  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.
