Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

Abstract : 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.
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Laurent Gosse. Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions. Annali dell'Universita di Ferrara, Springer Verlag, 2012, pp.81-116. ⟨10.1007/s11565-012-0159-3⟩. ⟨hal-00560962v2⟩

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