TITLE: Matrix Notation: <mat>, what is it useful for?
QUESTION [3 upvotes]: I have a 2 dimensional squared matrix: $Im \in \mathbb{R}^{c \times c}$ with $c$ a fixed integer.
What is the difference with refering to $\langle Im\rangle$ and just $Im$ ?
I found this notation in a research paper. It makes sense when refering to the following situation $\langle x, y\rangle = \sum\limits_{i=1}^{n}{x_i y_i}$ , but what's the benefit to use it when there is only one matrix inside the notation ?
Research Paper cited (Page 4): https://arxiv.org/abs/1703.05921
Extract of the article with the target of my question in bold:

We are given a set of $M$ medical images $\mathbf{I}_m$ showing healthy anatomy, with $m=1,2,\dots ,M$, where $\mathbf{I}_m \in \mathbb{R}^{a \times b}$ is an intensity image of size $a \times b$. From each image $\mathbf{I}_m$, we extract $K$ 2D image patches $x_{k,m}$ of size $c \times c$ from randomly sampled positions resulting in data $\mathbf{x}=x_{k,m} \in \mathcal{X}$, with $k=1,2,\dots ,K$. During training we are only given $\langle \mathbf{I}_m \rangle$ and train a generative adversarial model to learn the manifold $\mathcal{X}$ (blue region in Figure~2(b)), which represents the variability of the training images, in an unsupervised fashion. For testing, we are given $\langle \mathbf{y}_n, l_n\rangle$, where $\mathbf{y}_n$ are unseen images of size $c \times c$ extracted from new testing data $\mathbf{J}$ and $l_n \in \{0,1\}$ is an array of binary image-wise ground-truth labels, with $n=1,2,\dots ,N$. These labels are only given during testing, to evaluate the anomaly detection performance based on a given pathology.

Secondary question: Does it have any sense to talk about $\langle \mathbf{y}_n, l_n\rangle$ in this scenario ? Because it can't represent an inner-product ...

REPLY [1 votes]: This indeed has nothing to do with inner products.
It seems to me that they are using $\langle \mathbf{I}_m \rangle$ for the set
$$
\{ \mathbf{I}_1,\dots,\mathbf{I}_M \}
,
$$
or maybe more likely the vector (or $M$-tuple)
$$
( \mathbf{I}_1,\dots,\mathbf{I}_M )
,
$$
which might perhaps also be written as
$$
\langle \mathbf{I}_1,\dots,\mathbf{I}_M \rangle
$$
by some people.
And $\langle \mathbf{y}_n, l_n \rangle$ would then be $N$-tuple of pairs
$$
\bigl( (\mathbf{y}_1, l_1),\dots,(\mathbf{y}_N, l_N) \bigr)
.
$$
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