Elucidating Rectified Flow with Deterministic Sampler: Polynomial Discretization Complexity for Multi and One-step Models

Abstract

Recently, rectified flow (RF)-based models have achieved state-of-the-art performance in many areas for both the multi-step and one-step generation. However, only a few theoretical works analyze the discretization complexity of RF-based models. Existing works either focus on flow-based models with stochastic samplers or establish complexity results that exhibit exponential dependence on problem parameters. In this work, under the realistic bounded support assumption, we prove the first polynomial discretization complexity for multi-step and one-step RF-based models with a deterministic sampler simultaneously. For the multi-step setting, inspired by the predictor-corrector framework of diffusion models, we introduce a Langevin process as a corrector and show that RF-based models can achieve better polynomial discretization complexity than diffusion models. To achieve this result, we conduct a detailed analysis of the RF-based model and explain why it is better than previous popular models, such as variance preserving (VP) and variance exploding (VE)-based models. Based on the observation of multi-step RF-based models, we further provide the first polynomial discretization complexity result for one-step RF-based models, improving upon prior results for one-step diffusion-based models. These findings mark the first step toward theoretically understanding the impressive empirical performance of RF-based models in both multi-step and one-step generation.