Faithful variable screening for high-dimensional convex regression

Min Xu, Minhua Chen, John Lafferty

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2624-2660
Number of pages37
JournalAnnals of Statistics
Issue number6
StatePublished - Dec 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Additive model
  • Convex regression
  • Nonparametric regression
  • Quadratic programming
  • Variable selection


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