Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions
Résumé
Efficient recovery of smooth functions which are $s$-sparse with respect to the base of so--called Prolate Spheroidal Wave Functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform $L^\infty$ bound on the measurement ensembles which constitute the columns of the sensing matrix. Such a bound provides us with the Restricted Isometry Property for this rectangular random matrix, which leads to either the exact recovery property or the ``best $s$-term approximation" of the original signal by means of the $\ell^1$ minimization program. The first algorithm considers only a restricted number of columns for which the $L^\infty$ holds as a consequence of the fact that eigenvalues of the Bergman's restriction operator are close to 1 whereas the second one allows for a wider system of PSWF by taking advantage of a preconditioning technique. Numerical examples are spread throughout the text to illustrate the results.
Origine : Fichiers produits par l'(les) auteur(s)