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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which at- tempts to recover the ideal sparse signal by solving a convex pro- gram. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis. [ABSTRACT FROM AUTHOR]

This article presents new results on using a greedy ale gorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasie incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function Is Introduced to quantify the level of Incoherence. This analysis unifies all the recent results OMP and extends them to OMR Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP Is an approximation algorithm for the sparse problem over a quasi-Incoherent dictionary That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms. [ABSTRACT FROM AUTHOR]

This correspondence closes a gap in the proof of the main lemma from the paper "Just relax: Convex programming methods for identifying sparse signals in noise" (2006). [ABSTRACT FROM AUTHOR]