VAITER, Samuel; DELEDALLE, Charles-Alban; FADILI, Jalal M.; PEYRÉ, Gabriel; DOSSAL, Charles

doi:10.1007/s10463-016-0563-z

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VAITER, Samuel
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]

DELEDALLE, Charles-Alban
Institut de Mathématiques de Bordeaux [IMB]

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

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Annals of the Institute of Statistical Mathematics. 2017-08, vol. 69, n° 4, p. 791 – 832

Springer Verlag

Resumen en inglés

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.< Leer menos

Palabras clave en inglés

Manifold

O-minimal structures

Model selection

Sparsity

Degrees of freedom

Semi-algebraic sets

Total variation

Group Lasso

Partial smoothness

DOI

http://dx.doi.org/10.1007/s10463-016-0563-z

Proyecto europeo

Sparsity, Image and Geometry to Model Adaptively Visual Processings

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Importado de Hal

Centros de investigación

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251