On Theory of Compressive Sensing via L_1-Minimization: Simple Derivations and Extensions

Compressive (or compressed) sensing (CS) is an emerging methodology in computational signal processing that has recently attracted intensive research activities. At present, the basic CS theory includes recoverability and stability: the former quantifies the central fact that a sparse signal of length n can be exactly recovered from much less than n measurements via L_1-minimization or other recovery techniques, while the latter specifies how stable is a recovery technique in the presence of measurement errors and inexact sparsity. So far, most analyses in CS rely heavily on a matrix property called Restricted Isometry Property (RIP). In this paper, we present an alternative, non-RIP analysis for CS via L_1-minimization. Our purpose is three-fold: (a) to introduce an elementary treatment of the CS theory free of RIP and easily accessible to a broad audience, (b) to extend the current recoverability and stability results so that prior knowledge can be utilized to enhance recovery via L_1-minimization; and (c) to substantiate a property called uniform recoverability of L_1-minimization, that is, for almost all random measurement matrices recoverability is asymptotically identical. With the aid of two classic results, the non-RIP approach enables us to derive from scratch all basic results for the extended theory with short and simple derivations.