Necessary and sufficient conditions for sparsity pattern recovery

  • Authors:
  • Alyson K. Fletcher;Sundeep Rangan;Vivek K. Goyal

  • Affiliations:
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA;Qualcomm Technologies, Bedminster, NJ;Department of Electrical Engineering and Computer Science and the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2009

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Abstract

The paper considers the problem of detecting the sparsity pattern of a k-sparse vector in Rn 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.