Uniform Consistency of Generalized Fréchet Means
Abstract
We study a generalization of the Fréchet 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.07534
- arXiv:
- arXiv:2408.07534
- Bibcode:
- 2024arXiv240807534A
- Keywords:
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- Mathematics - Statistics Theory;
- Mathematics - Metric Geometry;
- Mathematics - Probability;
- 0F05;
- 60F15;
- 60D05;
- 62E20;
- 62F10;
- 58K05