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An ℓ1-oracle inequality for the Lasso in finite mixture Gaussian regression models

Published online by Cambridge University Press:  04 November 2013

Caroline Meynet
Caroline Meynet*
Affiliation:
Laboratoire de Mathématiques, Faculté des Sciences d’Orsay, Université Paris-Sud, 91405 Orsay, France. caroline.meynet@math.u-psud.fr
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Abstract

We consider a finite mixture of Gaussian regression models for high-dimensionalheterogeneous data where the number of covariates may be much larger than the sample size.We propose to estimate the unknown conditional mixture density by anℓ1-penalized maximum likelihood estimator. We shall providean ℓ1-oracle inequality satisfied by this Lasso estimator withthe Kullback–Leibler loss. In particular, we give a condition on the regularizationparameter of the Lasso to obtain such an oracle inequality. Our aim is twofold: to extendthe ℓ1-oracle inequality established by Massart and Meynet[12] in the homogeneous Gaussian linearregression case, and to present a complementary result to Städler et al.[18], by studying the Lasso for itsℓ1-regularization properties rather than considering it as avariable selection procedure. Our oracle inequality shall be deduced from a finite mixtureGaussian regression model selection theorem for ℓ1-penalizedmaximum likelihood conditional density estimation, which is inspired from Vapnik’s methodof structural risk minimization [23] and from thetheory on model selection for maximum likelihood estimators developed by Massart in [11].

Keywords

Finite mixture of Gaussian regressions modelLassoℓ1-oracle inequalitiesmodel selection by penalizationℓ1-balls
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 650 - 671
Copyright
© EDP Sciences, SMAI, 2013

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