Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Mathematical Programming Année : 2022

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

Résumé

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 functions 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 geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality 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 allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.
Fichier principal
Vignette du fichier
1703.09477.pdf (590.65 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03886199 , version 1 (06-12-2022)

Identifiants

Citer

Guillaume Garrigos, Lorenzo Rosasco, Silvia Villa. Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry. Mathematical Programming, In press, ⟨10.1007/s10107-022-01809-4⟩. ⟨hal-03886199⟩
7 Consultations
54 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More