A Generalized Forward-Backward Splitting - Archive ouverte HAL Access content directly
Journal Articles SIAM Journal on Imaging Sciences Year : 2013

A Generalized Forward-Backward Splitting

Abstract

This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators $B + \sum_{i=1}^{n} A_i$, where $B$ is cocoercive. It involves the computation of $B$ in an explicit (forward) step and of the parallel computation of the resolvents of the $A_i$'s in a subsequent implicit (backward) step. We prove its convergence in infinite dimension, and robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions $f + \sum_{i=1}^n g_i$, where $f$ has a Lipschitz-continuous gradient and each $g_i$ is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of $f$ in the forward step, and the proximity operators of the $g_i$'s are applied in parallel in the backward step. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, we generalize it to the case of arbitrary $n$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.
Fichier principal
Vignette du fichier
GFB-siam.pdf (1.32 Mo) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00613637 , version 1 (05-08-2011)
hal-00613637 , version 2 (22-08-2011)
hal-00613637 , version 3 (11-09-2011)
hal-00613637 , version 4 (12-01-2012)
hal-00613637 , version 5 (02-09-2012)

Identifiers

Cite

Hugo Raguet, Jalal M. Fadili, Gabriel Peyré. A Generalized Forward-Backward Splitting. SIAM Journal on Imaging Sciences, 2013, 6 (13), pp.1199-1226. ⟨10.1137/120872802⟩. ⟨hal-00613637v5⟩
1044 View
2400 Download

Altmetric

Share

Gmail Facebook X LinkedIn More