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：求解传输问题，即寻找将给定分布映射到另一个分布的映射，在机器学习中具有众多应用。近年来，受生成模型启发的新型质量传输方法被提出，例如去噪扩散模型和流匹配模型通过随机微分方程或常微分方程实现此类传输。然而，尽管在许多应用中期望逼近具有优良特性的确定性动态最优传输映射，但去噪扩散模型和流匹配模型不能保证提供接近最优传输映射的传输。相比之下，薛定谔桥计算随机动态映射，可恢复熵正则化的最优传输版本。遗憾的是，现有近似薛定谔桥的数值方法要么随维度扩展性差，要么在迭代过程中累积误差。在本工作中，我们提出迭代马尔可夫拟合——一种求解薛定谔桥问题的新方法，以及扩散薛定谔桥匹配——一种用于计算迭代马尔可夫拟合迭代的新型数值算法。扩散薛定谔桥匹配显著改进了先前的薛定谔桥数值方法，并将近期多种传输方法作为特例/极限情况恢复。我们在一系列问题上展示了扩散薛定谔桥匹配的性能。