Uniform Consistency of Generalized Fréchet Means
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916357161877504 |
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| author | Aveni, Andrea Mukherjee, Sayan |
| author_facet | Aveni, Andrea Mukherjee, Sayan |
| contents | We study a generalization of the Fréchet mean on metric spaces, which we call $ϕ$-means. Our generalization is indexed by a convex function $ϕ$. We find necessary and sufficient conditions for $ϕ$-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 $ϕ$-means coincide in a single point. Then, we prove the consistency of the sample $ϕ$-mean to its population analogue. We also find conditions under which classes of $ϕ$-means converge uniformly, providing a Glivenko-Cantelli result. Finally, we illustrate applications of our results and provide algorithms for the computation of $ϕ$-means. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_07534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform Consistency of Generalized Fréchet Means Aveni, Andrea Mukherjee, Sayan Statistics Theory Metric Geometry Probability 0F05, 60F15, 60D05, 62E20, 62F10, 58K05 We study a generalization of the Fréchet mean on metric spaces, which we call $ϕ$-means. Our generalization is indexed by a convex function $ϕ$. We find necessary and sufficient conditions for $ϕ$-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 $ϕ$-means coincide in a single point. Then, we prove the consistency of the sample $ϕ$-mean to its population analogue. We also find conditions under which classes of $ϕ$-means converge uniformly, providing a Glivenko-Cantelli result. Finally, we illustrate applications of our results and provide algorithms for the computation of $ϕ$-means. |
| title | Uniform Consistency of Generalized Fréchet Means |
| topic | Statistics Theory Metric Geometry Probability 0F05, 60F15, 60D05, 62E20, 62F10, 58K05 |
| url | https://arxiv.org/abs/2408.07534 |