Understanding Latent Diffusability via Fisher Geometry

arXiv cs.LG / 4/6/2026

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Key Points

The paper proposes a formal way to measure how well diffusion models remain “diffusable” in latent spaces by tracking the rate of change of MMSE along the diffusion trajectory.
It decomposes the MMSE change into contributions from Fisher Information (FI) and a Fisher Information Rate (FIR), showing that global isometry aligns FI while FIR depends on the encoder’s local geometry.
The analysis quantifies latent geometric distortion using three measurable penalties—dimensional compression, tangential distortion, and curvature injection—to explain latent diffusion degradation.
It derives theoretical conditions under which FIR (and thus diffusability) is preserved across spaces, providing criteria for designing/diagnosing latent autoencoders for diffusion.
Experiments across multiple autoencoding architectures validate the framework and present FI/FIR-based metrics as a diagnostic suite to identify and mitigate latent diffusion failure modes.

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.