Adaptive exact recovery in sparse nonparametric models

Abstract We observe an unknown function of d variables $$f(\textbf{t})$$ f ( t ) , $$\textbf{t}\in [0,1]^d$$ t ∈ [ 0 , 1 ] d , in the Gaussian white noise model of intensity $$\varepsilon >0$$ ε > 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 ( $$1\le s\le d$$ 1 ≤ s ≤ 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 $$d=d_\varepsilon \rightarrow \infty $$ d = d ε → ∞ as $$\varepsilon \rightarrow 0$$ ε → 0 and s is either fixed or $$s=s_\varepsilon \rightarrow \infty $$ s = s ε → ∞ , $$s=o(d)$$ s = o ( d ) as $$\varepsilon \rightarrow \infty $$ ε → ∞ . 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 $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) . We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.

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