Structured Variable Selection with Sparsity-Inducing Norms

Rodolphe Jenatton 1 Jean-Yves Audibert 1, 2 Francis Bach 1
1 SIERRA - Statistical Machine Learning and Parsimony
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 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 of Machine Learning Research, Journal of Machine Learning Research, 2011, 12, pp.2777-2824
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  • HAL Id : inria-00377732, version 3
  • ARXIV : 0904.3523

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Rodolphe Jenatton, Jean-Yves Audibert, Francis Bach. Structured Variable Selection with Sparsity-Inducing Norms. Journal of Machine Learning Research, Journal of Machine Learning Research, 2011, 12, pp.2777-2824. 〈inria-00377732v3〉

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