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The degrees of freedom of penalized l1 minimization

dc.contributor.authorChesneau, Christophe
dc.contributor.authorPeyré, Gabriel
dc.contributor.authorFadili, Jalal
dc.contributor.authorKachour, Maher
dc.contributor.authorDossal, Charles
dc.date.accessioned2011-11-09T14:44:10Z
dc.date.available2011-11-09T14:44:10Z
dc.date.issued2011
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/7429
dc.language.isoenen
dc.subjectSUREen
dc.subjectdegrees of freedomen
dc.subjectmodel selection criteriaen
dc.subjectLassoen
dc.subject.ddc519en
dc.titleThe degrees of freedom of penalized l1 minimizationen
dc.typeDocument de travail / Working paper
dc.contributor.editoruniversityotherLaboratoire de Mathématiques Nicolas Oresme (LMNO) http://www.math.unicaen.fr/lmno/ CNRS : UMR6139 – Université de Caen;France
dc.contributor.editoruniversityotherGroupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen (GREYC) http://www.greyc.unicaen.fr/ CNRS : UMR6072 – Université de Caen – Ecole Nationale Supérieure d'Ingénieurs de Caen;France
dc.contributor.editoruniversityotherInstitut de Mathématiques de Bordeaux (IMB) http://www.math.u-bordeaux.fr/IMB/ CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II;France
dc.description.abstractenIn this paper, we investigate the degrees of freedom (df) of penalized l1 minimization (also known as the Lasso) for linear regression models. We give a closed-form expression of the degrees of freedom of the Lasso response. Namely, we show that for any given Lasso regularization parameter$ \lambda$ and any observed data y belongs to a set of full measure, the cardinal of the support of a particular solution of the Lasso problem is an unbiased estimator of the degrees of freedom of the Lasso response. This work is achieved without any assumption on the uniqueness of the Lasso solution. Thus, our result remains true for both the underdetermined and the overdetermined case studied originally in Zou et al.. We also prove that a key result in Zou et al. is not true by providing a simple counterexample. 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.en
dc.publisher.nameUniversité Paris-Dauphineen
dc.publisher.cityParisen
dc.identifier.citationpages15en
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00638417/fr/en
dc.description.sponsorshipprivateouien
dc.subject.ddclabelProbabilités et mathématiques appliquéesen


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