Spacetime Geometry of Denoising in Diffusion Models

• URL: http://arxiv.org/abs/2505.17517v1
• Date: Fri, 23 May 2025 06:16:58 GMT
• Title: Spacetime Geometry of Denoising in Diffusion Models
• Authors: Rafał Karczewski, Markus Heinonen, Alison Pouplin, Søren Hauberg, Vikas Garg,
• Abstract summary: We present a novel perspective on diffusion models using the framework of information geometry.<n>We show that the set of noisy samples, taken across all noise levels simultaneously, forms a statistical manifold.<n>We demonstrate the practical value of this geometric viewpoint in transition path sampling.
• Score: 20.644091294762678
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a novel perspective on diffusion models using the framework of information geometry. We show that the set of noisy samples, taken across all noise levels simultaneously, forms a statistical manifold -- a family of denoising probability distributions. Interpreting the noise level as a temporal parameter, we refer to this manifold as spacetime. This manifold naturally carries a Fisher-Rao metric, which defines geodesics -- shortest paths between noisy points. Notably, this family of distributions is exponential, enabling efficient geodesic computation even in high-dimensional settings without retraining or fine-tuning. We demonstrate the practical value of this geometric viewpoint in transition path sampling, where spacetime geodesics define smooth sequences of Boltzmann distributions, enabling the generation of continuous trajectories between low-energy metastable states. Code is available at: https://github.com/Aalto-QuML/diffusion-spacetime-geometry.

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