Block-sparse signals: uncertainty relations and efficient recovery

  • Authors:

  • Yonina C. Eldar;Patrick Kuppinger;Helmut Bölcskei

  • Affiliations:

  • Department of Electrical Engineering, Technion, Haifa, Israel and Electrical Engineering and Statistics Departments, Stanford University, Stanford, CA;Department of Information Technology and Electrical Engineering, ETH Zurich, Zurich, Switzerland;Department of Information Technology and Electrical Engineering, ETH Zurich, Zurich, Switzerland

  • Venue:
  • IEEE Transactions on Signal Processing
  • Year:
  • 2010

Quantified Score

Hi-index 35.81

Visualization

Abstract

We consider efficient methods for the recovery of block-sparse signals--i.e., sparse signals that have nonzero entries occurring in clusters--from an underdetermined system of linear equations. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block k-sparse signals in no more than k steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed l2/l1-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.