Adaptive Rank Penalized Estimators in Multivariate Regression

We introduce a new criterion, the Rank Selection Criterion (RSC), for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models. The corresponding RSC estimator minimizes the Frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model. The rank of the RSC estimator provides a consistent estimator of the rank of the coefficient matrix. The consistency results are valid not only in the classic asymptotic regime, when the number of responses $n$ and predictors $p$ stays bounded, and the number of observations $m$ grows, but also when either, or both, $n$ and $p$ grow, possibly much faster than $m$. Our finite sample prediction and estimation performance bounds show that the RSC estimator achieves the optimal balance between the approximation error and the penalty term. Furthermore, our procedure has very low computational complexity, linear in the number of candidate models, making it particularly appealing for large scale problems. We contrast our estimator with the nuclear norm penalized least squares estimator (NNP). We show that NNP has estimation and prediction properties similar to those of RSC, albeit under stronger conditions. However, it is not as parsimonious as RSC. We offer a simple correction of the NNP estimator which leads to consistent rank estimation. We verify our theoretical findings via a simulation study. Applications to genomic and neuro-imaging data sets demonstrate the usage of our methods in practice.