The Spacetime of Diffusion Models: An Information Geometry Perspective

Conference: ICLR 2026 arXiv: 2505.17517 Code: GitHub Area: Diffusion Models / Information Geometry / Theoretical Analysis Keywords: Spacetime Geometry, Fisher-Rao Metric, Pullback Geometry, Diffusion Edit Distance, Transition Path Sampling

TL;DR

This paper proposes a "spacetime" framework for diffusion models from an information-geometric perspective. It proves that the standard pullback geometry degenerates to straight lines in diffusion models, and introduces instead a spacetime geometry based on the Fisher-Rao metric, from which practically computable diffusion edit distances (DiffED) and transition path sampling methods are derived.

Background & Motivation

Understanding the information evolution of intermediate noisy states \(\mathbf{x}_t\) in diffusion models remains an open problem:

Failure of pullback geometry: In generative models, the intrinsic geometry of data is typically studied via pullback of the ambient metric. However, this approach suffers from a fundamental issue in diffusion models.

Lack of understanding of intermediate-state geometry: Existing work focuses primarily on sampling and training, with little analysis of how information evolves through the noising process.

Need for principled notions of distance and path: Existing image similarity metrics (e.g., LPIPS) lack a geometric foundation grounded in the generative process.

Method

1. Degeneration of Pullback Geometry (Core Negative Result)

Theorem: The pullback metric of the deterministic PF-ODE decoder \(\mathbf{x}_T \mapsto \mathbf{x}_0(\mathbf{x}_T)\),

\[\mathbf{G}_{\text{PB}}(\mathbf{x}_T) = \left(\frac{\partial \mathbf{x}_0}{\partial \mathbf{x}_T}\right)^\top \left(\frac{\partial \mathbf{x}_0}{\partial \mathbf{x}_T}\right)\]

causes all geodesics to decode as straight line segments in data space.

Reason: In diffusion models, the latent and data spaces share the same dimensionality; the decoder operates in the ambient space and is thus unable to capture the intrinsic structure of the data manifold.

2. The Memorylessness Problem in Information Geometry

The Fisher-Rao metric of the stochastic decoder (reverse SDE) is:

\[\mathbf{G}_{\text{IG}}(\mathbf{x}_T) = \mathbb{E}_{\mathbf{x}_0 \sim p(\mathbf{x}_0|\mathbf{x}_T)}[\nabla_{\mathbf{x}_T}\log p(\mathbf{x}_0|\mathbf{x}_T) \nabla_{\mathbf{x}_T}\log p(\mathbf{x}_0|\mathbf{x}_T)^\top]\]

However, due to memorylessness: \(p(\mathbf{x}_T|\mathbf{x}_0) \approx p_T(\mathbf{x}_T)\), the Fisher-Rao metric collapses to zero at \(\mathbf{x}_T\).

3. Latent Spacetime

Core Innovation: A \((D+1)\)-dimensional spacetime \(\mathbf{z} = (\mathbf{x}_t, t) \in \mathbb{R}^D \times (0, T]\) is introduced to:

• Index the family of denoising distributions \(\{p(\mathbf{x}_0|\mathbf{x}_t)\}\) across all noise levels
• Recover a non-degenerate geometric structure
• Identify clean data as spacetime points \((\mathbf{x}, 0)\)

4. Exponential Family Structure and Computable Energy

Proposition: The denoising distributions form an exponential family, and the spacetime curve energy admits a closed-form approximation:

\[\mathcal{E}(\boldsymbol{\gamma}) \approx \frac{N-1}{2}\sum_{n=0}^{N-2}(\boldsymbol{\eta}(\mathbf{z}_{n+1}) - \boldsymbol{\eta}(\mathbf{z}_n))^\top(\boldsymbol{\mu}(\mathbf{z}_{n+1}) - \boldsymbol{\mu}(\mathbf{z}_n))\]

where the natural and expectation parameters are:

\[\boldsymbol{\eta}(\mathbf{x}_t, t) = \left(\frac{\alpha_t}{\sigma_t^2}\mathbf{x}_t, -\frac{\alpha_t^2}{2\sigma_t^2}\right)\]

\[\boldsymbol{\mu}(\mathbf{x}_t, t) = \left(\mathbb{E}[\mathbf{x}_0|\mathbf{x}_t], \mathbb{E}[\|\mathbf{x}_0\|^2|\mathbf{x}_t]\right)\]

Computation: Via the Tweedie formula and Hutchinson's trick, estimation requires only a single Jacobian-vector product (JVP).

5. Diffusion Edit Distance (DiffED)

\[\text{DiffED}(\mathbf{x}^a, \mathbf{x}^b) = \ell(\boldsymbol{\gamma})\]

where \(\boldsymbol{\gamma}\) is the spacetime geodesic connecting \((\mathbf{x}^a, 0)\) and \((\mathbf{x}^b, 0)\).

Intuition: The geodesic traces the minimal edit sequence — adding sufficient noise to forget the information unique to \(\mathbf{x}^a\), then denoising to introduce the information unique to \(\mathbf{x}^b\). The distance measures the total change in the denoising distribution along the path.

6. Transition Path Sampling

For a Boltzmann distribution \(q(\mathbf{x}) \propto \exp(-U(\mathbf{x}))\): - Estimate the spacetime geodesic between two low-energy states - Sample along the geodesic using annealed Langevin dynamics - Supports constrained variants (low-variance paths, region avoidance)

Key Experimental Results

Sampling Trajectory Comparison

• PF-ODE paths closely resemble energy-minimizing geodesics
• Geodesics curve slightly less during the early sampling phase

Diffusion Edit Distance

Property	Result
Correlation with LPIPS	~−7% (captures different information)
Correlation with SSIM	~53%
Less similar endpoints	Stronger intermediate noise

DiffED captures structural edit cost rather than perceptual similarity.

Transition Path Sampling (Alanine Dipeptide)

Method	MaxEnergy↓	Energy Evaluations↓
MCMC-Fixed Length	42.54±7.42	1.29B
MCMC-Variable Length	58.11±18.51	21.02M
Doob's Lagrangian	66.24±1.01	38.4M
Spacetime Geodesic (Ours)	37.36±0.60	16M (+16M)
Lower Bound	36.42	—

The proposed method most closely approaches the lower bound while requiring orders of magnitude fewer energy evaluations.

Constrained Paths

• Generated paths effectively avoid high-energy regions
• Unlike Doob's Lagrangian, paths do not collapse to a single trajectory

Highlights & Insights

1. Deep theoretical insight: Formally proves the fundamental failure of pullback geometry in diffusion models
2. Elegance of the spacetime concept: Unifies the geometric structure across all noise levels
3. Computability: Derives simulation-free estimators by exploiting the exponential family structure
4. Multi-domain applicability: Edit distance + molecular dynamics
5. Computational efficiency: Energy estimation requires only a single JVP

Limitations & Future Work

1. Spacetime geodesics cannot serve as an alternative sampling method, as both endpoints must be known in advance
2. The Hutchinson estimator may introduce variance in high-dimensional settings
3. The computational cost of DiffED remains higher than that of simple distance metrics
4. Results depend on the quality of the denoiser (approximation error in \(\hat{\mathbf{x}}_0\))
5. Transition path sampling requires a known energy function

Related Work & Insights

• Riemannian geometry + generative models: Arvanitidis (2018/2022), Park (2023)
• Geometry of diffusion models: Domingo-Enrich (2025), memorylessness analysis
• Transition path sampling: Holdijk (2023), Doob's Lagrangian (Du 2024)
• Information geometry: Fisher-Rao metric, Amari (2016)

Rating

• Novelty: ⭐⭐⭐⭐⭐ — The spacetime geometry concept is highly original and intellectually deep
• Utility: ⭐⭐⭐⭐ — DiffED and transition path sampling offer practical value
• Experimental Thoroughness: ⭐⭐⭐⭐ — Theoretical validation is thorough; molecular dynamics results are strong
• Writing Quality: ⭐⭐⭐⭐⭐ — Theoretically elegant with precise exposition

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