A Canonical Barycenter via Wasserstein Regularization

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 all optimal transport plans transporting ${\bf m}$ to any probability measure $\nu$ on $X$ are induced 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 $P_2(X)$, pushing a given measure $\mu$ forward to a measure on $P_2(X)$. We then regularize the solution to the latter problem by the squared 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 $B(\mu)$.