A pullback Riemannian metric yields diffusion tangent-space directions

Park et al. reframe the diffusion U-Net encoder's Jacobian as a pullback Riemannian metric on the latent x_t, whose eigenvectors form a local tangent basis [park-etal-2023-riemannian-diffusion-pullback] A power-spectral analysis of the tangent basis shifts from low- to high-frequency as t goes from T to 0, confirming coarse-to-fine generation [park-etal-2023-riemannian-diffusion-pullback] Grassmannian geodesic distance between samples' tangent spaces increases monotonically as t decreases, fastest for visually complex datasets [park-etal-2023-riemannian-diffusion-pullback] For Stable Diffusion v2.1, tangent-space geodesic distance negatively correlates with CLIP prompt similarity, collapsing below roughly t=0.7T [park-etal-2023-riemannian-diffusion-pullback] Single-timestep x-space editing is introduced as a causal application but validated only qualitatively [park-etal-2023-riemannian-diffusion-pullback]