Transformed Fr\'echet Means for Robust Estimation in Hadamard Spaces

Abstract

We establish finite-sample error bounds in expectation for transformed Fr\'echet means in Hadamard spaces under minimal assumptions. Transformed Fr\'echet 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-Fr\'echet 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 Fr\'echet median, power Fr\'echet 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.