Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature

2026-09-28

AI Summary

This study addresses the poor dimension dependence and excessive score function evaluations in existing convergence bounds for Riemannian diffusion models. To overcome these limitations, this work proposes a theoretical framework that decouples score discretization from Brownian motion simulation, alongside a multi-step geodesic random walk strategy. By leveraging Riemannian geometry, Kullback–Leibler divergence analysis, and total variation error control, the method achieves efficient sampling under non-negative Ricci curvature conditions. The primary contribution lies in significantly reducing the required frequency of score evaluations, demonstrating that only ~O(d/ε²) evaluations suffice to reach the target distribution. This matches the convergence rates established in Euclidean spaces, thereby providing a sharper theoretical characterization of the sampling complexity for Riemannian diffusion models.

Abstract

Diffusion models have emerged as state-of-the-art generative models, with recent extensions from Euclidean spaces to Riemannian manifolds. However, existing convergence guarantees for Riemannian diffusion models typically require $\tilde{O}(\mathrm{poly}(d,T)/ε^2)$ score evaluations, with potentially unfavorable dependence on the dimension. In this work, we develop a general framework that separates score discretization from Brownian-motion simulation and allows multiple geodesic random-walk steps per score evaluation. Under nonnegative Ricci curvature assumption and an exact Brownian-motion simulation oracle, we show that $\tilde{O}(d/ε^2)$ score evaluations suffice to achieve an $ε^2$ KL divergence from the target distribution, matching the existing convergence rate of Euclidean diffusion models. We further show that $\tilde{O}(d^4T/ε^2)$ geodesic random-walk steps suffice to approximate the required drifted Brownian motion to $ε$ total variation error. Combining these results yields a sampling scheme with $\tilde{O}(d/ε^2)$ score evaluations and $\tilde{O}(d^4T/ε^2)$ geodesic random-walk steps, motivating multiple random-walk steps between consecutive score evaluations. Our results provide a sharper characterization of the convergence and sampling complexity of Riemannian diffusion models.

Problem

• Riemannian diffusion models
• convergence guarantees
• sampling complexity
• score evaluations
• nonnegative Ricci curvature

Innovation

• Riemannian diffusion models
• score discretization
• geodesic random walk
• nonnegative Ricci curvature
• sampling complexity