标题： 扩散模型与流形假设：对数域平滑是几何自适应的 显示英文标题

标题： Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive

摘要： 扩散模型已实现了最先进的性能，在多个领域表现出显著的泛化能力。 然而，这些强大能力背后的机制仍然只有部分被理解。 一种主要的假设基于流形假设，将这种成功归因于它们适应数据中低维几何结构的能力。 本研究为这一假设提供了证据，重点探讨了通过分数匹配制定学习问题如何导致这种现象。 我们通过研究经验分数匹配目标的平滑最小值的影响，检查了隐式正则化的作用。 我们的理论和实证结果证实，对分数函数进行平滑处理——或等效地，在对数密度域中进行平滑处理——会产生沿着数据流形的平滑效果。 此外，我们表明，扩散模型泛化的流形可以通过选择适当的平滑来控制。 显示英文摘要

摘要： Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function -- or equivalently, smoothing in the log-density domain -- produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.