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The degrees of freedom of the Lasso for general design matrix

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Date
2013
Link to item file
http://hal.archives-ouvertes.fr/hal-00638417
Dewey
Probabilités et mathématiques appliquées
Sujet
SURE; degrees of freedom; model selection criteria; Lasso
Journal issue
Statistica Sinica
Volume
23
Number
2
Publication date
2013
Article pages
809-828
Publisher
Institute of Statistical Science, Academia Sinica
DOI
http://dx.doi.org/10.5705/ss.2011.281
URI
https://basepub.dauphine.fr/handle/123456789/9457
Collections
  • CEREMADE : Publications
Metadata
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Author
Chesneau, Christophe
Peyré, Gabriel
Fadili, Jalal
Kachour, Maher
Dossal, Charles
Type
Article accepté pour publication ou publié
Abstract (EN)
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.

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