Anℓ1-oracle inequality for the Lasso in finite mixture Gaussian regression models
Caroline Meynet · ESAIM Probability and Statistics · 2012
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].