The degrees of freedom of partly smooth regularizers - Archive ouverte HAL Access content directly
Journal Articles Annals of the Institute of Statistical Mathematics Year : 2017

The degrees of freedom of partly smooth regularizers


We study regularized regression problems where the regularizer is a proper, lower-semicontinuous, convex and partly smooth function relative to a Riemannian submanifold. This encompasses several popular examples including the Lasso, the group Lasso, the max and nuclear norms, as well as their composition with linear operators (e.g., total variation or fused Lasso). Our main sensitivity analysis result shows that the predictor moves locally stably along the same active submanifold as the observations undergo small perturbations. This plays a pivotal role in getting a closed-form expression for the divergence of the predictor w.r.t. observations. We also show that, for many regularizers, including polyhedral ones or the analysis group Lasso, this divergence formula holds Lebesgue a.e. When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and the prediction risk. Our results unify and go beyond those already known in the literature.
Fichier principal
Vignette du fichier
dof-partly-smooth.pdf (555.15 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-00981634 , version 1 (22-04-2014)
hal-00981634 , version 2 (23-07-2014)
hal-00981634 , version 3 (27-10-2015)
hal-00981634 , version 4 (10-02-2016)



Samuel Vaiter, Charles-Alban Deledalle, Jalal M. Fadili, Gabriel Peyré, Charles H Dossal. The degrees of freedom of partly smooth regularizers . Annals of the Institute of Statistical Mathematics, 2017, 69 (4), pp.791 - 832. ⟨10.1007/s10463-016-0563-z⟩. ⟨hal-00981634v4⟩
773 View
484 Download



Gmail Facebook X LinkedIn More