发表机构
Université de Genève; Universität Hamburg（日内瓦大学; 汉堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对幂律协方差谱数据，推导了截断扩散采样器所需保留方向数的匹配界，结合扩散收敛界得到采样步复杂度，提出用聚合谱尾误差预算选择保留子空间的实用方案。
AI 中文摘要
扩散采样器可通过生成选定的谱坐标并对剩余方向填充噪声来减少计算量，那么它们必须保留多少个方向？我们针对具有幂律协方差谱的数据研究该问题。对于与平滑目标对比的高斯数据，我们证明了在环境维度足够大时所需保留方向数的匹配界。截断误差取决于被省略方向的组合维纳增益，与采样器在保留坐标上的精度无关。因此，仅保留信号超过输出噪声水平的方向会留下非零误差：许多单独较弱的方向在整体上仍具有重要性。将该特性与扩散收敛界结合，可得到精确得分下的充分采样步复杂度。该上界还可扩展到估计的主成分，以及逐分量扩展到高斯混合模型。实际应用方案是使用聚合谱尾误差预算选择保留子空间，再据此选择扩散噪声水平。
英文摘要
Diffusion samplers can reduce computation by generating selected spectral coordinates and filling the remaining directions with noise. How many directions must they retain? We study this question for data with power-law covariance spectra. For Gaussian data compared to a smoothed target, we prove matching bounds on the required number of retained directions, provided that the ambient dimension is sufficiently large. The truncation error depends on the combined Wiener gains of the omitted directions, regardless of the accuracy of the sampler on the retained coordinates. Keeping only directions whose signal exceeds the output noise level can therefore leave a non-vanishing error: many individually weak directions remain significant in aggregate. Combining this characterization with a diffusion convergence bound yields sufficient sampling-step complexity under exact scores. The upper bounds also extend to estimated principal components and, componentwise, to Gaussian mixtures. The practical prescription is to select the retained subspace using an aggregate spectral-tail error budget, then to choose the diffusion noise level accordingly.
