Transformed Fréchet Means for Robust Estimation in Hadamard Spaces
Christof Schötz
TL;DR
This work introduces the transformed Fréchet mean, a unifying notion of central tendency for Hadamard spaces achieved by minimizing $\mathbb{E}[\tau(d(Y,q))]$ with a broad class of transformations $\tau$ that interpolate between the Fréchet median and the classical mean. The authors develop a cohesive framework—rooted in algorithm stability, the quadruple inequality, and a variance inequality—to derive finite-sample error bounds in expectation for transformed Fréchet means under minimal moment assumptions, applicable in infinite-dimensional Hilbert spaces and nonpositively curved geometries. They provide explicit rates for power Fréchet means ($\tau(x)=x^{\alpha}$, $\alpha\in(1,2]$), along with general transformation results, including tail-robust and contamination-robust variants, and special treatment of the Fréchet median with large-deviation bounds. Additionally, the paper demonstrates fast-rate possibilities when the data are highly concentrated, and it places the results in a broad context of robust central tendency in metric spaces. The resulting theory offers practical tools for robust estimation of central tendency in complex geometric settings, with implications for statistics on manifolds, trees, and other Hadamard-structured spaces.
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.