Reduced rank regression via adaptive nuclear norm penalization. (English) Zbl 1279.62115

Chen, Kun; Dong, Hongbo; Chan, Kung-Sik

Biometrika 100, No. 4, 901-920 (2013).

Summary: 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 nonconvex under the natural restriction that the weight decreases with the singular value. However, we show that the proposed nonconvex 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.

MSC:

62H12	Estimation in multivariate analysis
62J05	Linear regression; mixed models
65C60	Computational problems in statistics (MSC2010)

Keywords:

low-rank approximation; nuclear norm penalization; singular value decomposition