Intrinsic Analysis of the Sample Fr\'echet Mean and Sample Mean of Complex Wishart Matrices

TL;DR

This paper compares the intrinsic bias and risk of the sample mean and Fréchet mean for complex covariance matrices, finding the simple average generally performs better.

Contribution

It provides a geometric analysis of averaging methods for complex Wishart matrices, showing the simple average outperforms the Fréchet mean.

Findings

Simple average has lower intrinsic bias.

Simple average has lower asymptotic Riemannian risk.

Simple average is preferred overall for covariance estimation.

Abstract

We consider two types of averaging of complex covariance matrices, a sample mean (average) and the sample Fr\'echet mean. We analyse the performance of these quantities as estimators for the true covariance matrix via `intrinsic' versions of bias and mean square error, a methodology which takes account of geometric structure. We derive simple expressions for the intrinsic bias in both cases, and the simple average is seen to be preferable. The same is true for the asymptotic Riemannian risk, and for the Riemannian risk itself in the scalar case. Combined with a similar preference for the simple average using non-intrinsic analysis, we conclude that the simple average is preferred overall to the sample Fr\'echet mean in this context.