用恒定速度场实现分布精确映射，提升生成模型效率与一致性。
Beckmann Transport Models: From Autonomous Flows to One-Step Maps
- 采用时不变速度场构建精确的生成流，适用于低维数据流形。
- 一跳生成映射是守恒方程的唯一解，可直接从样本学习。
- 统一框架修复现有方法缺陷，适用于ImageNet 256x256生成任务。
我们提出一种基于时不变速度场（即自洽流）的流匹配实例，可在目标分布为奇异分布（支持于低维数据流形）时，精确映射两个分布。同时证明，该流对应的一步生成映射是简单守恒方程的唯一解，可直接从样本中学习。这些自洽流与映射为贝克曼运输问题的通量约束赋予了动态意义。其构造提供了一个统一框架，例如可恢复闭式泊松流生成模型和二次流匹配损失下的平衡匹配。我们展示了该理论如何纠正现有方法中的不一致性，并在ImageNet 256x256上验证了自洽流与一步映射的有效性。
原文摘要 · Abstract (English)
We propose an instantiation of flow matching that relies on a time-independent velocity field (an \emph{autonomous flow}) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.
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