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.

翻译：最优传输（Optimal Transport，OT）问题旨在寻找连接两个分布且最小化给定代价函数的传输映射。寻找此类传输映射在机器学习中具有多种应用，例如生成建模和图像到图像转换。本文提出了一种可扩展且无需仿真的方法来解决熵正则化非平衡最优传输（Entropic Unbalanced Optimal Transport，EUOT）问题。我们推导了该EUOT问题的动力学形式，该形式是薛定谔桥（Schrödinger bridges，SB）问题的推广。基于此，我们从随机最优控制的解释出发，推导了EUOT问题的对偶形式与最优性条件。利用这些性质，我们提出了一种无需仿真的算法来解决EUOT，称为无仿真EUOT（Simulation-free EUOT，SF-EUOT）。现有的SB模型在训练和评估阶段需要昂贵的仿真成本，而我们的模型通过利用互易性质实现了无仿真训练和一步生成。在生成建模和图像到图像转换任务中，与先前的SB方法相比，我们的模型展现出显著提升的可扩展性。