Algorithms for simultaneous sparse approximation: part I: Greedy pursuit
Signal Processing - Sparse approximations in signal and image processing
Block-sparsity: Coherence and efficient recovery
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
On the reconstruction of block-sparse signals with an optimal number of measurements
IEEE Transactions on Signal Processing
Theoretical Results on Sparse Representations of Multiple-Measurement Vectors
IEEE Transactions on Signal Processing
Sparse solutions to linear inverse problems with multiple measurement vectors
IEEE Transactions on Signal Processing
Uncertainty principles and ideal atomic decomposition
IEEE Transactions on Information Theory
On sparse representation in pairs of bases
IEEE Transactions on Information Theory
Sparse representations in unions of bases
IEEE Transactions on Information Theory
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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It has been known for a while that l1-norm relaxation can in certain cases solve an under-determined system of linear equations. Recently, [5], [11] proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that l1-norm relaxation succeeds in solving the system. In this paper we consider a modification of this standard setup, namely the case of so-called approximately block-sparse unknown vectors [2], [27]. We determine sharp lower bounds on the values of allowable approximate block-sparsity for any given number (proportional to the length of the unknown vector) of equations. Obtained lower bounds on the allowable sparsity are as expected functions of a parameter used to describe how close the approximately blocksparse unknown vectors are to the ideally block-sparse ones.