# Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds arXiv: https://arxiv.org/abs/2506.09681 ยท HTML: https://mlsanity.com/paper/2506.09681/ ## Key facts - Cited by 9 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 45 of its 66 references that are in the map. - First posted on arXiv 11 Jun 2025. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 45) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017) - [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015) - [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) - [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019) - [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022) - [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014) - [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015) - [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022) - [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023) - [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016) - [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018) - [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018) - [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017) - [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017) - [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019) - [Accelerating Convergence of Score-Based Diffusion Models, Provably](https://mlsanity.com/paper/2403.03852/) (2024) - [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024) - [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) - [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024) - [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024) - [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) - [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024) - [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022) - [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024) ## Built on by (most cited first, 9 of 9) - [Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities](https://mlsanity.com/paper/2604.06065/) (2026) - [Query Lower Bounds for Diffusion Sampling](https://mlsanity.com/paper/2604.10857/) (2026) - [Wasserstein bounds for denoising diffusion probabilistic models via the F"ollmer process](https://mlsanity.com/paper/2605.18069/) (2026) - [Factor-Based Conditional Diffusion Model for Contextual Portfolio Optimization](https://mlsanity.com/paper/2509.22088/) (2025) - [Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization](https://mlsanity.com/paper/2606.13796/) (2026) - [Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees](https://mlsanity.com/paper/2606.16610/) (2026) - [Diffusion Models for High-Dimensional Clustered Data: Intrinsic-Dimension Adaptivity via Bayesian Classification](https://mlsanity.com/paper/2608.19067/) (2026) - [Universal Local Error and Realized Amplification for the First-Order EDM Predictor](https://mlsanity.com/paper/2610.10190/) (2026) - [Radial Tail Repair: Post-Training Correction for Diffusion and Flow Models](https://mlsanity.com/paper/2610.16851/) (2026) ## Abstract (arXiv) Generative modeling aims to produce new random examples from an unknown target distribution, given access to a finite collection of examples. Among the leading approaches, denoising diffusion probabilistic models (DDPMs) construct such examples by mapping a Brownian motion via a diffusion process driven by an estimated score function. In this work, we first provide empirical evidence that DDPMs are robust to constant-variance noise in the score evaluations. We then establish finite-sample guarantees in Wasserstein-2 distance that exhibit two key features: (i) they characterize and quantify the robustness of DDPMs to noisy score estimates, and (ii) they achieve faster convergence rates than previously known results. Furthermore, we observe that the obtained rates match those known in the Gaussian case, implying their optimality.