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dc.contributor.authorFadili, Jalal*
dc.contributor.authorPeyré, Gabriel*
dc.contributor.authorVaiter, Samuel*
dc.contributor.authorDeledalle, Charles-Alban*
dc.subjectlow ranken
dc.subjectproximal splittingen
dc.subjectparameter selectionen
dc.subjectrisk estimationen
dc.subjectInverse problemen
dc.titleStein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selectionen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherGroupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC) CNRS : UMR6072 – Université de Caen Basse-Normandie – Ecole Nationale Supérieure d'Ingénieurs de Caen;France
dc.contributor.editoruniversityotherInstitut de Mathématiques de Bordeaux (IMB) CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II;France
dc.description.abstractenAlgorithms to solve variational regularization of ill-posed inverse problems usually involve operators that depend on a collection of continuous parameters. When these operators enjoy some (local) regularity, these parameters can be selected using the so-called Stein Unbiased Risk Estimate (SURE). While this selection is usually performed by exhaustive search, we address in this work the problem of using the SURE to efficiently optimize for a collection of continuous parameters of the model. When considering non-smooth regularizers, such as the popular l1-norm corresponding to soft-thresholding mapping, the SURE is a discontinuous function of the parameters preventing the use of gradient descent optimization techniques. Instead, we focus on an approximation of the SURE based on finite differences as proposed in (Ramani et al., 2008). Under mild assumptions on the estimation mapping, we show that this approximation is a weakly differentiable function of the parameters and its weak gradient, coined the Stein Unbiased GrAdient estimator of the Risk (SUGAR), provides an asymptotically (with respect to the data dimension) unbiased estimate of the gradient of the risk. Moreover, in the particular case of soft-thresholding, the SUGAR is proved to be also a consistent estimator. The SUGAR can then be used as a basis to perform a quasi-Newton optimization. The computation of the SUGAR relies on the closed-form (weak) differentiation of the non-smooth function. We provide its expression for a large class of iterative proximal splitting methods and apply our strategy to regularizations involving non-smooth convex structured penalties. Illustrations on various image restoration and matrix completion problems are given.en
dc.relation.isversionofjnlnameSIAM Journal on Imaging Sciences
dc.subject.ddclabelTraitement du signalen

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