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December 2019 Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models
Xin Bing, Marten H. Wegkamp
Ann. Statist. 47(6): 3157-3184 (December 2019). DOI: 10.1214/18-AOS1774

Abstract

We consider the multivariate response regression problem with a regression coefficient matrix of low, unknown rank. In this setting, we analyze a new criterion for selecting the optimal reduced rank. This criterion differs notably from the one proposed in Bunea, She and Wegkamp (Ann. Statist. 39 (2011) 1282–1309) in that it does not require estimation of the unknown variance of the noise, nor does it depend on a delicate choice of a tuning parameter. We develop an iterative, fully data-driven procedure, that adapts to the optimal signal-to-noise ratio. This procedure finds the true rank in a few steps with overwhelming probability. At each step, our estimate increases, while at the same time it does not exceed the true rank. Our finite sample results hold for any sample size and any dimension, even when the number of responses and of covariates grow much faster than the number of observations. We perform an extensive simulation study that confirms our theoretical findings. The new method performs better and is more stable than the procedure of Bunea, She and Wegkamp (Ann. Statist. 39 (2011) 1282–1309) in both low- and high-dimensional settings.

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Xin Bing. Marten H. Wegkamp. "Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models." Ann. Statist. 47 (6) 3157 - 3184, December 2019. https://doi.org/10.1214/18-AOS1774

Information

Received: 1 February 2018; Revised: 1 August 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1774

Subjects:
Primary: 62H15 , 62J07

Keywords: adaptive rank estimation , Dimension reduction , Multivariate response regression , Oracle inequalities , rank consistency , reduced rank estimator , self-tuning

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • December 2019
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