Regression for matrix-valued data via Kronecker products factorization

We study the matrix-variate regression problem Yi = sumk β1k Xi β2ktop + Ei for i=1,2dots,n in the high dimensional regime wherein the response Yi are matrices whose dimensions p1× p2 outgrow both the sample size n and the dimensions q1× q2 of the predictor variables Xi i.e., q1,q2 ll n ll p1,p2. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters β1k subset Rep1 × q1 and β2k subset Rep2 × q2 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 Yi. We establish perturbation bounds between hatβ1k -β1k and hatβ2k - β2k in spectral norm for the setting where either the rows of Ei or the columns of Ei 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.