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<p>In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. We propose an efficient numerical scheme to solve the latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.</p>

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Metric extrapolation in the Wasserstein space

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Thomas O. Gallouët,
  • Andrea Natale,
  • Gabriele Todeschi

摘要

In this article we study a variational problem providing a way to extend for all times minimizing geodesics connecting two given probability measures, in the Wasserstein space. This is simply obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem admits two equivalent convex formulations: the first can be seen as a particular instance of Toland duality and the second is a barycentric optimal transport problem. We propose an efficient numerical scheme to solve the latter formulation based on entropic regularization and a variant of Sinkhorn algorithm.