Phase transition of joint-sparse recovery from multiple measurements via convex optimization

Abstract

In sparse signal recovery of compressive sensing, the phase transition determines the edge, which separates successful recovery and failed recovery. Moreover, the width of phase transition determines the vague region, where sparse recovery is achieved in a probabilistic manner. Earlier works on phase transition analysis in either single measurement vector (SMV) or multiple measurement vectors (MMVs) is too strict or ideal to be satisfied in real world. Recently, phase transition analysis based on conic geometry has been found to close the gap between theoretical analysis and practical recovery result for SMV. In this paper, we explore a rigorous analysis on phase transition of MMVs. Such an extension is not intuitive at all since we need to redefine the null space and descent cone, and evaluate the statistical dimension for l 2,1 -norm. By presenting the necessary and sufficient condition of successful recovery from MMVs, we can have a boundary on the probability that the solution of a MMVs recovery problem by convex programming is successful or not. Our theoretical analysis is verified to accurately predict the practical phase transition diagram of MMVs.