# Wasserstein Bounds for generative diffusion models with Gaussian tail targets

arXiv: https://arxiv.org/abs/2412.11251 · HTML: https://mlsanity.com/paper/2412.11251/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 25 of its 51 references that are in the map.
- First posted on arXiv 15 Dec 2024.

## Placed in

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

## Builds on (most cited first, 25 of 25)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Accelerating Diffusion Models via Early Stop of the Diffusion Process](https://mlsanity.com/paper/2205.12524/) (2022)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models](https://mlsanity.com/paper/2311.11003/) (2023)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Continuous-Time Functional Diffusion Processes](https://mlsanity.com/paper/2303.00800/) (2023)
- [An analysis of the noise schedule for score-based generative models](https://mlsanity.com/paper/2402.04650/) (2024)
- [On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates](https://mlsanity.com/paper/2311.13584/) (2023)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors](https://mlsanity.com/paper/2405.14250/) (2024)
- [Chain of Log-Concave Markov Chains](https://mlsanity.com/paper/2305.19473/) (2023)
- [Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions](https://mlsanity.com/paper/2402.08082/) (2024)
- [Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates](https://mlsanity.com/paper/2405.16104/) (2024)

## Built on by (most cited first, 10 of 10)

- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities](https://mlsanity.com/paper/2604.06065/) (2026)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026)
- [Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces](https://mlsanity.com/paper/2603.14798/) (2026)
- [Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers](https://mlsanity.com/paper/2603.19670/) (2026)
- [Cold-Start Forecasting of New Product Life-Cycles via Conditional Diffusion Models](https://mlsanity.com/paper/2604.20370/) (2026)
- [Proof-of-Authorship for Diffusion-based AI Generated Content](https://mlsanity.com/paper/2603.17513/) (2026)
- [On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures](https://mlsanity.com/paper/2605.21388/) (2026)

## Abstract (arXiv)

We present an estimate of the Wasserstein distance between the data distribution and the generation of score-based generative models. The sampling complexity with respect to dimension is O(√d), with a logarithmic constant. In the analysis, we assume a Gaussian-type tail behavior of the data distribution and an ε-accurate approximation of the score. Such a Gaussian tail assumption is general, as it accommodates a practical target - the distribution from early stopping techniques with bounded support. The crux of the analysis lies in the global Lipschitz bound of the score, which is shown from the Gaussian tail assumption by a dimension-independent estimate of the heat kernel. Consequently, our complexity bound scales linearly (up to a logarithmic constant) with the square root of the trace of the covariance operator, which relates to the invariant distribution of the forward process.