{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bivariate-matrix-valued-linear-regression","title":"Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions","arxiv_id":"2412.17749","date":"2024-12-23","proceeding":null,"authors":["Nayel Bettache"],"abstract":"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 $Y_t = A^* X_t B^* + E_t$ for all $t = 1, \\ldots, T$. In this model, $A^* \\in \\mathbb{R}_+^{n \\times m}$ has $L_1$-normalized rows, $B^* \\in \\mathbb{R}^{q \\times p}$, and $(E_t)_{t=1}^T$ are independent noise matrices following a matrix Gaussian distribution. The primary objective is to estimate the unknown parameters $A^*$ 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.","url_abs":"https://arxiv.org/abs/2412.17749v2","url_pdf":"https://arxiv.org/pdf/2412.17749v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"bivariate-matrix-valued-linear-regression","repo_url":"https://github.com/nayelbettache/bmlr","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"denoising","task_name":"Denoising"}],"methods":[{"method_slug":"linear-regression","method_name":"Linear Regression"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}