is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easily-verifiable conditions under which optimally-sparse

Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known

enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving large-scale problems. We use phase diagrams to compare

sparse yet effective solutions. Our examples range from compressed sensing and process flexibility to queuing applications, and from equation systems and optimization problems to game theory models. Keywords sparse; complex; equations; quadratic program; game theory; two-stage stochastic pro

The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues

We first explain the research problem of finding the sparse solution of underdetermined linear systems with some applications. Then we explain three different approaches how to solve the sparse solution: the ℓ1 approach, the orthogonal greedy approach, and the ℓq approach with 0 < q ≤ 1. We

show that if P is k-neighborly, then if a system y = Ax has a solution with at most k nonzeros, that solution is also the unique solution of the convex optimization problem min �x�1 subject to y = Ax; the converse holds as well. This complete equivalence between the study of sparse solution by ℓ 1

This paper introduces a general Bayesian framework for obtaining sparse solutions to regression and classification tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance

Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a

An underdetermined linear system of equations Ax = b with nonnegativity constraint x 0 is considered. It is shown that for matrices A with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required