Mass transport problems arise in many areas of machine learning whereby one wants to compute a map transporting one distribution to another. Generative modeling techniques like Generative Adversarial Networks (GANs) and Denoising Diffusion Models (DDMs) have been successfully adapted to solve such transport problems, resulting in CycleGAN and Bridge Matching respectively. However, these methods do not approximate Optimal Transport (OT) maps, which are known to have desirable properties. Existing techniques approximating OT maps for high-dimensional data-rich problems, such as DDM-based Rectified Flow and Schr\"odinger Bridge procedures, require fully training a DDM-type model at each iteration, or use mini-batch techniques which can introduce significant errors. We propose a novel algorithm to compute the Schr\"odinger Bridge, a dynamic entropy-regularised version of OT, that eliminates the need to train multiple DDM-like models. This algorithm corresponds to a discretisation of a flow of path measures, which we call the Schr\"odinger Bridge Flow, whose only stationary point is the Schr\"odinger Bridge. We demonstrate the performance of our algorithm on a variety of unpaired data translation tasks.

翻译：质量传输问题在机器学习的诸多领域中频繁出现，其核心在于计算将一个分布传输至另一个分布的映射。生成对抗网络（GANs）和去噪扩散模型（DDMs）等生成建模技术已成功应用于解决此类传输问题，分别催生了CycleGAN和桥匹配方法。然而，这些方法并未逼近最优传输（OT）映射，而后者已知具有诸多理想特性。现有用于逼近高维数据丰富问题中OT映射的技术，例如基于DDM的整流流和薛定谔桥方法，需要在每次迭代中完整训练一个DDM类模型，或使用可能引入显著误差的小批量技术。我们提出了一种计算薛定谔桥（一种动态熵正则化的OT版本）的新颖算法，该算法无需训练多个DDM类模型。此算法对应于路径测度流的一种离散化，我们称之为薛定谔桥流，其唯一的稳态点即为薛定谔桥。我们在多种非配对数据转换任务上验证了所提算法的性能。