An ${\ell }_{1}$-oracle inequality for the Lasso in finite mixture gaussian regression models
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 650-671.

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].

DOI : https://doi.org/10.1051/ps/2012016
Classification : 62G08,  62H30
Mots clés : finite mixture of gaussian regressions model, Lasso, ℓ1-oracle inequalities, model selection by penalization, ℓ1-balls
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author = {Meynet, Caroline},
title = {An $\ell _1$-oracle inequality for the {Lasso} in finite mixture gaussian regression models},
journal = {ESAIM: Probability and Statistics},
pages = {650--671},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012016},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2012016/}
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Meynet, Caroline. An $\ell _1$-oracle inequality for the Lasso in finite mixture gaussian regression models. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 650-671. doi : 10.1051/ps/2012016. http://www.numdam.org/articles/10.1051/ps/2012016/

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