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

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

Received: 7 January 2012
Revised: 17 July 2012

Abstract

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