title: On the degrees of freedom of reduced-rank estimators in multivariate regression.
abstract: 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. A finite-sample exact unbiased estimator is derived that admits a closed-form expression in terms of the thresholded singular values of the least-squares solution and hence is readily computable. The results continue to hold in the high-dimensional setting where both the predictor and the response dimensions may be larger than the sample size. The derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator. 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.
authors.affiliation: Smart Forecasting Team, @WalmartLabs, 850 Cherry Avenue, San Bruno, California 94066, U.S.A.
authors.forename: A
authors.initials: A
authors.lastname: Mukherjee
authors.affiliation: Department of Statistics, University of Connecticut, 215 Glenbrook Road U-4120, Storrs, Connecticut 06269, U.S.A.
authors.forename: K
authors.initials: K
authors.lastname: Chen
authors.affiliation: Department of Statistics, University of Michigan, 1085 S. University Avenue, Ann Arbor, Michigan 48109, U.S.A.
authors.forename: N
authors.initials: N
authors.lastname: Wang
authors.affiliation: Department of Statistics, University of Michigan, 1085 S. University Avenue, Ann Arbor, Michigan 48109, U.S.A.
authors.forename: J
authors.initials: J
authors.lastname: Zhu
keywords: Adaptive nuclear norm; Degrees of freedom; Model selection; Multivariate regression; Reduced-rank regression; Singular value decomposition