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

arXiv: https://arxiv.org/abs/2609.36568 · HTML: https://mlsanity.com/paper/2609.36568/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 20 of its 25 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 20 of 20)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach](https://mlsanity.com/paper/2402.13901/) (2024)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions](https://mlsanity.com/paper/2508.16306/) (2025)
- [Riemannian Consistency Model](https://mlsanity.com/paper/2510.00983/) (2025)
- [Efficient Sampling on Riemannian Manifolds via Langevin MCMC](https://mlsanity.com/paper/2402.10357/) (2024)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds](https://mlsanity.com/paper/2502.07265/) (2025)

## Abstract (arXiv)

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.