We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector x from its lower-dimensional sketch Ax. A popular way of performing this recovery is by finding x* such that Ax=Ax*, and ||x*||_1 is minimal. It is known that this approach ``works'' if A is a random *dense* matrix, chosen from a proper distribution.In this paper, we investigate this procedure for the case where A is binary and *very sparse*. We show that, both in theory and in practice, sparse matrices are essentially as ``good'' as the dense ones. At the same time, sparse binary matrices provide additional benefits, such as reduced encoding and decoding time.
