On The Degrees of Freedom of Reduced-rank Estimators in Multivariate Regression

TL;DR

This paper derives an exact unbiased estimator for the degrees of freedom of reduced-rank estimators in multivariate regression, applicable in high-dimensional settings, enabling better risk estimation and model selection.

Contribution

It provides a finite-sample, closed-form unbiased estimator for degrees of freedom that works in high-dimensional multivariate regression, improving risk estimation and model selection.

Findings

The estimator is computationally efficient and exact in finite samples.

It performs well in high-dimensional scenarios where predictors exceed sample size.

Simulation studies demonstrate improved prediction risk estimation.

Abstract

In this paper we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of Stein's unbiased risk estimation (SURE). We derive a finite-sample exact unbiased estimator that admits a closed-form expression in terms of the singular values or thresholded singular values of the least squares solution and hence readily computable. The results continue to hold in the high-dimensional scenario when both the predictor and response dimensions are allowed to be larger than the sample size. The derived analytical form facilitates the investigation of its theoretical properties and provides new insights into the empirical behaviors of the degrees of freedom. In particular, we examine the differences and connections between the proposed estimator and a commonly-used naive estimator, i.e., the number of free parameters. The use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection, as demonstrated by simulation studies and a data example.