Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact d-dimensional smooth manifolds M subset [0,1]D with d > 2 and β-H"older densities strictly positive on M, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent tildeO(DOβ(d)n-(β+1)/(d+2β)), up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, H"older radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.