Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity

TL;DR

This paper advances the theoretical understanding of diffusion models for data on low-dimensional manifolds, analyzing score functions, curvature effects, and statistical complexity to improve generative modeling.

Contribution

It provides a novel analysis of score decomposition, curvature influence, and statistical rates for diffusion models on manifold-structured data.

Findings

Score functions decompose under different noise levels.

Manifold curvature impacts score function structures.

Statistical rates depend on intrinsic dimension and curvature.

Abstract

Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates how diffusion models learn such structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties. By modeling data as samples from a smooth Riemannian manifold, our analysis reveals crucial decompositions of score functions in diffusion models under different levels of injected noise. We also highlight the interplay of manifold curvature with the structures in the score function. These analyses enable an efficient neural network approximation to the score function, built upon which we further provide statistical rates for score estimation and distribution learning. Remarkably, the obtained statistical rates are governed by the intrinsic dimension of data and the manifold curvature. These results advance the statistical foundations of diffusion models, bridging theory and practice for generative modeling on manifolds.