The Annals of Statistics

Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models

Xin Bing and Marten H. Wegkamp

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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.

Article information

Ann. Statist., Volume 47, Number 6 (2019), 3157-3184.

Received: February 2018
Revised: August 2018
First available in Project Euclid: 31 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 62H15: Hypothesis testing 62J07: Ridge regression; shrinkage estimators

Multivariate response regression reduced rank estimator self-tuning adaptive rank estimation rank consistency dimension reduction oracle inequalities


Bing, Xin; Wegkamp, Marten H. Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models. Ann. Statist. 47 (2019), no. 6, 3157--3184. doi:10.1214/18-AOS1774.

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Supplemental materials

  • Supplement to “Adaptive estimation of the rank of the coefficient matrix in high-dimensional multivariate response regression models”. The supplementary document includes the oracle inequality for the fit, additional simulation results and all proofs.