  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.
