Geometry-Aware Diffusion through Anisotropic Perturbations

Score-based generative models perform remarkably well on data supported on lower-dimensional manifolds, despite relying on ambient forward processes that inject isotropic Gaussian noise in all directions. However, as the noise level vanishes, the score develops a singular normal component near the manifold, destabilizing score matching and hence complicating distribution estimation. We study a geometry-aware procedure that first perturbs data with Gaussian noise in the normal directions and then applies a standard ambient Ornstein-Uhlenbeck forward process, which regularizes the normal score without obscuring intrinsic density information. We establish non-asymptotic score estimation and distributional guarantees under $\gamma$-Holder integral probability metrics for $0<\gamma \le 1$. Under suitable geometric and smoothness regularity conditions, the resulting rates depend on the intrinsic dimension and match the corresponding minimax rate up to logarithmic factors, yielding better rates in the small $\gamma$ regime compared to the usual isotropic diffusion model. Empirical evaluations on synthetic manifolds and molecular dynamics validate our theoretical framework, demonstrating the practical advantages of geometry-aware perturbations.