Block-sparse signals: Uncertainty relations and efficient recovery


Yonina Eldar, Patrick Kuppinger, and Helmut Bölcskei


IEEE Transactions on Signal Processing, Vol. 58, No. 6, pp. 3042-3054, June 2010

DOI: 10.1109/TSP.2010.2044837

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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.


Basis pursuit, block-sparsity, compressed sensing, matching pursuit

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