# Score-based generative models are provably robust: an uncertainty quantification perspective

arXiv: https://arxiv.org/abs/2405.15754 · HTML: https://mlsanity.com/paper/2405.15754/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 24 May 2024.

## Placed in

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

## Builds on (most cited first, 23 of 23)

- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [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)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [Distributionally Robust Optimization and Generalization in Kernel Methods](https://mlsanity.com/paper/1905.10943/) (2019)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transport](https://mlsanity.com/paper/2206.11343/) (2022)

## Built on by (most cited first, 15 of 15)

- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration](https://mlsanity.com/paper/2502.04849/) (2025)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Bifidelity Parameter Estimation Using Conditional Diffusion Models](https://mlsanity.com/paper/2504.01894/) (2025)
- [Combining Wasserstein-1 and Wasserstein-2 proximals: robust manifold learning via well-posed generative flows](https://mlsanity.com/paper/2407.11901/) (2024)
- [When and how can inexact generative models still sample from the data manifold?](https://mlsanity.com/paper/2508.07581/) (2025)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [RoHyDR: Robust Hybrid Diffusion Recovery for Incomplete Multimodal Emotion Recognition](https://mlsanity.com/paper/2505.17501/) (2025)
- [Generative diffusion models from a PDE perspective](https://mlsanity.com/paper/2501.17054/) (2025)
- [GenUQ: Predictive Uncertainty Estimates via Generative Hyper-Networks](https://mlsanity.com/paper/2509.21605/) (2025)
- [Concentration Inequalities for Stochastic Optimization of Unbounded Objective Functions with Application to Denoising Score Matching](https://mlsanity.com/paper/2502.08628/) (2025)
- [Modeling Score Approximation Errors in Diffusion Models via Forward SPDEs](https://mlsanity.com/paper/2602.08579/) (2026)
- [OmniVLA-RL: A Vision-Language-Action Model with Spatial Understanding and Online RL](https://mlsanity.com/paper/2604.17706/) (2026)
- [A Generalized McDiarmid Inequality: Sample Reuse and Non-Uniform Unbounded Differences](https://mlsanity.com/paper/2610.17159/) (2026)

## Abstract (arXiv)

Through an uncertainty quantification (UQ) perspective, we show that score-based generative models (SGMs) are provably robust to the multiple sources of error in practical implementation. Our primary tool is the Wasserstein uncertainty propagation (WUP) theorem, a model-form UQ bound that describes how the L2 error from learning the score function propagates to a Wasserstein-1 (d1) ball around the true data distribution under the evolution of the Fokker-Planck equation. We show how errors due to (a) finite sample approximation, (b) early stopping, (c) score-matching objective choice, (d) score function parametrization expressiveness, and (e) reference distribution choice, impact the quality of the generative model in terms of a d1 bound of computable quantities. The WUP theorem relies on Bernstein estimates for Hamilton-Jacobi-Bellman partial differential equations (PDE) and the regularizing properties of diffusion processes. Specifically, PDE regularity theory shows that stochasticity is the key mechanism ensuring SGM algorithms are provably robust. The WUP theorem applies to integral probability metrics beyond d1, such as the total variation distance and the maximum mean discrepancy. Sample complexity and generalization bounds in d1 follow directly from the WUP theorem. Our approach requires minimal assumptions, is agnostic to the manifold hypothesis and avoids absolute continuity assumptions for the target distribution. Additionally, our results clarify the trade-offs among multiple error sources in SGMs.