通过截断KL展开降低扩散模型采样噪声，提升生成质量且无需重训练。
Improving Diffusion Generative Models via Truncated Karhunen--Loève Expansion
- 用截断KL展开重构前向过程，独立调节噪声水平。
- 在多个数据集上找到最优截断阶数，显著改善生成效果（如CIFAR-10 FID降至5.47）。
- 无需修改网络，可直接用于预训练模型，适合追求高效改进的开发者。
预训练扩散模型存在训练-采样不匹配问题，常归因于暴露偏差与分布偏移。我们通过有效噪声水平的概念量化解释该现象：经验上，预训练去噪器表现得像在略低于名义调度的噪声水平下训练。受此启发，我们提出一种无需训练的采样策略，基于布朗运动驱动前向随机微分方程的Karhunen--Loève（KL）展开截断。截断后得到低噪声水平的有限维前向过程，其噪声可独立于时间离散化调整。我们证明了截断过程对原扩散过程的统一收敛性。为解释结果，我们分析了一个理想模型：去噪器精确但作用于更低有效噪声水平。该模型预测采样噪声响应非单调，存在唯一内部最优解，可通过一维扫描截断阶数定位。我们实现对应截断反向时序与概率流方程，未改动网络结构。在CIFAR-10、CelebA、ImageNet及潜空间Stable Diffusion上，截断阶数均揭示出性能甜点，几乎在所有配置中改进预训练模型。从头训练时采用匹配截断阶数，使该阶数成为网络自身甜点，加速收敛约2.8倍，并降低生成误差（CIFAR-10 FID由7.14降至5.47，最佳检查点由6.74降至5.23）。勒维–切谢尔斯基对比验证有限展开具有广泛益处。
原文摘要 · Abstract (English)
Pretrained diffusion models exhibit a well-known training-sampling mismatch, often attributed to exposure bias and related distribution-shift effects. We provide a quantitative interpretation of this phenomenon through the notion of an effective noise level: empirically, a pretrained denoiser behaves as if trained at a noise level slightly below the nominal schedule. Motivated by this observation, we introduce a training-free sampling strategy based on truncating the Karhunen--Loève (KL) expansion of the Brownian motion driving the forward stochastic differential equation. Truncation yields a finite-dimensional forward process with a reduced noise level that can be adjusted independently of the time discretization. We prove uniform convergence of the truncated process to the original diffusion. To explain the resulting behaviour, we analyse a toy model in which the denoiser is exact but operates at a reduced effective noise level. The analysis predicts a non-monotone response to the sampling noise with a unique interior optimum, located by a one-dimensional sweep over the truncation order. We implement the approach through corresponding truncated reverse-time and probability-flow equations, without modifying the network architecture. Across CIFAR-10, CelebA, ImageNet, and latent-space Stable Diffusion, the truncation order consistently reveals a sweet spot, improving pretrained models in nearly all tested configurations. Training from scratch at a matched truncation order makes that order the network's own sweet spot, accelerates convergence by about $2.8\times$, and lowers the generation error on CIFAR-10 (Fréchet Inception Distance 7.14 to 5.47 at matched epochs; best checkpoint 6.74 to 5.23). A Lévy--Ciesielski comparison confirms that finite expansion is broadly beneficial.
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