Regularized Matrix Regression

TL;DR

This paper introduces a spectral regularization approach for matrix regression that effectively captures low-rank structures in high-dimensional matrix data, improving model estimation and selection.

Contribution

It proposes a novel regularized matrix regression method based on spectral regularization, addressing low-rank structures in complex matrix data.

Findings

Efficient and scalable estimation algorithm developed.

Degrees of freedom formula derived for model selection.

Superior performance demonstrated on synthetic and real data.

Abstract

Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry, and electroencephalography, matrix type covariates frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are crucial due to the ultrahigh dimensionality and complex structure of the matrix data. The popular lasso and related regularization methods hinge upon the sparsity of the true signal in terms of the number of its nonzero coefficients. However, for the matrix data, the true signal is often of, or can be well approximated by, a low rank structure. As such, the sparsity is frequently in the form of low rank of the matrix parameters, which may seriously violate the assumption of the classical lasso. In this article, we propose a class of regularized matrix regression methods based on spectral regularization. Highly efficient and scalable estimation algorithm is developed, and a degrees of freedom formula is derived to facilitate model selection along the regularization path. Superior performance of the proposed method is demonstrated on both synthetic and real examples.