Consistency of the group Lasso and multiple kernel learning

Francis Bach 1
1 WILLOW - Models of visual object recognition and scene understanding
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : We consider the least-square regression problem with regularization by a block 1-norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model misspecification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covariance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.
Type de document :
Pré-publication, Document de travail
2008
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https://hal.archives-ouvertes.fr/hal-00164735
Contributeur : Francis Bach <>
Soumis le : lundi 28 janvier 2008 - 10:49:21
Dernière modification le : jeudi 29 septembre 2016 - 01:22:07
Document(s) archivé(s) le : mardi 21 septembre 2010 - 15:44:37

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  • HAL Id : hal-00164735, version 2
  • ARXIV : 0707.3390

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Francis Bach. Consistency of the group Lasso and multiple kernel learning. 2008. <hal-00164735v2>

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