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

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 suboptimal 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. (C) 2005 Elsevier B.V. All rights reserved.