# Symmetry Breaking and Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2312.09016 · HTML: https://mlsanity.com/paper/2312.09016/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 14 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [Top-N: Equivariant set and graph generation without exchangeability](https://mlsanity.com/paper/2110.02096/) (2021)
- [Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks](https://mlsanity.com/paper/2007.02005/) (2020)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Multiset-Equivariant Set Prediction with Approximate Implicit Differentiation](https://mlsanity.com/paper/2111.12193/) (2021)
- [Expressive Sign Equivariant Networks for Spectral Geometric Learning](https://mlsanity.com/paper/2312.02339/) (2023)

## Built on by (most cited first, 25 of 25)

- [SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models](https://mlsanity.com/paper/2502.03638/) (2025)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Probing the effects of broken symmetries in machine learning](https://mlsanity.com/paper/2406.17747/) (2024)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [Denoising Hamiltonian Network for Physical Reasoning](https://mlsanity.com/paper/2503.07596/) (2025)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [Improved Canonicalization for Model Agnostic Equivariance](https://mlsanity.com/paper/2405.14089/) (2024)
- [Approximately-symmetric neural networks for quantum spin liquids](https://mlsanity.com/paper/2405.17541/) (2024)
- [R2Det: Exploring Relaxed Rotation Equivariance in 2D object detection](https://mlsanity.com/paper/2408.11760/) (2024)
- [Symmetry-Aware Generative Modeling through Learned Canonicalization](https://mlsanity.com/paper/2501.07773/) (2025)
- [Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems](https://mlsanity.com/paper/2509.03340/) (2025)
- [When GNNs meet symmetry in ILPs: an orbit-based feature augmentation approach](https://mlsanity.com/paper/2501.14211/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [Spontaneous symmetry breaking and Goldstone modes for deep information propagation](https://mlsanity.com/paper/2605.14685/) (2026)
- [Quasi-Equivariant Metanetworks](https://mlsanity.com/paper/2604.23720/) (2026)
- [Energy Loss Functions for Physical Systems](https://mlsanity.com/paper/2511.02087/) (2025)
- [Balancing Symmetry and Efficiency in Graph Flow Matching](https://mlsanity.com/paper/2602.18084/) (2026)
- [Group-Equivariant Poincar'e Convolutional Networks](https://mlsanity.com/paper/2607.00556/) (2026)
- [Reducing Symmetry Increase in Equivariant Neural Networks](https://mlsanity.com/paper/2608.12010/) (2026)
- [The parity gap in crystal tensor prediction](https://mlsanity.com/paper/2608.18714/) (2026)
- [Equivariance Breaks the Learning Rate](https://mlsanity.com/paper/2609.08381/) (2026)
- [Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria](https://mlsanity.com/paper/2609.35274/) (2026)
- [GLASS: Global Latent Aggregation with Slot-based Set Decoding for Scalable All-Atom Crystal Generation](https://mlsanity.com/paper/2609.37158/) (2026)

## Abstract (arXiv)

Using symmetry as an inductive bias in deep learning has been proven to be a principled approach for sample-efficient model design. However, the relationship between symmetry and the imperative for equivariance in neural networks is not always obvious. Here, we analyze a key limitation that arises in equivariant functions: their incapacity to break symmetry at the level of individual data samples. In response, we introduce a novel notion of 'relaxed equivariance' that circumvents this limitation. We further demonstrate how to incorporate this relaxation into equivariant multilayer perceptrons (E-MLPs), offering an alternative to the noise-injection method. The relevance of symmetry breaking is then discussed in various application domains: physics, graph representation learning, combinatorial optimization and equivariant decoding.