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

arXiv: https://arxiv.org/abs/2412.17749 · HTML: https://mlsanity.com/paper/2412.17749/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 50 references that are in the map.
- First posted on arXiv 23 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Autoregressive Models for Matrix-Valued Time Series](https://mlsanity.com/paper/1812.08916/) (2018)
- [Statistical Inference for High-Dimensional Matrix-Variate Factor Model](https://mlsanity.com/paper/2001.01890/) (2020)
- [Modeling Multivariate Spatial-Temporal Data with Latent Low-Dimensional Dynamics](https://mlsanity.com/paper/2002.01305/) (2020)
- [Matrix-Variate Regressions and Envelope Models](https://mlsanity.com/paper/1605.01485/) (2016)
- [Dynamic Matrix Factor Models for High Dimensional Time Series](https://mlsanity.com/paper/2407.05624/) (2024)
- [Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates](https://mlsanity.com/paper/2211.08580/) (2022)
- [Modeling longitudinal data using matrix completion](https://mlsanity.com/paper/1809.08771/) (2018)
- [Modeling and Learning on High-Dimensional Matrix-Variate Sequences](https://mlsanity.com/paper/2209.14846/) (2022)
- [Regression for matrix-valued data via Kronecker products factorization](https://mlsanity.com/paper/2404.19220/) (2024)
- [Matrix-valued Factor Model with Time-varying Main Effects](https://mlsanity.com/paper/2406.00128/) (2024)
- [Regularized Estimation of High-Dimensional Matrix-Variate Autoregressive Models](https://mlsanity.com/paper/2410.11320/) (2024)

## Abstract (arXiv)

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.