Block-Sparse Signals: Uncertainty Relations and Efficient Recovery
Abstract
We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. 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 $\ell_2/\ell_1$-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.
- Publication:
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IEEE Transactions on Signal Processing
- Pub Date:
- June 2010
- DOI:
- 10.1109/TSP.2010.2044837
- arXiv:
- arXiv:0906.3173
- Bibcode:
- 2010ITSP...58.3042E
- Keywords:
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- Computer Science - Information Theory
- E-Print:
- Submitted to the IEEE Trans. on Signal Processing, version 2 has updated figures