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

TL;DR

This paper establishes the intrinsic Wasserstein-1 convergence rates for score-based generative models on smooth manifolds, explicitly accounting for geometric and density factors in high-dimensional spaces.

Contribution

It provides a nonasymptotic bound for score-based models on manifolds, introducing a ReLU-based projection method and separating noise regimes for improved analysis.

Findings

Achieves intrinsic Wasserstein-1 sample exponent with explicit geometric and density factors.

Introduces a ReLU implementation of manifold projection via finite anchors and Gauss--Newton iterations.

Provides polynomial ambient dependence for score-network parameters under controlled geometry and density.

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 $\beta$-H\"older densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_\beta(d)}n^{-(\beta+1)/(d+2\beta)})$, 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.