In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. To this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation on why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

翻译：本文在流形假设下研究生成扩散模型的潜在几何结构。为此，我们分析了得分函数雅可比矩阵的特征值（及奇异值）谱，其不连续性（谱隙）揭示了不同子流形的存在及其维度。基于统计物理方法，我们在多种分布假设下推导了谱分布与谱隙的计算公式，并将理论预测结果与训练网络估计的谱进行比较。分析表明生成过程存在三个不同的定性相变：平凡相；流形覆盖相——扩散过程拟合流形内部分布；固化相——得分函数与流形正交，所有粒子被投影至数据支撑集。这种不同时间尺度上的“分工机制”优雅地解释了为何生成扩散模型不会受到困扰基于似然模型的流形过拟合现象的影响，因为内部分布与流形几何结构是在生成过程的不同时间点分别形成的。