Unique Fréchet means for broader distributional families

Determine which distributional families, and which parameter choices within specified parametric families, guarantee unique Fréchet means on the spaces considered in the paper, including symmetric spaces and related noncontractible metric spaces.

Background

The paper proves that, on complete finite-dimensional noncontractible Alexandrov spaces with curvature bounded from below, probability measures with unique Fréchet means and measures with nonunique Fréchet means are both dense in the quadratic Wasserstein and total variation metrics. Consequently, uniqueness is not stable under arbitrary small perturbations in these settings.

The authors note that certain concentrated distributions on Riemannian manifolds and particular radially symmetric distributions on symmetric spaces are known to have unique Fréchet means. They explicitly do not extend these results to general distributional families or identify all parameter regimes ensuring uniqueness, leaving the classification of such families and parameters unresolved.

References

In our contribution we do not extend this result to other distributions. In particular, we do not give an answer to questions of the following kind: Which distributional families or which parameter choices for certain parametric families guarantee unique means? Such questions are left for future research.

— Dense uniqueness and dense nonuniqueness of Fréchet means  (2610.11946 - Huckemann et al., 8 Oct 2026) in Section 1, Introduction, paragraph beginning “There are various results on uniqueness of Fréchet means in the literature.”