Multivariate regression shrinkage and selection by canonical correlation analysis

The problem of regression shrinkage and selection for multivariate regression is considered. The goal is to consistently identify those variables relevant for regression. This is done not only for predictors but also for responses. To this end, a novel relationship between multivariate regression and canonical correlation is discovered. Subsequently, its equivalent least squares type formulation is constructed, and then the well developed adaptive LASSO type penalty and also a novel BIC-type selection criterion can be directly applied. Theoretical results show that the resulting estimator is selection consistent for not only predictors but also responses. Numerical studies are presented to corroborate our theoretical findings.