Optimal selection of reduced rank estimators of high-dimensional matrices

TL;DR

This paper introduces the Rank Selection Criterion (RSC), a new method for selecting the optimal reduced rank estimator in high-dimensional multivariate regression, with proven consistency, optimal error bounds, and low computational complexity.

Contribution

The paper proposes the RSC, a novel criterion for rank selection that is consistent, computationally efficient, and applicable in high-dimensional settings, improving over existing methods like nuclear norm penalization.

Findings

RSC estimator is consistent in estimating the rank and effective rank.

RSC achieves minimax optimal bounds on mean squared error.

RSC has linear computational complexity, suitable for large-scale problems.

Abstract

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.