Adaptive Nuclear-Norm Penalization in Multivariate Regression

• Activity Number: 296
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Biometrics Section
• Author(s): Kun Chen*+ and Hongbo Dong and Kung-Sik Chan
• Companies: Kansas State University and University of Wisconsin-Madison and University of Iowa
• Keywords: adaptive nuclear-norm penalization ; reduced-rank regression ; low-rank matrix approximation ; singular value decomposition ; adaptive soft-thresholding

Abstract

The adaptive nuclear-norm penalization is proposed, based on which we develop a new method for simultaneous dimension reduction and coefficient estimation in high-dimensional multivariate regression. Different from the classical reduced-rank regression where the coefficient matrix is estimated via hard-thresholding the singular value decomposition of the least-squares estimator of the data matrix, the proposed non-convex weighted nuclear-norm penalized method leads to an adaptive soft-thresholding estimator (AST), which (i) is a global optimal solution of the proposed non-convex criterion, (ii) possesses better bias-variance property and (iii) enjoys low computational complexity. The rank consistency of the proposed AST estimator is shown for both classical and high-dimensional asymptotic regimes. The prediction and estimation performance bounds are also established. We contrast the AST estimator with the nuclear-norm penalized least-squares estimator and the reduced-rank regression estimator. The efficacy of the AST estimator is demonstrated by extensive simulation studies and an application in genetics.