# Structured Variable Selection with Sparsity-Inducing Norms

1 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique - ENS Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
2 imagine [Marne-la-Vallée]
LIGM - Laboratoire d'Informatique Gaspard-Monge, CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech
Abstract : We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual $\ell_1$-norm and the group $\ell_1$-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.
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Journal articles

Cited literature [50 references]

https://hal.inria.fr/inria-00377732
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Submitted on : Monday, March 29, 2010 - 7:16:07 PM
Last modification on : Thursday, March 17, 2022 - 10:08:43 AM
Long-term archiving on: : Thursday, September 23, 2010 - 6:10:13 PM

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### Identifiers

• HAL Id : inria-00377732, version 3
• ARXIV : 0904.3523

### Citation

Rodolphe Jenatton, Jean-yves Audibert, Francis Bach. Structured Variable Selection with Sparsity-Inducing Norms. Journal of Machine Learning Research, Microtome Publishing, 2011, 12, pp.2777-2824. ⟨inria-00377732v3⟩

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