Necessary and Sufficient Conditions for Sparsity Pattern Recovery
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Fletcher-2009-Necessary and Suffic.pdf
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Author(s)
Fletcher, Alyson K.
Goyal, Vivek K.
Rangan, Sundeep
Date Issued
November 2009
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers
Citation
Fletcher, A.K., S. Rangan, and V.K. Goyal. “Necessary and Sufficient Conditions for Sparsity Pattern Recovery.” Information Theory, IEEE Transactions on 55.12 (2009): 5758-5772. © 2009 IEEE
Version
Final published version
Abstract
he paper considers the problem of detecting the sparsity pattern of a k -sparse vector in BBR n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum-likelihood (ML) estimation and Gaussian measurement matrices is derived. This necessary condition for ML detection is compared against a sufficient condition for simple maximum correlation (MC) or thresholding algorithms. The analysis shows that the gap between thresholding and ML can be described by a simple expression in terms of the total signal-to-noise ratio (SNR), with the gap growing with increasing SNR. Thresholding is also compared against the more sophisticated Lasso and orthogonal matching pursuit (OMP) methods. At high SNRs, it is shown that the gap between Lasso and OMP over thresholding is described by the range of powers of the nonzero component values of the unknown signals. Specifically, the key benefit of Lasso and OMP over thresholding is the ability of Lasso and OMP to detect signals with relatively small components.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Research Laboratory of Electronics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://dx.doi.org/10.1109/tit.2009.2032726
