Tuning Parameter Selection for Underdetermined Reduced-Rank Regression

Multivariate regression is one of the most widely applied multivariate statistical methods with many uses across a range of disciplines. But the number of parameters increases exponentially with dimension and reduced-rank regression (RRR) is a well known approach to dimension reduction. But traditional RRR applies only to an overdetermined system. For increasingly common undetermined systems this issue can be managed by regularization, e.g., with a quadratic penalty. A significant problem is then the choice of the two tuning parameters: one discrete i.e., the rank; the other continuous i.e., the Tikhonov penalty parameter. In this paper we resolve this problem via Stein's unbiased risk estimator (SURE). We compare SURE to cross-validation and apply it on both simulated and real data sets.