Consistency of the group Lasso and multiple kernel learning

Francis Bach 1
1 WILLOW - Models of visual object recognition and scene understanding
CNRS - Centre National de la Recherche Scientifique : UMR8548, Inria Paris-Rocquencourt, DI-ENS - Département d'informatique de l'École normale supérieure
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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [47 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00164735
Contributor : Francis Bach <>
Submitted on : Monday, January 28, 2008 - 10:49:21 AM
Last modification on : Wednesday, January 30, 2019 - 11:07:38 AM
Long-term archiving on : Tuesday, September 21, 2010 - 3:44:37 PM

Files

grouplasso.pdf
Files produced by the author(s)

Identifiers

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

Collections

Citation

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

Share

Metrics

Record views

3274

Files downloads

541