arXiv：cs.LG· Samuel Hurault, Thomas Moreau, Gabriel Peyr\'e·· 4 小时前AI 评分40
扩散模型的几何感知离散化误差
Geometry-Aware Discretization Error of Diffusion Models
AI 导读
研究推导了通用平滑反向扩散过程 Euler-Maruyama 弱误差与 Frechet 误差的渐近精确小步长展开式，并给出高斯数据下的显式公式。该公式可用于按目标协方差谱优化噪声调度、缩放系数与随机性系数，理论预测更小步数预算下最优随机性更低，并据此提出新的有效噪声调度族。在真实图像数据集上的实验显示，FID 最优参数与理论定性预测一致。
正文
Abstract:Practical diffusion sampling requires simulating a reverse-time ODE or SDE with a limited number of denoising steps, making the choice of sampling parameters crucial for minimizing discretization error. Non-asymptotic convergence bounds characterize sampling complexity, but their worst-case constants can obscure target geometry and thereby limit guidance on parameter optimization. Rather than bounding the error, we derive asymptotically exact small-stepsize expansions of Euler-Maruyama weak and Frechet errors for general smooth reverse diffusions, with explicit formulas for Gaussian data. These formulas provide tractable objectives for optimizing diffusion parameters, including the noise and rescaling schedules and the stochasticity coefficient, according to the target's covariance spectrum. In particular, our theory predicts lower optimal stochasticity at smaller step budgets, shows how to adapt the rescaling coefficient to the data power spectrum, and motivates a new family of effective noise schedules. A perturbative extension to Gaussian scale mixtures (GSMs) quantifies how departures from Gaussianity shift the optimal parameters. Finally, experiments on different real image datasets show that FID-optimal parameters agree with the qualitative theoretical predictions.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.08392 [cs.LG] |
| (or arXiv:2605.08392v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.08392 arXiv-issued DOI via DataCite |
Submission history
From: Samuel Hurault [view email]
[v1]
Fri, 8 May 2026 19:02:21 UTC (2,250 KB)
[v2]
Wed, 7 Oct 2026 08:57:49 UTC (2,697 KB)
来源：arXiv：cs.LG · arxiv.org