# The Spacetime of Diffusion Models: An Information Geometry Perspective

arXiv: https://arxiv.org/abs/2505.17517 · HTML: https://mlsanity.com/paper/2505.17517/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 28 of its 41 references that are in the map.
- First posted on arXiv 23 May 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 28 of 28)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Paths](https://mlsanity.com/paper/2207.02149/) (2022)
- [Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling](https://mlsanity.com/paper/2410.07974/) (2024)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Riemannian Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2207.03024/) (2022)
- [Probability Density Geodesics in Image Diffusion Latent Space](https://mlsanity.com/paper/2504.06675/) (2025)
- [Image generation with shortest path diffusion](https://mlsanity.com/paper/2306.00501/) (2023)
- [Geometry of Score Based Generative Models](https://mlsanity.com/paper/2302.04411/) (2023)
- [Devil is in the Details: Density Guidance for Detail-Aware Generation with Flow Models](https://mlsanity.com/paper/2502.05807/) (2025)
- [Information Geometry for the Working Information Theorist](https://mlsanity.com/paper/2310.03884/) (2023)

## Built on by (most cited first, 6 of 6)

- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Exploring Time Conditioning in Diffusion Generative Models from Disjoint Noisy Data Manifolds](https://mlsanity.com/paper/2604.25289/) (2026)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)
- [Understanding Latent Diffusability via Fisher Geometry](https://mlsanity.com/paper/2604.02751/) (2026)
- [Stochastic Control Policies for Robust Molecular Transition Path Sampling](https://mlsanity.com/paper/2608.13800/) (2026)
- [The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr"odinger Bridges](https://mlsanity.com/paper/2608.23916/) (2026)

## Abstract (arXiv)

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of xT as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(xt,t) that indexes the family of denoising distributions p(x0 | xt) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.