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The degrees of freedom of the Lasso for general design matrix

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorDOSSAL, Charles
dc.contributor.authorKACHOUR, Maher
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.identifierLaboratoire de Mathématiques Nicolas Oresme [LMNO]
dc.contributor.authorCHESNEAU, Christophe
dc.date.accessioned2024-04-04T02:25:05Z
dc.date.available2024-04-04T02:25:05Z
dc.date.created2011-08-31
dc.date.issued2013
dc.identifier.issn1017-0405
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189860
dc.description.abstractEnIn this paper, we investigate the degrees of freedom ($\dof$) of penalized $\ell_1$ minimization (also known as the Lasso) for linear regression models. We give a closed-form expression of the $\dof$ of the Lasso response. Namely, we show that for any given Lasso regularization parameter $\lambda$ and any observed data $y$ belonging to a set of full (Lebesgue) measure, the cardinality of the support of a particular solution of the Lasso problem is an unbiased estimator of the degrees of freedom. This is achieved without the need of uniqueness of the Lasso solution. Thus, our result holds true for both the underdetermined and the overdetermined case, where the latter was originally studied in \cite{zou}. We also show, by providing a simple counterexample, that although the $\dof$ theorem of \cite{zou} is correct, their proof contains a flaw since their divergence formula holds on a different set of a full measure than the one that they claim. An effective estimator of the number of degrees of freedom may have several applications including an objectively guided choice of the regularization parameter in the Lasso through the $\sure$ framework. Our theoretical findings are illustrated through several numerical simulations.
dc.description.sponsorshipAdaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009
dc.language.isoen
dc.publisherTaipei : Institute of Statistical Science, Academia Sinica
dc.subject.enLasso
dc.subject.enmodel selection criteria
dc.subject.endegrees of freedom
dc.subject.enSURE
dc.title.enThe degrees of freedom of the Lasso for general design matrix
dc.typeArticle de revue
dc.identifier.doi10.5705/ss.2011.281
dc.subject.halMathématiques [math]/Statistiques [math.ST]
dc.subject.halStatistiques [stat]/Théorie [stat.TH]
dc.subject.halMathématiques [math]/Théorie de l'information et codage [math.IT]
dc.subject.halInformatique [cs]/Théorie de l'information [cs.IT]
bordeaux.journalStatistica Sinica
bordeaux.page809-828
bordeaux.volume23
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue2
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00638417
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00638417v1
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