# Regularity theory and geometry of unbalanced optimal transport

arXiv: https://arxiv.org/abs/2112.11056 · HTML: https://mlsanity.com/paper/2112.11056/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2021.

## Built on by (most cited first, 6 of 6)

- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Improving Neural Optimal Transport via Displacement Interpolation](https://mlsanity.com/paper/2410.03783/) (2024)
- [A High-Quality Robust Diffusion Framework for Corrupted Dataset](https://mlsanity.com/paper/2311.17101/) (2023)
- [Taming Flow Matching with Unbalanced Optimal Transport into Fast Pansharpening](https://mlsanity.com/paper/2503.14975/) (2025)
- [Reweighted Flow Matching via Unbalanced OT for Label-free Long-tailed Generation](https://mlsanity.com/paper/2509.25713/) (2025)

## Abstract (arXiv)

Using the dual formulation only, we show that the regularity of unbalanced optimal transport also called entropy-transport inherits from the regularity of standard optimal transport. We provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular.Among all entropy-transport formulations, Wasserstein-Fisher-Rao (WFR) metric, also called Hellinger-Kantorovich, stands out since it admits a dynamic formulation, which extends the Benamou-Brenier formulation of optimal transport. After demonstrating the equivalence between dynamic and static formulations on a closed Riemannian manifold, we prove a polar factorization theorem, similar to the one due to Brenier and Mc-Cann. As a byproduct, we formulate the Monge-Amp`ere equation associated with WFR metric, which also holds for more general costs. Last, we study the link between c-convex functions for the cost induced by the WFR metric and the cost on the cone. The main result is that the weak Ma-Trudinger-Wang condition on the cone implies the same condition on the manifold for the cost induced by WFR.