Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models

2024-05-23

arXiv.org

AI Summary

Score-based diffusion models (e.g., DDPM) suffer from low sampling efficiency on low-dimensional manifolds—such as natural images—and their theoretical convergence rates depend explicitly on the ambient dimension (d), which is overly pessimistic for intrinsically low-dimensional data. Method: This work introduces a novel deterministic dynamical systems analysis framework, integrating manifold assumptions and score estimation error modeling, to theoretically characterize how DDPM samplers adapt to unknown low-dimensional structures. It proposes an intrinsic-dimension-(k)-aware optimal diffusion coefficient design. Contribution/Results: We establish the first theoretical proof that DDPM sampling dynamics automatically adapt to the underlying manifold geometry. Our analysis yields a convergence rate of (O(k^2 / sqrt{T}))—where (T) is the number of steps—eliminating explicit dependence on (d). This reveals the decisive role of coefficient design in low-dimensional adaptation, substantially tightening existing theoretical bounds and establishing a new paradigm for manifold-aware diffusion modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• 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: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of $O(k^{2}/sqrt{T})$ (up to log factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.

Problem

• Fractional Diffusion Models
• Low-Dimensional Manifold Data
• Efficiency and Accuracy Optimization

Innovation

• Fractional-Diffusion-Model-Optimization
• High-Volume-Data-Efficiency
• Low-Dimensional-Manifold-Processing