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

This study explores the estimation of parameters in a matrix-valued linear regression model, where the $T$ responses $(Y_t{)}_{t=1}^T\in {\mathbb{R}}^{n\times p}$ and predictors $(X_t{)}_{t=1}^T\in {\mathbb{R}}^{m\times q}$ satisfy the relationship $Y_t=A^{\ast }{X}_t{B}^{\ast }+E_t$ for all $t=1,\dots ,T$. In this model, $A^{\ast }\in {\mathbb{R}}_{+}^{n\times m}$ has $L_1$-normalized rows, $B^{\ast }\in {\mathbb{R}}^{q\times p}$, and $(E_t{)}_{t=1}^T$ are independent noise matrices following a matrix Gaussian distribution. The primary objective is to estimate the unknown parameters $A^{\ast }$ and $B^{\ast }$ efficiently. We propose explicit optimization-free estimators and establish non-asymptotic convergence rates to quantify their performance. Additionally, we extend our analysis to scenarios where $A^{\ast }$ and $B^{\ast }$ exhibit sparse structures. To support our theoretical findings, we conduct numerical simulations that confirm the behavior of the estimators, particularly with respect to the impact of the dimensions $n,m,p,q$, and the sample size $T$ on finite-sample performances. We complete the simulations by investigating the denoising performances of our estimators on noisy real-world images.