Subspace methods for joint sparse recovery

Research output: Contribution to journalArticlepeer-review

Abstract

We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix. In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows. In this case, the MUltiple SIgnal Classification (MUSIC) algorithm, originally proposed by Schmidt for the direction of arrival estimation problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank defect or ill-conditioning. This situation arises with a limited number of measurement vectors, or with highly correlated signal components. In this case, MUSIC fails and, in practice, none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC such that the support is reliably recovered under such unfavorable conditions. Combined with a subspace-based greedy algorithm, known as Orthogonal Subspace Matching Pursuit, which is also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of the restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation step, which has been missing in the previous studies of MUSIC.

Original languageEnglish (US)
Article number6158602
Pages (from-to)3613-3641
Number of pages29
JournalIEEE Transactions on Information Theory
Volume58
Issue number6
DOIs
StatePublished - 2012

Keywords

  • Compressed sensing
  • joint sparsity
  • multiple measurement vectors (MMV)
  • restricted isometry property (RIP)
  • sensor array processing
  • spectrum-blind sampling
  • subspace estimation

ASJC Scopus subject areas

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

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