\subsection{Problem Statement Reformulation and Convergence}

As described in \cref{sec:gen_prob} we aim to construct $m$-periodic exploration sequences $u_\ell$ and $v_\ell$ and generating functions $f$ and $g$ such that \eqref{eq:algo} with transition map \eqref{eq:TM} yields \eqref{eq:grad_descent}. 
Our first result restates the problem in terms of solving a system of nonlinear equations.

\begin{theorem}
	\label{thm:cond_equ}
     Let \ref{assump:smooth_fg} hold. Then the $m$-th step of the evolution of \eqref{eq:algo} with transition map \eqref{eq:TM} is given by
     \begin{align}
	x_{k+m} &= x_k + \sqrt{h}(\alpha_1+\alpha_2)Y(f(J(x_k)),g(J(x_k)))W\mathds{1}  \nonumber \\
	&+ h \tilde{Y}(f(J(x_k)),g(J(x_k))) T(W) \nonumber \\
	&\times Y(f(J(x_k)),g(J(x_k)))^\top \nabla J(x_k) + \mathcal{O}(h^{3/2}) .
	\label{eq:taylor_matrix}
	\end{align}
	Here, $W = [w_k\ w_{k+1}\ \cdots \ w_{k+m-1}]\in\mathbb{R}^{2n\times m}$ with $w_i = [u_i^\top\ v_i^\top]^\top$ is the exploration sequence matrix and $T(W)\in\mathbb{R}^{2n\times 2n}$
	is given by
	\begin{align}
	&T(W) := \sum_{i=0}^{m-1}  \left( \alpha_2 w_iw_i^\top + (\alpha_1 + \alpha_2)^2 \sum_{j=0}^{i-1} w_iw_j^\top \right).
	
	\label{eq:cond_equ_gen} 
	\end{align} 
	Furthermore, $Y(f(z),g(z)):= [f(z)I \ \ g(z)I]\in \mathbb{R}^{n \times 2n}$ and $\tilde{Y}(f(z),g(z)) := [\frac{\partial f}{\partial z}(z)I \ \ \frac{\partial g}{\partial z}(z) I]= \frac{\partial}{\partial z}Y(f(z),g(z))\in \mathbb{R}^{n \times 2n}$.
	
\end{theorem} 

The proof of \cref{thm:cond_equ} is given in \cref{sec:proof_thm_cond_equ}. 

If there exist $m$-periodic exploration sequences $\{w_\ell\}_{\ell=0}^{m-1}$ (equivalently, an exploration sequence matrix $W$), and generating functions $f$ and $g$ such that 

\begin{align}
	\tilde{Y}(f(z),g(z)) T(W) Y(f(z),g(z))^\top &= -I,\ \forall z \in \mathbb{R}
	\label{eq:problem}
	\\
	 W\mathbb{1} &= 0
	\label{eq:cond_equ_gen_zero}
\end{align}

are satisfied, then \eqref{eq:grad_descent} holds.
Thus, this system of nonlinear ordinary differential equations (w.r.t. $f(z)$ or $g(z)$) with unknown coefficients is key in designing the algorithm.

The idea to solve this highly under-determined system of equations is now to proceed in two steps:
\begin{enumerate}[leftmargin=36pt, label=Step \arabic*)]
	\item For a class of normal (skew-symmetric) matrices $T_d$,
	we construct exploration sequence matrices $W$ such that 
	\eqref{eq:cond_equ_gen_zero} and
	$T(W)=T_d$ hold.
    \label{step1}
	\item We characterize gradient generating functions $f,g$ and normal (skew-symmetric) matrices $T_d$ 	such that \eqref{eq:problem} holds.
	\label{step2}
\end{enumerate}

These two constructions are presented in the following two subsections. We start with a remark on $T(W)$ and the convergence result of the proposed algorithms.

\begin{rem}
    \label{rem:T}
	To get a sense of equation	\eqref{eq:problem} and the role of $T(W)$, partition $T(W)$ as 
	
	\begin{align}
		T(W) = \begin{bmatrix}
			T_{11}(W) & T_{12}(W) \\ T_{21}(W) & T_{22}(W)
		\end{bmatrix},
		\label{eq:structure_T}
	\end{align}
	
	with $T_{11}(W),T_{12}(W),T_{21}(W), T_{22}(W) \in \mathbb{R}^{n\times n}$. Note that $T_{11}(W)$ is defined solely by $\{u_\ell\}_{\ell = 0}^{m-1}$, $T_{22}(W)$ solely by $\{v_\ell\}_{\ell = 0}^{m-1}$, and $T_{12}(W)$ and $T_{21}(W)$ by both $\{u_\ell\}_{\ell = 0}^{m-1}$ and $\{v_\ell\}_{\ell = 0}^{m-1}$. Then \eqref{eq:problem} with \eqref{eq:structure_T} yields
	
	\begin{align}
		\frac{\partial f}{\partial J}fT_{11} + \frac{\partial f}{\partial J}gT_{12} + \frac{\partial g}{\partial J}fT_{21} + \frac{\partial g}{\partial J}gT_{22} = -I
		\label{eq:problem_w_structure}
	\end{align}
	
	where the arguments $J(x_0)$ of the maps $f$ and $g$ and their derivatives and $W$ of $T_{ij}$ with $i,j\in \{1,2\}$ are omitted for the sake of readability. By plugging $\{u_\ell\}_{\ell = 0}^{m-1}$ and $\{v_\ell\}_{\ell = 0}^{m-1}$ from \eqref{eq:input_coord}, with $W_1$ and $W_2$ for $[\alpha_1\ \alpha_2] = [1\ 0]$ and $[\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}]$, respectively, into \eqref{eq:cond_equ_gen}, one obtains
	
	\begin{align}
		T(W_1) = \begin{bmatrix}	-I & -I \\ I & -I \end{bmatrix},
		\ \ \text{and} \ \
		T(W_2) = \begin{bmatrix} 0 & -I \\ I & 0 \end{bmatrix}.
		\label{eq:T_examp}
	\end{align} 
	
	Hence, the left hand side of \eqref{eq:problem_w_structure} translates into the terms in the curly brackets in \eqref{eq:taylor_E} and \eqref{eq:taylor_H}, respectively. A geometric interpretation of T(W) is discussed in \Cref{sec:numerical_results}.
	
\end{rem}


Due to 	property \eqref{eq:grad_descent}, for example, semi-global practical asymptotic convergence to the optimizer $x^*$ can be established \correct{(see \cite{moreau2000practical})}:

\begin{theorem}
	\label{thm:conv}
	
	Let \ref{assump:smooth_fg} and \ref{assump:smooth_J} hold. Assume that there exist generating functions $f(J(x))$ and $g(J(x))$ and an exploration sequence matrix $W$ such that \eqref{eq:problem} and \eqref{eq:cond_equ_gen_zero} are satisfied. Then, for all $\delta_1,\delta_2 \in \mathbb{R}_{>0}$ with $\delta_2 < \delta_1$, there exist an $h^* \in \mathbb{R}_{>0}$ and $N(h) \in \mathbb{N}$, such that for all $h\in \{\bar{h}\ |\ 0 < \bar{h} < h^*\}$ and $x_0 \in \mathcal{U}_{x^*}^{\delta_1}$, it holds $x_k \in \mathcal{U}_{x^*}^{\delta_2}$ for all $k\ge N(h)$.
	
\end{theorem}

The proof of \cref{thm:conv} follows along the lines of the proof of \cite[Theorem 2]{jfDerFree} by utilizing \cref{thm:taylor_gen} in \cref{sec:pre_lem} and \cref{thm:cond_equ}.


\begin{rem}
    \label{rem:var_h}
    \Cref{thm:conv} is based on a constant step size $h$. Applying a variable decreasing step size $h_k$, but constant over a period of length $m$, i.e., $h_0=h_{1}=\cdots=h_{m-1},\ h_{m}=h_{m+1}=\cdots=h_{2m-1},\ \ldots,$ with 
    
    \begin{align}
        \sum_{p=0}^{\infty} h_{pm} = \infty, \quad  \sum_{p=0}^{\infty} h_{pm}^2 < \infty,\quad 
    \end{align}
    e.g. $h_k = \nicefrac{1}{(\lfloor k/m\rfloor + 1)}$ (cf. Proposition 1 in \cite{spall1992multivariate}) lead to a semi-global asymptotic convergence result and a potential numerical performance improvement. 
    
    Note that the requirement of periodically $m$ constant steps preserves
    the $\mathcal{O}(\sqrt{h})$-order terms in \eqref{eq:taylor_matrix} (cf. proof of \Cref{thm:taylor_gen}).
\end{rem}


\subsection{Exploration Sequences}

In this part we characterize the conditions under which there exists an exploration sequence matrix $W$ for a given $T_d$ such that $T(W) = T_d$ together with $W\mathbb{1}=0$ are satisfied, hence, addressing \ref{step1} as stated above. The next lemma represents
$T(W)$, i.e. \eqref{eq:cond_equ_gen}, in combination with \eqref{eq:cond_equ_gen_zero} in
a more compact form. 


\begin{lem}
	
	\label{thm:matrices}
	
	Consider \eqref{eq:cond_equ_gen} and suppose
	the exploration sequence matrix $W\in \mathbb{R}^{2n\times m}$ satisfies
	\eqref{eq:cond_equ_gen_zero}. Then $T(W)$ can be expressed as
	
	\begin{align}
	T(W) = W P W^\top ,	\label{eq:w_mat_cond}
	\end{align}
	
	with $P \in \mathbb{R}^{m\times m}$ defined as
	
	\begin{align}
	P =
	\begin{bmatrix}
		c_1\! & \! c_2 \! &\! \cdots \!& \!c_2\! &\! 0 \\
		\alpha_2\! & \!\ddots\! & \!\ddots\! & \!\vdots\! & \!\vdots \\
		\vdots\! & \!\ddots\! & \!\ddots & \!c_2\! & \!\vdots  \\
		\alpha_2\! & \!\cdots\! & \!\alpha_2\! & \! c_1\! & \!0 \\
		0\! & \!\cdots\! & \!\cdots\! & \!0\! & \!0
	\end{bmatrix}
	\label{eq:def_mat}
	\end{align}
	where $c_1 = 2\alpha_2 - (\alpha_1+\alpha_2)^2$, $c_2 = \alpha_2-(\alpha_1+\alpha_2)^2$, and $\alpha_1,\alpha_2$ defined in \eqref{eq:TM}.
	
	
\end{lem}

The proof of \cref{thm:matrices} is given in \cref{sec:proof_matrices}. 
Consequently, when proceeding according to \ref{step1} and \ref{step2} described in the section above, the key equations for the design of the exploration sequences are 
\begin{align}
	\begin{split}
		WPW^\top &=  T_d \\
		W \allone{} &= 0.
	\end{split}
    \label{eq:svd_prob}
\end{align}


The following theorem, which provides a constructive design of the exploration sequence matrix $W$, is of central importance. It also provides structural insight in terms of obtaining lower bounds on the length (period) of the exploration sequence $m$, suitable choices of the parameters $\alpha_1$ and $\alpha_2$ and admissible structures for 
the desired target matrices $T_d$. 


\begin{theorem}
	
	\label{thm:existence_svd}
	
	Given $\alpha_1, \alpha_2$ and $T_d \in \mathbb{R}^{2n\times 2n}$.
	Suppose that either $T_d$ is normal, $(2\alpha_2-(\alpha_1+\alpha_2)^2) (T_d+T_d^\top)$ positive definite, and \cref{property:interlacing} (see below) is satisfied or that  $T_d$ is skew-symmetric with $2\alpha_2-(\alpha_1+\alpha_2)^2 = 0$. 
	Then there exists a $m\ge \text{rk}(T_d)+1$ such that $W\in\mathbb{R}^{2n\times m}$ satisfies the system of equations \eqref{eq:svd_prob}.
	
\end{theorem}


The role of \Cref{property:interlacing} in \Cref{thm:existence_svd} is discussed in \Cref{rem:pm}.

\begin{property}
\label{property:interlacing}

Let the skew-symmetric matrix $C(m) \in \mathbb{R}^{m\times m}$ be defined as 

\begin{align}
\begin{split}
    C(m) &:= A(m) + \epsilon(m+1) B(m)
\end{split}
\label{eq:pm}
\end{align}


\begin{align}
    A(m) &:= 
    \begin{bmatrix}
    0 & 1 & \cdots & 1 \\
    -1 & \ddots & \ddots & \vdots \\
    \vdots & \ddots & \ddots & 1 \\
    -1 & \cdots & \!\!\!\!-1 & 0
    \end{bmatrix} \in \mathbb{R}^{m \times m}, \\

	\begin{split}
	    B(m) &:=
	    \allone{} [0~2~4~\ldots~2(m-1)]\\
	    &-(\allone{} [0~2~4~\ldots~2(m-1)])^T\in \mathbb{R}^{m \times m} 
	\end{split}    
\end{align}
with 
$\epsilon(m) = (m-1)^{-1}(1-m^{-1/2})$.

Then for any $m \ge 2$, $C(m)$ and $C(m+1)$ satisfy the eigenvalue interlacing property (cf. \Cref{thm:fan_pall} in \Cref{sec:pre_lem}). 

\begin{align}
    \omega_{k}^{m+1}>\omega_k^m > \omega_{k+1}^{m+1}\ge 0,
    \label{eq:interlacing}
\end{align}

for $k = 1,\ldots,\lfloor m/2\rfloor$, where $\pm \omega_k^m i$ are the eigenvalues of $C(m)$, with $\{\omega^m_k\}_{k=1\ldots m}$ sorted in non-decreasing order \corrected{in $k$ for $m$ fixed}.
\end{property}

\begin{rem}
\label{rem:pm}
We verified numerically that \Cref{property:interlacing}
is always true (we verified it up
to dimension $m=10000$, see \Cref{thm:interlacing_P}), but a proof is still lacking.
Further notice, that $C(m)$ is part of the following equation:

\begin{align}
    \tilde{P}(m) &= 
    \big(P-\epsilon(m)(\mathds{1}\mathds{1}\!^\top\! P + P\mathds{1}\mathds{1}\!^\top\!) 
    \nonumber \\
    &+\epsilon^2(m) \mathds{1}\mathds{1}\!^\top\! P \mathds{1}\mathds{1}\!^\top\! \big)_{1:m-1} 
    \nonumber \\
    & = \Big( \frac{1}{2}(\alpha_1 + \alpha_2)^2-\alpha_2\Big) I 
    \nonumber \\
    &+\frac{1}{2}(\alpha_1+\alpha_2)^2 C(m-1)
    \label{eq:P_tilde}
\end{align}

with $P$ in \eqref{eq:def_mat} and $\epsilon(m)$ defined in \Cref{property:interlacing}.  Note that the interlacing property \eqref{eq:interlacing} holds also for $\tilde{P}(m)$, since it is arranged by a scaled unit matrix and the skew-symmetric matrix $C(m-1)$ (cf. \cite{connes1998noncommutative}). The 
interlacing property of  $\tilde{P}(m)$
is utilized in the proof of \cref{thm:existence_svd}.

\end{rem}

The proof of \cref{thm:existence_svd} is constructive and presented in \cref{sec:proof_existence_svd}, where in particular in  \eqref{pf:2_cond_normal}, \Cref{property:interlacing} enters. A step-by-step construction of $W$ for a
given $T_d$ is provided in \cref{sec:construction_svd}.
Moreover we obtain as a corollary (which follows by the proof of \cref{thm:existence_svd}):
\begin{corollary}
	
	\label{thm:W_sym}
	
	If $2\alpha_2-(\alpha_1+\alpha_2)^2=0$ and $T_d$ skew-symmetric, then there always exists an $W\in\mathbb{R}^{2n\times m}$ with $m=\text{rk}(T_d)+1$. 
	
\end{corollary}


\begin{rem}
	\label{rem:brockett_int2}
	It is worthwhile to point out an interesting connection between the equations in \eqref{eq:svd_prob} 
and nonlinear control theory, i.e. the controllability of the so-called nonholonomic integrator.
	Suppose $\{w_\ell\}_{\ell = 0}^{m-1}$ is a solution of \eqref{eq:svd_prob}, then 
	it can be verified by direct calculations (see also proof of \cref{thm:matrices} in \cref{sec:proof_matrices}) that it is
	also a solution of the two point boundary value problem 
	\begin{align}
		\begin{split}
		y_0 &= 0,\ \ Z_0 = 0,\ \ y_m = 0,\ \ Z_{m} = T_d\\
			y_{k+1} &= y_k+w_k
			\\
			Z_{k+1} &= Z_k + (\alpha_1+\alpha_2)^2w_k y_k^\top + \alpha_2w_kw_k^\top 
		\end{split}
		\label{eq:brockett_gen2}
	\end{align}
	with $k = 0,\ldots,m-1$, states $y_k \in \mathbb{R}^{2n}$, $Z_k \in \mathbb{R}^{2n \times 2n}$, input $w_k \in \mathbb{R}^{2n}$, and vice versa.
	
	In particular with $W \allone{} = 0$, i.e., $w_{m-1} = -\sum_{i=0}^{m-2}w_i$, \eqref{eq:brockett_gen2} translates into
	
	\begin{align}
		\begin{split}
		y_0 &= 0,\ \ Z_0 = 0,\ \ Z_{m-1} = T_d\\
			y_{k+1} &= y_k+w_k\\
			Z_{k+1} &= Z_k + \alpha_2w_ky_k^\top+(\alpha_2-(\alpha_1+\alpha_2)^2)y_kw_k^\top\\
			&+ (2\alpha_2-(\alpha_1+\alpha_2)^2)w_kw_k^\top 
		\end{split}.
		\label{eq:brockett_gen}
	\end{align}
	
	
	
	Considering now the case $[\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}]$
	shows that \eqref{eq:brockett_gen} is the state-transition of the generalized discrete-time nonholonomic integrator \cite{altafini2016nonintegrable} with given initial and final states. Problem \eqref{eq:brockett_gen2} with $[\alpha_1\ \alpha_2] = [1\ 0]$ has a  similar structure.
	Hence, \Cref{thm:existence_svd} provides an explicit solution to this 
	state transition problem. Moreover, this viewpoint underlines the relationship to non-commutative maps and flows
	as indicated in \Cref{sec:existing_results} (\Cref{fig:ncm}).
	
	Another, more geometric, interpretation of \eqref{eq:svd_prob}
	is also provided in \Cref{sec:numerical_results}.
	
\end{rem}


\subsection{Gradient Generating Functions}

This part addresses \ref{step2}, i.e. solving the (functional)  equation \eqref{eq:problem}
for $f,g$, and \corrected{$T_d \in \mathbb{R}^{2n \times 2n}$} with $T(W)=T_d$. 

First, solutions $(T_d,f,g)$ for the parameter setting $2\alpha_2 - (\alpha_1+\alpha_2)^2 = 0$ are presented. 

\begin{theorem}
	\label{thm:structure_W_H}
    Let $2\alpha_2 -(\alpha_1+\alpha_2)^2=0$ and $T_d$ skew-symmetric, then \eqref{eq:problem} is satisfied by the following triples $(T_d,f,g)$, where $a,b\in\mathbb{R}_{>0}$ and $c, \phi \in \mathbb{R}$:
    
	\begin{flalign}
		\bullet \quad	&T_d  = \begin{bmatrix}
				0 & -I \\ I & 0
			\end{bmatrix},
			\nonumber \\
			&g(z) = -f(z) \int f(z)^{-2} dz,\ f:\mathbb{R}\rightarrow\mathbb{R}
			\label{eq:cases_H1}
			&&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
		\bullet \quad &T_d  = \begin{bmatrix}
				aQ & -I \\ I & b Q
			\end{bmatrix},\ Q = -Q^\top, 
			\nonumber\\
			&f(z) = a^{-\nicefrac{1}{2}}\sin\Big(\sqrt{ab}z+\phi\Big),
			\nonumber \\ 
			&g(z) = b^{-\nicefrac{1}{2}}\cos\Big(\sqrt{ab}z+\phi\Big)
			\label{eq:cases_H2}
			 &&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
		\bullet \quad &T_d  = \begin{bmatrix}
				aQ & -I \\ I & -b Q
			\end{bmatrix},\ Q = -Q^\top, 
			\nonumber\\
			&f(z) = \pm a^{-\nicefrac{1}{2}}\cosh\Big(\sqrt{ab}z+\phi\Big),
			\nonumber \\ 
			&g(z) = \mp b^{-\nicefrac{1}{2}}\sinh\Big(\sqrt{ab}z+\phi\Big)
			\label{eq:cases_H3}
			 &&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
		\bullet \quad	& T_d  = \begin{bmatrix}
				Q & -I \\ I & 0
			\end{bmatrix},\  Q = -Q^\top, 
			\nonumber \\ 
			&f(z) = \pm \sqrt{a},\ g(z) = \mp \frac{z}{\sqrt{a}}
			\label{eq:cases_H4}
			&&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
		\bullet \quad	& T_d  = \begin{bmatrix}
				0 & -I \\ I & Q
			\end{bmatrix},\  Q = -Q^\top, 
			\nonumber \\ 
			&f(z) = \pm \frac{z}{\sqrt{a}},\ g(z) = \pm \sqrt{a}, 
			\label{eq:cases_H5}
			&&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
		\bullet \quad &T_d  = \begin{bmatrix}
		0 & -I-Q \\ I-Q & 0
		\end{bmatrix},\ Q = -Q^\top, 
		\nonumber \\ 
		&f(z) = \pm\frac{1}{\sqrt{a}}e^{-\frac{a}{2}z},
		\ g(z) = \mp \frac{1}{\sqrt{a}}e^{\frac{a}{2}z},
		\label{eq:cases_H6}
		&&
	\end{flalign}
	\vspace*{-1em}
	\begin{flalign}
		\bullet \quad &T_d  = \begin{bmatrix}
		\corrected{a}Q & -I-cQ \\ I-cQ & \corrected{b}Q
		\end{bmatrix},\ Q = -Q^\top, 
		\nonumber \\ 
		&f(z) = \sqrt{\frac{b}{ab-c^2}}\sin\Big(\sqrt{ab-c^2}z+\phi\Big),
		\nonumber \\
		&g(z) = b^{-\nicefrac{1}{2}}\cos\Big(\sqrt{ab-c^2}z+\phi\Big)
		\label{eq:cases_H7}
		&&
	\end{flalign}
	
	In addition, for each $T_d$ in \eqref{eq:cases_H1}-\eqref{eq:cases_H7} there exists an $W$, such that $T(W)=T_d$ in \eqref{eq:cond_equ_gen}.
	In~\eqref{eq:cases_H7}, we require that $a,b>c$. 
	
	
\end{theorem}

\begin{rem}
    Every pair $f,g$ in \eqref{eq:cases_H2}-\eqref{eq:cases_H7} satisfy \eqref{eq:cases_H1}, hence, these generating functions are valid for the given $T_d$ in \eqref{eq:cases_H1}, too. The advantage of the specified $T_d$'s are discussed in \Cref{sec:numerical_results}. 
    
\end{rem}

\begin{rem}
    Consider the indefinite integral in \eqref{eq:cases_H1}. Let $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be an anti-derivative of $f(z)^{-2}$. Then so is $F+\bar c$ for any $\bar c \in \mathbb{R}$. Set $g(z) = -f(z)(F(z)+\bar{c}).$ The constant  $\bar{c}$ is chosen such that $g'(z)f(z)-f'(z)g(z)=-1$.
    
\end{rem}

The proof of \cref{thm:structure_W_H} is given in \cref{sec:p_sructure_W_H}.


Solutions $(T_d,f,g)$ of \eqref{eq:problem} for the parameter setting $2\alpha_2 - (\alpha_1+\alpha_2)^2 \neq 0$ are presented next. 

\begin{theorem}
	\label{thm:structure_W_E}
	Let \corrected{$T_d \in \mathbb{R}^{2n \times 2n}$} be normal and $(2\alpha_2 -(\alpha_1+\alpha_2)^2) (T_d+T_d^\top)$ be positive definite, then \eqref{eq:problem} is satisfied by the following triples $(T_d,f,g)$, where $r:\mathbb{R}\rightarrow\mathbb{R}_{>0}$, $a\in\mathbb{R}\backslash\{0\}$, $b\in\mathbb{R}_{>0}$, and $\phi \in \mathbb{R}$:
	\begin{flalign}
			\bullet \ \	&T_d  = \begin{bmatrix}
				a I & -I \\ I & a I
			\end{bmatrix},\ a(2\alpha_2 - (\alpha_1+\alpha_2)^2)>0
			\nonumber \\
			&\! f(z) \!=\! \sqrt{r(z)}\sin(\varphi(z)),\ g(z)\! =\! \sqrt{r(z)}\cos(\varphi(z)), 
			\nonumber\\
			&\varphi(z) = \frac{a}{2}\ln(r(z))+\int \frac{1}{r(z)}dz+\phi 
			\label{eq:cases_E1}
			&&
	\end{flalign}
	\vspace*{-1em}
	
	\begin{flalign}
			\bullet \quad &T_d  = \begin{bmatrix}
				Q & -I \\ I & Q
			\end{bmatrix},
			\nonumber \\
			& (2\alpha_2 - (\alpha_1+\alpha_2)^2)(Q+Q^\top) \text{ pos. def. and normal}, 
			\nonumber \\
			&f(z) = b^{-\nicefrac{1}{2}}\sin\Big(bz+\phi\Big),
			\nonumber \\ 
			&g(z) = b^{-\nicefrac{1}{2}}\cos\Big(bz+\phi\Big)
			\label{eq:cases_E2}
			&&
	\end{flalign}
	
In addition for every $T_d$ in \eqref{eq:cases_E1}-\eqref{eq:cases_E2} there exists an $W$, such that $T(W)=T_d$ in \eqref{eq:cond_equ_gen}. 

\end{theorem}

The proof of \cref{thm:structure_W_E} is given in \cref{sec:p_sructure_W_E}.

\begin{rem}
    The list of triples $(T_d,f,g)$ in \cref{thm:structure_W_H} is essentially exhaustive, save for some scaled version of the presented cases. A case by case study is presented in the proof of \Cref{thm:structure_W_H} in \Cref{sec:p_sructure_W_H}. Whereas the list of triples $(T_d,f,g)$  in \cref{thm:structure_W_E} is not exhaustive (cf. \Cref{sec:p_sructure_W_E}).
    
\end{rem}


\cref{thm:structure_W_H} and \cref{thm:structure_W_E} together with \cref{thm:existence_svd} solve \eqref{eq:problem} and \eqref{eq:cond_equ_gen_zero} and thus ensure the existence of a exploration sequence $W$. Hence, a gradient descent step is approximated by the proposed algorithm \eqref{eq:algo} with transition maps \eqref{eq:TM}.
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