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

Many high-dimensional data distributions are supported on low-dimensional manifolds, whereas diffusion models learn score functions in the ambient space. This raises a basic question: can empirical risk minimization over ambient score-network classes attain a rate governed by intrinsic dimension without first estimating a manifold representation from the observations? We study this question for distributions with strictly positive $\beta$-H\"older densities on compact $d$-dimensional smooth manifolds embedded in ${\mathbb{R}}^D$. The main technical challenge is the low-noise regime, where the score becomes singular as the noise level vanishes. To address this challenge, we construct tangent-cell ReLU approximations at moderate and high noise and a projection-centered ReLU approximation at low noise, with explicit ambient-dimension dependence. For $d>2$, under the stated geometry, regularity, and optimization conditions, the cube-projected output of the continuous-time sampler has expected $1$-Wasserstein error $D^{{\mathcal{O}}_{\beta }(d)}{n}^{-(\beta +1)/(d+2\beta )}$ up to logarithmic factors. The result therefore establishes the intrinsic sample-size exponent convergence rate for ambient score networks, without a preliminary data-dependent dimension-reduction step.