Combining geometry and combinatorics: A unified approach to sparse signal recovery

There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix and then uses linear programming to decode information about the signal from its measurements. The combinatorial approach constructs the measurement matrix and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of high-quality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the l_p norm for p about 1, and then show that unbalanced expanders are essentially equivalent to RIP-p matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance.