On consistency and sparsity for sliced inverse regression in high dimensions

Qian Lin Harvard University Zhigen Zhao Temple University Jun S. Liu Temple University

Publications of CMSA of Harvard mathscidoc:1702.38034

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [1991]. Under mild conditions, the asymptotic ratio ρ = lim p/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ = 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 eigen-space of the covariance matrix of the conditional expectation var(E[x|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. 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.
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  author={Qian Lin, Zhigen Zhao, and Jun S. Liu},
Qian Lin, Zhigen Zhao, and Jun S. Liu. ON CONSISTENCY AND SPARSITY FOR SLICED INVERSE REGRESSION IN HIGH DIMENSIONS. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207031258215013234.
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