Transformed Fréchet Means for Robust Estimation in Hadamard Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908640612450304 |
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| author | Schötz, Christof |
| author_facet | Schötz, Christof |
| contents | We establish finite-sample error bounds in expectation for transformed Fréchet means in Hadamard spaces under minimal assumptions. Transformed Fréchet means provide a unifying framework encompassing classical and robust notions of central tendency in metric spaces. Instead of minimizing squared distances as for the classical 2-Fréchet mean, we consider transformations of the distance that are nondecreasing, convex, and have a concave derivative. This class spans a continuum between median and classical mean. It includes the Fréchet median, power Fréchet means, and the (pseudo-)Huber mean, among others. We obtain the parametric rate of convergence under fewer than two moments and a subclass of estimators exhibits a breakdown point of 1/2. Our results apply in general Hadamard spaces-including infinite-dimensional Hilbert spaces and nonpositively curved geometries-and yield new insights even in Euclidean settings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transformed Fréchet Means for Robust Estimation in Hadamard Spaces Schötz, Christof Statistics Theory Probability 62R20, 30L15 We establish finite-sample error bounds in expectation for transformed Fréchet means in Hadamard spaces under minimal assumptions. Transformed Fréchet means provide a unifying framework encompassing classical and robust notions of central tendency in metric spaces. Instead of minimizing squared distances as for the classical 2-Fréchet mean, we consider transformations of the distance that are nondecreasing, convex, and have a concave derivative. This class spans a continuum between median and classical mean. It includes the Fréchet median, power Fréchet means, and the (pseudo-)Huber mean, among others. We obtain the parametric rate of convergence under fewer than two moments and a subclass of estimators exhibits a breakdown point of 1/2. Our results apply in general Hadamard spaces-including infinite-dimensional Hilbert spaces and nonpositively curved geometries-and yield new insights even in Euclidean settings. |
| title | Transformed Fréchet Means for Robust Estimation in Hadamard Spaces |
| topic | Statistics Theory Probability 62R20, 30L15 |
| url | https://arxiv.org/abs/2511.06933 |