A simple performance analysis of l1 optimization in compressed sensing
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal 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
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors
IEEE Transactions on Signal Processing - Part I
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
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
IEEE Transactions on Information Theory
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It is well known that compressed sensing problems reduce to solving large under-determined systems of equations. If we choose the elements of the compressed measurement matrix according to some appropriate probability distribution and if the signal is sparse enough then the l1-optimization can recover it with overwhelming probability (see, e.g, [5], [9], (10)). In fact, [5], [9], [10] establish (in a statistical context) that if the number of measurements is proportional to the length of the signal then there is a sparsity of the unknown signal proportional to its length for which the success of the l1-optimization is guaranteed. In this paper we consider a modification of this standard setup, namely the case of the so-called approximately sparse unknown signals [7], [27]. We determine sharp lower bounds on the values of allowable approximate sparsity for any given number (proportional to the length of the unknown signal) of measurements. We introduce a novel, very simple technique which provides very good values for proportionality constants.