Contact Geometry for Generative Models: An Unbalanced Optimal Transport Formulation

Real-world generative problems in biology, medical imaging, or robotics, rarely come with perfectly paired source and target distributions. While unbalanced Optimal Transport (uOT) provides a robust framework to bridge unpaired datasets via minimum-length paths in probability space, existing formulations based on the Wasserstein–Fisher–Rao (WFR) geometry introduce a structural bias toward high density regions, often triggering mode collapse. We introduce Contact unbalanced Optimal Transport (CuOT), a novel uOT formulation grounded in contact geometry that decouples density transport from mass growth to eliminate structural bias. This allows CuOT to adapt sampling steps to the local data structure, naturally reducing step sizes in high-density regions to prevent overshooting and ensuring stable convergence. The result is a stable, scalable path-length minimization solver that consistently outperforms WFR-based methods on biological dynamics reconstruction, unpaired image-to-image translation, and video generation.