NDLI: Nearly Sharp Sufficient Conditions on Exact Sparsity Pattern Recovery

Content Provider	IEEE Xplore Digital Library
Author	Rahnama Rad, K.
Copyright Year	1963
Abstract	Consider the n-dimensional vector y=Xβ+ε where β ∈ \BBRp has only k nonzero entries and ε ∈ \BBRn is a Gaussian noise. This can be viewed as a linear system with sparsity constraints corrupted by noise, where the objective is to estimate the sparsity pattern of β given the observation vector y and the measurement matrix X. First, we derive a nonasymptotic upper bound on the probability that a specific wrong sparsity pattern is identified by the maximum-likelihood estimator. We find that this probability depends (inversely) exponentially on the difference of \|\|Xβ\|\|₂ and the l₂ -norm of Xβ projected onto the range of columns of X indexed by the wrong sparsity pattern. Second, when X is randomly drawn from a Gaussian ensemble, we calculate a nonasymptotic upper bound on the probability of the maximum-likelihood decoder not declaring (partially) the true sparsity pattern. Consequently, we obtain sufficient conditions on the sample size n that guarantee almost surely the recovery of the true sparsity pattern. We find that the required growth rate of sample size n matches the growth rate of previously established necessary conditions.
Sponsorship	IEEE Information Theory Society
Page Count	8
File Size	231084
Starting Page	4672
Ending Page	4679
File Format	PDF
ISSN	00189448
Volume Number	57
Issue Number	7
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2011-07-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Eigenvalues and eigenfunctions Upper bound Decoding Mathematical model Sufficient conditions Error probability Noise measurement underdetermined systems of equations Hypothesis testing random projections sparsity pattern recovery subset selection
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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