# Exact Support Recovery for Sparse Spikes Deconvolution

arXiv: https://arxiv.org/abs/1306.6909 · HTML: https://mlsanity.com/paper/1306.6909/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2013.

## Built on by (most cited first, 22 of 22)

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- [Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures](https://mlsanity.com/paper/2507.08444/) (2025)
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## Abstract (arXiv)

This paper studies sparse spikes deconvolution over the space of measures. We focus our attention to the recovery properties of the support of the measure, i.e. the location of the Dirac masses. For non-degenerate sums of Diracs, we show that, when the signal-to-noise ratio is large enough, total variation regularization (which is the natural extension of the L1 norm of vectors 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. We draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.