arXiv：cs.LG· Jing Gu, Morteza Mardani, Wonjun Lee, Dongmian Zou, Gilad Lerman·· 2 天前AI 评分36
基于 Fisher 几何理解潜在空间可扩散性
Understanding Latent Diffusability via Fisher Geometry
AI 导读
研究通过 MMSE 沿扩散轨迹的变化率量化潜在空间可扩散性，并将其分解为 Fisher 信息（FI）与 Fisher 信息率（FIR）两部分。理论证明等距嵌入保持内在 FI，并为受控弱体积失真的双 Lipschitz 编码器建立内在 FI 界，而 FIR 由编码器与数据几何的相互作用决定。多类自编码架构实验为理论识别的几何机制提供定性支持，FI 与 FIR 可追踪生成质量与潜在空间几何的多种度量。
正文
Abstract:Diffusion models often degrade in latent spaces, yet the formal causes remain poorly understood. We quantify latent-space diffusability via the rate of change of the Minimum Mean Squared Error (MMSE) along the diffusion trajectory. Our framework decomposes this MMSE rate into contributions from Fisher Information (FI) and Fisher Information Rate (FIR). We show that isometric embeddings preserve intrinsic FI and establish quantitative intrinsic-FI bounds for a broader class of bi-Lipschitz encoders with controlled weak volume distortion, whereas FIR is governed by the interplay between encoder and data geometries. Our analysis separates four geometric contributions in local stability bounds for Gaussian-smoothed FIR: dimensional compression, tangential distortion, high-frequency encoder curvature, and curvature of data manifold. Experiments across diverse autoencoding architectures provide qualitative support for the geometric mechanisms identified by the theory and show that empirical FI and FIR track several measures of generation quality and latent-space geometry in the settings tested. We establish FI and FIR as a comprehensive analytical framework for understanding latent diffusability.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2604.02751 [cs.LG] |
| (or arXiv:2604.02751v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.02751 arXiv-issued DOI via DataCite |
Submission history
From: Jing Gu [view email]
[v1]
Fri, 3 Apr 2026 05:52:09 UTC (4,133 KB)
[v2]
Fri, 12 Jun 2026 19:45:24 UTC (3,060 KB)
[v3]
Thu, 1 Oct 2026 05:01:56 UTC (3,369 KB)
来源：arXiv：cs.LG · arxiv.org