On estimation in the reduced-rank regression with a large number of responses and predictors

TL;DR

This paper investigates the statistical properties of reduced-rank multivariate regression with high-dimensional data, deriving the distribution of singular values and proposing algorithms for model rank selection and coefficient estimation.

Contribution

It introduces new asymptotic results for singular value distributions and develops algorithms for rank selection and coefficient estimation in high-dimensional reduced-rank regression.

Findings

Largest singular values follow a Tracy-Widom distribution.

Proposed algorithms outperform existing methods in rank selection.

Two consistent estimators for singular values are developed and analyzed.

Abstract

We consider a multivariate linear response regression in which the number of responses and predictors is large and comparable with the number of observations, and the rank of the matrix of regression coefficients is assumed to be small. We study the distribution of singular values for the matrix of regression coefficients and for the matrix of predicted responses. For both matrices, it is found that the limit distribution of the largest singular value is a rescaling of the Tracy-Widom distribution. Based on this result, we suggest algorithms for the model rank selection and compare them with the algorithm suggested by Bunea, She and Wegkamp. Next, we design two consistent estimators for the singular values of the coefficient matrix, compare them, and derive the asymptotic distribution for one of these estimators..