# Fr\'echet Means and Procrustes Analysis in Wasserstein Space

## TL;DR

This paper explores the computation of Fréchet means and optimal registration of measures in Wasserstein space, linking these problems through a geometric perspective and proposing consistent estimators for practical data analysis.

## Contribution

It introduces a novel connection between Fréchet means and Procrustes analysis in Wasserstein space, with methods for their computation and statistical inference.

## Key findings

- Proposed a gradient descent method for Fréchet mean computation.
- Established the equivalence between Fréchet mean and Procrustes registration.
- Proved consistency of nonparametric estimators for population mean and registration maps.

## Abstract

We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fr\'echet mean in the Wasserstein space of multivariate distributions; and the optimal registration of deformed random measures and point processes. We elucidate how the two problems are linked, each being in a sense dual to the other. We first study the finite sample version of the problem in the continuum. Exploiting the tangent bundle structure of Wasserstein space, we deduce the Fr\'echet mean via gradient descent. We show that this is equivalent to a Procrustes analysis for the registration maps, thus only requiring successive solutions to pairwise optimal coupling problems. We then study the population version of the problem, focussing on inference and stability: in practice, the data are i.i.d. realisations from a law on Wasserstein space, and indeed their observation is discrete, where one observes a proxy finite sample or point process. We construct regularised nonparametric estimators, and prove their consistency for the population mean, and uniform consistency for the population Procrustes registration maps.