Understanding Latent Diffusability via Fisher Geometry

TL;DR

This paper introduces a Fisher geometry-based framework to understand and diagnose why diffusion models degrade in latent spaces, providing theoretical insights and practical metrics for improved latent diffusion.

Contribution

It formalizes the causes of latent diffusion failure using Fisher Information geometry and offers measurable penalties and conditions to preserve diffusability across spaces.

Findings

FIR is influenced by local geometric properties of the encoder.

Global isometry aligns Fisher Information, aiding diffusion.

Proposed metrics effectively diagnose latent diffusion issues.

Abstract

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.