mlsanityData to 14 Oct 2026

Your diffusion model secretly knows the dimension of the data manifold

arXiv 2212.12611 ↗First posted 23 Dec 2022cs.LG stat.MLcited by 36

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

In this work, we propose a novel framework for estimating the dimension of the data manifold using a trained diffusion model. A diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, for small levels of corruption, the diffusion model provides us with access to an approximation of the normal bundle of the data manifold. This allows us to estimate the dimension of the tangent space, thus, the intrinsic dimension of the data manifold. To the best of our knowledge, our method is the first estimator of the data manifold dimension based on diffusion models and it outperforms well established statistical estimators in controlled experiments on both Euclidean and image data.

Key facts. Cited by 36 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 11 of its 44 references that are in the map. First posted on arXiv 23 Dec 2022.

Builds on

11 of 44 references.

Closest ancestors · 5

Same direction, newest first

  1. 2021Variational Diffusion Models1 Jul
  2. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  3. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  5. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 6

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2013Auto-Encoding Variational Bayescited by 4,053
  3. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,354
  4. 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 315
  5. 2020Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualizationcited by 173
  6. 2022Statistical Efficiency of Score Matching: The View from Isoperimetrycited by 64

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 5; 2024: 14; 2025: 9; 2026: 8.

36 papers.

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  1. 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 165
  2. 2023Manifold Preserving Guided Diffusioncited by 157
  3. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  4. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  5. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Modelscited by 42
  6. 2024A Geometric Framework for Understanding Memorization in Generative Modelscited by 40
  7. 2023Infinite-Dimensional Diffusion Modelscited by 39
  8. 2024Diffusion Models as Constrained Samplers for Optimization with Unknown Constraintscited by 35
  9. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusioncited by 32
  10. 2024Losing dimensions: Geometric memorization in generative diffusioncited by 29
  11. 2024On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion modelscited by 23
  12. 2024A Geometric Explanation of the Likelihood OOD Detection Paradoxcited by 22
24 more
  1. 2024Hardness of Learning Neural Networks under the Manifold Hypothesiscited by 18
  2. 2025Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modelingcited by 18
  3. 2025Learning normalized image densities via dual score matchingcited by 18
  4. 2023Manifold Learning by Mixture Models of VAEs for Inverse Problemscited by 17
  5. 2025Interpretable Machine Learning in Physics: A Reviewcited by 14
  6. 2024On the Local Complexity of Linear Regions in Deep ReLU Networkscited by 9
  7. 2025Anomaly Detection and Generation with Diffusion Models: A Surveycited by 8
  8. 2025Two-Steps Diffusion Policy for Robotic Manipulation via Genetic Denoisingcited by 8
  9. 2023Exploring the Representation Manifolds of Stable Diffusion Through the Lens of Intrinsic Dimensioncited by 7
  10. 2024What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Modelscited by 7
  11. 2025Iso-Riemannian Optimization on Learned Data Manifoldscited by 4
  12. 2024Response Theory via Generative Score Modelingcited by 3
  13. 2024Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flowscited by 3
  14. 2026Diffusion models recover accurate mixture weights despite score function insensitivitycited by 3
  15. 2025Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworkscited by 2
  16. 2026ELROND: Exploring and decomposing intrinsic capabilities of diffusion modelscited by 2
  17. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Modelscited by 2
  18. 2025State Combinatorial Generalization In Decision Making With Conditional Diffusion Modelscited by 1
  19. 2025RNNs perform task computations by dynamically warping neural representationscited by 1
  20. 2026Exploring Time Conditioning in Diffusion Generative Models from Disjoint Noisy Data Manifoldscited by 1
  21. 2026Style-CCL: Content-Preserving Style Transfer via Curriculum Continual Learningcited by 1
  22. 2026DDGAD: Disagreement-Driven Graph Anomaly Detection via Adapt-Then-Combinecited by 0
  23. 2026MUSE: Unlocking Timestep as Native Task Steering for One-Step Dense Predictioncited by 0
  24. 2026Cross-Time Directional Selection in Diffusion Samplingcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  2. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  3. 2025MAD: Manifold Attracted Diffusion
  4. 2026Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data
  5. 2025What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold
  6. 2025When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis
  7. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models
  8. 2026Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine