Order-Optimal Sample Complexity of Rectified Flows

TL;DR

This paper proves that rectified flow models, which restrict transport to linear paths, achieve order-optimal sample complexity of O(()()) and explain their empirical efficiency.

Contribution

The paper establishes the first theoretical sample complexity bounds for rectified flows, showing they are order-optimal and explaining their fast sampling in practice.

Findings

Rectified flows achieve O(()()) sample complexity.

They match the optimal rate for mean estimation.

The analysis uses localized Rademacher complexity to explain empirical performance.

Abstract

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.