Signed Support Recovery for Single Index Models in High- Dimensions

In this paper we study the support recovery problem for single index models Y = f(Xβ, ε), where f is an unknown link function, X ∼ Np(0, Ip) and β is an s-sparse unit vector such that βi ∈ {± 1 s , 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(p1−δ) for some δ > 0, we demonstrate that both procedures can succeed in recovering the support of β as long as the rescaled sample size Γ = n s log(p−s) is larger than a certain critical threshold. On the other hand, when Γ 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 β in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.