Bivariate Matrix-Valued Linear Regression (BMLR): Finite-Sample Performance Under Identifiability and Sparsity Assumptions

Abstract

This paper studies a bilinear matrix-valued regression model where both predictors and responses are matrices. For each observation $t$, the response \( Y_t \in \mathbb{R}^{n \times p} \) and predictor \( X_t \in \mathbb{R}^{m \times q} \) satisfy $Y_t = A^* X_t B^* + E_t,$ with \( A^* \in \mathbb{R}_+^{n \times m} \) (row-wise \(\ell_1\)-normalized), \( B^* \in \mathbb{R}^{q \times p} \), and \( E_t \) independent Gaussian noise matrices. The goal is to estimate \( A^* \) and \( B^* \) from the observed pairs \( (X_t, Y_t) \). We propose explicit, optimization-free estimators and establish non-asymptotic error bounds, including sparse settings. Simulations confirm the theoretical rates and demonstrate strong finite-sample performance. We further illustrate the practical utility of our method through an image denoising application on real data.