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

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 tildeO(poly(d,T)/ε2) 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 tildeO(d/ε2) score evaluations suffice to achieve an ε2 KL divergence from the target distribution, matching the existing convergence rate of Euclidean diffusion models. We further show that tildeO(d4T/ε2) 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 tildeO(d/ε2) score evaluations and tildeO(d4T/ε2) 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.