Statement

For a diffusion model, the family of denoising distributions $\{p(x_0\mid x_t, t)\}$, indexed by the joint “spacetime” coordinate $z=(x_t,t)$, is provably an exponential family (natural parameter $\eta(z)=\big(\tfrac{\alpha_t}{\sigma_t^2}x_t,\,-\tfrac{\alpha_t^2}{2\sigma_t^2}\big)$, sufficient statistic $T(x_0)=(x_0,\|x_0\|^2)$). This licenses a Fisher-Rao metric $\mathbf{G}_{\mathrm{IG}}(z) = \mathbb{E}_{x_0\sim p(x_0|z)}\big[\nabla_z\log p(x_0|z)\,\nabla_z\log p(x_0|z)^\top\big]$ on the spacetime manifold, measuring how much an infinitesimal step changes the entire denoising distribution (to second order, exactly the KL divergence between the distributions at $z$ and $z+dz$). Geodesics under this metric define the Diffusion Edit Distance: the minimal “noise-then-denoise” path connecting two clean data points, computed via a closed-form, simulation-free curve-energy estimator exploiting the family’s exponential form. Karczewski, Heinonen, Pouplin, Hauberg & Garg (2026, arXiv:2505.17517, ICLR 2026 Oral) introduce this construction and prove it strictly generalizes the naive approach of computing this metric only at the final noise level $x_T$: the memoryless property of diffusion ($p(x_0|x_T)\approx q(x_0)$ regardless of $x_T$) forces the metric to collapse to $\mathbf{0}$ unless time $t$ is included as an explicit coordinate alongside $x_t$.

Intuition

Instead of asking “how does the decoded output change as I nudge the noisy input” (the question a Jacobian-pullback metric answers), this asks “how does the model’s entire belief about what the clean image looks like change as I nudge the noisy input and/or the noise level.” The second question is answered by comparing probability distributions (via Fisher information), not by comparing points in an ambient Euclidean space — a fundamentally statistical, rather than pointwise, notion of “how far apart” two states of the diffusion process are.

Properties

• A different mathematical object from a pullback metric, not merely a different formula for the same one. A Riemannian pullback metric (e.g. $\mathbf{G}_{\mathrm{PB}}(x_T) = (\partial x_0/\partial x_T)^\top(\partial x_0/\partial x_T)$) measures how a decoded point changes; the Fisher-Rao metric here measures how an entire distribution changes — the two coincide only in degenerate special cases, and the originating paper proves the pullback metric provably collapses to a trivial (Euclidean, data-manifold-blind) form whenever latent and data space share the same dimension, which is the standard diffusion setting.
• Time must be an explicit coordinate, not a fixed parameter. Because diffusion’s forward process is memoryless in the limit, the metric computed at a single fixed noise level (treating only $x_t$ as varying) degenerates to zero; the “spacetime” move of jointly varying $(x_t,t)$ is what keeps the metric non-degenerate.
• Closed-form, simulation-free curve energy. Unlike an SDE/ODE integration along a candidate path, the exponential-family structure lets the curve energy be estimated directly from paired natural- and expectation-parameter differences at discretized path points — each requiring only a single Jacobian-vector product of the trained denoiser, not a full forward simulation.
• Validated with mixed results relative to perceptual similarity. On a real pretrained EDM2 ImageNet-512 model, 200 real ImageNet image pairs (10 classes x 20 pairs) show the Diffusion Edit Distance correlates only weakly with LPIPS (~-7%) but moderately with SSIM (~53%) — evidence the metric captures a genuinely different notion of “closeness” than standard perceptual-similarity metrics, not simply a re-derivation of them.
• Also validated on a real physical system beyond images. A custom-trained diffusion model over Alanine Dipeptide molecular conformations, using spacetime geodesics as transition paths between molecular states, achieves a lower mean maximum energy barrier (37.36 ± 0.60, versus a numerical lower bound of 36.42) than MCMC-based and Doob’s-Lagrangian transition-path-sampling baselines (42.54-66.24), at far fewer post-training energy evaluations (16M one-time training cost vs. up to 1.29B for MCMC).

Key papers

• Karczewski, Heinonen, Pouplin, Hauberg & Garg (2026). The Spacetime of Diffusion Models: An Information Geometry Perspective. arXiv:2505.17517, ICLR 2026 Oral. Introduces the spacetime Fisher-Rao construction and the Diffusion Edit Distance, proves the standard Riemannian pullback metric collapses to a trivial Euclidean form in same-dimension latent/data settings (Proposition B.1), and validates the resulting geodesics on a real pretrained EDM2 ImageNet-512 model (Section 6.1-6.2: qualitative PF-ODE-trajectory similarity, plus the quantified LPIPS/SSIM correlations above) and a custom-trained molecular diffusion model over Alanine Dipeptide conformations (Section 6.3, Table 1: outperforms MCMC and Doob’s-Lagrangian transition-path-sampling baselines on mean maximum energy barrier). See karczewski-etal-2026-diffed-vs-lpips-ssim-and-molecular-transition-paths.