Résumé en anglais

This article analyzes the recovery performance of two popular finitedimensional approximations of the sparse spikes deconvolution problem over Radonmeasures. We examine in a unified framework both theℓ1regularization (oftenreferred to asLassoor Basis-Pursuit) and the Continuous Basis-Pursuit (C-BP)methods. TheLassois the de-facto standard for the sparse regularization of inverseproblems in imaging. It performs a nearest neighbor interpolation ofthe spikeslocations on the sampling grid. The C-BP method, introduced by Ekanadham,Tranchina and Simoncelli, uses a linear interpolation of the locations toperform abetter approximation of the infinite-dimensional optimization problem, for positivemeasures. We show that, in the small noise regime, both methods estimate twice thenumber of spikes as the number of original spikes. Indeed, we showthat they bothdetect two neighboring spikes around the locations of an original spikes. These resultsfor deconvolution problems are based on an abstract analysis of the so-called extendedsupport of the solutions ofℓ1-type problems (including as special cases theLassoand C-BP for deconvolution), which are of an independent interest. They preciselycharacterize the support of the solutions when the noise is small and the regularizationparameter is selected accordingly. We illustrate these findings to analyze for thefirst time the support instability of compressed sensing recovery when the numberof measurements is below the critical limit (well documented in the literature) wherethe support is provably stable.