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

We study general M-estimators of location on Riemannian manifolds, extending classical notions such as the Fréchet 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 the associated population M-functional and the corresponding sample M-estimators. In particular, we provide sufficient conditions under which the population minimizer set is nonempty and reduces to a singleton, and under which the corresponding sample M-estimator is likewise 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.