aiqbench analysis
This research establishes a theoretical framework that significantly improves the sampling efficiency of diffusion models operating on Riemannian manifolds by decoupling score discretization from Brownian motion simulation. Under specific geometric constraints, the study demonstrates that convergence rates can match those of standard Euclidean models while optimizing the number of required score evaluations. These findings are critical for advancing generative AI applications in non-Euclidean domains where computational complexity and dimensionality have previously hindered practical deployment.
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 $\tilde{O}(\mathrm{poly}(d,T)/\epsilon^2)$ 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 $\tilde{O}(d/\epsilon^2)$ 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 $\tilde{O}(d^4T/\epsilon^2)$ 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 $\tilde{O}(d/\epsilon^2)$ score evaluations and $\tilde{O}(d^4T/\epsilon^2)$ 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.
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 $\tilde{O}(\mathrm{poly}(d,T)/\epsilon^2)$ 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 $\tilde{O}(d/\epsilon^2)$ 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 $\tilde{O}(d^4T/\epsilon^2)$ 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 $\tilde{O}(d/\epsilon^2)$ score evaluations and $\tilde{O}(d^4T/\epsilon^2)$ 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.
Machine Learning (cs.LG)
arXiv:2609.36568 [cs.LG]
(or arXiv:2609.36568v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.36568 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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