Order-Optimal Sample Complexity of Rectified Flows

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.

💡 Research Summary

This paper provides the first rigorous sample‑complexity analysis of Rectified Flow (RF), a recent flow‑based generative modeling framework that forces transport trajectories to be straight lines between a simple base distribution (e.g., a Gaussian) and the target data distribution. The authors show that, under standard smoothness, sub‑Gaussian, and Polyak‑Łojasiewicz (PL) assumptions on the data and on the neural‑network parameterization of the velocity field, RF achieves a sample complexity of (\tilde O(\varepsilon^{-2})) for approximating the target distribution in Wasserstein distance. This improves upon the best known (,O(\varepsilon^{-4})) bounds for general flow‑matching models and matches the information‑theoretic lower bound for mean estimation, establishing RF as order‑optimal.

The key technical insight stems from the special structure of RF: the training objective is a squared‑loss regression on linear interpolation paths (X_t = (1-t)X_0 + tX_1). Because the optimal velocity field is the conditional expectation (v^*(x,t)=\mathbb{E}