Over the several recent years, there has been a boom in development of Flow Matching (FM) 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 the fast integration (inference) of the learned flow's paths. Unfortunately, most existing flow straightening methods are based on non-trivial iterative FM procedures which accumulate the error during training or exploit heuristics based on minibatch OT. To address these issues, we develop and theoretically justify the novel Optimal Flow Matching approach which allows recovering the straight OT displacement for the quadratic transport in just one FM step. The main idea of our approach is the employment of vector field for FM which are parameterized by convex functions.

翻译：近年来，流匹配方法在生成建模领域的发展呈现爆发式增长。学界一直追求的一个引人入胜的特性，是学习具有直线轨迹的流，以实现最优传输位移。直线性对于所学流路径的快速积分（推理）至关重要。遗憾的是，现有的大多数流矫直方法基于非平凡的迭代流匹配过程，这些过程在训练期间会累积误差，或利用基于小批量最优传输的启发式方法。为了解决这些问题，我们提出并从理论上论证了新颖的最优流匹配方法，该方法仅需一步流匹配即可恢复二次传输情况下的直线最优传输位移。我们方法的核心思想是采用由凸函数参数化的向量场进行流匹配。