الملخص
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.
الكلمات المفتاحية
الموضوع
بيانات النشر
- المجلة
- غير متاح
- وصول مفتوح
- وصول مفتوح أخضر
اقتبس هذه المقالة
APA 7
Liu, Y., & Huang, L. (2026). Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature. https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature
MLA 9
Liu, Yuhao, and Longbo Huang. "Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature." https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature.
شيكاغو (المؤلف–التاريخ)
Liu, Yuhao, and Longbo Huang. 2026. "Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature." https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature.
هارفارد
Liu, Y. and Huang, L. (2026) 'Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature', Available at: https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature.
فانكوفر
Liu Y, Huang L. Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature. https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature
IEEE
Y. Liu, and L. Huang, "Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature," https://omanscience.com/ar/articles/sharp-convergence-and-sampling-trade-offs-for-riemannian-diffusion-under-nonnegative-ricci-curvature.