Bayesian sparse multiple regression for simultaneous rank reduction and variable selection

We develop a Bayesian methodology aimed at estimating low rank and row sparse matrices in a high dimensional multivariate response linear regression model. Starting with a full rank matrix and thus avoiding any prior specification on the rank, we let our estimate shrink towards the space of low rank matrices using continuous shrinkage priors. For selecting rows we propose a one step post processing scheme derived from putting group lasso penalties on the rows of the coefficient matrix with default choice of tuning parameters. We then provide an adaptive posterior estimate of the rank using a novel optimization function achieving dimension reduction in the covariate space. We exhibit the performance of the proposed methodology in an extensive simulation study and a real data example.