The Optimal Transport (OT) problem investigates a transport map that connects two distributions while minimizing a given cost function. Finding such a transport map has diverse applications in machine learning, such as generative modeling and image-to-image translation. In this paper, we introduce a scalable and simulation-free approach for solving the Entropic Unbalanced Optimal Transport (EUOT) problem. We derive the dynamical form of this EUOT problem, which is a generalization of the Schr\"odinger bridges (SB) problem. Based on this, we derive dual formulation and optimality conditions of the EUOT problem from the stochastic optimal control interpretation. By leveraging these properties, we propose a simulation-free algorithm to solve EUOT, called Simulation-free EUOT (SF-EUOT). While existing SB models require expensive simulation costs during training and evaluation, our model achieves simulation-free training and one-step generation by utilizing the reciprocal property. Our model demonstrates significantly improved scalability in generative modeling and image-to-image translation tasks compared to previous SB methods.

本研究通过提出一种可扩展且无需模拟的方法来解决熵不平衡最优传输（EUOT）问题，填补了传统最优传输模型在训练和评估过程中高昂模拟成本的空白。我们基于斯特克过程解释推导了EUOT的问题的对偶形式及最优性条件，并验证了该模型在生成建模和图像到图像转换任务中具有显著的扩展性和效率提升。

可扩展的无模拟熵不平衡最优传输