Hessian Geometry of Latent Space in Generative Models

TL;DR

This paper introduces a new approach to analyze the geometry of latent spaces in generative models by reconstructing the Fisher information metric, revealing complex phase transition structures and providing theoretical guarantees.

Contribution

It presents a novel method for estimating the Fisher metric in latent spaces, applicable to various models, with proven convergence and insights into phase transitions.

Findings

Outperforms existing methods in reconstructing thermodynamic quantities

Reveals fractal structure of phase transitions in diffusion models

Shows linear geodesics within phases but breakdown at phase boundaries

Abstract

This paper presents a novel method for analyzing the latent space geometry of generative models, including statistical physics models and diffusion models, by reconstructing the Fisher information metric. The method approximates the posterior distribution of latent variables given generated samples and uses this to learn the log-partition function, which defines the Fisher metric for exponential families. Theoretical convergence guarantees are provided, and the method is validated on the Ising and TASEP models, outperforming existing baselines in reconstructing thermodynamic quantities. Applied to diffusion models, the method reveals a fractal structure of phase transitions in the latent space, characterized by abrupt changes in the Fisher metric. We demonstrate that while geodesic interpolations are approximately linear within individual phases, this linearity breaks down at phase boundaries, where the diffusion model exhibits a divergent Lipschitz constant with respect to the latent space. These findings provide new insights into the complex structure of diffusion model latent spaces and their connection to phenomena like phase transitions. Our source code is available at https://github.com/alobashev/hessian-geometry-of-diffusion-models.