Exact Support Recovery for Sparse Spikes Deconvolution

This paper studies sparse spikes deconvolution over the space of measures. For non-degenerate sums of Diracs, we show that, when the signalto-noise ratio is large enough, total variation regularization (which the natural extension of ` norm of vector to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. Finally we draw connections between the performances of sparse recovery on a continuous domain and on a discretized grid.