用简单方法证明扩散模型可替换高斯噪声，提升计算效率
Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivity
- 基于格罗尼瓦尔不等式推导出收敛速度为O(T⁻¹ᐟ²)
- 只要均值方差匹配，离散噪声（如Rademacher）性能不降
- 适合关注理论简化与高效生成的科研人员
扩散模型通常被建模为随机微分方程（SDE）的离散化形式，在生成任务中表现优异。然而其理论分析常涉及复杂证明。本文提出一种简化框架，用于分析方差保持型SDE（VP-SDE）的欧拉-马鲁雅姆离散化。通过格罗尼瓦尔不等式，在标准利普希茨条件下，推导出收敛率为O(T⁻¹ᐟ²)，显著简化了先前分析。进一步证明，只要均值和方差匹配，标准高斯噪声可被计算成本更低的离散随机变量（如Rademacher）替代，且不破坏收敛性保证。实验验证表明：(i) 当方差正确匹配时，离散噪声生成样本质量与高斯噪声相当；(ii) 若噪声方差缩放错误，性能会下降。
原文摘要 · Abstract (English)
Diffusion models, typically formulated as discretizations of stochastic differential equations (SDEs), have achieved state-of-the-art performance in generative tasks. However, their theoretical analysis often involves complex proofs. In this work, we present a simplified framework for analyzing the Euler--Maruyama discretization of variance-preserving SDEs (VP-SDEs). Using Grönwall's inequality, we derive a convergence rate of $O(T^{-1/2})$ under standard Lipschitz assumptions, streamlining prior analyses. We then demonstrate that the standard Gaussian noise can be replaced by computationally cheaper discrete random variables (e.g., Rademacher) without sacrificing this convergence guarantee, provided the mean and variance are matched. Our experiments validate this theory, showing that (i) discrete noise achieves sample quality comparable to Gaussian noise provided the variance is matched correctly, and (ii) performance degrades if the noise variance is scaled incorrectly.
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