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Local Linear Convergence of Douglas-Rachford/ADMM for Low Complexity Regularization

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sparsdrrate.pdf (289.4Kb)
Date
2015
Dewey
Traitement du signal
Sujet
Douglas-Rachford; ADMM
Conference name
SPARS
Conference date
2015
Conference city
Cambridge
Conference country
UNITED KINGDOM
URI
https://basepub.dauphine.fr/handle/123456789/20867
Collections
  • CEREMADE : Publications
Metadata
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Author
Liang, Jingwei
150 Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen [GREYC]
Fadili, Jalal M.
150 Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen [GREYC]
Peyré, Gabriel
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Luke, Russell
100659 Mathematics Department [Gottingen] [NAM]
Type
Communication / Conférence
Abstract (EN)
The Douglas-Rachford (DR) and ADMM algorithms have become popular to solve sparse recovery problems and beyond. The goal of this work is to understand the local convergence behaviour of DR/ADMM which have been observed in practice to exhibit local linear convergence. We show that when the involved functions (resp. their Legendre-Fenchel conjugates) are partly smooth, the DR (resp. ADMM) method identifies their associated active manifolds in finite time. Moreover, when these functions are partly polyhedral, we prove that DR (resp. ADMM) is locally linearly convergent with a rate in terms of the cosine of the Friedrichs angle between the tangent spaces of the two active manifolds. This is illustrated by several concrete examples and supported by numerical experiments.

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