mlsanityData to 7 Oct 2026

Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective

arXiv 2405.16418 ↗First posted 26 May 2024cs.LG cs.AI cs.CVcited by 24

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models have made rapid progress in generating high-quality samples across various domains. However, a theoretical understanding of the Lipschitz continuity and second momentum properties of the diffusion process is still lacking. In this paper, we bridge this gap by providing a detailed examination of these smoothness properties for the case where the target data distribution is a mixture of Gaussians, which serves as a universal approximator for smooth densities such as image data. We prove that if the target distribution is a k-mixture of Gaussians, the density of the entire diffusion process will also be a k-mixture of Gaussians. We then derive tight upper bounds on the Lipschitz constant and second momentum that are independent of the number of mixture components k. Finally, we apply our analysis to various diffusion solvers, both SDE and ODE based, to establish concrete error guarantees in terms of the total variation distance and KL divergence between the target and learned distributions. Our results provide deeper theoretical insights into the dynamics of the diffusion process under common data distributions.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 54 of its 74 references that are in the map. First posted on arXiv 26 May 2024.

Builds on

54 of 74 references.

Closest ancestors · 29

Same direction, newest first

  1. 2024Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyond6 May
  2. 2024Learning general Gaussian mixtures with efficient score matching29 Apr
  3. 2024Learning Mixtures of Gaussians Using Diffusion Models29 Apr
  4. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models3 Mar
  5. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion models3 Mar
  6. 2024Trajectory Consistency Distillation: Improved Latent Consistency Distillation by Semi-Linear Consistency Function with Trajectory Mapping29 Feb
  7. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization28 Jan
  8. 2023Gaussian Mixture Solvers for Diffusion Models2 Nov
  9. 2023Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion1 Oct
  10. 2023Learning Mixtures of Gaussians Using the DDPM Objective3 Jul
  11. 2023Lipschitz Singularities in Diffusion Models20 Jun
  12. 2023Error Bounds for Flow Matching Methods26 May
17 more
  1. 2023The probability flow ODE is provably fast19 May
  2. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers6 Mar
  3. 2023Consistency Models2 Mar
  4. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  5. 2022Scalable Diffusion Models with Transformers19 Dec
  6. 2022Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance13 Dec
  7. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  8. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  9. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  10. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  11. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  12. 2021Non Gaussian Denoising Diffusion Models14 Jun
  13. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  14. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  15. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  16. 2020Denoising Diffusion Probabilistic Models19 Jun
  17. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 25

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2021Image Super-Resolution via Iterative Refinementcited by 1,506
  5. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  6. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 620
  7. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  8. 2021How to Train Your Energy-Based Modelscited by 271
  9. 2016Robust Estimators in High Dimensions without the Computational Intractabilitycited by 209
  10. 2021Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeedcited by 67
  11. 2022A Theoretical Analysis on Feature Learning in Neural Networks: Emergence from Inputs and Advantage over Fixed Featurescited by 49
  12. 2024Towards Few-Shot Adaptation of Foundation Models via Multitask Finetuningcited by 31
13 more
  1. 2020Robustly Learning any Clusterable Mixture of Gaussianscited by 28
  2. 2020Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Modelscited by 28
  3. 2023The Trade-off between Universality and Label Efficiency of Representations from Contrastive Learningcited by 27
  4. 2014Near-optimal-sample estimators for spherical Gaussian mixturescited by 26
  5. 2023Dolfin: Diffusion Layout Transformers without Autoencodercited by 23
  6. 2013Faster and Sample Near-Optimal Algorithms for Proper Learning Mixtures of Gaussianscited by 20
  7. 2020Outlier-Robust Clustering of Non-Spherical Mixturescited by 19
  8. 2015Sample-Optimal Density Estimation in Nearly-Linear Timecited by 16
  9. 2023Provable Guarantees for Neural Networks via Gradient Feature Learningcited by 13
  10. 2023Domain Generalization via Nuclear Norm Regularizationcited by 12
  11. 2022Tensor Moments of Gaussian Mixture Models: Theory and Applicationscited by 7
  12. 2023Feature Programming for Multivariate Time Series Predictioncited by 7
  13. 2021Beyond Parallel Pancakes: Quasi-Polynomial Time Guarantees for Non-Spherical Gaussian Mixturescited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 11; 2025: 8; 2026: 5.

24 papers.

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  1. 2024Why Larger Language Models Do In-context Learning Differently?cited by 44
  2. 2024LazyDiT: Lazy Learning for the Acceleration of Diffusion Transformerscited by 35
  3. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  4. 2024Multi-Layer Transformers Gradient Can be Approximated in Almost Linear Timecited by 22
  5. 2024Bypassing the Exponential Dependency: Looped Transformers Efficiently Learn In-context by Multi-step Gradient Descentcited by 21
  6. 2024On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimalitycited by 21
  7. 2024Looped ReLU MLPs May Be All You Need as Practical Programmable Computerscited by 19
  8. 2024HSR-Enhanced Sparse Attention Accelerationcited by 19
  9. 2024How much is a noisy image worth? Data Scaling Laws for Ambient Diffusioncited by 18
  10. 2025T2VWorldBench: A Benchmark for Evaluating World Knowledge in Text-to-Video Generationcited by 18
  11. 2025High-Order Matching for One-Step Shortcut Diffusion Modelscited by 15
  12. 2024Differential Privacy Mechanisms in Neural Tangent Kernel Regressioncited by 14
12 more
  1. 2024Provable Differentially Private Computation of the Cross-Attention Mechanismcited by 14
  2. 2025Force Matching with Relativistic Constraints: A Physics-Inspired Approach to Stable and Efficient Generative Modelingcited by 12
  3. 2025On Computational Limits of FlowAR Models: Expressivity and Efficiencycited by 10
  4. 2025Understanding Sampler Stochasticity in Training Diffusion Models for RLHFcited by 7
  5. 2026Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalitiescited by 6
  6. 2025HOFAR: High-Order Augmentation of Flow Autoregressive Transformerscited by 4
  7. 2026Weak Diffusion Priors Can Still Achieve Strong Inverse-Problem Performancecited by 4
  8. 2024Differentially Private Kernel Density Estimationcited by 2
  9. 2025Understanding Generalization in Diffusion Distillation via Probability Flow Distancecited by 1
  10. 2026Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structurescited by 1
  11. 2026Proof-of-Authorship for Diffusion-based AI Generated Contentcited by 0
  12. 2026A Quantitative Approximation Framework for Flow Distillation in Diffusion Modelscited by 0

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  4. 2024Structured Diffusion Models with Mixture of Gaussians as Prior Distribution
  5. 2025Diffusion Models: A Mathematical Introduction
  6. 2024The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications
  7. 2024Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat
  8. 2025Diffusion Models under Alternative Noise: Simplified Analysis and Sensitivity