title: Reduced rank regression via adaptive nuclear norm penalization.
abstract: We propose an adaptive nuclear norm penalization approach for low-rank matrix approximation, and use it to develop a new reduced rank estimation method for high-dimensional multivariate regression. The adaptive nuclear norm is defined as the weighted sum of the singular values of the matrix, and it is generally non-convex under the natural restriction that the weight decreases with the singular value. However, we show that the proposed non-convex penalized regression method has a global optimal solution obtained from an adaptively soft-thresholded singular value decomposition. The method is computationally efficient, and the resulting solution path is continuous. The rank consistency of and prediction/estimation performance bounds for the estimator are established for a high-dimensional asymptotic regime. Simulation studies and an application in genetics demonstrate its efficacy.
authors.affiliation: Department of Statistics, University of Connecticut, 215 Glenbrook Road, Storrs, Connecticut 06269, U.S.A.
authors.forename: Kun
authors.initials: K
authors.lastname: Chen
authors.affiliation: Wisconsin Institutes for Discovery, University of Wisconsin, 330 N. Orchard St., Madison, Wisconsin 53715, U.S.A.
authors.forename: Hongbo
authors.initials: H
authors.lastname: Dong
authors.affiliation: Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa 52242, U.S.A.
authors.forename: Kung-Sik
authors.initials: KS
authors.lastname: Chan
keywords: Low-rank approximation; Nuclear norm penalization; Reduced rank regression; Singular value decomposition