# Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

arXiv: https://arxiv.org/abs/2409.18804 · HTML: https://mlsanity.com/paper/2409.18804/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 1 of its 48 references that are in the map.
- First posted on arXiv 27 Sep 2024.

## Placed in

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

## Builds on (most cited first, 1 of 1)

- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)

## Built on by (most cited first, 30 of 65)

- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [Provable Acceleration for Diffusion Models under Minimal Assumptions](https://mlsanity.com/paper/2410.23285/) (2024)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Confidence-Based Decoding is Provably Efficient for Diffusion Language Models](https://mlsanity.com/paper/2603.22248/) (2026)
- [Faster Diffusion Models via Higher-Order Approximation](https://mlsanity.com/paper/2506.24042/) (2025)
- [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [A solvable model of learning generative diffusion: theory and insights](https://mlsanity.com/paper/2501.03937/) (2025)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Shallow diffusion networks provably learn hidden low-dimensional structure](https://mlsanity.com/paper/2410.11275/) (2024)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds](https://mlsanity.com/paper/2506.09681/) (2025)
- [Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models](https://mlsanity.com/paper/2506.13061/) (2025)
- [Nonparametric estimation of a factorizable density using diffusion models](https://mlsanity.com/paper/2501.01783/) (2025)
- [Adaptivity and Convergence of Probability Flow ODEs in Diffusion Generative Models](https://mlsanity.com/paper/2501.18863/) (2025)
- [Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces](https://mlsanity.com/paper/2505.02508/) (2025)
- [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025)
- [Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis](https://mlsanity.com/paper/2507.03756/) (2025)
- [Diffusion and Flow-based Copulas: Forgetting and Remembering Dependencies](https://mlsanity.com/paper/2509.19707/) (2025)
- [Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity](https://mlsanity.com/paper/2603.20645/) (2026)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Localized Diffusion Models](https://mlsanity.com/paper/2505.04417/) (2025)
- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees](https://mlsanity.com/paper/2601.06715/) (2026)
- [Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity](https://mlsanity.com/paper/2503.12966/) (2025)

## Abstract (arXiv)

Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The textitmanifold hypothesis states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning. In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.