Signed Support Recovery for Single Index Models in High-Dimensions

Abstract

In this paper we study the support recovery problem for single index models Y=f(X^ ,), where f is an unknown link function, X N_p(0,I_p) and is an s-sparse unit vector such that _i \1s,0\. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When s=O(p^1-) for some >0, we demonstrate that both procedures can succeed in recovering the support of as long as the rescaled sample size =ns(p-s) is larger than a certain critical threshold. On the other hand, when is smaller than a critical value, any algorithm fails to recover the support with probability at least 12 asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.

Tasks

Reproductions