研究流模型推理加速的极限行为，发现其收敛到最优传输映射。
Limit Points of Reflow with Minibatch Optimal Transport
- 用小批量最优传输交替更新流向量场，迭代收敛
- 极限点具有N-循环单调性，结构更优且路径更直
- 在特定条件下，极限即为最优传输映射，适合理论分析
修正流（Rectified flows），又称流匹配或随机插值，是通过学习时间依赖的向量场，将概率分布从隐空间引导至目标分布的生成模型。Reflow通过迭代拉直该向量场诱导的轨迹来加速推理。本文研究此迭代的渐近行为，刻画其极限点性质。首先定义了始终存在的弱修正耦合。当修正流更新与固定批量大小的小批量最优传输步骤交替进行时，证明任意极限均为N-循环单调耦合，其中N为批次大小。这类耦合具有良好的结构性和稳定性，如可修正性和路径直性。进一步在速度限制为梯度场并满足额外支撑条件时，证明Reflow极限等于端点分布间的最优传输映射。
原文摘要 · Abstract (English)
Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.
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