Definition

For a diffusion model’s score function $s_\theta(x_t,t)\approx\nabla_{x_t}\log p(x_t)$, the Jacobian $J_{x_t}=\partial s_\theta/\partial x_t$ defines a Riemannian metric tensor at each point directly in data/noise space:

$G_{x_t} = J_{x_t}^\top J_{x_t}$

This metric lives natively where the model itself operates — no assumption that some separate linear latent space carries the “true” geometry, and no metric pulled back from an ambient activation space through an assumed-linear encoder. Length-minimizing geodesics under $G_{x_t}$ between two real data points (computed via DDIM inversion to locate the corresponding noise-space endpoints) define a genuinely curved interpolation path, distinct from a straight line (Lerp) or a constant-norm spherical arc (Slerp) between the same two endpoints.

Relative to diffusion-spacetime-information-geometry

Diffusion Spacetime Information Geometry is a different metric construction on a diffusion model’s own manifold: it treats the model’s family of denoising distributions $\{p(x_0\mid x_t,t)\}$ as an exponential-family statistical manifold and derives a Fisher-Rao metric from that distributional structure, indexed by the joint “spacetime” coordinate $(x_t,t)$. This node’s metric is instead built directly from the score function’s own Jacobian at a fixed diffusion step, with no distributional/exponential-family argument involved — same broad theme (a metric intrinsic to the trained denoiser, not pulled back from an assumed ambient/linear structure), different mathematical object and different derivation. The two are linked via related: rather than merged, since neither reduces to the other.

Relative to pullback-of-assumed-linear-latent approaches

Park et al. (2023, already in this map under Linear Direction) pull back an assumed linear latent-space metric to define local geometry for diffusion-model latent-space edits. This node’s metric instead requires no such assumption: it is derived directly from the real model’s own score-Jacobian in data/noise space, with the paper below explicitly contrasting the two approaches.

Key evidence

Saito & Matsubara (2025) construct $G_{x_t}=J_{x_t}^\top J_{x_t}$ from the Jacobian of a real, pretrained Stable Diffusion v1-5 model’s own score function, and use it to compute length-minimizing geodesics between real image pairs (via DDIM inversion). Geodesic-guided interpolation improves CLIP-IQA “Reality” (0.716 vs. 0.389-0.704 for Lerp/Slerp/NAO/NoiseDiffusion baselines) and “Noisiness” (0.818) while matching the best reconstruction error (MSE, LPIPS) among all baselines tested. A secondary experiment repeats the same score-Jacobian metric construction and geodesic computation on a small custom diffusion model trained from scratch on MNIST, confirming the method is not specific to one model’s architecture or training scale. See a-riemannian-metric-derived-directly-from-stable-diffusions-own-score-function-jacobian-not-pulled-back-from-any-assumed-linear-latent-and-geodesic-guided-interpolation-outperforms-lerp-slerp-baselines.

Key papers

• Saito, S. & Matsubara, T. (2025). Image Interpolation with Score-based Riemannian Metrics of Diffusion Models. arXiv:2504.20288 — origin of the score-Jacobian pullback metric and its use for geodesic-guided image interpolation.