Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures

2026-06-18

AI Summary

This work addresses the limitations of existing score approximation theories for diffusion models, which rely on strong smoothness assumptions and struggle with irregular structures in real-world data—such as singularities, sharp boundaries, and disconnected clusters. The authors propose a discrete mixture formulation and establish a universal score approximation theorem applicable to any compactly supported distribution. They prove that ReLU networks whose complexity depends only on the Minkowski dimension \(d\) of the support set suffice to accurately approximate the score function. This approach overcomes the traditional exponential dependence on ambient dimension and, for the first time, achieves dimension-aware score approximation without assuming smooth manifolds or Lipschitz densities. The result theoretically substantiates the inherent adaptability of diffusion models to non-smooth and irregular data structures, providing a rigorous foundation for their remarkable generative capabilities.

Abstract

The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive assumptions like Lipschitz continuous densities or smooth manifold supports, which are routinely violated by the singularities, sharp boundaries, and disjoint clusters inherent to real-world perceptual data. This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension $d$. Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with $d$, thus breaking the exponential curse of ambient dimensionality. Combined with existing theories on accurately solving the backward diffusion SDE for arbitrary compact distributions, our work shows that diffusion models readily adapt to irregular, non-smooth data structures, explaining their competence in real-world generative tasks.

Problem

• score approximation
• diffusion models
• low-dimensional structures
• singularities
• non-smooth data

Innovation

• score approximation
• diffusion models
• Minkowski dimension
• ReLU networks
• dimensionality curse