Dynamic Conditional Optimal Transport through Simulation-Free Flows
This provides a new approach to conditional generative modeling with theoretical guarantees for Bayesian inverse problems.
The authors tackled the problem of conditional generative modeling by developing a simulation-free flow-based method based on conditional optimal transport theory, which proved competitive on several challenging conditional generation tasks including infinite-dimensional inverse problems.
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.