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December 2016 Faithful variable screening for high-dimensional convex regression
Min Xu, Minhua Chen, John Lafferty
Ann. Statist. 44(6): 2624-2660 (December 2016). DOI: 10.1214/15-AOS1425


We study the problem of variable selection in convex nonparametric regression. Under the assumption that the true regression function is convex and sparse, we develop a screening procedure to select a subset of variables that contains the relevant variables. Our approach is a two-stage quadratic programming method that estimates a sum of one-dimensional convex functions, followed by one-dimensional concave regression fits on the residuals. In contrast to previous methods for sparse additive models, the optimization is finite dimensional and requires no tuning parameters for smoothness. Under appropriate assumptions, we prove that the procedure is faithful in the population setting, yielding no false negatives. We give a finite sample statistical analysis, and introduce algorithms for efficiently carrying out the required quadratic programs. The approach leads to computational and statistical advantages over fitting a full model, and provides an effective, practical approach to variable screening in convex regression.


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Min Xu. Minhua Chen. John Lafferty. "Faithful variable screening for high-dimensional convex regression." Ann. Statist. 44 (6) 2624 - 2660, December 2016.


Received: 1 November 2014; Revised: 1 December 2015; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1360.62197
MathSciNet: MR3576556
Digital Object Identifier: 10.1214/15-AOS1425

Primary: 62G08
Secondary: 52A41

Keywords: Additive model , convex regression , Nonparametric regression , quadratic programming , Variable selection

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 6 • December 2016
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