A Simple Homotopy Proximal Mapping Algorithm for Compressive Sensing

In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping for l1 norm regularization at each iteration and gradually reduces the regularization parameter of the l1 norm. We prove a global linear convergence for the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing under three different settings (i) sparse signal recovery under noiseless measurements, (ii) sparse signal recovery under noise measurements, and (iii) nearlysparse signal recovery under sub-gaussian noise measurements. In particular, we show that when the measurement matrix satisfies Restricted Isometric Properties (RIP), our theoretical results in settings (i) and (ii) almost recover the best condition on the RIP constants for compressive sensing. In addition, in setting (iii), our results for sparse signal recovery are better than the previous results, and furthermore our analysis explicitly exhibits that more observations lead to not only more accurate recovery but also faster convergence. Compared with previous studies on linear convergence for sparse signal recovery, our algorithm is simple and efficient, and our results are better and provide more insights. Finally our empirical studies provide further support for the proposed homotopy proximal mapping algorithm and verify the theoretical results.