Adaptive exact recovery in sparse nonparametric models.

Abstract

We observe an unknown function of d variables $f\left(t\right)$ , $t\in {[0,1]}^d$ , in the Gaussian white noise model of intensity $\epsilon >0$ . We assume that the function f is regular and that it is a sum of k-variate functions, where k varies from 1 to s ( $1\le s\le d$ ). These functions are unknown to us and only a few of them are nonzero. In this article, we address the problem of identifying the nonzero components of f in the case when $d=d_{\epsilon }\to \infty$ as $\epsilon \to 0$ and s is either fixed or $s=s_{\epsilon }\to \infty$ , $s=o\left(d\right)$ as $\epsilon \to \infty$ . This may be viewed as a variable selection problem. We derive the conditions when exact variable selection in the model at hand is possible and provide a selection procedure that achieves this type of selection. The procedure is adaptive to a degree of model sparsity described by the sparsity parameter $\beta \in (0,1)$ . We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.