Abstract: This study examines matrix quantile regression where the covariate is a matrix and the response is a scalar. Although the statistical estimation of matrix regression is an active field of research, few studies examine quantile regression with matrix covariates. We propose an estimation procedure based on convex regularizations in a high-dimensional setting. In order to reduce the dimensionality, the coefficient matrix is assumed to be low rank and/or sparse. Thus, we impose two regularizers to encourage different low-dimensional structures. We develop the asymptotic properties and an implementation based on the incremental proximal gradient algorithm. We then apply the proposed estimator to quadratic quantile regression, and demonstrate its advantages using simulations and a real-data analysis.

Key words and phrases: Dual norm, interaction effects, matrix regression, penalization.