Abstract
This article introduces Huber means on Riemannian manifolds, providing a robust alternative to the Fréchet
mean by integrating elements of both L2 and L1 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 Fréchet mean under heavy-tailed distributions.

http://dx.doi.org/10.1093/jrsssb/qkaf054