arXiv：cs.LG· Adam Perbost, Francis Bach, Pierre Marion·· 4 小时前
扩散模型为何摆脱 Langevin 采样的条件数依赖：一项尖锐的高斯分析
Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis
AI 导读
一项高斯分析给出了扩散模型与经典 score-based 采样器的 2-Wasserstein 收敛界对比：扩散过程采样误差为 O(√(dλ_max)·log N/N)，而 Unadjusted 与 Underdamped Langevin 动力学额外多出 √κ 因子（κ 为条件数）。
正文
Abstract:Despite their empirical success, why diffusion models overcome the bottlenecks of classical score-based samplers remains unclear. In this work, we leverage Gaussian distributions to isolate this phenomenon. We establish 2-Wasserstein convergence bounds for optimized hyperparameters, showing that diffusion processes achieve a sampling error of $O(\sqrt{d\lambda_{\max}}\log N/N)$, where $d$ is the dimension, $N$ the number of sampling steps, and $\lambda_{\max}$ the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional $\sqrt\kappa$ factor, where $\kappa$ is the condition number. These rates follow from spectral bounds which are sharp: we confirm them via matching first-order asymptotics as $N\rightarrow\infty$. Our analysis provides a rigorous characterization, in the Gaussian setting, of how time-dependent score trajectories remove condition-number dependence during sampling. By contrast, in the learning phase, we show that estimating the unnoised score by gradient descent leads to essentially the same estimator as estimating a noisy score, which suggests that the benefits of noising do not come from the learning phase.
| Comments: | 49 pages (10 main + appendix), 3 figures |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.12052 [stat.ML] |
| (or arXiv:2610.12052v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.12052 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Adam Perbost [view email]
[v1]
Thu, 8 Oct 2026 14:36:38 UTC (75 KB)
来源：arXiv：cs.LG · arxiv.org