非负Ricci曲率下黎曼扩散的尖锐收敛性与采样权衡

中文摘要 AI 辅助

扩散模型已成为最先进的生成模型，近期其应用从欧几里得空间扩展到了黎曼流形。然而，现有的黎曼扩散模型收敛保证通常需要$\tilde{O}(\mathrm{poly}(d,T)/\epsilon^2)$次分数评估，且对维度的依赖可能不利。在本工作中，我们开发了一个通用框架，将分数离散化与布朗运动模拟分离，并允许在每次分数评估之间进行多次测地线随机游走步骤。在非负Ricci曲率假设和精确布朗运动模拟预言机下，我们证明$\tilde{O}(d/\epsilon^2)$次分数评估足以实现与目标分布的$\epsilon^2$ KL散度，与欧几里得扩散模型的现有收敛速率相匹配。我们进一步证明，$\tilde{O}(d^4T/\epsilon^2)$步测地线随机游走足以将所需的漂移布朗运动近似到$\epsilon$总变差误差。结合这些结果，我们得到一个采样方案，其需要$\tilde{O}(d/\epsilon^2)$次分数评估和$\tilde{O}(d^4T/\epsilon^2)$步测地线随机游走，这激励了在连续分数评估之间进行多次随机游走步骤。我们的结果为黎曼扩散模型的收敛性和采样复杂度提供了更尖锐的表征。

英文摘要

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.

发表机构

• Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）

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