Necessary and sufficient conditions for sparsity pattern recovery

Alyson K. Fletcher, Sundeep Rangan, Vivek K. Goyal

Research output: Contribution to journalArticlepeer-review


The paper considers the problem of detecting the sparsity pattern of a κ-sparse vector in Rnfrom ampmu; 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.

Original languageEnglish (US)
Pages (from-to)5758-5772
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2009


  • Compressed sensing
  • Convex optimization
  • Lasso
  • Maximum-likelihood (ML) estimation
  • Orthogonal matching pursuit (OMP)
  • Random matrices
  • Random projections
  • Sparse approximation
  • Subset selection
  • Thresholding

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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