Sparse spikes super-resolution on thin grids II: the continuous basis pursuit

Abstract

This article analyzes the performance of the continuous basis pursuit (C-BP) method for sparse super-resolution. The C-BP has been recently proposed by Ekanadham, Tranchina and Simoncelli as a refined discretization scheme for the recovery of spikes in inverse problems regularization. One of the most well known discretization scheme, the basis pursuit (BP, also known as ) makes use of a finite dimensional norm on a grid. In contrast, the C-BP uses a linear interpolation of the spikes positions to enable the recovery of spikes between grid points. When the sought-after solution is constrained to be positive, a remarkable feature of this approach is that it retains the convexity of the initial problem. The present paper shows how the C-BP is able to recover the spikes locations with sub-grid accuracy in the favorable case. We also prove that this regime generally breaks when the grid is too thin, and we describe precisely the artifacts that appear: each spike is approximated by a pair of Dirac masses. We show numerical illustrations of these phenomena, and evaluate numerically the validity of the technical assumptions of our analysis.