Metric extrapolation in the Wasserstein space

TL;DR

This paper introduces a novel variational approach to extend Wasserstein geodesics for all times by allowing negative coefficients, with two convex formulations and an efficient numerical scheme using entropic regularization.

Contribution

It presents a new variational formulation for Wasserstein geodesic extension, including duality and barycentric transport perspectives, and develops a practical Sinkhorn-based algorithm.

Findings

Two equivalent convex formulations established

Efficient numerical scheme with entropic regularization proposed

Method enables continuous geodesic extension in Wasserstein space

Abstract

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.