Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schr\"odinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schr\"odinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.

翻译：解决运输问题，即寻找将一个给定分布映射到另一个分布的映射，在机器学习中有众多应用。受生成模型启发，近年来提出了新型质量运输方法，例如去噪扩散模型（DDMs）和流匹配模型（FMMs）通过随机微分方程（SDE）或常微分方程（ODE）实现这种运输。然而，尽管在许多应用中希望近似具有吸引性质的确定性动态最优传输（OT）映射，但DDMs和FMMs并不能保证提供接近OT映射的传输。相比之下，薛定谔桥（SBs）计算随机动态映射，恢复熵正则化的OT版本。遗憾的是，现有的近似SBs的数值方法要么在高维情况下扩展性差，要么在迭代过程中累积误差。在这项工作中，我们引入了迭代马尔可夫拟合（IMF），一种解决SB问题的新方法论，以及扩散薛定谔桥匹配（DSBM），一种用于计算IMF迭代的新型数值算法。DSBM显著改进了先前的SB数值方法，并将多种近期运输方法作为特殊/极限情况恢复。我们在各种问题上展示了DSBM的性能。