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 (Yt)t=1T ∈ ℝn × p and predictors (Xt)t=1T ∈ ℝm × q satisfy the relationship Yt = A^* Xt B^* + Et for all t = 1, ldots, T. In this model, A^* ∈ ℝ_+n × m has L1-normalized rows, B^* ∈ ℝq × p, and (Et)t=1T are independent noise matrices following a matrix Gaussian distribution. The primary objective is to estimate the unknown parameters A^* and B^* 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^* and B^* 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.