Abstract: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.
From: Christof Schötz [view email]
[v1]
Mon, 10 Nov 2025 10:27:15 UTC (53 KB)
[v2]
Sun, 2 Aug 2026 08:47:18 UTC (103 KB)
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