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

arXiv: https://arxiv.org/abs/2605.15822 · HTML: https://mlsanity.com/paper/2605.15822/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 7 of its 59 references that are in the map.
- First posted on arXiv 15 May 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 7 of 7)

- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [Mathematical analysis of singularities in the diffusion model under the submanifold assumption](https://mlsanity.com/paper/2301.07882/) (2023)
- [Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity](https://mlsanity.com/paper/2505.09922/) (2025)
- [Provable Maximum Entropy Manifold Exploration via Diffusion Models](https://mlsanity.com/paper/2506.15385/) (2025)
- [Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces](https://mlsanity.com/paper/2505.02508/) (2025)
- [Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity](https://mlsanity.com/paper/2603.20645/) (2026)
- [Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2603.03700/) (2026)

## Built on by (most cited first, 2 of 2)

- [Optimal score function estimation via derivatives constraints](https://mlsanity.com/paper/2606.19084/) (2026)
- [Diffusion Transformers are Provably Optimal In-context Generators](https://mlsanity.com/paper/2610.05333/) (2026)

## Abstract (arXiv)

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 M subset [0,1]D with d > 2 and β-H"older densities strictly positive on M, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent tildeO(DOβ(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, 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.