ON CONSISTENCY AND SPARSITY FOR SLICED INVERSE REGRESSION IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Lin, Qian; Zhao, Zhigen; Liu, Jun S.
• 署名单位: Tsinghua University; Harvard University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Temple University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1561
• 发表日期: 2018
• 页码: 580-610
• 关键词: VARIABLE SELECTION; index models; reduction; regularization; asymptotics; PURSUIT

摘要

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [J. Amer. Statist. Assoc. 86 (1991) 316-342]. Under mild conditions, the asymptotic ratio rho = lim p/n is the phase transition parameter and the SIR estimator is consistent if and only if rho = 0. When dimension p is greater than n, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigenspace of var(E [x vertical bar y]), the covariance matrix of the conditional expectation. The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigenspace. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high-dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.