# Dynamical Regimes of Diffusion Models

arXiv: https://arxiv.org/abs/2402.18491 · HTML: https://mlsanity.com/paper/2402.18491/

## Key facts

- Cited by 142 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 28 Feb 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Lumiere: A Space-Time Diffusion Model for Video Generation](https://mlsanity.com/paper/2401.12945/) (2024)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [AUC Maximization in the Era of Big Data and AI: A Survey](https://mlsanity.com/paper/2203.15046/) (2022)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization](https://mlsanity.com/paper/2203.05093/) (2022)
- [The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability](https://mlsanity.com/paper/2310.17467/) (2023)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)
- [Sampling with flows, diffusion and autoregressive neural networks: A spin-glass perspective](https://mlsanity.com/paper/2308.14085/) (2023)

## Built on by (most cited first, 30 of 142)

- [Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models](https://mlsanity.com/paper/2404.07724/) (2024)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [CleanDIFT: Diffusion Features without Noise](https://mlsanity.com/paper/2412.03439/) (2024)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [Losing dimensions: Geometric memorization in generative diffusion](https://mlsanity.com/paper/2410.08727/) (2024)
- [Dynamic Negative Guidance of Diffusion Models](https://mlsanity.com/paper/2410.14398/) (2024)
- [Probing the Latent Hierarchical Structure of Data via Diffusion Models](https://mlsanity.com/paper/2410.13770/) (2024)
- [Shielded Diffusion: Generating Novel and Diverse Images using Sparse Repellency](https://mlsanity.com/paper/2410.06025/) (2024)
- [On the Edge of Memorization in Diffusion Models](https://mlsanity.com/paper/2508.17689/) (2025)
- [Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curves](https://mlsanity.com/paper/2502.00336/) (2025)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling](https://mlsanity.com/paper/2502.05743/) (2025)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Models](https://mlsanity.com/paper/2404.18444/) (2024)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusion](https://mlsanity.com/paper/2502.21278/) (2025)
- [Entropic Time Schedulers for Generative Diffusion Models](https://mlsanity.com/paper/2504.13612/) (2025)
- [Feedback Guidance of Diffusion Models](https://mlsanity.com/paper/2506.06085/) (2025)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [Detecting Out-Of-Distribution Earth Observation Images with Diffusion Models](https://mlsanity.com/paper/2404.12667/) (2024)
- [A solvable model of learning generative diffusion: theory and insights](https://mlsanity.com/paper/2501.03937/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [The Information Dynamics of Generative Diffusion](https://mlsanity.com/paper/2508.19897/) (2025)
- [Cascade of phase transitions in the training of Energy-based models](https://mlsanity.com/paper/2405.14689/) (2024)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [Analysis of Diffusion Models for Manifold Data](https://mlsanity.com/paper/2502.04339/) (2025)

## Abstract (arXiv)

Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The generative dynamics, starting from pure noise, encounters first a 'speciation' transition where the gross structure of data is unraveled, through a mechanism similar to symmetry breaking in phase transitions. It is followed at later time by a 'collapse' transition where the trajectories of the dynamics become attracted to one of the memorized data points, through a mechanism which is similar to the condensation in a glass phase. For any dataset, the speciation time can be found from a spectral analysis of the correlation matrix, and the collapse time can be found from the estimation of an 'excess entropy' in the data. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models. Analytical solutions for simple models like high-dimensional Gaussian mixtures substantiate these findings and provide a theoretical framework, while extensions to more complex scenarios and numerical validations with real datasets confirm the theoretical predictions.