A Riemannian metric derived directly from Stable Diffusion's own score-function Jacobian -- not pulled back from any assumed linear latent -- and geodesic-guided interpolation outperforms Lerp/Slerp baselines

Saito & Matsubara (2025) construct a genuine Riemannian metric tensor $G_{x_t}=J_{x_t}^\top J_{x_t}$ directly from the Jacobian of a real, pretrained diffusion model's own score function, explicitly contrasting this with the existing Park et al. (2023) approach (already in this map, structure `linear-direction`) of pulling back an assumed linear latent-space metric -- here the metric lives natively in data/noise space with no linearity assumption. Using this metric to compute length-minimizing geodesics between real image pairs (via DDIM inversion) in real Stable Diffusion (v1-5), 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. 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. Both criteria clear: (a) a genuinely derived, quantified Riemannian metric on real model activations; (b) using that metric to guide generation (geodesic interpolation) with a measured, baseline-beating image-quality effect.