Variable selection in high-dimensional sparse multiresponse linear regression models

We consider variable selection in high-dimensional sparse multiresponse linear regression models, in which a q-dimensional response vector has a linear relationship with a p-dimensional covariate vector through a sparse coefficient matrix B∈Rp×q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\in R^{p\times q}$$\end{document}. We propose a consistent procedure for the purpose of identifying the nonzeros in B. The procedure consists of two major steps, where the first step focuses on the detection of all the nonzero rows in B, the latter aims to further discover its individual nonzero cells. The first step is an extension of Orthogonal Matching Pursuit (OMP) and the second step adopts the bootstrap strategy. The theoretical property of our proposed procedure is established. Extensive numerical studies are presented to compare its performances with available representatives.