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Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

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3 Author(s)
Yonina C. Eldar ; Department of Electrical Engineering, Technion, Haifa, Israel ; Patrick Kuppinger ; Helmut Bolcskei

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 -sparse signals in no more than steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed -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.

Published in:

IEEE Transactions on Signal Processing  (Volume:58 ,  Issue: 6 )