This paper studies a fundamental problem that arises in sparse representation and compressed sensing community: can greedy algorithms give us a stable recovery from incomplete and contaminated observations ? Using the Regularized Orthogonal Matching Pursuit (ROMP) algorithm, a modified version of Orthogonal Matching Pursuit (OMP) [1], which was recently introduced by D.Needell and R.Vershynin [...

Polytope Faces Pursuit is an algorithm that solves the standard sparse recovery problem. In this paper, we consider the case of block structured sparsity, and propose a novel algorithm based on the Polytope Faces Pursuit which incorporates this prior knowledge. The so-called Group Polytope Faces Pursuit is a greedy algorithm that adds one group of dictionary atoms at a time and adopts a path fo...

This report studies the effect of introducing a priori knowledge to recover sparse representations when overcomplete dictionaries are used. We focus mainly on Greedy algorithms and Basis Pursuit as for our algorithmic basement, while a priori is incorporated by suitably weighting the elements of the dictionary. A unique sufficient condition is provided under which Orthogonal Matching Pursuit, M...

Compressed sensing is a developing field aiming at reconstruction of sparse signals acquired in reduced dimensions, which make the recovery process under-determined. The required solution is the one with minimum l0 norm due to sparsity, however it is not practical to solve the l0 minimization problem. Commonly used techniques include l1 minimization, such as Basis Pursuit (BP) and greedy pursui...

This report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m) measurements. The new results for OMP are comparable with recent results for a...

In the present paper the method of Basis Pursuit with an overcomplete dictionary is applied to a set of phonemes chosen from the TIMIT database. Emphasis is placed on an analysis of the sparseness of the representation, and its relation to the preservation of the important acoustic cues for the phoneme group. A comparison with other methods, such as Method of Frames, Matching Pursuit and Best O...

Suppose we have a signal y which we wish to represent using a linear combination of a number of basis atoms ai, y = ∑ i xiai = Ax. The problem of finding the minimum l0 norm representation for y is a hard problem. The Basis Pursuit (BP) approach proposes to find the minimum l1 norm representation instead, which corresponds to a linear program (LP) that can be solved using modern LP techniques, ...

In this paper, we analyze the uniqueness of the sparse time frequency decomposition and investigate the efficiency of the nonlinear matching pursuit method. Under the assumption of scale separation, we show that the sparse time frequency decomposition is unique up to an error that is determined by the scale separation property of the signal. We further show that the unique decomposition can be ...

The role of sparse representations in the context of structured noise filtering is discussed. A strategy, especially conceived so as to address problems of an ill posed nature, is presented. The proposed approach revises and extends the Oblique Matching Pursuit technique. It is shown that, by working with an orthogonal projection of the signal to be filtered, it is possible to apply orthogonal ...