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.

Sparse analysis for mesoscale convective systems tracking

“…In [25], [28], the problem is recast as a semi-definite program, whereas the ADCG solver proposed in [26], [29] relies on an alternating gradient based method which progressively adds Dirac masses. Recently, a variant of the ADCG called Sliding Frank-Wolfe (SFW) appeared in [30], which is guaranteed to converge in a finite number of steps under suitable assumptions.…”

“…To optimize (5), we use the Sliding Frank Wolfe algorithm (SFW), a variant of [26], [29] proposed recently in [30]. This greedy algorithm iteratively adds new Dirac masses to the current set, then optimizes only the weights w t in a classical LASSO setting, and then optimizes locally both w t and θ t using a quasi-Newton method such as BFGS [31].…”

“…Details on its implementation are given in Appendix B. We refer the reader to [30] for further detail on the SFW algorithm.…”

“…Local variations. As suggested in [30], we use a limited-memory bounded BFGS algorithm [34] to restrict the search space to D.…”

Iterative Discretization of Optimization Problems Related to Superresolution

“…A convergence rate beyond the one known to hold generically for any instance of the Frank-Wolfe algorithm is yet to be proven. Here should also be mentioned the recent sliding Frank-Wolfe algorithm, which consists of alternating between a conjugate gradient and continuously changing the positions and amplitudes of the δ x -components [2], [8]. The latter are known to converge in a finite, but as of now unbounded.…”