mlsanityData to 14 Oct 2026

Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions

arXiv 2402.15602 ↗First posted 23 Feb 2024math.ST cs.IT cs.LGcited by 47

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of widetildeOleft(n-1 t^-fracd+22(t^fracd2 vee 1)right) for the score function of p0*N(0,tboldsymbolId), where n and d represent the sample size and the dimension, t is bounded above and below by polynomials of n, and p0 is an arbitrary sub-Gaussian distribution. As a consequence, this yields an widetildeOleft(n-1/2 t^-fracd4right) upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian assumption. If in addition, p0 belongs to the nonparametric family of the β-Sobolev space with β≤ 2, by adopting an early stopping strategy, we obtain that the diffusion model is nearly (up to log factors) minimax optimal. This removes the crucial lower bound assumption on p0 in previous proofs of the minimax optimality of the diffusion model for nonparametric families.

Key facts. Cited by 47 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 28 of its 46 references that are in the map. First posted on arXiv 23 Feb 2024.

Builds on

28 of 46 references.

Closest ancestors · 12

Same direction, newest first

  1. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  2. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  3. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  4. 2022Convergence of score-based generative modeling for general data distributions26 Sep
  5. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  6. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  7. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  8. 2022Score-Based Generative Models Detect Manifolds2 Jun
  9. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  10. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  11. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  12. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 16

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2021GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Modelscited by 3,614
  5. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 447
  6. 2022Brain Imaging Generation with Latent Diffusion Modelscited by 268
  7. 2023Geometric Latent Diffusion Models for 3D Molecule Generationcited by 186
  8. 2021Diff-TTS: A Denoising Diffusion Model for Text-to-Speechcited by 157
  9. 2022ProDiff: Progressive Fast Diffusion Model For High-Quality Text-to-Speechcited by 149
  10. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  11. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64
  12. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
4 more
  1. 2022Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localizationcited by 48
  2. 2023MUDiff: Unified Diffusion for Complete Molecule Generationcited by 42
  3. 2023Sampling, Diffusions, and Stochastic Localizationcited by 40
  4. 2022Accelerating Material Design with the Generative Toolkit for Scientific Discoverycited by 16

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 13; 2025: 17; 2026: 17.

47 papers.

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  1. 2025Low-dimensional adaptation of diffusion models: Convergence in total variationcited by 39
  2. 2024Stochastic Runge-Kutta Methods: Provable Acceleration of Diffusion Modelscited by 35
  3. 2024Flow matching achieves almost minimax optimal convergencecited by 29
  4. 2024From optimal score matching to optimal samplingcited by 28
  5. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  6. 2024Harnessing Earnings Reports for Stock Predictions: A QLoRA-Enhanced LLM Approachcited by 24
  7. 2025Minimax Optimality of the Probability Flow ODE for Diffusion Modelscited by 23
  8. 2025Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixturescited by 23
  9. 2025Breaking AR's Sampling Bottleneck: Provable Acceleration via Diffusion Language Modelscited by 23
  10. 2026Confidence-Based Decoding is Provably Efficient for Diffusion Language Modelscited by 18
  11. 2024Score-based generative models are provably robust: an uncertainty quantification perspectivecited by 15
  12. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
35 more
  1. 2025A solvable model of learning generative diffusion: theory and insightscited by 14
  2. 2024Shallow diffusion networks provably learn hidden low-dimensional structurecited by 12
  3. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flowscited by 10
  4. 2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 10
  5. 2026Adaptation to Intrinsic Dependence in Diffusion Language Modelscited by 10
  6. 2025Domain Guidance: A Simple Transfer Approach for a Pre-trained Diffusion Modelcited by 8
  7. 2024Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplerscited by 7
  8. 2025Non-asymptotic error bounds for probability flow ODEs under weak log-concavitycited by 7
  9. 2024Statistical guarantees for denoising reflected diffusion modelscited by 6
  10. 2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 6
  11. 2025DiffEM: Learning from Corrupted Data with Diffusion Models via Expectation Maximizationcited by 6
  12. 2024Dimension-independent rates for structured neural density estimationcited by 5
  13. 2025Algorithm- and Data-Dependent Generalization Bounds for Diffusion Modelscited by 5
  14. 2024Score-Based Metropolis-Hastings Algorithmscited by 4
  15. 2026Quantifying Error Propagation and Model Collapse in Diffusion Modelscited by 4
  16. 2024Breaking the curse of dimensionality in structured density estimationcited by 3
  17. 2026Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionscited by 3
  18. 2026Benign Overfitting Does Not Occur in Diffusion Modelscited by 3
  19. 2025Regularization can make diffusion models more efficientcited by 2
  20. 2025Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiencycited by 2
  21. 2025Improved Sample Complexity For Diffusion Model Training Without Empirical Risk Minimizer Accesscited by 2
  22. 2026Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errorscited by 2
  23. 2024Semi-Implicit Functional Gradient Flow for Efficient Samplingcited by 1
  24. 2026Provably Reliable Classifier Guidance via Cross-Entropy Controlcited by 1
  25. 2026On the Robustness of Distribution Support under Diffusion Guidancecited by 1
  26. 2026Diffusion Models for Adaptive Sequential Data Generationcited by 1
  27. 2026Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structurescited by 1
  28. 2026Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choicescited by 1
  29. 2025A Black-Box Debiasing Framework for Conditional Samplingcited by 0
  30. 2026Proof-of-Authorship for Diffusion-based AI Generated Contentcited by 0
  31. 2026Statistical Convergence of Spherical First Hitting Diffusion Modelscited by 0
  32. 2026Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterizationcited by 0
  33. 2026Breaking the Curse with BAND: Nonparametric Distribution Estimation in High Dimensionscited by 0
  34. 2026Finite-Sample Guarantees for Fully Discrete Conditional Diffusion Modelscited by 0
  35. 2026A Stochastic Analysis Perspective on Diffusion Modelscited by 0

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  1. 2026Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees
  2. 2026Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data
  3. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
  4. 2026Tightening the Score Matching Gap for Diffusion Models
  5. 2024Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis
  6. 2026Query Lower Bounds for Diffusion Sampling
  7. 2026Robustness of Diffusion Models under Distribution Shift
  8. 2026Minimax Optimality of Score-Entropy Discrete Diffusion