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.

翻译：我们研究了条件最优传输（COT）的几何结构，并证明了一种推广了Benamou-Brenier定理的动态形式化表述。借助这些工具，我们提出了一种基于无模拟流的条件生成建模方法。该方法通过三角COT方案将任意源分布与指定目标分布耦合起来。我们基于流匹配框架，通过近似该COT方案所诱导的测度测地线路径来训练条件生成模型。我们的理论和方法适用于无限维设定，使其特别适合处理逆问题。在实验中，我们在两个图像到图像的翻译任务以及一个无限维贝叶斯逆问题上展示了所提出方法的性能。