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

• URL: http://arxiv.org/abs/2605.15822v1
• Date: Fri, 15 May 2026 10:20:05 GMT
• Title: Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
• Abstract summary: We show that Score-based generative models are trained in high-dimensional ambient spaces.<n>We use a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss Newton.
• Score: 61.14405512940818
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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