mlsanityData to 7 Oct 2026

Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion

arXiv 2410.05898 ↗First posted 8 Oct 2024stat.ML cs.LGcited by 31

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions, and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation of why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 24 of its 37 references that are in the map. First posted on arXiv 8 Oct 2024.

Builds on

24 of 37 references.

Closest ancestors · 16

Same direction, newest first

  1. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions4 Sep
  2. 2024Exploring Low-Dimensional Subspaces in Diffusion Models for Controllable Image Editing4 Sep
  3. 2024Dynamical Regimes of Diffusion Models28 Feb
  4. 2024A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data26 Feb
  5. 2024On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models6 Feb
  6. 2023Score-based generative models learn manifold-like structures with constrained mixing16 Nov
  7. 2023The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability26 Oct
  8. 2023Generalization in diffusion models arises from geometry-adaptive harmonic representations4 Oct
  9. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  10. 2023Spontaneous Symmetry Breaking in Generative Diffusion Models31 May
  11. 2023Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation1 Mar
  12. 2022Your diffusion model secretly knows the dimension of the data manifold23 Dec
4 more
  1. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  2. 2022Score-Based Generative Models Detect Manifolds2 Jun
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  4. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 8

  1. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  2. 2022Video Diffusion Modelscited by 2,280
  3. 2022Make-A-Video: Text-to-Video Generation without Text-Video Datacited by 1,725
  4. 2023Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Modelscited by 1,465
  5. 2017PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modificationscited by 415
  6. 2023Sampling, Diffusions, and Stochastic Localizationcited by 38
  7. 2022Diagnosing and Fixing Manifold Overfitting in Deep Generative Modelscited by 28
  8. 2024Varying Manifolds in Diffusion: From Time-varying Geometries to Visual Saliencycited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 18; 2026: 10.

31 papers.

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  1. 2024An analytic theory of creativity in convolutional diffusion modelscited by 103
  2. 2025Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Trainingcited by 78
  3. 2025Memorization to Generalization: Emergence of Diffusion Models from Associative Memorycited by 35
  4. 2024Losing dimensions: Geometric memorization in generative diffusioncited by 28
  5. 2025On the Edge of Memorization in Diffusion Modelscited by 22
  6. 2024Understanding and Mitigating Memorization in Generative Models via Sharpness of Probability Landscapescited by 20
  7. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  8. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  9. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 14
  10. 2025Provable Separations between Memorization and Generalization in Diffusion Modelscited by 14
  11. 2025When Scores Learn Geometry: Rate Separations under the Manifold Hypothesiscited by 13
  12. 2025Carr'e du champ flow matching: better quality-generalisation tradeoff in generative modelscited by 13
19 more
  1. 2025The Information Dynamics of Generative Diffusioncited by 12
  2. 2025Analysis of Diffusion Models for Manifold Datacited by 11
  3. 2026Video Models Reason Early: Exploiting Plan Commitment for Maze Solvingcited by 9
  4. 2025Landing with the Score: Riemannian Optimization through Denoisingcited by 7
  5. 2025Image Interpolation with Score-based Riemannian Metrics of Diffusion Modelscited by 4
  6. 2025Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Studycited by 3
  7. 2025CoVAE: Consistency Training of Variational Autoencoderscited by 3
  8. 2026Emergence of Distortions in High-Dimensional Guided Diffusion Modelscited by 3
  9. 2026An exact information theory of generalization phase transitions in Bayesian diffusion modelscited by 3
  10. 2025On Convolutions, Intrinsic Dimension, and Diffusion Modelscited by 2
  11. 2026Diffusion Models Memorize in Training -- and Generalize in Inferencecited by 2
  12. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Modelscited by 1
  13. 2026Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimationcited by 1
  14. 2026Recovering Weighted Tangent Geometry from a Single-Scale Score Fieldcited by 1
  15. 2026Double Descent and Malign Overfitting in Diffusion Modelscited by 1
  16. 2025Determinism of Randomness: Prompt-Residual Seed Shaping for Diffusion Generationcited by 0
  17. 2026One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversioncited by 0
  18. 2026A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifoldscited by 0
  19. 2026Matched Queries for Curvature and Density at Branching Junctionscited 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. 2023Infinite-Dimensional Diffusion Models
  3. 2024What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models
  4. 2025MAD: Manifold Attracted Diffusion
  5. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Models
  6. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
  7. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  8. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis