A Framework of Regularized Low-Rank Matrix Models for Regression and Classification

While matrix-covariate regression models have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional matrix-valued covariates. To address these issues, this paper proposes a framework of matrix-covariate regression models based on a low-rank constraint and an additional regularization term for structured signals, with considerations of models of both continuous and binary responses. We propose an efficient Riemannian-steepest-descent algorithm for regression coefficient estimation. We prove that the consistency of the proposed estimator is in the order of $O(\sqrt{r(q+m)+p}/\sqrt{n})$, where $r$ is the rank, $p\times m$ is the dimension of the coefficient matrix and $p$ is the dimension of the coefficient vector. When the rank $r$ is small, this rate improves over $O(\sqrt{qm+p}/\sqrt{n})$, the consistency of the existing work \citep{li2021double} that does not apply a rank constraint. In addition, we prove that all accumulation points of the iterates have similar estimation errors asymptotically and substantially attaining the minimax rate. We validate the proposed method through a simulated dataset on two-dimensional shape images and two real datasets of brain signals and microscopic leucorrhea images.