Polynomial Convergence of Riemannian Diffusion Models

2026-01-05

arXiv.org

AI Summary

This work addresses the limitation of existing diffusion models, which typically assume data reside in Euclidean space and thus struggle with generative tasks under manifold constraints. The authors investigate diffusion processes on Riemannian manifolds by analyzing the perturbed heat equation and its logarithmic gradient—i.e., the score function—and establish a convergence theory for diffusion models in non-Euclidean settings through Li–Yau estimates and the Minakshisundaram–Pleijel parametrix expansion. Under mild conditions—requiring only L²-accurate score estimation and without assuming smoothness or positivity of the data distribution—they prove that polynomially small step sizes suffice to achieve low sampling error in total variation distance, relying solely on standard curvature assumptions of the manifold. This significantly relaxes the theoretical requirements for Riemannian diffusion models and provides a more general foundation for generative modeling on non-Euclidean spaces.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Diffusion models have demonstrated remarkable empirical success in the recent years and are considered one of the state-of-the-art generative models in modern AI. These models consist of a forward process, which gradually diffuses the data distribution to a noise distribution spanning the whole space, and a backward process, which inverts this transformation to recover the data distribution from noise. Most of the existing literature assumes that the underlying space is Euclidean. However, in many practical applications, the data are constrained to lie on a submanifold of Euclidean space. Addressing this setting, De Bortoli et al. (2022) introduced Riemannian diffusion models and proved that using an exponentially small step size yields a small sampling error in the Wasserstein distance, provided the data distribution is smooth and strictly positive, and the score estimate is $L_\infty$-accurate. In this paper, we greatly strengthen this theory by establishing that, under $L_2$-accurate score estimate, a {\em polynomially small stepsize} suffices to guarantee small sampling error in the total variation distance, without requiring smoothness or positivity of the data distribution. Our analysis only requires mild and standard curvature assumptions on the underlying manifold. The main ingredients in our analysis are Li-Yau estimate for the log-gradient of heat kernel, and Minakshisundaram-Pleijel parametrix expansion of the perturbed heat equation. Our approach opens the door to a sharper analysis of diffusion models on non-Euclidean spaces.

Problem

• Riemannian diffusion models
• non-Euclidean spaces
• sampling error
• score estimation
• manifold constraints

Innovation

• Riemannian diffusion models
• polynomial convergence
• score estimation
• heat kernel
• manifold generative modeling