从高斯视角解析生成扩散模型的分布差异，给出优化噪声调度的方法。
A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Models
- 基于高斯源推导出扩散过程的闭式解和KL散度演化轨迹。
- 提出由噪声调度主导的KL散度最小化方法，得到正切律噪声调度。
- 可作为评估预训练模型时间离散化策略的理论依据，适合模型优化研究者。
本文提出一种分析生成扩散模型中分布差异的解析方法。针对多变量高斯源，我们显式推导了扩散过程的闭式演化轨迹及源数据与逆向采样数据分布间的KL散度。通过Euler-Maclaurin展开进行渐近分析，揭示该KL散度的收敛行为，提取其主导项为噪声调度的显式函数。利用变分法最小化该主导项，得到由源协方差谱决定的正切律噪声调度。进一步证明，在给定协方差下，高斯源使KL散度达到极值。还将该解析KL散度用于指导预训练扩散模型的高效时间离散化策略，实验表明在多种数据集上，所识别策略在受限函数评估预算下均显著优于现有基线。
原文摘要 · Abstract (English)
This paper introduces an analytical approach to quantifying and optimizing the distributional discrepancy in generative diffusion models. For a multivariate Gaussian source, we explicitly derive the closed-form evolution trajectory and the resulting Kullback-Leibler (KL) divergence between the distributions of the source data and the reversely sampled data. Asymptotic analysis via the Euler-Maclaurin expansion characterizes the convergence behavior of this KL divergence, extracting its dominant term as an explicit functional of the noise schedule. Minimizing this dominant term via the calculus of variations yields a noise schedule described by a tangent law, inherently determined by the source covariance spectrum. We further prove that the Gaussian source exhibits an extremal property for the KL divergence among general source distributions with a given covariance. We also utilize the analytical KL divergence as a principled metric to identify efficient time discretization strategies for pretrained diffusion models, and demonstrate via experiments over diverse datasets that the identified strategies consistently outperform established baselines, particularly under constrained function evaluation budgets.
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