Within this chapter we present recent results on sparse recovery algorithms for inverse and ill-posed problems, i.e. we focus on those inverse problems in which we can assume that the solution has a sparse series expansion with respect to a preassigned basis or frame. The presented approaches to approximate solutions of inverse problems are limited to iterative strategies that essentially rely ...

The problem of jointly sparse support recovery is to determine the common support of jointly sparse signal vectors from multiple measurement vectors (MMV) related to the signals by a linear transformation. The fundamental limit of performance has been studied in terms of a so-called algebraic bound, relating the maximum recoverable sparsity level to the spark of the sensing matrix and the rank ...

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 l1 norm at each iteration and gradually reduces the regularization parameter for the l1 norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing under three diff...

We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses ...

Consider n data sequences, each consisting of independent and identically distributed elements drawn from one of the two possible zero-mean Gaussian distributions with variances A0 and A1. The problem of quickly identifying all of the sequences with varianceA1 is considered and an adaptive two-stage experimental design and testing procedure is proposed. The agility and reliability gains in comp...

We consider the problem of exact support recovery of sparse signals via noisy measurements. The main focus is the sufficient and necessary conditions on the number of measurements for support recovery to be reliable. By drawing an analogy between the problem of support recovery and the problem of channel coding over the Gaussian multiple access channel, and exploiting mathematical tools develop...

We consider the problem of recovering the parameter α ∈ R of a sparse function f (i.e. the number of non-zero entries of α is small compared to the numberK of features) given noisy evaluations of f at a set of well-chosen sampling points. We introduce an additional randomization process, called Brownian sensing, based on the computation of stochastic integrals, which produces a Gaussian sensing...

These lecture notes are an introduction to methods recently developed for performing numerical optimizations with linear model constraints and additional sparsity conditions to solutions, i.e. we expect solutions which can be represented as sparse vectors with respect to a prescribed basis. Such a type of problems has been recently greatly popularized by the development of the field of nonadapt...

In this paper, we introduce a subspace-augmented MUSIC technique for recovering the joint sparse support of a signal ensemble corrupted by additive impulsive noise. Our approach uses multiple vectors of random compressed measurements and employs fractional lower-order moments stemming from modeling the underlying signal statistics with symmetric alpha-stable distributions. We show through simul...