Faithful Variable Screening for High-Dimensional Convex Regression

Shape restrictions such as monotonicity, convexity, and concavity provide a natural way of limiting the complexity of many statistical estimation problems. Shape-constrained estimation is not as well understood as more traditional nonparametric estimation involving smoothness constraints. For instance, the minimax rate of convergence for multivariate convex regression has yet to be rigorously established in full generality. Even the one-dimensional case is challenging, and has been of recent interest (Guntuboyina and Sen, 2013). In this paper we study the problem of variable selection in multivariate convex regression. Assuming that the regression function is convex and sparse, our goal is to identify the relevant variables. We show that it suffices to estimate a sum of onedimensional convex functions, leading to significant computational and statistical advantages. This is in contrast to general nonparametric regression, where fitting an additive model can result in false negatives. Our approach is based on a twostage quadratic programming procedure.