Enhancing Sparsity by Reweighted ℓ(1) Minimization

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing

An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition]

This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Our intent in this article is to overview the basic CS theory that emerged in the works [1]–[3], present the key mathematical ideas underlying this theory, and survey a couple of important results in the field. Our goal is to explain CS as plainly as possible, and so our article is mainly of a tutorial nature. One of the charms of this theory is that it draws from various subdisciplines within the applied mathematical sciences, most notably probability theory. In this review, we have decided to highlight this aspect and especially the fact that randomness can — perhaps surprisingly — lead to very effective sensing mechanisms. We will also discuss significant implications, explain why CS is a concrete protocol for sensing and compressing data simultaneously (thus the name), and conclude our tour by reviewing important applications

Matching Pursuit Shrinkage in Hilbert Spaces

International audienceIn this paper, we study a variant of the Matching Pursuit named Matching Pursuit Shrinkage. Similarly to the Matching Pursuit it seeks for an approximation of a datum living in a Hilbert space by a sparse linear expansion in an enumerable set of atoms. The difference with the usual Matching Pursuit is that, once an atom has been selected, we do not erase all the information along the direction of this atom. Doing so, we can evolve slowly along that direction. The goal is to attenuate the negative impact of bad atom selections. We analyse the link between the shrinkage function used by the algorithm and the fact that the result belongs to an lp space