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Support Recovery for Sparse Super-Resolution of Positive Measures

Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel (2016), Support Recovery for Sparse Super-Resolution of Positive Measures, Journal of Fourier Analysis and Applications, 23, 5, p. 1153–1194. 10.1007/s00041-016-9502-x

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Type
Article accepté pour publication ou publié
Date
2016
Journal name
Journal of Fourier Analysis and Applications
Volume
23
Number
5
Pages
1153–1194
Publication identifier
10.1007/s00041-016-9502-x
Metadata
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Author(s)
Denoyelle, Quentin
Duval, Vincent cc
Peyré, Gabriel
Abstract (EN)
We study sparse spikes super-resolution over the space of Radon measures on \(\mathbb {R}\) or \(\mathbb {T}\) when the input measure is a finite sum of positive Dirac masses using the BLASSO convex program. We focus on the recovery properties of the support and the amplitudes of the initial measure in the presence of noise as a function of the minimum separation t of the input measure (the minimum distance between two spikes). We show that when \({w}/\lambda \), \({w}/t^{2N-1}\) and \(\lambda /t^{2N-1}\) are small enough (where \(\lambda \) is the regularization parameter, w the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level small enough with respect to the minimum separation, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We show that the amplitudes and positions of the spikes of the solution both converge toward those of the input measure when the noise and the regularization parameter drops to zero faster than \(t^{2N-1}\).
Subjects / Keywords
Radon measure; Sparse Signal Processing; Super-resolution; Sparsity; Deconvolution; Convex optimization; LASSO; BLASSO

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