# Matrix-Variate Regressions and Envelope Models

arXiv: https://arxiv.org/abs/1605.01485 · HTML: https://mlsanity.com/paper/1605.01485/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2016.

## Built on by (most cited first, 5 of 5)

- [Errors-in-variables Fr'echet Regression with Low-rank Covariate Approximation](https://mlsanity.com/paper/2305.09282/) (2023)
- [Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model](https://mlsanity.com/paper/2307.00575/) (2023)
- [Regression for matrix-valued data via Kronecker products factorization](https://mlsanity.com/paper/2404.19220/) (2024)
- [Coefficient Shape Alignment in Multiple Functional Linear Regression](https://mlsanity.com/paper/2312.01925/) (2023)
- [Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions](https://mlsanity.com/paper/2412.17749/) (2024)

## Abstract (arXiv)

Modern technology often generates data with complex structures in which both response and explanatory variables are matrix-valued. Existing methods in the literature are able to tackle matrix-valued predictors but are rather limited for matrix-valued responses. In this article, we study matrix-variate regressions for such data, where the response Y on each experimental unit is a random matrix and the predictor X can be either a scalar, a vector, or a matrix, treated as non-stochastic in terms of the conditional distribution Y|X. We propose models for matrix-variate regressions and then develop envelope extensions of these models. Under the envelope framework, redundant variation can be eliminated in estimation and the number of parameters can be notably reduced when the matrix-variate dimension is large, possibly resulting in significant gains in efficiency. The proposed methods are applicable to high dimensional settings.