Fourier transform sparse inverse regression estimators for sufficient variable selection

Abstract

Sufficient dimension reduction aims to reduce the dimension of predictors while maintaining the regression information. Recently, researchers have studied an impressive range of sparse inverse regression estimators. Nonetheless, conspicuously less attention has been given to the multivariate response with high-dimensional covariates settings. To fill the gap, a Fourier transform inverse regression approach via regularized quadratic discrepancy functions is investigated. Theoretically, consistency and oracle properties for the proposed estimators are established. An iterated alternating direction method of multipliers (ADMM) algorithm is employed to estimate two target parameters simultaneously. Each step of the ADMM algorithm has an explicit solution yielding computational efficiency gain. Numerical studies and real data analysis confirm the theoretical properties and yield superior performance of our proposed methods. In specific, the proposal has higher support recovery rates compared to the state-of-the-art approach. An open-source Python package called ADMMFTIRE accompanying this paper is available online.1