Factor Modeling of a High-Dimensional Matrix-Variate and Statistical Learning for Matrix-Valued Sequences

We propose a new matrix factor model, named RaDFaM, the latent structure of which is strictly derived based on a hierarchical rank decomposition of a matrix. Hierarchy is in the sense that all basis vectors of the column space of each multiplier matrix are assumed the structure of a vector factor model. Compared to the most commonly used matrix factor model that takes the latent structure of a bilinear form, RaDFaM involves additional row-wise and column-wise matrix latent factors. This yields modest dimension reduction and stronger signal intensity from the sight of tensor subspace learning, though poses challenges of new estimation procedure and concomitant inferential theory for a collection of matrix-valued observations. We develop a class of estimation procedure that makes use of the separable covariance structure under RaDFaM and approximate least squares, and derive its superiority in the merit of the peak signal-to-noise ratio. We also establish the asymptotic theory when the matrix-valued observations are uncorrelated or weakly correlated. Numerically, in terms of image/matrix reconstruction, supervised learning, and so forth, we demonstrate the excellent performance of RaDFaM through two matrix-valued sequence datasets of independent 2D images and multinational macroeconomic indices time series, respectively.