The Sliding Frank-Wolfe Algorithm for the BLASSO

This paper showcases the Sliding Frank-Wolfe (SFW), which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is the continuous (i.e. off-thegrid or grid-less) counterpart of 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). We prove theoretically that this algorithm terminates in a finite number of steps under a mild nondegeneracy hypothesis.