Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry

We provide a comprehensive study of the convergence 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 fine geometry objective functions relevant in inverse problems signal processing, that have nice behaviour on manifolds, sets open with respect weak topology. Motivated this observation, we revisit those over arbitrary sets. In turn, allows us present several new results well collect unified view variety scattered literature. Our contributions include analysis infinite dimensional convex minimization problems, showing first inequality for quadratic function associated compact operator, derivation linear rates arising from low-complexity priors. approach establish unexpected connections between priori conditions source restricted isometry