# A Diagrammatic Approach to Improve Computational Efficiency in Group Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2412.10837 · HTML: https://mlsanity.com/paper/2412.10837/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 74 references that are in the map.
- First posted on arXiv 14 Dec 2024.

## Builds on (most cited first, 7 of 7)

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Permutation-equivariant neural networks applied to dynamics prediction](https://mlsanity.com/paper/1612.04530/) (2016)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [Stochastic Neural Network Symmetrisation in Markov Categories](https://mlsanity.com/paper/2406.11814/) (2024)
- [On the Anatomy of Attention](https://mlsanity.com/paper/2407.02423/) (2024)
- [Position Paper: Generalized grammar rules and structure-based generalization beyond classical equivariance for lexical tasks and transduction](https://mlsanity.com/paper/2402.01629/) (2024)

## Abstract (arXiv)

Group equivariant neural networks are growing in importance owing to their ability to generalise well in applications where the data has known underlying symmetries. Recent characterisations of a class of these networks that use high-order tensor power spaces as their layers suggest that they have significant potential; however, their implementation remains challenging owing to the prohibitively expensive nature of the computations that are involved. In this work, we present a fast matrix multiplication algorithm for any equivariant weight matrix that maps between tensor power layer spaces in these networks for four groups: the symmetric, orthogonal, special orthogonal, and symplectic groups. We obtain this algorithm by developing a diagrammatic framework based on category theory that enables us to not only express each weight matrix as a linear combination of diagrams but also makes it possible for us to use these diagrams to factor the original computation into a series of steps that are optimal. We show that this algorithm improves the Big-O time complexity exponentially in comparison to a na"ive matrix multiplication.