Regression for matrix-valued data via Kronecker products factorization

Abstract

We study the matrix-variate regression problem Y_i = _k _1k X_i _2k^ + E_i for i=1,2,n in the high dimensional regime wherein the response Y_i are matrices whose dimensions p_1 p_2 outgrow both the sample size n and the dimensions q_1 q_2 of the predictor variables X_i i.e., q_1,q_2 n p_1,p_2. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters \_1k\ ^p_1 q_1 and \_2k\ ^p_2 q_2 that utilizes the Kronecker product factorization and rearrangement operations from Van Loan and Pitsianis (1993). The KRO-PRO-FAC algorithm is computationally efficient as it does not require estimating the covariance between the entries of the _i\. We establish perturbation bounds between _1k -_1k and _2k - _2k in spectral norm for the setting where either the rows of E_i or the columns of E_i are independent sub-Gaussian random vectors. Numerical studies on simulated and real data indicate that our procedure is competitive, in terms of both estimation error and predictive accuracy, compared to other existing methods.

Tasks

Reproductions