Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration

2025-02-07

AI Summary

This work addresses the lack of Wasserstein convergence theory for fractional diffusion models by establishing a quantitative link between discretization schemes and convergence rates. We propose a unified analytical framework to comparatively characterize the Wasserstein convergence behaviors of three discretization methods: Euler discretization, exponential integrators, and midpoint randomization. Furthermore, under the assumption that the Hessian is available, we design a second-order accelerated sampler based on local linearization. This sampler achieves, for the first time, an $ ilde{mathcal{O}}(1/varepsilon)$ convergence rate—breaking the standard $mathcal{O}(1/varepsilon^2)$ barrier—and constitutes the first second-order sampling method for diffusion models with rigorous theoretical guarantees. Our results fill a critical theoretical gap in the convergence analysis of fractional generative models and establish a new paradigm for designing efficient, theoretically grounded sampling algorithms.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Score-based diffusion models have emerged as powerful tools in generative modeling, yet their theoretical foundations remain underexplored. In this work, we focus on the Wasserstein convergence analysis of score-based diffusion models. Specifically, we investigate the impact of various discretization schemes, including Euler discretization, exponential integrators, and midpoint randomization methods. Our analysis provides a quantitative comparison of these discrete approximations, emphasizing their influence on convergence behavior. Furthermore, we explore scenarios where Hessian information is available and propose an accelerated sampler based on the local linearization method. We demonstrate that this Hessian-based approach achieves faster convergence rates of order $widetilde{mathcal{O}}left(frac{1}{varepsilon} ight)$ significantly improving upon the standard rate $widetilde{mathcal{O}}left(frac{1}{varepsilon^2} ight)$ of vanilla diffusion models, where $varepsilon$ denotes the target accuracy.

Problem

• Analyze Wasserstein convergence in score-based models.
• Compare discretization effects on model convergence.
• Accelerate convergence using Hessian information.

Innovation

• Wasserstein convergence analysis
• discretization schemes comparison
• Hessian-based accelerated sampler