A simple test to check the optimality of a sparse signal approximation

Authors:
R. Gribonval;R. M. Figueras i Ventura;P. Vandergheynst
Affiliations:
IRISA-INRIA, Rennes Cedex, France;Signal Processing Institute, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland;Signal Processing Institute, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland
Venue:
Signal Processing - Sparse approximations in signal and image processing
Year:
2006

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Abstract

Approximating a signal or an image with a sparse linear expansion from an overcomplete dictionary of atoms is an extremely useful tool to solve many signal processing problems. Finding the sparsest approximation of a signal from an arbitrary dictionary is a NP-hard problem. Despite this, several algorithms have been proposed that provide sub-optimal solutions. However, it is generally difficult to know how close the computed solution is to being "optimal", and whether another algorithm could provide a better result. In this paper we provide a simple test to check whether the output of a sparse approximation algorithm is nearly optimal, in the sense that no significantly different linear expansion from the dictionary can provide both a smaller approximation error and a better sparsity. As a by-product of our theorems, we obtain results on the identifiability of sparse overcomplete models in the presence of noise, for a fairly large class of sparse priors.