Theory of Compressive Sensing via ℓ1-Minimization: A Non-RIP Analysis and Extensions

Abstract

Compressive 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 far fewer than n measurements via ℓ 1-minimization or other recovery techniques, while the latter specifies 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 ℓ 1-minimization. Our purpose is three-fold: (a) to introduce an elementary and RIP-free treatment of the basic CS theory; (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 quickly derive from scratch all basic results for the extended theory. © 2013 Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg.