The degrees of freedom of the Lasso for general design matrix - Archive ouverte HAL Access content directly
Journal Articles Statistica Sinica Year : 2013

The degrees of freedom of the Lasso for general design matrix

Charles H Dossal
  • Function : Author
  • PersonId : 1069990
Maher Kachour
  • Function : Author
Jalal M. Fadili


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.
Fichier principal
Vignette du fichier
version_rev.pdf (630.79 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00638417 , version 1 (04-11-2011)
hal-00638417 , version 2 (27-03-2012)
hal-00638417 , version 3 (28-05-2012)



Charles H Dossal, Maher Kachour, Jalal M. Fadili, Gabriel Peyré, Christophe Chesneau. The degrees of freedom of the Lasso for general design matrix. Statistica Sinica, 2013, 23 (2), pp.809-828. ⟨10.5705/ss.2011.281⟩. ⟨hal-00638417v3⟩
579 View
639 Download



Gmail Facebook X LinkedIn More