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The degrees of freedom of partly smooth regularizers

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorVAITER, Samuel
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorDELEDALLE, Charles-Alban
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
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorDOSSAL, Charles
dc.date.created2015-10-23
dc.date.issued2017-08
dc.identifier.issn0020-3157
dc.description.abstractEnWe study regularized regression problems where the regularizer is a proper, lower-semicontinuous, convex and partly smooth function relative to a Riemannian submanifold. This encompasses several popular examples including the Lasso, the group Lasso, the max and nuclear norms, as well as their composition with linear operators (e.g., total variation or fused Lasso). Our main sensitivity analysis result shows that the predictor moves locally stably along the same active submanifold as the observations undergo small perturbations. This plays a pivotal role in getting a closed-form expression for the divergence of the predictor w.r.t. observations. We also show that, for many regularizers, including polyhedral ones or the analysis group Lasso, this divergence formula holds Lebesgue a.e. When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and the prediction risk. Our results unify and go beyond those already known in the literature.
dc.language.isoen
dc.publisherSpringer Verlag
dc.subject.enManifold
dc.subject.enO-minimal structures
dc.subject.enModel selection
dc.subject.enSparsity
dc.subject.enDegrees of freedom
dc.subject.enSemi-algebraic sets
dc.subject.enTotal variation
dc.subject.enGroup Lasso
dc.subject.enPartial smoothness
dc.title.enThe degrees of freedom of partly smooth regularizers
dc.typeArticle de revue
dc.identifier.doi10.1007/s10463-016-0563-z
dc.subject.halMathématiques [math]/Statistiques [math.ST]
dc.subject.halMathématiques [math]/Théorie de l'information et codage [math.IT]
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.halStatistiques [stat]/Théorie [stat.TH]
dc.subject.halInformatique [cs]/Théorie de l'information [cs.IT]
dc.identifier.arxiv1404.5557
dc.description.sponsorshipEuropeSparsity, Image and Geometry to Model Adaptively Visual Processings
bordeaux.journalAnnals of the Institute of Statistical Mathematics
bordeaux.page791 – 832
bordeaux.volume69
bordeaux.issue4
bordeaux.peerReviewedoui
hal.identifierhal-00981634
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00981634v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Annals%20of%20the%20Institute%20of%20Statistical%20Mathematics&rft.date=2017-08&rft.volume=69&rft.issue=4&rft.spage=791%20%E2%80%93%20832&rft.epage=791%20%E2%80%93%20832&rft.eissn=0020-3157&rft.issn=0020-3157&rft.au=VAITER,%20Samuel&DELEDALLE,%20Charles-Alban&FADILI,%20Jalal%20M.&PEYR%C3%89,%20Gabriel&DOSSAL,%20Charles&rft.genre=article


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