On consistency and sparsity for sliced inverse regression in high dimensions

We provide here a framework to analyze the phase transition phe- nomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by Li [1991]. 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 dimen- sion 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 expec- tation 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 covari- ance 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 exper- iments demonstrate superior performances of the proposed method in comparison to its competitors