  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 minimal conditions for ensuring the existence and
uniqueness of the Huber mean and discuss regularity conditions for
unbiasedness. The Huber means are statistically consistent and enjoy the
central limit theorem. Additionally, we propose a 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. Huber means are shown to be highly robust and efficient in the
presence of outliers or under heavy-tailed distribution. To be more specific,
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 distribution. Numerical examples on spheres and the set of
symmetric positive-definite matrices further illustrate the efficiency and
reliability of the proposed Huber means on Riemannian manifolds.
