The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01921604
Contributor : Gabriel Peyré <>
Submitted on : Tuesday, November 13, 2018 - 9:19:41 PM
Last modification on : Friday, April 19, 2019 - 4:55:08 PM
Long-term archiving on : Thursday, February 14, 2019 - 4:55:59 PM

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  • HAL Id : hal-01921604, version 1
  • ARXIV : 1811.06416

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Quentin Denoyelle, Vincent Duval, Gabriel Peyré, Emmanuel Soubies. The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy. 2018. ⟨hal-01921604⟩

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