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arXiv · 2402.13176

$h$-Wasserstein barycenters

Camilla Brizzi·Gero Friesecke·Tobias Ried
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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.

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Camilla Brizzi, Gero Friesecke, Tobias Ried. 2024-02-20. $h$-Wasserstein barycenters. https://arxiv.org/abs/2402.13176

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