Uniform Consistency of Generalized Fr\'echet Means

Abstract

We study a generalization of the Fr\'echet mean on metric spaces, which we call $\phi$-means. Our generalization is indexed by a convex function $\phi$. We find necessary and sufficient conditions for $\phi$-means to be finite and provide a tight bound for the diameter of the intrinsic mean set. We also provide sufficient conditions under which all the $\phi$-means coincide in a single point. Then, we prove the consistency of the sample $\phi$-mean to its population analogue. We also find conditions under which classes of $\phi$-means converge uniformly, providing a Glivenko-Cantelli result. Finally, we illustrate applications of our results and provide algorithms for the computation of $\phi$-means.