In this paper, we consider the compressive sensing (CS) problem in the presence of noise. Theproblem is to recover a K-sparse signal s ∈ R from noisy linear measurements y = As+w. Wepropose the Fast-Continuous algorithm with large fraction recovery guarantee and full recoveryguarantee. Specifically, with high probability, to recover an arbitrarily large fraction of the sup-p...

We investigate conditions under which the solution of an underdetermined linear system with minimal lp norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Our results highlight the pessimistic nature of sparse recovery analysis when recovery is predicted based on the restricted isometry constants (RIC) of the associated matrix. We construct matrices with RIC δ2m arbitrarily close to 1/ ...

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 unde...

The main shortcoming of sparse recovery with a convex regularizer is that it is a biased estimator and therefore will result in a suboptimal performance in many cases. Recent studies have shown, both theoretically and empirically, that non-convex regularizer is able to overcome the biased estimation problem. Although multiple algorithms have been developed for sparse recovery with non-convex re...

Simultaneous sparse approximation is a generalization of the standard sparse approximation, for simultaneously representing a set of signals using a common sparsity model. Generalizing the compressive sensing concept to the simultaneous sparse approximation yields distributed compressive sensing (DCS). DCS finds the sparse representation of multiple correlated signals using the common + innovat...

Recovery of arbitrarily positioned samples that are missing in sparse signals recently attracted significant research interest. Sparse signals with heavily corrupted arbitrary positioned samples could be analyzed in the same way as compressive sensed signals by omitting the corrupted samples and considering them as unavailable during the recovery process. The reconstruction of missing samples i...

We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block k-sparse signals in no more than k steps if the block-c...

In this paper, we consider the recovery of block sparse signals, whose nonzero entries appear in blocks (or clusters) rather than spread arbitrarily throughout the signal, from incomplete linear measurement. A high order sufficient condition based on block RIP is obtained to guarantee the stable recovery of all block sparse signals in the presence of noise, and robust recovery when signals are ...

In compressive sensing, sparse signals are recovered from underdetermined noisy linear observations. One of the interesting problems which attracted a lot of attention in recent times is the support recovery or sparsity pattern recovery problem. The aim is to identify the non-zero elements in the original sparse signal. In this article we consider the sparsity pattern recovery problem under a p...

Classical results in sparse recovery guarantee the exact reconstruction of a sparse signal under assumptions on the dictionary that are either too strong or NP hard to check. Moreover, such results may be too pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the fo...