In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping of the @math norm at each iteration and gradually reduces the regularization parameter for the @math norm. We prove a global linear convergence of 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 noisy measurements, and (iii) nearly-sparse signal recovery under sub-gaussian noisy 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.
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key technical tool, which represents points in a polytope by convex combinations of sparse vectors. The technique is elementary while yielding sharp results. It is shown that for any given constant t ≥ 4 3, in compressed sensing, δtkA 0, δtkA <; √(t-1 t) + e is not sufficient to guarantee the exact recovery of all k-sparse signals for large k. Similar results also hold for matrix recovery. In addition, the conditions δtkA <; √((t-)1 t) and δtrM <; √((t-1) t) are also shown to be sufficient, respectively, for stable recovery of approximately sparse signals and low-rank matrices in the noisy case.