An l1-oracle inequality for the lasso in finite mixture gaussian regression models

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]. © EDP Sciences, SMAI 2013.