DOSSAL, Charles; KACHOUR, Maher; FADILI, Jalal M.; PEYRÉ, Gabriel; CHESNEAU, Christophe

doi:10.5705/ss.2011.281

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DOSSAL, Charles
Institut de Mathématiques de Bordeaux [IMB]

KACHOUR, Maher

FADILI, Jalal M.
Equipe Image - Laboratoire GREYC - UMR6072

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Article de revue

Ce document a été publié dans

Statistica Sinica. 2013, vol. 23, n° 2, p. 809-828

Taipei : Institute of Statistical Science, Academia Sinica

Résumé en anglais

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.< Réduire

Mots clés en anglais

Lasso

model selection criteria

degrees of freedom

SURE

URI

https://oskar-bordeaux.fr/handle/20.500.12278/189860

DOI

http://dx.doi.org/10.5705/ss.2011.281

Project ANR

Adaptivité pour la représentation des images naturelles et des textures - ANR-08-EMER-0009

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251