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 $\mathcal{M}\subset [0,1{]}^D$ with $d>2$ and $\beta$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein–1 sample exponent $\tilde{\mathcal{O}}(D^{{\mathcal{O}}_{\beta }(d)}{n}^{-(\beta +1)\mathrm{/}(d+2\beta )})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder 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.