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Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting (2007)
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@TECHREPORT{Wainwright07information-theoreticlimits,
author = {Martin J. Wainwright},
title = {Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting},
institution = {},
year = {2007}
}
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Abstract
Abstract—The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an un-3 p known vector 2 based on a set of n noisy observations. It arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. The sample complexity of a given method for subset recovery refers to the scaling of the required sample size n as a function of the signal dimension p, sparsity index k (number of non-zeroes in 3), as well as the minimum value min of 3 over its support and other parameters of measurement matrix. This paper studies the information-theoretic limits of sparsity recovery: in particular, for a noisy linear observation model based on random measurement matrices drawn from general Gaussian measurement matrices, we derive both a set of sufficient conditions for exact support recovery using an exhaustive search decoder, as well as a set of necessary conditions that any decoder, regardless of its computational complexity, must satisfy for exact support recovery. This analysis of fundamental limits complements our previous work on sharp thresholds for support set recovery over the same set of random measurement ensembles using the polynomial-time Lasso method (`1-constrained quadratic programming). Index Terms—Compressed sensing, `1-relaxation, Fano’s method, high-dimensional statistical inference, information-theoretic
Keyphrases
information-theoretic limit sparsity recovery noisy setting exact support recovery random measurement ensemble exhaustive search decoder random measurement matrix nonzero coefficient required sample size graphical model selection measurement matrix sharp threshold sample complexity necessary condition previous work paper study noisy linear observation model signal dimension constructive approximation recovery refers subset recovery refers sparsity index compressive sensing signal denoising high-dimensional statistical inference index term computational complexity general gaussian measurement matrix fundamental limit complement minimum value min sufficient condition noisy observation 1-constrained quadratic programming polynomial-time lasso method sparsity pattern
