Regression for matrix-valued data via Kronecker products factorization

Abstract

We study the matrix-variate regression problem $Y_i = \sum_{k} \beta_{1k} X_i \beta_{2k}^{\top} + E_i$ for $i=1,2\dots,n$ in the high dimensional regime wherein the response $Y_i$ are matrices whose dimensions $p_{1}\times p_{2}$ outgrow both the sample size $n$ and the dimensions $q_{1}\times q_{2}$ of the predictor variables $X_i$ i.e., $q_{1},q_{2} \ll n \ll p_{1},p_{2}$. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters $\{\beta_{1k}\} \subset \Re^{p_1 \times q_1}$ and $\{\beta_{2k}\} \subset \Re^{p_2 \times q_2}$ 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 $\{Y_i\}$. We establish perturbation bounds between $\hat{\beta}_{1k} -\beta_{1k}$ and $\hat{\beta}_{2k} - \beta_{2k}$ 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.