Dynamic Conditional Optimal Transport through Simulation-Free Flows
Gavin Kerrigan Giosue Migliorini Padhraic Smyth
We study the geometry of conditional optimal transport (COT) and prove a
dynamical formulation which generalizes the Benamou-Brenier Theorem. 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. We build on the
framework of flow matching to train a conditional generative model by
approximating the geodesic path of measures induced by this COT plan. Our
theory and methods are applicable in the infinite-dimensional setting, making
them well suited for inverse problems. Empirically, we demonstrate our proposed
method on two image-to-image translation tasks and an infinite-dimensional
Bayesian inverse problem.