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This research provides a rigorous theoretical framework using Fisher Information geometry to diagnose why diffusion models degrade in latent spaces, identifying specific geometric penalties like curvature and distortion as root causes. By decoupling these factors into four distinct components, the study offers actionable insights for improving encoder stability without relying on empirical trial-and-error tuning. Such analytical depth is critical for advancing the mathematical foundations of generative AI beyond black-box experimentation.
Understanding Latent Diffusability via Fisher Geometry
arXiv:2604.02751v2 Announce Type: replace 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 demonstrate that while global isometry ensures FI alignment, FIR is governed by the interplay between encoder and data geometries. Our analysis decouples diffusion degradation into four penalties: dimensional compression, tangential distortion, high-frequency encoder curvature, and intrinsic data curvature. We derive theoretical conditions for FIR preservation to ensure stable diffusability. Experiments across diverse autoencoding architectures demonstrate the implications of our theoretical bounds. We establish FI and FIR as a comprehensive analytical framework for understanding latent diffusability.
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 demonstrate that while global isometry ensures FI alignment, FIR is governed by the interplay between encoder and data geometries. Our analysis decouples diffusion degradation into four penalties: dimensional compression, tangential distortion, high-frequency encoder curvature, and intrinsic data curvature. We derive theoretical conditions for FIR preservation to ensure stable diffusability. Experiments across diverse autoencoding architectures demonstrate the implications of our theoretical bounds. We establish FI and FIR as a comprehensive analytical framework for understanding latent diffusability.
Machine Learning (cs.LG)
arXiv:2604.02751 [cs.LG]
(or arXiv:2604.02751v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2604.02751 Focus to learn more arXiv-issued DOI via DataCite
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