Fr\'echet Mean Set Estimation in the Hausdorff Metric, via Relaxation

TL;DR

This paper establishes conditions under which the Fréchet mean set can be consistently estimated in non-Euclidean spaces using relaxed empirical estimators, with theoretical analysis and practical algorithms demonstrated on phylogenetic data.

Contribution

It introduces a framework for consistent estimation of the Fréchet mean set via relaxation, including adaptive rates and an algorithm for phylogenetic tree spaces.

Findings

Relaxed estimators achieve weak and strong consistency under specific relaxation rates.

An adaptive estimator optimally finds the fastest relaxation rate for strong consistency.

The proposed algorithm effectively estimates Fermat-Weber points in phylogenetic tree space.

Abstract

This work resolves the following question in non-Euclidean statistics: Is it possible to consistently estimate the Fr\'echet mean set of an unknown population distribution, with respect to the Hausdorff metric, when given access to independent identically-distributed samples? Our affirmative answer is based on a careful analysis of the "relaxed empirical Fr\'echet mean set estimators" which identify the set of near-minimizers of the empirical Fr\'echet functional and where the amount of "relaxation" vanishes as the number of data tends to infinity. On the theoretical side, our results include exact descriptions of which relaxation rates give weak consistency and which give strong consistency, as well as a description of an estimator which (assuming only the finiteness of certain moments and a mild condition on the metric entropy of the underlying metric space) adaptively finds the fastest possible relaxation rate for strongly consistent estimation. On the applied side, we consider the problem of estimating the set of Fermat-Weber points of an unknown distribution in the space of equidistant trees endowed with the tropical projective metric; in this setting, we provide an algorithm that provably implements our adaptive estimator, and we apply this method to real phylogenetic data.