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Stein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorDELEDALLE, Charles-Alban
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorVAITER, Samuel
hal.structure.identifierEquipe Image - Laboratoire GREYC - UMR6072
dc.contributor.authorFADILI, Jalal M.
hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorPEYRÉ, Gabriel
dc.date.accessioned2024-04-04T03:20:59Z
dc.date.available2024-04-04T03:20:59Z
dc.date.created2014-08-06
dc.date.issued2014-11-25
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/194629
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, it is proved to be also a consistent estimator. This gradient estimate 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 methods including proximal splitting ones and apply our strategy to regularizations involving non-smooth convex structured penalties. Illustrations on various image restoration and matrix completion problems are given.
dc.language.isoen
dc.publisherSociety for Industrial and Applied Mathematics
dc.subject.ensparsity
dc.subject.enproximal splitting
dc.subject.enparameter selection
dc.subject.enSURE
dc.subject.enInverse problem
dc.subject.enrisk estimation
dc.title.enStein Unbiased GrAdient estimator of the Risk (SUGAR) for multiple parameter selection
dc.typeArticle de revue
dc.subject.halInformatique [cs]/Traitement des images
dc.subject.halInformatique [cs]/Traitement du signal et de l'image
dc.subject.halSciences de l'ingénieur [physics]/Traitement du signal et de l'image
dc.subject.halMathématiques [math]/Statistiques [math.ST]
dc.subject.halStatistiques [stat]/Théorie [stat.TH]
dc.subject.halStatistiques [stat]/Applications [stat.AP]
dc.identifier.arxiv1405.1164
bordeaux.journalSIAM Journal on Imaging Sciences
bordeaux.page2448–2487
bordeaux.volume7
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue4
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00987295
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00987295v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=SIAM%20Journal%20on%20Imaging%20Sciences&rft.date=2014-11-25&rft.volume=7&rft.issue=4&rft.spage=2448%E2%80%932487&rft.epage=2448%E2%80%932487&rft.au=DELEDALLE,%20Charles-Alban&VAITER,%20Samuel&FADILI,%20Jalal%20M.&PEYR%C3%89,%20Gabriel&rft.genre=article


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