mlsanityData to 14 Oct 2026

Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures

arXiv 2504.05300 ↗First posted 7 Apr 2025cs.LG cs.NA math.NAcited by 23

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models are distinguished by their exceptional generative performance, particularly in producing high-quality samples through iterative denoising. While current theory suggests that the number of denoising steps required for accurate sample generation should scale linearly with data dimension, this does not reflect the practical efficiency of widely used algorithms like Denoising Diffusion Probabilistic Models (DDPMs). This paper investigates the effectiveness of diffusion models in sampling from complex high-dimensional distributions that can be well-approximated by Gaussian Mixture Models (GMMs). For these distributions, our main result shows that DDPM takes at most widetildeO(1/varepsilon) iterations to attain an varepsilon-accurate distribution in total variation (TV) distance, independent of both the ambient dimension d and the number of components K, up to logarithmic factors. Furthermore, this result remains robust to score estimation errors. These findings highlight the remarkable effectiveness of diffusion models in high-dimensional settings given the universal approximation capability of GMMs, and provide theoretical insights into their practical success.

Key facts. Cited by 23 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 49 of its 78 references that are in the map. First posted on arXiv 7 Apr 2025.

Builds on

49 of 78 references.

Closest ancestors · 37

Same direction, newest first

  1. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Models12 Mar
  2. 2025Low-dimensional adaptation of diffusion models: Convergence in total variation22 Jan
  3. 2025Nonparametric estimation of a factorizable density using diffusion models3 Jan
  4. 2024Provable Acceleration for Diffusion Models under Minimal Assumptions30 Oct
  5. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality24 Oct
  6. 2024Instance-dependent Convergence Theory for Diffusion Models17 Oct
  7. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis11 Oct
  8. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Models7 Oct
  9. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions27 Sep
  10. 2024What does guidance do? A fine-grained analysis in a simple setting19 Sep
  11. 2024From optimal score matching to optimal sampling11 Sep
  12. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions4 Sep
25 more
  1. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models5 Aug
  2. 2024Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective26 May
  3. 2024Reverse Transition Kernel: A Flexible Framework to Accelerate Diffusion Inference26 May
  4. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models23 May
  5. 2024Learning general Gaussian mixtures with efficient score matching29 Apr
  6. 2024Learning Mixtures of Gaussians Using Diffusion Models29 Apr
  7. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Models15 Apr
  8. 2024Accelerating Convergence of Score-Based Diffusion Models, Provably6 Mar
  9. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions23 Feb
  10. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial12 Feb
  11. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances31 Jan
  12. 2023Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models20 Sep
  13. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  14. 2023Learning Mixtures of Gaussians Using the DDPM Objective3 Jul
  15. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models15 Jun
  16. 2023The probability flow ODE is provably fast19 May
  17. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  18. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  19. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  20. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  21. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  22. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models31 Aug
  23. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  24. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  25. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov

Other references · 12

  1. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  2. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  3. 2017Mixture Models, Robustness, and Sum of Squares Proofscited by 81
  4. 2017List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussianscited by 70
  5. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  6. 2018Singularity, Misspecification, and the Convergence Rate of EMcited by 38
  7. 2020Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Modelscited by 28
  8. 2021Clustering Mixtures with Almost Optimal Separation in Polynomial Timecited by 28
  9. 2017On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoisingcited by 19
  10. 2020Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Modelscited by 17
  11. 2020Optimal estimation of high-dimensional location Gaussian mixturescited by 16
  12. 2024Optimal convex M-estimation via score matchingcited by 9

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 9; 2026: 14.

23 papers.

Sort:

  1. 2025Breaking AR's Sampling Bottleneck: Provable Acceleration via Diffusion Language Modelscited by 23
  2. 2026Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guaranteescited by 19
  3. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 18
  4. 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
  5. 2025Stage-wise Dynamics of Classifier-Free Guidance in Diffusion Modelscited by 16
  6. 2025Optimal Convergence Analysis of DDPM for General Distributionscited by 16
  7. 2026High-accuracy sampling for diffusion models and log-concave distributionscited by 16
  8. 2026Adaptation to Intrinsic Dependence in Diffusion Language Modelscited by 10
  9. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  10. 2025Provable Efficiency of Guidance in Diffusion Models for General Data Distributioncited by 8
  11. 2025Towards a unified framework for guided diffusion modelscited by 8
  12. 2026High-accuracy and dimension-free sampling with diffusionscited by 7
11 more
  1. 2026Provably adaptive sampling with uniform and remasking discrete diffusion modelscited by 4
  2. 2025Dimension-free error estimate for diffusion model and optimal schedulingcited by 3
  3. 2026Denoising growth complexity: Data geometry and certified schedules for diffusion samplingcited by 2
  4. 2026ConvergeFlow: Language Flow with Provable Convergence to Token Embeddingscited by 2
  5. 2026Entropy-Based Dimension-Free Convergence and Loss-Adaptive Schedules for Diffusion Modelscited by 1
  6. 2026Why DDIM Hallucinates More Than DDPM: A Theoretical Analysis of Reverse Dynamicscited by 1
  7. 2026Diffusion Models for Adaptive Sequential Data Generationcited by 1
  8. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1
  9. 2025Are First-Order Diffusion Samplers Really Slower? A Fast Forward-Value Approachcited by 0
  10. 2026When Diffusion Model Can Ignore Dimension: An Entropy-Based Theorycited by 0
  11. 2026Diffusion Models for High-Dimensional Clustered Data: Intrinsic-Dimension Adaptivity via Bayesian Classificationcited by 0

Similar, not cited

Close in wording, no citation either way

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  2. 2023To smooth a cloud or to pin it down: Guarantees and Insights on Score Matching in Denoising Diffusion Models
  3. 2023Improved DDIM Sampling with Moment Matching Gaussian Mixtures
  4. 2026Blind denoising diffusion models and the blessings of dimensionality
  5. 2023Fast Inference in Denoising Diffusion Models via MMD Finetuning
  6. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  7. 2023Star-Shaped Denoising Diffusion Probabilistic Models
  8. 2024Diffusion Model Conditioning on Gaussian Mixture Model and Negative Gaussian Mixture Gradient