# Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations

arXiv: https://arxiv.org/abs/2410.06665 · HTML: https://mlsanity.com/paper/2410.06665/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 40 references that are in the map.
- First posted on arXiv 9 Oct 2024.

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

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Optimizing Mode Connectivity via Neuron Alignment](https://mlsanity.com/paper/2009.02439/) (2020)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Permutation Equivariant Neural Functionals](https://mlsanity.com/paper/2302.14040/) (2023)
- [Re-basin via implicit Sinkhorn differentiation](https://mlsanity.com/paper/2212.12042/) (2022)
- [Graph Neural Networks for Learning Equivariant Representations of Neural Networks](https://mlsanity.com/paper/2403.12143/) (2024)
- [Graph Metanetworks for Processing Diverse Neural Architectures](https://mlsanity.com/paper/2312.04501/) (2023)
- [Neural Functional Transformers](https://mlsanity.com/paper/2305.13546/) (2023)
- [Equivariant Deep Weight Space Alignment](https://mlsanity.com/paper/2310.13397/) (2023)
- [The general theory of permutation equivarant neural networks and higher order graph variational encoders](https://mlsanity.com/paper/2004.03990/) (2020)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformers](https://mlsanity.com/paper/2404.09411/) (2024)
- [Equivariant Maps for Hierarchical Structures](https://mlsanity.com/paper/2006.03627/) (2020)
- [Connecting Permutation Equivariant Neural Networks and Partition Diagrams](https://mlsanity.com/paper/2212.08648/) (2022)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions](https://mlsanity.com/paper/2308.00273/) (2023)

## Built on by (most cited first, 2 of 2)

- [Permutation Equivariant Neural Networks for Symmetric Tensors](https://mlsanity.com/paper/2503.11276/) (2025)
- [Finite Probes Suffice: Identifiability and Universality for Weight-Space Learning](https://mlsanity.com/paper/2609.33901/) (2026)

## Abstract (arXiv)

This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches, which address these problems using parameter-sharing, we consider an alternative methodology based on irreducible representations and Schur's lemma. Using this methodology, we obtain an alternative derivation for existing models like DeepSets, 2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks. The derivation for DWS networks is significantly simpler than that of previous results. Next, we extend our approach to unaligned symmetric sets, where equivariance to the wreath product of groups is required. Previous works have addressed this problem in a rather restrictive setting, in which almost all wreath equivariant layers are Siamese. In contrast, we give a full characterization of layers in this case and show that there is a vast number of additional non-Siamese layers in some settings. We also show empirically that these additional non-Siamese layers can improve performance in tasks like graph anomaly detection, weight space alignment, and learning Wasserstein distances. Our code is available at hrefhttps://github.com/yonatansverdlov/Irreducible-Representations-of-Deep-Weight-SpacesGitHub.