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

Joint rank and variable selection for parsimonious estimation in a high-dimensional finite mixture regression model