Average performance of the approximation in a dictionary using an l0 objective

We consider the minimization of the number of non-zero coefficients (the eo "norm") of the representation of a data set in a general dictionary under a fidelity constraint. This (nonconvex) optimization problem leads to the sparsest approximation. The average performance of the model consists in the probability (on the data) to obtain a K-sparse solution-involving at most K nonzero components-from data uniformly distributed on a domain. These probabilities are expressed in terms of the parameters of the model and the accuracy of the approximation. We comment the obtained formulas and give a simulation. To cite this article: F Malgouyres, M. Nikolova, C R. Acad. Sci. Paris, Ser. 1347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.