An \ell_1-oracle inequality for the Lasso in finite mixture of multivariate Gaussian regression models

We consider a multivariate finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates and the size of the response may be much larger than the sample size. We provide an ${\mathrm{\ell }}_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback-Leibler loss. This result is an extension of the ${\mathrm{\ell }}_1$-oracle inequality established by Meynet in \cite{Meynet} in the multivariate case. We focus on the Lasso for its ${\mathrm{\ell }}_1$-regularization properties rather than for the variable selection procedure, as it was done in St"adler in \cite{Stadler}.