Robust factorization for high-dimensional matrix-variate observations

Abstract

Large-dimensional matrix-variate observations have been ubiquitous in the big data era, while unsupervised low-rank approximation technique would help reveal their hidden patterns and structures. In this paper, we study a hierarchical Canonical Polyadic (CP) product matrix factor model under the elliptical framework, which essentially assumes that the matrix-variate observations are from a matrix elliptical distribution. The proposed model not only incorporates the row-wise and column-wise interrelated information, but also adapts to the tail properties of the matrix-variate observations. We resort to the matrix Kendall’s tau introduced in the recent literature to recover the loading spaces, and minimize the square loss function to estimate the factor scores. We also propose an eigenvalue-ratio method to estimate the pair of factor numbers. Thorough theories for the model estimation, including statistical consistency and rates of convergence, are established under regular conditions. It is worth highlighting that the proposed method exhibits superior performance compared to other methods for estimating the signal part, particularly in the heavy-tailed cases. This superiority has been thoroughly validated through extensive simulations. The effectiveness in matrix reconstruction of the proposed method is demonstrated by applying it to a macroeconomic dataset of China.