Faithful Variable Screening for High-Dimensional Convex Regression
This provides an effective, practical approach for variable screening in convex regression, addressing a domain-specific problem in nonparametric statistics.
The paper tackles variable selection in high-dimensional convex regression by developing a two-stage quadratic programming method that ensures no false negatives in the population setting, offering computational and statistical advantages over full model fitting.
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.