mlsanityData to 7 Oct 2026

Analysis of Diffusion Models for Manifold Data

arXiv 2502.04339 ↗First posted 1 Feb 2025math.ST cond-mat.dis-nn cs.ITcited by 11

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

We analyze the time reversed dynamics of generative diffusion models. If the exact empirical score function is used in a regime of large dimension and exponentially large number of samples, these models are known to undergo transitions between distinct dynamical regimes. We extend this analysis and compute the transitions for an analytically tractable manifold model where the statistical model for the data is a mixture of lower dimensional Gaussians embedded in higher dimensional space. We compute the so-called speciation and collapse transition times, as a function of the ratio of manifold-to-ambient space dimensions, and other characteristics of the data model. An important tool used in our analysis is the exact formula for the mutual information (or free energy) of Generalized Linear Models.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 15 of its 27 references that are in the map. First posted on arXiv 1 Feb 2025.

Builds on

15 of 27 references.

Closest ancestors · 9

Same direction, newest first

  1. 2024Losing dimensions: Geometric memorization in generative diffusion11 Oct
  2. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion8 Oct
  3. 2024Critical windows: non-asymptotic theory for feature emergence in diffusion models3 Mar
  4. 2024Dynamical Regimes of Diffusion Models28 Feb
  5. 2024A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data26 Feb
  6. 2023Spontaneous Symmetry Breaking in Generative Diffusion Models31 May
  7. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  8. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  9. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 6

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  3. 2020Generalisation error in learning with random features and the hidden manifold modelcited by 126
  4. 2017Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Modelscited by 115
  5. 2020The Gaussian equivalence of generative models for learning with shallow neural networkscited by 113
  6. 2017Global Guarantees for Enforcing Deep Generative Priors by Empirical Riskcited by 37

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 7; 2026: 4.

11 papers.

  1. 2025Memorization to Generalization: Emergence of Diffusion Models from Associative Memorycited by 35
  2. 2025Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modelingcited by 18
  3. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 14
  4. 2025Generalization Dynamics of Linear Diffusion Modelscited by 9
  5. 2026A theory of learning data statistics in diffusion models, from easy to hardcited by 5
  6. 2025Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Studycited by 3
  7. 2026Emergence of Distortions in High-Dimensional Guided Diffusion Modelscited by 3
  8. 2026Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Datacited by 3
  9. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Modelscited by 1
  10. 2025Optimal Stopping in Latent Diffusion Modelscited by 1
  11. 2026Memorisation, convergence and generalisation in generative modelscited by 1

Similar, not cited

Close in wording, no citation either way

  1. 2026A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds
  2. 2023Infinite-Dimensional Diffusion Models
  3. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis
  4. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  5. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
  6. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  7. 2025Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction
  8. 2026Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold