# The degrees of freedom of the Lasso for general design matrix

3 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : 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.
Keywords :
Type de document :
Article dans une revue
Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2013, 23 (2), pp.809-828. 〈10.5705/ss.2011.281〉
Domaine :

Littérature citée [32 références]

https://hal.archives-ouvertes.fr/hal-00638417
Soumis le : lundi 28 mai 2012 - 22:42:36
Dernière modification le : vendredi 13 octobre 2017 - 19:38:04
Document(s) archivé(s) le : jeudi 15 décembre 2016 - 10:03:21

### Fichier

version_rev.pdf
Fichiers produits par l'(les) auteur(s)

### Citation

Charles Dossal, Maher Kachour, Jalal M. Fadili, Gabriel Peyré, Christophe Chesneau. The degrees of freedom of the Lasso for general design matrix. Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2013, 23 (2), pp.809-828. 〈10.5705/ss.2011.281〉. 〈hal-00638417v3〉

Consultations de
la notice

## 638

Téléchargements du document