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.

翻译：我们研究了条件最优传输（COT）的几何结构，并证明了一个推广Benamou-Brenier定理的动力学表述。基于这些理论工具，我们提出了一种用于条件生成建模的仿真无流方法。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，并通过逼近该COT方案导出的测地路径获得条件生成模型。我们的理论与方法适用于无限维场景，使其能够广泛应用于贝叶斯反问题类别。实验表明，本方法在多个具有挑战性的条件生成任务（包括无限维反问题）中均表现出竞争力。