Optimal Dual Certificates for Noise Robustness Bounds in Compressive Sensing

Abstract

The paper deals with optimizing Lipschitz bounds relating locally the reconstruction error to the measurement error in the RIPless compressive sensing framework. Most recent theoretical papers in the field parametrize such bounds relative to certain families of vectors called dual certificates, which are fundamental to several reconstruction criteria. We show in the paper that such a family of bounds admits a unique minimizer that has a deep geometric meaning and can be explicitly constructed via a convex projection algorithm that we describe. We also give a faster greedy algorithm that provides approximate solutions. The algorithms are numerically illustrated and analyzed for different types of sensing matrices, such as random matrices or deterministic matrices issued from tomography and super-resolution.