Spacetime Geometry of Denoising in Diffusion Models

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.