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.

Cited by in corpus (28)

• On the linear convergence rates of exchange and continuous methods for total variation minimization
• A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization
• Projected gradient descent for non-convex sparse spike estimation
• Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon's Theorem Meets Compressive Sensing
• Fast Convolutional Dictionary Learning off the Grid
• TV-based Reconstruction of Periodic Functions
• Asymptotic linear convergence of fully-corrective generalized conditional gradient methods
• When does OMP achieve exact recovery with continuous dictionaries?
• Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems
• Optimal Influencer Marketing Campaign Under Budget Constraints Using Frank-Wolfe
• Functional Penalised Basis Pursuit on Spheres
• A Fast and Scalable Polyatomic Frank-Wolfe Algorithm for the LASSO
• Towards Off-the-grid Algorithms for Total Variation Regularized Inverse Problems
• Sketching Datasets for Large-Scale Learning (long version)
• Off-the-grid learning of mixtures from a continuous dictionary
• On the Uniqueness of Solutions for the Basis Pursuit in the Continuum
• Towards optimal sensor placement for inverse problems in spaces of measures
• Proximal methods for point source localisation
• MARS via LASSO
• Localization of point scatterers via sparse optimization on measures
• FastPart: Over-Parameterized Stochastic Gradient Descent for Sparse optimisation on Measures
• A fast Primal-Dual-Active-Jump method for minimization in $\mathrm{BV}((0,T);{\mathbb{R}}^d)$
• Boosting the Sliding Frank-Wolfe solver for 3D deconvolution
• COL0RME: Super-resolution microscopy based on sparse blinking/fluctuating fluorophore localization and intensity estimation
• Sparse Source Identification in Transient Advection-Diffusion Problems with a Primal-Dual-Active-Point Strategy
• Gridless 3D Recovery of Image Sources from Room Impulse Responses
• Atomic Super-Resolution Tomography
• Sketch and shift: a robust decoder for compressive clustering