A Generalized Forward-Backward Splitting
Peyré, Gabriel; Fadili, Jalal; Raguet, Hugo (2013), A Generalized Forward-Backward Splitting, SIAM Journal on Imaging Sciences, 6, 3, p. 1199–1226. http://dx.doi.org/10.1137/120872802
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00613637Date
2013Journal name
SIAM Journal on Imaging SciencesVolume
6Number
3Publisher
SIAM
Pages
1199–1226
Publication identifier
Metadata
Show full item recordAbstract (EN)
This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit 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. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Subjects / Keywords
wavelets; total variation; image processing; convex optimization; proximal; splitting; Forward-backward algorithmRelated items
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