Open Access
November 2020 Fundamental limits of exact support recovery in high dimensions
Zheng Gao, Stilian Stoev
Bernoulli 26(4): 2605-2638 (November 2020). DOI: 10.3150/20-BEJ1197


We study the support recovery problem for a high-dimensional signal observed with additive noise. With suitable parametrization of the signal sparsity and magnitude of its non-zero components, we characterize a phase-transition phenomenon akin to the signal detection problem studied by Ingster in 1998. Specifically, if the signal magnitude is above the so-called strong classification boundary, we show that several classes of well-known procedures achieve asymptotically perfect support recovery as the dimension goes to infinity. This is so, for a very broad class of error distributions with light, rapidly varying tails which may have arbitrary dependence. Conversely, if the signal is below the boundary, then for a very broad class of error dependence structures, no thresholding estimators (including ones with data-dependent thresholds) can achieve perfect support recovery. The proofs of these results exploit a certain concentration of maxima phenomenon known as relative stability. We provide a complete characterization of the relative stability phenomenon for Gaussian triangular arrays in terms of their correlation structure. The proof uses classic Sudakov–Fernique and Slepian lemma arguments along with a curious application of Ramsey’s coloring theorem.

We note that our study of the strong classification boundary is in a finer, point-wise, rather than minimax, sense. We also establish results on the finite-sample Bayes optimality and sub-optimality of thresholding procedures. Consequently, we obtain a minimax-type characterization of the strong classification boundary for errors with log-concave densities.


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Zheng Gao. Stilian Stoev. "Fundamental limits of exact support recovery in high dimensions." Bernoulli 26 (4) 2605 - 2638, November 2020.


Received: 1 April 2019; Revised: 1 September 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256154
MathSciNet: MR4140523
Digital Object Identifier: 10.3150/20-BEJ1197

Keywords: concentration of maxima , high–dimensional inference , Ramsey theory , rapid variation , Relative stability , Sudakov–Fernique inequality , support recovery

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 4 • November 2020
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