发表机构
Inria; École normale supérieure – PSL Research University（法国国家信息与自动化研究所; 巴黎高等师范学院-PSL研究大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高斯设定建立了扩散模型的2-Wasserstein收敛界，揭示其采样误差无额外条件数依赖，且加噪的益处源于采样阶段而非学习阶段。
AI 中文摘要
尽管扩散模型在经验上取得了成功，但它们为何能克服经典基于得分的采样器的瓶颈仍不清楚。在这项工作中，我们利用高斯分布来分离这一现象。我们针对优化后的超参数建立了2- Wasserstein收敛界，表明扩散过程的采样误差为$O(\sqrt{d\lambda_{max}}\log N/N)$，其中$d$为维度，$N$为采样步数，$\lambda_{max}$为目标协方差矩阵的最大特征值。未调整的欠阻尼朗之万动力学则会遭受额外的$\sqrt{\kappa}$因子影响，其中$\kappa$为条件数。这些速率源于严格的谱界：我们通过$N\rightarrow\infty$时匹配一阶渐近性来验证它们。我们的分析在高斯设定下提供了严格刻画，说明时变得分轨迹如何在采样过程中消除条件数依赖。相比之下，在学习阶段，我们证明通过梯度下降估计无噪得分会得到与估计有噪得分基本相同的估计量，这表明加噪的益处并非来自学习阶段。
英文摘要
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 $O(\sqrt{dλ_{\max}}\log N/N)$, where $d$ is the dimension, $N$ the number of sampling steps, and $λ_{\max}$ the largest eigenvalue of the target covariance matrix. Unadjusted and underdamped Langevin dynamics suffer from an additional $\sqrtκ$ factor, where $κ$ 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.
Comments49 pages (10 main + appendix), 3 figures
