mlsanityData to 7 Oct 2026

Score-Based Generative Models Detect Manifolds

arXiv 2206.01018 ↗First posted 2 Jun 2022stat.ML cs.LG cs.NAcited by 121

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models (SGMs) need to approximate the scores nabla log pt of the intermediate distributions as well as the final distribution pT of the forward process. The theoretical underpinnings of the effects of these approximations are still lacking. We find precise conditions under which SGMs are able to produce samples from an underlying (low-dimensional) data manifold M. This assures us that SGMs are able to generate the "right kind of samples". For example, taking M to be the subset of images of faces, we find conditions under which the SGM robustly produces an image of a face, even though the relative frequencies of these images might not accurately represent the true data generating distribution. Moreover, this analysis is a first step towards understanding the generalization properties of SGMs: Taking M to be the set of all training samples, our results provide a precise description of when the SGM memorizes its training data.

Key facts. Cited by 121 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 33 of its 65 references that are in the map. First posted on arXiv 2 Jun 2022.

Builds on

33 of 65 references.

Closest ancestors · 16

Same direction, newest first

  1. 2022Riemannian Score-Based Generative Modelling6 Feb
  2. 2021Score-Based Generative Modeling with Critically-Damped Langevin Diffusion14 Dec
  3. 2021Conditional Image Generation with Score-Based Diffusion Models26 Nov
  4. 2021Variational Diffusion Models1 Jul
  5. 2021Score-based Generative Modeling in Latent Space10 Jun
  6. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation10 Jun
  7. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  8. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  9. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  10. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  11. 2021UNIT-DDPM: UNpaired Image Translation with Denoising Diffusion Probabilistic Models12 Apr
  12. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
4 more
  1. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  3. 2020Denoising Diffusion Probabilistic Models19 Jun
  4. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 17

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  3. 2021Image Super-Resolution via Iterative Refinementcited by 1,506
  4. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  5. 2021Diffusion Probabilistic Models for 3D Point Cloud Generationcited by 734
  6. 2020WaveGrad: Estimating Gradients for Waveform Generationcited by 620
  7. 20213D Shape Generation and Completion through Point-Voxel Diffusioncited by 516
  8. 2021Grad-TTS: A Diffusion Probabilistic Model for Text-to-Speechcited by 445
  9. 2021SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Modelscited by 415
  10. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  11. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 307
  12. 2020The Creation and Detection of Deepfakes: A Surveycited by 246
5 more
  1. 2020Learning Gradient Fields for Shape Generationcited by 245
  2. 2021Diff-TTS: A Denoising Diffusion Model for Text-to-Speechcited by 156
  3. 2020Adversarial score matching and improved sampling for image generationcited by 95
  4. 2017Learning to Generate Samples from Noise through Infusion Trainingcited by 15
  5. 2020A Note on Data Biases in Generative Modelscited by 13

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 5; 2023: 19; 2024: 34; 2025: 38; 2026: 25.

121 papers.

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  1. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
  2. 2022Convergence of denoising diffusion models under the manifold hypothesiscited by 237
  3. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
  4. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 171
  5. 2023The probability flow ODE is provably fastcited by 145
  6. 2023Error Bounds for Flow Matching Methodscited by 88
  7. 2023Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Modelscited by 84
  8. 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 74
  9. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  10. 2023Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplerscited by 73
  11. 2023On the Generalization Properties of Diffusion Modelscited by 67
  12. 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 64
38 more
  1. 2024Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorialcited by 60
  2. 2023Learning Mixtures of Gaussians Using the DDPM Objectivecited by 56
  3. 2024On the Trajectory Regularity of ODE-based Diffusion Samplingcited by 51
  4. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 48
  5. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  6. 2023Closed-Form Diffusion Modelscited by 46
  7. 2024Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Modelscited by 46
  8. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptionscited by 45
  9. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesiscited by 42
  10. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion modelscited by 41
  11. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Modelscited by 41
  12. 2024A Geometric Framework for Understanding Memorization in Generative Modelscited by 40
  13. 2023Infinite-Dimensional Diffusion Modelscited by 38
  14. 2023Analysis of learning a flow-based generative model from limited sample complexitycited by 38
  15. 2023A Theoretical Justification for Image Inpainting using Denoising Diffusion Probabilistic Modelscited by 37
  16. 2022FP-Diffusion: Improving Score-based Diffusion Models by Enforcing the Underlying Score Fokker-Planck Equationcited by 36
  17. 2024Diffusion Models as Constrained Samplers for Optimization with Unknown Constraintscited by 33
  18. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusioncited by 31
  19. 2024Contractive Diffusion Probabilistic Modelscited by 29
  20. 2024Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallelcited by 29
  21. 2024Losing dimensions: Geometric memorization in generative diffusioncited by 28
  22. 2024Provable Acceleration for Diffusion Models under Minimal Assumptionscited by 27
  23. 2024Towards a mathematical theory for consistency training in diffusion modelscited by 25
  24. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correctioncited by 23
  25. 2024A Geometric Explanation of the Likelihood OOD Detection Paradoxcited by 22
  26. 2023The Emergence of Reproducibility and Generalizability in Diffusion Modelscited by 21
  27. 2024Understanding and Mitigating Memorization in Generative Models via Sharpness of Probability Landscapescited by 20
  28. 2025Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curvescited by 20
  29. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  30. 2024Deep MMD Gradient Flow without adversarial trainingcited by 18
  31. 2024Hardness of Learning Neural Networks under the Manifold Hypothesiscited by 18
  32. 2025Selective Underfitting in Diffusion Modelscited by 17
  33. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  34. 2024Wasserstein proximal operators describe score-based generative models and resolve memorizationcited by 14
  35. 2024Score-based generative models are provably robust: an uncertainty quantification perspectivecited by 14
  36. 2024When are dynamical systems learned from time series data statistically accurate?cited by 14
  37. 2025A solvable model of learning generative diffusion: theory and insightscited by 14
  38. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 14

50 most cited of 121 All 121 (JSON)

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  1. 2023Score-based generative models learn manifold-like structures with constrained mixing
  2. 2024A Good Score Does not Lead to A Good Generative Model
  3. 2024Moderating the Generalization of Score-based Generative Model
  4. 2024Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions
  5. 2022Convergence for score-based generative modeling with polynomial complexity
  6. 2022Convergence of score-based generative modeling for general data distributions
  7. 2024An analysis of the noise schedule for score-based generative models
  8. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions