mlsanityData to 7 Oct 2026

Losing dimensions: Geometric memorization in generative diffusion

arXiv 2410.08727 ↗First posted 11 Oct 2024stat.ML cs.LGcited by 28

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models power leading generative AI, but when and how they memorize training data, especially on low-dimensional manifolds, remains unclear. We find memorization emerges gradually, not abruptly: as data become scarce, diffusion models experience a smooth collapse where their capacity to vary across independent directions diminishes. Measuring latent dimensionality via the learned score field, we reveal how generative behavior increasingly centers on a few examples while other variations "freeze out". We propose a geometric memorization theory, showing that salient features collapse first, then finer details, leading to near point-wise replication. This mirrors physical systems condensing into a few low-energy configurations. Our theoretical predictions align with both synthetic and real data, identifying geometric memorization as a distinct phase between generalization and exact copying.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 31 of its 55 references that are in the map. First posted on arXiv 11 Oct 2024.

Builds on

31 of 55 references.

Closest ancestors · 16

Same direction, newest first

  1. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion8 Oct
  2. 2024Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions4 Sep
  3. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models5 Jun
  4. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesis5 Mar
  5. 2024Dynamical Regimes of Diffusion Models28 Feb
  6. 2024On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models6 Feb
  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. 2023In search of dispersed memories: Generative diffusion models are associative memory networks29 Sep
  10. 2023Understanding and Mitigating Copying in Diffusion Models31 May
  11. 2023Spontaneous Symmetry Breaking in Generative Diffusion Models31 May
  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 · 15

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2017Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithmscited by 4,283
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  5. 2022Video Diffusion Modelscited by 2,280
  6. 2022Make-A-Video: Text-to-Video Generation without Text-Video Datacited by 1,725
  7. 2023Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Modelscited by 1,465
  8. 2015LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loopcited by 1,350
  9. 2022Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Modelscited by 418
  10. 2017PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modificationscited by 415
  11. 2016Dense Associative Memory for Pattern Recognitioncited by 230
  12. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 62
3 more
  1. 2023A Reproducible Extraction of Training Images from Diffusion Modelscited by 60
  2. 2023Sampling, Diffusions, and Stochastic Localizationcited by 38
  3. 2024A Geometric Explanation of the Likelihood OOD Detection Paradoxcited by 22

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 15; 2026: 13.

28 papers.

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  1. 2025Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Trainingcited by 78
  2. 2025Memorization to Generalization: Emergence of Diffusion Models from Associative Memorycited by 35
  3. 2025Bigger Isn't Always Memorizing: Early Stopping Overparameterized Diffusion Modelscited by 24
  4. 2025Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curvescited by 20
  5. 2025Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modelingcited by 18
  6. 2025On the Interpolation Effect of Score Smoothing in Diffusion Modelscited by 15
  7. 2025A solvable model of learning generative diffusion: theory and insightscited by 14
  8. 2025Carr'e du champ flow matching: better quality-generalisation tradeoff in generative modelscited by 13
  9. 2025TCFG: Tangential Damping Classifier-free Guidancecited by 12
  10. 2025The Information Dynamics of Generative Diffusioncited by 12
  11. 2025Analysis of Diffusion Models for Manifold Datacited by 11
  12. 2025Generalization of Diffusion Models Arises with a Balanced Representation Spacecited by 10
16 more
  1. 2026Local Coverage Governs Memorization in Diffusion Modelscited by 5
  2. 2025When and how can inexact generative models still sample from the data manifold?cited by 4
  3. 2026Emergence of Distortions in High-Dimensional Guided Diffusion Modelscited by 3
  4. 2026Steering Dynamical Regimes of Diffusion Models by Breaking Detailed Balancecited by 3
  5. 2026Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Datacited by 3
  6. 2026Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regimecited by 3
  7. 2026Diffusion Models Memorize in Training -- and Generalize in Inferencecited by 2
  8. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Modelscited by 2
  9. 2025Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problemscited by 1
  10. 2026The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Modelscited by 1
  11. 2026On the Limits of Latent Reuse in Diffusion Modelscited by 1
  12. 2026Evaluating the Representation Space of Diffusion Models via Self-Supervised Principlescited by 1
  13. 2026Double Descent and Malign Overfitting in Diffusion Modelscited by 1
  14. 2025Latent Iterative Refinement Flow: A Geometric Constrained Approach for Few-Shot Generationcited by 0
  15. 2026Language Diffusion Models are Associative Memories Capable of Retrieving Unseen Datacited by 0
  16. 2026First Learn, Then Memorize: The Spectral Bias of Diffusion Modelscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2024Understanding and Mitigating Memorization in Generative Models via Sharpness of Probability Landscapes
  2. 2026Two Calm Ends and the Wild Middle: A Geometric Picture of Memorization in Diffusion Models
  3. 2024A Geometric Framework for Understanding Memorization in Generative Models
  4. 2025On the Edge of Memorization in Diffusion Models
  5. 2026Localizing Memorized Regions in Diffusion Models via Coordinate-Wise Curvature Differences
  6. 2025A Closer Look at Model Collapse: From a Generalization-to-Memorization Perspective
  7. 2026Understanding diffusion models requires rethinking (again) generalization
  8. 2025How Diffusion Models Memorize