  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 $\ell_1$-oracle
inequality satisfied by the Lasso estimator according to the Kullback-Leibler
loss. This result is an extension of the $\ell_1$-oracle inequality established
by Meynet in \cite{Meynet} in the multivariate case. We focus on the Lasso for
its $\ell_1$-regularization properties rather than for the variable selection
procedure, as it was done in St\"adler in \cite{Stadler}.
