
The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy
Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel (2018), The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy. https://basepub.dauphine.fr/handle/123456789/18449
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Type
Document de travail / Working paperExternal document link
https://hal.archives-ouvertes.fr/hal-01921604Date
2018Series title
Cahier de recherche CEREMADE, Université Paris-DauphinePages
42
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Show full item recordAuthor(s)
Denoyelle, QuentinDuval, Vincent

CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Peyré, Gabriel
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Soubies, Emmanuel

Abstract (EN)
This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.Subjects / Keywords
Frank-Wolfe Algorithm; Super-Resolution MicroscopyRelated items
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Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel (2019) Article accepté pour publication ou publié
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Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel (2016) Article accepté pour publication ou publié
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Duval, Vincent; Peyré, Gabriel (2017) Document de travail / Working paper
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