Modeling and Learning on High-Dimensional Matrix-Variate Sequences

TL;DR

This paper introduces RaDFaM, a novel matrix factor model for high-dimensional matrix sequences that leverages low-rank tensor structures, improves PCA-based estimation, and demonstrates superior signal recovery in various applications.

Contribution

RaDFaM provides a new low-rank tensor-based matrix factor model with improved PCA estimation and theoretical guarantees for high-dimensional data analysis.

Findings

RaDFaM achieves higher peak signal-to-noise ratios than traditional PCA methods.

The model demonstrates strong consistency and convergence properties.

Numerical experiments show effective matrix reconstruction, learning, and clustering.

Abstract

We propose a new matrix factor model, named RaDFaM, which is strictly derived based on the general rank decomposition and assumes a structure of a high-dimensional vector factor model for each basis vector. RaDFaM contributes a novel class of low-rank latent structure that makes tradeoff between signal intensity and dimension reduction from the perspective of tensor subspace. Based on the intrinsic separable covariance structure of RaDFaM, for a collection of matrix-valued observations, we derive a new class of PCA variants for estimating loading matrices, and sequentially the latent factor matrices. The peak signal-to-noise ratio of RaDFaM is proved to be superior in the category of PCA-type estimations. We also establish the asymptotic theory including the consistency, convergence rates, and asymptotic distributions for components in the signal part. Numerically, we demonstrate the performance of RaDFaM in applications such as matrix reconstruction, supervised learning, and clustering, on uncorrelated and correlated data, respectively.