Variance Inequalities for Transformed Fr\'echet Means in Hadamard Spaces

Abstract

The Fr\'echet mean (or barycenter) generalizes the expectation of a random variable to metric spaces by minimizing the expected squared distance to the random variable. Similarly, the median can be generalized by its property of minimizing the expected absolute distance. We consider the class of transformed Fr\'echet means with nondecreasing, convex transformations that have a concave derivative. This class includes the Fr\'echet median, the Fr\'echet mean, the Huber loss-induced Fr\'echet mean, and other statistics related to robust statistics in metric spaces. We study variance inequalities for these transformed Fr\'echet means. These inequalities describe how the expected transformed distance grows when moving away from a minimizer, i.e., from a transformed Fr\'echet mean. Variance inequalities are useful in the theory of estimation and numerical approximation of transformed Fr\'echet means. Our focus is on variance inequalities in Hadamard spaces - metric spaces with globally nonpositive curvature. Notably, some results are new also for Euclidean spaces. Additionally, we are able to characterize uniqueness of transformed Fr\'echet means, in particular of the Fr\'echet median.