  We study a dimensionality reduction technique for finite mixtures of
high-dimensional multivariate response regression models. Both the dimension of
the response and the number of predictors are allowed to exceed the sample
size. We consider predictor selection and rank reduction to obtain
lower-dimensional approximations. A class of estimators with a fast rate of
convergence is introduced. We apply this result to a specific procedure,
introduced in [11], where the relevant predictors are selected by the
Group-Lasso.
