What is multilinear principal components analysis?

Here are some exerpts from Lu, Plataniotis and Venetsanopoulos 2006.

Let $\{ \mathcal{A}_m, m=1,\cdots, M \}$ be a set of $M$ tensor samples in $\mathbb{R}^{I_1} \otimes \mathbb{R}^{I_2} \otimes \cdots \mathbb{R}^{I_N}$. The total scatter of these tensors is defined as: $\Psi_{\mathcal{A}} = \sum_{m=1}^M \vert| \mathcal{A}_m - \bar{\mathcal{A}} \vert|_F^2$, where $\bar{\mathcal{A}}$ is the mean tensor calculated as $\bar{\mathcal{A}} = \frac{1}{M} \sum_{m=1}^M \mathcal{A}_m$. The $n$-mode total scatter matrix of these samples is then defined as: $\mathbf{S}_{T_{\mathcal{A}}}^{(n)} = \sum_{m=1}^M (\mathbf{A}_{m(n)} - \bar{\mathbf{A}_{(n)}}) (\mathbf{A}_{m(n)} - \bar{\mathbf{A}_{(n)}})^T$, where $\mathbf{A}_{m(n)}$ is the $n$-node unfolded matrix of $\mathcal{A}_m$. The statement above leads to the following formal definition of the problem to be solved:

A set of $M$ tensor objects $\{ \chi_1, \chi_2, \cdots, \chi_M \}$ is available for training. Each tensor object $\chi_m \in \mathbb{R}^{I_1 \times I_2 \times \cdots \times I_N}$ assumes values in a tensor space $\mathbb{R}^{I_1} \otimes \mathbb{R}^{I_2} \otimes \cdots \mathbb{R}^{I_N}$, where $I_n$ is the $n$-mode dimension of the tensor. The MPCA objective is to define a multilinear transformation $\{ \tilde{\mathbf{U}}^{(n)} \in \mathbb{R}^{I_n \times P_n, n = 1, \cdots, N} \}$ that maps the original tensor space $\mathbb{R}^{I_1} \otimes \mathbb{R}^{I_2} \otimes \cdots \mathbb{R}^{I_N}$ into a tensor subspace $\mathbb{R}^{P_1} \otimes \mathbb{R}^{P_2} \otimes \cdots \mathbb{R}^{P_N}$ (with $P_n < I_n$, for $n=1,\cdots,N$): $\mathcal{Y}_m = \chi_m \times _1\tilde{\mathbf{U}}^{(1)^T} \times _2\tilde{\mathbf{U}}^{(2)^T} \cdots \times _N \tilde{\mathbf{U}}^{(N)^T}$, $m=1, \cdots, M$, such that $\{ \mathcal{Y}_m \in \mathbb{R}^{P_1} \otimes \mathbb{R}^{P_2} \otimes \cdots \mathbb{R}^{P_N}, m=1, \cdots, M \}$ captures most of the variation observed in the original tensor objects, assuming that these variation are measured by the total scatter.

In other words, the MPCA objective is the determination of the $N$ projection matrices $\{ \tilde{\mathbf{U}}^{(n)} \in \mathbb{R}^{I_n \times P_n}, n=1,\cdots,N \}$ that maximizes the total tensor scatter $\Psi_{\mathcal{Y}}$: $$\{ \tilde{\mathbf{U}}^{(n)}, n=1, \cdots, N \} = \arg\max_{\tilde{\mathbf{U}^{(1)}}, \tilde{\mathbf{U}^{(2)}}, \cdots, \tilde{\mathbf{U}^{(N)}}} \Psi_{\mathcal{Y}}.$$

I guess this technically answers the question, but I am going to have to ruminate on this some more.

A different way to think about it is to write down the factor model for the tensor $$Y = \sum_{r=1}^Rv_{1,r}\otimes v_{2,r}\otimes\dots\otimes v_{d,r} + U$$ This factor model is the CP decomposition + noise. Then it is possible to estimate the loadings and factors in this model applying the standard PCA (or SVD) to the flattened tensors, see https://junsupan.github.io and https://arxiv.org/abs/2212.12981 for more details.

Here is a gentle introduction to Multilinear Principal Component Analysis (2002) and Multilinear Independent Component analysis(2005).

MPCA is computed with the N-mode SVD algorithm: https://www.media.mit.edu/~maov/tensorfaces/eccv02_corrected.pdf

MICA is computed with the N-mode ICA algorithm: https://www.media.mit.edu/~maov/mica/mica05.pdf

N-mode SVD algorithm is employed to perform Multilinear Principal Component Analysis (MPCA).

MPCA computed with the M-mode SVD algorithm when images are vectorized and organized into a data tensor versus when images are treated as "matrix-valued". Most arguments in favor of treating an image as a "matrix-valued" do not stand up to analytical scrutiny.

"Causal Deep Learning", M. Alex O. Vasilescu, In the Proceedings of the 2024 27h International Conference on Pattern Recognition (ICPR 2024)