Permutation Equivariant Neural Networks for Symmetric Tensors

Edward Pearce-Crump
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:48595-48634, 2025.

Abstract

Incorporating permutation equivariance into neural networks has proven to be useful in ensuring that models respect symmetries that exist in data. Symmetric tensors, which naturally appear in statistics, machine learning, and graph theory, are essential for many applications in physics, chemistry, and materials science, amongst others. However, existing research on permutation equivariant models has not explored symmetric tensors as inputs, and most prior work on learning from these tensors has focused on equivariance to Euclidean groups. In this paper, we present two different characterisations of all linear permutation equivariant functions between symmetric power spaces of $\mathbb{R}^{n}$. We show on two tasks that these functions are highly data efficient compared to standard MLPs and have potential to generalise well to symmetric tensors of different sizes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-pearce-crump25b, title = {Permutation Equivariant Neural Networks for Symmetric Tensors}, author = {Pearce-Crump, Edward}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {48595--48634}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/pearce-crump25b/pearce-crump25b.pdf}, url = {https://proceedings.mlr.press/v267/pearce-crump25b.html}, abstract = {Incorporating permutation equivariance into neural networks has proven to be useful in ensuring that models respect symmetries that exist in data. Symmetric tensors, which naturally appear in statistics, machine learning, and graph theory, are essential for many applications in physics, chemistry, and materials science, amongst others. However, existing research on permutation equivariant models has not explored symmetric tensors as inputs, and most prior work on learning from these tensors has focused on equivariance to Euclidean groups. In this paper, we present two different characterisations of all linear permutation equivariant functions between symmetric power spaces of $\mathbb{R}^{n}$. We show on two tasks that these functions are highly data efficient compared to standard MLPs and have potential to generalise well to symmetric tensors of different sizes.} }
Endnote
%0 Conference Paper %T Permutation Equivariant Neural Networks for Symmetric Tensors %A Edward Pearce-Crump %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-pearce-crump25b %I PMLR %P 48595--48634 %U https://proceedings.mlr.press/v267/pearce-crump25b.html %V 267 %X Incorporating permutation equivariance into neural networks has proven to be useful in ensuring that models respect symmetries that exist in data. Symmetric tensors, which naturally appear in statistics, machine learning, and graph theory, are essential for many applications in physics, chemistry, and materials science, amongst others. However, existing research on permutation equivariant models has not explored symmetric tensors as inputs, and most prior work on learning from these tensors has focused on equivariance to Euclidean groups. In this paper, we present two different characterisations of all linear permutation equivariant functions between symmetric power spaces of $\mathbb{R}^{n}$. We show on two tasks that these functions are highly data efficient compared to standard MLPs and have potential to generalise well to symmetric tensors of different sizes.
APA
Pearce-Crump, E.. (2025). Permutation Equivariant Neural Networks for Symmetric Tensors. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:48595-48634 Available from https://proceedings.mlr.press/v267/pearce-crump25b.html.

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