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

arXiv: https://arxiv.org/abs/1811.06416 · HTML: https://mlsanity.com/paper/1811.06416/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2018.

## Built on by (most cited first, 16 of 16)

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- [Gridless 2D Recovery of Lines using the Sliding Frank-Wolfe Algorithm](https://mlsanity.com/paper/2403.11649/) (2024)
- [On the Logic Elements Associated with Round-Off Errors and Gaussian Blur in Image Registration: A Simple Case of Commingling](https://mlsanity.com/paper/2502.11992/) (2025)
- [Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures](https://mlsanity.com/paper/2507.08444/) (2025)
- [Unlocking Off-the-Grid Sparse Recovery with Unlimited Sensing: Simultaneous Super-Resolution in Time and Amplitude](https://mlsanity.com/paper/2510.16948/) (2025)
- [Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances](https://mlsanity.com/paper/2609.30220/) (2026)

## Abstract (arXiv)

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.