}$ Aghanim:2018eyx , $\omega_{int}=1/3$ and $\chi_{end}\simeq
0.94\>M_{P}$, and replacing the above results in (47) we find
$\displaystyle\chi_{k}=5.38~{}M_{P}~{},~{}~{}~{}~{}~{}N_{k}=61.7~{}~{}~{}~{}~{}~{}for~{}~{}~{}~{}~{}\delta=1.02~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}e^{-\delta(\phi-\phi_{0})/M_{P}}=2.25$
(54)
and
$\displaystyle\chi_{k}=5.37~{}M_{P}~{},~{}~{}~{}~{}~{}N_{k}=61.2~{}~{}~{}~{}~{}~{}for~{}~{}~{}~{}~{}\delta=0.35~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}e^{-\delta(\phi-\phi_{0})/M_{P}}=1.08$
(55)
Comparing these result to the single field Higgs inflation with $N_{k}=58.7$,
we obtain a shift in the number of e-folds $\Delta N_{k}\simeq 3$ for our two-
field model.
### II.2 Reheating
For $h\ll M_{P}/\sqrt{\xi}$, the conformal transformation factor
$\Omega^{2}\rightarrow 1$. For detailed analysis, one can divide this region
to two epochs. Defining a critical value Bezrukov:2008ut
$\displaystyle X_{cr}=\sqrt{\frac{2}{3}}\frac{M_{P}}{\xi}~{},$ (56)
and then consider an intermediate regime between inflation and radiation eras,
called reheating in which $X_{cr}<h\ll M_{P}/\sqrt{\xi}$ and the Higgs field
oscillates around its potential minima to reheat the universe. In this epoch,
one obtains
$\displaystyle\chi\simeq\sqrt{\frac{3}{2}}\>\frac{\xi}{M_{P}}h^{2}~{},$ (57)
and the reheating approximated potential of quintessential Higgs model is
$\displaystyle\hat{V}(\chi,\phi)=\frac{1}{2}\>\omega^{2}\chi^{2}~{},$ (58)
where
$\displaystyle\omega^{2}\equiv\dfrac{\lambda}{3}\>e^{-\delta(\phi-\phi_{0})}\>\frac{M_{P}^{2}}{\xi^{2}}~{}.$
(59)
The behavior of the potential near its minima is shown in Fig. 2. As the value
of $e^{-\delta(\phi-\phi_{0})}\sim 1$, therefore, this term does not affect on
the value of the oscillation frequency $\omega$ of the reheating and also on
its temperature.
Figure 2: The behavior of the potential near its minima.
In the second epoch and for small values of $h<X_{cr}\ll M_{P}/\sqrt{\xi}$ ,
we are in the radiation-dominated epoch. For this period, $\chi\simeq h$ the
related two-field potential which is the initial quintessential Higgs
potential (5), transmits its energy to relativistic particles via coherent
oscillations of the Higgs field, rapidly. One could suppose that the the ratio
of the energy density relative to the critical energy density of the model at
the EW symmetry breaking scale changes from $\lambda\nu^{4}/\rho_{cr}$ to
$\Omega_{\Lambda_{0}}$ rapidly and then the evolution of this two-field
potential after coherent oscillation can be described by an attractor
solution, which is ultra-slow-roll in the radiation-dominated and matter-
dominated eras. Finally, in the the dark energy-dominated universe, the
quintessence field undergoes the slow-roll evolution.
## III Conclusion
In this work, we studied a new two-field model for inflation and drak energy
as quintessential inflation. The quintessence field was coupled with Higgs
field from inflationary era and the Higgs field had non-minimal coupling with
gravity in the Jordan frame. Moving to the Einstein frame through conformal
transformation, we found a plateau-like two-filed potential for inflation
which its cosmological parameters had very good consistency with recent
cosmological data.
Furthermore, this model removes the instability problem of single field Higgs
model during inflation and also satisfies the Swampland criterias, well.
## Acknowledgement
M.E would like to thank Y. Akrami, M. Sasaki, A. Hebecker and A. Linde for
various comments and useful discussions. This project funded by Iran Science
Elites Federation.
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|
[1]Nantheera Anantrasirichai
[1]Visual Information Laboratory, University of Bristol, Bristol, BS8 1UB, UK
# A Comprehensive Study of Object Tracking in Low-Light Environments
Anqi Yi<EMAIL_ADDRESS><EMAIL_ADDRESS>*
###### Abstract
Accurate object tracking in low-light environments is crucial, particularly in
surveillance and ethology applications. However, achieving this is
significantly challenging due to the poor quality of captured sequences.
Factors such as noise, color imbalance, and low contrast contribute to these
challenges. This paper presents a comprehensive study examining the impact of
these distortions on automatic object trackers. Additionally, we propose a
solution to enhance tracking performance by integrating denoising and low-
light enhancement methods into the transformer-based object tracking system.
Experimental results show that the proposed tracker, trained with low-light
synthetic datasets, outperforms both the vanilla MixFormer and Siam R-CNN.
###### keywords:
tracking, low-light, enhancement, denoising
## 1 Introduction
The task of visual-based object tracking has been a core research area in
computer vision for decades, focusing on determining the state of a designated
target within video sequences, starting from its initial state. Its
applications include surveillance, security, robotics, automotive,
transportation, ethology, etc. However, tracking objects in low-light
environments presents significant challenges due to the poor sequence quality
captured. The presence of noise, motion blur, color imbalance, and low
contrast in these sequences makes it difficult for traditional algorithms to
accurately track objects. This paper explores methods aimed at enhancing the
performance of visual object tracking in low-light conditions and analyzing
how various factors, such as noise, color imbalance, and low contrast, impact
tracking effectiveness.
Similar to other computer vision tasks, deep learning has emerged as an
effective tool for object tracking. Early learning-based methods adapted
object recognition techniques to individual frames within a video [1].
Subsequently, recurrent neural networks (RNNs) were integrated to track
detected objects over time. In 2017, the Transformer was introduced [2],
proposing a novel architecture for natural language processing tasks that
solely relies on attention mechanisms, eliminating the need for recurrent or
convolutional neural networks (CNNs). The key innovation of the Transformer
architecture is its ability to process input sequences in parallel, rather
than sequentially as in traditional RNN models. This property enables more
efficient training and faster convergence. Additionally, the Transformer model
effectively handles long-range dependencies in input sequences, addressing a
common issue faced by RNNs, which has made it an attractive option for object
tracking.
This paper employs a state-of-the-art approach for transformer-based object
tracking, MixFormer [3], known for its superiority over several CNNs, RNNs,
and earlier Transformer models. MixFormer simplifies the traditional multi-
stage pipeline and integrates feature extraction and target information
integration within a unified transformer-based framework. In contrast to
existing trackers, such as Siam R-CNN [4], which relies on CNNs pretrained for
generic object recognition, MixFormer leverages the flexibility and global
modeling capacity of attention operations to capture target-specific features
and promote wide range communication between the target and search area. By
introducing a Mixed Attention Module (MAM) with hybrid interaction schemes,
MixFormer enables simultaneous feature extraction and target integration,
which results in a more compact and neat tracking pipeline. This approach
overcomes the limitations of traditional trackers that use separate components
for feature extraction, integration, and target-aware localization.
In low-light conditions, using the tracker faces several limitations: i) Lack
of specialized modules for low-light tracking: The Mixformer is specifically
designed to capture target features in daylight conditions. However, these
features may become indistinct or distorted in low-light conditions due to
insufficient lighting and noise, thereby limiting its performance. ii) Limited
training data in low-light conditions: The model relies on a substantial
amount of labelled data to effectively conduct visual object tracking. The
inadequacy of training data specific to low-light conditions contributes to
diminished performance in such scenarios. To address these issues, we propose
integrating denoiser and enhancement module to the framework and using
synthetic low-light datasets to train the model.
This paper presents a comprehensive study on object tracking in low-light
environments. We investigate the distinct types of distortions present in low-
light content and their individual impacts on tracking performance.
Subsequently, we enhance the trackers by employing preprocessing techniques
involving denoising and brightness enhancement. Finally, we discuss the
limitations of the current approach.
## 2 Related work
### 2.1 Object tracking
The early work in learning-based object detection focused on fully
convolutional networks (FCNs), which have demonstrated effectiveness in
capturing both local and global contextual information during the tracking
process [5]. Subsequently, more sophisticated methods gained popularity, such
as the fully-convolutional Siamese network [6]. Further enhancements include
the Siamese Region Proposal Network (SiamRPN) [7], which integrates the
Siamese network with a region proposal mechanism for high-performance
tracking. Additionally, the Discriminative Model Prediction (DiMP) [8] was
proposed to address object deformations and occlusions during tracking.
The transformer architecture and attention mechanisms have recently emerged as
powerful techniques in various computer vision tasks, including object
tracking. The transformer architecture has successfully replaced traditional
convolutional layers with self-attention mechanisms. A well-known example of a
transformer-based object tracking model is DETR (DEtection TRansformer) [9],
which captures both local and global context information, enabling it to
handle complex scenarios effectively. Although DETR is primarily focused on
object detection, it can be adapted for object tracking tasks by gathering
information from multiple frames. Another example is TrackFormer [10] for
multi-object tracking. This single unified transformer architecture performs
both detection and tracking in an end-to-end manner. The model demonstrates
exceptional performance in multi-object tracking benchmarks. Similarly, the
MOTR model [11] employs a transformer-based architecture with temporal
aggregation network for multiple object tracking.
In addition to transformer-based models, attention mechanisms have been
integrated into other object tracking models to enhance their performance.
This includes the Distractor-aware Siamese Networks (DaSiamRPN) [12], where a
distractor-aware training strategy incorporates attention mechanisms to search
objects effectively. This strategy improves the tracker’s robustness against
distractors. The Attentional Correlation Filter Network (ACFN), proposed in
[13], incorporates an attention mechanism into a correlation filter-based
tracker to adaptively weigh different spatial regions based on their
importance during tracking.
### 2.2 Low-light enhancement
The advancements in deep learning have significantly progressed image
enhancement, yet learning-based video enhancement remains relatively new. Some
methods show promise in extending low-light image enhancement techniques to
videos. These strategies involve estimating noise maps to guide attention
mechanisms, implementing self-calibrated illumination frameworks, and
utilizing adaptive total variation regularization (e.g., [14], [15]).
Additionally, there’s growing interest in techniques that integrate the
Retinex theory model with learnable physical priors for reflectance and
illumination (e.g., [16]).
For video processing, various methods utilize alignment modules (e.g., [17])
to synchronize feature maps of neighboring frames with the current frame,
aiding in motion handling. Despite their purpose, these alignment modules
occasionally fall short in compensating for motion adequately, leading to
artifacts in feature combinations. Some approaches leverage Siamese Networks
with shared weights to reduce noise in videos [18]. To cope with limited
paired datasets, certain methods resort to unpaired training strategies, such
as employing CycleGAN [19].
## 3 Methods for object tracking in low-light environments
Despite the existence of specific methods proposed for low-light enhancement,
we chose to adopt separate denoising and light correction approaches. This
decision allows us to investigate the distinct impacts of various distortions
on the tracker’s performance within low-light environments. The workflow,
depicted in Fig. 1, integrates the MixFormer tracker with two preprocessing
modules. Depending on the specific cases studied, one or both of these
preprocessors might be omitted. Moreover, with the utilization of synthetic
low-light datasets, we have access to clean, daylight ground truth data,
enabling us to fine-tune the networks.
Figure 1: The diagram used for our study on object tracking in low-light scene
### 3.1 Preprocessing with denoising
In visual object tracking tasks, noise is inevitable and can significantly
impact tracking efficiency. A common solution to address this issue is to
preprocess tracking data before inputting it into the tracking network.
Denoising techniques, such as filtering, temporal accumulation, and learning-
based methods, are widely used in practice [20, 21, 22].
In this paper, we adopt the state-of-the-art method, SUNet [23] for denoising.
This model, although simple, effectively combines the Swin Transformer and
UNet architectures, enhancing feature extraction and hierarchical
representation capabilities. Its dual up-sample block architecture, employing
subpixel and bilinear up-sampling methods, helps prevent checkerboard
artifacts and enhances overall performance. Demonstrating competitive results
on widely-used denoising datasets, the SUNet model proves its practical
effectiveness in addressing real-world image denoising issues, setting it
apart from existing models. In this project, a pretrained SUNet model is
utilized to preprocess the input dataset via denoising, aiming for improved
tracking performance.
### 3.2 Preprocessing with enhancement
In the previous sections, a methodology was discussed for addressing noise in
low-light sequences. However, other low-light features, such as color
imbalance and low contrast, also contribute to the degradation of tracking
performance. Various light enhancement methods have been proposed in the past,
ranging from histogram based ones to learning based ones.
Here, we adopt EnlightenGAN [24] which is a deep learning based generative
adversarial network. The model represents a significant advancement in the
field, introducing a pioneering unpaired training strategy that eliminates the
need for paired training data and improves real-world generalisation. Its
innovative global-local discriminator structure addresses spatially-varying
light conditions effectively, while self-regularisation techniques, including
self feature preserving loss and self-regularised attention mechanism,
contribute to the model’s success in the unpaired setting. EnlightenGAN offers
superior performance and adaptability in comparison to state-of-the-art
methods. From the diagram in Fig. 1, when the EnlightenGAN is fine-tuned, the
least-square GAN loss $L_{G}$ is applied to the generator of the EnlightenGAN.
### 3.3 MixFormer
MixFormer [3] tracks the target object by progressively extracting coupled
features for the target template and search area while deeply integrating the
information between them. This architecture consists of two main components:
i) a backbone, which comprises iterative target-search MAMs (Mixed Attention
Mechanism), and ii) a localization head, which is responsible for producing
the target bounding box. The MAM blocks allow for the simultaneous extraction
and integration of features from the target template and search area. The
localization head simplifies the process of localising the tracked object
within the search area, making the overall pipeline more efficient.
One of the key advantages of the MixFormer model is its compact and neat
tracking pipeline. Unlike other trackers that typically decouple the steps of
feature extraction and information integration, MixFormer combines these steps
within its single backbone. This design choice results in a more efficient and
streamlined architecture. Additionally, the MixFormer model does not require
an explicit integration module or any post-processing steps, further
simplifying the overall tracking pipeline. This simplification can lead to
reduced computational complexity and faster inference times, making the
MixFormer model a more suitable option for real-time tracking applications.
Mixed Attention Module (MAM) processes the input target template and search
area with the aim of simultaneously extracting their long-range features and
fusing the information interaction between them. This module enhances the
tracker’s ability to capture and integrate essential information from both the
target and search area smoothly. Unlike the original Multi-Head Attention
mechanism [2], the MAM operates on two separate token sequences corresponding
to the target template and search area. It achieves this through dual
attention operations. Self-attention is performed on the tokens (image
patches) in each sequence (target and search) themselves to capture target or
search specific information. Cross-attention is conducted between tokens from
two sequences to allow communication between the target template and the
search area. A concatenated token sequence is used to implement the mixed
attention mechanism. Let vector $q_{t}$, $k_{t}$ and $v_{t}$ to represent
target, $q_{s}$, $k_{s}$ and $v_{s}$ to represent search region. The mixed
attention can be defined as:
$k_{m}=Concat(k_{t},k_{s}),\ v_{m}=Concat(v_{t},v_{s}),$ (1)
$Attention_{t}=softmax(\frac{q_{t}k_{m}^{T}}{\sqrt{d}})v_{m},$ (2)
$Attention_{s}=softmax(\frac{q_{s}k_{m}^{T}}{\sqrt{d}})v_{m},$ (3)
where $d$ denotes the dimension of the key vectors, $Attention_{t}$ and
$Attention_{s}$ are the attention maps of the target and search respectively.
To achieve additional modeling of local spatial context, a separable depth-
wise convolutional projection layer is performed on each feature map (i.e.,
query, key and value). Then each feature map of the target and search is
flattened and processed by a linear projection to produce queries, keys, and
values of the attention operation. Finally, the target token and search token
are concatenated and processed by a linear projection.
Online Template Update plays a crucial role in capturing temporal information,
as well as addressing object deformation and appearance variations in visual
tracking. However, it is widely acknowledged that poor-quality templates may
result in inferior tracking performance. Consequently, the authors introduce a
score prediction module (SPM) to select reliable online templates based on
their predicted confidence scores.
The SPM comprises two attention blocks and a three-layer perceptron.
Initially, a learnable score token serves as a query to attend to the search
ROI (region of interest) tokens. This process enables the score token to
encode the extracted target information. Subsequently, the score token attends
to all positions of the initial target token, implicitly comparing the
extracted target with the first target. Finally, the score is generated by the
MLP (multi-layer perceptron) layer and a sigmoid activation function.
The online template is considered negative when its predicted score falls
below 0.5. By filtering out low-confidence templates, the SPM helps improve
the overall tracking performance. The introduction of the SPM ensures that the
tracker utilises high-quality templates for tracking, which in turn enhances
its ability to adapt to object deformation and appearance changes. This
approach enables more accurate and robust tracking performance in various
challenging scenarios.
Loss Function used by the MixFormer model is a combination of L1 loss and GIoU
loss. It is denoted as follows:
$L_{loc}=\lambda_{L1}L1(B_{i},\hat{B})+\lambda_{GIoU}L_{GIoU}(B_{i},\hat{B})\\\
$ (4)
where $\lambda_{L1}=5$ and $\lambda_{GIoU}=2$ are the weights of the two
losses, $B_{i}$ is the ground-truth bounding box and $\hat{B}$ is the
predicted bounding box. L1 Loss is commonly used because of its robustness and
insensitivity to outliers. Object tracking often involves dealing with
occlusions, sudden motion changes, and noisy measurements. L1 Loss is less
sensitive to outliers because it considers the absolute differences, thus it
is more robust and makes it an ideal choice for visual object tracking tasks.
Generalised Intersection over Union (GIoU) loss $L_{GIoU}$ is designed to
address the limitations of the commonly used Intersection over Union (IoU)
metric as it does not provide meaningful gradients for non-overlapping
bounding boxes [25]. GIoU loss addresses this issue by extending the IoU
metric to account for the non-overlapping bounding boxes as well. It is
computed as follows:
$GIoU=IoU-\frac{|C\setminus(A\cup B)|}{|C|}=\frac{|A\cap B|}{|A\cup
B|}-\frac{|C\setminus(A\cup B)|}{|C|}$ (5)
where $A$ and $B$ are the prediction and ground truth bounding boxes, $C$
represents the area of the smallest enclosing box containing both boxes.
For the online training stage, a standard cross-entropy loss is used to train
the SPM. It is defined as follows:
$L_{score}=y_{i}\log(p_{i})+(1-y_{i})\log(1-p_{i})\\\ $ (6)
where $y_{i}$ is the ground-truth label and $p_{i}$ is the predicted
confidence score.
## 4 Experiments and discussion
### 4.1 Synthetic Low-Light Dataset
We used the GOT-10K dataset [26], which is a large-scale, high-diversity
benchmark for visual object tracking, comprising a wide variety of sources,
such as YouTube, Vimeo, and Dailymotion. GOT10K contains more than 10,000
videos and covers 560 distinct object classes. The predefined testing set
consists of 420 videos, including 84 different object classes and 31 forms of
motion. To prevent larger-scale classes from dominating the evaluation
results, the maximum number of videos for each class was limited to 8, which
accounts for only 1.9% of the test set size. The validation set was created by
randomly sampling 180 videos from the training subset, with a uniform
probability distribution across different object classes.
GOT-10K dataset was captured in normal light and good conditions, whereas
video sequences taken in poor lighting conditions often display attributes
like low brightness and contrast, a limited grayscale spectrum, color
distortion, and considerable noise. To synthesize low light, we followed the
image degradation model proposed in [27] and we included the color imbalance
effect $\mathcal{C}$ in the model as shown in Equation 7:
$g(x,y)=\mathcal{C}\left(\alpha\cdot{f(x,y)^{\gamma}}+\beta\right)+\epsilon_{n},$
(7)
where $g(x,y)$ is the output image, $f(x,y)$ is the input image, $\alpha$ is
the contrast adjustment parameter, $\beta$ is the brightness adjustment
parameter, $\gamma$ is the gamma factor, and $\epsilon_{n}$ represents
Gaussian noise.
An $\alpha$ value above 1 boosts image contrast, darkening dark areas and
brightening bright areas. An $\alpha$ value below 1 reduces image contrast,
lightening dark areas and darkening bright areas. An $\alpha$ value of 1
maintains the image’s contrast unchanged. A positive $\beta$ increases image
brightness, a negative $\beta$ decreases it, and $\beta$ at 0 maintains the
brightness. The $\gamma$ value describes the nonlinearity of the imaging
system to different input brightness levels. Typically ranging from 0.1 to 5,
a gamma of 1 signifies a linear relationship between input and output
brightness. A gamma above 1 accentuates sensitivity to darker areas, while
below 1 emphasizes brighter areas.
To create a color imbalance effect $\mathcal{C}$ in dark images, it can be
done by selectively manipulating the saturation channel ($S$) without altering
the hue ($H$) or value ($V$) channels. This is achieved by applying a scaling
factor to the saturation channel, which can be represented by the equation
$S^{\prime}=S\cdot\alpha_{S}$, where $S^{\prime}$ is the adjusted saturation,
$S$ is the original saturation, and $\alpha_{S}$ is the scaling factor. By
selectively adjusting the saturation of specific color channels, an imbalance
in color distribution can be created that mimics the appearance of color
imbalance often observed in real-world low-light conditions. Note that
modifying the $V$ channel alone will not achieve color imbalance, as it only
affects the overall brightness of the image without altering the color
relationships. We refrained from adjusting $H$ as it tended to alter white
balance, a task already effectively addressed by commercial software.
Finally, we added Gaussian noise with the mean $\mu$=0 and the standard
deviation $\sigma$, determining the spread or the variability of the noise
added to the image. A larger standard deviation implies that the image is more
“grainy” or “fuzzy” due to the presence of more random noise values.
### 4.2 Training setting
We trained the models with various synthetic low-light data with diverse
parameters, including Gaussian noise ($\sigma$), gamma ($\gamma$) adjustment
and saturation adjustment scaling factor ($\alpha_{S}$). These trackers, along
with the tracker attained by normal light, were tested on a single synthesised
dark test set to evaluate the tracking results of training with different
parameters and to assess the impact of different low-light features on the
tracking accuracy of the tracker. The range of the parameters were set as
follows:
* •
For Gaussian noise, the mean value was constant at 128 for all datasets, while
the $\sigma$ was set to 10, 25, 40, 55 and 70, with the default being 10.
* •
For contrast adjustment, the linear intensity factor maintained at 0.4 for all
datasets, while the $\gamma$ was set to 0.2, 0.3, 0.4, 0.5 and 0.6, with the
default being 0.5.
* •
For saturation adjustment, the scaling factor $\alpha_{S}$ was set to 0.2,
0.3, 0.4, 0.5 and 0.6, with the default being 0.4.
It is worth noting that to assess the impact of each parameter on the tracking
results, only one parameter was altered at each time, with the others set to
their default values. Other training parameters remained the same as they were
set for normal light to avoid the results being affected by other factors.
### 4.3 Metrics
Intersection over Union (IoU) is calculated as the ratio of the intersection
of the predicted and ground-truth regions to their union. In other words, it
measures the overlap between the two regions, where a value of 1 indicates a
perfect match and a value of 0 indicates no overlap.
Area Under the Curve(AUC) refers to the area under the curve, plotting the
fraction of successfully predicted frames against a threshold of IoU values. A
higher AUC value suggests a better tracking performance as it indicates that
the tracker is able to successfully track objects with larger IoU threshold
$t$, which ranges from 0 to 1. The AUC can be calculated using numerical
integration and defined as follows:
$AUC=\int_{0}^{1}\frac{Number\>of\>frames\>with\>IoU>=t}{Total\>number\>of\>frames}\,dt\\\
$ (8)
OP50 and OP75 are the Overlap Percentages when the thresholds (as percentage)
50 and 75 are considered successful, respectively. Typically, OP75 is
considered a more strict criterion for measuring the performance as it sets a
higher threshold for the overlap percentage. A higher OP50 or OP75 implies a
better performance.
Precision measures the accuracy of the predicted position of the tracked
object. It calculates the average distance between the center of the ground
truth bounding box and the center of the predicted bounding box for each
sequence [28]. The Precision is the proportion of frames of which the distance
is below a threshold $d$:
$precision(d)=\frac{Number\>of\>frames\>with\>distance_{i}<=d}{Total\>number\>of\>frames}\\\
$ (9)
where $distance_{i}$ is the Euclidean distance between the center points in
frame $i$:
$distance_{i}=\sqrt{(x_{gt_{i}}-x_{pred_{i}})^{2}+(y_{gt_{i}}-y_{pred_{i}})^{2}}$
(10)
where $x_{gt_{i}}$ and $y_{gt_{i}}$ are the coordinates of the center of
ground truth bounding box, and $x_{pred_{i}}$ and $y_{pred_{i}}$ are the
coordinates of the center of the predicted bounding box.
Normalized Precision takes into account the differences in object sizes and
frame resolution by normalizing the distance between the ground truth and
predicted bounding box centers. The normalisation is commonly done by dividing
the $distance_{i}$ by the diagonal length of the ground truth bounding box,
which can be defined respectively as follows.[28], where where $d$ is the
threshold:
$Normalized\>Distance_{i}=\frac{distance_{i}}{Diagonal\>length\>of\>ground\>truth\>bounding\>box_{i}}\\\
$ (11)
$Normalized\>Precision(d)=\frac{Number\>of\>frames\>with\>Normalized\>distance_{i}<=d}{Total\>number\>of\>frames}$
(12)
### 4.4 Impact of low-light distortions on tracking performance
This section investigates the impact of individual distortions observed in
low-light environments—such as noise, gamma, and saturation changes—on
tracking performance. This demonstrates the parameters that should be set to
generate synthetic low-light videos for training the model to achieve optimal
performance when used in a general scenario.
#### 4.4.1 Noise levels
We explored the impact of varying noise levels on the tracker’s performance.
While maintaining normal lighting conditions, we adjusted noise levels by
generating test sets with different sigma values: 10, 25, 40, 55, and 70 for
each set. All other parameters were maintained at their default as specified
in section 4.2. The results are shown in Figure 2. When the model was trained
in normal light without noise, it showed poor robustness to noise (as
indicated by the blue line in the plots). Surprisingly, the model trained with
a noise level of 25 demonstrates the highest performance in object tracking
across varying noise levels. Conversely, models trained with higher noise
levels failed to achieve optimal tracking performance, even when tested under
similar noise conditions. This struggle may indicate that the network faces
difficulties in capturing features from very noisy inputs.
Figure 2: Test results of trackers trained with different noise level. The x
axis shows the noise level of the test sets, while the Y axis shows the values
of testing metrics.
#### 4.4.2 Gamma values
Figure 3 illustrates the test results of the three trackers trained with
different gamma values. A notable observation is that the testing result of
model trained with daylight dataset exhibits a non linear decrease.
Specifically, in between gamma gain of 0.2 and 0.3 the model’s precision
experienced a sharp decline. The same trend can be found in trackers trained
with gamma gain of 0.3 and 0.5. This phenomenon can be explained by the
characteristics of gamma correction. When the gamma gain decreases to a
certain level, the image becomes extremely dark, hence the object is not
visually distinguishable and it is also difficult for machine to extract
useful features. Unlike noise, which distorts edges and destroys certain
features, low brightness and low contrast cause the object to blend into the
background, making it impossible to identify edges or features. Figure 4 shows
the original daylight image in comparison to the synthesised outcome of images
with gamma gains 0.6, 0.3 and 0.2 respectively.
Figure 3: Test results of trackers trained with different gamma gains. The x
axis shows the gamma value of the test sets, while the Y axis shows the values
of testing metrics.
Figure 4: Images with different levels of gamma gain. Top left shows the image
in original daylight environment. Other images has gamma gains of 0.6, 0.3 and
0.2 respectively.
Figure 5: Test results of trackers trained with different saturation gains.
The x axis shows the saturation gain values of the test sets, while the Y axis
shows the values of testing metrics.
#### 4.4.3 Saturation values
Showing in Figure 5, a descending pattern can be found in the curves as the
saturation gain reduces. Furthermore, the trackers trained with saturation
gains of 0.3 and 0.5 display a significantly improved performance compared to
the track trained on the daylight dataset. This observation aligns with the
previously mentioned findings regarding the impact of noise and gamma gain,
where trackers trained on the synthetic low-light dataset show better
robustness when tested on various dark dataset. This consistency suggests that
training the model on dataset with diverse low-light features can improve
their versatility and effectiveness when handling visual object tracking tasks
in a wide range of low-light scenarios.
The impact of saturation on the model’s tracking efficiency is relatively
smaller than that of noise and gamma gain. As the saturation gain drops from 0
to 0.2, these five metrics only decrease by approximately 3%. For the impact
of noise, the AUC decreases from 79.30% to 73.86% as the noise level rises
from 0 to 40. Similarly, OP50, OP75, Precision, and Normalized Precision
decline by 5.69%, 9.55%, 9.37%, and 5.48% respectively. Similarly, when the
gamma value drops from 1 to 0.3, while the features remain recognizable, the
AUC, OP50, OP75, Precision, and Normalized Precision decrease by 6.13%, 6.8%,
10.68%, 10.08%, and 6.58% respectively.
While changes in saturation can alter the appearance of an image by adjusting
the color intensity, they do not have massive impact on the overall image
quality, or the visibility of the objects and their edges. This implies that
alterations in shapes and edges deteriorate tracking performance more
significantly than saturation changes do. Thus, the model is able to maintain
high performance and is less affected by the changes in saturation compared to
noise and gamma, where the features in the object are greatly impacted.
### 4.5 Performance improvement with denoising and image enhancement
Table 1 shows the test results of trackers trained using synthetic dark
datasets (sigma = 40, gamma = 0.5, saturation = 0.4), denoised datasets, and
enhanced datasets. These trackers were then tested on dark, denoised, and
enhanced data, respectively. The outcomes on the denoised dataset surpass
those on the enhanced dataset, confirming that pre-processing with denoising
significantly enhances the model’s performance compared to enlightening.
Moreover, upon applying denoising techniques to both the training and testing
sets, the AUC increased by 5.36% compared to training and testing on the
original dark dataset. Conversely, applying preprocessing solely to the
testing set resulted in a 3.21% increase in AUC. This suggests that applying
preprocessing to both training and testing procedures yields a more
substantial improvement in the model’s performance.
Table 1: Test results of trackers trained on dark, denoised and enhanced datasets and tested on corresponding test sets. Trackers | AUC | OP50 | OP75 | Precision | Norm Precision
---|---|---|---|---|---
Dark | 61.29 | 73.43 | 57.60 | 51.23 | 72.84
Denoised | 66.65 | 76.33 | 59.98 | 53.01 | 74.43
Enhanced | 64.32 | 74.93 | 58.56 | 52.25 | 73.21
Denoised+Enhanced | 67.15 | 77.12 | 60.72 | 53.68 | 75.18
| | | | |
### 4.6 Visualised Tracking Results
In this section, the visualised tracking results are discussed to further
investigate the model’s ability in handing challenging conditions.
Specifically, the reasons to why the tracking failed in certain cases are
examined to provide insights into the future improvements of the model. The
main reasons for the model’s failure to track objects are classified into
three categories: i) ambiguity caused by the background, ii) presence of
multiple, visually similar objects within the scene, and iii) occlusion or
obstruction of the object.
#### 4.6.1 Ambiguity caused by the background
The background in the image can sometimes have similar features as the object.
In low-light images, the visibility of the edges and textures of the object is
degraded, making it more challenging for the tracker to distinguish between
the object and the background when they share similar features, hence leading
to tracking errors.
Figure 6 displays two tracking failures, where the tracker struggles to
differentiate the black squirrel from the background. The left shows when both
the normal and dark trackers fail to identify the object in the frame. The
cause of this error is that the door mat is mistaken as the squirrel, as they
both appear black and have a slender shape. However, as presented in the image
on the right, after the squirrel moves to the doorstep, where its features
contract with the background, more features are captured by the dark tracker,
hence it is able to correct the tracking result. On the contrary, the normal
tracker continues to fail on recognising the object in this frame, indicating
that its ability in feature extraction in low-light conditions is weaker than
that of the dark tracker.
Figure 6: The image on the left show the example when both trackers fail to
track a black cat in the dark due to confusion caused by background. The image
on the right shows improved. Green, blue and red boxes are ground truth and
the results from the models trained by daylight and dark datasets,
respectively.
#### 4.6.2 Multiple Objects
The challenges include consistently recognizing individual objects when
multiple objects are present in a scene, managing interactions between
objects, and coping with the appearance changes of each object.
Figure 8 presents an example where the trackers are unsuccessful in
maintaining the identity of the object. In the two frames, both trackers
trained by normal light and low light manage to identify the object (as shown
on the left) – a small black bear – when there is only one such bear walking
on the ground. Nevertheless, in the right image, both trackers incorrectly
identify the original black bear (bear 1) on the right and instead
misinterpret the one on the left (bear 2) as the initial bear. This may occur
due to the trackers’ inability to accurately follow the object’s movements.
Specifically, as bear 1 moves to the right, bear 2 takes its original
position. Consequently, despite the trackers’ initial success in tracking the
object and the minor changes in the visual appearance of bear 1, they still
confuse bear 2 with the originally tracked bear 1. This highlights the
trackers’ lack of ability in accurately capturing the temporal information
within the sequence. This limitation seems reasonable, considering the score
prediction head in the original model design, which is crucial for capturing
temporal information, is trained in the online stage of MixFormer. However, to
reduce training complexity in this project, the online training step is
excluded when training these trackers. As a result, the trackers may exhibit a
diminished ability to capture temporal information.
Figure 8 displays further examples where the trackers fail to track an object
due to the presence of multiple objects in the scene. In both cases presented
in the frames, the target object is interacting with another object in the
scene. Consequently, some features of the other object involved in the
interaction are incorrectly attributed to the target object. For example, in
the left image, as the calf manatee interacts with the adult manatee, the dark
tracker falsely includes the adult manatee’s head as part of the calf. This is
probably because the head of adult manatee has more distinct features, such as
the eyes and head shape, which the dark tracker can easily capture. Similarly,
in the right image, the border collie is interacting with the black sheep.
Since the border collie is in a position where its head is not visible in the
picture, the sheep’s head and neck, which have more distinguishable and
pronounced features, are mistakenly identified by the trackers as part of the
border collie. This mistake can also be attributed to the missing edges of the
border collie’s head, making it difficult for the tracker to identify the
boundary of the object.
Figure 7: Examples when the normal tracker or both trackers failed to track an
object in the dark due to multiple objects existing in the scene. Green, blue
and red boxes are ground truth and the results from the models trained by
daylight and dark datasets, respectively. The images are shown in the normal
light for better visualisation.
Figure 8: Examples when one or both the trackers fail to track an object in a
frame due to interaction of objects. Green, blue and red boxes are ground
truth and the results from the models trained by daylight and dark datasets,
respectively. The images are shown in the normal light for better
visualisation.
#### 4.6.3 Occlusion
Occlusion poses a considerable challenge in object tracking, as it can impede
the tracker’s capacity to maintain a precise representation of the object
throughout a sequence [29]. An illustration of this issue can be found in
Figure 9, where the trackers struggle to track an object, a black dog, during
occlusion events. On the left, the object, a dog, is partially obscured by a
person’s legs. In this frame, the dark tracker is able to capture the object’s
features and define its edges despite the occlusion. In contrast, the daylight
tracker inappropriately identifies the person’s head, which shares similar
features, such as brown and fuzzy appearance, with the dog in the image, as
the target. This observation aligns with the previous finding that the dark
tracker is more capable at capturing features and defining edges in low-light
conditions than the normal tracker, allowing it to identify the object even
when occlusion occurs in the dark.
The dark tracker’s ability to handle occlusion has its limitations. In the
right frame, where the dog is entirely obscured by the structure, both
trackers fail to recognise the animal. This failure can potentially be
attributed to the model’s inability to handle temporal information effectively
due to the absence of online training, as previously discussed. When a model
is adept at processing temporal information, it can leverage the object’s
motion patterns, trajectory, and appearance changes observed in previous
frames to make predictions about the object’s position and appearance during
occlusion [30]. Hence, when the model lacks this ability, it may fail to
continuously track the object when the object is partially or entirely hidden
from the view.
Figure 9: Left: When the object, a black dog, is partially occluded, the dark
tracker is able to identify the part which is visible, while the normal
tracker cannot identify the object. Right: when the object is fully
obstructed, both trackers are unable to track the object in the frame. Green,
blue and red boxes are ground truth and the results from the models trained by
daylight and dark datasets, respectively.
### 4.7 MixFormer versus Siam R-CNN
This section compares our MixFormer-based tracker with Siam R-CNN [4], a
state-of-the-art object tracking model that merges Siamese networks with
Region-based Convolutional Neural Networks (R-CNN). Siam R-CNN excels in
robust and accurate visual tracking by effectively matching the target object
across frames using a Siamese architecture. The R-CNN component aids in
precise object localization and classification. This fusion enhances tracking
performance, especially in challenging scenarios involving occlusions,
deformations, and appearance changes. The following segment illustrates the
performance comparison between these two methods.
Table 2 displays the outcomes from testing sets generated using various
parameters (e.g., sigma, gamma, saturation gain), while other settings remain
default (see Section 4.2 for an in-depth explanation of parameter
configurations). A noticeable observation in Table 2 is the consistent lower
performance of the Siam R-CNN model compared to the MixFormer model in both
daylight and low-light tracking scenarios. This observation emphasizes the
effectiveness of the MixFormer’s MAM architecture, significantly enhancing
tracking performance even under challenging lighting conditions. These results
underscore the superiority of the transformer-based architecture over the
conventional CNN network and underscore the advantages of the Mixed Attention
Module.
Table 2: Test results of MixFormer and Siam R-CNN on different test sets | AUC | Norm Precision
---|---|---
Setting | MixFormer | Siam R-CNN | MixFormer | Siam R-CNN
Normal | 79.30 | 77.21 | 88.93 | 86.83
Sigma=10 | 76.31 | 74.92 | 85.54 | 84.02
Sigma=40 | 73.36 | 70.24 | 83.45 | 81.73
Gamma=0.3 | 73.17 | 71.58 | 82.35 | 80.57
Saturation=0.3 | 76.18 | 74.86 | 85.79 | 85.13
| | | |
## 5 Conclusion
This paper examines the performance of object tracking algorithms in low-light
conditions. The strategies involve training the model using synthetic datasets
and applying denoising and image enhancement techniques in preprocessing. Our
findings demonstrate that training the model on synthetic dark datasets
notably improves its performance in low-light settings, particularly under
varying noise and brightness levels. Our comprehensive study on the effects of
low-light distortions reveals that noise has the most detrimental impact on
tracking performance, followed by non-linear brightness changes. Training the
model with a noise level of 25 and a gamma of 0.3 yields the best overall
performance across various low-light conditions. Additionally, we propose and
evaluate two preprocessing methods, SUNet for denoising and EnlightenGAN for
image enhancement, to enhance tracking accuracy. Implementing both techniques
on the test set results in a 4.41% improvement (AUC) in tracking accuracy
compared to performance on the noisy dark dataset. Furthermore, utilizing
denoising for both training and testing stages on the dark dataset leads to a
5.36% improvement in tracking accuracy compared to models trained and tested
on the original dark dataset.
### Funding Information
This work was supported by UKRI MyWorld Strength in Places Programme
(SIPF00006/1), BRISTOL+BATH CREATIVE R+D (AH/S002936/1).
### Data availability
The datasets generated and analysed during the current study are available
from the corresponding author on reasonable request.
### Competing interests
The authors declare that we have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
### Author contribution
All authors contributed to the study conception and design. Material
preparation, data collection and analysis were performed by Anqi Yi. The first
draft of the manuscript was written by Anqi Yi and all authors commented on
previous versions of the manuscript. All authors read and approved the final
manuscript.
### Research Involving Human and/or Animals
Not Applicable
### Informed Consent
Not Applicable
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# Analysis of HARQ-IR over Time-Correlated Rayleigh Fading Channels
Zheng Shi, Haichuan Ding, Shaodan Ma, and Kam-Weng Tam Manuscript received
January 14, 2015; revised May 29, 2015; accepted July 19, 2015. The associate
editor coordinating the review of this paper and approving it for publication
was M. Elkashlan.Zheng Shi, Shaodan Ma and Kam-Weng Tam are with the
Department of Electrical and Computer Engineering, University of Macau, Macao
<EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]).Haichuan
Ding was with University of Macau, and is now with the Department of
Electrical and Computer Engineering, University of Florida, U.S.A. (email:
[email protected]).This work was supported by the Research Committee of
University of Macau under grants: MYRG078(Y1-L2)-FST12-MSD and
MYRG101(Y1-L3)-FST13-MSD.
###### Abstract
In this paper, performance of hybrid automatic repeat request with incremental
redundancy (HARQ-IR) over Rayleigh fading channels is investigated. Different
from prior analysis, time correlation in the channels is considered. Under
time-correlated fading channels, the mutual information in multiple HARQ
transmissions is correlated, making the analysis challenging. By using
polynomial fitting technique, probability distribution function of the
accumulated mutual information is derived. Three meaningful performance
metrics including outage probability, average number of transmissions and long
term average throughput (LTAT) are then derived in closed-forms. Moreover,
diversity order of HARQ-IR is also investigated. It is proved that full
diversity can be achieved by HARQ-IR, i.e., the diversity order is equal to
the number of transmissions, even under time-correlated fading channels. These
analytical results are verified by simulations and enable the evaluation of
the impact of various system parameters on the performance. Particularly, the
results unveil the negative impact of time correlation on the outage and
throughput performance. The results also show that although more transmissions
would improve the outage performance, they may not be beneficial to the LTAT
when time correlation is high. Optimal rate design to maximize the LTAT is
finally discussed and significant LTAT improvement is demonstrated.
###### Index Terms:
Hybrid automatic repeat request, incremental redundancy, time correlation,
Rayleigh fading channels, outage probability.
## I Introduction
Over the last decade, wireless data traffic has experienced an explosive
growth. Contrary to this boom in data traffic, transmission reliability and
throughput of wireless channels are limited by unideal propagation environment
[1]. As a combination of forward error control and automatic repeat request
(ARQ), hybrid automatic repeat request (HARQ) has been proved as an effective
technique to improve transmission reliability and boost system throughput. It
thus has been adopted in various wireless standards, such as High Speed Packet
Access (HSPA) and Long Term Evolution (LTE) [2, 3]. Generally, there exist
three kinds of HARQ schemes, i.e., Type-I HARQ, HARQ with chase combining
(HARQ-CC) and HARQ with incremental redundancy (HARQ-IR). In Type-I HARQ, the
erroneously received packets are discarded and each retransmitted packet is
decoded independently. In HARQ-CC and HARQ-IR, the previously failed packets
are stored and combined with the packets received in subsequent
retransmissions for decoding. Specifically, the same packet is retransmitted
in each transmission attempt in HARQ-CC scheme, while redundant information is
incrementally transmitted in each HARQ round in the case of HARQ-IR. By
exploiting additional coding gain, HARQ-IR is able to achieve a higher link
throughput than Type-I HARQ and HARQ-CC [2]. The focus of this paper is thus
turned to HARQ-IR scheme.
Performance of HARQ-IR under various wireless systems has been investigated in
the literature [4, 5, 6, 7, 8, 9, 10]. To name a few, delay and throughput of
a HARQ-IR enabled multicast system are investigated and scaling laws with
respect to the number of users are discovered based on an upper bound of
outage probability in [4]. Aiming at throughput maximization, rate allocation
and adaptation for HARQ-IR are discussed in [5]. For quasi-static fading
environments (i.e., channel coefficients are constant during multiple HARQ
rounds), power efficiency of HARQ-IR is concerned and optimal power allocation
is obtained to minimize outage-limited average transmission power in [6].
Considering the same quasi-static environments as [6], average rate of HARQ-IR
enabled spectrum sharing networks is analyzed in [7]. By expressing outage
probability through k-fold convolution, throughput of network coded HARQ-IR
with arbitrary number of users is derived in [8]. In [9], an optimal rate
adaptation policy is proposed for cooperative HARQ-IR with a multi-bit
feedback channel. Dynamic programming is then employed to find the optimal
rate for maximizing the throughput of the outage-constrained transmission.
Moreover, based on dominant term approximation and upper bounds of outage
probability, energy-delay-tradeoff (EDT) is analyzed for both one-way and two-
way relaying systems with HARQ-IR in [10]. Noticing the lack of exact
analytical result on outage probability of HARQ-IR, an analytical approach is
proposed to derive the outage probability in a closed form through the
generalized Fox’s H function in [11]. Unfortunately, all of the prior studies
are conducted for either quasi-static fading channels or fast fading channels
(i.e., channel coefficients in multiple HARQ rounds are independent and
identical distributed). The results are not applicable to time-correlated
fading channels, which usually occur when the transceiver has low-to-medium
mobility [12, 13]111In a dense scattering environment, the time-correlation
between two channel amplitudes with time spacing of $\tau$ is quantified by
$\rho={J_{0}}^{2}\left({2\pi{f_{c}}\tau v{c^{-1}}}\right)$ where
$J_{0}(\cdot)$ denotes the zero-th order Bessel function of the first kind,
$f_{c}$ represents the carrier frequency, $c$ is the speed of light and $v$
refers to the moving speed of a mobile terminal [14]. Taking a 3GPP LTE system
as an example, the successive transmissions are not carried out in adjacent
time slots, and the time spacing between two successive HARQ transmissions is
$\tau=8$ms [15]. When the LTE system is operated at a carrier frequency of
$f_{c}=2.6$GHz [16], the time correlation coefficient $\rho$ between the
channel amplitudes in two HARQ transmissions is 0.83 and 0.45 for moving
speeds of 5 km/h and 10 km/h, respectively..
Considering the wide occurrence of time correlation in fading channels in
practice, it is necessary and meaningful to analyze the performance of HARQ-IR
operating over time-correlated fading channels. However the analysis is very
challenging because of the difficulty in handling a product of multiple
correlated random variables (RVs). Notice that the analysis is essentially
different from [12, 13] where HARQ-CC is analyzed and the sum of multiple
correlated RVs is concerned. In this paper, we consider HARQ-IR operating over
time-correlated Rayleigh fading channels. The accumulated mutual information
after multiple HARQ rounds is first expressed as a logarithm function of a
product of multiple shifted correlated signal-to-noise ratios. By using
polynomial fitting technique, probability distribution function (PDF) of the
accumulated mutual information is derived as a product of Gamma distribution
and a correction polynomial. Outage probability, average number of
transmissions and long term average throughput (LTAT) are then obtained in
closed-forms. Moreover, diversity order of HARQ-IR is also analyzed. It is
proved that full diversity can be achieved, i.e., the diversity order is equal
to the number of transmissions, even under time-correlated fading channels.
The impact of channel time correlation on the performance is also investigated
and optimal rate design to maximize the throughput is finally discussed. The
results reveal that time correlation of the channel causes negative effect on
the outage and throughput performance. More HARQ rounds may not be beneficial
to the throughput under highly correlated channels, although they do improve
the outage performance.
The remainder of this paper is organized as follows. In Section II, a point-
to-point HARQ-IR enabled system operating over time-correlated Rayleigh fading
channels is introduced. In Section III, outage probability, average number of
transmission and LTAT are derived in closed-forms by using polynomial fitting
technique. Section IV analyzes the diversity order of HARQ-IR over time-
correlated fading channels. The analytical results are verified through Monte-
Carlo simulations, and the impact of time correlation on the performance of
HARQ-IR and optimal rate design are then discussed in Section V. Section VI
finally concludes this paper.
## II System Model
Consider a point-to-point system with one source and one destination, as shown
in Fig. 1. To enhance the transmission reliability, HARQ-IR protocol is
adopted here. Notice that most of the prior research on HARQ-IR is carried out
over quasi-static or fast fading channels [10, 11, 17]. Different from the
prior analysis, time-correlated fading channels are considered in this paper.
Specifically, the HARQ-IR protocol and channel model are introduced in the
following.
Figure 1: System model.
### II-A HARQ-IR Protocol
Following the HARQ-IR protocol, prior to transmit a message with $b$ bits, the
source first encodes the message into $M$ packets, each with $L$ symbols. The
$M$ packets are denoted as $B_{1},B_{2},\cdots,B_{M}$, as shown in Fig. 1.
Thus the maximum allowable number of transmissions for this message is limited
to $M$. Then the source transmits the $M$ packets one by one in multiple HARQ
rounds till the destination succeeds to decode the message. If the destination
succeeds/fails to decode the message, a positive/negative acknowledgement
(ACK/NACK) message will be fed back to the source. At the source, after
receiving an NACK message from the destination, the subsequent packet will be
delivered in the next HARQ round until the maximum allowable number of
transmissions is reached or an ACK message is received. Once either of these
two events happens, the source will initiate the transmission of a new message
following the same procedure.
At the destination, the received packets are corrupted by fading channels and
additive white Gaussian noises. The corrupted packets associated with
$B_{1},B_{2},\cdots,B_{M}$ are denoted as $\\{C_{1},C_{2},...,C_{M}\\}$. The
channel decoder attempts to recover the message based on all the previously
received packets. More specifically, after $k$ HARQ rounds, the received
packets from $C_{1}$ to $C_{k}$ are utilized for decoding the message at the
destination. If the message can be successfully recovered, an ACK message will
be fed back to the source. Otherwise, a feedback of failure notification will
be sent to the source, namely, NACK message.
### II-B Channel Model
Denote the modulated signal from the $l$th packet $B_{l}$ as ${\bf{x}}_{l}$.
The signal received at the destination in the $l$th HARQ round is written as
${\bf{y}}_{l}=h_{l}{\bf{x}}_{l}+{\bf{n}}_{l}$ (1)
where ${\bf{n}}_{l}$ represents complex additive white Gaussian noise with
zero mean and variance $\mathfrak{N}_{l}$, i.e.,
${\bf{n}}_{l}\sim\mathcal{CN}({\bf{0}},\mathfrak{N}_{l}{{\bf{I}}})$, $h_{l}$
denotes the block Rayleigh fading channel coefficient in the $l$th HARQ round,
i.e., the magnitude of $h_{l}$ obeys a Rayleigh distribution, such that
${\left|{{h_{l}}}\right|}\sim Rayleigh({2{\sigma_{l}}^{2}})$ and the
expectation of the squared channel magnitude is
${\mathrm{E}}\\{{{\left|{{h_{l}}}\right|}^{2}}\\}=2\sigma_{l}^{2}$. The PDF of
$|h_{l}|$ is given by
${f_{\left|{{h_{l}}}\right|}}\left(x\right)=\frac{x}{{{\sigma_{l}}^{2}}}\exp\left({-\frac{{{x^{2}}}}{{2{\sigma_{l}}^{2}}}}\right),x\in[0,+\infty).$
(2)
In this paper, time correlation of the channels is considered. Herein, a
widely used correlated channel model [18] is adopted, that is, the channel
coefficients $\left|{\bf{h}}\right|=\\{|h_{1}|,|h_{2}|,\cdots,|h_{K}|\\}$ are
modeled as a multivariate Rayleigh distribution with generalized correlation.
The joint PDF of $\left|{\bf{h}}\right|$ follows as 222In fact, the joint PDF
of channel amplitudes (II-B) is consistent with the conditional PDF of channel
power gains given in [12, Eq. 5]. However, our channel model is different from
[19, Eq. 11] where fast fading channels with imperfect channel state
information are considered.
$\displaystyle{f_{\left|{\bf{h}}\right|}}\left({\left|{{h_{1}}}\right|={x_{1}},\cdots,\left|{{h_{K}}}\right|={x_{K}}}\right)=\prod\limits_{k=1}^{K}{\frac{1}{{{\sigma_{k}}^{2}\left({1-{\lambda_{k}}^{2}}\right)}}}$
$\displaystyle\times\int\nolimits_{t=0}^{\infty}{{e^{-\left({1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}\right)t}}\prod\limits_{k=1}^{K}{{x_{k}}{e^{-\frac{{{x_{k}}^{2}}}{{2{\sigma_{k}}^{2}\left({1-{\lambda_{k}}^{2}}\right)}}}}}}$
$\displaystyle\times{}_{0}{F_{1}}\left({;1;\frac{{{x_{k}}^{2}{\lambda_{k}}^{2}t}}{{2{\sigma_{k}}^{2}{{\left({1-{\lambda_{k}}^{2}}\right)}^{2}}}}}\right)dt,\,|{\lambda_{k}}|<1,$
(3)
where ${}_{0}{F_{1}}\left({\cdot}\right)$ denotes the confluent hypergeometric
limit function and $\lambda_{k}$ indicates the correlation degree of the
channels. Under this model, the time correlation coefficient between channels
associated with the $k$th and the $l$th HARQ rounds is given as
$\displaystyle{\rho_{{{{{k}}}},{{{{l}}}}}}$
$\displaystyle=\frac{{{\rm{E}}\left({{{\left|{{h_{k}}}\right|}^{2}}{{\left|{{h_{l}}}\right|}^{2}}}\right)-{\rm{E}}\left({{{\left|{{h_{k}}}\right|}^{2}}}\right){\rm{E}}\left({{{\left|{{h_{l}}}\right|}^{2}}}\right)}}{{\sqrt{{\rm{Var}}\left({{{\left|{{h_{k}}}\right|}^{2}}}\right){\rm{Var}}\left({{{\left|{{h_{l}}}\right|}^{2}}}\right)}}}$
$\displaystyle={\lambda_{k}}^{2}{\lambda_{l}}^{2}<1,\quad k,l\in[1,K],$ (4)
where the notation ${\rm{Var}}(x)$ denotes the variance of $x$. Notice that
this channel model is applicable to the time-correlated channels with
correlation coefficients of ${\rho_{k,l}}<1$. For fully correlated Rayleigh
fading channels (i.e., quasi-static Rayleigh fading channels), the channel
remains static in all HARQ rounds, i.e., $|h_{1}|=|h_{2}|=\cdots=|h_{M}|\sim
Rayleigh(2{\sigma}^{2})$ and $\rho_{k,l}=1$. The analysis of HARQ-IR over
fully correlated fading channels has been conducted in [6, 7] and our analysis
focuses on the time-correlated channels with ${\rho_{k,l}}<1$.
From (1), the received signal-to-noise ratio (SNR) at the destination in the
$l$th HARQ round is written as
$\gamma_{l}=\frac{{{{\left|{h_{l}}\right|}^{2}}P_{l}}}{{\mathfrak{N}_{l}}}$
(5)
where $P_{l}$ is the transmitted signal power in the $l$th HARQ round. The
accumulated mutual information at the destination after $K$ HARQ rounds is
then given by
$I_{K}^{IR}=\sum\limits_{l=1}^{K}{{I_{l}}}$ (6)
where $I_{l}$ represents the mutual information acquired from the $l$th HARQ
round and is given as
${I_{l}}={\log_{2}}\left({1+\gamma_{l}}\right).$ (7)
## III Performance Analysis
### III-A Performance Metrics
To investigate the performance of HARQ-IR over time-correlated Rayleigh fading
channels, three widely adopted metrics including outage probability, average
number of transmissions and long term average throughput (LTAT) are discussed
here.
#### III-A1 Outage Probability
In each HARQ round, the destination combines the current received packet with
all the previously received packets for joint decoding. When the accumulated
mutual information at the destination is less than the transmission rate
$\mathcal{R}$, an outage (i.e., the failure of the decoding) would occur. The
outage probability after $K$ HARQ rounds $P_{out}^{IR}\left(K\right)$ is then
given as
$P_{out}^{IR}\left(K\right)=\Pr\left({I_{K}^{IR}<\mathcal{R}}\right).$ (8)
This outage probability can well approximate the error probability when a
capacity achieving coding is adopted, and is of great importance in the
analysis of HARQ schemes [11].
#### III-A2 Average Number of Transmissions
HARQ scheme is a combination of forward error control and automatic repeat
request. To enhance the transmission reliability, each message may be
retransmitted through multiple HARQ rounds. When the channel condition is
good, few retransmissions are sufficient for successful decoding, while more
retransmissions are needed over a poor channel. From statistical point of
view, it is meaningful to know the average transmission time for each message,
which can been well characterized by the average number of transmissions.
Given the maximum allowable number of transmissions $M$, the average number of
transmissions $\bar{\mathcal{N}}$ is expressed as [20]
$\bar{\mathcal{N}}=1+\sum\limits_{K=1}^{M-1}{P_{out}^{IR}\left(K\right)}.$ (9)
#### III-A3 LTAT
As an effective metric to characterize the system throughput of HARQ schemes,
the LTAT given the transmission rate $\mathcal{R}$ and the maximum number of
transmissions $M$ is defined as [21]
$\bar{\mathcal{R}}=\frac{{\mathcal{R}\left({1-P_{out}^{IR}\left(M\right)}\right)}}{{\bar{\mathcal{N}}}}.$
(10)
Clearly, the average number of transmissions and LTAT only depend on the
outage probability, when the transmission rate and the maximum number of
transmissions are given. Meanwhile, the outage probability is equivalent to
the cumulative distribution function (CDF)
${{F_{I_{K}^{IR}}}\left(\mathcal{R}\right)}$ of the accumulated mutual
information $I_{K}^{IR}$, i.e.,
$P_{out}^{IR}\left(K\right)={{F_{I_{K}^{IR}}}\left(\mathcal{R}\right)}$. As
further proved in [22], the essential parameter to characterize the
performance of HARQ schemes is the CDF of the accumulated mutual information
in each round. It will then be particularly investigated in the next
subsection.
### III-B Analysis of Outage Probability
By substituting (5) and (7) into (8), the outage probability becomes
$\displaystyle P_{out}^{IR}\left(K\right)$
$\displaystyle=\Pr\left({{{\log}_{2}}\left({\prod\limits_{l=1}^{K}{\left({1+{\gamma_{l}}}\right)}}\right)<{\cal
R}}\right)$ $\displaystyle=\int_{0}^{\cal
R}{{f_{I_{K}^{IR}}}\left(x\right)dx}$ (11)
where ${{f_{I_{K}^{IR}}}\left(x\right)}$ stands for the PDF of $I_{K}^{IR}$.
Since the considered channel is Rayleigh fading, it is easy to get that the
received SNR ${\gamma_{l}}$ follows exponential distribution with a PDF of
${f_{{\gamma_{l}}}}\left(x\right)=\frac{1}{{{2\sigma_{l}^{\prime}}{{}^{2}}}}\exp\left({-\frac{{{x}}}{{2{\sigma_{l}^{\prime}}{{}^{2}}}}}\right),x\in[0,+\infty)$
(12)
where
${\sigma_{l}^{\prime}}={\left({{{{P_{l}}}}/{{{\mathfrak{N}_{l}}}}}\right)^{\frac{1}{2}}}{\sigma_{l}}$.
Due to the time correlation in the channels, the SNRs ${\gamma_{l}}$ are
correlated. By making simple substitutions of variables on (II-B), the joint
distribution of
${\boldsymbol{\gamma}_{1:K}}=\left\\{{{\gamma_{1}},{\gamma_{2}},\cdots,{\gamma_{K}}}\right\\}$
can be readily derived as
$\displaystyle{{f_{\boldsymbol{\gamma}_{1:K}}}\left({{\gamma_{1}}={x_{1}},\cdots,{\gamma_{K}}={x_{K}}}\right)=\prod\limits_{k=1}^{K}{\frac{1}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)}}}}$
$\displaystyle\times{\int\nolimits_{t=0}^{\infty}{{{\rm{e}}^{-\left({1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}\right)t}}\prod\limits_{k=1}^{K}{{e^{{-\frac{{{x_{k}}}}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)}}}}}}}}$
$\displaystyle\times{}_{0}{F_{1}}\left({;1;\frac{{{x_{k}}{\lambda_{k}}^{2}t}}{{2{\sigma_{k}^{\prime}}{{}^{2}}{{\left({1-{\lambda_{k}}^{2}}\right)}^{2}}}}}\right)dt.$
(13)
It is clear from (III-B) that the distribution of a product of time-correlated
shifted-exponential RVs is necessary to derive the outage probability. As
reported in the literature, Mellin transform can be exploited to derive the
distribution of the product of independent RVs [23, 24, 25, 26, 27, 28, 11].
Unfortunately, it is inapplicable for the case with correlated RVs due to the
involvement of multiple integral. In fact, the presence of time correlation
makes the derivation of the exact distribution of ${I_{K}^{IR}}$ intractable.
To proceed with the analysis, we resort to find a good approximation of the
distribution of ${I_{K}^{IR}}$ based on polynomial fitting technique which
will be introduced in the following.
As shown in [21], the accumulated mutual information in HARQ-IR systems over
independent fading channels can be well approximated as a Gamma RV by using
Laguerre series. Inspired by this result, the PDF of the accumulated mutual
information $I_{K}^{IR}$ over correlated Rayleigh fading channels can be
written as the product of a Gamma PDF $\varphi(x)$ and a correction term
$\psi\left(x\right)$ as 333Notice that when the channels are independent
fading, the correction term can be approximated as $\psi\left(x\right)\approx
1$ and the PDF is reduced as ${f_{I_{K}^{IR}}}(x)\approx\varphi(x)$ [21].
${f_{I_{K}^{IR}}}(x)=\varphi(x)\psi\left(x\right).$ (14)
In (14), the Gamma PDF $\varphi(x)$ serves as a basis function and is given by
$\varphi(x)=\frac{{{x^{\zeta-1}}{e^{-\frac{x}{\theta}}}}}{{{\theta^{\zeta}}\Gamma\left(\zeta\right)}},x\geq
0$ (15)
where $\Gamma(\cdot)$ denotes Gamma function, and the parameters $\zeta$ and
$\theta$ are determined by matching the first two moments of
$f_{I_{K}^{IR}}(x)$ with that of $\varphi(x)$, thus leading to [29]
$\zeta=\frac{{{{\mathcal{M}}^{2}}\left(1\right)}}{{{\mathcal{M}}\left(2\right)-{{\mathcal{M}}^{2}}\left(1\right)}}$
(16)
$\theta=\frac{{{\mathcal{M}}\left(2\right)-{{\mathcal{M}}^{2}}\left(1\right)}}{{{\mathcal{M}}\left(1\right)}}$
(17)
where ${\mathcal{M}}(i)$ denotes the $i$th moment with respect to
$f_{I_{K}^{IR}}(x)$. As shown in Appendix A, the $i$th moment can be derived
using Gaussian Quadrature as
$\displaystyle{\mathcal{M}}\left(i\right)\approx\frac{{i!}}{{1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}}\sum\limits_{\sum\limits_{l=1}^{K}{i_{l}}=i,{i_{l}}\geq
0}{\frac{1}{{{i_{1}}!{i_{2}}!\cdots{i_{K}}!}}}$
$\displaystyle\times\sum\limits_{{q_{k}}\in\left[{1,{N_{Q}}}\right],k\in\left[{1,K}\right]}{\prod\limits_{k=1}^{K}{{\varrho_{{q_{k}}}}{{\log}_{2}}^{{i_{k}}}\left({1+{w_{k}}{\xi_{{q_{k}}}}}\right)}}$
$\displaystyle\times\Psi_{2}^{\left(K\right)}\left({1;1,1,\cdots,1;{\varpi_{1}}{\xi_{{q_{1}}}},{\varpi_{2}}{\xi_{{q_{2}}}},\cdots,{\varpi_{K}}{\xi_{{q_{K}}}}}\right).$
(18)
where
${\varpi_{k}}=\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}{\left({1+\sum\nolimits_{l=1}^{K}{\frac{{{\lambda_{l}}^{2}}}{{1-{\lambda_{l}}^{2}}}}}\right)^{-1}}$,
${w_{k}}=2{\sigma_{k}^{\prime}}^{2}\left({1-{\lambda_{k}}^{2}}\right)$,
$N_{Q}$ is the quadrature order, the weights $\varrho_{p}$ and the abscissas
${{\xi_{p}}}$ are tabulated in [30, Table 25.9], and
$\Psi_{2}^{\left(K\right)}(\cdot)$ is defined as the confluent form of
Lauricella hypergeometric function [31, Definition A.20], [32]. Specifically,
${\mathcal{M}}(0)=1$.
On the other hand, the correction term $\psi\left(x\right)$ in (14) is used to
compensate the difference between ${f_{I_{K}^{IR}}}(x)$ and the basis function
$\varphi(x)$. Apparently, it is very difficult to derive the exact expression
for $\psi\left(x\right)$. However, $\psi\left(x\right)$ can be generally
approximated as a polynomial $\hat{\psi}_{N}(x)\in\mathbb{P}_{N}$ with degree
$N$ by means of polynomial fitting technique [33], where the fitting error is
characterized as $e\left(x\right)=\psi\left(x\right)-\hat{\psi}_{N}(x)$. The
remaining problem here is then to find the optimal polynomial
$\hat{\psi}_{N}(x)$ which can minimize the mean square error (MSE), i.e.,
${\rm{E}}\\{{e\left(x\right)^{2}}\\}{\rm{=}}\int_{0}^{\infty}{\varphi\left(x\right){{\left({\psi\left(x\right)-\hat{\psi}_{N}(x)}\right)}^{2}}dx}$.
It is widely known that any polynomial can be written as a unique linear
combination of orthogonal polynomials. Denote the monic orthogonal polynomials
$\boldsymbol{\mathcal{P}}(x)=[\mathcal{P}_{0}(x),\mathcal{P}_{1}(x),\cdots,\mathcal{P}_{N}(x)]^{\rm{T}}$
with respect to a measure $d\mu(x)$ be the basis in the space of polynomials
of degree less than or equal to $N$, where
${{\mathcal{P}}_{n}}\left(x\right)=\sum\nolimits_{k=0}^{n}{{C_{n,k}}{x^{k}}}\in{{\mathbb{P}}_{n}}$
and $C_{n,n}=1$. As pointed out in [34, Theorem 1.27],
$\boldsymbol{\mathcal{P}}(x)$ is uniquely determined given the measure
$d\mu(x)$. Moreover, the monic orthogonal polynomials obey the orthogonality
$\left\langle{{\mathcal{P}_{n}}\left(x\right),{\mathcal{P}_{k}}\left(x\right)}\right\rangle={\delta_{n,k}}{{\cal
D}_{n}}=\left\\{{\begin{array}[]{*{20}{c}}{{{\cal D}_{n}},}&{n=k};\\\
{0,}&{else}.\end{array}},\right.$ (19)
where $\delta_{n,k}$ indicates Kronecker delta function,
$\left\langle{g\left(x\right),h\left(x\right)}\right\rangle$ denotes an inner
product defined on $2$-norm Lebesgue space $L^{2}(\mathbb{R},\mathcal{F},\mu)$
444Herein, $(\mathbb{R},\mathcal{F},\mu)$ is a measure space, where
$\mathcal{F}$ is $\delta$-algebra over $\mathbb{R}$. with the measure
$d\mu(x)$, that is,
$\left\langle{g\left(x\right),h\left(x\right)}\right\rangle=\int_{-\infty}^{+\infty}{g\left(x\right)h(x)d\mu(x)},$
(20)
and ${\cal D}_{n}$ is a non-zero parameter which will be specified later. With
the monic orthogonal polynomials $\boldsymbol{\mathcal{P}}(x)$,
$\hat{\psi}_{N}(x)$ can be written as
$\hat{\psi}_{N}(x){\rm{=}}\sum\limits_{i=0}^{N}{{\eta_{i}}{\mathcal{P}_{i}}\left(x\right)}={\boldsymbol{\eta}}^{\rm
T}\boldsymbol{\mathcal{P}}(x)$ (21)
where the column vector
${\boldsymbol{\eta}}=[\eta_{0},\eta_{1},\cdots,\eta_{N}]^{\rm{T}}$ can be
regarded as the corresponding coordinate vector of $\hat{\psi}_{N}(x)$ in the
space $\mathbb{P}_{N}$. Substituting (21) into (14), the PDF
${f_{I_{K}^{IR}}}(x)$ can be approximated as
${f_{I_{K}^{IR}}}(x){\approx}f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)=\varphi(x)\hat{\psi}_{N}\left(x\right)=\varphi(x){\boldsymbol{\eta}}^{\rm
T}\boldsymbol{\mathcal{P}}(x).$ (22)
To guarantee the approximation accuracy, we need to find the optimal
polynomial correction term $\hat{\psi}_{N}(x)$ (i.e., the optimal
${\boldsymbol{\eta}}$ and $\boldsymbol{\mathcal{P}}(x)$) which can minimise
the MSE ${\rm{E}}\\{{e\left(x\right)^{2}}\\}$. It is noteworthy that the
resulting approximated PDF $f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)$
should be normalized, such that
$\int_{0}^{\infty}f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)dx=1.$ (23)
Considering this constraint and the fact that the monic orthogonal polynomials
$\boldsymbol{\mathcal{P}}(x)$ are uniquely determined by the measure
$d\mu(x)$, given the measure $d\mu(x)$ and $N$, the approximation problem can
be formulated as
$\displaystyle\underset{{\boldsymbol{\eta}}}{\text{min}}$ $\displaystyle{{\cal
S}_{mse}}({\boldsymbol{\eta}}|d\mu(x),N)$ (24)
$\displaystyle=\int\nolimits_{0}^{\infty}{\varphi\left(x\right){{\left({\sum\limits_{i=0}^{N}{{\eta_{i}}{{\cal
P}_{i}}\left(x\right)}-\psi\left(x\right)}\right)}^{2}}dx}$ s.t.
$\displaystyle\int_{0}^{\infty}f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)dx=1$
This minimization problem can be solved by adopting the method of Lagrange
multiplier. The Lagrangian function corresponding to the minimization problem
can be written in matrix form as
$\displaystyle\Lambda({\boldsymbol{\eta}},\varsigma|d\mu(x),N)$
$\displaystyle=\int_{0}^{\infty}{\varphi\left(x\right){{\left({\sum\limits_{i=0}^{N}{{\eta_{i}}{{\cal
P}_{i}}\left(x\right)}-\psi\left(x\right)}\right)}^{2}}dx}$
$\displaystyle+\varsigma\left({\int_{0}^{\infty}{f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)dx}-1}\right)$
$\displaystyle={{\boldsymbol{\eta}}^{\rm{T}}}{\bf{A\boldsymbol{\eta}}}-2{{\boldsymbol{\eta}}^{\rm{T}}}{\bf{b}}+\varsigma{{\boldsymbol{\eta}}^{\rm{T}}}{\bf{d}}+c-\varsigma$
(25)
where $\varsigma$ represents the Lagrange multiplier, $\bf A$, $\bf b$ and
$\bf d$ are given by (30)-(34) as shown on the top of next page, respectively,
$\displaystyle{{\bf{A}}=\left[{\begin{array}[]{*{20}{c}}{\int_{-\infty}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{0}}^{2}\left(x\right)dx}}&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{0}}\left(x\right){{\mathcal{P}}_{1}}\left(x\right)dx}}&\cdots&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{0}}\left(x\right){{\mathcal{P}}_{N}}\left(x\right)dx}}\\\
{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{1}}\left(x\right){{\mathcal{P}}_{0}}\left(x\right)dx}}&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{1}}^{2}\left(x\right)dx}}&\cdots&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{1}}\left(x\right){{\mathcal{P}}_{N}}\left(x\right)dx}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{N}}\left(x\right){{\mathcal{P}}_{0}}\left(x\right)dx}}&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{N}}\left(x\right){{\mathcal{P}}_{1}}\left(x\right)dx}}&\cdots&{\int_{0}^{\infty}{\varphi\left(x\right){{\mathcal{P}}_{N}}^{2}\left(x\right)dx}}\end{array}}\right]},$
(30)
$\displaystyle{\bf{b}}={\left[{\begin{array}[]{*{20}{c}}{\int_{0}^{\infty}{{f_{I_{K}^{IR}}}(x){{\cal
P}_{0}}\left(x\right)dx}}&{\int_{0}^{\infty}{{f_{I_{K}^{IR}}}(x){{\cal
P}_{1}}\left(x\right)dx}}&\cdots&{\int_{0}^{\infty}{{f_{I_{K}^{IR}}}(x){{\cal
P}_{N}}\left(x\right)dx}}\end{array}}\right]^{\rm{T}}},$ (32)
$\displaystyle{\bf{d}}={\left[{\begin{array}[]{*{20}{c}}{\int_{0}^{\infty}{\varphi(x){\mathcal{P}_{0}}\left(x\right)dx}}&{\int_{0}^{\infty}{\varphi(x){\mathcal{P}_{1}}\left(x\right)dx}}&\cdots&{\int_{0}^{\infty}{\varphi(x){\mathcal{P}_{N}}\left(x\right)dx}}\end{array}}\right]^{\rm{T}}},$
(34)
and $c$ is
$c=\int_{0}^{\infty}{\varphi\left(x\right){\psi^{2}}\left(x\right)dx}.$ (35)
According to the Karush-Kuhn-Tucker (KKT) conditions, the optimal solutions
${\boldsymbol{\eta}}$ and $\varsigma$ should satisfy the following conditions
$\left\\{{\begin{array}[]{*{20}{l}}{\frac{{\partial\Lambda\left({{\boldsymbol{\eta}},\varsigma|d\mu(x),N}\right)}}{{\partial{{\boldsymbol{\eta}}}}}{\rm{=2}}{\bf{A{\boldsymbol{\eta}}}}-2{\bf{b}}+\varsigma{\bf{d}}=0}\\\
{\frac{{\partial\Lambda\left({{\boldsymbol{\eta}},\varsigma|d\mu(x),N}\right)}}{{\partial{\varsigma}}}{={\boldsymbol{\eta}}^{\rm{T}}}{{\bf{d}}-1}=0}\end{array}}\right..$
(36)
As proved in Appendix B, the matrix $\bf A$ is invertible. Then the solution
to (36) is unique and follows as
$\left\\{{\begin{array}[]{*{20}{l}}{{\boldsymbol{\eta}}={{\bf{A}}^{-1}}\left({{\bf{b}}-\frac{\varsigma}{2}{\bf{d}}}\right)}\\\
{\varsigma=\frac{{2\left({{{\bf{b}}^{\rm{T}}}{{\bf{A}}^{-1}}{\bf{d}}-1}\right)}}{{{{\bf{d}}^{\rm{T}}}{{\bf{A}}^{-1}}{\bf{d}}}}}\end{array}}\right.$
(37)
Clearly from (30), (32), (34) and (37), the monic orthogonal polynomials
${{\mathcal{P}}_{n}}\left(x\right)$ need to be determined before the
calculation of the coefficient ${\boldsymbol{\eta}}$. In other words, the
measure $d\mu(x)$ should be determined first. In what follows, the selection
of the measure and the orthogonal polynomials and the calculation of the
coefficient ${\boldsymbol{\eta}}$ are then discussed in detail.
#### III-B1 Selection of $d\mu(x)$ and $\boldsymbol{\mathcal{P}}(x)$
After analyzing the MSE of the fitting error
${\rm{E}}\\{{e\left(x\right)^{2}}\\}$, we have the following property.
###### Property 1.
For any measure $d\mu(x)$, the same minimal MSE can be attained. Moreover, the
optimal polynomial $\hat{\psi}_{N}(x)$ is unique irrespective of the choice of
$d\mu(x)$.
###### Proof:
Please see Appendix C. ∎
Although the choice of the measure $d\mu(x)$ would not affect the solution of
the optimal polynomial $\hat{\psi}_{N}(x)$ as shown in Property 1, it does
affect the computational complexity in deriving the optimal polynomial. More
specifically, the major computation comes from the inverse operation of the
matrix $\bf A$ as seen from (37). It is clear from the definition of $\bf A$
(30) that the complexity of the inverse operation depends on the choice of the
orthogonal polynomials ${{\mathcal{P}}_{n}}\left(x\right)$ and thus depends on
the measure $d\mu(x)$. It is generally known that the complexity in computing
the inverse of a $N\times N$ matrix could be significantly reduced from
$\mathcal{O}(N^{3})$ to $\mathcal{O}(N)$ when the matrix is diagonal.
Therefore, to reduce the computational complexity, the measure $d\mu(x)$ is
suggested to be chosen such that the matrix $\bf A$ is diagonal. Clearly from
the structure of the matrix $\bf A$ (30), when the measure is chosen as
$d\mu(x)={\varphi\left(x\right)}dx$, with the orthogonality of the polynomials
in (19), the matrix $\bf A$ will reduce to a diagonal matrix as
${\bf{A}}=\rm{diag}\left({{\mathcal{D}_{0}},{\mathcal{D}_{1}},\cdots,{{\mathcal{D}}_{N}}}\right).$
(38)
Now with the measure $d\mu(x)={\varphi\left(x\right)}dx$, the monic orthogonal
polynomials
${{\mathcal{P}}_{n}}\left(x\right)=\sum\nolimits_{k=0}^{n}{{C_{n,k}}{x^{k}}}$
with $C_{n,n}=1$ can be uniquely determined by the method introduced in [34].
Specifically, the polynomial coefficients $C_{n,k}$ can be determined as
follows. Following the three-term recurrence relation (TTRR) in [34, Theorem
1.27], the monic orthogonal polynomials should satisfy
$\displaystyle{{{\mathcal{P}}_{n+1}}\left(x\right)=\left({x-{\alpha_{n}}}\right){{\mathcal{P}}_{n}}\left(x\right)-{\beta_{n}}{{\mathcal{P}}_{n-1}}\left(x\right),\,n=0,1,2,\cdots,}$
$\displaystyle{{{\mathcal{P}}_{-1}}\left(x\right)=0,\;\;\;{\kern
1.0pt}{{\mathcal{P}}_{0}}\left(x\right)=1,}$ (39)
where
$\displaystyle{\alpha_{n}}$
$\displaystyle=\frac{\left<x{\mathcal{P}}_{n}(x),{\mathcal{P}}_{n}(x)\right>}{\left<{\mathcal{P}}_{n}(x),{\mathcal{P}}_{n}(x)\right>}$
$\displaystyle=\frac{{\sum\limits_{i=0}^{n}{\sum\limits_{j=0}^{n}{{C_{n,i}}{C_{n,j}}{\nu_{i+j+1}}}}}}{{\sum\limits_{i=0}^{n}{\sum\limits_{j=0}^{n}{{C_{n,i}}{C_{n,j}}{\nu_{i+j}}}}}},\,n=0,1,2,\cdots,$
(40) $\displaystyle{\beta_{n}}$
$\displaystyle=\frac{\left<{\mathcal{P}}_{n}(x),{\mathcal{P}}_{n}(x)\right>}{\left<{\mathcal{P}}_{n-1}(x),{\mathcal{P}}_{n-1}(x)\right>}$
$\displaystyle=\frac{{\sum\limits_{i=0}^{n}{\sum\limits_{j=0}^{n}{{C_{n,i}}{C_{n,j}}{\nu_{i+j}}}}}}{{\sum\limits_{i=0}^{n-1}{\sum\limits_{j=0}^{n-1}{{C_{n-1,i}}{C_{n-1,j}}{\nu_{i+j}}}}}},\,n=1,2,\cdots.$
(41)
and $\nu_{n}$ denotes the $n$th moment with respect to the cumulative
distribution function (CDF) $\mu(x)$, that is,
$\displaystyle{\nu_{n}}$
$\displaystyle=\int_{0}^{\infty}{{x^{n}}d\mu\left(x\right)}=\int_{0}^{\infty}{{x^{n}}\varphi\left(x\right)dx}$
$\displaystyle=\int_{0}^{\infty}{\frac{{{x^{n+\zeta-1}}{e^{-\frac{x}{\theta}}}}}{{{\theta^{\zeta}}\Gamma\left(\zeta\right)}}dx}=\frac{{{\theta^{n}}\Gamma\left({n+\zeta}\right)}}{{\Gamma\left(\zeta\right)}}$
(42)
With
${{\mathcal{P}}_{n}}\left(x\right)=\sum\nolimits_{k=0}^{n}{{C_{n,k}}{x^{k}}}$
and the relation in (III-B1), the polynomial coefficients $C_{n,k}$ can be
obtained recursively as
${C_{n+1,k}}=\left\\{{\begin{array}[]{*{20}{c}}{{C_{n,k-1}}-{\alpha_{n}}{C_{n,k}}-{\beta_{n}}{C_{n-1,k}},}&{0\leq
k\leq n+1;}\\\ {0,}&{else.}\end{array}}\right.$ (43)
with ${C_{0,0}}=1$. Then the parameter ${\mathcal{D}}_{n}$ in (19) can be
determined as
${{\mathcal{D}}_{n}}=\sum\limits_{i=0}^{n}{\sum\limits_{j=0}^{n}{{C_{n,i}}{C_{n,j}}{\nu_{i+j}}}}.$
(44)
Specifically, ${{\mathcal{D}}_{0}}=1$.
#### III-B2 Calculation of $\boldsymbol{\eta}$
To compute $\boldsymbol{\eta}$ in (37), the vectors $\bf b$, $\bf d$ and
$\varsigma$ should be determined first. By the definition of $\mathcal{M}(i)$
and integrating out $x$ for (32), the vector $\bf{b}$ can be consequently
written as
${\bf{b}}=\left[{\begin{array}[]{*{20}{c}}{\sum\limits_{k=0}^{0}{{C_{0,k}}}{\cal
M}\left(k\right)}&{\sum\limits_{k=0}^{1}{{C_{0,k}}}{\cal
M}\left(k\right)}\end{array}}\right.\\\
{\left.{\begin{array}[]{*{20}{c}}\cdots&{\sum\limits_{k=0}^{N}{{C_{N,k}}}{\cal
M}\left(k\right)}\end{array}}\right]^{\rm{T}}}.$ (45)
Considering $\mathcal{P}_{0}(x)=1$ and the orthogonality among the polynomials
$\mathcal{P}_{n}(x)$, the vector d given in (34) is directly reduced as
${\bf{d}}={\left[{1,\quad\overbrace{0,\quad\cdots\quad,0}^{N-terms}}\right]^{\rm
T}}.$ (46)
On the other hand, by putting (38), (45) and (46) into (37), the Lagrange
multiplier $\varsigma$ is calculated as
$\varsigma=\frac{{{C_{0,0}}{\mathcal{M}}\left(0\right){D_{0}}-1}}{{{D_{0}}}}=0$
(47)
by recalling that $C_{0,0}=1$, ${\mathcal{M}}(0)=1$ and $\mathcal{D}_{0}=1$.
Accordingly, the coefficients $\boldsymbol{\eta}$ can be finally computed as
${\boldsymbol{\eta}}={{\bf{A}}^{-1}}{\bf{b}}\\\
={\left[{\begin{array}[]{*{20}{c}}{\frac{{\sum\limits_{k=0}^{0}{{C_{0,k}}}{\mathcal{M}}\left(k\right)}}{{{{\mathcal{D}}_{0}}}}}&{\frac{{\sum\limits_{k=0}^{1}{{C_{1,k}}}{\mathcal{M}}\left(k\right)}}{{{{\mathcal{D}}_{1}}}}}&\cdots&{\frac{{\sum\limits_{k=0}^{N}{{C_{N,k}}}{\mathcal{M}}\left(k\right)}}{{{{\mathcal{D}}_{N}}}}}\end{array}}\right]^{\rm
T}}.$ (48)
### III-C Discussions
With the approximated PDF of the accumulated mutual information in (22), the
outage probability which is equivalent to the CDF of the accumulated mutual
information can be easily obtained. From the expression of the outage
probability, some interesting insights could be found in the following.
Moreover, as shown in (24), the MSE of the fitting error also depends on the
degree of the polynomials $N$. The selection of the degree will also be
briefly discussed here.
#### III-C1 Insights of Outage Probability
After determining $\boldsymbol{\eta}$, the approximated PDF
$f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)$ is expressed with the expansion
of ${{{\mathcal{P}}_{n}}\left(x\right)}$ as
$f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)=\varphi(x)\sum\limits_{n=0}^{N}{{\eta_{n}}\sum\limits_{i=0}^{n}{{C_{n,i}}{x^{i}}}}.$
(49)
Henceforth, the approximated CDF for $I_{K}^{IR}$ can be obtained as
$\displaystyle{F_{I_{K}^{IR}}}(x)$ $\displaystyle\approx
F_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)=\int_{0}^{x}{f_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(t)dt}$
$\displaystyle=\sum\limits_{n=0}^{N}{{\eta_{n}}\sum\limits_{i=0}^{n}{{C_{n,i}}\int_{0}^{x}{{t^{i}}\varphi(t)dt}}}.$
(50)
To facilitate the analysis, a family of functions $W_{i}(x)$ is defined as
$\displaystyle{W_{i}}\left(x\right)$
$\displaystyle=\frac{1}{{{\nu_{i}}}}\int_{0}^{x}{{t^{i}}\varphi\left(t\right)dt}=\frac{1}{{{\theta^{i+\zeta}}\Gamma\left({i+\zeta}\right)}}\int_{0}^{x}{{t^{i+\zeta-1}}{e^{-\frac{t}{\theta}}}dt}$
$\displaystyle=\frac{{\gamma\left({i+\zeta,\frac{x}{\theta}}\right)}}{{\Gamma\left({i+\zeta}\right)}}$
(51)
where $\gamma(\cdot)$ represents the lower incomplete Gamma function. It is
clear that $W_{i}(x)$ is the CDF of a Gamma RV with parameters
$(i+\zeta,\theta)$. Consequently, by exchanging the order of summations, the
CDF in (III-C1) can be rewritten as
$\displaystyle F_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)$
$\displaystyle=\sum\limits_{n=0}^{N}{{\eta_{n}}\sum\limits_{i=0}^{n}{{C_{n,i}}{\nu_{i}}{W_{i}}\left(x\right)}}$
$\displaystyle=\sum\limits_{i=0}^{N}{{W_{i}}\left(x\right){\nu_{i}}\sum\limits_{n=i}^{N}{{\eta_{n}}}{C_{n,i}}}$
$\displaystyle=\sum\limits_{i=0}^{N}{{\kappa_{i}}}{W_{i}}\left(x\right)$ (52)
where
${\kappa_{i}}={\nu_{i}}\sum\limits_{n=i}^{N}{{\eta_{n}}}{C_{n,i}}.$ (53)
Meanwhile, the parameters $\\{\kappa_{i}\\}$ satisfy
$\mathop{\lim}\limits_{x\to\infty}F_{{{\boldsymbol{\gamma}_{1:K}}},N}^{IR}(x)=\mathop{\lim}\limits_{x\to\infty}\sum\limits_{i=0}^{N}{{\kappa_{i}}}{W_{i}}\left(x\right)=1\\\
\Rightarrow\sum\limits_{i=0}^{N}{{\kappa_{i}}}=1.$ (54)
Therefore, under time-correlated Rayleigh fading channels, the CDF of the
accumulated mutual information $F_{I_{K}^{IR}}(x)$ can be written as the
weighted sum of the CDFs of Gamma RVs, such that
${F_{I_{K}^{IR}}}(x)\approx\sum\limits_{i=0}^{N}{{\kappa_{i}}}{W_{i}}\left(x\right).$
(55)
It means that the outage probability
$P_{out}^{IR}\left(K\right)=F_{I_{K}^{IR}}({\cal R})$ can be written as the
weighted sum of a number of outage probabilities, each associated with one
Gamma RV555The source code of our approximation is available at
https://sourceforge.net/projects/matlabgammaapproximation/files/Gamma%20Approximation/..
Specifically, if $N=0$, the approximation reduces to the ordinary Gamma
approximation which is valid for the case with independent fading channels
[21]. The approximation in (55) can ease the analysis of the system behaviors
with respect to various system parameters and facilitate system design to
achieve various objectives, e.g., the optimal rate design to maximize the long
term average throughput. This will be further illustrated in Section V.
#### III-C2 Choice of $N$
As shown in Property 1, the same minimal MSE ${{\cal
S}_{mse}}({\boldsymbol{\eta}}|d\mu(x),N)$ can be attained whatever the measure
$d\mu(x)$ is. Hereby, we define
${{\mathcal{S}}_{min\\_mse}}({\boldsymbol{\eta}}|N)$ as the minimal MSE given
$N$. By substituting (37) into the objective function of MSE in (24), it
yields the minimal MSE as
$\displaystyle{{\mathcal{S}}_{min\\_mse}}({\boldsymbol{\eta}}|N)$
$\displaystyle={{\boldsymbol{\eta}}^{\rm{T}}}\left({{\bf{A}\boldsymbol{\eta}}-2{\bf{b}}}\right)+c$
$\displaystyle=c-{{\boldsymbol{\eta}}^{\rm{T}}}\left({{\bf{b}}+\frac{\varsigma}{2}{\bf{d}}}\right)$
$\displaystyle=c-\left({{{\bf{b}}^{\rm
T}}-\frac{\varsigma}{2}{{\bf{d}}^{\rm{T}}}}\right){{\bf{A}}^{-1}}\left({{\bf{b}}+\frac{\varsigma}{2}{\bf{d}}}\right).$
(56)
Putting (38), (45), (46) and (47) into (III-C2), the minimal MSE is rewritten
as
${{\mathcal{S}}_{min\\_mse}}({\boldsymbol{\eta}}|N)=\int_{0}^{\infty}{\varphi\left(x\right){\psi^{2}}\left(x\right)dx}\\\
-\sum\limits_{n=0}^{N}{{{\mathcal{D}}_{n}}^{-1}{{\left({\sum\limits_{k=0}^{n}{{C_{n,k}}}{\mathcal{M}}\left(k\right)}\right)}^{2}}}\geq
0.$ (57)
Clearly, the minimum MSE decreases as the degree $N$ increases, i.e.
${{\mathcal{S}}_{min\\_mse}}({\boldsymbol{\eta}}|N)\geq{{\mathcal{S}}_{min\\_mse}}({\boldsymbol{\eta}}|N+1)$,
since
${\mathcal{D}_{n}}=\left\langle{{{\mathcal{P}}_{n}}\left(x\right),{{\mathcal{P}}_{n}}\left(x\right)}\right\rangle>0$.
It implies that the accuracy of the PDF approximation is limited by the degree
$N$ and would be improved as the degree $N$ increases. This result can be
further demonstrated by Fig. 2 where the approximated CDFs of $I_{K}^{IR}$
using different degrees are compared with the true CDFs obtained from Monte-
Carlo simulations, by taking a system with the following setting as an
example: the channel correlation coefficient $\rho_{k,l}=0.5$ and the mean of
the SNR ${\rm{E}}(\gamma_{l})=2{\sigma_{l}^{\prime}}^{2}=5$, where $1\leq
k\neq l\leq K$.
Figure 2: Comparison between the approximated CDFs ${F_{I_{K}^{IR}}}(x)$ with
different $N$ and the true CDFs obtained from Monte-Carlo simulations.
However, as shown in Fig. 2, the improvement of the approximation accuracy
becomes minor when the degree becomes relatively large. Moreover, the increase
of the degree $N$ would also cause the increase of the computational
complexity. It is thus necessary to properly choose the degree to balance the
approximation accuracy and computational complexity. To quantify the
contribution of increasing the degree in terms of the approximation accuracy,
a metric of MSE reduction is defined as
$\displaystyle{\Delta_{N}}$ $\displaystyle\triangleq{{\cal
S}_{min\\_mse}}({\boldsymbol{\eta}}|0)-{{\cal
S}_{min\\_mse}}({\boldsymbol{\eta}}|N)$
$\displaystyle=\sum\limits_{n=1}^{N}{{{\mathcal{D}}_{n}}^{-1}{{\left({\sum\limits_{k=0}^{n}{{C_{n,k}}}{\mathcal{M}}\left(k\right)}\right)}^{2}}}.$
(58)
For a good balance between approximation accuracy and computational
complexity, the degree is suggested to be chosen as
${N}=\min\left\\{{\min\left\\{{n\left|{{r_{n}}\leq\epsilon}\right.}\right\\},\hat{N}}\right\\}$
(59)
where $\hat{N}$ is a pre-determined upper bound of the degree to limit the
computational complexity, and $\epsilon$ denotes the tolerance for normalized
MSE reduction defined as
${r_{n}}=\frac{{{\Delta_{n}-{\Delta_{n-1}}}}}{{{\Delta_{n}}}}$. Clearly,
${{r_{N}}\leq\epsilon}$ holds. It roughly indicates that no significant
improvement on MSE can be expected if the degree is larger than $N$ in (59).
Therefore the degree of $N$ in (59) is sufficient for a good approximation.
Notice that $\Delta_{1}={\Delta_{2}}=0$ by using the orthogonality of the
polynomials. Thus the upper bound $\hat{N}$ and the degree $N$ should be set
greater than $2$ without doubt.
## IV Diversity Order
Basically, HARQ-IR schemes exploit not only coding gain but also time
diversity to improve the transmission reliability. To better understand the
behavior of HARQ-IR schemes, diversity order which is another important
performance metric is also analyzed in this paper. To facilitate the analysis,
the transmit SNR in each HARQ round is set equal, i.e.
$P_{1}/\mathfrak{N}_{1}=P_{2}/\mathfrak{N}_{2}=\cdots=P_{M}/\mathfrak{N}_{M}=\gamma_{T}$.
The diversity order $d$ for the HARQ scheme is defined as [21, 35]
$d=-\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({P_{e}}\right)}}{{\log\left({{\gamma_{T}}}\right)}},$
(60)
where $P_{e}$ denotes the error probability. As shown in [21, 35], the error
probability can be well approximated as the outage probability
${P_{out}^{IR}\left(M\right)}$ when a capacity achieving code is applied.
Therefore the diversity order $d$ can be well approximated as
$d=-\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({P_{out}^{IR}\left(M\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}.$
(61)
Namely, diversity order quantifies the slope of
$-\log\left({P_{out}^{IR}\left(M\right)}\right)$ with respect to
$\log\left({{\gamma_{T}}}\right)$ when ${{\gamma_{T}}\to\infty}$.
Owing to the following inequalities
${\log_{2}}\left({1+\sum\limits_{l=1}^{M}{\gamma_{l}}}\right)\leq{\log_{2}}\left({\prod\limits_{l=1}^{M}{\left({1+\gamma_{l}}\right)}}\right)\\\
\leq M{\log_{2}}\left({1+{M^{-1}}\sum\limits_{l=1}^{M}{\gamma_{l}}}\right),$
(62)
the outage probability can be bounded as
$\Pr\left({M{{\log}_{2}}\left({1+{M^{-1}}\sum\limits_{l=1}^{M}{\gamma_{l}}}\right)<{\mathcal{R}}}\right)\leq
P_{out}^{IR}\left(M\right)\\\
\leq\Pr\left({{{\log}_{2}}\left({1+\sum\limits_{l=1}^{M}{\gamma_{l}}}\right)<{\mathcal{R}}}\right),$
(63)
where the right inequality in (62) holds by using the Jensen’s inequality.
Defining $Y=\sum\nolimits_{l=1}^{M}{\gamma_{l}}$, the bounds in (63) can be
rewritten as
$\Pr\left({Y<M({{2^{{M^{-1}}{\cal R}}}-1})}\right)\leq
P_{out}^{IR}\left(M\right)\leq\Pr\left({Y<{2^{\cal R}}-1}\right).$ (64)
Clearly, $Y$ represents a sum of correlated exponential RVs
$\\{\gamma_{l}\\}_{l=1}^{M}$. The CDF of a sum of correlated exponential RVs
has been derived in [36] and the result is summarized as the following
theorem.
###### Theorem 1.
[36] Given the joint PDF regarding to exponential RVs
$\\{\gamma_{l}\\}_{l=1}^{M}$ in (III-B), the CDF of
$Y=\sum\nolimits_{l=1}^{M}{\gamma_{l}}$ can be obtained as
${F_{Y}}\left(y\right)=\Pr\left({Y<y}\right)=\frac{{{y^{M}}}}{{\det\left({\bf{B}}\right)\Gamma\left({M+1}\right)}}\times\\\
\Phi_{2}^{\left(M\right)}\left({1,\cdots,1;M+1;-{\delta_{1}}^{-1}y,\cdots,-{\delta_{M}}^{-1}y}\right)$
(65)
where the notation $\rm det(\cdot)$ represents the determinant operation,
$\Phi_{2}^{\left(M\right)}\left(\cdot\right)$ denotes the confluent Lauricella
function [31, Def. A.19], $\\{\delta_{k}\\}_{k=1}^{M}$ are defined as the
eigenvalues of the matrix $\bf B=FE$, where $\bf F$ is an $M\times M$ diagonal
matrix with diagonal entries as $\\{2{\sigma_{k}^{\prime}}^{2}\\}_{k=1}^{M}$,
and $\bf E$ is an $M\times M$ positive definite matrix given by
${\bf{E}}=\left[{\begin{array}[]{*{20}{c}}1&{\sqrt{{\rho_{1,2}}}}&\cdots&{\sqrt{{\rho_{1,M}}}}\\\
{\sqrt{{\rho_{2,1}}}}&1&\cdots&{\sqrt{{\rho_{2,M}}}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{\sqrt{{\rho_{M,1}}}}&{\sqrt{{\rho_{M,2}}}}&\cdots&1\end{array}}\right],\,0\leq\rho_{k,l}<1.$
(66)
Substituting (64) into (61), the bounds of the diversity order can be found as
$-\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({{F_{Y}}\left({{2^{\mathcal{R}}}-1}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}\leq
d\\\
\leq-\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({{F_{Y}}\left({M\left({{2^{{M^{-1}}{\cal
R}}}-1}\right)}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}.$ (67)
By using (65), the left inequality in (67) can be rewritten as
$\displaystyle
d\geq\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\left[{\frac{{\log\left({\det\left({\bf{B}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}}\right.-$
$\displaystyle{\left.{\frac{{\log\left({\Phi_{2}^{\left(M\right)}\left({1,\cdots,1;M+1;-{\delta_{1}}^{-1}y,\cdots,-{\delta_{M}}^{-1}y}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}}\right]_{y={2^{R}}-1}}.$
(68)
The first term on the right hand side of (IV) can be simplified by using the
property of determinants as
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({\det\left({\bf{B}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}=\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({\det\left({\bf{F}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}+\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({\det\left({\bf{E}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}.$
(69)
Recalling that
${\bf{F}}={\gamma_{T}}{\rm{diag}}\left({2{\sigma_{1}}^{2},\cdots,2{\sigma_{M}}^{2}}\right)$
and ${\bf{E}}$ is irrelevant with ${\gamma_{T}}$, (69) can be reduced to
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({\det\left({\bf{B}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}=M.$
(70)
To derive the second term on the right hand side of the inequality (IV), we
define ${\bf
G}={\rm{diag}}\left({2{\sigma_{1}}^{2},\cdots,2{\sigma_{M}}^{2}}\right)\bf{E}$
for notational simplicity, such that ${\bf{B}}=\gamma_{T}\bf{G}$. By using the
series representation of the confluent Lauricella function [37], the confluent
Lauricella function in the second term on the right hand side of the
inequality (IV) can be rewritten as
$\Phi_{2}^{\left(M\right)}\left({1,\cdots,1;M+1;-{\delta_{1}}^{-1}y,\cdots,-{\delta_{M}}^{-1}y}\right)\\\
=1+\sum_{{m_{1}+\cdots+m_{M}}>0}{{\frac{{\prod\limits_{k=1}^{M}{{{\left({-{\delta_{k}}^{-1}y}\right)}^{{m_{k}}}}}}}{{{{\left(M+1\right)}_{{m_{1}}+\cdots+{m_{M}}}}}}}},$
(71)
where $(\cdot)_{n}$ denotes Pochhammer symbol. Since
$\displaystyle\left|\sum_{{m_{1}+\cdots+m_{M}}>0}{{\frac{{\prod\limits_{k=1}^{M}{{{\left({-{\delta_{k}}^{-1}y}\right)}^{{m_{k}}}}}}}{{{{\left(M+1\right)}_{{m_{1}}+\cdots+{m_{M}}}}}}}}\right|$
$\displaystyle\leq\sum_{{m_{1}+\cdots+m_{M}}>0}{{\frac{{{m_{1}}!\cdots{m_{M}}!}}{{{{\left({M+1}\right)}_{{m_{1}}+\cdots+{m_{M}}}}}}\frac{{\prod\limits_{k=1}^{M}{{{\left|{-{\delta_{k}}^{-1}y}\right|}^{{m_{k}}}}}}}{{{m_{1}}!\cdots{m_{M}}!}}}}$
$\displaystyle\underset{(a)}{\leq}\sum_{{m_{1}+\cdots+m_{M}}>0}{\frac{{\prod\limits_{k=1}^{M}{{{\left|{-{\delta_{k}}^{-1}y}\right|}^{{m_{k}}}}}}}{{{m_{1}}!\cdots{m_{M}}!}}}$
$\displaystyle=\sum\limits_{L=1}^{\infty}{\frac{1}{{L!}}\sum_{\sum\limits_{k=1}^{M}{{m_{k}}=L}}{\frac{{L!}}{{{m_{1}}!\cdots{m_{M}}!}}\prod\limits_{k=1}^{M}{{{\left|{-{\delta_{k}}^{-1}y}\right|}^{{m_{k}}}}}}}$
$\displaystyle=\sum\limits_{L=1}^{\infty}{\frac{1}{{L!}}{{\left({y\sum\limits_{k=1}^{M}{\left|{\delta_{k}}\right|^{-1}}}\right)}^{L}}}$
$\displaystyle={e^{y\sum\limits_{k=1}^{M}{\left|{\delta_{k}}\right|^{-1}}}}-1\underset{(b)}{=}{e^{y\sum\limits_{k=1}^{M}{\left|{\gamma_{T}\beta_{k}}\right|^{-1}}}}-1$
$\displaystyle={e^{y\gamma_{T}^{-1}\sum\limits_{k=1}^{M}{\left|{\beta_{k}}\right|^{-1}}}}-1$
(72)
where $(a)$ follows from
${m_{1}}!\cdots{m_{M}}!\leq{\left({M+1}\right)_{{m_{1}}+\cdots+{m_{M}}}}$,
$\beta_{k}$ denotes the eigenvalues of $\bf G$ and $(b)$ comes from
${\bf{B}}=\gamma_{T}\bf{G}$, together with
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}{e^{y\gamma_{T}^{-1}\sum\nolimits_{k=1}^{M}{\left|{\beta_{k}}\right|^{-1}}}}=1$,
we have
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\sum_{{m_{1}+m_{2}+\cdots+m_{M}}>0}{{\frac{{\prod\limits_{k=1}^{M}{{{\left({-{\delta_{k}}^{-1}y}\right)}^{{m_{k}}}}}}}{{{{\left(M+1\right)}_{{m_{1}}+\cdots+{m_{M}}}}}}}}=0.$
(73)
It follows the limit of the confluent Lauricella function as
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\Phi_{2}^{\left(M\right)}\left({1,\cdots,1;M+1;-{\delta_{1}}^{-1}y,\cdots,-{\delta_{M}}^{-1}y}\right)=1.$
(74)
and then the second term on the right hand side of the inequality (IV) reduces
as
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({\Phi_{2}^{\left(M\right)}\left({1,\cdots,1;M+1;-{\delta_{1}}^{-1}y,\cdots,-{\delta_{M}}^{-1}y}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}\\\
=0.$ (75)
Substituting (70) and (75) into (IV) leads to $d\geq M$. Following the same
approach, the second inequality of (67) can also be derived as $d\leq M$. As a
result, under time correlated fading channels with $0\leq\rho_{k,l}<1$, the
diversity order $d$ is equal to the number of transmissions $M$, i.e., $d=M$.
Equivalently,
${P_{out}^{IR}\left(M\right)}\propto\frac{1}{{\gamma_{T}}^{M}}$ (76)
for large SNR. More precisely, the outage probability
${P_{out}^{IR}\left(M\right)}$ can be expressed as [1, 3.158]
$P_{out}^{IR}\left(M\right)=c\left({{\gamma_{T}},\sigma_{k},{\rho_{k,l}}},\mathcal{R},M\right){{{\gamma_{T}}^{-M}}},$
(77)
where the coefficient $c\left({{\gamma_{T}},{\sigma_{k}},{\rho_{k,l}},{\cal
R},M}\right)$ satisfies
$\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({c\left({{\gamma_{T}},\sigma_{k},{\rho_{k,l}},{\cal
R},M}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}=0.$ (78)
From (77) and (78), we can find that when $\gamma_{T}$ is large, the slope of
$-\log\left({P_{out}^{IR}\left(M\right)}\right)$ with respect to
$\log\left({{\gamma_{T}}}\right)$ tends to be constant as $M$. In other words,
when $\rho_{k,l}<1$, the time correlation ${\rho_{k,l}}$ would not affect the
diversity order and the diversity order is constant as $M$. However the time
correlation would influence the coefficient
$c\left({{\gamma_{T}},{\sigma_{k}},{\rho_{k,l}},{\cal R},M}\right)$ and thus
affect the outage performance. This effect will be further investigated in
Section V.
The above analysis indicates that full diversity can be achieved by HARQ-IR
schemes even under time-correlated fading channels with $0\leq\rho_{k,l}<1$,
which further justifies the benefit of HARQ-IR.
Remark 1: The result of the diversity order is not applicable to the case with
fully correlated fading channels. Under fully correlated Rayleigh fading
channels, $|h_{1}|=|h_{2}|=\cdots=|h_{M}|\sim Rayleigh(2{\sigma}^{2})$ and
$\rho_{k,l}=1$. The outage probability $P_{out}^{IR}\left(M\right)$ can be
easily derived as
$P_{out}^{IR}\left(M\right)=1-\exp\left({-\frac{{{2^{\mathcal{R}/M}}-1}}{{{2\sigma^{2}\gamma_{T}}}}}\right).$
(79)
Putting (79) into (61), and by applying L’Hôpital’s rule and the method of
replacement with equivalent infinitesimal, the diversity order follows as
$\displaystyle d$
$\displaystyle=-\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\log\left({1-\exp\left({-\frac{{{2^{{\cal
R}/M}}-1}}{{{2\sigma^{2}\gamma_{T}}}}}\right)}\right)}}{{\log\left({{\gamma_{T}}}\right)}}$
$\displaystyle=\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{\left({{2^{{\cal
R}/M}}-1}\right)\exp\left({-\frac{{{2^{{\cal
R}/M}}-1}}{{{2\sigma^{2}\gamma_{T}}}}}\right)}}{{{2\sigma^{2}\gamma_{T}}\left({1-\exp\left({-\frac{{{2^{{\cal
R}/M}}-1}}{{{2\sigma^{2}\gamma_{T}}}}}\right)}\right)}}$
$\displaystyle=\mathop{\lim}\limits_{{\gamma_{T}}\to\infty}\frac{{{2^{{\cal
R}/M}}-1}}{{{2^{{\cal R}/M}}-1}}=1.$ (80)
It means that under fully correlated fading channels, the diversity order is
reduced to one.
## V Numerical Results and Discussions
With the above analytical results, the performance of HARQ-IR over time-
correlated fading channels can be evaluated and optimal design of transmission
scheme is also enabled. In the following, we take systems with
${\rm{E}}(|h_{l}|^{2})=2\sigma_{l}^{2}=1$ and $\rho_{k,l}=\rho$ for $1\leq
k\neq l\leq M$ as examples for performance evaluation and optimal design.
### V-A Verification of Analytical Results
To verify our analytical expressions for the performance metrics of HARQ-IR
over time-correlated Rayleigh fading channels, Monte-Carlo simulations are
conducted for comparison. For illustration, we set $\mathcal{R}=2\rm{bps/Hz}$
and $\rho=0.5$, and the outage probability versus transmit SNR $\gamma_{T}$ is
shown in Fig. 3. Apparently, there is a perfect match between the analytical
results and simulation results, which demonstrates the correctness of our
analytical results. Moreover, as expected, the outage probability
$P_{out}^{IR}(M)$ decreases as $M$ increases. For example, the outage
probability for $M=1$ is about $2.6*10^{-1}$ for a transmit SNR of
$10~{}\rm{dB}$. When $M$ is increased to $4$, the outage probability
significantly drops to $3*10^{-4}$. It demonstrates a notable performance gain
of HARQ-IR schemes.
Figure 3: Outage Probability $P_{out}^{IR}(M)$ versus transmit SNR
$\gamma_{T}$.
### V-B Impact of Time Correlation
The impact of time correlation on the performance of HARQ-IR including the
outage probability, the average number of transmissions and the LTAT is now
investigated, and the results are shown in Fig. 4-6, respectively. Notice that
for fully correlated fading channels with $\rho=1$, the outage probability is
obtained as (79). Correspondingly, the average number of transmissions
$\bar{\mathcal{N}}$ and the LTAT $\bar{\mathcal{R}}$ can be obtained by
putting (79) into (9) and (10), respectively.
The outage probability versus the time correlation coefficient under various
$\gamma_{T}$ and $M$ is shown in Fig. 4. It is readily seen that the outage
probability increases with time correlation coefficient $\rho$. Specifically,
for the case with $M=4$ and $\gamma_{T}=7~{}\rm{dB}$, $P_{out}^{IR}(M)$
increases from $3*10^{-4}$ to $8*10^{-2}$ when $\rho$ increases from $0$ to
$1$. It indicates that time correlation does cause negative impact on the
outage performance. Additionally, it is observed that the gap between the
outage probabilities for two different $M$ becomes narrower when $\rho$ gets
higher. In other words, under highly correlated channels, further increase of
the number of transmissions will only lead to slight improvement on the outage
probability.
Figure 4: Outage Probability $P_{out}^{IR}\left(M\right)$ versus correlation
coefficient $\rho$.
In Fig. 5, the average number of transmissions $\bar{\mathcal{N}}$ versus
correlation coefficient $\rho$ is plotted. It can be seen that
$\bar{\mathcal{N}}$ increases with $\rho$ for $M>2$. This is also because of
the negative impact of the correlation in the channels. When the time
correlation increases, outage probability will increase, thus more HARQ rounds
are required to successfully deliver a message. Notice that when $M=1,2$, the
average number of transmissions $\bar{\mathcal{N}}$ is irrelevant to the time
correlation, which directly follows from (9).
Figure 5: Average number of transmissions $\bar{\mathcal{N}}$ versus
correlation coefficient $\rho$. ($\gamma_{T}=7$dB)
With respect to the effect of time correlation on the LTAT, similarly we can
find that the LTAT $\bar{\mathcal{R}}$ decreases with the increase of $\rho$
as shown in Fig. 6. For instance, the LTAT $\bar{\mathcal{R}}$ for the case
with $M=4$ decreases from $1.30~{}\rm{bps/Hz}$ to $1.05~{}\rm{bps/Hz}$ as
$\rho$ increases from $0$ to $1$. Interestingly, it can also be easily
observed that the LTAT shows opposite trends with the increase of the number
of transmissions under different correlation regions. More specifically, under
low-to-median correlation, the LTAT is improved when the number of
transmissions increases. However, when the time correlation is high,
additional transmission causes degradation of the LTAT when $M\geq 2$. This is
due to the twofold impact of increasing the number of transmissions. On one
hand, the increase of the number of transmissions would decrease the outage
probability. On the other hand, It would also increase the average number of
transmissions. The first impact dominates under low-to-median correlation,
while the second one dominates under high correlation. Therefore, from the
LTAT’s point of view, more transmissions may not be better when time
correlation is high.
Figure 6: LTAT $\bar{\mathcal{R}}$ versus correlation coefficient.
($\gamma_{T}=7$dB)
### V-C Optimum Rate Design
As defined in (10), the LTAT is a complicated function of the transmission
rate $\mathcal{R}$ due to the implicit involvement of $\mathcal{R}$ in outage
probability and the average number of transmissions. To maximize the LTAT, the
rate should be properly designed. Mathematically, the problem of optimal rate
design can be formulated as
$\displaystyle\underset{\mathcal{R}}{\text{min}}$
$\displaystyle\bar{\mathcal{R}}=\frac{{\mathcal{R}\left({1-P_{out}^{IR}\left(M\right)}\right)}}{{\bar{\mathcal{N}}}}$
(81) s.t. $\displaystyle P_{out}^{IR}\left(M\right)\leq\varepsilon.$
With the approximation in (55) where $\kappa_{i}$, ${{\zeta}}$, and
${{\theta}}$ are irrelevant to the rate $\mathcal{R}$, the optimal rate can be
easily solved by using optimization tools.
Fig. 7 shows the optimal rate given various outage constraints $\varepsilon$
under different correlation scenarios. It is clear that the optimal
transmission rate increases with the transmit SNR and the allowable outage
probability $\varepsilon$. For examples, the optimum rate $\mathcal{R}$
increases by $3.2~{}\rm{bps/Hz}$ when the transmit SNR $\gamma_{T}$ is
increased from $0~{}\rm{dB}$ to $10~{}\rm{dB}$ for the case with
$\varepsilon=10^{-2}$ and $\rho=0.5$. It also increases from
$3.87~{}\rm{bps/Hz}$ to $6.57~{}\rm{bps/Hz}$ when the allowable outage
probability $\varepsilon$ increases from $0.01$ to $0.1$ for the case with
$\rho=0.5$ and $\gamma_{T}=10~{}\rm{dB}$. On the other hand, time correlation
of the channels has negative effect on the optimal rate.
Figure 7: Optimum transmission rate $R$ versus transmit SNR $\gamma_{T}$ for
different target outage probability $\varepsilon$ for $M=4$.
To further investigate the improvement of LTAT through optimal rate design,
the LTAT $\bar{\mathcal{R}}$ versus transmit SNR $\gamma_{T}$ for the schemes
with optimal rate design and a constant rate (which is set as the optimal rate
corresponding to the case with $\gamma_{T}=0~{}\rm{dB}$), is depicted in Fig.
8 by setting $\varepsilon=0.01$ and $M=4$. Apparently, notable improvement of
LTAT can be observed through optimal rate design and the contribution of the
optimal rate design becomes more significant when the transmit SNR gets
higher.
Figure 8: LTAT $\bar{\mathcal{R}}$ for the schemes with optimum rate design
and a constant rate.
## VI Conclusions
Performance of HARQ-IR scheme operating over time-correlated Rayleigh fading
channels has been analyzed in this paper. By using polynomial fitting
technique, the PDF of the accumulated mutual information has been derived,
which enables the derivation of outage probability, average number of
transmission and LTAT in closed-forms. It has been found that the outage
probability can be written as a weighted sum of outage probabilities
corresponding to a number of Gamma RVs. Moveover, diversity order has been
analyzed and it has been revealed that full diversity can be achieved even
under time-correlated fading channels. The impact of time correlation in the
channels has also been investigated. It has been demonstrated that time
correlation has negative effect on the performance. Under highly correlated
channels, more transmissions would not necessarily lead to a higher LTAT and a
few transmissions may be sufficient. Finally, the analytical results have
enabled the optimal design of HARQ-IR scheme and optimal rate design has been
particularly discussed to demonstrate the significance of our analytical
results on HARQ-IR.
## Appendix A Derivation of moments ${\mathcal{M}}(i)$
By definition, the $i$th moment of ${I_{K}^{IR}}$ is expressed as
${\cal
M}\left(i\right)=\int_{{x_{1}}=0}^{\infty}\cdots\int_{{x_{K}}=0}^{\infty}{{\left({\sum\limits_{k=1}^{K}{{{\log}_{2}}\left({1+{x_{k}}}\right)}}\right)}^{i}}\\\
\times{f_{{{\bf{\gamma}}_{1:K}}}}\left({{x_{1}},\cdots,{x_{K}}}\right)d{x_{1}}\cdots
d{x_{K}}.$ (82)
By substituting (III-B) into (82), it follows that
${\cal
M}\left(i\right)=\prod\limits_{k=1}^{K}{\frac{1}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)}}}\int\limits_{t=0}^{\infty}{{\rm{e}}^{-\left({1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}\right)t}}\\\
\times{\int\limits_{0}^{\infty}{\cdots\int\limits_{0}^{\infty}{{{\left({\sum\limits_{k=1}^{K}{{{\log}_{2}}\left({1+{x_{k}}}\right)}}\right)}^{i}}\prod\limits_{k=1}^{K}{e^{-\frac{{{x_{k}}}}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)}}}}}}}\\\
\times{}_{0}{F_{1}}\left({;1;\frac{{{x_{k}}{\lambda_{k}}^{2}t}}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)^{2}}}}\right)d{x_{1}}\cdots
d{x_{k}}dt.$ (83)
Using binomial expansion, (83) can be rewritten as
${{\cal
M}\left(i\right)=\prod\limits_{k=1}^{K}{\frac{1}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)}}}\sum\limits_{{i_{1}}=0}^{i}{\sum\limits_{{i_{2}}=0}^{i-{i_{1}}}{\cdots\sum\limits_{{i_{K-1}}=0}^{i-\sum\limits_{l=1}^{K-2}{{i_{l}}}}}C_{i}^{{i_{1}}}}}\\\
\times C_{i-{i_{1}}}^{{i_{2}}}\cdots
C_{i-\sum\limits_{l=1}^{K-1}{{i_{l}}}}^{{i_{K}}}\int\limits_{t=0}^{\infty}{{\rm{e}}^{-\left({1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}\right)t}}\\\
\times\prod\limits_{k=1}^{K}\int\limits_{0}^{\infty}\left({{\log}_{2}}\left({1+{x_{k}}}\right)\right)^{{i_{k}}}{e^{{-\frac{{{x_{k}}}}{{2{\sigma_{k}^{\prime}}{{}^{2}}{{\left({1-{\lambda_{k}}^{2}}\right)}}}}}}}\\\
\times{{{}_{0}{F_{1}}\left({;1;\frac{{{x_{k}}{\lambda_{k}}^{2}t}}{{2{\sigma_{k}^{\prime}}{{}^{2}}\left({1-{\lambda_{k}}^{2}}\right)^{2}}}}\right)}d{x_{k}}}dt.$
(84)
Then making change of variables yields
${\cal
M}\left(i\right)=\frac{{i!}}{{1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}}\sum\limits_{\sum\limits_{l=1}^{K}{i_{l}}=i,{i_{l}}\geq
0}{\frac{1}{{{i_{1}}!{i_{2}}!\cdots{i_{K}}!}}}\\\
\times\int\limits_{t=0}^{\infty}{{{\rm{e}}^{-t}}\prod\limits_{k=1}^{K}{\int\limits_{0}^{\infty}{{e^{-y}}{{\log}_{2}}^{{i_{k}}}\left({1+{w_{k}}y}\right){}_{0}{F_{1}}\left({;1;{\varpi_{k}}yt}\right)}dy}}dt$
(85)
where
${\varpi_{k}}=\frac{{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}{{1+\sum\limits_{l=1}^{K}{\frac{{{\lambda_{l}}^{2}}}{{1-{\lambda_{l}}^{2}}}}}}$,
${w_{k}}=2{\sigma_{k}^{\prime}}^{2}\left({1-{\lambda_{k}}^{2}}\right)$.
By applying Gaussian quadrature into (84) [33], it produces
${\mathcal{M}}\left(i\right)\approx\frac{{i!}}{{1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}}\sum\limits_{\sum\limits_{l=1}^{K}{i_{l}}=i,{i_{l}}\geq
0}{\frac{1}{{{i_{1}}!{i_{2}}!\cdots{i_{K}}!}}}\\\
\times\int\limits_{t=0}^{\infty}{{{\mathop{\rm
e}\nolimits}^{-t}}\prod\limits_{k=1}^{K}{\sum\limits_{{q_{k}}=1}^{{N_{Q}}}{{\varrho_{{q_{k}}}}}{{\log}_{2}}^{{i_{k}}}\left({1+{w_{k}}{\xi_{{q_{k}}}}}\right){}_{0}{F_{1}}\left({;1;{\varpi_{k}}{\xi_{{q_{k}}}}t}\right)}}dt\\\
=\frac{{i!}}{{1+\sum\limits_{k=1}^{K}{\frac{{{\lambda_{k}}^{2}}}{{1-{\lambda_{k}}^{2}}}}}}\sum\limits_{\sum\limits_{l=1}^{K}{i_{l}}=i,{i_{l}}\geq
0}{\frac{1}{{{i_{1}}!{i_{2}}!\cdots{i_{K}}!}}\sum\limits_{{q_{k}}\in\left[{1,{N_{Q}}}\right],k\in\left[{1,K}\right]}}\\\
{\prod\limits_{k=1}^{K}{{\varrho_{{q_{k}}}}{{\log}_{2}}^{{i_{k}}}\left({1+{w_{k}}{\xi_{{q_{k}}}}}\right)}\int\limits_{t=0}^{\infty}{{{\mathop{\rm
e}\nolimits}^{-t}}\prod\limits_{k=1}^{K}{{}_{0}{F_{1}}\left({;1;{\varpi_{k}}{\xi_{{q_{k}}}}t}\right)}}dt}$
(86)
where $N_{Q}$ is the quadrature order, and the weights $\varrho_{q_{k}}$ and
abscissas ${{\xi_{q_{k}}}}$ are tabulated in [30, Table 25.9]. The residue
error becomes negligible if $N_{Q}$ is sufficiently large. By using the
following formula
$\displaystyle{\int\limits_{z=0}^{\infty}{{e^{-z}}\prod\limits_{l=1}^{K}{{}_{0}{F_{1}}\left({;1;{a_{l}}z}\right)}}dz}$
$\displaystyle=\int\limits_{z=0}^{\infty}{{e^{-z}}\prod\limits_{l=1}^{K}{\frac{1}{{2\pi{\rm
j}}}\int\limits_{{{\cal
C}_{l}}}{\frac{{\Gamma\left(s_{l}\right)}}{{\Gamma\left({1-s_{l}}\right)}}{{\left({-{a_{l}}z}\right)}^{-{s_{l}}}}d{s_{l}}}}}dz$
$\displaystyle={{\left({\frac{1}{{2\pi{\rm
j}}}}\right)}^{K}}\int\limits_{{{\cal C}_{1}}}{{\cdots\int\limits_{{{\cal
C}_{K}}}{\frac{{\Gamma\left({1-\sum\limits_{l=1}^{K}{{s_{l}}}}\right)\prod\limits_{l=1}^{K}{\Gamma\left({{s_{l}}}\right)}}}{{\prod\limits_{l=1}^{K}{\Gamma\left({1-{s_{l}}}\right)}}}}}}$
$\displaystyle\times{{\left({-{a_{1}}}\right)}^{-{s_{1}}}}\cdots{{\left({-{a_{K}}}\right)}^{-{s_{K}}}}d{s_{1}}\cdots
d{s_{K}}$
$\displaystyle=\Psi_{2}^{\left(K\right)}\left({1;\underbrace{1,1,\cdots,1}_{K-terms};{a_{1}},{a_{2}},\cdots,{a_{K}}}\right)$
(87)
where ${\rm j}=\sqrt{-1}$, and $\Psi_{2}^{\left(K\right)}(;;)$ denotes
confluent form of Lauricella hypergeometric function [31, Definition A.20],
[32], the final expression for (86) then follows as (III-B).
## Appendix B Proof of Invertibility of Matrix $\bf{A}$
To complete the proof, it suffices to show that $\bf{A}$ is a positive
definite matrix. For arbitrary $(N+1)\times 1$ real vector $\bf u$, a function
$\bf{u^{\rm T}}{\bf A}{{\bf{u}}}$ can be derived as
${\bf{u^{\rm T}}}{\bf
A}{{\bf{u}}}=\int_{0}^{\infty}{\varphi\left(x\right)\left({{\bf{u}}^{\rm{T}}\boldsymbol{\mathcal{P}}(x)}\right)^{2}dx}.$
(88)
It is straightforward that
$\int_{0}^{\infty}{\varphi\left(x\right)\left({{\bf{u}}^{\rm{T}}\boldsymbol{\mathcal{P}}(x)}\right)^{2}dx}\geq
0$ and the equality holds if and only if $\bf{u}=\bf{0}$. It follows that
$\bf{A}\succ 0$, thus $\bf{A}$ is non-singular, that is, $\bf{A}$ is
invertible.
## Appendix C Proof of Property 1
The proof is completed by contradiction. First, denote two measures as
$d\mu_{1}(x)$ and $d\mu_{2}(x)$, and their corresponding monic orthogonal
polynomials as $\boldsymbol{\mathcal{P}}_{1}(x)$ and
$\boldsymbol{\mathcal{P}}_{2}(x)$, respectively. As aforementioned, given the
measure $d\mu(x)$, the optimal solution to the problem of (24) is unique. We
then represent the unique optimal solutions corresponding to the two measures
$d\mu_{1}(x)$ and $d\mu_{2}(x)$ as $\boldsymbol{\eta}_{1}$ and
$\boldsymbol{\eta}_{2}$, respectively. Meanwhile, the corresponding optimal
polynomials are denoted as
$\hat{\psi}_{N}^{(1)}(x)={\boldsymbol{\eta}_{1}}^{\rm
T}\boldsymbol{\mathcal{P}}_{1}(x)$ and
$\hat{\psi}_{N}^{(2)}(x)={\boldsymbol{\eta}_{2}}^{\rm
T}\boldsymbol{\mathcal{P}}_{2}(x)$, while the corresponding minimal MSEs are
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{1}|d\mu_{1}(x),N)$ and
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{2}|d\mu_{2}(x),N)$.
Assume that
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{1}|d\mu_{1}(x),N)\neq{{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{2}|d\mu_{2}(x),N)$,
which implies that $\hat{\psi}_{N}^{(1)}(x)\neq\hat{\psi}_{N}^{(2)}(x)$.
Without loss of generality, we consider the case with
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{1}|d\mu_{1}(x),N)>{{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{2}|d\mu_{2}(x),N)$.
It is known that any polynomial $\hat{\psi}_{N}(x)\in\mathbb{P}_{N}$ has a
unique coordinate $\boldsymbol{\eta}$ regarding to a given basis of orthogonal
polynomials. Thus by taking $\boldsymbol{\mathcal{P}}_{1}(x)$ as a basis, we
have $\hat{\psi}_{N}^{(2)}(x)={\boldsymbol{\eta}_{3}}^{\rm
T}\boldsymbol{\mathcal{P}}_{1}(x)$, where ${\boldsymbol{\eta}}_{3}$ is the
unique coordinate associated with $\hat{\psi}_{N}^{(2)}(x)$ on the basis of
$\boldsymbol{\mathcal{P}}_{1}(x)$. Considering the unique optimality of
${\boldsymbol{\eta}}_{1}$ given the measure $d\mu_{1}(x)$, we have
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{1}|d\mu_{1}(x),N)<{{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{3}|d\mu_{1}(x),N)={{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{2}|d\mu_{2}(x),N)$.
This is contradictory with our assumption. Therefore,
${{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{1}|d\mu_{1}(x),N)={{\mathcal{S}}_{mse}}({\boldsymbol{\eta}}_{2}|d\mu_{2}(x),N)$
and $\hat{\psi}_{N}^{(1)}(x)=\hat{\psi}_{N}^{(2)}(x)$. Then the proof follows.
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| Zheng Shi received the B.S. degree in communication engineering from Anhui
Normal University, China, in 2010 and the M.S. degree in communication and
information system from Nanjing University of Posts and Telecommunications
(NUPT), China, in 2013. Since Sep. 2013, he has been a Ph.D. student in
Department of Electrical and Computer Engineering, University of Macau, Macao.
His research interests include hybrid automatic repeat request (HARQ)
protocols, cooperative communications, full-duplex communications, massive
MIMO, and heterogeneous wireless networks.
---|---
| Haichuan Ding is currently a Ph.D. student at the University of Florida. He
received the B.Eng. and M.S. degrees in electrical engineering from Beijing
Institute of Technology (BIT), Beijing, China, in 2011 and 2014, respectively.
From 2012 to 2014, he was with the Department of Electrical and Computer
Engineering, University of Macau, as a visiting student. During his M.S.
studies, he mainly worked on the analysis of HARQ techniques using the tools
of stochastic geometry. His current research is focused on cognitive radio
networks and security and privacy in distributed systems.
---|---
| Shaodan Ma received her double Bachelor degrees in Science and Economics,
and her Master degree of Engineering, from Nankai University, Tianjin, China.
She obtained her Ph. D. degree in electrical and electronic engineering from
the University of Hong Kong, Hong Kong, in 2006. After graduation, she joined
the University of Hong Kong as a Postdoctoral Fellow. Since August 2011, she
has been with the University of Macau and is now an Associate Professor there.
She was a visiting scholar in Princeton University in 2010 and is currently an
Honorary Assistant Professor in the University of Hong Kong. Her research
interests are in the general areas of signal processing and communications,
particularly, transceiver design, resource allocation and performance
analysis.
---|---
| Kam-Weng Tam (S’91-M’01-SM’05) received the B.Sc. and joint Ph.D. degrees
in electrical and electronics engineering from the University of Macau, Taipa,
Macao, China, and the University of Macau and Instituto Superior Técnico
(IST), Technical University of Lisbon, Lisbon, Portugal, in 1993 and 2000,
respectively. From 1993 to 1996, he was with the Instituto de Engenharia de
Sistemas e Computadores (INESC), Lisbon, Portugal, where he participated in
research and development on a broad range of applied microwave technologies
for satellite communication systems. From July 2000 to December 2001, he was
the Director of the Instituto de Engenharia de Sistemas e Computadores
(INESC)-Macau. In 2001, he cofounded the microelectronic design house Chipidea
Microelectrónica, Macau, China, where until 2003 he was the General Manager.
Since 1996, he has been with the University of Macau, where he is currently a
Professor and the Associate Dean (Research and Graduate Studies) with the
Faculty of Science and Technology. He has authored or coauthored over 100
journal and conference papers. His research interests have concerned
multifunctional microwave circuits, RFID, UWB for material analysis and
terahertz technology. Dr. Tam was interim secretary for the establishment of
the Macau Section in 2003. He supervised two IEEE Microwave Theory and
Techniques Society (IEEEMTT-S) Undergraduate Scholarship recipients in 2002
and 2003. He was founder of the IEEE Macau AP/MTT Joint Chapter in 2010 and
was chair in 2011–2012. He was a member of the organizing committees of 21
international and local conferences including co-chair of APMC2008, co-chair
of the Technical Program, IEEE MTT-S International Microwave Workshop Series
on Art of Miniaturizing RF and Microwave Passive Components (2008), and co-
chair of ISAP2010.
---|---
|
# Hardware Trojan Threats to Cache Coherence in Modern 2.5D Chiplet Systems
Gino A. Chacon, Charles Williams, Johann Knechtel, Ozgur Sinanoglu, and Paul
V. Gratz G. A. Chacon, C. Williams, and P. V. Gratz are with Texas A&M
University (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]).J. Knechtel and O. Sinanoglu are with New York University
Abu Dhabi (e-mail<EMAIL_ADDRESS>[email protected]).
###### Abstract
As industry moves toward chiplet-based designs, the insertion of hardware
Trojans poses a significant threat to the security of these systems. These
systems rely heavily on cache coherence for coherent data communication,
making coherence an attractive target. Critically, unlike prior work, which
focuses only on malicious packet modifications, a Trojan attack that exploits
coherence can modify data in memory that was never touched and is not owned by
the chiplet which contains the Trojan. Further, the Trojan need not even be
physically between the victim and the memory controller to attack the victim’s
memory transactions. Here, we explore the fundamental attack vectors possible
in chiplet-based systems and provide an example Trojan implementation capable
of directly modifying victim data in memory. This work aims to highlight the
need for developing mechanisms that can protect and secure the coherence
scheme from these forms of attacks.
## 1 Introduction
Computing systems are moving toward 2.5D designs that source various hard IPs,
called chiplets, from multiple vendors and integrate them using an interposer.
Industry has demonstrated that 2.5D designs lower manufacturing costs,
enabling further scaling post-Moore’s Law [1]. Future 2.5D designs may
leverage standards such as Compute Express Link [2] to interoperate via a
shared memory system. While 2.5D designs provide many benefits, we show they
also increase the risk of Trojan attacks, specifically targeting the coherence
system. Here, we demonstrate several novel Trojans attacking cache coherence
in 2.5D designs. We illustrate the risks for these systems and hope to excite
the architecture community to address these risks. Though we focus on 2.5D
integrated systems, note that these attacks also apply to general cache-
coherent systems integrating closed-source or hard IP blocks from various
vendors.
_Hardware Trojans_ , or Trojans for short [3], are a threat in which an
attacker infiltrates some level of the design or fabrication process to insert
malicious circuitry into a design. Trojans can cause disastrous system
failures via confidentiality, integrity, and/or availability violations. Prior
work has shown that Trojans can leak data from memory [4], disrupt
cryptographic security features [5], and induce denial-of-service attacks [6].
As industry moves towards 2.5D designs integrating multiple vendor chiplets,
specific chiplets used in building these systems may be untrustworthy. Even if
the IP vendor is trustworthy, the manufacturing process may not be, leading to
infiltration and the insertion of Trojans.
In 2.5D designs, memory coherence is crucial to allow each component and
chiplet to maintain an up-to-date view of the system’s memory. We identify
this system as an ideal target for Trojan attacks as coherence mechanisms
control how all components communicate data updates. Existing coherence
schemes do not enforce existing virtual/physical memory permissions, thus, a
Trojan connected to the coherence scheme can directly manipulate any memory
region in the full system regardless of memory permissions or physical
location. Unlike prior packet-level NoC attacks, Trojans on cache coherence do
not need to be physically on the path between the victim and the memory
controller to launch effective attacks. Despite this attractiveness, there is
a lack of works deeply exploring coherence exploits and their defense in 2.5D
systems or otherwise.
Here, we propose several new Trojan attacks that leverage the coherence system
protocol to maliciously manipulate the victim process’ memory. We first
describe a set of new fundamental attacks that a Trojan can mount on coherence
systems, _passive reading_ , _masquerading_ , _modifying_ , and _diverting_
attacks, according to Basak _et al._ ’s taxonomy [7]. Here we assume an
attacker implements these coherence system attacks in hardware through
compromised design or manufacturing. While each of these attacks individually
violates a system’s security, we further show that adversaries can orchestrate
them together to perform complex _Forging_ attacks that modify _any_ process’
memory. These purely hardware attacks cannot be thwarted by contemporary
software defense mechanisms since all exploited coherence interactions are
transparent to software and legal within the coherence protocol. Further, no
prior work considers such attacks on coherence systems, neither in the context
of 2.5D systems with chiplets nor traditional 2D systems.
_Contributions._ This work provides new insights into how Trojans can
manipulate coherence systems to violate the security of a chiplet system. We
present a simulated example of a substantial attack that can directly
manipulate memory in an address space other than that of the compromised
chiplet. This work makes the following contributions:
* •
We present potential attack stages available to a Trojan designers exploiting
coherence systems.
* •
We demonstrate how to use these fundamental stages to orchestrate a complex
Trojan attack in a chiplet-based system.
* •
We provide suggestions for future work on hardening modern chiplet designs.
## 2 Background
### 2.1 Hardware Trojans
Hardware Trojans are malicious hardware inserted by an attacker during a
device’s design or manufacturing process. Chiplet-based devices have a complex
multistage design flow that can be compromised at many levels, especially as
multiple 3rd party vendors emerge to provide chiplets from separate foundries
and design teams. This design flow makes verification much more difficult and
expensive than traditional System-on-Chip (SoC) manufacturing.
Detecting Trojans is challenging as chiplet-based systems contain multiple
complex IPs sourced from various vendors. Testing a chiplet’s functionality
may occur during the manufacturing or integration stage, which requires a
reference model or device [3]. However, if the 3rd party IP’s source is
untrustworthy, the reference model itself may incorporate the Trojan, or the
IP may camouflage the Trojan as a correct implementation. Attackers can
conceal a Trojan by only allowing it to trigger under specific conditions. For
example, the Trojan we describe in Sec. 3.2 only activates when it observes
references to a specific address. These properties make the Trojan difficult
to detect by simply testing the functionality of the chiplet. For this work,
we assume that an attacker infiltrates some stage within the design or
manufacturing process to target the system’s coherence mechanisms.
Prior art focuses on infiltrating the NoC of a target design to cause
deadlocks [6], leak information [4], or disrupt security features [5].
However, NoC-based attacks require the Trojan to directly attack NoC packets,
limiting the Trojan to only packets traversing a particular path in the NoC.
Prior attacks would not work in a 2.5D integrated system because attacking
chiplets are not on the path between victim chiplets and the memory
controller.
### 2.2 Coherence Protocols
Multi-processor systems incorporate cache coherence protocols to ensure data
coherency across processors’ private caches. The coherence system has a
complete vantage point and control over the memory system. All communication
between cores and main memory follows the coherence protocol, making it an
ideal target for a Trojan co-located with a processor’s private caches. At
this location, a Trojan can snoop on coherence messages produced by other
processors, manipulate those messages, or generate messages without incurring
exceptions and remaining invisible to software running on the system. Further,
coherence schemes do not enforce virtual/physical memory permissions, thus any
Trojan connected to the coherence scheme can directly manipulate any memory
region in the full system, without regard to memory permissions or physical
location.
We target the _MOESI Hammer_ coherence protocol, a hybrid broadcast-directory
system used in many AMD processors [8]. Though our focus is MOESI Hammer, our
attack scenarios can easily be ported to other coherence schemes. In schemes
without broadcast, the Trojan simply needs to register “S” or “X” state with
the directory to ensure that update requests on the given line are seen.
## 3 Trojans Targeting Coherence Systems
Here we propose a new set of “basic” Trojan attacks on the coherence scheme
that follows the general, not coherence specific, taxonomy for Trojan attacks
by Basak _et al._ [7]: _passive reading_ , _masquerading_ , _modifying_ , and
_diverting_. These basic attacks can adversely affect the system but cannot
provide an attacker with control the memory system. We then propose a novel,
more sophisticated and powerful ‘ _‘Forging”_ attack to modify data belonging
to another core, even on a different chiplet than that holding the Trojan.
Target system: We demonstrate our attacks on a 64-core processor with eight
chiplets, eight cores per chiplet, based on the Rocket-64 architecture [9].
Each core has private L1 and L2 caches with a unified cache controller. An NoC
connects each chiplet and four memory controllers that maintain a portion of
the global state directory. The cache controllers generate coherence messages
that are injected into the NoC as network packets.
### 3.1 Basic Trojan Attacks on the Coherence Systems
Figure 1 illustrates our proposed basic coherence attacks. We assume a Trojan
is placed at a core’s cache controller and can intercept coherence messages
from the network interface ahead of the state directory. While these attacks
target the the MOESI Hammer hybrid protocol, the basic principles of the
attacks, are consistent with any coherence protocol.
(a) Passive Reading: Trojan passively observes write traffic for other
chiplets. (1) Misses from Chiplet A cause (2) broadcast invalidations to all
chiplets; (3) Trojan snoops invalidation addresses.
(b) Masquerading: Trojan acts as another core. (1) Miss causes GETX to
directory; (2) broadcast invalidations to each chiplet; (3) Trojan blocks
local observation, replies with different core ID; (4) requesting core
proceeds, leaving local caches incoherent.
(c) Modifying: Trojan modifies a message to achieve incoherent state. (1)
Chiplet A sends GETS to directory; (2) directory forwards request to Trojan’s
core which has line in ‘E’ state. Trojan blocks GETS and (3) replies with GETX
to requestor, (4) invalidating Chiplet A’s cache entry, leaving attacker in
control of another cache’s contents.
(d) Diverting: Trojan diverts invalidation requests. (1) Chiplet A sends GETX
to the directory; (2) directory broadcasts invalidations. (3) Trojan blocks
message and diverts a request to another core, (4) which responds with a
negative-acknowledge or acknowledgment resulting in (5) the directory allowing
original requestor to continue.
Figure 1: Coherence Trojan attacks in interposer-based systems in which
Chiplet A is the victim of a Trojan attack from Chiplet B.
Passive Reading (Fig. 1(a)): Trojans passively reading (snooping) observe
incoming coherence messages from the chiplet’s network-on-chip (NoC) sub-
system as they reach the L2’s state directory. The Trojan may buffer messages,
identify specific request patterns, and facilitate a covert communication
channel. The Trojan does not affect the system’s state but may trigger a more
complex Trojan.
Masquerading (Fig. 1(b)): Masquerading (spoofing) occurs when a Trojan
modifies the packet’s sender field such that the packet appears to originate
from a different core. If the target packet is a request, such an attack can
result in a deadlock since all responses from the directory or other cores are
sent to the incorrect core. If the target packet is a response, the Trojan may
block it and respond with an acknowledgment that appears to be from a
different core, resulting in an incoherent memory state.
Modification (Fig. 1(c)): Such attacks occur when the Trojan directly modifies
the message type of a coherence message. This attack may result in a deadlock
since the Trojan may cause the memory controller’s directory to assume the
data is in one state, due to a modified packet, while the local directory
holds the data in a different—incorrect—state.
Diverting (Fig. 1(d)): Trojans can launch diverting attacks by blocking the
local state directory from observing a request and then resending the request
with a different destination field. This results in the compromised core and
the original requestor becoming incoherent with respect to the rest of the
memory system.
Limitations of Basic Attacks: Any of the above attacks can individually result
in incoherence or deadlocks but cannot directly manipulate a victim’s data.
Only combining these attacks allows for a more complex set of attack vectors
that would enable a Trojan to pose a significant security threat.
### 3.2 Trojan Design
In the remainder of this paper, we propose the _Forging Attack_ , a novel
attack that manipulates legal coherence transactions to allow a Trojan to
write to a target address in a different process operating in a different
chiplet. The compromised chiplet containing the Trojan does not have access to
the victim process’ address space but can observe coherence interactions
broadcasted by the MOESI Hammer protocol. Since the Trojan resides between the
network interface and a core’s state directory in the compromised chiplet
(Figure 2), the Trojan has a complete view of incoming or outgoing coherence
messages, enabling it to block the core from observing specific interactions.
The Trojan holds few registers to track the target data’s current state
relative to the Trojan. These registers imitate the core’s state directory to
ensure the Trojan correctly responds to the global directory.
Figure 2: Our proposed Trojan attack on the coherence system that forges
messages to gain control and modify specific addresses accessed by a process
operating in a different chiplet. The attack executes in two phases. The first
phase (top) allows the Trojan to gain control of the target address and the
second phase (bottom) enables it to mimic the steps required to write back
maliciously formed data to main memory.
### 3.3 Forging Attack Demonstration
Here we assume the Trojan has a predefined target address. In a real-world
scenario, the Trojan can observe coherence messages broadcasted to the
compromised chiplet of the network to select its target. The coherence
protocol requires that the global directory sends invalidation messages each
time a core sends a write request, or GETX, to a line that it does not own.
The invalidation broadcast removes all copies in other cores before updating
the line with new data.
The Trojan operates in two phases. During the first phase, the Trojan deceives
the global directory into giving the Trojan access to the data. During the
second phase, the Trojan follows the protocol’s required transactions to write
to the target address, which the victim will later read. The interactions
caused by the Trojan in both phases are legal from the perspective of the
global directory. Furthermore, they are transparent to the software executing
in the victim process and all other security software in the system.
Phase 1, Acquiring Access to Target Data: Figure 2(top) illustrates the
initial steps the Trojan must take to gain access permissions to the target
address before it can maliciously write to it. 1 The Trojan observes coherence
requests, waiting for a specific address to trigger the attack. 2 The Trojan
generates a malicious GETX packet for the target address. 3 The directory
receives the GETX request, broadcasts an invalidation to all cores, and waits
for all cores to send acknowledgments (ACKs). 4 The directory forwards the
data and all ACKs to the compromised core. 5 The Trojan blocks the local
directory from seeing any response from the directory or cores, waiting to
receive all ACKs. 6 Once all ACKs are received, the Trojan can access the
data, since the directory considers the compromised core as the data owner.
Phase 2, Writing Malicious Data: Once access permissions are acquired, the
global directory assumes that the Trojan’s core is the exclusive owner of the
data. Figure 2(bottom) illustrates Phase 2 of the attack. This phase allows
the Trojan to mimic the legal operations that enable writing to main memory as
if the core was evicting the data after modifying it. The steps of the attack
are as follows: 1 Once the Trojan receives the final ACK, the requests to the
target address are unblocked. 2 The Trojan immediately sends a PUTX to the
directory to indicate that it is “evicting” modified data. 3 The directory
responds with a WRITEBACK_ACKNOWLEDGEMENT, allowing the Trojan to proceed with
“evicting” the maliciously changed dirty data. 4 The Trojan responds to the
WRITEBACK_ACKNOWLEDGEMENT with a WRITEBACK_EXCLUSIVE_DIRTY response containing
the malicious data. 5 The data is written to memory.
### 3.4 Results
We evaluate the Trojans in gem5, targeting a victim which iterates over an
array to set each value to ’1’ or ’0’ and then reads the array to compute a
sum. Figure 3(a) shows the data the victim process observes without the Trojan
enabled. The victim writes ‘0’ or ‘1’ to various locations in its data array
and then re-reads these locations, seeing the expected data values. Figure
3(b) shows the data the victim receives when it attempts to read the data
array after writing to all indexes. The _Forging Attack_ successfully modifies
the data array’s first value, which the victim then reads unknowingly of the
manipulation. This demonstrates our Trojan can manipulate the coherence system
to modify data that another application is operating on, even without
requiring shared memory access.
Unlike prior work, which focuses on Trojans modifying packets [10, 11, 12,
13], we leverage the coherence mechanism itself to modify data in memory that
is _never touched_ and _not owned_ by the chiplet containing the Trojan. Our
attack does not require the data to be in the compromised core’s caches.
Generating and blocking specific coherence messages allows the Trojan to
mislead the global directory about the state and ownership of the targeted
data.
(a) Data received by the victim when the Trojan is not activated. The
application reads an alternating sequence of ‘1’ and ‘0.’
(b) Data received after the Trojan has completed its attack. The first entry
in the array is now set to ‘5,’ instead of the expected ‘1.’
Figure 3: Data values as seen by the victim.
## 4 Conclusion and Future Defenses
As industry moves toward chiplet-based designs, hardware Trojans pose a
significant threat to security. These systems will rely heavily on coherence
to ensure that data remains up-to-date in all components, making the coherence
protocol an attractive target. Critically, unlike prior work, which focuses
only on packet modifications, we show that a coherence-centric Trojan attack
can modify memory that is not even owned by the compromised chiplet. We
provide an example of a complex Trojan implementation that modifies memory
without relying on malicious software components. This work highlights the
need for mechanisms to protect the coherence scheme from these novel attacks.
Detecting Trojans during chiplet manufacturing is challenging considering the
complexity of individual IPs. Defenses against hardware Trojan exploiting a
system’s coherence mechanisms could implement runtime monitoring to identify
malicious behavior originating from a particular chiplet. A benefit of 2.5D
integration is that the components are usually sourced from vendors and then
integrated onto an interposer layer at a separate foundry than each IP’s
manufacturing [9]. Requiring an interposer’s manufacturing and integration by
a trustworthy facility could allow the 2.5D interposer to act as a hardware
root of trust that can embed security features. Embedding the security
features into the interposer layer could allow defenders to observe coherence
packets and securely control data flow freely. We plan to explore these themes
in our future work.
## References
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* [2] “Compute Express Link (CXL), www.computeexpresslink.org.” Accessed: 2022-05-18.
* [3] S. Bhunia and M. Tehranipoor, The Hardware Trojan War. Springer, 2018.
* [4] M. N. I. Khan, A. De, and S. Ghosh, “Cache-out: Leaking cache memory using hardware Trojan,” IEEE TVSLI, vol. 28, no. 6, pp. 1461–1470, 2020\.
* [5] M. Bidmeshki, G. R. Reddy, L. Zhou, J. Rajendran, and Y. Makris, “Hardware-based attacks to compromise the cryptographic security of an election system,” in IEEE ICCD, pp. 153–156, 2016.
* [6] M. Kim, S. Kong, B. Hong, L. Xu, W. Shi, and T. Suh, “Evaluating coherence-exploiting hardware Trojan,” in IEEE DATE, pp. 157–162, 2017\.
* [7] A. Basak, S. Bhunia, T. Tkacik, and S. Ray, “Security assurance for system-on-chip designs with untrusted IPs,” IEEE TIFS, vol. 12, no. 7, pp. 1515–1528, 2017.
* [8] P. Conway, N. Kalyanasundharam, G. Donley, K. Lepak, and B. Hughes, “Cache hierarchy and memory subsystem of the AMD Opteron processor,” IEEE Micro, vol. 30, no. 2, pp. 16–29, 2010.
* [9] J. Kim et al., “Architecture, chip, and package co-design flow for 2.5D IC design enabling heterogeneous IP reuse,” in ACM/IEEE DAC, pp. 1–6, 2019.
* [10] D. M. Ancajas, K. Chakraborty, and S. Roy, “Fort-NoCs: Mitigating the threat of a compromised NoC,” in ACM/EDAC/IEEE DAC, pp. 1–6, 2014.
* [11] T. Boraten and A. K. Kodi, “Mitigation of denial of service attack with hardware Trojans in NoC architectures,” in IEEE IPDPS, pp. 1091–1100, 2016.
* [12] M. H. Khan, R. Gupta, J. Jose, and S. Nandi, “Dead flit attack on NoC by hardware Trojan and its impact analysis,” in ACM NoCArc, pp. 10–15, 2021\.
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|
11institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, 1090 GE Amsterdam, The Netherlands
# A systematic search for double eclipsing binaries in Zwicky Transient
Facility data
T. Vaessen<EMAIL_ADDRESS>J. van Roestel<EMAIL_ADDRESS>
(Received 31 October, 2023; accepted 11 December, 2023)
###### Abstract
Context. Double eclipsing binaries are gravitationally bound quadruple systems
in a ‘2+2’ configuration where both of the binaries are eclipsing. These
systems are interesting objects to better understand stellar formation, to
investigate the dynamical interaction between the two binary systems or to
study certain stages of stellar evolution, such as common-envelope events or
Type Ia Supernovae.
Aims. With this work, we aim to determine if double eclipsing binaries can be
found using ZTF data and what the difficulties are in doing so. Secondly, we
aim to significantly increase the number of known double eclipsing systems and
determine how this sample differs from samples of double eclipsing binaries
found with other telescopes.
Methods. We develop a new method to systematically search for double eclipsing
binaries in sparsely sampled light curves. For this we use box-least-squares
(BLS) to search for the period of the first binary in the system. We then
remove that signal from the light curves, and search the residual light curve
again with BLS to find the second period. We applied this method to ZTF light
curves of 575 526 eclipsing binaries known in the Gaia eclipsing binary
catalogue.
Results. We report the discovery of 198 new double eclipsing binary systems.
The shortest and longest orbital periods of the newly detected systems are
0.11 days to 323 days respectively.
Conclusions. We successfully implemented a method that systematically searches
for double eclipsing binary systems in sparsely sampled data. In total 198 new
double eclipsing binary systems have been found in 575 526 light curves
($\approx$ 0.034$\%$). The ZTF sample typically contains more short period
binaries compared to the TESS sample, but is also able to find systems with
longer periods than what is currently known. We expect that at least three to
four times more quadruples can be found by applying this method to all ZTF
stellar light curves, by increasing the number of data points as a result of
longer observations, and by implementing an automatic detection mechanism that
replaces visual inspection.
###### Key Words.:
(Stars:) binaries: eclipsing - (Stars:) binaries (including multiple): close -
Methods: data analysis
## 1 Introduction
Most stars in the Universe can be found in gravitationally bound binaries or
higher-order hierarchies (Tokovinin, 2014). It is estimated that approximately
$4\%$ of all solar-type stars can be found in quadruple systems (Tokovinin,
2021), either in a 3+1 or 2+2 configuration. A small fraction of 2+2 quadruple
systems are double eclipsing binaries, in which two eclipsing binaries also
orbit each other (Zasche et al., 2019). If the motion of the inner pair is not
strongly perturbed by the outer companions, the motions of the stars are
approximated by stable Keplerian orbits and can survive for a long time
(Tokovinin, 2014).
Compact double eclipsing binaries are interesting objects of study to better
understand stellar formation or to investigate the dynamical interaction
between the two binary systems. An important example of such a dynamical
interaction is the Kozai-Lidov mechanism where, on long timescales, a periodic
exchange between the binaries’ eccentricities and inclination can take place
(Kozai, 1962; Lidov, 1962). Furthermore, these objects serve as an
astrophysical laboratory when studying certain stages of stellar evolution,
such as common-envelope events or Type Ia Supernovae (Kostov et al., 2022a).
Therefore, the identification of more double eclipsing binaries will lead to a
larger sample that can be used to study the origin and life cycle of these
objects. Quadruples are intrinsically rare and require a favourable alignment
of both binaries with the observer in order to be detected. However, in the
case of fortunate alignment, the objects can be identified by periodic
eclipses in the light curve. Zasche et al. (2019) systematically searched the
OGLE-LMC data by visually inspecting the light curves of eclipsing binary
systems and found 72 systems of double eclipsing binaries. More recently,
light curves of the Transient Exoplanet Survey Satellite (TESS, Ricker et al.
2015) have been used to identify double eclipsing binaries in this way. Zasche
et al. (2022b) identified 116 double eclipsing binaries using TESS, and Kostov
et al. (2022a, b) detected in total 199 double eclipsing binaries, 18 of which
overlapped with Zasche et al. (2022b).
In this paper, we present the search for double eclipsing binaries in Zwicky
Transient Facility light curves. In Sect. 2 we present the target preselection
and the ZTF data. In Sect. 3 we discuss the method we used to identify double
eclipsing binaries. In Sect. 4 we present the analysis of the data. The
results are presented in Sect. 5 and we discuss them further in Sect. 6. We
end the paper with a summary and recommendations for future work in Sect. 7.
## 2 Data
As part of the Zwicky Transient Facility (ZTF), the Palomar 48-inch (P48)
telescope has been imaging the sky every night since 2018 (Graham et al.,
2019; Bellm et al., 2019; Dekany et al., 2020). Most of the time, ZTF uses the
$g$ and $r$ bands, but a small fraction of the observations are also made in
the $i$ band. The exposure times are predominantly 30 seconds for $g$ and $r$
and 60 or 90 seconds for $i$ exposures. The median limiting magnitude,
averaged over the lunar cycle, is $\approx 20.5$ in all three bands. We used
the PSF-photometry-based light curves which are automatically generated for
all persistent sources detected in the ZTF reference images (for a full
description see Masci et al., 2019) and are publicly available.
In order to avoid searching billions of ZTF light curves, we make a
preselection of objects. We process only light curves of objects that have
been identified as eclipsing binaries by Gaia (Mowlavi et al., 2023). Although
this preselection inevitably means that we will miss any system that has not
been identified as an eclipsing star by Gaia first, the advantage is that it
strongly reduces the number of light curves we need to search.
The Gaia eclipsing binary catalogue contains 2 184 477 sources distributed
over the whole sky and down to magnitude $G\approx 20.5$. While ZTF goes
slightly deeper than Gaia, ZTF only covers the sky from the celestial north
pole to a declination of $-30\deg$. This means there are 1 210 001 sources in
the Gaia eclipsing binary catalogue that are also in the ZTF footprint and are
fainter than $G=12$ (the approximate ZTF saturation limit). To download ZTF
light curves for these objects, we used
ZTFquery111https://github.com/MickaelRigault/ztfquery (Rigault, 2018).
## 3 Method
Light curves of double eclipsing binaries contain the signal of two eclipsing
binaries with different periods. If the light curve is well sampled (e.g. the
continuous light curves of TESS), multiple epochs sample each individual
eclipse and the individual eclipses are easy to identify. This makes the
identification of double eclipsing binaries relatively straightforward (Zasche
et al., 2019; Kostov et al., 2022a; Zasche et al., 2022a). However, if a light
curve is sparsely sampled (the typical time between observations is much
larger than the duration of an eclipse) individual eclipses are sampled by a
few or just a single epoch only and the light curve needs to be phase-folded
in order to identify eclipses. Therefore, searching for double eclipsing
binaries requires a slightly different approach for sparsely sampled light
curves like ZTF. Instead of identifying individual eclipses, we use the fact
that both eclipse signals are periodic and use period-finding methods to
identify both periods.
We implemented this method using box-least-squares (BLS, Kovács et al., 2002)
which is available through the Python package Astrobase (Bhatti et al., 2017).
First the data from the different bands ($g$ and $r$) were combined by
normalising the data in each band to its corresponding mean. Then we performed
the BLS period search on the light curve to find the first period, which we
define as period A. In our experience, this period was often half the orbital
period and thus the primary and secondary eclipse overlapped. To avoid this
overlap, we phase-folded the light curve to twice period A. To remove the
signal of period A, we binned the phase-folded light curve using an
empirically determined number of 100 bins. We then divided each flux
measurement in the light curve by the mean of its corresponding bin. After the
removal of the signal at period A, we applied the BLS method again to find the
second period; period B. At each point in this process, we saved the phase-
folded and binned light curves for visual inspection. An example of a (phase-
folded) light curve can be seen in Fig. 1.
## 4 Analysis
To ensure that the method worked correctly, we used a test dataset consisting
of 68 double eclipsing binaries found in prior research (Zasche et al., 2022b)
for which ZTF data is available. Furthermore, this test sample also helped to
establish a method for identifying double eclipsing binaries quantitatively.
For this, the relative peak height (RPH) was introduced as
$\mathrm{RPH}\equiv\frac{\mathrm{peak\
periodpower}-\overline{\mathrm{periodpower}}}{\sigma_{\mathrm{periodpower}}}$
(1)
where max periodpower is the BLS-periodogram peak associated with the best
period, $\overline{\mathrm{periodpower}}$ is the mean of powers and
$\sigma_{\mathrm{periodpower}}$ is the standard deviation. In Astrobase these
periodpowers are referred to as lspvals (Bhatti et al., 2017). For readers not
familiar with the BLS algorithm, an example of a periodogram is shown in the
appendix in Fig. 8. Here we see the periodogram power value of each orbital
period. Additionally, Astrobase offers a function to calculate the signal-to-
noise ratio (S/N) for each periodic signal (Bhatti et al., 2017). From the
test sample we empirically determined that a RPH ¿ 10 and S/N ¿ 10 may
indicate the presence of a _double_ eclipsing binary. As it was not feasible
to visually inspect all objects in the sample, we selected only candidates
with RPH ¿ 10 and S/N ¿ 10. This significantly reduced the number of light
curves that needed visual inspection to identify double eclipsing binary
candidates. These selection criteria are discussed further in Sect. 6.
Some objects were, based on a high relative peak height and S/N, initially
identified as quadruple candidate but appeared to exhibit a sinusoidal rather
than an eclipsing signal. These objects were associated with relatively short
periods $\mathrm{P_{B}}<0.30$ days, which could be an indication of a variable
star in the binary system that lies on the instability strip, such as $\delta$
Scuti stars (Pietrukowicz et al., 2020). Another common false positive is
associated with the imperfect removal of signal A. This would result in in
period B being the same as period A, or an integer multiple of period A. We
therefore removed any candidate for which period B was a multiple of period A.
When testing the method, it appeared that light curves containing high cadence
observations (nights with more than 30 epochs) had a negative impact on the
ability to find the right periods. This is probably caused by correlated noise
introduced by the calibration of the data. Therefore, observations with more
than 30 epochs per night were removed. A similar effect was observed by
outliers in high and low flux. For this reason, outliers in normalised flux
were removed using the interquartile range with a cut-off value of 5. Other
unreliable data, flagged by the ZTF data process pipeline, was also removed.
After this data pre-processing and cleaning, a minimum threshold of 500 data
points per light curve was set for analysis. This decreased the sample size
from 1 210 001 to 575 526 objects.
Based on the objects in the test sample, we set the period search range for
new objects between 0.1 and 150 days with a stepsize determined by the minimum
transit duration and observational period. We chose this lower limit as it is
not expected to find any objects with shorter orbital periods (Drake et al.,
2014). The upper limit was based on the five-year observational period of the
ZTF and the decreasing likelihood of finding longer periods. For the transit
duration, the BLS performed a search for a periodic signal lasting between
0.01 and 0.2 in phase.
Finally, we applied the pipeline to the 575 526 ZTF light curves. We used the
HELIOS computer cluster to process the light curves. The average processing
time per object is approximately 10 seconds.
We used the BLS output statistics to selected 5350 promising candidates. These
were visually inspected in order to confirm the double eclipsing nature of the
objects. At this point, we also determined, to the best of our abilities, what
the two orbital periods are, since the BLS algorithm typically finds half the
orbital period. Finally, we also checked if the photocenter position was
correlated with the eclipse periods to identify chance alignments of two
unrelated eclipsing binaries. The typical RMS positional accuracy is $\approx
0.1\arcsec$. No chance alignments were found in this manner.
Figure 1: ZTF light curves of Gaia dr3 source ID 2216420454386370176. Upper
panel: Normalised flux as a function of time. It shows two sets of eclipses
with $\mathrm{P_{A}}=1.072780$ days and $\mathrm{P_{B}}=0.404213$ days. Middle
panel: Phase folded light curve for binary A with $\mathrm{P_{A}}=1.072780$
days. We also see that not all data points fall nicely on the curve,
indicating the presence of another signal. Lower panel: Phase folded light
curve for binary B, after removal of the signal at period A, with
$\mathrm{P_{B}}=0.404213$ days.
## 5 Results
Of the 575 526 light curves analysed (500 data points or more), 203 ($\approx$
0.035$\%$) are identified as double-eclipsing binaries candidates. They are
summarised in the appendix in Table 1, and an example is shown in Fig. 1.
Almost all candidates for double eclipsing binaries are not currently known in
any scientific literature (i.e. they are not listed as such on SIMBAD; Wenger
et al., 2000). Only four of these 203 objects have been found in prior
research and marked in Table 1 by an asterisk. One of these four objects, Gaia
dr3 source ID 2003428628134264448, has previously been identified as a triple
star system (TIC 388459317; Borkovits et al., 2021) but the ZTF data suggests
that it is a double-eclipsing binary. The other three objects, Gaia dr3 source
ID’s 2060949991271681664 (Zasche et al., 2022a), 407849617690288128 (TIC
354314226; Kostov et al., 2022b) and 158123726424621824 (TIC 150055835; Kostov
et al., 2022b) have been identified as double eclipsing binary systems by
previous studies that used TESS data. Using the ZTF data, we find that the
periods of these objects agree between both studies.
The periods of the double eclipsing binaries in the ZTF sample range from 0.11
days to 323.20 days and the mean of periods A and B are 1.21 days and 10.52
days respectively, see Fig. 2. The figure shows that most objects have both
periods shorter than a few days. Using a kernel density estimate, this
distribution seems to be the same for the candidates and the test objects.
Figure 2: Periods A and B of 13 test objects and 203 quadruple candidates.
Here $P_{short}$ is defined as the shortest period of A and B and $P_{long}$
as the longest period of A and B. The figure shows that most objects have both
periods shorter than a few days. Using a kernel density estimate, this
distribution seems to be the same for the candidates and test objects.
To understand if there is any bias in our search, all 1 210 001 binaries in
the sample were plotted in an HR-diagram, see the left panel of Fig. 3. Here
we see that the double eclipsing binaries, plotted separately as dots, are
scattered throughout the diagram and do not seem to have any preferred
location.
Since it is interesting to not only compare the double eclipsing binaries to
the input sample but also to a general star population within the Milky Way,
we refer to the HR-diagram of all stars in the Gaia dr3 archive within 200 pc
of the sun. This can be seen in the right panel of Fig. 3. The quadruple
candidates and test objects are systematically above the single stars in this
diagram and are mostly Sun-like stars. We note, however, that quadruple
systems on the HR-diagram have a lower magnitude of $2.5\log_{10}(4)=1.505$
compared to single stars, assuming all stars in the quadruple are similar and
equally bright.
Figure 3: Left: HR-diagram of all 1 210 001 binary objects in the sample. The
double eclipsing binary candidates and test objects, separately plotted as
dots, are scattered throughout the diagram and do not seem to have any
preferred location. Using the sun as a reference with $M_{G}=5$ and
$G_{BP}-G_{RP}=1$, a larger number of objects seem to be similar to the sun.
Right: An HR-diagram of all stars in the Gaia dr3 archive within 200 pc of the
Sun. The candidates and test objects are systematically above the single stars
in this diagram and are mostly sun-like stars. The arrow with a length of
1.505 magnitude indicates where one star in the quadruple system would lie,
assuming all stars in the system are equally bright.
## 6 Discussion
### 6.1 Confusion with triple eclipsing systems
As we already noted; Gaia dr3 source ID 2003428628134264448 was previously
identified as a triply eclipsing triple system (Borkovits et al., 2021) (a
multistar system where all three stars eclipse each other). We initially
identified this system as a quadruple because we detected a primary and
secondary eclipse at a period of 88.8 days. However, as is shown by Borkovits
et al. (2021) using well-sampled TESS and ground-based follow-up light curves;
the eclipse shapes are complex and only consistent with a tertiary star that
eclipses, and is being eclipsed, by a short period binary. Triple eclipsing
stars are also extremely interesting for reasons similar to those of quadruple
systems. For recent work on triple eclipsing systems, see Carter et al.
(2011); Derekas et al. (2011); Hajdu et al. (2017); Mitnyan et al. (2020);
Borkovits et al. (2021); Powell et al. (2022); Rappaport et al. (2022, 2023);
Czavalinga et al. (2023)
The fact that one of our double eclipsing binary candidates turned out to be a
triply eclipsing triple system, prompted us to reconsider the long period
systems that we recovered. We focus on long period systems since, for triple
systems, the outer orbit has to be at least 5 times the inner orbit, while a
factor of 10 is required for slightly eccentric orbits (Mardling & Aarseth,
2001; Borkovits et al., 2021). With just the ZTF data, individual eclipses are
not well-resolved, which makes an unambiguous identification of triples
difficult. However, an eclipse of a triple eclipsing system is more noisy in a
folded light curve because of the complex and changing shape of individual
eclipses. We use this to determine whether some systems might be triple-
eclipsing systems instead of double-eclipsing binaries.
Objects 2020970648976206208, 2210495392378660864, and 2038100043701450112 show
somewhat irregular, long-duration eclipses in ZTF data and could be eclipsing
triple systems. Objects 2058085351143518464, 2003428628134264448,
2201723866572302976, and 413930359370384384 show a primary and a shallow
secondary eclipse at period B, with the primary eclipse somewhat irregular,
suggesting that they could also be triply eclipsing systems. Object
2041659197183314304 shows well-behaved eclipses, and we judge that this system
is unlikely to be a triply eclipsing system.
Objects 207943285475761280 and 2032118077650924672 are two systems for which
the eclipse of the long period signal is much deeper in $g$ than $r$ and maybe
a shallow secondary eclipse can be seen. This suggests that a large, cold star
is eclipsed by a smaller, hotter star. We suspect that these are double
eclipsing systems where one of the components has evolved off the main-
sequence. For 260945106052343296 and 2012994482376100736 the period is too
uncertain to draw any conclusions.
Since we cannot definitively classify these objects as triple eclipsing
systems, we consider all these objects double eclipsing binary candidates in
the rest of this paper. Follow-up light curve observations of the eclipses are
needed for each object to definitively determine their nature.
### 6.2 Orbital period distribution
In general, binaries with shorter periods seem easier to detect than binaries
with longer periods. One explanation is that stars in binaries with longer
orbital periods are further apart from each other. As the orbital distance
increases, the eclipse probability
$\text{Eclipse probability}=\frac{R_{1}+R_{2}}{a}\propto P^{-2/3}$ (2)
decreases since this requires a more precise alignment with the observer for
the eclipse to be detected. In addition, longer orbital periods thus lead to
shorter eclipse durations as fraction of the orbit. Some light curves showed
that the eclipse duration of binary B was either very short (¡ 0.01 in phase),
or contained very few data points, which makes it difficult for the algorithm
to identify the eclipse. This agrees with the results shown in Fig. 2, where
most double eclipsing binary candidates have both periods shorter than a few
days.
Figure 4: The orbital periods of double eclipsing binaries of the three
largest samples: the sample from OGLE (Zasche et al., 2019), the sample from
TESS by Kostov et al. (2022b, a) and Zasche et al. (2022b) and the ZTF sample
from this work. Here $P_{short}$ is defined as the shortest period of A and B
and $P_{long}$ as the longest period of A and B
The orbital periods of the candidates found in this research seem fairly
similar to those found by Kostov et al. (2022a); Zasche et al. (2022b), see
Fig. 4. However, one difference is that we find some shorter periods in this
research. Whereas previous research shows that most objects show a period
ratio of $\mathrm{P_{B}}/\mathrm{P_{A}}$ between 1 and 2 (Kostov et al.,
2022a), in Fig. 5 we see that here also much smaller ratios are found,
indicating a larger difference between period A and period B. An explanation
for this can be the data cadence of TESS which is 26 days, meaning it most
likely does not find any significant periods larger than approximately 13
days. Furthermore, short 30 minute exposures by TESS may lead to the spreading
out of short eclipses, which is why TESS might not find very short periods
either. We also note that all stars in our sample are listed in the Gaia
catalogue. Any bias of Gaia towards shorter periods may therefore reflect in
our results.
Figure 5: Period ratios $\mathrm{P_{short}}/\mathrm{P_{long}}$ of each object.
Here $P_{short}$ is defined as the shortest period of A and B and $P_{long}$
as the longest period of A and B. The vertical lines indicate where potential
harmonics would fall. We note that integer multiple harmonics were removed in
the object search. However, non integer harmonics were not removed. Here we
see that most objects do not fall on any of the harmonics lines indicating
that there does not seem to be any particular correlation between the periods.
### 6.3 Detection efficiency and completeness
In this section, we briefly discuss the detection efficiency and completeness
of our search. Of the 575 526 analysed objects, at least 203 ($\approx$
0.035%) passed the visual inspection tests and showed to be a double eclipsing
binary candidate rather than a binary system. This is approximately a factor
4.5 lower success rate than Zasche et al. (2022b) (116 out of 70.000) and a
factor 7 lower than Pawlak et al. (2013) (15 out of 6138). However, we note
that OGLE-III has observed all stars in a dense area in the Small Magellanic
Cloud for a longer period of time (Pawlak et al., 2013). Zasche et al. (2022b)
have, like this research, only looked at eclipsing binaries, but a difference
with this research is the data cadence, which for ZTF typically is one
observation per night only for five years. TESS, on the other hand, provides
continuous undisturbed photometry for 26 days (Zasche et al., 2022b). Lastly,
the angular resolution of ZTF, OGLE, and TESS are very different. This,
combined with the new method used here, makes it difficult to compare success
rates, but could explain the difference.
In the rest of this section, we briefly discuss a few causes that limit the
completeness of our search. First, we consider the efficiency of our quadruple
detection method. Our method was able to retrieve the correct periods for most
known objects in the test dataset. For periods that could not be recovered,
further investigation revealed why this was not possible. In some cases, many
data points were flagged as unreliable, leaving only very few data points in
the light curve for period finding. In other instances, the eclipse depth of
one binary was very small, on the order of a few percent, making it difficult
for the algorithm to detect any periodic signal. Other reasons why the
algorithm was not successful included that some light curves showed a very
small trend in flux over time, resulting in the cadence of the telescope being
found as significant period; this typically being one or two days. Even light
from the Moon coming in as background noise could lead to the period of the
Moon being found to be a significant period. For compact double eclipsing
binaries, eclipse-timing variations (ETV) can smear periodic signals which may
affect the period finding. However, when comparing our results with known
systems with ETV, we found this not to be the case. We therefore expect this
method to be suitable for finding compact double eclipsing binaries.
The completeness of our search for double eclipsing binaries is also limited
by our use of the Gaia eclipsing binary catalogue. When we look at the results
of other studies (Kostov et al., 2022a, b; Zasche et al., 2022b), we see that
approximately 30% of the found quadruples are in the Gaia catalogue. In this
research we have only looked at ZTF objects that are known in the Gaia
catalogue. This means that potentially three times more quadruples can be
found in the ZTF data that are not in the Gaia catalogue.
As it was not practical to look at 575 526 light curves, we used automatically
calculated statistics to significantly decrease the number of light curves to
inspect which could also affect the completeness of our search of the ZTF
data. We used the test objects to empirically determine cuts; relative peak
height $>10$ and S/N $>10$. As can be seen in Fig. 9, the wide range of values
of relative peak height and S/N cannot reliably indicate whether an object
might be a quadruple candidate. This suggests that more double eclipsing
binaries can be found for values of relative peak height and S/N below 10. It
can also be seen in the figure that some objects did have a high relative peak
height and S/N but were not identified as a quadruple candidate. This could
have several reasons. In some cases, the algorithm was not able to completely
remove the first periodic signal, which means that the signal, or an harmonic,
was returned again for period B. Other explanations include, but are not
limited to, an outlier in the light curve, the cadence of the telescope, or
the influence of the Moon. Sometimes the algorithm did not find an eclipsing
signal but a sinusoidal signal, possibly due to the pulsation of one of the
stars in the eclipsing binary. Even though this is not a double eclipsing
binary, a strong periodic signal was still found, resulting in a high relative
peak height and S/N.
Figure 6: Left: Cumulative histogram for the number of data points per light
curve. After passing a threshold of approximately 1000 data points there is a
significant increase in the quadruple detection efficiency. Note that the
plateau and discontinuity at around 2000 data points is a result of the ZTF
survey strategy. Right: Cumulative histogram for the magnitude of stars for
the complete sample and quadruple candidates in the magnitude range 13-20.5.
We see that for stars fainter than $G\gtrsim 18$ the quadruple detection
efficiency rapidly approaches zero. The sharp increase at $G\approx 15$ is not
fully understood.
We suspected that the detection efficiency of double eclipsing binary stars is
also strongly affected by the number of epochs in the light curve. If we look
at the left panel in Fig. 6 we see a cumulative histogram for the number of
data points per light curve. It shows that, only after passing a threshold of
approximately 1000 data points, the quadruple detection efficiency
significantly increases. After that, there are more double eclipsing binaries
as the number of data points increases. The difference, however, is small.
This means that, as ZTF keeps collecting data (or ZTF is combined with
external data), we can expect the number of detectable double eclipsing
binaries to double.
We also briefly consider to what brightness we can reliably find double
eclipsing binaries in ZTF data. On the right panel in Fig. 6 we see a
cumulative histogram for the magnitude of stars, both for the complete sample
and for the quadruple candidates. The ZTF has looked at stars in the magnitude
range 13-20.5. Since the curve of the quadruples is constantly above the
overall sample curve, this indicates that it is easier to find quadruples for
bright stars compared to faint stars. Almost all quadruples have a magnitude
of $G\lesssim 18$, while only half of the input sample is fainter than this
limit. We conclude that the light curve signal-to-noise ratio is too low for
sources fainter than $G\approx 18$. As can be seen in Masci et al. (2019), the
RMS precision is $\approx 1\%$ for sources brighter than 18 magnitude, but the
precision degrades rapidly for fainter sources.
To summarise, we expect that future research can find at least three to four
times more quadruples by including the following considerations. First, the
search should be not limited to the Gaia eclipsing binary catalogue but the
entire ZTF database can be searched instead. Secondly, systems with a relative
peak height and S/N lower than 10 should also be considered. Thirdly, longer
observation time of the ZTF will lead to more data points which has a positive
effect on quadruple finding. Lastly, by using an automatic detection method
that can replace visual detection, for instance machine learning, all the
light curves can be analysed more efficiently.
## 7 Conclusions and future work
We have developed a method to systematically search for double eclipsing
binaries in sparsely sampled data. Using this method we found 198 new double
eclipsing binaries candidates in the ZTF data. The periods of the objects
ranged from 0.11 days to 323 days with a mean of periods A and B of 1.21 days
and 10.52 days respectively. By searching the entire ZTF database, increasing
the number of data points per light curve, and implementing an automatic
detection mechanism that replaces visual inspection, we expect that at least
three to four times more quadruples can be found using this method. Other
recommendations for future work are; to apply this method to data collected by
other telescopes such as Gaia. In addition, we recommend that the multi star
systems are included in the target lists of large multiplex spectroscopic
surveys such as SDSS, DESI, and WEAVE in order to obtain phase-resolved
spectra of the quadruple candidates to further investigate their properties.
Lastly, detection of eclipse-timing variations in the double eclipsing
binaries can definitively prove their quadruple nature.
###### Acknowledgements.
We thank Silvia Toonen for a useful discussion about multi-star systems. This
publication is part of the project ”The life and death of white dwarf binary
stars” (with project number VI.Veni.212.201) of the research programme NWO
Talent Programme Veni Science domain 2021 which is financed by the Dutch
Research Council (NWO). Based on observations obtained with the Samuel Oschin
48-inch Telescope at the Palomar Observatory as part of the Zwicky Transient
Facility project. ZTF is supported by the National Science Foundation under
Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the
Weizmann Institute for Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, the University of Washington,
Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National
Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at
Milwaukee, and Lawrence Berkeley National Laboratories. Operations are
conducted by COO, IPAC, and UW. Based on observations obtained with the Samuel
Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory
as part of the Zwicky Transient Facility project. ZTF is supported by the
National Science Foundation under Grants No. AST-1440341 and AST-2034437 and a
collaboration including current partners Caltech, IPAC, the Weizmann Institute
for Science, the Oskar Klein Center at Stockholm University, the University of
Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO
Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity
College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of
Warwick, Ruhr University Bochum, Northwestern University and former partners
the University of Washington, Los Alamos National Laboratories, and Lawrence
Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.
The ztfquery code was funded by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement n°759194 - USNAC, PI: Rigault). This research has made use of the
VizieR catalogue access tool, CDS, Strasbourg, France. This research made use
of NumPy (Harris et al., 2020) This research made use of matplotlib, a Python
library for publication quality graphics (Hunter, 2007) This research made use
of Astroquery (Ginsburg et al., 2019) This research made use of Astropy, a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2022)
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## Appendix A Additional figures
For the ZTF, the periods of objects that lie in the Galactic plane can be more
easily retrieved. This is can also be seen in Fig. 7 where the location of all
quadruple candidates and test objects are plotted. These results agree with
the results found in earlier research (Zasche et al. 2022b; Kostov et al.
2022a). It is most likely that the number of double eclipsing binaries is
higher in the Galactic plane as a result of higher star density.
Figure 7: Right ascension (RA) and declination (DEC) of the double eclipsing
binary candidates and test objects in this research. As expected, most
candidates lie in the Galactic plane. No objects are seen below a declination
of -30° as this is not visible by the ZTF. Figure 8: A periodogram of Gaia dr3
source ID 2216420454386370176 showing the lspvals as a function of orbital
period. Left: The periodogram power values for period A. Right: The periodgram
power values for period B after removing the signal of period A. Figure 9: The
relative peak height and S/N of all objects including double eclipsing binary
candidates and test objects. Left: Relative peak heights A and B for all
objects. Middle: S/N A and S/N B results for all objects. Right: S/N B and
relative peak height B of all objects. We see that quadruples can have a wide
range of RPH and S/N values. However, this also means that more can be found
at values of RPH ¡ 10 & S/N ¡10.
## Appendix B Summary of double eclipsing binary candidates
Table 1 in this appendix lists all new double eclipsing binaries we found in
ZTF.
Table 1: Summary of the double eclipsing binary candidates. For each candidate the location (RA & DEC), Gaia dr3 source ID, periods ($\mathrm{P_{A}[d]}$ & $\mathrm{P_{B}[d]}$) and eclipse depths ($\mathrm{D_{A}}$ & $\mathrm{D_{B}}$) are given. Four objects have been found in previous research and are marked here with an asterisk. The object marked with a ”T” was initially identified as a quadruple, but further investigation revealed that this is, in fact, a triple system. RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
1.461502 | 44.447714 | 386474080852740992 | 2.599973 | 0.178 | 0.114405 | 0.017
2.702797 | 43.426465 | 384686137507666944 | 0.561156 | 0.057 | 0.30431 | 0.02
3.255083 | 66.065778 | 528173641885393792 | 1.377078 | 0.007 | 1.958498 | 0.067
5.530266 | 63.915121 | 431071367690614016 | 1.993434 | 0.207 | 2.83458 | 0.12
6.850741 | 42.275447 | 382118159381427840 | 0.302332 | 0.045 | 11.331499 | 0.063
8.806599 | 48.41824 | 390942874067575936 | 0.254808 | 0.066 | 100.03476 | 0.159
13.007138 | 60.631402 | 427296611137250432 | 1.39752 | 0.108 | 1.233086 | 0.034
13.938753 | 62.946494 | 523702890174595456 | 0.620434 | 0.129 | 0.766964 | 0.071
14.457916 | 46.742013 | 377967807928167552 | 1.557556 | 0.059 | 0.569134 | 0.181
17.516555 | 60.901287 | 522445255030705920 | 0.357662 | 0.183 | 0.653228 | 0.098
18.801672 | 57.606248 | 413446544200880896 | 0.951178 | 0.076 | 7.547427 | 0.073
20.433859 | 58.575703 | 413930359370384384 | 1.745764 | 0.211 | 74.595949 | 0.184
23.839866 | 48.216903 | 399464707656254976 | 6.934429 | 0.026 | 8.655612 | 0.066
24.594285 | 62.368258 | 511572184543165184 | 1.111992 | 0.017 | 0.893415 | 0.099
25.46978 | 48.453185 | 405343761970536320 | 3.214925 | 0.181 | 0.475366 | 0.051
26.105725 | 57.363963 | 505854376178706048 | 0.395542 | 0.283 | 0.613647 | 0.075
26.99833 | 58.484187 | 506382008620308352 | 0.469946 | 0.117 | 1.607418 | 0.082
27.345582 | 51.957608 | 407610302112852992 | 0.27613 | 0.067 | 4.742124 | 0.105
27.478311 | 61.054208 | 511202954794627072 | 2.522958 | 0.088 | 2.118876 | 0.037
27.619643 | 53.266914 | 407849617690288128∗ | 1.388514 | 0.182 | 1.842243 | 0.133
29.369827 | 36.938288 | 330442487264141952 | 2.165996 | 0.148 | 3.201325 | 0.069
32.660282 | 56.361212 | 457045959812411520 | 2.907525 | 0.249 | 1.766785 | 0.083
35.274278 | 59.921344 | 507367789513371392 | 4.733592 | 0.119 | 1.111969 | 0.06
35.979435 | 63.411351 | 514335841379134336 | 1.718536 | 0.264 | 1.220503 | 0.075
36.881203 | 63.214977 | 514291963992396928 | 3.006382 | 0.093 | 0.212537 | 0.019
38.031578 | 62.343023 | 513987498047116544 | 1.019208 | 0.086 | 0.948252 | 0.071
38.292867 | 59.484742 | 465028723463664384 | 3.460204 | 0.192 | 0.92533 | 0.045
39.195205 | 60.984813 | 465274395594682240 | 3.887904 | 0.148 | 1.896361 | 0.083
42.417307 | 58.265701 | 461012963401936256 | 1.060832 | 0.048 | 10.593118 | 0.034
44.301404 | 59.454303 | 461553472144934656 | 0.270063 | 0.053 | 1.385304 | 0.049
49.178341 | -10.308834 | 5166225811703486464 | 0.275808 | 0.248 | 2.432133 | 0.098
54.711333 | 44.047482 | 244408588611828096 | 0.178865 | 0.145 | 0.243028 | 0.022
57.458651 | 56.117922 | 445534966419218560 | 1.292364 | 0.246 | 1.088451 | 0.051
59.624908 | 56.534336 | 469545650369412864 | 0.224785 | 0.178 | 1.551227 | 0.05
64.168902 | 50.317876 | 271381051952218496 | 2.245528 | 0.108 | 1.40276 | 0.05
67.213782 | 42.115243 | 228205115816360832 | 1.125492 | 0.13 | 2.201828 | 0.02
67.217417 | 48.572915 | 258090121033397248 | 5.48996 | 0.156 | 25.643685 | 0.035
68.629624 | 48.698441 | 257683439171736576 | 0.81849 | 0.107 | 4.499209 | 0.08
69.925315 | 29.134575 | 158123726424621824∗ | 1.58676 | 0.128 | 13.783777 | 0.06
72.308262 | 52.265466 | 260945106052343296 | 3.070683 | 0.241 | 37.120622 | 0.061
73.106774 | 45.534572 | 206454920393197440 | 0.583875 | 0.365 | 0.715742 | 0.089
76.053644 | 42.807356 | 202201601392851072 | 0.624014 | 0.05 | 14.132366 | 0.087
77.972018 | 51.672042 | 262779881722935424 | 1.637614 | 0.193 | 2.266959 | 0.101
80.461508 | 37.37017 | 184468231186516352 | 2.345842 | 0.133 | 6.86879 | 0.09
80.679986 | 37.664808 | 184497570107735296 | 2.3771 | 0.094 | 2.839768 | 0.055
82.738969 | 44.803406 | 207943285475761280 | 52.829036 | 0.274 | 1.184305 | 0.066
83.858615 | 33.004921 | 3448886540015107584 | 2.989964 | 0.115 | 0.52143 | 0.06
86.591414 | 28.923839 | 3443402279093860096 | 0.53534 | 0.144 | 0.541528 | 0.079
87.5282 | 21.330875 | 3400120484904001408 | 3.572446 | 0.166 | 1.759766 | 0.066
88.00886 | 8.805691 | 3335190578070549632 | 0.36877 | 0.083 | 1.890454 | 0.068
96.842245 | 21.72127 | 3376140681762631168 | 0.39992 | 0.093 | 0.842531 | 0.049
98.587002 | 7.745466 | 3325963098535865728 | 0.584642 | 0.307 | 0.6136 | 0.053
Table 2: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
101.872246 | -0.444575 | 3113400013097881600 | 0.86539 | 0.121 | 2.182992 | 0.048
103.057739 | -1.125556 | 3112366227355971328 | 0.948312 | 0.155 | 0.632336 | 0.069
104.069404 | 15.809035 | 3355042672829484032 | 2.502663 | 0.177 | 0.530042 | 0.06
105.184549 | 32.479562 | 890716293806690816 | 0.263998 | 0.105 | 3.144128 | 0.136
105.242128 | -16.522209 | 2935716391431248256 | 0.424388 | 0.076 | 7.412474 | 0.067
109.571295 | 1.765508 | 3111928351141425792 | 0.274758 | 0.174 | 0.144463 | 0.054
109.612639 | -10.149447 | 3047190824491991168 | 0.45911 | 0.163 | 0.25037 | 0.05
111.089388 | -2.032067 | 3061770932787038336 | 0.422212 | 0.082 | 0.966995 | 0.054
111.47435 | -19.281677 | 2930398740886024064 | 0.347152 | 0.219 | 2.294894 | 0.144
117.697495 | 28.467895 | 875709132614498560 | 0.442358 | 0.158 | 1.466414 | 0.069
118.096034 | -5.697484 | 3044312990239139968 | 1.452632 | 0.176 | 1.779009 | 0.072
202.102553 | 61.876817 | 1663741485747221248 | 2.770754 | 0.15 | 8.163296 | 0.08
223.42518 | 52.715832 | 1594082407606370176 | 2.700472 | 0.074 | 1.488501 | 0.043
242.649013 | 34.62108 | 1323756753679794048 | 7.041158 | 0.114 | 0.128904 | 0.014
257.277242 | 49.948796 | 1414515322519168640 | 0.301544 | 0.197 | 0.231617 | 0.019
259.250732 | -20.297582 | 4115999620890627968 | 0.621141 | 0.24 | 0.773402 | 0.114
263.775976 | 38.311866 | 1343353448205456256 | 3.048348 | 0.095 | 11.792556 | 0.033
273.743299 | 39.207529 | 2109307852667550464 | 2.205617 | 0.187 | 4.373204 | 0.029
276.008132 | -5.774128 | 4161007815803011968 | 3.070536 | 0.211 | 1.371504 | 0.102
278.026086 | 54.402996 | 2147458221793949184 | 3.109478 | 0.097 | 1.853905 | 0.079
279.117985 | -8.525243 | 4156431820198005376 | 1.407928 | 0.281 | 1.331908 | 0.085
279.719282 | 13.163625 | 4508288602094506112 | 0.399492 | 0.061 | 1.528379 | 0.102
280.635591 | -5.252461 | 4256607950976483968 | 1.907376 | 0.222 | 0.903688 | 0.036
280.635808 | 13.120684 | 4505600296166989312 | 4.870614 | 0.097 | 4.70215 | 0.019
280.713417 | 34.65467 | 2091761983553949696 | 0.271486 | 0.204 | 0.372731 | 0.028
281.121588 | 11.13593 | 4504043245957168640 | 10.015578 | 0.077 | 1.042907 | 0.022
282.145741 | 30.399305 | 2041659197183314304 | 0.283664 | 0.061 | 55.568657 | 0.06
282.312661 | 13.469479 | 4505497904155844992 | 0.28221 | 0.303 | 3.150768 | 0.103
283.996449 | 28.282368 | 2040256804464235648 | 5.739774 | 0.104 | 0.139815 | 0.012
284.492451 | -11.193995 | 4201927347951103488 | 0.513114 | 0.405 | 0.453093 | 0.05
285.268359 | 18.064422 | 4517289341731165568 | 1.056042 | 0.07 | 0.545708 | 0.031
286.391765 | 16.558561 | 4513882917294832384 | 0.875846 | 0.043 | 2.838248 | 0.022
286.904493 | -9.457319 | 4204113619406564480 | 0.33327 | 0.097 | 0.867417 | 0.047
287.114664 | -8.543055 | 4204326997705570304 | 0.272314 | 0.3 | 0.170204 | 0.061
287.445354 | 19.597742 | 4516377056329377664 | 1.035768 | 0.171 | 6.325886 | 0.082
287.648865 | 16.355321 | 4513263857861420672 | 0.45636 | 0.166 | 2.129335 | 0.1
287.884621 | 46.028573 | 2130268255146656128 | 0.422266 | 0.269 | 1.029215 | 0.035
288.026347 | 23.298684 | 4521240642906912128 | 3.00593 | 0.011 | 1.616295 | 0.071
288.248394 | 0.080909 | 4264206469687918848 | 3.166976 | 0.208 | 1.852653 | 0.103
288.471817 | -1.094539 | 4263034768239285632 | 1.862082 | 0.16 | 0.592354 | 0.053
288.648651 | 22.157332 | 4520164564651728128 | 0.355088 | 0.09 | 0.32902 | 0.045
288.825262 | 4.312846 | 4293109022582320640 | 0.390062 | 0.277 | 0.164066 | 0.11
289.517888 | 15.160579 | 4320727174167431296 | 2.163692 | 0.033 | 1.607176 | 0.042
290.184501 | 28.541396 | 2038100043701450112 | 0.325552 | 0.122 | 79.87107 | 0.122
290.604938 | 9.66816 | 4308597705466532224 | 5.097978 | 0.173 | 1.412602 | 0.057
290.991691 | 35.958626 | 2050141310916611584 | 3.877374 | 0.021 | 4.766349 | 0.047
291.640789 | 4.331223 | 4292639977810529152 | 1.64215 | 0.063 | 1.061804 | 0.058
291.714347 | 18.228635 | 4323257219102940288 | 1.588804 | 0.186 | 2.302579 | 0.077
291.820676 | 35.97582 | 2049954600104034432 | 4.030792 | 0.113 | 14.062535 | 0.062
293.462031 | 19.846805 | 1825483906846429056 | 6.45557 | 0.145 | 2.467705 | 0.047
293.504188 | 27.432478 | 2025279046615280512 | 2.759184 | 0.134 | 0.863998 | 0.026
293.592507 | 11.006839 | 4314657904341880960 | 3.349092 | 0.109 | 1.32064 | 0.027
293.657387 | 19.881442 | 1825578567909317760 | 4.770348 | 0.152 | 2.118271 | 0.038
293.823697 | 25.752582 | 2021549301357843328 | 0.417266 | 0.168 | 0.182446 | 0.089
Table 3: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
294.156333 | 14.316215 | 4318042922678224768 | 1.840298 | 0.023 | 0.915486 | 0.087
294.332664 | 27.158424 | 2025146250574308480 | 0.430934 | 0.184 | 0.698726 | 0.062
294.518815 | -7.404433 | 4207090654517833344 | 3.426032 | 0.186 | 0.315937 | 0.04
294.832597 | 26.871716 | 2025081478136260480 | 3.072568 | 0.066 | 2.212912 | 0.052
294.839444 | 35.975154 | 2048268017993738624 | 0.234828 | 0.043 | 4.944031 | 0.052
294.933937 | 20.202009 | 1825745006505362048 | 7.807304 | 0.051 | 3.649778 | 0.042
295.309706 | 42.822037 | 2077913565886105344 | 0.373692 | 0.344 | 0.322663 | 0.03
295.479516 | 34.635703 | 2047323949824485376 | 2.346872 | 0.18 | 2.481432 | 0.055
295.527859 | 22.054362 | 1827708802964917888 | 0.545902 | 0.187 | 0.96728 | 0.047
295.87929 | 25.881643 | 2021774220180193536 | 1.357318 | 0.0 | 2.09345 | 0.026
296.235173 | 29.583425 | 2031977619399032448 | 8.147016 | 0.064 | 8.083514 | 0.023
296.585034 | 23.395937 | 2020056405032433152 | 0.812118 | 0.18 | 0.753871 | 0.086
296.895517 | 25.636448 | 2020970648976206208 | 6.375142 | 0.168 | 323.201568 | 0.109
296.955946 | 30.241753 | 2032118077650924672 | 171.076078 | 0.28 | 2.335024 | 0.089
297.00612 | 24.058537 | 2020490437267187712 | 1.35866 | 0.148 | 1.729201 | 0.093
297.034812 | 30.08806 | 2031925182082532608 | 3.590404 | 0.001 | 0.789163 | 0.032
297.056926 | 25.787105 | 2026976864358731904 | 1.425158 | 0.0 | 0.89166 | 0.105
298.002327 | 31.866931 | 2033828574151981184 | 1.768354 | 0.231 | 0.571495 | 0.122
298.172845 | 26.864218 | 2027154955201119488 | 1.647592 | 0.093 | 1.191313 | 0.026
298.530047 | 24.682368 | 1834469768675202560 | 2.881026 | 0.081 | 4.353022 | 0.036
298.820248 | 32.01454 | 2033656156987971840 | 7.548416 | 0.327 | 0.159107 | 0.049
298.881534 | 29.89156 | 2030312580776346240 | 0.462184 | 0.179 | 7.651529 | 0.06
298.908525 | 32.746538 | 2034277144881784704 | 2.452324 | 0.245 | 0.558699 | 0.026
299.522067 | 30.246542 | 2030426861313716224 | 1.305256 | 0.061 | 1.607371 | 0.019
299.620393 | 33.527522 | 2034419218049672704 | 1.411226 | 0.099 | 1.614413 | 0.072
299.976835 | 29.83971 | 2030212490900354176 | 1.189566 | 0.022 | 3.314311 | 0.025
301.355473 | 34.989363 | 2058617549147113728 | 2.08353 | 0.045 | 15.704131 | 0.051
301.37899 | 35.97206 | 2059117002318116096 | 1.741608 | 0.114 | 1.064947 | 0.033
301.391172 | 42.318524 | 2074882590295669376 | 1.127028 | 0.197 | 2.46579 | 0.132
302.250175 | 21.436492 | 1829670778385777920 | 5.414028 | 0.109 | 3.970943 | 0.085
302.637062 | 38.400325 | 2061735248734886400 | 1.079346 | 0.226 | 0.468297 | 0.029
302.98334 | 36.463072 | 2059178162649303808 | 5.993378 | 0.266 | 0.563785 | 0.056
303.229266 | 27.908953 | 1836995514370597120 | 0.374674 | 0.076 | 3.126105 | 0.05
303.523133 | 38.382872 | 2060949991271681664∗ | 1.144666 | 0.129 | 0.481849 | 0.017
304.151811 | 41.930677 | 2068658632915468672 | 4.363316 | 0.078 | 1.242846 | 0.052
304.756649 | 37.920034 | 2061050802729418496 | 1.325726 | 0.172 | 0.813421 | 0.097
304.795015 | 30.560399 | 1861612037845705728 | 5.302642 | 0.194 | 0.323736 | 0.037
306.408352 | 37.894974 | 2058085351143518464 | 1.152476 | 0.193 | 85.498447 | 0.139
308.681181 | 48.20847 | 2167733048719069312 | 2.17693 | 0.17 | 0.584749 | 0.086
308.779214 | 35.826622 | 2056704914305732096 | 5.152764 | 0.146 | 0.910183 | 0.069
309.435623 | 30.852742 | 1862347439319475072 | 0.920364 | 0.145 | 1.417808 | 0.094
311.55134 | 31.682545 | 1859824197571796736 | 0.283018 | 0.133 | 0.394832 | 0.064
311.97923 | 34.771038 | 1869489385821202816 | 0.488582 | 0.115 | 2.110629 | 0.109
312.564502 | 12.412033 | 1760943570684334208 | 3.047312 | 0.209 | 15.580245 | 0.076
312.603366 | 47.202077 | 2166511040319614208 | 0.38052 | 0.147 | 1.264149 | 0.127
313.727051 | 39.848684 | 1872913471188045184 | 2.26941 | 0.054 | 0.678956 | 0.023
314.203035 | 66.204173 | 2245787855903677952 | 1.313048 | 0.21 | 1.250967 | 0.228
315.998617 | 47.984506 | 2165439879783209344 | 3.822778 | 0.095 | 2.650119 | 0.037
316.499668 | 36.761395 | 1868409321799840768 | 0.543036 | 0.063 | 0.725643 | 0.036
316.935639 | 47.04712 | 2165141946503725056 | 1.272084 | 0.103 | 3.086064 | 0.035
317.167097 | -8.177796 | 6897094092939104768 | 0.223992 | 0.027 | 0.622656 | 0.041
317.354427 | 55.485175 | 2176903937763785088 | 3.446506 | 0.256 | 3.158925 | 0.236
318.20909 | 56.26039 | 2177329891137899904 | 0.90333 | 0.253 | 0.515231 | 0.087
318.442107 | 51.071054 | 2166229226047556096 | 15.0598 | 0.169 | 3.180452 | 0.086
Table 4: Continued RA [J2000.0] | DEC [J2000.0] | Gaia dr3 source ID | $\mathrm{P_{A}[d]}$ | $\mathrm{D_{A}}$ | $\mathrm{P_{B}[d]}$ | $\mathrm{D_{B}}$
---|---|---|---|---|---|---
321.274386 | 49.322575 | 2170863908089217280 | 3.714414 | 0.154 | 2.643588 | 0.078
322.785455 | 37.809557 | 1952168090372267008 | 1.158128 | 0.035 | 1.25026 | 0.023
322.795044 | 41.61673 | 1967117802081173504 | 4.672324 | 0.102 | 3.584245 | 0.036
324.064461 | 40.289056 | 1966045232784781952 | 0.264192 | 0.119 | 0.779663 | 0.061
325.003123 | 56.109079 | 2178057359827382144 | 3.296832 | 0.225 | 1.291317 | 0.036
325.081161 | 48.44538 | 1978029359778790016 | 0.678389 | 0.01 | 0.746095 | 0.054
325.126003 | 39.737305 | 1953928099249228032 | 1.068532 | 0.22 | 1.123977 | 0.027
325.523023 | 58.222165 | 2178581380197081216 | 1.810504 | 0.115 | 0.707697 | 0.062
325.791627 | 27.217748 | 1800509737128120320 | 0.85369 | 0.239 | 3.215801 | 0.166
326.593483 | 49.781513 | 1978958103506727424 | 0.421342 | 0.068 | 1.667499 | 0.039
327.020592 | 62.78985 | 2216420454386370176 | 2.14556 | 0.228 | 0.404213 | 0.082
327.271647 | 58.943552 | 2202597664775578112 | 1.06056 | 0.143 | 2.05528 | 0.107
328.521806 | 57.784601 | 2199284050276697216 | 0.433052 | 0.121 | 1.459645 | 0.106
329.000459 | 52.050747 | 1981103972253289088 | 0.685961 | 0.114 | 0.739756 | 0.027
329.356711 | 45.001855 | 1973338392191011584 | 0.397442 | 0.112 | 6.522002 | 0.093
329.904956 | 43.718061 | 1961000256819235200 | 0.473181 | 0.095 | 0.964035 | 0.103
330.10459 | 57.341933 | 2199104211409698816 | 0.979846 | 0.194 | 0.443703 | 0.067
330.629145 | 55.984692 | 2198180415480756096 | 1.328192 | 0.0 | 22.393498 | 0.086
332.366702 | 56.166443 | 2197966251234148352 | 2.705704 | 0.168 | 2.822163 | 0.04
332.720979 | 59.64723 | 2199925099913441920 | 2.15344 | 0.175 | 1.57226 | 0.067
332.943294 | 54.42099 | 2005474711889658624 | 3.234134 | 0.101 | 0.617324 | 0.135
333.801196 | 53.749768 | 2004590532743751424 | 0.340078 | 0.058 | 2.522543 | 0.034
334.655366 | 55.538438 | 2005911802117955968 | 1.508362 | 0.197 | 0.947192 | 0.097
335.103223 | 36.014911 | 1905994992912248448 | 0.834723 | 0.19 | 0.422142 | 0.116
335.115083 | 24.563415 | 1878929101147587200 | 2.107929 | 0.256 | 2.068346 | 0.048
337.436575 | 54.279883 | 2001849763802035840 | 2.263908 | 0.235 | 0.545817 | 0.03
337.832153 | 60.857037 | 2201723866572302976 | 2.627162 | 0.107 | 101.237102 | 0.125
339.088072 | 47.538959 | 1986716807300365696 | 6.451769 | 0.218 | 0.164641 | 0.043
339.963254 | 54.98005 | 2003428628134264448∗T | 2.184788 | 0.145 | 88.853882 | 0.143
340.921211 | 57.261299 | 2007240188275431936 | 1.625996 | 0.064 | 1.379002 | 0.038
341.302946 | 56.012542 | 2003907912126462592 | 0.404544 | 0.102 | 2.944626 | 0.058
341.422596 | 53.230019 | 2002122850696945664 | 3.484958 | 0.063 | 2.227952 | 0.036
342.74 | 58.714103 | 2007474143733297152 | 1.63917 | 0.116 | 1.617598 | 0.023
344.803515 | 48.306854 | 1984790600366568576 | 0.699564 | 0.452 | 1.513966 | 0.034
347.564145 | 65.090617 | 2208692193312331904 | 5.095412 | 0.059 | 1.198208 | 0.03
349.093506 | 66.228219 | 2210495392378660864 | 2.568998 | 0.241 | 285.579588 | 0.218
350.55902 | 56.248535 | 1997226175657082496 | 0.351738 | 0.091 | 1.855745 | 0.043
351.050509 | 59.819433 | 2010769070836151424 | 1.312382 | 0.17 | 0.89549 | 0.057
352.521122 | 57.484618 | 1998851258144250112 | 0.280512 | 0.094 | 1.619602 | 0.047
352.593979 | 48.961381 | 1942119756681849216 | 4.303021 | 0.4 | 0.149352 | 0.039
353.524226 | 36.870485 | 1918839728265382912 | 0.366704 | 0.208 | 4.201864 | 0.116
358.069196 | 62.459358 | 2012994482376100736 | 1.950456 | 0.256 | 34.404796 | 0.119
358.221987 | 61.525403 | 2012695312133566208 | 3.21884 | 0.179 | 2.533716 | 0.056
|
# Spectral representation of Matsubara n-point functions:
Exact kernel functions and applications
Johannes Halbinger, Benedikt Schneider and Björn Sbierski Department of
Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-
Maximilians-Universität München, Theresienstr. 37, München D-80333, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
D-80799 München, Germany
(August 28, 2024)
###### Abstract
In the field of quantum many-body physics, the spectral (or Lehmann)
representation simplifies the calculation of Matsubara $n$-point correlation
functions if the eigensystem of a Hamiltonian is known. It is expressed via a
universal kernel function and a system- and correlator-specific product of
matrix elements. Here we provide the kernel functions in full generality, for
arbitrary $n$, arbitrary combinations of bosonic or fermionic operators and an
arbitrary number of anomalous terms. As an application, we consider bosonic 3-
and 4-point correlation functions for the fermionic Hubbard atom and a free
spin of length $S$, respectively.
## I Introduction and definitions
Multi-point correlation functions of $n$ quantum mechanical operators, also
known as $n$-point functions, are a central concept in the study of quantum
many-body systems and field theory (Negele and Orland, 1988). They generalize
the well-known 2-point functions, which, for the example of electrons in the
solid state, are routinely measured by scanning tunneling spectroscopy or
angle-resolved photon emission spectroscopy (Bruus and Flensberg, 2004). For
magnetic systems, the 2-point spin correlators can be probed in a neutron
scattering experiment. Higher order correlation functions with $n=3,4,5...$
can for example be measured in non-linear response settings (Kappl et al.,
2023). In the emerging field of cold atomic quantum simulation, (equal-time)
$n$-point functions are even directly accessible (Semeghini et al., 2021).
On the theoretical side the study of higher order correlation functions gains
traction as well. One motivation is the existence of exact relations between
correlation functions of different order $n$ (Hedin, 1965; Bickers, ).
Although these relations can usually not be solved exactly, they form a
valuable starting point for further methodological developments like the
parquet approximation (Roulet et al., 1969). Thus even if the 4-point
correlator (or, in that context, its essential part, the one-line irreducible
vertex (Negele and Orland, 1988)) might not be the primary quantity of
interest in a calculation, it appears as a building block of the method.
Another example is the functional renormalization group method (fRG) in a
vertex expansion (Kopietz et al., 2010; Metzner et al., 2012). It expresses
the many body problem as a hierarchy of differential equations for the
vertices that interpolate between a simple solvable starting point and the
full physical theory (Wetterich, 1993). Whereas experiments measure
correlation functions in real time (or frequency), in theory one often is
concerned with the related but conceptually simpler versions depending on
imaginary time (Negele and Orland, 1988). In the following, we will focus on
these Matsubara correlation functions, which, nevertheless feature an
intricate frequency dependence.
Whereas the above theoretical methods usually provide only an approximation
for the $n$-point functions, an important task is to calculate these objects
exactly. This should be possible for simple quantum many body systems. We
consider systems simple if they are amenable to exact diagonalization (ED),
i.e. feature a small enough Hilbert space, like few-site clusters of
interacting quantum spins or fermions. Also impurity systems, where
interactions only act locally, can be approximately diagonalized using the
numerical renormalization group (Lee et al., 2021).
Knowing the exact $n$-point functions for simple systems is important for
benchmark testing newly developed methods before deploying them to harder
problems. Moreover, $n$-point functions for simple systems often serve as the
starting point of further approximations like in the spin-fRG (Reuther and
Thomale, 2014; Krieg and Kopietz, 2019; Rückriegel et al., 2022), or appear
intrinsically in a method like in diagrammatic extensions of dynamical mean
field theory (Rohringer et al., 2018) with its auxiliary impurity problems.
Another pursuit enabled by the availability of exact $n$-point functions is to
interpret the wealth of information encoded in these objects, in particular in
their rich frequency structure. For example, Ref. (Chalupa et al., 2021)
studied the fingerprints of local moment formation and Kondo screening in
quantum impurity models.
In this work we complete the task to calculate exact $n$-point functions by
generalizing the spectral (or Lehmann) representation (Lehmann, 1954; Negele
and Orland, 1988) for Matsubara $n$-point correlation functions to arbitrary
$n$. We assume that a set of eigenstates and -energies is given. Following
pioneering work of Refs. (Shvaika, 2006; Hafermann et al., 2009; Shvaika,
2016) and in particular the recent approach by Kugler _et al._ (Kugler et al.,
2021), we split the problem of calculating imaginary frequency correlators
into the computation of a universal kernel function and a system- and
correlator-specific part (called partial spectral function in Ref. (Kugler et
al., 2021)). We provide the kernel functions in full generality for an
arbitrary number $n$ of bosonic or fermionic frequencies. Previously, these
kernel functions were known exactly only up to the 3-point case (Shvaika,
2006), for the fermionic 4-point case (Hafermann et al., 2009; Shvaika, 2016;
Kugler et al., 2021) or for the general $n$-point case (Kugler et al., 2021)
but disregarding anomalous contributions to the sum that the kernel function
consists of. These anomalous contributions are at the heart of the complexity
of Matsubara $n$-point functions. They occur when certain combinations of
eigenenergies and external frequencies vanish individually, see the anti-
diagonal rays in Fig. 1(c). Physically, they correspond to long-term memory
effects, are related to non-ergodicity and, in the case of bosonic two-point
functions reflect the difference between static isothermal susceptibilities
and the zero-frequency limit of the dynamical Kubo response function (Kwok and
Schultz, 1969; Watzenböck et al., 2022).
The structure of the paper is as follows: In Sec. II we define the Matsubara
$n$-point function
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n-1}\bigr{)}$ and review some
of its properties. The spectral representation is derived in Sec. III with Eq.
(15) being the central equation written in terms of the kernel function
$K_{n}(\Omega_{1},...,\Omega_{n-1})$. Our main result is an exact closed-form
expression of this most general kernel function which is given in Sec. IV.
Examples for $n=2,3,4,5$ are given in Sec. V where we also discuss
simplifications for the purely fermionic case. We continue with applications
to two particular systems relevant in the field of condensed matter theory: In
Sec. VI, we consider the Hubbard atom and the free spin of length $S$, for
which we compute $n$-point functions not previously available in the
literature. We conclude in Sec. VII.
Figure 1: (a) Ordering convention for imaginary times in Eq. (9). (b)
Eigenstates and energies of the Hubbard atom. (c) Matsubara correlation
function $G_{S^{x}S^{y}S^{z}}(\omega_{1},\omega_{2})$ with $\omega_{j}=2\pi
m_{j}/\beta$ ($m_{j}\in\mathbb{Z}$, $j=1,2$) for the Hubbard atom (35) at
$\beta=10$, $h=0.1$, $\epsilon=-2$, $U=2$, see Eq. (45). The sharp anti-
diagonal ray $\propto\delta_{\omega_{1}+\omega_{2},0}$ represents an anomalous
term of order $a=1$. The other broadened rays become sharp and anomalous for
$h\rightarrow 0$, see Eq. (49).
## II Definition of Matsubara $n$-point function
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n}\bigr{)}$
We consider a set of $n=2,3,4,...$ operators $\\{A_{1},A_{2},...,A_{n}\\}$
defined on the Hilbert space of a quantum many-body Hamiltonian $H$. The
operators can be fermionic, bosonic or a combination of both types, with the
restriction that there is an even number of fermionic operators. As an
example, $A_{1}=d^{\dagger}d\equiv n$, $A_{2}=d$, $A_{3}=d^{\dagger}$ where
$d^{\dagger}$ and $d$ are canonical fermionic creation and annihilation
operators. A subset of operators is called bosonic if they create a closed
algebra under the commutation operation. They are called fermionic if the
algebra is closed under anti-commutation, see Sec. 1 of Ref. (Tsvelik, 2007).
Spin operators are thus bosonic.
We define the imaginary time-ordered $n$-point correlation functions for
$\tau_{k}\in[0,\beta]$ (Rohringer et al., 2012; Rohringer, 2013),
$G_{A_{1}A_{2}...A_{n}}\left(\tau_{1},\tau_{2},...,\tau_{n}\right)\equiv\left\langle\mathcal{T}A_{1}(\tau_{1})A_{2}(\tau_{2})...A_{n}(\tau_{n})\right\rangle,$
(1)
where $A_{k}(\tau_{k})=e^{\tau_{k}H}A_{k}e^{-\tau_{k}H}$ denotes Heisenberg
time evolution. Here and in the following, $k=1,2,...,n$. The expectation
value is calculated as $\bigl{\langle}...\bigr{\rangle}=\mathrm{tr}[\rho...]$
where $\rho=\exp(-\beta H)/Z$ is the thermal density operator at temperature
$\beta=1/T$ and $Z=\mathrm{tr}\exp(-\beta H)$ denotes the partition function.
Note that other conventions for the $n$-point function differing by a
prefactor are also used in the literature, e.g. Ref. (Kugler et al., 2021)
multiplies with $(-1)^{n-1}$. In Eq. (1), the imaginary time-ordering operator
$\mathcal{T}$ orders the string of Heisenberg operators,
$\mathcal{T}A_{1}(\tau_{1})A_{2}(\tau_{2})...A_{n}(\tau_{n})\equiv\boldsymbol{\zeta}(p)A_{p(1)}(\tau_{p(1)})A_{p(2)}(\tau_{p(2)})...A_{p(n)}(\tau_{p(n)}),$
(2)
where $p$ is the permutation $p\in S_{n}$ such that
$\tau_{p(1)}>\tau_{p(2)}>...>\tau_{p(n)}$ [see Fig. 1(a)] and the sign
$\boldsymbol{\zeta}(p)$ is $-1$ if the operator string
$A_{p(1)}A_{p(2)}...A_{p(n)}$ differs from $A_{1}A_{2}...A_{n}$ by an odd
number of transpositions of fermionic operators, otherwise it is $+1$. The
special case $n=2$, with $\boldsymbol{\zeta}(12)=1$ and
$\boldsymbol{\zeta}(21)=\zeta$ ($\zeta=1$ for $A_{1,2}$ bosonic, $\zeta=-1$
for $A_{1,2}$ fermionic), simplifies to
$\mathcal{T}A_{1}(\tau_{1})A_{2}(\tau_{2})=\begin{cases}A_{1}(\tau_{1})A_{2}(\tau_{2})&:\tau_{1}>\tau_{2},\\\
\zeta A_{2}(\tau_{2})A_{1}(\tau_{1})&:\tau_{2}>\tau_{1}.\end{cases}$ (3)
Imaginary time-ordered correlation functions (1) fulfill certain properties
which we review in the following, see e.g. (Rohringer, 2013) for a more
extensive discussion. First, they are invariant under translation of all time
arguments,
$G_{A_{1}A_{2}...A_{n}}\left(\tau_{1},\tau_{2},...,\tau_{n}\right)=G_{A_{1}A_{2}...A_{n}}\left(\tau_{1}+\tau,\tau_{2}+\tau,...,\tau_{n}+\tau\right),$
(4)
with $\tau\in\mathbb{R}$ such that $\tau_{k}+\tau\in[0,\beta]$. They also
fulfill periodic or anti-periodic boundary conditions for the individual
arguments $\tau_{k}$,
$G_{A_{1}...A_{n}}\left(\tau_{1},...,\tau_{k}=0,...,\tau_{n}\right)=\zeta_{k}G_{A_{1}...A_{n}}\left(\tau_{1},...,\tau_{k}=\beta,...,\tau_{n}\right)$
(5)
where $\zeta_{k}=+1$ or $-1$ if $A_{k}$ is from the bosonic or fermionic
subset of operators, respectively. This motivates the use of a Fourier
transformation,
$\displaystyle G_{A_{1}...A_{n}}\left(\tau_{1},...,\tau_{n}\right)$
$\displaystyle\equiv$
$\displaystyle\beta^{-n}\sum_{\omega_{1},...,\omega_{n}}e^{-i(\omega_{1}\tau_{1}+...+\omega_{n}\tau_{n})}G_{A_{1}...A_{n}}\left(\omega_{1},...,\omega_{n}\right),$
(6) $\displaystyle G_{A_{1}...A_{n}}\left(\omega_{1},...,\omega_{n}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\beta}\mathrm{d}\tau_{1}\cdots\int_{0}^{\beta}\mathrm{d}\tau_{n}e^{+i(\omega_{1}\tau_{1}+...+\omega_{n}\tau_{n})}G_{A_{1}...A_{n}}\left(\tau_{1},...,\tau_{n}\right),$
(7)
where $\omega_{k}=2\pi m_{k}/\beta$ or $\omega_{k}=2\pi(m_{k}+1/2)/\beta$ with
$m_{k}\in\mathbb{Z}$ bosonic or fermionic Matsubara frequencies, respectively,
and $\sum_{\omega_{k}}$ is shorthand for $\sum_{m_{k}\in\mathbb{Z}}$. Note
that fermionic Matsubara frequencies are necessarily nonzero, a property that
will become important later. As we will not discuss the real-frequency
formalisms, we will not write the imaginary unit in front of Matsubara
frequencies in the arguments of
$G_{A_{1}...A_{n}}(\omega_{1},...,\omega_{n})$. Again, note that in the
literature, different conventions for the Fourier transformation of $n$-point
functions are in use. In particular some authors pick different signs in the
exponent of Eq. (7) for fermionic creation and annihilation operators, or
chose these signs depending on operator positions.
Time translational invariance (4) implies frequency conservation at the left
hand side of Eq. (7),
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n-1},\omega_{n}\bigr{)}\equiv\beta\delta_{0,\omega_{1}+...+\omega_{n}}G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n-1}\bigr{)},$
(8)
where on the right hand side we skipped the $n$-th frequency entry in the
argument list of $G$. Note that we do not use a new symbol for the correlation
function when we pull out the factor $\beta$ and the Kronecker delta function.
## III Spectral representation of
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n-1}\bigr{)}$
The integrals involved in the Fourier transformation (7) generate all $n!$
different orderings of the time arguments $\tau_{k}$. As in Ref. (Kugler et
al., 2021) it is thus convenient to use a sum over all $n!$ permutations $p\in
S_{n}$ and employ a product of $n-1$ step-functions $\theta$, with
$\theta(x)=1$ for $x>0$ and $0$ otherwise, to filter out the unique ordering
for which $\beta>\tau_{p(1)}>\tau_{p(2)}>...>\tau_{p(n-1)}>\tau_{p(n)}>0$, see
Fig. 1(a),
$\displaystyle G_{A_{1}...A_{n}}(\tau_{1},...,\tau_{n})$ $\displaystyle=$
$\displaystyle\sum_{p\in
S_{n}}\boldsymbol{\zeta}(p)\left[\prod_{i=1}^{n-1}\theta(\tau_{p(i)}-\tau_{p(i+1)})\right]\left\langle
A_{p(1)}(\tau_{p(1)})A_{p(2)}(\tau_{p(2)})...A_{p(n)}(\tau_{p(n)})\right\rangle.$
(9)
To expose explicitly the time dependence of the Heisenberg operators, we
insert $n$ times the basis of eigenstates and -energies of the many-body
Hamiltonian $H$. Instead of the familiar notation
$\bigl{|}j_{1}\bigr{\rangle},\bigl{|}j_{2}\bigr{\rangle},...$ and
$E_{j_{1}},E_{j_{2}},...$ we employ
$\bigl{|}\underline{1}\bigr{\rangle},\bigl{|}\underline{2}\bigr{\rangle},...$
and $E_{\underline{1}},E_{\underline{2}}$,…. for compressed notation and
denote operator matrix elements as
$A^{\underline{1}\underline{2}}=\left\langle\underline{1}|A|\underline{2}\right\rangle$.
We obtain
$\displaystyle G_{A_{1}...A_{n}}(\tau_{1},...,\tau_{n})=\sum_{p\in
S_{n}}\boldsymbol{\zeta}(p)\left[\prod_{i=1}^{n-1}\theta(\tau_{p(i)}-\tau_{p(i+1)})\right]$
(10)
$\displaystyle\times\frac{1}{Z}\sum_{\underline{1}...\underline{n}}e^{-\beta
E_{\underline{1}}}e^{\tau_{p(1)}E_{\underline{1}}}A_{p(1)}^{\underline{1}\,\underline{2}}e^{(-\tau_{p(1)}+\tau_{p(2)})E_{\underline{2}}}A_{p(2)}^{\underline{2}\,\underline{3}}e^{(-\tau_{p(2)}+\tau_{p(3)})E_{\underline{3}}}...e^{(-\tau_{p(n-1)}+\tau_{p(n)})E_{\underline{n}}}A_{p(n)}^{\underline{n}\,\underline{1}}e^{-\tau_{p(n)}E_{\underline{1}}},$
and apply the Fourier transform according to the definition (7),
$\displaystyle
G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n}\bigr{)}=\frac{1}{Z}\sum_{p\in
S_{n}}\boldsymbol{\zeta}(p)\sum_{\underline{1}...\underline{n}}e^{-\beta
E_{\underline{1}}}A_{p(1)}^{\underline{1}\,\underline{2}}A_{p(2)}^{\underline{2}\,\underline{3}}...A_{p(n)}^{\underline{n}\,\underline{1}}$
(11)
$\displaystyle\times\left[\int_{0}^{\beta}\\!\\!\mathrm{d}\tau_{p(1)}e^{\Omega_{p(1)}^{\underline{1}\,\underline{2}}\tau_{p(1)}}\right]\left[\int_{0}^{\tau_{p(1)}}\\!\\!\mathrm{d}\tau_{p(2)}e^{\Omega_{p(2)}^{\underline{2}\,\underline{3}}\tau_{p(2)}}\right]...\left[\int_{0}^{\tau_{p(n-2)}}\\!\\!\mathrm{d}\tau_{p(n-1)}e^{\Omega_{p(n-1)}^{\underline{n-1}\,\underline{n}}\tau_{p(n-1)}}\right]\left[\int_{0}^{\tau_{p(n-1)}}\\!\\!\mathrm{d}\tau_{p(n)}e^{\Omega_{p(n)}^{\underline{n}\,\underline{1}}\tau_{p(n)}}\right],$
where we defined
$\Omega_{k}^{\underline{a}\,\underline{b}}\equiv
i\omega_{k}+E_{\underline{a}}-E_{\underline{b}}\in\mathbb{C}.$ (12)
In Eq. (11), the first line carries all the information of the system and the
set of operators $\\{A_{1},A_{2},...,A_{n}\\}$. The second line can be
regarded as a universal kernel function defined for general
$\\{\Omega_{1},\Omega_{2},...,\Omega_{n}\\}$ probed at
$\Omega_{k}\in\mathbb{C}$ which depends on the system and correlators via
(12). Upon renaming the $\tau$-integration variables
$\tau_{p(k)}\rightarrow\tau_{k}$, this kernel function is written as follows:
$\displaystyle\mathcal{K}_{n}\left(\Omega_{1},...,\Omega_{n}\right)$
$\displaystyle\equiv$
$\displaystyle\left[\int_{0}^{\beta}\mathrm{d}\tau_{1}e^{\Omega_{1}\tau_{1}}\right]\left[\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}e^{\Omega_{2}\tau_{2}}\right]...\left[\int_{0}^{\tau_{n-2}}\mathrm{d}\tau_{n-1}e^{\Omega_{n-1}\tau_{n-1}}\right]\left[\int_{0}^{\tau_{n-1}}\mathrm{d}\tau_{n}e^{\Omega_{n}\tau_{n}}\right]$
(13) $\displaystyle\equiv$
$\displaystyle\beta\delta_{0,\Omega_{1}+\Omega_{2}+...+\Omega_{n}}K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)+R_{n}\left(\Omega_{1},...,\Omega_{n}\right).$
(14)
In the second line we split $\mathcal{K}_{n}$ into a part $K_{n}$ proportional
to $\beta\delta_{0,\Omega_{1}+\Omega_{2}+...+\Omega_{n}}$ and the rest
$R_{n}$. We dropped $\Omega_{n}$ from the argument list of $K_{n}$ which can
be reconstructed from $\\{\Omega_{1},...,\Omega_{n-1}\\}$.
Finally, we express
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n}\bigr{)}$ of Eq. (11) using
the kernel $\mathcal{K}_{n}$ so that the general $\Omega_{k}\in\mathbb{C}$ get
replaced by $\Omega_{k}^{\underline{a}\,\underline{b}}$ of Eq. (12). For
these,
$\Omega_{p(1)}^{\underline{1}\,\underline{2}}+\Omega_{p(2)}^{\underline{2}\,\underline{3}}+...+\Omega_{p(n)}^{\underline{n}\,\underline{1}}=i(\omega_{1}+\omega_{2}+...+\omega_{n})$,
since the $E_{\underline{k}}$ cancel pairwise. The structure of Eq. (8) (which
followed from time translational invariance) implies that the terms
proportional to $R_{n}$ are guaranteed to cancel when summed over permutations
$p\in S_{n}$, so that only the terms proportional to $K_{n}$ remain. We drop
the $\beta\delta_{0,\omega_{1}+\omega_{2}+...+\omega_{n}}$ from both sides
[c.f. Eq. (8)] and find the spectral representation of the $n$-point
correlation function in the Matsubara formalism,
$\boxed{G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n-1}\bigr{)}=\frac{1}{Z}\sum_{p\in
S_{n}}\boldsymbol{\zeta}(p)\sum_{\underline{1}...\underline{n}}e^{-\beta
E_{\underline{1}}}A_{p(1)}^{\underline{1}\,\underline{2}}A_{p(2)}^{\underline{2}\,\underline{3}}...A_{p(n)}^{\underline{n}\,\underline{1}}\times
K_{n}\left(\Omega_{p(1)}^{\underline{1}\,\underline{2}},\Omega_{p(2)}^{\underline{2}\,\underline{3}},...,\Omega_{p(n-1)}^{\underline{n-1}\,\underline{n}}\right).}$
(15)
An equivalent expression was derived in the literature before (Kugler et al.,
2021), see also Refs. (Shvaika, 2006; Hafermann et al., 2009; Shvaika, 2016)
for the cases of certain small $n$. However, the kernel functions $K_{n}$
where previously only known approximately, for situations involving only a low
order of anomalous terms, see the discussion in Sec. V. We define an
_anomalous_ term of order $a=1,2,...n-1$ as a summand contributing to
$K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)$ that contains a product of $a$
Kronecker delta functions $\delta_{0,x}$, where $x$ is a sum of a subset of
$\\{\Omega_{1},...,\Omega_{n-1}\\}$. As can be seen in Fig. 1(c), these
anomalous contributions to
$G_{A_{1}...A_{n}}\bigl{(}\omega_{1},...,\omega_{n}\bigr{)}$ correspond to
qualitatively important sharp features.
In the next section, we present a simple, exact expression for general
$K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)$. Readers not interested in the
derivation can directly skip to the result in Eq. (26) or its explicit form
for $n=2,3,4,5$ in Sec. (V).
## IV General kernel function
$K_{n}\bigl{(}\Omega_{1},...,\Omega_{n-1}\bigr{)}$
Assuming the spectrum and matrix elements entering Eq. (15) are known, the
remaining task is to find expressions for the kernel function
$K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)$ defined via Eqns. (13) and
(14) as the part of
$\mathcal{K}_{n}\left(\Omega_{1},\Omega_{2},...,\Omega_{n}\right)$ multiplying
$\beta\delta_{0,\Omega_{1}+\Omega_{2}+...+\Omega_{n}}$. To facilitate the
presentation in this section, in Eq. (13) we rename the integration variables
$\tau_{k}\rightarrow\tau_{n-k+1}$ and define new arguments
$z_{n-j+1}=\Omega_{j}$ for $j=1,2,...,n-1$,
$\displaystyle\mathcal{K}_{n}\left(\Omega_{1}=z_{n},\Omega_{2}=z_{n-1},...,\Omega_{n}=z_{1}\right)$
$\displaystyle=\Bigl{[}\int_{0}^{\beta}\mathrm{d}\tau_{n}e^{z_{n}\tau_{n}}\Bigr{]}\Bigl{[}\int_{0}^{\tau_{n}}\mathrm{d}\tau_{n-1}e^{z_{n-1}\tau_{n-1}}\Bigr{]}...\Bigl{[}\int_{0}^{\tau_{3}}\mathrm{d}\tau_{2}\underset{\equiv
h_{2}(\tau_{2})}{\underbrace{e^{z_{2}\tau_{2}}\Bigr{]}\Bigl{[}\int_{0}^{\tau_{2}}\mathrm{d}\tau_{1}\underset{\equiv
h_{1}(\tau_{1})}{\underbrace{e^{z_{1}\tau_{1}}}}\Bigr{]}}}$ (16)
$\displaystyle=\beta\delta_{0,z_{1}+z_{2}+...+z_{n}}K_{n}\left(z_{n},z_{n-1},...,z_{2}\right)+R\left(z_{n},z_{n-1},...,z_{1}\right).$
(17)
As indicated in Eq. (16), we call $h_{k}(\tau_{k})$ the integrand of the
$\int_{0}^{\tau_{k+1}}\mathrm{d}\tau_{k}$ integral for $k=1,2,...,n$. At $k=1$
this integrand is given by $h_{1}(\tau_{1})=e^{z_{1}\tau_{1}}$ and we will
find $h_{k}$ for $k=2,3,...,n$ iteratively. For $z\in\mathbb{C}$, we define
the abbreviations $\delta_{z}\equiv\delta_{0,z}$ and
$\Delta_{z}\equiv\begin{cases}0&\textnormal{{if} }z=0\\\
\frac{{1}}{z}&\textnormal{{if} }z\neq 0\end{cases}$ (18)
and consider the integral (for $p=0,1,2,...$ and $\tilde{\tau}\geq 0$, proof
by partial integration and induction)
$\int_{0}^{\tilde{\tau}}\mathrm{d}\tau\,\tau^{p}e^{z\tau}=\left[\frac{\tilde{\tau}^{p+1}}{p+1}\delta_{z}+p!\left(-1\right)^{p}\Delta_{z}^{1+p}\sum_{l=0}^{p}\frac{(-1)^{l}}{l!}\Delta_{z}^{-l}\tilde{\tau}^{l}\right]e^{z\tilde{\tau}}-p!\left(-1\right)^{p}\Delta_{z}^{p+1}.$
(19)
Recall that we are only interested in the contribution
$K_{n}\left(z_{n},z_{n-1},...,z_{2}\right)$ that fulfills frequency
conservation, see Eq. (17). The $\delta_{z_{1}+z_{2}+...+z_{n}}$ in front of
this term arises from the final $\tau_{n}$ integration of
$h_{n}(\tau_{n})\propto e^{(z_{1}+z_{2}+...+z_{n})\tau_{n}}$ via the first
term in Eq. (19). This however requires that all $z_{k}$ (except the vanishing
ones, of course) remain in the exponent during the iterative integrations.
This requirement is violated by the last term in the general integral (19)
(which comes from the lower boundary of the integral). All terms in
$\mathcal{K}_{n}$ that stem from this last term in Eq. (19) thus contribute to
$R_{n}$ and can be dropped in the following (Kugler et al., 2021). Note
however, that it is straightforward to generalize our approach and keep these
terms if the full $\mathcal{K}_{n}$ is required.
To define the iterative procedure to solve the $n$-fold integral in Eq. (16),
we make the ansatz
$h_{k}(\tau_{k})=\sum_{l=0}^{k-1}f_{k}(l)\tau_{k}^{l}e^{\left(z_{k}+z_{k-1}+...+z_{1}\right)\tau_{k}},$
(20)
which follows from the form of the integral (19) and our decision to disregard
the terms contributing to $R_{n}$. The ansatz $\eqref{eq:h_k-Ansatz}$ is
parameterized by the numbers $f_{k}(l)$ with $l=0,1,...,k-1$. These numbers
have to be determined iteratively, starting from $f_{k=1}(l=0)=1$, read off
from $h_{1}(\tau_{1})=e^{z_{1}\tau_{1}}$, c.f. Eq. (16). Iteration rules to
obtain the $f_{k}(l)$ from $f_{k-1}(l)$ are easily derived from Eqns. (16),
(19) and $\eqref{eq:h_k-Ansatz}$. We obtain the recursion relation
$f_{k}(l)=\sum_{p=0}^{k-1}\tilde{M}_{k-1}(l,p)f_{k-1}(p)$ (21)
written as a matrix-vector product of
$\mathbf{f}_{k-1}=(f_{k-1}(0),f_{k-1}(1),...,f_{k-1}(k-2))^{\mathrm{T}}$ with
the $k\times(k-1)$-matrix
$\tilde{M}_{k-1}(l,p)=\frac{p!}{l!}\left[\delta_{l,p+1}\tilde{\delta}_{k-1}+\theta\left(p-l+1/2\right)\left(-1\right)^{l+p}\tilde{\Delta}_{k-1}^{1+p-l}\right],$
(22)
where $\tilde{\Delta}_{k}\equiv\Delta_{z_{k}+...+z_{2}+z_{1}}$,
$\tilde{\delta}_{k}\equiv\delta_{z_{k}+...+z_{2}+z_{1}}$. The tilde on top of
the $\tilde{\delta}_{k}$ and $\tilde{\Delta}_{k}$ signals the presence of a
sum of $z_{j}$ in the arguments (below we will define related quantities
without tilde for the sum of $\Omega_{j}$). Note that the first (second) term
in brackets of Eq. (22) comes from the first (second) term in square brackets
of Eq. (19).
The next step is to find $K_{n}\left(z_{n},z_{n-1},...,z_{2}\right)$. This
requires to do the integral
$\int_{0}^{\beta}\mathrm{d}\tau_{n}h_{n}(\tau_{n})$ which can be again
expressed via Eq. (19) but with the replacement
$\tilde{\tau}\rightarrow\beta$. Only the first term provides a
$\beta\delta_{z_{1}+z_{2}+...+z_{n}}$ and is thus identified with $K_{n}$. We
find:
$K_{n}\left(z_{n},z_{n-1},...,z_{2}\right)=\sum_{l=0}^{n-1}\frac{\beta^{l}f_{n}(l)}{l+1}.$
(23)
The argument $z_{1}$ that the right hand side of Eq. (23) depends on is to be
replaced by $z_{1}=-z_{2}-z_{3}-...-z_{n}$, in line with the arguments in
$K_{n}\left(z_{n},z_{n-1},...,z_{2}\right)$. Then, to conform with Eq. (15),
we reinstate $\Omega_{j}=z_{n-j+1}$ for $j=1,2,...,n-1$. This amounts to
replacing the terms $\tilde{\delta}_{j}$ and $\tilde{\Delta}_{j}$ that appear
in $f_{n}(l)$ as follows,
$\displaystyle\tilde{\delta}_{j}$ $\displaystyle=$
$\displaystyle\delta_{z_{j}+...+z_{2}+z_{1}}=\delta_{\Omega_{1}+\Omega_{2}+...+\Omega_{n-j}}\equiv\delta_{n-j},$
(24) $\displaystyle-\tilde{\Delta}_{j}$ $\displaystyle=$
$\displaystyle-\Delta_{z_{j}+...+z_{2}+z_{1}}=\Delta_{\Omega_{1}+\Omega_{2}+...+\Omega_{n-j}}\equiv\Delta_{n-j},$
(25)
where we used $\Omega_{1}+\Omega_{2}+...+\Omega_{n}=0=z_{n}+...+z_{2}+z_{1}$.
Finally, we can express Eq. (23) using a product of $n-1$ matrices $\tilde{M}$
multiplying the initial length-1 vector with entry $f_{1}(0)=1$. Transferring
to the $\Omega$-notation by using Eqns. (24) and (25), we obtain
$\boxed{K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)=\\!\\!\\!\sum_{i_{n-1}=0}^{n-1}\sum_{i_{n-2}=0}^{n-2}\\!\cdots\\!\sum_{i_{2}=0}^{2}\sum_{i_{1}=0}^{1}\frac{\beta^{i_{n-1}}}{i_{n-1}+1}M_{1}(i_{n-1},i_{n-2})M_{2}(i_{n-2},i_{n-3})\cdots
M_{n-2}(i_{2},i_{1})M_{n-1}(i_{1},0)}$ (26)
with
$M_{j}(l,p)\equiv\frac{p!}{l!}\left[\delta_{l,p+1}\delta_{j}-\theta\left(p-l+1/2\right)\Delta_{j}^{1+p-l}\right].$
(27)
The closed form expression (26) of the universal kernel, to be used in the
spectral representation (15), is our main result. By definition it is free of
any singularities as the case of vanishing denominators is explicitly excluded
in Eq. (18).
## V Explicit kernel functions
$K_{n}\bigl{(}\Omega_{1},...,\Omega_{n-1}\bigr{)}$ for $n=2,3,4,5$
While the previous section gives a closed form expression for kernel functions
of arbitrary order, we here evaluate the universal kernel functions
$K_{n}\left(\Omega_{1},...,\Omega_{n-1}\right)$ defined in Eq. (14) from Eq.
(26) for $n=2,3,4,5$ and show the results in Tab. 1. In each column, the
kernel function denoted in the top row is obtained by first multiplying the
entries listed below it in the same column by the common factor in the
rightmost column and then taking the sum. The symbols $\delta_{j}$ and
$\Delta_{j}$ for $j=1,2,...,n-1$ which appear in Tab. 1 are defined by
$\delta_{j}$ $\displaystyle\equiv$
$\displaystyle\delta_{\Omega_{1}+\Omega_{2}+...+\Omega_{j},0},$ (28)
$\Delta_{j}$ $\displaystyle\equiv$
$\displaystyle\Delta_{\Omega_{1}+\Omega_{2}+...+\Omega_{j}}\equiv\begin{cases}0&\textnormal{{if}
$\Omega_{1}$+$\Omega_{2}$+...+$\Omega_{j}$=0}\\\
\frac{1}{\Omega_{1}+\Omega_{2}+...+\Omega_{j}}&\textnormal{{if}
$\Omega_{1}$+$\Omega_{2}$+...+$\Omega_{j}$$\neq 0$}\end{cases},$ (29)
compare also to the previous section. As an example, for $n=2$ and $n=3$ we
obtain from Tab. 1
$\displaystyle K_{2}(\Omega_{1})$ $\displaystyle=$
$\displaystyle-\Delta_{1}+\frac{\beta}{2}\delta_{1},$ (30) $\displaystyle
K_{3}(\Omega_{1},\Omega_{2})$ $\displaystyle=$
$\displaystyle+\Delta_{1}\Delta_{2}-\frac{\beta}{2}\delta_{1}\Delta_{2}-\Delta_{1}\delta_{2}\left(\frac{\beta}{2}+\Delta_{1}\right)+\delta_{1}\delta_{2}\frac{\beta}{2}\frac{\beta}{3},$
(31)
respectively. The rows of Tab. 1 are organized with respect to the number $a$
of factors $\delta_{l}$ in the summands. Here, $a=0$ indicates the regular
part and $a=1,2,...,n-1$ indicates anomalous terms. There are $n-1$ _choose
$a$_ anomalous terms of order $a$. Our results are exact and go substantially
beyond existing expressions in the literature – these are limited to $n\leq 3$
(Shvaika, 2006) or to fermionic $n=4$ (Hafermann et al., 2009; Shvaika, 2016;
Kugler et al., 2021) with $a=0,1$ (and $a=2,3$ guaranteed to vanish, see
below) or arbitrary $n$ with $a=0$ (Kugler et al., 2021). Alternative
expressions for the $n=3,4$ kernel functions with $a\leq 1$ were given in
(Kugler et al., 2021), but they are consistent with our kernel functions as
they yield the same correlation functions, see the Appendix.
#anom. | $K_{2}(\Omega_{1})$ | $K_{3}(\Omega_{1},\Omega_{2})$ | $K_{4}(\Omega_{1},\Omega_{2},\Omega_{3})$ | $K_{5}(\Omega_{1},\Omega_{2},\Omega_{3},\Omega_{4})$ | factor for entire row
---|---|---|---|---|---
$a=0$ | $-\Delta_{1}$ | $+\Delta_{1}\Delta_{2}$ | $-\Delta_{1}\Delta_{2}\Delta_{3}$ | $+\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}$ | $1$
$a=1$ | $+\delta_{1}$ | $-\delta_{1}\Delta_{2}$ | $+\delta_{1}\Delta_{2}\Delta_{3}$ | $-\delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}$ | $\frac{\beta}{2}$
| $-\Delta_{1}\delta_{2}$ | $+\Delta_{1}\delta_{2}\Delta_{3}$ | $-\Delta_{1}\delta_{2}\Delta_{3}\Delta_{4}$ | $\frac{\beta}{2}+\Delta_{1}$
| | $+\Delta_{1}\Delta_{2}\delta_{3}$ | $-\Delta_{1}\Delta_{2}\delta_{3}\Delta_{4}$ | $\frac{\beta}{2}+\Delta_{1}+\Delta_{2}$
| | | $-\Delta_{1}\Delta_{2}\Delta_{3}\delta_{4}$ | $\frac{\beta}{2}+\Delta_{1}+\Delta_{2}+\Delta_{3}$
$a=2$ | | $+\delta_{1}\delta_{2}$ | $-\delta_{1}\delta_{2}\Delta_{3}$ | $+\delta_{1}\delta_{2}\Delta_{3}\Delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}$
| | $-\delta_{1}\Delta_{2}\delta_{3}$ | $+\delta_{1}\Delta_{2}\delta_{3}\Delta_{4}$ | $\frac{\beta}{2}\left(\frac{\beta}{3}+\Delta_{2}\right)$
| | $-\Delta_{1}\delta_{2}\delta_{3}$ | $+\Delta_{1}\delta_{2}\delta_{3}\Delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}+\Delta_{1}\left(\frac{\beta}{2}+\Delta_{1}\right)$
| | | $+\delta_{1}\Delta_{2}\Delta_{3}\delta_{4}$ | $\frac{\beta}{2}\left(\frac{\beta}{3}+\Delta_{2}+\Delta_{3}\right)$
| | | $+\Delta_{1}\delta_{2}\Delta_{3}\delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}+\left(\Delta_{1}+\Delta_{3}\right)\left(\frac{\beta}{2}+\Delta_{1}\right)$
| | | $+\Delta_{1}\Delta_{2}\delta_{3}\delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}+\left(\Delta_{1}+\Delta_{2}\right)\left(\frac{\beta}{2}+\Delta_{2}\right)+\Delta_{1}^{2}$
$a=3$ | | | $+\delta_{1}\delta_{2}\delta_{3}$ | $-\delta_{1}\delta_{2}\delta_{3}\Delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}\frac{\beta}{4}$
| | | $-\delta_{1}\delta_{2}\Delta_{3}\delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}\left(\frac{\beta}{4}+\Delta_{3}\right)$
| | | $-\delta_{1}\Delta_{2}\delta_{3}\delta_{4}$ | $\frac{\beta}{2}\left(\frac{\beta}{3}\frac{\beta}{4}+\Delta_{2}\left(\frac{\beta}{3}+\Delta_{2}\right)\right)$
| | | $-\Delta_{1}\delta_{2}\delta_{3}\delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}\frac{\beta}{4}+\Delta_{1}\left(\frac{\beta}{2}\frac{\beta}{3}+\Delta_{1}\left(\frac{\beta}{2}+\Delta_{1}\right)\right)$
$a=4$ | | | | $+\delta_{1}\delta_{2}\delta_{3}\delta_{4}$ | $\frac{\beta}{2}\frac{\beta}{3}\frac{\beta}{4}\frac{\beta}{5}$
Table 1: Universal kernel functions
$K_{n}\bigl{(}\Omega_{1},...,\Omega_{n-1}\bigr{)}$ for $n=2,3,4,5$ defined in
Eq. (14) and calculated from Eq. (26) in Sec. IV. In each column, the kernel
function in the top row is obtained by first multiplying the entries listed
below it in the same column by the common factor in the rightmost column and
then taking the sum, see Eqns. (30) and (31) as examples. The symbols
$\delta_{j}$ and $\Delta_{j}$ are defined in Eqns. (28) and (29). The rows are
organized with respect to the number $a$ of appearances of $\delta_{j}$, i.e.
the order of the anomalous terms.
In the case of purely fermionic correlators (all $A_{k}$ fermionic),
individual Matsubara frequencies $\omega_{k}$ cannot be zero. Thus the complex
numbers
$\Omega_{k}^{\underline{a}\,\underline{b}}=i\omega_{k}+E_{\underline{a}}-E_{\underline{b}}$
of Eq. (12) always have a finite imaginary part, regardless of the
eigenenergies. In this case, only sums of an even number of frequencies can be
zero, and we can simplify $\delta_{1}=\delta_{3}=\delta_{5}=...=0$. The
expressions for the kernels in Tab. 1, now denoted by $K_{n}|_{F}$ for the
fermionic case, simplify to
$\displaystyle K_{2}(\Omega_{1})|_{F}$ $\displaystyle=$
$\displaystyle-\Delta_{1},$ (32) $\displaystyle
K_{4}(\Omega_{1},\Omega_{2},\Omega_{3})|_{F}$ $\displaystyle=$
$\displaystyle\Delta_{1}\Delta_{3}\left[\delta_{2}\left(\frac{\beta}{2}+\Delta_{1}\right)-\Delta_{2}\right],$
(33) $\displaystyle K_{6}(\Omega_{1},...,\Omega_{5})|_{F}$ $\displaystyle=$
$\displaystyle\Delta_{1}\Delta_{3}\Delta_{5}\biggl{\\{}-\Delta_{2}\Delta_{4}-\delta_{2}\delta_{4}\left[\frac{\beta}{2}\frac{\beta}{3}+\left(\Delta_{1}+\Delta_{3}\right)\left(\frac{\beta}{2}+\Delta_{1}\right)\right]$
$\displaystyle+\delta_{4}\Delta_{2}\left(\frac{\beta}{2}+\Delta_{1}+\Delta_{2}+\Delta_{3}\right)+\delta_{2}\Delta_{4}\left(\frac{\beta}{2}+\Delta_{1}\right)\biggr{\\}}.$
This concludes the general part of this work. Next, we consider two example
systems frequently discussed in the condensed matter theory literature. Using
our formalism, we provide analytical forms of correlation functions that to
the best of our knowledge were not available before.
## VI Applications: Hubbard atom and Free Spin
### VI.1 Fermionic Hubbard atom
The Hubbard atom (HA) describes an isolated impurity or otherwise localized
system with Hamiltonian
$H=\epsilon(n_{\uparrow}+n_{\downarrow})+Un_{\uparrow}n_{\downarrow}-h(n_{\uparrow}-n_{\downarrow}),$
(35)
see Fig. 1(b) for a sketch. The HA corresponds to the limit of vanishing
system-bath coupling of the Anderson impurity model (AIM), or vanishing
hopping in the Hubbard model (HM). The particle number operators
$n_{\sigma}=d_{\sigma}^{\dagger}d_{\sigma}$ count the number of fermionic
particles with spin $\sigma\in\\{\uparrow,\downarrow\\}$, each contributing an
onsite energy $\epsilon$ shifted by an external magnetic field $h$ in
$z$-direction. An interaction energy $U$ is associated to double occupation.
Due to its simplicity and the four-dimensional Hilbert space, the correlation
functions for the HA can be found analytically using the spectral
representation. It is therefore often used for benchmarking (Krien and Valli,
2019; Krien et al., 2021; Kappl et al., 2023). The presence of the interaction
term leads to a non-vanishing $n=4$ one-line irreducible vertex function. The
HA serves as an important reference point to study and interpret properties of
the AIM and HM beyond the one-particle level, for example divergences of two-
line irreducible vertex functions (Schäfer et al., 2016; Thunström et al.,
2018; Chalupa et al., 2018; Pelz et al., 2023) and signatures of the local
moment formation in generalized susceptibilities (Chalupa et al., 2021; Adler
et al., 2022). Using the fermionic kernels in Eqns. (32) and (33), we have
checked that our formalism reproduces the results for the 2-point and 4-point
correlators given in Refs. (Hafermann et al., 2009; Rohringer, 2013; Kugler et
al., 2021) for half-filling, $\epsilon=-U/2$ and $h=0$.
Correlation functions including bosonic operators describe the asymptotic
behaviour of the $n=4$ fermion vertex for large frequencies (Wentzell et al.,
2020) or the interaction of electrons by the exchange of an effective boson
(Krien et al., 2019; Gievers et al., 2022). These relations involve
correlation functions of two bosonic operators or of one bosonic and two
fermionic operators, giving rise to expressions possibly anomalous in at most
one frequency argument, i.e. $a\leq 1$.
For the HA, AIM and HM, bosonic correlation functions for $n>2$ have not been
considered thoroughly so far. Only recently, steps in this direction were
taken, particularly in the context of non-linear response theory (Kappl et
al., 2023). The response of a system to first and second order in an external
perturbation is described by $2$\- and $3$-point correlation functions,
respectively. For the HA, physically motivated perturbations affect the onsite
energy via a term $\delta_{\epsilon}n$ or take the form of a magnetic field
$\boldsymbol{\delta}_{h}\cdot\mathbf{S}$. Here, the parameters
$\delta_{\varepsilon}$ and $\boldsymbol{\delta}_{h}$ denote the strength of
the perturbation and we define
$\begin{aligned} n&=n_{\uparrow}+n_{\downarrow}\end{aligned},\quad
S^{x}=\frac{1}{2}\left(d_{\uparrow}^{\dagger}d_{\downarrow}+d_{\downarrow}^{\dagger}d_{\uparrow}\right),\quad
S^{y}=\frac{-i}{2}\left(d_{\uparrow}^{\dagger}d_{\downarrow}-d_{\downarrow}^{\dagger}d_{\uparrow}\right),\quad
S^{z}=\frac{1}{2}\left(n_{\uparrow}-n_{\downarrow}\right).$ (36)
The resulting changes of the expectation values of the density or
magnetization in arbitrary direction are described in second order of the
perturbation by the connected parts of the correlation functions
$G_{A_{1}A_{2}A_{3}}(\tau_{1},\tau_{2},\tau_{3})$, with
$A_{i}\in\\{n,S_{x},S_{y},S_{z}\\}$, where the time-ordered expectation value
is evaluated with respect to the unperturbed system (35) and Fourier
transformed to the frequencies of interest. These objects have been studied
numerically in Ref. (Kappl et al., 2023). In the following, we give explicit,
analytic expressions of the full correlation functions
$G_{A_{1}A_{2}A_{3}}(\omega_{1},\omega_{2})$ (i.e. including disconnected
parts), for arbitrary parameters $\epsilon$, $U$ and $h$ and for all possible
operator combinations using the (bosonic) kernel function $K_{3}$, see Eq.
(31). To the best of our knowledge, these expressions have not been reported
before.
The eigenstates of the HA Hamiltonian (35) [see Fig. 1(b)] describe an empty
($|0\rangle$), singly occupied
($d_{\uparrow}^{\dagger}|0\rangle=|\uparrow\rangle$,
$d_{\downarrow}^{\dagger}|0\rangle=|\downarrow\rangle$) or doubly occupied
($d_{\uparrow}^{\dagger}d_{\downarrow}^{\dagger}|0\rangle=|\uparrow\downarrow\rangle$)
impurity with eigenenergies $E_{0}=0$, $E_{\uparrow}=\epsilon-h$,
$E_{\downarrow}=\epsilon+h$ and $E_{\uparrow\downarrow}=2\epsilon+U$,
respectively. The partition function is
$Z=1+e^{-\beta(\epsilon-h)}+e^{-\beta(\epsilon+h)}+e^{-\beta(2\epsilon+U)}.$
We define
$s=\frac{e^{-\beta\epsilon}}{Z}\sinh(\beta h),\quad
c=\frac{e^{-\beta\epsilon}}{Z}\cosh(\beta h),$ (37)
and obtain all non-vanishing bosonic 3-point correlation functions (where
$\omega_{3}=-\omega_{1}-\omega_{2}$):
$\displaystyle G_{nnn}(\omega_{1},\omega_{2})$
$\displaystyle=2\beta^{2}\delta_{\omega_{1}}\delta_{\omega_{2}}\left(\frac{4e^{-\beta(2\epsilon+U)}}{Z}+c\right),$
(38) $\displaystyle G_{nnS^{z}}(\omega_{1},\omega_{2})$
$\displaystyle=\beta^{2}\delta_{\omega_{1}}\delta_{\omega_{2}}s,$ (39)
$\displaystyle G_{nS^{x}S^{y}}(\omega_{1},\omega_{2})$
$\displaystyle=-\beta\delta_{\omega_{1}}s\frac{\omega_{2}}{\omega_{2}^{2}+4h^{2}},$
(40) $\displaystyle
G_{nS^{x}S^{x}}(\omega_{1},\omega_{2})=G_{nS^{y}S^{y}}(\omega_{1},\omega_{2})$
$\displaystyle=2\beta\delta_{\omega_{1}}\frac{h\ s}{\omega_{2}^{2}+4h^{2}},$
(41) $\displaystyle G_{nS^{z}S^{z}}(\omega_{1},\omega_{2})$
$\displaystyle=\frac{\beta^{2}}{2}\delta_{\omega_{1}}\delta_{\omega_{2}}c,$
(42) $\displaystyle
G_{S^{z}S^{x}S^{x}}(\omega_{1},\omega_{2})=G_{S^{z}S^{y}S^{y}}(\omega_{1},\omega_{2})$
$\displaystyle=-s\frac{\omega_{2}\omega_{3}+4h^{2}}{(\omega_{2}^{2}+4h^{2})(\omega_{3}^{2}+4h^{2})}+\beta\delta_{\omega_{1}}\frac{h\
c}{\omega_{2}^{2}+4h^{2}},$ (43) $\displaystyle
G_{S^{z}S^{z}S^{z}}(\omega_{1},\omega_{2})$
$\displaystyle=\frac{\beta^{2}}{4}\delta_{\omega_{1}}\delta_{\omega_{2}}s,$
(44) $\displaystyle G_{S^{x}S^{y}S^{z}}(\omega_{1},\omega_{2})$
$\displaystyle=2h\
s\frac{\omega_{1}-\omega_{2}}{(\omega_{1}^{2}+4h^{2})(\omega_{2}^{2}+4h^{2})}-\frac{\beta}{2}\delta_{\omega_{3}}c\frac{\omega_{1}}{\omega_{1}^{2}+4h^{2}}.$
(45)
We observe that each conserved quantity, in this case $n$ and $S_{z}$,
contributes an anomalous term $\propto\delta_{\omega_{k}}$in its respective
frequency argument $\omega_{k}$. If an operator $A_{k}$ is conserved
$[H,A_{k}]=0$, the basis over which we sum in Eq. (15) can be chosen such that
both $H$ and $A_{k}$ are diagonal,
$A_{k}^{\underline{1}\,\underline{2}}=A_{k}^{\underline{1}\,\underline{1}}\delta_{\underline{1},\underline{2}}$.
If $A_{k}^{\underline{1}\,\underline{1}}\neq 0$ for some state $\underline{1}$
the vanishing eigenenergy difference leads to the appearance of an anomalous
contribution. If the operators in the correlator additionally commute with
each other, in our case for example $[n,S^{z}]=0$, there exists a basis in
which all operators and the Hamiltonian are diagonal, giving rise to
correlation functions anomalous in all frequency arguments.
In the limit of vanishing field $h\rightarrow 0$, we introduce an additional
degeneracy $E_{\uparrow}=E_{\downarrow}=\epsilon$ in the system, potentially
resulting in additional anomalous contributions. The corresponding correlation
functions can then be obtained in two ways. Either we recompute them using the
kernel function $K_{3}$ or we take appropriate limits, for example
$\lim_{h\rightarrow 0}\frac{h\ \sinh(\beta
h)}{\omega_{k}^{2}+4h^{2}}=\frac{\beta}{4}\delta_{\omega_{k}},$ (46)
resulting in
$\displaystyle G_{nnn}(\omega_{1},\omega_{2})$
$\displaystyle=\beta^{2}\delta_{\omega_{1}}\delta_{\omega_{2}}\frac{2(4e^{-\beta(2\epsilon+U)}+e^{-\beta\epsilon})}{Z},$
(47) $\displaystyle G_{nS^{\alpha}S^{\alpha}}(\omega_{1},\omega_{2})$
$\displaystyle=\beta^{2}\delta_{\omega_{1}}\delta_{\omega_{2}}\frac{e^{-\beta\epsilon}}{2Z}\quad(\alpha\in\\{x,y,z\\}),$
(48) $\displaystyle G_{S^{x}S^{y}S^{z}}(\omega_{1},\omega_{2})$
$\displaystyle=\beta\frac{e^{-\beta\epsilon}}{2Z}(-\delta_{\omega_{1}}\Delta_{\omega_{2}}+\delta_{\omega_{2}}\Delta_{\omega_{1}}-\delta_{\omega_{1}+\omega_{2}}\Delta_{\omega_{1}}),$
(49)
with all other correlation functions vanishing. As already pointed out in Ref.
(Kappl et al., 2023), only the last correlation function retains a nontrivial
frequency dependence due to non-commuting operators.
### VI.2 Free spin $S$
We now consider correlation functions of a free spin of length $S$, without a
magnetic field, so that temperature $T=1/\beta$ is the only energy scale. The
operators $\\{S^{\alpha}\\}_{\alpha=x,y,z}$ fulfill
$S^{x}S^{x}+S^{y}S^{y}+S^{z}S^{z}=S(S+1)$ and the SU(2) algebra
$\left[S^{\alpha_{1}},S^{\alpha_{2}}\right]=i\sum_{\alpha_{3}=\\{x,y,z\\}}\epsilon^{\alpha_{1}\alpha_{2}\alpha_{3}}S^{\alpha_{3}}$,
thus they are bosonic. Since the Hamiltonian vanishes and therefore all
eigenenergies are zero, every $\Omega_{k}^{\underline{a}\,\underline{b}}$ in
the spectral representation (15) can vanish and a proper treatment of all
anomalous terms is essential. As the Heisenberg time dependence is trivial,
$S^{\alpha}(\tau)=S^{\alpha}$, the non-trivial frequency dependence of the
correlators, which can be can be non-vanishing at any order $n$, derives
solely from the action of imaginary time-ordering.
The correlators are required, for example, as the non-trivial initial
condition for the spin-fRG recently suggested by Kopietz et al., Refs. (Krieg
and Kopietz, 2019; Goll et al., 2019, 2020; Tarasevych and Kopietz, 2021;
Tarasevych et al., 2022). However, for $n>3$ they are so far only partially
available: They are either given for restricted frequency combinations, or for
the purely classical case $S^{\alpha_{1}}=S^{\alpha_{2}}=...=S^{\alpha_{n}}$
where the SU(2) algebra does not matter, or for finite magnetic field via an
equation of motion (Goll et al., 2019) or diagrammatic approach (Vaks and
Pikin, 1968; Vaks et al., 1968).
We define the spin raising and lowering operators,
$S^{\pm}=\left(S^{x}\pm iS^{y}\right)/\sqrt{2},$ (50)
which have to appear in pairs for a non-vanishing correlator due to spin-
rotation symmetry. As for the HA, we do not consider connected correlators in
this work for brevity. The classical $S^{z}$-correlator can be found from its
generating functional with source field $h$ (Krieg and Kopietz, 2019),
$\displaystyle\mathcal{G}\left(y=\beta h\right)$ $\displaystyle=$
$\displaystyle\frac{\sinh\left[y(S+1/2)\right]}{(2S+1)\sinh\left[y/2\right]},$
(51) $\displaystyle\left\langle(S^{z})^{l}\right\rangle$ $\displaystyle=$
$\displaystyle\underset{y\rightarrow
0}{\mathrm{lim}}\partial_{y}^{l}\mathcal{G}(y)\equiv b_{l-1},$ (52)
for example $b_{1}=\frac{S}{3}(S+1)$ and
$b_{3}=\frac{S}{15}\left(3S^{3}+6S^{2}+2S-1\right)$ and vanishing $b_{l}$ for
even $l$. For all other correlators involving $\alpha_{k}=\pm$, we adapt Eq.
(15) for the free spin case,
$G_{S^{\alpha_{1}}S^{\alpha_{2}}...S^{\alpha_{n}}}\left(\omega_{1},...,\omega_{n-1}\right)=\sum_{p\in
S_{n}}\left\langle
S^{\alpha_{p(1)}}S^{\alpha_{p(2)}}...S^{\alpha_{p(n)}}\right\rangle
K_{n}\left(i\omega_{p(1)},i\omega_{p(2)},...,i\omega_{p(n-1)}\right),$ (53)
where we made use of the fact that all eigenenergies are zero and the
Heisenberg time evolution is trivial. It is convenient to evaluate the equal-
time correlators in Eq. (53) as
$\left\langle
S^{\alpha_{1}}S^{\alpha_{2}}...S^{\alpha_{n}}\right\rangle=\frac{1}{2S+1}\sum_{m=-S}^{S}\left\langle
m\right|S^{\alpha_{1}}S^{\alpha_{2}}...S^{\alpha_{n}}\left|m\right\rangle\equiv\frac{1}{2S+1}\sum_{m=-S}^{S}\sum_{l=0}^{n}p_{l}m^{l}=p_{0}+\sum_{l=2}^{n}p_{l}b_{l-1}$
(54)
where in the last step we used Eq. (52). We find the real expansion
coefficients $\\{p_{l}\\}_{l=0,1,...,n}$ iteratively by moving through the
string $\alpha_{1}\alpha_{2}...\alpha_{n}$ from the right and start from
$p_{l}=\delta_{0,l}$. Based on the $S^{z}$ eigenstates
$\\{\bigl{|}m\bigr{\rangle}\\}_{m=-S,...,S-1,S}$ we obtain the iteration rules
from $S^{z}\bigl{|}m\bigr{\rangle}=m\bigl{|}m\bigr{\rangle}$ and
$S^{\pm}\bigl{|}m\bigr{\rangle}=\sqrt{1/2}\sqrt{S(S+1)-m(m\pm 1)}\bigl{|}m\pm
1\bigr{\rangle}$. We define an auxiliary integer $c$ that keeps track of the
intermediate state $\left|m+c\right\rangle$, initially $c=0$. Depending on the
$\alpha_{j}$ that we find in step $j=n,n-1...,1$ we take one of the following
actions: (i) For $\alpha_{j}=z$, we update $p_{l}\leftarrow
p_{l-1}+cp_{l}\;\forall l$ and leave $c$ unchanged. It is understood that
$p_{l<0}=0$. (ii) For $\alpha_{j}=+$, we combine the square-root factor
brought by the raising operator with the factor that comes from the necessary
$\alpha_{j^{\prime}}=-$ at another place in the string. We replace
$p_{l}\leftarrow-\frac{1}{2}p_{l-2}-\frac{2c+1}{2}p_{l-1}+\left(\frac{3}{2}b_{1}-c\frac{c+1}{2}\right)p_{l}\;\forall
l$ and then let $c\leftarrow c+1$. (iii) For $\alpha_{j}=-$, we update
$c\leftarrow c-1$ and keep $p_{l}$ unchanged, $p_{l}\leftarrow p_{l}\;\forall
l$.
$n=2$ | $G_{S^{+}S^{-}}(\omega)=G_{S^{z}S^{z}}(\omega)=\beta\delta_{\omega}b_{1}$
---|---
$n=3$ | $G_{S^{+}S^{-}S^{z}}(\omega_{1},\omega_{2})=\beta b_{1}(-\delta_{\omega_{1}}\Delta_{i\omega_{2}}+\delta_{\omega_{2}}\Delta_{i\omega_{1}}+\delta_{\omega_{1}+\omega_{2}}\Delta_{i\omega_{2}})=-iG_{S^{x}S^{y}S^{z}}(\omega_{1},\omega_{2})$
$n=4$ | $G_{S^{z}S^{z}S^{z}S^{z}}\left(\omega_{1},\omega_{2},\omega_{3}\right)=\delta_{\omega_{1}}\delta_{\omega_{2}}\delta_{\omega_{3}}\beta^{3}b_{3}$
$G_{S^{+}S^{+}S^{-}S^{-}}\left(\omega_{1},\omega_{2},\omega_{3}\right)=\beta
b_{1}[2\times\delta_{\omega_{1}}\delta_{\omega_{2}}\delta_{\omega_{3}}\times\frac{\beta^{2}}{5}\left(3b_{1}-\frac{1}{3}\right)+r]$
$G_{S^{+}S^{-}S^{z}S^{z}}\,\left(\omega_{1},\omega_{2},\omega_{3}\right)\,=\beta
b_{1}[1\times\delta_{\omega_{1}}\delta_{\omega_{2}}\delta_{\omega_{3}}\times\frac{\beta^{2}}{5}\left(3b_{1}-\frac{1}{3}\right)-r]$
$r=\Delta_{i\omega_{1}}\Delta_{i\text{$\omega_{2}$}}\left(\delta_{\omega_{1}+\omega_{3}}+\delta_{\omega_{2}+\omega_{3}}-\delta_{\omega_{3}}-\delta_{\omega_{4}}\right)-\left(\delta_{\omega_{1}}\Delta_{i\text{$\omega_{2}$}}^{2}+\delta_{\omega_{2}}\Delta_{i\text{$\omega_{1}$}}^{2}\right)\left(\delta_{\omega_{3}}+\delta_{\text{$\omega_{4}$}}\right)-\Delta_{i\omega_{3}}\Delta_{i\omega_{4}}\left(\delta_{\omega_{1}}+\delta_{\omega_{2}}\right)$
Table 2: Matsubara correlation functions for a free spin-$S$ up to order
$n=4$. Here, $\omega_{4}=-\omega_{1}-\omega_{2}-\omega_{3}$.
Our final results for the free spin correlators are reported in Tab. 2. We
reproduce the known spin correlators for $n=2,3$ and determine the non-
classical correlators $G_{S^{+}S^{+}S^{-}S^{-}}$ and
$G_{S^{+}S^{-}S^{z}S^{z}}$ at order $n=4$, which to the best of our knowledge
were not available in the literature (tha, ). We also confirmed the classical
result for $G_{S^{z}S^{z}S^{z}S^{z}}$, which in our full quantum formalism
requires some non-trivial cancellations. To arrive at our results, we used the
identity
$\displaystyle\Delta_{a+b}\left(\Delta_{a}+\Delta_{b}\right)-\Delta_{a}\Delta_{b}$
$\displaystyle=$
$\displaystyle\delta_{a}\Delta_{b}^{2}+\delta_{b}\Delta_{a}^{2}-\delta_{a+b}\Delta_{a}\Delta_{b}.$
(55)
We finally comment on the relation between the $n=3$ free spin-$S$correlator
$G_{S^{+}S^{-}S^{z}}$ from Tab. (2) and the result for $G_{S^{x}S^{y}S^{z}}$
found for the zero-field limit of the HA in Eq. (49). The operators
$S^{x,y,z}$ for the Hubbard model [c.f. Eq. (36)] project to the singly-
occupied $S=1/2$ subspace spanned by the states
$\bigl{|}\uparrow\bigr{\rangle},\bigl{|}\downarrow\bigr{\rangle}$. Thus, using
$G_{S^{x}S^{y}S^{z}}=iG_{S^{+}S^{-}S^{z}}$ and specializing the free spin
result from Tab. (2) to $S=1/2$ (where $b_{1}=1/4$) we find agreement with the
HA result (49) up to the factor $2e^{-\beta\epsilon}/Z$. This factor
represents the expectation value of the projector to the singly-occupied
sector in the HA Hilbert space and goes to unity in the local-moment regime.
## VII Conclusion
In summary, we have provided exact universal kernel functions for the spectral
representation of the $n$-point Matsubara correlator. Our results are an
efficient alternative to equation-of-motion approaches which often have
difficulties to capture anomalous terms related to conserved or commuting
operators. We expect our results to be useful for various benchmarking
applications, as starting points for emerging many-body methods and for
unraveling the physical interpretation of $n$-point functions in various
settings. Our results also apply in the limit $T\rightarrow 0$ where the
formally divergent anomalous contributions are to be understood as
$\beta\delta_{\omega,0}\rightarrow 2\pi\delta(\omega)$. Some of these Dirac
delta-functions will vanish after subtracting the disconnected contributions,
others indicate truely divergent susceptibilities like the $1/T$ Curie law for
the spin-susceptiblity of the Hubbard atom in the local moment regime
(Rohringer, 2013). Although our work has focused on imaginary frequency
(Matsubara) correlators, with analytical expressions now at hand, it is also
interesting to study the intricacies of analytical continuation to real
frequencies and thus to further explore the connection of Matsubara and
Keldysh correlators (Ge et al., ).
###### Acknowledgements.
We acknowledge useful discussions with Karsten Held, Friedrich Krien, Seung-
Sup Lee, Peter Kopietz, Fabian Kugler, Nepomuk Ritz, Georg Rohringer, Andreas
Rückriegel. We thank Andreas Rückriegel for sharing unpublished results on
4-point free spin correlators and pointing out further simplifications of the
analytical expressions. BS and BS are supported by a MCQST-START fellowship.
We acknowledge funding from the International Max Planck Research School for
Quantum Science and Technology (IMPRSQST) for JH, from the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy EXC-2111 (Project
No. 390814868), and from the Munich Quantum Valley, supported by the Bavarian
state government with funds from the Hightech Agenda Bayern Plus.
## Appendix: Equivalence to convention of Ref. [21]
In Ref. (Kugler et al., 2021) by Kugler, Lee and von Delft (KLD), only regular
($a=0$) and anomalous terms of order $a=1$ have been considered for $n=3,4$.
The corresponding kernel functions were derived from only $(n-1)!$
permutations by setting $\tau_{n}=0$ and $\tau_{i\neq n}>0$, but still applied
to all $n!$ permutations to obtain the correlation functions. For $n=3$, the
resulting kernel function (Eq. (46) in Ref. (Kugler et al., 2021)) reads
$K_{3,\textnormal{KLD}}(\Omega_{1},\Omega_{2})=\Delta_{1}\Delta_{2}-\Delta_{1}\delta_{2}\frac{1}{2}\left(\beta+\Delta_{1}\right)-\delta_{1}\Delta_{2}\frac{1}{2}\left(\beta+\Delta_{2}\right).$
(56)
This can be compared to the corresponding kernel function for $n=3$ found in
our Eq. (31) truncated to $a\leq 1$,
$K_{3}^{a\leq
1}(\Omega_{1},\Omega_{2})=\Delta_{1}\Delta_{2}-\Delta_{1}\delta_{2}\left(\frac{\beta}{2}+\Delta_{1}\right)-\frac{\beta}{2}\delta_{1}\Delta_{2}.$
(57)
Both approaches are equally valid and should yield the same correlation
functions (consistently discarding terms with $a=2$), yet the kernel functions
are obviously different. To resolve this issue, we define the difference of
the kernel functions
$K_{3,\textnormal{diff}}(\Omega_{1},\Omega_{2})=K_{3,\textnormal{KLD}}(\Omega_{1},\Omega_{2})-K_{3}^{a\leq
1}(\Omega_{1},\Omega_{2})=\frac{1}{2}\left(\Delta_{1}^{2}\delta_{2}-\delta_{1}\Delta_{2}^{2}\right)$
(58)
and show that the corresponding contributions to the correlation function
vanishes when summed over cyclically related permutations $p=123,231,312$.
These contributions are given by
$\displaystyle\frac{1}{Z}\sum_{p=123,231,312}\zeta(p)\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{p(1)}^{\underline{1}\underline{2}}A_{p(2)}^{\underline{2}\underline{3}}A_{p(3)}^{\underline{3}\underline{1}}K_{3,\textnormal{diff}}(\Omega_{p(1)}^{\underline{1}\,\underline{2}},\Omega_{p(2)}^{\underline{2}\,\underline{3}})$
(59)
$\displaystyle=\frac{\zeta(123)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{1}^{\underline{1}\underline{2}}A_{2}^{\underline{2}\underline{3}}A_{3}^{\underline{3}\underline{1}}\left(\frac{(1-\delta_{\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{2}}})\delta_{\omega_{1}+\omega_{2}}\delta_{E_{\underline{1}}-E_{\underline{3}}}}{(i\omega_{1}+E_{\underline{1}}-E_{\underline{2}})^{2}}-\frac{\delta_{\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{2}}}(1-\delta_{\omega_{1}+\omega_{2}}\delta_{E_{\underline{1}}-E_{\underline{3}}})}{(i\omega_{1}+i\omega_{2}+E_{\underline{1}}-E_{\underline{3}})^{2}}\right)$
$\displaystyle+\frac{\zeta(231)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{2}^{\underline{1}\underline{2}}A_{3}^{\underline{2}\underline{3}}A_{1}^{\underline{3}\underline{1}}\left(\frac{(1-\delta_{\omega_{2}}\delta_{E_{\underline{1}}-E_{\underline{2}}})\delta_{\omega_{2}+\omega_{3}}\delta_{E_{\underline{1}}-E_{\underline{3}}}}{(i\omega_{2}+E_{\underline{1}}-E_{\underline{2}})^{2}}-\frac{\delta_{\omega_{2}}\delta_{E_{\underline{1}}-E_{\underline{2}}}(1-\delta_{\omega_{2}+\omega_{3}}\delta_{E_{\underline{1}}-E_{\underline{3}}})}{(i\omega_{2}+i\omega_{3}+E_{\underline{1}}-E_{\underline{3}})^{2}}\right)$
$\displaystyle+\frac{\zeta(312)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{3}^{\underline{1}\underline{2}}A_{1}^{\underline{2}\underline{3}}A_{2}^{\underline{3}\underline{1}}\left(\frac{(1-\delta_{\omega_{3}}\delta_{E_{\underline{1}}-E_{\underline{2}}})\delta_{\omega_{3}+\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{3}}}}{(i\omega_{3}+E_{\underline{1}}-E_{\underline{2}})^{2}}-\frac{\delta_{\omega_{3}}\delta_{E_{\underline{1}}-E_{\underline{2}}}(1-\delta_{\omega_{3}+\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{3}}})}{(i\omega_{3}+i\omega_{1}+E_{\underline{1}}-E_{\underline{3}})^{2}}\right).$
Considering the second term of permutation $p=312$ and renaming the summation
variables $\underline{2}\rightarrow\underline{1}$,
$\underline{3}\rightarrow\underline{2}$,
$\underline{1}\rightarrow\underline{3}$ yields
$\displaystyle-\frac{\zeta(312)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{3}^{\underline{1}\underline{2}}A_{1}^{\underline{2}\underline{3}}A_{2}^{\underline{3}\underline{1}}\frac{\delta_{\omega_{3}}\delta_{E_{\underline{1}}-E_{\underline{2}}}(1-\delta_{\omega_{3}+\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{3}}})}{i(\omega_{3}+\omega_{1})+E_{\underline{1}}-E_{\underline{3}}}$
(60)
$\displaystyle=-\frac{\zeta(312)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{3}}}A_{1}^{\underline{1}\underline{2}}A_{2}^{\underline{2}\underline{3}}A_{3}^{\underline{3}\underline{1}}\frac{\delta_{\omega_{3}}\delta_{E_{\underline{3}}-E_{\underline{1}}}(1-\delta_{\omega_{3}+\omega_{1}}\delta_{E_{\underline{3}}-E_{\underline{2}}})}{(i\omega_{3}+i\omega_{1}+E_{\underline{3}}-E_{\underline{2}})^{2}}$
$\displaystyle=-\frac{\zeta(123)}{2Z}\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{1}^{\underline{1}\underline{2}}A_{2}^{\underline{2}\underline{3}}A_{3}^{\underline{3}\underline{1}}\frac{\delta_{\omega_{1}+\omega_{2}}\delta_{E_{\underline{1}}-E_{\underline{3}}}(1-\delta_{\omega_{1}}\delta_{E_{\underline{1}}-E_{\underline{2}}})}{(i\omega_{1}+E_{\underline{1}}-E_{\underline{2}})^{2}},$
where we used $\omega_{3}=-\omega_{1}-\omega_{2}$ and the fact that
$\delta_{\omega_{3}}$ enforces the third operator to be bosonic, such that
$\zeta(312)=\zeta(123)$. This term exactly cancels the first contribution of
permutation $p=123$ in (59). Repeating similar steps for the remaining terms,
we find that the the second term of $p=123$ and the first term of $p=231$ as
well as the second term of $p=231$ and the first term of $p=312$ cancel,
leading to
$\frac{1}{Z}\sum_{p\in\\{123,231,312\\}}\zeta(p)\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{p(1)}^{\underline{1}\underline{2}}A_{p(2)}^{\underline{2}\underline{3}}A_{p(3)}^{\underline{3}\underline{1}}K_{3,\textnormal{diff}}(\Omega_{p(1)}^{\underline{1}\underline{2}},\Omega_{p(2)}^{\underline{2}\underline{3}})=0.$
(61)
Similarly, summing over the second set of cyclically related permutations
$p=132,213,321$ leads to a vanishing result, leading to the conclusion that
$\frac{1}{Z}\sum_{p\in
S_{3}}\zeta(p)\sum_{\underline{1}\underline{2}\underline{3}}e^{-\beta
E_{\underline{1}}}A_{p(1)}^{\underline{1}\underline{2}}A_{p(2)}^{\underline{2}\underline{3}}A_{p(3)}^{\underline{3}\underline{1}}K_{3,\textnormal{diff}}(\Omega_{p(1)}^{\underline{1}\underline{2}},\Omega_{p(2)}^{\underline{2}\underline{3}})=0.$
(62)
Thus we have shown that both kernel functions in Eqns. (56) and (57) are
equivalent as they yield the same correlation functions after summing over all
permutations. The same statement holds true for case of $n=4$ and $a=1$.
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* [43] We thank Andreas Rückriegel for sharing unpublished results on 4-point free spin correlators and pointing out further simplifications of the analytical expressions.
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|
11institutetext: Istituto Nazionale di Astrofisica (INAF) – Osservatorio
Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo, Italy
11email<EMAIL_ADDRESS>
22institutetext: Universidad de Rio Negro, Sede Atlántica - CONICET, Viedma
CP8500, Río Negro, Argentina
33institutetext: Instituto de Astrofísica e Ciências do Espaço, Faculdade de
Ciências, Universidade de Lisboa, Ed. C8, Campo Grande, 1749-016 Lisbon,
Portugal
44institutetext: European Southern Observatory, Karl-Schwarzschild-Strasse 2,
D-85748 Garching bei München, Germany
55institutetext: Department of Physics and Chemistry, University of Palermo,
Palermo, Italy
66institutetext: Donostia International Physics Center (DIPC), Paseo Manuel de
Lardizabal, 4, E-20018 Donostia-San Sebastián, Guipuzkoa, Spain;
77institutetext: IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao,
Spain;
88institutetext: Astrophysics Research Institute, Liverpool John Moores
University, IC2 Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF,
UK
99institutetext: Space Sciences, Technologies and Astrophysics Research (STAR)
Institute, University of Liège, Quartier Agora, 19c, Allée du 6 Aôut, B5c,
B-4000 Sart Tilman, Belgium
1010institutetext: Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain
1111institutetext: Departamento de Astrofísica, Universidad de La Laguna,
E-38206 La Laguna, Tenerife, Spain
1212institutetext: Departamento de Astrofísica, Centro de Astrobiología,
(CSIC-INTA), Ctra. Torrejón a Ajalvir, km 4, Torrejón de Ardoz, E-28850
Madrid, Spain
1313institutetext: School of Physical Sciences, The Open University, Walton
Hall, Milton Keynes MK7 6AA, UK
1414institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, UK
1515institutetext: Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer
de Can Magrans s/n, E-08193, Barcelona, Spain
1616institutetext: Institut d’Estudis Espacials de Catalunya (IEEC), Carrer
Gran Capitá 2-4, E-08034 Barcelona, Spain
1717institutetext: Center for Astrophysics $|$ Harvard & Smithsonian, 60
Garden Street, Cambridge, MA 02138, USA
1818institutetext: Department of Astronomy, University of Florida, P.O. Box
112055, Gainesville, FL 32611-2055, USA
1919institutetext: Istituto Nazionale di Astrofisica (INAF) - Osservatorio
Astronomico di Roma, Via Frascati 33, I-00078 Monte Porzio Catone, Italy
2020institutetext: Universität Heidelberg, Zentrum für Astronomie, Institut
für Theoretische Astrophysik, Albert-Ueberle-Str. 2, D–69120 Heidelberg,
Germany
2121institutetext: Physics and Astronomy Department Galileo Galilei,
University of Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
2222institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
2323institutetext: Departamento de Física Aplicada, Facultad de Ciencias,
Universidad de Alicante, Carretera de San Vicente s/n, E-03690, San Vicente
del Raspeig, Spain
2424institutetext: Dipartimento di Fisica e Astronomia, Università di Bologna,
Via Gobetti 93/2, Bologna I-40129, Italy;
2525institutetext: Istituto Nazionale di Astrofisica (INAF) - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, Bologna
I-40129, Italy
2626institutetext: Space Telescope Science Institute, 3700 San Martin Dr,
Baltimore, MD, 21218, USA
2727institutetext: School of Physics & Astronomy, University of St Andrews,
North Haugh, St Andrews KY16 9SS, UK
2828institutetext: Istituto Nazionale di Astrofisica (INAF) – Osservatorio
Astrofisico di Catania, Via Santa Sofia 78, I-95123 Catania, Italy
2929institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, F-06300 Nice, France; Université Grenoble Alpes,
CNRS, IPAG, F-38000 Grenoble, France
3030institutetext: Astrophysics Group, Keele University, Keele, Staffordshire
ST5 5BG, United Kingdom
3131institutetext: University of Split, Faculty of Science, Department of
Physics, Rudera Boškovića 33, 21000, Split, Croatia
3232institutetext: Johns Hopkins University, 3400 N. Charles Street,
Baltimore, MD 21218, USA
3333institutetext: Department of Astronomy, University of Massachusetts, 710
North Pleasant Street, Amherst, MA 01003, USA
3434institutetext: AURA for the European Space Agency (ESA), ESA Office, Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
3535institutetext: The William H. Miller III Department of Physics &
Astronomy, Bloomberg Center for Physics and Astronomy, Johns Hopkins
University, 3400 N. Charles Street, Baltimore, MD 21218, USA
3636institutetext: Department of Astronomy, University of Texas at Austin,
Austin, TX 78712, USA
3737institutetext: Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, PO Box 218, Hawthorn VIC 3122, Australia
3838institutetext: Department of Physics & Astronomy, University College
London, Gower Street, London WC1E 6BT, UK
3939institutetext: Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg, Mönchhofstr. 12–14, 69120 Heidelberg, Germany
# EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended
Westerlund 1 and 2 Open Clusters Survey††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.
M. G. Guarcello 11 E. Flaccomio 11 J. F. Albacete-Colombo 22 V. Almendros-Abad
11 K. Anastasopoulou 11 M. Andersen 44 C. Argiroffi 1155 A. Bayo 44 E. S.
Bartlett 44 N. Bastian 667788 M. De Becker 99 W. Best 3636 R. Bonito 11 A.
Borghese 10101111 D. Calzetti 3333 R. Castellanos 1212 C. Cecchi-Pestellini 11
J. S. Clark 1313 C. J. Clarke 1414 F. Coti Zelati 15151616 F. Damiani 11 J. J.
Drake 1717 M. Gennaro 26263535 A. Ginsburg 1818 E. K. Grebel 3939 J. L. Hora
1717 G. L. Israel 1919 G. Lawrence 37373838 D. Locci 11 M. Mapelli 20202121 J.
R. Martinez-Galarza 1717 G. Micela 11 M. Miceli 1155 E. Moraux 2222 K. Muzic
33 F. Najarro 1212 I. Negueruela 2323 A. Nota 2626 C. Pallanca 24242525 L.
Prisinzano 11 B. Ritchie 1313 M. Robberto 26263232 T. Rom 22223131 E. Sabbi
2626 A. Scholz 2727 S. Sciortino 11 C. Trigilio 2828 G. Umana 2828 A. Winter
2929 N. J. Wright 3030 P. Zeidler 3434
###### Abstract
Context. With a mass exceeding several $10^{4}$ M⊙ and a rich and dense
population of massive stars, supermassive young star clusters represent the
most massive star-forming environment that is dominated by the feedback from
massive stars and gravitational interactions among stars.
Aims. In this paper we present the ”Extended Westerlund 1 and 2 Open Clusters
Survey” (EWOCS) project, which aims to investigate the influence of the
starburst environment on the formation of stars and planets, and on the
evolution of both low and high mass stars. The primary targets of this project
are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with
the _Chandra_ and JWST observatories. Specifically, the _Chandra_ survey of
Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a
total exposure time of 1 Msec. Additionally, we included 8 archival _Chandra_
/ACIS-S observations. This paper presents the resulting catalog of X-ray
sources within and around Westerlund 1. Sources were detected by combining
various existing methods, and photon extraction and source validation were
carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the
9420 initially provided to ACIS-Extract, reaching a photon flux threshold of
approximately $2\times 10^{-8}$ photons cm${}^{-2}\,$s-1. The X-ray sources
exhibit a highly concentrated spatial distribution, with 1075 sources located
within the central 1 arcminute. We have successfully detected X-ray emissions
from 126 out of the 166 known massive stars of the cluster, and we have
collected over 71000 photons from the magnetar CXO J164710.20-455217.
###### Key Words.:
Galaxies: star clusters: individual: Westerlund 1; Stars: formation; X-rays:
stars
## 1 Introduction
The star formation rate and the properties of the most common star-forming
environments in galaxies vary over time. When considering cosmological
timescales, the star formation rate is known to reach its peak at
approximately z$\sim$2–3 and then gradually decline (e.g., Hopkins & Beacom,
2006; Dunlop, 2011; Madau & Dickinson, 2014). In the local Universe, mergers
play a dominant role in shaping the star formation process in galaxies (Rieke
& Rujopakarn, 2011) as they influence the overall properties of the
interstellar medium. Such interactions happen frequently, and various studies
have demonstrated that interacting galaxies undergo periods of intense star
formation (e.g., Larson & Tinsley, 1978; Smith & Struck, 2010) because of the
considerable impact interactions have on the stars formation process, for
instance from the enhancement of the star cluster formation rate due to close
encounters (e.g. in the Magellanic Clouds) and the ram pressure stripping
enhancing star formation (e.g., jellyfish galaxies). Noticeable examples are
the nearby interacting galaxies M51 and M82, where we observe extreme star
formation taking place in very massive young clusters with masses reaching
several times $10^{5}\,$M⊙ (known as super star clusters; de Grijs et al.,
2001, 2003b, 2003a). Generally, these highly massive star clusters constitute
the dominant star-forming environments in starburst galaxies and are likely
prevalent during the peak era of cosmic star formation (e.g., Figer, 2008;
Adamo et al., 2020).
In the Milky Way, current estimates of the star forming heavily rely on the
methods employed. For example, Robitaille & Whitney (2010) derived a range of
0.68–1.45 M${{}_{\odot}}\,$yr-1 based on the population of young stellar
objects identified in the _Spitzer_ /IRAC survey of the Galactic plane GLIMPSE
(Benjamin et al., 2003). On the other hand, Licquia & Newman (2015) applied a
hierarchical Bayesian statistical method to previous analyses and determined a
star formation rate of about 1.6 M⊙/yr. For comparison, recent estimates of
the star formation rate in M51 range from 4.8 M${{}_{\odot}}\,$yr-1 (from a
158$\mu m$ map of the galaxy Pineda et al., 2018) to 2.7 M${{}_{\odot}}\,$yr-1
(from combined UV+optical spectral energy distribution fitting; Eufrasio et
al., 2017), while in M82 star formation rates of 2-4 M${{}_{\odot}}\,$yr-1
were observed (de Grijs et al., 2001). Nevertheless, all these studies
indicate that our Galaxy does not currently have a high star formation rate.
Consequently, it is not surprising that the Milky Way lacks a prominent
population of super star clusters with masses exceeding 104 M⊙. In order of
distance from the Sun, the most massive clusters known are Westerlund 1 (2.6-5
kpc; Aghakhanloo et al., 2020, Clark et al., 2005), Westerlund 2 ($\sim$4.2
kpc; Vargas Álvarez et al., 2013), NGC 3603 (7.6 kpc; Melena et al., 2008),
the Arches and Quintuplet clusters (both at $\sim$8.5 kpc; Figer et al., 2002,
Figer et al., 1999), Mercer 81 (11 kpc; Davies et al., 2012), and Mercer 30
(12 kpc; de la Fuente et al., 2016). Similar regions in terms of mass, but
with with a low stellar density, are the Cygnus OB2 association (1.4 kpc; Rygl
et al., 2012) and the W3 complex (about 2 kpc; Hachisuka et al., 2006).
Slightly older supermassive star clusters (10-20 Myrs) are found in the
Scutum-Crux arm (about 6 kpc from the Sun; Figer et al., 2006; Davies et al.,
2007; Clark et al., 2009). Despite their limited number, these super star
clusters hold significant importance as they enable the study of star and
planet formation, as well as early stellar evolution, in a star-forming
environment that was characteristic of epochs when the Milky Way had higher
rates of star formation than today and most of the field stars in our Galaxy
formed.
In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey
(EWOCS) project, which is focused on studying star and planet formation and
early stellar evolution in compact starbursts, using Westerlund 1 and 2 as
first science cases. In particular, this paper focuses on the catalog of X-ray
sources detected in the deep _Chandra_ observations of Westerlund 1. The paper
is organized as follows: We present Westerlund 1 and the EWOCS project in
Sect. 2. The EWOCS observations are described in Sect. 3, the procedure for
source detection is described in Sect. 4 and that of source validation and
extraction in Sect. 5. The final catalog of the X-ray sources in Westerlund 1
is described in Sect. 6.
## 2 Westerlund 1 and the EWOCS project
Westerlund 1 is located at RAJ2000=16h47m04s and dec.J2000=-45∘51′05′′,
corresponding to Galactic coordinates l=339.55∘ and b=-00.40∘. The cluster was
discovered by Westerlund (1961) through observations made with the 26-inch
Uppsala-Schmidt telescope at Mt. Stromlo Observatory in Australia. From these
initial observations, it became evident that Westerlund 1 is a very massive
cluster. Today, it is considered to be the most massive young cluster known
within the Milky Way, with mass estimates ranging from approximately
5$\times$104 M⊙ to over 105 M⊙ (Clark et al., 2005; Brandner et al., 2008;
Gennaro et al., 2011; Lim et al., 2013; Andersen et al., 2017).
Despite over 60 years of studies and observations of Westerlund 1, the tension
regarding the parameters of this distinctive cluster remains unresolved. This
is primarily due to its compact nature and the significant extinction that has
long hindered the ability to resolve its low-mass stars.
The distance to the cluster has been a subject of long-standing debate. The
initial estimate by Westerlund (1961) was of 1.4 kpc. However, the same
authors later presented a more distant estimate of 5 kpc based on photographic
observations in the $VRI$ bands (Westerlund, 1968). The first study utilizing
CCD imaging of the cluster (Piatti et al., 1998) reported a distance estimate
of 1.0$\pm$0.4 kpc. However, this estimate was based on the incorrect
assumption that all cluster members were on the main sequence.
Several authors have made distance estimates for Westerlund 1 based on the
analysis of its rich population of massive stars. For example, Clark et al.
(2005) based their estimate on six yellow hypergiants (YHGs), assuming that
these stars have the standard luminosity for this class of objects
(log(L/L⊙)$\sim$5.7; Smith et al., 2004), and adopting an extinction of
AV=11m, resulting in a distance range between 2 kpc and 5.5 kpc. A similar
value was found by Crowther et al. (2006) through infrared analysis of WN and
WC stars. Koumpia & Bonanos (2012) derived a distance of 3.7$\pm$0.6 kpc from
the analysis of the dynamics and geometry of the eclipsing binary W13. By
comparing the cluster locus in color-magnitude diagrams with suitable
isochrones, Brandner et al. (2008) determined a distance of 3.55$\pm$0.17 kpc,
while Gennaro et al. (2011) found a distance of 4.0$\pm$0.2 kpc, and Lim et
al. (2013) reported a distance of about 3.8 kpc. An independent estimate
(3.9$\pm$0.7 kpc) was provided by Kothes & Dougherty (2007) using the radial
velocity of HI clouds in the direction of the cluster, assuming they were
physically connected to Westerlund 1.
More recently, _Gaia_ data have been extensively utilized to measure the
distances of star clusters, providing precise values up to distances of about
1 kpc (Gaia Collaboration et al., 2016). However, for more distant clusters,
careful analysis and assumptions are required to obtain reliable distance
measurements. Consequently, it is not surprising that different estimates of
the distance to Westerlund 1 have emerged from authors who have analyzed
_Gaia_ data. Aghakhanloo et al. (2020) conducted a Bayesian analysis of _Gaia_
data along the line of sight to Westerlund 1 and obtained a mean cluster
parallax of 0.35${}^{+0.07}_{-0.06}$ mas, which corresponds to a distance of
2.6${}^{+0.6}_{-0.4}$ kpc and is in tension with the previous estimate of
approximately 0.19 mas provided by Clark et al. (2020). Focusing on known
members of Westerlund 1, Davies & Beasor (2019) found a distance of
3.9${}^{+1.0}_{-0.64}$ kpc. More recently, Negueruela et al. (2022) carried
out a detailed determination of candidate members in Westerlund 1 using _Gaia_
Early Third Data Release (EDR3; van Leeuwen et al., 2021) data and obtained a
distance of 4.23${}^{+0.23}_{-0.21}$ kpc, suggesting that the cluster is
located in the Norma arm. A similar estimate from the _Gaia_ /EDR3 was
obtained by Navarete et al. (2022).
Given the uncertainty surrounding the distance to Westerlund 1, it is not
surprising that estimates of the cluster’s age provided by different authors
also vary significantly. Age estimates in the range of 3.2 to 5 million years
have been derived using isochrone fitting on the high-mass sequence and
arguments based on the diverse population of massive stars, including Wolf-
Rayet (WR) stars, YHGs, and red supergiants (RSGs; Clark et al., 2005;
Crowther et al., 2006; Brandner et al., 2008; Ritchie et al., 2010; Gennaro et
al., 2011; Koumpia & Bonanos, 2012; Kudryavtseva et al., 2012; Mackey et al.,
2015). These authors found a relatively narrow age spread, with an upper limit
of 0.4 million years indicating that Westerlund 1 likely formed in a single
burst of star formation (Kudryavtseva et al., 2012). However, some of these
estimates are based on arguments that strictly apply to single stars, whereas
it is known that the binary fraction among the massive members of Westerlund 1
is very high (Crowther et al., 2006). More recent studies suggest a more
complex star formation history and a slightly older age (Aghakhanloo et al.,
2020; Beasor et al., 2021; Navarete et al., 2022; Negueruela et al., 2022). In
particular, arguments based on spectral energy distribution fitting and the
luminosity of individual RSGs support an age estimate exceeding 10 Myrs
(Beasor et al., 2021; Navarete et al., 2022), although this estimate is in
tension with other properties of the cluster.
Figure 1: Contours of the pre-EWOCS and EWOCS _Chandra_ observations of
Westerlund 1 overlaid on the combined ACIS event file (left panel) and on an
image in the $Ks$ band obtained with the FourStar infrared camera mounted on
the Magellan 6.5 m telescopes (right panel).
There is a general consensus in the literature regarding other important
properties of Westerlund 1, including its high extinction, large mass, and
notably, its impressive population of massive stars. The significant
extinction toward Westerlund 1 has been acknowledged since the initial
publication on this cluster, where an approximate visual extinction of
AV$\sim$12m has been found (Westerlund, 1961). Subsequent estimates range from
10m to 13m of visual extinction (Negueruela et al., 2010; Lim et al., 2013;
Damineli et al., 2016). There is some disagreement regarding the extinction
law in the direction of Westerlund 1: While according to Negueruela et al.
(2010) it follows the standard law in the $VRI$ bands, Lim et al. (2013) and
Damineli et al. (2016) suggested a steeper extinction law in the near-IR
(RV=2.50$\pm$0.04).
The most remarkable characteristic of Westerlund 1 is its large population of
massive stars (Clark et al., 2005; Ritchie et al., 2009; Clark et al., 2020),
which includes 24 WR stars (Clark & Negueruela, 2002; Negueruela & Clark,
2005; Skinner et al., 2006; Groh et al., 2006; Crowther et al., 2006), the
luminous blue variable (LBV) Wd1-243 (Clark & Negueruela, 2004), ten YHGs with
spectral classes ranging from A5Ia+ to F8Ia+ (Clark et al., 2005)111A
different classification for six YHGs has recently been presented by Beasor et
al. (2023), two blue stragglers (Clark et al., 2019), four RSGs (Wright et
al., 2014b), seven blue hypergiants (BHGs), and over 100 bright OB supergiants
dominated by spectral classes O9-B1 (Negueruela et al., 2010). Most of these
sources are concentrated in the inner region of the cluster, spanning
approximately 1 arcminute, with only a few more isolated massive stars, such
as WR77.
In particular, Westerlund 1 hosts examples of every known transitional
evolutionary phase between H-rich OB supergiants and H-depleted WR stars. This
makes the cluster a unique target for studying massive stars and,
specifically, for understanding how binarity and mass loss impact the
evolutionary paths of these stars and how the initial stellar masses are
linked to the types of compact objects that form at the end of their
evolution. Winds and mass loss in these stars have been extensively studied
with radio and millimeter-continuum observations, which have detected
individual bright sources, such as W9, surrounded by extended nebulae,
providing evidence of intense mass loss in the past (up to several
10${}^{-4}\,$M⊙ per year, Dougherty et al., 2010; Fenech et al., 2018; Andrews
et al., 2019). These short-lived and episodic mass-loss events appear to be
necessary to explain the diversity of evolved massive stars in Westerlund 1.
Westerlund 1 is rich in binary systems. A high binary fraction has been
identified in massive stars through spectroscopic (Ritchie et al., 2022),
radio (Dougherty et al., 2010), infrared (Crowther et al., 2006), and X-ray
(Skinner et al., 2006; Clark et al., 2008, 2019) observations. For instance,
the WR population of Westerlund 1 has an estimated binary fraction of at least
70% (Crowther et al., 2006; Clark et al., 2008). Isolated stars are primarily
found among the mid-B to F hypergiants, with the exception of the LBV star
W243, whose binarity is supported by interferometric (Clark et al., 2019),
X-ray (Mahy et al., 2022), and spectroscopic (Ritchie et al., 2009)
observations.
This harsh environment is expected to have effects on the star formation
process, the evolution and dispersal of protoplanetary disks, and the
formation and early evolution of planets and their atmospheres. While no
studies to date have been able to identify the population of protoplanetary
disks in Westerlund 1 and explore the feedback provided by the starburst
environment on their evolution and dispersal, several authors have attempted
to quantify the cluster’s initial mass function (IMF) to investigate possible
deviations from the universal law. An IMF consistent with the Salpeter (1955)
law has been found by Brandner et al. (2008) in the 3.4–27 M⊙ range, by
Gennaro et al. (2011) extrapolated in the 0.5–120 M⊙ range, and Andersen et
al. (2017) down to 0.15 M⊙ in the outer cluster, while a shallower IMF slope
was found by Lim et al. (2013), integrated in the 0.08-85 M⊙ range.
### 2.1 Previous X-ray observations
X-ray observations of young clusters provide valuable diagnostics for
selecting pre-main-sequence (PMS) stars independently of the presence of
circumstellar disks (e.g., Montmerle, 1996), down to low stellar masses (e.g.,
Getman et al., 2005; Barrado et al., 2011). Additionally, in a cluster rich in
massive stars with a very compact configuration, X-ray observations can reveal
a plethora of processes and physical mechanisms that play an important role in
the evolution of massive stars (Seward et al., 1979; Berghoefer et al., 1996).
It is also worth mentioning that only the _Chandra_ X-Ray Observatory
(Weisskopf et al., 2002) can currently provide the high spatial resolution
required to resolve individual X-ray faint sources in a crowded cluster like
Westerlund 1. Given the designs of future X-ray missions currently in
development, such observations will likely be challenging for quite some time
after the _Chandra_ era.
Both _Chandra_ and XMM have been used in the past to observe Westerlund 1. The
initial observations performed with Chandra reached a depth of approximately
58 ksec (P.I. Skinner) and resolved numerous X-ray sources (Skinner et al.,
2006; Muno et al., 2006; Clark et al., 2008). Skinner et al. (2006) focused on
the WR stars and their spectral properties, detecting 12 out of 24 known stars
and finding strong evidence for the existence of very hot plasma in the
circumstellar environment in the two brightest objects (W72/A and WRB),
strongly suggesting the presence of a colliding winds in these binary systems;
Muno et al. (2006) studied the diffuse X-ray emission and its dominating hard
spectral component, which was later confirmed by Kavanagh et al. (2011) from
48 ksec XMM/Newton observations. These authors detected a strong Fe 6.7 keV
line in the diffuse emission spectrum, indicating its thermal nature. Clark et
al. (2008) found that 46 known high-mass members of Westerlund 1 were detected
in X-rays, and they supposed that the remaining $\sim$60 X-ray sources
detected in these images are likely PMS stars with masses $\leq$1.5 M⊙.
Since its discovery by Muno et al. (2006), the magnetar CXO J164710.2-455216
(CXOU J16) in Westerlund 1 — the brightest X-ray source in the cluster — has
garnered significant attention. Dedicated observations using XMM-Newton and
_Chandra_ /ACIS-S have accumulated a total exposure of 273.14 ksec (P.I.s
Israel, Muno and Schartel) and 94.65 ksec (P.I.s Israel and Rea),
respectively. A typical property of this class of pulsars is their frequent
bursts and recurrent outbursts. In fact, three distinct outbursts from CXOU
J16 have been observed in the past 17 years (Borghese et al., 2019). The first
one occurred in September 2006 and was triggered by a short burst that
released an energy of approximately $10^{39}\,$erg in the 15-150 keV band
(Krimm et al., 2006). It was followed by a second outburst in September 2011
(Israel et al., 2011), during which the pulsar exhibited a peculiar behavior:
The pulse profile evolved from a single peak in the pre-outburst phase to an
energy-dependent tri-peaked profile post-outburst. The overall spectrum
evolved from a single blackbody to a more complex shape that was well modeled
by including an additional hotter blackbody component. The most recent
outburst was again triggered by a short burst detected by the _Swift_ Burst
Alert Telescope in May 2017 (D’Ai et al., 2017). During these intense outburst
activities, the magnetic field strength was estimated to range from
7$\times$10${}^{13}\,$G (a value typical for low-field magnetars, Perna &
Pons, 2011) to $\sim$10${}^{14}\,$G (An et al., 2013; Israel et al., 2007).
Table 1: Pre-EWOCS observations of Westerlund 1. Obs.ID. | Instrument | Exposure | Roll Angle | RA | Dec | Date | P.I.
---|---|---|---|---|---|---|---
| | ksec | degrees | J2000 | J2000 | |
5411 | ACIS-S | 38.47 | 326 | 16:47:05.40 | -45:50:36.70 | 2005-06-18 | Skinner
6283 | ACIS-S | 18.81 | 25 | 16:47:05.40 | -45:50:36.70 | 2005-05-22 | Skinner
14360 | ACIS-S | 19.06 | 242 | 16:47:10.20 | -45:52:16.90 | 2011-10-23 | Israel
19135 | ACIS-S | 9.13 | 22 | 16:47:10.20 | -45:52:17.00 | 2017-05-25 | Rea
19136 | ACIS-S | 13.67 | 331 | 16:47:10.20 | -45:52:17.00 | 2017-06-16 | Rea
19137 | ACIS-S | 18.2 | 295 | 16:47:10.20 | -45:52:17.00 | 2017-07-10 | Rea
19138 | ACIS-S | 18.2 | 86 | 16:47:10.20 | -45:52:17.00 | 2018-02-24 | Rea
20976 | ACIS-S | 16.39 | 86 | 16:47:10.20 | -45:52:17.00 | 2018-02-25 | Rea
### 2.2 The EWOCS project
The pre-EWOCS _Chandra_ observations of Westerlund 1 have been analyzed by
Townsley et al. (2018) in the framework of the Second Installment of the
Massive Star-forming Regions Omnibus X-ray Catalog (MOXC2), identifying 1721
X-ray sources. This work has confirmed that Westerlund 1 is rich in X-ray
bright sources, even though its low-mass stellar content remained undetected
in the pre-EWOCS observations. According to these authors, the X-ray
luminosity limit in the broad band where half of the brighter population is
detected was log(Lx)=30.69, with Lx in erg/s, corresponding to a 1.5 M⊙
star222As stated by Townsley et al. (2018), the corresponding X-ray flux has
been calculated using PIMMS6 assuming a limit of five-counts detection on-
axis, for a source with an APEC thermal plasma with kT=2.7 keV and abundance
0.4$\times$Z⊙, which are typical values for a PMS star (Preibisch et al.,
2005)..
The need to unveil the low-mass population of Westerlund 1 has motivated the 1
Msec observation of Westerlund 1 (P.I. Guarcello) with the _Chandra_ Advanced
CCD Imaging Spectrometer (ACIS-I, Garmire et al., 2003), which, together with
a 18.9 hours Cycle 1 JWST/MIRI and NIRCam observation (program ID 1905, P.I.
Guarcello) and a 48 ksec NICER observation (P.I. Borghese) of CXOU J16,
constitutes the set of new observations of the EWOCS
project333https://Westerlund1survey.wordpress.com/. The main objective of
EWOCS is to use Westerlund 1 and 2 as a test cases for understanding how star
and planet formation, early stellar evolution, and the production of compact
objects occur in a starburst environment. Specifically, the project aims to
achieve the following objectives:
* •
Unveil the low-mass stellar population of Westerlund 1 and 2, both in their
core and halo. X-ray observations are expected to be critically important for
selecting cluster members in the halo, where contamination from background and
foreground sources could affect membership determination based on photometric
data.
* •
Determine the actual stellar content of the clusters, down to the low-mass
regime, mainly thanks to the JWST observations; calculate their IMF down to
the brown dwarf regime, and understand whether the starburst environment
impact the formation of low-mass and very-low mass stars.
* •
Study the clusters properties, particularly age, age spread, morphology and
dynamics. The project aims to understand whether the clusters formed in a
single burst of star formation or through a process spanning several million
years, as well as how and if they will disperse.
* •
Identify the disk-bearing population of the clusters, mainly though the JWST
observations. Combining this with the detection of disk-less stars from the
_Chandra_ /ACIS-I observations and modeling of disks dispersal, we will
finally assess how disks evolve and how planet formation proceeds in a
starburst environment.
* •
If planets can form, understand how they evolve while immersed in such an
environment characterized by high local fluxes of UV and X-ray radiation and
relativistic particles.
* •
Study how binarity and mass-loss affect the evolution of the massive stars in
the clusters, and how their initial mass is mapped into the type of compact
objects formed at the end of their evolution.
* •
Determine whether binarity across stellar masses is different in a starburst
environment.
* •
Study for the first time the status of CXOU J16 far from bursts, which will
allow us to estimate the intrinsic properties of the pulsar.
* •
Search for the expected population of compact objects that have been suggested
to exist in Westerlund 1, since, under specific assumptions, up to $\sim$65
core-collapse supernovae could have already occurred in the cluster (Muno et
al., 2006; Brandner et al., 2008). Besides, Westerlund 1 is one of the few
known star clusters meeting the properties required for the formation of
intermediate mass black holes from runaway coalescence (Portegies Zwart et
al., 2004). As estimated by Clark et al. (2008), such objects, if present and
if currently accreting mass, should be observable with a very deep _Chandra_
observation.
* •
Understanding how stellar winds from massive evolved stars can affect the ISM
to produce diffuse X-ray emission, whether this hot gas could affect star
formation throughout the region, and whether we can prove ongoing accumulation
of polluted material in the cluster core.
Figure 1 shows the contours of the pre-EWOCS and EWOCS observations of
Westerlund 1 and CXOU J16, plotted over the combined _Chandra_ event file and
a $K_{S}$ band image of the cluster and the surrounding area obtained with the
FourStar infrared camera mounted on the Magellan 6.5 m telescopes.
## 3 _Chandra_ observations and data reduction
The EWOCS survey also includes eight pre-EWOCS observations performed with
ACIS-S, two of which were pointed at Westerlund 1 and six at CXOU J16. These
observations were conducted between June 2005 and February 2018 (Table 1).
Additionally, 36 EWOCS observations were carried out with the ACIS-I detector
from June 2020 to August 2021. The aim point of each ACIS-I observation was
adjusted based on the nominal roll angle, as indicated in Table 2. This
adjustment was crucial to for avoiding gaps that cover the cluster core, the
pulsar, or some of the brightest massive members, while ensuring the cluster
remained within the inner arcminute. By adopting this design, we maximized the
benefits of the subarcsecond spatial resolution and sensitivity in the central
part of the ACIS-I detector to observe the cluster core, which is highly
compact and crowded.
Table 2: EWOCS observations Obs.ID. | Exposure | Roll Angle | RA | Dec | Date
---|---|---|---|---|---
| ksec | degrees | J2000 | J2000 |
22316 | 39.55 | 245 | 16:46:59.97 | -45:51:13.70 | 2020-10-04
22317 | 24.75 | 272 | 16:47:00.55 | -45:51:29.59 | 2021-08-14
22318 | 26.72 | 312 | 16:47:03.24 | -45:51:45.84 | 2020-06-25
22319 | 46.45 | 243 | 16:46:59.97 | -45:51:13.70 | 2020-10-09
22320 | 37.58 | 321 | 16:47:05.45 | -45:51:39.01 | 2020-06-20
22321 | 37.58 | 1 | 16:47:04.93 | -45:51:14.41 | 2020-06-02
22977 | 37.57 | 236 | 16:46:59.97 | -45:51:13.70 | 2020-10-22
22978 | 24.75 | 340 | 16:47:05.45 | -45:51:39.01 | 2021-06-12
22979 | 21.79 | 14 | 16:47:07.63 | -45:51:13.62 | 2021-05-28
22980 | 24.75 | 331 | 16:47:05.45 | -45:51:39.01 | 2021-06-16
22981 | 21.85 | 314 | 16:47:03.24 | -45:51:45.84 | 2021-06-24
22982 | 16.85 | 335 | 16:47:05.45 | -45:51:39.01 | 2020-06-13
22983 | 27.72 | 340 | 16:47:05.45 | -45:51:39.01 | 2021-06-09
22984 | 22.61 | 303 | 16:47:03.24 | -45:51:45.84 | 2021-07-02
22985 | 24.75 | 52 | 16:47:07.13 | -45:50:48.14 | 2021-05-01
22986 | 17.67 | 335 | 16:47:05.45 | -45:51:39.01 | 2020-06-11
22987 | 24.75 | 1 | 16:47:04.93 | -45:51:14.41 | 2021-06-04
22988 | 17.85 | 280 | 16:47:02.20 | -45:51:31.62 | 2021-07-27
22989 | 21.79 | 14 | 16:47:07.63 | -45:51:13.62 | 2021-05-27
22990 | 24.75 | 288 | 16:47:01.97 | -45:51:40.91 | 2020-07-17
23272 | 11.92 | 1 | 16:47:04.93 | -45:51:14.41 | 2020-06-03
23279 | 29.69 | 335 | 16:47:05.45 | -45:51:39.01 | 2020-06-12
23281 | 30.49 | 321 | 16:47:05.45 | -45:51:39.01 | 2020-06-21
23287 | 34.61 | 312 | 16:47:03.24 | -45:51:45.84 | 2020-06-26
23288 | 29.18 | 312 | 16:47:03.24 | -45:51:45.84 | 2020-06-26
24827 | 24.75 | 269 | 16:47:00.55 | -45:51:14.83 | 2021-08-21
24828 | 24.75 | 1 | 16:47:04.93 | -45:51:14.41 | 2021-06-04
25051 | 31.66 | 14 | 16:47:07.63 | -45:51:13.62 | 2021-05-28
25055 | 29.68 | 1 | 16:47:04.93 | -45:51:14.41 | 2021-06-05
25057 | 25.25 | 340 | 16:47:05.45 | -45:51:39.01 | 2021-06-13
25058 | 27.22 | 333 | 16:47:05.45 | -45:51:39.01 | 2021-06-10
25073 | 34.62 | 314 | 16:47:03.24 | -45:51:45.84 | 2021-06-25
25096 | 18.14 | 280 | 16:47:02.20 | -45:51:31.62 | 2021-07-29
25097 | 23.59 | 280 | 16:47:02.20 | -45:51:31.62 | 2021-07-30
25098 | 25.43 | 280 | 16:47:02.20 | -45:51:31.62 | 2021-08-01
25683 | 24.74 | 272 | 16:47:00.55 | -45:51:29.59 | 2021-08-15
The total exposure for the pre-EWOCS observations is 151.93 ksec, while for
the EWOCS observations it is 967.80 ksec. The exposure times for the
individual EWOCS observations range from 11.92 ksec (Obs.ID 23272) to 39.55
ksec (Obs.ID 22319), with a mean exposure of 26.33 ksec. The EWOCS
observations span over one year, providing a robust baseline for studying the
X-ray variability of the brightest sources. All EWOCS observations were
conducted using the ACIS-I detector in imaging mode, utilizing all four chips
(I0–I3). The observations were performed in the VERY FAINT mode, which employs
telemetry in 5$\times$5 pixel event islands for improved background
suppression444http://cxc.harvard.edu/cal/Acis/Cal_prods/vfbkgrnd/index.html.
When combined with the pre-EWOCS observations, the total time baseline exceeds
16 years, which is particularly valuable for studying certain sources in the
cluster, such as the magnetar. Figure 2 displays a composite RGB ACIS image of
Westerlund 1, where colors represent different photon energies (red: soft
band, green: medium band, blue: hard band). The image shows both the entire
EWOCS field and a central region of approximately $\sim$3′. In the right
panel, it is evident that the source density is high in the cluster core and
it reveals that the majority of faint sources are predominantly hard, likely
due to high absorption or since they have been observed during periods of
intense magnetic activity such as flares. The list of Chandra datasets used in
this paper, and obtained by the Chandra X-ray Observatory, are contained in
the Chandra Data Collection (CDC) 153 https://doi.org/10.25574/cdc.153
Figure 2: RGB images of the whole ACIS-I field (left panel) and the central
area (right panel) of the composite _Chandra_ images. Soft band (0.5-1.0 keV)
photons are marked in red, medium band (1.0-2.0 keV) photons in green, and
hard band (2-7.9 keV) photons in blue. The brightest source in the southeast
direction is CXOU J16. The two images were smoothed adopting a Gaussian kernel
with a radius of 2 pixels.
### 3.1 Data reduction
The _Chandra_ observations were analyzed using the ”pre-_ACIS_ Extract
workflow” procedure outlined in Townsley et al. (2003) and Broos et al.
(2010). This procedure utilizes various tools integrated within the _Chandra_
Interactive Analysis of Observations (CIAO) software (Fruscione et al., 2006).
We employed versions 4.13 of CIAO along with the CALDB 4.9.5 calibration
files.
The L1-to-L2 processing flow aims to generate calibrated _Chandra_ event files
from the L1 products provided by the _Chandra_ X-Ray Center (CXC). It includes
event energy calibration, refinement of event positions, and correction for
contamination caused by bad pixels and cosmic-ray afterglow. This workflow
utilizes a less aggressive bad pixel table compared to the one produced by
CIAO and it incorporates the $clean55$ algorithm for background reduction.
Additionally, cosmic-ray afterglows are removed, and the source point spread
function (PSF) is improved by disabling the random $\pm$0.25 pixel
randomization. The standard grade filter is applied to events, retaining only
$ASCA$ grades 0, 2, 3, 4, and 6. However, events are not filtered for the
standard _status=0_ requirement, which may result in the exclusion of a
significant number of reliable events.
Afterglows, which are groups of events appearing at the same location in
consecutive CCD frames, can often be mistaken for faint sources. To address
this, the CIAO tool _acis_detect_afterglow_ is typically employed to remove
afterglows. This tool applies a relatively aggressive cleaning approach,
eliminating several false positives. Another tool, _acis_run_hotpix_ , is less
aggressive but it may fail to detect afterglow series with fewer than ten
counts. In this L1-to-L2 procedure, a bifurcated workflow is adopted, where we
applied an aggressive cleaning to the files used for source detection and
validation, and a less aggressive cleaning for the files used in spectral
analysis.
The background light curves were examined to identify and exclude intervals
with intense and fluctuating background. This correction was required only for
Obs.ID 5411, as the background remained relatively stable throughout the other
observations.
The astrometry of the event files was corrected in three steps. In the first
step, we addressed the offset of each Obs.ID relative to Obs.ID 22319, which
is the deepest observation. We utilized _Wavdetect_ to identify the brightest
sources in each observation and cross-matched their positions with those
detected in the Obs.ID 22319 image. Subsequently, we employed the CIAO tools
_wcs_match_ and _wcs_update_ to update the astrometry for each observation. In
the second step, which was part of the L1-to-L2 workflow, we corrected the
astrometry of each event file using the _Gaia_ Third Data Release (DR3; Gaia
Collaboration et al., 2023) astrometric system. This process was repeated as
the third step, but using the brightest sources from the final list of
validated sources (Sect. 6).
Exposure maps were calculated using the standard CIAO tools implemented in the
pre-_ACIS_ Extract workflow for each observation in the broad (0.5–7.9 keV),
soft (0.5–1.0 keV), medium (1.0–2.0 keV), hard (2.0–7.9 keV), and very hard
(4.0–7.9 keV) bands, and subsequently combined. The resulting combined
exposure map in the broad band is displayed in Fig. 3, revealing a deep and
nearly uniform exposure in the central region. This region is sufficiently
large to encompass both the core of Westerlund 1 and a portion of the expected
halo of the cluster (as recently discovered, extended haloes are typically
associated with stellar clusters; Meingast et al., 2021; Prisinzano et al.,
2022).
Figure 3: Combined exposure map in the broad band.
## 4 Source detection
The strategy we employed for source detection aims to maximize the depth of
the EWOCS catalog, even in the core of Westerlund 1. This region presents
challenges due to source confusion and a bright, irregular background, making
the detection of faint sources a complex task (see Figure 4).
Figure 4: Inner region of Westerlund 1 observed with ACIS (left panel) in the
broad band and with HST (right panel) using the F160W filter. In the ACIS
image, source confusion and a high background intensity dominate the cluster
core.
Source detection is implemented using four different methods:
* •
The wavelet-based algorithm _PWDetect_ (Damiani et al., 1997) is applied in
the broad, soft, medium, hard, and very hard energy bands. The detection
threshold we adopted roughly corresponds to 50 spurious sources. We excluded
the outermost regions where the selection resulted in a very large number of
false positives, resulting in 2306 detected sources.
* •
The wavelet-based algorithm _Wavdetect_ (Freeman et al., 2002) is applied to
images in the broad, soft, medium, hard, and very hard energy bands. We set
the _sigmathreshold_ parameter equal to 10-4 and used only two small detection
scales, resulting in 2509 detected sources.
* •
The maximum likelihood reconstruction method developed by Townsley et al.
(2006) is applied in the broad, soft, hard, and very hard energy bands. This
algorithm operates over small tiles across the observed field, making it more
sensitive to the spatial variation in the PSF and background and thus more
capable of detecting faint sources in crowded fields (Broos et al., 2010). The
reconstructed image is first calculated using the Lucy-Richardson algorithm
(Lucy, 1974), and then searched for peaks that identify the positions of point
sources. This method resulted in 7585 detected sources.
* •
A time-resolved deployment of _PWDetect_ , described in more details in the
following, is performed over segments of 10 ksec of the observations. This
method is aimed at detecting faint and variable sources that may only be
significant during specific short time segments in which they were detected.
This method produced a list of 1147 detected sources.
Figure 5 shows a comparison of the spatial distribution of candidate sources
detected using the four methods. In all cases, the cluster appears highly
crowded, with the image reconstruction method being the only one capable of
detecting a large number of sources in the central region of Westerlund 1, as
expected.
Figure 5: Spatial distribution of candidate sources detected with the four
methods (from the top: _Pwdetect_ , _Wavdetect_ , image reconstruction, and
time-resolved Pwdetect). The left panels show the whole ACIS field, those on
the right the inner region. Different colors in the first and second rows mark
sources detected at different energy bands.
### 4.1 Time-resolved PWDetect
We devised a simple time-resolved detection method, tailored to faint
transient sources such as magnetically flaring low-mass stars. These may
remain undetected in the full dataset because of the high background, but may
be detected in a shorter time slice that includes the transient emission,
thanks to the enhanced source-counts/background contrast.
We started by considering 10 ks time slices from each observation segment.
Since exposure times are not multiple of 10 ks, the exposure time of the last
frame was forced to range between 7 ks and 17 ks. Moreover, in order to fully
capture transients that would otherwise be split between two frames, we also
considered intervals shifted in time by half a frame (5 ks).
The 44 EWOCS and pre-EWOCS observations were thus split into 194 frames: 106
were 10 ks long, while the duration of the remaining ones are quite uniformly
distributed between 5 and 16 ks. For each frame, event lists (and exposure
maps) were then extracted in the following five energy bands: 0.5-7.0 keV,
4.0-7.0 keV, 0.5-1.2 keV, 1.2-2.0 keV, and 2.0-7.0 keV, resulting in a total
of 970 event files.
We ran _PWDetect_ twice on each of these 970 event files, once to estimate the
background level and thus evaluate the significance thresholds to obtain the
desired number of spurious sources, and once more for the final detection. For
the first run we adopted a low significance threshold, 4.9$\sigma$, so to
detect as many sources as possible, but also resulting in several spurious
faint sources. The number of background photons was then estimated by
subtracting the detected source photons from the total; we then chose the
final detection threshold so to yields, on average, 0.1 spurious sources per
frame. This was derived from the appropriate significance versus background
curve provided by Damiani et al. (1997).
The final _PWDetect_ runs produced 944 lists of sources555For the observations
in standard ACIS-I configuration, detection was performed only on the most on-
axis CCD (CCD.ID=7). In 26 cases _PWdetect_ crashed or no sources were found.
We did not investigate these cases further for a total of 14178 sources, most
of which are, obviously, repeated detections of the same source in multiple
frames and/or bands. We started cleaning up this large sample by removing
$\sim$1600 extended sources, many of which were unresolved detections of
multiple point sources (extent parameter, as given by _PWDetect_ , larger than
2). We then screened for the remaining sources for possible cosmic rays
afterglow events: for each source we extracted photons from a circle with
radius twice the ”detection scale” provided by _PWDetect_. In 284 cases the
arrival times of all extracted photons were in subsequent 3.14s-long readout
frames, and the detection, a likely afterglow artifact was discarded.
All detection lists were cross-identified and merged in a final source list
using an iterative procedure: first we cross-identified and merged the first
two catalogs. The resulting catalog was then merged with the third original
catalog, and so on for all the 944 catalogs. Identifications were performed
searching the close spatial coincidences with identification radii of each
original detection taken as the 1$\sigma$ uncertainties as estimated by
_PWDetect_ (rounded up to 0.5 arcsec if smaller). The coordinates and
uncertainties or identification radii of merged sources were computed, at each
step, as the uncertainty-weighted means of the coordinates/radii of cross-
identified sources666Since most detections are not independent (they may share
the same photons because of energy band or overlapping time frames), we
computed weighted means only among values belonging to independent groups of
positions. Within each group of dependent detections we chose the coordinates
and radii of the source with the smallest positional uncertainty. At the end
of this process we are left with 1262 cross-matched sources.
Finally, we inspected all the final sources by eye, examining individual
detections in the original event file, and the positions in the _Hubble_ Space
Telescope (HST) H-band image (when available, in the field center) and images
from the Digitized Sky Survey (DSS) and the Two Micron All-Sky Survey (2MASS).
Some cross-identifications were adjusted and a number of ”sources,” which were
not merged by the automatic process above, where merged as they clearly
referred to the same star. The final list counts 1147 sources.
### 4.2 The merged list of candidate sources
At this stage, we generated a list of candidate sources that includes all the
sources detected using the adopted methods. Contamination of this list by
false positives is expected to be large, but we relied on the source-
validation step, described in the next sections, to prune the catalog from
these false and not significant sources.
The lists generated by the four detection methods were cross-matched by eye.
In cases where it was difficult to confidently determine the presence of one
or more nearby sources, we left those entries unmatched between the catalogs.
Additionally, we included in the input list the following sources that were
not detected by any of the aforementioned methods:
* •
446 faint sources from the catalog presented by Townsley et al. (2018), which
are likely not detected in EWOCS observations because of the intrinsic
variability of young stars;
* •
21 massive stars of Westerlund 1 from the list published by Clark et al.
(2020);
* •
47 candidate sources added by eye corresponding to the positions of _Gaia_
sources in or nearby the cluster center.
The final list of candidate sources, which was used as input for the source
validation process in $ACIS$-Extract (AE), consists of 9420 sources.
## 5 Source extraction, validation, and photometry
Source validation and photometry were performed using the AE software in IDL
(Broos et al., 2010)777http://www.astro.psu.edu/xray/acis/acis_analysis.html,
which has been successfully employed in previous X-ray surveys including the
_Chandra_ Carina Complex Project (Townsley et al., 2011), the Massive Young
Star-forming Complex Study in Infrared and X-Rays (MYStIX) survey (Feigelson
et al., 2013), the three Massive Star-forming Regions Omnibus X-ray Catalog
(MOCX) data releases (Townsley et al., 2014, 2018, 2019), the _Chandra_ Cygnus
OB2 Legacy Survey (Wright et al., 2014a), and the Star Formation In Nearby
Clouds (SFiNCs) project (Getman et al., 2017). AE enables the extraction and
validation of sources across multiple observations, generating individual
source spectra and light curves. It utilizes various data analysis software
packages including CIAO, MARX (Davis et al., 2012),
HEASoft888https://heasarc.gsfc.nasa.gov/lheasoft, and the IDL Astronomy User’s
Library (Landsman, 1993).
Following the guidelines provided by the authors and available on the AE
website, we adopted a three-step procedure to compile the X-ray EWOCS catalog:
* •
Initially, sources were extracted and validated using a parameter defined by
AE, which helps distinguish between genuine and false sources. This step was
repeated iteratively until no more false sources were identified and removed
(Sect. 5.1).
* •
Subsequently, source positions were updated, followed by another round of
source validation process (Sect. 5.2).
* •
Once the catalog reached a stable state, we performed the photometric
procedure, to extract source events comprehensively and calculate the primary
spectral and temporal properties for each source across multiple energy bands
(Sect. 5.3).
### 5.1 Source validation
The AE procedure assesses the local PSF at the given position of each source
and defines extraction regions based on the 1.5 keV local PSF, ensuring they
do not overlap with neighboring sources. In the case of close pairs, the
extraction region of the fainter source is progressively reduced to prevent
overlap until it reaches 40% of its original size. Once this threshold is
reached, if the two extraction regions still overlap, AE further reduces the
size of the brighter source until the regions no longer overlap. If overlap
persists even when both extraction regions are reduced to 40%, AE either
discards the specific observation or automatically removes the fainter source.
The local background is determined within an optimized region surrounding the
source. For isolated sources, this region is delimited by an inner radius,
which is 1.1 times the radius encompassing 99% of the PSF, and an outer radius
large enough to collect at least 100 background events not associated with
nearby sources. AE adjusts the size of the background-extraction region to
ensure that Poissonian noise contributes no more than 3% to the background
uncertainty. However, in crowded regions, defining a region with 100 events
may not be feasible. In such cases, AE employs a different calculation that
incorporates the contribution from nearby bright sources and a model
accounting for the spatial variation of the background.
Source validation relies on a parameter provided by AE called _prob_no_source_
(PB), which represents the probability that there is no real source at a given
position. In our case, where multiple observations of a source are available,
AE calculates PB based on the extractions with the highest source
significance. To differentiate between valid and spurious sources, we applied
a threshold of PB=0.01, consistent with previous studies. Since the removal of
not valid sources could potentially impact the extraction region and
background of valid sources, the procedure is iterated until the catalog
reaches convergence and no further spurious sources are detected.
After the first iteration, we conducted a visual inspection of sources flagged
by AE as potentially resulting from the hook-shaped feature of the
PSF999http://cxc.harvard.edu/ciao/caveats/psf_artifact.html. This feature can
account for up to 5% of the source flux and its position is influenced by the
roll-angle, making it distinguishable from the actual source only in a few
cases where the real source is both on-axis and sufficiently bright. Figure 6
illustrates an example of a source (MOXC2) that was flagged by AE as a
potential PSF hook and subsequently removed after visual inspection.
Figure 6: Example of a source (label MOXC2) that was excluded as a potential
product of a PSF hook near a brighter source (label c10200). The green
contours outline the extraction regions of MOXC2 in all observations, while
the red polygons indicate the locations where the PSF hook may appear in each
observation, depending on the roll-angle.
AE also identifies sources that are expected to suffer from significant
pileup101010http://cxc.harvard.edu/ciao/why/pileup_intro.html (which is the
loss of information due to different incident photons registered as a unique
event by the detector). In our case, the only source affected by piled source
is the magnetar.
### 5.2 Positions update and visual review
For each source, AE calculates three different position estimates. The first
estimate is obtained by taking the mean value of the positions of the events
associated with the source (mean-data position). However, this estimate may be
inaccurate for large off-axis angles and in cases where there are significant
offsets between the true source position and the extraction region (which can
happen when the PSF is asymmetric). To obtain a more accurate estimate in
these cases, AE correlates the source PSF with the spatial distribution of
extracted events (PSF position). This calculation takes into account the
combination of several Obs.IDs by using the PSF calculated in each
observation. Both of these estimates can be influenced by nearby sources. In
crowded fields, a third estimate is provided by AE using the reconstructed
image of the source’s neighborhood. It identifies the position of the closest
peak in the reconstructed image. According to AE’s recommendations, the mean-
data position is used for on-axis sources, the PSF position is used for off-
axis sources, and the image reconstruction position is used for sources in
crowded regions. The repositioning of sources was performed twice, with each
step followed by a new sequence of iterations for source validation, as
described in the previous section.
Before conducting the visual review of validated sources, the astrometry of
both the X-ray sources and the main products file was corrected using the
_Gaia_ /DR3 astrometric system. After this step, and when catalog stability
was achieved again, we conducted a visual review of specific critical sources,
including very faint sources that could affect the size of the extraction
region of nearby bright sources and suspected afterglows. The decisions made
during the visual review were guided also by the presence of high-probability
optical and/or infrared counterparts. After the visual review, a new round of
source validation was performed.
### 5.3 Spectral extraction
After 21 iterations of the source validation process, the catalog reached
stability, with a total of 5963 validated X-ray sources. The final step
involved the extraction of X-ray events and the estimation of X-ray properties
in 17 energy bands, merging all available observations in a consistent manner.
In addition, AE generates light curves and spectra for each source, although
these will not be discussed in this paper. AE performs this calculation by
excluding observations where the sources are observed off-axis to improve the
overall signal-to-noise ratio. However, in our case, this correction was not
necessary due to the design of our survey. The calculated quantities include
source counts, net counts, photon flux in photon/cm2/s, and the quartiles of
photons energy.
Figure 7: Extraction regions of the validated sources across the entire merged
ACIS image (left panel), and extraction regions in the central area of
approximately $3^{\prime}$ in size (right panel).
Figure 7 depicts the spatial distribution of the validated sources in the
merged ACIS event files. In the left panel it is evident that there is a high
concentration of validated sources toward the center of the cluster, as well
as a significant number of sources surrounding the cluster core. This
indicates that we have detected stars associated with the extended halo of
Westerlund 1. This will be further investigated in upcoming papers of this
series, which will focus on source classification and the identification of
optical/infrared (OIR) counterparts. The right panel also highlights how this
survey has pushed to the limits of _Chandra_ in resolving individual stars
within such a densely populated stellar cluster with a bright and irregular
background. In fact, in the actual core of the cluster, where the background
is both intense and variable, a few tens of sources that were initially
included as input to AE were subsequently discarded during the validation
process (see Fig. 8). The limited number of validated sources in the central
region can be attributed to the intense background, and it is likely that many
of these discarded sources are indeed genuine X-ray sources. Although we did
not attempt to recover these stars, in future papers of this series their
candidate OIR counterparts will be analyzed in order to estimate the fraction
of real sources that we have excluded.
Figure 8: Extraction regions of the validated sources and positions of the
input candidate sources (crosses) within the central 1 arcmin region
## 6 The final catalog
It is informative to analyze the number of sources detected using the various
methods we employed and assess how many have survived the pruning process.
Table 3 presents the total number of input sources for each detection method,
as well as the fraction of these sources within 1′′ and 3′′ of a source in the
final catalog (source positions changed during the pruning process and thus an
exact position match was not possible). The image reconstruction method is the
only one that experienced significant pruning of the input catalog, as it
selects sources that are too faint according to the adopted PB threshold.
According to Table 3, and considering also that the number of sources in the
input _PWDetect_ list of candidate sources more distant than 3′′ from any
source in the Image reconstruction input catalog is low, but not negligible
(314, 688 for _Wavdetect_), it is evident that in complex fields like these,
deploying different detection methods is crucial for optimizing the number of
detected sources.
Table 3: EWOCS sources and detection methods. Detection method | Input N | within 1′′ | within 3′′
---|---|---|---
Image reconstruction | 7585 | 0.29 | 0.30
_PWDetect_ | 2306 | 0.87 | 0.92
_Wavdetect_ | 2509 | 0.77 | 0.82
Time resolved _PWDetect_ | 1147 | 0.77 | 0.82
Massive stars | 21 | 0.34 | 0.81
Townsley et al. (2018) | 446 | 0.48 | 0.66
Added by eye | 47 | 0.38 | 0.95
Given the design of the EWOCS survey and the compact nature of Westerlund 1,
it is not surprising that the majority of sources are observed at low off-axis
angles, as depicted in Fig. 9. Specifically, 63.7% of the sources (3485/5464)
are located within 1 arcminute from the field center, and 87.7% are within 3
arcminutes. Consequently, source positions are generally well-determined, with
a median position error of 0.17′′ and a 75% quantile position error of 0.27′′.
Position errors are estimated from the single-axis standard deviations of the
PSF inside the extraction region and the number of counts extracted. This
precision is crucial for the search of OIR counterparts and for dynamics
studies.
Figure 9: Distribution of the off-axis angles of the EWOCS X-ray sources.
As depicted in Fig. 10, the catalog is predominantly composed of faint
sources. In the broad band, the median value of the net counts is 12.9 counts.
There are 607 sources (10.2%) with fewer than 5 net counts and only 69 sources
(1.2%) with fewer than 3 counts. It is well known that the sensitivity of
ACIS-I decreases with the off-axis angle. This must be taken in consideration
when comparing the spatial distribution of X-ray sources detected with ACIS-I
to those detected with other instruments. Fig. 11 illustrates the spatial
distributions of EWOCS X-ray sources with fewer than 12.9 net counts and those
with more net counts. The former sample exhibits a higher concentration in the
center of the field, with only 37 sources having an off-axis angle larger than
7′. This region is considerably large compared to the size of Westerlund 1, so
studies based on the spatial distribution of cluster members would not be
significantly affected by the decline in sensitivity with the off-axis angle.
Given the design of the EWOCS observations and the intricate procedure we
employed for source detection and validation, it is not currently feasible to
provide a reliable estimate of catalog completeness without making strong and
unverified assumptions about cluster properties, its morphology, and both mass
and LX distributions. Instead, we prefer to discuss the achieved completeness
in future papers of this series, once the identification of OIR counterparts
and the determination of true cluster members have been accomplished. In
Appendix C, however, we present a simplified analysis of completeness based on
different assumptions regarding cluster morphology, along with simulations
conducted using the MARX simulator.
Figure 10: Distributions of source net counts in the broad, soft, medium,
hard, and very hard energy bands. The number of sources with more than 50
counts is indicated in the top-right corner of each panel. Figure 11: Spatial
distribution of EWOCS X-ray sources, sorted into two bins based on their net
counts in the broad band.
The distribution of the median photon energy for the EWOCS X-ray sources is
shown in Fig. 12. The median value of the distribution is 2.8 keV. In the case
of young stellar populations in clusters with low extinction, the median
photon energy serves as a reliable indicator of membership since the coronal
plasma temperature in young stars is typically higher than in older stars.
However, since interstellar absorption is significant in the direction of
Westerlund 1, it becomes challenging to differentiate between the absorbed
background population and the young stars within the cluster. However, the
secondary peak observed at energies below 2 keV in the Emed distribution could
potentially be attributed to a foreground population. Nevertheless, the
spatial distribution of these soft sources does not differ significantly from
that of the more energetic sources. Additionally, Fig. 12 includes a
comparison between the photon median energy distribution of the EWOCS X-ray
sources and the X-ray catalog published by Townsley et al. (2018) based on
pre-EWOCS observations, which clearly exhibits a peak below 2 keV. The evident
differences in the two distributions can be due to a combination of factors:
the presence of a large population of stars associated with Westerlund 1 in
the EWOCS catalog (which is a factor of $\sim$5 deeper in X-ray photon flux
compared to the catalog published by Townsley et al. 2018, considering the
faintest sources in the two catalogs), as well as the decline in sensitivity
in the soft band of the ACIS detector with the years, and the better
sensitivity toward soft events of the ACIS-S detector compared with ACIS-I.
Figure 12: Distribution of the median photon energy in the broad band for the
validated EWOCS X-ray sources (black) and the catalog published by Townsley et
al. (2018), in red.
The catalog also includes a measure of source flux provided by AE: the photon
flux (Fphotons), which is calculated as the ratio of the source net counts to
the product of the mean effective area and nominal exposure time (thus
expressed in units of photons cm${}^{-2}\,$s-1). A model-independent estimate
of the apparent source energy flux can be calculated as
1.602$\times$10${}^{-9}\times$E$\rm{}_{med}\times$Fphotons. The coefficient is
derived from the conversion between keV and erg, as determined by Getman et
al. (2010).
In Appendix A, we show ten rows of the X-ray EWOCS catalog, which is available
in full at the CDS. We have also made the output table produced by AE
available on the EWOCS website in its original IDL
format.111111https://westerlund1survey.wordpress.com/.
### 6.1 Specific sources
Not surprisingly, CXOU J16 is the brightest source in the EWOCS X-ray catalog,
with a total of 71601$\pm$268 net counts collected and a photon flux of
3.45$\times$10${}^{-4}\,$photons cm${}^{-2}\,$s-1. The source, which is
strongly piled-up, is quite isolated and produces a surrounding bright
background, with the closest source being at about 8 arcsec. The pulsar will
be analyzed in detail in future papers of this project.
As explained in Sect. 2, Westerlund 1 hosts a unique ensemble of massive stars
caught in different evolutionary stages. Understanding the mechanisms
responsible for the emission of X-rays and studying both binarity and the
circumstellar environment in these stars is of primary importance for EWOCS.
We visually inspected the X-ray counterparts of massive stars published by
Clark et al. (2020) and found 126 coincidences out of the 166 listed massive
stars. The results are listed in Table LABEL:tab_massive. In the vast majority
of cases, there was a clear one-to-one correspondence between the sources in
the two catalogs. There are a few uncertain cases, which can be easily
identified by repeated massive star IDs, EWOCS objects, or large separations.
The brightest X-ray massive star in the EWOCS catalog is the SgB[e] star W9
(Clark et al., 2014), with nearly 8000 net counts collected in the broad band.
The intense X-ray brightness (L$\rm{}_{X}\sim$3.6$\times 10^{33}\,$erg/s) and
the hardness of the spectrum, as previously reported by Clark et al. (2008),
are consistent with the evidence of intense mass loss rate, estimated to be
around 10${}^{-5}\,$M⊙/yr (Andrews et al., 2019), and strong indications of
binarity (Ritchie et al., 2022). W9 is the brightest source in the cluster
also at radio wavelengths (Andrews et al., 2019), millimeter band (Fenech et
al., 2018), and it shows very bright mid-IR emission (Clark et al., 1998).
In terms of X-ray luminosity, W9 is followed by the post-binary blue straggler
W30 (O4-5Ia+ Clark et al., 2019, 2008), for which a putative orbital period of
approximately 6.2 days has been identified from a radial velocity series
analyzed by Ritchie et al. (2022). We have detected nearly 6000 net counts in
the broad band for W30. After W9 and W30, the list of sources with net counts
ranging between 190 and 5400 photons includes most of the known WR stars and
OB supergiant binary systems.
The deep EWOCS observations provide, for the first time, candidate X-ray
detections for some normal giant and subgiant stars, such as W50b and W1051.
The lack of detection in the pre-EWOCS observations has been explained as a
natural consequence of the lower intrinsic bolometric luminosity of these
stars compared to more evolved massive stars in the cluster (Clark et al.,
2019). The detection in the EWOCS observations supports this hypothesis. Faint
X-ray counterparts have also been found for the two YHGs W4 and W8 (with 22
and 73 net counts, respectively), whose nature has been recently discussed by
Beasor et al. (2023), who classified them as yellow supergiants. Additionally,
a faint counterpart has been detected for the BHG W1049 (with
$14.16^{20.4}_{8.4}$ net counts). For the first time, faint counterparts have
been found for the four O9.5II SB1 stars W1022, W1050, W1056, and W1060, as
well as for the B1.5II star W1048. We also confirm the relatively faint
($75.75^{85.7}_{65.8}$ net counts) and soft (median photon energy of 1.9 keV)
X-ray emission from the SB2 star (B0.5I+OB) W10, as previously reported by
Clark et al. (2008). Ritchie et al. (2022) attributed the X-ray properties of
this star to the possibility that the pre-EWOCS observations were made at a
phase where the wind collision zone was weak or obscured. However, given the
length of the EWOCS observations, it is more likely that these properties are
intrinsic to the star.
It is quite interesting that out of the 124 sources in our catalog with more
than 100 net counts, 94 do not readily match any known massive stars in the
cluster. This subset will be studied in detail in future works of this series
to determine their nature. It is intriguing that this sample does not seem to
follow the distribution of photon energy shown in Fig. 12. In fact, its photon
median energy distribution exhibits three distinct peaks: one below 2 keV
(which may be dominated by foreground stars), one between 2.5 keV and 3 keV
(compatible with cluster stars), and one between 3.8 keV and 4.3 keV (which
could be influenced by background sources or flaring low-mass cluster
members).
We also provide a list of positions for the unvalidated candidate X-ray
sources on the EWOCS website121212https://westerlund1survey.wordpress.com/.
This list will be cross-matched with existing optical and infrared catalogs of
Westerlund 1 to determine the fraction of rejected sources that could
potentially be true counterparts of cluster members. Likewise, identifying
optical and infrared counterparts will enable us to assess the level of
contamination and the fraction of expected spurious sources in the EWOCS X-ray
source catalog, as well as determine the completeness limit achieved by our
survey.
## 7 Conclusions
In this paper, we present the EWOCS project and a new list of X-ray sources in
the young supermassive star cluster Westerlund 1 and its surrounding area. The
EWOCS project aims to investigate the impact of the starburst environment on
the formation process of stars and planets, the dispersal of protoplanetary
disks, and the evolutionary pathway of massive stars.
Here we present the 1 Msec _Chandra_ /ACIS-I EWOCS observations of Westerlund
1, the workflow for data reduction, the procedure for source detection and
validation, and the spectral extraction of the validated sources. Initially,
we generated a preliminary list of 9420 candidate X-ray sources using the
image reconstruction method, _PWDetect_ , _WAVDETECT_ , and a specific
deployment of _PWDetect_ focused on identifying flaring stars that exhibited a
significant signal above the background for a brief duration. Additionally, a
few sources were manually added or obtained from existing catalogs of
Westerlund 1 sources. From these input sources, we compiled the EWOCS catalog
of X-ray sources in Westerlund 1 of 5963 sources successfully validated using
the IDL-based software AE.
The median value of net counts in the EWOCS X-ray catalog is approximately 13
counts, with about 10% of sources having fewer than 5 net counts detected in
the broad energy band. The distribution of the median photon energy of the
sources peaks at approximately 2.8 keV, with a contribution from unrelated
(foreground and background) sources that is challenging to distinguish from
the candidate cluster members. The brightest source in the catalog is the
magnetar CXO J164710.2-455216, with over 70000 net counts detected in the
broad band. It is followed by several massive stars in Westerlund 1, including
the SgB[e] star W9, the post-binary blue straggler W30, and some WR stars and
supergiants in binary systems. Out of the 166 known very massive stars in
Westerlund 1, we have identified a reliable X-ray counterpart for 126 of them.
Additionally, we have made the first detection of an extended and rich halo
surrounding the core of Westerlund 1, which will be crucial in assessing the
cluster’s true mass content, formation, and evolution.
###### Acknowledgements.
We acknowledge the referee for his/her careful reading of our paper and
suggestions. M.G.G. is also indebted to Leisa K. Townsley and Patrick S. Broos
for their valuable suggestions and advice on data reduction and ACIS
extraction. M. G. G., C. A., R. B., E. F., G. L. I., L. P., and S. S.
acknowledge the INAF grant 1.05.12.05.03. K.M. acknowledges support from the
Fundação para a Ciência e a Tecnologia (FCT) through the CEEC-individual
contract 2022.03809.CEECIND and research grants UIDB/04434/2020 and
UIDP/04434/2020. Support for this work was also provided by the National
Aeronautics and Space Administration through _Chandra_ Proposal 21200267
issued by the _Chandra_ X-ray Center, which is operated by the Smithsonian
Astrophysical Observatory for and on behalf of the National Aeronautics Space
Administration under contract NAS8-03060. I.N. is partially supported by the
Spanish Government Ministerio de Ciencia e Innovación (MCIN) and Agencia
Estatal de Investigación (MCIN/AEI/10.130 39/501 100 011 033/FEDER, UE) under
grant PID2021-122397NB-C22, and also by MCIN with funding from the European
Union NextGenerationEU and Generalitat Valenciana in the call Programa de
Planes Complementarios de I+D+i (PRTR 2022), project HIAMAS, reference
ASFAE/2022/017. M. G. G. and R. B. also acknowledge partial support from the
Grant INAF 2022 YODA; R. B. also from the project PRIN-INAF 2019
”Spectroscopically Tracing the Disk Dispersal Evolution”. The scientific
results reported in this article are based on observations made by the
_Chandra_ X-ray Observatory. M.M. acknowledges financial support from the
European Research Council for the ERC Consolidator grant DEMOBLACK, under
contract no. 770017, and from the German Excellence Strategy via the
Heidelberg Cluster of Excellence (EXC 2181 - 390900948) STRUCTURES.
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## Appendix A Extract of the EWOCS X-ray Westerlund 1 sources catalog
Ten rows of the EWOCS catalog of the X-ray sources in Westerlund 1 are shown
here. The full catalog is available at the CDS.
Table 4: Ten rows extracted from the EWOCS catalog of the X-ray sources in Westerlund 1 Astrometry | Photometry |
---|---|---
EWOCS-X ID | Catalog name(a) | $\alpha$ | $\delta$ | $\sigma_{\alpha}^{b}$ | $\sigma_{\delta}^{b}$ | $\Theta^{(c)}$ | Ct | Cs | Cm | Ch | |
| | J2000 | J2000 | arcsec | arcsec | arcmin | counts | counts | counts | counts | |
3001 | 164704.14-454957.4 | 251.767277 | -45.832630 | 0.06 | 0.06 | 1.6 | 38 | 0 | 11 | 27 | |
3002 | 164704.14-455100.2 | 251.767284 | -45.850057 | 0.06 | 0.06 | 0.8 | 32 | 1 | 11 | 20 | |
3003 | 164704.15-455133.6 | 251.767302 | -45.859349 | 0.05 | 0.05 | 0.6 | 47 | 0 | 13 | 34 | |
3004 | 164704.15-455118.1 | 251.767311 | -45.855035 | 0.05 | 0.05 | 0.6 | 50 | 0 | 11 | 39 | |
3005 | 164704.16-455320.2 | 251.767343 | -45.888959 | 0.09 | 0.10 | 1.9 | 14 | 1 | 5 | 8 | |
3006 | 164704.16-455002.9 | 251.767353 | -45.834155 | 0.10 | 0.10 | 1.6 | 7 | 0 | 2 | 5 | |
3007 | 164704.16-455135.0 | 251.767366 | -45.859734 | 0.06 | 0.06 | 0.6 | 19 | 0 | 5 | 14 | |
3008 | 164704.17-455010.5 | 251.767382 | -45.836274 | 0.08 | 0.08 | 1.4 | 23 | 0 | 5 | 18 | |
3009 | 164704.18-455025.7 | 251.767450 | -45.840474 | 0.06 | 0.05 | 1.2 | 36 | 1 | 13 | 22 | |
3010 | 164704.19-455126.4 | 251.767479 | -45.857345 | 0.07 | 0.07 | 0.6 | 27 | 0 | 10 | 17 | |
Photometry
Cnet,t | Cnet,s | Cnet,m | Cnet,h | Emedian | Fphotons,t | Fphotons,s | Fphotons,m | Fphotons,h | log(PB,best) | | |
counts | counts | counts | counts | keV | photons cm${}^{-2}\,$s-1 | photons cm${}^{-2}\,$s-1 | photons cm${}^{-2}\,$s-1 | photons cm${}^{-2}\,$s-1 | | | |
25.9${}^{+7.2}_{-6.1}$ | -0.5${}^{+1.8}_{NaN}$ | 8.1${}^{+4.4}_{-3.2}$ | 18.2${}^{+6.3}_{-5.1}$ | 3.22 | 1.25$\times$10${}^{-}7$ | NaN | 3.00$\times$10${}^{-}8$ | 8.68$\times$10${}^{-}8$ | -8.94 | | |
12.9${}^{+6.8}_{-5.7}$ | 0.7${}^{+2.3}_{-0.8}$ | 3.7${}^{+4.4}_{-3.3}$ | 8.4${}^{+5.6}_{-4.4}$ | 2.77 | 6.97$\times$10${}^{-}8$ | 2.32$\times$10${}^{-}8$ | 1.50$\times$10${}^{-}8$ | 4.49$\times$10${}^{-}8$ | -2.23 | | |
37.1${}^{+7.9}_{-6.8}$ | -0.1${}^{+1.8}_{NaN}$ | 8.9${}^{+4.7}_{-3.5}$ | 28.2${}^{+6.9}_{-5.8}$ | 2.95 | 2.19$\times$10${}^{-}7$ | NaN | 3.93$\times$10${}^{-}8$ | 1.65$\times$10${}^{-}7$ | -15.2 | | |
19.1${}^{+8.3}_{-7.2}$ | 0.0${}^{+1.8}_{NaN}$ | -0.1${}^{+4.5}_{-3.4}$ | 19.2${}^{+7.4}_{-6.3}$ | 4.15 | 8.95$\times$10${}^{-}8$ | NaN | NaN | 8.85$\times$10${}^{-}8$ | -5.1 | | |
7.4${}^{+4.8}_{-3.7}$ | 0.7${}^{+2.3}_{-0.8}$ | 3.2${}^{+3.4}_{-2.1}$ | 3.4${}^{+3.9}_{-2.7}$ | 2.02 | 4.15$\times$10${}^{-}8$ | 2.14$\times$10${}^{-}8$ | 1.36$\times$10${}^{-}8$ | 1.90$\times$10${}^{-}8$ | -6.61 | | |
3.1${}^{+3.7}_{-2.5}$ | -0.1${}^{+1.8}_{NaN}$ | 0.9${}^{+2.6}_{-1.2}$ | 2.2${}^{+3.4}_{-2.1}$ | 2.24 | 2.49$\times$10${}^{-}8$ | NaN | 5.53$\times$10${}^{-}9$ | 1.80$\times$10${}^{-}8$ | -3.53 | | |
9.7${}^{+5.4}_{-4.3}$ | -0.1${}^{+1.8}_{NaN}$ | 1.2${}^{+3.4}_{-2.1}$ | 8.6${}^{+4.8}_{-3.7}$ | 3.46 | 5.56$\times$10${}^{-}8$ | NaN | 5.34$\times$10${}^{-}9$ | 4.84$\times$10${}^{-}8$ | -2.92 | | |
11.1${}^{+5.9}_{-4.7}$ | -0.2${}^{+1.8}_{NaN}$ | 0.9${}^{+3.4}_{-2.1}$ | 10.4${}^{+5.3}_{-4.2}$ | 4.99 | 5.31$\times$10${}^{-}8$ | NaN | 3.62$\times$10${}^{-}9$ | 4.87$\times$10${}^{-}8$ | -4.68 | | |
19.4${}^{+7.1}_{-6.0}$ | 0.6${}^{+2.3}_{-0.8}$ | 7.2${}^{+4.7}_{-3.6}$ | 11.6${}^{+5.8}_{-4.7}$ | 2.17 | 1.06$\times$10${}^{-}7$ | 2.04$\times$10${}^{-}8$ | 2.94$\times$10${}^{-}8$ | 6.27$\times$10${}^{-}8$ | -3.70 | | |
9.2${}^{+6.3}_{-5.2}$ | -0.5${}^{+1.8}_{NaN}$ | 3.7${}^{+4.3}_{-3.1}$ | 6.0${}^{+5.2}_{-4.1}$ | 4.16 | 4.66$\times$10${}^{-}8$ | NaN | 1.44$\times$10${}^{-}8$ | 2.98$\times$10${}^{-}8$ | -2.60 | | |
Columns 1–11 are shown in the top table; columns 12–21 in the bottom table.
a: IAU designation.
b: single axis position error, representing only the random component of the
position uncertainty.
c: Off-axis angle.
Photometric quantities are given in broad ($t$), soft ($s$), medium ($m$), and
hard ($h$) bands.
CX indicate the total counts in the X band, CX,net the net counts.
## Appendix B EWOCS X-ray counterparts of the massive stars in Westerlund 1
Table LABEL:tab_massive shows the EWOCS X-ray counterparts of the massive
stars in Westerlund 1 listed in Clark et al. (2020).
Table 5: Known massive stars in the EWOCS X-ray catalog ID | Spectral type | Catalog name | Cnet,t | Fphotons,t | Sep.
---|---|---|---|---|---
| | | counts | photons cm${}^{-2}\,$s-1 | arcsec
W9 | sgB[e] | 164704.13-455031.3 | $7975.3_{7885.7}^{8065.8}$ | 3.7$\times$10-5 | 0.2
W30 | O4-5Ia | 164704.10-455039.2 | $5930.2_{5852.8}^{6008.4}$ | 2.7$\times$10-5 | 0.2
W72 | WN7b | 164708.35-455045.4 | $5412.8_{5338.8}^{5487.5}$ | 4.5$\times$10-5 | 0.3
WRB | WN7o | 164705.36-455104.8 | $3046.2_{2990.6}^{3102.4}$ | 1.4$\times$10-5 | 0.1
WRU | WN6o | 164706.53-455039.1 | $1996.9_{1952.0}^{2042.3}$ | 9.6$\times$10-6 | 0.1
W44 | WN9h | 164704.19-455107.2 | $1221.9_{1186.5}^{1257.6}$ | 5.6$\times$10-6 | 0.3
W239 | WC9d | 164705.20-455225.0 | $903.2_{873.0}^{933.7}$ | 6.0$\times$10-6 | 0.0
W53 | OBIa+OBIa | 164700.38-455131.8 | $515.5_{492.4}^{538.7}$ | 2.5$\times$10-6 | 1.0
W36 | OBIa+OBIa | 164705.07-455055.2 | $514.7_{491.0}^{538.7}$ | 2.4$\times$10-6 | 0.4
WRO | WN6o | 164707.65-455236.0 | $387.6_{367.7}^{407.7}$ | 2.5$\times$10-6 | 0.1
WRN | WC9d | 164659.91-455525.6 | $375.2_{355.3}^{395.2}$ | 2.2$\times$10-6 | 0.4
W27 | O7-8Ia+ | 164705.14-455041.4 | $340.1_{320.8}^{359.6}$ | 1.5$\times$10-6 | 0.1
W13 | B0.5Ia++OB | 164706.44-455026.1 | $258.0_{241.4}^{274.7}$ | 1.2$\times$10-6 | 0.1
WRW | WN6h | 164707.61-454922.1 | $243.0_{227.1}^{259.1}$ | 1.3$\times$10-6 | 0.3
WRJ | WN5h | 164702.47-455059.9 | $233.4_{217.5}^{249.5}$ | 1.1$\times$10-6 | 0.1
W14c | WN5o | 164706.09-455022.4 | $192.9_{178.1}^{207.8}$ | 9.2$\times$10-7 | 0.3
W24 | O9Iab | 164702.15-455112.6 | $191.7_{177.2}^{206.4}$ | 9.3$\times$10-7 | 0.2
W43c | O9Ib | 164703.75-455058.5 | $188.5_{173.8}^{203.3}$ | 1.0$\times$10-6 | 0.2
1041 | O9.5Iab | 164704.45-455109.4 | $170.4_{156.2}^{184.8}$ | 7.9$\times$10-7 | 1.0
WRX | WN5o | 164714.13-454832.0 | $154.2_{141.3}^{167.3}$ | 8.6$\times$10-7 | 0.3
WRG | WN7o | 164704.00-455125.1 | $146.8_{133.8}^{159.9}$ | 6.8$\times$10-7 | 0.0
W50b | O9III | 164701.21-455027.6 | $136.2_{124.1}^{148.4}$ | 7.4$\times$10-7 | 1.0
W38 | O9Iab | 164702.88-455046.2 | $118.9_{106.9}^{130.9}$ | 5.6$\times$10-7 | 0.3
W37 | O9Ib | 164706.01-455047.5 | $118.1_{106.1}^{130.2}$ | 5.5$\times$10-7 | 0.1
W35 | O9Iab | 164704.20-455053.7 | $110.3_{98.5}^{122.3}$ | 5.9$\times$10-7 | 0.2
W25 | O9Iab | 164705.77-455033.4 | $108.4_{96.9}^{120.1}$ | 5.1$\times$10-7 | 0.1
W232 | B0Iab | 164701.43-455235.2 | $104.7_{94.1}^{115.3}$ | 6.4$\times$10-7 | 0.4
W6a | B0.5Iab | 164703.04-455023.7 | $98.5_{87.9}^{109.2}$ | 4.6$\times$10-7 | 0.1
W17 | O9Iab | 164706.23-455049.3 | $96.7_{85.7}^{107.7}$ | 4.8$\times$10-7 | 0.1
W74 | O9.5Iab | 164707.07-455013.0 | $93.7_{83.5}^{104.0}$ | 4.9$\times$10-7 | 0.0
W15 | O9Ib | 164706.62-455029.6 | $92.1_{81.7}^{102.5}$ | 4.5$\times$10-7 | 0.0
W47 | O9.5Iab | 164702.61-455117.8 | $89.0_{78.5}^{99.5}$ | 4.4$\times$10-7 | 0.3
W57c | WN7o | 164701.59-455145.2 | $88.9_{78.9}^{98.9}$ | 5.3$\times$10-7 | 0.2
WRI | WN8o | 164700.87-455120.6 | $86.6_{76.6}^{96.6}$ | 4.0$\times$10-7 | 0.1
WRQ | WN6o | 164655.54-455134.5 | $86.6_{76.9}^{96.3}$ | 5.6$\times$10-7 | 0.4
1027 | O9.5Iab | 164701.02-455007.0 | $85.6_{75.8}^{95.5}$ | 4.2$\times$10-7 | 0.7
1051 | O9III | 164706.98-454940.1 | $79.5_{70.0}^{88.9}$ | 4.6$\times$10-7 | 0.2
1056 | O9.5II | 164708.69-455101.7 | $76.3_{66.9}^{85.8}$ | 3.7$\times$10-7 | 0.6
W10 | B0.5I+OB | 164703.34-455034.6 | $75.8_{65.8}^{85.7}$ | 3.5$\times$10-7 | 0.3
W8a | F8Ia+ | 164704.83-455025.5 | $73.7_{63.9}^{83.6}$ | 3.4$\times$10-7 | 0.8
W1 | O9.5Iab | 164659.39-455046.7 | $69.8_{60.9}^{78.8}$ | 3.4$\times$10-7 | 1.2
WRD | WN7o | 164706.25-455126.4 | $69.3_{59.8}^{78.8}$ | 3.1$\times$10-7 | 0.1
W62a | B0.5Ib | 164702.52-455138.0 | $69.1_{60.0}^{78.1}$ | 4.0$\times$10-7 | 0.2
W65 | O9Ib | 164703.88-455146.5 | $67.9_{58.9}^{76.9}$ | 3.7$\times$10-7 | 0.2
WRV | WN8o | 164703.79-455038.7 | $66.8_{57.8}^{75.9}$ | 4.9$\times$10-7 | 0.1
1037 | O9.5II | 164702.84-455006.4 | $64.8_{56.1}^{73.3}$ | 3.2$\times$10-7 | 0.1
W28 | B2Ia | 164704.66-455038.5 | $63.2_{53.1}^{73.3}$ | 2.9$\times$10-7 | 0.1
W61b | O9.5Iab | 164702.56-455141.9 | $61.3_{52.5}^{70.0}$ | 3.1$\times$10-7 | 0.3
1030 | O9.5Iab | 164701.67-455258.0 | $60.1_{51.9}^{68.4}$ | 3.4$\times$10-7 | 0.3
1040 | O9-9.5I-III | 164704.59-455008.1 | $59.9_{51.7}^{68.0}$ | 4.0$\times$10-7 | 1.0
1061 | O9-9.5III | 164709.61-455040.4 | $59.7_{51.3}^{68.1}$ | 3.1$\times$10-7 | 1.3
W84 | O9.5Ib | 164659.03-455028.3 | $57.0_{49.1}^{64.8}$ | 4.4$\times$10-7 | 0.1
1064 | O9.5Iab | 164711.50-455000.0 | $56.4_{48.3}^{64.5}$ | 2.9$\times$10-7 | 0.6
W241 | WC9 | 164705.96-455208.3 | $56.3_{48.2}^{64.3}$ | 3.9$\times$10-7 | 0.9
1060 | O9.5II | 164709.19-455048.4 | $56.2_{47.8}^{64.5}$ | 2.8$\times$10-7 | 0.1
1036 | O9.5Ia | 164702.78-455212.7 | $55.8_{47.8}^{63.8}$ | 3.8$\times$10-7 | 0.3
1004 | OeBe star | 164653.44-455300.3 | $53.9_{45.9}^{61.8}$ | 2.7$\times$10-7 | 0.8
1058 | O9III | 164708.89-455124.5 | $53.7_{45.8}^{61.6}$ | 3.3$\times$10-7 | 0.1
W56b | O9.5Ib | 164658.87-455145.9 | $52.5_{44.9}^{60.1}$ | 4.8$\times$10-7 | 0.2
W29 | O9Ib | 164704.40-455039.9 | $51.5_{43.5}^{59.5}$ | 3.7$\times$10-7 | 0.1
1023 | O9III | 164700.14-455110.3 | $49.7_{42.0}^{57.8}$ | 2.3$\times$10-7 | 1.0
W53 | OBIa+OBIa | 164700.55-455132.0 | $47.6_{39.4}^{56.3}$ | 2.3$\times$10-7 | 0.7
1034 | O9.5Iab | 164702.52-455148.3 | $47.4_{40.1}^{55.3}$ | 2.9$\times$10-7 | 0.1
1063 | O9III | 164710.74-454947.8 | $46.8_{39.5}^{54.6}$ | 2.3$\times$10-7 | 0.6
1005 | B0Iab | 164654.28-455154.8 | $43.8_{37.0}^{51.3}$ | 2.7$\times$10-7 | 0.4
1047 | O9.5II | 164706.12-455232.2 | $43.4_{36.4}^{51.0}$ | 2.9$\times$10-7 | 0.2
W41 | O9Iab | 164702.70-455057.1 | $43.0_{35.7}^{50.8}$ | 2.3$\times$10-7 | 0.2
1033 | O9-9.5I-III | 164702.37-455234.2 | $42.8_{36.1}^{50.1}$ | 2.7$\times$10-7 | 0.2
1018 | O9.5Iab | 164658.28-455057.0 | $41.8_{35.1}^{49.1}$ | 2.5$\times$10-7 | 0.4
W11 | B2 | 164702.24-455046.8 | $41.4_{34.1}^{49.2}$ | 1.9$\times$10-7 | 0.2
1040 | O9-9.5I-III | 164704.54-455009.0 | $36.0_{29.8}^{42.8}$ | 2.4$\times$10-7 | 0.3
1038 | O9III | 164703.49-454857.1 | $34.8_{28.3}^{41.9}$ | 1.7$\times$10-7 | 1.1
1007 | O9-9.5III | 164654.90-455005.8 | $34.5_{28.3}^{41.2}$ | 2.1$\times$10-7 | 0.5
W243 | LBV | 164707.50-455229.0 | $33.7_{27.5}^{40.4}$ | 2.4$\times$10-7 | 0.7
1043 | O9.5II-III | 164704.56-455059.5 | $32.3_{25.8}^{39.2}$ | 2.2$\times$10-7 | 0.2
W86 | O9.5Ib | 164657.15-455010.0 | $30.3_{24.6}^{36.5}$ | 1.8$\times$10-7 | 0.1
W61a | B0.5Ia | 164702.27-455141.7 | $28.6_{22.0}^{35.6}$ | 1.5$\times$10-7 | 0.2
W46b | O9.5Ib | 164703.67-455120.5 | $28.5_{21.4}^{36.2}$ | 1.3$\times$10-7 | 0.9
1066 | O9III | 164712.60-455055.6 | $28.3_{22.8}^{34.4}$ | 2.0$\times$10-7 | 1.2
1050 | O9.5II | 164706.77-454955.2 | $26.4_{20.8}^{32.6}$ | 1.3$\times$10-7 | 0.0
WRH | WC9d | 164704.23-455120.2 | $26.3_{19.3}^{33.9}$ | 1.2$\times$10-7 | 0.1
1029 | O9-9.5III | 164701.50-454950.1 | $25.5_{19.9}^{31.6}$ | 1.2$\times$10-7 | 0.6
W46a | B1Ia | 164703.90-455119.9 | $24.5_{17.2}^{32.4}$ | 1.1$\times$10-7 | 0.4
W21 | B0.5Ia | 164701.10-455113.7 | $24.4_{18.3}^{31.0}$ | 1.1$\times$10-7 | 0.1
W5 | WN10/B0.5Ia+WRS | 164702.98-455018.5 | $23.7_{18.2}^{29.7}$ | 1.2$\times$10-7 | 1.0
1055 | B0Ib(+O?) | 164707.82-455147.1 | $23.1_{17.8}^{28.8}$ | 1.9$\times$10-7 | 1.2
W4 | F3Ia+ | 164701.54-455037.1 | $22.3_{16.6}^{28.6}$ | 1.0$\times$10-7 | 1.3
1065 | B0Ib | 164711.60-454922.6 | $22.2_{16.8}^{28.2}$ | 1.1$\times$10-7 | 0.2
1048 | B1.5 | 164706.28-455104.0 | $21.4_{16.5}^{26.9}$ | 1.7$\times$10-7 | 0.3
W34 | B0Ia | 164704.39-455047.3 | $21.3_{13.6}^{29.4}$ | 1.0$\times$10-7 | 0.1
W228b | O9Ib | 164658.13-455301.2 | $21.1_{16.4}^{26.5}$ | 1.5$\times$10-7 | 0.9
1059 | O9III? | 164709.08-455320.7 | $21.1_{16.3}^{26.4}$ | 1.4$\times$10-7 | 0.3
W43c | O9Ib | 164703.70-455057.7 | $21.0_{15.6}^{27.0}$ | 2.1$\times$10-7 | 0.8
1044 | O9-9.5III | 164705.56-454951.8 | $19.8_{14.6}^{25.5}$ | 1.0$\times$10-7 | 0.3
W43b | B1Ia | 164703.52-455056.6 | $19.8_{13.2}^{26.8}$ | 1.1$\times$10-7 | 0.1
1059 | O9III? | 164709.11-455319.4 | $18.7_{14.2}^{23.7}$ | 1.4$\times$10-7 | 1.3
1042 | O9.5II | 164704.66-455206.8 | $17.9_{13.0}^{23.3}$ | 1.3$\times$10-7 | 1.1
W2a | B2Ia | 164659.77-455051.8 | $17.3_{12.3}^{22.8}$ | 8.1$\times$10-8 | 0.9
1024 | O9.5Iab | 164700.78-455102.0 | $16.6_{11.8}^{21.9}$ | 8.6$\times$10-8 | 0.6
W50b | O9III | 164701.11-455026.6 | $16.5_{11.5}^{22.0}$ | 8.7$\times$10-8 | 0.7
W228b | O9Ib | 164658.02-455301.1 | $16.3_{12.0}^{21.0}$ | 1.2$\times$10-7 | 0.3
W243 | LBV | 164707.62-455228.4 | $15.9_{11.4}^{21.1}$ | 1.2$\times$10-7 | 0.7
W1 | O9.5Iab | 164659.20-455045.4 | $15.8_{11.0}^{21.2}$ | 7.9$\times$10-8 | 1.4
W4 | F3Ia+ | 164701.35-455036.5 | $15.6_{10.5}^{21.2}$ | 7.4$\times$10-8 | 0.8
1032 | O9-9.5III | 164702.32-455017.1 | $15.3_{10.4}^{20.7}$ | 7.3$\times$10-8 | 0.7
1016 | O9-9.5III | 164658.09-455247.1 | $15.2_{10.7}^{20.2}$ | 8.2$\times$10-8 | 0.2
W54 | B0.5Iab | 164703.14-455131.2 | $14.7_{9.30}^{20.7}$ | 6.9$\times$10-8 | 1.2
1014 | O9-9.5III | 164657.81-455119.3 | $14.4_{10.1}^{19.3}$ | 8.7$\times$10-8 | 0.4
1010 | O+O? | 164655.99-455210.1 | $14.4_{10.2}^{19.2}$ | 8.7$\times$10-8 | 0.7
1015 | O9III | 164657.97-455141.0 | $14.2_{10.3}^{18.6}$ | 1.5$\times$10-7 | 0.3
1049 | B1-2Ia+ | 164706.66-454738.8 | $14.2_{8.4}^{20.4}$ | 8.5$\times$10-8 | 0.3
1031 | O9III | 164701.90-455056.1 | $14.0_{9.8}^{18.6}$ | 9.9$\times$10-8 | 0.2
1043 | O9.5II-III | 164704.63-455059.4 | $13.1_{8.6}^{18.1}$ | 1.2$\times$10-7 | 0.7
W23a | B2Ia+BI? | 164702.56-455108.8 | $12.9_{7.1}^{19.2}$ | 6.0$\times$10-8 | 0.1
W63a | B0Iab | 164703.41-455157.4 | $12.7_{8.3}^{17.6}$ | 9.0$\times$10-8 | 0.3
W55 | B0Ia | 164658.40-455131.1 | $12.5_{8.5}^{17.0}$ | 8.9$\times$10-8 | 0.0
1012 | O9-9.5III | 164656.95-455055.6 | $12.4_{8.3}^{17.1}$ | 7.1$\times$10-8 | 0.3
W238 | B1Iab | 164704.41-455227.7 | $12.1_{7.8}^{17.1}$ | 7.5$\times$10-8 | 0.1
1046 | O+O? | 164705.98-454955.4 | $11.6_{7.3}^{16.6}$ | 5.6$\times$10-8 | 1.4
1045 | O9.5II | 164705.83-455155.1 | $11.6_{8.0}^{15.8}$ | 1.0$\times$10-7 | 0.2
W75 | M4Ia | 164708.96-454958.7 | $11.6_{7.3}^{16.4}$ | 6.3$\times$10-8 | 0.4
1021 | O9-9.5III | 164658.77-455432.0 | $11.5_{6.8}^{16.8}$ | 6.2$\times$10-8 | 0.1
1013 | O+O? | 164657.54-455231.0 | $11.5_{7.4}^{16.2}$ | 6.4$\times$10-8 | 0.6
1035 | O9-9.5III | 164702.67-455151.2 | $11.3_{7.1}^{16.1}$ | 7.6$\times$10-8 | 0.4
1046 | O+O? | 164706.09-454957.7 | $11.1_{6.8}^{15.9}$ | 5.4$\times$10-8 | 1.3
1017 | O9-9.5III | 164658.24-455033.8 | $10.8_{6.7}^{15.5}$ | 6.0$\times$10-8 | 0.1
W20 | M5Ia | 164703.11-455218.9 | $8.8_{5.1}^{13.3}$ | 5.4$\times$10-8 | 0.3
1028 | O9-9.5 | 164701.32-455137.5 | $8.7_{4.5}^{13.6}$ | 4.5$\times$10-8 | 0.6
1054 | O9-9.5 | 164707.64-455141.1 | $8.0_{5.0}^{11.7}$ | 8.3$\times$10-8 | 0.3
W78 | B1Ia | 164701.48-454957.4 | $7.8_{4.0}^{12.2}$ | 3.8$\times$10-8 | 0.6
W373 | B0Iab | 164657.72-455320.0 | $7.7_{4.1}^{11.9}$ | 4.6$\times$10-8 | 0.1
1026 | O9-9.5III | 164701.01-454948.8 | $7.0_{3.3}^{11.3}$ | 3.5$\times$10-8 | 0.5
W71 | B2.5Ia | 164708.57-455049.8 | $5.5_{2.1}^{9.4}$ | 4.5$\times$10-8 | 1.5
1020 | O9-9.5+O? | 164658.49-455228.4 | $5.2_{2.5}^{8.5}$ | 3.7$\times$10-8 | 1.3
1008 | O9.5II | 164655.45-455154.2 | $5.1_{2.1}^{8.7}$ | 3.3$\times$10-8 | 0.1
1062 | O+O? | 164710.65-455047.2 | $5.0_{1.9}^{8.7}$ | 3.2$\times$10-8 | 0.9
1022 | O9.5II | 164659.88-455025.1 | $4.6_{1.6}^{8.1}$ | 3.0$\times$10-8 | 0.5
1045 | O9.5II | 164705.86-455154.2 | $4.4_{2.1}^{7.5}$ | 4.8$\times$10-8 | 0.8
W29 | O9Ib | 164704.47-455039.5 | $4.3_{1.4}^{7.8}$ | 4.8$\times$10-8 | 0.7
## Appendix C Estimate of catalog completeness
The resulting completeness of our survey depends not only on the total
exposure, but also on source crowding and the bright and irregular background.
A full understanding of completeness will only be possible after the
identification and classification of the OIR counterparts of the X-ray sources
in order to distinguish between cluster members and sources in the foreground
and background. However, we conducted some simple simulations using MARX,
which, despite being based on strong assumptions, can provide some hints about
completeness.
In order to simulate the cluster population, since the true shape of the IMF
of Westerlund 1 is still a subject of debate, particularly in the low-mass
regime, we made the assumption that the cluster IMF follows the law proposed
by Kroupa (2001), which is applicable to most known young stellar clusters. We
understand that the starburst environment can influence the distribution of
stellar masses, leading to different mass functions. However, at this level of
approximation, this is considered a secondary effect. To accommodate the
compact morphology of the cluster, we assumed that cluster members are
distributed according to a Gaussian function with a full width at half maximum
of 4 arcminutes. Therefore, we did not account for the asymmetric morphology
of Westerlund 1, as suggested by previous authors (e.g., Gennaro et al. 2011).
Additionally, we assumed a total cluster mass of 45000 solar masses,
encompassing stars with masses as low as 0.08 solar masses.
To convert the mass distribution into an LX distribution, we utilized the LX
versus mass distribution derived from the _Chandra_ Orion Ultradeep Project
(COUP) conducted in the Orion Nebula Cluster (Preibisch & Feigelson 2005),
accounting for its observed spread. We chose this distribution because the
COUP survey provides the most complete X-ray observation of a young stellar
cluster. However, it should be noted that this distribution may not accurately
represent the population of Westerlund 1 due to differences in age and the
presence of a distinct massive stellar population in this cluster. To account
for this massive stellar population, we simply added the massive sources
identified by Clark et al. (2005) with their corresponding measured LX values
to the simulated cluster population. Additionally, we normalized the COUP LX
versus mass distribution to account for the decline in stellar X-ray
luminosity with age (Preibisch & Feigelson 2005), and we used the specified
values for cluster distance and absorption to convert luminosity into flux.
We simulated a 1 Msec ACIS-I observation of this fake cluster, taking into
account instrumental
background131313https://cxc.harvard.edu/cal/Acis/detailed_info.html, and
performed source detection using _Wavdetect_ (thus not accounting for the
source validation procedure we adopted with AE). By comparing the input and
output lists of sources, we determined that the completeness in the 0.8-2
solar mass range is approximately 40% within the central 4 arcminute region,
decreasing by approximately 10% in the inner 1 arcminute region. For more
massive stars, the estimated completeness is around 85% regardless of the
distance from the cluster center. It is important to note that this is a
preliminary estimation of the completeness of the EWOCS X-ray catalog, which
will be further validated through the identification of OIR counterparts and
source classification.
|
# Novel Chapter Abstractive Summarization using Spinal Tree Aware Sub-
Sentential Content Selection
Hardy* Miguel Ballesteros Faisal Ladhak* Muhammad Khalifa*
Vittorio Castelli Kathleen McKeown
###### Abstract
††*Work done while at Amazon AWS AI Labs
Summarizing novel chapters is a difficult task due to the input length and the
fact that sentences that appear in the desired summaries draw content from
multiple places throughout the chapter. We present a pipelined extractive-
abstractive approach where the extractive step filters the content that is
passed to the abstractive component. Extremely lengthy input also results in a
highly skewed dataset towards negative instances for extractive summarization;
we thus adopt a margin ranking loss for extraction to encourage separation
between positive and negative examples. Our extraction component operates at
the constituent level; our approach to this problem enriches the text with
spinal tree information which provides syntactic context (in the form of
constituents) to the extraction model. We show an improvement of 3.71 Rouge-1
points over best results reported in prior work on an existing novel chapter
dataset.
## 1 Introduction
Research on summarizing novels (Mihalcea and Ceylan, 2007; Wu et al., 2017;
Ladhak et al., 2020; Kryściński et al., 2021; Wu et al., 2021) has recently
gained popularity following advancements in sequence-to-sequence pre-trained
models (Zhang et al., 2019a; Lewis et al., 2019; Raffel et al., 2019) and in
summarization of newswire datasets (Narayan et al., 2018; Hermann et al.,
2015; Grusky et al., 2018). Novel chapters present challenges not commonly
encountered when summarizing news articles. Phrases from multiple, non-
contiguous sentences within the chapter are often fused to form new sentences
for the summary. One would be inclined to use an abstractive approach, but the
length of chapters (on average, seven times longer than news articles (Ladhak
et al., 2020)) makes it unfeasible to use state of the art generative models,
such as BART (Lewis et al., 2019) and even Longformer (Beltagy et al., 2020).
Chapter length causes the additional problem of an imbalanced dataset, as a
much higher percentage of the input will not be selected for the summary than
is typical in domains such as news.
To address these challenges, we adopt an extractive-abstractive architecture,
where content is first selected by extracting units from the input and then an
abstractive model is used on the filtered input to produce fluent text.
Kryściński et al. (2021) benchmarked the extractive-abstractive architecture,
first proposed by Chen and Bansal (2018), for novel summarization, but did not
extend it. In this work, we propose several novel extensions to improve its
performance on the novel chapter summarization task.
First, we address the issue of imbalanced dataset where the large amount of
compression in novel chapter summarization (372 summary words per 5,165
chapter words on average) creates an extreme imbalance in the training data; a
successful extractive summarization algorithm would have to discard most of
the text. The standard practice of using Cross-Entropy loss (Good, 1992) when
training a neural network model backfires in our case: a network that opts to
discard everything will achieve near-perfect performance. We alleviate the
issue by improving the margin structure of the minority class boundary using
the Margin Ranking loss (Rosasco et al., 2004), which encourages separation
between the two classes. Other studies, such as Cruz et al. (2016), also shows
that a pairwise ranking improves model performances on imbalanced data.
Second, in order to model the fusion of chapter phrases into summary
sentences, we carry out extraction at the constituent level. Ladhak et al.
(2020) also tried this approach, but with mixed results. They noted that
sometimes the sub-sentential unit can be too small and, therefore, lack
meaningful content (e.g., phrases such as “what has?” in the extractive
summary, Table 1). These small unintelligible pieces can negatively affect the
performance of the extractive model and, more importantly, the subsequent
abstractive model. We hypothesize that we can improve the performance of the
extractive model—and, consequently, that of the downstream abstractive
model—by augmenting the meaning of the extracted sub-sentential units using
additional information from the sentence. To that end, we propose an
enrichment process, during model training, where we augment the sub-sentential
units with linguistic information. For this purpose, we use a spinal tree
(Carreras et al., 2008; Ballesteros and Carreras, 2015) which carries
information about both the dependency and the constituent structure of the
segment. We encode the spine’s information using a recurrent network and
concatenate its output to the embedding of the token, as illustrated in Figure
1. We choose spinal tree since it
Figure 1: The encoding part of the model for spinal tree encoding. Each token
is represented by its corresponding spine from the spinal tree and is encoded
by bidirectional-GRU networks (Cho et al., 2014) before being concatenated
with its token embedding. We don’t show the CLS and SEP tokens here for space-
saving purposes, but they are treated as in BertSumm (Liu and Lapata, 2019).
Our contributions are threefold: (1) we adopt an extractive-abstractive
architecture, improving the decision boundary of the content selection by
using a Margin Ranking loss, (2) we perform extraction at the constituent
level, introducing an enrichment process that uses spinal tree information and
(3) we show that our approach improves over the state-of-the-art with a 3.71
gain in Rouge-1 points.
Extracts from our best performance Extractive Model
---
tess went down the hill to trantridge cross , and inattentively waited to take
her seat in the van returning from chaseborough to shaston .<q>her mother had
advised her to stay here for the night , at the house of a cottage-
woman<q>what has ? ”<q>“ they say – mrs d’urberville says –<q>that she wants
you to look after a little fowl-farm which is her hobby .<q>cried joan to her
husband .
Abstracts from our best performance Abstractive Model
tess goes down the hill to trantridge cross and waits to take her seat in the
van returning from chaseborough to shaston . her mother has advised her to
stay for the night at the house of a cottage-woman who has a fowl-farm . joan
tells her husband that mrs. d’urberville has written a letter asking her
daughter to look after her poultry-farm
Reference
when tess returns home the following day. a letter from mrs. d’urberville
offering her a job tending fowl awaits her . despite her mother ’s ecstatic
eagerness , tess is displeased and looks instead for local jobs to earn money
to replace the family ’s horse .alec d’urberville stops by and prompts her
mother for an answer about the job . her efforts to find alternative work
prove fruitless and so tess accepts d’urberville ’s offer . she remarks that
mrs. d’urberville ’s handwriting looks masculine .
Table 1: The outputs from two different models. The extract is obtained
through a content selection model while the abstract is obtained by passing
the extract into BART (Lewis et al., 2019) language generation model. The <q>
tokens in the extract are the delimiters for constituents.
## 2 Related Work
Several previous works on novel chapter summarization, such as Mihalcea and
Ceylan (2007), Wu et al. (2017), Ladhak et al. (2020), Kryściński et al.
(2021) and Wu et al. (2021), are closely related to ours. Mihalcea and Ceylan
(2007) uses MEAD, an unsupervised extractive summarization described in Radev
et al. (2004); this approach includes features focusing on terms weighting
that take into account the different topics in the text. In this work, topic
boundaries are determined using a graph-based segmentation algorithm that uses
normalized cuts (Malioutov, 2006). A similar line of work, including Mihalcea
and Ceylan (2007) and Wu et al. (2017), also performs topic modelling with
Latent Dirichlet Allocation (Blei et al., 2003) followed by greedy
unsupervised extraction.
Conversely, Ladhak et al. (2020) experiment with extracting information at the
sentence and at the syntactic constituent level, via a supervised learning
approach. To train their model, they use an aligning process based on the
weighted ROUGE scores between the reference and novel text to assign proxy
extract labels, in the absence of manually annotated ground truth. Their
results at the constituent level are mixed; human evaluation shows a lower
performance of constituent extraction models presumably because the summaries
are not very readable. Kryściński et al. (2021) construct a novel chapter
dataset that is slightly larger than that of Ladhak et al. (2020) and
benchmark existing summarization algorithms on the dataset.
Wu et al. (2021), on the other hand, use a human-in-the-loop approach to
obtain summaries via behaviour cloning and reward modelling.
## 3 Novel Chapter Summarization
We use a two-step process where we first run an _extractive_ model (Mihalcea
and Ceylan, 2007; Wu et al., 2017; Ladhak et al., 2020) to select informative
content and then run a separate _abstractive_ model (Lewis et al., 2019; Zhang
et al., 2019a; Raffel et al., 2019) to produce a coherent and readable version
of this content.
### 3.1 Dataset and Pre-processing
For our novel dataset, we use summary-chapter pairs collected by Ladhak et al.
(2020) from Project Gutenberg and various study guide sources. The size of the
dataset is 8,088 chapter/summary pairs 111Train/dev/test splits are
6,288/938/862. The average length of the chapters is 5,165 words with the
longest being 33,167 words222We are aware that there is a larger dataset
called BookSum (Kryściński et al., 2021), which uses similar sources; however,
due to licensing issues, we are unable to use it in our work..
In order to prepare the data for the experiments, we follow the same pre-
processing steps as Ladhak et al. (2020) to obtain the sub-sentential units
and their alignment to reference summaries. In addition, we truncate chapters
to 30k tokens to fit into the GPU memory333We use Amazon AWS EC2 P4dn 40GB GPU
memory; as a result, a single chapter of the dataset is actually truncated.
### 3.2 Extractive Model
The extractive summarization task can be posed as a classic regression and
ranking problem where the model produces a score for each of a given set of
units and then ranks them based on that score. The top $k$ units are then used
as an extract. The input of our model is the sub-sentential units of the novel
chapter text. We train the model with the oracle labels which we obtain from
the alignment between sub-sentential units and reference summaries.
#### Baseline
Our baseline is BertSummExt model (Liu and Lapata, 2019) modified as follows.
First, we replaced the underlying Transformer models (Vaswani et al., 2017)
with Longformers, which can better capture long context and requires less
computing memory than BERT (Devlin et al., 2019). Second, we removed the
inter-sentence Transformer layers stacked on top of the BERT output, to
further reduce memory usage. To avoid confusion with Liu and Lapata (2019)’s
model, we named this baseline as Longformer Ext.
#### Spinal Tree
A spinal tree is a dependency structure of a sentence that is augmented with
constituent information (Carreras et al., 2008; Ballesteros and Carreras,
2015). For each sub-sentential unit, we retrieve the spinal tree parse by
first using the constituency parser (Manning et al., 2014) and then apply
Collins Head-Word Finder (Collins, 1997) to calculate the spines. We then
encode444We use the hidden size of 512 the spinal tree using bidirectional-GRU
networks (Cho et al., 2014)555We experimented with other architectures
including bi-LSTM and found that bidirectional-GRU were the best..
We construct the input of the Longformer by concatenating the embeddings of
the tokens666We use the embedding size of 768, the corresponding positional
embeddings per token, and the encoding of the spines for each token via the
bidirectional-GRU encoders, as illustrated in Figure 1.
#### Ranking Loss
The baseline model uses the Cross-Entropy (CE) loss function and minimizes the
loss via gradient descent. However, the CE loss function focuses on optimizing
both the negative and positive labels at the same time. To compensate for the
imbalance in our dataset, we add a Margin Ranking (MR) loss that gives the
positive labels higher ranks than the negative labels 777We also have tried
the weighted CE loss function but we get worse results. We also found that
training our model first with the CE loss function until convergence and then
continuing using the MR loss gives the best result..
#### Re-ordering Scheme
The default baseline of Liu and Lapata (2019) produces extracts with sub-
sentential units that are ordered based on their score. This scheme, however,
destroys the plot of the story. Hence, we re-order the units according to the
original positional order in the source text, thus preserving the correct plot
order in the story.
### 3.3 Abstractive Model
Since the extractive model outputs are sometimes incoherent and hard to read,
we forward them to an abstractive model, with the goal to produce a more
fluent and coherent result.
We use BART (Lewis et al., 2019) as our engine for abstractive summarization.
To train BART, we use the oracle extracts as the input source and the
reference summaries as the target. During prediction, we use the output of our
content selection model as the input source.
Model | R1 | R2 | RL | WMD | BERTScore
---|---|---|---|---|---
Extractive
Oracle Ext | 46.75 | 14.27 | 45.64 | 0.633 | 0.823
CB const R-wtd (Ladhak, 2020) | 36.62 | 6.9 | 35.4 | N/A | N/A
Longformer Ext | 39.24 | 7.61 | 38.29 | 0.712 | 0.803
(Modified Liu and Lapata (2019)) | | | | |
\+ Spinal Information | 39.35 | 7.62 | 38.45 | 0.711 | 0.802
\+ Ranking Loss | 39.48 | 7.63 | 38.58 | 0.708 | 0.802
\+ Re-ordering | 39.48 | 7.70 | 38.58 | 0.708 | 0.806
Abstractive
Oracle Abs | 45.82 | 14.14 | 42.74 | 0.641 | 0.828
BART Abs | 39.77 | 9.28 | 37.56 | 0.693 | 0.807
\+ Spinal Information | 39.83 | 9.33 | 37.61 | 0.691 | 0.807
\+ Ranking Loss | 39.88 | 9.35 | 37.68 | 0.691 | 0.807
\+ Re-ordering | 40.33 | 9.10 | 37.95 | 0.690 | 0.810
Table 2: ROUGE, Word Mover Distance and BERTScore for extractive and
abstractive models
## 4 Results
Examples of outputs from our best abstractive and extractive models are shown
in Table 1. Here we report results from an automatic and a manual evaluation.
We compare our approach with and without the different extensions to the prior
best model from Ladhak et al. (2020) We also included the oracle for both the
extractive and abstractive models.
### 4.1 Automatic Evaluation
We use three different metrics for automatic evaluation: ROUGE (Lin, 2004),
BERTScore (Zhang et al., 2019b) and Word Mover Distance (WMD) (Kusner et al.,
2015). ROUGE measures syntactic similarities between system and reference
summaries and BERTScore and WMD measure semantic similarities. BERTScore
measures similarities at the sentence level while WMD does at the token level.
We run each experiment three times using different random seeds and we report
the mean score.
Table 2 shows our models performance against the baseline and previous works.
Our best extractive model (Longformer Ext+spinal+Ranking+Re-ordering)
outperforms previous work (CB const R-wtd) by 2.86 ROUGE-1, 0.8 ROUGE-2, and
3.18 ROUGE-L points. Meanwhile, the abstractive model (BART
Abs+spinal+Ranking+Re-ordering) outperforms previous work (CB const R-wtd) by
3.71 ROUGE-1, 2.2 ROUGE-2, and 2.55 ROUGE-L points. We have also shown that
both the best abstractive and extractive models exceed their corresponding
baselines (Longformer Ext and BART Abs) in all metrics. Our models still have
room to grow as shown by the oracle results.
### 4.2 Human Evaluation
For human evaluation, we use the lightweight Pyramid (Shapira et al., 2019).
We randomly selected 99 samples 888We prepared 100 samples, but one sample got
corrupted during the evaluation. from the test dataset for human evaluation.
We also re-run Ladhak et al. (2020)’s output’s using the same samples in order
to compare ours with their work.
Model | Pyramid
---|---
CB Const R-wtd (Ladhak, 2020) | 17.91
BART Abs | 22.03
BART Abs+Spinal+Rank+Re-ordering | 22.86
Table 3: Pyramid score for our best abstractive performance model compared to
previous works
Table 3 shows that our models outperform previous work by at least 2 points.
We also show that the application of spinal tree enrichment, ranking loss and
re-ordering show an improvement of 0.83 points in the human evaluation.
## 5 Conclusion and Future Work
We have built a novel chapter summarization that produces abstract summaries
using a spinal tree aware sub-sentential content selection method. Our results
show that we have improved over the state-of-the-art of an existing novel
chapter dataset in both automatic and human evaluations.
For future work, we propose an approach where the segmentation of sub-
sentential units is jointly trained with the content selection instead of pre-
processed before the training process. We hypothesize that this could improve
the alignment with reference summaries, therefore, increasing the performance
of the overall models.
## Limitation
The limitation of our work is that the dataset is small. It is also difficult
to show significance using a small dataset. Investigation on larger datasets
would be necessary to further validate our conclusions.
## Ethical Impact
We don’t foresee any ethical issues with our approach. One could argue that
our system might ultimately take jobs away from the people who currently write
such summaries. However, given the number of books being written, it is more
likely that some summaries would never be written and a good system for novel
chapter summarization might help to increase the amount of summaries that are
available online.
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## Appendix A Human Evaluation
For performing Human Evaluation, we use crowdsourcing service provided by
Appen.999https://appen.com/solutions/crowd-management/ We follow Ladhak et al.
(2020)’s approach including the instructions and number of crowdworkers. For
each crowdworker, we calculate the payment based on the minimum wage in the US
($ 15 per hour).
|
# A tight bound on the stepsize of the Decentralized gradient descent
Woocheol Choi Department of Mathematics, Sungkyunkwan University, Suwon
440-746, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.
In this paper, we consider the decentralized gradinet descent (DGD) given by
$x_{i}(t+1)=\sum_{j=1}^{m}w_{ij}x_{j}(t)-\alpha(t)\nabla f_{i}(x_{i}(t)).$
We find a sharp range of the stepsize $\alpha(t)>0$ such that the sequence
$\\{x_{i}(t)\\}$ is uniformly bounded when the aggregate cost $f$ is assumed
be strongly convex with smooth local costs which might be non-convex.
Precisely, we find a tight bound $\alpha_{0}>0$ such that the states of the
DGD algorithm is unfiromly bounded for non-increasing sequence $\alpha(t)$
satisfying $\alpha(0)\leq\alpha_{0}$. The theoretical results are also
verified by numerical experiments.
###### Key words and phrases:
Distributed optimization, Gradient descent, Sharp range
###### 2010 Mathematics Subject Classification:
Primary 65K10, 90C26
## 1\. Introduction
In this work, we consider the distributed optimization
$\min_{x}~{}f(x):=\frac{1}{m}\sum_{k=1}^{m}f_{k}(x),$ (1.1)
where $m$ denotes the number of agents and
$f_{k}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a differentiable local cost
only known to agent $k$ for each $1\leq k\leq m$. The decetralized gradient
descent is given as
$x_{k}(t+1)=\sum_{j=1}^{m}w_{kj}x_{j}(t)-\alpha(t)\nabla f_{k}(x_{k}(t)).$
(1.2)
Here $\alpha(t)>0$ is a stepsize and $x_{k}(t)$ denotes the variable of agent
$k$ at time instant $t\geq 0$. The communication pattern among agents in (1.1)
is determined by an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$,
where each node in $\mathcal{V}$ represents each agent, and each edge
$\\{i,j\\}\in\mathcal{E}$ means $i$ can send messages to $j$ and vice versa.
The value $w_{ij}$ is a nonnegative weight value such that $w_{ij}>0$ if and
only if $\\{i,j\\}\in\mathcal{E}$ and the matrix $W=\\{w_{ij}\\}_{1\leq
i,j\leq m}$ is doubly stochastic.
This algorithm has recieved a lot of attentions from researchers in various
fields. In particular, the algorithm has been a pivotal role in the
development of several methods, containing online distributed gradient descent
method [13, 3], the stochastic decentralized gradient descent [11], and multi-
agent Reinforcement Learning [7, 15]. It was also extended to nested
communication-local computation algorithms [1, 5].
For the fast convergence of the algorithm (1.2), it is important to choose a
suitable sequence of the stepsize $\alpha(t)$. It is often advantageous to
choose a possibly large stepsize as the convergence may become faster as the
stepsize gets larger in a stable regime.
When it comes to the cases $m=1$, the algorithm is reduced to the gradient
descent algorithm given as
$x(t+1)=x(t)-\alpha(t)\nabla f(x(t)),$ (1.3)
and we recall a well-known convergence result in the following theorem.
###### Theorem 1.1.
Assume that $f$ is $\mu$-strongly convex and $L$-smooth. Suppose that the
stepsize of (1.3) satisfies $\alpha(t)\leq\frac{2}{\mu+L}$. Then the sequence
$\\{x(t)\\}_{t\geq 0}$ is bounded. Moreover,
$\|x(t+1)-x_{*}\|^{2}\leq\Big{(}1-\frac{2L\mu\alpha(t)}{L+\mu}\Big{)}\|x(t)-x_{*}\|^{2}.$
Although the convergence property of the algorithm (1.2) has been studied
extensively ([8, 9, 12, 14, 6]), the sharp range of $\alpha(t)$ for the
convergence of the algorithm (1.2) has not been completely understood, even
when each cost $f_{k}$ is a quadratic form.
In the early stage, the convergene of the algorithm (1.2) was studied with
assuming that $\|\nabla f_{i}\|_{\infty}<\infty$ and each function $f_{i}$ is
convex for $1\leq i\leq m$. Nedić-Ozdaglar [8] showed that for the algorithm
(1.2) with the stepsize $\alpha(t)\equiv\alpha$, the cost value $f(\cdot)$ at
an average of the iterations converges to an $O(\alpha)$-neighborhood of an
optimal value of $f$. Nedić-Ozdaglar [9] proved that the algorithm (1.2)
converges to an optimal point if the stepsize satisfies
$\sum_{t=1}^{\infty}\alpha(t)=\infty$ and
$\sum_{t=1}^{\infty}\alpha(t)^{2}<\infty$. In the work of Chen [4], the
algorithm (1.2) with stepsize $\alpha(t)=c/t^{p}$ with $0<p<1$ was considered
and the convergene rate was achieved as $O(1/t^{p})$ for $0<p<1/2$, $O(\log
t/\sqrt{t})$ for $p=1/2$, and $O(1/t^{1-p})$ for $1/2<p<1$.
Recently, Yuan-Ling-Yin [14] established the convergence property of the
algorithm (1.2) without the gradient bound assumption. Assuming that each
local cost function is convex and the total cost is strongly convex, they
showed that the algorithm (1.2) with constant stepsize $\alpha(t)\equiv\alpha$
converges exponentially to an $O(\alpha)$-neighborhood of an optimizer $x_{*}$
of (1.1). Recently, the work [6] obtained the convergence property of the
algorithm (1.2) for a general class of non-increasing stepsize
$\\{\alpha(t)\\}_{t\in\mathbb{N}_{0}}$ given as $\alpha(t)=a/(t+w)^{p}$ for
$a>0$, $w\geq 1$ and $0<p\leq 1$ assuming the strong convexity on the total
cost function $f$, with cost functions $f_{i}$ not necessarily being convex.
To discuss the convergence property of (1.2), it is convenient to state the
following definition.
###### Definition 1.2.
The sequence $\\{x_{k}(t)\\}$ of (1.2) is said to be uniformly bounded if
there exists a value $R>0$ such that
$\|x_{i}(t)\|\leq R$
for all $t\geq 0$ and $1\leq i\leq m$.
We state the following convergence results established in the works [14, 6].
###### Theorem 1.3 ([14, 6]).
Assume that each cost $f_{k}$ is $L$-smooth and the aggregate cost $f$ is
$\mu$-strongly convex. Suppose also that the sequence $\\{x_{k}(t)\\}_{t\geq
0}$ of (1.2) is uniformly bounded and $\alpha(0)\leq\frac{2}{\mu+L}$. Then we
have the following results:
1. (1)
If $\alpha(t)\equiv\alpha$, then the sequence $x_{k}(t)$ converges
exponentially to an $O\Big{(}\frac{\alpha}{1-\beta}\Big{)}$ neighborhood of
$x_{*}$.
2. (2)
If $\alpha(t)=\frac{a}{(t+w)^{p}}$ for some $a>0,w\geq 0$ and $p\in(0,1]$,
then the sequence $x_{k}(t)$ converges to $x_{*}$ with the following rate
$\|x_{i}(t)-x_{*}\|=O\Big{(}\frac{1}{t^{p}}\Big{)}.$
We remark that a uniform bound assumption for the sequence
$\\{x_{k}(t)\\}_{t\geq 0}$ is required in the above result, which is contrast
to the result of Theorem 1.1. In fact, the boundedness property of (1.2) has
been obtained under an additional restriction on the stepsize as in the
following results.
###### Theorem 1.4 ([14]).
Assume that $f_{j}$ is convex and $L$-smooth. Suppose that the stepsize is
constant $\alpha(t)=\alpha$ and $\alpha\leq\frac{(1+\lambda_{m}(W))}{L}$, then
the sequence $\\{x(t)\\}$ is uniformly bounded. Here $\lambda_{m}(W)$ denotes
the smallest eigenvalue of the matrix $W$.
###### Theorem 1.5 ([6]).
Assume that $f_{j}$ is $L$-smooth and $f$ is $\mu$-strongly convex. Let
$\eta=\frac{\mu L}{\mu+L}$ and suppose that
$\alpha(t)\leq\frac{\eta(1-\beta)}{L(\eta+L)}$. Then the sequence
$\\{x_{i}(t)\\}$ is uniformly bounded.
In the above results, we note the following inequality
$\frac{\eta(1-\beta)}{L(\eta+L)}<\frac{1+\lambda_{m}(W)}{L}.$
Therefore, the result of Theorem 1.4 establishes the boundedness property for
a broader range of the constant stepsize with assuming the convexity for each
local cost. Meanwhile, the result of Theorem 1.5 establishes the boundedness
property for time varying stepsize without assuming the convexity on each
local cost, but the range of the stepsize is more restrictive. Having said
this, it is natural to consider the following questions:
Question 1: _Does the result of Theorem 1.4 hold for time-varying stepsize?
and can we moderate the convexity assumption on each cost?_
Question 2: _Can we extend the range of $\alpha(t)$ in the result of Theorem
1.5 with an additional information of the cost functions?_
We may figure out the importance of these questions by considering an example:
Consider $m=3$ and the functions
$f_{1},f_{2},f_{3}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ defined as
$f_{1}(x,y)=f_{2}(x,y)=\frac{L}{2}x^{2}+\frac{\mu}{2}y^{2}\quad\textrm{and}\quad
f_{3}(x,y)=-\frac{\epsilon}{2}x^{2}+\frac{\mu}{2}y^{2},$ (1.4)
where $L\gg\mu_{0}>0$ and $\epsilon\geq 0$ is small enough. Then $f_{k}$ is
$L$-smooth for $1\leq k\leq 3$ and the aggregate cost $f$ is $\mu$-strongly
convex. If we apply the above results, we have the following results:
If $\epsilon=0$, then the boundedness property holds by Theorem 1.4 for
constant stepsize $\alpha(t)\equiv\alpha$ satisfying
$\alpha\leq\frac{1+\lambda_{m}(W)}{L}.$
If $\epsilon>0$, then the boundedness property holds by Theorem 1.5 for the
stepsize $\alpha(t)$ satisfying
$\alpha(t)\leq\frac{\eta(1-\beta)}{L(\eta+L)}.$
Here we see that the range guaranteed by the above theorems drastically
changes as the value $\epsilon$ becomes positive from zero.
The purpose of this paper is to provide reasonable answers to the above-
mentioned questions. For this we consider the function
$G_{\alpha}:\mathbb{R}^{nm}\rightarrow\mathbb{R}$ for $\alpha>0$ defined by
$\begin{split}G_{\alpha}(\mathbf{x})&=\frac{\alpha}{m}\sum_{k=1}^{m}f_{k}(x_{k})+\frac{1}{2}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}\\\
&=:\alpha\mathbf{F}(\mathbf{x})+\frac{1}{2}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}.\end{split}$
As well-known, the algorithm (1.2) is then written as
$\mathbf{x}(t+1)=\mathbf{x}(t)-\nabla G_{\alpha(t)}(\mathbf{x}(t)).$ (1.5)
For each $\alpha>0$ we define the following function class
$\mathbf{G}_{\alpha}=\Big{\\{}(f_{1},\cdots,f_{m})\in(C_{1}(\mathbb{R}^{n}))^{m}~{}\Big{|}~{}\textrm{the
function $G_{\alpha}$ is strongly convex }\Big{\\}}.$ (1.6)
The following is the main result of this paper.
###### Theorem 1.6.
Suppose that $F=(f_{1},\cdots,f_{m})\in A_{\alpha_{0}}$ for some
$\alpha_{0}>0$ and $f_{k}$ is $L$-smooth for each $1\leq k\leq m$. Assume that
the sequence of stepsize $\\{\alpha(t)\\}_{t\in\mathbb{N}}$ is a non-
increasing sequence satisfying
$\alpha(0)\leq\min\Big{\\{}\frac{1+\lambda_{m}(W)}{L},~{}\alpha_{0}\Big{\\}}.$
(1.7)
Then the sequence $\\{\mathbf{x}(t)\\}$ is uniformly bounded.
This result naturally extends the previous works [14, 6]. We mention that the
result [14] was obtained for constant stepsize
$\alpha(t)\equiv\alpha\leq\frac{1+\lambda_{m}(W)}{L}$ under the assumption
that each cost $f_{k}$ is convex. Meanwhile, the result [6] was obtained only
assuming that the aggregate cost $f$ is strongly convex but the stepsize range
is more conservative. We summarize the results of [14, 6] and this work in
Table 1.
| Convexity condition | Type of stepsize | Bound on stepsize
---|---|---|---
[14] | Each $f_{j}$ is convex | constant $\alpha(t)\equiv\alpha$ | $\alpha\leq\frac{(1+\lambda_{m}(W))}{L}$
[6] | $f$ is $\mu$-strongly convex | any sequence | $\alpha(t)\leq\frac{\eta(1-\beta)}{L(\eta+L)}$
This work | $G_{\alpha_{0}}$ is strongly convex | non-increasing sequence | $\alpha(t)\leq\min\Big{\\{}\frac{1+\lambda_{m}(W)}{L},~{}\alpha_{0}\Big{\\}}.$
Table 1. This table summarizes the conditions for the uniform boundedness of
the sequence $\mathbf{x}(t)$ of DGD algorithm (1.5).
For the example (1.4), we set $L=10$, $\mu=1$, and
$W=\begin{pmatrix}0.4&0.3&0.3\\\ 0.3&0.3&0.4\\\ 0.3&0.4&0.3\end{pmatrix}.$
Then for $0\leq\epsilon\leq 10$ we have
$\alpha_{S}:=\frac{\eta(1-\beta)}{L(\eta+L)}=0.0075,\quad\frac{1+\lambda_{m}(W)}{L}=0.2.$
Finally the value $\alpha_{0}$ of Theorem 1.6 is computed for each
$\epsilon\in(0,10]$. The graph of Figure 1 shows that the value $\alpha_{0}$
is larger than $(1+\lambda_{m}(W))/L$ for small $\epsilon>0$, but it becomes
smllaer than the latter value $\epsilon>0$ is larger than $3$. For the detail
computing these values, we refer to Section 5.
Figure 1. The figure corresponds to the graph of values $\alpha_{0}$,
$\alpha_{S}$ and $\frac{1+\lambda_{m}(W)}{L}$ with respect to $\epsilon\geq
0$.
The function class $\mathbf{G}_{\alpha}$ is closely related to the function
class that the aggregate cost $f$ is strongly convex. To explain the relation,
for each $\alpha>0$ we let
$S_{\alpha}=\\{(f_{1},\cdots,f_{m})\in(C_{1}(\mathbb{R}^{n})^{m}~{}\mid~{}\textrm{the
function $f$ is $\alpha$-strongly convex}\\}.$
Then, for given $F=(f_{1},\cdots,f_{m})\in(C_{1}(\mathbb{R}^{n})^{m}$, we will
prove that the following relation between the class $\mathbf{G}_{\alpha}$ and
$S_{\alpha}$ in Section 2:
1. (1)
If $f_{k}$ is quadratic and convex for each $1\leq k\leq m$, then
$F\in S_{\mu}\textrm{~{}for some~{}}\mu>0\Longrightarrow
F\in\mathbf{G}_{\alpha}\textrm{~{}for all~{}}\alpha>0.$
2. (2)
If $f_{k}$ is quadratic for each $1\leq k\leq m$, then
$F\in S_{\mu}\textrm{~{}for some~{}}\mu>0\Longrightarrow
F\in\mathbf{G}_{\alpha}\textrm{~{}for some~{}}\alpha>0.$
3. (3)
It always holds that
$F\in\mathbf{G}_{\alpha}\textrm{~{}for
some~{}}\alpha>0~{}\Longrightarrow~{}F\in S_{\mu}~{}\textrm{ for
some}~{}\mu>0.$
For given $F\in\mathbf{G}_{\alpha}$, let us denote the optimal point of
$G_{\alpha}$ by $\mathbf{x}_{*}^{\alpha}$. In order to prove the above result,
we exploit the fact that the algorithm (1.2) at step $t$ is interpreted as the
gradient descent of $G_{{\alpha}(t)}$ as in (1.5). Using this fact, it is not
difficult to derive the result of Theorem 1.6 if the stepsize is given by a
constant $\alpha(t)\equiv\alpha>0$ since the gradient descent descent (1.5)
converges to the minimizer $\mathbf{x}_{*}^{\alpha}$. However, when the
stepsize is varying, then we may not interprete (1.5) as a gradient descent
algorithm of a single objective function. In order to handle this case, we
prove the continuity and boundedness property of $x_{\alpha}$ with respect to
$\alpha$. Precisely, we obtain the following result.
###### Theorem 1.7.
Assume that $G_{\alpha_{0}}$ is $\mu$-strongly convex for some $\alpha_{0}>0$
and $\mu>0$. Then the following results hold:
1. (1)
For $\alpha\in(0,\alpha_{0}]$ we have
$\|\mathbf{x}_{*}^{\alpha}\|^{2}\leq\frac{2\alpha_{0}}{\mu}f(0).$
2. (2)
For all $\alpha,\beta\in(0,\alpha_{0}]$ we have
$\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|\leq\frac{2\alpha_{0}C_{1}|\beta-\alpha|}{\mu\beta},$
where
$C_{1}=\sup_{s\in[0,1]}\sup_{\alpha,\beta\in(0,\alpha_{0}]}\Big{\|}\nabla\mathbf{F}(\mathbf{x}_{*}^{\beta}+s(\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}))\Big{\|}.$
In the above result, we remark that the constant $C_{1}$ is bounded since
$\mathbf{F}$ is smooth and $\mathbf{x}_{*}^{\alpha}$ are uniformly bounded for
$\alpha\in(0,\alpha_{0}]$. We will make use of this result to prove Theorem
1.6.
The rest of this paper is organized as follows. Section 2 is devoted to study
the function class $\mathbf{G}_{\alpha}$. In Section 3 we exploit the
boundedness and continuity property of the optimizer of $G_{\alpha}$ with
respect to $\alpha$. Based on the property, we prove the main result in
Section 4. Section 5 provides some numerical experiments supporting the result
of this paper.
## 2\. Properties of the class $\mathbf{G}_{\alpha}$
In this section, we study the function classe $\mathbf{G}_{\alpha}$ defined in
(1.6). Before this, we introduce two standard assumptions on the graph
$\mathcal{G}$ ans its associated weight $W=\\{w_{ij}\\}_{1\leq i,j\leq m}$.
###### Assumption 1.
The communication graph $\mathcal{G}$ is undirected and connected, i.e., there
exists a path between any two agents.
We define the mixing matrix $W=\\{w_{ij}\\}_{1\leq i,j\leq m}$ as follows. The
nonnegative weight $w_{ij}$ is given for each communication link
$\\{i,j\\}\in\mathcal{E},$ where $w_{ij}\neq 0$ if $\\{i,j\\}\in\mathcal{E}$
and $w_{ij}=0$ if $\\{i,j\\}\notin\mathcal{E}$. In this paper, we make the
following assumption on the mixing matrix $W$.
###### Assumption 2.
The mixing matrix $W=\\{w_{ij}\\}_{1\leq i,j\leq m}$ is doubly stochastic,
i.e., $W\mathbf{1}=\mathbf{1}$ and $\mathbf{1}^{T}W=\mathbf{1}^{T}$. In
addition, $w_{ii}>0$ for all $i\in\mathcal{V}$.
In the following theorem, we study the function class $\mathbf{G}_{\alpha}$
when each local cost is a quadratic form.
###### Theorem 2.1.
Assume that each cost is a quadratic form
$f_{k}(x)=\frac{1}{2}x_{k}^{T}A_{k}x_{k}$.
1. (1)
Assume that each $f_{k}$ is convex and $f$ is $\mu$-strongly convex. Then
$G_{\alpha}$ is strongly convex for any $\alpha>0$.
2. (2)
Assume that $f$ is $\mu$-strongly convex. Then there exists a value
$\alpha_{0}>0$ such that for $\alpha>\alpha_{0}$, $G_{\alpha}$ is strongly
convex.
###### Proof.
Then $G_{\alpha}$ is given as
$G_{\alpha}(\mathbf{x})=\frac{\alpha}{2m}\sum_{k=1}^{m}x_{k}^{T}A_{k}x_{k}+\frac{1}{2}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}.$
Choose a constant $Q>0$ and $\bar{Q}>0$ such that
$\Big{\|}\sum_{k=1}^{n}A_{k}u_{k}\Big{\|}\leq
Q\|u\|\quad\forall~{}u\in\mathbb{R}^{mn}$ (2.1)
and
$\Big{\|}(A_{1}u_{1},\cdots,A_{m}u_{m})\Big{\|}\leq\bar{Q}\|u\|\quad\forall~{}u\in\mathbb{R}^{nm}.$
(2.2)
Let $\mathbf{x}=1\otimes\bar{x}+u$ with $u=\mathbf{x}-1\otimes\bar{x}$ and
$\bar{x}=\frac{1}{n}\sum_{k=1}^{n}x_{k}$. Using that $W1_{m}=1_{m}$, we find
$G_{\alpha}(\mathbf{x})=\frac{\alpha}{2m}\sum_{k=1}^{n}(\bar{x}+u_{k})^{T}A_{k}(\bar{x}+u_{k})+\frac{1}{2}u^{T}(I-W\otimes
1_{n})u.$ (2.3)
Take a constant $c>0$ and consider
$Y_{c}=\\{\mathbf{x}=1\bar{x}+u~{}:~{}\|u\|\leq
c\|\bar{x}\|\\}\quad\textrm{and}\quad
Z_{c}=\\{\mathbf{x}=1\bar{x}+u~{}:~{}\|u\|\geq c\|\bar{x}\|\\}.$
If $\mathbf{x}\in Y_{c}$, then we have
$\|\mathbf{x}\|^{2}=n\|\bar{x}\|^{2}+\|u\|^{2}\leq(n+c^{2})\|\bar{x}\|^{2}.$
(2.4)
For $\mathbf{x}\in Z_{c}$, it holds that
$\|\mathbf{x}\|^{2}=n\|\bar{x}\|^{2}+\|u\|^{2}\leq\frac{(n+c^{2})}{c^{2}}\|u\|^{2}.$
(2.5)
Now we proceed to prove (1). For $\mathbf{x}\in Y_{c}$ we use (2.1) to
estimate (2.3) as
$\begin{split}G_{\alpha}(\mathbf{x})&=\frac{\alpha}{2m}\Big{[}\bar{x}^{T}\Big{(}\sum_{k=1}^{m}A_{k}\Big{)}\bar{x}+2\bar{x}\Big{(}\sum_{k=1}^{m}A_{k}u_{k}\Big{)}+\sum_{k=1}^{m}u_{k}^{T}A_{k}u_{k}\Big{]}+\frac{1}{2}u^{T}(I-W\otimes
I_{n})u\\\
&\geq\frac{\alpha}{2}\Big{[}\mu\|\bar{x}\|^{2}-2Q\|\bar{x}\|\|u\|\Big{]}\\\
&\geq\frac{\alpha}{2}[\mu-2Qc]\|\bar{x}\|^{2}.\end{split}$
Using (2.4) here, we find
$G_{\alpha}(\mathbf{x})\geq\frac{\alpha(\mu-2Qc)}{2(n+c^{2})}\|\mathbf{x}\|^{2}.$
For $\mathbf{x}\in Z_{c}$ we have
$G_{\alpha}(\mathbf{x})\geq(1-\beta)\|u\|^{2}\geq\frac{(1-\beta)c^{2}}{n+c^{2}}\|\mathbf{x}\|^{2}.$
Here we used the fact that
$y^{T}(I-W)y\geq(1-\beta)\|y\|^{2}$
for any $y\in\mathbb{R}^{m}$ satisfying $\langle y,1_{m}\rangle=0$, where
$\beta\in(0,1)$ is the second largest eigenvalue of $W$.
Combining the above two estimates, we find
$G_{\alpha}(\mathbf{x})\geq\min\Big{\\{}\frac{(1-\beta)c^{2}}{n+c^{2}},\frac{\alpha(\mu-2Qc)}{2(n+c^{2})}\Big{\\}}\|\mathbf{x}\|^{2},$
which provesthe first assertion (1).
Next we prove (2). For $\mathbf{x}\in Y_{c}$ we have
$\begin{split}G_{\alpha}(\mathbf{x})&\geq\frac{\alpha}{2}\Big{[}\mu\|\bar{x}\|^{2}-2\|\bar{x}\|\Big{\|}\sum_{k=1}^{m}A_{k}u_{k}\Big{\|}-\|u\|\Big{\|}\sum_{k=1}^{n}A_{k}u_{k}\Big{\|}\Big{]}\\\
&\geq\frac{\alpha}{2}\Big{[}\mu\|\bar{x}\|^{2}-2Q\|\bar{x}\|\|u\|-\bar{Q}\|u\|^{2}\Big{]}\\\
&\geq\frac{\alpha}{2}\Big{[}\mu\|\bar{x}\|^{2}-(2cQ+c^{2}\bar{Q})\|\bar{x}\|^{2}\Big{]}.\end{split}$
For $\mathbf{x}\in Z_{c}$ we estimate
$\begin{split}G_{\alpha}(\mathbf{x})&\geq\frac{\alpha}{2}\sum_{k=1}^{m}\bar{x}^{T}A_{k}\bar{x}+\alpha\sum_{k=1}^{m}\bar{x}(A_{k}u_{k})+\frac{\alpha}{2}\sum_{k=1}^{m}u_{k}^{T}A_{k}u_{k}+(1-\beta)\|u\|^{2}\\\
&\geq\frac{\alpha\mu}{2}\|\bar{x}\|^{2}-\alpha
Q\|\bar{x}\|\|u\|-\frac{\alpha\bar{Q}}{2}\|u\|^{2}+(1-\beta)\|u\|^{2}\\\
&\geq\frac{\alpha\mu}{2}\|\bar{x}\|^{2}-\Big{(}\frac{\alpha
Q}{c}+\frac{\alpha\bar{Q}}{2}\Big{)}\|u\|^{2}+(1-\beta)\|u\|^{2}.\end{split}$
This gives the following estimate
$G_{\alpha}(\mathbf{x})\geq\frac{c^{2}}{n+c^{2}}\Big{[}(1-\beta)-\Big{(}\frac{\alpha
Q}{c}+\frac{\alpha\bar{Q}}{2}\Big{)}\Big{]}\|\mathbf{x}\|^{2}.$
This completes the proof of the second assertion (2). ∎
We also have the following result.
###### Theorem 2.2.
If $G_{\alpha}$ is $\mu$-stronlgy convex for some $\alpha>0$, then $f$ is
$\frac{\mu}{\alpha}$-stronlgy convex.
###### Proof.
Let $\mathbf{x}=(x,\cdots,x)$ and $\mathbf{y}=(y,\cdots,y)$. The strongly
convexity of $G_{\alpha}$ yields that
$G_{\alpha}(\mathbf{y})\geq
G_{\alpha}(\mathbf{x})+(\mathbf{y}-\mathbf{x})\nabla
G_{\alpha}(x)+\frac{\mu}{2}\|\mathbf{y}-\mathbf{x}\|^{2}.$ (2.6)
We have
$G_{\alpha}(\mathbf{x})=\frac{\alpha}{m}\sum_{k=1}^{m}f_{k}(x),\quad
G_{\alpha}(\mathbf{y})=\frac{\alpha}{m}\sum_{k=1}^{m}f_{k}(y).$
Also,
$\begin{split}\nabla G_{\alpha}(\mathbf{x})&=\frac{\alpha}{m}(\nabla
f_{1}(x),\cdots,\nabla f_{m}(x))+(I-W)\mathbf{x}\\\ &=\frac{\alpha}{m}(\nabla
f_{1}(x),\cdots,\nabla f_{m}(x)).\end{split}$
Using these equalities in (2.6) we find
$f(y)\geq f(x)+(y-x)\nabla f(x)+\frac{\mu}{2\alpha}\|y-x\|^{2},$
and so the function $f$ is $\frac{\mu}{2\alpha}$-strongly convex. The proof is
done. ∎
## 3\. Uniform bound and smoothness property of $x_{*}^{\alpha}$ with respect
to $\alpha>0$
In this section, we exploit the property of the minimizer
$\mathbf{x}_{\alpha}$ of the function $G_{\alpha}$. Under the strongly
convexity assumption on $G_{\alpha_{0}}$ for some $\alpha_{0}$, we will show
that the optimizers $\mathbf{x}_{*}^{\alpha}\in\mathbb{R}^{n}$ is uniformly
bounded for $\alpha\in(0,\alpha_{0}]$ and also locally Lipschitz continuous
with respect to $\alpha$.
###### Lemma 3.1.
We assume that $G_{\alpha}$ is $\mu$-strongly convex for some $\alpha>0$.
Then, the function $G_{\beta}$ is $\frac{\beta\mu}{\alpha}$-strongly convex
for any $\beta\in(0,\alpha]$.
###### Proof.
For $\beta\in(0,\alpha]$, we express the function $G_{\beta}$ as
$\begin{split}G_{\beta}(\mathbf{x})&=\beta\sum_{k=1}^{n}f_{k}(x_{k})+\frac{1}{2}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}\\\
&=\frac{\beta}{\alpha}\Big{[}\alpha\sum_{k=1}^{n}f_{k}(x_{k})+\frac{1}{2}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}\Big{]}+\frac{(\alpha-\beta)}{2\alpha}\mathbf{x}^{T}(I-W\otimes
1_{n})\mathbf{x}.\end{split}$
Since the last term is convex, and $G_{\alpha}$ is $\mu$-strongly convex, the
above formula yields that $G_{\beta}$ is $\frac{\beta\mu}{\alpha}$-strongly
convex. The proof is done. ∎
We now prove Theorem 1.7.
###### Proof of Theorem 1.7.
For $0<\alpha\leq\alpha_{0}$ we know that $G_{\alpha}$ is
$\Big{(}\frac{\alpha}{\alpha_{0}}\Big{)}\mu$-strongly convex by Lemma 3.1.
Therefore
$G_{\alpha}(\mathbf{x})\geq\frac{\alpha\mu}{2\alpha_{0}}\|\mathbf{x}-\mathbf{x}_{*}^{\alpha}\|^{2}$
for all $\mathbf{x}\in\mathbb{R}^{nm}$. Taking $\mathbf{x}=0$ here, we get
$\frac{\alpha\mu}{2\alpha_{0}}\|\mathbf{x}_{*}^{\alpha}\|^{2}\leq
G_{\alpha}(0)=\alpha f(0),$
which gives
$\|\mathbf{x}_{*}^{\alpha}\|^{2}\leq\frac{2\alpha_{0}}{\mu}f(0).$
This proves the first assertion.
Next we are concerned with the smoothness property of
$\mathbf{x}_{*}^{\alpha}\in\mathbb{R}^{d}$ with respect to the parameter
$\alpha>0$. The function $G_{\beta}$ is $\frac{\mu\beta}{\alpha_{0}}$-strongly
convex by Lemma 3.1. Using this fact and that $\mathbf{x}_{*}^{\beta}$ is a
minimizer of $F_{\beta}$, we find
$\begin{split}&\beta\mathbf{F}(\mathbf{x})+\frac{1}{2}\mathbf{x}^{T}(I-W)\mathbf{x}\\\
&\geq\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}-\mathbf{x}_{*}^{\beta}\|^{2}+\beta\mathbf{F}(\mathbf{x}_{*}^{\beta})+\frac{1}{2}(\mathbf{x}_{*}^{\beta})^{T}(I-W)\mathbf{x}_{*}^{\beta}\\\
&=\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}-\mathbf{x}_{*}^{\beta}\|^{2}+(\beta-\alpha)\mathbf{F}(\mathbf{x}_{*}^{\beta})+\Big{[}\alpha\mathbf{F}(\mathbf{x}_{*}^{\beta})+\frac{1}{2}(\mathbf{x}_{*}^{\beta})^{T}(I-W)\mathbf{x}_{*}^{\beta}\Big{]}.\end{split}$
Using the minimality of $\mathbf{x}_{*}^{\alpha}$ for $G_{\alpha}$ in the
right hand side, we find
$\begin{split}&\beta\mathbf{F}(\mathbf{x})+\frac{1}{2}\mathbf{x}^{T}(I-W)\mathbf{x}\\\
&\geq\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}-\mathbf{x}_{*}^{\beta}\|^{2}+(\beta-\alpha)\mathbf{F}(\mathbf{x}_{*}^{\beta})+\Big{[}\alpha\mathbf{F}(\mathbf{x}_{*}^{\alpha})+\frac{1}{2}({\mathbf{x}_{*}^{\alpha}})^{T}(I-W)\mathbf{x}_{*}^{\alpha}\Big{]}.\end{split}$
Taking $\mathbf{x}=\mathbf{x}_{*}^{\alpha}$, we find
$\begin{split}&\beta\mathbf{F}(\mathbf{x}_{*}^{\alpha})+\frac{1}{2}(\mathbf{x}_{*}^{\alpha})^{T}(I-W)\mathbf{x}_{*}^{\alpha}\\\
&\geq\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|^{2}+(\beta-\alpha)\mathbf{F}(\mathbf{x}_{*}^{\beta})+\alpha\mathbf{F}(\mathbf{x}_{*}^{\alpha})+\frac{1}{2}(\mathbf{x}_{*}^{\alpha})^{T}(I-W)\mathbf{x}_{*}^{\alpha},\end{split}$
which gives
$(\beta-\alpha)(\mathbf{F}(\mathbf{x}_{*}^{\alpha})-\mathbf{F}(\mathbf{x}_{*}^{\beta}))\geq\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|^{2}.$
(3.1)
Notice that
$\mathbf{F}(\mathbf{x}_{*}^{\alpha})-\mathbf{F}(\mathbf{x}_{*}^{\beta})=(\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta})\cdot\int_{0}^{1}\nabla\mathbf{F}(\mathbf{x}_{*}^{\beta}+s(\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}))ds.$
Therefore we have
$|\mathbf{F}(\mathbf{x}_{*}^{\alpha})-\mathbf{F}(\mathbf{x}_{*}^{\beta})|\leq
C_{1}\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|.$
This together with (3.1) gives
$\frac{\mu\beta}{2\alpha_{0}}\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|^{2}\leq
C_{1}|\beta-\alpha|\|\mathbf{x}_{*}^{\alpha}-\mathbf{x}_{*}^{\beta}\|.$
The proof is done. ∎
## 4\. Boundedness property
In this section, we make use of the properties of $\mathbf{x}_{*}^{\alpha}$
obtained in the previous section to study the sequence
$\\{\mathbf{x}(t)\\}_{t\geq 0}$ of the decentralized gradient descent (1.5).
###### Lemma 4.1.
Suppose that $G_{\alpha_{0}}$ is $\mu$-strongly convex for some
$\alpha_{0}>0$. Assume that
$\alpha(t)\leq\min\Big{\\{}\frac{1+\sigma_{n}(W)}{L},~{}\alpha_{0}\Big{\\}}$.
Then we have
$\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha(t+1)}\|\leq\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha(t)}\|+\frac{2\alpha_{0}C_{1}}{\mu\alpha(t)}|\alpha(t+1)-\alpha(t)|.$
###### Proof.
Note that $G_{\alpha}$ is $L_{G}$-smooth with $L_{G}=\alpha
L+(1-\sigma_{n}(W))$. Thus, if $\alpha\leq\frac{1+\sigma_{n}(W)}{L}$, then we
have $L_{G}\leq 2$. Note that
$\begin{split}&\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha}\|^{2}\\\
&=\|\mathbf{x}_{t}-\nabla
G_{\alpha}(\mathbf{x}_{t})-\mathbf{x}_{*}^{\alpha}\|^{2}\\\
&=\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha}\|^{2}-2\Big{\langle}\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha},~{}\nabla
G_{\alpha}(\mathbf{x}_{t})-\nabla
G_{\alpha}(\mathbf{x}_{*}^{\alpha})\Big{\rangle}+\|\nabla
G_{\alpha}(\mathbf{x}_{t})-\nabla
G_{\alpha}(\mathbf{x}_{*}^{\alpha})\|^{2}.\end{split}$
Since $G_{\alpha}$ is convex and $L_{G}$-smooth, we have
$\langle\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha},~{}\nabla
G_{\alpha}(\mathbf{x}_{t})-\nabla
G_{\alpha}(\mathbf{x}_{*}^{\alpha})\rangle\geq\frac{1}{L_{G}}\|\nabla
G_{\alpha}(\mathbf{x}_{t})-\nabla G_{\alpha}(\mathbf{x}_{*}^{\alpha})\|^{2}.$
Combining the above two estimates, we get
$\begin{split}\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha}\|^{2}&\leq\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha}\|^{2}-\Big{(}\frac{2}{L_{G}}-1\Big{)}\|\nabla
G_{\alpha}(\mathbf{x}_{t})-\nabla G_{\alpha}(\mathbf{x}_{*}^{\alpha})\|^{2}\\\
&\leq\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha}\|^{2}.\end{split}$
Using this with the triangle inequality and Theorem 1.7, we deduce
$\begin{split}\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha(t+1)}\|&\leq\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha(t)}\|+\|\mathbf{x}_{*}^{\alpha(t)}-\mathbf{x}_{*}^{\alpha(t+1)}\|\\\
&\leq\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha(t)}\|+\frac{2\alpha_{0}C_{1}}{\mu\alpha(t)}|\alpha(t)-\alpha(t+1)|.\end{split}$
The proof is done.
∎
In order to prove Theorem 1.6, we recall the following result from [6].
###### Theorem 4.2 ([6]).
Suppose that the total cost function $f$ is $\mu$-strongly convex for some
$\mu>0$ and each $f_{i}$ is $L$-smooth for $1\leq i\leq m$. If
$\\{\alpha(t)\\}_{t\in\mathbb{N}_{0}}$ is non-increasing stepsize satisfying
$\alpha(0)<\frac{\eta(1-\beta)}{L(\eta+L)}$, then we have
$\|\bar{\mathbf{x}}(t)-\mathbf{x}_{*}\|\leq R\quad
and\quad\|\mathbf{x}(t)-\bar{\mathbf{x}}(t)\|\leq\frac{\eta
R}{L}<R,\quad\forall t\geq 0.$ (4.1)
Here $\eta=\frac{\mu L}{\mu+L}$ and a finite value $R>0$ is defined as
$R=\max\left\\{\|\bar{\mathbf{x}}(0)-\mathbf{x}_{*}\|,~{}\frac{L}{\eta}\|\mathbf{x}(0)-\bar{\mathbf{x}}(0)\|,~{}\frac{\sqrt{m}D\alpha(0)}{\eta(1-\beta)/L-(\eta+L)\alpha(0)}\right\\}.$
Using the above lemma, we give the proof of the main theorem on the
boundedness property of the sequence $\\{\mathbf{x}(t)\\}_{t\geq 0}$.
###### Proof of Theorem 1.6.
Scine $F\in\mathbf{G}_{\alpha}$, the function $G_{\alpha}$ is $\mu$-strongly
convex for some $\mu>0$. Then we know that $f$ is
$\frac{\mu}{\alpha_{0}}$-strongly convex by Theorem 2.2. We set the following
constants
$\eta=\frac{(\mu/\alpha_{0})L}{(\mu/\alpha_{0})+L}\quad\textrm{and}\quad\mathbf{a}_{c}=\frac{\frac{\mu}{\alpha_{0}}(1-\beta)}{2L\Big{(}\frac{\mu}{\alpha_{0}}+L\Big{)}}=\frac{\mu(1-\beta)}{4L(\mu+\alpha_{0}L)}.$
Take a number $t_{0}\in\mathbb{N}$ such that $\alpha(t_{0})\geq\mathbf{a}_{c}$
and $\alpha(t_{0}+1)<\mathbf{a}_{c}$.
For $t\leq t_{0}$ we apply Lemma 4.1 to find
$\begin{split}\|\mathbf{x}_{t+1}-\mathbf{x}_{*}^{\alpha(t+1)}\|&\leq\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha(t)}\|+\frac{2C_{1}}{\alpha(t)}|\alpha(t+1)-\alpha(t)|\\\
&\leq\|\mathbf{x}_{t}-\mathbf{x}_{*}^{\alpha(t)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}|\alpha(t+1)-\alpha(t)|.\end{split}$
Thus,
$\begin{split}\|\mathbf{x}_{t_{0}+1}-\mathbf{x}_{*}^{\alpha(t)}\|&\leq\|\mathbf{x}_{0}-\mathbf{x}_{*}^{\alpha(1)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}\sum_{s=0}^{t_{0}}|\alpha(s+1)-\alpha(s)|\\\
&=\|\mathbf{x}_{0}-\mathbf{x}_{*}^{\alpha(0)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}(\alpha(0)-\alpha(t_{0}+1)).\end{split}$
Note that
$\|\mathbf{x}_{t_{0}+1}\|\leq\|\mathbf{x}_{0}-\mathbf{x}_{*}^{\alpha(0)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}\alpha(0)+\|\mathbf{x}_{*}^{\alpha(t_{0}+1)}\|.$
Therefore
$\|\bar{\mathbf{x}}_{t_{0}+1}-\mathbf{x}_{t_{0}+1}\|\leq\|\mathbf{x}_{0}-\mathbf{x}_{*}^{\alpha(0)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}\alpha(0)+\|\mathbf{x}_{*}^{\alpha(t_{0}+1)}\|$
and
$\begin{split}\|\bar{\mathbf{x}}_{t_{0}+1}-\mathbf{x}_{*}\|&\leq\|\bar{\mathbf{x}}_{t_{0}+1}\|+\|\mathbf{x}_{*}\|\\\
&\leq\|\mathbf{x}_{0}-\mathbf{x}_{*}^{\alpha(0)}\|+\frac{2C_{1}}{\mathbf{a}_{c}}\alpha(0)+\|\mathbf{x}_{*}^{\alpha(t_{0}+1)}\|+\|\mathbf{x}_{*}\|.\end{split}$
Now we use the result of Theorem 4.2 to deduce that
$R=\max\left\\{\|\bar{\mathbf{x}}_{t_{0}+1}-\mathbf{x}_{*}\|,~{}\frac{L}{\eta}\|\mathbf{x}_{t_{0}+1}-\bar{\mathbf{x}}_{t_{0}+1}\|,~{}\frac{\sqrt{m}D\alpha(t_{0}+1)}{\eta(1-\beta)/L-(\eta+L)\alpha(t_{0}+1)}\right\\},$
then we have
$\|\bar{\mathbf{x}}(t)-\mathbf{x}_{*}\|\leq R\quad
and\quad\|\mathbf{x}(t)-\bar{\mathbf{x}}(t)\|\leq\frac{\eta
R}{L}<R,\quad\forall t\geq t_{0}+1.$ (4.2)
Here $\eta=\frac{\mu L}{\mu+L}$. We also check that
$\begin{split}\frac{\sqrt{m}D\alpha(t_{0}+1)}{\eta(1-\beta)/L-(\eta+L)\alpha(t_{0}+1)}&=\frac{\sqrt{m}D}{\frac{\eta(1-\beta)}{L\alpha(t_{0}+1)}-(\eta+L)}\\\
&\leq\frac{\sqrt{m}D}{\frac{\eta(1-\beta)}{L\mathbf{a}_{c}}-(\eta+L)}=\frac{\sqrt{m}D}{2(\eta+L)}.\end{split}$
The proof is finished. ∎
## 5\. Numerical experiments
In this section, we provide numerical experiments supporting the theoretical
results of this paper.
First we set $m=3$ and $n=2$ for problem (1.1). For each $1\leq k\leq m$, we
choose the local cost $f_{k}$ as
$f_{k}(x)=\frac{1}{2}x^{T}A_{k}x+B_{k}^{T}x.$
Here the matrix $A_{k}$ is an $n\times n$ symmetrix matrix chosen as
$A_{k}=\epsilon I_{n\times n}+(R_{k}+R_{k}^{T}),$
where $\epsilon>0$ and each element of $R_{k}$ is chosen randomly from
$[-1,1]$ with uniform distribution. Also, each element of
$B_{k}\in\mathbb{R}^{n}$ is randomly chosen in $[-1,1]$ with uniform
distribution. We choose $W$ as
$W=\begin{pmatrix}1/2&1/4&1/4\\\ 1/4&1/2&1/4\\\ 1/4&1/4&1/2\end{pmatrix}.$
In order to verify the result of Theorem 1.6, we compute the following
constants:
$\begin{split}\alpha_{A}&=\textrm{a possibly large value $\alpha_{0}>0$ such
that $F\in\mathbf{G}_{\alpha_{0}}$}\\\
\alpha_{L}&=\frac{1+\lambda_{m}(W)}{L}.\end{split}$
The detail for computing the above values are explained below:
* •
To compute $\alpha_{A}$ we choose a large $N\in\mathbb{N}$ and set
$\alpha_{A}=\sup_{k\in\mathbb{N}}\Big{\\{}\frac{k}{N}>0~{}|~{}F\in\mathbf{G}_{(k/N)}\Big{\\}}.$
Since $G_{(k/N)}(x)$ is a quadratic function, we may check the positivity of
all eigenvalues of $\nabla^{2}G_{(k/N)}(x)$ to determine if the function
$G_{(k/N)}$ is strongly convex.
* •
To find the smallest value of $L>0$, we compute $L_{k}=\|A_{k}\|_{\infty}$ by
using the eigenvalues of $A_{k}$. Then we set $L=\sup_{1\leq k\leq m}L_{k}$.
In our experiment, the constants are computed as
$\alpha_{A}\simeq 0.0799\quad\textrm{and}\quad\alpha_{L}\simeq 0.1721$
with $L\simeq 7.2615$ and $\lambda_{m}(W)=0.25$.
Since $\alpha_{A}<\alpha_{L}$, the range (1.7) of the stepsize $\alpha(t)$
guaranteed by Theorem 1.6 is given as
$\alpha(0)\leq\alpha_{A}.$
We take the constant stepsize $\alpha(t)\equiv\alpha>0$ with various choices
of $\alpha$ given as
$\\{0.5\alpha_{A},~{}0.95\alpha_{A},~{}0.99\alpha_{A},~{}1.01\alpha_{A},~{}1.01\\}.$
For each time step $t\geq 0$, we measure the following error
$R(t)=\sum_{k=1}^{m}\|x_{k}(t)-x_{*}\|,$
where $x_{k}(t)$ is the state of $k$ in (1.2) and $x_{*}$ is the optimizer of
(1.1).
Figure 2. The left (resp., right) figure corresponds to the graph of value
$\log R(t)$ (resp., $R(t)$) with respect to $t\geq 0$.
The result shows that the states $\\{x_{k}(t)\\}_{k=1}^{m}$ are uniformly
bounded for the three cases
$\alpha\in\\{0.5\alpha_{A},0.95\alpha_{A},0.99\alpha_{A}\\}$ as expected by
Theorem 1.6. Meanwhile, the states $\\{x_{k}(t)\\}$ diverges as
$t\rightarrow\infty$ for the choices
$\alpha\in\\{1.01\alpha_{A},1.02\alpha_{A}\\}$ which are larger than the value
$\alpha_{A}$. This shows the sharpness of the result of Theorem 1.6.
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# A high sensitivity tool for geophysical applications: A geometrically locked
Ring Laser Gyroscope.
E. Maccioni 1,2 N. Beverini 1 G. Carelli 1,2,* G. Di Somma 1,2 A. Di
Virgilio 2 and P. Marsili1,2 1University of Pisa, Pisa, Italy, 2INFN Sezione
di Pisa, Pisa, Italy<EMAIL_ADDRESS>
###### Abstract
This work demonstrates that a middle size ring laser gyroscope (RLG) can be a
very sensitive and robust instrument for rotational seismology, even if it
operates in a quite noisy environment. The RLG has a square cavity,
$1.60\times 1.60$ m2, and it lies in a plane orthogonal to the Earth
rotational axis. The Fabry-Perot optical cavities along the diagonals of the
square were accessed and their lengths were locked to a reference laser.
Through a quite simple locking circuit, we were able to keep the sensor fully
operative for 14 days. The obtained long term stability is of the order of 3
nanorad/s and the short term sensitivity close is to 2 nanorad/s$\cdot$Hz-1/2.
These results are limited only by the noisy environment, our laboratory is
located in a building downtown.
††journal: osajournal††articletype: Research Article
## 1 Introduction
Large frame ring laser gyroscopes (RLG) are the most sensitive absolute
angular rotation sensors, they are able to combine the status of art
sensitivity with a bandwidth above 100 Hz, long term operation and very large
dynamic range. The same device can efficiently records extremely low amplitude
signals and large shocks. They are based on the Sagnac effect that appears as
a difference in the optical path between waves propagating in opposite
direction in a rotating closed loop. As a consequence, in a rotating RLG a
difference arises between the frequencies emitted by the laser in the two
opposite directions (Sagnac frequency). For a RLG rigidly connected to the
ground, the Earth rotation velocity is by far the dominant component of
$\Omega$.
Eq. (1) gives the general relation connecting the Sagnac frequency $f_{s}$,
and the modulus of the local angular rotation rate $\Omega$:
$f_{s}=4\frac{A}{P\lambda}\Omega\cdot\cos(\theta)$ (1)
where $A$ is the area enclosed by the optical path, $P$ is the ring perimeter
length, $\lambda$ the intracavity laser wavelength, and $\theta$ is the angle
between the area vector and the rotational axis.
Our prototype, GP2, is a square laser gyroscope with a perimeter of 6.40 m,
that is operative in a laboratory placed inside the basement of the Pisa INFN
building, see Fig.1.
Figure 1: GP2 gyroscope inside the building of the INFN Pisa section
It was built as a test bench to develop the optical and electronic
technologies that will be implemented in GINGER. The last is a very high
sensitivity three dimensional array of large frame RLGs, currently in the
design phase, which has the aim of verifying general relativity theories on a
laboratory scale [1]. Nevertheless, GP2 has also the aim of contributing to
the development of reduced scale RLGs (4 - 8 m in perimeter) with a bit lower
performance, devoted to geophysical and seismic applications.
The quality of the performances of a RLG are strictly related to the control
of the geometrical scale factor $S=4\frac{A}{P\lambda}\cos(\theta)$, and of
the laser kinematics. In a previous papers we addressed the complex problem of
the reduction of the non-linear laser dynamics perturbation [2, 3],
elaborating a procedure that was applied to the $3.60\times 3.60~{}$m2 RLG
"Gingerino", placed in the deep underground Gran Sasso laboratory.
In spite of its limitation, due to the reduced dimension and to the noisy
location, GP2 can give important information about the procedures for the
control of the scale factor, thanks to its peculiar properties. First of all,
the ring is oriented with its axis nearly parallel to the Earth axis, so that
the accuracy is improved and the disturbances induced on the Sagnac signal by
local tilting are minimized. Furthermore, the vacuum chamber includes also
additional pipes connecting the opposite mirrors along the two diagonals . In
this way, the four mirrors define, besides the square optical cavity, also two
Fabry-Pérot (FP) linear resonators along the diagonals, that can be used to
better define and control the ring geometry.
We already demonstrated in [4, 5] to be able to measure and to stabilize the
lengths of the diagonals of GP2 with a statistical accuracy of some tens of
nanometers. This results was obtained through an interferometric technique, by
using for each diagonal of a double Pound-Drever-Hall (PDH) control loop. In
this work we investigate the geometrical properties of GP2, by testing new
simplified procedures to measure and control the geometrical shape and the
scale factor that could be profitably applied to the reduced scale RLGs.
## 2 The apparatus
### 2.1 The mechanical structure
The optical cavity of GP2 is a square defined by four high-quality dielectric
mirrors. It is mounted inside a vacuum room composed by four corner chambers,
hosting the mirrors, connected, through bellows, by pipes along the sides and
the diagonals of the square. Each mirror is rigidly fixed to the chambers that
are mounted on a piezoelectric slide (PZT), screwed on a massive granite table
that warrants the stiffness of the apparatus. The PZTs can drive displacements
along the direction of the diagonals within an $80\ \mu$m range.
The whole vacuum volume is filled by a 6.4 millibar of He and 0.2 millibar of
a mixture of 20Ne and 22Ne, with a 1:1 ratio of the two isotopes. A pyrex
capillary, 5 mm inner diameter, is inserted halfway of one side where the gas
is excited by a capacitive rf discharge. In order to minimize the losses and
maximize the resonator quality factor, no windows are inserted inside the
laser optical paths. A getter-pump provides to keep clean from any
contamination the noble gas mixtures.
The granite slab is fixed on a concrete base, tilted in the N-S direction by
about $46.6^{\circ}$, which is the value of the local colatitude. Besides, in
order to have the ring axis oriented in the local meridian plane, it was taken
specific care of the positioning of the base .
### 2.2 The optics
We mounted very high quality mirrors 1" in diameter on GP2. Recently, we
changed the optics with respect to that reported in Ref. [4, 5], mounting a
more performing set of mirrors and realigning the optical cavity. The new
mirrors were tested, demonstrating at 45° s-polarized incidence a reflectivity
of 0.999995(1) and a transmission of 0.35(5) ppm, in accordance with the
supplier’s specifications. By means of ring-down measurements, the quality
factor of the ring cavity was obtained equal to approx$10^{12}$, corresponding
to overall losses of 100 ppm. Note that overall losses include also the losses
for diffraction in the capillary. The reflectivity of the mirrors at normal
incidence is quite lower, so that the Q factor of the FP optical cavities
along the diagonals is of the order of $10^{10}$. Also the reference
metrological laser (RfL) is presently a Winters Electro-Optics Helium-Neon
laser locked to a saturated absorption transition of molecular iodine.
Following the specification, the emission frequency is
$f_{Winter}=473~612~622.97\pm 0.01~$ MHz. By means of an optical hetherodyne
technique between the RLG emission beam and the reference laser on a fast
photo-diode we can measure the RLG frequency in the new alignment with an
accuracy of 1 MHz, limited by the RLG vibration noise.
### 2.3 Data acquisition and elaboration
The Sagnac interferometric signal is the beat note, obtained by a photo-diode,
between the clock-wise and counter-clock-wise beams. The two beams are
extracted by the same mirror, and the beat note is stored continuously at a
rate of 5 kSample/s. In parallel at the same rate, we also record the
intensity of the two counter-propagating beams ($monobeam~{}signals$) exiting
from a second mirror, see Fig.2. Data are elaborated off-line by a Matlab®
procedure in order to correct the interferometric signal for mirrors back-
scattering effects. Following the algorithms described in [2], we identify and
correct these effects by using the monobeam signals. The procedure can be
elaborated quickly, producing the reduced data quasi on-line, in a few
seconds, and includes the largest correction terms, usually referred as
backscatter noise. In a second step, [3] linear regression methods could be
applied to evaluate and correct the null-shift, related to the time evolution
of the losses of the laser optical resonator. This procedure requires the off-
line elaboration of extended sets of data, of the order of $10^{4}-10^{5}$ s.
However, in RLG equipped with high quality mirrors the effects are quite small
with a slow evolution time, and can be neglected at the Fourier frequencies of
interest in seismic applications, frequency window spanning from 0.01 to 100
Hz.
Figure 2: Operational setup for geometrical stabilization. AOM Acoustic
Optical Modulator; VCO Voltage Controlled Oscillator; PZT piezoelectric
device; PD photo diode; $I_{1t}$, $I_{2t}$ transmission signal of the diagonal
1 and 2 resonators; $V_{1}$, $V_{2}$ error signal of diagonal 1 and 2.
## 3 Stabilization of the RLG scale factor
In order to stabilize the scale factor $S$, the standard techniques are to
lock the ring perimeter length by comparing the RLG emission frequency with an
optical frequency standard or the RLG cavity FSR with a radio-frequency
standard. The presence of the two FP resonators along the square ring cavity
allows the implementation of a procedure that optimise the ring shape and
stabilize $S$ [6] by stabilizing the length of the two diagonals. Our
procedure acts symmetrically on the PZTs, moving the four mirrors along the
diagonal direction. It gives the advantage of keeping invariant the ring shape
and, as a consequence, the phase of the radiation back scattered by the
mirrors, which is the principal source of error due the non-linearity of the
laser kinematics.
The new simplified scheme is shown in Fig.2. The radiation coming from our RfL
is sent through two polarization maintaining single-mode fibers into the
Fabry-Pérot resonators build by the diagonal opposite mirrors to interrogate
the two diagonal FP. Into the fiber path is inserted an acousto-optics
modulator (AOM), operating around 200 MHz. Each FP transmission signal is
collected on a photodiode, then it’s amplified by a transimpendance modulus
and finally it’s sent to a data acquisition card National Instrument USB-6363.
A VCO sweeps the in-fiber AOM every 100 ms of some MHz. A LabVIEW program
visualises the two transmission signals. At the beginning of each run, the
PZTs that move the mirrors are manually driven by a DC voltage in order to
center the transmission peaks inside the AOM sweep. Eventually, the sweep is
reduced to about 1 MHz and the LabView program will calculate in two
independent parallel ways the voltage of the PZTs corresponding to the maxima
of the transmission signals, sending the error signal to the PZTs to close the
loop. The high frequency response of the servo loop is limited to a few tenths
of Hz, because each PZT slide must move the whole chamber mass, bigger than 2
kg.
We underline that the correction signal is sent symmetrically on the two PZTs
of each diagonal. This solution allows a correction of the temperature
dilation without deforming the geometry of the optical path, which would
affect the back-scattering phase.
Figure 3: Allan plot of the Sagnac frequency relative to a run with active
locking condition 14 days long (blue points) and to a selected free running
3-hours recording (red circles)
## 4 Discussion of the experimental results
We compared Sagnac signal of two 2-days long runs, recorded respectively with
and without the active stabilization of the diagonals, in slightly different
environmental condition. The raw data were in both cases reduced for
backscattering. For the run without stabilization, we obtained a mean Sagnac
frequency value of 184.1593 Hz with a standard deviation of 0.0734 Hz, while
for the locked diagonals run, we find a mean Sagnac frequency value of
184.1327 Hz with a standard deviation of 0.0536 Hz. Moreover, in locked
condition no data point were lost in the whole run, while in the free running
case it was necessary to discard the 26% of the data.
The efficacy of the stabilization is further demonstrated in Fig. 3 that shows
the Allan plot of the Sagnac frequency relative to a 14 days long locked run.
No data were discarded and the Allan is certainly affected by the
environmental disturbances produced by the everyday activity. At high
frequency the signal is dominated by the environment vibration noise, while it
is achieved a long time stability better than $3\times 10^{-9}~{}$rad/s over
one day. By comparison, we superimpose the Allan plot of a selected time
interval (about two hours) in unlocked operation. Note that also in the best
condition of the free running mode it was possible to acquire a continuous set
of data without spikes or jumps for no more than two or three hours.
The location guesting GP2 is a building downtown and it’s quite noisy. An
evaluation of its instrumental sensitivity in locked configuration can be
deduced from the plots of Amplitude Spectral Density shown in Fig. 4,
calculated in selected lime interval with the lowest environment noise.
Figure 4: Amplitude Spectral Density calculated on two selected time intervals
A level of 2 nanorad/s/Hz-1/2 is achieved in 0.02-0.3 Hz range. Note that in
present configuration the high frequency locking efficiency is limited by the
PZT, as stated in section 3.
The technique described in the previous chapter allows the stabilization of
the ring geometry and of the scale factor of the RLG, but it does not give the
opportunity of optimizing the ring geometry following the procedure suggested
in [6]. For this purpose it is necessary to measure the absolute values of the
ring optical path and of the diagonals length. Knowing this quantities, it is
possible to evaluate an expected value of the scale factor $S$ and to compare
it with the measured one.
The perimeter of the optical path can be calculated as $p=nc/$FSR, where the
refraction index of the intracavity medium $n$ can be estimated as
$1+2.1\times 10^{-7}$ [7]. The ring laser FSR is measured by observing on a
fast photo-diode the frequency spectrum of the RLG emission while increasing a
bit the laser excitation up to the onset of a second longitudinal mode. We
find a ring $p=6.3963216$ m.
To measure the absolute diagonal length we described in Ref [4, 5] a complex
procedure requiring for each diagonal a double PDH locking circuit. Here we
used an alternative more simple method by eliminating the second PDH locking.
It is a less performing method, but it’s accurate enough for our purpose, . As
in [4], the diagonal are measured one at a time in sequence. The light beam
coming from the RfL is injected through polarization maintaining single-mode
fibers in the two diagonal resonators and the single diagonal resonance
frequency is locked to the RfL frequency through a PDH circuit. Then, through
an in-fiber EOM the laser radiation is frequency swept around a multiple of
the resonator FSR. A LabVIEW software analyzes the transmitted signal fitting
the data in order to find the central FSR resonance frequency. The diagonal
length is then given by $L=c/(2n$ FSR). We found a value of the lengths of
$2~261~341~\mu$m and $2~261~541~{}\mu$m, with an estimated error of the order
of 1 $\mu$m.
We can observe that the perimeter $p^{\prime}$ of a rhombus whose diagonals
are equal to the measured one, which is $p^{\prime}=6.396~{}320$ m,is almost
identical to the value $p=6.396~{}321~{}6$ found before.
By using the observed values of the diagonals and of the perimeter, we have
also calculated the expected value of the scale factor $S$ of the GP2,
assuming a perfect rhomboid shape, perfectly aligned to Earth rotation axis,
as: $S=4A/(\lambda p)$ where $A$ is the area enclose by the optical path. The
intracavity wavelength $\lambda$ is calculated as $c/f_{0}$, where
$f_{0}=473~612~683$ MHz is the frequency emitted by the RLG, measured beating
on a fast photodiode the RLG radiation against the RfL. Then, the scale factor
of the RLG is $S=4A/(\lambda p)=2.52768\times 10^{6}$, giving a theoretical
Sagnac frequency due to the Earth rotation rate $\Omega_{E}=7.29211586\times
10^{-5}$ rad/s : $f_{th}=S~\Omega_{E}=184.216Hz$.
This can be compared with an experimentally detected Sagnac frequency of
184.133 Hz. The difference could be consistent with a misalignment of the ring
resonator axis of $1.7^{\circ}$, but also with residual non compensation of
the laser non-linear dynamics.
## 5 Conclusions
Rotational seismology requires instrumentation able to provide long term
operation with sensitivity close to nrad/s and suitable response in the
frequency range from 0.01 to 100 Hz. Middle size RLGs, such as our GP2 RLG
prototype, have a simple and compact control apparatus and are suitable to be
installed in common geophysical observatories. This paper shows that they can
satisfy the listed requirements. The duty cycle of a RLG is limited in
principle by the dynamics of the laser, which is affected by mode jumps and
split mode operation. The duty cycle of our free running prototype is
effectively less than $80\%$. The control strategy, tested on GP2, allows to
recover the $100\%$ of the duty cycle in 14 days long continuous unattended
operation, without affecting the sensitivity response. Note that the present
work make use of an expensive metrological laser with an accuracy better then
$10^{-10}$ as reference length standard, but more simple commercially
available reference lasers with a stability of 1 part on $10^{7}$ can provide
a likewise efficient control for geophysical application.
## 6 Acknowledgments
Work funded by INFN
## 7 Disclosures
The authors declare no conflicts of interest.
## References
* [1] A. D. V. Di Virgilio, J. Belfi, W.-T. Ni, N. Beverini, G. Carelli, E. Maccioni, and A. Porzio, “Ginger: A feasibility study,” The European Physical Journal Plus 132, 157 (2017).
* [2] A. D. V. Di Virgilio, N. Beverini, G. Carelli, D. Ciampini, F. Fuso, and E. Maccioni, “Analysis of ring laser gyroscopes including laser dynamics,” The European Physical Journal C 79, 573 (2019).
* [3] A. D. V. Di Virgilio, N. Beverini, G. Carelli, D. Ciampini, F. Fuso, U. Giacomelli, and E. Maccioni, “Identification and correction of sagnac frequency variations,” The European Physical Journal C 80 (2020).
* [4] F. Stefani, N. Beverini, G. Carelli, D. Ciampini, A. Di Virgilio, F. Fuso, and E. Giacomelli, U.and Maccioni, “Long term stabilization of large frame laser gyroscopes,” in _3020nd European Frequency and Time Forum (EFTF),_ (IEEE, Torino, Italy, 2018), pp. 119–121.
* [5] N. Beverini, G. Carelli, A. D. Virgilio, U. Giacomelli, E. Maccioni, F. Stefani, and J. Belfi, “Length measurement and stabilization of the diagonals of a square area laser gyroscope,” Classical and Quantum Gravity 37, 065025 (2020).
* [6] R. Santagata, R. Beghi, J. Belfi, N. Beverini, D. Cuccato, A. Di Virgilio, A. Ortolan, A. Porzio, and S. Solimeno, “Optimization of the geometrical stability in square ring laser gyroscopes,” Classical and Quantum Gravity 32, 055013 (2015).
* [7] R. B. Hurst, M. Mayerbacher, A. Gebauer, K. U. Schreiber, and J.-P. R. Wells, “High-accuracy absolute rotation rate measurements with a large ring laser gyro: establishing the scale factor,” Applied Optics 56, 1124 (2017).
|
[1]Department of Physics, Indian Institute of Technology Roorkee, Roorkee
247667, India
[2]School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore 637371, Singapore
[3]Centre for Quantum Technologies, National University of Singapore,
Singapore 117543, Singapore
[4]Centre for Photonics and Quantum Communication Technology, Indian Institute
of Technology Roorkee, Roorkee 247667, India
[5]Institute of Theoretical Physics and Astrophysics, University of Gdańsk,
80-308 Gdańsk, Poland
[6]School of Mathematics and Physics, Xiamen University Malaysia, 43900
Sepang, Malaysia
# Continuous-Variable Entanglement through Central Forces: Application to
Gravity between Quantum Masses
Ankit Kumar<EMAIL_ADDRESS>Tanjung Krisnanda P. Arumugam Tomasz
Paterek
###### Abstract
We describe a complete method for a precise study of gravitational interaction
between two nearby quantum masses. Since the displacements of these masses are
much smaller than the initial separation between their centers, the
displacement-to-separation ratio is a natural parameter in which the
gravitational potential can be expanded. We show that entanglement in such
experiments is sensitive to initial relative momentum only when the system
evolves into non-Gaussian states, i.e., when the potential is expanded at
least up to the cubic term. A pivotal role of force gradient as the dominant
contributor to position-momentum correlations is demonstrated. We establish a
closed-form expression for the entanglement gain, which shows that the
contribution from the cubic term is proportional to momentum and from the
quartic term is proportional to momentum squared. From a quantum information
perspective, the results find applications as a momentum witness of non-
Gaussian entanglement. Our methods are versatile and apply to any number of
central interactions expanded to any order.
## 1 Introduction
Due to the weakness of gravitational coupling, all quantum experiments up to
date in which gravity plays a role utilized the field of the Earth, see Refs.
[1, 2, 3, 4, 5, 6] for milestone examples. Since this field undergoes
practically undetectable back-action from quantum particles, it effectively
admits a classical description either in terms of a fixed background Newtonian
field [3, 4, 5] or as a fixed background spacetime [1, 2, 6]. This argument
strongly motivates theoretical and experimental research towards a
demonstration of gravitation between two quantum masses, as this is one of the
simplest scenarios where quantum properties of gravity could be observed.
Along this line, several proposals studied the possibility of the generation
of quantum entanglement between two massive objects [7, 8, 9, 10, 11, 12, 13,
14, 15, 16]. Our aim here is to build upon these ideas and provide a simple
precision test of gravitational coupling between two nearby quantum particles.
The methods we develop are generic and apply to arbitrary central
interactions, including situations where many of them are present at the same
time.
Figure 1: Setup under consideration. Two spheres of mass $m$ are released from
the ground state of identical harmonic traps with an equal and opposite
momentum $p_{0}$ along the line joining their centers. The centers are
initially separated by a distance $L$, and displacements from them are denoted
by $x_{A}$ and $x_{B}$. After time $t$ entanglement is estimated with the help
of the probing lasers.
The experiment we have in mind could be realized within the field of
optomechanics [17], which already succeeded in cooling individual massive
particles near their motional ground state [18, 19, 20], and in entangling
cantilevers to light and themselves [21, 22, 23]. In such a setup the
particles are separated much more than their displacements. For example, two
Osmium spheres (the densest natural material) each of mass 100 $\mu$g (radius
$0.1$ mm) with an initial inter-surface distance of $0.1$ mm move by less than
a nanometer within $1$ second of evolution [11], vividly illustrating the
weakness of gravity. Since the situation is non-relativistic, the relevant
interaction is characterised by quantum Newtonian potential. Given that the
displacements are small compared to the initial separation between the two
spheres, a natural parameter in which the potential can be expanded is the
displacement-to-separation ratio [11, 12, 24, 25, 26]. We propose to identify
phenomena that can only occur if the potential is expanded to a particular
order, thus witnessing the relevance of at least this order in experiments.
From this perspective, the gravitational entanglement proposals, in addition
to providing clues about the quantum nature of gravity, also supply tests of
the form of gravitational interaction. For example, entangling two initially
disentangled masses requires at least the second-order term [7, 8, 11]. Here
we show a method that witnesses the third- and fourth-order terms and has an
advantage of a simple modification of the entanglement scheme with confined
particles. Hence, an experiment designed to probe gravitational entanglement
can also be used to witness even weaker gravitational coupling.
Our basic idea is to push the particles towards each other as it is
intuitively expected that such obtained stronger gravity will lead to higher
accumulated entanglement. Yet, we demonstrate that the quantum entanglement
generated by gravitational potential truncated at the second order is
_insensitive_ to any relative motion of the two masses. This is shown
explicitly with an exact analytical solution for the time evolution of the
corresponding covariance matrix [27, 28, 29]. Our intuition is only recovered
with the potential containing at least the third-order term, i.e., when the
system evolves into non-Gaussian states. Closed-form expressions for the
amount of entanglement are established for the potential expanded to any
order. The introduced methods apply to any central interaction, even when many
are present side by side. They also show remarkable robustness, e.g., even the
impact of the fourth-order term on the non-Gaussianity quantifier and the
amount of entanglement can be captured numerically despite an astonishingly
weak gravitational interaction. Moreover, the derived closed forms can be
extrapolated for expansions to arbitrary order. They are in remarkable
quantitative agreement with numerical simulations, which show that the
contribution from the cubic term is proportional to momentum. In contrast, the
contribution from the quartic term is proportional to momentum squared.
Accordingly, the cubic correction decreases the entanglement gain when the two
particles are moving away, and the quartic correction increases the
entanglement irrespective of whether they are moving towards or away from each
other.
## 2 Experimental setup
Figure 2: From LAB frame to COM frame. Gaussianity of the initial state is
preserved as well as the product form. The widths, however, are different in
different frames as marked.
Consider the setup schematically represented in Fig. 1, where we introduce our
notation. The initial wave function is assumed to describe two independent
masses, each in a natural Gaussian state with position spread $\sigma$:
$\Psi(x_{A},x_{B},t=0)=\psi_{A}(x_{A})\ \psi_{B}(x_{B})$, where
$\displaystyle\psi_{A}(x_{A})=\quantity(\frac{1}{2\pi\sigma^{2}})^{1/4}\exp(-\frac{x_{A}^{2}}{4\sigma^{2}}+i\frac{p_{0}}{\hbar}x_{A}),$
(1)
$\displaystyle\psi_{B}(x_{B})=\quantity(\frac{1}{2\pi\sigma^{2}})^{1/4}\exp(-\frac{x_{B}^{2}}{4\sigma^{2}}-i\frac{p_{0}}{\hbar}x_{B}).$
(2)
Note that without loss of generality we chose the momenta to be opposite and
equal. The Hamiltonian in the non-relativistic regime is given by
$\hat{H}=\frac{\hat{p}_{A}^{2}}{2m}+\frac{\hat{p}_{B}^{2}}{2m}-\frac{Gm^{2}}{L+(\hat{x}_{B}-\hat{x}_{A})}.$
(3)
Since this is a two-body problem, it is well-known that the center-of-mass
(COM) motion separates from the relative movement. Accordingly, we introduce
the usual change of variables from the LAB frame to the COM frame:
$R=(x_{A}+x_{B})/2$ and $r=x_{B}-x_{A}$, where $R$ and $r$ denote the
respective displacements of the COM and the reduced mass from their initial
average positions (see Appendix A for details). As a result, the initial wave
function separates as $\Psi(x_{A},x_{B},t=0)=\phi(R,t=0)\ \psi(r,t=0)$, where
$\displaystyle\phi(R,t=0)=\quantity(\frac{1}{\pi\sigma^{2}})^{1/4}\exp(-\frac{R^{2}}{2\sigma^{2}}),$
(4)
$\displaystyle\psi(r,t=0)=\quantity(\frac{1}{4\pi\sigma^{2}})^{1/4}\exp(-\frac{r^{2}}{8\sigma^{2}}-i\frac{p_{0}}{\hbar}r).\
$ (5)
The wave functions $\phi$ and $\psi$ describe the motion of the COM and the
reduced mass, respectively. Compared to the original wave packets, the COM
wave packet admits a smaller width of $\sigma/\sqrt{2}$, and the reduced mass
wave packet has a larger width of $\sigma\sqrt{2}$. The corresponding
relations are illustrated in Fig. 2. In this frame the Hamiltonian decouples
as
$\hat{H}=\hat{H}_{R}+\hat{H}_{r}=\quantity(\frac{\hat{P}^{2}}{4m})+\quantity(\frac{\hat{p}^{2}}{m}-\frac{Gm^{2}}{L+\hat{r}}),$
(6)
where $\hat{P}=-i\hbar\partial/\partial R$ and
$\hat{p}=-i\hbar\partial/\partial r$ are the momentum operators for the COM
and the reduced mass, respectively. A separable Hamiltonian implies that the
two-body wave function retains its product form at all times, i.e.,
$\Psi(x_{A},x_{B},t)=\phi(R,t)\ \psi(r,t)$. Furthermore, the COM wave packet
evolves like a free particle, i.e., its Gaussianity is preserved [30, 31, 32].
The first two statistical moments characterize the quantum state fully, and
for completeness, they are given in Appendix B.
The state $\psi$ evolves in the gravitational potential, which we now expand
in the powers of the displacement-to-separation ratio $r/L$:
$V(\hat{r})=-\frac{Gm^{2}}{L+\hat{r}}\approx-\frac{1}{4}m\omega^{2}\sum_{n=0}^{N}\frac{(-1)^{n}}{L^{n-2}}\hat{r}^{n},$
(7)
where $N$ is the order of approximation, and we defined $\omega^{2}=4Gm/L^{3}$
for later convenience. In Appendix B, we derive exact analytical expressions
for the statistical moments of $\psi$ by solving the related Ehrenfest
equations in the case of $N=2$. Together with the statistical moments for the
COM, these determine the covariance matrix in an exact closed-form [see
Appendix C for the methodology].
With the inclusion of higher-order terms in the potential, i.e., $N>2$, the
corresponding Ehrenfest’s equations cannot be solved analytically due to the
emergence of an infinite set of coupled differential equations involving ever-
increasing statistical moments. We therefore resort to numerical methods and
calculate the time evolution of $\psi$ by implementing Cayley’s form of
evolution operator [33]. The numerical evaluations for $\psi$ are combined
with analytical solutions for the COM to construct the covariance matrix and
the two-body wave function. In order to deal with weak gravitational
interaction, we improve the accuracy of Cayley’s method by utilizing the five-
point stencil and discretise onto a pentadiagonal Crank-Nicolson scheme, which
is further solved by implementing the LU factorization techniques. The code is
publicly available at Zenodo [34], with the corresponding documentation in
Ref. [35] where we demonstrate our superior accuracy as compared to the
standard tridiagonal solutions. We also implemented a dynamic grid allocation,
as described in Ref. [36], to avoid any reflections from numerical boundaries.
## 3 Results
The methodology just described returns an analytical form of the covariance
matrix at time $t$ for potentials truncated at $N=2$ and a numerical form of
the two-body wave function for all $N$. These are thereafter used for
computing the entanglement between two masses [see Appendix C for the
methodology]. In particular, we use logarithmic negativity and entropy of
entanglement as entanglement quantifiers. While logarithmic negativity is
known to be a faithful entanglement quantifier for Gaussian states [27, 28,
29], we will also discuss non-Gaussian pure states and hence the inclusion of
the entropy of entanglement. We first give the results for $N=2$, emphasizing
the independence of relative momentum and its origin. Then we move to $N=3$
and demonstrate that entanglement is linearly dependent on the initial
momentum. We also analyze an indicator of non-Gaussianity (skewness) and
demonstrate the precision of our methods by calculating the marginal impacts
of the fourth-order term in the potential expansion. A methodology to obtain
closed-form expressions for the entanglement gain through potentials expanded
to arbitrary order is presented. Quantitative comparisons are made with
numerical simulations for the fourth-order potential, which show that the
contribution from the quartic term is proportional to relative momentum
squared.
### 3.1 Quadratic interactions
Figure 3: Accumulation of entanglement with the gravitational potential
truncated at the quadratic term ($N=2$). The configuration consists of
identical Osmium spheres with $m=0.25$ pg, $L=2.5$ times their radius, and
$\sigma=2.5$ nm. $p_{0}$ is the initial momentum. Analytical results are
calculated with the closed form of the covariance matrix in Eq. (8). $E$
denotes the logarithmic negativity, and $S$ is the entanglement entropy. The
values of $p_{0}/mL$ in the legends are written in multiples of
$6.18082292\times 10^{-3}$ s-1.
Consider first the gravitational potential truncated at the second order. We
obtained exact analytical forms for the independent elements of the covariance
matrix $\bm{\sigma}$. The solutions simplify if they are written in terms of
already introduced $\omega$ and in terms of $\omega_{0}=\hbar/2m\sigma^{2}$,
which is the frequency of harmonic trap for which the initial state is the
ground state:
$\displaystyle\bm{\sigma}_{00}=\frac{\hbar}{4m\omega_{0}}\quantity[2+\omega_{0}^{2}t^{2}+\quantity(1+\frac{\omega_{0}^{2}}{\omega^{2}})\sinh^{2}(\omega
t)],$
$\displaystyle\bm{\sigma}_{02}=\frac{\hbar}{4m\omega_{0}}\quantity[\omega_{0}^{2}t^{2}-\quantity(1+\frac{\omega_{0}^{2}}{\omega^{2}})\sinh^{2}(\omega
t)],$
$\displaystyle\bm{\sigma}_{11}=\frac{m\hbar\omega_{0}}{4}\quantity[2+\quantity(1+\frac{\omega^{2}}{\omega_{0}^{2}})\sinh^{2}(\omega
t)],$
$\displaystyle\bm{\sigma}_{13}=-\frac{m\hbar\omega_{0}}{4}\quantity(1+\frac{\omega^{2}}{\omega_{0}^{2}})\sinh^{2}(\omega
t),$
$\displaystyle\bm{\sigma}_{01}=\frac{\hbar}{8}\quantity[2\omega_{0}t+\quantity(\frac{\omega_{0}}{\omega}+\frac{\omega}{\omega_{0}})\sinh(2\omega
t)],$
$\displaystyle\bm{\sigma}_{03}=\frac{\hbar}{8}\quantity[2\omega_{0}t-\quantity(\frac{\omega_{0}}{\omega}+\frac{\omega}{\omega_{0}})\sinh(2\omega
t)].$ (8)
The logarithmic negativity for $p_{0}=0$, in the regime of
$\omega\ll\omega_{0}$ and $\omega t\ll 1$, was already approximated to [11]
$E(\bm{\sigma})\approx-\log_{2}\sqrt{1+2\Omega^{6}t^{6}-2\Omega^{3}t^{3}\sqrt{1+\Omega^{6}t^{6}}},$
(9)
where $\Omega^{3}=\omega_{0}\omega^{2}/6\equiv\hbar G/3\sigma^{2}L^{3}$. We
verified that this formula indeed matches our results and emphasize that the
solutions obtained in this work are _exact_. Hence, they can quantify
entanglement outside the demanding constraints that led to Eq. (9). An
example is presented below.
The most striking feature of the covariance matrix is its insensitivity to the
initial momentum $p_{0}$. Accordingly, all quantities derived from the
covariance matrix, say entanglement or squeezing [24, 37], are independent of
the initial momentum. In this approximation, the two initially moving masses
accumulate the same amount of entanglement as when they start from rest.
Furthermore, the amount of entanglement is the same irrespective of whether
the masses are moving toward or away from each other. This is confirmed by the
numerical simulations presented in Fig. 3. Not only there is no momentum
dependence in the dynamics of logarithmic negativity and entropy of
entanglement, they also perfectly overlap with analytical results showing that
our methods are reliable and consistent. We emphasize that the configurations
considered here are non-relativistic. Field theory calculations imply
momentum-dependent relativistic corrections to the Newtonian potential [38,
39], and accordingly, we verify that second-order quantum entanglement
generated by relativistic particles is, in principle, momentum dependent.
However, for the parameters in Fig. 3, such corrections to the Newtonian
potential energy are sixteen orders of magnitude smaller, hence not discussed
in this work. We also note that Eq. (9) is not applicable to the
configuration considered in Fig. 3 because $\omega_{0}\approx 25\omega$.
### 3.2 Relevance of force gradient
We now move to explanations of the observed results. Mathematically, it is
clear that a non-zero force gradient across the size of the wave packet is a
necessary condition for entanglement. Without it the potential is effectively
truncated at $N=1$, and the total Hamiltonian is the sum of local terms.
Physically, entanglement is caused by correlations in complementary variables,
here between positions and momenta. Due to a force gradient, the parts of the
wave packets that are closer are gravitationally attracted more than the parts
which are further apart. Hence a moment later, different momentum is developed
across different positions within the wave packets, leading to quantum
entanglement.
Furthermore, assuming that the force gradient is the dominant contributor to
entanglement gain explains the insensitivity to the initial momentum. Since
the potential is truncated at $N=2$, the force gradient is constant in space.
Therefore, it is irrelevant if the particle moves to a different location in
the meantime; hence the initial momentum does not play a role in entanglement
dynamics. Quantitatively, the force gradient is
$F_{2}^{\prime}=m\omega^{2}/2$, and therefore it fully describes entanglement
in Eq. (9) since now
$\Omega^{3}=\omega_{0}\omega^{2}/6\equiv(\omega_{0}/3m)F_{2}^{\prime}$. In the
following section we provide further evidence for the pivotal role of force
gradient in the entanglement dynamics due to higher-order interactions.
(a)
(b)
Figure 4: Accumulation of entanglement with gravitational potential truncated
at the cubic term ($N=3$). The same physical configuration as in Fig. 3. $E$
denotes the logarithmic negativity, and $S$ is the entanglement entropy.
Panels (a) on the left show the dependence of entanglement on the initial
momentum. Analytical results are calculated from the closed-form of the
covariance matrix for $N=2$, and coincide with the numerical results for $N=3$
and $p_{0}=0$. Panels (b) on the right compare entanglement accumulated with
non-zero momentum to entanglement gained from rest. The ratios show a linear
dependence on the momentum and is very well approximated in the regime of
positive $p_{0}$ (masses moving towards each other) with Eqs. (11) and (12).
The values of $p_{0}/mL$ in the legends are written in multiples of
$6.18082292\times 10^{-3}$ s-1.
### 3.3 Higher-order interactions
Let us continue with the working hypothesis that the force gradient is the
dominant factor in entanglement dynamics. For the cubic potential, $N=3$, the
gradient is given by $F_{3}^{\prime}({\hat{r}})=(1-3\hat{r}/L)m\omega^{2}/2$
and importantly it admits a position dependence. Accordingly, entanglement
should be sensitive to the initial momentum as the gradients differ at
different locations. This is indeed observed in Fig. 4a for gravitational
potential truncated at the cubic term. Furthermore, when the two masses move
towards each other, $p_{0}>0$ and $\expectationvalue{\hat{r}}<0$, the gradient
increases, matching the growing entanglement. Conversely, when the masses move
away, $p_{0}<0$ and $\expectationvalue{\hat{r}}>0$, the force gradient
decreases, matching the slower entanglement gain. Quantitative statements can
also be achieved.
Fig. 4b shows experimentally friendly plots of the ratio of entanglement
accumulated within time $t$ with non-zero initial momentum to entanglement
gained from rest. The numerically calculated linear dependence (solid lines)
can be explained with closed expressions (dotted lines) that we now explain.
The force gradients for the quadratic and the cubic interactions are related
by the following factor:
$\hat{F}_{3}^{\prime}(\hat{r})=(1-3\hat{r}/L)F_{2}^{\prime}$. The average
factor therefore reads
$\frac{\expectationvalue{F_{3}^{\prime}}}{F_{2}^{\prime}}=1-\frac{3}{L}\expectationvalue{\hat{r}}\approx
1+\frac{6p_{0}t}{mL}\equiv 1+\epsilon_{3}(t),$ (10)
where we utilized the fact that $p_{0}$ is much larger than the momenta
generated by gravity and the wave packet, on average, practically follows a
free particle trajectory: $\expectationvalue{\hat{r}}\approx
r_{\text{cl}}=-2p_{0}t/m$.
Fig. 4a shows that for vanishing initial momentum, $p_{0}=0$, the entanglement
obtained with cubic and quadratic potentials is practically the same. We
therefore extrapolate that entanglement for non-zero initial momentum is
related to entanglement from rest by a simple function of the conversion
factor. The plots of Fig. 4b are fitted with
$\displaystyle S(\rho_{A})=\Big{[}1+\epsilon_{3}(t)\Big{]}\
S(\rho_{A},p_{0}=0),$ (11) $\displaystyle
E(\bm{\sigma})=\Bigg{[}1+\frac{1}{2}\epsilon_{3}(t)\Bigg{]}\
E(\bm{\sigma},p_{0}=0).$ (12)
Note that the factor of $1/2$ next to $\epsilon_{3}$ in the logarithmic
negativity is causing a departure from the exact conversion factor between the
force gradients. These formulae are in remarkable agreement with the
computational results in the regime of positive initial momentum (masses
moving towards each other, the regime of experimental interest) and also work
quite well for negative initial momenta. This again affirms that the force
gradient is the primary driver of gravitational entanglement. Furthermore,
these closed forms can now be used in many configurations to estimate the
amplification of entanglement for a non-zero initial momentum given
entanglement from rest.
For the potentials expanded to even higher-order terms, their contribution can
also be incorporated with an appropriate conversion factor between the force
gradients. Note that the entanglement entropy in Eq. (11) is amplified in the
same way as the force gradient in Eq. (10). Assuming that this holds for
higher-order terms, the entropy amplification factor can be written as
$\frac{S(\rho_{A})}{S(\rho_{A},p_{0}=0)}=\frac{\expectationvalue{F_{N}^{\prime}}}{F_{2}^{\prime}}=1+\sum_{n=3}^{N}\epsilon_{n}(t),$
(13)
where the corrections for gravity-like interactions (inverse-square forces)
arising due to the $n^{\text{th}}$ term in the potential expansion is
$\epsilon_{n}(t)=\frac{(-1)^{n}}{2L^{n-2}}\
n(n-1)\expectationvalue{\hat{r}^{n-2}}.$ (14)
Since the gravitational force between two quantum masses is weak, for the
estimation of $\expectationvalue{\hat{r}^{n}}$ we approximate the reduced mass
wave packet to be a Gaussian, with the average position following classical
trajectory and the width following the free evolution:
$\absolutevalue{\psi_{0}(r,t)}^{2}\approx\frac{1}{\bm{\Delta}r_{0}\sqrt{2\pi}}\exp\quantity(-\frac{\quantity(r-r_{\text{cl}})^{2}}{2\bm{\Delta}r_{0}^{2}}),$
(15)
where $\bm{\Delta}r_{0}^{2}=2\sigma^{2}\quantity(1+\omega_{0}^{2}t^{2})$. With
this approximation one obtains the correction terms for $n\geq 3$ as
$\displaystyle\epsilon_{n}(t)=\frac{(-1)^{n}}{2\sqrt{\pi}L^{n-2}}\
n(n-1)\hskip 71.13188pt$ (16)
$\displaystyle\times\sum_{m=0,2,}^{n-2}{n-2\choose m}r_{\text{cl}}^{n-m-2}\
\quantity(\sqrt{2}\bm{\Delta}r_{0})^{m}\ \Gamma\quantity(\frac{m+1}{2}),$
where $\Gamma$ is the gamma function, and the summation is only over even $m$.
Note that the gravitational interaction is already included in
$F_{2}^{\prime}$, and the present estimation is for the ratio of the force
gradients of different orders only,
$\expectationvalue{F_{N}^{\prime}}/F_{2}^{\prime}$. Since $\epsilon_{n}\propto
1/L^{n-2}$, each consecutive term is diminished by a factor of $L$. Hence, a
cubic order correction should be sufficient for practical applications in the
near future. Nevertheless, one can see that the fourth-order correction to
entanglement entropy is given by
$\epsilon_{4}(t)=24\frac{p_{0}^{2}t^{2}}{m^{2}L^{2}}+12\frac{\sigma^{2}}{L^{2}}\quantity(1+\omega_{0}^{2}t^{2}).$
(17)
Unlike the third-order term, which was sensitive to the direction of momentum,
the fourth-order one depends on the momentum squared, leading to a positive
correction in both the scenarios of masses moving towards and away from each
other. This prediction is confirmed in Fig. 5, where we show the entanglement
accumulated with the gravitational potential expanded up to the fourth order.
The derived formulae exactly recover the entanglement gain in the regime of
positive momentum, and they work quite well in the case of negative momentum.
Note that $\epsilon_{4}$ also depends on the position spread, hence it might
be important even for stationary configurations where the wave packet
undergoes a fast expansion.
Figure 5: Comparison of entanglement accumulated with the gravitational
potential expanded up to quadratic ($N=2$), cubic ($N=3$) and quartic ($N=4$)
term, respectively. Solid lines show the results for positive momentum (masses
moving toward each other), and dashed lines are for negative momentum. The
dots represent the entanglement entropy ($S$) computed with the closed
formulae derived in this work. Compared to the quadratic case, the cubic term
lowers entanglement between particles that move away from each other. Compared
to the cubic case, the quartic term adds a positive correction irrespective of
the particles moving towards or away from each other. The values of $p_{0}/mL$
in the legends are written in multiples of $6.18082292\times 10^{-3}$ s-1.
### 3.4 Galilean relativity and a drifting COM
We made a change of reference frames to dissect the bipartite evolution into
two independent single-particle dynamics. The first one is the free evolution
of the COM, and the second one is the evolution of reduced mass in
gravitational potential. Under the assumption that the two spheres are
imparted with equal and opposite momentum, the COM is stationary on average.
While this simplifies our theoretical framework substantially, such a
configuration may be difficult to achieve in experiments. It is much easier to
push one of the masses while the other is at rest. In such a case the COM
moves rectilinearly with a constant velocity.
The Galilean principle of relativity demands that the laws of non-relativistic
physics must be invariant in all inertial frames of reference. Consequently,
the centered moments of the moving COM should evolve in the same way as for
the stationary COM [40]. This is readily cross-checked as we get exactly the
same correlation and variances after incorporating a non-zero momentum in the
initial conditions for solving COM Ehrenfest’s equations. In conclusion, a
uniformly moving COM has no role in generating quantum (or, for that matter,
classical) correlations. Only the relative momentum matters, and as long as it
remains the same, the individual momenta can be tweaked as per convenience.
## 4 Discussion
The results presented so far could also be perceived as a simple momentum-
based witness of non-Gaussianity in a quantum state. Indeed the cubic term is
responsible for non-Gaussian evolution that we now quantify in more detail.
Fig. 6 presents the skewness $\tilde{\mu}_{3}$ in the evolution of the reduced
mass wave function $\psi$. While $\tilde{\mu}_{3}$ vanishes for $N=2$, as it
should, it rises steeply for $N=3$. The physical reason is clear from Fig. 2
describing the change of variables between LAB and COM frames. The left end of
wave function $\psi$ is attracted towards the COM much more than the right
end. Over time this makes the probability density function negatively skewed,
which is indicated by $\tilde{\mu}_{3}<0$. Just like Fig. 5, Fig. 6 also
demonstrates the precision of our numerical methods, which even capture
marginal contributions of the fourth-order term.
Let us summarise the interplay between non-Gaussianity, initial momentum
dependence of entanglement, and the amount of entanglement. For the quadratic
Hamiltonian the skewness of course vanishes, and we derived an exact
analytical solution for the covariance matrix that shows no dependence on the
initial momentum. For the cubic Hamiltonian the skewness rises but its value
is small, and within the first few seconds of the evolution the wave function
is very close to a Gaussian. Nevertheless, this is already sufficient for non-
zero force gradient, and we obtained closed-form equations for the amount of
entanglement that show a linear dependence on the initial momentum and do not
feature skewness, see Eqs. (10) and (12). Therefore, while non-zero skewness
enables entanglement dependence on the initial momentum, it plays a marginal
role in the amount of entanglement. Quantitative estimation of this small
contribution of skewness to the amount of entanglement is left as an open
problem.
All these mutual dependencies can be seen in our data. Fig. 6 shows that
skewness is practically the same for the two considered relative momenta. For
the same momenta, Fig. 4a demonstrates that entanglement entropy accumulated
after 5 seconds is different by 30$\%$ and linear in initial momentum. The
amount of entanglement is therefore determined by the initial momentum only.
Similarly, skewness is non-zero for initially stationary particles but
entanglement dynamics with and without non-Gaussianity look practically the
same. To give quantitative values, consider the stationary configuration of
two Osmium spheres with $m=1$ pg separated by a distance of $L=2.1$ times
their radius. After an evolution for 5 seconds, the entanglement gain with
cubic potential is larger than the entanglement accumulated with quadratic
potential by only $\approx 0.001,\ 0.002$, and $0.003\%$, for an initial
spread of $\sigma=5.00,\ 0.50$, and $0.05$ nm, respectively. We emphasize that
the force gradient plays a pivotal role in entanglement dynamics.
We would also like to address the question of whether a simpler method for
detecting the third-order coupling exists than based on measurements of
entanglement. Indeed, note that solely the mean relative momentum signal could
be used for such purposes. One verifies that by truncating the potential in
Eq. (7) at $N=2$, the relative momentum satisfies the condition
$\ddot{\expectationvalue{p}}/\expectationvalue{p}=\omega^{2}$, i.e., it is
time-independent. Any time dependence of this ratio reveals third-order
coupling. In cases where the center of mass is stationary, instead of the
relative momentum, the local momentum of any particle could be used.
Finally, a word on decoherence effects is in place. The common decoherence
mechanisms, due to thermal photons and air molecules, have already been
studied in the considered setup [7, 8, 11, 13, 24]. The experiment was found
to be challenging, but the required coherence times are in principle
realisable, e.g., for freely-falling particles in a high vacuum. The
calculations presented here only relax these requirements as the entanglement
is improved when the two masses are pushed toward each other. For example, in
the configuration considered in this work, the entanglement gain of $E\approx
1.75\times 10^{-4}$ is relaxed from 5 seconds to 4 seconds with an initial
momentum of $p_{0}/mL\approx+0.022$ s-1. Note that an entanglement detection
scheme achieving a precision of $E\sim 10^{-4}$ has recently been put forward
in Ref. [41].
Figure 6: Non-Gaussian reduced mass dynamics. The physical situation is as in
Fig. 3. Skewness ($\tilde{\mu}_{3}$) is computed for the position space
distribution. $N$ denotes the order of approximation, see Eq. (7). The left
panel is for particles initially at rest, and the right panel is for masses
moving toward each other. The values of $p_{0}/mL$ in the legends are written
in multiples of $6.18082292\times 10^{-3}$ s-1.
## 5 Versatility
While our discussions were mainly focused on gravity-induced entanglement, the
methods we have presented are applicable more generally. First of all, they
hold for arbitrary central interactions. We need to expand the potential in a
binomial series similar to Eq. (7) and the entanglement is characterised by
the parameters $\omega$ and $\epsilon_{3}$. As we derived, for identical
masses coupled via Newtonian gravity,
$\omega^{2}=\frac{4Gm}{L^{3}},\hskip
14.22636pt\epsilon_{3}(t)=\frac{6p_{0}t}{mL}.$ (18)
The general rule is quite simple. Once the potential is expanded in a binomial
series of the relative displacement, the coefficient of $r^{2}$ is to be
compared with $-m\omega^{2}/4$, and $\epsilon_{3}$ is to be calculated by
comparing the force gradients as
$\expectationvalue{F_{3}^{\prime}}/F_{2}^{\prime}=1+\epsilon_{3}(t)$. One can
then verify that for the Coulomb potential between charges $q_{1}$ and $q_{2}$
embedded into the masses, we obtain
$\omega^{2}=\frac{4q_{1}q_{2}\alpha\hbar c}{e^{2}mL^{3}},\hskip
14.22636pt\epsilon_{3}(t)=\frac{6p_{0}t}{mL},$ (19)
where $\alpha$ is the fine structure constant and $e$ is the electronic
charge. For the Casimir interaction between their surfaces (under proximity
force approximation [42]) we arrive at
$\omega^{2}=\frac{\pi^{3}\hbar cR_{0}}{120m(L-2R_{0})^{4}},\hskip
5.69046pt\epsilon_{3}(t)=\frac{8p_{0}t}{m(L-2R_{0})},$ (20)
where $R_{0}$ is the radius of each sphere.
For an arbitrary central interaction with a potential of the form
$V(x_{A},x_{B})=-\frac{C}{(X+x_{B}-x_{A})^{j}},$ (21)
we obtain
$\omega^{2}=\frac{2j(j+1)C}{mX^{j+2}},\hskip
14.22636pt\epsilon_{3}(t)=\frac{2(j+2)p_{0}t}{mX}.$ (22)
In some situations the force is known, but solving for the potential is
difficult or uncertain due to non-unique boundary conditions. In those cases
the general rule would be to expand the force in a binomial series and compare
the coefficient of $r$ with $m\omega^{2}/2$. For example, if the force is of
the form
$F(x_{A},x_{B})=-\frac{C}{(X+x_{B}-x_{A})^{j}},$ (23)
one arrives at
$\omega^{2}=\frac{2jC\ }{mX^{j+1}},\hskip
14.22636pt\epsilon_{3}(t)=\frac{2(j+1)p_{0}t}{mX}.$ (24)
Note that the functional forms of $\epsilon_{3}$ in this section are valid
only for weak interactions. For stronger potentials one has to take a step
back and use $\epsilon_{3}=-3\expectationvalue{\hat{r}}/L$, where
$\expectationvalue{\hat{r}}$ has to be approximated either analytically or
numerically.
Furthermore, the methods established also work for multiple central forces
acting simultaneously. If we write the interaction as a sum
$V=\sum_{k}V_{k}(x_{B}-x_{A}),\hskip 7.11317ptF=\sum_{k}F_{k}(x_{B}-x_{A}),$
(25)
the total $\omega$ characterising the Gaussian covariance matrix is given by a
Pythagoras-like theorem, and the equivalent $\epsilon_{3}$ governing the
entanglement amplification due to the cubic-order term is calculated to be a
weighted sum:
$\omega^{2}=\sum_{k}\omega^{2}_{k},\hskip
14.22636pt\epsilon_{3}(t)=\frac{1}{\omega^{2}}\sum_{k}\omega_{k}^{2}\epsilon_{k3}(t).$
(26)
where $\omega_{k}$ and $\epsilon_{k3}(t)$ characterise the individual
interactions. This is particularly useful from an experimental point of view
as, in practice, it might be difficult to screen all interactions except
gravity. For example, the gravitational and the Casimir interaction will
likely act side by side.
## 6 Conclusions
We have shown that experiments aimed at observing gravitational entanglement
can also be used as precision tests of gravitational coupling. In particular,
entanglement dependence on the relative momentum of interacting particles
indicates a coupling which is higher than quadratic. Furthermore, for the
potential expanded to the cubic term, the amount of entanglement accumulated
in a fixed time interval grows linearly with the relative momentum when the
particles are pushed toward each other. We presented a closed expression for
the amount of entanglement as a function of relative momentum based on the
derived exact covariance matrix for Gaussian dynamics extended to higher-order
couplings. The methods introduced apply to arbitrary central interactions,
even when many are present side by side.
## Acknowledgements
This work is jointly supported by (i) NAWA, Poland, via project
PPN/PPO/2018/1/00007/U/00001, (ii) XMUM, Malaysia, via project
XMUMRF/2022-C10/IPHY/0002, and (iii) DORA office of IIT Roorkee, India. A.K.
thanks the IIT Roorkee Heritage Foundation, USA, for the ‘Pledge a Dream’
grant. Discussions with Andy Chia (CQT-NUS, Singapore) are acknowledged. T.K.
thanks Timothy Liew for their hospitality at NTU, Singapore, and Yvonne Gao
for their hospitality at NUS, Singapore. We acknowledge the National
Supercomputing Mission (NSM) for providing computing resources of ‘PARAM
Ganga’ at IIT Roorkee, India, which is implemented by C-DAC and supported by
MeitY and DST, Govt. of India. The authors gratefully thank the reviewers for
their comments and suggestions which have significantly improved the
presentation of this article.
## Appendix A Coordinate transformations
The time-dependent Schrödinger equation (TDSE) for two identical particles $A$
and $B$ is given by
$\displaystyle\quantity(-\frac{\hbar^{2}}{2m}\partialderivative[2]{x_{A}}-\frac{\hbar^{2}}{2m}\partialderivative[2]{x_{B}}+V(\hat{x}_{A},\hat{x}_{B}))\Psi(x_{A},x_{B},t)$
$\displaystyle=i\hbar\partialderivative{t}\Psi(x_{A},x_{B},t),\hskip
14.22636pt$ (27)
where $\hat{p}_{A}=-i\hbar\partial/\partial x_{A}$ and
$\hat{p}_{B}=-i\hbar\partial/\partial x_{B}$ are their respective momentum
operators. Coordinate transformations between the LAB the COM frame of
reference are defined by
$R=\frac{x_{A}+x_{B}}{2},\hskip 14.22636ptr=x_{B}-x_{A},$ (28)
where $R$ and $r$ are the displacements of the COM (mass $2m$) and the reduced
mass (mass $m/2$), respectively. One can take time derivatives to arrive at
their respective momenta as
$P=p_{A}+p_{B},\hskip 14.22636ptp=\frac{p_{B}-p_{A}}{2}.$ (29)
Accordingly, the inverse transformations are
$x_{A}(x_{B})=R-\\!(\\!+\\!)\ \frac{r}{2},\hskip
14.22636ptp_{A}(p_{B})=\frac{P}{2}-\\!(\\!+\\!)\ p.$ (30)
The displacements $x_{A}$ and $x_{B}$ are functions of $R$ and $r$, and the
rules of differentiation imply
$\partialderivative[2]{x_{A}}\quantity(\partialderivative[2]{x_{B}})=\frac{1}{4}\partialderivative[2]{R}+\partialderivative[2]{r}-\\!(\\!+\\!)\partialderivative{}{R}{r},$
(31)
which means that the kinetic energy is equivalent to
$-\frac{\hbar^{2}}{2m}\partialderivative[2]{x_{A}}-\frac{\hbar^{2}}{2m}\partialderivative[2]{x_{B}}=-\frac{\hbar^{2}}{4m}\partialderivative[2]{R}-\frac{\hbar^{2}}{m}\partialderivative[2]{r}.$
(32)
For central potentials $V(x_{A},x_{B})=V(x_{B}-x_{A})\equiv V(r)$, and hence
the TDSE is transformed to
$\displaystyle\quantity(-\frac{\hbar^{2}}{4m}\partialderivative[2]{R}-\frac{\hbar^{2}}{m}\partialderivative[2]{r}+V(\hat{r}))\Psi(x_{A},x_{B},t)$
$\displaystyle=i\hbar\partialderivative{t}\Psi(x_{A},x_{B},t).$ (33)
In this work the initial wave function is $\Psi(x_{A},x_{B},t=0)=\phi(R,t=0)\
\psi(r,t=0)$, and the separation of variables in Eq. (33) ensures that the
product form is maintained at all times. Accordingly, the problem decouples
into two independent single-particle TDSEs:
$\displaystyle-\frac{\hbar^{2}}{4m}\partialderivative[2]{R}\phi(R,t)=i\hbar\partialderivative{t}\phi(R,t),$
(34)
$\displaystyle\quantity(-\frac{\hbar^{2}}{m}\partialderivative[2]{r}+V(\hat{r}))\psi(r,t)=i\hbar\partialderivative{t}\psi(r,t),$
(35)
where $\hat{P}=-i\hbar\partial/\partial R$ and
$\hat{p}=-i\hbar\partial/\partial r$ can now be identified as the momentum
operators for the COM and the reduced mass, respectively. The COM is a free
particle (but not a plane wave), and the reduced mass undergoes an evolution
under the influence of the interaction. The bipartite wave function is simply
the product of the two wave functions
$\Psi(x_{A},x_{B},t)=\phi\quantity(\frac{x_{A}+x_{B}}{2},t)\
\psi\quantity(x_{B}-x_{A},t).$ (36)
## Appendix B The Ehrenfest dynamics
Ehrenfest’s theorem relates the time derivative of the expectation value of an
operator $\hat{A}$ to the expectation of its commutator with the Hamiltonian
$\hat{H}$:
$\derivative{t}\expectationvalue{\hat{A}}=\frac{1}{i\hbar}\expectationvalue{\commutator{\hat{A}}{\hat{H}}}+\expectationvalue{\partialderivative{\hat{A}}{t}}.$
(37)
In this work we focus on the noiseless dynamics, i.e., when the system is
isolated from the environment. Accordingly, the COM and the reduced mass
evolve exclusively into pure states, with their covariance matrices satisfying
$\text{Det}(\bm{\sigma}_{R})=\text{Det}(\bm{\sigma}_{r})=(\hbar/2)^{2}$ [43].
### B.1 Evolution of the COM
The COM evolution corresponds to the free expansion of a Gaussian wave packet:
$\hat{H}_{R}=\hat{P}^{2}/4m$. The initial is characterised by
$\expectationvalue{\hat{R}}=0$, $\expectationvalue{\hat{P}}=0$,
$\expectationvalue{\anticommutator{\hat{R}}{\hat{P}}}=0$,
$\expectationvalue{\hat{R}^{2}}=\sigma^{2}/2$, and
$\expectationvalue{\hat{P}^{2}}=\hbar^{2}/2\sigma^{2}$ [see Fig. 2 or Eq.
(4)]. The solution to the corresponding Ehrenfest’s differential equations
for the first two statistical moments,
$\displaystyle\derivative{t}\expectationvalue{\hat{R}}=\frac{1}{2m}\expectationvalue{\hat{P}},$
$\displaystyle\derivative{t}\expectationvalue{\hat{P}}=0,$
$\displaystyle\derivative{t}\expectationvalue{\anticommutator{\hat{R}}{\hat{P}}}=\frac{1}{m}\expectationvalue{P^{2}},$
$\displaystyle\derivative{t}\expectationvalue{\hat{R}^{2}}=\frac{1}{2m}\expectationvalue{\anticommutator{\hat{R}}{\hat{P}}},$
$\displaystyle\derivative{t}\expectationvalue{\hat{P}^{2}}=0,$ (38)
imply
$\displaystyle\bm{\Delta}R^{2}=\frac{1}{2}\sigma^{2}(1+\omega_{0}^{2}t^{2}),$
$\displaystyle\bm{\Delta}P^{2}=\frac{\hbar^{2}}{2\sigma^{2}},$
$\displaystyle\textbf{Cov}({R},{P})=\frac{1}{2}\hbar\omega_{0}t.$ (39)
Alternatively, one arrives at the same results by utilizing the functional
form of the wave function [40]:
$\phi(R,t)=\frac{1}{\sqrt{\sigma(1+i\omega_{0}t)\sqrt{\pi}}}\exp\quantity(-\frac{R^{2}}{2\sigma^{2}(1+i\omega_{0}t)}).$
(40)
### B.2 Evolution of the reduced mass
The reduced mass Hamiltonian can be represented by a binomial series as in Eq.
(7). The initial state is characterised by $\expectationvalue{\hat{r}}=0$,
$\expectationvalue{\hat{p}}=p_{0}$,
$\expectationvalue{\anticommutator{\hat{r}}{\hat{p}}}=0$,
$\expectationvalue{\hat{r}^{2}}=2\sigma^{2}$, and
$\expectationvalue{\hat{p}^{2}}=p_{0}^{2}+\hbar^{2}/8\sigma^{2}$ [see Fig. 2
or Eq. (5)]. For $N=2$, i.e., a quadratic Hamiltonian,
$\hat{H}_{r}=\frac{\hat{p}^{2}}{m}-\frac{1}{4}m\omega^{2}\quantity(L^{2}-Lr+r^{2}),$
(41)
the relevant Ehrenfest’s coupled differential equations for the first two
statistical moments are given by
$\displaystyle\derivative{t}\expectationvalue{\hat{r}}=\frac{2}{m}\expectationvalue{\hat{p}},$
$\displaystyle\derivative{t}\expectationvalue{\hat{p}}=-\frac{1}{4}m\omega^{2}L+\frac{1}{2}m\omega^{2}\expectationvalue{\hat{r}},$
$\displaystyle\derivative{t}\expectationvalue{\anticommutator{\hat{r}}{\hat{p}}}=\frac{4}{m}\expectationvalue{\hat{p}^{2}}-\frac{1}{2}m\omega^{2}L\expectationvalue{\hat{r}}+m\omega^{2}\expectationvalue{\hat{r}^{2}},$
$\displaystyle\derivative{t}\expectationvalue{r^{2}}=\frac{2}{m}\expectationvalue{\anticommutator{\hat{r}}{\hat{p}}},$
$\displaystyle\derivative{t}\expectationvalue{\hat{p}^{2}}=\frac{1}{2}m\omega^{2}\expectationvalue{\anticommutator{\hat{r}}{\hat{p}}}-\frac{1}{2}m\omega^{2}L\expectationvalue{\hat{p}},$
(42)
Their exact analytical solution reads
$\displaystyle\expectationvalue{\hat{r}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}L\Big{(}1-\cosh(\omega
t)\Big{)}-\frac{2p_{0}}{m\omega}\sinh(\omega t),$
$\displaystyle\expectationvalue{\hat{p}}$ $\displaystyle=$ $\displaystyle-
p_{0}\cosh(\omega t)-\frac{1}{4}m\omega L\sinh(\omega t),$
$\displaystyle\expectationvalue{\anticommutator{\hat{r}}{\hat{p}}}$
$\displaystyle=$ $\displaystyle Lp_{0}\Big{(}\cosh(2\omega t)-\cosh(\omega
t)\Big{)}$
$\displaystyle+\frac{2}{m\omega}\quantity(p_{0}^{2}+\frac{\hbar^{2}}{8\sigma^{2}}+\frac{1}{2}m^{2}\omega^{2}\sigma^{2})\sinh(2\omega
t)$ $\displaystyle+\frac{1}{8}m\omega L^{2}\Big{(}\sinh(2\omega
t)-2\sinh(\omega t)\Big{)},$ $\displaystyle\expectationvalue{\hat{r}^{2}}$
$\displaystyle=$ $\displaystyle 2\sigma^{2}\Big{(}1+\sinh[2](\omega t)\Big{)}$
$\displaystyle+\frac{1}{8}L^{2}\Big{(}3+\cosh(2\omega t)-4\cosh(\omega
t)\Big{)}$ $\displaystyle+\frac{Lp_{0}}{m\omega}\Big{(}\sinh(2\omega
t)-2\sinh(\omega t)\Big{)}$
$\displaystyle+\frac{4}{m^{2}\omega^{2}}\quantity(p_{0}^{2}+\frac{\hbar^{2}}{8\sigma^{2}})\sinh[2](\omega
t),$ $\displaystyle\expectationvalue{\hat{p}^{2}}$ $\displaystyle=$
$\displaystyle\quantity(p_{0}^{2}+\frac{\hbar^{2}}{8\sigma^{2}})\Big{(}1+\sinh[2](\omega
t)\Big{)}$ (43)
$\displaystyle+\frac{1}{4}m^{2}\omega^{2}\quantity(2\sigma^{2}+\frac{1}{4}L^{2})\sinh[2](\omega
t)$ $\displaystyle+\frac{1}{4}m\omega Lp_{0}\sinh(2\omega t),$
which implies that variances and the correlation are
$\displaystyle\bm{\Delta}r^{2}=2\sigma^{2}\quantity(\cosh[2](\omega
t)+\frac{\omega_{0}^{2}}{\omega^{2}}\sinh[2](\omega t)),$
$\displaystyle\bm{\Delta}p^{2}=\frac{\hbar^{2}}{8\sigma^{2}}\quantity(\cosh[2](\omega
t)+\frac{\omega^{2}}{\omega_{0}^{2}}\sinh[2](\omega t)),$
$\displaystyle\textbf{Cov}({r},{p})=\frac{\hbar}{4}\quantity(\frac{\omega_{0}}{\omega}+\frac{\omega}{\omega_{0}})\sinh(2\omega
t).$ (44)
## Appendix C Quantification of entanglement
We have employed the formalism based on the covariance matrix to quantify
entanglement gain via logarithmic negativity and additionally used the density
matrix to compute the entropy of entanglement.
### C.1 Covariance matrix
The covariance matrix formalism is based on the first two statistical moments
of a quantum state. Given a bipartite system $AB$ with
$\hat{u}=(\hat{x}_{A},\hat{p}_{A},\hat{x}_{B},\hat{p}_{B})^{T}$, the
covariance matrix is defined as [27, 28, 29]:
$\bm{\sigma}_{jk}=\frac{1}{2}\expectationvalue{\anticommutator{\hat{u}_{j}}{\hat{u}_{k}}}-\expectationvalue{\hat{u}_{j}}\expectationvalue{\hat{u}_{k}}.$
(45)
In the block form we can write
$\bm{\sigma}\equiv\matrixquantity(\bm{\alpha}&\bm{\gamma}\\\
\bm{\gamma}^{T}&\bm{\beta}),$ (46)
where $\bm{\alpha}(\bm{\beta}$) contains the local mode correlation for
$A(B)$, and $\bm{\gamma}$ describes the intermodal correlation. In our setting
the local modes are identical, i.e., $\bm{\alpha}=\bm{\beta}$, and a
coordinate change to the COM frame implies
$\displaystyle\bm{\sigma}_{00}(\bm{\sigma}_{02})=\bm{\Delta}R^{2}+\\!(\\!-\\!)\
\frac{1}{4}\bm{\Delta}r^{2},$
$\displaystyle\bm{\sigma}_{11}(\bm{\sigma}_{13})=\frac{1}{4}\bm{\Delta}P^{2}+\\!(\\!-\\!)\
\bm{\Delta}p^{2},$
$\displaystyle\bm{\sigma}_{01}(\bm{\sigma}_{03})=\frac{1}{2}\textbf{Cov}({R},{P})+\\!(\\!-\\!)\
\frac{1}{2}\textbf{Cov}({r},{p}).$ (47)
Given the symmetry of the problem we have $\bm{\sigma}_{22}=\bm{\sigma}_{00}$,
$\bm{\sigma}_{33}=\bm{\sigma}_{11}$, $\bm{\sigma}_{23}=\bm{\sigma}_{01}$,
$\bm{\sigma}_{12}=\bm{\sigma}_{03}$. The rest of the elements are constrained
due to the symmetry property $\bm{\sigma}_{jk}=\bm{\sigma}_{kj}$.
### C.2 Logarithmic negativity
The negativity of partially transposed density matrix is a necessary and
sufficient condition for entanglement in two–mode Gaussian states [44]. As a
result of partial transposition, the covariance matrix is transformed to
$\tilde{\bm{\sigma}}$, which differs from $\bm{\sigma}$ by a sign-flip of
$\text{Det}(\bm{\gamma})$ [29]. The symplectic eigenvalues of the covariance
matrix, $\tilde{\nu}_{\pm}(\bm{\sigma})$, are given by [28, 27]
$2\tilde{\nu}^{2}_{\pm}(\bm{\sigma})=\tilde{\Sigma}(\bm{\sigma})\pm\sqrt{\tilde{\Sigma}^{2}(\bm{\sigma})-4\
\text{Det}\quantity(\bm{\sigma})},$ (48)
where
$\tilde{\Sigma}(\bm{\sigma})=\text{Det}(\bm{\alpha})+\text{Det}(\bm{\beta})-2\
\text{Det}(\bm{\gamma})$. For the symmetric problem as considered in this work
the local modes $\bm{\alpha}$ and $\bm{\beta}$ are identical, and hence
$\tilde{\Sigma}(\bm{\sigma})=2\
\quantity[\text{Det}(\bm{\alpha})-\text{Det}(\bm{\gamma})]$. Entanglement is
quantified by the minimum symplectic eigenvalue via logarithmic negativity:
$E(\bm{\sigma})=\max\Bigg{[}0,\
-\log_{2}\quantity(\frac{\tilde{\nu}_{-}(\bm{\sigma})}{\hbar/2})\Bigg{]}.$
(49)
### C.3 Entropy of entanglement
For a pure bipartite system described by a density matrix $\rho_{AB}$, the
entanglement entropy is defined as the von Neumann entropy for any one of the
subsystems, e.g., $S(\rho_{A})=-\Tr\quantity[\rho_{A}\log_{2}(\rho_{A})]$,
where $\rho_{A}=\Tr_{B}\quantity(\rho_{AB})$ is the reduced density matrix for
subsystem $A$. In order to calculate $S(\rho_{A})$ we start with the two-body
wave function of Eq. (36). The COM wave function $\phi(R,t)$ is derived
analytically in Eq. (40), and we calculate $\psi(r,t)$ numerically by
implementing the improved Cayley’s propagator [34]. Once this is available at
a given time $t$, we perform a singular value decomposition [45, 46]:
$\Psi(x_{A},x_{B},t)=\sum_{j}\sqrt{\lambda_{j}(t)}\ \chi^{(A)}_{j}(x_{A},t)\
\chi^{(B)}_{j}(x_{B},t),$ (50)
where $\quantity{\chi^{(A)}_{j}}$ and $\quantity{\chi^{(B)}_{j}}$ are
orthonormal states in subsystems $A$ and $B$, respectively, and $\lambda_{j}$
are the Schmidt coefficients. A numerical implementation utilizes the
algorithms in Google TensorNetwork [47, 48] with the help of open source
libraries hosted at GitHub [49]. Note that the total number of Schmidt
coefficients is not fixed. At any given time, the number is dynamically
increased until the norm bipartite wave function is recovered up to seven
decimal places, i.e., until
$1-\absolutevalue{\Psi(x_{A},x_{B},t)}^{2}=1-\sum_{j}\lambda_{j}(t)\lesssim
10^{-7}$. With this decomposition, the entanglement entropy reduces to
$S(\rho_{A})=-\sum_{j}\lambda_{j}\log_{2}(\lambda_{j}).$ (51)
In the case of Gaussian evolution, the entanglement entropy is calculable
using the covariance matrix [43]:
$S(\bm{\alpha})=f\quantity(\frac{1}{\hbar}\sqrt{\text{Det}(\bm{\alpha})})\equiv
S(\rho_{A}),$ (52)
where
$f(x)=\quantity(x\\!+\frac{1}{2})\log_{2}\quantity(x\\!+\frac{1}{2})-\quantity(x\\!-\frac{1}{2})\log_{2}\quantity(x\\!-\frac{1}{2}).$
(53)
## Appendix D Numerical details
Numerical calculations are performed in natural units of $c=1$, hence the
conversion constant $\hbar c=197.3269804$ keV pm. The density of Osmium is
$22.5872$ g/cm3. An error analysis implies that, in the numerical time
evolution of the reduced mass wave function, a grid size of $\lesssim 0.2$ pm
with a time step of $\lesssim 10\ \mu$s is required to maintain accuracy in
the extreme cases of the largest momentum considered in this work.
Accordingly, we set a grid size of $0.1$ pm and a time step of $5\ \mu$s
throughout this work. Note that the first term in the gravitational potential
of Eq. (7) is just a constant energy offset, which only contributes to an
irrelevant global phase in the quantum dynamics. Moreover, this term is the
most prominent of all magnitude-wise. We have therefore ignored it in
numerical simulations to maximally utilize the computer precision for the
(relevant) higher-order terms.
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|
# The G-dynamics of the QCPB theory
Gen WANG 111720195765863, email<EMAIL_ADDRESS>
sterces ofo neila aiv siunew gang-pwc-zgy.
( School of Mathematical Sciences, Xiamen University,
Xiamen, 361005, P.R.China )
###### Abstract
In this paper, we further study the G-dynamics newly emerged in the covariant
dynamics defined by the QCPB theory. Then we want to seek the precise
formulations of the G-dynamics to calculate the practical quantum problems. We
explicitly verify how the role of G-dynamics plays in quantum mechanics, and
then we propose three new operators based on the G-dynamics: D-operator,
T-operator and G-operator, we find that the non-Hermitian operators are
inevitable to appear in the framework of the QCPB. Then the eigenvalue
equation of them are discussed accordingly by using the Schrödinger equation
and geometrinetic energy operator (GEO) which leads to the Ri-operator,
respectively, we obtain some useful results between two quantum situations.
Meanwhile, we present generalized covariant wave equation based on the
G-operator. The GEO for quantum harmonic oscillator as an application is
further calculated.
###### Contents
1. 1 Introduction
1. 1.1 The Schrödinger equation
2. 2 Generalized geometric commutator and GAC
1. 2.1 QCPB and quantum geobracket (QGB)
2. 2.2 Quantum covariant Hamiltonian system
3. 2.3 Covariant dynamics, generalized Heisenberg equation, G-dynamics
4. 2.4 Imaginary geomenergy
5. 2.5 The QCPB for quantum harmonic oscillator
6. 2.6 Geomentum operator
7. 2.7 The geoperator
3. 3 G-dynamics
1. 3.1 D-operator, T-operator and G-operator
4. 4 Geometrinetic energy operator (GEO) and Ri-operator
1. 4.1 The G-dynamics with Schrödinger equation
5. 5 The cases of G-dynamics
1. 5.1 For classical Hamiltonian operator
2. 5.2 For Ri-operator
3. 5.3 G-operator for Ri-operator
4. 5.4 Generalized covariant wave equation
6. 6 GEO for generalized quantum harmonic oscillator
7. 7 Conclusions
## 1 Introduction
### 1.1 The Schrödinger equation
In quantum mechanics, the form of the Schrödinger equation depends on the
physical situation. The most general form is the time-dependent Schrödinger
equation, which gives a description of a system evolving with time [1, 2]
$\sqrt{-1}\hbar\frac{\partial\psi(r,t)}{\partial
t}={{\widehat{H}}^{\left(cl\right)}}\psi=\left({-\frac{\hbar^{2}}{2m}}\nabla^{2}+V\right)\psi(r,t)$
(1)
where $\partial/\partial t$ symbolizes a partial derivative with respect to
time $t$, $\psi$ is the wave function of the quantum system,
${{\widehat{H}}^{\left(cl\right)}}=-\frac{{{\hbar}^{2}}}{2m}{{\nabla}^{2}}+V$
(2)
is the Hamiltonian operator in terms of the coordinates. The time-dependent
Schrödinger equation described above predicts that wave functions can form
standing waves, called stationary states. Stationary states can also be
described by a simpler form of the Schrödinger equation, the time-independent
Schrödinger equation
${\displaystyle{{{\widehat{H}}^{\left(cl\right)}}}\psi=\left({-\frac{\hbar^{2}}{2m}}\nabla^{2}+V\right)\psi=E\psi}$
The energy and momentum operators, which are respectively shown as
${{\widehat{H}}^{\left(cl\right)}}=\sqrt{-1}\hbar\frac{\partial}{\partial
t}\,,\quad\widehat{p}^{\left(cl\right)}=-\sqrt{-1}\hbar\nabla$ (3)
The Schrödinger equation in its general form is shown as (1).
With the non-universal validity of the QPB that has been replaced by the
complete QCPB theory. In this paper, on the foundation of the Schrödinger
equation, we mainly explore the properties of the G-dynamics in the covariant
dynamics defined by the QCHS in QCPB, we obtain the more specific expressions
of the G-dynamics in terms of the coordinates to make the G-dynamics practical
in the quantum mechanics as it develops.
## 2 Generalized geometric commutator and GAC
Let $a$ and $b$ be any two operators, their commutator is formally defined by
${\displaystyle[a,b]_{cr}=ab-ba}$. Note that the operator for commutator can
be any mathematical form to be appeared for the calculation, such as a
function, vector, differential operator, partial differential operator, even a
number in a number field, and so on, it can be arbitrarily chosen according to
our needs.
Let $a$ and $b$ be any elements of any algebra, or any operators, their
generalized geometric commutator is formally defined by the following.
###### Definition 1 (GGC).
[3] A generalized geometric commutator (GGC) of arity two $a,b$ is formally
given by
$\left[a,b\right]={{\left[a,b\right]}_{cr}}+G\left(s;a,b\right)$
The geomutator is
$G\left(s;a,b\right)=a{{\left[s,b\right]}_{cr}}-b{{\left[s,a\right]}_{cr}}$
satisfying $G\left(s;a,b\right)=-G\left(s;b,a\right)$, where $s$ is a
geometric structure function given by domain.
Some properties of the geomutator are given by
$\displaystyle
G\left(s,a+b,c+d\right)=G\left(s,a,c\right)+G\left(s,b,d\right)+G\left(s,b,c\right)+G\left(s,a,d\right)$
$\displaystyle G\left(s,a+b,c\right)=G\left(s,a,c\right)+G\left(s,b,c\right)$
$\displaystyle G\left(s,a,c+d\right)=G\left(s,a,c\right)+G\left(s,a,d\right)$
for operators $a,b,c,d$.
Obviously, in some sense, the anticommutator also needs to be generalized as
the commutator does, Analogously, denote by
${{\left\\{a,b\right\\}}_{ir}}=ab+ba$ the anticommutator.
###### Definition 2.
The geometric anticommutator (GAC) of any two elements $a$ and $b$ is defined
by
$\left\\{a,b\right\\}={{\left\\{a,b\right\\}}_{ir}}+Z\left(s,a,b\right)$
where $Z\left(s,a,b\right)=Z\left(s,b,a\right)$ is called anti-geomutator, $s$
is geometric function.
As the definition above stated,
###### Definition 3.
The anti-geomutator can be taken as
$Z\left(s,a,b\right)=\left(a:s:b\right)+{{\left\\{a,b\right\\}}_{ir}}s$
where $\left(a:s:b\right)=asb+bsa$, and $s$ is the geometric function created
by the environment.
As a result of the symmetry of anti-geomutator, geometric anticommutator then
follows the symmetry $\left\\{a,b\right\\}=\left\\{b,a\right\\}$. Actually,
the anti-geomutator can be expressed in the form
$\displaystyle Z\left(s,a,b\right)$
$\displaystyle=a{{\left\\{s,b\right\\}}_{ir}}+b{{\left\\{s,a\right\\}}_{ir}}$
$\displaystyle=\left(a:s:b\right)+{{\left\\{a,b\right\\}}_{ir}}s$
As seen, this form is very similar to the quantum geometric bracket, this is
why we need to generalize the anticommutator. There has a clear property of
the anti-geomutator given by
$\displaystyle Z\left(s,a+b,c+d\right)$
$\displaystyle=\left(a+b\right){{\left\\{s,c+d\right\\}}_{ir}}+\left(c+d\right){{\left\\{s,a+b\right\\}}_{ir}}$
$\displaystyle=Z\left(s,a,c\right)+Z\left(s,a,d\right)+Z\left(s,b,c\right)+Z\left(s,b,d\right)$
for operators $a,b,c,d$.
### 2.1 QCPB and quantum geobracket (QGB)
As [3] stated, the quantum covariant Poisson bracket (QCPB) is defined by
generalized geometric commutator (GGC) while quantum geometric bracket (QGB)
is given based on the geomutator. More precisely,
###### Definition 4 (QCPB).
[3] The QCPB is generally defined as
$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$
in terms of quantum operator $\hat{f},~{}\hat{g}$, where
$G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$ is called
quantum geometric bracket.
It is zero if and only if $\hat{f}$ and $\hat{g}$ covariant commute, i,e.
$\left[\hat{f},\hat{g}\right]=0$. It is remarkable to see that the QCPB
representation admits a dynamical geometric bracket formula on the manifold.
###### Definition 5 (Quantum geometric bracket (QGB)).
[3] The quantum geometric bracket is
$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
where $s$ represents the globally condition of space.
Let’s assert the role of the structure function $s$ that is a geometric
structure function given by domain based on the generalized geometric
commutator in definition 4, it means that structure function $s$ is only
determined by the environment, or spacetime, or manifolds, the domain, ect,
from this viewpoint, the environment joins the physical process, the influence
of the environment now based on new theory can’t be ignored, it’s naturally
necessary to be considered in a physical process.
###### Definition 6.
[3] The covariant equilibrium equation is given by
$\left[\hat{f},\hat{g}\right]=0$, i.e,
${{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=0$
for operators $\hat{f},~{}\hat{g}$.
### 2.2 Quantum covariant Hamiltonian system
This section will give the covariant dynamics which contains two different
sub-dynamics: the generalized Heisenberg equation and G-dynamics [3]. It tells
that the generalized Heisenberg equation has improved the classical Heisenberg
equation by considering the quantum geometric bracket. Firstly, let’s give the
definition of quantum covariant Hamiltonian system (QCHS) based on the QCPB as
definition 4 defined.
In the beginning, let us start with the QCPB by taking $\hat{g}=\hat{H}$ into
consideration, then it formally yields below definition.
###### Definition 7 (QCHS).
[3] The QCHS defined by QCPB in terms of quantum operator $\hat{f},~{}\hat{H}$
is generally given by
$\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$
where $G\left(s,\hat{f},\hat{H}\right)$ is quantum geobracket .
###### Definition 8.
[3] The quantum geobracket for a Hamiltonian operator $\hat{H}$ and operator
$\hat{f}$ is given by
$G\left(s,\hat{f},\hat{H}\right)=\hat{f}{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$
and $s$ is a real structural function only associated with the structure
space.
Manifestly, this operator $\hat{f}$ covariantly commutes with $H$. By our
construction, the QCHS defined by the QCPB is a transition for the further
development of a complete and covariant theory that can naturally generalize
the Heisenberg equation. Since $\hat{f}$ is an observable, $s$ is assumed to
be real function as well. In this way, we have successfully given a complete
description of the Heisenberg equation.
###### Definition 9.
[3] The covariant equilibrium equation is given by
$\left[\hat{f},\hat{H}\right]=0$, i.e,
${{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)=0$
for operators $\hat{f},~{}\hat{H}$, then $\hat{f}$ is called quantum covariant
conserved quantity.
Furthermore, there is a certain case given by
$\left[\hat{H},\hat{H}\right]=0$, when $\hat{f}=\hat{H}$ based on definition
9. It is clear to see that $\hat{H}$ is a quantum covariant conserved
quantity.
### 2.3 Covariant dynamics, generalized Heisenberg equation, G-dynamics
In this section, we will briefly review the entire theoretical framework of
quantum covariant Hamiltonian system defined by the quantum covariant Poisson
bracket totally based on the paper [3]. More precisely, the time covariant
evolution of any observable $\hat{f}$ in the covariant dynamics is given by
both the generalized Heisenberg equation of motion and G-dynamics.
###### Theorem 1.
[3] The covariant dynamics, the generalized Heisenberg equation, G-dynamics
can be formally formulated as
The covariant dynamics:
$\frac{\mathcal{D}\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]$
The generalized Heisenberg equation:
$\frac{d\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}{{\left[\hat{f},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$
G-dynamics:
$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}$.
respectively, where $\frac{\mathcal{D}}{dt}=\frac{d}{dt}+\hat{w}$ is covariant
time operator, and ${{\hat{H}}}$ is the Hamiltonian and $[\cdot,\cdot]$
denotes the GGC of two operators.
### 2.4 Imaginary geomenergy
In this section, the imaginary geomenergy is defined based on the G-dynamics,
by using this new concept, we can better give a presentation for the covariant
dynamics, ect.
###### Definition 10.
[3] The imaginary geomenergy is defined by
${{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)=\sqrt{-1}\hbar\hat{w}$,
where $\hat{w}$ means G-dynamics.
It is clear that the covariant dynamics leads us to a complete quantum system
in which the state of the system is represented by two separated dynamic
quantum system, the covariant dynamics can be totally pictured in the form
below
$\displaystyle\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}$
$\displaystyle=\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$
$\displaystyle=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\sqrt{-1}\hbar\hat{f}\hat{w}$
With the imaginary geomenergy defined above, then covariant dynamics is
rewritten in the form
$\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}=\left[\hat{f},\hat{H}\right]=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\hat{f}{{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)$
As a consequence of the imaginary geomenergy, we can say that imaginary
geomenergy is a new kind of Hamiltonian operator.
### 2.5 The QCPB for quantum harmonic oscillator
In this section, we simply review some basic results given by the paper [3] in
QCPB. As quantum mechanics illustrated, the commutator relations may look
different than in the Schrödinger picture, because of the time dependence of
operators. The time evolution of those operators depends on the Hamiltonian of
the system. Considering the one-dimensional quantum harmonic oscillator,
$\hat{H}^{\left(cl\right)}={\frac{{{\hat{p}}}^{{\left(cl\right)2}}}{2m}}+{\frac{m\omega^{{2}}x^{{2}}}{2}}$
(4)
the evolution of the position and momentum operators is given by:
${d\over
dt}x(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(cl\right)},x(t)]_{QPB}={\frac{{{\hat{p}}}^{{\left(cl\right)}}}{m}}$
(5) ${d\over
dt}{{\hat{p}}}^{{\left(cl\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(cl\right)},{{\hat{p}}}^{{\left(cl\right)}}(t)]_{QPB}=-m\omega^{{2}}x$
As for the application of the QPB or the commutation, there are some many
fields including the physics and mathematics, and so on.
As a certainly example, we will now start by briefly reviewing the quantum
mechanics of a one-dimensional quantum harmonic oscillator (4), and see how
the QCPB can be incorporated using GGC in the covariant quantization
procedure. With the Hamiltonian given by (4), let’s use the QCPB to
recalculate the (5), we can see how difference emerges. More specifically, the
covariant dynamics in terms of the position reads
$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}^{\left(cl\right)},x\left(t\right)\right]=\frac{{{\hat{p}}^{\left(cl\right)}}\left(t\right)}{m}+\frac{\sqrt{-1}}{\hbar}x\left(t\right)\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$
By direct computation, the G-dynamics is given by
${{\widehat{w}}^{\left(cl\right)}}=\frac{\sqrt{-1}}{\hbar}\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$
(6)
And the generalized Heisenberg equation with respect to $x$ follows
$\displaystyle\frac{d}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H}^{\left(cl\right)},x\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}^{\left(cl\right)}{{\left[s,x\left(t\right)\right]}_{QPB}}$
$\displaystyle=\frac{{{\widehat{p}}^{\left(cl\right)}}\left(t\right)}{m}$
In the same way, the covariant dynamics for the classical momentum operator is
$\displaystyle\frac{\mathcal{D}}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}^{\left(cl\right)},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}^{\left(cl\right)}\frac{ds}{dx}+{{\widehat{p}}^{\left(cl\right)}}\left(t\right){{\widehat{w}}^{\left(cl\right)}}$
Accordingly, the generalized Heisenberg equation in terms of the
${{\hat{p}}}^{{\left(cl\right)}}$ appears
$\displaystyle\frac{d}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H}^{\left(cl\right)},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}^{\left(cl\right)}{{\left[s,{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}^{\left(cl\right)}\frac{ds}{dx}$
### 2.6 Geomentum operator
###### Definition 11.
[3] Let $M$ be a smooth manifold represented by structural function $s$, then
geomentum operator is defined as
$\hat{p}=-\sqrt{-1}\hbar D$
where $D=\nabla+\nabla s$. The component is
${{\hat{p}}_{j}}=-\sqrt{-1}\hbar{{D}_{j}}$ in which
${{D}_{j}}={{\partial}_{j}}+{{\partial}_{j}}s$ holds,
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$.
Note that the geomentum operator is a revision of the classical momentum
operator given by (3).
###### Theorem 2 (Geometric canonical quantization rules).
[3] Geometric equal-time canonical commutation relation is
$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}$
where
$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s,\cdot,~{}\cdot\right)$
is QCPB.
Geometric canonical commutation relation can be expressed in a specific form
$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar\left({{\delta}_{ij}}+{{x}_{i}}\frac{\partial}{\partial{{x}_{j}}}s\right)$
In other words, it also can be rewritten as
$\left[{{x}_{i}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{\theta}_{ij}}$, where
${{\theta}_{ij}}={{\delta}_{ij}}+{{x}_{i}}{{\partial}_{j}}s$, and
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$.
### 2.7 The geoperator
###### Definition 12 (Geoperator).
[4] Let $\hat{X}$ be a Hermitian operator, then geoperator can be defined as
${{\hat{X}}^{\left(s\right)}}=\hat{X}+u$
where $u=\hat{X}s=I\left(\hat{X},s\right)$ is coupling interaction between the
observable $\hat{X}$ and the environment $s$.
Obviously, the geometric operator is a extension of the Hermitian operator.
Note that the interaction term $u=\hat{X}s=I\left(\hat{X},s\right)\neq 0$
actually is a precise function with respect to the spacetime, as we state, the
structure function $s$ generated by the space or manifolds as an environment
variable only associate with the space or the manifolds, and it’s independent
to the wave function.
Note that the definition of the geoperator is reasonable, in particular, the
evidence comes from the coupling interaction between the observable $\hat{X}$
and the environment variable $s$ that is described by
$u=\hat{X}s=I\left(\hat{X},s\right)$ in which the environment variable $s$
satisfies the G-dynamics in the theorem 1, it also can be seen from (6) in the
one-dimensional quantum harmonic oscillator. In fact, geomentum operator is
one of the geoperator if the Hermitian operator
$\hat{X}=\widehat{p}^{\left(cl\right)}=-\sqrt{-1}\hbar\nabla$.
## 3 G-dynamics
In this section, G-dynamics as a new dynamical form appears in the quantum
mechanics, we mainly focus on the G-dynamics expressed by theorem 1, that is,
$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}$. As [3]
already stated, the G-dynamics is only induced by the structure function $s$,
and meanwhile, it’s independent to the other observables. There is a clear
fact of the G-dynamics taken as a different form by choosing different
Hamiltonian operators, in one word, various Hamiltonian operators correspond
to the multiple G-dynamics. Therefore, for a given wave function $\psi$, the
eigenvalue equation of operator is accordingly given by
$\widehat{w}\psi={{w}}\psi$, where ${{w}}$ is the eigenvalue. By using
definition 10, it gets corresponding imaginary geomenergy given by definition
10, that is, ${{E}^{\left(\operatorname{Im}\right)}}=\sqrt{-1}\hbar\hat{w}$,
thusly, it then has energy spectrum given by
${{E}^{\left(\operatorname{Im}\right)}}\psi=\sqrt{-1}{{E}^{\left(g\right)}}\psi$,
where ${E}^{\left(g\right)}=\hbar w$. Obviously, we can see that the
G-dynamics is completely determined by the structure function $s$, it implies
that the G-dynamics represents the properties of the spatial manifolds, in
other words, the spatial manifolds has abundant activities. As a result of
this point, we need to seek more clues to unlock this quantum characters. The
classical Hamiltonian operator (3) or (2) is the first to bear the brunt to be
considered in above formula.
### 3.1 D-operator, T-operator and G-operator
In this section, the discussions on the most general form will be done, we use
the G-dynamics $\widehat{w}$ to define the T-operator, and then the imaginary
geomenergy is used with the classical Hamiltonian operator (3) to construct
the G-operator as a non-Hermitian operator to deeply study some properties of
the G-dynamics.
To start with the definition of the D-operator and the T-operator,
###### Definition 13.
The D-operator is given by
$\widehat{{{D}_{t}}}={{\partial}_{t}}\pm\widehat{w}$, and then T-operator is
defined as $\widehat{{{N}_{t}}}=\sqrt{-1}\widehat{{{D}_{t}}}$, where
$\widehat{w}$ is the G-dynamics.
Note that D-operator $\widehat{{{D}_{t}}}={{\partial}_{t}}\pm\widehat{w}$
means that there are two kinds of D-operators. More precisely, we take one of
them into consideration as reality needed. Its eigenvalue equation of
D-operator is given by $\widehat{{{D}_{t}}}\psi={{D}_{t}}\psi$.
As a result of the definition 13, that naturally leads to bond the imaginary
geomenergy and the classical Hamiltonian operator (3) together to construct a
new energy operator, accordingly, we can derive following G-operator.
###### Definition 14.
The G-operator can be defined as
${{\widehat{H}}^{\left(gr\right)}}=\hbar\widehat{{{N}_{t}}}$.
Thusly, then the G-operator in details is given by
$\displaystyle{{\widehat{H}}^{\left(gr\right)}}$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}+{{E}^{\left(\operatorname{Im}\right)}}={{\widehat{H}}^{\left(cl\right)}}+\sqrt{-1}\hbar\widehat{w}$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}+\widehat{w}\right)$
$\displaystyle=\sqrt{-1}\hbar\widehat{{{D}_{t}}}$
That T-operator is then expressed as
${{\widehat{H}}^{\left(gr\right)}}/\hbar=\widehat{{{N}_{t}}}$. Thusly, the
eigenvalue equation $\widehat{{{N}_{t}}}\psi={{N}_{t}}\psi$ follows. As we
stated, G-dynamics $\widehat{w}$ is a Hermitian operator in the most of time,
it implies that the the G-operator is a non-Hermitian operator. As a result of
this point, we say that the eigenvalue of the G-operator must be in a complex
form.
###### Theorem 3.
The eigenvalue of G-operator is ${{E}^{\left(gr\right)}}=\hbar{{N}_{t}}$,
where the eigenvalue of T-operator is
$\displaystyle{{N}_{t}}$
$\displaystyle={{E}^{\left(gr\right)}}/\hbar={{E}^{\left(cl\right)}}/\hbar+\sqrt{-1}{{E}^{\left(g\right)}}/\hbar$
where ${{E}^{\left(g\right)}}=\hbar{{w}}$.
###### Proof.
According to the definition 14 of G-operator, the eigenvalue equation can be
more specifically expressed below
$\displaystyle{{\widehat{H}}^{\left(gr\right)}}\psi$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}\psi+{{E}^{\left(\operatorname{Im}\right)}}\psi={{\widehat{H}}^{\left(cl\right)}}\psi+\sqrt{-1}\hbar\widehat{w}\psi$
(7)
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}+\widehat{w}\right)\psi=\sqrt{-1}\hbar\widehat{{{D}_{t}}}\psi$
$\displaystyle=\sqrt{-1}\hbar{{D}_{t}}\psi$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}\psi+{{w}}\psi\right)$
$\displaystyle=\left({{E}^{\left(cl\right)}}+\sqrt{-1}\hbar{{w}}\right)\psi$
$\displaystyle={{E}^{\left(gr\right)}}\psi$
Hence, we obtain the eigenvalue of the G-operator that is given by
${{E}^{\left(gr\right)}}=\sqrt{-1}\hbar{{D}_{t}}$, it’s rewritten as
${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}+\sqrt{-1}{{E}^{\left(g\right)}}$
by denoting ${{E}^{\left(g\right)}}=\hbar{{w}}$, and the eigenvalue of the
D-operator $\widehat{{{D}_{t}}}$ follows
${{D}_{t}}={{E}^{\left(gr\right)}}/\sqrt{-1}\hbar={{E}^{\left(cl\right)}}/\sqrt{-1}\hbar+{{w}}$
Similarly, the eigenvalue of the T-operator $\widehat{{{N}_{t}}}$ also emerges
as follows ${{N}_{t}}={{E}^{\left(gr\right)}}/\hbar$, where
${{E}^{\left(g\right)}}/\hbar={{w}}$, therefore, the eigenvalue of the
operator $\widehat{{{N}_{t}}}$ can be rewritten as
$\displaystyle{{N}_{t}}$
$\displaystyle={{E}^{\left(gr\right)}}/\hbar={{E}^{\left(cl\right)}}/\hbar+\sqrt{-1}{{E}^{\left(g\right)}}/\hbar$
Note that the eigenvalue of the G-operator can be rewritten in a form
${{E}^{\left(gr\right)}}=\hbar{{N}_{t}}$. Therefore, we complete the proof as
desired. ∎
Notice that the eigenvalue of G-operator and T-operator are complex form,
these are the features of the non-Hermitian operator. As G-dynamics stated,
it’s generated by the structure function, and it forms the G-operator, it
reveals that there exists a complete Schrödinger equation such that the
structure function is naturally involved in the equation.
## 4 Geometrinetic energy operator (GEO) and Ri-operator
As a result of the G-operator, we use geomentum operator 11 to reconstruct the
new Hamiltonian operator. In order to smoothly build such complete new
Hamiltonian operator, we firstly need to define a corresponding kinetic energy
operator associated with the structure function $s$, therefore, let’s rewrite
geomentum operator as ${{\hat{P}}^{\left(ri\right)}}=-\sqrt{-1}\hbar D$ from
definition 11, then we give the definition below based on the geomentum
operator.
###### Definition 15 (Geometrinetic energy operator (GEO)).
Geometrinetic energy operator (GEO) is defined as
${{\hat{T}}^{\left(ri\right)}}=\frac{{{\hat{P}}^{\left(ri\right)2}}}{2m}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla\right)+{{T}^{\left(c\right)}}$
where ${{T}^{\left(c\right)}}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta s+\nabla
s\cdot\nabla s\right)$ is called structural $c$-energy.
More precisely, the derivation of the geometrinetic energy operator can be
given by logically computation. Since the square of generalized gradient
operator that is calculated as
${{D}^{2}}\psi=\Delta\psi+2\nabla s\cdot\nabla\psi+\psi\left(\Delta s+\nabla
s\cdot\nabla s\right)$
Accordingly, it yields an second order operator
${{D}^{2}}=\Delta+2\nabla s\cdot\nabla+\left(\Delta s+\nabla s\cdot\nabla
s\right)$ (8)
Then the eigenvalue equation of the geometrinetic energy operator
${{\hat{T}}^{\left(ri\right)}}$ with respect to the wave function $\psi$
follows
${{\hat{T}}^{\left(ri\right)}}\psi=\frac{{{\hat{P}}^{\left(ri\right)2}}}{2m}\psi=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta\psi+2\nabla
s\cdot\nabla\psi\right)+\psi{{T}^{\left(c\right)}}$
###### Definition 16.
Ri-operator is defined as
${{\hat{H}}^{\left(ri\right)}}={{\widehat{T}}^{\left(ri\right)}}+V$
where ${{\hat{T}}^{\left(ri\right)}}$ is the geometrinetic energy operator.
As a result, Ri-operator can be rewritten in the form
${{\hat{H}}^{\left(ri\right)}}={{\hat{H}}^{\left(cl\right)}}-\frac{{{\hbar}^{2}}}{m}\nabla
s\cdot\nabla+{{T}^{\left(c\right)}}$ (9)
by using the classical Hamiltonian operator (2).
### 4.1 The G-dynamics with Schrödinger equation
This section, we will mainly discuss the eigenvalue equation of G-dynamics
based on the Schrödinger equation (1), it nicely proves that G-operator and
T-operator are natural results on G-dynamics. Rewriting the G-dynamics that is
originally proposed as
${{E}^{\left(\operatorname{Im}\right)}}/\sqrt{-1}\hbar=\widehat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\widehat{H}\right]}_{QPB}}$
where the Hamiltonian operator $\widehat{H}$ in it is completely determined by
the physical truth.
###### Theorem 4.
The eigenvalue of the G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$ based on
Schrödinger equation (1) in time parameter or Hamiltonian operator (3) is
given by ${{w}^{\left(q\right)}}=-{{\partial}_{t}}s$.
###### Proof.
Let’s take $H={{\widehat{H}}^{\left(cl\right)}}$, then, more precisely,
${{\widehat{w}}^{\left(cl\right)}}\psi=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi=\frac{1}{\sqrt{-1}\hbar}\left(s{{\widehat{H}}^{\left(cl\right)}}\psi-{{\widehat{H}}^{\left(cl\right)}}\left(s\psi\right)\right)$
Plugging the Schrödinger equation (1) into above eigenvalue equation, we
obtain
$\sqrt{-1}\hbar{{\partial}_{t}}\psi={{\widehat{H}}^{\left(cl\right)}}\psi,~{}~{}\sqrt{-1}\hbar{{\partial}_{t}}\left(s\psi\right)={{\widehat{H}}^{\left(cl\right)}}\left(s\psi\right)$
and the eigenvalue equation of G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$
as an operator is
$\displaystyle{{\widehat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi$
(10)
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}\left(s\sqrt{-1}\hbar{{\partial}_{t}}\psi-\sqrt{-1}\hbar{{\partial}_{t}}\left(s\psi\right)\right)$
$\displaystyle=\left(s{{\partial}_{t}}\psi-{{\partial}_{t}}\left(s\psi\right)\right)$
$\displaystyle=-\psi{{\partial}_{t}}s$
We denote ${{\widehat{w}}^{\left(cl\right)}}\psi={{w}^{\left(q\right)}}\psi$
here. Thusly, we certainly get the specific eigenvalue for
${{\widehat{w}}^{\left(cl\right)}}$ given by
${{w}^{\left(q\right)}}=-{{\partial}_{t}}s$. ∎
Note that theorem 4 has indicated a fact that for any function form given for
the G-dynamics $\widehat{w}$ in such condition, the eigenvalue is always
dependent on the structure function.
###### Theorem 5.
The heat equation for the structure function $s$ based on the Schrödinger
equation (1) is given by
${{\partial}_{t}}s=\frac{\sqrt{-1}\hbar}{m}\left(\Delta s/2+\nabla
s\cdot\nabla\ln\psi\right)$
###### Proof.
In the one hand, theorem 4 has evidently given. In another hand, we take the
(2) into the G-dynamics $\widehat{w}$, and it deduces a result
$\displaystyle{{\hat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{H}}^{\left(cl\right)}}\right]}_{QPB}}\psi$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,-\frac{{{\hbar}^{2}}}{2m}\Delta+V\right]}_{QPB}}\psi$
$\displaystyle=-\frac{\hbar}{2\sqrt{-1}m}{{\left[s,\Delta\right]}_{QPB}}\psi$
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla s\cdot\nabla+\Delta
s\right)\psi$
According to (10), it yields
${{\hat{w}}^{\left(cl\right)}}\psi=-\psi{{\partial}_{t}}s=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)\psi$
It’s definite to form a heat equation associated with the structure function
$\displaystyle{{\partial}_{t}}s$
$\displaystyle=\frac{\sqrt{-1}\hbar}{2m\psi}\left(\psi\Delta s+2\nabla
s\cdot\nabla\psi\right)=\frac{\sqrt{-1}\hbar}{m}\left(\Delta s/2+\nabla
s\cdot\nabla\ln\psi\right)$
Then, the proof is completed. ∎
###### Corollary 1.
The G-dynamics in terms of the Hamiltonian operator (2) in coordinates can be
expressed as
${{\widehat{w}}^{\left(cl\right)}}=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)$ (11)
As we can see, the G-dynamics on such expression given by the (11) is totally
determined by the structure function. Geometrinetic energy operator by using
the G-dynamics ${{\widehat{w}}^{\left(cl\right)}}$ can be rewritten as
${{\widehat{T}}^{\left(ri\right)}}={{\widehat{T}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)-{E}^{\left(s\right)}/2$
where the imaginary geomenergy is
${{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\hat{w}}^{\left(cl\right)}}$,
and ${{\widehat{T}}^{\left(cl\right)}}=-\frac{{{\hbar}^{2}}}{2m}\Delta$, and
${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla s$. As a
consequence, based on the theorem 5, we can get a corollary shown as follows.
###### Corollary 2.
If wave function $\psi$ satisfies both (1) and heat equation 5, then
$\left\\{\begin{matrix}{{\partial}_{t}}s=\frac{\sqrt{-1}\hbar}{m}\left(\Delta
s/2+\nabla s\cdot\nabla\ln\psi\right)\\\
\sqrt{-1}\hbar{{\partial}_{t}}\psi=-\frac{{{\hbar}^{2}}}{2m}\Delta\psi+V\psi\\\
\end{matrix}\right.$
Take a deep analysis to above two equations, obviously, it yields
$\sqrt{-1}\hbar\psi{{\partial}_{t}}s=-\frac{{{\hbar}^{2}}}{2m}\left(\psi\Delta
s+2\nabla s\cdot\nabla\psi\right)$
It follows the bonding equation
$\sqrt{-1}\hbar\left({{\partial}_{t}}\psi+\psi{{\partial}_{t}}s\right)=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta\psi+2\nabla
s\cdot\nabla\psi+\psi\Delta s\right)+V\psi$
Definitely, it can be rewritten as
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}+{{\partial}_{t}}s\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+\nabla s\cdot\nabla s\right)\psi$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+\left(V+\frac{{{\hbar}^{2}}}{2m}\nabla s\cdot\nabla s\right)\psi$
$\displaystyle={{\widehat{H}}^{\left(ri\right)}}\psi+\frac{{{\hbar}^{2}}}{2m}\psi\nabla
s\cdot\nabla s$
Or in the form
$\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi={{\hat{H}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
(12)
where ${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla s$,
it evidently implies that ${{E}^{\left(s\right)}}$ plays the role of potential
energy. In fact, the (12) supports a statement to the definition 14. In other
words, such definition 14 is a natural extension for the classical results, in
the later discussions, we can surely say about this points.
###### Definition 17.
D-operator based on the Schrödinger equation (1) is
$\widehat{{{D}_{t}}}^{\left(cl\right)}={{\partial}_{t}}-{{\hat{w}}^{\left(cl\right)}}$.
Actually, the D-operator also can be taken in another form
$\widehat{{{D}_{t}}}^{\left(cl\right)}={{\partial}_{t}}+{{\hat{w}}^{\left(cl\right)}}$
(13)
that is, $-{{\hat{w}}^{\left(cl\right)}}\to{{\hat{w}}^{\left(cl\right)}}$ on
above definition. According to the G-operator given by definition 14, thusly,
it yields
###### Definition 18.
G-operator is then defined as
${{\widehat{H}}^{\left(gr\right)}}=\hbar{{\widehat{N}}_{t}}^{\left(cl\right)}={{\widehat{H}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)$
based on definition 17, where the imaginary geomenergy is
${{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\hat{w}}^{\left(cl\right)}}$.
As a consequence of the definition 14, we figure out that eigenvalue theorem
below.
###### Theorem 6.
The eigenvalue of the G-operator in definition 18 on Schrödinger equation (1)
is
${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}-\sqrt{-1}\hbar{{w}^{\left(q\right)}}$
###### Proof.
For a given wave function $\psi$, the eigenvalue equation of the G-operator on
Schrödinger equation (1) is expressed as
$\displaystyle{{\widehat{H}}^{\left(gr\right)}}\psi$
$\displaystyle=\hbar{{\widehat{N}}_{t}}^{\left(cl\right)}\psi={{\widehat{H}}^{\left(cl\right)}}\psi-{{E}^{\left(\operatorname{Im}\right)}}\left({{{\hat{w}}}^{\left(cl\right)}}\right)\psi$
$\displaystyle={{\widehat{H}}^{\left(cl\right)}}\psi-\sqrt{-1}\hbar{{{\hat{w}}}^{\left(cl\right)}}\psi$
$\displaystyle={{E}^{\left(cl\right)}}\psi-\sqrt{-1}\hbar\psi{{w}^{\left(q\right)}}$
$\displaystyle=\left({{E}^{\left(cl\right)}}-\sqrt{-1}\hbar{{w}^{\left(q\right)}}\right)\psi$
$\displaystyle={{E}^{\left(gr\right)}}\psi$
where ${{\widehat{w}}^{\left(cl\right)}}\psi={{w}^{\left(q\right)}}\psi$ has
been used for the proof. Then, we complete the proof. ∎
In fact, if D-operator is taken according to (13), then eigenvalue of the
G-operator on Schrödinger equation (1) is
${{E}^{\left(gr\right)}}={{E}^{\left(cl\right)}}+\sqrt{-1}\hbar{{w}^{\left(q\right)}}$
This is exactly the characters of conjugate complex numbers.
###### Corollary 3.
Based on theorem 6, there exists a pure imaginary geomenergy form
${{E}^{\left(\operatorname{Im}\right)}}\left(\pm{{w}^{\left(q\right)}}\right)=\pm\sqrt{-1}\hbar{{w}^{\left(q\right)}}$
induced by G-dynamics.
## 5 The cases of G-dynamics
### 5.1 For classical Hamiltonian operator
To consider quantum harmonic oscillator (4), as the G-dynamics of it shown by
(6),
${{\widehat{w}}^{\left(cl\right)}}=\frac{\sqrt{-1}}{\hbar}\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}+\frac{\sqrt{-1}}{m\hbar}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}$
The pure imaginary geomenergy form is
${{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}=-\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}-\frac{1}{m}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}$
Let’s see how its characteristic equation shows, it’s given by
$\displaystyle{{\widehat{w}}^{\left(cl\right)}}\psi$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\psi\frac{{{\widehat{p}}^{\left(cl\right)2}}s}{2m}+\frac{\sqrt{-1}}{m\hbar}{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}\psi$
$\displaystyle=-\frac{\sqrt{-1}\hbar}{2m}\left(\psi\Delta s+2\nabla
s\cdot\nabla\psi\right)$ $\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta
s+2\nabla s\cdot\nabla\right)\psi$
where $\Delta={{\nabla}^{2}}$. Subsequently, it derives the G-dynamics (11),
or in the form
$\frac{2\sqrt{-1}m}{\hbar}{{\widehat{w}}^{\left(cl\right)}}=\Delta s+2\nabla
s\cdot\nabla$, by using (8), we get its another expression in terms of the
G-dynamics shown as
${{D}^{2}}=\frac{2\sqrt{-1}m}{\hbar}{{\widehat{w}}^{\left(cl\right)}}+\Delta+\nabla
s\cdot\nabla s$
### 5.2 For Ri-operator
###### Theorem 7.
The G-dynamics in ${{\hat{H}}^{\left(ri\right)}}$ in coordinates is
$\widehat{w}^{\left(ri\right)}={{\widehat{w}}^{\left(cl\right)}}+w^{\left(s\right)}$
where $w^{\left(s\right)}=\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$ is
denoted.
###### Proof.
For this case, let’s consider
${{\hat{H}}^{\left(ri\right)}}={{\widehat{T}}^{\left(ri\right)}}+V$ given by
Ri-operator 16, then the G-dynamics can be calculated as
$\widehat{w}^{\left(ri\right)}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\hat{H}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{T}}^{\left(ri\right)}}+V\right]}_{QPB}}$
where $V$ is a real potential function. Thusly, we obtain
$\displaystyle{{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}{{\left[s,\Delta+2\nabla
s\cdot\nabla\right]}_{QPB}}\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}{{\left[s,\Delta\right]}_{QPB}}\psi-\frac{{{\hbar}^{2}}}{m}{{\left[s,\nabla
s\cdot\nabla\right]}_{QPB}}\psi$
where
$\displaystyle{{\left[s,\Delta\right]}_{QPB}}\psi$
$\displaystyle=s\Delta\psi-\Delta\left(s\psi\right)$ $\displaystyle=-2\nabla
s\cdot\nabla\psi-\psi\Delta s$ $\displaystyle=-\left(2\nabla
s\cdot\nabla+\Delta s\right)\psi$
and
${{\left[s,\nabla s\cdot\nabla\right]}_{QPB}}\psi=-\psi\nabla s\cdot\nabla s$
Therefore, we get
${{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)$
Thusly,
$\displaystyle\widehat{w}^{\left(ri\right)}$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[s,{{\widehat{T}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)$ (14)
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla s\cdot\nabla+\Delta
s\right)+\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$
$\displaystyle={{\widehat{w}}^{\left(cl\right)}}+w^{\left(s\right)}$
where $w^{\left(s\right)}=\frac{\hbar}{\sqrt{-1}m}\nabla s\cdot\nabla s$ is
denoted. ∎
Obviously,
${{\hat{w}}^{\left(ri\right)}}-{{\hat{w}}^{\left(cl\right)}}={{w}^{\left(s\right)}}$
holds. The theorem 7 can be rewritten in a form
${{E}^{\left(\operatorname{Im}\right)}}\left({\hat{w}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}$
where ${{E}^{\left(s\right)}}=\frac{{{\hbar}^{2}}}{m}\nabla s\cdot\nabla
s=\sqrt{-1}\hbar{{w}^{\left(s\right)}}$, or expressed by the imaginary
geomenergy
${{E}^{\left(\operatorname{Im}\right)}}\left({\hat{w}^{\left(ri\right)}}\right)={{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)+{{E}^{\left(s\right)}}$
where
${{E}^{\left(\operatorname{Im}\right)}}\left({{\widehat{w}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{\widehat{w}}^{\left(cl\right)}}$
is the imaginary geomenergy in terms of ${{\widehat{w}}^{\left(cl\right)}}$.
Correspondingly, the structural $c$-energy can be given in a form
$\displaystyle{{T}^{\left(c\right)}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta s+{{\left|\nabla
s\right|}^{2}}\right)=-\frac{{{\hbar}^{2}}}{2m}\Delta
s-\frac{{{\hbar}^{2}}}{2m}{{\left|\nabla s\right|}^{2}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\Delta s-{{E}^{\left(s\right)}}/2$
where it results in
${{T}^{\left(c\right)}}+{{E}^{\left(s\right)}}/2=-\frac{{{\hbar}^{2}}}{2m}\Delta
s$. There is more we need to say clearly,
${{\widehat{w}}^{\left(ri\right)}}={{\widehat{w}}^{\left(cl\right)}}+{{w}^{\left(s\right)}}={{\widehat{w}}^{\left(cl\right)}}-\sqrt{-1}a\nabla
s\cdot\nabla s$
where $a=\hbar/m$. Then the conjugate transpose of the
${{\widehat{w}}^{\left(ri\right)}}$ is given by
${{\widehat{w}}^{\left(ri\right)\dagger}}={{\widehat{w}}^{\left(cl\right)\dagger}}+\sqrt{-1}a\nabla
s\cdot\nabla s$
Accordingly, it yields
${{\widehat{w}}^{\left(ri\right)}}+{{\widehat{w}}^{\left(ri\right)\dagger}}={{\widehat{w}}^{\left(cl\right)}}+{{\widehat{w}}^{\left(cl\right)\dagger}}=2{{\widehat{w}}^{\left(cl\right)}}$.
Meanwhile, let ${{\widehat{w}}^{\left(g\right)}}$ be denoted as NG operator to
fit
${{\widehat{w}}^{\left(g\right)}}={{\widehat{w}}^{\left(ri\right)}}-{{\widehat{w}}^{\left(ri\right)\dagger}}=-2\sqrt{-1}a\nabla
s\cdot\nabla s\neq 0$
Obviously, this NG operator breaks the Hermiticity as usually understood
### 5.3 G-operator for Ri-operator
To analyze G-dynamics for both situations together, we can obtain some useful
results.
$\displaystyle\widehat{w}^{\left(ri\right)}\psi$
$\displaystyle={{\widehat{w}}^{\left(cl\right)}}\psi+w^{\left(s\right)}\psi$
$\displaystyle=\frac{\hbar}{2\sqrt{-1}m}\left(\Delta s+2\nabla
s\cdot\nabla\right)\psi+\frac{\hbar}{\sqrt{-1}m}\psi\nabla s\cdot\nabla s$
Accordingly, it leads to the imaginary geomenergy
$\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}\psi=\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)\psi$
and with the Schrödinger equation (1), it generates
$\displaystyle\sqrt{-1}\hbar{{\partial}_{t}}\psi-\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}\psi$
$\displaystyle=\sqrt{-1}\hbar\left({{\partial}_{t}}-\widehat{w}^{\left(ri\right)}\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\Delta\psi+V\psi-\frac{{{\hbar}^{2}}}{2m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla s\right)\psi$
$\displaystyle=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+\nabla s\cdot\nabla
s\right)\psi+\left(V-{{E}^{\left(s\right)}}/2\right)\psi$
$\displaystyle={{\hat{H}}^{\left(ri\right)}}\psi-{{E}^{\left(s\right)}}\psi/2$
As a result of equation (12) together, we get a similar equations given by
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi={{{\hat{H}}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
(15)
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(ri\right)}}\right)\psi={{{\hat{H}}}^{\left(ri\right)}}\psi-{{E}^{\left(s\right)}}\psi/2$
And then it rightly reflects the theorem 7.
###### Corollary 4.
Ri-operator expressed by the G-operator is given by
${{\hat{H}}^{\left(ri\right)}}={{\hat{H}}^{\left(gr\right)}}+{{E}^{\left(s\right)}}/2$
###### Proof.
Since the (9) and theorem 7 hold, then
$\displaystyle{{{\hat{H}}}^{\left(ri\right)}}$
$\displaystyle={{{\hat{H}}}^{\left(cl\right)}}-\frac{{{\hbar}^{2}}}{m}\nabla
s\cdot\nabla+{{{{T}}}^{\left(c\right)}}$
$\displaystyle={{{\hat{H}}}^{\left(cl\right)}}-\sqrt{-1}\hbar{{\widehat{w}}^{\left(ri\right)}}+{{E}^{\left(s\right)}}/2$
With the definition 14 of G-operator, it has
${{\hat{H}}^{\left(gr\right)}}={{\hat{H}}^{\left(cl\right)}}-\sqrt{-1}\hbar{{\widehat{w}}^{\left(ri\right)}}=\sqrt{-1}\hbar{{\widehat{D}}_{t}}^{\left(ri\right)}$
where D-operator here is
${{\widehat{D}}_{t}}^{\left(ri\right)}={{\partial}_{t}}-{{\widehat{w}}^{\left(ri\right)}}$,
thus, then we complete the proof. ∎
Conversely, note that the G-operator in corollary 4 is also expressed as
${{\hat{H}}^{\left(gr\right)}}={{\hat{H}}^{\left(ri\right)}}-{{E}^{\left(s\right)}}/2$
(16)
By using Ri-operator in corollary 4, (15) can be simplified as
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(cl\right)}}\right)\psi={{{\hat{H}}}^{\left(gr\right)}}\psi+{{E}^{\left(s\right)}}\psi$
(17)
$\displaystyle\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}\right)\psi={{{\hat{H}}}^{\left(gr\right)}}\psi$
where G-operator in terms of the Ri-operator is definitely chosen as
${{\hat{H}}^{\left(gr\right)}}=\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar{{\hat{D}}_{t}}^{\left(ri\right)}$
and
${{\hat{D}}_{t}}^{\left(ri\right)}={{\partial}_{t}}-{{\hat{w}}^{\left(ri\right)}}$.
As a matter of fact, these two equations are equivalent to each other based on
the theorem 7.
### 5.4 Generalized covariant wave equation
Correspondingly, the definition of generalized covariant wave equation as a
revision of classic Schrödinger equation follows based on geometrinetic energy
operator.
###### Theorem 8.
The generalized covariant wave equation is defined as
$\hbar\widehat{{{N}_{t}}}^{\left(ri\right)}\psi={{{\hat{H}}}^{\left(gr\right)}}\psi$
by using (16), where the D-operator is taken as
$\widehat{{{D}_{t}}}^{\left(ri\right)}={{\partial}_{t}}-{{{\hat{w}}}^{\left(ri\right)}}$.
Apparently, the generalized covariant wave equation can be given by employing
(17),
$\sqrt{-1}\hbar\left({{\partial}_{t}}-{{{\hat{w}}}^{\left(cl\right)}}-{{E}^{\left(s\right)}}/\sqrt{-1}\hbar\right)\psi={{\hat{H}}^{\left(gr\right)}}\psi$
where
${{\hat{w}}^{\left(ri\right)}}={{\hat{w}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}/\sqrt{-1}\hbar$
is distinct to be seen from the theorem 7. More precisely about the
generalized covariant wave equation are listed
$\displaystyle{{{\hat{H}}}^{\left(gr\right)}}=-\frac{{{\hbar}^{2}}}{2m}\left(\Delta+2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla
s\right)={{{\hat{H}}}^{\left(ri\right)}}-{{E}^{\left(s\right)}}/2$
$\displaystyle{{{\hat{w}}}^{\left(ri\right)}}=\frac{\hbar}{2\sqrt{-1}m}\left(2\nabla
s\cdot\nabla+\Delta s+2\nabla s\cdot\nabla
s\right)={{{\hat{w}}}^{\left(cl\right)}}+{{E}^{\left(s\right)}}/\sqrt{-1}\hbar$
Obviously, the generalized covariant wave equation is a natural complete
extension of the classic Schrödinger equation.
Supported by the definition 12, we consider geoperator in terms of the
Hamiltonian $\hat{H}$, let’s see what we get in this way. Thusly, we have
${{\widehat{H}}^{\left(s\right)}}=\widehat{H}+u_{H}$ that can be called the
geomenergy operator, where $u_{H}=\widehat{H}s$ is coupling interaction
between the observable $\hat{H}$ and the environment $s$. In order to better
understand this case, we precisely consider the classical Hamiltonian operator
(3) ${{\hat{H}}^{\left(cl\right)}}$ for the geomenergy operator that is
expressed as
${{\widehat{H}}^{\left(s\right)}}={{\widehat{H}}^{\left(cl\right)}}+u_{H^{(cl)}}$,
where $u_{H^{(cl)}}={{\widehat{H}}^{\left(cl\right)}}s$, more specifically,
the coupling interaction for the classical Hamiltonian operator (3)
${{\widehat{H}}^{\left(cl\right)}}$ and environment is given by
$u_{H^{(cl)}}={{\widehat{H}}^{\left(cl\right)}}s=-{{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)$,
then,
${{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)=-u_{H^{(cl)}}$,
as a result, the geomenergy operator can be rewritten as
${{\widehat{H}}^{\left(s\right)}}={{\widehat{H}}^{\left(cl\right)}}-{{E}^{\left(\operatorname{Im}\right)}}\left({{w}^{\left(q\right)}}\right)$,
when we refer to the (12), we can obtain self-consistent result
${{\widehat{H}}^{\left(s\right)}}\psi=\sqrt{-1}\hbar\left({{\partial}_{t}}-{{\widehat{w}}^{\left(cl\right)}}\right)\psi$
for a given wave function $\psi$, that is to say,
${{\widehat{H}}^{\left(s\right)}}\psi={{\hat{H}}^{\left(ri\right)}}\psi+{{E}^{\left(s\right)}}\psi/2$
holds, accordingly, it yields
${{\widehat{H}}^{\left(s\right)}}\psi-{{E}^{\left(s\right)}}\psi/2={{\hat{H}}^{\left(ri\right)}}\psi$
which is exactly the generalized covariant wave equation as theorem 8 given.
## 6 GEO for generalized quantum harmonic oscillator
The covariant time evolution of operators
$x(t),{{\hat{p}}}^{{\left(ri\right)}}(t)$ depends on the Hamiltonian of the
system. Considering the one-dimensional generalized quantum harmonic
oscillator based upon geomentum operator 11, it’s given by
$\hat{H}^{\left(ri\right)}={\frac{{{\hat{p}}}^{{\left(ri\right)2}}}{2m}}+{\frac{m\omega^{{2}}x^{{2}}}{2}}$
(18)
where $V\left(x\right)=\frac{1}{2}m{{\omega}^{2}}{{x}^{2}}$, the geomentum
operator in one-dimensional is
${{\widehat{p}}^{\left(ri\right)}}=-\sqrt{-1}\hbar\frac{\rm{D}}{dx}$, and
$\frac{\rm{D}}{dx}=d/dx+ds/dx$. Accordingly, the covariant evolution of the
position and geomentum operator based on the QCHS given by definition 7 are
respectively given by:
${\mathcal{D}\over
dt}x(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},x(t)]$ (19)
${\mathcal{D}\over
dt}{{\hat{p}}}^{{\left(ri\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},{{\hat{p}}}^{{\left(ri\right)}}(t)]$
where $[\cdot,\cdot]$ denotes the GGC of two operators. More precisely, the
concrete computational process of the first covariant evolution terms of the
position is shown as
$\left[{{\widehat{H}}^{\left(ri\right)}},x\right]={{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$
Hence, the direct computation leads to a series of results
$\displaystyle{{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}$
$\displaystyle={{\left[\frac{{{\widehat{p}}^{\left(ri\right)2}}}{2m},x\right]}_{QPB}}=\frac{1}{2m}{{\left[{{\widehat{p}}^{\left(ri\right)2}},x\right]}_{QPB}}$
$\displaystyle=\frac{-{{\hbar}^{2}}}{2m}{{\left[\frac{{{d}^{2}}}{d{{x}^{2}}}+2\frac{ds}{dx}\frac{d}{dx},x\right]}_{QPB}}$
$\displaystyle=\frac{-{{\hbar}^{2}}}{m}\frac{\rm{D}}{dx}$
where
${{\widehat{p}}^{\left(ri\right)2}}=-{{\hbar}^{2}}\left(\frac{{{d}^{2}}}{d{{x}^{2}}}+2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}s}{d{{x}^{2}}}+{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
and the QGB in terms of ${{\widehat{H}}^{\left(ri\right)}},x$ is
$\displaystyle G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$
$\displaystyle=-x{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}=x{{\left[{{\widehat{H}}^{\left(ri\right)}},s\right]}_{QPB}}$
$\displaystyle=-\frac{{{\hbar}^{2}}}{m}x\frac{ds}{dx}\frac{d}{dx}-\frac{{{\hbar}^{2}}}{2m}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
Then the QCPB about ${{\widehat{H}}^{\left(ri\right)}},x$ specifically shows
$\displaystyle\left[{{\widehat{H}}^{\left(ri\right)}},x\right]$
$\displaystyle={{\left[{{\widehat{H}}^{\left(ri\right)}},x\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},x\right)$
$\displaystyle=\frac{-{{\hbar}^{2}}}{m}\frac{\rm{D}}{dx}-\frac{{{\hbar}^{2}}}{m}x\frac{ds}{dx}\frac{d}{dx}-\frac{{{\hbar}^{2}}}{2m}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
$\displaystyle=-\frac{{{\hbar}^{2}}}{m}\left(\frac{\rm{D}}{dx}+x\frac{ds}{dx}\frac{d}{dx}+\frac{1}{2}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)\right)$
As a result, the covariant evolution of the position is given by
$\displaystyle\frac{\mathcal{D}}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},x\right]$
$\displaystyle=-\frac{\sqrt{-1}\hbar}{m}\left(\frac{\rm{D}}{dx}+x\frac{ds}{dx}\frac{d}{dx}+\frac{1}{2}x\left(\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)\right)$
In other words, the covariant evolution of the position based on the QCHS is
$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},x\right]=\frac{{{\widehat{p}}^{\left(ri\right)}}}{m}+x\widehat{w}^{\left(ri\right)}$
where G-dynamics is equal to
$\widehat{w}^{\left(ri\right)}=-\frac{\sqrt{-1}\hbar}{2m}\left(2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
(20)
Definitely, G-dynamics (20) in terms of $\widehat{w}^{\left(ri\right)}$ is the
equivalent expression of the (14) in one dimensional case. Accordingly, the
imaginary geomenergy follows
${{E}^{\left(\operatorname{Im}\right)}}\left(\widehat{w}^{\left(ri\right)}\right)=\sqrt{-1}\hbar\widehat{w}^{\left(ri\right)}=\frac{{{\hbar}^{2}}}{2m}\left(2\frac{ds}{dx}\frac{d}{dx}+\frac{{{d}^{2}}}{d{{x}^{2}}}s+2{{\left(\frac{ds}{dx}\right)}^{2}}\right)$
Similarly, the covariant evolution of the geomentum operator is
${\mathcal{D}\over
dt}{{\hat{p}}}^{{\left(ri\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H}^{\left(ri\right)},{{\hat{p}}}^{{\left(ri\right)}}(t)]$
The calculation process is as follows
$\displaystyle{{\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}$
$\displaystyle={{\left[V,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\frac{1}{2}m{{\omega}^{2}}{{\left[{{x}^{2}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}$
$\displaystyle=\frac{1}{2}m{{\omega}^{2}}{{\left[{{x}^{2}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\sqrt{-1}\hbar
m{{\omega}^{2}}x$
The QGB in terms of
${{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}$ is
$G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)={{\widehat{H}}^{\left(ri\right)}}{{\left[s,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$
where
${{\left[s,{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}=\sqrt{-1}\hbar\frac{ds}{dx}$
has been used, and the QGB in details becomes
$G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)=\sqrt{-1}\hbar{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$
As a consequence, the QCPB here is given by
$\displaystyle\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]$
$\displaystyle={{\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]}_{QPB}}+G\left(s,{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right)$
$\displaystyle=\sqrt{-1}\hbar
m{{\omega}^{2}}x+\sqrt{-1}\hbar{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$
It turns out that the covariant evolution of the geomentum operator is derived
as follows
$\displaystyle\frac{\mathcal{D}}{dt}{{\widehat{p}}^{\left(ri\right)}}$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[{{\widehat{H}}^{\left(ri\right)}},{{\widehat{p}}^{\left(ri\right)}}\right]$
$\displaystyle=-m{{\omega}^{2}}x-{{\widehat{H}}^{\left(ri\right)}}\frac{ds}{dx}-\frac{\sqrt{-1}}{\hbar}{{\widehat{p}}^{\left(ri\right)}}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$
where the G-dynamics reads
$\widehat{w}^{\left(ri\right)}=-\frac{\sqrt{-1}}{\hbar}{{\left[s,{{\widehat{H}}^{\left(ri\right)}}\right]}_{QPB}}$
which is equivalent to the (20) as a precise expression. The G-dynamics seen
from the (20) can evidently say how it forms and works associated with the
structure function.
## 7 Conclusions
In this paper, we have proposed a series of new concepts based on the analysis
of the G-dynamics included in the covariant dynamics defined by the QCHS in
QCPB. On the solid foundation of the QCPB, we have gained some valuable
consequences centred on the G-dynamics. By using the G-dynamics, we define new
operators such as the D-operator, T-operator, G-operator, Ri-operator etc, we
find their complex connections, in particular, we mainly focus on the their
eigenvalues equation. With the help of the Schrödinger equation, we analyze
the G-dynamics and its related properties. We concretely discuss the different
cases of the G-dynamics, in this way, we present the extension of the
Schrödinger equation, naturally. As an implication, we derive the generalized
quantum harmonic oscillator and the covariant dynamics on it based on the QCHS
theory.
## References
* [1] Greiner W. Relativistic quantum mechanics. wave equations [M]. Beijing: World Book Inc, 2003.
* [2] Nikolenko N V. On the complete integrability of the nonlinear Schrödinger equation [J]. Functional Analysis and Its Applications, 1976, 10(3): 209-220.
* [3] Wang G. Generalized geometric commutator theory and quantum geometric bracket and its uses [J]. arXiv:2001.08566.
* [4] Wang G. A revision for Heisenberg uncertainty relation based on environment variable in the QCPB theory [J]. arXiv:2003.07203.
|
# HINT: Hypernetwork Instruction Tuning
for Efficient Zero- & Few-Shot Generalisation
Hamish Ivison α &Akshita Bhagia α &Yizhong Wang ω Hannaneh Hajishirzi αω
&Matthew Peters α αAllen Institute for AI
ωPaul G. Allen School of Computer Science & Engineering, University of
Washington
<EMAIL_ADDRESS>
###### Abstract
Recent NLP models have shown the remarkable ability to effectively generalise
‘zero-shot’ to new tasks using only natural language instructions as guidance.
However, many of these approaches suffer from high computational costs due to
their reliance on concatenating lengthy instructions with every input example,
resulting in costly reprocessing of the instruction. To avoid this, we
introduce Hypernetworks for INstruction Tuning (HINT), which convert task
instructions and examples into parameter-efficient modules inserted into an
underlying model using a pretrained text encoder, eliminating the need to
include instructions in the model input. The hypernetwork in HINT also
produces an encoded instruction, which we concatenate with encoded inputs
during decoding to further improve performance. HINT models outperform strong
state-of-the-art baselines by over 10% when controlling for compute (measured
in FLOPs). By converting instructions into modules, HINT models can
effectively disregard the length of instructions and few-shot example inputs
in terms of compute usage. As a result, HINT can enhance its performance by up
to 25% by incorporating additional few-shot data, while utilizing only up to
5% more compute. This combines the strengths of parameter-efficient fine-
tuning and in-context learning. We release our code publicly111Our code is
available at:
https://github.com/allenai/hyper-task-descriptions..
## 1 Introduction
Figure 1: Overview of HINT. (1) We feed an instruction into a HyperEncoder to
produce an encoded instruction, use it to generate prefix and adapter weights,
and then insert them into the underlying model. (2) We run the underlying
model encoder as usual and optionally concatenate the encoded input with the
previously encoded instruction, before running the underlying model decoder to
generate the answer. We only use the hypernetwork once per task.
Large pretrained language models have demonstrated a striking ability to
perform new tasks through the use of in-context examples or instructions alone
(Brown et al., 2020), or after training on input instances augmented with
instructions (Weller et al., 2020; Mishra et al., 2022; Sanh et al., 2022; Wei
et al., 2022; Chung et al., 2022; Wang et al., 2022b). This ability allows a
single model to adapt to many tasks where training data is difficult to
collect or task-specific fine-tuning is impractical (i.e., ‘zero-shot’
settings): models trained on instructions need only a single instruction to
achieve non-trivial performance on the task at hand. The most common method to
achieve this zero-shot ability is to meta-train the model with task
instructions concatenated with every input, allowing the model to learn to
associate instructions with tasks. While empirically highly successful, this
is inefficient and requires reprocessing lengthy task instructions and any
additional task data (e.g., few-shot examples) with every input example.
In this paper, we introduce Hypernetworks222Hypernetworks are neural networks
trained to generate neural networks (Ha et al., 2017). for Instruction Tuning
(HINT), which directly generate task-specific parameter-efficient modules
given only an instruction, combining the benefits of instruction-based
learning with parameter-efficient modules. HINT models convert instructions
and other task data (e.g., few-shot examples) into efficient modules within a
pretrained language model, enabling cheaper inference and better compute
scaling with few-shot data for an underlying instruction-based meta-learning
approach. Additionally, fusing hypernetwork-encoded instructions with the
encoded input at the underlying model decoder greatly improves the performance
while using minimal extra compute. An important benefit of HINT is that it
processes instructions and other task information only once, making the
compute used by our method almost independent of the amount of task data
available, unlike both regular finetuning and input concatenation-based
approaches (see Figure 3).
We find that our hypernetwork-based approach (‘HINT’), is able to achieve
similar performance to baselines that receive the full instruction with every
input example while using significantly less compute (as measured by FLOPs),
due to the greatly reduced input length. When controlling for inference
budget, we find that HINT models outperform strong baselines in zero- and few-
shot settings. This validates our assumption that we can significantly reduce
inference costs by avoiding reprocessing the instruction with every input, and
instead saving it for repeated use. Furthermore, we find that including
additional few-shot information alongside task instructions significantly
improves HINT model performance while using minimal additional compute during
inference. Ultimately, our work pushes towards directly generating cheap,
customised models from task data, without requiring any expensive task-
specific finetuning.
In summary, our findings are:
* •
We introduce HINT models, which make use of a text-conditioned hypernetwork to
generate parameter-efficient modules based on task descriptions and few-shot
examples.
* •
HINT models, by reducing input lengths, are able to achieve similar
performance to strong full-input baselines while reducing inference cost
(measured in FLOPs) by up to 4$\times$.
* •
As the compute used by HINT models is effectively independent of the length of
the instruction and amount of few-shot data provided with the instruction,
HINT models provided with additional few-shot data simultaneously outperform
and use up to 4$\times$ fewer FLOPs than baselines without few-shot data.
* •
HINT models outperform strong decoder-only baselines. While decoder-only
models allow for input caching, we find that instruction-tuned GPT-2 models
significantly underperform HINT models (8-9 point difference), matching prior
work suggesting that encoder-decoder models work better for instruction-tuning
(Wang et al., 2022a; Iyer et al., 2022).
## 2 Related Work
##### Instruction Following
Further finetuning large pretrained language models on instructions has been
found to greatly improve zero-shot generalisation, as the finetuned model
learns to make use of the instructions to perform the given task (Weller et
al., 2020; Wei et al., 2022; Mishra et al., 2022; Chung et al., 2022; Wang et
al., 2022b). Additionally, Sanh et al. (2022) found that training models on
multiple prompts per task also resulted in improved performance, suggesting
that further increasing prompt diversity aids generalisation, even when using
the same pool of underlying tasks. The majority of these popular instruction-
tuning approaches involve concatenating the instruction with the input
directly and training a text-to-text model on these combined inputs. As the
instruction can be as long as, if not longer, than the input333As is the case
for Super-Natural Instructions, see Appendix A.1., this can greatly increase
the computation needed to process inputs compared to task-specific models.
##### In-Context Learning
Similar to instruction-based models, in-context learning (Brown et al., 2020),
where example instances are used in place of or in addition to instructions,
also requires extremely long and expensive-to-process inputs for every test
example, with Liu et al. (2022) showing that parameter-efficient finetuning
(PEFT) can be cheaper and more effective when dealing with many test examples.
In this work, we propose a halfway step between PEFT and instruction
concatenation, where we train a model to predict parameter-efficient modules
based on instructions, avoiding the few-shot training required by Liu et al.
(2022) while also avoiding repeatedly processing lengthy inputs.
##### Hypernetworks In NLP
Hypernetworks (Ha et al., 2017; Schmidhuber, 1992) in NLP have recently gained
popularity in multitask and multilingual setups due to their ability to softly
share parameters while avoiding negative interference through the use of
shared parameter generation module. Several approaches (Karimi Mahabadi et
al., 2021; Tay et al., 2021; He et al., 2022b) learn per-task embeddings along
with a shared hypernetwork to generate task-specific adapter or prefix
modules. This means making the model perform new tasks requires at least few-
shot learning to learn a task embedding. Recent work has explored using text-
conditioned hypernetworks for parameter-efficient multitasking (Ivison and
Peters, 2022) or improving out-of-domain generalisation (Volk et al., 2022),
removing the need to train task-specific embeddings. Hypernetwork-based
methods have also been highly successful in multilingual settings, where
generating language-specific models via shared hypernetworks often results in
improved performance across various tasks (Platanios et al., 2018; Baziotis et
al., 2022; Ustun et al., 2022, inter alia)
Our work primarily builds on Ye and Ren (2021), which explored generating
adapters from task descriptions. We expand their approach to larger models and
datasets and find that pretraining and a significantly different hypernetwork
architecture are important for achieving strong performance.
Our work is also similar to the concurrently developed Phang et al. (2022) and
Deb et al. (2022), both of which examine how well hypernetwork-based meta-
learning can improve model performance in zero- and few-shot settings. Deb et
al. (2022) examine hypernetworks and model-agnostic meta-learning for
instruction-finetuning and find that they can yield improved performance on
difficult unseen tasks in Super-Natural Instructions. However, they still
struggle to achieve overall good zero-shot performance and do not investigate
eliminating task descriptions from the model input itself. Phang et al. (2022)
find that training a hypernetwork to produce model adaptations provides an
initialisation better than pretraining for parameter-efficient adaptations and
that this initialisation improves with more few-shot examples provided to the
hypernetwork. They also explore eliminating the instruction from the
underlying model input but find this severely underperforms baseline
approaches. We have similar findings, but find that our novel hypernetwork
design and use of instruction fusion closes the gap with baseline approaches.
We also perform further analysis of the hypernetwork-based models and show
that when controlling for inference compute budgets, our hypernetwork-based
model still outperforms strong baselines.
## 3 HINT Model
Here, we introduce the main elements of our HINT model. The model has two core
parts: the hypernetwork, which takes in text instructions and outputs
parameter-efficient modules, and the underlying model, into which we insert
the generated parameter weights. The underlying model is simply an encoder-
decoder444We use encoder-decoder models as they generally outperform decoder-
only models for zero-shot generalisation – see Section 5.1. transformer model
(Vaswani et al., 2017) with additional parameter-efficient adaptations
inserted in, while the hypernetwork has a more complex architecture which we
describe below. Figure 1 provides a visual overview.
### 3.1 Hypernetwork
The first step in our model is to make use of a hypernetwork to convert an
instruction to parameter-efficient modules. Our hypernetwork consists of three
core elements: an encoder (or ‘hyperencoder’) to transform instruction and
few-shot text into continuous (contextual) representations, saving the encoded
instructions for instruction fusion during decoding, and a parameter generator
to then convert these embeddings into the parameter efficient modules.
##### HyperEncoder
To encode our text, we use a pretrained language model encoder. We initially
experimented with using different encoder configurations, and find that re-
using the encoder from the underlying model we wish to augment works well, and
tying the hypernetwork and underlying encoder model weights works best.
##### Instruction Fusion
We save the instruction representations produced by the hyperencoder and allow
the decoder of the underlying model access to them by concatenating them with
input examples during inference and training. This is inspired by the fusion-
in-decoder method used in open-domain QA (Izacard and Grave, 2021).
#### 3.1.1 Efficient Parameter Generators
##### Parameter Generators
Our generator design consists of two parts. First, we use a trainable set of
embeddings and perform multi-head cross-attention with the encoded instruction
and these embeddings. Each embedding represents a unique column or token in
each parameter-efficient module (e.g., prefix, adapter - see below) for each
layer. This allows us to effectively collect the information required for
different parameters in different embeddings via cross-attention with the
instruction:
embed
$\displaystyle=[\bm{\alpha}_{e^{1}_{1}},\bm{\alpha}_{e^{1}_{2}},...,\bm{\alpha}_{e^{2}_{1}},\bm{\alpha}_{e^{2}_{2}},...,\bm{\pi}_{e^{1}_{1}},...]$
$\displaystyle\text{embed}^{\prime}$ $\displaystyle=\text{Cross-
Attention}(\text{embed},\text{instr.})$
Where $\bm{\alpha}_{e^{1}_{1}}$ refers to an embedding we will use as the
first column of the first layer adapter weight, $\bm{\alpha}_{e^{1}_{2}}$ is
the second column, $\bm{\alpha}_{e^{2}_{1}}$ is the first column for the
second layer adapter, $\bm{\pi}_{e^{1}_{1}}$ is the first token of the first
layer prefix, etc. We then take the subset of the embedding representing all
columns/tokens for a particular model adaptation and pass it through a two-
layer MLP to generate parameters. We use a unique network for each adaptation
and share between layers (i.e. one network for prefixes for all layers, one
for all adapter weights for all layers, etc.).
$\displaystyle\text{Adapter}_{1}$
$\displaystyle=\text{reshape}[\text{MLP}_{a}(\bm{\alpha}^{\prime}_{e^{1}_{1}});\text{MLP}_{a}(\bm{\alpha}^{\prime}_{e^{1}_{2}});...]$
$\displaystyle\text{Prefix}_{1}$
$\displaystyle=\text{reshape}[\text{MLP}_{p}(\bm{\pi}^{\prime}_{e^{1}_{1}});\text{MLP}_{p}(\bm{\pi}^{\prime}_{e^{1}_{2}});...]$
Where Adapter1 and Prefix1 are the first layer adapter and prefix,
respectively.
##### Generated Parameters
We generate two types of parameter-efficient modules: adapters (Houlsby et
al., 2019) and prefixes (Li and Liang, 2021). Adapters (Houlsby et al., 2019)
are small bottleneck networks inserted into a transformer model. We follow He
et al. (2022a) in placing our adapters parallel to the feed-forward layer:
$\displaystyle\bm{x}=\text{FFN}(\bm{x})+f_{1}(\text{GELU}(f_{2}(\bm{x})))$ (1)
Where $f_{1},f_{2}$ are linear layers that project an input $\bm{x}$ to a
small bottleneck size $n_{a}$ and then back up to the hidden size of the model
respectively. Prefixes (Li and Liang, 2021) are short continuous sequences
concatenated with the key and values in the self- and cross-attention modules
in every layer of the underlying model.
##### Scaling Down Parameters
A naïve hypernetwork implementation may suffer from poor scaling with the size
of the parameter-efficient modules. Consider a case where we wish to convert a
single embedding of size $n_{e}$ to an adapter weight matrix of size
$n_{d}\times n_{a}$ (the model hidden dimension size by adapter bottleneck
size). Our hypernetwork generator will have $n_{d}*n_{e}*n_{a}$ parameters,
and so increasing the adapter bottleneck $n_{a}$ quickly becomes extremely
expensive, especially if $n_{d}$ is large - as is the case for large language
models. We address this by decomposing the adapter weight into columns and
assigning an embedding per column. Thus, our hypernetwork now has to convert a
sequence of embeddings with size $n_{a}\times n_{e}$ to an adapter weight of
size $n_{a}\times n_{d}$, meaning that the network only needs $n_{e}*n_{d}$
parameters. This means that the size of our parameter-efficient modules is
independent of the size of the hypernetwork, and we can effectively scale the
size of our adapters or number of prefixes without extreme parameter blowup.
Note that we set $n_{e}=n_{d}$ in our experiments for simplicity.
### 3.2 Underlying Model
Once our hypernetwork has produced a set of parameter-efficient modules, we
then insert these into our underlying network, and can then perform training
and inference as normal. The underlying model can be any pretrained encoder-
decoder model that works with our parameter-efficient modules. We make use of
T5 (Raffel et al., 2022) as our underlying model in our experiments.
### 3.3 Training and Inference
##### Hypernetwork Pretraining
Figure 2: Overview of our proposed pretraining scheme.
To help better generalization, we pretrain the hypernetwork on a large corpus
(C4; Raffel et al., 2022) before finetuning on multitask prompted datasets.
Given a single input string, we split our input string into random-length
chunks $a$, $b$, $c$, and feed $a$ to the hypernetwork, $b$ to the main
network, and predict $c$. This resembles the input in used in instruction
finetuning (as the instruction precedes the input in the default prompt used
for Super-Natural Instructions). We fully finetune all parameters during
pretraining.
##### Training HINT
HINT training looks similar to pretraining, except we replace $a$ with the
task instruction (and any few-shot examples), $b$ with the main input, and $c$
with the gold generation. We used mixed-task batches such that a unique
adaptor and prefix set is generated for every input in each batch.
This means, for every batch, we first generate a set of adapters, prefixes,
and encoded instructions from a batch of tasks using the hypernetwork. The
adapters and prefixes are placed within the underlying model to act as
parameter-efficient modules (i.e., insert them into the model), and the
encoded instruction is concatenated with encoded inputs during decoding. We
then perform a forward pass of the underlying model with the inputs associated
with each task in the batch and perform backpropagation using cross-entropy
loss as standard for text-to-text models. As we fully finetune all parameters,
the parameter generator will produce different weights for the same task
inputs after a gradient step, meaning that we have to rerun the hypernetwork
for every batch. This means that HINT requires more compute to train than a
baseline transformer - although it provides significant compute reductions
during inference, as we will see.
##### Inference
The inference process is similar to training, but we do not use mixed-batch
inputs: instead, we generate the parameters for one task, insert them into the
underlying model, and then process all test-time inputs for that task. This
prevents redundant processing of the instruction.
We also consider the cost of HINT models during inference. We consider a case
where we have to process $n$ samples from a single task. Assume each sample
has length $i$ and the task instruction has length $t$. We will ignore the
cost of processing (typically short) output sequences. Following prior work
(Kaplan et al., 2020; Liu et al., 2022), we use FLOPs as an estimate of the
amount of compute required to run particular models and estimate that
processing a token with an encoder-decoder model takes $N$ FLOPs to process a
single token, where $N$ is the total number of model parameters.
In this scenario, a standard instruction-trained model which concatenates
every input with the instruction (e.g., Tk-Instruct) uses $Nn(t+i)$ FLOPs to
process all examples. Meanwhile, HINT models process the task instruction only
once and so use roughly $N(t+ni)$ FLOPs555When reporting FLOPs, we use a more
detailed formula described in Appendix C.1 that takes into account extra
(albeit small) hypernetwork costs.. This makes clear that HINT models (a)
scale better with more same-task inference examples than input concatenation
approaches (increasing $n$), and (b) require relatively few extra FLOPs to
process long instructions (large $t$), allowing them to benefit from adding
more few-shot examples without incurring significant compute increases.
## 4 Experimental Details
We evaluate our approach on two popular instruction-based datasets: Super-
Natural Instructions (SNI) (Wang et al., 2022b) and the T0 split of P3 (Sanh
et al., 2022; Bach et al., 2022). We use t5x and seqio (Roberts et al., 2022)
to handle data preprocessing and model training. We use T5 v1.1 + LM
adaptation (Lester et al., 2021) as our base models, using the 3B size unless
otherwise stated. Unless otherwise stated, the hypernetwork generates prefixes
of length 30 and adapters with a bottleneck size of 512, matching the sizes
recommended by He et al. (2022a). We use the Adafactor optimizer (Shazeer and
Stern, 2018) with a constant learning rate of 0.001. Unless otherwise stated,
we report results from single runs.
##### Pretraining
We pretrain all models for 10,000 steps (7,000 for 11B size models) using C4
(Raffel et al., 2022) with a batch size of 1,024 samples and sequences of
length 512.
##### Super-Natural Instructions (SNI)
For SNI, we examine two settings: providing the hypernetwork with the task
definition and the underlying network with the instance input only (‘Def’),
and providing the hypernetwork with the task definition and two few-shot task
examples (‘Def + 2 Pos.’). To train, we finetune our pretrained HINT models
for 1,000 steps with a batch size of 1,024, with a maximum sequence length of
1,024 for both the underlying model and the hypernetwork input. We then
evaluate the final checkpoint on the test split of SNI, which is a set of 119
unseen tasks. We use v2.6 of Super-Natural Instructions.
Figure 3: (Left) SNI RougeL against FLOPs for varying model sizes using task
definitions with two positive examples; (Centre) SNI RougeL against FLOPs for
differing numbers of additional few-shot information; (Right) FLOPs against
instruction and task data length (in number of tokens). FLOPs calculations are
based on processing 100 examples from the same task during inference.
##### P3
For P3, we explore two settings: (a) ‘joint’, where we give the hypernetwork a
templated form of the prompt with instance information removed and give the
underlying model the full prompted input. (b) ‘split’, where we give the
hypernetwork the templated prompt without instance information and give the
underlying model only the instance information without the prompt. In both
cases, we fully finetune our model for 10,000 steps with a batch size of
2,048. We use a maximum input length of 1,024 for the underlying model and 512
for the hypernetwork. We train and evaluate on the same tasks and splits as T0
(Sanh et al., 2022).
##### Baselines
We primarily compare against T0 and Tk-Instruct, models fully-finetuned on P3
and SNI respectively with all task information concatenated with the input. We
replicate these models, matching the finetuning settings used for HINT models,
and find that our replications significantly outperform previously reported
results, making these baselines extremely strong. We note where results are
our replications or reported from prior work. We additionally compare against
‘X + PEFT’, the prior models with adapters and prefixes added in before
finetuning, HyperTune (Phang et al., 2022), a concurrent work that primarily
makes use of a pretrained hypernetwork but without instruction fusion, Hypter
(Ye and Ren, 2021), a prior hypernetwork-based model that does not use
pretraining, GPT-2 (Radford et al., 2019), a strong decoder-only model, which
we fully finetune, and OPT (Zhang et al., 2022), another strong decoder-only
model, which we also fully finetune.
## 5 Model Performance and Efficiency
Def Def + 2 Pos. Model Model Size RougeL Rel. FLOPs RougeL Rel. FLOPs Tk-
Instruct (our replication) 250 mil. 35.3 $\times$1.0 42.9 $\times$1.5 Tk-
Instruct (Wang et al., 2022b) 250 mil. - - 42.1 $\times$1.5 Tk-Instruct + PEFT
250 mil. 33.3 $\times$1.1 42.9 $\times$1.6 Hypter (our replication) 250 mil.
12.1 $\times$0.4 10.6 $\times$0.4 HINT (ours) 250 mil. 33.3 $\times$0.4 41.8
$\times$0.4 Tk-Instruct (our replication) 3B 48.9 $\times$12.0 56.6
$\times$17.9 Tk-Instruct (Wang et al., 2022b) 3B 45.0 $\times$12.0 54.3
$\times$17.9 Tk-Instruct + PEFT 3B 49.8 $\times$12.4 56.2 $\times$18.5 No-
Instruct 3B 12.4 $\times$3.9 - - GPT-2 XL 1.5B 38.2 $\times$4.1 45.3
$\times$4.2 Hypter (our replication) 3B 16.8 $\times$4.3 14.2 $\times$4.4
HyperTune (Phang et al., 2022) 3B 38.9 $\times$4.1 48.6 $\times$4.3 HINT
(ours) 3B 47.2 $\times$4.5 53.2 $\times$4.6 Tk-Instruct (our replication) 11B
53.6 $\times$44.0 60.5 $\times$65.7 Tk-Instruct (Wang et al., 2022b) 11B - -
62.0 $\times$65.7 Tk-Instruct + PEFT 11B 54.6 $\times$44.0 60.3 $\times$65.7
OPT-13B 13B 44.8 $\times$15.9 51.5 $\times$16.4 Hypter (our replication) 11B
15.5 $\times$15.3 13.4 $\times$15.7 HINT (ours) 11B 51.1 $\times$16.1 56.4
$\times$16.5
Table 1: Super-Natural Instructions RougeL and relative number of FLOPs used
when given task definition only (‘Def’), and task definition along with 2
labelled examples (‘Def + 2 Pos.’). Where noted, results are taken directly
from Phang et al. (2022) and Wang et al. (2022b). Relative FLOPs cost is
calculated relative to the base-size Tk-Instruct with task definition only. We
calculate the values using the number of FLOPs required to process 1 task with
100 examples for each model. Model size is given by the number of parameters.
### 5.1 Super-Natural Instructions
We report the performance and inference costs of HINT models and baselines in
Table 1 and Figure 3 and find that:
HINT models outperform baselines when FLOPs-matched. As seen in Figure 3
(left), when FLOPs-matched, HINT models outperform Tk-Instruct, a strong
baseline that fully concatenates the instruction with every input. This holds
for both ‘Def’ and ‘Def + 2 Pos.’ settings.
HINT models are up to 4$\times$ more efficient than similarly-sized baselines.
we find that HINT models use 2–4x fewer FLOPs than similarly-sized state-of-
the-art Tk-Instruct baselines (Table 1). While other hypernetwork-based models
are able to achieve similar compute savings, their performance is
significantly worse than HINT ($\geq 8$ points). HINT has similar cost to a
model trained without including instructions in the input (‘No-Instruct’),
while performing over 30 points better.
HINT models improve performance with few-shot examples, but do not cost more
FLOPs. When introducing additional few-shot data (‘Def + 2 Pos.’), HINT models
improve dramatically (5-8 points) but the compute used barely increases
(Figure 3, centre), as HINT models only need to encode the task data
(instruction and few-shot examples) once per task. In contrast, while Tk-
Instruct similarly improves with few-shot examples, the compute needed during
inference increases dramatically, usually costing around $1.5\times$ more.
Overall, we find that HINT models require much less compute to deal with
longer instruction and few-shot data inputs than Tk-Instruct (Figure 3,
right).
HINT models outperform a strong decoder-only baseline. HINT significantly
outperforms GPT-2 and OPT-13B, in line with prior work that shows encoder-
decoder models often significantly outperform even much larger decoder-only
equivalents (Wang et al., 2022a; Iyer et al., 2022). In particular, 11B-size
HINT outperforms OPT-13B by 5 points or more despite using a similar number of
FLOPs. This highlights the utility of improving efficiency for encoder-
decoder-based models. We also note that caching key/value attention pairs, the
simplest way to reduce inference costs with decoder-only models, scales worse
than HINT. The size of cached key/value pairs for GPT-2 is $\propto lds$,
where $l$ is the number of layers, $d$ is the size of the model hidden
dimension, and $s$ is the cached sequence length. In contrast, the size of the
saved PEFT parameters for HINT is $\propto sd+ld$, which scales better with
respect to sequence length (larger $s$) and model size (larger $d$, $l$)666We
provide details for these calculations in Appendix C.2..
Model Avg Rel. FLOPs T0-3B 54.9 $\times$1.0 T0-3B (our replication) 64.4
$\times$1.0 T0-3B + PEFT 65.5 $\times$1.0 No Prompt 57.5 $\times$0.8 Hypter
(Joint) 64.6 $\times$1.0 HINT (Joint) 65.4 $\times$1.1 Hypter (Split) 56.2
$\times$0.8 HINT (Split) 60.3 $\times$0.8
Table 2: Avg performance over T0 evaluation tasks after training on the T0 P3
train set. FLOPs are calculated assuming we are processing 100 examples of a
single task. The ‘Joint’ and ‘Split’ HINT variants refer to the two input
formats for P3 described in Section 4.
# Shots T0-3B (our repl.) HINT (split) HINT FLOPs Reduction 1 66.4 66.4
$\times$2.3 2 67.1 66.6 $\times$3.2 4 67.1 67.2 $\times$5.1 5 67.9 67.1
$\times$6.0
Table 3: P3 performance with differing numbers of few-shot examples using 3B
size models and the FLOPs reduction when using HINT instead of T0-3B for that
number of shots. In few-shot settings, HINT always remains wihin 1 point of
T0-3B despite the greatly reduced FLOPs cost.
### 5.2 P3
We report results on the T0 evaluation set in Table 2, with full results in
Appendix B. We find that:
Our T0-3B replication significantly outperforms the results reported by Sanh
et al. (2022). This matches prior suggestions that T0 is undertrained (Phang
et al., 2022; Wu et al., 2022). We provide further details in Appendix E.
HINT outperforms hypernetwork baselines. The HINT model consistently
outperforms Hypter, a prior hypernetwork-based approach, and learns to make
use of the P3 prompts as evidenced by its improved performance over a baseline
model trained without prompts (‘No Prompt’).
HINT remains cheaper than T0 for inference. HINT uses significantly less flops
than T0-3B, albeit with smaller savings compared to SNI, likely due to the
different style of prompts: P3 prompts tend to be shorter, and interleave task
inputs (e.g. ‘Does <sentence 1> entail <sentence 2>?’). Despite this, HINT
still provides reasonable FLOPs savings. We suggest that the performance of
HINT could be greatly improved by leveraging additional few-shot information,
further exploiting the efficiency of HINT models in encoding task data.
We investigate if HINT models can provide benefits even when the input and
instruction are concatenated through training and evaluating in the ‘joint’
setting of P3, and find that HINT performs similarly to T0-3B with additional
parameter-efficient modules, which suggests that the hypernetwork is unable to
improve on the baseline model through additional customisation, and so is
primarily useful as a mechanism for reducing inference costs and cheaply
incorporating few-shot data.
HINT performs similarly to the baseline in few-shot settings. In Table 3, we
show that HINT remains within 1 point performance of T0 in few-shot settings,
despite the large reductions in FLOPs cost, using up to 6$\times$ fewer FLOPs.
This makes HINT especially useful in few-shot scenarios.
## 6 Analysis
### 6.1 Pretraining
Model Pretraining SNI RougeL HINT None 44.0 HINT Ours 46.3 HINT CACLM 45.8
HINT - No Instr. Fus. None 27.4 HINT - No Instr. Fus. Ours 32.1 HINT - No
Instr. Fus. CACLM 30.4
Table 4: SNI performance for HINT models with and without instruction fusion
after 10,000 steps of the given pretraining scheme and 1,000 steps of
finetuning on SNI. CACLM is the pretraining scheme proposed by Phang et al.
(2022).
We compare using no pretraining, our pretraining scheme, and the pretraining
scheme proposed by Phang et al. (2022) (‘CACLM’) in Table 4. As the
pretraining scheme is primarily for improving the parameter generators, we
evaluate its effect both with and without using instruction fusion (‘HINT’ and
‘HINT - No Instr. Fus.’, respectively).
We find that: (a) using pretraining gives a large boost in performance for
hypernetwork-only models, showing that pretraining is essential to good
hypernetwork performance, and (b) using our pretraining scheme works best
overall. We hypothesise this reflects the fact that our scheme is closer to
the Super-Natural Instructions format than CACLM. Unlike Phang et al. (2022),
we found that further pretraining did not aid performance. This is likely due
to the fact that we tie the underlying model encoder and hypernetwork encoder
weights together, meaning that the model weights must balance between acting
as the hypernetwork and underlying model encoder.
### 6.2 Inference Speed
Figure 4: Time taken to process 100 examples with average SNI lengths with
varying numbers of shots on CPU and GPU for 3B HINT and baseline (i.e.,
vanilla T5) models.
While HINT provides significant FLOPs reductions compared to baselines, these
do not necessarily translate to real-world inference speedups. We examine this
by measuring the average speed of HINT to process 100 samples of the same
task, assuming the average input lengths given in Appendix A.1.
As seen in Figure 4, while baseline decoding remains faster for small input
lengths on GPU777This is likely due to the small additional overhead of
running the HyperEncoder, which must be run before the rest of the model., it
lags compared to HINT for longer sequences. In fact, HINT’s inference latency
increases at a much slower rate compared to the baseline as the input size
increases (with the number of shots), highlighting that HINT is especially
effective in few-shot scenarios and scenarios with lengthy inputs.
### 6.3 Architecture Ablations
Model SNI RougeL Adapters + Prefixes 32.1 Adapters ($a=512$) 30.1 Prefixes
($l=30$) 12.1 Prefixes ($l=512$) 15.1 LoRA ($r=128$) 12.1 LoRA ($r=512$) 12.6
Table 5: SNI performance of different parameter-efficient modules in a HINT
model without instruction fusion. $a$, $l$, $r$ are the bottleneck size,
number of tokens, and rank used for each experiment respectively.
We experiment with a series of ablations to determine the best architecture
for HINT, and find that:
##### Adapters and prefixes work best together.
We consider alternatives to using adapters and prefixes together: using
adapters alone, using prefixes alone, and using LoRA (Hu et al., 2022) instead
of either. In order to isolate the effect of these choices, we test without
using instruction fusion. We find that adapters and prefixes provide the best
overall performance, with prefixes-only and LoRA-only performance
substantially worse, even when increasing the number of parameters generated.
This suggests that our hypernetwork approach is more adept at generating
certain types of PEFT modules.
Model SNI RougeL HINT 47.2 \+ Decoder 42.6 \- Instr. Fus. 32.1 \- PEFT Gen.
40.9
Table 6: HINT model ablations. All models are pretrained for 10,000 steps,
except for - PEFT Gen., which contains no new parameters and requires no
pretraining.
##### PEFT and instruction fusion are complementary.
We find that using just the generated parameter-efficient modules or the
encoded instruction alone (‘-Instr. Fus.’ and ‘-PEFT Gen.’ in Table 6) perform
significantly worse than using both methods together, suggesting that these
methods provide complementary improvements.
##### Cross-Attention Layer wins over Full Decoder.
We compare using a full T5 decoder (with self-attention removed) as the
hypernetwork weight generator as in Phang et al. (2022) with our approach, and
find that our single multi-head cross-attention layer performs better at a
much cheaper cost than using the full decoder (‘+ Decoder’ in Table 6).
## 7 Conclusion
We introduce Hypernetworks for INstruction Tuning (HINT) models and show that
they consistently outperform strong full-input baselines when controlling for
inference compute. This is primarily due to the fact that HINT models process
their task instructions once per task, while current state-of-the-art models
re-encode instructions with every task input. We show that the success of HINT
models relies on a pretrained hypernetwork, which converts task instructions
into parameter-efficient modules and an encoded instruction, both of which we
insert into the underlying model.
Future work could investigate how HINT aids in few-shot settings, further
building on HINT’s strong few-shot efficiency and taking advantage of the
improved initialisation provided by hypernetworks (Phang et al., 2022).
Overall, HINT models combine the benefits of parameter-efficient learning with
the benefits of instruction-based learning, allowing one to easily turn
pretrained language models into efficient, task-customised models.
## Limitations
While promising, HINT comes with several drawbacks related to its ease of use.
First, HINT takes advantage of the fact that (a) instructions are often long,
and (b) often we want to perform inference over a larger ($>100$) amount of
examples with the same instruction. If either of these items are not true in a
setup, then HINT is unlikely to provide a large benefit over simply including
the instruction with the input text. This can be seen in the smaller compute
savings provided by HINT for P3 in Table 2. Second, while HINT is compute-
efficient at inference time, it is far more costly to train, as it effectively
requires running the underlying model together with the hypernetwork for every
batch. This means that while HINT may be useful for practitioners with limited
compute budgets, it may be difficult to train HINT models with the same
limited budget. Finally, we train and test on English data only, and do not
explore the generalisation of our approach to multilingual setups. Considering
the success of hypernetworks in multilingual settings (Platanios et al., 2018;
Baziotis et al., 2022; Ustun et al., 2022), we believe this is a promising
direction for future research. As such, while promising, HINT is limited by
certain assumptions made about the length and format of instruction-augmented
data, and we hope further improvements of the method work towards loosening
these assumptions.
## Ethics Statement
We believe that the impact of our work is largely beneficial, examining a
novel method to make instruction-based models cheaper to use. This may aid in
reducing the carbon footprint of large language models running in inference
(Schwartz et al., 2019) and in making these models more accessible to people
with limited compute budgets. However, we also note that our approach requires
unsupervised pretraining on a large corpus, making it difficult to document
exactly the data it has seen during training and making it likely to reflect
problematic or even dangerous biases within the corpus (Bender et al., 2021).
We believe that future research could investigate reducing the need for
hypernetwork pretraining and further investigate the behaviour of
hypernetwork-augmented language models.
## Acknowledgements
Research supported with Cloud TPUs from Google’s TPU Research Cloud (TRC). We
thank AllenNLP members and Jonas Pfeiffer for encouraging and useful
coversations in earlier stages of this project.
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## Appendix A Dataset Details
### A.1 Input Lengths
When calculating FLOPs estimates, we use the median sequence length of the
inputs and outputs to calculate inference costs. We compute the median over
the train split of Super-Natural Instructions and over 10,000 random samples
from the T0 train split of P3. We calculate the medians for each format
separately, rather than adding the instance and instruction-only values
together (hence the mismatch in values). We provide the calculated values in
Table 7. We find that P3 inputs mostly consist of the instance, with prompt
templates consisting of relatively few tokens, while SNI inputs consist mostly
of instructions. This explains why HINT models are much cheaper than Tk-
Instruct models, but not that much cheaper than T0 models, as HINT models
reduce FLOPs by avoiding reprocessing the instruction with every input.
Median # Tokens Text Sequence SNI P3 Instance only 44 81 Instruction only 69
24 Instruction + Instance 133 103 Instruction + 2 positives 197 - Instruction
+ 2 pos. + instance 199 - Output 1 6
Table 7: Median sequence length, given in number of T5 tokens, for Super-
Natural Instructions and P3.
### A.2 Split Sizes
We report the sizes of splits here. For Super-Natural Instructions, we use the
default setting from Wang et al. (2022b) where 100 examples are provided for
each task in train and test splits. We also note that we follow the sampling
procedure used by Sanh et al. (2022), where we “treat any dataset with over
500,000 examples as having 500,000 / num templates examples” during training.
Taking this sampling into account results in the much smaller dataset size
seen in Table 8. We refer readers to Sanh et al. (2022) for more details on
P3.
Dataset Train Test Super-Natural Instructions 75,417 11,810 P3 90,897,454
2,940,068 P3 (adjusted for sampling) 17,277,532 2,940,068
Table 8: Number of samples in given splits for each dataset.
## Appendix B Full P3 Results
We report the full results of models on the P3 dataset from Table 2 in Table
9.
Model Avg Rel. FLOPs ANLI HellaSwag StoryCloze CB COPA RTE WiC WSC WinoGrande
T0-3B 54.9 $\times$1.0 33.4 27.3 84.0 45.4 72.8 64.6 50.6 64.9 50.9 T0-3B (our
replication) 64.4 $\times$1.0 41.7 30.1 96.9 72.7 89.1 81.2 51.7 57.2 59.2
T0-3B + PEFT 65.5 $\times$1.0 41.5 30.1 96.6 76.9 92.2 82.1 54.2 56.6 59.2 No
Prompt 57.5 $\times$0.8 34.4 27.7 88.8 69.4 66.3 56.5 52.5 61.3 60.6 Hypter
(Joint) 64.6 $\times$1.0 41.1 29.4 96.7 76.3 87.4 79.6 52.1 58.3 60.9 HINT
(Joint) 65.4 $\times$1.1 41.6 30.3 96.6 76.0 88.8 84.2 51.4 59.5 60.1 Hypter
(Split) 56.2 $\times$0.8 34.2 28.1 86.8 58.0 67.3 65.0 50.5 60.0 55.7 HINT
(Split) 60.3 $\times$0.8 37.0 29.1 85.6 67.6 71.0 77.2 51.0 64.2 60.0
Table 9: Results on T0 evaluation tasks after training on the T0 P3 train set.
We report averaged accuracy across prompts for each task, and the overall
average performance (‘avg’). FLOPs are calculated assuming we are processing
100 examples of a single task. The ‘Joint’ and ‘Split’ HINT variants refer to
the two input formats for P3 described in Section 4.
## Appendix C Model Compute Calculations
We provide a more thorough description of the compute and memory costs
associated with various models we discuss here.
### C.1 Compute Costs
We will let $i$ be the sample length, $t$ be the task instruction length, $o$
be the output sequence length, $n$ be the number of same-task samples we wish
to process, and $N$ be the number of parameters in the model. We will assume
we are only processing examples from the same task.
##### Tk-Instruct
As Tk-Instruct concatenates instruction and sample together as input, it uses
roughly $Nn(i+t+o)$ FLOPs.
##### HINT
The cost of the HINT model is more complicated. Let $N^{\prime}$ be the cost
of the hypernetwork generator, and $A$ be the cost of the parameter-efficient
modules inserted into the underlying model. The cost of running the
hypernetwork is $t(N+N^{\prime})$ (since the hypernetwork encoder is the same
size as the underlying model). The cost of then running the underlying model
with parameter-efficient modules is $n(N+A)(i+o)$. We sum these two terms to
get the total cost of HINT: $t(N+N^{\prime})+n(N+A)(i+o)$. We do not consider
the additional cost of inserting the instruction in the decoder as this only
affects the few (usually 1-2) output tokens and the decoder cross-attention
only, and so is negligible. We can simplify the HINT compute cost down by
observing that in most cases $N>>A$ and $N>>N^{\prime}$, resulting in the cost
of HINT being roughly $tN+nN(i+o)$. This simpler formulation highlights the
main benefit of HINT: the instruction no longer is processed with every
sample, and so compute cost is $\propto t+n$ as opposed to $\propto tn$.
### C.2 Memory Costs
Here, we will let $l$ be the number of layers, $d$ the model hidden dimension,
$h$ the number of heads, $k$ the size of the keys/values, and $s$ be the
length of the sequence we want to save. We ignore bias terms for simplicity.
##### Decoder-only Models
If we want to cache the key/value pairs for a given sequence, we will store
$2lhks$ values - a key and value for every head in every layer, for each item
in the sequence. We note that typically $kh=d$ in models, and so in the main
text we simplify this to $2lds\propto lds$.
##### HINT
In the default HINT setup, we save three elements: the processed instruction
sequence, which contains $ds$ values (one vector per token); the adapter
weights, $2*512ld$ values (one adapter comprising of two weight matrices per
layer, where each weight matrix has size $512\times d$); the prefix values,
$2*30lhk$ values (a 30-length prefix and key per layer per head) ). This sums
to give a total memory cost of $ds+1024ld+60lhk$. Note that in the default
HINT settings, we use prefixes of length 30 and adapters with bottleneck size
512, but these settings could be adjusted to reduce memory costs. Applying the
simplification $kh=d$, we get that the HINT memory cost is $\propto ds+ld$.
## Appendix D GPT-2 Instruction Finetuning
When finetuning GPT-2 for Table 1, we trained for [3, 5, 10] epochs with a
batch size of 32. We use AdamW (Loshchilov and Hutter, 2019) and swept
learning rates of [$1\times 10^{-5}$, $2\times 10^{-5}$, $5\times 10^{-5}$],
using a linear warmup and decay schedule with 1,000 steps of warmup. We report
the highest overall results in Table 1. Following Iyer et al. (2022), we
minimise the loss only over the target tokens (with EOS token added after the
target answer), not the inputs, since these are always provided during test
time. Note that we calculate the FLOPs used by GPT-2 during inference based on
the estimates provided by Kaplan et al. (2020) that GPT-series models use $2N$
FLOPs per token, where $N$ is the number of parameters in the model.
## Appendix E T0 Replication
During initial experiments, we replicated the T0 training in the t5x
framework, using the same training set and mixing proportions as Sanh et al.
(2022). We found that our replications performed significantly better than the
reported T0 performance when trained for longer. We train 3B and 11B size
models on the T0 training mixture for 20,000 steps using a batch size of 2048,
a maximum input sequence length of 1024, a maximum output sequence length of
256, and the Adafactor optimizer with a constant learning rate of 0.001. We
start from the T5 v1.1 + LM adaptation checkpoints and fully finetune the
model. As seen in Table 10, our replications significantly outperform both T0
models, suggesting that T0 was undertrained. We also compare the variances in
prompt performance in Figure 5.
Model ANLI HellaSwag StoryCloze CB COPA RTE WiC WSC WinoGrande AVG T0-3B 33.4
27.2 84.0 45.4 75.9 64.6 50.7 65.1 51.0 55.2 T0-3B (ours) 41.7 30.1 96.9 72.7
89.1 81.2 51.7 57.2 59.2 64.4 T0-11B 41.0 33.6 92.4 70.1 91.5 81.0 56.1 61.1
59.9 65.2 T0-11B (ours) 46.8 34.1 98.2 81.2 96.6 84.0 52.1 62.6 64.8 68.9
Table 10: T0 evaluation task accuracy, comparing results using models from
Sanh et al. (2022) and our own replications. We report accuracy averaged
across prompts for the same dataset. For ANLI, we first average performance
across prompts for ANLI R1/2/3 separately, and then average the three results.
Figure 5: T0 evaluation set accuracy across tasks. Each dot represents the
performance from a different prompt. ‘T03Bp’ and ‘T011Bp’ refer to our T0
replications, while ‘T03B’ and ‘T011B’ are the models released by Sanh et al.
(2022).
|
# A Formal Analysis of Iterated TDD
Hemil Ruparel Pune<EMAIL_ADDRESS>and Nabarun Mondal Hyderabad
<EMAIL_ADDRESS>
###### Abstract.
In this paper we formally analyze the software methodology called (iterated)
Test Driven Development (TDD). We formally define Specification, Software,
Testing, Equivalence Partitions, Coupling, to argue about the nature of the
software development in terms of TDD. We formalize Iterative TDD and find a
context in which iterated TDD “provably produce” “provably correct code” from
“specifications” while being stable in terms of iterated code churns. We
demonstrate that outside this context iterated TDD will exhibit chaotic
behavior, implying unpredictable messy amount of code churn. We argue that the
research finding of “ineffective” iterated TDD found by earlier researches are
due to missing this context, while the findings of “effective” iterated TDD is
due to accidentally falling into the context or simply placebo.
###### Key words and phrases:
Software ; Testing ; Test Driven Development; Formal Specification;
Equivalence Class Partitioning; Dynamical Systems ; Chaotic Dynamics; System
Stability; Lyapunov Exponent
###### 2010 Mathematics Subject Classification:
Primary 68N30 ; Secondary 37B99, 68Q99, 93C99, 93D99, 68Q30
Hemil Ruparel : Dedicated to my parents and family without their presence we
are nothing.
Nabarun Mondal : Dedicated to my late professor Dr. Prashanta Kumar Nandi.
Dedicated to my parents.
In Memory of : Dhrubajyoti Ghosh. Dear Dhru, rest in peace.
## 1\. Canonical Definition of Iterated TDD
### 1.1. Canon Definition
We define TDD [1] as it is written in the canon article taken as the
“Definition of TDD” [2] :
1. (1)
Write a list of the test scenarios you want to cover
2. (2)
Turn exactly one item on the list into an actual, concrete, runnable test
3. (3)
Change the code to make the test (& all previous tests) pass (adding items to
the list as you discover them)
4. (4)
Optionally refactor to improve the implementation design
5. (5)
Until the list is empty, go back to [2].
### 1.2. Narrative
The previous definition does not talk about any formal goals for iterative
TDD. Hence, we formalize the objective of TDD as follows:
To ensure that we end up having a formally verifiable software in each step
and in the end when all the “scenarios” are exhausted. Another “optional”
objective is given as to “Improve the implementation design”.
Note that it is not defined anywhere between one implementation to other what
can be “improvement”.
This is not a good starting point to formally analyze the methodology, as
success metrics are not possible to be created on top of it. It is very
imprecise, and open to interpretations.
In this paper we propose a formal methodology and provably demonstrate how
“provably correct software” can emerge with clear metric of “amount of code
churn” was done to attain it over the iterations - albeit in a very narrow
context.
This practice we shall call formal Iterated TDD. We are calling the “canon”
practice followed in the industry as “Iterated TDD” for reasons which would be
apparent shortly.
## 2\. Definitions
We would need some definitions to formalize the (Iterated) TDD pseudo
algorithm.
### 2.1. Specification of Functions via Point Pairs
Any function, computable or not, can be imagined to be pairs ( potentially
$\aleph_{1}$ [3] ) of input and output points in some abstract space. It makes
sense to describe functions by defining their specific outputs at specific
points or a large set of equivalent points. This list of pairs we shall call
point specification or “specification” for brevity for the function it is
trying to describe.
#### 2.1.1. Consistency
A function is formally defined as a relation where it is impossible to have
“re-mapping” e.g. same input point mapped to two different output points. The
set of pair points must not have such spurious points, this we shall call
consistency criterion. This will become a key point in case of software
specification.
#### 2.1.2. Completeness
For functions which are well behaved this makes some sense. But even for well
behaved functions this is not a good enough approximation.
Take a nice function like $f(x)=x$ , identify function, but one can not define
this function by keeping on adding pairs of specification values.
A much more interesting function like $f(x)=sin(x)$ is much harder to
describe, although we can always define them pointwise, and that would ensure
the resulting “sampling” looks much and much like the target function, one
must understand infinite pairs would be required to specify $sin(x)$. Even
with $\aleph_{0}$ points specified, there would be set of infinite family of
functions who are not $sin(x)$ but just gives off the exact same value at all
those specific points. This has a name, called pointwise convergence [4].
Outside those fixed set of points the family of functions can take arbitrary
values, and thus specification via point pairs arguably pose a problem.
Luckily, for software we can do much better, which is the topic for the next
section.
### 2.2. Software
A software is defined to be a Computable Function - mapping abstract vector
space of input to the output vector space. The notion of using vectors is due
to all real software works with many inputs and hence the state space is
multidimensional which is the exact same space as output.
$S:\hat{I}\to\hat{O}$
where $\hat{I}:=<x_{i}>$ is the input vector while $\hat{O}:=<y_{j}>$ is the
output vector. These vectors are defined not in physics sense, but pure
mathematical sense. The only change between the pointwise defined function vs
specified software is about being “Computable” [5].
### 2.3. Software Test
A “Software” test is defined as a higher order function [6] :
$T:t<\hat{I}_{t},S_{t},\hat{O}_{e}>\to(S_{t}(\hat{I}_{t}):=\hat{O}_{t})=\hat{O}_{e}$
In plain English, a test is comprise of Input vector $\hat{I}_{t}$, the
software under test $S_{t}$, and the expected output vector $\hat{O}_{t}$ , it
runs the $S_{t}$ with the input, and checks whether or not the expected output
$\hat{O}_{t}$ matches against the actual output of the system
$S_{t}(\hat{I}_{t}):=\hat{O}_{t}$, and it simply checks whether or not
$\hat{O}_{t}=\hat{O}_{e}$ , hence the range of the test is Boolean.
A software test, then contains a single point specification for the desired
Software, this is the test vector [7].
A software test does not need to be computable in general. Unfortunately, any
automated test, by definition needs to be computable. This also pose a problem
for testing in general. Example of a test that is not computable [8] [9] can
be : a human reporting software has hung or went into infinite loop. This is
impossible to do algorithmically, unless we bound the time. This sort of
scenarios comes under Oracles in computation [10].
### 2.4. Code : Control Flow Graph, Branches
Software is written essentially using arithmetic logic and then conditional
jump - this being the very definition of Turing Complete languages [11]. This
structure with conditional jump ensures that the different inputs takes
different code paths. A code path is a path ( even having cycle ) in the
control flow graph [12] (CFG) of the software which starts at the top layer of
the directed graph that is the code and ends in the output or bottom later.
Formally we can always create a single input node and output node in any
control flow graph.
Treating multiple iterations of the same cycle as a single cycle, we can
evidently say given the nodes of the graph is finite, there would be finite
(but incredibly high) number of flow paths in the graph.
### 2.5. Partitions : Equivalence Classes
At this point we introduce the notion of equivalence class of input vectors to
software. If two inputs $\hat{I}_{x}$ and $\hat{I}_{y}$ takes the same path
$P$ in the control flow graph, then they are equivalent.
This has immense implication in testing and finding tests. Because this
induces an equivalence partitioning on the input space itself, because all
$\hat{I}_{x}$ in the same equivalence class can be treated as exactly
equivalent, because all of them would follow the exact same code path [13] in
the control flow graph. There is another related concept called boundary value
analysis (BVA) [14], but we would not go there, because that is not going to
alter the subsequent analysis in any significant way.
This effectively means by isolating all equivalence partitions and choosing
one input member from each of them we can test the system the most optimal way
- by restricting the number of “Software Test”s, as well as providing a full
“coverage” in terms of specification.
For example, if there are $A,B,C,D$ equivalent classes [15], then choosing
$\hat{I}_{A}\in A$, only one would test the code path for $A$, similarly for
the rest. So instead of infinite inputs, only 4 inputs would suffice. Notice
that these are the most optimal set of inputs, the bare minimum to ensure that
the system works in a provably correct manner.
This formally brings the problem to finding the exhaustive set of equivalent
classes ( let’s call it $\mathbb{E}$ ) that completely describes one
implementation of a “Software” system.
That is impossible without the implementation. It is wrong to perceive that
this technique is driven by specification alone. EQCP is a gray box testing
[16] technique as it requires assuming some implementation details[17].
What would be an upper bound of the number of such equivalent classes ? This
depends on the number of the conditional jumps. It is easy to prove that if
there are $B$ branches, then the bound for the number of the equivalence class
is $O(2^{B})$ where $O(.)$ is “Big-Oh” one of the Bachmann Landau asymptotic
notations [18], This also would be very important for a pragmatic discussion
later.
The Equivalent classes would be called EQCP from now on because they partition
the input set into Equivalent Classes. There would be many EQCP for individual
“features” in “Software”.
### 2.6. Coupling in Software
At this point we introduce the phenomenon of coupling [19] between Equivalent
Classes, when seen with respect to code implementation.
Given individual EQCP are depicting unique paths in the control flow graph
(CFG), then coupling said to exists between EQCPs $E_{x}$ with path $P_{x}$
and $E_{y}$ with path $P_{y}$ if and only if $P_{x}\cap P_{y}\neq\varnothing$.
That is, if paths [13] $P_{x},P_{y}$ has some common nodes, then $E_{x},E_{y}$
are coupled. In fact we can define the amount of coupling using similarity
measures now, most easy one would be a Jaccard distance [20] like measure:
$C(E_{x},E_{y})=\frac{|P_{x}\cap P_{y}|}{|P_{x}\cup P_{y}|}$ (2.1)
This essentially says - “Measure of the coupling between two equivalent
classes is the amount of code shared between them relative to all the unique
code path they have together”. We need to understand that even code shared for
good reason, like applying DRY [21] and not doing it even methodically also
would create coupling via this definition. Any shared function between two
EQCP would mean coupling exists. As we shall see Coupling becomes a key
phenomenon while analyzing the stability of software under Iterative TDD.
### 2.7. Test Driven Development as Equivalent Class Specification
We can now formally define a software system specification in a finite, and
provably correct way.
If we can just specify the equivalence classes, then we can just fix the
software output at those specification points and the resulting tests
precisely, and correctly defines the software behavior. This must be taken as
the formal definition of (non iterative, formal) TDD with absolute minimal
test inputs:
> Given an abstract (not written) Software $S_{a}$, let’s imagine the
> equivalence classes $E_{x}$ such that $E_{x},E_{y}$ are independent and
> specify the input and output expected from each equivalence classes. Now,
> ensure all of these tests pass by writing the implementation.
This system is provably complete and correct, by construction. Every test just
ensures all individual EQCP behavior is passed via construction. Given that
was the entire specification, this means the system passes all criterion for
the specification, and thus becomes provably correct.
The input output specifications can be immediately translated into tests, and
that gives the formal provable meaning to TDD. Any random tests on features
won’t do, it have to be (at bare minimum) spanning the entire EQCP (the formal
specification points).
This is the real superpower of TDD, formal verification baked into
development. Although, truth to be told, this way of constructing software has
been known for many decades.
And this is why the canonical TDD was called out as “iterated TDD” because
this formal non iterative TDD model does not include change of specification,
thereby does not follow any iteration and thus does not consider code churn
thereof. This formal non iterated model is one single shot transformation of
bunch of specifications points into code via transforming them into EQCP.
### 2.8. Practical Correctness of TDD
The correctness of TDD for a practical application hinges on the following :
1. (1)
Is the specification complete enough ( to take care of all the equivalent
classes )?
2. (2)
Is the specification non contradictory ?
That it is impossible to get (1,2) done together follows from Godel’s
Incompleteness theorems [22], but that is applicable to any specification, not
only Software. Thus this argument should not be admissible as failure of TDD
in itself.
Now we ignore the notion of contradiction and focus on completeness and
stability when one tests gets added at one time ( iterated or incrementally
changed specification TDD ).
### 2.9. Practical Completeness of TDD Spec
The business specification should be such that the formal specification of all
possible Equivalence classes must be drawn from it. As it is bounded by
$O(2^{B})$ \- this itself is not remotely possible. To understand how this
bound works, a simple program unix cat has more than 60 branches [23]. The
equivalent class specification of this program is bounded by $2^{60}$ and the
total stars in the universe are estimated to be $2\times 10^{24}$ for
comparison.
But this huge numbers does not disprove the crux of TDD, it only points to the
fact that formal EQCP is a practical challenge and to be handled
pragmatically, probably via reducing the specification scope further and
further.
## 3\. Analysis of Iterated TDD
### 3.1. Development under TDD
Note that the methodology does not specify how to implement the paths of each
equivalent classes in the code. Hence evidently there is no way it can ever
improve on the “non correct aspect of quality” of software, one of them would
be to lower coupling. In fact if not controlled this would bring in way more
coupling than it was required due to application of other principles like DRY.
Because there are infinite way to conform to the “point wise convergence” but
then the methodology does not specify any family of approach to do so. These
are some of the key open problems of the methodology as it formally stands as
of now.
A trivial non coupled way to construct code would be such that no equivalence
class share any code path. This would solve the coupling problem, but code
would be massively bloated. Any other way would reduce the code but ensure the
classes would be coupled to some extent.
This is a choice. We want to simultaneously minimize two metrics:
$C_{S}=\sum\limits_{x\neq y}C(E_{x},E_{y})$ (3.1)
along with:
$S_{S}=min_{n}\\{K(S_{n})\\}$ (3.2)
where $S_{S}$ stands for “source code size” where $K(S_{n})$ defines the
optimal code size of the System $S$ at $n$’th implementation trial. This is a
very hard problem as Chaitin Solomonoff Kolmogorov Complexity (CSK) [24] is
non Computable [5]. We do not even know if such a problem can be solved in
formal setting. We posit it as an open problem in Software Development.
In lieu of that we continue in our analysis where we imagine a bit of
necessary code coupling and try to reduce the code churn in terms of EQCPs.
This coupling would have implication in iterated TDD, and we show a provable
methodology that can reduce code churn in the later sections.
### 3.2. Iterated TDD
An Iterated ( incremental) TDD is when we add more specification to the mix of
already existing ones one step at a time under practical setting. This
incrementally added test based iterative TDD methodology is what we discuss in
the next sections as this is the one which proponents of TDD talks about. We
note down it is different from the formal TDD we have established before -
canon TDD is an iterated version of the formal TDD with specifications being
added per iteration.
### 3.3. Stability of EQCP under Iterated TDD
Suppose, there is already an existing system in place with tests done the
right way - following the EQCP method discussed earlier, e.g. following TDD.
Is it possible to add more specification w/o rewriting existing equivalent
classes in a stable manner?
The sort of stability we are looking for is called BIBO Bounded input Bounded
Output stability [25], that is, for a small change in specification, not much
change would happen in the EQCP space.
This is the iterative TDD, applying this again and again. The answer to this
is key to the prospect of iterative TDD.
Formally, Software $S_{r}$, has the equivalent classes
$E_{x}\in\mathbb{E}_{r}$ , and now more specification augmentation is
happening. The following questions need to be asked:
1. (1)
How many of the existing EQCP will not be effected by this?
2. (2)
How many new EQCP needs to be added?
3. (3)
How many EQCP needs to be removed?
As one can surmise, this is the transformation step of a fixed point iteration
on the abstract space of the EQCP. We shall get back to it slightly later.
### 3.4. Additional Branching
The answer to the question [2] is in isolation if there would be $K$ branches
to implement the delta specification \- new feature then, the isolated
equivalent classes would be in $O(2^{K})$ , thus, the minimum new classes
needed would be bounded by this value.
At most it can impact every equivalence class and at least it adds $O(2^{K})$
classes and hence tests. So, at the best case scenario, the total branches
would become $O(2^{B}+2^{K})=O(2^{B})$ given $B>>K$. The complexity increases,
but not drastically, unless $B=O(K)$.
### 3.5. Impact of Coupling
What happens when there is coupling? Instead of adding the terms, now because
of dependency, the terms gets multiplied. Thus, with coupling the resulting
complexity becomes $O(2^{B}\times 2^{K})=O(2^{B+K})$ . The delta change
results in exponential growth even if $B\neq O(K)$.
This is a problem.
If the implementation of those equivalent class was such a way that there was
minimal coupling, then less classes would be impacted via this step in the
iteration. But this is not a principle of TDD in the first place in any form
in any practical application of software development. In fact software
principle like DRY and modular programming would mandate code sharing, and
hence there would always be some coupling.
### 3.6. Iterated TDD as a Dynamical System
At this point we can formally represent iterated TDD as a dynamical system
[26].
As discussed, this EQCP merging culminates into a lot of those equivalence
classes being thrown out, new classes being created - a fixed point iteration
on the abstract space of the EQCP itself, which we can now formally define as
follows:
$\mathbb{E}_{n+1}=\tau(\mathbb{E}_{n},\delta_{n})$ (3.3)
Where at step $n$, $\mathbb{E}_{n}$ is the current set of EQCPs, while based
on new specification ( $\delta_{n}$ ) and the $\mathbb{E}_{n}$ TDD system
$\tau$ produces new set of EQCPs ( $\mathbb{E}_{n+1}$ ) for the next step
$n+1$.
This is the fixed point iteration of incremental software development from
point pair specification or incremental, iterated TDD.
It is obvious that the first ever specification was done with empty equivalent
classes ( $\mathbb{E}_{0}=\varnothing$ ) and initial specification of
$\delta_{0}$:
$\mathbb{E}_{1}=\tau(\varnothing,\delta_{0})$
This is how formally iterated or incremental TDD looks like. These equations
now depicts a dynamical, complex system with am initial boundary value or
starting condition.
### 3.7. Stability Space
While EQCP space is nice to visualize what is happening for real in terms of
Software Specification and Test cases, it is not descriptive enough to
translate into numbers so that we can track the trajectory of the Dynamical
System.
How much change in the EQCP space is happening on each iteration of iterated
TDD? It is impossible to comprehend that in the EQCP space.
For gaining this insight we would need a metric, that would define how stable
the system is over the iterations in terms of retaining past EQCPs - how much
code remained same between iterations.
We define the stability metric as follows :
$\Sigma_{n+1}=1-\frac{|\mathbb{E}_{n}\cap\mathbb{E}_{n+1}|}{|\mathbb{E}_{n}\cup\mathbb{E}_{n+1}|}\;;\;\Sigma_{n}\in\mathbb{Q}\cap(0,1)$
(3.4)
The stability metric $\Sigma$ also depicts a metric space [27] with distance
between two stability points $a,b\in\Sigma$ as defined to be : $d(a,b)=|a-b|$.
#### 3.7.1. Stable Point : 0
Observe the following, if we ensure that no EQCP has any shared code, then the
only way to make change is to simply add new code, and thus
$\mathbb{E}_{n}\subset\mathbb{E}_{n+1}$, and that gives minimum value of
$\Sigma$ if and only if $|\mathbb{E}_{n+1}\setminus\mathbb{E}_{n}|$ can be
minimized .
A value of $\Sigma$ close to 0 shows the system has been very stable between
last to the current iteration. This is when “very loose” coupling ensured that
we can create branches which do not interact with existing branches that much.
We present order of magnitude estimates for “highly stable” uncoupled
${}^{U}\Sigma$ value as follows:
$^{U}\Sigma_{n+1}\approx 1-\frac{|\mathbb{E}_{n}|}{|\mathbb{E}_{n+1}|}\approx
1-\frac{O(2^{B})}{O(2^{B}+2^{K})}\approx 1-\frac{1}{1+2^{K-B}}\approx
0\;;\;B>>K$ (3.5)
We note that it is impossible to reach value 0 under any circumstances other
than when $\mathbb{E}_{n}=\mathbb{E}_{n+1}$ which means, the specification
$\delta_{n}$ did not change anything in EQCP space, e.g. a complete dud or
spurious specification.
Importantly, there can be cases where even without coupling, as demonstrated
by : $|\mathbb{E}_{n}|<<|\mathbb{E}_{n+1}|$ , then even though
$\mathbb{E}_{n}\subset\mathbb{E}_{n+1}$, the stability would be going for a
toss - this is driven by having $B=O(K)$.
#### 3.7.2. Unstable Point : 1
Now the other side of the coin is when
$\mathbb{E}_{n}\cap\mathbb{E}_{n+1}\approx\varnothing$, in this case the value
of $\Sigma$ goes to 1.
A value of $\Sigma$ close to 1 shows the system has been very unstable between
last to the current iteration. This is when “strong” coupling ensured that we
need to rewrite a lot of the EQCP implementations in code.
The “reasonably coupled” ${}^{C}\Sigma$ estimate would be as follows:
$^{C}\Sigma_{n+1}\approx
1-\frac{O(|\mathbb{E}_{n}\cap\mathbb{E}_{n+1}|)}{O(|\mathbb{E}_{n}\cup\mathbb{E}_{n+1}|)}\approx
1-\frac{O(2^{B})}{O(2^{B+K})}\approx 1-\frac{1}{2^{K}}\approx 1\;;\;K>>1$
(3.6)
Where $K$ is some constant estimating the branch changes due to $\delta$ as
depicted in previous section.
### 3.8. Guiding Stability Algorithm
Assuming coupling would almost always be present, one way for us to avoid
unpredictable jumps in the stability, we can device our development strategy
such that the $\Sigma$ does not change drastically towards 1.
At this point, if there were many alternative way to program ( $P_{i}$ ) the
$\delta_{n}$ change, we may want to chose the alternative $P_{x}$ way to
program which minimizes $\Sigma_{n+1}$. If we do, then the system remains
stable in the short term. But this is a direct anti thesis of “less code
change and faster changing ability”, as it minimizing $\Sigma_{n+1}$ culminate
into more code change, because it would inherently try to lose some coupling!
More importantly, this computation of minimizing the $\Sigma$ post applying
the $\delta$ change can be greedy, but it is evident that here is where hill
climbing creeps up, there can be a minima hidden somewhere else.
At this point, in the worst case it would boil down to applying all
specification changes $\\{\delta_{i}\\}$ which would have have a factorial
runtime or, would be in NP. This is anti agile, and definitely not “small
incremental change”, this is a lot of change, pre-computed, and applied to
minimize code churn.
By this time, we have understood that practically following guided stability
is already very hard, however, worse is yet to be seen by us. Unfortunately
even with this guided approach there would be some problems which would not go
away, in the long term, that is the discussion of the next section.
### 3.9. Chaos in Stability space
We now proceed to demonstrate that the iteration driven by $(\tau,\delta_{n})$
in Stability Space $\Sigma$ has characteristics of a system capable of
showcasing chaotic dynamical behavior [28].
Given there is no universally agreed definition of chaos - we - like most
people would accept the following working definition [29] [30]:
> Chaos is aperiodic time-asymptotic behavior in a deterministic system which
> exhibits sensitive dependence on initial conditions.
These characteristics would now be demonstrated for iterated TDD.
1. (1)
_Aperiodic time-asymptotic behavior_ : this implies the existence of phase-
space trajectories which do not settle down to fixed points or periodic
orbits. For practical reasons, we insist that these trajectories are not too
rare. We also require the trajectories to be _bounded_ : _i.e._ , they should
not go off to infinity.
The sequence $\Sigma_{n}\in\mathbb{Q}\cap(0,1)$ is bounded by definition. The
trajectories are not rare, and it is practically impossible for the sequence
to settle down to periodic orbits or converging sequence. Note that w/o the
presence of coupling this sequence can be made to orbit around approximating 0
most of the time.
2. (2)
_Deterministic_ : this implies that the equations of motion of the system
possess no random inputs. In other words, the irregular behavior of the system
arises from non-linear dynamics and not from noisy driving forces.
One can argue that the sequence is driven by $\delta_{n}$ \- an external
input, but it is not. Iterative TDD has this baked in, as part of the system
iteration description , and the processing of it is algorithmic in the formal
methodology which we present for formal correctness for the software. In fact
we can argue that the sequence $\delta_{n}$ can be specified beforehand, and
it would make it fully deterministic and it would not impact our analysis.
3. (3)
_Sensitive dependence on initial conditions_ : this implies that nearby points
can be spread further over time while distant points can come close over time
- e.g. stretching and folding of the space. In fact it is said to be:
> Chaos can be understood as a dynamical process in which microscopic
> information hidden in the details of a system’s state is dug out and
> expanded to a macroscopically visible scale (_stretching_), while the
> macroscopic information visible in the current system’s state is
> continuously discarded (_folding_). The system has a positive Lyapunov
> exponent [31].
This is evident in case of coupling.
CFG comprise of the micro details which culminates into the the space of EQCP,
and merging further specification over that produce the sequence $\Sigma_{n}$.
Inherently a lot of micro details are being pushed into visibility and then
again being discarded as in the $\Sigma$ space, the information about current
complexity of the system ( EQCP space $\mathbb{E}$ ) does not exist.
We shall now proceed to formally demonstrate that Lyapunov exponent is
positive for $\Sigma$.
Given two nearby points in $\Sigma_{n}$ , say $a,b:|a-b|<\epsilon$ , there is
no guarantee that in next iteration how further apart the sequence would go,
given even exactly same specification of $\delta_{n}$ . Let $\Sigma(p,\delta)$
be the next iteration sequence after starting from $p$ in $\Sigma_{n}$ post
applying the same specification change $\delta$.
Then $\textbar{}\Sigma(a,\delta)-\Sigma(b,\delta)\textbar{}\neq 0$ holds true
almost always for all practical purposes.
Let us define the function $\Delta(a,b,\delta)$ as follows:
$\Delta(a,b,\delta)=\frac{|\Sigma(a,\delta)-\Sigma(b,\delta)|}{|a-b|}$ (3.7)
Then, a stretch happens when $\Delta(x,y,\delta)>1$ and a fold happens when
$\Delta(x,y,\delta)<1$.
This is to say, stretch increases the distance between the trajectories
starting with $(a,b)$ while fold reduces it. We notice that the definition of
Lyapunov exponent of the $\Sigma$ would be as follows:
$\lambda=ln(\Delta(a,b,\delta))$ (3.8)
We can approximate $\Sigma(x,\delta)$ in presence of some coupling - where
$B_{x}$ is the branching at $x$ and $K_{x}$ is the addition of branching due
to application of $\delta$ as follows ( estimating from previous section):
${}^{C}\Sigma(x,\delta)\approx
1-\frac{O(2^{B_{x}})}{O(2^{B_{x}+K_{x}})}\approx 1-\frac{1}{2^{K_{x}}}$
This when substituted reduces to:
$\Delta(a,b,\delta)\approx\frac{|\frac{1}{2^{K_{a}}}-\frac{1}{2^{K_{b}}}|}{|a-b|}\approx\frac{|2^{K_{a}}-2^{K_{b}}|}{2^{K_{a}+K_{b}}|a-b|}$
Now we choose a suitable $\epsilon$ for our purpose to simplify the expression
as well as minimize it:
$\epsilon<\frac{1}{2^{K_{a}+K_{b}}}$
Thus making the smallest bound possible for $\Delta$ as :
$\Delta(a,b,\delta)\approx|2^{K_{a}}-2^{K_{b}}|\approx\theta(2^{L})\;;\;\forall(K_{a}\neq
K_{b})\;L>1$
And this immediately demonstrates that Lyapunov Exponent for the system is
positive ( $\lambda>0$ ) :
$\lambda=ln(\Delta(a,b,\delta))\approx L\times ln(2)\;;\;\forall(K_{a}\neq
K_{b})\;L>1$ (3.9)
thereby proving that the $\Sigma$ map is expansive and hence Chaotic under the
influence of coupling.
We can argue the same in a semi formal way.
Evidently, if only folding happens, then every sequence would converge. This
is an extreme view. In the same way if only stretching happens, then because
the sequence is bound, it must converge again to 0 or 1. This is another
extreme view.
We can safely say the probability that for every tuple $(a,b,\delta)$ that the
$\lambda>1$ would be $0$. So goes the same for $\lambda<1$.
It is much more plausible that a function like this would have some intervals
where it would stretch and some intervals where it would fold depends on the
$\delta$. This is the most likely phenomenon which invariably would generate a
sequences diverging and converging in $\Sigma$ thereby producing the dynamic
process that stretches and folds - and thus creating sensitive dependence on
initial condition, the hallmark of chaos.
The above points make it very clear that the sequence $\Sigma$ may show all
properties of chaotic dynamics. Which proves that iteration of iterated TDD
can and would show chaotic dynamics.
## 4\. Practical Considerations for Software Development Under Iterative TDD
### 4.1. Identifying Chaotic Trajectory
Is there a guarantee that chaotic patterns would emerge on each case? No one
knows. Chaos in software development [32] has been discussed about although
not in much formal details like this. If we are very lucky it would not, but
it is hard to tell. Only by carefully monitoring the sequences we would be
able to claim whether we entered any chaotic sequence or not and this
formalism gives a metric such that the sequence can be tested for emergence of
chaos - by following Kantz [33]. That would be the empirical way of measuring
on each iteration how the progress is happening. Given agility is the name of
the game now, we can add 52 data points a year for each project if weekly
shipping of software is followed.
### 4.2. Domain of Stability for Iterated TDD
Let’s imagine the worst case, almost all of the sequences would be chaotic.
What is so problematic about chaotic dynamics appearing in the phase of
“stability” of EQCP ? This means there might a unpredictable amount of churn
in terms of the changes in the EQCP. And that means churns in the “pair points
specifications” e.g tests which were to “hold the correctness of the
software”, implying a unpredictable, possibly a very high implementation
change MUST happen.
If in one iteration which was created by a tiny change in specification
impacted 50% of the test cases to refactor source code and tests thoroughly,
evidently this would become a huge problem.
The chaotic thesis suggests that not this is only possible, but also highly
likely due to the mixing of EQCPs in terms of coupling, and a direct result of
code refactoring trying to apply DRY principle.
Hence the formal idea of just fixing input output points and rapid, small
iteration on specification can not work in general unless we keep on reducing
the scope of the specification.
It is only guaranteed to work (produce provably correct software and
predictable amount of code churn) at the lowest abstraction level if there are
very less coupling by definition. Unfortunately the proponents of TDD want to
make it work even at user specification level - where it entirely lose out its
rigor and has no provable applicability to either improve the quality of the
product or the code itself.
### 4.3. Uncertainty Principle of Iterated TDD
We have uncovered an uncertainty principle [34] of sorts here:
> With coupling at play, if we try to fix more specification by specifying
> more EQCP, then the code churn becomes unpredictable. And if we do not go
> exhaustive on EQCP, then the formal correctness software producing
> characteristics of the methodology disappears.
It seems in the presence of coupling, we can either choose formal correctness
or choose code churn stability, not both.
This insight is unheard of, but the theory points us in this direction. If the
chaotic thesis is correct, this is to be taken as a foundational law of
Software Engineering.
While this demonstrates why coupling is a problem, however, this is much
stronger thesis, this tantamount to any shared code is a problem if the code
supposed to change later.
### 4.4. Revisiting Guided Approach
Readers may argue that how then this analysis does not apply to any other
software development process? The answer lies in the guided approach. In case,
if one does not make the software fixed via hard test driven specification,
then there is loss of “correctness” - granted, but there is a lot of “wiggle”
room to build the system.
With the guided approach one can even try to avoid the entire chaotic
trajectories by prioritizing specifications or even rejecting it for the time
being, till a suitable time comes to apply such that the stability is not
changed that much.
This, evidently is what non agile waterfall, or iterated waterfall [35]was all
about. In fact we are formally defining prototypical development at this point
[36].
Would they avoid the unstable paths? Sometimes. But mostly they would make the
system “slower” in the stability space. Here, we are not talking about the
slowness of delivery, we are talking about slow movement of the system in the
stability space. This way, it would take a very long time to reach a chaotic
state.
### 4.5. Path Forward - Approaches
From the last section to avoid these chaotic sequences we can try avoiding all
of these by either:
1. (1)
Making the specification more relaxed - at that point it would specify almost
nothing and there would be almost no chaotic behavior because of the state
space of EQCP being reduced drastically. This is the a cargo cult approach,
producing only placebo, the application of TDD w/o any formalism.
2. (2)
Or, we can try to decrease coupling, in which case it would bloat the software
by not having shared code path - this would result is unimaginable bloat in
the software - given we are looking at very large dimension of EQCP state
space.
Evidently, then via [2] iterated TDD, therefore, can only be effectively done
in practice when the $\mathbb{E}_{n}$ space is extremely small and the context
of “Software” is very narrow.
### 4.6. Context Of Applicability
Not all is lost however. As it is proven, if we can go narrower and narrower,
to the point when EQCPs stop effectively sharing code with one another, TDD
becomes formally correct, also the methodology to develop software in regular
iteration with predictable churn. This narrow specification contexts are in
fact the unit tests with very less coupling which guarantee of becoming chaos
free!
We can now formally define scope for formal iterated TDD, which is guaranteed
to work - e.g. create formal verifiable correct software as follows without
ever destabilizing source code:
> Unit like tests where implementation of such features do not share any
> source code, e.g. Independent (completely decoupled) - such that in every
> iteration the decoupling holds true guarantee to hold to verifiable correct
> behavior.
And it is in this context TDD reigns supreme. Anything other than that -
correctness or stability can not be guaranteed. Just like one can try to use a
scalpel to dig a canal, it just won’t work. Any effort of using the scalpel to
create a canal is not only misguided, but futile, and not even wrong.
Do iterative TDD, just ensure all EQCPs are completely decoupled, this, now
becomes a formally correct software producing code churn wise stable
methodology. Now, in practice it is hard to do, even for Unit tests, so a
small amount coupling should not really harm the effectiveness via that much -
but at that point Chaotic behavior stems in.
Principles like AHA, WET [21] comes in extremely handy in this regard. Even
with very less coupling there is no absolute guarantee of code stability, due
to emergence of chaos but at least we are in the right track by being formally
correct, and the resulting chaos can be tamed.
### 4.7. A Perspective on Popular “Business Specification based” TDD
The previous issues culminates into less and less specific specifications used
in the industry. At that point they cover so less equivalence classes that TDD
would lose all it’s effectiveness which is to be found rigorously at the unit
test level. Thus we do have a problem, if we specify more and more, the
resulting software has high coupling thereby ensuring the iterations are
destabilized. If we specify less and less the resulting diluted TDD is just
homeopathy, water in the name of medicine but peoples believe making it “work”
- a placebo [37].
This is not hard to understand, as TDD mandates writing the tests first, there
are some tests, for sure, better than none, and this essentially ensures there
is at least some correctness in the mix. The fear of failing tests ensures
code is often correctly written. It has been well understood that developers
tend to write better code just because there would be testers who would test
it. This however does not consider the “cost” of stability in code churn. This
metric, surprisingly was never studied!
Interestingly “Business Specification Driven TDD” is the most popular TDD in
the industry. This “Some input,output are verified” is not really an effective
methodology, given the nature of the number of tests required runs in
exponential numbers in terms of the EQCP for the features.
However, it gives a lot of people something to talk about and mental peace
just like Homeopathy sans effectiveness other than placebo as it was found out
in another research : [38].
We can also safely say, any low level, low coupled, EQCP based formal TDD
method would be reasonably successful, if those practices were to be followed,
iterated TDD would definitely be very effective. There are some publications
where it has been shown to do exactly that [37].
### 4.8. Cargo Cult “Software” Engineering?
We can therefore conclude that iterated TDD without understanding the
applicability context is like washing your hand with water before you eat,
while the “washing hand” would be a good practice, but if the water used was
filthy, it would degenerate to numerable problems. This is the status of
industry with respect to TDD, for those who are into the right context, it
works, give or take. Those who are not, it does not.
We conclude by making a much more starker remark, the proponents of TDD, or
“industry best practices” stopped asking “is this effective or provable” a
long time ago. Their new established position is : “No evidence required for
common sense practices”. In fact, this is the verbatim response when asked
about efficacy and provability of some of the best practices:
> You want to debate seriously? Then you have to drop the ridiculous sense
> that “Good Practices” require scientific evidence before they can be
> realized to work - which would disprove much of the “Good Practices” which
> are “successfully used” in the industry.
Even if we ignore the irony of the previous quote, one but just wonder if
evidently Software had become entirely cargo cult [39], the above quote proves
it beyond doubt. Very few admit it openly, but it is what it has become.
## 5\. Closing Remarks
Formal iterated TDD, as presented here, is shown to produce correct software
code. The issue with such production requires a lot more formal and practical
considerations.
When done correctly (by EQCP and reducing coupling between them) it ensures we
can further add more features to the existing software while maintaining
stability as well as correctness as we go.
If that reduction of coupling is not followed, then the addition of more
equivalence classes could and most definitely would modify a significant
amount EQCP mapping by ensuring one must rewrite a very significant amount of
tests, as well as implementations. This is also seen in reality. Anything at
any further higher level of abstraction that Unit like tests would have impact
like placebo.
Hence we propose iterated TDD is to be done at the Unit Testing level only,
where it works correctly and satisfactorily because of Units should be
essentially maximally decoupled keeping an constant eye on the coupling
generated by those tests being constantly added, which is hard, but not
impossible to do and shows provable theoretical efficacy: provably correct
software production along with predictable code churn.
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|
type of motion, thus verifying in a concrete way one of the main consequences
of complete integrability of the system.
In this chapter, we will study Kerr polar motion in full generality, but leave
the radial motion aside. Its study is actually much more involved, since the
associated potential is of degree four, whereas it is only of degree two for
the polar motion. Instead, we will analyze in details the radial geodesic
motion in both NHEK and near-NHEK spacetimes. This analysis turns out to be
relevant for at least three reasons: (i) it will provide us with a deep
comprehension of the peculiarities of the motion near highly spinning black
holes, and in particular of the behaviour of their spherical geodesics. (ii)
As we will see, any geodesic can be obtained by applying a well-chosen
conformal transformation to a spherical geodesic. This enables to use the
computational scheme extensively applied in [58] to generate the waveforms
emitted by objects moving along geodesics in the near-horizon region. Such an
analysis can allow to unravel “smoking gun” signatures for the existence of
highly spinning black holes in Nature. Finally, (iii) this radial
classification was also the work which paved the road to the one of generic
Kerr radial motion, which was completed by G. Compère, Y. Liu and J. Long
[64].
All along this chapter, we will adopt a different notation than the one used
in all the rest of this thesis for Kerr geodesic conserved quantities. The
main reason for this switch is to match with the conventions used in [20]. We
also take the opportunity to restore the dependence of the equations upon
$\mu$. This enables our classification scheme to hold for null geodesics as
well as for timelike ones. The temporary convention is given by the following
dictionary
$\displaystyle\begin{split}&\hat{t}\to t,\quad,\hat{r}\to
r\quad,\hat{\varphi}\to\phi,\quad E_{0}\to\frac{\hat{E}}{\mu},\quad
L_{0}\to\frac{\ell}{\mu},\\\ &L_{*}\to\frac{\ell_{*}}{\mu},\quad
L_{\circ}\to\frac{\ell_{\circ}}{\mu},\quad
Q_{0}\to\frac{Q}{\mu^{2}}.\end{split}$ (3.2)
#### 3.1 Classification of polar motion
This section aims to describe in full detail the classification of the polar
geodesic motion around a Kerr black hole. The classification scheme used has
been first introduced in [59], and subsequently completed in [20]. The
taxonomy is depicted in Figs. 3.2 and 3.2. The phenomenology of the polar
behavior is principally governed by the sign of $Q$. The details of the
classification are summarized in Tables 3.1 and 3.2.
Let us recall that, due to mirror symmetry between the two hemispheres, the
natural variable to describe the polar motion is $z\triangleq\cos^{2}\theta$.
Let us define
$\epsilon_{0}(\hat{E},\mu)\triangleq a^{2}(\hat{E}^{2}-\mu^{2}),$ (3.3)
and
$z_{\pm}\triangleq\Delta_{\theta}\pm\text{sign}\,{\epsilon_{0}}\sqrt{\Delta_{\theta}^{2}+\frac{Q}{\epsilon_{0}}},\qquad\Delta_{\theta}\triangleq\frac{1}{2}\quantity(1-\frac{Q+\ell^{2}}{\epsilon_{0}}),\qquad
z_{0}\triangleq\frac{Q}{Q+\ell^{2}}.$ (3.4)
The classification is determined by the roots of the polar potential (1.25c).
Assuming $a\neq 0$, we can rewrite it as
$\Theta(z)=-\ell^{2}z+(Q+\epsilon_{0}z)(1-z)=\left\\{\begin{array}[]{ll}\epsilon_{0}(z_{+}-z)(z-z_{-}),&\epsilon_{0}\neq
0;\\\ (Q+\ell^{2})(z_{0}-z),&\epsilon_{0}=0.\end{array}\right.$ (3.7)
Our definition of the roots $z_{\pm}$ implies the ordering $z_{-}<z_{+}$ (and
respectively $z_{+}<z_{-}$) for $\epsilon_{0}>0$ (respectively
$\epsilon_{0}<0$). This is a convenient convention because in both cases the
maximal angle will be related to $z_{+}$. The positivity of the polar
potential implies that the poles $z=1$ ($\theta=0,\pi$) can only be reached if
$\ell=0$. Note that when a geodesic crosses a pole, its $\varphi$ coordinates
discontinuously jump by $\pi$. The invariance of the polar geodesic equation
under $(\hat{E},\ell)\to(-\hat{E},-\ell)$ allows us to reduce the analysis to
prograde $\ell\geq 0$ orbits. We distinguish the orbits with angular momentum
$\ell\neq 0$ and without, $\ell=0$:
###### I. Nonvanishing angular momentum $\ell\neq 0$.
We must consider the following cases:
1. 1.
$\boldsymbol{-(\ell-a\hat{E})^{2}\leq Q<0}$ can only occur if
$\epsilon_{0}>0$, otherwise leading to $\Theta<0$. For $\epsilon_{0}>0$, the
motion is vortical; i.e., it takes place only in one of the two hemispheres
without crossing the equatorial plane and is bounded by
$0<z_{-}\leq z\leq z_{+}<1.$ (3.8)
This vortical motion can only occur provided
$\ell^{2}\leq\quantity(\sqrt{\epsilon_{0}}-\sqrt{-Q})^{2}$.
2. 2.
$\boldsymbol{Q>0}$ leads to motion crossing the equator and symmetric with
respect to it, bounded by
$\displaystyle 0$ $\displaystyle\leq z\leq z_{+}<1\qquad(\epsilon_{0}\neq 0),$
(3.9a) $\displaystyle 0$ $\displaystyle\leq z\leq
z_{0}<1\qquad(\epsilon_{0}=0).$ (3.9b)
We will refer to such a motion as pendular;
3. 3.
$\boldsymbol{Q=0}$ allows us to write
$\Theta(z)=\epsilon_{0}\,z\,(1-\frac{\ell^{2}}{\epsilon_{0}}-z).$ (3.10)
If $\epsilon_{0}\leq 0$, the positivity of the polar potential enforces the
motion to be equatorial. For $\epsilon_{0}\geq 0$, equatorial motion exists at
$z=0$. For $\epsilon_{0}\geq 0$ and $\ell^{2}\leq\epsilon_{0}$, another motion
exists bounded by
$0<z\leq 1-\frac{\ell^{2}}{\epsilon_{0}}<1,$ (3.11)
which is a marginal case separating the pendular and vortical regimes; the
motion then admits only one turning point and asymptotes to the equator both
at future and at past times. Since we could not find a terminology for such a
motion in the literature, we propose to call it _equator-attractive_ 111This
neologism accurately reflects the fact that the motion is polar and that the
equator is an attractor. The terminology “homoclinic” is already used in the
literature to refer to radial motion.. In the special case where $z=0$ at the
initial time, the motion remains $z=0$ at all times: it is _equatorial_.
###### II. Vanishing angular momentum $\ell=0$.
The polar potential reduces to
$\Theta(z)=\left\\{\begin{array}[]{ll}\epsilon_{0}\quantity(\frac{Q}{\epsilon_{0}}+z)\quantity(1-z),&\epsilon_{0}\neq
0\\\ Q\quantity(1-z),&\epsilon_{0}=0.\end{array}\right.$ (3.14)
We distinguish the following cases:
1. 1.
$\boldsymbol{\epsilon_{0}=0}$ leads to motion over the whole polar range
$0\leq z\leq 1$ for $Q>0$; we called it polar motion. The only turning point
is located at $z=1$. For $Q=0$, the potential vanishes identically and the
polar angle remains constant; we call it azimuthal motion; for $Q<0$ the
potential is positive only if the motion takes place along the black hole axis
$z=1$; we call it axial motion.
2. 2.
$\boldsymbol{\epsilon_{0}>0}$ leads to a polar motion $0\leq z\leq 1$ for
$Q>0$. For $Q=0$ and $z=0$, the motion is equatorial. For $Q=0$ and $z\neq 0$,
$z=0$ is an asymptotic attractor of the motion which only takes place in one
of the hemispheres. It is therefore a special case of equator-attractive
motion where the turning point is at the pole $z=1$. For $Q<0$, the motion is
either vortical ($0<-\frac{Q}{\epsilon_{0}}\leq z\leq 1$) for
$-\epsilon_{0}<Q<0$ or axial with $z=1$ for $Q\leq-\epsilon_{0}<0$.
3. 3.
$\boldsymbol{\epsilon_{0}<0}$ leads to a polar motion $0\leq z\leq 1$ for
$Q\geq-\epsilon_{0}>0$ and to a pendular one ($0\leq
z\leq-\frac{Q}{\epsilon_{0}}<1$) for $0<Q<-\epsilon_{0}$. For $Q=0$, the
motion is either equatorial or axial for the potential to be positive. For
$Q<0$, the motion also has to take place along the axis.
Let us finally notice that, for any value of $\epsilon_{0}$ and
$Q\geq-(aE_{0})^{2}$, an axial motion is always possible.
Energy | Carter constant | Polar range | Denomination
---|---|---|---
$\epsilon_{0}<0$ ($|\hat{E}|<\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $\emptyset$ | $\emptyset$
| $Q=0$ | $z=0$ | Equatorial$(\hat{E})$
| $Q>0$ | $0\leq z\leq z_{+}<1$ | Pendular$(\hat{E},Q)$
$\epsilon_{0}=0$ ($|\hat{E}|=\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $\emptyset$ | $\emptyset$
| $Q=0$ | $z=0$ | Equatorial∘
| $Q>0$ | $0\leq z\leq z_{0}<1$ | Pendular${}_{\circ}(Q)$
$\epsilon_{0}>0$ ($|\hat{E}|>\mu$) | $-(\ell-a\hat{E})^{2}\leq Q<0$ | $0<z_{-}\leq z\leq z_{+}<1$ | Vortical$(\hat{E},Q)$
| $Q=0$ and $\epsilon_{0}\geq\ell^{2}$ | | $0<z\leq 1-\frac{\ell^{2}}{\epsilon_{0}}<1$
---
($\text{sign}\,{\cos\theta}$ fixed)
| Equator-
---
attractive$(\hat{E})$
| | $z=0$ | Equatorial$(\hat{E})$
| $Q>0$ | $0\leq z\leq z_{+}<1$ | Pendular$(\hat{E},Q)$
Table 3.1: Polar taxonomy of Kerr geodesics with $\ell\geq 0$. The orbits with
$\ell<0$ are obtained from $\ell>0$ by flipping the signs of both $\hat{E}$
and $\ell$.
a
Energy | Carter constant | Polar range | Denomination
---|---|---|---
$\epsilon_{0}<0$ ($|\hat{E}|<\mu$) | $-(a\hat{E})^{2}\leq Q<0$ | $z=1$ | Axial${}^{0}(\hat{E},Q)$
| $Q=0$ | $z=0,1$ | | Equatorial${}^{0}(\hat{E})$
---
Axial${}^{0}(\hat{E})$
| $0<Q<-\epsilon_{0}$ | $0\leq z\leq-\frac{Q}{\epsilon_{0}}<1$ | Pendular${}^{0}(\hat{E},Q)$
| $0<-\epsilon_{0}\leq Q$ | $0\leq z\leq 1$ | Polar${}^{0}(\hat{E},Q)$
$\epsilon_{0}=0$ ($|\hat{E}|=\mu$) | $-(a\hat{E})^{2}\leq Q<0$ | $z=1$ | Axial${}^{0}_{\circ}(Q)$
| $Q=0$ | $z=\text{constant}$ | Azimuthal${}^{0}_{\circ}$
| $Q>0$ | $0\leq z\leq 1$ | Polar${}^{0}_{\circ}(Q)$
$\epsilon_{0}>0$ ($|\hat{E}|>\mu$) | $-(a\hat{E})^{2}\leq Q\leq-\epsilon_{0}<0$ | $z=1$ | Axial${}^{0}(\hat{E},Q)$
| $-\epsilon_{0}<Q<0$ | $0<-\frac{Q}{\epsilon_{0}}\leq z\leq 1$ | Vortical${}^{0}(\hat{E},Q)$
| $Q=0$ | $0<z\leq 1$ ($\text{sign}\,{\cos\theta}$ fixed) | | Equator-
---
attractive${}^{0}(\hat{E})$
| | $z=0$ | Equatorial${}^{0}(\hat{E})$
| $Q>0$ | $0\leq z\leq 1$ | Polar${}^{0}(\hat{E},Q)$
$\epsilon_{0}\in\mathbb{R}$ | $Q\geq-(a\hat{E})^{2}$ | $z=1$ | Axial${}^{0}(E_{0},Q)$
Table 3.2: Polar taxonomy of Kerr geodesics with $\ell=0$.
Figure 3.1: Polar taxonomy of $\ell\neq 0$ Kerr geodesics. Equator-
attractive$(\hat{E})$ orbits become Equatorial$(\hat{E})$ orbits when the
initial angle is at the equator.
Figure 3.2: Polar taxonomy of $\ell=0$ Kerr geodesics. In addition to the
possible motions depicted in the figure, an axial motion is always possible
for any value of $\epsilon_{0}$ and $Q\geq-(aE_{0})^{2}$.
##### 3.1.1 Solution to the polar integrals
After having classified the different types of motion allowed, we will provide
manifestly real and positive explicit solutions in terms of elliptic integrals
for each type of polar motion with $\ell\neq 0$, in line with the recent
analysis [59]. All such integrals will turn out to agree with Ref. [59], but
our presentation will be slightly simpler.
The solution to the polar integrals (1.28a) and (1.28b) can be organized in
terms of the categories of polar motion with $\ell\neq 0$:
| Vortical | Equator-attractive | Pendular
---|---|---|---
$\boldsymbol{\epsilon_{0}<0}$ | $\emptyset$ | $\emptyset$ | Pendular$(\hat{E},Q)$
$\boldsymbol{\epsilon_{0}=0}$ | $\emptyset$ | $\emptyset$ | Pendular${}_{*}(Q)$
$\boldsymbol{\epsilon_{0}>0}$ | Vortical$(\hat{E},Q)$ | Equator-attractive$(\hat{E})$ | Pendular$(\hat{E},Q)$
Each type of motion yields to a specific decomposition of the line integrals
$\scriptsize\rotatebox[origin={c}]{-90.0}{$\backslash$}\int$ in terms of basic
integrals. In order to simplify the notations, we drop the “$f$” indices
labeling the final event and define $h\equiv\text{sign}\,{\cos\theta}$,
$\theta_{a}\triangleq\arccos\sqrt{z_{a}}$ ($a=+,-,0$), as well as the initial
and final signs $\eta_{i}$, $\eta$:
$\displaystyle\eta_{i}\triangleq-
s_{\theta}^{i}\,\text{sign}\,{\cos\theta_{i}},\qquad\eta\triangleq-(-1)^{m}s_{\theta}^{i}\,\text{sign}\,{\cos\theta}.$
(3.52)
We are now ready to perform the explicit decomposition:
1. 1.
Pendular motion. We have $0<z_{+}\leq 1$, and $\theta$ therefore belongs to
the interval $\theta_{+}\leq\theta\leq\pi-\theta_{+}$. The polar integral can
be written (see Ref. [59])
$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}$
$\displaystyle=2m\left|\int_{0}^{\cos\theta_{+}}\right|-\eta\left|\int_{0}^{\cos\theta}\right|+\eta_{i}\left|\int_{0}^{\cos\theta_{i}}\right|,\qquad\epsilon_{0}\neq
0,$ (3.53a)
$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}$
$\displaystyle=2m\left|\int_{0}^{\cos\theta_{0}}\right|-\eta\left|\int_{0}^{\cos\theta}\right|+\eta_{i}\left|\int_{0}^{\cos\theta_{i}}\right|,\qquad\epsilon_{0}=0.$
(3.53b)
It is useful to note that our definitions of the roots imply
$\epsilon_{0}\,z_{-}<0,\qquad\epsilon_{0}(z-z_{-})>0,\qquad\frac{z_{+}}{z_{-}}\leq
1.$ (3.54)
2. 2.
Vortical motion. We have $\epsilon_{0}>0$ and $0<z_{-}\leq\cos^{2}\theta\leq
z_{+}<1$. The motion therefore never reaches the equator. The sign of
$\cos\theta$ is constant and determines whether the motion takes place in the
northern or the southern hemisphere. Without loss of genericity, let us focus
on the northern hemisphere:
$0\leq\theta_{+}\leq\theta\leq\theta_{-}<\frac{\pi}{2}$; we denote again as
$m$ the number of turning points at Mino time $\lambda$. The polar integral
can be written (see Ref. [59] and Appendix A of Ref. [60]):
$\displaystyle\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}=\left(m-\eta_{i}\frac{1-(-1)^{m}}{2}\right)\left|\int_{\cos\theta_{-}}^{\cos\theta_{+}}\right|-\eta\left|\int_{\cos\theta_{-}}^{\cos\theta}\right|+\eta_{i}\left|\int_{\cos\theta_{-}}^{\cos\theta_{i}}\right|.$
(3.55)
3. 3.
Equator-attractive motion. This is a limit case of the vortical motion reached
in the limit $z_{-}\to 0$, $z_{+}\to 2\Delta_{\theta}$. As detailed in Ref.
[59], the turning point $z_{-}=0$ corresponds to a non-integrable singularity
of the polar integrals and the motion exhibits consequently at most one
turning point at $z_{+}=2\Delta_{\theta}$, leading to the line-integral
decomposition
$\rotatebox[origin={c}]{-90.0}{$\backslash$}\int_{\cos\theta_{i}}^{\cos\theta}=\eta\absolutevalue{\int_{\cos\theta_{+}}^{\cos\theta}}-\eta_{i}\absolutevalue{\int_{\cos\theta_{+}}^{\cos\theta_{i}}}.$
(3.56)
In all cases but the equator-attractive case, the polar motion is periodic.
Denoting by $\Lambda_{\theta}$ its period, one can easily give an explicit
formula for the number of turning points $m$ as a function of the Mino time:
$m(\lambda)=\left\\{\begin{array}[]{ll}\rule{0.0pt}{6.0pt}\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda-\lambda_{i}^{\theta})+\frac{1}{2}\right\rfloor,&Q>0\\\
\rule{0.0pt}{16.0pt}\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda-\lambda_{i}^{\theta})\right\rfloor+\left\lfloor\frac{2}{\Lambda_{\theta}}(\lambda_{i}^{\theta}-\lambda_{i})\right\rfloor+\frac{3-s^{i}_{\theta}}{2},&Q<0\end{array}\right.$
(3.59)
with
$\lambda_{i}^{\theta}\triangleq\lambda_{i}-s^{i}_{\theta}\int_{0}^{\cos\theta_{i}}\frac{\differential\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}$
and where the floor function is defined as $\left\lfloor
x\right\rfloor\triangleq\max\quantity{n\in\mathbb{Z}|n\leq x}$. For the
equator-attractive case, one has simply
$m(\lambda)=\theta(\lambda-\lambda_{i}^{\theta})$ where $\theta$ is here the
Heaviside step function.
The integrals introduced above are solved explicitly in Appendix B.1. For each
case, the corresponding solutions are detailed below and schematically
depicted in Fig. 3.3.
|
---|---
(a) Pendular$(\hat{E},Q)$ | (b) Pendular${}_{\circ}(Q)$
|
(c) Vortical$(\hat{E},Q)$ | (d) Equator-attractive$(\hat{E})$
Figure 3.3: Angular taxonomy of $\ell\neq 0$ Kerr geodesics. The angular
behavior is depicted in spherical coordinates on the unit sphere: the polar
angle is $\theta(\lambda)$, and the azimuthal angle is the purely angular part
of the Kerr azimuthal angle
$(\ell-a\hat{E})(\lambda-\lambda_{i})+\ell\Phi_{\theta}(\lambda)$.
###### Pendular$(\hat{E},Q)$ motion.
The motion exhibits a positive Carter constant $Q$ and can occur for any
$\epsilon_{0}\neq 0$; our definition of the roots $z_{\pm}$ allows us to treat
simultaneously the two cases $\epsilon_{0}<0$ and $\epsilon_{0}>0$, which is a
simplification with respect to the analysis carried out in Ref. [59]. The
period of the polar motion (comprising two turning points) in Mino time is
given by
$\Lambda_{\theta}=4\int_{0}^{\cos\theta_{+}}\frac{\text{d}\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}\triangleq
4I^{(0)}(\sqrt{z_{+}})=\frac{4}{\sqrt{-\epsilon_{0}z_{-}}}K\quantity(\frac{z_{+}}{z_{-}}).$
(3.64)
Using the basic integrals of Appendix A, one can write (1.28a) as
$\displaystyle\lambda-\lambda_{i}$
$\displaystyle=\frac{1}{\sqrt{-\epsilon_{0}z_{-}}}\left[2mK\quantity(\frac{z_{+}}{z_{-}})+s^{i}_{\theta}(-1)^{m}F\quantity(\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle~{}\left.-s_{\theta}^{i}F\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right]$
(3.65)
where we define
$\Psi^{+}(x)\triangleq\arcsin\quantity(\frac{x}{\sqrt{z_{+}}})$. Using
(9.126n), one can invert (3.65) as
$\cos\theta=s^{i}_{\theta}(-1)^{m}\sqrt{z_{+}}\text{sn}\quantity(\sqrt{-\epsilon_{0}z_{-}}\quantity(\lambda-\lambda_{i}^{\theta})-2mK\quantity(\frac{z_{+}}{z_{-}}),\frac{z_{+}}{z_{-}})$
(3.66)
where we introduce
$\displaystyle\lambda_{i}^{\theta}$
$\displaystyle\triangleq\lambda_{i}-\frac{s^{i}_{\theta}}{\sqrt{-\epsilon_{0}z_{-}}}F\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}}).$
(3.67)
This expression matches with Eq. (38) of Ref. [54]. Using the periodicity
property (9.126da) of the elliptic sine, we can further simplify it to
$\cos\theta(\lambda)=s^{i}_{\theta}\sqrt{z_{+}}\text{sn}\quantity(\sqrt{-\epsilon_{0}z_{-}}(\lambda-\lambda_{i}^{\theta}),\frac{z_{+}}{z_{-}}).$
(3.68)
It consistently obeys $\cos\theta(\lambda_{i})=\cos\theta_{i}$ and
$\text{sign}\,{\cos\theta^{\prime}(\lambda_{i})}=s_{\theta}^{i}$. This formula
agrees with (53) of Ref. [59] but it is written in a simpler form. We also
obtain
$\displaystyle T_{\theta}$
$\displaystyle=\frac{-2z_{+}}{\sqrt{-\epsilon_{0}z_{-}}}\left[2mE^{\prime}\quantity(\frac{z_{+}}{z_{-}})+(\pm_{\theta})E^{\prime}\quantity(\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle\left.~{}-s_{\theta}^{i}E^{\prime}\quantity(\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right],$
(3.69) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{1}{\sqrt{-\epsilon_{0}z_{-}}}\left[2m\Pi\quantity(z_{+},\frac{z_{+}}{z_{-}})+(\pm_{\theta})\Pi\quantity(z_{+},\Psi^{+}(\cos\theta),\frac{z_{+}}{z_{-}})\right.$
$\displaystyle~{}\left.-s_{\theta}^{i}\Pi\quantity(z_{+},\Psi^{+}(\cos\theta_{i}),\frac{z_{+}}{z_{-}})\right]-(\lambda-\lambda_{i}).$
(3.70)
where $\lambda-\lambda_{i}$ is given by (3.65). All quantities involved are
manifestly real. These final expressions agree with Ref. [59].
###### Pendular${}_{\circ}(Q)$ motion.
We now consider the critical case $|\hat{E}|=\mu$. The period of the polar
motion is
$\Lambda_{\theta}=4I^{(0)}(\sqrt{z_{0}})=2\pi\sqrt{\frac{z_{0}}{Q}}.$ (3.71)
In this critical case, (1.28a) leads to
$\lambda-\lambda_{i}=\sqrt{\frac{z_{0}}{Q}}\quantity[m\pi+s^{i}_{\theta}(-1)^{m}\arcsin{\frac{\cos\theta}{\sqrt{z_{0}}}}-s^{i}_{\theta}\arcsin{\frac{\cos\theta_{i}}{\sqrt{z_{0}}}}],$
(3.72)
which can be simply inverted as
$\cos\theta=s_{\theta}^{i}\sqrt{z_{0}}\,\sin\quantity(\sqrt{\frac{Q}{z_{0}}}(\lambda-\lambda_{i}^{\theta})),\qquad\lambda_{i}^{\theta}\triangleq\lambda_{i}-\sqrt{\frac{z_{0}}{Q}}\arcsin{\frac{\cos\theta_{i}}{\sqrt{z_{0}}}}.$
(3.73)
The other polar integrals are
$\displaystyle T_{\theta}$
$\displaystyle=\frac{1}{2}\quantity{z_{0}(\lambda-\lambda_{i})-\sqrt{\frac{z_{0}}{Q}}\quantity[(\pm_{\theta})\cos\theta\sqrt{z_{0}-\cos^{2}\theta}-s^{i}_{\theta}\cos\theta_{i}\sqrt{z_{0}-\cos^{2}\theta_{i}}]},$
$\displaystyle\Phi_{\theta}$
$\displaystyle=\sqrt{\frac{z_{0}}{Q(1-z_{0})}}\bigg{[}m\pi+(\pm_{\theta})\arcsin\quantity(\sqrt{\frac{1-z_{0}}{z_{0}}}\cot\theta)$
$\displaystyle\quad-s^{i}_{\theta}\arcsin\quantity(\sqrt{\frac{1-z_{0}}{z_{0}}}\cot\theta_{i})\bigg{]}-(\lambda-\lambda_{i}).$
(3.74)
###### Vortical$(\hat{E},Q)$ motion.
The period in Mino time is given by
$\Lambda_{\theta}=2\left|\int_{\cos\theta_{-}}^{\cos\theta_{+}}\frac{\text{d}\cos\theta}{\sqrt{\Theta(\cos^{2}\theta)}}\right|=\frac{2}{\sqrt{\epsilon_{0}z_{+}}}K\quantity(1-\frac{z_{-}}{z_{+}}).$
(3.75)
Using the basic integrals of Appendix B.1, one has
$\displaystyle\lambda-\lambda_{i}$
$\displaystyle=\frac{1}{\sqrt{\epsilon_{0}z_{+}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})K(\tilde{m})-s^{i}_{\theta}(-1)^{m}F\quantity(\Psi^{-}(\cos\theta),\tilde{m})\right.$
$\displaystyle~{}\left.+s^{i}_{\theta}F\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m})\right]$
(3.76)
where
$\tilde{m}\triangleq
1-\frac{z_{-}}{z_{+}},\qquad\Psi^{-}(x)=\arcsin{\sqrt{\frac{z_{+}-x^{2}}{z_{+}-z_{-}}}}.$
(3.77)
Using the inversion formula (9.126t) and the periodicity property (9.126db),
we obtain
$\cos\theta=h\sqrt{z_{+}}\text{dn}\quantity(\sqrt{\epsilon_{0}z_{+}}(\lambda-\lambda_{\theta}^{i}),\tilde{m})$
(3.78)
with
$\lambda_{i}^{\theta}\triangleq\lambda_{i}+\frac{s^{i}_{\theta}h}{\sqrt{\epsilon_{0}z_{+}}}F\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m}).$
(3.79)
Again, one has $\cos\theta(\lambda_{i})=\cos\theta_{i}$ and
$\text{sign}\,{\cos\theta^{\prime}(\lambda_{i})}=s_{\theta}^{i}$. The two
other polar integrals are
$\displaystyle T_{\theta}$
$\displaystyle=\sqrt{\frac{z_{+}}{\epsilon_{0}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})E(\tilde{m})-(\pm_{\theta})E\quantity(\Psi^{-}(\cos\theta),\tilde{m})\right.$
$\displaystyle~{}\left.+s^{i}_{\theta}E\quantity(\Psi^{-}(\cos\theta_{i}),\tilde{m})\right],$
(3.80) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{1}{(1-z_{+})\sqrt{\epsilon_{0}z_{+}}}\left[\quantity(m-hs^{i}_{\theta}\frac{1-(-1)^{m}}{2})\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\tilde{m})\right.$
$\displaystyle~{}\left.-(\pm_{\theta})\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\Psi^{-}(\cos\theta),\tilde{m})+s^{i}_{\theta}\Pi\quantity(\frac{z_{-}-z_{+}}{1-z_{+}},\Psi^{-}(\cos\theta_{i}),\tilde{m})\right]$
$\displaystyle~{}-(\lambda-\lambda_{i})$ (3.81)
in agreement with the results of Ref. [59].
###### Equator-attractive$(\hat{E})$ motion.
This is the only polar motion which is not periodic. One has
$\lambda-\lambda_{i}=\frac{h}{\sqrt{\epsilon_{0}z_{+}}}\quantity[-(\pm_{\theta})\,\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta}{z_{+}}}+s^{i}_{\theta}\,\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta_{i}}{z_{+}}}]$
(3.82)
leading to
$\displaystyle\cos\theta$
$\displaystyle=h\sqrt{z_{+}}\,\text{sech}\quantity(\sqrt{\epsilon_{0}z_{+}}(\lambda-\lambda_{i}^{\theta})),$
(3.83a) $\displaystyle\lambda_{i}^{\theta}$
$\displaystyle\triangleq\lambda_{i}+\frac{s^{i}_{\theta}h}{\sqrt{\epsilon_{0}z_{+}}}\text{arctanh}\sqrt{1-\frac{\cos^{2}\theta_{i}}{z_{+}}}.$
(3.83b)
The polar integrals are
$\displaystyle T_{\theta}$
$\displaystyle=\frac{h}{\sqrt{\epsilon_{0}}}\quantity[-(\pm_{\theta})\sqrt{z_{+}-\cos^{2}\theta}+s^{i}_{\theta}\sqrt{z_{+}-\cos^{2}\theta_{i}}],$
(3.84a) $\displaystyle\Phi_{\theta}$
$\displaystyle=\frac{h}{\sqrt{\epsilon_{0}(1-z_{+})}}\bigg{[}-(\pm_{\theta})\arctan\sqrt{\frac{z_{+}-\cos^{2}\theta}{1-z_{+}}}$
$\displaystyle\quad+s^{i}_{\theta}\arctan\sqrt{\frac{z_{+}-\cos^{2}\theta_{i}}{1-z_{+}}}\bigg{]}.$
(3.84b)
This agrees with the results of Ref. [59].
#### 3.2 Classification of near-horizon motion for high spin Kerr black holes
In this section, we derive a complete classification of timelike and null
geodesic trajectories lying in the near-horizon region of a quasi extremal
Kerr black hole. We will provide explicit manifestly real analytic expressions
for all geodesic trajectories. We will present the classification in terms of
the geodesic energy, angular momentum, and Carter constant $Q$. We will also
illustrate each radial motion in NHEK with a Penrose diagram.
Partial classifications were performed in Refs. [58] and [59]. In Ref. [58],
equatorial timelike prograde incoming (i.e. that originate from the Kerr
exterior geometry) geodesics were classified. Such geodesics reach the spatial
boundary of the near-horizon region at infinite past proper time and therefore
physically reach the asymptotically flat Kerr region once the near-horizon is
glued back to the exterior Kerr region. It turns out that bounded geodesics in
the near-horizon Kerr region also arise in the study of gravitational waves
since they correspond to the end point of the transition motion [96]. Timelike
outgoing geodesics originating from the white hole horizon and reaching the
near-horizon boundary are also relevant for particle emission within the near-
horizon region [80]. In addition, null outgoing geodesics are relevant for
black hole imaging around high-spin black holes [37].
The generic non-equatorial geodesics were obtained in Ref. [59]. In
particular, real forms were obtained for each angular integral involved in
geodesic motion. However, zero-measure sets of parameters were discarded.
These zero-measure sets include in particular the separatrix between bounded
and unbounded radial motion which plays a key role in EMRIs.
In the following, we do not make any assumption on the geodesic parameters. We
will treat both timelike and null geodesics, prograde or retrograde, and with
any boundary conditions. Without loss of genericity, we will consider future-
directed orbits. Past-directed geodesics can be obtained from future-directed
geodesics using the $\mathbb{Z}_{2}$ map:
$\displaystyle T\rightarrow-T,\quad\Phi\rightarrow-\Phi,\quad
E\rightarrow-E,\quad\ell\rightarrow-\ell,$ (3.85)
which will play an important role in Sec. 3.4. We will denote it as the
$\uparrow\\!\downarrow$-flip.
##### 3.2.1 NHEK
Future orientation of the geodesic is equivalent to $\differential
T/\differential\lambda>0$ or
$\displaystyle E+L_{0}R>0.$ (3.86)
Future-oriented geodesics with $L_{0}=0$ have $E>0$. For $L_{0}\neq 0$, we
define the critical radius as in [59]:
$\displaystyle R_{c}=-\frac{E}{L_{0}}.$ (3.87)
Future-orientation of the orbit requires
$\displaystyle R<R_{c}\quad\text{for}\quad L_{0}<0,\quad\text{and}\quad
R>R_{c}\quad\text{for}\quad L_{0}>0.$ (3.88)
###### Polar behavior
The results derived in Sec. 3.1 in the context of generic Kerr still hold in
the near-horizon high-spin limit which is obtained by the scaling limit
$\lambda\rightarrow 0$ taken in the near-horizon coordinates (1.42). We
anticipate that the results also hold in the distinct near-NHEK limit
$\lambda\rightarrow 0$ taken in the near-horizon coordinates (1.63). Due to
the high-spin limit, the following substitution can be made:
$\displaystyle a\mapsto
M,\qquad\hat{E}\mapsto\frac{\ell}{2M},\qquad\epsilon_{0}\mapsto\mathcal{C}_{\circ}\triangleq\frac{\ell^{2}-\ell_{\circ}^{2}}{4},$
(3.89) $\displaystyle\Theta(z)\mapsto
v_{\theta}(z),\qquad\hat{\Phi}_{\theta}-\frac{1}{4}\hat{T}_{\theta}\mapsto\Phi_{\theta}.$
(3.90)
Notice that the dependence on $\hat{E}$ of $\epsilon_{0}$ has been changed
into a dependence in $\ell$, the Kerr energy being the same at zeroth order on
$\lambda$ for all trajectories. Therefore, the quadratic term of the polar
potential vanishes at the critical value $\ell_{\circ}$ of the angular
momentum $\ell$.
One of the most striking features of the near-horizon polar motion is that $Q$
is non-negative as a consequence of the reality of polar motion, as noticed in
Ref. [59]:
###### Proposition 3.1.
$\forall z\in[0,1]:v_{\theta}(z)\geq 0\Rightarrow Q\geq 0.$ (3.91)
_Proof._ This property is a consequence of the dependence on $Q$ of
$\mathcal{C}$ defined in (1.54). Indeed, using the fact that
$z=\cos^{2}\theta\in[0,1]$ one can write
$\displaystyle
Q=\mathcal{C}+\frac{3}{4}\ell^{2}-M^{2}\mu^{2}\geq\mathcal{C}+(1-\Lambda^{-2})\ell^{2}-M^{2}\mu^{2}\geq
v_{\theta}(z)\geq 0.$ (3.92)
A direct consequence is that the near-horizon polar motion cannot be vortical
and is consequently either equatorial, pendular, polar or axial. We note that
the condition $\epsilon_{0}\geq\ell^{2}$ is never obeyed in the near-horizon
case after using the definition (3.3), $a=M$ and $\hat{E}=\frac{\ell}{2M}$.
The equator-attractive class is therefore discarded. The resulting polar
classes are listed in Table 3.3 and the phase space is represented in Fig.3.4.
###### Radial behavior
The radial behavior for generic inclined orbits can be solved using the
equatorial results [58] thanks to the following observation:
###### Proposition 3.2.
The radial integrals $T_{R}^{(i)}(R)$ $(i=0,1,2)$ only depend upon the NHEK
energy $E$ and angular momentum $\ell$ while all the dependence upon the mass
$\mu$ and Carter constant $Q$ is through
$\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.
This simple observation has far-reaching consequences. For any timelike
geodesic with $Q\neq 0$, one could directly reuse the classification
established in Ref. [58], modulo the substitution
$\frac{2}{\sqrt{3}}M\mu\to\ell_{*}$ in every expression encountered. Moreover,
null geodesics with $\mu=0$ have $Q\geq 0$ from Proposition 3.1. We can
therefore reuse the classification established in Ref. [58] to classify null
geodesics modulo the substitution $M\mu\to Q$ in every expression encountered.
Overall, all radial integrals can be described in closed form for all cases by
keeping the dependence upon $\ell_{*}$ or, equivalently, upon the Casimir
invariant $\mathcal{C}$.
Since the equatorial taxonomy of Ref. [58] did not consider bounded orbits and
only considered $\ell>0$, we will expand the taxonomy to the generic case. The
generic classification can be achieved by studying the roots of $v_{R}$ and
the range of $R$ where $v_{R}\geq 0$. We only consider orbits outside the
horizon, $R>0$. There are three broad categories depending on the angular
momentum: the supercritical case $|\ell|>\ell_{*}$ or equivalently
$\mathcal{C}<0$, the critical case $|\ell|=\ell_{*}$ or equivalently
$\mathcal{C}=0$ and the subcritical case $0\leq|\ell|<\ell_{*}$ or
$\mathcal{C}>0$.
The relative position of the critical radius (3.87) with respect to the roots
of $v_{R}$ may restrict the allowed classes of future-oriented orbits. As a
result of (3.88), subcritical $\ell^{2}<\ell^{2}_{*}$ orbits have either
$R_{+}<R_{c}$ for $\ell<0$ or $R_{c}<0$ for $\ell>0$, and all orbits are
future oriented. Critical orbits $\ell^{2}=\ell^{2}_{*}$ have either $R_{c}<0$
for $\ell=\ell_{*}$ or $R_{c}>R_{0}$ for $\ell=-\ell_{*}$. This restricts the
classes of orbits. Supercritical orbits $\ell^{2}>\ell^{2}_{*}$ with
$E,\ell>0$ are future directed. Supercritical orbits with $E>0$, $\ell<0$
admit $R_{-}<R_{c}<R_{+}$, and only bounded orbits with $R\leq R_{-}$ are
admissible. Finally, supercritical orbits with $E<0$ and $\ell>0$ obey $R\geq
R_{+}>R_{c}$ and are therefore deflecting.
After a simple analysis, we reach the following taxonomy, displayed in Table
3.4 and in Fig. 3.5. In comparison with Ref. [58], the classes
Outward$(E,\ell)$, Outward${}_{*}(E)$, Bounded${}_{>}(E,$ $\ell)$,
Bounded${}^{-}_{*}(E)$, and Bounded${}_{<}(E,\ell)$ are new, while all other
classes with $\ell>0$ appeared in Ref. [58]. The class
$\text{Osculating}(E,\ell)$ is now better called
$\text{Def}\text{lecting}(E,\ell)$. The classes with $\ell=\pm\ell_{*}$ will
be denoted with a subscript ∗. The Spherical∗ orbit with $\ell=\ell_{*}$ is
also the prograde ISSO. For $\ell\geq 0$, the conformal diagrams corresponding
to those orbits are depicted in Fig. 3.6 and their explicit forms are given in
Appendix B.2. Past-oriented geodesics (not depicted) are obtained from a
central symmetry around the origin $E=\ell=0$ as a result of the
$\uparrow\\!\downarrow$-flip (3.85).
Angular momentum | Carter constant | Polar range | Denomination
---|---|---|---
$\ell=0$ ($\mathcal{C}_{\circ}=-\ell_{\circ}^{2}/4$) | $Q=0$ | $z=0,~{}1$ | | Equatorial0
---
Axial0
| $Q>0$ | $z=1$ | Axial${}^{0}(Q)$
$0<\ell<\ell_{\circ}$ ($-\frac{\ell_{\circ}^{2}}{4}<\mathcal{C}_{\circ}<0$) | $Q=0$ | $z=0$ | Equatorial$(\ell)$
| $Q>0$ | $0\leq z\leq z_{+}$ | Pendular$(Q,\ell)$
$\ell=\ell_{\circ}$ ($\mathcal{C}_{\circ}=0$) | $Q=0$ | $z=0$ | Equatorial∘
| $Q>0$ | $0\leq z\leq z_{0}$ | Pendular${}_{\circ}(Q)$
$\ell>\ell_{\circ}$ ($\mathcal{C}_{\circ}>0$) | $Q>0$ | $0\leq z\leq z_{+}$ | Pendular$(Q,\ell)$
Table 3.3: Polar taxonomy of near-horizon geodesics with $\ell\geq 0$. The
orbits with $\ell<0$ are obtained from $\ell>0$ by flipping the sign of $\ell$
with the rest unchanged. Figure 3.4: Polar taxonomy of near-horizon geodesics.
For clarity, the scale is not respected on the horizontal axis. The dashed
blue curves $\ell^{2}=\ell^{2}_{*}$ represent the position of the spherical
orbits in parameter space. This figure contrasts with Figure 3.2.
Angular momentum (and Casimir) | Energy | Radial range | Denomination
---|---|---|---
Supercritical: $\ell>\ell_{*}$ ($-\ell^{2}<\mathcal{C}<0$) | $E>0$ | $0\leq R\leq\infty$ | | Plunging$(E,\ell)$
---
Outward$(E,\ell)$
| $E=0$ | $0<R\leq\infty$ | Marginal$(\ell)$
| $E<0$ | $R_{+}\leq R\leq\infty$ | Deflecting$(E,\ell)$
Critical: $\ell=\ell_{*}$ ($\mathcal{C}=0$) | $E>0$ | $0\leq R\leq\infty$ | | Plunging${}_{*}(E)$
---
Outward${}_{*}(E)$
| $E=0$ | $0<R\leq\infty$ | Spherical∗ (ISSO)
Subcritical: $0\leq\ell^{2}<\ell_{*}^{2}$ ($0<\mathcal{C}\leq\frac{3\ell_{*}^{2}}{4}$) | $E>0$ | $0\leq R\leq R_{+}$ | Bounded${}_{<}(E,\ell)$
Critical: $\ell=-\ell_{*}$ ($\mathcal{C}=0$) | $E>0$ | $0\leq R\leq R_{0}$ | Bounded${}^{-}_{*}(E)$
Supercritical: $\ell<-\ell_{*}$ ($-\ell^{2}<\mathcal{C}<0$) | $E>0$ | $0\leq R\leq R_{-}$ | Bounded${}_{>}(E,\ell)$
Table 3.4: Radial taxonomy of future-oriented geodesics in NHEK. Figure 3.5: Radial taxonomy of future oriented geodesics in NHEK. For equatorial geodesics, $\ell_{*}=\frac{2}{\sqrt{3}}M\mu$, while for orbits with inclination, $\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$. | | |
---|---|---|---
(a) Spherical${}_{*}(ISSO)$ | (b) Plunging${}_{*}(E)$ | (c) Bounded${}^{-}_{*}(E)$ | | (d) Outward${}_{*}(E)$,
---
Outward$(E,\ell)$
| | |
| | |
(e) Bounded${}_{<}(E,\ell)$ | | (f) Bounded${}_{>}(E,\ell)$,
---
Deflecting$(E,\ell)$
(g) Marginal$(\ell)$ | (h) Plunging$(E,\ell)$
Figure 3.6: Taxonomy of NHEK geodesics depicted in the global NHEK conformal
diagram. The upper (or respectively, lower) blue line represent the future
(respectively, past) event horizon $R=0$ and the dashed/dotted lines are the
roots of the radial potential. We used $M=1$, $E=\pm 1$ and $\ell=\pm
2\ell_{*}$ (and $\ell=\pm\frac{1}{2}\ell_{*}$, respectively) for supercritical
(and subcritical, respectively) trajectories.
a
| Angular momentum
---
(and Casimir)
Energy | Radial range | Denomination
Supercritical: $\ell>\ell_{*}$ | $e>-\kappa\sqrt{-\mathcal{C}}$ | $\kappa\leq R\leq\infty$ | | Plunging$(e,\ell)$
---
Outward$(e,\ell)$
($-\ell^{2}<\mathcal{C}<0$) | $e=-\kappa\sqrt{-\mathcal{C}}<0$ | $R=\frac{\kappa\ell}{\sqrt{-\mathcal{C}}}$ | Spherical$(\ell)$
| $e<-\kappa\sqrt{-\mathcal{C}}<0$ | $R_{+}\leq R\leq\infty$ | Deflecting$(e,\ell)$
Critical: $\ell=\ell_{*}$ ($\mathcal{C}=0$) | $e>0$ | $\kappa\leq R\leq\infty$ | | Plunging${}_{*}(e)$
---
Outward${}_{*}(e)$
| $e=0$ | $\kappa\leq R\leq\infty$ | | Plunging∗
---
Outward∗
| $-\kappa\ell<e<0$ | $\kappa\leq R\leq R_{0}$ | Bounded${}_{*}(e)$
| Subcritical: $0\leq\ell^{2}<\ell_{*}^{2}$
---
($0<\mathcal{C}\leq\frac{3\ell_{*}^{2}}{4}$)
$e>-\kappa\ell$ | $\kappa\leq R\leq R_{+}$ | Bounded${}_{<}(e,\ell)$
Critical: $\ell=-\ell_{*}$ ($\mathcal{C}=0$) | $e>-\kappa\ell>0$ | $\kappa\leq R\leq R_{0}$ | Bounded${}_{*}^{-}(e)$
| Supercritical: $\ell<-\ell_{*}$
---
($-\ell^{2}<\mathcal{C}<0$)
$e>-\kappa\ell>0$ | $\kappa\leq R\leq R_{-}$ | Bounded${}_{>}(e,\ell)$
Table 3.5: Taxonomy of future-directed geodesics in near-NHEK.
##### 3.2.2 Near-NHEK
The only difference between NHEK and near-NHEK geodesic solutions lies in the
terms involving the radial coordinate. The proposition stating the equivalence
relation between the equatorial and inclined radial parts of the geodesic
motion takes the same form as in NHEK:
###### Proposition 3.3.
For a given normalization $\kappa$, the radial integrals
$t^{(i)}_{R;\kappa}(R)$ $(i=0,1,2)$ only depend upon the near-NHEK energy $e$
and angular momentum $\ell$ while all the dependence upon the mass $\mu$ and
Carter constant $Q$ is through
$\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.
As in NHEK, the radial taxonomy of Ref. [58] is easily extended to bounded,
outward and/or retrograde orbits by studying the roots and the sign of
$v_{R;\kappa}(R)$. This leads to the classification displayed in Table 3.5 and
Figure 3.7.
The future-orientation condition (3.88) implies $e>-\kappa\ell$ for each orbit
that reaches the horizon at $R=\kappa$. In the case $\ell>\ell_{*}$ and $e<0$,
the condition $e\leq-\kappa\sqrt{-\mathcal{C}}$ implies $e+\kappa\ell\geq 0$
and therefore the parabola does not intersect the line. Past-oriented
geodesics (not depicted here) are obtained from a central symmetry around the
origin $e=\ell=0$ as a result of the $\uparrow\\!\downarrow$-flip (3.85). The
explicit expressions of all near-NHEK geodesics are listed in Appendix B.3.
Also notice that the energy range of the Deflecting$(e,\ell)$ class has been
corrected in this thesis with respect to the original derivation [20],
accordingly to the observation of [64].
Figure 3.7: Radial taxonomy of geodesics in near-NHEK. For equatorial
geodesics, $\ell_{*}=\frac{2}{\sqrt{3}}M\mu$, while for orbits with
inclination, $\ell_{*}=\frac{2}{\sqrt{3}}\sqrt{M^{2}\mu^{2}+Q}$.
##### 3.2.3 High-spin features of geodesic motion
Let us now discuss a few generic and universal features of near-horizon
geodesic motion holding in the high-spin case.
###### Radial motion
A first straightforward conclusion one can derive from the analysis of the
near-horizon radial geodesic motion is that
###### Proposition 3.4.
All radially unbounded NHEK or near-NHEK geodesics are prograde and either
critical or supercritical; _i.e._ , they satisfy $\ell\geq\ell_{*}$.
This feature of the near-horizon radial motion is directly visible in Figs.
3.5 and 3.7 and leads to remarkable consequences concerning the polar behavior
of such trajectories that we will derive in the following section.
The separatrix between bound and unbound motion is clearly visible in Figs.
3.5 and 3.7. It consists of the geodesic classes Plunging${}_{*}(E)$ and
Outward${}_{*}(E)$ for NHEK and the geodesic classes Plunging${}_{*}(e)$,
Outward${}_{*}(e)$, and Bounded${}_{*}(e)$ for near-NHEK that each lie at the
critical angular momentum line $\ell=\ell_{*}$.
###### Polar motion
The polar motion of both NHEK and near-NHEK trajectories is bounded in an
interval around the equator,
$\theta_{\text{\text{min}}}\leq\theta\leq\pi-\theta_{\text{\text{min}}}$,
where $\cos\theta_{\text{\text{min}}}=\sqrt{z_{+}}$ or
$\cos\theta_{\text{\text{min}}}=\sqrt{z_{0}}$. The maximal polar angle is
determined for $\ell^{2}\neq\ell^{2}_{\circ}=4M^{2}\mu^{2}$ as
$z_{+}(\ell,Q)=\frac{3\ell^{2}+4(Q+M^{2}\mu^{2})-\sqrt{9\ell^{4}+16(M^{2}\mu^{2}-Q)^{2}+8\ell^{2}(3M^{2}\mu^{2}+5Q)}}{2(4M^{2}\mu^{2}-\ell^{2})}$
and for $\ell=\pm\ell_{\circ}$ as
$\displaystyle z_{0}(Q)=\lim_{\ell\rightarrow\pm
2M\mu}z_{+}=\frac{Q}{Q+4M^{2}\mu^{2}}.$ (3.147)
Remember that $Q\geq 0$ by consistency of polar motion. The asymptotic values
are
$\displaystyle\lim_{\scriptsize{\begin{array}[]{l}Q\to 0\\\ \ell\text{
fixed}\end{array}}}z_{+}(\ell,Q)$
$\displaystyle=0,\qquad\qquad\qquad\qquad\qquad\;\;\;\;\;\;\;\;\lim_{\scriptsize{\begin{array}[]{l}Q\to\infty\\\
\ell\text{ fixed}\end{array}}}z_{+}(\ell,Q)=1,$ (3.152)
$\displaystyle\lim_{\scriptsize{\begin{array}[]{l}\ell\to 0\\\ Q\text{
fixed}\end{array}}}z_{+}(\ell,Q)$
$\displaystyle=\left\\{\begin{array}[]{cl}\frac{Q}{M^{2}\mu^{2}}&\text{ if
}Q<M^{2}\mu^{2}\\\ 1&\text{ if }Q\geq
M^{2}\mu^{2}\end{array}\right.,\qquad\lim_{\scriptsize{\begin{array}[]{l}\ell\to\infty\\\
Q\text{ fixed}\end{array}}}z_{+}(\ell,Q)=0.$ (3.159)
For fixed $\ell$, $z_{+}$ is a monotonic function of $Q$, and reciprocally
$z_{+}$ is monotonic in $\ell$ at fixed $Q$. The pendular oscillation around
the equatorial plane will explore a larger range of $\theta$ when
$\theta_{\text{\text{min}}}$ is smallest or $z_{+}$ closer to 1, which occurs
either for small $\ell$ and $Q\geq M^{2}\mu^{2}$ or large $Q$.
Now, one can check that for critical or supercritical angular momentum
$\ell^{2}\geq\ell^{2}_{*}(Q)$, one has $z_{+}<2\sqrt{3}-3$ for
$\ell\neq\ell_{\circ}(Q)$ and $z_{0}<2\sqrt{3}-3$ for
$\ell^{2}=\ell^{2}_{\circ}$. The special angle
$\displaystyle\theta_{\text{VLS}}\triangleq\arccos{\sqrt{2\sqrt{3}-3}}\approx
47^{\circ}$ (3.160)
is in fact the velocity-of-light surface in the NHEK geometry (1.43) (or near-
NHEK geometry) defined as the polar angle such that $\partial_{T}$ is null. It
obeys $\Lambda(\theta_{\text{VLS}})=1$. The polar region closer to either the
north or south poles admits a timelike Killing vector, namely $\partial_{T}$.
On the contrary, the polar region around the equator
$\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ does not admit a
timelike Killing vector. The velocity-of-light surface separates these two
polar regions. We have therefore proven the following property:
###### Proposition 3.5.
All critical or supercritical orbits $\ell^{2}\geq\ell^{2}_{*}$ in (near-)NHEK
geometry lie in the polar region
$\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ where there is no
timelike Killing vector. This applies in particular to all spherical orbits.
The subcritical orbits $\ell^{2}<\ell^{2}_{*}$ can explore all polar regions
of the (near)-NHEK geometry. As a consequence of Propositions 3.4 and 3.5, we
have
###### Proposition 3.6.
All radially unbounded geodesics in (near-)NHEK geometry lie in the polar
region $\theta\in]\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}[$ bounded by the
velocity-of-light surface.
In particular, for null geodesics, this feature provides the “NHEKline” in the
imaging of light sources around a nearly extreme Kerr black hole [30, 80]. In
[79, 57], Proposition 3.6 was proven for null geodesics. Here, we show that it
is a generic property of all timelike geodesics as well.
#### 3.3 Spherical geodesics
The spherical (near-)NHEK geodesics take a distinguished role among all
geodesics. First, a subclass of spherical geodesics in NHEK and near-NHEK
constitute the innermost stable spherical orbits (ISSOs) and the innermost
spherical bound orbits (ISBOs) in the high-spin limit, respectively. Our first
motivation is to fully characterize the ISSO, in order to generalize the
analysis of the inspiral/merger transition performed around the equatorial
plane in the high-spin limit [96, 97] to inclined orbits.
Second, as noticed in Ref. [58], the equatorial NHEK (resp. near-NHEK) orbits
are the simplest representatives for each equivalence class of prograde
incoming critical (respectively, supercritical) equatorial orbits under
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\mathbb{Z}_{2}$ symmetry.
We will show in Sec. 3.4 that the spherical (near-)NHEK orbits are the
simplest representatives for each equivalence class of arbitrary timelike
(near-)NHEK geodesics under
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times(\mathbb{Z}_{2})^{3}$
symmetry without any restriction. These two reasons justify the comprehensive
study of the spherical geodesics.
##### 3.3.1 Innermost stable spherical orbits
The ISSOs are defined as the last stable spherical orbits of Kerr. They are
defined from the solutions to
$\displaystyle R(r)=R^{\prime}(r)=R^{\prime\prime}(r)=0$ (3.161)
where $R$ is defined in (1.25b). They admit a constant radius $r$ and a fixed
$\hat{E}$ and $\ell$, which can be obtained as solutions of polynomial
equations which we will not give explicitly. There are two branches at
positive $\hat{E}$ corresponding to prograde ($\ell\geq 0$) and retrograde
orbits ($\ell<0$). For the Schwarzschild black hole, the parameters on the two
branches of the ISSO are
$\displaystyle
r_{\text{ISSO}}=6M,\qquad\frac{\hat{E}_{\text{ISSO}}}{\mu}=\frac{2\sqrt{2}}{3},\qquad\frac{\ell_{\text{ISSO}}}{\mu
M}=\pm\sqrt{12-\frac{Q}{M^{2}\mu^{2}}},$ (3.162)
which implies the bound $Q\leq 12M^{2}\mu^{2}$.
For arbitary spin, the innermost stable circular orbit (ISCO) is defined as
the prograde ISSO equatorial orbit, i.e. restricted to $Q=0$
($\theta=\frac{\pi}{2}$). The parameters are [30]
$\displaystyle\frac{\hat{E}_{\text{ISCO}}}{M\mu}=\frac{1-2/\tilde{r}_{\text{ISCO}}-\tilde{a}/\tilde{r}_{\text{ISCO}}^{3/2}}{\sqrt{1-3/\tilde{r}_{\text{ISCO}}-2\tilde{a}/\tilde{r}_{\text{ISCO}}^{3/2}}},\qquad\frac{\ell_{\text{ISCO}}}{\mu
M}=\frac{2}{\sqrt{3\tilde{r}_{\text{ISCO}}}}(3\sqrt{\tilde{r}_{\text{ISCO}}}+2\tilde{a}),$
(3.163)
where $\tilde{a}=a/M$ and
$\displaystyle\tilde{r}_{\text{ISCO}}$
$\displaystyle\triangleq\frac{r_{\text{ISCO}}}{M}=3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$
(3.164a) $\displaystyle Z_{1}$ $\displaystyle\triangleq
1+(1-\tilde{a}^{2})^{1/3}[(1+\tilde{a})^{1/3}+(1-\tilde{a})^{1/3}],\qquad
Z_{2}\triangleq\sqrt{3\tilde{a}^{2}+(Z_{1})^{2}}.$ (3.164b)
###### Minimal polar angle
In the generic case $\ell\neq 0$, the polar motion is pendular – i.e.,
oscillating around the equator in the interval
$[\theta_{\text{\text{min}}},\pi-\theta_{\text{\text{min}}}]$. The minimal
angle as a function of the spin $a$ and ISCO radius $r_{\text{ISSO}}$ can
simply be found by solving numerically the three equations (3.161) that define
the ISSO together with the condition that there is a polar turning point,
$\Theta(\cos\theta_{\text{\text{min}}})=0$ where $\Theta(\cos^{2}\theta)$ is
defined in (1.25c). The resulting minimal angle is displayed in Fig. 3.8 for a
large range of spins including nearly extremal. This completes a similar plot
drawn in Ref. [98] for spins far from extremality.
Figure 3.8: $\cos\theta_{\text{\text{min}}}$ as a function of ISSO radius for
several black hole spins $a$. Figure 3.9: Euclidean embedding of the ISSO
using the Boyer-Lindquist radius $r$, azimuthal angle $\varphi$ and polar
angle $\theta$ for $a=0.9999$. A cone is drawn at the critical polar angle
beyond which the ISSO lies in the NHEK region. In the extremal limit, this
polar angle is $\theta\approx 65^{\circ}$.
We note that for high-spins, the radius asymptotes to $r=M$ and the minimal
angle reaches a critical value around $0.42$ radians or $65^{\circ}$. When the
motion reaches regions sufficiently far from the equatorial plane, the ISSO
radius increases steeply and leaves the near-horizon region $r\simeq M$.
Another graphical representation of this behavior is shown in Fig. 3.9. We
will explain these features in the next section.
##### 3.3.2 The NHEK spherical orbit and the high-spin ISSOs
In the high-spin limit $a\rightarrow M$, the prograde ISSOs are characterized
by the following Boyer-Lindquist energies and angular momentum:
$\displaystyle\hat{E}_{\text{ISSO}}=\frac{1}{\sqrt{3}M}\sqrt{M^{2}\mu^{2}+Q},\qquad\ell_{\text{ISSO}}=+2M\hat{E}_{\text{ISSO}}$
(3.165)
and the following Boyer-Lindquist radius:
$\displaystyle
r_{\text{ISSO}}=M+M\left(\frac{Q+M^{2}\mu^{2}}{-Q+\frac{M^{2}\mu^{2}}{2}}\right)^{1/3}\lambda^{2/3}+\mathcal{O}(\lambda^{4/3}).$
(3.166)
Given the scaling in $\lambda$, for the range
$\displaystyle 0\leq Q\leq\frac{M^{2}\mu^{2}}{2},$ (3.167)
the ISSOs belong to the NHEK geometry and admit the NHEK radius
$\displaystyle
R=R_{\text{ISSO}}\triangleq\left(\frac{Q+M^{2}\mu^{2}}{-Q+\frac{M^{2}\mu^{2}}{2}}\right)^{1/3}.$
(3.168)
In particular, the ISCO has the minimal radius $R_{\text{ISCO}}=2^{1/3}$. In
terms of NHEK quantities, the orbits admit a critical angular momentum and a
vanishing NHEK energy,
$\displaystyle\ell=\ell_{*}\triangleq\frac{2}{\sqrt{3}}\sqrt{Q+M^{2}\mu^{2}},\qquad
E=0.$ (3.169)
In the high-spin limit, the prograde ISSOs in the range (3.167) are therefore
exactly the $\text{Spherical}_{*}(Q)$ orbits in the classification of Sec.
1.3.2. The prograde ISSOs outside the range (3.167) and the retrograde ISSOs
do not belong to the near-horizon geometry and will not be described here.
In terms of polar behavior, $\text{Spherical}_{*}(Q)$ orbits are instances of
$\text{Pendular}(Q,\ell_{*})$ motion (except for $Q=0$, where they are just
equatorial orbits). In the range (3.167), they admit an $\epsilon_{0}$ as
defined in (3.3) given by $\epsilon_{0}=\frac{Q-2M^{2}\mu^{2}}{3}<0$, and the
angular momentum lies below the value $\ell_{\circ}$:
$\displaystyle\ell_{*}\leq\sqrt{2}M\mu<\ell_{\circ}.$ (3.170)
The main property of $\text{Pendular}(Q,\ell_{*})$ motion is that the polar
angle $\theta$ is bounded in an interval around the equator (see (3.68) and
(3.4)) :
$\theta\in[\theta_{\text{min}},\pi-\theta_{\text{min}}]$ (3.171)
where
$\cos\theta_{\text{min}}=\sqrt{z_{+}}=\sqrt{\frac{Q}{\frac{3}{4}\ell^{2}_{*}+\sqrt{\frac{9}{16}\ell_{*}^{4}-\frac{\ell_{*}^{2}Q}{2}+Q^{2}}}}.$
(3.172)
At fixed $M\mu$, $\theta_{\text{min}}(Q)$ is a monotonic function
interpolating between the equator $\theta=90^{\circ}$ at $Q=0$ and
$\theta_{\text{VLS}}\triangleq\arccos{\sqrt{2\sqrt{3}-3}}\approx 47^{\circ}$
for $Q\rightarrow\infty$. The special angle $\theta_{\text{VLS}}$ is the
velocity-of-light surface in the NHEK geometry (1.43) as described in Sec.
3.2.3. The ISSO therefore always lies in the region of NHEK spacetime around
the equator, where there is no timelike Killing vector. This is depicted in
Fig. 3.10.
| | | … |
---|---|---|---|---
$Q$
Figure 3.10: For increasing $Q\geq 0$, $\text{Spherical}_{*}(Q)$ orbits can
explore a equator-centered band whose width becomes larger, finally reaching
for $Q\rightarrow\infty$ the angular range
$\theta\in[\theta_{\text{VLS}},\pi-\theta_{\text{VLS}}]$ bounded by the
velocity-of-light surface. The prograde IBSOs lie in the near-NHEK region for
$Q\leq 2M^{2}\mu^{2}$, which further bounds the angular range.
However, since the ISSO admits the range (3.167) due to its relationship to
the asymptotically flat Boyer-Lindquist radius (3.166), the limiting angle is
reached first for $Q=\frac{M^{2}\mu^{2}}{2}$ at
$\arccos{\sqrt{3-2\sqrt{2}}}\approx 65^{\circ}$. This explains the behavior
depicted in Fig. 3.8. This result was discovered simultaneously in [62, 20].
The limiting angle of the ISSO is given by
$\arcsin{\sqrt{2(\sqrt{2}-1)}}=\arccos{\sqrt{3-2\sqrt{2}}}\approx 65^{\circ}$.
##### 3.3.3 The near-NHEK spherical orbits and the high-spin IBSOs
The innermost bound spherical orbits (IBSOs) are determined by the equations
$\displaystyle R(r)=R^{\prime}(r)=0,\qquad\hat{E}=\mu.$ (3.174)
In the high-spin limit $\lambda\rightarrow 0$, the angular momentum and Boyer-
Lindquist radius of the prograde IBSOs are given by
$\displaystyle\ell$
$\displaystyle=\ell_{\circ}\left(1+\frac{\lambda}{\sqrt{2}}\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}+\mathcal{O}(\lambda^{2})\right),$
(3.175a) $\displaystyle r$
$\displaystyle=M\left(1+\frac{\sqrt{2}\lambda}{\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}}+\mathcal{O}(\lambda^{2})\right)$
(3.175b)
where $\ell_{\circ}\equiv 2M\mu$. In particular, for $Q=0$ we recover the
scaling of the innermost bound circular orbit (IBCO) [30]. Given the scaling
$\sim\lambda$, the prograde IBCOs therefore lie in the near-NHEK region for
all $Q<2M^{2}\mu^{2}$. Using (1.63)–(1.72), the angular momentum, near-NHEK
energy and near-NHEK radius are given in the high-spin limit by
$\displaystyle\ell$ $\displaystyle=\ell_{\circ},$ (3.176a)
$\displaystyle\frac{e}{\kappa}$ $\displaystyle=-\sqrt{2M^{2}\mu^{2}-Q},$
(3.176b) $\displaystyle\frac{r}{\kappa}$
$\displaystyle=\frac{\sqrt{2}\lambda}{\sqrt{1-\frac{Q}{2M^{2}\mu^{2}}}}.$
(3.176c)
The prograde IBCOs in the range $0\leq Q<2M^{2}\mu^{2}$ are described by
instances of Spherical$(\ell)$ orbits. In terms of polar motion, $Q=0$ are
equatorial and $Q>0$ are pendular of class Pendular${}_{\circ}(Q)$; see Table
3.3. The polar range is determined as
$\theta_{\text{\text{min}}}\leq\theta\leq\pi-\theta_{\text{\text{min}}}$ where
$\displaystyle\theta_{\text{\text{min}}}=\arccos\sqrt{\frac{Q}{Q+\ell_{\circ}^{2}}}.$
(3.177)
The maximal polar angle reachable within the near-NHEK region by IBSOs is
obtained for the limiting value $Q=2M^{2}\mu^{2}$ at
$\displaystyle\theta_{\text{\text{min}}}=\arccos\sqrt{1/3}=\arcsin{\sqrt{2/3}}\approx
55^{\circ}.$ (3.178)
This critical angle was also previously obtained in Refs. [78, 62]. Finally,
note that spherical photon orbits in the high-spin limit were also discussed
in Refs. [99, 100].
#### 3.4 Conformal mappings between radial classes
The near-horizon region of near-extremal Kerr black holes admits four Killing
vectors forming the group $\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$,
hereafter denoted as the conformal group $G$. The geodesic equations are
invariant under $G$ and the geodesics therefore transform under the action of
$G$. Moreover, a group generated by four $\mathbb{Z}_{2}$ symmetries exists
that preserve the geodesic equations. The subgroup preserving the domain $R>0$
for NHEK (or $r>0$ for near-NHEK) is generated by the
$\uparrow\\!\downarrow$-flip (3.85), which flips the geodesic orientation, and
two additional $\mathbb{Z}_{2}$ transformations that preserve the geodesic
orientation: namely, the parity flip
$\displaystyle\theta\rightarrow\pi-\theta,\qquad\Phi\rightarrow\Phi+\pi,\qquad
s_{\theta}^{i}\rightarrow-s_{\theta}^{i},$ (3.179)
and the $\rightleftarrows$-flip
$\displaystyle
T\rightarrow-T,\qquad\Phi\rightarrow-\Phi,\qquad\lambda\rightarrow-\lambda,\qquad
s_{R}^{i}\rightarrow-s_{R}^{i},\qquad s_{\theta}^{i}\rightarrow-
s_{\theta}^{i}.$ (3.180)
The last discrete transformation that we use as a basis is the
$\rightleftarrows$ -flip
$\displaystyle
R\rightarrow-R,\qquad\Phi\rightarrow-\Phi,\qquad\ell\rightarrow-\ell,\qquad
s^{i}_{R}\rightarrow-s^{i}_{R}.$ (3.181)
The parity transformation defined in (3.179) leaves each motion invariant and
will not be considered further. The $\rightleftarrows$-flip changes the
boundary conditions of the geodesics, which may affect their denomination. It
maps bounded orbits to bounded orbits, and deflecting orbits to deflecting
orbits, but plunging orbits to outward orbits, as illustrated in Fig. 3.11.
For bounded orbits, the part before the turning point is mapped to the part
after the turning point, and vice-versa. The $\rightleftarrows$ -flip can be
used as follows: one first continues a geodesic defined in $R>0$ beyond the
horizon $R=0$ and the resulting geodesic with $R<0$ is then mapped to a
geodesic in the $R>0$ region using the $\rightleftarrows$ -flip. Together
with the action of (3.180), it allows us to map plunging orbits with $\ell>0$
to bounded orbits with $\ell<0$. This process is illustrated in Fig. 3.12.
|
---|---
(a) | (b)
Figure 3.11: Penrose diagram of NHEK spacetime depicting the action of the $\rightleftarrows$-flip on (a) plunging and (b) bounded geodesics. Under this transformation, a trajectory belonging to the patch I is mapped to an orbit of the patch I’. While plunging geodesics become outward ones, bounded motion remains bounded. The energy and angular momentum of the trajectory are unchanged. | |
---|---|---
(a) | (b) | (c)
Figure 3.12: Penrose diagram representation of the construction of a critical
NHEK bounded geodesic from a plunging one. (a) Continuation of the trajectory
beyond the horizon in patch I’ until the radial potential root (depicted with
dashes); (b)
$\rightleftarrows$
-flip which brings the part of the bounded geodesic before the turning point in the NHEK Poincaré patch I; (c) $\rightleftarrows$-flip, which maps the part of the bounded geodesic before the turning point to the part after the turning point.
The equivalence classes of equatorial critical and supercritical prograde
timelike geodesics under the action of
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow$
symmetry were derived in Ref. [58] following earlier work [73, 74, 75, 77]. In
this section, we will perform the decomposition of arbitrary geodesics into
equivalence classes under the action of
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow\times\rightleftarrows\times$
$\rightleftarrows$ .
The Casimir $\mathcal{C}$ of $\textsf{SL}(2,\mathbb{R})$ cannot vary upon
acting with $G\triangleq\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$
transformations. Moreover, the action of the group $G$ acts trivially on the
polar coordinate $\theta$. These two properties imply that both $Q$ and $\ell$
are invariant under the action of $G$. In particular, critical, supercritical
or subcritical geodesics form distinct classes under $G$. On the contrary, the
(near-)NHEK energy $E$ (or $e$) can vary under conformal transformations.
Conformal transformations can map NHEK to near-NHEK orbits, and vice-versa. As
a result of Propositions 3.2 and 3.3, null geodesics can be treated on the
same footing as timelike geodesics.
A conformal transformation belonging to
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)$ maps (near)-NHEK spacetime
parametrized by $(T,R,\theta,\Phi)$ to (near-)NHEK spacetime parametrized by
$(\bar{T},\bar{R},\theta,\bar{\Phi})$222We denote here without distinction
NHEK and near-NHEK coordinates with capital letters. where
$\displaystyle\overline{T}$ $\displaystyle=\overline{T}(T,R),$
$\displaystyle\overline{R}$ $\displaystyle=\overline{R}(T,R),$ (3.186)
$\displaystyle\overline{\Phi}$ $\displaystyle=\Phi+\delta\bar{\Phi}(T,R).$
The geodesic equations in (near)-NHEK imply $T=T(R)$. Therefore, the action of
conformal symmetries reduces to an action on the radial motion, leaving the
polar motion unchanged. More precisely, in the decomposition of
$\Phi(\lambda)$ (1.60c)–(1.74c) in terms of a radial part and a polar part,
the polar part will remain untouched by conformal transformations.
It was shown in Ref. [58] that each equivalence class of equatorial prograde
critical (respectively, supercritical) geodesics with incoming boundary
conditions under $G\times\uparrow\\!\downarrow$ admits a distinguished simple
representative, namely the NHEK (respectively, near-NHEK) circular orbits.
After analysis, we obtain that each geodesic equivalence class under
$G\times\uparrow\\!\downarrow\times\rightleftarrows\times$ $\rightleftarrows$
admits a spherical orbit as the simplest representative as illustrated in Fig.
3.13. Past directed geodesics must be considered as intermediate steps in
order to relate each future directed geodesic to spherical geodesics.
Supercritical orbits ($\ell^{2}>\ell^{2}_{*}$) admit the near-NHEK
Spherical$(\ell)$ orbit as a representative and critical orbits
($\ell=\pm\ell_{*}$) admit the NHEK Spherical∗ orbit as a representative. No
subcritical spherical geodesic exists. However, we introduce an analytically
continued complex subcritical geodesic by continuing the radius $R_{0}\mapsto
iR_{0}$ and show that it generates the subcritical class.
The explicit formulas for the three categories of equivalence classes of
orbits under $G\times\uparrow\\!\downarrow\times\rightleftarrows\times$
$\rightleftarrows$ are given in the following sections. We will denote the
final coordinates and orbital parameters reached by the conformal mappings
with bars.
##### 3.4.1 Critical $\mathcal{C}=0$
###### Spherical∗ $\Leftrightarrow$ Plunging${}_{*}(E)$ (NHEK/NHEK).
The conformal mapping is given by
$\displaystyle\bar{T}$ $\displaystyle=-\frac{R^{2}T}{R^{2}T^{2}-1},$
$\displaystyle\bar{R}$ $\displaystyle=\frac{R^{2}T^{2}-1}{R},$ (3.187)
$\displaystyle\bar{\Phi}$ $\displaystyle=\Phi+\log\frac{RT+1}{RT-1}-i\pi.$
It maps a (future-directed) NHEK spherical trajectory of radius $R_{0}$ to a
(future-directed) critical plunge of energy
$\bar{E}=\frac{2\ell_{*}}{R_{0}}>0$.
###### Spherical∗ $\Leftrightarrow$ Plunging∗ (NHEK/near-NHEK).
One performs the NHEK/near-NHEK diffeomorphism
$(T,R,\theta,\Phi)\to(\bar{t},\bar{R},\theta,\bar{\phi})$, whose explicit form
is
$\displaystyle T$
$\displaystyle=-\exp\quantity(-\kappa\bar{t})\frac{\bar{R}}{\sqrt{\bar{R}^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\frac{1}{\kappa}\exp\quantity(\kappa\bar{t})\sqrt{\bar{R}^{2}-\kappa^{2}},$
(3.188) $\displaystyle\Phi$
$\displaystyle=\phi-\frac{1}{2}\log\frac{\bar{R}-\kappa}{\bar{R}+\kappa}.$
Its inverse is
$\displaystyle\bar{t}$
$\displaystyle=\frac{1}{\kappa}\log\frac{R}{\sqrt{R^{2}T^{2}-1}},$
$\displaystyle\bar{R}$ $\displaystyle=-\kappa RT,$ (3.189)
$\displaystyle\bar{\phi}$
$\displaystyle=\Phi+\frac{1}{2}\log\frac{RT+1}{RT-1}$
for $R>0$ and $RT<-1$. The orbital parameters are related as
$\displaystyle R_{0}=\frac{1}{\kappa}\exp\quantity(\kappa
t_{0}),\qquad\Phi_{0}=\phi_{0}-\frac{3}{4}.$ (3.190)
###### Plunging∗ $\Leftrightarrow$ Outward∗ (near-NHEK/near-NHEK).
The orbits are related by the $\rightleftarrows$-flip (3.180).
###### Plunging∗ $\Leftrightarrow$ Plunging${}_{*}(e)$ (near-NHEK/near-NHEK).
The two (future-directed) orbits are related via the diffeomorphism
$\displaystyle\bar{t}$
$\displaystyle=\frac{1}{2\kappa}\log\frac{\sqrt{R^{2}-\kappa^{2}}\cosh{\kappa
t}-R}{\sqrt{R^{2}-\kappa^{2}}\cosh{\kappa t}+R}-\frac{i\pi}{\kappa},$
$\displaystyle\bar{R}$ $\displaystyle=\sqrt{R^{2}-\kappa^{2}}\sinh{\kappa t},$
(3.191) $\displaystyle\bar{\phi}$
$\displaystyle=\phi+\frac{1}{2}\log\frac{R\sinh\kappa t+\kappa\cosh\kappa
t}{R\sinh\kappa t-\kappa\cosh\kappa t}.$
The energy of the new trajectory is a function of the initial time $t_{0}$ of
the former one:
$\bar{e}=\kappa^{2}\ell_{*}\exp\quantity(-\kappa t_{0})>0.$ (3.192)
###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Outward${}_{*}(e)$ (near-
NHEK/near-NHEK).
The orbits are related by the $\rightleftarrows$-flip.
###### Plunging${}_{*}(E)$ $\Leftrightarrow$ Bounded${}_{*}^{-}(E)$
(NHEK/NHEK).
The critical bounded orbit is obtained from the plunging orbit by a
continuation of the trajectory beyond the horizon ($R<0$) combined with
$\mathbb{Z}_{2}$ flips. One must proceed in three steps:
1. 1.
Continue the plunge defined from the physical domain $0\leq R\leq\infty$ to
its whole domain of definition $R_{0}\leq R\leq\infty$ (i.e., up to the root
of the radial potential $R_{0}=-\frac{e}{2\ell_{*}}$) and consider now only
the part of the trajectory located beyond the horizon $R_{0}\leq R\leq 0$.
2. 2.
Apply the $\rightleftarrows$ -flip to the latter part of the solution. This
transformation restores the positivity of the radial coordinate. It preserves
the time orientation of the geodesic but flips the sign of its angular
momentum $\ell_{*}\to-\ell_{*}$. The new domain of definition of the
trajectory is consequently $0\leq R\leq\frac{E}{2\ell_{*}}$.
3. 3.
The procedure outlined above only leads to the part of the geodesic with
$R^{\prime}(\lambda)>0$, which is located before the turning point. As
outlined in Appendix B.2, the part of a bounded trajectory located after the
turning point can be obtained from the one located before it by a
$\rightleftarrows$-flip.
This whole procedure is represented in Fig. 3.12.
###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Bounded${}_{*}^{-}(e)$ (near-
NHEK/near-NHEK).
The mapping is similar to the one outlined above using the $\rightleftarrows$
-flip. One subtlety is that one should start with the Plunging${}_{*}(e)$
orbit with $e>\kappa\ell_{*}$ in order to obtain the future-directed
Bounded${}_{*}^{-}(e)$ orbit.
###### Plunging${}_{*}(e)$ $\Leftrightarrow$ Bounded${}_{*}(-e)$ (near-
NHEK/near-NHEK).
We apply the $\rightleftarrows$ -flip as outlined in the previous paragraph,
but now choosing $0<e<\kappa\ell_{*}$. This leads to a retrograde past-
directed bounded orbit. The future-directed prograde geodesic is then reached
using the $\uparrow\\!\downarrow$-flip.
##### 3.4.2 Supercritical $\mathcal{C}<0$
###### Spherical$(\ell)$ $\Leftrightarrow$ Marginal$(\ell)$ (near-NHEK/NHEK).
One applies the NHEK/near-NHEK diffeomorphism
$\displaystyle T$
$\displaystyle=-\exp\quantity(-\kappa\bar{t})\frac{\bar{R}}{\sqrt{\bar{R}^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\frac{1}{\kappa}\exp\quantity(\kappa\bar{t})\sqrt{\bar{R}^{2}-\kappa^{2}},$
(3.193) $\displaystyle\Phi$
$\displaystyle=\phi-\frac{1}{2}\log\frac{\bar{R}-\kappa}{\bar{R}+\kappa}$
which maps the orbit Spherical$(\ell)$ on the past-directed Marginal$(-\ell)$
orbit. The future-directed Marginal$(\ell)$ orbit is recovered by composing
this transformation with a $\uparrow\\!\downarrow$-flip.
###### Marginal$(\ell)$ $\Leftrightarrow$ Plunging$(E,\ell)$ or
Deflecting$(E,\ell)$ (NHEK/NHEK).
One performs the transformation ($\zeta\neq 0$)
$\displaystyle\bar{T}$
$\displaystyle=\frac{1}{\bar{R}}\frac{2R^{2}T\cos\zeta-(1+R^{2}(1-T^{2}))\sin\zeta}{2R},$
$\displaystyle\bar{R}$
$\displaystyle=\frac{R^{2}(1+T^{2})-1+(1+R^{2}(1-T^{2}))\cos\zeta+2R^{2}T\sin\zeta}{2R},$
(3.194) $\displaystyle\bar{\Phi}$
$\displaystyle=\Phi+\log\frac{\cos\frac{\zeta}{2}R+\sin\frac{\zeta}{2}(RT+1)}{\cos\frac{\zeta}{2}R+\sin\frac{\zeta}{2}(RT-1)}.$
As outlined in Ref. [58], this mapping can be viewed as the action on Poincaré
NHEK coordinates of a shift of the global NHEK time $\tau\to\tau-\zeta$. The
energy of the final orbit is
$\displaystyle\bar{E}=\sqrt{-\mathcal{C}}\quantity(\sin\zeta+T_{0}(\cos\zeta-1)).$
(3.195)
We directly see that any energy $E\neq 0$ can be reached by conveniently
choosing the values of $T_{0}$ and $\zeta$.
###### Plunging$(E,\ell)$ $\Leftrightarrow$ Outward$(E,\ell)$ (NHEK/NHEK).
The orbits are related by the $\rightleftarrows$-flip.
###### Plunging$(E,\ell)$ $\Leftrightarrow$ Bounded${}_{>}(E,-\ell)$
(NHEK/NHEK).
The mapping consists in extending the radial range of the plunging orbit
beyond the horizon, $R<0$, then using the $\rightleftarrows$ -flip, which
leads to the Bounded${}_{>}(E,-\ell)$ orbit.
###### Spherical$(\ell)$ $\Leftrightarrow$ Plunging$(e,\ell)$ or
Deflecting$(e,\ell)$ (near-NHEK/ near-NHEK).
One uses the diffeomorphism ($\chi\neq\pm 1$)
$\displaystyle t$
$\displaystyle=\frac{1}{\kappa}\log\frac{\sqrt{\bar{R}^{2}-\kappa^{2}}\cosh\kappa\bar{t}-\bar{R}}{\sqrt{R^{2}-\kappa^{2}}},$
$\displaystyle R$
$\displaystyle=\sqrt{\bar{R}^{2}-\kappa^{2}}\quantity(\sinh\kappa\bar{t}+\chi\cosh\kappa\bar{t})-\chi\bar{R},$
(3.196) $\displaystyle\phi$
$\displaystyle=\bar{\phi}-\frac{1}{2}\log\quantity[\frac{\sqrt{\bar{R}^{2}-\kappa^{2}}-\bar{R}\cosh\kappa\bar{t}+\kappa\sinh\kappa\bar{t}}{\sqrt{\bar{R}^{2}-\kappa^{2}}-\bar{R}\cosh\kappa\bar{t}-\kappa\sinh\kappa\bar{t}}\frac{R+\kappa}{R-\kappa}].$
This mapping can be seen as a NHEK global time shift written in near-NHEK
coordinates; see Refs. [58, 77]. The explicit inversion formula can be found
in Ref. [77]. The energy of the new trajectory reads as
$\displaystyle\bar{e}=\kappa\sqrt{-\mathcal{C}}\,\chi.$ (3.197)
For $-\frac{\ell}{\sqrt{-\mathcal{C}}}<\chi<-1$, the orbit reached is future-
directed and deflecting. The trajectory becomes plunging for $\chi>-1$. Note
that for $\absolutevalue{\chi}>1$,
$\bar{t}_{0}=-\frac{1}{2\kappa}\log\frac{1+\chi}{1-\chi}$ is complex and one
has to perform an additional shift on $\bar{t}$ to make it real.
###### Plunging$(e,\ell)$ $\Leftrightarrow$ Outward$(e,\ell)$ (near-NHEK/near-
NHEK).
The orbits are related by the $\rightleftarrows$-flip.
###### Plunging$(e,\ell)$ $\Leftrightarrow$ Bounded${}_{>}(e,-\ell)$ (near-
NHEK/near-NHEK).
The mapping consists in extending the radial range of the plunging orbit with
$e>\kappa\ell$ beyond the horizon, $r<0$, then using the $\rightleftarrows$
-flip, which leads to the Bounded${}_{>}(e,-\ell)$ orbit.
##### 3.4.3 Subcritical $\mathcal{C}>0$
There is no near-NHEK spherical geodesic for $\mathcal{C}>0$. We can
nevertheless introduce the formal class of complex spherical trajectories
$\displaystyle\begin{split}t(\lambda)&=-i\frac{\ell}{R_{0}}\lambda,\\\
R(\lambda)&=iR_{0},\qquad
R_{0}\triangleq\frac{\kappa\ell}{\sqrt{\mathcal{C}}},\\\
\phi(\lambda)&=\phi_{0}-\frac{3}{4}\ell\lambda+\ell\Phi_{\theta}(\lambda)\end{split}$
(3.198)
which is a formal (but nonphysical) solution of the near-NHEK geodesic
equations, of complex near-NHEK “energy” $e=-i\kappa\sqrt{\mathcal{C}}$. We
will denote this class of solutions as Spherical${}_{\mathbb{C}}(\ell)$ and
show that it can be used to generate all subcritical bounded trajectories by
acting on it with properly chosen conformal transformations. The parametrized
form of the orbit reads as
$\displaystyle\begin{split}R&=iR_{0},\\\
\phi(t)&=\phi_{0}-\frac{3}{4}iR_{0}t+\ell\Phi_{\theta}(\lambda(t)).\end{split}$
(3.199)
###### Spherical${}_{\mathbb{C}}(\ell)$ $\Leftrightarrow$
Bounded${}_{<}(E,\ell)$.
One has to proceed in two steps, mimicking the procedure used to obtain the
NHEK Plunging$(E,\ell)$ class:
* $\blacktriangleright$
We apply the near-NHEK/NHEK diffeomorphism (3.193) to a
Spherical${}_{\mathbb{C}}(\ell)$ orbit, leading to another complex NHEK
geodesic of null energy parametrized by
$\displaystyle\begin{split}T(R)&=-\frac{i\ell}{\sqrt{C}R},\\\
\Phi(R)&=\Phi_{0}-\frac{3i\ell}{8\sqrt{C}}\log\frac{\mathcal{C}R^{2}}{\mathcal{C}+\ell^{2}}\end{split}$
(3.200)
with the initial azimuthal angle
$\Phi_{0}\triangleq\phi_{0}-\frac{3\pi\ell}{8\sqrt{C}}-\frac{1}{2}\log\quantity(1-\frac{2\sqrt{C}}{\sqrt{C}+i\ell})$.
We denote this class as Marginal${}_{\mathbb{C}}(\ell)$.
* $\blacktriangleright$
Second, we apply to the trajectory found above the global time shift (3.194),
but upgraded with an imaginary parameter $\zeta\to i\zeta$. This leads to the
Bounded${}_{<}(E,\ell)$ class with orbital parameters
$\displaystyle\bar{E}$ $\displaystyle=\sqrt{\mathcal{C}}\sinh\zeta,$ (3.201a)
$\displaystyle\bar{\Phi}_{0}$
$\displaystyle=\phi_{0}-\frac{3\pi\ell}{8\sqrt{\mathcal{C}}}-\log\quantity(\sqrt{\mathcal{C}}-i\ell)+\frac{3i\ell}{8\sqrt{\mathcal{C}}}\log\quantity[\mathcal{C}(\mathcal{C}+\ell^{2})\quantity(1+\sqrt{\frac{\mathcal{C}+E^{2}}{\mathcal{C}}})^{2}]$
$\displaystyle~{}-\frac{3\ell}{8\sqrt{C}}\log\quantity[E^{2}(\mathcal{C}+\ell^{2})]+\arctan\frac{\sqrt{\mathcal{C}}}{\ell}.$
(3.201b)
Note that choosing $\zeta>0$ is sufficient to reach the full range of energies
allowed for such a geodesic ($E>0$). Any geodesic of orbital parameters
($T_{0},\tilde{\Phi}_{0}$) can finally be obtained by performing the
transformation $T\to T+T_{0}$, $\Phi\to\Phi-\bar{\Phi}_{0}+\tilde{\Phi}_{0}$,
which also removes the unphysical imaginary part of the azimuthal coordinate.
###### Spherical${}_{\mathbb{C}}(\ell)$ $\Leftrightarrow$
Bounded${}_{<}(e,\ell)$.
We apply to the Spherical${}_{\mathbb{C}}(\ell)$ class the near-NHEK global
time shift (3.196) upgraded with an imaginary parameter $\chi\to i\chi$
($\chi\neq\pm 1$), leading to a Bounded${}_{<}(e,\ell)$ orbit of parameters
$\displaystyle\begin{split}\bar{e}&=\kappa\sqrt{\mathcal{C}}\,\chi,\\\
\bar{t}_{0}&=t_{0}+\frac{i}{\kappa}\arctan\frac{\kappa\sqrt{\mathcal{C}}}{e},\\\
\bar{\phi}_{0}&=\bar{\phi}_{0}(\phi_{0},e,\ell,\mathcal{C},\kappa).\end{split}$
(3.202)
The explicit value of $\bar{\phi}_{0}$ is easily calculable, but too long to
be reproduced here. To reach a manifestly real orbit of orbital parameters
$(\tilde{t}_{0},\tilde{\phi}_{0})$, one has to perform the final shift
$t\to
t-\bar{t}_{0}+\tilde{t}_{0},\qquad\phi\to\phi-\bar{\phi}_{0}+\tilde{\phi}_{0}.$
(3.203)
Figure 3.13: Schematic overview of the action of the group
$\textsf{SL}(2,\mathbb{R})\times\textsf{U}(1)\times\uparrow\\!\downarrow\times\rightleftarrows\times\\!$
$\rightleftarrows$
on near-horizon geodesics.
## Part II Test bodies in Curved Spacetime:
Theoretical Foundations
Figure II.1: The two main pictures for the description of extended test bodies
used in this text: the gravitational skeletonization (left) and the Lagrangian
formulation (right).
Let us consider the motion of a object described by some smooth stress-energy
tensor $T_{\mu\nu}$ in a fixed background metric $g_{\mu\nu}$, thus neglecting
self-force effects. Provided that the body has a finite spatial extension, its
stress-energy tensor is supported on compact slices for any 3+1 decomposition
of the spacetime. Such an object will be referred to as an extended test body.
We have in mind the motion of a “small” astrophysical object (stellar mass
black hole or neutron star) around a hypermassive black hole. In this
situation, the former can be viewed as a perturbation of the spacetime
geometry created by the later.
While the geodesic equations describe the motion of a structureless monopole
test body in a fixed background spacetime, an important generalization is to
allow the test body (while still having negligible mass and thus negligible
influence on the gravitational field) to have a finite size and a nontrivial
structure. All these effects – departing from a bare geodesic motion – are
known as finite size effects.
###### Worldline description of extended test bodies
In the case where the extended test body is compact, that is, if its typical
size $\ell$ is small compared to the radius of curvature $r$ of the background
($\ell\ll r$), there exists various equivalent approximation schemes for
describing its motion in a somehow simpler way than considering its full
stress-energy tensor. Both approaches end into characterizing the body by a
centroid worldline $\gamma=\quantity{z^{\mu}(\lambda)}$ along which a tower of
gravitational multipole moments $I^{\mu\nu\alpha_{1}\ldots\alpha_{n}}$
replacing the original stress-energy tensor are defined, see Figure II.1.
These moments can be understood as spatial integrals of $T^{\mu\nu}$,
$I^{\mu\nu\alpha_{1}\ldots\alpha_{n}}\triangleq\int_{x^{0}=\text{cst}}\differential[3]{x}\sqrt{-g}T^{\mu\nu}\delta
x^{\alpha_{1}}\ldots\delta x^{\alpha_{n}},$
where $\delta x^{\mu}\triangleq x^{\mu}-z^{\mu}(\lambda)$.
The first of these schemes is known as the gravitational skeletonization: the
body is described by a distributional stress-energy tensor, which is non-
vanishing only on a certain worldline, and contains the aforementioned tower
of multipole moments. This tensor must be conserved within the background,
$\nabla_{\mu}T^{\mu\nu}=0$, and one can show that it implies that the monopole
$p_{\mu}$ and dipole $S_{\mu\nu}$ must evolve according to the Mathisson-
Papapetrou-Dixon (MPD) equations [101, 102, 103]
$\displaystyle\frac{\text{D}p^{\mu}}{\differential\tau}$
$\displaystyle=-\frac{1}{2}R^{\mu}_{\phantom{\mu}\nu\alpha\beta}v^{\nu}S^{\alpha\beta}+\ldots,\qquad\frac{\text{D}S^{\mu\nu}}{\differential\tau}=2p^{[\mu}v^{\nu]}+\ldots,$
where the dots represent corrections due to the quadrupole and higher
multipole moments. The monopole $p^{\mu}$ takes the interpretation of the
linear momentum of the body, whereas the dipole $S^{\mu\nu}$ can be seen as
its skew-symmetric spin tensor, describing the relativistic angular momentum
of the body. This last object will play a central role in our description,
which is the reason why we will sometimes refer to extended test bodies as
spinning test bodies. In terms of the original smooth stress-energy
distribution, one can show that the two first moments are related to the
original stress-energy tensor by
$\displaystyle p^{\mu}$
$\displaystyle\triangleq\int_{x^{0}=\text{constant}}\differential[3]{x}\sqrt{-g}\,T^{\mu
0},$ $\displaystyle S^{\mu\nu}$
$\displaystyle\triangleq\int_{x^{0}=\text{constant}}\differential[3]{x}\sqrt{-g}\,\quantity(\delta
x^{\mu}T^{\nu 0}-\delta x^{\nu}T^{\mu 0}).$
This approach has been investigated since the late thirties. The leading-order
EOMs were first derived in the seminal works of M. Mathisson [101, 104] and A.
Papapetrou [102]. They have been subsequently generalized to higher multipolar
orders by W.G. Dixon [105, 106, 107, 108]. Despite its elegance and rigour,
this approach appears to be quite long to perform and technically involved,
discarding it from being a well-suited viewpoint for exposing comprehensively
the problem in an introductory text like the present one.
We will instead follow the Lagrangian approach, whose generic formulation in
curved spacetime is due to I. Bailey and W. Israel in 1975 [109].
Nevertheless, this method was pioneered for Special Relativity in earlier
works, notably by A.J. Hanson and T. Regge (see e.g. [110, 111]). It consists
in formulating a generic action principle for the extended body modelled as a
worldline, representing the motion of some “center” of its stress-energy
distribution, and endowed with an orthonormal tetrad rigidly attached to it,
whose evolution describes the orientation of the body. This approach also
leads to the very same MPD equations. Having two equivalent descriptions of
the same problem is extremely fruitful, since each of them turns out to be
more appropriated for different purposes: as we will see, the skeletonization
will be powerful for providing us with physical insights about the
interpretation of multipole moments, while the Lagrangian approach will reveal
particularly useful when we will turn to the Hamiltonian description of
extended test bodies.
Others routes yielding the same equations of motion have also been followed. A
supersymmetric description of classical spinning particles has been provided
in 1993 by G.W. Gibbons, R.H. Rietdijk and J.W. van Holten [112], and recently
extended to include quadrupole effects [113]. Another recent (and somehow
elegant) formulation accounting for the description of finite size effects is
due to A. Harte [114, 115], using the concept of “generalized Killing vector
fields”. Its main interest it that it allows naturally to account for the
inclusion of gravitational back-reaction effects.
###### Spin supplementary conditions
There is a technical subtlety arising when studying the motion of test bodies
described by the MPD equations. In order to obtain a closed system of
equations, they have to be supplemented by an algebraic condition of the form
$\mathcal{V}_{\mu}S^{\mu\nu}=0$, for some timelike vector field
$\mathcal{V}^{\mu}$. Such conditions are known as spin supplementary
conditions. Physically, enforcing this kind of condition amounts to fix a
choice of centroid worldline, setting to zero the mass-dipole moment in the
proper frame whose timelike vector is aligned $\mathcal{V}^{\mu}$. The
discussion of what is really happening is quite subtle, the main reason being
that the notion of center of mass is observer-dependent in relativity.
###### Truncation of the multipole expansion
Another point of concern is to understand if the multipole expansion
introduced above can be consistently truncated as some desired order, that is,
if there exists a small parameter such that the magnitude of the successive
multipoles decreases when the order of the multipoles increases. As we will
see in Chapter 5, for compact objects, this small parameter will be the ratio
between the typical size of the object and the typical radius of the
background curvature, which is small by assumption.
As discussed in the introduction of the thesis, we will always truncate the
expansion at the quadrupole order. This is the first order in the multipole
expansion where the internal structure of the body begins to matter. At pole-
dipole order, the motion of finite size bodies is universal, in the sense that
it is independent of the nature of the object. Because we exclude the self-
force in our description, our expansion will thus be valid at zeroth order in
the mass ratio and at second order in the spin.
MPD equations promote the two first multipole $p^{\mu}$ and $S^{\mu\nu}$ to
the rank of dynamical variables, but leave the higher order multipoles acting
as sources. The latter shall consequently be prescribed depending on the
internal structure of the test body. In this thesis, we will only be concerned
with multipole moments induced by the proper rotation of the object, also
known as spin-induced multipoles. We therefore discard tidal and other type of
contributions to the multipole structure. As we will see, this is the relevant
description for modelling a binary black hole system evolving in vacuum, and
this is the choice of multipole structure that will allow the existence of the
largest number of conserved quantities along the motion.
Actually, for compact test bodies, they are several equivalent way of thinking
about the truncation of the multipole expansion, which are consistent one to
another, as we will check explicitly: (i) as a truncation of the number of
multipoles that we use to describe the stress-energy tensor, which is the
viewpoint of gravitational skeletonization that will be discussed in Chapter
5; (ii) as a truncation of the number of the derivatives of the background
Riemann tensor upon which the action of the Lagrangian formulation can depend
upon, as will be described in Chapter 4; and finally (iii) as an expansion in
integer powers of the magnitude of the spin dipole $\mathcal{S}$. This latter
viewpoint turns out to be consistent with the two former for spin-induced
multipoles, as will be reviewed in Chapter 7.
###### Plan of the text
This part of the thesis is organized as follows: Chapter 4 will describe the
Lagrangian formulation for extended test bodies in full generality, up to
quadrupole order. In Chapter 5, we will discuss a particularly simple form of
the gravitational skeletonization up to dipole order, which will enable to
gain more intuition about the physical meaning of the monopole and the dipole
moments. Our discussion will be however specialized to a specific choice of
coordinates. Chapter 6 will describe the problem of enforcing the
aforementioned spin supplementary conditions, as well as their physical
interpretation. Finally, Chapter 7 will be devoted to spin-induced multipoles,
and will focus on the explicit construction of the spin-induced quadrupole
moment.
### Chapter 4 Lagrangian Formulation
This chapter discusses the Lagrangian formalism for spinning test bodies in
General Relativity. In this text, we will always restrict our derivations up
to quadrupole order. Nevertheless, higher orders can be reached, see e.g.
[116]. This chapter mainly follows the excellent exposition of Marsat [116].
The core idea of the Lagrangian approach is to construct the most generic
worldline Lagrangian action $S=\int L\,\differential\lambda$ describing the
motion of a spinning test body in curved spacetime. As we will see, the form
of the allowed Lagrangian $L$ can be highly constrained from very generic
symmetry arguments. Like in any classical mechanics problem, the equations of
motion can then be derived from the associated first order variational
principle $\delta S=0$ [66, 67].
#### 4.1 Rotational degrees of freedom
The two main questions one should ask for building an action are
1. 1.
What are the relevant degrees of freedom that should be introduced for
describing a spinning body in curved spacetime ?
2. 2.
What are the symmetries under which the action should be invariant?
This section aims to tackle the first of them.
Let us denote $z^{\mu}(\lambda)$ the body’s worldline. Here, $\lambda$ is an
arbitrary “time” parameter describing the evolution of the motion. We also
define the four-velocity
$\displaystyle v^{\mu}\triangleq\derivative{z^{\mu}}{\lambda}.$ (4.1)
For any physical massive object, the four-velocity will be a timelike vector,
$v_{\mu}v^{\mu}<0$. In the canonical language of Lagrangian mechanics, the
four components $v^{\mu}$ will play the role of the velocities describing the
position of the test body and associated to the coordinates $z^{\mu}$.
All along this discussion, the specific form of $\lambda$ as well as the
normalization of the four-velocity will be left arbitrary at the level of the
action; they will only be constrained later at the level of the equations of
motion, setting $\lambda$ to be the proper time and consequently yielding the
standard normalization $v_{\mu}v^{\mu}=-1$.
We are left with the problem of choosing the degrees of freedom that will
account for the rotational orientation of the test body. Following the
proposition of Hanson and Regge for Special Relativity [111], the spin (that
is, the rotational) degrees of freedom of the body will be represented by an
orthonormal tetrad $e_{A}^{\phantom{A}\mu}(\lambda)$ rigidly attached to the
body’s worldline. Its orientation at any value of $\lambda$ will be measured
thanks to the introduction of another background orthonormal tetrad frame
$\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)$. At any point
of the worldline, these two tetrads are related by a Lorentz transformation
$\Lambda^{A}_{\phantom{A}\underline{A}}(\lambda)$:
$\displaystyle\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}\quantity(z(\lambda))=\Lambda^{A}_{\phantom{A}\underline{A}}(\lambda)e_{A}^{\phantom{A}\mu}(\lambda).$
(4.2)
Of course, we have the standard relations for Lorentz matrices
$\displaystyle\Lambda_{A}^{\phantom{A}\underline{A}}(\lambda)\Lambda_{B\underline{A}}(\lambda)=\eta_{AB},\qquad\Lambda_{A\underline{A}}(\lambda)\Lambda^{A}_{\phantom{A}\underline{B}}(\lambda)=\eta_{\underline{A}\,\underline{B}},$
(4.3)
and for tetrad frames
$\displaystyle e_{A}^{\phantom{A}\mu}(\lambda)e_{B\mu}(\lambda)$
$\displaystyle=\eta_{AB},\quad
e_{A}^{\phantom{A}\mu}(\lambda)e^{A\nu}(\lambda)=g^{\mu\nu}(z(\lambda)),$
(4.4a)
$\displaystyle\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)\underline{e}_{\underline{B}\mu}(x)$
$\displaystyle=\eta_{\underline{A}\,\underline{B}},\quad\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}(x)\underline{e}^{\underline{A}\nu}(x)=g^{\mu\nu}(x).$
(4.4b)
The evolution of the body’s tetrad will be described using the standard
antisymmetric rotation coefficients $\Omega^{\mu\nu}$ (see e.g. [43])
$\displaystyle\frac{\text{D}e_{A}^{\phantom{A}\mu}}{\differential\lambda}\triangleq-\Omega^{\mu\nu}e_{A\nu}\qquad\Leftrightarrow\qquad\Omega^{\mu\nu}\triangleq
e^{A\mu}\frac{\text{D}e_{A}^{\phantom{A}\nu}}{\differential\lambda}.$ (4.5)
Here and in the remaining of this text, we use the notation
$\text{D}/\differential\lambda\triangleq v^{\alpha}\nabla_{\alpha}$.
The Lorentz matrices
$\Lambda^{\underline{A}}_{\phantom{\underline{A}}A}(\lambda)$ encode all the
informations regarding the orientation of the body’s tetrad with respect to
the background. As any homogeneous Lorentz transformation, they contain 6
degrees of freedom: three of them represent spatial rotations, and the three
others relativistic boosts. Intuitively, one can see the three rotational
degrees of freedom (DOFs) as being the spin ones, whereas the three boosts
originate from the fact that one has not chosen yet the exact position of the
worldline $z^{\mu}(\lambda)$ inside the body’s worldtube. This ambiguity will
be extensively discussed and resolved in Chapter 6, by enforcing a so-called
spin supplementary condition (SSC).
#### 4.2 Constraining the action
It is now time to write down an action for our theory. It seems natural to
require the following symmetry requirements to hold [117]:
* $\blacktriangleright$
Spacetime diffeomorphisms: as any GR scalar expression, the action should be
invariant under any generic spacetime diffeomorphism $x^{\mu}\to
x^{\mu^{\prime}}\quantity(x^{\mu})$. The Lagrangian should consequently be a
tensorial scalar, in the sense that all the spacetime indices of the objects
it is built from should be properly contracted between themselves;
* $\blacktriangleright$
Lorentz transformations: the action should be invariant under local Lorentz
transformations, which transform the body and the background tetrad as
$\displaystyle
e_{A}^{\phantom{A}\mu}\to\Lambda^{A}_{\phantom{A}B}e_{B}^{\phantom{B}\mu},\qquad\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}\to\bar{\Lambda}^{\underline{A}}_{\phantom{\underline{A}}\underline{B}}\underline{e}_{\underline{B}}^{\phantom{\underline{B}}\mu}.$
(4.6)
It amounts to require all the Lorentz indices of the tetrads to be properly
contracted;
* $\blacktriangleright$
Reparametrization invariance: the time parameter $\lambda$ being arbitrary,
the action (4.7) should be invariant under any reparametrization
$\lambda\to\lambda^{\prime}(\lambda)$ of the trajectory.
In order to actually describe a spinning body, the Lagrangian should
kinematically depend on the worldline velocity $v^{\mu}$ and on the rotation
coefficients $\Omega^{\mu\nu}$, but not on the “positions” ($z^{\mu}$ and
$e_{A}^{\phantom{A}\mu}$) themselves, for the purpose of ensuring general
covariance. Moreover, we forbid any dependence in the background structure
$\underline{e}_{\underline{A}}^{\phantom{\underline{A}}\mu}$, so that our
description depends only on degrees of freedom intrinsic to the body. The
prescribed Lagrangian should account for finite size effects, that is,
dynamical effects originating from the coupling between the body’s spin and
the background’s curvature. The later is accounted for by the Riemann tensor
and its derivatives. Notice that the background metric $g_{\mu\nu}$ is assumed
to appear only for the purpose of contracting indices, thus allowing to
construct scalars from the other tensorial objects in a natural way. We
however forbid any dependence upon first derivatives of the metric (that is,
upon Christoffel symbols). Derivatives of the metric are only allowed to enter
in the action through the Riemann tensor and its derivatives.
Excluding any coupling with other external fields and given the discussion
above, the generic action for an extended test body is then assumed takes the
form:
$\displaystyle
S\quantity[z^{\mu},e_{A}^{\phantom{A}\mu}]=\int_{\gamma}L\quantity(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu}(z),R_{\mu\nu\rho\sigma}(z),\nabla_{\lambda}R_{\mu\nu\rho\sigma}(z),\ldots)\differential\lambda.$
(4.7)
The subscript $\gamma$ just refers to the fact that the integration over
$\lambda$ is actually an integration over the worldline $\gamma$.
###### Homogeneity condition
The next step will be to constrain the generic form of the action Eq. (4.7).
Actually, a very simple argument allows to provide a simple explicit – but
still non-uniquely fixed – expression for the Lagrangian. As we have just
mentioned, the action Eq. (4.7) should be invariant under any
reparametrization of the trajectory; in particular, it should be invariant
under a scaling $\lambda\to\Delta\lambda$ ($\Delta\neq 0)$. This implies that
the Lagrangian must be homogeneously linear in $v^{\mu}$ and
$\Omega^{\mu\nu}$, which both scale as $\Delta^{-1}$ under this
transformation. Euler’s theorem on homogeneous functions111 Let
$f:\mathbb{R}^{n}\to\mathbb{R}$ be a positively homogeneous function of degree
$k\in\mathbb{Z}$, i.e. $\displaystyle\forall\Delta>0:f(\Delta
x_{1},\ldots\Delta x_{n})=\Delta^{k}f(x_{1},\ldots,x_{n})$ which is
continuously differentiable in some open subset
$\mathcal{U}\subset\mathbb{R}^{n}$. Then, $\displaystyle
kf(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}x_{i}\partialderivative{f}{x_{i}}\quantity(x_{1},\ldots,x_{n}),\qquad\forall\quantity(x_{1},\ldots,x_{n})\in\mathcal{U}.$
then implies that
$\displaystyle
L(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu},R_{\mu\nu\rho\sigma},\nabla_{\lambda}R_{\mu\nu\rho\sigma},\ldots)=\partialderivative{L}{v^{\mu}}v^{\mu}+\partialderivative{L}{\Omega^{\mu\nu}}\Omega^{\mu\nu}.$
(4.8)
Defining the conjugate momenta222The factor $2$ in the definition of
$S_{\mu\nu}$ is present for consistency with the conventions used in the
literature. (respectively referred to as the linear momentum and the spin
tensor)
$\displaystyle p_{\mu}\triangleq\partialderivative{L}{v^{\mu}},\qquad
S_{\mu\nu}\triangleq 2\partialderivative{L}{\Omega^{\mu\nu}},$ (4.9)
one can write
$\displaystyle L=p_{\mu}v^{\mu}+\frac{1}{2}S_{\mu\nu}\Omega^{\mu\nu}.$ (4.10)
Be careful: we have not provided a unique expression for the Lagrangian of our
theory. The momenta $p_{\mu}$ and $S_{\mu\nu}$ remain here arbitrary functions
of $v^{\mu}$, $\Omega^{\mu\nu}$, the Riemann tensor and its derivatives. They
will be fixed when a spin supplementary condition will be enforced, which will
provide us with an explicit relation between the linear momentum $p_{\mu}$ and
the four-velocity $v^{\mu}$. Nevertheless, the form of the Lagrangian (4.10)
is relevant to mention for later purposes (e.g. Hamiltonian description of the
present problem); in particular, it is the same regardless to the finite size
interactions allowed in the theory, i.e. regardless to the dependence of $L$
in the Riemann tensor and its derivatives that is allowed.
###### Quadrupole approximation
We will now constrain our theory by restricting the functional dependency of
the Lagrangian in the (derivatives of the) Riemann tensor. In the continuation
of this text, we will always restrict ourselves to the so-called quadrupole
approximation which consists into allowing $L$ to depends on the Riemann
tensor, but not on its derivatives:
$\displaystyle
L=L\quantity(v^{\mu},\Omega^{\mu\nu},g_{\mu\nu},R_{\mu\nu\rho\sigma}).$ (4.11)
More generically, allowing the Lagrangian to depend in the Riemann tensor up
to its $n^{\text{th}}$ derivative will lead to the appearance of
$2^{n+2}$-pole moments terms in the equations of motion. For an action of the
form (4.7), the multipole moments are only sourced by the spin of the body.
For spin-induced moments, one can show (see Chapter 7) that the $2^{n}$-pole
moment scales as $\mathcal{O}\quantity(\mathcal{S}^{n})$, where the spin
magnitude $\mathcal{S}$ is defined as
$\displaystyle\mathcal{S}^{2}\triangleq\frac{1}{2}S_{\mu\nu}S^{\mu\nu}.$
(4.12)
The aforementioned approximation makes sense, because (i) the spin magnitude
will turn out to be a constant of motion, regardless to the multipole order we
are working with and because (ii) $\mathcal{S}$ can be assumed to be small in
astrophysically realistic situations. It can consequently be used as an
expansion parameter for setting up a perturbative treatment of the generic
problem.
From a physical viewpoint, the quadrupole approximation amounts to consider
deformations induced by the proper rotation of the object in our description
up to quadrupole order, while neglecting higher order corrections.
Finally, let us notice that only the (mass) monopole moment $p_{\mu}$ and the
(spin) dipole moment $S_{\mu\nu}$ are dynamical variables, because they are
the only multipole moments present in the explicit form of the Lagrangian
(4.10). The higher moments are non-dynamical and entirely written in terms of
these two first moments. They will act as sources in the equations of motion.
#### 4.3 Equations of motion
Before deriving the equations of motion, it is useful to explicit some results
concerning the first-order variations of the Lagrangian. First, in the
quadrupole approximation, a generic variation of $L$ takes the form
$\displaystyle\delta L=p_{\mu}\delta
v^{\mu}+\frac{1}{2}S_{\mu\nu}\delta\Omega^{\mu\nu}+\partialderivative{L}{g_{\mu\nu}}\delta
g_{\mu\nu}-\frac{1}{6}J^{\mu\nu\rho\sigma}\delta R_{\mu\nu\rho\sigma}.$ (4.13)
Here, we have defined – up to a proportionality factor – the quadrupole moment
$J^{\mu\nu\rho\sigma}$ as the conjugate moment to the Riemann tensor:
$\displaystyle
J^{\mu\nu\rho\sigma}\triangleq-6\partialderivative{L}{R_{\mu\nu\rho\sigma}}.$
(4.14)
Notice that the variation (4.13) is independent of the explicit form (4.10) of
the Lagrangian.
In particular, the action should be invariant under an infinitesimal change of
coordinates $z^{\mu}\to z^{\mu}+\epsilon\xi^{\mu}$
($\absolutevalue{\epsilon}\ll 1$). Particularizing the variation (4.13) to
this case and working in a locally inertial frame yields
$\displaystyle\delta_{\xi}L$
$\displaystyle=\epsilon\quantity(p^{\nu}v^{\mu}+S^{\nu}_{\phantom{\nu}\lambda}\Omega^{\mu\lambda}-2\partialderivative{L}{g_{\mu\nu}}+\frac{2}{3}J^{\nu\alpha\beta\gamma}R^{\mu}_{\phantom{\mu}\alpha\beta\gamma})\partial_{\mu}\xi_{\nu}.$
(4.15)
This variation must vanish regardless to the value of $\xi^{\mu}$; therefore,
the following constraint must hold:
$\displaystyle
p_{\nu}v_{\mu}+S_{\nu\lambda}\Omega_{\mu}^{\phantom{\mu}\lambda}-2\partialderivative{L}{g_{\mu\nu}}+\frac{2}{3}J_{\nu}^{\phantom{\nu}\alpha\beta\gamma}R_{\mu\alpha\beta\gamma}=0.$
(4.16)
Taking the antisymmetric part of this expression allows to write
$\displaystyle
S^{\lambda[\mu}\Omega^{\nu]}_{\phantom{\nu]}\lambda}=p^{[\mu}v^{\nu]}+\frac{2}{3}R^{[\mu}_{\phantom{[\mu}\alpha\beta\gamma}J^{\nu]\alpha\beta\gamma},$
(4.17)
which is valid in any frame. This last relation will become extremely useful
in the following derivations.
###### Evolution equation for the spin tensor
The evolution equation for the spin tensor, also known as the precession
equation, is obtained by varying the action with respect to the body’s tetrad
$e_{A}^{\phantom{A}\mu}$. In the background frame, the variation of the
rotation coefficients takes the form
$\displaystyle\delta\Omega^{\underline{A}\,\underline{B}}=e^{\underline{A}}_{\phantom{\underline{A}}\mu}e^{\underline{B}}_{\phantom{\underline{B}}\nu}\frac{\text{D}\,\delta\theta^{\mu\nu}}{\differential\lambda}+\Omega^{\underline{A}}_{\phantom{\underline{A}}\underline{C}}\delta\theta^{\underline{C}\,\underline{B}}-\Omega^{\underline{B}}_{\phantom{\underline{B}}\underline{C}}\delta\theta^{\underline{C}\,\underline{A}}.$
(4.18)
For convenience, the variation of the tetrad has been entirely expressed in
terms of the object
$\displaystyle\delta\theta^{\underline{A}\,\underline{B}}\triangleq\Lambda^{A\underline{A}}\delta\Lambda_{A}^{\phantom{A}\underline{B}}.$
(4.19)
Plugging this result either in the explicit expression of the Lagrangian
(4.10) or in the generic variation (4.13), one obtains
$\displaystyle\delta_{\theta}L=\frac{1}{2}\quantity(-\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}+S^{\nu\rho}\Omega^{\mu}_{\phantom{\mu}\rho}-S^{\mu\rho}\Omega^{\nu}_{\phantom{\nu}\rho})\delta\theta_{\mu\nu}.$
(4.20)
Requiring this variation to be vanishing yields the evolution equation
$\displaystyle\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}=S^{\nu\rho}\Omega^{\mu}_{\phantom{\mu}\rho}-S^{\mu\rho}\Omega^{\nu}_{\phantom{\nu}\rho}.$
(4.21)
Before going further on, let us stress some points useful for the continuation
of this work.
1. 1.
A direct computations shows that the identity
$\displaystyle S^{\rho}_{\phantom{\rho}\mu}\Omega^{\mu\nu}S_{\nu\rho}=0$
(4.22)
holds. It implies that the spin magnitude is conserved,
$\displaystyle\derivative{\mathcal{S}}{\lambda}=0.$ (4.23)
This conservation equation actually holds at any multipole order and is
independent of the spin supplementary condition (see e.g. [116] and references
therein).
2. 2.
In the object’s frame, the evolution equation becomes
$\displaystyle\frac{\text{D}\,S^{AB}}{\differential\lambda}=0.$ (4.24)
The components of the spin tensor in the object’s frame $S^{AB}$ are thus
constant. This provides a posteriori a justification to the statement that the
tetrad $e_{A}^{\phantom{A}\mu}$ is “rigidly attached” to the compact object.
3. 3.
Using Eq. (4.17), one can eliminate the dependence of the precession equation
in the rotation coefficients and make explicit its dependence in the Riemann
tensor. A straightforward computation yields
$\displaystyle\frac{\text{D}\,S^{\mu\nu}}{\differential\lambda}=2p^{[\mu}v^{\nu]}+\mathcal{L}^{\mu\nu},\qquad\mathcal{L}^{\mu\nu}\triangleq\frac{4}{3}R^{[\mu}_{\phantom{[\mu}\alpha\beta\gamma}J^{\nu]\alpha\beta\gamma}.$
(4.25)
This is the standard form of the precession equation that can be found in the
literature.
4. 4.
Finally, contracting Eq. (4.25) with $v_{\nu}$, we remark that the momentum
and the four velocity are not aligned anymore when spin is present, by
contrast to the geodesic case:
$\displaystyle-v^{2}p^{\mu}=\mathfrak{m}v^{\mu}+p_{\perp}^{\mu},\qquad
p_{\perp}^{\mu}\triangleq\quantity(-\frac{\text{D}S^{\mu\nu}}{\differential\lambda}+\mathcal{L}^{\mu\nu})v_{\nu}.$
(4.26)
Here, $\mathfrak{m}\triangleq-v_{\alpha}p^{\alpha}$ denotes the body’s mass in
the frame attached to the worldline, also known as the kinetic mass. The norm
of the four-velocity can be set to $-1$ if the time evolution parameter is
chosen to be the body proper time. The orthogonal component of the momentum
$p^{\mu}_{\perp}$ can be expressed as a function of $x^{\mu}$, $v^{\mu}$ and
$S^{\mu\nu}$ solely when a spin supplementary condition has been enforced, see
Chapter 6.
###### Evolution equation for the linear momentum
The method for finding the evolution equation for the linear momentum is to
vary the action with respect to the worldline. The procedure is the very same
that the one which can be used for the derivation of the geodesic deviation
equation [83, 84]: let us consider an infinitesimal change of the worldline,
parametrized by a displacement vector $\xi^{\mu}(\lambda)$ which is Lie-
dragged along the worldline:
$\displaystyle\mathcal{L}_{v}\xi^{\mu}=0\quad\Leftrightarrow\quad\xi^{\lambda}\nabla_{\lambda}v^{\mu}=v^{\lambda}\nabla_{\lambda}\xi^{\mu}.$
(4.27)
In this case, the variation of the action takes the form
$\displaystyle\delta_{\xi}S=\int_{\gamma}\delta_{\xi}L\,\differential\lambda=\int_{\gamma}\xi^{\lambda}\partial_{\lambda}L\,\differential\lambda=\int_{\gamma}\xi^{\lambda}\nabla_{\lambda}L\,\differential\lambda.$
(4.28)
It is useful to notice that the following identities hold [116]:
$\displaystyle\xi^{\lambda}\nabla_{\lambda}v^{\mu}v^{\mu}$
$\displaystyle=\frac{\text{D}\,\xi^{\mu}}{\differential\lambda},$ (4.29a)
$\displaystyle\xi^{\lambda}\nabla_{\lambda}\Omega^{\mu\nu}$
$\displaystyle=-\frac{\text{D}}{\differential\lambda}\quantity(e^{A\mu}\delta_{\xi}e_{A}^{\phantom{A}\nu})+e^{A\mu}\frac{\text{D}\,\delta_{\xi}e_{A}^{\phantom{A}\nu}}{\differential\lambda}-e^{A\nu}\frac{\text{D}\,\delta_{\xi}e_{A}^{\phantom{A}\mu}}{\differential\lambda}-\xi^{\alpha}v^{\beta}R^{\mu\nu}_{\phantom{\mu\nu}\alpha\beta}.$
(4.29b)
Here, we have defined
$\displaystyle\delta_{\xi}e_{A}^{\phantom{A}\mu}\triangleq\xi^{\lambda}\nabla_{\lambda}e_{A}^{\phantom{A}\mu}$
(4.30)
The value of this quantity is actually is arbitrary, since we are left with
the freedom of choosing the way the tetrad is transported from the original
worldline $z^{\mu}(\lambda)$ to the perturbed one
$z^{\mu}(\lambda)+\xi^{\mu}(\lambda)$. Choosing the tetrad to be parallelly
transported between the two worldlines allows to set
$\displaystyle\delta_{\xi}e_{A}^{\phantom{A}\mu}=0.$ (4.31)
Gathering all the previous pieces, the evolution equation for the linear
momentum can then be derived – after integration by parts – from the
variational problem $\delta_{\xi}S=0$, yielding
$\displaystyle\frac{\text{D}\,p^{\mu}}{\differential\lambda}=-\frac{1}{2}R^{\mu}_{\phantom{\mu}\nu\alpha\beta}v^{\nu}S^{\alpha\beta}+\mathcal{F}^{\mu},\qquad\mathcal{F}^{\mu}\triangleq-\frac{1}{6}J^{\alpha\beta\gamma\delta}\nabla^{\mu}R_{\alpha\beta\gamma\delta}.$
(4.32)
This is the standard evolution equation of the linear momentum at quadrupole
order. Together with Eq. (4.25), these equations are the Mathisson-Papapetrou-
Dixon equations, restricted to quadrupole order. At higher orders, the
structure of the equations remains the same, the contribution of the higher
order multipoles being only contained in the force $\mathcal{F}^{\mu}$ and
torque $\mathcal{L}^{\mu\nu}$ terms [118].
#### 4.4 Stress-energy tensor
An interesting computation to be performed at this stage of the discussion is
to write out the stress-energy tensor of the theory. It is advantageously
computed from the variation of the action with respect to the body’s tetrad
frame: first expressing the metric in terms of the tetrad in the action thanks
to Eq. (4.4a), the stress-energy tensor as be computed from
$\displaystyle T_{\mu\nu}\triangleq\frac{1}{\sqrt{-g}}e_{a(\mu}\frac{\delta
S}{\delta e_{a}^{\phantom{a}\nu)}}.$ (4.33)
The stress-energy tensor can be split as a sum over all the multipole orders
involved:
$\displaystyle
T_{\mu\nu}=T^{\text{pole}}_{\mu\nu}+T^{\text{dipole}}_{\mu\nu}+T^{\text{quad}}_{\mu\nu}.$
(4.34)
As usually when computing stress-energy tensors for point-like objects moving
along worldlines, the action should be written as an integral over the
spacetime by introducing a Dirac delta:
$\displaystyle
S=\int\differential[4]x\sqrt{-g}\int_{\gamma}\,L\quantity(v^{\mu},\Omega^{\mu\nu},e_{A\mu}e^{A}_{\phantom{A}\nu},R_{\mu\nu\rho\sigma})\delta_{4}(x,z)\differential\lambda.$
(4.35)
Here, the symbol $\delta_{4}(x,z)$ stands for the diffeomorphism invariant
Dirac distribution
$\displaystyle\delta_{4}(x,z)\triangleq\frac{\delta^{(4)}(x-z)}{\sqrt{-g}},$
(4.36)
where $\delta^{(4)}(x-z)$ is the standard four-dimensional Dirac distribution
[119]. After computation, the contributions of the right hand side are found
to be given by
$\displaystyle T^{\text{pole}}_{\mu\nu}$
$\displaystyle=\int_{\gamma}p_{(\mu}v_{\nu)}\delta_{4}(x,z)\differential\lambda,$
(4.37a) $\displaystyle T^{\text{dipole}}_{\mu\nu}$
$\displaystyle=-\nabla_{\lambda}\int_{\gamma}S^{\lambda}_{\phantom{\lambda}(\mu}v_{\nu)}\delta_{4}(x,z)\differential\lambda,$
(4.37b) $\displaystyle T^{\text{quad}}_{\mu\nu}$
$\displaystyle=\frac{1}{3}\int_{\gamma}R_{(\mu}^{\phantom{(\mu}\alpha\beta\gamma}J_{\nu)\alpha\beta\gamma}\delta_{4}(x,z)\differential\lambda$
$\displaystyle\quad-\frac{2}{3}\nabla_{\lambda}\nabla_{\rho}\int_{\gamma}J^{\lambda\phantom{(\mu\nu)}\rho}_{\phantom{\lambda}(\mu\nu)}\delta_{4}(x,z)\differential\lambda.$
(4.37c)
As expected, the stress energy tensor is not a function, but a distribution
which is only non-vanishing on the body’s worldline.
### Chapter 5 Skeletonization of the stress-energy tensor
Until now, we have obtained the Mathisson-Papapetrou-Dixon equations governing
the motion of spinning test bodies in curved spacetime at quadrupole order,
which are given by Eqs. (4.25) and (4.32). This was performed through writing
down an action principle for our theory and deriving the associated equations
of motion from the associated variational principle.
In this chapter, we will see that a totally different method allows to recover
the very same equations of motion. It consists into replacing the smooth
stress-energy tensor of the physical extended body by a distributional one,
which is only supported on a single worldline encompassed in the body’s
worldtube. The equations of motion then follow from the stress-energy
conservation equation. Justifying rigorously this approximation from first
principles in GR is however very involved and technical. We refer the
interested reader to the references mentioned in the introduction of Part II
for more details (especially Dixon ones). In the present text, we will give
some insights about the coherence of this approximation by comparing the
(involved) GR situation to the (simpler) Newtonian one.
At quadrupole order, the computations associated to gravitational
skeletonization turn out to be very cumbersome. This chapter aiming to provide
a pedagogical introduction, we will restrict ourselves to the dipole order.
Explicit computations for the quadrupole may be found in [118, 120].
This chapter is organized as follows: Section 5.1 reviews the gravitational
skeletonization in Newtonian theory, which is then generalized to General
Relativity in Section 5.2. As we will see, they are several decompositions
that can be chosen for performing the skeletonization. In Section 5.3, we use
the Ellis decomposition to recover the MPD equations at dipole order. The
computations are carried out in a specific coordinates system, the adapted
coordinates, which enable to reduce dramatically the length and the
technicality of the derivation.
#### 5.1 Invitation: gravitational skeleton in Newtonian gravity
This section is mainly based on [121, 118].
In order to acquire some feeling about the form of the Ansatz of the GR’s
gravitational skeleton of the stress-energy tensor, let us have a look at the
equivalent problem in Newtonian gravity. The gravitational potential
$U(t,\mathbf{x})$ created by an object of mass density $\rho(t,\mathbf{x})$
enclosed in a volume $\mathcal{V}\subset\mathbb{R}^{3}$ is solution of the
Poisson equation
$\displaystyle\Delta U(t,\mathbf{x})=-4\pi\rho(t,\mathbf{x}).$ (5.1)
This equation can be solved analytically, and its solution reads (up to a
trivial additive constant)
$\displaystyle
U(t,\mathbf{x})=\int_{\mathcal{V}}\differential[3]x^{\prime}\,\frac{\rho(t,\mathbf{x})}{\norm{\mathbf{x}-\mathbf{x}^{\prime}}}.$
(5.2)
This potential admit a convenient rewriting under the form of a multipole
decomposition above an arbitrary point $\mathbf{x}_{0}\in\mathcal{V}$. For any
$\mathbf{x}\notin\mathcal{V}$, one can write
$\displaystyle
U(t,\mathbf{x})=\sum_{l=0}^{+\infty}\frac{\quantity(-)^{l}}{l!}I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\partial_{i_{1}}\ldots\partial_{i_{l}}\norm{\mathbf{x}-\mathbf{x}_{0}}^{-1}$
(5.3)
where we have defined the multipole moments
$\displaystyle I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\triangleq\int_{\mathcal{V}}\differential[3]x\,\quantity(x-x_{0})^{i_{1}}\ldots\quantity(x-x_{0})^{i_{l}}\rho(t,\mathbf{x}).$
(5.4)
The proof of Eq. (5.3) is easily carried out by performing a Taylor expansion
of $\norm{\mathbf{x}-\mathbf{x}^{\prime}}$ with respect to
$\mathbf{x}^{\prime}$ above some point $\mathbf{x}_{0}\in\mathcal{V}$.
The gravitational skeletonization consists here in replacing the smooth mass
density distribution $\rho(t,\mathbf{x})$ (which is supported on a finite-size
region of space, $\text{supp}\,\rho\subseteq\mathcal{V}$) by a singular mass
density distribution – say $\rho_{\text{skel}}$ – which is supported on a
single point of space,
$\text{supp}\,\rho_{\text{skel}}=\quantity{\mathbf{x}_{0}}\in\mathcal{V}$. The
key result allowing such a skeletonization to be performed can be stated as
follows:
###### Proposition 5.1.
Let $\mathbf{x}_{0}$ be a point of $\mathcal{V}$. For any point
$\mathbf{x}\notin\mathcal{V}$ outside of the object, the distributional mass
density
$\displaystyle\rho_{\text{skel}}(t,\mathbf{x})\triangleq\sum_{l=0}^{+\infty}\frac{(-)^{l}}{l!}I^{i_{1}\ldots
i_{l}}(t,\mathbf{x}_{0})\partial_{i_{1}}\ldots\partial_{i_{l}}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{0})$
(5.5)
generates the same potential $U(t,\mathbf{x})$ as the smooth mass density
$\rho(t,\mathbf{x})$.
###### Proof.
It is enough to recall ourselves that the identity
$\displaystyle\Delta\norm{\mathbf{x}-\mathbf{x}_{0}}^{-1}=-4\pi\delta^{(3)}\quantity(\mathbf{x}-\mathbf{x}_{0}).$
(5.6)
holds in the sense of distributions [119]. ∎
Of course, the explicit expression of $\rho_{\text{skel}}$ and all the related
equations have to be understood in the sense of distributions (see e.g. [119]
for a clear reminder of the meaning of this assertion).
In others words, when observed from outside, any localized gravitating object
can be replaced by a particle located at a single point of spacetime and
possessing an infinite tower of multipole moments. This replacement holds in
the sense that the gravitational potentials generated by these two systems are
identical as long as we remain outside of the object.
#### 5.2 General Relativist Skeletons
We now consider an extended body within the framework of General Relativity.
This object is assumed to be described by a smooth stress-energy tensor
supported on some worldtube $\mathcal{T}$. In the same spirit, we replace its
smooth stress-energy tensor $T^{\mu\nu}(x)$ by a distributional stress-energy
tensor $T^{\mu\nu}_{\text{skel}}(x)$ supported on a single timelike worldline
$\gamma\subset\mathcal{T}$. By analogy with Eq. (5.5), we assume this stress-
energy tensor to take the form
$\displaystyle
T^{\mu\nu}_{\text{skel}}(x)=\sum_{l=0}^{+\infty}\frac{1}{l!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}(z)\mathcal{D}^{(l)}_{\alpha_{1}\ldots\alpha_{l}}\delta_{4}(x,z).$
(5.7)
Here, $\mathcal{D}^{(k)}_{\alpha_{1}\ldots\alpha_{l}}$ is some differential
operator which contains at most $l$ derivatives. $z^{\mu}(\tau)$ are
coordinates parametrizing the worldline $\gamma$ with respect to an affine
time parameter $\lambda$. We denote the tangent vector to the worldline
$v^{\mu}=\derivative{z^{\mu}}{\lambda}$. For $l=0$, we use the conventions
$\mathcal{D}^{(0)}=\text{Id}$ and
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}=I^{\mu\nu}$. At this level, the
multipoles $I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}$ are still arbitrary
functions.
###### Perturbative treatment
One can show that it makes sense to treat the expansion (5.7) perturbatively,
and consequently to truncate it at any desired order. By analogy with the non-
relativistic case, we do expect the multipole to scale as
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}\sim\mu\ell^{l}$, with $\mu$ and $\ell$
being respectively the typical mass and size of the extended body under
consideration. Moreover, we expect the multiple covariant derivative to scale
as $\nabla_{\alpha_{1}\ldots\alpha_{l}}\sim r^{-l}$, with $r$ the typical
curvature radius of the background metric. All in all, the $l^{\text{th}}$
term of the expansion (5.7) scales as $\mu\quantity(\frac{\ell}{r})^{l}$.
Then, if the object is assumed to be compact in the sense mentioned in the
introduction, we have $r\gg\ell$ and consequently $\frac{\ell}{r}\ll 1$. The
expansion (5.7) can thus be truncated in a perturbative sense. A truncation at
$l=0$ will correspond to the monopole approximation, $l=1$ to the pole-dipole
(or simply dipole) one, $l=2$ to the quadrupole, etc.
###### Dixon and Ellis representations
To go further, we shall choose an explicit form for the operator
$\mathcal{D}^{l}_{\alpha_{1}\ldots\alpha_{l}}$. Two equivalent choices have
been used in the literature [120]. The most common is the Dixon representation
[106, 107, 108]
$\displaystyle\mathcal{D}^{(k)}_{\alpha_{1}\ldots\alpha_{k}}$
$\displaystyle\triangleq\nabla_{\alpha_{1}}\ldots\nabla_{\alpha_{k}}.$ (5.8)
In this text, we will instead make another choice, the Ellis representation,
defined by [122]
$\displaystyle\mathcal{D}^{(l)}_{\alpha_{1}\ldots\alpha_{l}}$
$\displaystyle=\left\\{\ \begin{array}[]{ll}0&\text{ if }l<N\\\
\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}&\text{ if
}l=l_{\text{max}}\end{array}\right.,$ (5.11)
with $l_{\text{max}}$ the order in $l$ at which the multipole expansion Eq.
(5.7) is chosen to be truncated.
Both of these representations have advantages and drawbacks, which are nicely
reviewed in [120]. For our purposes, it will be more convenient to work in the
Ellis representation, since the computations involved at dipole order turn out
to be less technical, and thus more suited for an introductory exposition.
###### Reduction of the stress-energy tensor
The above skeletonization amounts to replace the smooth object by a collection
of multipole moments supported on a single worldline contained in the object
worldtube. Actually proving the validity of the decomposition (5.7) would
require to show that there exists some expressions of the multipole moments
$I^{\mu\nu\alpha_{1}\ldots\alpha_{l}}$ such that both the smooth and the
distributional stress-energy tensors generate the same spacetime curvature
though the Einstein field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$. This task
being extremely involved, we will not attempt to tackle it in the present
text, but rather consider (5.7) for granted. The interested reader would
fruitfully refer to [108] for a more formal exposition of the subject.
In this text, we will instead discuss the so-called reduction of the stress-
energy tensor: as it is always the case in GR, our stress-energy tensor must
be conserved and consequently obeys [83, 84]
$\displaystyle\nabla_{\mu}T^{\mu\nu}=0.$ (5.12)
As we will show, this conservation equation highly constrains the form of the
stress-energy tensor. At quadrupole order, one can show that it implies that
there must exist a vector $p^{\mu}$, and antisymmetric tensor $S^{\mu\nu}$ and
a rank four tensor $J^{\mu\nu\rho\sigma}$ exhibiting the same symmetries than
the Riemann tensor such that [118, 120]
$\displaystyle T^{\mu\nu}(x)$
$\displaystyle=\int_{\gamma}\differential\lambda\,\quantity[v^{(\mu}p^{\nu)}+\frac{1}{3}R^{(\mu}_{\phantom{(\mu}\alpha\beta\gamma}J^{\nu)\alpha\beta\gamma}]\delta_{4}(x,z)+\nabla_{\alpha}\int_{\gamma}\differential\lambda\,v^{(\mu}S^{\nu)\alpha}\delta_{4}(x,z)$
$\displaystyle\quad-\frac{2}{3}\nabla_{\alpha}\nabla_{\beta}\int_{\gamma}\differential\lambda\,J^{\alpha(\mu\nu)\beta}\delta_{4}(x,z).$
(5.13)
Moreover, during the reduction process, we find additional constraints taking
the form of differential equations for the quantities $p^{\mu}$, $S^{\mu\nu}$
and $J^{\mu\nu\rho\sigma}$ that turn out to be precisely the MPD equations.
Notice that Eq. (5.2) agrees exactly with the form of the stress-energy tensor
which has been found through the Lagrangian formulation, given in Eq. (4.37).
#### 5.3 Ellis skeleton in adapted coordinates: to dipole order
This section mainly follows the exposition of [120].
Performing the explicit reduction of the stress-energy tensor at quadrupole
order is computationally quite involved, and can be found in [118, 120]. As a
proof of principle, we will instead perform the reduction at pole-dipole level
and show that it leads to the MPD equations at the corresponding order. In the
following, we will work with Ellis representation of multipoles. Explicitly,
Ellis representation truncated at order $N$ takes the form
$\displaystyle T^{\mu\nu}_{\text{skel}}(x)$
$\displaystyle=\frac{1}{N!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}(z)\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}\delta_{4}\quantity(x,z)$
(5.14)
with $z^{\mu}=z^{\mu}(\lambda)$ the coordinates of the worldline.
Since $T_{\text{skel}}^{\mu\nu}$ is a distributional stress-energy tensor, Eq.
(5.14) shall be formally understood in the sense of distributions: for any
symmetric test function $\phi_{\mu\nu}$, the quantity
$\displaystyle\int_{\mathcal{T}}\differential[4]x\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}(x)\phi_{\mu\nu}(x)$
(5.15)
is a real number. Integrating distributional quantities against test functions
will be the main tool we will use to show that the conservation of the stress-
energy tensor leads to the MPD equations.
Notice that since $T^{\mu\nu}$ is symmetric and since the partial derivatives
commute, the moment $I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ must obey the
algebraic symmetries
$\displaystyle
I^{\mu\nu\alpha_{1}\ldots\alpha_{k}}=I^{(\mu\nu)\alpha_{1}\ldots\alpha_{k}}=I^{\mu\nu(\alpha_{1}\ldots\alpha_{k})}.$
(5.16)
Finally, let us comment on two drawbacks of Ellis representation: (i) from
their very definitions, the moments $I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ are
not tensors, since they are contracted on their last $N$ indices with partial
derivatives, and since the total stress-energy tensor must be itself a
tensorial object. Moreover, (ii) for a given order of truncation $N$, the full
multipolar structure of the test object is represented by a single moment
$I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$. There is thus no a priori
decomposition of this moment between a hierarchy of moments (monopole,
dipole…). As we will see in the following, such a split can be obtained by
introducing a specific system of coordinates, the so-called adapted
coordinates.
###### Ellis skeleton in adapted coordinates
We now turn to coordinates adapted to the worldline, $x^{\mu}\to
X^{\mu}=(X^{0},\mathbf{X})$, with
$\mathbf{X}\triangleq\quantity(X^{1},X^{2},X^{3})$. They are required to
satisfy
$\displaystyle X^{0}\evaluated{}_{\gamma}$ $\displaystyle=\lambda,\quad
X^{i}\evaluated{}_{\gamma}=0\qquad\Rightarrow\qquad
v^{0}\evaluated{}_{\gamma}=\derivative{X^{0}}{\lambda}\evaluated{}_{\gamma}=1,\quad
v^{i}\evaluated{}_{\gamma}=\derivative{X^{i}}{\lambda}\evaluated{}_{\gamma}=0.$
(5.17)
An example of explicit construction of this type of coordinates are Fermi
normal coordinates [123]. In what follows, we will systematically assume that
such coordinates can be constructed over the worldtube $\mathcal{T}$ of the
body. Using these coordinates, we can write
$\displaystyle\begin{split}&\int_{\mathcal{T}}\differential[4]X\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}\phi_{\mu\nu}\\\
&=\frac{(-1)^{N}}{N!}\int_{\gamma}\differential\lambda\,I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}\partial_{\alpha_{1}}\ldots\partial_{\alpha_{N}}\phi_{\mu\nu}\evaluated{}_{\gamma}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{N}}{N!}\frac{N!}{k!(N-k)!}\int_{\gamma}\differential\lambda\,I^{\mu\nu
i_{1}\ldots i_{k}0\ldots
0}\partial_{i_{1}}\ldots\partial_{i_{k}}\partial^{N-k}_{0}\phi_{\mu\nu}\evaluated{}_{\gamma}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{k}}{k!(N-k)!}\int_{\gamma}\differential\lambda\,\partial^{N-k}_{0}I^{\mu\nu
i_{1}\ldots i_{k}0\ldots
0}\partial_{i_{1}}\ldots\partial_{i_{k}}\phi_{\mu\nu}\evaluated{}_{\gamma}.\end{split}$
(5.18)
It is therefore useful to define a new collection of moments
$\displaystyle\gamma_{(N)}^{i_{1}\ldots
i_{k}}\triangleq\frac{1}{(N-k)!}\partial^{N-k}_{0}I^{\mu\nu i_{1}\ldots
i_{k}0\ldots 0},\qquad k\leq N.$ (5.19)
They still satisfy the algebraic symmetries
$\displaystyle\gamma_{(N)}^{\mu\nu i_{1}\ldots
i_{k}}=\gamma_{(N)}^{(\mu\nu)i_{1}\ldots
i_{k}}=\gamma_{(N)}^{\mu\nu(i_{1}\ldots i_{k})}.$ (5.20)
In terms of the new moments (and still in adapted coordinates), Ellis
decomposition is equivalently given by
$\displaystyle
T^{\mu\nu}_{\text{skel}}=\frac{1}{\sqrt{-g}}\mathcal{T}^{\mu\nu},\qquad\mathcal{T}^{\mu\nu}\triangleq\sum_{k=0}^{N}\frac{1}{k!}\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}(\lambda)\partial_{i_{1}}\ldots\partial_{i_{k}}\delta^{(3)}\quantity(\mathbf{X}).$
(5.21)
Notice that since $T^{\mu\nu}_{\text{skel}}$ is a tensor,
$\mathcal{T}^{\mu\nu}$ is a tensor density of weight $-1$. The proof of Eq.
(5.21) consists into integrating this expression against an arbitrary,
symmetric test function $\phi_{\mu\nu}$:
$\displaystyle\begin{split}\int_{\mathcal{T}}\differential[4]X\sqrt{-g}\,T_{\text{skel}}^{\mu\nu}\phi_{\mu\nu}&=\sum_{k=0}^{N}\frac{1}{k!}\int_{\gamma}\differential\lambda\int\differential[3]X\,\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}\partial_{i_{1}}\ldots\partial_{i_{k}}\delta^{(3)}(\mathbf{X})\phi_{\mu\nu}\\\
&=\sum_{k=0}^{N}\frac{(-1)^{k}}{k!}\int_{\gamma}\differential\lambda\,\gamma_{(N)}^{\mu\nu
i_{1}\ldots
i_{k}}\partial_{i_{1}}\ldots\partial_{i_{k}}\phi_{\mu\nu}.\end{split}$ (5.22)
We therefore recover the expression obtained in Eq. (5.18) using the
definition Eq. (5.19).
Before turning to the derivation of the MPD equations using the conservation
of the stress-energy tensor, let us make a couple of remarks. As we will check
explicitly at the dipole level, the multipoles $\gamma$ are fully determined
by the distribution of stress-energy, whereas the $I$s are not, thus leading
to the appearance of a gauge freedom in their definition. This can be seen
from Eq. (5.19), since the construction of the $I$s from the $\gamma$s require
to integrate with respect to time, thus leading to the appearance of arbitrary
integration constants.
Moreover, as we will see when discussing the relation between distributional
and smooth stress-energy tensors, the split of
$I^{\mu\nu\alpha_{1}\ldots\alpha_{N}}$ in $k$ moments $\gamma^{\mu\nu
i_{1}\ldots i_{k}}$ actually corresponds to a physical split between a
monopole, a dipole, etc. Notice that the price to pay for obtaining simple
computations in Ellis representation is that we are forced to work in a
specific coordinate system, the adapted coordinates. Moreover, the multipole
decomposition tuned to this system is not explicitly covariant. Nevertheless,
this framework allows to recover the results obtained with more involved
approaches, e.g. Dixon representation. Finally, notice that a coordinate-free
approach to multipoles can be formulated, see [120] for more details.
###### Conservation equation for stress-energy tensor density
In adapted coordinates, since the decomposition Eq. (5.21) is valid, it is
more convenient to write the conservation of the stress-energy tensor Eq.
(5.12) in terms of the tensor density $\mathcal{T}^{\mu\nu}$ introduced in Eq.
(5.21). Since $T^{\mu\nu}_{\text{skel}}$ is a tensor, the covariant derivative
appearing in Eq. (5.12) can be expanded in terms of partial derivatives and
Christoffel symbols. We get
$\displaystyle\begin{split}\nabla_{\mu}T^{\mu\nu}_{\text{skel}}&=\partial_{\mu}T^{\mu\nu}_{\text{skel}}+2\Gamma^{(\mu}_{\nu\rho}T^{\nu)\rho}_{\text{skel}}\\\
&=\partial_{\mu}\quantity(\frac{1}{\sqrt{-g}})\mathcal{T}^{\mu\nu}+\frac{1}{\sqrt{-g}}\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+2\Gamma^{(\mu}_{\nu\rho}\mathcal{T}^{\nu)\rho})\\\
&=\frac{1}{\sqrt{-g}}\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho}).\end{split}$
(5.23)
The last equality uses the identity
$\displaystyle\partial_{\mu}\quantity(\sqrt{-g})=\sqrt{-g}\Gamma^{\alpha}_{\mu\alpha}.$
(5.24)
At the end of the day, the conservation equation for the stress-energy tensor
Eq. (5.12) is equivalent to the following equation for the stress-energy
tensor density
$\displaystyle\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho}=0.$
(5.25)
###### MPD equations in adapted coordinates
Before investigating the consequences of this conservation equation, it is
useful to write down the form that the MPD equations take in adapted
coordinates. Since we will derive only the pole-dipole equations in next
section, we set the force and torque terms to zero in all the equations below.
Recalling that $v^{\mu}=\delta^{\mu}_{0}$ on the worldline, the evolution
equation for the spin (4.25) takes the form
$\displaystyle\nabla_{0}S^{\mu\nu}=2p^{[\mu}v^{\nu]}.$ (5.26)
Separating the spatial coordinates from the temporal one, it is equivalent to
the two following equations
$\displaystyle\nabla_{0}S^{0i}$
$\displaystyle=-p^{i},\qquad\nabla_{0}S^{ij}=0.$ (5.27)
Using the definition of the Riemann tensor, the evolution equation for the
momentum (4.32) takes the form
$\displaystyle\begin{split}\nabla_{0}p^{\mu}&=-\frac{1}{2}R^{\mu}_{\phantom{\mu}0\alpha\beta}S^{\alpha\beta}\\\
&=-\quantity(\partial_{\alpha}\Gamma^{\mu}_{0\beta}+\Gamma^{\mu}_{\alpha\lambda}\Gamma^{\lambda}_{0\beta})S^{\alpha\beta}\\\
&=-\dot{\Gamma}^{0}_{0\mu}S^{0\mu}-\partial_{i}\Gamma^{0}_{0\mu}S^{i\mu}-\Gamma^{\mu}_{\alpha\lambda}\Gamma^{\lambda}_{0\beta}S^{\alpha\beta},\end{split}$
(5.28)
where we use the notation $\dot{}\triangleq\partial_{0}$.
###### Recovering MPD equations from stress-energy conservation
We will now truncate the multipole expansion to dipole order, and study the
constraints enforced by the conservation equation (5.25). At dipole order,
Ellis decomposition takes the form
$\displaystyle\mathcal{T}^{\mu\nu}=\gamma^{\mu\nu}\delta^{(3)}(\mathbf{X})+\gamma^{\mu\nu
i}\partial_{i}\delta^{(3)}(\mathbf{X})$ (5.29)
To lighten the notations, we drop the subscript “$(N)$” from the multipoles
$\gamma$ in the continuation of this section. For such a choice of stress-
energy density, the conservation equation (5.25) is a distributional identity.
It will be satisfied provided that
$\displaystyle\int\differential[3]X\,\quantity(\partial_{\mu}\mathcal{T}^{\mu\nu}+\Gamma^{\nu}_{\mu\rho}\mathcal{T}^{\mu\rho})\phi_{\nu}=0$
(5.30)
for any arbitrary test function $\phi_{\nu}$. Inserting the decomposition Eq.
(5.29) in this equation and integrating by parts yields
$\displaystyle\begin{split}&\quantity(\dot{\gamma}^{0\nu}+\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho}-\partial_{i}\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho
i})\phi_{\nu}\\\ &\quad-\quantity(\dot{\gamma}^{0\nu
i}+\gamma^{i\nu}+\Gamma^{\nu}_{\mu\rho}\gamma^{\mu\rho
i})\partial_{i}\phi_{\nu}+\gamma^{j\nu
i}\partial_{i}\partial_{j}\phi_{\nu}=0\end{split}$ (5.31)
Because this identity shall be valid for any choice of test function
$\phi_{\nu}$, it is equivalent to the set of three equations
|
email<EMAIL_ADDRESS>email<EMAIL_ADDRESS>
# Assessing quantum thermalization in physical and configuration spaces
via many-body weak values
Carlos F. Destefani1 [ Xavier Oriols1 [ 1 Department of Electronic
Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona,
Spain
###### Abstract
We explore the origin of the arrow of time in an isolated quantum system
described by the Schrödinger equation. We provide an explanation from weak
values in the configuration space, which are understood as operational
properties obtained in the laboratory following a well-defined protocol. We
show that quantum systems satisfying the eigenstate thermalization hypothesis
can simultaneously provide thermalized ensemble expectation values and
nonthermalized weak values of the momentum, both from the same operational
probability distribution. The reason why weak values of the momentum may
escape from the eigenstate thermalization hypothesis is because they are
linked only to off-diagonal elements of the density matrix in the energy
representation. For indistinguishable particles, however, operational
properties can not be defined in the configuration space. Therefore, we state
that the origin of the arrow of time in isolated quantum systems described by
the Schrödinger equation comes from dealing with properties obtained by
averaging (tracing out) some degrees of freedom of the configuration space. We
then argue that thermalization does not occur in the properties defined in the
configuration space, and our argument is compatible with defending that
thermalization is a real phenomenon in the properties defined in the physical
space. All of these conclusions are testable in the laboratory through many-
body weak values.
## I Introduction
The arrow of time has always been a topic of lively debate [1, 2, 3, 4, 5, 6,
7, 8]. It appears in many disciplines as, for example, a cosmological arrow of
time pointing in the direction of the Universe expansion [2]. A related arrow
of time appears in the time evolution of time-irreversible macroscopic systems
governed by the second law of thermodynamics, where entropy always increases
with time [1, 7]. The puzzle implicit in such irreversibility is that most
fundamental microscopic laws have no arrow of time. They are time-reversible
laws, in the sense that time appears as a variable, just like position,
without any privileged direction. But, if macroscopic laws emerge from
microscopic laws, what can make them so different? A possible explanation is
that such fundamental laws are in fact not time-reversal. For example, it has
been argued that the time-reversible Schrödinger equation is not the true law
at a fundamental level, and that it should be substituted by laws from
spontaneous collapse theories which, by construction, are time-irreversible
[3]. Another explanation argues that real systems are never perfectly
isolated, so that time-reversible fundamental laws, despite being the true
laws, are not directly applicable [4]. And yet another argumentation claims
that the evolution of a real system depends, apart from the true time-
reversible microscopic laws, on the initial conditions which provide an
irreversible time evolution [6].
In this paper we explore a different path, via weak values in the
configuration space, to understand the physical origins of the arrow of time
in perfectly isolated nonrelativistic systems described by the Schrödinger
equation, assumed as a true reversible law where initial conditions are not
relevant to explain the observed time-irreversibility in our results. Our goal
is to link the evolution of microscopic (or macroscopic) properties of a model
system with the configuration (or physical) space where such properties are
defined. The wave function solution of the Schrödinger equation is defined in
a $3N$-dimensional configuration space, while typical properties where time-
irreversibility is observed, are defined in smaller spaces where some degrees
of freedom of the configuration space are integrated. Thus, the question that
motivates us is whether the presence or absence of an arrow of time in the
time evolution of the properties of a quantum system is a consequence of
defining them in the full configuration space or in a smaller space.
For our goal, the renewed interest in statistical mechanics of closed quantum
systems [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
provides the perfect scenario. Such interest has been generated by the
successful experimental ability to isolate and manipulate bosonic [25, 26, 27,
28, 29] and fermionic [30, 31, 32, 33, 34] many-body systems built on ultra-
cold atomic gases subjected to optical lattices. Such quantum systems can be
described by the many-particle Schrödinger equation; an arrow of time appears
because, despite these systems being expected to present unitary evolution,
some of their initial nonequilibrium nonthermalized properties may later
thermalize.
A preliminary consideration is that it is not at all obvious whether the
configuration space is more or less fundamental than the ordinary physical
space. The nonrelativistic Schrödinger equation can be seen as a sort of
approximation to the relativistic quantum field theory [35]. For
indistinguishable particles, quantum field theory does not require knowledge
of the exact position of each particle in the configuration space, but only of
how many particles are present in a position of the physical space and, as
such, configuration space seems less fundamental than physical space.
Another consideration is that thermalization is typically reported in
properties mensurable in a laboratory, so that one needs to discuss, within
similar empirical protocols, nonthermalized properties also mensurable in a
laboratory. But making conclusions testable in a laboratory opens new
difficulties since, strictly speaking, a measured closed system is no longer a
closed system because of its interaction with the measuring apparatus [36].
Such measurement produces a collapse of the quantum state of the isolated
system, which is at the origin of quantum randomness and backaction. Within
the orthodox theory, such a collapse requires a time-asymmetric law, different
from the time-symmetric Schrödinger equation, so that an “orthodox quantum-
mechanical arrow of time” seems to enter into play [37]. We will see how the
weak values protocol provides operational properties of the quantum system
without backaction or quantum randomness (avoiding the role of the collapse in
our discussion).
It is important to differentiate between operational (empirical) and
ontological (real) properties of a system. Operational properties are those
whose definition comes exclusively from operations done on the laboratory over
the system (with or without ontological meaning for that property), which are
defined independently on any quantum theory. Ontological properties, on the
other hand, are those that a specific quantum theory postulates to exist.
Therefore, it is possible that a given operational property coincides with an
ontological property in one theory, but not in another. A typical example is
the velocity of a particle, known to be an operational property computed from
weak values, independently on any quantum theory; such operational property
coincides with an ontological property in the Bohmian theory (the velocity
itself), but it is not an ontological property in the Orthodox theory. It is
far from our scope to discuss whether or not thermalization is an ontological
phenomenon or not occurring in the configuration space or in the physical
space, since this depends on which quantum theory is invoked. Our less
controversial focus here is to approach thermalization in closed systems from
operational properties testable in laboratory.
The structure of the paper is the following. Sect. II presents the many-body
generalization of the single-particle weak values, stating them as operational
properties in configuration space without explicit dependence on quantum
randomness and backaction, for both distinguishable and indistinguishable
particles. Sect. III discusses thermalization and equilibration concepts as
found in the literature for closed quantum systems, and address the eigenstate
thermalization hypothesis [38, 39]. Sect. IV defines our model system and its
nonequilibrium dynamics, and summarizes our results for both expectation
values and weak values from the Schrödinger equation dynamics. In Sect. V we
conclude.
## II Operational properties in the configuration space
To simplify notation, along the paper we use natural units and consider a
$1$-dimensional physical space with degree of freedom $x$, so that
$\mathbf{x}=\\{x_{1},...,x_{N}\\}$ is the position in the $N$-dimensional
configuration space; the extension to a $3N$-dimensional space should be
straightforward. As already mentioned, a measured closed system is no longer a
closed system, and a strong measurement, for example, of the momentum operator
$\hat{p}$ yields the eigenvalue $p$, and produces the initial state to be
converted into the momentum eigenstate. On the other hand, expectation values
and weak values yield operational information of the system without backaction
and quantum randomness.
### II.1 Expectation values
We define the expectation value of the momentum $\langle p(t)\rangle$ as an
operational property of the system in the sense that it is linked to a well
defined protocol in the laboratory,
$\langle p(t)\rangle=\int dp\;p\;\mathbb{P}(p,t),$ (1)
where the probability distribution $\mathbb{P}(p,t)$ can be obtained as
follows: i) many identical initial states $|\Psi(t)\rangle$ are prepared at
time $t$; ii) for each initial state, a (weak or strong) measurement of
momentum is done, yielding the value $p$ at time $t$; iii) $\mathbb{P}(p,t)$
is constructed by counting how many $p$ occurs when repeating ii) on ensemble
i).
When then applying Born law to predict the value of $\mathbb{P}(p,t)$ one can
easily identifies
$\langle p(t)\rangle=\int
dp\;p\;\mathbb{P}(p,t)=\langle\Psi(t)|\hat{p}|\Psi(t)\rangle.$ (2)
Notice that the right hand side of (2) depends on the state of the system
$|\Psi(t)\rangle$ before a measurement is done, without neither randomness nor
backaction. In fact, $\langle p(t)\rangle$ is a typical property used to
analyze when an isolated quantum system thermalizes. We are here interested in
discussing thermalization from operational properties of the isolated quantum
system requiring the measurement of both momentum and position simultaneously.
Let us then start by discussing weak values in physical space.
### II.2 Weak values in the physical space
It has recently been shown that weak values [40] are able to yield dynamic
information on two noncommuting operators at a single time avoiding backaction
[41, 42] and quantum randomness. Weak values have attracted a lot of
theoretical [41, 42, 43, 44, 45, 46] and experimental [47, 48, 49] interests
in many research fields.
At the laboratory the single-particle weak value of momentum $p_{W}(x,t)$ is
given by
$p_{W}(x,t)=\frac{\int dp\;p\;\mathbb{P}(p,x,t)}{\int dp\;\mathbb{P}(p,x,t)},$
(3)
computed from the probability distribution $\mathbb{P}(p,x,t)$ via the
following procedure: i) many identical initial states $|\Psi(t)\rangle$ are
prepared at time $t$; ii) for each initial state, a weak measurement of the
momentum is done, yielding the value $p$ at time $t$; iii) subsequently, a
strong measurement of the position is done, yielding the value $x$ at time
$t$; iv) $\mathbb{P}(p,x,t)$ is constructed by counting how many $p$ and $x$
occurs when repeating ii) and iii) on ensemble i).
Since $x$ is post-selected in (3), but not integrated out, one gets
information on how the expectation value of the momentum is distributed in the
physical space. Again, when applying Born law to predict the value of
$\mathbb{P}(p,x,t)$ one can easily identifies
$p_{W}(x,t)=\frac{\int dp\;p\;\mathbb{P}(p,x,t)}{\int
dp\;\mathbb{P}(p,x,t)}=\text{Real}\left(\frac{\langle
x|\hat{p}|\Psi(t)\rangle}{\langle x|\Psi(t)\rangle}\right).$ (4)
Similarly to (2), the ensemble-over-identical-experiments in the right hand
side in (4) eliminates the undesired backaction and quantum randomness induced
by the first measuring apparatus [40, 42, 46, 50, 51], so that $p_{W}(x,t)$ in
(4) depends only on the initial state before the measurements took place.
Expression (4) allows us to give the weak value a very simple interpretation.
In a single-particle system, from $\mathbb{I}=\int dx|x\rangle\langle x|$, one
can rewrite the expectation value of the momentum in (2) (see Appendix A) as
$\langle p\rangle(t)=\int dx\langle\Psi(t)|x\rangle\langle
x|\hat{p}|\Psi(t)\rangle=\int dx|\Psi(x,t)|^{2}p_{W}(x,t),$ (5)
so that the same probability distribution $\mathbb{P}(p,x,t)$ used to compute
$p_{W}(x,t)$ in (4) can be employed to obtain $\mathbb{P}(p,t)$ used to
compute $\langle p(t)\rangle$ in (2), since
$\mathbb{P}(p,t)=\int dx\;\mathbb{P}(p,x,t).$ (6)
Using the mathematics (and not necessarily the ontology) of Bohmian mechanics,
one can also re-interpret $p_{W}(x,t)$ as the (operational) velocity of the
particle at position $x$ and time $t$ [43, 44, 45, 46, 52, 53, 54, 55, 56,
57],
$p_{W}(x,t)=\text{Imag}\left(\frac{1}{\Psi(x,t)}\frac{\partial\Psi(x,t)}{\partial
x}\right)=\frac{J(x,t)}{|\Psi(x,t)|^{2}},$ (7)
with $J(x,t)=\text{Imag}(\Psi^{*}(x,t)\partial\Psi(x,t)/\partial x)$ the
current density (see Appendix B).
### II.3 Weak values in the configuration space for distinguishable particles
Notice that $p_{W}(x,t)$ is an operational property in the ordinary physical
space, but we now need to define an operational property in the configuration
space for dealing with an isolated quantum system with $N$ particles.
Therefore, in this paper, we extend the original single-particle weak values
in (4) to $N$-particle scenarios for both distinguishable and
indistinguishable cases. For the former case, the weak values for the
$j$-particle is
$p_{W}^{j}(\mathbf{x},t)=\frac{\int
dp_{j}\;p_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}{\int
dp_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)},$ (8)
where now the probability distribution $\mathbb{P}(p_{j},\mathbf{x},t)$ is
obtained as follows: i) many identical initial states $|\Psi(t)\rangle$ are
prepared at time $t$; ii) for each initial state, a weak measurement of the
momentum of the $j$-particle is done, yielding the value $p_{j}$ at time $t$;
iii) subsequently, a strong measurement of the positions of particles
$1$,$2$,…,$N$ yielding respectively the values $x_{1}$,$x_{2}$,…,$x_{N}$ is
done; iv) $\mathbb{P}(p_{j},\mathbf{x},t)$ is constructed by counting how many
$p_{j}$,$x_{1}$,$x_{2}$,…,$x_{N}$ occurs when repeating ii) and iii) on
ensemble i).
Our definition of distinguishable particles above is operational in the sense
that the measuring apparatus is somehow able to distinguish particles, for
example, by measuring their masses but, to avoid unnecessary notation, we have
not indicated this extra measurement in the above protocol. Again Born law
allows us to rewrite (8) as
$p_{W}^{j}(\mathbf{x},t)=\frac{\int
dp_{j}\;p_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}{\int
dp_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}=\text{Real}\left(\frac{\langle\mathbf{x}|\hat{p}_{j}|\Psi(t)\rangle}{\langle\mathbf{x}|\Psi(t)\rangle}\right),$
(9)
where once more from the mathematics (and not necessarily from the ontology)
of Bohmian mechanics, one can also re-interpret $p_{W}^{j}(\mathbf{x},t)$ as
the (operational) velocity of the $j$-particle at the position $\mathbf{x}$ in
the configuration space and time $t$ [43, 44, 45, 46, 52, 53, 54, 55, 56, 57],
$p_{W}^{j}(\mathbf{x},t)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}},$
(10)
with
$J^{j}(\mathbf{x},t)=\text{Imag}(\Psi^{*}(\mathbf{x},t)\partial\Psi(\mathbf{x},t)/\partial
x_{j})$ the current density in the $x_{j}$ direction.
### II.4 Weak values for indistinguishable particles
The most common situation in the laboratory however relates to identical
particles, for which a proper many-body wave function should implicitly
include the exchange symmetry among the particles. Equation (9) then becomes
inaccessible in a laboratory because it is no longer possible to know,
operationally, which position belongs to each particle. To deal with
indistinguishable particles, one needs to construct a many-body weak value
defined in physical space coordinate $x$ by averaging (integrating) all
degrees of freedom (see Appendix A). By doing so one obtains
$\tilde{p}_{W}(x,t)=\frac{\int dp\;p\;\mathbb{\tilde{P}}(p,x,t)}{\int
dp\;\mathbb{\tilde{P}}(p,x,t)},$ (11)
where the probability distribution $\mathbb{\tilde{P}}(p,x,t)$ is now obtained
as follows: i) many identical initial states $|\Psi(t)\rangle$ are prepared at
time $t$; ii) for each initial state, a weak measurement of the momentum of
one nonidentified particle is done, yielding the value $p$ at time $t$; iii)
subsequently, a strong measurement of the position of the same or another
nonidentified particle is done, yielding the value $x$ at time $t$; iv)
$\mathbb{\tilde{P}}(p,x,t)$ is constructed by counting how many $p$ and $x$
occurs when repeating ii) and iii) on ensemble i).
Born law again allows us to rewrite (11) as (see Appendix A)
$\tilde{p}_{W}(x,t)=\frac{\int dp\;p\;\mathbb{\tilde{P}}(p,x,t)}{\int
dp\;\mathbb{\tilde{P}}(p,x,t)}=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$
(12)
with
$p_{W}^{j,k}(x,t)=\frac{\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;p_{W}^{j}(..,x_{k-1},x,x_{k+1},..,t)|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}{\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}.$ (13)
Notice that $\tilde{p}_{W}(x,t)$ is, in fact, the local velocity as used in
quantum hydrodynamic models [52, 53, 54, 55], being empirically accessible in
both distinguishable and indistinguishable scenarios. The operational protocol
for computing $\mathbb{\tilde{P}}(p,x,t)$ for indistinguishable particles is
related to $\mathbb{P}(p_{k},\mathbf{x},t)$ for distinguishable particles as
$\mathbb{\tilde{P}}(p,x,t)=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;\mathbb{P}(p_{j},..,x_{k-1},x,x_{k+1},..,t).$ (14)
## III Thermalization in isolated systems from expectation values
Our main contribution in the study of quantum thermalization, as detailed in
next section, is the inclusion of many-body weak values in the configuration
space as operational properties. However, such a study has usually been done
in the literature in terms of expectation values as in (2), and as such we
summarize in this section the role of expectation values to characterize
thermalization.
For an initial nonequilibrium pure state $|\Psi(0)\rangle$, the Schrödinger
equation provides its unitary evolution as
$|\Psi(t)\rangle=\sum_{n}c_{n}e^{-iE_{n}t}|n\rangle$, where $|n\rangle$ is an
energy eigenstate with eigenvalue $E_{n}$, and $c_{n}=\langle
n|\Psi(0)\rangle$ is defined by the initial conditions. The expectation value
of some observable $\hat{A}$ is given by
$\langle A\rangle(t)=\sum_{n}\rho_{n,n}A_{n,n}+\sum_{n,m\neq
n}\rho_{n,m}(t)A_{m,n},$ (15)
with the time-dependent off-diagonal elements of the density matrix in the
energy representation defined as
$\rho_{n,m}(t)=c_{m}^{*}c_{n}e^{i(E_{m}-E_{n})t},$ (16)
and the time-independent diagonal elements as
$\rho_{n,n}(t)=c_{n}^{*}c_{n}=\rho_{n,n}(0),$ (17)
with the operator $\hat{A}$ in the energy representation being
$A_{m,n}=\langle m|\hat{A}|n\rangle.$ (18)
A system is said to equilibrate if, after some time $t_{eq}$ enough for full
dephasing between different energy eigenstates, the off-diagonal terms
(coherences) cancel out so that (15) can be computed solely from the time-
independent diagonal terms (populations), that is, $\langle
A\rangle(t)\approx\sum_{n}\rho_{n,n}A_{n,n}$ for most times $t>t_{eq}$ (except
for some recurrence times). The properties of the system after equilibration
are fully determined by the initial conditions $\rho_{n,n}(0)=|c_{n}|^{2}$,
since the density matrix populations are time-independent. The closed system
is then said to thermalize when $\langle A\rangle(t)$ becomes roughly equal to
the expectation value as computed from the classical density matrix in the
microcanonical ensemble, $\rho_{cl}$, in which equal probabilities are
attached to each microstate within an energy window defined by the initial
conditions; that is, a subset $N_{act}$ of the energy eigenstates are
initially activated at $t=0$, and they will remain as the only states
dictating the system dynamics at any $t$. It is important to notice that
$N_{act}$ refers to a given number of relevant elements in a basis set, but it
has no relation with the number of particles $N$ of the system. In other
words, one can also expect thermalization even in few-particle systems, as
detailed in next section; in fact, thermalization has also been studied in
laboratories in small systems with as little as 6 [27], 5 [14], or 2-4 bosons
[58, 59], 3 qubits [60], and even single-particle systems [61, 62, 63].
The eigenstate thermalization hypothesis (ETH) [38, 39] has become the
standard theory dealing with quantum thermalization in closed systems. It
states that the dephasing above mentioned is typical to nondegenerate and
chaotic many-body nonintegrable systems, where the off-diagonals $A_{m,n}$ in
(18) become exponentially smaller than $A_{n,n}$. In recent years a large
amount of numerical experiments has successfully tested such a hypothesis by
directly diagonalizing some sort of short range many-body lattice Hamiltonian,
like Fermi- or Bose-Hubbard [26, 31, 32, 64, 65, 66], and XXZ- or XYZ-
Heisenberg [14, 23, 27, 67, 68, 69, 70], in the search of chaotic signatures
in the statistics of their spectra, as in general induced by local impurities,
without the need to explicitly evolve $|\Psi(0)\rangle$. The ETH states that
nonintegrable systems (where total energy may be the only conserved quantity),
after a quench (which may create a nonequilibrium initial state by activating
a subset $N_{act}$ of excited eigenstates), can present a ‘chaotic’ spectrum
ruled by the Wigner-Dyson statistics (which contains level repulsion), and
that the long time average of the expectation value of some observable roughly
equals its thermal equilibrium value in an (microcanonical) ensemble. Such a
hypothesis claims that thermalization is indeed hidden in the chaotic initial
nature of the Hamiltonian eigenstates themselves. Lattice models typically
handle $\approx 24$ sites with $\approx 1/3$ filling, and the above local
impurities are added to break their otherwise integrable character, as the ETH
overall claims that integrable systems are not expected to thermalize.
On the other hand, our time evolution dwells in true configuration space with
an antisymmetrized wave function and full long range electron-electron
interaction; due to the tensorial nature of our problem, and since we need to
employ a grid with $M\approx 10^{3}$ points per degree of freedom for decent
position and momentum resolutions, we can realistically only deal with
$N\lesssim 4$ particles ($N=3$ already implies $M\approx 10^{9}$ grid points
at each time step).
## IV Numerical results for expectation values and weak values
We now apply the many-body weak values machinery for the analysis of quantum
thermalization in a model with $N$ spinless electrons [20, 71, 72, 73, 74, 75,
76, 77], typical of condensates in harmonic oscillator traps under a speckle
field. Such a field translates to a ‘chaotic’ random disorder potential, which
can yield a chaotic energy spectrum as requested by the ETH and so induce
thermalization even in systems with small $N$. Our model can attach a disorder
to each grid point, and an initial velocity at $t=0$ is given to the electrons
as to simulate the initial quench of the confining potential, for each of the
considered $N=1,2,3$ systems. In fact, most of the thermalization literature
dealing with identical particles employs some lattice-based model, since such
models avoid the need of explicit knowledge of the exact position of each
particle in a point of the configuration space; instead, they only need to
know how many particles are present in each site of the physical space. From a
computational point of view lattice models have unquestionable advantages but
are inappropriate for our goal of discussing whether or not thermalization
occurs in configuration space, a goal that forces us to directly solve the
time-evolution of the few-body Schrödinger equation in configuration space,
and to analyze thermalization by monitoring the time-evolution of both
expectation values and weak values.
### IV.1 Initial state
The pure initial $N$-electron nonequilibrium antisymmetric state is
$\langle\mathbf{x}|\Psi(0)\rangle=\frac{1}{\mathcal{C}}\sum_{n=1}^{N!}\text{sign}(\vec{p}(n))\prod_{j=1}^{N}\psi_{j}(x_{p(n)_{j}},0),$
(19)
with $\mathcal{C}$ a normalization constant and $\text{sign}(\vec{p}(n))$ the
sign of the permutation $\vec{p}(n)=\\{p(n)_{1},..,P(n)_{N}\\}$. Each initial
Gaussian state in (19) is
$\psi_{j}(x)=\exp\left[-\frac{(x-x_{0j})^{2}}{2\sigma_{j}^{2}}\right]\exp\left[ip_{0j}(x-x_{0j})\right],$
(20)
with spatial dispersion $\sigma_{j}$, central position $x_{0j}$, and central
velocity $p_{0j}$. The dynamical evolution of $|\Psi(t)\rangle$ is determined
by the Schrödinger equation, $i\partial|\Psi(t)\rangle/\partial
t=\hat{H}|\Psi(t)\rangle$, where the Hamiltonian $\hat{H}$ is described in
next section.
### IV.2 Full Hamiltonian
The $N$-electron Hamiltonian of our model system is
$\hat{H}=\hat{H}_{0}+\hat{D}$, with
$\hat{H}_{0}=\sum_{j=1}^{N}\left[\hat{k}_{j}+\hat{v}_{j}+\sum_{k<j}^{N}\hat{e}_{k,j}\right],\;\;\;\;\;\;\hat{D}=\sum_{j=1}^{N}\hat{d}_{j}.$
(21)
In $\hat{H}_{0}$, $\langle
x_{j}|\hat{e}_{k,j}|x_{k}\rangle=1/\sqrt{(x_{j}-x_{k})^{2}+\alpha^{2}}$ takes
care of the Coulomb repulsion with a smooth parameter $\alpha$, $\langle
x_{j}|\hat{k}_{j}|x_{j}\rangle=-\partial^{2}/(2\partial x_{j}^{2})$ stands for
the kinetic energy, and $\langle
x_{j}|\hat{v}_{j}|x_{j}\rangle=\omega^{2}x_{j}^{2}/2$ is the harmonic trap
potential. On the other hand, $\hat{D}$ introduces random disorder at every
grid point, where $\langle
x_{j}|\hat{d}_{j}|x_{j}\rangle=\gamma_{D}\sum_{k=1}^{M}a_{k}\exp[-4(x_{j}-g_{k})^{2}/\sigma_{D}^{2}]$,
with $\gamma_{D}$ its strength and $\sigma_{D}$ its spatial dispersion, while
$g_{k}$ runs through $M$ grid points; the set of random numbers $a_{k}$
satisfies $\langle a_{k}\rangle=0$ and $\langle a_{k}^{2}\rangle=1$, and the
disorder potential is normalized via $\int\langle
x_{j}|\hat{d}_{j}|x_{j}\rangle^{2}dx_{j}=\gamma_{D}^{2}$. Such random disorder
can be mapped onto speckle field potentials typical in some optical lattice
experiments. All simulation parameters are found in [78].
Figure 1: (a): Zoom of a disordered harmonic potential for $N=1$, $\langle
x_{1}|\hat{v}_{1}+\hat{d}_{1}|x_{1}\rangle$, from a larger simulation box.
(b): Successive energy splittings $\Delta E=E_{n}-E_{n-1}$ from a grid
diagonalization of the respective Hamiltonian $\hat{H}$. (d): zoom around the
peak of the corresponding density matrix modulus $|\rho_{n,m}(t=0)|$, which
remains the same at any time $t$ in an unitary evolution. (c): initial
positions $x_{0j}$ for $N=1,2,3$, and initial velocity $p_{0j}$ which is the
same in every case.
Figure 1(a) exemplifies, for $N=1$, a typical shape of the disordered harmonic
potential, $\langle x_{1}|\hat{v}_{1}+\hat{d}_{1}|x_{1}\rangle$, while figure
1(b) shows the corresponding successive energy splittings $\Delta
E=E_{n}-E_{n-1}$, as obtained from a direct diagonalization of $\hat{H}$,
which oscillate around the pure harmonic oscillator value of $1$. Figure 1(d)
shows the related shape of the density matrix modulus $|\rho_{n,m}(t=0)|$,
from where one identifies $N_{act}\approx 70$ (counting every level above
$10\%$ of peak value), while figure 1(c) shows initial positions $x_{0j}$,
$j=1...N$, and the initial velocity $p_{0}$, the same for any $j$ and $N$, for
the $N=1,2,3$ systems. Notice that $p_{0}$ not only takes the role of
simulating the initial quench of the trap and so to initiate the
nonequilibrium dynamics, but it is also responsible for determining the size
of the energy window and so the value of $N_{act}$. The initial energy $E_{0}$
of the wave packet defines, in the language of the ETH, the center of the
microcanonical energy window; since $E_{0}\approx\omega+p_{0}^{2}/2=201$, the
peak is at $n=m=201$. As mentioned in (15), the diagonal terms are time-
independent and, although real and imaginary parts of the off-diagonals terms
oscillate in time, their modulus remain constant so that, in an unitary
evolution, the modulus seen in figure 1(d) remains the same at any $t$ (see
Appendix B).
Both initial wave packet and Hamiltonian in (19)-(21) have many adjustable
parameters in [78] upon which the value of $t_{eq}$ depends on: (i) Coulomb
correlation by varying $\omega$ or $\alpha$; (ii) initial Gaussian by varying
$\sigma_{j}$, $x_{0j}$, or $p_{0j}$; (iii) disorder potential by varying
$\gamma_{D}$ or $\sigma_{D}$. All of them play a role in determining the
$\Delta E$ splittings of the involved $N_{act}$ activated eigenstates, which
is what drives the nonequilibrium dynamics of both expectation values and weak
values. It is not our goal to fully characterize the equilibration process as
a function of all those parameters. Neither to address the topic of many-body
localization [79, 80, 81], which should work against thermalization, nor to go
deeper in the issue of quantum-to-classical transition at $t\gg t_{eq}$; these
two latter issues are beyond the scope of our work and are focus of extensive
studies elsewhere. The main goal of our paper is to present a distinct
perspective in the understanding of quantum thermalization by looking at the
many-body weak values of the momentum in the configuration space.
Each numerical experiment corresponds to a static realization of a disorder
pattern; $t_{eq}$ hardly changes among different runs, given that all other
model parameters remain unchanged. The disorder amplitude should not be too
strong, to avoid localization due to all small wells created on the top of the
trap, neither too weak, to avoid too long simulation times until reaching
$t_{eq}$. We emphasize that even in the absence of electron-electron
collision, disorder collision is able to create correlation among distinct
degrees of freedom in configuration space when $N>1$.
### IV.3 Time evolution
Figure 2: Wave packet dynamics. Upper panels for $N=3$: initial
$|\Psi(x_{1},x_{2},[x_{3}],t=0)|^{2}$ (a) and final
$|\Psi(x_{1},x_{2},[x_{3}],t=150)|^{2}$ (b) shapes. Middle panels for $N=2$:
initial $|\Psi(x_{1},x_{2},t=0)|^{2}$ (c) and final
$|\Psi(x_{1},x_{2},t=150)|^{2}$ (d) shapes. Lower panels: (e) for $N=1$:
initial (dotted) $|\Psi(x_{1},t=0)|^{2}$ and final (solid)
$|\Psi(x_{1},t=150)|^{2}$ shapes; (f): $1$D-view for the three systems:
$|\Psi(x_{1},[x_{2}],[x_{3}],t)|^{2}$ for $N=3$ (red),
$|\Psi(x_{1},[x_{2}],t)|^{2}$ for $N=2$ (blue), and $|\Psi(x_{1},t)|^{2}$ for
$N=1$ (green), with solid (dotted) lines for $t=150$ ($t=0$); also shown the
classical microcanonical harmonic oscillator distribution $\rho_{cl}(x_{1})$
(black dashed line). All plots are in log-scale, and the horizontal axis in
(a),(c) ((b),(d)) is the same as in (e) ((f)).
Figure 2 plots in configuration space the time evolution of the $N$-electron
wave function for the three $N=1,2,3$ systems from the initial nonequilibrium
state. We show $|\Psi(x_{1},x_{2},[x_{3}],t)|^{2}$ for $N=3$ and
$|\Psi(x_{1},x_{2},t)|^{2}$ for $N=2$ at initial ($t=0$, panels (a),(c)) and
final ($t=150$, panels (b),(d)) simulation times, while panel (e) shows
$|\Psi(x_{1},t)|^{2}$ for $N=1$ for both initial ($t=0$, dotted) and final
($t=150$, solid) simulation times; the notation $[x_{j}]$ means that the
degree $j$ is integrated out, with results independent on chosen $j$ due to
the antisymmetry of the problem. The initially localized wave packets fully
spread out due to random scattering generated by both disorder potential and
Coulomb repulsion, with such spreading becoming more homogeneous as $N$
increases. In panel (f) we show the respective $1$D plots of
$|\Psi(x_{1},[x_{2}],[x_{3}],t)|^{2}$ ($N=3$, red),
$|\Psi(x_{1},[x_{2}],t)|^{2}$ ($N=2$, blue), and $|\Psi(x_{1},t)|^{2}$ ($N=1$,
green), with initial (final) results at dotted (solid) lines; the
probabilities at large $t$, as one expects to have crossed $t_{eq}$, approach
the microcanonical distribution of a classical harmonic oscillator (dashed
black line), $\rho_{cl}(x)=[\pi l_{0}\sqrt{p_{0}^{2}-x^{2}/l_{0}^{2}}]^{-1}$,
so that the dynamics develops in between the classical turning points at
$x_{TP}=\pm 20$ ($=p_{TP}$ since $\omega=1/l_{0}^{2}=1$); the deviation from
$\rho_{cl}(x)$ decreases as $N$ increases. In Appendix C we analyze the same
nonequilibrium dynamics in momentum representation.
### IV.4 Thermalized expectation values
Figures 3 and 4 show time evolution and thermalization of some typical
expectation values $\langle
A\rangle(t)=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle$, for $N=1,2,3$ respectively
in panels (a), (b), (c). On one hand, figure 3 focus on energy terms
normalized by $N$: kinetic $\langle K\rangle(t)$, with
$\hat{K}=\sum_{j}\hat{k}_{j}$, potential $\langle V\rangle(t)=\langle
V_{HO}\rangle(t)+\langle V_{D}\rangle(t)+\langle V_{Cou}\rangle(t)$, with
$\hat{V}_{HO}=\sum_{j}\hat{v}_{j}$, $\hat{V}_{D}=\sum_{j}\hat{d}_{j}$, and
$\hat{V}_{Cou}=\sum_{j,k<j}\hat{e}_{k,j}$, and half of total energy $\langle
E\rangle(t)=\langle K\rangle(t)+\langle V\rangle(t)$. On other hand, figure 4
focus on position $\langle x_{j}\rangle(t)$ and momentum $\langle
p_{j}\rangle(t)$ for the $j$-electron, and on their RMS values
$z_{j,RMS}(t)=\sqrt{\langle z^{2}_{j}\rangle(t)-\langle z_{j}\rangle^{2}(t)}$,
with $z=x,p$, and results independing on chosen $j$. _Without_ random
disorder, such expectation values would only exhibit harmonic oscillations
with period $2\pi/\omega$: while $\langle x_{j}\rangle(t)$ and $\langle
p_{j}\rangle(t)$ would respectively oscillate within position $\pm x_{TP}=\pm
20$ and momentum $\pm p_{TP}=\pm 20$ turning points, $\langle V\rangle(t)/N$
and $\langle K\rangle(t)/N$ would respectively oscillate within $0$ and
$x_{TP}^{2}/2=200$ and within 0 and $p_{TP}^{2}/2=200$; these latter values
increase a little upon $N$ due to Coulomb repulsion, whose isolated
contribution is shown ($100\times$-magnified) in figure 3.
Figure 3: Energy expectation values from the dynamics in figure 2. Panels (a),
(b), (c) respectively for $N=1,2,3$. Kinetic $\langle K\rangle(t)$, potential
$\langle V\rangle(t)$, half of total energy $\langle E\rangle(t)=\langle
K\rangle(t)+\langle V\rangle(t)$, and isolated contribution of $\langle
V_{Cou}\rangle(t)$ ($100\times$-magnified) are all normalized by $N$. Inset
for $N=2$ shows a longer propagation time, $t=[150-300]$. Legend in (a) and
horizontal axis in (c) apply to all panels.
The _presence_ of random disorder, even though $\langle V_{D}\rangle(t)\approx
0$ at any $t$, brings the initial nonequilibrium state into a final
_equilibrium_ state after a relaxation time $t_{eq}$. We know from the
discussions in (15) and in figure 1(d) that thermalization of an observable
$\hat{A}$ is determined by the diagonal populations of the density matrix,
while its off-diagonal coherences should dephase and only yield small
fluctuations around the relaxed value (see Appendix B). So one may estimate
the value of $t_{eq}$ either from figure 3, when the virial theorem $\langle
K\rangle\approx\langle V\rangle\approx\langle E\rangle/2$ is roughly satisfied
(since $\langle V_{Cou}\rangle\ll\langle V_{HO}\rangle$) [74], or from figure
4, when $\langle p_{j}\rangle\approx\langle x_{j}\rangle\approx 0$ seemingly
indicating a frozen dynamics after thermalization. From this latter result the
RMS values become $z_{j,RMS}(t)\approx\sqrt{\langle z^{2}_{j}\rangle(t)}$,
becoming also constant in figure 4 at $t>t_{eq}$ (from $\approx 14.2$ for
$N=1$ to $\approx 14.4$ for $N=3$), so that the values of
$p^{2}_{j,RMS}/2=\langle p^{2}_{j}\rangle/2$ and $x^{2}_{j,RMS}/2=\langle
x^{2}_{j}\rangle/2$ roughly yield the respective values of $\langle
K\rangle/N$ and $\langle V\rangle/N$ at $t>t_{eq}$ in figure 3. As expected
for an unitary evolution, $\langle E\rangle(t)/N$ remains conserved at any
$t$, from $\approx 201$ for $N=1$ to $\approx 207$ for $N=3$.
Figure 4: Position $\langle x_{j}\rangle(t)$ and momentum $\langle
p_{j}\rangle(t)$ expectation values from the dynamics in figure 2. Panels (a),
(b), (c) respectively for $N=1,2,3$. Their respective RMS values, as defined
in the text, are also shown, where results do not depend on chosen $j$. Inset
for $N=2$ shows a longer propagation time, $t=[150-300]$. Legend in (a) and
horizontal axis in (c) apply to all panels.
The value of $t_{eq}$ depends on all parameters [78] in (19)-(21), e.g., the
smaller is $p_{0j}$ or the higher is $\gamma_{D}$ the smaller is $t_{eq}$. By
increasing the influence of $\langle V_{Cou}\rangle$ in comparison to $\langle
K\rangle$ (by decreasing $\omega$ or $\alpha$), $t_{eq}$ increases since the
oscillation period and so $x_{TP}$ increases. The plots of $\langle
V_{Cou}\rangle(t)$ in figure 3 show that the Coulomb repulsion is more
effective at the turning points for $t\ll t_{eq}$, where electrons spend more
time reversing their movements, while the disorder potential overall acts
through a whole oscillation, although one may take it as more effective at the
origin. Coulomb correlation has a striking influence on the thermalization
process: in configuration space the only scattering mechanism for $N=1$ is due
to disorder, while for $N>1$ Coulomb scattering also makes more difficult for
the system to relax. This is seen by the slightly increasing values of
$t_{eq}$ as one moves in figure 3 from (a) ($t_{eq}\approx 70$) to (b)
($t_{eq}\approx 80$) to (c) ($t_{eq}\approx 90$). The vanishing of $\langle
p\rangle(t)$ in figure 4 seems more effective as one moves from (a) to (b) to
(c) but, however, it does not necessarily imply that electrons have achieved a
stationary-state null velocity at $t\gg t_{eq}$, as our following analysis on
weak values of the momentum will clarify (see also ‘phase-space’ in Appendix
C).
### IV.5 Nonthermalized weak values
We can at last elaborate on how the many-body weak values of the momentum may
improve our understanding on thermalization. In table 1 we summarize the five
types of operational properties accessible for the three different types of
quantum systems considered in this section: single-particle, distinguishable
particles, indistinguishable particles.
$N=1$ | $\langle p(t)\rangle$ (2) | $p_{W}(x,t)$ (4) | | |
---|---|---|---|---|---
$N>1$ (Dis) | $\langle p(t)\rangle$ (2) | | $p_{W}^{j}(\mathbf{x},t)$ (9) | $\tilde{p}_{W}(x,t)$ (12) | $p_{W}^{j,k}(x,t)$ (13)
$N>1$ (Ind) | $\langle p(t)\rangle$ (2) | | | $\tilde{p}_{W}(x,t)$ (12) |
Table 1: Five operational properties accessible in the laboratory for each of
the three quantum systems considered in our work: (i) with $N=1$ particles,
(ii) with $N>1$ distinguishable particles, and (iii) with $N>1$
indistinguishable particles.
The plot in figure 5(a) corresponds to the quantum system $N=1$ in table 1.
Although the expectation value $\langle p_{j}\rangle(t)$ seems to indicate
that the quantum behavior at $t\gg t_{eq}$ roughly equals the behavior of a
diagonal density matrix in (17), the weak values $p_{W}^{1,1}(0,t)$ (which
obviously corresponds to $p_{W}(0,t)$ in (4)) certifies that the off-diagonal
terms in (16) do not vanish after thermalization. For $N=1$, the fact that
expectation values thermalize is just a result that positive and negative off-
diagonals elements can roughly compensate each other, but they certainly do
not disappear as indicated by $p_{W}^{1,1}(x,t)$. We remind that $\langle
p\rangle(t)=\int dx\int dp\;p\;\mathbb{P}(p,x,t)$ and $p_{W}(x,t)$ can, both,
be computed from the same empirical probability $\mathbb{P}(p,x,t)$. In other
words, $\mathbb{P}(p,x,t)$ provides simultaneous thermalized and
nonthermalized results depending on how it is treated.
Figure 5: Local-in-position many-body weak values of the momentum from the
dynamics in figure 2 for $N=1$ in (a), and for distinguishable $N=2$ particles
in (b),(c) where Coulomb/exchange terms are disconsidered. Panels (a), (b)
show $p_{W}^{j,j}(x,t)$ from (13), which does not depend on $j$. Panel (c)
shows both $p_{W}^{j,k}(x,t)$ from (13) and $\tilde{p}_{W}(x,t)$ from (12).
Values of $x$ are the respective initial $x_{0j}$ values. The expectation
value of the momentum $\langle p_{j}\rangle(t)$ is also shown. Horizontal axis
in (a),(b) the same as in (c).
The plots in figures 5(b) and 5(c) correspond to the quantum system
$N>1(\text{Dis})$ in table 1, because neither exchange nor Coulomb interaction
among the particles are included; that is, the many-body wave function here
could be written as $\Psi(\mathbf{x},t)=\psi_{1}(x_{1},t)\psi_{2}(x_{2},t)$.
The weak values $p_{W}^{1,1}(0,t)$ and $p_{W}^{1,1}(-4,t)$ in panel (b)
confirm the nonthermalized operational properties even at $t\gg t_{eq}$. In
panel (c), one notices that $p_{W}^{1,2}(0,t)$, which corresponds to the weak
measurement of the momentum of particle 1 and strong measurement of the
position of particle 2, shows a thermalized behavior as it overlaps $\langle
p_{j}\rangle(t)$; but this is because here, as the particles have no
correlations among them, one gets $p_{W}^{j,k}(x,t)=\langle p_{j}\rangle(t)$
when $j\neq k$, as discussed in (32). Also in panel (c) $\tilde{p}_{W}(0,t)$,
from (12), which here is just a particle-average of the oscillating term
$p_{W}^{1,1}(0,t)$ and the non-oscillating term $p_{W}^{1,2}(0,t)$, presents
less oscillations but still clearly showing a non-thermalized behavior of
these operational properties.
Figure 6: Local-in-position many-body weak values of the momentum from the
dynamics in figure 2 for indistinguishable particles with $N=2$ in (a),(b) and
$N=3$ in (c). Panels (a), (c) show $p_{W}^{j,j}(x,t)$ from (13), which does
not depend on $j$, for the respective initial $x_{0j}$ values. Panel (b) shows
both $p_{W}^{j,k}(x,t)$ from (13) and $\tilde{p}_{W}(x,t)$ from (12). The
expectation value of the momentum $\langle p_{j}\rangle(t)$ is also shown.
Inset in (a) shows a longer propagation time, $t=[150-300]$. Horizontal axis
in (a),(b) the same as in (c).
Figure 6 corresponds to the (most common) quantum system $N>1(\text{Ind})$ in
table 1, in which one only has operational access in the laboratory to the
expectation value $\langle p\rangle(t)$ in (2) and to the weak value
$\tilde{p}_{W}(x,t)$ in (12). Panels (a) and (b) for $N=2$; panel (c) for
$N=3$. Since we also have mathematical (not operational) access in our
simulations (directly from configuration space) to the weak value
$p_{W}^{j,k}(x,t)$, we have also plotted it to help us to understand the
behavior of the operational weak value $\tilde{p}_{W}(x,t)$, which is a
particle average over different $p_{W}^{j,k}(x,t)$. We see in Fig. 6(a) that
$p_{W}^{1,1}(x,t)$ has smaller oscillations than in Fig. 5(b), but it still
does not thermalize, while $\langle p_{j}\rangle(t)$ effectivelly thermalize
(this later result is independent on $j$); the larger time window in the inset
so confirms. In figure 6(b) one notices that both $p_{W}^{1,2}(0,t)$ and
$p_{W}^{1,1}(0,t)$ have a similar non-thermalizing behavior. From these two
latter values we obtain
$\tilde{p}_{W}(0,t)=(p_{W}^{1,1}(0,t)+p_{W}^{1,2}(0,t))/2$ (thanks to the
properties in (31)), whose plot in panel (b) shows that this operational
parameter for identical particles does not fully thermalize. The $N=3$ case in
panel (c) starts to show the trend that, at higher $N$, $p_{W}^{1,1}(x,t)$
will approach the expectation value $\langle p_{1}\rangle(t)$ and so will also
thermalize, which is seem for all used $x_{0j}$ values. This is understood
from the fact that, for identical particles, $p_{W}^{j,k}(x,t)$ (and
$p_{W}^{j,j}(x,t))$ contains $N-1$ spatial integrals, while $\langle
p_{j}\rangle(t)$ contains $N$ and so, as one increases $N$, one gets
$p_{W}^{j,k}(x,t)\approx\langle p_{j}\rangle(t)$ because $N-1\approx N$.
The fact that the weak value of the momentum does not thermalize in the
configuration space can be understood when $p_{W}^{j}(\mathbf{x},t)$ in (9) is
mathematically interpreted as a Bohmian velocity, satisfying all of its
mathematical properties without any ontologic implications. Without external
perturbation, the initial state $|\Psi(0)\rangle$ of a closed system cannot
change with time its condition of being or not an energy eigenstate [36]. In
other words, the only energy eigenstates are the ones that are so at all
times. We know that for a closed system with $\Psi(\mathbf{x},t)$ vanishing at
the boundaries, the only Bohmian velocities in (9) that are zero are those
linked to an energy eigenstate [56]. From (10) we see that the weak value of
the momentum depends on the current density, and a time-dependent
$J^{j}(\mathbf{x},t)$ (strictly different from zero) is required for a time-
dependent probability presence in such a space, due to the continuity equation
in the configuration space implicit in the Schrödinger equation. Thus, since
our initial nonequilibrium state is not an energy eigenstate, from all
previous arguments, we conclude that $p_{W}^{j}(\mathbf{x},t)$ will never
vanish in the configuration space no matter the value of $N$, independently on
the thermalization or not of the expectation value of the momentum $\langle
p_{j}\rangle(t)$; that is, $p_{W}^{j}(\mathbf{x},t)$ will never thermalize
when evaluated at a point $\mathbf{x}$ of the configuration space (the same
conclusions apply to the presence probability in such space). Most of the
developments done in this paper come from the fact that, in most cases, the
weak value $p_{W}^{j}(\mathbf{x},t)$ in the configuration space is not
empirically accessible in the laboratory, and as such it is not an operational
property for indistinguishable particles; in such cases, though, the weak
value $\tilde{p}_{W}(x,t)$ in the physical space tends to thermalize as $N$
increases.
## V Conclusions
We have seen how, from the empirical knowledge of a unique distribution
probability $\mathbb{P}(p,\textbf{x},t)$, it is possible to get,
simultaneously, both thermalized (expectation values) and nonthermalized (weak
values) operational properties of a closed quantum system described by the
Schrödinger equation. A possible origin of the arrow of time in such systems
comes from dealing with operational properties obtained by averaging (tracing
out) some of the degrees of freedom defined in configuration space. As such
there is no contradiction in that there are properties defined in the
configuration space that do not thermalize, while some properties defined in
the physical space (by averaging or tracing out some degrees of freedom) have
their own time irreversible equations of motion. Such conclusion can be tested
in the laboratory through the many-body weak values of the momentum for
distinguishable particles where, at least conceptually, it is possible to get
information of the position of all particles in the laboratory after their
strong measurement.
However, for indistinguishable particles, a position measurement at $x$ cannot
be linked to a specific particle since there is no experimental protocol to
tag identical particles and, as such, instead of $\mathbb{P}(p,\textbf{x},t)$
in configuration space, one has operational access to $\mathbb{P}(p,x,t)$ in
physical space. But we have shown that $\mathbb{P}(p,x,t)$ can be understood
as an averaging of $\mathbb{P}(p,\textbf{x},t)$ over degrees of freedom in the
configuration space. For the simpler single-particle case, where obviously the
distinction between distinguishable/indistinguishable particles and between
configuration/physical spaces makes no sense, we have also seen that
expectation values can thermalize while weak values may not. The simple
explanation is that the expectation value has one integration over the single
valid coordinate while the weak value has not. Even for a system with $N=2$
identical particles, we see different behaviors for the expectation value
$\langle p\rangle(t)$ and the weak value $\tilde{p}_{W}(x,t)$ in the physical
space. In general, for identical particles, $\tilde{p}_{W}(x,t)$ contains
$N-1$ spatial integrals, while $\langle p_{j}\rangle(t)$ contains $N$ and so,
as one increases $N$, one gets $p_{W}^{j,k}(x,t)\approx\langle
p_{j}\rangle(t)$ because $N-1\approx N$; that is, the former thermalizes when
the latter also does. But such thermalization is seen in the physical space,
not in the configuration space.
So does thermalization occur in configuration space? No thermalization occurs
in the most common operational properties of a quantum system when evaluated
in a point $\mathbf{x}$ of the configuration space, even when other properties
defined in simpler spaces are thermalized. Notice that we have avoided along
the paper to discuss ontologic properties. Instead, our operational approach
has the advantage that the conclusions do not depend on which quantum theory
is invoked, but it also forbids the discussion whether or not thermalization
is a real (ontologic) phenomenon occurring in physical space. This latter
discussion depends on the ontology of each quantum theory; there are quantum
theories where the configuration space is not the fundamental space.
In summary, as mentioned in our Intro [1, 2, 3, 4, 5, 6, 7, 8], there are many
arrows of time which could require different explanations. We here only
discuss the origin of the arrow of time in the non-relativistic many-body
Schrödinger equation in closed quantum systems: a non-thermalized property in
the configuration space, when some of its degrees of freedom are averaged,
leads to a thermalized property in the physical space. We have also developed
the operational many-body weak values of the momentum to make such a
conclusion testable in the laboratory.
###### Acknowledgements.
This research was funded by Spain’s Ministerio de Ciencia, Innovación y
Universidades under Grant No. RTI2018-097876-B-C21 (MCIU/AEI/FEDER, UE), Grant
PID2021-127840NB-I00 (MICINN/AEI/FEDER, UE), the “Generalitat de Catalunya”
and FEDER for the project 001-P-001644 (QUANTUMCAT), the European Union’s
Horizon 2020 research and innovation programme under Grant No. 881603
GrapheneCore3 and under the Marie Skłodowska-Curie Grant No. 765426 TeraApps.
## Appendix A Weak values equations in many-body systems
In this appendix we develop the main weak values equations in the paper.
### A.1 Development of equations (4) and (9) in the paper
For a single-particle system described by the wave function $\psi(x,t)=\langle
x|\psi(t)\rangle$, the expression for the weak values of the momentum,
$p_{W}(x,t)$, can be obtained from the position distribution of the mean
momentum $\langle p\rangle(t)$ of the single-particle operator,
$\hat{p}=p|p\rangle\langle p|$, as
$\langle p\rangle(t)=\langle\psi(t)|\left(\int dx|x\rangle\langle
x\right)|\hat{p}|\psi(t)\rangle=\int dx|\psi(x,t)|^{2}\frac{\langle
x|\hat{p}|\psi(t)\rangle}{\langle x|\psi(t)\rangle}=\int
dx|\psi(x,t)|^{2}p_{W}(x,t),$ (22)
where we have defined the weak values as
$p_{W}(x,t)=\text{Real}\left(\frac{\langle x|\hat{p}|\psi(t)\rangle}{\langle
x|\psi(t)\rangle}\right)$. Since $\langle p\rangle(t)$ is real, as $\hat{p}$
is an hermitian operator, we have $\int
dx|\psi(x,t)|^{2}\text{Imag}\left(\frac{\langle
x|\hat{p}|\psi(t)\rangle}{\langle x|\psi(t)\rangle}\right)=0$. Thus, only the
real part is considered for defining the weak values and so we reproduce
equation (4) in the paper.
From a similar development we can find the weak values for an $N$-particle
system, with $\Psi(\mathbf{x},t)=\langle\mathbf{x}|\psi(t)\rangle$. The mean
momentum $\langle p_{j}\rangle(t)$ of degree of freedom $j$ belonging to
operator
$\hat{P}_{j}\equiv\hat{1}\otimes...\otimes\hat{p}_{j}\otimes...\otimes\hat{1}$
with $\hat{p}_{j}=p_{j}|p_{j}\rangle\langle p_{j}|$ is
$\displaystyle\langle p_{j}\rangle(t)$ $\displaystyle=$
$\displaystyle\langle\Psi(t)|\hat{P}_{j}|\Psi(t)\rangle=\int
d\mathbf{x}\langle\Psi(t)|\mathbf{x}\rangle\langle\mathbf{x}|\hat{P}_{j}|\Psi(t)\rangle=\int
d\mathbf{x}\langle\Psi(t)|\mathbf{x}\rangle\langle
x_{1}|\otimes...\otimes\left(\langle
x_{j}|\hat{p}_{j}\right)\otimes...\otimes\langle x_{N}|\Psi(t)\rangle$ (23)
$\displaystyle=$ $\displaystyle\int
d\mathbf{x}\Psi^{*}(\mathbf{x},t)(-i)\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}=\int
d\mathbf{x}|\Psi(\mathbf{x},t)|^{2}\text{Imag}\left(\frac{\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}}{\Psi(\mathbf{x},t)}\right)=\int
d\mathbf{x}|\Psi(\mathbf{x},t)|^{2}p_{W}^{j}(\mathbf{x},t),$
where we have used $\int d\mathbf{x}=\int dx_{1}...\int dx_{N}$,
$|\mathbf{x}\rangle=|x_{1}\rangle\otimes...\otimes|x_{N}\rangle$, and
$\left(\langle x_{j}|\hat{p}_{j}\right)=\int dx_{j}^{\prime}\langle
x_{j}|\hat{p}_{j}|x_{j}^{\prime}\rangle\langle x_{j}^{\prime}|=\langle
x_{j}|(-i)\frac{\partial}{\partial x_{j}}$. This result shows that $\langle
p_{j}\rangle(t)$ can be decomposed into different components along the
positions $\mathbf{x}$ on the configuration space. Each component
$p_{W}^{j}(\mathbf{x},t)$ is the many-body weak values of the momentum of the
$j$-th particle,
$p_{W}^{j}(\mathbf{x},t)=\text{Real}\left(\frac{\langle\mathbf{x}|\hat{P}_{j}|\Psi(t)\rangle}{\langle\mathbf{x}|\Psi(t)\rangle}\right)=\text{Imag}\left(\frac{\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}}{\Psi(\mathbf{x},t)}\right)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}},$
(24)
with $|\Psi(\mathbf{x},t)|^{2}$ the probability of finding a particle at the
given configuration position $\mathbf{x}$. Expression (24) so reproduces
equation (9) in the paper. The last identity in (24) shows that the weak
values of the momentum is just the Bohmian velocity of the $j$-th particle in
the configuration position $\mathbf{x}$. If one deals with few particles well-
separated in the physical space, then the measurement of the many-body weak
values of each particle in the laboratory is unproblematic.
### A.2 Development of equation (13) in the paper
The problem with the weak values in (24) appears in the laboratory when we
consider $N$ particles in the same region of the physical space, since
$p_{W}^{j}(\mathbf{x},t)$ depends on all positions of the $N$ particles. Then,
it seems impossible for practical purposes to develop a measurement protocol
identifying the $N$ positions of the particles simultaneously. Thus, we want
to rewrite (23) in a way that it only depends on one position of one of the
$N$ particles. We are interested in an expression for computing $\langle
p_{j}\rangle(t)$ as a product of a probability in the physical space,
$\mathbb{P}^{k}(x,t)$, by a weak values which is also local in the physical
space, $p_{W}^{j,k}(x,t)$, which goes like
$\langle p_{j}\rangle(t)=\int dx\mathbb{P}^{k}(x,t)\;p_{W}^{j,k}(x,t),$ (25)
where
$\mathbb{P}^{k}(x,t)=\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;|\Psi(\mathbf{x},t)|^{2}.$ (26)
From (23), (24), and (26), one exactly gets equation (13) in the paper,
$p_{W}^{j,k}(x,t)=\frac{\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;p_{W}^{j}(..,x_{k-1},x,x_{k+1},..,t)|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}{\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}.$ (27)
It is straightforward to show that $p_{W}^{j,k}(x,t)$ in (27) pondered by
$\mathbb{P}^{k}(x,t)$ in (26) exactly gives $\langle p_{j}\rangle(t)$.
### A.3 Development of equation (12) in the paper
The problem with the weak values in (27) is that it seems very difficult to
identify on which $j$-th particle the momentum is (weakly) measured, and on
which $k$-particle the position is (strongly) measured. Even worst, it seems
not possible to repeat the experiment and get the position and momentum of the
same two particles as in the previous measurement. In fact, the evaluation of
$\langle p_{j}\rangle(t)$ is independent on which $k$-th particle one does the
position measurement. If there are $N$ particles in the system we can write
$\langle p_{j}\rangle(t)=\frac{1}{N}\sum_{k=1}^{N}\int
dx\;\mathbb{P}^{k}(x,t)p_{W}^{j,k}(x,t)=\int
dx\;\mathbb{P}(x,t)\frac{1}{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$ (28)
since $\mathbb{P}^{k}(x,t)=\mathbb{P}(x,t)$ for identical particles. Finally,
if we assume that we will not identify the $j$-th particle for the momentum
measurement, then instead of trying to get $\langle p_{j}\rangle(t)$ we will
get an average over the $N$ particles,
$\displaystyle\langle p\rangle(t)\equiv\frac{1}{N}\sum_{j=1}^{N}\langle
p_{j}\rangle(t)=\frac{1}{N}\frac{1}{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\int
dx\;\mathbb{P}(x,t)p_{W}^{j,k}(x,t)=\int
dx\;\mathbb{P}(x,t)\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t).$
(29)
As such, we arrive to the weak values of the momentum as in equation (12) in
the paper,
$\tilde{p}_{W}(x,t)=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$
(30)
which satisfies $\langle p\rangle(t)=\int
dx\;\mathbb{P}(x,t)\;\tilde{p}_{W}(x,t)$.
For identical particles, either fermions or bosons, we have
$\mathbb{P}^{k}(x,t)=\mathbb{P}^{j}(x,t)\equiv\mathbb{P}(x,t)$ for all $j,k$
since $|\Psi(..,x_{k},..,x_{j},..,t)|^{2}=|\Psi(..,x_{j},..,x_{k},..,t)|^{2}$.
We also have
$p_{W}^{j}(..,x_{l},..,x_{j},..,t)=p_{W}^{l}(..,x_{j},..,x_{l},..,t)$ when
$j,l\neq k$ because
$J^{j}(..,x_{l},..,x_{j},..,t)=J^{l}(..,x_{j},..,x_{l},..,t)$ when $j,l\neq
k$. As such, we obtain
$\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$ $\displaystyle
p_{W}^{l,k}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,l\neq k,$ $\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{j,l}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }k,l\neq j,$ $\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{k,j}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j\neq k,$ $\displaystyle p_{W}^{j,j}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{k,k}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,k.$ (31)
For a separable wave function,
$\Psi(\mathbf{x},t)=\psi_{1}(x_{1},t)...\psi_{N}(x_{N},t)$, we obtain
$\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$ $\displaystyle\frac{\int
dx\;p_{W}^{j}(x,t)|\psi_{j}(x,t)|^{2}}{\int dx|\psi_{j}(x,t)|^{2}}=\int
dx\;J^{j}(x,t)=\langle p_{j}\rangle(t)\;\;\;\;\;\;\;\text{for all }j\neq k,$
(32) $\displaystyle p_{W}^{j,j}(x,t)$ $\displaystyle=$ $\displaystyle
p_{W}^{j}(x,t)=\frac{J^{j}(x,t)}{|\psi_{j}(x,t)|^{2}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,$ (33)
where $J^{j}(x,t)$ and $p_{W}^{j}(x,t)$ are the current density and the weak
values, respectively, linked to the single-particle $\psi_{j}(x_{j},t)$. So,
for separable systems, the terms $p_{W}^{j,k}(x,t)$ are spatially uniform,
while $p^{j}_{W}(x,t)$ depend strongly on the position. For nonseparable
systems such differences are not true. Thus, $p_{W}^{j,k}(x,t)$ also provides
a procedure to quantify the interaction between distinct particles.
### A.4 Time-averaging of weak values
Figure 7: Time-averaged many-body weak values of the momentum,
$\bar{f}_{W}(\\{x_{0j}\\},t,T)=\bar{p}_{W}^{j,k}(\\{x_{0j}\\},t,T)$, from the
weak values presented in figure 6 in the paper. Only $j=k=1$ is plotted, which
is the same as $j=k=2$ for $N=2$ and $j=k=2,3$ for $N=3$. The values of
$\\{x_{0j}\\}$ are taken as the set of initial values of the respective wave
packets for $N=1$ in (a),(b), $N=2$ in (c),(d), and $N=3$ in (e),(f). Left and
right panels consider respectively a time $t=20<t_{eq}$ and $t=120>t_{eq}$,
while the integration period $T/2$ in all panels runs from $0$ to $20$.
We have seen in figure 6 in the paper that, contrary to the momentum
expectation value in figure 4 in the paper, the weak values of the momentum
only approach $0$ at higher $N$. For $N=2$ they remain within a finite range
of values after $t_{eq}$, while for $N=1$ such range is much larger. We have
also discussed how the weak values (as well as the density matrix coherences)
have a random nature for $t>t_{eq}$. Thus we compute the time-average of such
weak values on a period of time $T$,
$\bar{f}_{W}(x,t,T)=\frac{1}{T}\int_{t-T/2}^{t+T/2}dt^{\prime}\;f_{W}(x,t^{\prime}),$
(34)
where $f_{W}(x,t)$ can be any of the $4$ different types of weak values
expressed in the paper, that is, $p_{W}(x,t)$, $p_{W}^{j}(x,t)$,
$p_{W}^{j,k}(x,t)$, and $\tilde{p}_{W}(x,t)$. We consider in figure 7 the case
$f_{W}(x,t)=p_{W}^{j,k}(x,t)$, and then plot
$\bar{f}_{W}(x,t,T)=\bar{p}_{W}^{j,k}(x,t,T)$ from the weak values presented
in figure 6 in the paper, as a function of $T$. Upper, middle, lower panels
respectively for $N=1$, $N=2$, $N=3$; panels in left and right column
respectively related to a time before ($t=20<t_{eq}$) and after
($t=120>t_{eq}$) thermalization. For $N>1$, the time-average of the weak
values always vanish at $t>t_{eq}$, even at small $T$ ((d),(f)); for
$t<t_{eq}$ this may only happen at much higher $T$ ((c),(e)). The $N=1$ case
is, once more, much less smooth due to the many nodes of the wave function.
## Appendix B Diagonal and off-diagonal elements of the density matrix in the
energy representation and its connection with weak values of the momentum
The dynamics presented in our paper is based on an initial nonequilibrium
_pure_ state evolving in a closed system, while being affected by some
‘chaotic’ disorder potential. Without disorder, such a pure state simply
evolves periodically in the underneath harmonic oscillator potential. With
disorder, we have seen its role in bringing the nonequilibrium state into an
equilibrium regime characterized e.g. by the behaviour of the expectation
values shown in figures 3 and 4 in the paper. But what is the role of disorder
on the density matrix in the energy representation?
### B.1 Density matrix populations and coherences
We consider a pure state as a superposition of (single-particle or many-
particle) energy eigenstates,
$\langle x_{1}|\otimes...\otimes\langle
x_{N}|\Psi(t)\rangle=\langle\mathbf{x}|\Psi(t)\rangle=\sum_{n}c_{n}\langle\mathbf{x}|n\rangle
e^{-iE_{n}t}=\sum_{n}\;c_{n}R_{n}(\mathbf{x})e^{-iE_{n}t},$ (35)
being $|n\rangle$ an energy eigenstate with eigenvalue $E_{n}$, $c_{n}=\langle
n|\Psi(0)\rangle$, and $\langle\mathbf{x}|n\rangle=R_{n}(\mathbf{x})$ a real
function. The $n$-sum runs within a set $N_{act}$ of activated states, which
is defined by some quench responsible for creating the nonequilibrium initial
state; in our model, the quench is a sudden shift of the harmonic trap
translated to an initial velocity for the electrons at $t=0$. The pure state
density matrix, $\hat{\rho}(t)=|\Psi(t)\rangle\langle\Psi(t)|$, in the energy
basis becomes
$\rho_{n,m}(t)=\langle m|\hat{\rho}(t)|n\rangle=\langle
m|\Psi(t)\rangle\langle\Psi(t)|n\rangle=c_{n}\;c_{m}^{*}\;e^{i(E_{m}-E_{n})t},$
(36)
where the diagonal elements, $\rho_{n,n}$ (_populations_), are clearly time-
independent, while the off-diagonal terms (_coherences_) are not. When time-
averaged, within a time interval $T\to\infty$, the off-diagonal elements
vanish and the density matrix becomes diagonal,
$\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\;\rho_{n,m}(t)\;dt=c_{n}\;c_{m}^{*}\;\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}e^{i(E_{m}-E_{n})t}dt=c_{n}\;c_{m}^{*}2\pi\delta_{n,m},$
(37)
with $\delta_{n,m}$ the Kronecker delta. This does not imply that the density
matrix in (36) (without time averaging) becomes diagonal as $t\to\infty$, but
only that the off-diagonal elements oscillate around zero.
In figure 8 we show the evolution of real and imaginary parts of a few off-
diagonal elements $\rho_{n,m}(t)$ ($N=1$), which can be written from (36) by
employing $c_{l}=|c_{l}|e^{i\theta_{l}}$ as
$\rho_{n,m}(t)=|c_{n}|\;|c_{m}|\text{cos}(\theta_{n}-\theta_{m}+(E_{m}-E_{n})t)+i|c_{n}|\;|c_{m}|\text{sin}(\theta_{n}-\theta_{m}+(E_{m}-E_{n})t),$
(38)
such that the modulus $|\rho_{n,m}(t)|=|c_{n}||c_{m}|$ is also time-
independent, as seen in panel (a). The oscillation period depends inversely on
their level splittings, $\Delta E=E_{m}-E_{n}$, as shown in panels (b)-(d),
which relate to the same level $n_{p}=201$ at the peak of $\rho_{n,m}(t)$: as
one increases the splitting, by considering a matrix element with $1$ (b), $3$
(c), and $6$ (d) levels apart, the oscillation period decreases. Also, if
considering both $n,m$ values away from the peak at $n_{p}=201$, the
respective matrix elements have exponentially smaller amplitudes. In a pure
harmonic oscillator all level splittings are simply proportional to the
difference in number of levels, but the disorder potential creates _random_
splittings. The full density matrix is given in figure 1(d) in the paper,
which shows $N_{act}\approx 70$ activated states at $t=0$, which will remain
as the main states determining the system unitary evolution at any $t$. From
these results we conclude that the thermalized system keeps memory of its
initial state through the propagation of its off-diagonal coherences (in
particular by keeping the information $c_{l}=|c_{l}|e^{i\theta_{l}}$), which
never disappear. Most importantly for the current operator as discussed in the
paper, since its diagonal populations are zero by construction.
Figure 8: Density matrix coherences for $N=1$. From the grid-diagonalization
as depicted in the inset (b) of figure 1 in the paper, the time-evolution of a
few of the off-diagonal coherences $\rho_{n,m}(t)$ is shown with respect to
the peak at $n_{p}=201$, with real (imaginary) part in blue (red). As the
energy splitting, $\Delta E=E_{m}-E_{n}$, increases from (b) to (c) to (d),
the respective oscillation period decreases. In (a) one verifies that the
respective moduli $|\rho_{n,m}(t)|^{2}$ remain constant.
### B.2 Connection between density matrix coherences and weak values
First, we need the presence probability density $|\Psi(\mathbf{x},t)|^{2}$,
given by
$\displaystyle|\Psi(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\text{trace}\left[\hat{\rho}(|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle x_{N}|)\right]$ (39) $\displaystyle=$
$\displaystyle\sum_{n}\sum_{m}c_{n}c_{m}^{*}\;e^{i(E_{m}-E_{n})\;t}R_{n}(\mathbf{x})R^{*}_{m}(\mathbf{x}),$
and that can be decomposed into time-independent populations and time-
dependent coherences as
$|\Psi(\mathbf{x},t)|^{2}=|\Psi_{dia}(\mathbf{x},t)|^{2}+|\Psi_{off}(\mathbf{x},t)|^{2},$
(40)
with
$\displaystyle|\Psi_{dia}(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\rho_{n,n}|R_{n}(\mathbf{x})|^{2},$
$\displaystyle|\Psi_{off}(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)R_{n}(\mathbf{x})R^{*}_{m}(\mathbf{x}).$ (41)
We also need the current density $J^{j}(\mathbf{x},t)$ due to the $j$-th
particle at one given position in the configuration space,
$\displaystyle J^{j}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\text{trace}\left[\hat{\rho}\left(\hat{p}_{j}|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle
x_{N}|+|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle x_{N}|\hat{p}_{j}\right)\right]$ (42)
$\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\sum_{m}c_{n}c_{m}^{*}\;e^{i(E_{m}-E_{n})\;t}\left[R_{n}(\mathbf{x},t)\;\frac{\partial
R_{m}(\mathbf{x},t)}{\partial x_{j}}-R_{m}(\mathbf{x},t)\;\frac{\partial
R_{n}(\mathbf{x},t)}{\partial x_{j}}\right],$
that can also be decomposed in terms of diagonal and off-diagonal components
as
$J^{j}(\mathbf{x},t)=J^{j}_{dia}(\mathbf{x},t)+J^{j}_{off}(\mathbf{x},t),$
(43)
with
$\displaystyle J^{j}_{dia}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\rho_{n,n}\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}-R_{n}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial
x_{j}}\right]=\sum_{n}\rho_{n,n}(t)\;J^{j}_{n,n}(\mathbf{x}),$ $\displaystyle
J^{j}_{off}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{m}(\mathbf{x})}{\partial x_{j}}-R_{m}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}\right]=\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x}),$ (44)
where
$J^{j}_{m,n}(\mathbf{x})=\frac{i}{2}\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{m}(\mathbf{x})}{\partial x_{j}}-R_{m}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}\right]$. The relevant point is that only
off-diagonal elements $\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x})$ for $n\neq m$
provide current densities, since the contribution of diagonal elements vanish,
$J^{j}_{dia}(\mathbf{x},t)=0$, in closed systems with wave function vanishing
at the boundaries. The diagonal terms $\rho_{n,n}(t)\;J^{j}_{n,n}(\mathbf{x})$
do not contribute to the total current because energy eigenstates are pure
real (or pure imaginary), so that their current $J^{j}_{n,n}(\mathbf{x})=0$ in
first equation in (44).
Both results from (40) and (43) are in agreement with the well-known
continuity equation,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial|\Psi(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}(\mathbf{x},t)}{\partial x_{j}}$ (45)
$\displaystyle=$
$\displaystyle\frac{\partial|\Psi_{dia}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{dia}(\mathbf{x},t)}{\partial
x_{j}}+\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{off}(\mathbf{x},t)}{\partial x_{j}}$
$\displaystyle=$
$\displaystyle\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{off}(\mathbf{x},t)}{\partial x_{j}},$
where we have used the trivial results
$\frac{\partial|\Psi_{dia}(\mathbf{x},t)|^{2}}{\partial t}=0$ (because
$|\Psi_{dia}(\mathbf{x},t)|$ is time-independent) and
$\sum_{j=1}^{N}\frac{\partial J^{j}_{dia}(\mathbf{x},t)}{\partial x_{j}}=0$
(because $J^{j}_{dia}(\mathbf{x},t)=0$).
Since we know from (36) and (38) that the coherences never vanish
($\rho_{n,m}(t)\neq 0$ for $n\neq m$) and always oscillate, we conclude that
$\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial t}\neq 0$, so that
(45) means that the off-diagonal probability presence
$|\Psi_{off}(\mathbf{x},t)|^{2}$ and the off-diagonal current density
$J^{j}_{off}(\mathbf{x},t)$ are dynamically changing during the whole
simulation, before and after $t_{eq}$. We notice now that
$|\Psi(\mathbf{x},t)|^{2}$ and $J^{j}_{off}(\mathbf{x},t)$ are the elements
that define the weak values in equation (3) in the paper,
$p_{W}^{j}(\mathbf{x},t)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}}=\frac{J^{j}_{off}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}}=\frac{1}{|\Psi(\mathbf{x},t)|^{2}}\left(\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x})\right).$ (46)
This provides the required connection between weak values and coherences.
## Appendix C Momentum representation and ‘phase-space’
In this appendix we provide the dynamics evolution in momentum representation
and construct a pseudo phase-space of the system.
Figure 9: Wave packet dynamics in momentum representation. This figure is
similar to figure 2 in the paper, which employed position representation, but
instead it considers momentum representation. Upper panels: wave packet
$|\Psi(p_{1},p_{2},[p_{3}],t)|^{2}$ for $N=3$ at $t=0$ in (a) and $t=150$ in
(b). Middle panels: $|\Psi(p_{1},p_{2},t)|^{2}$ for $N=2$ at $t=0$ in (c) and
$t=150$ in (d). Panel (e): wave packet $|\Psi(p_{1},t)|^{2}$ for $N=1$ at
$t=0$ (dotted) and at $t=150$ (solid). Panel (f): $1$D-view of
$|\Psi(p_{1},[p_{2}],[p_{3}],t)|^{2}$ for $N=3$ (red),
$|\Psi(p_{1},[p_{2}],t)|^{2}$ for $N=2$ (blue), and $|\Psi(p_{1},t)|^{2}$ for
$N=1$ (green), with solid (dotted) lines for the wave packet at $t=150$
($t=0$), where all $t=0$ plots overlap at $p_{0j}=20$. All plots are in log-
scale. Horizontal axis in (a),(c) ((b),(d)) is the same as in (e) ((f)).
Figure 10: ‘Phase-space’ analysis. From position $\langle x_{1}\rangle$(t) and
momentum $\langle p_{1}\rangle(t)$ expectation values in figure 4 in the
paper, we compile a respective ‘phase-space’ for $N=1$ in (a), $N=2$ in (c),
and $N=3$ in (e). Panels (b),(d),(f) are a zoom at the origin for the
respective $N$. The initial value of $\langle x_{1}\rangle(t=0)$ for each $N$
is the average of the set of respective initial positions $\\{x_{0j}\\}$,
while $\langle p_{1}\rangle(t=0)=20$ for any $N$. As one approaches the final
simulation time, $t=150>t_{eq}$, the ‘phase-space’ looks noisier for $N=1,2$
as a consequence of the recurrences as seen in the respective expectation
values in the paper. Horizontal axis in (a),(c) ((b),(d)) is the same as in
(e) ((f)).
### C.1 Dynamics in momentum representation
Figure 2 in the paper summarizes the dynamics for the systems with $N=1,2,3$
by showing the initial and final snapshots of the respective wave functions in
_position_ representation. Since our algorithm for propagating the Schrödinger
equation uses a split-operator method where the kinetic energy is handled in
momentum representation, where it is diagonal, and then Fourier-transformed
back to position representation (where the potential terms are diagonal), for
consistency, we show in figure 9 exactly the same plots as in figure 2 in the
paper, but in _momentum_ representation. That is, we plot in the upper panels
$|\Psi(p_{1},p_{2},[p_{3}],t)|^{2}$, with $[p_{3}]$ integrated out, for $N=3$
at $t=0$ in (a) and $t=150$ in (b) (results would be the same if integrating
on $[p_{1}]$ or $[p_{2}]$). The middle panels show $|\Psi(p_{1},p_{2},t)|^{2}$
for $N=2$ at $t=0$ in (c) and $t=150$ in (d). Panel (e) shows
$|\Psi(p_{1},t)|^{2}$ for $N=1$ at $t=0$ ($t=150$) in dotted (solid) lines,
while panel (f) compiles all respective $1$D plots of
$|\Psi(p_{1},[p_{2}],[p_{3}],t)|^{2}$ for $N=3$, $|\Psi(p_{1},[p_{2}],t)|^{2}$
for $N=2$, and $|\Psi(p_{1},t)|^{2}$ for $N=1$, with solid (dotted) lines for
$t=150$ ($t=0$). Since the initial velocity is the same and centered at
$p_{0j}=20$ for each degree of freedom and for every $N$, all $1$D plots at
$t=0$ overlap. One also notices in momentum representation the same full
spread of the wave function after thermalization, at $t\gg t_{eq}$, which then
remains in between the ’turning points’ $p_{TP}=\pm 20$ (since $\omega=1$). As
in position representation, the higher the $N$ the more homogeneous is the
wave packet spread.
### C.2 Pseudo phase-space
We have shown in the paper that thermalization provides time-independent
expectation values, so that another useful plot is a pseudo ‘phase-space’ as
if that could be simply compiled from the expectation values of $\langle
p_{j}\rangle(t)$ and $\langle x_{j}\rangle(t)$ in figure 4 in the paper.
Figure 10 shows the ‘phase space’ for $N=1,2,3$ respectively in upper, middle,
lower panels; left column the full range, right column a zoom at the origin
$(0,0)$. The figures look the same no matter which $j$-electron is considered
at each $N$. For any $N$ the curves start at $\langle p_{1}\rangle(0)=20$ and
$\langle x_{1}\rangle(0)=0$, only exception being $N=2$ which starts at
$\langle x_{1}\rangle(0)=-2$. The curves after thermalization (right column)
are about the same at any $N$, with $\langle p_{1}\rangle(t)\approx\langle
x_{1}\rangle(t)\approx 0$; they look visually different since some recurrences
may happen causing a noisier ‘phase space’ as time keeps rolling after
$t_{eq}$; however, once more, the behaviour is more smooth as $N$ increases.
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|
# TREET: TRansfer Entropy Estimation via Transformer
Omer Luxembourg, , Dor Tsur, , Haim Permuter All authors are with the
Department of Electrical and Computer Engineering of Ben-Gurion University of
the Negev.
###### Abstract
Transfer entropy (TE) is a measurement in information theory that reveals the
directional flow of information between processes, providing valuable insights
for a wide range of real-world applications. This work proposes Transfer
Entropy Estimation via Transformers (TREET), a novel transformer-based
approach for estimating the TE for stationary processes. The proposed approach
employs Donsker-Vardhan (DV) representation to TE and leverages the attention
mechanism for the task of neural estimation. We propose a detailed theoretical
and empirical study of the TREET, comparing it to existing methods. To
increase its applicability, we design an estimated TE optimization scheme that
is motivated by the functional representation lemma. Afterwards, we take
advantage of the joint optimization scheme to optimize the capacity of
communication channels with memory, which is a canonical optimization problem
in information theory, and show the memory capabilities of our estimator.
Finally, we apply TREET to real-world feature analysis. Our work, applied with
state-of-the-art deep learning methods, opens a new door for communication
problems which are yet to be solved.
###### Index Terms:
Deep Learning, Information Theory, Transfer Entropy, Transformers, Neural
Estimation, Communication Channels
## I Introduction
Transfer entropy (TE), introduced by Schreiber [1], stands as a pivotal
information-theoretic measurement that captures the coupling dynamics within
temporally evolving systems [2]. Derived from the principles of mutual
information (MI), TE is distinctive for its inherent asymmetry, strategically
employed in diverse applications for causal analysis [3]. TE serves as a
robust measure of the directed, asymmetric information flow between two
stochastic processes. Specifically, TE quantifies the reduction in uncertainty
about prospective values, incorporating the past values of one process to
predict the future values of another [4].
Transformers [5] are a neural network (NN) architecture that in recent years
became prevalent in many fields of research, including time series forecasting
[6, 7, 8]. With the attention mechanism, transformers are avoiding traditional
recurrent flow of states, such in recurrent NNs (RNNs), with a bounded length
of input sequence, yet manage to achieve great results [9].
The versatility of TE is evident in various domains. In neuroscience, it has
proven to be effective in deciphering functional connectivity among neurons
and between neurons to various physical tasks [10, 11, 12]. Moreover, analysis
between visual sensors and movement actuators in embodied cognitive systems
can be one via TE [13]. TE proves indispensable as it empowers social networks
to delve into the intricate dynamics of social influence and helps to
encompass the spread of misinformation and scrutinize statistical causality in
various text-related events, including the spread of misinformation [14]. [15]
utilizes a variant of TE in conjunction with an RNN to predict the direction
of the US stock market, incorporating TE as input feature. Lately [16]
suggested a greedy algorithm for feature selection while leveraging the
connection between each feature and the target with TE.
### I-A Estimation and Optimization of Transfer Entropy
Estimating information theoretic measures such as TE is challenging, due to
lack of samples and unknown underlying distribution of the data, that may lead
to larger errors in estimation [4]. Among the estimation methods, Kernel
Density Estimation (KDE) [17], $k$-nearest neighbors (KNN) [18, 19], and the
Kraskov-Stögbauer-Grassberger (KSG) technique are noteworthy, despite their
struggle with the bias-variance trade-off. Notably, Granger causality [20]
also serves as a foundational method for TE estimation, particularly in
Gaussian joint processes where it is equivalent to TE, highlighting its
effectiveness in quantifying information flow between sequences [21, 22].
Recent advancements have embraced neural estimation methods for information
theory measures, leveraging the Donsker-Varadhan (DV) variational formula for
Kullback-Leibler (KL) divergence to accurately estimate metrics like MI,
directed information (DI) rate, and TE [23, 24, 25, 26]. These methods,
including MI neural estimation (MINE), DI neural estimation (DINE), and the
intrinsic TE neural estimator (ITENE) for MI, DI rate and variant of TE
respectively, utilize NNs to solve optimization problems, facilitating the
estimation process even in complex scenarios, such as estimating the channel
capacity of continuous channels in the presence of memory. However, challenges
arise with larger context windows, necessitating adjustments for accurate
estimation in broader historical contexts.
NNs have revolutionized fields such as computer vision and natural language
processing, with transformers now leading time series analysis, overtaking
RNNs due to their superior performance [27, 7, 6, 8]. Despite the widespread
use of MINE for information measure estimation, it subjects to certain
limitations that undergone comprehensive scrutiny [28, 29, 30]. To address
these issues, our novel approach, inspired by DINE, utilizes the capabilities
of transformers. This method specifically tailors TE estimation to the
sequential dynamics of data, offering a significant improvement over
conventional techniques that overlook temporal relationships.
### I-B Contributions
In this work, we introduce TREET, a novel approach for estimating TE using
transformers. Our method is grounded in the DV representation, leveraging
attention-based NNs adapted to meet the structural constraints of DV
optimization. Theoretically, we establish the consistency of TREET and develop
an auxiliary neural distribution generator (NDG) module to facilitate TE
optimization, utilizing transformers.
Empirically, we showcase the versatility of TREET across various applications.
We present our findings on channel capacity estimation, illustrating the
efficacy of the TREET and NDG in a joint optimization and estimation process,
supported by theoretical validations. Additionally, we highlight the memory
capabilities of TREET through a long memory channel example. A key empirical
contribution is our feature analysis of the Apnea dataset, illustrating the
potential of TREET for comprehensive real-world data analysis, paving the way
for broader applications in future research.
### I-C Organizations
The paper is structured as follows, Section II provides background on
information theory and NNs. Section III presents the TREET, it’s theoretical
guarantees and practical implementations, while the optimization of TREET is
described in Section IV. Experimental results for channel capacity estimation,
memory analysis and features analysis on physiological data are shown in
Section V. Section VI concludes the paper and discuss future research
potentials.
## II Background
In this section, we elaborate on the preliminaries necessary to present our
method. We familiarize the reader with our notation and provide the formal
definition of TE, then relate it to DI. Subsequently, we introduce the concept
of transformer NN, define them, and discuss the theorem of universal
approximation. Lastly, we present the use of NN as estimators for information
theory measurements.
### II-A Notation
Calligraphic letters, such as $\mathcal{X}$, denote subsets of the
$d$-dimensional Euclidean space, $\mathbb{R}^{d}$. Expectations are
represented by $\mathbb{E}$, with all random variables defined in the
probability space $(\Omega,\mathcal{F},\mathbb{P})$. The collection of Borel
probability measures on $\mathcal{X}$ is indicated as
$\mathcal{P}(\mathcal{X})$, and $\mathcal{P}_{\mathsf{ac}}(\mathcal{X})$
specifically refers to those measures that are absolutely continuous with
respect to Lebesgue measure, with their densities denoted by lowercase $p$.
Random variables and vectors are uppercase, e.g., $X$, and stochastic
processes are in blackboard bold, e.g., $\mathbb{X}:=(X_{t})_{t\in\mathbb{Z}}$
for discrete time $t$.
The sequence of $l$ samples from time $t$ in process $\mathbb{X}$ is
$X_{t}^{t+l}:=[X_{t},\ldots,X_{t+l}]^{\top}$, and for stationary processes,
$X^{l}:=[X_{1},\ldots,X_{l}]^{\top}$. For a measure
$\mathcal{Q}\in\mathcal{P}_{\mathsf{ac}}(\mathcal{X})$ with PDF $q$, cross
entropy between $P$ and $Q$ is
$\mathsf{h}_{\mathsf{CE}}(P,Q):=-\mathbb{E}_{P}[\log q]$. Differential entropy
of $X\sim P$ is $\mathsf{h}(X):=\mathsf{h}_{\mathsf{CE}}(P,P)$. If $Q\ll P$,
the KL divergence
$\mathsf{D}_{\mathsf{KL}}(P\|Q):=\mathbb{E}_{P}[\log\frac{\mathrm{d}P}{\mathrm{d}Q}]$.
MI between $(X,Y)\sim P_{XY}$ is
$\mathsf{I}(X;Y):=\mathsf{D}_{\mathsf{KL}}(P_{XY}\|P_{X}\otimes P_{Y})$.
Conditional KL divergence for $P_{Y|X},Q_{Y|X}$ given $X\sim P_{X}$ is
$\mathsf{D}_{\mathsf{KL}}(P_{Y|X}\|Q_{Y|X}|P_{X})$, and conditional MI for
$(X,Y,Z)\sim P_{XYZ}$ is
$\mathsf{I}(X;Y|Z):=\mathsf{D}_{\mathsf{KL}}(P_{XY|Z}\|P_{X|Z}\otimes
P_{Y|Z}|P_{Z})$.
### II-B Transfer Entropy
TE quantifies causal influence from the past of one sequence on the present of
another, formally given by
###### Definition 1 (Transfer Entropy)
For jointly distibuted processes $\mathbb{X}$ and $\mathbb{Y}$ and
$k,l<\infty$, the TE, with parameters $(k,l)$, is given by
$\mathsf{TE}_{X\to
Y}(t;k,l):=\mathsf{I}\left(X^{t}_{t-l};Y_{t}|Y^{t-1}_{t-k}\right).$ (1)
When the considered processes are jointly stationary, we can omit the temporal
index $t$ in(1) and write TE as
$\mathsf{TE}_{X\to Y}(k,l):=\mathsf{I}\left(X^{l};Y_{l}|Y^{l-1}_{l-k}\right).$
(2)
Note that our definition of TE includes $X_{t}$, whereas other definitions [1,
31, 21, 26] define TE as $\mathsf{I}(X^{l-1};Y_{l}|Y^{l-1}_{l-k})$. The main
difference of this change is that our definition reduces to MI when the
processes are jointly independent and identically distributed (i.i.d.), i.e.,
$\mathsf{TE}_{X\to Y}(0,0)=I(X;Y)$. When $k=l$, we use the abbreviated
notation $\mathsf{TE}_{X\to Y}(l)$.
### II-C Relation to Directed Information
DI [32, 33] is given by
$\displaystyle\mathsf{I}\left(X^{n}\rightarrow{}Y^{n}\right)$
$\displaystyle:=\sum_{i=1}^{n}\mathsf{I}\left(X^{i};Y_{i}|Y^{i-1}\right)$
$\displaystyle=\sum_{i=1}^{n}\mathsf{TE}_{X\to Y}(i).$ (3)
When $(X^{n},Y^{n})$ are $n$-fold projections of some stochastic processes
$\mathbb{X}$ and $\mathbb{Y}$ and from (2) we can observe that DI is the sum
of TE. We can define the DI rate, which is given by
$\displaystyle\mathsf{I}(\mathbb{X}\to\mathbb{Y})$
$\displaystyle:=\lim_{n\to\infty}\frac{1}{n}\mathsf{I}\left(X^{n}\rightarrow{}Y^{n}\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\mathsf{I}(X^{n};Y_{n}|Y^{n-1}),$
(4)
and $(a)$ holds due to stationarity [25]. The DI rate can be therefore
observed as a limit case of TE, where the limit of TE exists since
$\mathsf{TE}_{X\to
Y}(l)=\mathsf{h}(Y_{l}|Y^{l-1})-\mathsf{h}(Y_{l}|X^{l},Y^{l-1})$ and each
conditional entropy is non-negative, decreasing for increasing $l$
(conditioning reduces entropy). Moreover, under appropriate Markov
assumptions,
$Y_{l}-Y^{l-1}-Y^{-1}_{-\infty},Y_{l}-(X^{l},Y^{l-1})-(X^{-1}_{-\infty},Y^{-1}_{-\infty})$,
we can form an equality between TE and DI (refer to Appendix VII-A for further
details).
### II-D Attention Mechanism and Transformers
NNs serve as a universal function approximators [34], excelling in capturing
complex relationships within the data. NNs capabilities can be enhanced by the
attention111This paper considers the dot-product attention. mechanism [27],
which selects various combinations of inputs according to their significance
to the predictions. The attention can address time series datasets, while
weighting over each time input. Attention comprises queries, which represent
the current temporal focus, keys, which match against the queries to determine
relevance, and values, that contain the NN inputs to be weighted and act as a
memory function in time-series. Attention is formally given by
###### Definition 2 (Attention)
Let $Q=XW_{Q},K=XW_{K}$ and $V=XW_{V}$
($W_{Q},W_{K},W_{V}\in\mathbb{R}^{x\times d}$) be the queries, keys and
values, which are linear projections of the input
$X=(x_{1},x_{2},\ldots,x_{n})$, $\forall i:x_{i}\in\mathbb{R}^{x}$. Then the
attention is as follows,
$\mathsf{Attn}(X)=\mathsf{softmax}\left(QK^{\top}\right)V,$
where $\mathsf{softmax}(Z)_{j}={\exp{[Z_{j}]}}/{\sum_{m=1}^{n}\exp{[Z_{m}]}}$
where $Z\in\mathbb{R}^{n}$, is performed for each column of the dot-product
between queries and keys separately. The weights are calculated from the
softmax operation and the output is weighted from the values $V$.
Transformers utilize positional encoding (PE), which is a maps each input to
the attention according to its ordinal index, and is usually applied before
the attention for time series applications to deformation of the sequence
structure. In addition, a generalization of the transformer uses multi-head
attention,
$\mathsf{MultiHead}\text{-}\mathsf{Attn}(X)=[H_{1},\ldots,H_{h}]W_{0},$ (5)
where
$H_{i}=\mathsf{softmax}\left(Q^{(i)}{K^{(i)}}^{\top}\right)V^{(i)},\quad
i=1,\ldots,n,$ (6)
where $W_{0}\in\mathbb{R}^{dh\times d}$ is learnable parameter, and
$Q^{(i)},K^{(i)},V^{(i)}$ are the $i^{\text{th}}$ learnable projections of
queries keys and values [27]. A transformer block is a sequence-to-sequence
function, which maps input sequence of to an output sequence. Each transformer
block consists of attention layer and time-wise feed-forward layer.
Formally, a transformer is given as follows.
###### Definition 3 (Transformer function class)
Let $d_{i},d_{o},l,v\in\mathbb{N}$. The class of transformers with $v$
neurons, denoted
$\mathcal{G}_{\mathsf{tf}}^{(d_{i},d_{o},l,v)}:\mathbb{R}^{d_{i}\times
l}\to\mathbb{R}^{d_{o}\times l}$, is the set of discrete-time with the
following structure:
$\displaystyle X_{\mathsf{pe}}=W_{1}\cdot X+E,$ (7a)
$\displaystyle\mathsf{Attn}(X_{\mathsf{pe}})=X_{\mathsf{pe}}+\sum_{i=1}^{h}W_{O}^{i}W_{V}^{i}X_{\mathsf{pe}}$
$\displaystyle\qquad\qquad\cdot\mathsf{softmax}\left[(W_{K}^{i}X_{\mathsf{pe}})^{\top}(W_{Q}^{i}X_{\mathsf{pe}})\right],$
(7b) $\displaystyle
Y=\mathsf{FF}(X_{\mathsf{pe}})=\mathsf{Attn}(X_{\mathsf{pe}})+W_{3}\cdot\sigma_{\mathsf{R}}$
$\displaystyle\qquad\qquad\left(W_{2}\cdot\mathsf{Attn}(X_{\mathsf{pe}})+b_{1}\mathbf{1}_{l}^{\top}\right)+b_{2}\mathbf{1}_{l}^{\top},$
(7c)
where $X\in\mathbb{R}^{d_{i}\times l}$ is the input sequence of $l$ samples,
$Y\in\mathbb{R}^{d_{o}\times l}$ is the transformer output,
$W_{1}\in\mathbb{R}^{d_{e}\times d_{i}},W_{O}^{i}\in\mathbb{R}^{d_{e}\times
d_{m}},W_{Q}^{i},W_{K}^{i},W_{V}^{i}\in\mathbb{R}^{d_{m}\times
d_{e}},W_{2}\in\mathbb{R}^{d_{r}\times d_{e}},W_{3}\in\mathbb{R}^{d_{e}\times
d_{r}},b_{1}\in\mathbb{R}^{d_{r}},b_{2}\in\mathbb{R}^{d_{e}}$ are the weights
and biases of the network, $E\in\mathbb{R}^{d_{e}\times l}$ is the PE of the
input. The number of heads $h$ and the head size $d_{m}$ are the parameters of
the multi-head attention while $d_{r}$ is the hidden dimension of the feed-
forward (FF) layer. Assuming that the input is an additive product with it’s
positional encoding product, before the transformer blocks. The class of
transformers with dimensions $(d_{i},d_{o},l)$ is thus given by
$\mathcal{G}_{\mathsf{tf}}^{(d_{i},d_{o},l)}:=\bigcup_{v\in\mathbb{N}}\mathcal{G}_{\mathsf{tf}}^{(d_{i},d_{o},l,v)}.$
(8)
Transformers are a universal approximation class of sqeuence to sequence
mappings:
###### Theorem 1 (Universal approx. for transformers)
Let $\epsilon>0$, $l\in\mathbb{N}$. $\mathcal{U}\subset\mathbb{R}^{T\times
d_{i}},\mathcal{Z}\subset\mathbb{R}^{l\times d_{o}}$ be open sets, and
$f:\mathcal{U}\to\mathcal{Z}$ be a continuous vector-valued function. Then,
there exist $v\in\mathbb{N}$ and a $v$-neuron transformer
$g\in\mathcal{G}_{\mathsf{tf}}^{(d_{i},d_{o},l,v)}$ (as in Definition 3, such
that for any sequence of inputs $\\{u^{l}\\}\in\mathcal{U}$ and sequence of
outputs $\\{z^{l}\\}\in\mathcal{Z}$, we have
$\left\|f(u^{l})-g(u^{l})\right\|_{1}\leq\epsilon,$ (9)
This paper utilizes the class of causal transformers,
$\mathcal{G}_{\mathsf{ctf}}$, which is built upon $\mathcal{G}_{\mathsf{tf}}$.
To this end, we use the notion of causal functions
###### Definition 4 (Causal Function)
Let $\mathcal{F}:\mathbb{R}^{d_{u}\times L}\to\mathbb{R}^{d_{z}\times L}$ be a
function for $d_{u},d_{z},L\in\mathbb{N}$. For a series of inputs
$U=\\{u_{t}\\}_{t=1}^{L}$ and function outputs $Z=\\{z_{t}\\}_{t=1}^{L}$, the
function $\mathcal{F}$ is defined as a causal function iff, for any
$t_{0},t\in\\{1,\ldots,L\\},t\leq t_{0}$, the output $z_{t_{0}}$ depends only
on the inputs $u_{t}$.
In order to achieve a causal mapping function, we apply causal mask on the
attention scores, to result with a mapping that overlooks dependence on future
elements as in Definition 4 (note that only the attention perform time mixing,
thus leaving the rest of the transformers intact is valid). The causal mask,
denoted as $M\in\mathbb{R}^{l\times l}$, multiplied element-wise with the dot-
product of keys and queries, before the softmax operation, and is given by
$M_{[i,j]}=\begin{cases}1&\text{if}j\leq i\\\
-\infty&\text{otherwise}\end{cases}$ (10)
where $-\infty$ nullifies the corresponding entries after the softmax
operation. Hence, (7b) is written as,
$\displaystyle\mathsf{Attn}(X_{\mathsf{pe}})=X_{\mathsf{pe}}+\sum_{i=1}^{h}W_{O}^{i}W_{V}^{i}X_{\mathsf{pe}}$
$\displaystyle\qquad\cdot\mathsf{softmax}\left[(W_{K}^{i}X_{\mathsf{pe}})^{\top}(W_{Q}^{i}X_{\mathsf{pe}})\odot
M\right].$ (11)
Although changing the dot-product operation of the attention, the model
remains consistent with the universality framework, which is grounded in the
architecture’s capacity for pair-wise operations rather than being constrained
by the specifics of the attention mechanism [5]. Thus, zeroing out elements in
the input sequence aligns with the transformers function class.
### II-E Neural Estimation
Neural estimation is a methodology that utilizes NN optimization for the
optimization of various functionals. First introduced by the MINE, neural
estimators often utilize the DV representation [23, Theorem 3.2].
###### Theorem 2 (DV representation)
For any, $P,Q\in\mathcal{P}(\mathcal{X})$, we have
$\mathsf{D}_{\mathsf{KL}}(P\|Q)=\sup_{f:\mathcal{X}\rightarrow{}\mathbb{R}}\mathbb{E}_{P}[f]-\log\left(\mathbb{E}_{Q}[e^{f}]\right),$
(12)
where the supremum is taken over all measurable functions $f$ with finite
expectations.
The MINE is then obtained by approximating the DV optimization function class
with the class of NNs, and estimating expectations with samples means. The
MINE was generalized to additional information theoretic quantities, such as
conditional MI [35], total correlation [36], and TE with its variations [26].
Furthermore, a generalization of the MINE to DI was proposed in [25]. Drawing
inspiration from the MI and DI neural estimators, we develop a provably
consistent estimator of TE that utilizes the power of the attention mechanism.
## III Estimation of Transfer Entropy
In this paper we harness to computational power of transformers architecture
and modify the attention mechanism to result with TREET, a new estimator of TE
for high dimensional continuous data. We begin by deriving the estimator, then
account for its theoretical guarantees. Finally, we provide implementation
details, outlining the modifications of attention to neural estimation.
### III-A Estimator Derivation
The TREET provides an estimate of the TE (1) from a set of samples
$D_{n,l}=(X^{n+l},Y^{n+l})\sim P_{X^{n+l},Y^{n+l}}$. We assume that $l\ll n$
and this omit the dependence on $l$ in the dataset notation. The TREET
decomposes TE into two KL terms, each of which taken between the the data
distribution and a corresponding reference distribution $\widetilde{P}_{Y}$,
such that $\widetilde{P}_{Y}$ can be sampled freely. To derive TREET, first we
represent TE as subtraction of KL divergences w.r.t. and absolutely continuous
$\widetilde{P}_{Y}$. We propose the following.
###### Lemma 1 (TE as KL Divergences)
TE decomposes as
$\mathsf{TE}_{X\to
Y}(l)=\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}},$
(13)
where
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}:=\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}}\right),$
(14a) $\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}:=$
$\displaystyle\qquad\qquad\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}X^{l}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}X^{l}}\right),$
(14b)
and the conditional KL divergence is
$\mathsf{D}_{\mathsf{KL}}(P_{X|Z}\|P_{Y|Z}|P_{Z}):=\mathbb{E}_{Z}[\mathsf{D}_{\mathsf{KL}}(P_{X|Z}\|P_{Y|Z})]$.
Lemma 1 is proved in Section VII-C, and follows basic information theoretic
properties of TE and KL divergences. Utilizing the DV variational
representation (12) and performing the optimization over a set of causal
transformer architectures
$\mathcal{G}_{\mathsf{ctf}}^{Y}:=\mathcal{G}_{\mathsf{ctf}}^{(d_{y},1,l,v_{y})},\mathcal{G}_{\mathsf{ctf}}^{XY}:=\mathcal{G}_{\mathsf{ctf}}^{(d_{x}+d_{y},1,l,v_{xy})}$
for given $l,d_{y},d_{x},v_{y},v_{xy}\in\mathbb{N}$, and let
$g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y},g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}$.
Each KL divergence can be approximated with transformers by
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}=$
$\displaystyle\sup_{g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y}}\mathbb{E}\left[g_{y}\left(Y^{l}\right)\right]$
$\displaystyle-\log\left(\mathbb{E}\left[e^{g_{y}\left(\widetilde{Y}_{l},Y^{l-1}\right)}\right]\right),$
(15a) $\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}=$
$\displaystyle\sup_{g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}}\mathbb{E}\left[g_{xy}\left(Y^{l},X^{l}\right)\right]$
$\displaystyle-\log\left(\mathbb{E}\left[e^{g_{xy}\left(\widetilde{Y}_{l},Y^{l-1},X^{l}\right)}\right]\right).$
(15b)
Finally, replacing expectations with sample means in (15), we result with the
TREET, given by
$\displaystyle\widehat{\mathsf{TE}}_{X\to Y}(D_{n};l)$
$\displaystyle:=\sup_{g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}}\sup_{g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y}}\widehat{\mathsf{TE}}_{X\to
Y}(D_{n},g_{y},g_{xy};l)$ (16a)
$\displaystyle=\sup_{g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n},g_{xy})$
$\displaystyle\quad-\sup_{g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n},g_{y}),$
(16b)
where
$\displaystyle\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n},g_{y}):=\frac{1}{n}\sum_{i=1}^{n}g_{y}\left(Y_{i}^{i+l}\right)$
$\displaystyle\quad\qquad\qquad\qquad-\log\left(\frac{1}{n}\sum_{i=1}^{n}e^{g_{y}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)}\right),$
(17a)
$\displaystyle\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n},g_{xy}):=\frac{1}{n}\sum_{i=1}^{n}g_{xy}\left(Y_{i}^{i+l},X_{i}^{i+l}\right)$
$\displaystyle\quad\qquad\qquad-\log\left(\frac{1}{n}\sum_{i=1}^{n}e^{g_{xy}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1},X_{i}^{i+l}\right)}\right).$
(17b)
where $\widetilde{Y}$ is i.i.d. under absolutely continuous reference
measurement under the alphabet $\mathcal{Y}$, and
$\sup_{g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n},g_{xy})=\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$,
$\sup_{g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n},g_{y})=\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}$.
The TREET (16b) is consequently given as the subtraction of the solutions of
two optimization problems (17), each optimizing its own NN. The optimization
is then performed via gradient ascent with the corresponding model
$g_{y},g_{xy}$ and the dataset $D_{n}$. Note that Definition 15 states that
each KL divergence we estimate with DV representation, achieved by function of
$(Y^{l},\widetilde{Y}_{l})$ and $(Y^{l},X^{l},\widetilde{Y}_{l})$, thus we use
a causal transformer to opt $g_{y},g_{xy}$ in order to ignore relative future
values of the process. In the following section, we prove that the TREET
implemented with causal transformers is a consistent estimator of TE.
### III-B Theoretical Guarantees
To further understand the capabilities of TREET, we explore its theoretical
guarantees. Assuming joint stationarity of $(\mathbb{X},\mathbb{Y})$ and using
the causal transformers function class, we introduce the consistency of the
proposed TREET.
###### Theorem 3 (TREET consistency)
Let $\mathbb{X}$ and $\mathbb{Y}$ be jointly stationary, ergodic stochastic
processes. TREET is strongly consistent estimator of $\mathsf{TE}_{X\to Y}(l)$
for $l\in\mathbb{N}$, i.e. $\mathbb{P}-a.s.$ for every $\epsilon>0$ there
exists and $N\in\mathbb{N}$ such that for every $n>N$ we have
$\left|\widehat{\mathsf{TE}}_{X\to Y}(D_{n};l)-\mathsf{TE}_{X\to
Y}(l)\right|\leq\epsilon$ (18)
where $l$ is the memory parameter of the TE.
The proof following the steps of representation step - represents TE as a
subtraction of two DV potentials, estimation step - proves that the DV
potentials is achievable by empirical mean of a given set of samples, and
approximation step - shows that the estimator built upon causal transformers
converges to TE with the corresponding memory parameter. The proof is given in
the Appendix VII-B.
ReferenceSamplerTREET$\theta_{y}$DV
Loss$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n},g_{\theta_{y}})$Eq.
(17a)$\widetilde{Y}_{t}$$Y^{t}_{t-l}$$g_{\theta_{y}}(Y_{t},Y_{t-l}^{t-1})$$g_{\theta_{y}}(\widetilde{Y}_{t},Y_{t-l}^{t-1})$$\nabla_{\theta_{y}}\widehat{\mathsf{D}}_{Y}(D_{n},g_{\theta_{y}})$
Figure 1: The estimator architecture for the calculation of
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n},g_{\theta_{y}})$.
### III-C Algorithm and Implementation
This section describes the TREET implementation. We present an overview scheme
for estimation of TE and describe the algorithm. Afterwards we show the
architecture of the TREET and going deeper to its implementation. Formally,
the transformer function class is parametric models whose finitely many
parameters are a subset of a parameter space
$\Theta\subset\mathbb{R}^{d},d\in\mathbb{N}$. For a fixed $v<\infty$ neurons,
denote the functions from the above classes as $g\in\mathcal{G}_{v}$, or their
corresponding parameterized form $g_{\theta},\theta\in\Theta$. We therefore
denote the corresponding networks with $g_{\theta_{y}},g_{\theta_{xy}}$, where
$\theta_{y}$ and $\theta_{xy}$ are the network parameters, respectively. We
now describe the algorithm design and transformer modifications.
#### III-C1 Overview and Algorithm
The TREET algorithm follows an iterative joint optimization of
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$ and
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$ through
iterative mini-batch gradient optimization. The algorithm inputs are
$l,D_{n}$, which are the TE parameter and the dataset, respectively. Every
iteration begins with feeding mini-batch sized $m<n$ with sequences length $l$
in each model, followed by the calculation of both DV potentials (17), that
construct $\mathsf{TE}_{X\to Y}(l)$ (16a). The calculated objective is then
used for gradient-based optimization of the NN parameters. The iterative
process continues until a stopping criteria is met, typically defined as the
convergence of $\mathsf{TE}_{X\to Y}(l)$ within a specified tolerance
parameter $\epsilon>0$. For evaluation, we produce the estimated TE again for
many samples as possible, and taking the averaged results. The full pipeline
for estimating
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n},g_{\theta_{y}})$
is presented in Figure 1, and the complete list of steps is given in Algorithm
1.
$g_{y}({Y_{l},Y^{l-1}})$ Dense Add & Norm Feed Forward Concat & Norm Fixed-
Past- Causal-Attention Embeddings+ Position Value
$\begin{bmatrix}Y_{1},\cdots,Y_{l}\end{bmatrix}$
KVQ$g_{y}({\widetilde{Y}_{l},Y^{l-1}})$ Dense Add & Norm Feed Forward Concat &
Norm Modified Fixed-Past- Causal-Attention Embeddings+ Position Value
$\begin{bmatrix}\smash{\widetilde{Y}_{1},\cdots,\widetilde{Y}_{l}}\end{bmatrix}$
$\widetilde{\textbf{K}}$$\widetilde{\textbf{V}}$$\smash{\widetilde{\textbf{Q}}}$K,V
Figure 2: The TREET architecture for $g_{y}$, with memory parameter $l$. It
illustrated using a single sequence as an example. However, it is capable of
parallel processing for sequences lengthed $L>l$, and in such cases, the
number of outputs for the function will be $L-l+1$. Both transformer share the
same weights. The FPCA and the modified FPCA are as elaborated in Section
III-C2. Algorithm 1 TREET
Input: Joint process samples $D_{n}$; Observation length $l\in\mathbb{N}$.
Output: $\widehat{\mathsf{TE}}_{X\to Y}(D_{n};l)$ \- TE estimation.
1:NNs initialization $g_{\theta_{y}}$, $g_{\theta_{xy}}$ with corresponding
parameters $\theta_{y},\theta_{xy}$.
2:Step 1 – Optimization:
3:repeat
4: Draw a batch $B_{m}$: $m<n$ sub-sequences, length $L>l$ from $D_{n}$, with
reference samples $P_{\widetilde{Y}}$ for each.
5: Compute both potentials
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(B_{m},g_{\theta_{xy}})$,
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(B_{m},g_{\theta_{y}})$
via (17).
6: Update parameters:
7:
$\theta_{xy}\leftarrow\theta_{xy}+\nabla_{\theta_{xy}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(B_{m},g_{\theta_{xy}})$
8:
$\theta_{y}\leftarrow\theta_{y}+\nabla_{\theta_{y}}\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(B_{m},g_{\theta_{y}})$
9:until convergence criteria.
10:Step 2 – Evaluation: Evaluate for a sub-sequence (17) and (16a) to obtain
$\widehat{\mathsf{TE}}_{X\to Y}(D_{n};l)$.
$\displaystyle QK^{\top}=\begin{bmatrix}{q}_{(t+l)}\cdot
k_{(t)}&\ldots&{q}_{(t+l)}\cdot{k}_{(t+l)}&-\infty&\ldots&-\infty\\\
-\infty&{q}_{(t+l+1)}\cdot
k_{(t+1)}&\ldots&{q}_{(t+l+1)}\cdot{k}_{(t+l+1)}&-\infty&\vdots\\\
\vdots&-\infty&\ddots&&\ddots&-\infty\\\
-\infty&\ldots&-\infty&{q}_{(t+L)}\cdot
k_{(t+L-l)}&\ldots&{q}_{(t+L)}\cdot{k}_{(t+L)}\\\ \end{bmatrix}$
(19)
$\scalebox{0.9}{\mbox{$\displaystyle\widehat{QK^{\top}}=\begin{bmatrix}\widetilde{q}_{(t+l)}\cdot
k_{(t)}&\ldots&\widetilde{q}_{(t+l)}\cdot\widetilde{k}_{(t+l)}&-\infty&\ldots&-\infty\\\
-\infty&\widetilde{q}_{(t+l+1)}\cdot
k_{(t+1)}&\ldots&\widetilde{q}_{(t+l+1)}\cdot\widetilde{k}_{(t+l+1)}&-\infty&\vdots\\\
\vdots&-\infty&\ddots&&\ddots&-\infty\\\
-\infty&\ldots&-\infty&\widetilde{q}_{(t+L)}\cdot
k_{(t+L-l)}&\ldots&\widetilde{q}_{(t+L)}\cdot\widetilde{k}_{(t+L)}\\\
\end{bmatrix}$}}.$ (20)
The DV representation, (12), suggest that the function $f$ is the same
function for both terms that construct it, i.e. the weights are shared for
both network propagation. Hence, our transformer in TREET, which constructed
by (17), is the same for both terms, for both DV representation. Exemplifying
with $\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}$, both
$g_{y}(Y^{l})$ and $g_{y}(\widetilde{Y}^{l},Y^{l-1})$ use the same learning
parameters, from positional encoding layers and attention to FF layers. The
only difference we have between the two terms, is in how we operate the
attention mechanism, which essentially re-use keys and values generated for
the first term, to generate the later one. This model for $g_{y}$ is
visualized in Figure 2. The following sections will elaborate about the
proposed fixed past causal attention (FPCA) that constructs the TREET and its
variation, modified fixed past causal attention which is required for the
reference measurement sampling.
#### III-C2 Fixed Past Causal Attention
In this section we describe the calculation of $g_{y}(Y^{l})$ (17a). For
$L>l$, the causal attention architecture is constructed by queries
$Q\in\mathbb{R}^{L\times d_{o}}$, keys $K\in\mathbb{R}^{L\times d_{o}}$ and
values $V\in\mathbb{R}^{L\times d_{o}}$, and given by
$\displaystyle Q=\begin{bmatrix}q_{1},\ldots,q_{L}\end{bmatrix}$
${}^{\top},\quad
K=\begin{bmatrix}k_{1},\ldots,k_{L}\end{bmatrix}^{\top},\quad$ $\displaystyle
V=$ $\displaystyle\begin{bmatrix}v_{1},\ldots,v_{L}\end{bmatrix}^{\top},\quad$
(21)
where $L\in\mathbb{N}$ is the temporal length of the model’s input and
$l\in\mathbb{N}$ is the memory parameter which determines $\mathsf{TE}_{X\to
Y}(l)$. The $i^{\text{th}}$ step queries, keys and values in the are given by
$q_{i}=W_{Q}x_{i},k_{i}=W_{K}x_{i},v_{i}=W_{V}x_{i}$, and
$M\in\mathbb{R}^{L\times L}$, thus we denote
$\mathsf{Causal}\text{-}\mathsf{Attention}:=\mathsf{softmax}(QK^{\top}\odot
M)V.$ (22)
To result in a valid causal attention mechanism and fixed length input size
$l$, we propose the FPCA, where each query will be multiplied only by its
relative time key and past $l-1$ keys. Thus the dot-product operation is
between $[t+l,t+L]$ queries time steps and $[t,t+L]$ keys time steps. In
addition, we change the causal mask (i.e. for the above case
$M\in\mathbb{R}^{L-l\times L}$) to a Toeplitz-like band matrix mask
$M^{\prime}\in\mathbb{R}^{L-l\times L}$,
$M^{\prime}_{[i,j]}=\begin{cases}1&\text{if}j-i<l\text{and}j\geq i\\\
-\infty&\text{otherwise}\end{cases}$ (23)
The resulting queries and keys dot-product with the given mask yields (19),
and the FPCA is given by
$\mathsf{FPCA}:=\mathsf{softmax}(QK^{\top}\odot M^{\prime})V$ (24)
Note that we use only the last $L-l+1$ results from the transformer outputs
sequence, in order to keep the past information fix to length $l$. With the
given FPCA, the required DV output $g_{y}(Y^{l})$ can be calculated.
#### III-C3 Reference Sampling with FPCA
This section describes the calculation of $g_{y}(\widetilde{Y}_{l},Y^{l-1})$
(17a). As mentioned in Section III-A, the reference distribution,
$\widetilde{Y}$, is independently drawn from some absolute continuous PDF over
the bounds of $\mathcal{Y}$222If $\mathcal{Y}$ is not bounded, we set manually
the maximal bounds regarding to the dataset properties.. Denote the scoring
value after the softmax operation of causal attention for query $q_{i}$ with
key $k_{j}$ as
$\mathcal{R}_{(i,j)}:=\frac{e^{q_{i}\cdot k_{j}}}{\sum_{m=1}^{i}e^{q_{i}\cdot
k_{m}}},\quad j\leq i.$ (25)
Since we use FPCA with memory parameter $l$, the output at time $t$ can be
written as
$\mathsf{FPCA}_{t}=\mathcal{R}_{(t,t-l)}v_{t-l}+\mathcal{R}_{(t,t-l+1)}v_{t-l+1}+\ldots+\mathcal{R}_{(t,t)}v_{t}$.
In order to calculate $g_{y}(\widetilde{Y}_{t},Y^{t-1}_{t-l})$, the output of
the time mixing must contain the original distribution from past information
up to time step $t-1$. This information is stored in the keys and values of
the previous resulted $\mathsf{FPCA}_{t}$, for $g_{y}(Y^{t}_{t-l})$. Thus, the
modified FPCA should be constructed by
$\widetilde{\mathcal{R}}_{(i,j)}:=\begin{cases}\frac{e^{\widetilde{q}_{i}\cdot
k_{j}}}{e^{\widetilde{q}_{i}\cdot\widetilde{k}_{i}}+\sum_{m=1}^{i-1}e^{\widetilde{q}_{i}\cdot
k_{m}}}&,i\neq j\\\\[5.0pt]
\frac{e^{\widetilde{q}_{i}\cdot\widetilde{k}_{i}}}{e^{\widetilde{q}_{i}\cdot\widetilde{k}_{i}}+\sum_{m=1}^{i-1}e^{\widetilde{q}_{i}\cdot
k_{m}}}&,i=j.\end{cases}$ (26)
In this case, the modified FPCA can be written as
$\mathsf{Modified}\text{-}\mathsf{FPCA}_{t}=\widetilde{\mathcal{R}}_{(t,t-l)}v_{t-l}+\widetilde{\mathcal{R}}_{(t,t-l+1)}v_{t-l+1}+\ldots+\widetilde{\mathcal{R}}_{(t,t)}\widetilde{v}_{t}$,
for time $t$. Summarizing, the second term (17a) generated by the modified
FPCA, contains all keys and values of the relative past and query, key and
value for the current present, which is a function of the reference
distribution. The dot-product matrix between queries and keys can be written
as (20), for $l\leq L$, and modified FPCA is written as
$\mathsf{Modified}\text{-}\mathsf{FPCA}=\mathsf{softmax}(\widehat{QK^{\top}}\odot
M^{\prime})V$.
The extension of FPCA and modified FPCA to multi-head version is immediate. In
our implementation, the reference inputs, for the second term in (17a) are
drawn from the uniform measure on the bounding box of the current batch of $Y$
samples, while theoretically, our method allows to draw from any positive
continuous distribution measure; for further details check Appendix VII-B. The
implementation of
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n},g_{xy})$
(17b) is obtained by concatenating the $X^{l}$ values with the corresponding
$Y^{l}$ or $\widetilde{Y}^{l}$ values for both the FPCA and the modified FPCA,
respectively.
## IV Optimization of Estimated Transfer Entropy
Many applications can leverage the optimization of TE, in a given data-driven
setting. The optimization of which can be done via controlling the
distribution of the process $\mathbb{X}$ and $\mathbb{Y}$, independently or
jointly. An important example for such a setting would be communication
channels, whose capacity can be characterized with a certain TE term.
$\begin{bmatrix}X_{t-L}\\\
0\end{bmatrix}$$\begin{matrix}\cdots\end{matrix}$$\cdots$$\begin{bmatrix}X_{t-1}\\\
0\end{bmatrix}$$\begin{bmatrix}0\\\ U_{t}\end{bmatrix}$$X_{t}$NDGTransformer
Figure 3: The recursive process for the NDG with transformers. If feedback
presents, the input includes the past channel output, concatenated with the
corresponding value $X_{i}$ at time $i$.
###### Remark 1 (Channel capacity)
As presented in [25], consider channels with and without feedback links from
the channel output back to the encoder. The feedforward capacity of a channel
sequence $\left\\{P_{Y^{n}\|X^{n}}\right\\}$, for $n\in\mathbb{N}$ is
$C_{\mathsf{FF}}=\lim_{n\to\infty}\sup_{P_{X^{n}}}\frac{1}{n}\mathsf{I}(X^{n};Y^{n}),$
(27)
while the feedback capacity is
$C_{\mathsf{FB}}=\lim_{n\to\infty}\sup_{P_{X^{n}\|Y^{n-1}}}\frac{1}{n}\mathsf{I}(X^{n}\to
Y^{n}).$ (28)
The achievability of the capacities is further discussed in [37], [38]. [32]
showed that for non-feedback scenario, the optimization problem over
$P_{X^{n}\|Y^{n}}$ can be translated to $P_{X^{n}}$, which support the use of
DI rate for both optimization problems. Under some conditions TE can estimate
the capacity as well, as presented in Section II-C.
Focusing on channel capacity, we assume that we can control the input
distribution sampling mechanism and propose an algorithm for the optimization
of estimated TE with respect to the input generator. We refer to this model as
the Neural Distribution Generator (NDG). The estimated TE optimization
methodology is inspired by the proposed methods from [25]. However, the
adaptation to transformer architectures considers a different implementation
of the proposed scheme. As the TREET estimates TE from samples, the NDG is
defined as a generative model of the input distribution samples, and is
optimized with the goal of maximizing the downstream estimated TE.
###### Lemma 2 (Optimal TE)
Let $(\mathbb{X},\mathbb{Y})$ be jointly stationary processes, and the TE with
memory parameter $l\in\mathbb{N}$. Then, the maximal TE, $\mathsf{TE}_{X\to
Y}^{\star}(l)$, by $\mathbb{X}$ is
$\mathsf{TE}_{X\to
Y}^{\star}(l):=\sup_{P_{X^{l}}}\mathsf{I}(X^{l};Y_{l}|Y^{l-1}).$ (29)
The proof of the lemma is given in Appendix VII-D. The lemma suggests that NDG
with input sequence length $l$ is enough to achieve maximum TE with memory
parameter $l$, for independently controlling the distribution of $\mathbb{X}$.
The NDG calculates a sequence of channel input $X^{l}$ through the mapping
$h_{\phi}:(U_{i},Z^{i-1}_{i-l})\rightarrow X_{i}^{\phi},\qquad i=1,\ldots,n,$
(30)
where $U_{i}$ is the random noise drawn from
$P_{U}\in\mathcal{P}_{\mathsf{ac}}(\mathcal{U}),\mathcal{U}\subset\mathbb{R}^{d_{x}}$,
which cause the stochasticity, $Z_{i-l}^{i-1}$ are the past observation of the
generated process created by the model, and the channel corresponding outputs
if feedback exists, and $h_{\phi}$ is the parametric NDG mapping with
parameters $\phi\in\Phi$. By the functional representation lemma [39] and the
restated lemma in [25], we can achieve the distribution of $X$ from an NN
function. After $l$ iterations, with $Z_{i-l}^{i-1}$ storing the relevant
information of previous iterations, the NDG generates the whole sequence.
Transformers need access to the whole sequence at once, in contrast to RNNs
where a single state can theoretically represent the past sequence.
Thus, the input sequence to the NDG with transformer is created via past
outputs of the transformer itself (and corresponding channel outputs if
feedback exists) as depicted in Figure 3, and the past observations are taken
without gradients to prevent backpropagation through iterations. In our
implementation, the input convention for each time step is a concatenated
vector of $[X_{i},U_{i}]^{\top}$ to maintain the input structure of samples
and random noise for every projection in the network, while in relative
history time steps, the noise is replaced with zero, and for the present time,
the input $X_{i}$ is replaced with a zero vector of the same dimension.
NDG Transformer $\phi$Channel$P_{Y_{i}|X^{i}_{i-l},Y^{i-1}_{i-l}}$TREET
Transformer
$g_{\theta_{y}},g_{\theta_{xy}}$$X_{i-l}^{i-1}$$X_{i-l}^{i-1},Y_{i-l}^{i-1}$DV
Loss$U_{i}$$X_{i}$$Y_{i},X_{i}$$\nabla g_{\theta_{y}},\nabla
g_{\theta_{xy}}$$\nabla\phi$$\star$
Figure 4: Complete system for estimating and optimizing TREET with NDG while
Altering between the models to train on. ($\star$) If feedback presents, past
channel output realizations included. Algorithm 2 Continuous TREET
optimization
Input: Continuous sequence-to-sequence system $\mathcal{S}$; Observation
length $l\in\mathbb{N}$.
Output: $\widehat{\mathsf{TE}}_{X\to Y}^{\star}(U^{n};l)$, optimized NDG.
1:NNs initialization $g_{\theta_{y}}$, $g_{\theta_{xy}}$ and $h_{\phi}$ with
corresponding parameters $\theta_{y},\theta_{xy},\phi$.
2:repeat
3: Draw noise $U^{m}$, $m<n$.
4: Compute batch $B_{m}^{\phi}$ sized $m$ using NDG, $\mathcal{S}$
5: if training TREET then
6: Perform TREET optimization - Step 1 in Algorithm 1.
7: else Train NDG
8: Compute $\widehat{\mathsf{TE}}_{X\to
Y}(B_{m}^{\phi},g_{\theta_{y}},g_{\theta_{xy}},h_{\phi};l)$ using (16a).
9: Update NDG parameters:
10: $\quad\phi\leftarrow\phi+\nabla_{\phi}\widehat{\mathsf{TE}}_{X\to
Y}(B_{m}^{\phi},g_{\theta_{y}},g_{\theta_{xy}},h_{\phi};l)$
11:until convergence criteria.
12:Draw $U^{m}$ to produce $l$ length sequence and evaluate
$\widehat{\mathsf{TE}}_{X\to Y}(D_{n}^{\phi};l)$.
13:return $\widehat{\mathsf{TE}}_{X\to Y}^{\star}(U^{n};l)$, optimized NDG.
To estimate and optimize the TE at once, we jointly training the TREET and the
NDG. As described in Algorithm 2 and Figure 4, in each iteration we only
update one of those models by maximizing each DV potentials,
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}},\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$,
for TREET model and by maximizing $\widehat{\mathsf{TE}}_{X\to Y}$ for NDG
model. The entire pipeline of one iteration will create a sequence length $l$
from the NDG, by iterating it $l$ times with some initiated zero values -
which creates the dataset, $D_{n}^{\phi}=(X^{\phi,n},Y^{\phi,n})$. Afterwards,
feeding each sample sequence from the dataset to TREET for achieving the
corresponding networks’ outputs as mentioned back in Algorithm 1, which
constructs the loss that will generate gradients backward according to each
model. Next, we demonstrate the power of the proposed method for estimating
the channel capacity.
## V Experimental Results
In this section we demonstrate the utility and performance of the proposed
algorithms to several tasks, encompasing both estimation and optimization of
TE. Additionally, we perform a comparison with the RNN-based DI rate
estimation scheme from [25] and discusses the results. While the experiments
are about continuous spaces of distributions, discrete spaces can easily be
applied to the algorithm of TREET, but not for the optimization part with NDG
as presented. The TREET consists of transformer with a single fixed-past-
causal-attention layer follows by a single FF layer, and the NDG also consists
of transformer with a single causal-attention layer and a single FF layer.
Further details on the implementation parameters are provided in Appendix
VII-E. All simulations are implemented in PyTorch and the code implementation
can be found at this repository: https://github.com/omerlux/TREET.
$-1$$-0.5$$0$$0.5$$1$$0$$0.2$$0.4$EmpiricalProbabilityInput
Noise$-2$$0$$2$$0$$0.1$$0.2$$0.3$EmpiricalProbabilityOutput $X$
Figure 5: NDG input noise and output $X$, $0$ SNR case.
$-20$$-10$$0$$10$$0$$0.5$$1$$1.5$$\displaystyle
P/\sigma^{2}\mathrm{[dB]}$$\displaystyle\mathrm{Capacity}$TheoreticalDINE
est.TREET est.
(a) Capacity as function of SNR.
$1$$2$$3$$4$$5$$6$$0.5$$1$$1.5$$2$Vector Dimension
$\displaystyle\mathrm{(0dB)}$CapacityTheoreticalDINE est.TREET est.
(b) Capacity as function of vector dimension.
Figure 6: Channel capacity estimation of AWGN channel. 6(a) presents the
estimation of TREET and DINE in comparison to theoretical capacities for
various SNR rates. $P$ represents the power constraint and the noise parameter
is $\sigma^{2}$. 6(b) shows the capacity estimation with variate vector
process dimension for the $0$ dB case. For the case of power constraint $P=1$
theoretically the input $P_{X}$ distributes $\mathcal{N}(0,1)$.
$-20$$-10$$0$$10$$0$$0.5$$1$$1.5$$\displaystyle
P/\sigma^{2}\mathrm{[dB]}$Feedback CapacityTheoreticalDINE est.TREET est.
(a) Feedback Capacity.
$-20$$-10$$0$$10$$0$$0.5$$1$$1.5$$\displaystyle
P/\sigma^{2}\mathrm{[dB]}$Feedforward CapacityTheoreticalDINE est.TREET est.
(b) Feedforward Capacity.
Figure 7: Channel capacity estimation of Gaussian MA(1) channel with variate
SNR.
$-20$$-10$$0$$10$$0$$0.5$$1$$1.5$$\displaystyle
P/\sigma^{2}\mathrm{[dB]}$Feedback CapacityTheoreticalDINE est.TREET est.
(a) Feedback Capacity.
$-20$$-10$$0$$10$$0$$0.5$$1$$1.5$$\displaystyle
P/\sigma^{2}\mathrm{[dB]}$Feedforward CapacityTheoreticalDINE est.TREET est.
(b) Feedforward Capacity.
Figure 8: Channel capacity estimation of Gaussian AR(1) channel with variate
SNR.
### V-A Empirical Capacity Estimation for Finite Memory Processes
As shown in Section II-C, under some conditions TE is equal to the DI rate and
converges to it. DINE [25] proved that it can estimate a capacity of
stationary channels by optimizing the DI rate estimator and the input
distribution of the channel $P_{X}$, as Remark 1 mentions about channel
capacities. Our experiments has shown that for memory-less channels and for
memory channels with and without feedback, TREET can approximate the capacity
with the joint optimization procedure of the input distribution (NDG) and TE
estimation. Important to note that channel input constraints are essential for
ensuring that the transmitted signals are well-suited to the channel’s
characteristics and limitations. In our experiment we applied the power
constraint on the input signal of the channel, which is implemented via
normalizing a batch of samples to a certain statistics according to the power
constraint.
#### V-A1 AWGN Channel
Consider additive white Gaussian noise (AWGN) channel with i.i.d. noise,
$\displaystyle Z_{i}$ $\displaystyle\sim\mathcal{N}(0,\sigma^{2}),$
$\displaystyle Y_{i}$ $\displaystyle=X_{i}+Z_{i},\qquad i\in\mathbb{Z},$
$X_{i}$ is the channel’s input sequence, coupled with the average power
constraint $\mathbb{E}[X_{i}^{2}]\leq P$. The capacity of this channel is
simple for analytical calculation, and is given by the following formula
$\text{C}=0.5\log\left(1+{P}/{\sigma^{2}}\right)$. Since the process is
memoryless, maximized both DI rate and TE (for any $l\geq 0$) coincide with
the channel capacity. We estimated and optimized the TREET according to
Algorithm 2 and compared the model performance with the DI rate estimation and
optimization scheme from [25]. Results are presented in Figure 6. It can be
seen that both are estimating the right capacities which are the MI, although
their access to multiple past observations, and in addition, the dimension of
the input vector and output vector of a channel changes the capacities, as
expected. Note that larger dimensions cause error of in estimation and still
is an open academic research [28]. To further analyse the learned
distribution, we visualize the optimized NDG mapping in Figure 5. It can be
seen that the optimized NDG maps the uniform ($U\sim[-1,1]$) inputs into
Gassian samples. This observation meets our expectations, as the capacity
achieving AWGN distribution is Gaussian [40].
#### V-A2 Gaussian MA(1) Channel
Given a Moving Average (MA) Gaussian noise channel with order 1
$\displaystyle Z_{i}$ $\displaystyle=N_{i}+\alpha N_{i-1},$ $\displaystyle
Y_{i}$ $\displaystyle=X_{i}+Z_{i},\qquad i\in\mathbb{Z},$
where $N_{i}\sim\mathcal{N}(0,\sigma^{2})$ are i.i.d. and $X_{i}$ is the input
to the channel with power constraint $\mathbb{E}[X_{i}^{2}]\leq P$. We apply
Algorithm 2 to both feedforward and feedback settings, comparing with ground
truth solutions and the DI-based scheme. The feedforward capacity can be
calculated with the water-filling algorithm [34], while the feedback capacity
can be calculated by the root of forth order polynomial equation [41]. As seen
in Figure 7 our method successfully estimated the capacity for a wide range of
SNR values.
$0$$20$$40$$60$$80$$100$$120$$5$$10$$15$$20$$25$Attention Relative Time Step
($t-i$)Series Index$0.02$$0.07$$0.12$Attention weight
(a) 130 input length.
$0$$15$$30$$45$$60$$75$$90$$5$$10$$15$$20$$25$Attention Relative Time Step
($t-i$)Series Index$0.010$$0.015$$0.020$Attention weight
(b) 90 input length.
Figure 9: Attention weights at training convergence of TREET optimized by NDG.
Each row in the matrices represent a different input sequence and the columns
are the weights of past values $i$ from current prediction $t$ (i.e. $i=0$
represent the present prediction $t$). For the GMA(100) process, it can be
observed that giving enough time-steps, TREET easily observes at the related
time $i=100$ where the information needed to be gathered from, while shorter
length lead to instability of training.
#### V-A3 Gaussian AR(1) Channel
The case of autoregressive (AR) Gaussian noise channel of order 1 is similar,
$\displaystyle Z_{i}$ $\displaystyle=N_{i}+\alpha Z_{i-1},$ $\displaystyle
Y_{i}$ $\displaystyle=X_{i}+Z_{i},\qquad i\in\mathbb{Z},$
where $N_{i}\sim\mathcal{N}(0,\sigma^{2})$ are i.i.d. and $X_{i}$ is the input
to the channel with power constraint $\mathbb{E}[X_{i}^{2}]\leq P$. The
capacity is also affected by the existence of feedback. Feedforward capacity
can be solved with the water-filling algorithm [34] and [42] prove how to
achieve the feedback capacity analytically. Figure 8 compares the results of
DINE and TREET estimators.
### V-B Memory Capabilities Analysis
Previous experiments showed the capability of correctly estimating TE. Delving
deeper into the memory effectiveness of TREET, we tested Gaussian MA channel,
as presented before, but with time delay of 100 steps.
$\displaystyle Z_{i}$ $\displaystyle=N_{i}+\alpha N_{i-100},$ $\displaystyle
Y_{i}$ $\displaystyle=X_{i}+Z_{i},\qquad i\in\mathbb{Z}.$
It is notable that to achieve the correct capacity, the information from
$t-100$ steps must be an input to the TREET. We tested the estimation
optimization algorithm with shorter input length and longer input length, than
the demanded one. Figure 9(a) shows the analysis of the attention weights
related to the $\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$
model with $l=130$ input steps for the estimation, at the late stages of
training close to convergence, while Figure 9(b) considers the same for $l=90$
input steps. It is evident from Figure 9 that TREET captures relevant
information even from very long time series, while shorter length will lead
larger error of information extraction which cause a worse estimation result,
which is rational due to trimming of critical information.
$60$$70$$80$$90$TimeAmplitudeHeart
Rate$200$$220$$240$$260$$280$$300$$320$$340$$360$$380$$400$$-0.5$$0$$0.5$$1$$1.5$$\cdot
10^{4}$TimeAmplitudeBreath Rate
(a) Sampled sequence of Apnea dataset.
$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$\cdot
10^{-2}$Label Length $\displaystyle[l]$$\displaystyle TE_{X\to
Y}(k,l=2)$Breath $\rightarrow$ HeartHeart $\rightarrow$ Breath
(b) TE estimation for variable length history of $Y$.
Figure 10: Transfer Entropy estimation on physiological data. Apnea dataset
consists of heart rate and breath rate and we seek to find the information
flow for patient diagnosing. Patient with Apnea suffer from breathing
cessation which leads to alterations in heart rate during sleep.
Additionaly, we compare TREET with DINE for different input lengths on the
GMA(100) channel. Important to note that DINE can technically deal with long
sequences even if given a shorter backpropgation through time (bptt) input
length then the process memory is due to state propagation. However, as seen
in Table I, the DINE struggles to estimate the right capacity under long
memory, while TREET successfully estimates capacity with low error rate.
Nonetheless, when the TREET memory is shorter than the channel memory, its
performance significantly degrades.
| Memory | Estimated | Absolute
---|---|---|---
Model | Length | Capacity [nits] | Error (%)
DINE | 90 | 0.3435 | 15.3 (%)
TREET | 90 | 0.2962 | 26.9 (%)
DINE | 130 | 0.3333 | 17.8 (%)
TREET | 130 | 0.3851 | 5.0 (%)
TABLE I: GMA(100), with 0 SNR, capacity estimation with different lengths.
Note that memory length in DINE is the bptt length for LSTM, while in TREET it
is the sequence input length.
### V-C Transfer Entropy Estimation in Physiological Data
Motivated by the results in [4, Chapter 7.1], we tested the TREET on the Apnea
dataset from Santa Fe Time Series
Competition333https://physionet.org/content/santa-fe/1.0.0/ [43, 44]. This is
a multivariate data that has been recorded from a diseased patient of Apnea in
a sleep laboratory, which is a condition characterized by brief, involuntary
pauses in breathing, particularly during sleep. Each sample consists of three
different variables from a specific time, with a sample rate of 2 Hz. The
three features are heart rate, chest volume (which is the respiration force)
and blood oxygen concentration. A sampled sequence of heart rate and breath
rate (chest volume) is presented in Figure 10(a).
Determining the interaction between different physiological features is
crucial for diagnosing diseases by revealing causal connections within the
human body, enabling targeted diagnostics and personalized treatments
according to identified risk factors. Therefore, we applied the TREET to
determine the magnitude and direction of information transfer in the given
setting. We used both heart rate and breath features and compared the TE
measurements, $\mathsf{TE}_{\mathsf{Breath}\to\mathsf{Heart}}(k,2)$ and
$\mathsf{TE}_{\mathsf{Heart}\to\mathsf{Breath}}(k,2)$ for variable length $k$
that is the $Y$ process’s history observations length. The results are
presented in Figure 10(b) and are aligned with the results of [4] (for
$k,l=2$) and extend it further with tests on variate history length of $Y$
process.
Notably, for every considered $k$, the TE from the breath process to the heart
process is consistently higher, aligning with the diagnosis of Apnea disorder
(abrupt cessation of breathing during sleep). In terms of information flow,
our results indicate that the breathing process transfers more information
regarding the behavior of the heart rate process, than the opposite direction,
since the value of the estimated TE is consistently higher in comparison,
while in information theory it essentially reflect that the $X$ process has
more to reveal about $Y$’s current value than $Y$’s past itself.
Furthermore, in the influence direction
$\mathsf{TE}_{\mathsf{Breath}\to\mathsf{Heart}}$, we can infer that increasing
the of visible heart rate history samples decreases the information transfer,
since a longer history of heart rate provides more insight into future
outcomes than the instantaneous breath value. However,
$\mathsf{TE}_{\mathsf{Heart}\to\mathsf{Breath}}$ shows that for variate $k$
values, the majority of TE results indicate that the heart process barely
affect the breathing process in Apnea patients.
The conclusion of this experiment is very valuable and validates what is known
to science - although it is known that when the heart rate increases the
breath rate increases also, for Apnea patients, the sudden cessation of muscle
movement during inhalation, i.e., the stopping of breathing, affects the heart
rate, in contrast to what is observed in healthy individuals. This experiment
supports the results of [4] and add a new insight about the information
transfer decreasing for increasing context of $Y$ process (which reveals more
information about $Y$, thus will lead to lower value of TE as mentioned
before).
## VI Conclusions and Future Work
This work presented a modified attention-based architecture to estimate TE,
for a class of ergodic and stationary processes. We devise a DV-based neural
TE estimator, proved its consistency and described the proposed novel modified
attention mechanism for the task at hand. We then developed an optimizer of
estimated TE, which was then leveraged for the estimation of channel capacity.
We concluded by studying the TREET application in causal features analysis on
the Apnea dataset and compared its performance with RNN-based mechanisms.
With the increasing popularity of sequential data to most contemporary fields
of machine learning, we plan to leverage the TREET for information theoretic
analysis and design of architectures, through the lenses of causal information
transfer. Examples of such applications include enhancing predictive models,
refining feature selection processes, reconstructing complex networks,
improving anomaly detection capabilities, and optimizing decision-making in
dynamic environments. Furthermore, building upon the work in [45], we plan to
extend the TREET optimization scheme to sequential data compression tasks.
## VII Appendix
### VII-A Lemma TE Equals to DI
###### Lemma 3
Define markov property $Z_{n}-Z_{n-1}-Z^{n-2}$ as
$P_{Z_{n}|Z^{n-1}}=P_{Z_{n}|Z_{n-1}}$. Let $\mathbb{X}$ and $\mathbb{Y}$ be
two jointly stationary processes such that the following markov property
holds,
$\displaystyle Y_{l}-Y^{l-1}-Y^{0}_{-\infty},$ $\displaystyle
Y_{l}-(X^{l},Y^{l-1})-(X^{0}_{-\infty},Y^{0}_{-\infty}),$
for $l\in\mathbb{N}$. Then,
$\mathsf{TE}_{X\to Y}(m)=\mathsf{TE}_{X\to Y}(l)$ (31)
for $m\geq l$ and
$\mathsf{TE}_{X\to Y}(m)=\mathsf{I}\left(\mathbb{X}\to\mathbb{Y}\right).$ (32)
proof: Let $\mathbb{X}$, $\mathbb{Y}$ be two jointly stationary processes that
pose the markov property with $l\in\mathbb{N}$ and $m\geq l$, then
$\displaystyle\mathsf{TE}_{X\to Y}(l)$
$\displaystyle=\mathsf{I}\left(X^{l};Y_{l}|Y^{l-1}\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathsf{I}\left(X^{l}_{l-m};Y_{l}|Y^{l-1}_{l-m}\right)$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathsf{I}\left(X^{m};Y_{m}|Y^{m-1}\right)$
$\displaystyle=\mathsf{TE}_{X\to Y}(m),$ (33)
where (a) is true from the markovity, and (b) transition is index shift, and
is valid due to stationarity of the process. Observing the DI rate,
$\displaystyle\mathsf{I}(\left(\mathbb{X}\to\mathbb{Y}\right)$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\mathsf{I}\left(X^{i};Y_{i}|Y^{i-1}\right)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\left[\mathsf{h}\left(Y_{i}|Y^{i-1}\right)-\mathsf{h}\left(Y_{i}|X^{i},Y^{i-1}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\lim_{n\to\infty}\mathsf{h}\left(Y_{n}|Y^{n-1}\right)-\mathsf{h}\left(Y_{n}|X^{n},Y^{n-1}\right)$
$\displaystyle=\lim_{n\to\infty}\mathsf{I}\left(X^{n};Y_{n}|Y^{n-1}\right)$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\lim_{n\to\infty}\mathsf{TE}_{X\to
Y}(n)$ $\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\mathsf{TE}_{X\to
Y}(m),$ (34)
where the limit in (a) exists whenever the joint process is stationary,
transition (b) is valid because the limit exists for each series of
conditional entropies, and since conditioning reduces entropy, the limit for
each normalized sum of series is
$\lim_{n\to\infty}\mathsf{h}\left(Y_{n}|Y^{n-1}\right),\lim_{n\to\infty}\mathsf{h}\left(Y_{n}|X^{n},Y^{n-1}\right)$,
respectively [40, Theorem 4.2.1]. Since the limit exists for the conditional
MI, transition (c) is valid by definition of TE, and the TE with limit of
parameter $m$ exists, and (d) is from (VII-A) for $n\geq m$. Concluding the
proof. $\square$
### VII-B Proof of theorem 3
The proof following the steps of representation step - represents TE as a
subtraction of two DV potentials, estimation step - proves that the DV
potentials is achievable by empirical mean of a given set of samples, and
approximation step - shows that the estimator converges to TE with the
corresponding memory parameter $l$. Thus concluding that our estimator is a
consistent estimator for the TE.
Let $\\{X_{i},Y_{i}\\}_{i\in\mathbb{Z}}$ be the two values of a process
$\mathbb{X},\mathbb{Y}$ respectively, and $\mathbb{P}$ be the stationary
ergodic measure over $\sigma(\mathbb{X},\mathbb{Y})$. Define
$P_{X^{n},Y^{n}}:=\mathbb{P}|_{\sigma(X^{n},Y^{n})}$ as the $n$-coordinate
projection of $\mathbb{P}$, where $\sigma(X^{n},Y^{n})$ is the
$\sigma$-algebra generated by $(X^{n},Y^{n})$. Let $D_{n}=(X^{n},Y^{n})\sim
P_{X^{n},Y^{n}}$. Let $\widetilde{Y}\sim{Q}_{Y}$ be an absolutely continuous
PDF, independent of $\\{(X_{i},Y_{i})\\}_{i\in\mathbb{Z}}$ and its
distribution noted as $\widetilde{P}_{Y}$. The proof is divided to three steps
- variational representation, estimation from samples and functional
approximation.
#### VII-B1 Representation of TE
In order to write TE as a difference between two KL divergences, recall Lemma
1 \- let,
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}:=\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}}\right),$
(35a)
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}:=\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}X^{l}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}X^{l}}\right).$
(35b)
Then
$\mathsf{TE}_{X\to
Y}(l)=\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}.$
(36)
This lemma is proved in Section VII-C. Estimating the KL divergence is
applicable with the DV representation 2
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}=$
$\displaystyle\sup_{f_{y}:\Omega_{\mathcal{Y}}\to\mathbb{R}}\mathbb{E}\left[f_{y}\left(Y^{l}\right)\right]$
$\displaystyle-\log\mathbb{E}\left[e^{f_{y}\left(Y^{l-1},\widetilde{Y}_{l}\right)}\right],$
(37a) where $\Omega_{\mathcal{Y}}=\mathcal{Y}^{l}$. For the other term, the DV
representation is,
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}=$
$\displaystyle\sup_{f_{xy}:\Omega_{\mathcal{X}\times\mathcal{Y}}\to\mathbb{R}}\mathbb{E}\left[f_{xy}\left(X^{l},Y^{l}\right)\right]$
$\displaystyle-\log\mathbb{E}\left[e^{f_{xy}\left(X^{l},Y^{l-1},\widetilde{Y}_{l}\right)}\right],$
(37b)
where
$\Omega_{\mathcal{X}\times\mathcal{Y}}=\mathcal{X}^{l}\times\mathcal{Y}^{l}$.
The next section refers to (37a) but the claims are the same for (37b).
#### VII-B2 Estimation
By the DV representation, the supremum in (37a) achieved for
$\displaystyle f_{y,l}^{\star}$
$\displaystyle:=\log\left(\frac{dP_{Y^{l}}}{d(P_{Y^{l-1}}\otimes\widetilde{P}_{Y_{l}})}\right)$
$\displaystyle=\log p_{Y_{l}|Y^{l-1}}-\log\widetilde{p}_{Y_{l}},$ (38)
where the last equality holds due to $P_{Y^{l}}\ll
P_{Y^{l-1}}\otimes\widetilde{P}_{Y_{l}}$, both measures have Lebesgue
densities. While it is not mandatory to select the reference measurement as
uniform, choosing a uniform reference measurement can result in a constant
that can be neutralized to obtain the likelihood function of $Y_{l}|Y^{l-1}$.
This approach allows for simplification and facilitates the estimation
process. Empirical means can estimate the expectations in (37a), while
applying the generalized Birkhoff theorem [46], stated next:
###### Theorem 4 (The generalized Birkhoff theorem)
Let $T$ be a metrically transitive 1 - 1 measure preserving transformation of
the probability space $(\Omega,\mathcal{F},\mathbb{P})$ onto itself. Let
$g_{0}(\omega),g_{1}(\omega),\ldots$ be a sequence of measurable functions on
$\Omega$ converging a.s. to the function $g(\omega)$ such that
$\mathbb{E}[\sup_{i}|g_{i}|]\leq\infty$. Then,
$\frac{1}{n}\sum_{i=1}^{n}g_{i}(T^{i}\omega)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow{}}}\mathbb{E}[g],\quad\mathbb{P}-a.s.$ (39)
By applying Theorem 4 and for any $\epsilon>0$ and sufficiently large $n$, we
have
$\displaystyle\left|\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]-\frac{1}{n}\sum_{i=0}^{n-1}f_{y,l}^{\star}\left(Y_{i}^{i+l}\right)\right|<\frac{\epsilon}{8},$
(40a)
$\displaystyle\left|\log\left(\mathbb{E}\left[e^{f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)}\right]\right)\right.$
$\displaystyle\quad\left.-\log\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{f_{y,l}^{\star}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right)\right|<\frac{\epsilon}{8},$
(40b)
where $\\{f_{y,l}^{\star}\\}$ is the function of $l$ time steps, that achieves
the supremum of
$\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}}\right)$.
Convergence achieved from the generalized Brikhoff theorem, where the series
of functions is the fixed function $f_{y,l}^{\star}$,
$\displaystyle\mathbb{E}_{n}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]:=\frac{1}{n}\sum_{i=0}^{n-1}f_{y,l}^{\star}\left(Y_{i}^{i+l}\right),$
(41a)
$\displaystyle\mathbb{E}_{n}\left[e^{f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)}\right]:=\frac{1}{n}\sum_{i=0}^{n-1}e^{f_{y,l}^{\star}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}.$
(41b)
#### VII-B3 Approximation
Last step is to approximate the functional space with the space of
transformers. Recall that set of causal transformer architectures
$\mathcal{G}_{\mathsf{ctf}}^{Y}:=\mathcal{G}_{\mathsf{ctf}}^{(d_{y},1,l,v_{y})},\mathcal{G}_{\mathsf{ctf}}^{XY}:=\mathcal{G}_{\mathsf{ctf}}^{(d_{x}+d_{y},1,l,v_{xy})}$
for given $l,d_{y},d_{x},v_{y},v_{xy}\in\mathbb{N}$. Define
$\displaystyle\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n}):=\sup_{g_{y}\in\mathcal{G}_{\mathsf{tf}}^{Y}}\frac{1}{n}\sum_{i=0}^{n-1}g_{y}\left(Y_{i}^{i+l}\right)$
$\displaystyle\qquad\qquad\quad-\log\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{y}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right),$
(42)
where the DV functions are transformers
$g_{y}\in\mathcal{G}_{\mathsf{ctf}}^{Y},g_{xy}\in\mathcal{G}_{\mathsf{ctf}}^{XY}$.
We want to prove that for a given $\epsilon>0$
$\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n})-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}\right|\leq\frac{\epsilon}{2}.$
(43)
From Theorem 2, we obtain
$\displaystyle\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]=\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}},$
(44a)
$\displaystyle\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)\right]=1.$
(44b)
Thus, we bound the term following
$\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n})-\mathbb{E}\left[f_{y,l}^{\star}\left(Y_{i}^{i+l}\right)\right]\right|$.
By the identity $\log(x)\leq x-1,\forall x\in\mathbb{R}_{\geq 0}$ we gain
$\displaystyle\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n})-\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]\right|$
$\displaystyle=\left|-\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]+\sup_{g_{y}\in\mathcal{G}_{\mathsf{tf}}^{Y}}\left\\{\frac{1}{n}\sum_{i=0}^{n-1}g_{y}\left(Y_{i}^{i+l}\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad\left.\left.-log\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{y}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right)\right\\}\right|$
$\displaystyle\leq\left|1-\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]+\sup_{g_{y}\in\mathcal{G}_{\mathsf{tf}}^{Y}}\left\\{\frac{1}{n}\sum_{i=0}^{n-1}g_{y}\left(Y_{i}^{i+l}\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.\left.-\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{y}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right)\right\\}\right|$
$\displaystyle\leq\left|+\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)\right]-\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]\right.$
$\displaystyle\quad\left.+\sup_{g_{y}\in\mathcal{G}_{\mathsf{tf}}^{Y}}\left\\{\frac{1}{n}\sum_{i=0}^{n-1}g_{y}\left(Y_{i}^{i+l}\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.\left.-\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{y}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right)\right\\}\right|.$
(45)
Due to (40), there exists $N\in\mathbb{N}$ such that $\forall n>\mathbb{N}$
$\displaystyle\left|\mathbb{E}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]-\mathbb{E}_{n}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]\right|<\frac{\epsilon}{8},$
(46a)
$\displaystyle\left|\left(\mathbb{E}\left[e^{f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)}\right]\right)-\mathbb{E}_{n}\left[e^{f_{y,l}^{\star}\left(Y^{l-1},\widetilde{Y}_{l}\right)}\right]\right|<\frac{\epsilon}{8}.$
(46b)
Plugging (46) to (45) gives
$\displaystyle\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n})-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}\right|$
$\displaystyle\leq\frac{\epsilon}{4}+\Bigg{|}-\mathbb{E}_{n}\left[e^{f_{y,l}^{\star}(Y^{l-1},\widetilde{Y}_{l})}\right]-\mathbb{E}_{n}\left[f_{y,l}^{\star}\left(Y^{l}\right)\right]$
$\displaystyle\quad+\sup_{g_{y}\in\mathcal{G}_{\mathsf{tf}}^{Y}}\left\\{\frac{1}{n}\sum_{i=0}^{n-1}g_{y}\left(Y_{i}^{i+l}\right)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{y}\left(Y_{i}^{i+l-1},\widetilde{Y}_{i+l}\right)}\right)\right\\}\Bigg{|}.$
(47)
Since the empirical mean of $f_{y,l}^{\star}$ is converging to the expected
mean, it is uniformly bounded by some $M\in\mathbb{R}_{\geq 0}$. Since the
exponent function is Lipschitz continuous with Lipschitz constant $\exp{[M]}$
on the interval $(-\infty,M]$, we obtain
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}e^{f_{y,l}^{\star}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)}-e^{g_{y}\left(\widetilde{Y}_{l},Y_{i}^{i+l-1}\right)}$
$\displaystyle\leq
e^{M}\frac{1}{n}\sum_{i=1}^{n}\Big{|}f_{y,l}^{\star}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-
g_{y}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)\Big{|}.$ (48)
Definition 4 is a sub-functions class of the continuous sequence to sequence
functions class, and applies to the causal transformer. Thus, concludes that
the causal transformer $g\in\mathcal{G}_{\mathsf{ctf}}^{(d_{i},d_{o},l,v)}$
also applies for the universal approximation theorem, for continuous vector-
values functions. Moreover, for our case the output sequence is a scalar
value, $\mathcal{U}\subset\mathbb{R}^{l\times
d_{i}},\mathcal{Z}\subset\mathbb{R}^{1\times d_{o}}$. For given $\epsilon,M,l$
and $n$, denote $g_{y}^{\star}\in\mathcal{G}_{\mathsf{ctf}}^{(d_{y},1,l,v)}$
the Causal transformer, such that the approximation error is uniformly bounded
$\exp{[-M]}\times{\epsilon}/{4}$ for the final time prediction out from the
model. Finally, combining Theorem 1 we have
$\displaystyle\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}(D_{n})-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}\right|$
$\displaystyle\leq\left(1+e^{M}\right)\frac{1}{n}\sum_{i=1}^{n}\Big{|}f_{y,l}^{\star}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad-
g_{y}^{\star}\left(\widetilde{Y}_{i+l},Y_{i}^{i+l-1}\right)\Big{|}+\frac{\epsilon}{4}$
$\displaystyle\leq\frac{\epsilon}{2}.$ (49)
This concludes the proof of (37a). For (37b), note that
$\displaystyle f_{y,l}^{\star}$
$\displaystyle:=\log\left(\frac{dP_{Y^{l}}}{d(P_{X^{l}Y^{l-1}}\otimes\widetilde{P}_{Y_{l}})}\right)$
$\displaystyle=\log p_{Y_{l}|X^{l}Y^{l-1}}-\log\widetilde{p}_{Y_{l}},$ (50)
achieves the supremum. Following the same claims for
$\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n})$
$\left|\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n})-\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}\right|\leq\frac{\epsilon}{2},$
(51)
where
$\displaystyle\widehat{\mathsf{D}}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}(D_{n})$
$\displaystyle:=\sup_{g_{xy}\in\mathcal{G}_{\mathsf{tf}}^{XY}}\frac{1}{n}\sum_{i=0}^{n-1}g_{xy}\left(Y_{i}^{i+l},X_{i}^{i+l}\right)$
$\displaystyle\qquad\qquad-\log\left(\frac{1}{n}\sum_{i=0}^{n-1}e^{g_{xy}\left(Y_{i}^{i+l-1},X_{i}^{i+l},\widetilde{Y}_{i+l}\right)}\right).$
(52)
With (51) and (49) we end the proof. $\blacksquare$
### VII-C Proof of lemma 1
Recall
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}:=\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}}\right),$
(53a)
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}:=\mathsf{D}_{\mathsf{KL}}\left(P_{Y_{l}|Y^{l-1}X^{l}}\|\widetilde{P}_{Y_{l}}\Big{|}P_{Y^{l-1}X^{l}}\right).$
(53b)
By expanding the first term we obtain,
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}$
$\displaystyle=\mathbb{E}_{P_{Y^{l-1}}}\left[\mathsf{D}_{\mathsf{KL}}(P_{Y_{l}|Y^{l-1}}\|P_{\widetilde{Y}_{l}})\right]$
$\displaystyle=\int_{\mathcal{Y}^{l-1}}\Bigg{[}\int_{\mathcal{Y}_{l}}\log\left(\frac{P(y_{l}|y^{l-1})}{\widetilde{P}(y^{l})}\right)$
$\displaystyle\qquad\qquad\qquad
P(y_{l}|y^{l-1})d(y_{l})\Bigg{]}P(y^{l-1})d(y^{l-1})$
$\displaystyle=\int_{\mathcal{Y}^{l}}\log\left(\frac{P(y_{l}|y^{l-1})}{\widetilde{P}(y_{l})}\right)P(y^{l})d(y^{l})$
$\displaystyle=\int_{\mathcal{X}^{l}\mathcal{Y}^{l}}\log\left(\frac{P(y_{l}|y^{l-1})}{\widetilde{P}(y_{l})}\right)P(x^{l}y^{l})d(x^{l}y^{l}),$
(54)
and the second term,
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}$
$\displaystyle=\mathbb{E}_{P_{X^{l}Y^{l-1}}}\left[\mathsf{D}_{\mathsf{KL}}(P_{Y_{l}|X^{l}Y^{l-1}}\|P_{\widetilde{Y}_{l}})\right]$
$\displaystyle=\int_{\mathcal{X}^{l}\mathcal{Y}^{l-1}}\Bigg{[}\int_{\mathcal{Y}_{l}}\log\left(\frac{P(y_{l}|x^{l}y^{l-1})}{\widetilde{P}(y_{l})}\right)$
$\displaystyle\qquad\qquad\quad
P(y_{l}|x^{l}y^{l-1})d(y_{l})\Bigg{]}P(x^{l}y^{l-1})d(x^{l}y^{l-1})$
$\displaystyle=\int_{\mathcal{X}^{l}\mathcal{Y}^{l}}\log\left(\frac{P(y_{l}|x^{l}y^{l-1}}{\widetilde{P}(y_{l})}\right)P(x^{l}y^{l})d(x^{l}y^{l}),$
(55)
where $\widetilde{P}_{Y}$ is a reference density function and is absolutely
continuous on $\mathcal{Y}$. Subtructing the two terms resulting,
$\displaystyle\mathsf{D}_{Y_{l}|Y^{l-1}X^{l}\|\widetilde{Y}_{l}}-\mathsf{D}_{Y_{l}|Y^{l-1}\|\widetilde{Y}_{l}}$
$\displaystyle=\int_{\mathcal{X}^{l}\mathcal{Y}^{l}}\log\left(\frac{P(y_{l}|x^{l}y^{l-1})}{P(y_{l}|y^{l-1}}\right)P(x^{l}y^{l})d(x^{l}y^{l})$
$\displaystyle=h(Y_{l}|Y^{l-1})-h(Y_{l}|X^{l}Y^{l-1})$
$\displaystyle=\mathsf{TE}_{X\to Y}(l).\quad\square$ (56)
### VII-D Proof of lemma 2
The optimal TE, $\mathsf{TE}_{X\to Y}^{\star}(l)$, by non-finite memory
process $\mathbb{X}$, is given by
$\mathsf{TE}_{X\to
Y}^{\star}(l):=\lim_{n\to\infty}\sup_{P_{X^{l}_{-n}}}\mathsf{TE}_{X\to Y}(l).$
(57)
For any $n>0$ we have,
$\displaystyle\sup_{P_{X^{l}_{-n}}}\mathsf{TE}_{X\to Y}(l)$
$\displaystyle=\sup_{P_{X^{0}_{-n}}P_{X^{l}|X^{0}_{-n}}}\mathsf{I}\left(X^{l};Y_{l}|Y^{l-1}\right)$
$\displaystyle=\sup_{P_{X^{0}_{-n}}P_{X^{l}|X^{0}_{-n}}}\int_{\mathcal{X}^{l}_{-n}\mathcal{Y}^{l}}P(x^{0}_{-n})P(x^{l},y^{l}|x^{0}_{-n})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\underbrace{\mathsf{I}\left(X^{l};Y_{l}|Y^{l-1}\right)}_{\text{independent
of }x^{0}_{-n}}d(x^{l}x^{0}_{-n}y^{l})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sup_{P_{X^{l}}}\int_{\mathcal{X}^{l}\mathcal{Y}^{l}}P(x^{l},y^{l})\mathsf{I}\left(X^{l};Y_{l}|Y^{l-1}\right)d(x^{l}y^{l})$
$\displaystyle=\sup_{P_{X^{l}}}\mathsf{TE}_{X\to Y}(l),$ (58)
where (a) follows from the fact that the conditional MI does not depend on
$x^{0}_{-n}$, thus we can eliminate the integral of $\mathcal{X}_{-n}^{0}$
that does not affect the conditional MI, as for the supremum. Since (VII-D) is
true for any $n\in\mathbb{N}$, with (57) we get,
$\mathsf{TE}_{X\to Y}^{\star}(l)=\sup_{P_{X^{l}}}\mathsf{TE}_{X\to
Y}(l).\quad\square$ (59)
### VII-E Implementation Parameters
For the channel capacity estimation model, we trained the optimization
procedure with a limit of 200 epochs. We utilized a batch size of 1024, a
learning rate of $8\times 10^{-3}$, and the Adam optimizer [47]. The
transformer architecture comprises one attention layer followed by a FF layer.
The attention mechanism is implemented with a single head, having a dimension
of 32 neurons. The dimension of the FF layer is 64, featuring ELU activation
[48]. The input sequence length is set to 59, and we back-propagate through
sequences of length 30, corresponding to the parameter $l$ in TE. Each epoch
generates 100K samples of channel input-output tuples, with random noise
uniformly distributed over the bounds of a given batch. The noise distribution
generator (NDG) also employs a transformer structure with one attention layer
and one FF layer, maintaining the same dimensional specifications. The
learning rate for the NDG is set to $8\times 10^{-4}$. It is trained at a rate
of once every four epochs.
For the Apnea dataset feature analysis, the learning rate was adjusted to
$1\times 10^{-4}$. Additionally, we configured the model to use 1 head with a
dimension of 16 for the attention mechanism, and the FF layer dimension was
set to 32, with ELU activation. All experiments were conducted on Nvidia RTX
3090 GPUs.
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|
# Towards a Participatory and Social Justice-Oriented Measure of Human-Robot
Trust
Raj Korpan<EMAIL_ADDRESS>0000-0003-0431-9134 Hunter College,
City University of New York695 Park AveNew YorkNew YorkUSA10065
(2024)
###### Abstract.
Many measures of human-robot trust have proliferated across the HRI research
literature because each attempts to capture the factors that impact trust
despite its many dimensions. None of the previous trust measures, however,
address the systems of inequity and structures of power present in HRI
research or attempt to counteract the systematic biases and potential harms
caused by HRI systems. This position paper proposes a participatory and social
justice-oriented approach for the design and evaluation of a trust measure.
This proposed process would iteratively co-design the trust measure with the
community for whom the HRI system is being created. The process would
prioritize that community’s needs and unique circumstances to produce a trust
measure that accurately reflects the factors that impact their trust in a
robot.
human-robot trust, trust measure, participatory design, social justice
††copyright: rightsretained††doi: ††conference: Workshop on Trust in HRI;
March 11, 2024; Boulder, CO††isbn: ††ccs: Computer systems organization
Robotics††ccs: Human- centered computing Participatory design††ccs: Human-
centered computing Collaborative and social computing design and evaluation
methods††ccs: Human- centered computing HCI design and evaluation
methods††ccs: Social and professional topics User characteristics
## 1\. Introduction
In many Human-Robot Interaction (HRI) studies the most important evaluation
metric is often trust in the robot (Law and Scheutz, 2021) because it is
believed that trust is the channel through which robots will be accepted and
successfully deployed in real-world contexts (Kok and Soh, 2020). A human’s
trust in a robot, however, cannot be directly observed (Kok and Soh, 2020). So
there is no objective way to directly measure trust during interactions
(Schaefer, 2013) 111Although some use certain observable behaviors during an
interaction as a proxy objective measure, such as compliance with the robots
suggestions (Salem et al., 2015) or attention focused on the robot as measured
by eye tracking (Jenkins and Jiang, 2010), these behaviors do not directly
observe a human’s mental state (Kohn et al., 2021).. Furthermore, it has been
suggested that trust is multi-faceted (Knowles et al., 2022; Khalid et al.,
2021), dynamic (Kaplan et al., 2021), subjective (Schaefer, 2016), and
affected by the robot’s physical form (Bernotat et al., 2021; Fischer et al.,
2023) and performance (Hancock et al., 2011). These characteristics (lack of
observability and complexity) have resulted in the proliferation of many
different measures of trust in the HRI community, each focused on different
aspects of trust (Chita-Tegmark et al., 2021; Khavas, 2021). This position
paper argues that the development and evaluation of a reliable measure of
human-robot trust should be participatory and justice-oriented.
## 2\. Prior Work in Measure Development
Measures of trust are often a self-report questionnaire that consists of a
series of Likert-style questions that ask about different aspects of trust
(Schaefer, 2016; Baker et al., 2018). The questions attempt to represent a
“universal” experience of trust that applies in all situations and contexts
(e.g., (Muir, 1989; Jian et al., 2000; Madsen and Gregor, 2000)). For example,
the Merritt trust scale asks participants to respond on a 5-point Likert
rating scale to 6 statements, such as “I trust the [technology]” (Merritt,
2011). In contrast, the HRI Trust Scale uses a 7-point Likert rating scale
with 37 statements across 5 attributes: team configuration, team process,
context, task, and system (Yagoda and Gillan, 2012). The Trust Perception
Scale-HRI (TPS-HRI) consists of 40 questions all proceeded by “What percentage
of the time will this robot…” and participants could select between 0% and
100% in increments of 10% (Schaefer, 2016). The Multi-Dimensional Measure of
Trust (MDMT) measures trust with 20 items across five dimensions: reliability,
competency, ethicality, transparency and benevolence (Ullman and Malle, 2019;
Malle and Ullman, 2021).
The differences in these measures demonstrate the disparate ways in which they
have been developed and evaluated. For example, the Merritt trust scale was
created by the author based on their expertise and then only evaluated to show
that trust was related to but distinct from the propensity to trust machines
or like them (Merritt, 2011). This approach is susceptible to encoding the
individual biases of the author. It was also not validated for its
generalizability.
Yagoda and Gillan’s HRI Trust Scale started with a preliminary list of items,
conducted an exploratory study where 11 HRI subject matter experts (SMEs)
provided feedback on this initial list, generated a list of HRI trust scale
items based on the feedback, conducted an online crowdsourced quality
assessment with 100 participants on Amazon Mechanical Turk (MTurk), and then
used factor analysis to determine the final HRI Trust Scale (Yagoda and
Gillan, 2012). Although this approach is an improvement over an individual’s
independent creation of a scale, it still has several limitations. First, the
HRI SMEs used to provide feedback were required to have at least 5 years of
professional experience and have published contributions in the field of HRI
(Yagoda and Gillan, 2012). The authors do not mention any other demographic
information about these experts, so there is no way to know what biases they
may have introduced in their feedback. Second, the use of these experts
reinforces the structures of power in the robotics community (Williams, 2024).
Reliance on these experts maintains the historical inequities of who robotic
systems are built by and for because the robotics community has often excluded
Black people (Howard and Kennedy III, 2020), women (Graesser et al., 2021),
queer people (OrganizersOfQueerin et al., 2023; Korpan, 2023), and other
underrepresented groups (Tanevska et al., 2023).
The TPS-HRI began with an initial list of 156 items identified from a
literature review of over 700 papers, which contained 86 trust scales, and two
initial experiments to identify physical attributes that affect
trustworthiness (Schaefer, 2016). Each item on this initial list was evaluated
on a 7-point Likert rating scale by 159 undergraduate students and statistical
analysis (e.g., principal component analysis) was used to reduce the number of
items to 73 (Schaefer, 2016). The scale was also changed to use percentage
increments at this point so that participants rate items on a range from no
trust (0%) to complete trust (100%) (Schaefer, 2016). Next, 11 SMEs were
surveyed to evaluate each item’s importance to include in a trust scale and
their feedback resulted in the reduction of items to 42 (Schaefer, 2016).
These SMEs were recruited from the United States Army and Air Force Research
Laboratories and university research laboratories and had 4 to 30 years of
experience in robotics research (Schaefer, 2016). Similar to the HRI Trust
Scale, the use of SMEs can be problematic for the same reasons. Finally, two
studies were conducted with undergraduate students to evaluate the validity of
the 42-item scale to see if it was able to capture the dynamics of trust over
time and whether it actually measures trust (Schaefer, 2016). The results of
these final studies eliminated 2 items from the scale and showed that the
final 40 items successfully measured trust across multiple interactions
(Schaefer, 2016).
Although the final TPS-HRI is validated, it mainly relied on convenience
samples of undergraduate students with only two demographic characteristics
reported: binary gender (male or female) and mean age (for only the first
validation study) (Schaefer, 2016). This is likely not a representative sample
for many of the other applications in which this scale may be applied to
measure trust (Baxter et al., 2016). Another issue with the validation studies
was that it was conducted in simulation in the context of a soldier’s
interaction with a robot (Schaefer, 2016). A scale validated to measure trust
in this scenario may not necessarily work in other tasks or physical
interactions.
The MDMT began with a list of 62 words related to trust collected from
dictionaries, related literature, and other trust scales (Malle and Ullman,
2021). An initial MTurk study evaluated whether each term related more to
capacity trust or personal trust, two dimensions of trust identified from the
human-robot trust literature (Malle and Ullman, 2021). Principal component
analysis and clustering were then used to identify a reduced list of 20 items
but split into 4 dimensions (reliable, capable, sincere, and ethical) (Malle
and Ullman, 2021). A second MTurk study then asked participants to sort an
expanded list of 32 items into the 4 dimensions or an “other” category (Malle
and Ullman, 2021). The results were used to select the top three items for
each dimension and a face-valid item was added for each, which resulted in the
initial scale with 16 items that is evaluated on a 7-point Likert rating scale
(Malle and Ullman, 2021). A MTurk study was then used to validate the
sensitivity of each dimension to respond to a change in trust related to that
dimension and the internal consistency of the items in each dimension (Malle
and Ullman, 2021). This validation, however, was done where participants were
presented with a sentence that described a robot’s behavior to evaluate their
initial trust ratings, and then new information would be given and trust
ratings evaluated again (Malle and Ullman, 2021). Similar to the TPS-HRI, this
approach to validation may not result in a trust measure that reliably works
in actual human-robot interactions (simulated or physical).
Subsequent work revised the MDMT to update the dimensions (reliable,
competent, ethical, transparent, and a fifth dimension, benevolence) for a
total of 20 items, again based on an online study where participants had to
sort and group terms, and then validated it in a similar way as the initial
MDMT (Ullman, 2021; Malle and Ullman, 2023). Although an online crowdsourcing
tool may produce a more representative sample than a set of undergraduate
students, other research has investigated ethical concerns regarding the use
of MTurk (Moss et al., 2023), shown significant levels of inattentiveness
among MTurk workers (Saravanos et al., 2021), and explored the many data
quality concerns with MTurk workers (Hauser et al., 2019). Furthermore, the
representativeness of MTurk samples is also questionable – in the MDMT
validation study, for example, 76.5% of the sample self-reported as “Non-
Hispanic White or Euro-American” (Ullman, 2021) which is certainly not
representative of the world population or the United States population
(Bureau, [n. d.]).
The development of both the TPS-HRI and the MDMT took a different approach to
create their initial list of items compared to the Merritt trust scale and the
HRI Trust Scale – they drew items from the existing literature and trust
scales. Although they did not use SMEs to generate this initial list, the
power structures and norms are still reinforced in this approach because the
previous research that was used was developed by the established robotics
community (Williams, 2024; Tanqueray and Larsson, 2023). The next section
proposes an alternative approach for the development and evaluation of a
measure of human-robot trust that incorporates principles of participatory
design and aligns with a social justice-oriented framework for human-robot
interaction.
## 3\. Proposed Approach
Because robots have the opportunity to shape our society, their designers can
reinforce or disrupt systems of inequity (Šabanović, 2010) and force their
cultural norms on the communities where these technologies are deployed
(Hipólito et al., 2023). Beyond interpersonal power, robots can wield
structural, disciplinary, and cultural power that can reinforce the structures
of White Patriarchy (Williams, 2024). Recent work has shown how discrimination
and bias exist at multiple levels in the robotics and artificial intelligence
research communities (Fosch-Villaronga and Poulsen, 2022; Fosch-Villaronga and
Drukarch, 2023). As a result, there have been several calls for a more
equitable HRI, such as the Feminist HRI framework (Winkle et al., 2023b), the
HRI Equitable Design framework (Ostrowski et al., 2022), and the Robots for
Social Justice framework (Zhu et al., 2024). Inspired by these calls, this
position paper suggests a participatory and social justice-oriented human-
robot trust measure.
### 3.1. Participatory Design
Participatory design is a collaborative process that directly involves the
people affected by technology in its creation (Spinuzzi, 2005). Participatory
design approaches in HRI have been used to design systems that meet the needs
of specific users, such as the elderly (Rogers et al., 2022) or people with
physical or mental disabilities (Lee et al., 2017; Azenkot et al., 2016;
Axelsson et al., 2019; Arevalo Arboleda et al., 2021). Participatory design
has also been successfully used to develop autonomous social robots (Winkle et
al., 2021; Tian et al., 2021; Axelsson et al., 2021). Participatory design,
however, has not been used in the development of trust measures. As described
in Section 2, prior approaches were created based initially on an author’s
expertise, from SMEs input, or a literature review of past trust scales.
Subsequently, some of those were evaluated for their validity with
undergraduate students or online MTurk workers. A participatory approach would
instead engage individuals in the community for whom a robot system is
intended to benefit to help co-design the trust measure.
First, the community members would identify the relevant attributes that could
affect their trust in a robot, which would be used to produce an initial set
of questions. Next, the community members would use the trust measure to
evaluate their trust after a physical interaction task with a robot. A
questionnaire would then ask them to evaluate the appropriateness of each
question (e.g., on a 3-point Likert rating scale “Not necessary”, “Useful, but
not essential”, and “Essential” (Lawshe et al., 1975)). A semi-structured
interview would also be used to analyze the trust measure’s validity – do the
questions allow the participants to express the level of trust they felt
during the interaction? Together, the quantitative and qualitative results
would be used to revise the initial measure to reflect the participants’
feedback. This process would then be iteratively repeated in a cycle with the
participants using the revised measure to evaluate their trust, followed by a
quantitative and qualitative examination of their experience used to further
revise the measure. The participatory design process would stop when the
measure converges to a consistent set of questions that accurately represent
the attributes that affect trust in that community.
Only one work used a similar participatory process but to develop an initial
understanding of trust in the context of robot assistants for elderly care
(Schwaninger et al., 2021). In this work, a card game was designed and used
during interviews to elicit trust-related factors from older adults
(Schwaninger et al., 2021). The card game allowed researchers to ask about
specific trust characteristics but also facilitated open conversation about
trust (Schwaninger et al., 2021). The results showed that this methodology
allowed them to identify the specific dimensions of trust that were relevant
to the community they were investigating, such as privacy, control,
companionship, and skepticism (Schwaninger et al., 2021). As the authors of
that work state, this research represents the initial ideation phase of
participatory design, which corresponds to the initial step of the
participatory process proposed here. They did not, however, use the findings
to produce a trust measure, as proposed here
Although there are potential risks in the use of participatory design, such as
possible physical or mental harm, risk of exploitation, and removal of agency
(Zytko and Louie, 2022), there are also potential benefits, such as bringing
visibility to marginalized people in HRI, addressing social justice issues,
and revealing HRI researchers internal biases (Lee et al., 2022). For example,
a participatory design process that involves queer people could result in a
culturally-competent tool (Fine and Torre, 2019), more inclusive language
(Meyer and Elias, 2022), center their experiences (Korpan, 2023; Stolp-Smith
and Williams, 2024), and ultimately produce a technology that they trust more
(Haimson et al., 2020, 2023).
The use of participatory design would also support a recent recommendation to
center the human experience of trust in the development of a framework for
human-robot trust (Lange et al., 2023). It could allow for the development of
ad hoc trust measures that address other circumstances that impact trust:
repeated interactions, imperfect interactions, subconscious influences on
perception, conformity to social norms, and environmental factors (Holthaus
and Rossi, 2023). It could also produce context-based questions that address
different individual identity factors that affect trust (Korpan, 2023) and how
those relate to aspects of the robot, task, environment, and other agents
(Holthaus and Rossi, 2023). It could also potentially weigh the importance of
trust items differently based on community differences (Korpan, 2023).
### 3.2. Social Justice Framework
Robotics researchers’ choices reflect the values and beliefs of society, both
the good and bad (Michalec et al., 2021). For example, the deployment of
language-capable robots poses several potential risks, such as overtrust being
manipulated to influence human morals, reinforcement of gendered and
racialized biases, and perpetuation of harmful assumptions about human
identity (Williams et al., 2023). HRI studies also often fail to broadly
address diversity and inclusion in both the human subjects used and the
systems created (Seaborn, 2023; Winkle et al., 2023a). A systematic review of
HRI research from 2006 to 2022 found that over 90% of papers used “WEIRD”
(Western, Educated, Industrial, Rich, and Democratic) participants (Seaborn et
al., 2023). Furthermore, a robot’s stereotypical (gendered) appearance have
been shown to activate human stereotypes, which subsequently affects people’s
behavior with those robots and their trust in them (Perugia et al., 2023;
Perugia and Lisy, 2023). As a result of these pervasive challenges in how HRI
research is conducted, robots are designed and deployed, and who they are
evaluated with, a social justice-oriented approach could result in a more
equitable HRI (Zhu et al., 2024).
The Robots for Social Justice framework recommends five considerations for HRI
researchers: what the community the research is intended to benefit, what
human capabilities would be enhanced, how those capabilities are valued and
prioritized, what structural conditions are present in that community, and
what are the power structures that surround and influence the community (Zhu
et al., 2024). A social justice-oriented process for the development of a
human-robot trust measure should then be grounded in the context in which it
will be used. It should begin with a specification of the community for which
trust will be measured, consider what human capabilities are being enhanced by
the robot, and prioritize the needs of that community as defined by them. As
the community is engaged in this development process, the structural and
systematic limitations and the power structures that exist in the community
should be critically examined. This would ensure that the developed trust
measure considers the potential risks and harms that could result if it fails
to capture important trust attributes for that community, given its unique
circumstances. A participatory design process could easily integrate with this
social justice framework to produce a measure of trust that addresses the
social and ethical challenges of HRI research.
## 4\. Conclusion
Human trust in a robot is a difficult construct to measure because of its
complexity. Many different measurement tools have been created and validated
in a variety of ways. None of the past approaches, however, addressed the
power structures or systems of inequity that HRI research can reinforce. This
position paper proposes the use of a participatory design process in the
context of a social justice framework would result in a validated trust
measure that captures the relevant factors that affect trust for a community.
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Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.