We observe an unknown function of d variables \(f(\textbf{t})\) , \(\textbf{t}\in [0,1]^d\) , in the Gaussian white noise model of intensity \(\varepsilon >0\) . 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 \(d=d_\varepsilon \rightarrow \infty \) as \(\varepsilon \rightarrow 0\) and s is either fixed or \(s=s_\varepsilon \rightarrow \infty \) , \(s=o(d)\) as \(\varepsilon \rightarrow \infty \) . 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.
