Equivariant Estimation of Fr\'echet Means

TL;DR

This paper develops a framework for equivariant estimation of Fréchet means on Riemannian manifolds, deriving optimal estimators that respect the space's symmetries and demonstrating their advantages through simulations.

Contribution

It introduces the general form of the minimum risk equivariant estimator for Fréchet means on Riemannian manifolds and proposes an adaptive approach when a minimum risk estimator does not exist.

Findings

Optimal equivariant estimators outperform common estimators in certain models.

Explicit expressions for the minimum risk equivariant estimator are derived in some cases.

Adaptive estimators perform favorably in simulations.

Abstract

The Fr\'echet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fr\'echet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space is encoded by its isometry group. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models there exists an optimal equivariant estimator, which necessarily will perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fr\'echet mean. We derive the general form of this minimum risk equivariant estimator and in a few cases provide explicit expressions for it. In other models the isometry group is not large enough relative to the parametric family of distributions for there to exist a minimum risk equivariant estimator. In such cases, we introduce an adaptive equivariant estimator that uses the data to select a submodel for which there is an MRE. Simulations results show that the adaptive equivariant estimator performs favorably relative to alternative estimators.