# Regression for matrix-valued data via Kronecker products factorization

arXiv: https://arxiv.org/abs/2404.19220 · HTML: https://mlsanity.com/paper/2404.19220/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 44 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 12 of 12)

- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Practical sketching algorithms for low-rank matrix approximation](https://mlsanity.com/paper/1609.00048/) (2016)
- [Parsimonious Tensor Response Regression](https://mlsanity.com/paper/1501.07815/) (2015)
- [High-dimensional vector autoregressive time series modeling via tensor decomposition](https://mlsanity.com/paper/1909.06624/) (2019)
- [Constrained Factor Models for High-Dimensional Matrix-Variate Time Series](https://mlsanity.com/paper/1710.06075/) (2017)
- [Statistical Inference for High-Dimensional Matrix-Variate Factor Model](https://mlsanity.com/paper/2001.01890/) (2020)
- [Factor Models for Matrix-Valued High-Dimensional Time Series](https://mlsanity.com/paper/1610.01889/) (2016)
- [Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation](https://mlsanity.com/paper/1411.1352/) (2014)
- [Matrix-Variate Regressions and Envelope Models](https://mlsanity.com/paper/1605.01485/) (2016)
- [Scalable Interpretable Multi-Response Regression via SEED](https://mlsanity.com/paper/1608.03686/) (2016)
- [Adaptive estimation of the rank of the coefficient matrix in high dimensional multivariate response regression models](https://mlsanity.com/paper/1704.02381/) (2017)
- [KoPA: Automated Kronecker Product Approximation](https://mlsanity.com/paper/1912.02392/) (2019)

## Built on by (most cited first, 1 of 1)

- [Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions](https://mlsanity.com/paper/2412.17749/) (2024)

## Abstract (arXiv)

We study the matrix-variate regression problem Yi = sumk β1k Xi β2ktop + Ei for i=1,2dots,n in the high dimensional regime wherein the response Yi are matrices whose dimensions p1× p2 outgrow both the sample size n and the dimensions q1× q2 of the predictor variables Xi i.e., q1,q2 ll n ll p1,p2. We propose an estimation algorithm, termed KRO-PRO-FAC, for estimating the parameters β1k subset Rep1 × q1 and β2k subset Rep2 × q2 that utilizes the Kronecker product factorization and rearrangement operations from Van Loan and Pitsianis (1993). The KRO-PRO-FAC algorithm is computationally efficient as it does not require estimating the covariance between the entries of the Yi. We establish perturbation bounds between hatβ1k -β1k and hatβ2k - β2k in spectral norm for the setting where either the rows of Ei or the columns of Ei are independent sub-Gaussian random vectors. Numerical studies on simulated and real data indicate that our procedure is competitive, in terms of both estimation error and predictive accuracy, compared to other existing methods.