Open Access
April 2011 Optimal selection of reduced rank estimators of high-dimensional matrices
Florentina Bunea, Yiyuan She, Marten H. Wegkamp
Ann. Statist. 39(2): 1282-1309 (April 2011). DOI: 10.1214/11-AOS876


We introduce a new criterion, the Rank Selection Criterion (RSC), for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models. The corresponding RSC estimator minimizes the Frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model.

The rank of the RSC estimator provides a consistent estimator of the rank of the coefficient matrix; in general, the rank of our estimator is a consistent estimate of the effective rank, which we define to be the number of singular values of the target matrix that are appropriately large. The consistency results are valid not only in the classic asymptotic regime, when n, the number of responses, and p, the number of predictors, stay bounded, and m, the number of observations, grows, but also when either, or both, n and p grow, possibly much faster than m.

We establish minimax optimal bounds on the mean squared errors of our estimators. Our finite sample performance bounds for the RSC estimator show that it achieves the optimal balance between the approximation error and the penalty term.

Furthermore, our procedure has very low computational complexity, linear in the number of candidate models, making it particularly appealing for large scale problems. We contrast our estimator with the nuclear norm penalized least squares (NNP) estimator, which has an inherently higher computational complexity than RSC, for multivariate regression models. We show that NNP has estimation properties similar to those of RSC, albeit under stronger conditions. However, it is not as parsimonious as RSC. We offer a simple correction of the NNP estimator which leads to consistent rank estimation.

We verify and illustrate our theoretical findings via an extensive simulation study.


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Florentina Bunea. Yiyuan She. Marten H. Wegkamp. "Optimal selection of reduced rank estimators of high-dimensional matrices." Ann. Statist. 39 (2) 1282 - 1309, April 2011.


Published: April 2011
First available in Project Euclid: 9 May 2011

zbMATH: 1216.62086
MathSciNet: MR2816355
Digital Object Identifier: 10.1214/11-AOS876

Primary: 62H15 , 62J07

Keywords: adaptive estimation , Dimension reduction , low rank matrix approximation , Multivariate response regression , nuclear norm , Oracle inequalities , rank selection , reduced rank estimators

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 2 • April 2011
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