Flow and bridge matching are a novel class of processes which encompass diffusion models. One of the main aspect of their increased flexibility is that these models can interpolate between arbitrary data distributions i.e. they generalize beyond generative modeling and can be applied to learning stochastic (and deterministic) processes of arbitrary transfer tasks between two given distributions. In this paper, we highlight that while flow and bridge matching processes preserve the information of the marginal distributions, they do \emph{not} necessarily preserve the coupling information unless additional, stronger optimality conditions are met. This can be problematic if one aims at preserving the original empirical pairing. We show that a simple modification of the matching process recovers this coupling by augmenting the velocity field (or drift) with the information of the initial sample point. Doing so, we lose the Markovian property of the process but preserve the coupling information between distributions. We illustrate the efficiency of our augmentation in learning mixture of image translation tasks.

翻译：流动与桥接匹配是一类新型过程，涵盖了扩散模型。其灵活性增强的主要方面在于，这些模型可以在任意数据分布之间进行插值，即它们超越了生成式建模的范畴，可应用于学习两个给定分布之间任意迁移任务的随机（及确定性）过程。本文指出，尽管流动与桥接匹配过程保留了边际分布的信息，但除非满足额外的更严格最优性条件，否则它们并不必然保留耦合信息。若旨在保持原始经验配对，这可能引发问题。我们证明，通过向速度场（或漂移）添加初始样本点的信息，对匹配过程进行简单修改即可恢复该耦合。如此操作后，过程失去了马尔可夫性质，但保留了分布间的耦合信息。我们通过在图像翻译任务混合学习中的实例，展示了所提增强方法的有效性。