arXiv：cs.LG· Radmehr Karimian, Ali Mohades, Johannes Lederer·· 3 小时前
截断扩散采样器需保留多少方向？幂律谱下的匹配边界
How Many Directions Must a Truncated Diffusion Sampler Retain? Matching Bounds Under Power-Law Spectra
AI 导读
针对具有幂律协方差谱的数据，研究者证明了在环境维度足够大时，截断扩散采样器所需保留方向数的匹配边界。截断误差取决于被省略方向的组合 Wiener 增益，只保留信号超过输出噪声水平的方向会留下非消失误差，因为许多单独微弱的方向在总体上仍然显著。结合扩散收敛边界，该分析给出了精确得分下足够的采样步复杂度，并将上界扩展至估计主成分与高斯混合模型。
正文
Abstract: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.
| Subjects: | Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.10640 [math.ST] |
| (or arXiv:2610.10640v1 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2610.10640 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Radmehr Karimian [view email]
[v1]
Wed, 7 Oct 2026 14:32:37 UTC (605 KB)
来源：arXiv：cs.LG · arxiv.org