Random matrix theory improved Fr\'echet mean of symmetric positive definite matrices

TL;DR

This paper introduces a random matrix theory-based method for computing Fréchet means of symmetric positive definite matrices, improving accuracy especially with limited samples, and demonstrates superior performance on synthetic and real datasets.

Contribution

The paper presents a novel random matrix theory approach for estimating Fréchet means on SPD manifolds, enhancing robustness in low-sample scenarios.

Findings

Outperforms existing methods on synthetic datasets.

Achieves better results on EEG and hyperspectral data.

Effective in low sample support conditions.

Abstract

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fr\'echet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine-learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory-based method that estimates Fr\'echet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.