The degrees of freedom of the Lasso in underdetermined linear regression models

Maher Kachour (1) , Jalal M. Fadili (2) , Christophe Chesneau (3) , Charles H Dossal (4) , Gabriel Peyré (5)

1 Equipe Electronique - Laboratoire GREYC - UMR6072
2 Equipe Image - Laboratoire GREYC - UMR6072
3 LMNO - Laboratoire de Mathématiques Nicolas Oresme
4 IMB - Institut de Mathématiques de Bordeaux
5 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper, we investigate the degrees of freedom (df) of penalized l1 minimization (also known as the Lasso) for an un-derdetermined linear regression model. We show that under a suitable condition on the design matrix, the number of nonzero coefficients of the Lasso solution is an unbiased estimate for the degrees of freedom. 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 or GCV frameworks.

Mots clés: SURE; Lasso; degrees of freedom; SURE.

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Maher Kachour, Jalal M. Fadili, Christophe Chesneau, Charles H Dossal, Gabriel Peyré. The degrees of freedom of the Lasso in underdetermined linear regression models. SPARS 2011, Jul 2011, Edinburgh, United Kingdom. pp.56. ⟨hal-00625219⟩