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

TL;DR

This paper introduces a new, data-driven method for accurately estimating the rank of the coefficient matrix in high-dimensional multivariate regression, avoiding complex variance estimation and tuning parameter selection.

Contribution

The authors develop an iterative, adaptive procedure that reliably identifies the true rank with minimal steps, outperforming previous methods in stability and accuracy.

Findings

Method accurately estimates rank in high-dimensional settings.

Procedure is fully data-driven and adapts to signal-to-noise ratio.

Simulation results confirm superior performance over existing approaches.

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 [7] in that it does not require estimation of the unknown variance of the noise, nor depends 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 more stable than that in [7] in both low- and high-dimensional settings.