Huber means on Riemannian manifolds

TL;DR

This paper introduces Huber means on Riemannian manifolds as a robust alternative to the Fréchet mean, demonstrating their statistical properties, robustness, and efficiency through theoretical analysis and numerical examples.

Contribution

It develops the concept of Huber means on Riemannian manifolds, establishing their existence, uniqueness, robustness, and efficiency, along with practical estimators and testing procedures.

Findings

Huber means are highly robust with a breakdown point of at least 0.5.

They are more efficient than Fréchet means under heavy-tailed distributions.

Numerical examples confirm their effectiveness on spheres and positive-definite matrices.

Abstract

This article introduces Huber means on Riemannian manifolds, providing a robust alternative to the Frechet mean by integrating elements of both square and absolute loss functions. The Huber means are designed to be highly resistant to outliers while maintaining efficiency, making it a valuable generalization of Huber's M-estimator for manifold-valued data. We comprehensively investigate the statistical and computational aspects of Huber means, demonstrating their utility in manifold-valued data analysis. Specifically, we establish nearly minimal conditions for ensuring the existence and uniqueness of the Huber mean and discuss regularity conditions for unbiasedness. The Huber means are consistent and enjoy the central limit theorem. Additionally, we propose a novel moment-based estimator for the limiting covariance matrix, which is used to construct a robust one-sample location test procedure and an approximate confidence region for location parameters. The Huber mean is shown to be highly robust and efficient in the presence of outliers or under heavy-tailed distributions. Specifically, it achieves a breakdown point of at least 0.5, the highest among all isometric equivariant estimators, and is more efficient than the Frechet mean under heavy-tailed distributions. Numerical examples on spheres and the space of symmetric positive-definite matrices further illustrate the efficiency and reliability of the proposed Huber means on Riemannian manifolds.