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

arXiv:2609.36568v1 Announce Type: new 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 ilde{O}(poly(d,T)/epsilon^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 ilde{O}(d/epsilon^2) score evaluations suffice to achieve an epsilon^2 KL divergence from the target distribution, matching the existing convergence rate of Euclidean diffusion models. We further show that ilde{O}(d^4T/epsilon^2) geodesic random-walk steps suffice to approximate the required drifted Brownian motion to epsilon total variation error. Combining these results yields a sampling scheme with ilde{O}(d/epsilon^2) score evaluations and ilde{O}(d^4T/epsilon^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.

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Source: arXiv cs.LG | 2026-09-30