# Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance

arXiv: https://arxiv.org/abs/2508.03210 · HTML: https://mlsanity.com/paper/2508.03210/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 39 of its 53 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Placed in

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

## Builds on (most cited first, 30 of 39)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [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)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024)
- [Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers](https://mlsanity.com/paper/2303.03384/) (2023)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [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 of Probability Flow ODE for Score-based Generative Models](https://mlsanity.com/paper/2404.09730/) (2024)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022)
- [Contractive Diffusion Probabilistic Models](https://mlsanity.com/paper/2401.13115/) (2024)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)

## Built on by (most cited first, 21 of 21)

- [Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities](https://mlsanity.com/paper/2604.06065/) (2026)
- [Non-asymptotic error bounds for probability flow ODEs under weak log-concavity](https://mlsanity.com/paper/2510.17608/) (2025)
- [Parallel Sampling via Autospeculation](https://mlsanity.com/paper/2511.07869/) (2025)
- [Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching](https://mlsanity.com/paper/2601.06514/) (2026)
- [On Forgetting and Stability of Score-based Generative models](https://mlsanity.com/paper/2601.21868/) (2026)
- [Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2603.03700/) (2026)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)
- [Understanding diffusion models requires rethinking (again) generalization](https://mlsanity.com/paper/2605.06077/) (2026)
- [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers](https://mlsanity.com/paper/2603.19670/) (2026)
- [Why DDIM Hallucinates More Than DDPM: A Theoretical Analysis of Reverse Dynamics](https://mlsanity.com/paper/2605.06831/) (2026)
- [Factor-Based Conditional Diffusion Model for Contextual Portfolio Optimization](https://mlsanity.com/paper/2509.22088/) (2025)
- [Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEs](https://mlsanity.com/paper/2601.08527/) (2026)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)
- [Tightening the Score Matching Gap for Diffusion Models](https://mlsanity.com/paper/2607.04442/) (2026)
- [Robustness of Diffusion Models under Distribution Shift](https://mlsanity.com/paper/2609.27546/) (2026)
- [Universal Local Error and Realized Amplification for the First-Order EDM Predictor](https://mlsanity.com/paper/2610.10190/) (2026)
- [Diffusion Removes Langevin's Conditioning Dependence: A Sharp Gaussian Analysis](https://mlsanity.com/paper/2610.12052/) (2026)
- [A Stochastic Analysis Perspective on Diffusion Models](https://mlsanity.com/paper/2610.14133/) (2026)

## Abstract (arXiv)

We provide new convergence guarantees in Wasserstein distance for diffusion-based generative models, covering both stochastic (DDPM-like) and deterministic (DDIM-like) sampling methods. We introduce a simple framework to analyze discretization, initialization, and score estimation errors. Notably, we derive the first Wasserstein convergence bound for the Heun sampler and improve existing results for the Euler sampler of the probability flow ODE. Our analysis emphasizes the importance of spatial regularity of the learned score function and argues for controlling the score error with respect to the true reverse process, in line with denoising score matching. We also incorporate recent results on smoothed Wasserstein distances to sharpen initialization error bounds.