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Local Linear Convergence of Inertial Forward-Backward Splitting for Low Complexity Regularization

Jingwei Liang 1 Jalal M. Fadili 1 Gabriel Peyré 2
1 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : In this abstract, we consider the inertial Forward-Backward (iFB) splitting method and its special cases (Forward-Backward/ISTA and FISTA). Under the assumption that the non-smooth part of the objective is partly smooth relative to an active smooth manifold, we show that iFB-type methods (i) identify the active manifold in finite time, then (ii) enter a local linear convergence regime that we characterize precisely. This gives a grounded and unified explanation to the typical behaviour that has been observed numerically for many low-complexity regularizers, including 1 , 1,2-norms, total variation (TV) and nuclear norm to name a few. The obtained results are illustrated by concrete examples.
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Jingwei Liang, Jalal M. Fadili, Gabriel Peyré. Local Linear Convergence of Inertial Forward-Backward Splitting for Low Complexity Regularization. SPARS, 2015, Cambridge, France. ⟨hal-02456434⟩

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