Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions

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 $Y_t=A^{\ast }{X}_t{B}^{\ast }+E_t$, with $A^{\ast }\in {\mathbb{R}}_{+}^{n\times m}$ (row-wise ${\ell }_1$-normalized), $B^{\ast }\in {\mathbb{R}}^{q\times p}$, and $E_t$ independent Gaussian noise matrices. The goal is to estimate $A^{\ast }$ and $B^{\ast }$ 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.