# Spontaneous Symmetry Breaking in Generative Diffusion Models

arXiv: https://arxiv.org/abs/2305.19693 · HTML: https://mlsanity.com/paper/2305.19693/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 27 of its 48 references that are in the map.
- First posted on arXiv 31 May 2023.

## Placed in

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

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)

## Built on by (most cited first, 30 of 80)

- [MACE: Mass Concept Erasure in Diffusion Models](https://mlsanity.com/paper/2403.06135/) (2024)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Critical windows: non-asymptotic theory for feature emergence in diffusion models](https://mlsanity.com/paper/2403.01633/) (2024)
- [In search of dispersed memories: Generative diffusion models are associative memory networks](https://mlsanity.com/paper/2309.17290/) (2023)
- [The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability](https://mlsanity.com/paper/2310.17467/) (2023)
- [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)
- [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)
- [Diffusion Models are Evolutionary Algorithms](https://mlsanity.com/paper/2410.02543/) (2024)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (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)
- [TRCE: Towards Reliable Malicious Concept Erasure in Text-to-Image Diffusion Models](https://mlsanity.com/paper/2503.07389/) (2025)
- [Detecting Out-Of-Distribution Earth Observation Images with Diffusion Models](https://mlsanity.com/paper/2404.12667/) (2024)
- [CoDe: Blockwise Control for Denoising Diffusion Models](https://mlsanity.com/paper/2502.00968/) (2025)
- [Memory in Plain Sight: Surveying the Uncanny Resemblances of Associative Memories and Diffusion Models](https://mlsanity.com/paper/2309.16750/) (2023)
- [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)
- [Analysis of Diffusion Models for Manifold Data](https://mlsanity.com/paper/2502.04339/) (2025)
- [Theory of Speciation Transitions in Diffusion Models with General Class Structure](https://mlsanity.com/paper/2602.04404/) (2026)
- [How do Minimum-Norm Shallow Denoisers Look in Function Space?](https://mlsanity.com/paper/2311.06748/) (2023)
- [Neural cellular automata: applications to biology and beyond classical AI](https://mlsanity.com/paper/2509.11131/) (2025)
- [The Entropic Signature of Class Speciation in Diffusion Models](https://mlsanity.com/paper/2602.09651/) (2026)

## Abstract (arXiv)

Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data. In this paper, we show that the dynamics of these models exhibit a spontaneous symmetry breaking that divides the generative dynamics into two distinct phases: 1) A linear steady-state dynamics around a central fixed-point and 2) an attractor dynamics directed towards the data manifold. These two "phases" are separated by the change in stability of the central fixed-point, with the resulting window of instability being responsible for the diversity of the generated samples. Using both theoretical and empirical evidence, we show that an accurate simulation of the early dynamics does not significantly contribute to the final generation, since early fluctuations are reverted to the central fixed point. To leverage this insight, we propose a Gaussian late initialization scheme, which significantly improves model performance, achieving up to 3x FID improvements on fast samplers, while also increasing sample diversity (e.g., racial composition of generated CelebA images). Our work offers a new way to understand the generative dynamics of diffusion models that has the potential to bring about higher performance and less biased fast-samplers.