Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective
\nfunctions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those
\ngeometric notions over arbitrary sets. In turn, this allows us to present several new results as well
\nas collect in a unified view a variety of results scattered in the literature. Our contributions include
\nthe analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz
\ninequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach
\nallows to establish unexpected connections between geometry and a priori conditions in inverse
\nproblems, such as source conditions, or restricted isometry properties.