This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counterexamples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.

翻译：本文探讨了整流流、流匹配与最优传输之间的关联。流匹配是一种通过估计引导从源分布到目标分布转换的速度场来学习生成模型的新方法。整流流匹配旨在拉直已学习的传输路径，从而在分布间产生更直接的流动。我们的首要贡献是建立了一组整流流与显式速度场的不变性性质。此外，我们还在高斯（未必独立）与高斯混合场景中提供了显式构造与分析，并研究了其与最优输运的关系。我们的第二项贡献针对近期某些观点展开探讨，这些观点认为当对整流流施加约束使得学习到的速度场为梯度场时，其可（渐近地）得到最优输运问题的解。我们研究了该问题解的存在性，并证明仅在远强于以往认知的假设条件下，这些解才与最优输运相关。特别地，我们提出了若干反例，否定了文献中早期的等价性结论，并论证了在整流流上强制施加梯度约束通常并非计算最优输运映射的可靠方法。