Publication

On the exponential convergence of matching pursuits in quasi-incoherent dictionaries

Pierre Vandergheynst
2004
Reports

Abstract

The purpose of this paper is to extend results by Villemoes about exponential convergence of Matching Pursuit with some structured dictionaries for simple functions in finite or infinite dimension. Our results are based on an extension of Tropp s results about Orthogonal Matching Pursuit in finite dimension, with the observation that it does not only work for OMP but also for MP. Our main contribution is a detailed analysis of the approximation and stability properties of MP with quasi-incoherent dictionaries, and a bound on the number of steps sufficient to reach an error no larger than a penalization factor times the best m-term approximation error.

Official source

https://infoscience.epfl.ch/entities/publication/cc6f0dec-b3b0-49fe-ad6e-a981caa6fb7d

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Pierre Vandergheynst
2004
Reports

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/cc6f0dec-b3b0-49fe-ad6e-a981caa6fb7d

About this result

Related concepts (32)

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.

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.

Basis pursuit

Basis pursuit is the mathematical optimization problem of the form where x is a N-dimensional solution vector (signal), y is a M-dimensional vector of observations (measurements), A is a M × N transform matrix (usually measurement matrix) and M < N. It is usually applied in cases where there is an underdetermined system of linear equations y = Ax that must be exactly satisfied, and the sparsest solution in the L1 sense is desired. When it is desirable to trade off exact equality of Ax and y in exchange for a sparser x, basis pursuit denoising is preferred.