Signed Support Recovery for Single Index Models in High-Dimensions

Neykov Matey Princeton University Qian Lin Harvard University Jun S. Liu Harvard University

Statistics Theory and Methods mathscidoc:1701.333183

Annals of Mathematical Sciences and Applications, 1, (2), 379-426, 2016
In this paper we study the support recovery problem for single index models Y=f(X⊺β,ε), where f is an unknown link function, X∼Np(0,𝕀p) and β is an s-sparse unit vector such that βi∈{±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(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 κ=nslog(p−s) is larger than a certain critical threshold. On the other hand, when κ 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.
sliced inverse regression, single index models, high dimensinal statistics
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@inproceedings{neykov2016signed,
  title={ Signed Support Recovery for Single Index Models in High-Dimensions},
  author={Neykov Matey, Qian Lin, and Jun S. Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170121193654341940118},
  booktitle={Annals of Mathematical Sciences and Applications},
  volume={1},
  number={2},
  pages={379-426},
  year={2016},
}
Neykov Matey, Qian Lin, and Jun S. Liu. Signed Support Recovery for Single Index Models in High-Dimensions. 2016. Vol. 1. In Annals of Mathematical Sciences and Applications. pp.379-426. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170121193654341940118.
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