# KL Convergence Guarantees for Score diffusion models under minimal data assumptions

arXiv: https://arxiv.org/abs/2308.12240 · HTML: https://mlsanity.com/paper/2308.12240/

## Key facts

- Cited by 63 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 58 references that are in the map.
- First posted on arXiv 23 Aug 2023.

## 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)
- [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)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [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)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [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)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [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)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Metropolis-Hastings Generative Adversarial Networks](https://mlsanity.com/paper/1811.11357/) (2018)
- [MadMiner: Machine learning-based inference for particle physics](https://mlsanity.com/paper/1907.10621/) (2019)
- [Improved Convergence of Score-Based Diffusion Models via Prediction-Correction](https://mlsanity.com/paper/2305.14164/) (2023)
- [VIME: Variational Information Maximizing Exploration](https://mlsanity.com/paper/1605.09674/) (2016)

## Built on by (most cited first, 30 of 63)

- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Optimal Inference Schedules for Masked Diffusion Models](https://mlsanity.com/paper/2511.04647/) (2025)
- [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)
- [Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance](https://mlsanity.com/paper/2508.03210/) (2025)
- [Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach](https://mlsanity.com/paper/2402.13901/) (2024)
- [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026)
- [Convergence Analysis of Flow Matching in Latent Space with Transformers](https://mlsanity.com/paper/2404.02538/) (2024)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [On the Asymptotic Mean Square Error Optimality of Diffusion Models](https://mlsanity.com/paper/2403.02957/) (2024)
- [Latent Schr"odinger Bridge Diffusion Model for Generative Learning](https://mlsanity.com/paper/2404.13309/) (2024)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Blind denoising diffusion models and the blessings of dimensionality](https://mlsanity.com/paper/2602.09639/) (2026)
- [Conditional Stochastic Interpolation for Generative Learning](https://mlsanity.com/paper/2312.05579/) (2023)
- [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)
- [Wasserstein Convergence of Critically Damped Langevin Diffusions](https://mlsanity.com/paper/2511.02419/) (2025)
- [Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates](https://mlsanity.com/paper/2405.16104/) (2024)
- [DiffEM: Learning from Corrupted Data with Diffusion Models via Expectation Maximization](https://mlsanity.com/paper/2510.12691/) (2025)
- [Localized Diffusion Models](https://mlsanity.com/paper/2505.04417/) (2025)
- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold](https://mlsanity.com/paper/2602.06021/) (2026)
- [Optimal Denoising in Score-Based Generative Models: The Role of Data Regularity](https://mlsanity.com/paper/2503.12966/) (2025)
- [Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based models](https://mlsanity.com/paper/2512.06702/) (2025)
- [A Jump-Diffusion Framework for Irregular Time Series Generation](https://mlsanity.com/paper/2505.23215/) (2025)
- [Exponential Convergence Guarantees for Iterative Markovian Fitting](https://mlsanity.com/paper/2510.20871/) (2025)
- [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025)
- [Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training](https://mlsanity.com/paper/2605.13175/) (2026)

## Abstract (arXiv)

Diffusion models are a new class of generative models that revolve around the estimation of the score function associated with a stochastic differential equation. Subsequent to its acquisition, the approximated score function is then harnessed to simulate the corresponding time-reversal process, ultimately enabling the generation of approximate data samples. Despite their evident practical significance these models carry, a notable challenge persists in the form of a lack of comprehensive quantitative results, especially in scenarios involving non-regular scores and estimators. In almost all reported bounds in Kullback Leibler (KL) divergence, it is assumed that either the score function or its approximation is Lipschitz uniformly in time. However, this condition is very restrictive in practice or appears to be difficult to establish. To circumvent this issue, previous works mainly focused on establishing convergence bounds in KL for an early stopped version of the diffusion model and a smoothed version of the data distribution, or assuming that the data distribution is supported on a compact manifold. These explorations have led to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However, the question remains about the relevance of such early-stopping procedure or compactness conditions. In particular, if there exist a natural and mild condition ensuring explicit and sharp convergence bounds in KL. In this article, we tackle the aforementioned limitations by focusing on score diffusion models with fixed step size stemming from the Ornstein-Uhlenbeck semigroup and its kinetic counterpart. Our study provides a rigorous analysis, yielding simple, improved and sharp convergence bounds in KL applicable to any data distribution with finite Fisher information with respect to the standard Gaussian distribution.