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October 2012 Tight conditions for consistency of variable selection in the context of high dimensionality
Laëtitia Comminges, Arnak S. Dalalyan
Ann. Statist. 40(5): 2667-2696 (October 2012). DOI: 10.1214/12-AOS1046

Abstract

We address the issue of variable selection in the regression model with very high ambient dimension, that is, when the number of variables is very large. The main focus is on the situation where the number of relevant variables, called intrinsic dimension, is much smaller than the ambient dimension $d$. Without assuming any parametric form of the underlying regression function, we get tight conditions making it possible to consistently estimate the set of relevant variables. These conditions relate the intrinsic dimension to the ambient dimension and to the sample size. The procedure that is provably consistent under these tight conditions is based on comparing quadratic functionals of the empirical Fourier coefficients with appropriately chosen threshold values.

The asymptotic analysis reveals the presence of two quite different re gimes. The first regime is when the intrinsic dimension is fixed. In this case the situation in nonparametric regression is the same as in linear regression, that is, consistent variable selection is possible if and only if $\log d$ is small compared to the sample size $n$. The picture is different in the second regime, that is, when the number of relevant variables denoted by $s$ tends to infinity as $n\to\infty$. Then we prove that consistent variable selection in nonparametric set-up is possible only if $s+\log\log d$ is small compared to $\log n$. We apply these results to derive minimax separation rates for the problem of variable selection.

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Laëtitia Comminges. Arnak S. Dalalyan. "Tight conditions for consistency of variable selection in the context of high dimensionality." Ann. Statist. 40 (5) 2667 - 2696, October 2012. https://doi.org/10.1214/12-AOS1046

Information

Published: October 2012
First available in Project Euclid: 4 February 2013

zbMATH: 1373.62154
MathSciNet: MR3097616
Digital Object Identifier: 10.1214/12-AOS1046

Subjects:
Primary: 62G08
Secondary: 62H12 , 62H15

Keywords: Nonparametric regression , set estimation , sparsity pattern , Variable selection

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • October 2012
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