Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
For practitioners and theorists of generative modeling, this provides the first rigorous nonasymptotic guarantee that SGMs adapt to low-dimensional manifold structure, resolving a key theoretical gap.
This work proves that score-based generative models achieve the intrinsic Wasserstein-1 sample rate on compact smooth manifolds, with a bound of Õ(D^{O_β(d)} n^{-(β+1)/(d+2β)}), up to logarithmic factors. The result shows that the ambient dimension dependence can be polynomial for families with controlled geometry and density lower bounds.
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 $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, 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.