Understanding Latent Diffusability via Fisher Geometry
This work addresses a specific issue for researchers and practitioners using diffusion models in latent spaces, providing a diagnostic suite to identify and mitigate failure, but it is incremental as it builds on existing understanding of geometric properties.
The paper tackles the problem of diffusion models degrading when trained in latent spaces by quantifying latent-space diffusability through the rate of change of Minimum Mean Squared Error along the diffusion trajectory, decomposing it into contributions from Fisher Information and Fisher Information Rate, and validating the framework with experiments across autoencoding architectures.
Diffusion models often degrade when trained in latent spaces (e.g., VAEs), yet the formal causes remain poorly understood. We quantify latent-space diffusability through 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 encoder's local geometric properties. Our analysis explicitly decouples latent geometric distortion into three measurable penalties: dimensional compression, tangential distortion, and curvature injection. We derive theoretical conditions for FIR preservation across spaces, ensuring maintained diffusability. Experiments across diverse autoencoding architectures validate our framework and establish these efficient FI and FIR metrics as a robust diagnostic suite for identifying and mitigating latent diffusion failure.