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On combinatorial approaches to compressed sensing

In this paper, we look at combinatorial algorithms for Compressed Sensing from a different perspective. We show that certain combinatorial solvers are in fact recursive implementations of convex relaxation methods for solving compressed sensing, under the assumption of sparsity for the projection matrix. We extend the notion of sparse binary projection matrices to sparse real-valued ones. We prove that, contrary to their binary counterparts, this class of sparse-real matrices has the Restricted Isometry Property. Finally, we generalize the voting mechanism (employed in combinatorial algorithms) to notions of isolation/alignment and present the required solver for real-valued sparse projection matrices based on such isolation/alignment mechanisms.