用费舍尔度量揭示生成模型潜空间的几何结构，发现相变边界处奇异行为
Hessian Geometry of Latent Space in Generative Models
- 通过后验近似重构费舍尔度量，刻画潜空间曲率
- 在扩散模型中发现相变边界的分形结构与利普希茨常数发散
- 适合研究生成模型内在几何与相变现象的学者
本文提出一种新方法，分析生成模型（包括统计物理模型和扩散模型）潜空间的几何结构，通过重构费舍尔信息度量实现。该方法近似给定生成样本下潜变量的后验分布，并利用其学习对数分区函数，从而定义指数族中的费舍尔度量。理论证明了收敛性，验证于伊辛模型和TASEP模型，在重构热力学量方面优于现有基线。应用于扩散模型时，揭示潜空间存在分形结构的相变特征，表现为费舍尔度量的突变。我们发现：在单个相内测地线插值近似线性，但在相变边界处线性失效，扩散模型对潜空间的利普希茨常数发散。这些发现为扩散模型潜空间的复杂结构及其与相变现象的关联提供了新见解。代码已开源。
原文摘要 · Abstract (English)
This paper presents a novel method for analyzing the latent space geometry of generative models, including statistical physics models and diffusion models, by reconstructing the Fisher information metric. The method approximates the posterior distribution of latent variables given generated samples and uses this to learn the log-partition function, which defines the Fisher metric for exponential families. Theoretical convergence guarantees are provided, and the method is validated on the Ising and TASEP models, outperforming existing baselines in reconstructing thermodynamic quantities. Applied to diffusion models, the method reveals a fractal structure of phase transitions in the latent space, characterized by abrupt changes in the Fisher metric. We demonstrate that while geodesic interpolations are approximately linear within individual phases, this linearity breaks down at phase boundaries, where the diffusion model exhibits a divergent Lipschitz constant with respect to the latent space. These findings provide new insights into the complex structure of diffusion model latent spaces and their connection to phenomena like phase transitions. Our source code is available at https://github.com/alobashev/hessian-geometry-of-diffusion-models.
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