Rectified Flow 用线性路径加速采样，理论证明样本复杂度达最优。
Order-Optimal Sample Complexity of Rectified Flows
- 通过线性路径约束速度场，实现单步快速采样。
- 在标准假设下，样本复杂度达到约 ε⁻²，优于此前的 ε⁻⁴。
- 理论解释其高效性，适合关注生成模型效率的研究者。
近年来，基于流的生成模型相比扩散模型展现出更高的效率。本文研究了修正流（rectified flow）模型，该模型将传输轨迹限制为从先验分布到数据分布的直线路径。这一结构约束极大加速了采样过程，常可在单次欧拉步长下实现高质量生成。在对用于参数化速度场的神经网络类和数据分布的标准假设下，我们证明修正流达到样本复杂度 $ ilde{O}(ar{ heta}^{-2})$，优于此前流匹配模型的最佳已知 $O(ar{ heta}^{-4})$ 界，并达到均值估计的最优率。分析利用了修正流的特殊结构：因模型沿直线路径以平方损失训练，其对应的假设类具有被严格控制的局部 Rademacher 复杂度，从而获得阶最优的样本复杂度，为修正流模型的优异实证表现提供了理论解释。
原文摘要 · Abstract (English)
Recently, flow-based generative models have shown superior efficiency compared to diffusion models. In this paper, we study rectified flow models, which constrain transport trajectories to be linear from the base distribution to the data distribution. This structural restriction greatly accelerates sampling, often enabling high-quality generation with a single Euler step. Under standard assumptions on the neural network classes used to parameterize the velocity field and data distribution, we prove that rectified flows achieve sample complexity $\tilde{O}(\varepsilon^{-2})$. This improves on the best known $O(\varepsilon^{-4})$ bounds for flow matching model and matches the optimal rate for mean estimation. Our analysis exploits the particular structure of rectified flows: because the model is trained with a squared loss along linear paths, the associated hypothesis class admits a sharply controlled localized Rademacher complexity. This yields the improved, order-optimal sample complexity and provides a theoretical explanation for the strong empirical performance of rectified flow models.
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