$h$-Wasserstein barycenters

TL;DR

This paper extends the concept of Wasserstein barycenters to general convex costs, proving uniqueness and Monge form of optimal plans using a novel injectivity approach.

Contribution

It generalizes Wasserstein barycenters beyond quadratic costs, establishing equivalence of formulations and proving uniqueness and Monge structure with a new injectivity method.

Findings

Multi-marginal optimal plan is unique.

Optimal plan is of Monge form.

Introduces a new injectivity-based approach.

Abstract

We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs $h$ with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $h$-barycenter on the support of an optimal multi-marginal plan.