带小批量最优传输的重流（Reflow）的极限点

中文摘要 AI 辅助

整流流（也称为流匹配或随机插值器）是生成模型，用于学习时变向量场，引导概率曲线在两个概率分布（通常称为潜在分布和目标分布）之间演化。重流（Reflow）通过迭代拉直该向量场诱导的轨迹来加速推理。本文研究该迭代的渐近行为并刻画其极限点：首先，定义始终存在的弱整流耦合；其次，当整流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，其中N为批量大小，此类N-循环单调耦合具备可整流性和直性等良好结构与稳定性性质；最后，将速度限制为梯度场并假设额外支撑条件时，证明重流极限与端点分布间的最优传输映射一致。

英文摘要

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.

发表机构

• Université Paris Dauphine - PSL（巴黎多芬大学 - 巴黎文理研究大学）
• Inria（法国国家信息与自动化研究所）
• ENS Paris（巴黎高等师范学院）

机构由 AI 辅助整理，请以论文原文为准。