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 1-penalized maximum likelihood estimator. We shall provide an 1-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 1-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 1-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 1-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] . Mathematics Subject Classification. 62G08, 62H30.
REF provided an ℓ 1 -oracle inequality for a Lasso estimator in finite mixture of Gaussian regression models.
Anℓ1-oracle inequality for the Lasso in finite mixture Gaussian regression models
Esaim: Probability and Statistics
Mathematics