Optimal Flow Matching: Learning Straight Trajectories in Just One Step
Nikita Kornilov Alexander Gasnikov Alexander Korotin
Over the several recent years, there has been a boom in development of flow
matching methods for generative modeling. One intriguing property pursued by
the community is the ability to learn flows with straight trajectories which
realize the optimal transport (OT) displacements. Straightness is crucial for
fast integration of the learned flow's paths. Unfortunately, most existing flow
straightening methods are based on non-trivial iterative procedures which
accumulate the error during training or exploit heuristic minibatch OT
approximations. To address this issue, we develop a novel optimal flow matching
approach which recovers the straight OT displacement for the quadratic cost in
just one flow matching step.