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Algorithms for Weak Optimal Transport with an Application to Economics
The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear.
Regularized Optimal Transport is Ground Cost Adversarial
In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial.
Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport
On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth.
Subspace Robust Wasserstein Distances
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge.
