AN l(1)-ORACLE INEQUALITY FOR THE LASSO IN FINITE MIXTURE GAUSSIAN REGRESSION MODELS

作者：Meynet Caroline^*

来源：ESAIM: Probability and Statistics , 2013, 17: 650-671.

DOI：10.1051/ps/2012016

• Finite mixture of Gaussian regressions model
• Lasso
• l(1)-oracle inequalities
• model selection by penalization
• l(1)-balls

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 l(1)-penalized maximum likelihood estimator. We shall provide an l(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 l(1)-oracle inequality established by Massart and Meynet [12] in the homogeneous Gaussian linear regression case, and to present a complementary result to Stadler et al. [18], by studying the Lasso for its l(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 l(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].

• 出版日期2013-1