Sparse Reduced-Rank Regression for Simultaneous Dimension Reduction and Variable Selection in Multivariate Regression

The reduced-rank regression is an effective method to predict multiple response variables from the same set of predictor variables, because it can reduce the number of model parameters as well as take advantage of interrelations between the response variables and therefore improve predictive accuracy. We propose to add a new feature to the reduced-rank regression that allows selection of relevant variables using sparsity inducing penalties. By treating each row of the matrix of the regression coefficients as a group, we propose a group-lasso type penalty and show that this penalty satisfies certain desirable invariance property. We develop two numerical algorithms to solve the penalized regression problem and establish the asymptotic consistency of the proposed method. In particular, the manifold structure of the reduced-rank regression coefficient matrix is respected and carefully studied in our theoretical analysis. In a simulation study and real data analysis, the new method is compared with several existing variable selection methods for multivariate regression and exhibits competitive performance in prediction and variable selection.