Information-Theoretic Limits on Sparsity Recovery in the High-Dimensional and Noisy Setting

The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an unknown vector beta* isin Ropf p based on a set of n noisy observations. It arises in a variety of settings, including subset selection in regression, graphical model selection, signal...

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Veröffentlicht in:	IEEE transactions on information theory Jg. 55; H. 12; S. 5728 - 5741
1. Verfasser:	Wainwright, M.J.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York, NY IEEE 01.12.2009 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	Applied sciences Beta Coding, codes Complexity Compressed sensing Computational complexity Decoders Decoding Detection, estimation, filtering, equalization, prediction ell_1 -relaxation Estimating techniques Exact sciences and technology Fano's method Gaussian noise Graphical models high-dimensional statistical inference Information theory Information, signal and communications theory information-theoretic bounds Lasso Mathematical analysis Mathematical models Matrices Matrix Matrix methods Measurement Miscellaneous model selection Particle measurements Polynomials Quadratic programming Recovery Sample size Signal and communications theory Signal denoising Signal processing Signal, noise Size measurement sparsity pattern sparsity recovery subset selection Sufficient conditions support recovery Telecommunications and information theory Parameter estimation Noise reduction Lasso Polynomial method information-theoretic bounds Linear model Relaxation ℓ Fano's method subset selection Model selection Decoding Quadratic programming Signal denoising Computational complexity Polynomial time sparsity recovery Selection problem Necessary condition Sufficient condition Regression model Signal processing sparsity pattern support recovery Compressed sensing high-dimensional statistical inference Information theory
ISSN:	0018-9448, 1557-9654
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Beschreibung
Zusammenfassung:	The problem of sparsity pattern or support set recovery refers to estimating the set of nonzero coefficients of an unknown vector beta* isin Ropf p 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 beta), as well as the minimum value beta min of beta 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 (lscr 1 -constrained quadratic programming).
Bibliographie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23
ISSN:	0018-9448 1557-9654
DOI:	10.1109/TIT.2009.2032816