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

arXiv:2603.20645v2 Announce Type: replace 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.

Related

• Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data
• Diffusion Processes on Implicit Manifolds
• Conditional flow matching for physics-constrained inverse problems with finite training data
• A Review of Diffusion-based Simulation-Based Inference: Foundations and Applications in Non-Ideal Data Scenarios

Source: arXiv cs.LG | 2026-04-29