Publication

Sparse Approximation by Linear Programming using an L1 Data-Fidelity Term

Pierre Vandergheynst
2005
Conference Papers

Abstract

This paper studies the problem of sparse signal approximation over redundant dictionaries. Our attention is focused on the minimization of a cost function where the error is measured by using the L1 norm, giving thus less importance to outliers. We show a constructive equivalence between the proposed minimization problem and Linear Programming. A recovery condition is then provided and an example illustrates the use of such a technique for denoising.

Official source

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Pierre Vandergheynst
2005
Conference Papers

Abstract

Official source

Related concepts (32)

Sparse dictionary learning

Sparse dictionary learning (also known as sparse coding or SDL) is a representation learning method which aims at finding a sparse representation of the input data in the form of a linear combination of basic elements as well as those basic elements themselves. These elements are called atoms and they compose a dictionary. Atoms in the dictionary are not required to be orthogonal, and they may be an over-complete spanning set. This problem setup also allows the dimensionality of the signals being represented to be higher than the one of the signals being observed.

Sparse approximation

Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in , signal processing, machine learning, medical imaging, and more. Consider a linear system of equations , where is an underdetermined matrix and . The matrix (typically assumed to be full-rank) is referred to as the dictionary, and is a signal of interest.

Matching pursuit

Matching pursuit (MP) is a sparse approximation algorithm which finds the "best matching" projections of multidimensional data onto the span of an over-complete (i.e., redundant) dictionary . The basic idea is to approximately represent a signal from Hilbert space as a weighted sum of finitely many functions (called atoms) taken from . An approximation with atoms has the form where is the th column of the matrix and is the scalar weighting factor (amplitude) for the atom . Normally, not every atom in will be used in this sum.

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