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

2005; Elsevier BV; Volume: 86; Issue: 3 Linguagem: Inglês

10.1016/j.sigpro.2005.05.026

1872-7557

Rémi Gribonval, Rosa Ventura, Pierre Vandergheynst,

Blind Source Separation Techniques

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.