Adaptive exact recovery in sparse nonparametric models

TL;DR

This paper studies how to identify important variables in a high-dimensional function model with noise, focusing on when and how exact recovery is possible.

Contribution

The paper introduces an adaptive procedure for exact variable selection in sparse nonparametric models with increasing dimensionality.

Findings

Exact variable selection is possible under certain sparsity and noise conditions.

A selection procedure is proposed that adapts to the model's sparsity level.

Conditions for when exact recovery is impossible are also established.

Abstract

We observe an unknown function of d variables \documentclass[12pt]{minimal}
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\begin{document}$$f(\textbf{t})$$\end{document}f(t), \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{t}\in [0,1]^d$$\end{document}t∈[0,1]d, in the Gaussian white noise model of intensity \documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document}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 (\documentclass[12pt]{minimal}
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\begin{document}$$1\le s\le d$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$d=d_\varepsilon \rightarrow \infty $$\end{document}d=dε→∞ as \documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow 0$$\end{document}ε→0 and s is either fixed or \documentclass[12pt]{minimal}
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\begin{document}$$s=s_\varepsilon \rightarrow \infty $$\end{document}s=sε→∞, \documentclass[12pt]{minimal}
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\begin{document}$$s=o(d)$$\end{document}s=o(d) as \documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow \infty $$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\beta \in (0,1)$$\end{document}β∈(0,1). We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.