A diffusion score-Jacobian splits into tangent and normal subspaces
measured in 1 paperSaito & Matsubara define a Riemannian metric on diffusion noise space via the score function Jacobian and prove (Proposition 1) that minimizing ||J v|| for fixed norm pushes v into the small-singular-value tangent subspace, with large singular values forming the normal space [saito-matsubara-2025-tangential-manifold-diffusion] They argue density-based geodesic interpolation biases paths toward high-density regions, producing over-smoothed detail-losing images [saito-matsubara-2025-tangential-manifold-diffusion] Their tangent-constrained geodesics achieve the best FID across MorphBench/CelebA-HQ/AFHQ (e.g. AFHQ 21.01 vs 25.80 for GeodesicDiffusion) [saito-matsubara-2025-tangential-manifold-diffusion] On video frame interpolation they achieve the lowest MSE/LPIPS against ground truth everywhere [saito-matsubara-2025-tangential-manifold-diffusion]