We study the geometry of conditional optimal transport (COT) and prove a dynamical formulation which generalizes the Benamou-Brenier Theorem. Equipped with these tools, we propose a simulation-free flow-based method for conditional generative modeling. Our method couples an arbitrary source distribution to a specified target distribution through a triangular COT plan, and a conditional generative model is obtained by approximating the geodesic path of measures induced by this COT plan. Our theory and methods are applicable in infinite-dimensional settings, making them well suited for a wide class of Bayesian inverse problems. Empirically, we demonstrate that our method is competitive on several challenging conditional generation tasks, including an infinite-dimensional inverse problem.
@article{arxiv.2404.04240,
title = {Dynamic Conditional Optimal Transport through Simulation-Free Flows},
author = {Gavin Kerrigan and Giosue Migliorini and Padhraic Smyth},
journal= {arXiv preprint arXiv:2404.04240},
year = {2024}
}