# On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates

arXiv: https://arxiv.org/abs/2311.13584 · HTML: https://mlsanity.com/paper/2311.13584/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 30 of its 58 references that are in the map.
- First posted on arXiv 22 Nov 2023.

## Placed in

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

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [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)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [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)
- [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)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [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)
- [Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance](https://mlsanity.com/paper/2212.06359/) (2022)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023)
- [On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case](https://mlsanity.com/paper/1812.02709/) (2018)
- [Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks](https://mlsanity.com/paper/2105.13937/) (2021)

## Built on by (most cited first, 25 of 25)

- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (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)
- [Convergence of Continuous Normalizing Flows for Learning Probability Distributions](https://mlsanity.com/paper/2404.00551/) (2024)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Discrete Diffusion Models: Novel Analysis and New Sampler Guarantees](https://mlsanity.com/paper/2509.16756/) (2025)
- [Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach](https://mlsanity.com/paper/2402.13901/) (2024)
- [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)
- [Generating High Dimensional User-Specific Wireless Channels using Diffusion Models](https://mlsanity.com/paper/2409.03924/) (2024)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds](https://mlsanity.com/paper/2506.09681/) (2025)
- [Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers](https://mlsanity.com/paper/2410.13746/) (2024)
- [Non-asymptotic error bounds for probability flow ODEs under weak log-concavity](https://mlsanity.com/paper/2510.17608/) (2025)
- [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Wasserstein Convergence of Critically Damped Langevin Diffusions](https://mlsanity.com/paper/2511.02419/) (2025)
- [Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity](https://mlsanity.com/paper/2411.01629/) (2024)
- [Multi-Subspace Multi-Modal Modeling for Diffusion Models: Estimation, Convergence and Mixture of Experts](https://mlsanity.com/paper/2601.01475/) (2026)
- [Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Models](https://mlsanity.com/paper/2601.21943/) (2026)
- [Flatness-Aware Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2510.02174/) (2025)
- [Adaptive Auxiliary Prompt Blending for Target-Faithful Diffusion Generation](https://mlsanity.com/paper/2603.19158/) (2026)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)
- [Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees](https://mlsanity.com/paper/2606.16610/) (2026)
- [Tightening the Score Matching Gap for Diffusion Models](https://mlsanity.com/paper/2607.04442/) (2026)
- [A Stochastic Analysis Perspective on Diffusion Models](https://mlsanity.com/paper/2610.14133/) (2026)

## Abstract (arXiv)

We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation is made of Lipschitz continuous functions avoiding any Lipschitzness assumption on the score function. We demonstrate via a motivating example, sampling from a Gaussian distribution with unknown mean, the powerfulness of our approach. In this case, explicit estimates are provided for the associated optimization problem, i.e. score approximation, while these are combined with the corresponding sampling estimates. As a result, we obtain the best known upper bound estimates in terms of key quantities of interest, such as the dimension and rates of convergence, for the Wasserstein-2 distance between the data distribution (Gaussian with unknown mean) and our sampling algorithm. Beyond the motivating example and in order to allow for the use of a diverse range of stochastic optimizers, we present our results using an L2-accurate score estimation assumption, which crucially is formed under an expectation with respect to the stochastic optimizer and our novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.