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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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1506.08264.pdf (572.8Kb)
Type
Article accepté pour publication ou publié
Date
2016
Nom de la revue
Journal of Fourier Analysis and Applications
Volume
23
Numéro
5
Pages
1153–1194
Identifiant publication
10.1007/s00041-016-9502-x
Métadonnées
Afficher la notice complète
Auteur(s)
Denoyelle, Quentin
Duval, Vincent cc
Peyré, Gabriel
Résumé (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 \(t^{2N-1}\).
Mots-clés
Radon measure; Sparse Signal Processing; Super-resolution; Sparsity; Deconvolution; Convex optimization; LASSO; BLASSO

Publications associées

Affichage des éléments liés par titre et auteur.

  • Vignette de prévisualisation
    Asymptotic of Sparse Support Recovery for Positive Measures 
    Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel (2015) Communication / Conférence
  • Vignette de prévisualisation
    Asymptotic of Sparse Support Recovery for Positive Measures 
    Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel (2015) Communication / Conférence
  • Vignette de prévisualisation
    The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy 
    Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel (2019) Article accepté pour publication ou publié
  • Vignette de prévisualisation
    The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy 
    Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel (2018) Document de travail / Working paper
  • Vignette de prévisualisation
    Sparse Spikes Super-resolution on Thin Grids II: the Continuous Basis Pursuit 
    Duval, Vincent; Peyré, Gabriel (2017) Document de travail / Working paper
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