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

2025-08-12

AI Summary

Existing rectified flow (RF)-based generative models lack theoretical analysis of discretization complexity for discrete-time sampling; prior results either assume stochastic samplers or exhibit exponential dependence on problem parameters. Method: Under a bounded-support assumption, we establish the first polynomial upper bounds on discretization complexity for both multi-step and single-step RF models under deterministic sampling. Our approach introduces a Langevin corrector to construct a predictor-corrector framework and incorporates a refined error propagation analysis tailored to flow-based models. Results: We prove that RF models achieve $mathrm{poly}(1/varepsilon, d)$ discretization complexity in both multi-step and single-step settings—significantly improving upon the exponential dependence typical of diffusion models. This is the first rigorous theoretical explanation for the empirically observed superior performance of RF models, bridging a critical gap between theory and practice in flow-based generative modeling.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• Analyzing discretization complexity of rectified flow models
• Proving polynomial complexity for deterministic sampler models
• Comparing RF models with VP and VE diffusion models

Innovation

• Polynomial discretization complexity for rectified flow
• Deterministic sampler for multi-step and one-step models
• Langevin corrector improves upon diffusion models