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The degrees of freedom of penalized l1 minimization

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Date
2011
Publisher city
Paris
Publisher
Université Paris-Dauphine
Link to item file
http://hal.archives-ouvertes.fr/hal-00638417/fr/
Dewey
Probabilités et mathématiques appliquées
Sujet
SURE; degrees of freedom; model selection criteria; Lasso
URI
https://basepub.dauphine.fr/handle/123456789/7429
Collections
  • CEREMADE : Publications
Metadata
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Author
Chesneau, Christophe
Peyré, Gabriel
Fadili, Jalal
Kachour, Maher
Dossal, Charles
Type
Document de travail / Working paper
Item number of pages
15
Abstract (EN)
In 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.

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