A Fast Noniterative Algorithm for Compressive Sensing Using Binary Measurement Matrices

In this paper we present a new algorithm for compressive sensing that makes use of binary measurement matrices and achieves exact recovery of sparse vectors, in a single pass, without any iterations. Our algorithm is hundreds of times faster than $\ell_1$-norm minimization, and methods based on expander graphs (which require multiple iterations). Moreover, our method requires the fewest measurements amongst all methods that use binary measurement matrices. The algorithm can accommodate nearly sparse vectors, in which case it recovers the largest components, and can also Numerical experiments with randomly generated sparse vectors indicate that the sufficient conditions for our algorithm to work are very close to being necessary. In contrast, the best known sufficient condition for $\ell_1$-norm minimization to recover a sparse vector, namely the Restricted Isometry Property (RIP), is about thirty times away from being necessary. Therefore it would be worthwhile to explore alternate and improved sufficient conditions for $\ell_1$-norm minimization to achieve the recovery of sparse vectors.