# Permutation-Invariant Spectral Learning via Dyson Diffusion

arXiv: https://arxiv.org/abs/2510.08535 · HTML: https://mlsanity.com/paper/2510.08535/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 27 of its 59 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Permutation Invariant Graph Generation via Score-Based Generative Modeling](https://mlsanity.com/paper/2003.00638/) (2020)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators](https://mlsanity.com/paper/2204.01613/) (2022)
- [Discrete-state Continuous-time Diffusion for Graph Generation](https://mlsanity.com/paper/2405.11416/) (2024)
- [A Survey on Audio Diffusion Models: Text To Speech Synthesis and Enhancement in Generative AI](https://mlsanity.com/paper/2303.13336/) (2023)
- [Position: Graph Learning Will Lose Relevance Due To Poor Benchmarks](https://mlsanity.com/paper/2502.14546/) (2025)
- [GraphGDP: Generative Diffusion Processes for Permutation Invariant Graph Generation](https://mlsanity.com/paper/2212.01842/) (2022)
- [Fast Graph Generation via Spectral Diffusion](https://mlsanity.com/paper/2211.08892/) (2022)
- [Graph Isomorphism in Quasipolynomial Time](https://mlsanity.com/paper/1512.03547/) (2015)
- [Generating Graphs via Spectral Diffusion](https://mlsanity.com/paper/2402.18974/) (2024)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [Structural Balance and Random Walks on Complex Networks with Complex Weights](https://mlsanity.com/paper/2307.01813/) (2023)
- [Do Graph Diffusion Models Accurately Capture and Generate Substructure Distributions?](https://mlsanity.com/paper/2502.02488/) (2025)

## Abstract (arXiv)

Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to n! such representations for graphs with n nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features. This shortcoming stems from enforcing the inductive bias within the learning architecture. In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix. Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level. We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.