Title: Convergence of the Forward-Backward Algorithm: Beyond the Worst Case with the Help of Geometry,
Abstract: We provide a comprehensive study of the convergence of forward-backward
algorithm under suitable geometric conditions leading to fast rates. We present
several new results and collect in a unified view a variety of results
scattered in the literature, often providing simplified proofs. Novel
contributions include the analysis of infinite dimensional convex minimization
problems, allowing the case where minimizers might not exist. Further, we
analyze the relation between different geometric conditions, and discuss novel
connections with a priori conditions in linear inverse problems, including
source conditions, restricted isometry properties and partial smoothness.