# A canonical barycenter via Wasserstein regularization

## TL;DR

This paper introduces a new notion of barycenter in metric measure spaces using Wasserstein regularization, providing a well-defined, unique measure supported on classical barycenter points under certain conditions.

## Contribution

It proposes a novel Wasserstein regularization approach to define a canonical barycenter in metric measure spaces, ensuring uniqueness and support on classical barycenter points.

## Key findings

- The barycenter $B()$ is well-defined and supported on classical barycenter points.
- Regularization with Wasserstein distance yields a unique measure.
- Various properties of the barycenter are analyzed.

## Abstract

We introduce a weak notion of barycenter of a probability measure $\mu$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(\mu)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $\mu$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given measure $\mu$ forward to a measure on $P(X)$. We then regularize the measure by the Wasserstein distance to the reference measure ${\bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $\mu$. We investigate various properties of $B(\mu)$