揭示扩散模型如何学习流形数据的统计规律
Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
- 将数据视为流形上的样本，分解得分函数在不同噪声下的结构
- 发现统计误差由数据内在维度和流形曲率决定
- 为流形上生成建模提供理论支持，适合研究生成模型的学者
扩散模型已成为生成建模的主流框架，但对其在高维低结构数据上的理论理解仍不充分。本文聚焦于数据几何特性与统计复杂度的影响，将数据建模为光滑黎曼流形上的样本，揭示了扩散模型中得分函数在不同噪声水平下的关键分解结构。分析表明，流形曲率与得分函数结构存在相互作用。基于此，我们提出了高效的神经网络得分函数近似方法，并给出了得分估计与分布学习的统计收敛速率。值得注意的是，这些速率由数据的内在维度和流形曲率共同决定。该研究推进了扩散模型在流形数据上的统计基础，弥合了理论与实践的差距。
原文摘要 · Abstract (English)
Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates how diffusion models learn such structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties. By modeling data as samples from a smooth Riemannian manifold, our analysis reveals crucial decompositions of score functions in diffusion models under different levels of injected noise. We also highlight the interplay of manifold curvature with the structures in the score function. These analyses enable an efficient neural network approximation to the score function, built upon which we further provide statistical rates for score estimation and distribution learning. Remarkably, the obtained statistical rates are governed by the intrinsic dimension of data and the manifold curvature. These results advance the statistical foundations of diffusion models, bridging theory and practice for generative modeling on manifolds.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。