# 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.

## Key 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.

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Source: https://tomesphere.com/paper/PMC12559153