On the Degrees of Freedom of Reduced-Rank Estimators in Multivariate Regression

• Activity Number: 474
• Type: Invited
• Date/Time: Wednesday, August 6, 2014 : 10:30 AM to 12:20 PM
• Sponsor: WNAR
• Author(s): Ji Zhu*+ and Ashin Mukherjee and Kun Chen and Naisyin Wang
• Companies: University of Michigan and Walmart Labs and University of Connecticut and University of Michigan
• Keywords: adaptive nuclear norm ; degrees of freedom ; model selection ; multivariate regression ; singular value decomposition ; reduced rank regression

Abstract

In this talk we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of Stein's unbiased risk estimation (SURE). We derive a finite-sample exact unbiased estimator that admits a closed-form expression in terms of the singular values or thresholded singular values of the least squares solution and hence readily computable. The results continue to hold in the high-dimensional scenario when both the predictor and response dimensions are allowed to be larger than the sample size. The derived analytical form facilitates the investigation of its theoretical properties and provides new insights into the empirical behaviors of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator, i.e., the number of free parameters. The use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection, as demonstrated by simulation studies and a data example.