Sparse and Low-Rank Matrix Quantile Estimation With Application to Quadratic Regression
Wenqi Lu, Zhongyi Zhu, Heng Lian · Statistica Sinica · 2021
This paper studies matrix quantile regression where the covariate is a matrix and the response is a scalar.Statistical estimation of matrix regression is an active field of research.However, quantile regression with matrix covariates is rarely studied.We propose an estimation procedure based on convex regularizations in the 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 the different low-dimensional structures.The asymptotic properties and implementation based on incremental proximal gradient algorithm are developed.We then apply the proposed estimator to quadratic quantile regression.The advantages of the proposed method in its applications to quadratic regression are also illustrated by simulations and real data analysis.