General M-estimators of location on Riemannian manifolds: existence and uniqueness

TL;DR

This paper develops a general theoretical framework for robust location estimation on Riemannian manifolds using M-estimators, establishing conditions for their existence and uniqueness in non-Euclidean spaces.

Contribution

It extends classical Euclidean M-estimators to Riemannian manifolds, providing new existence and uniqueness results under broad loss functions and minimal regularity assumptions.

Findings

Established conditions for existence of M-estimators on manifolds

Proved uniqueness of estimators under convex loss functions

Unified prior results within a general geometric framework

Abstract

We study general M-estimators of location on Riemannian manifolds, extending classical notions such as the Frechet mean by replacing the squared loss with a broad class of loss functions. Under minimal regularity conditions on the loss function and the underlying probability distribution, we establish theoretical guarantees for the existence and uniqueness of such estimators. In particular, we provide sufficient conditions under which the population and sample M-estimators exist and are uniquely defined. Our results offer a general framework for robust location estimation in non-Euclidean geometric spaces and unify prior uniqueness results under a broad class of convex losses.