Selection of variables and dimension reduction in high-dimensional non-parametric regression

Abstract

We consider a l₁-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension d of the input variable X is very large (sometimes depending on the number of observations). Estimation of a β-regular regression function f cannot be faster than the slow rate n^{−2β/(2β+d)}. Hopefully, in some situations, f depends only on a few numbers of the coordinates of X. In this paper, we construct two procedures. The first one selects, with high probability, these coordinates. Then, using this subset selection method, we run a local polynomial estimator (on the set of interesting coordinates) to estimate the regression function at the rate n^{−2β/(2β+d*)}, where d^*, the “real” dimension of the problem (exact number of variables whom f depends on), has replaced the dimension d of the design. To achieve this result, we used a l₁ penalization method in this non-parametric setup.