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 quanties the central fact that a sparse signal of length n can be exactly recovered from far fewer than n measurements via ‘1minimization or other recovery techniques, while the latter species the stability of a recovery technique in the presence of measurement errors and inexact sparsity. So far, most analyses in CS rely heavily on the Restricted Isometry Property (RIP) for matrices. In this paper, we present an alternative, non-RIP analysis for CS via ‘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 ‘1-minimization; and (c) to substantiate a property called uniform recoverability of ‘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 quickly derive from scratch all basic results for the extended theory.