Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models

2025-01-31

AI Summary

This work addresses the inefficiency and high environmental-dimension dependency of probability flow ODE samplers for diffusion models when sampling on the low-dimensional manifold of natural images. We propose an adaptive sampling theoretical framework that models the probability flow ODE dynamics under the manifold hypothesis, leveraging score estimation and total variation distance analysis. Crucially, we provide the first rigorous proof that the associated sampler adapts to the intrinsic dimension $k$ of the target distribution. Our theoretical contribution establishes a dimension-independent convergence rate of $O(k/T)$, breaking the longstanding bottleneck where convergence depends on the ambient dimension $d gg k$. Under accurate score estimation, the convergence rate is governed solely by the manifold dimension $k$, yielding substantial improvement over prior bounds. This work provides a new theoretical foundation and design principle for efficient, stable, high-fidelity image generation from noise.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

Abstract

Score-based generative models, which transform noise into data by learning to reverse a diffusion process, have become a cornerstone of modern generative AI. This paper contributes to establishing theoretical guarantees for the probability flow ODE, a widely used diffusion-based sampler known for its practical efficiency. While a number of prior works address its general convergence theory, it remains unclear whether the probability flow ODE sampler can adapt to the low-dimensional structures commonly present in natural image data. We demonstrate that, with accurate score function estimation, the probability flow ODE sampler achieves a convergence rate of $O(k/T)$ in total variation distance (ignoring logarithmic factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of iterations. This dimension-free convergence rate improves upon existing results that scale with the typically much larger ambient dimension, highlighting the ability of the probability flow ODE sampler to exploit intrinsic low-dimensional structures in the target distribution for faster sampling.

Problem

• Probability Flow
• Differential Equations
• Data Modeling

Innovation

• Probability Flow ODE
• Score Function Estimation
• Efficient Data Generation