The degrees of freedom of the Lasso for general design matrix
hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
dc.contributor.author | DOSSAL, Charles | |
dc.contributor.author | KACHOUR, Maher | |
hal.structure.identifier | Equipe Image - Laboratoire GREYC - UMR6072 | |
dc.contributor.author | FADILI, Jalal M. | |
hal.structure.identifier | CEntre de REcherches en MAthématiques de la DEcision [CEREMADE] | |
dc.contributor.author | PEYRÉ, Gabriel | |
hal.structure.identifier | Laboratoire de Mathématiques Nicolas Oresme [LMNO] | |
dc.contributor.author | CHESNEAU, Christophe | |
dc.date.accessioned | 2024-04-04T02:25:05Z | |
dc.date.available | 2024-04-04T02:25:05Z | |
dc.date.created | 2011-08-31 | |
dc.date.issued | 2013 | |
dc.identifier.issn | 1017-0405 | |
dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189860 | |
dc.description.abstractEn | In 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.sponsorship | Adaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009 | |
dc.language.iso | en | |
dc.publisher | Taipei : Institute of Statistical Science, Academia Sinica | |
dc.subject.en | Lasso | |
dc.subject.en | model selection criteria | |
dc.subject.en | degrees of freedom | |
dc.subject.en | SURE | |
dc.title.en | The degrees of freedom of the Lasso for general design matrix | |
dc.type | Article de revue | |
dc.identifier.doi | 10.5705/ss.2011.281 | |
dc.subject.hal | Mathématiques [math]/Statistiques [math.ST] | |
dc.subject.hal | Statistiques [stat]/Théorie [stat.TH] | |
dc.subject.hal | Mathématiques [math]/Théorie de l'information et codage [math.IT] | |
dc.subject.hal | Informatique [cs]/Théorie de l'information [cs.IT] | |
bordeaux.journal | Statistica Sinica | |
bordeaux.page | 809-828 | |
bordeaux.volume | 23 | |
bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
bordeaux.issue | 2 | |
bordeaux.institution | Université de Bordeaux | |
bordeaux.institution | Bordeaux INP | |
bordeaux.institution | CNRS | |
bordeaux.peerReviewed | oui | |
hal.identifier | hal-00638417 | |
hal.version | 1 | |
hal.popular | non | |
hal.audience | Internationale | |
hal.origin.link | https://hal.archives-ouvertes.fr//hal-00638417v1 | |
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