Geometry and Machine Learning 2, Paris, France

Date: Wednesday, 28th May 2025, 4:20pm CEST (3:20pm BST)

Title: Permutation Equivariant Neural Networks for Symmetric Tensors

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 talk, we present a complete characterisation of all linear permutation equivariant functions between symmetric power spaces of $\mathbb{R}^{n}$ using a combinatorial, diagrammatic approach. We further introduce an efficient implementation method based on map label notation, and demonstrate its practical benefits on a toy example.

Erlangen AI Hub Meeting, Oxford, UK

Date: Tuesday, 20th May 2025, 11:00am GMT

Title: TBC

PDRA Seminar Series, EPSRC AI Hub, Imperial College London

Date: Wednesday, 12th March 2025, 2:00pm GMT

Title: Around the Equivariant World in 45 Minutes

Imperial College London

Date: Monday, 2nd December 2024, 10:30am GMT

Title: Diagrammatic Algebra for Equivariant Neural Network Architectures

Twenty-Seventh European Conference on Artificial Intelligence, Santiago de Compostela, Spain

Date: Thursday, 24th October 2024, 11:00am CET

Title: Connecting Permutation Equivariant Neural Networks and Partition Diagrams

Permutation equivariant neural networks are often constructed using tensor powers of $\mathbb{R}^{n}$ as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl duality between the symmetric group and the partition algebra. In particular, we adapt Schur-Weyl duality to derive a simple, diagrammatic method for calculating the weight matrices themselves.

Workshop on Symmetry and Equivariance in Deep Learning, Paris, France

Date: Wednesday, 4th September 2024, 2:00pm CET

Title: Diagrammatic Mathematics for Equivariant Deep Learning Architectures 

In deep learning, we would like to develop principled approaches for constructing neural network architectures. One important approach involves encoding symmetries into neural network architectures using representations of groups such that the learned functions are equivariant to the group. In this talk, we show how certain group equivariant neural network architectures can be built using set partition diagrams. In many cases, we can establish a category theory framework both for the set partition diagrams and for the equivariant linear maps between layer spaces. We extend this framework to characterise the weight matrices that appear in neural networks that are equivariant to the automorphism group of a graph.

Quantum Groups Seminar (Online)

Date: Monday, 20th May 2024, 4:00pm CET

Compact Matrix Quantum Group Equivariant Neural Networks

In deep learning, we would like to develop principled approaches for constructing neural networks. One important approach involves identifying symmetries that are inherent in data and then encoding them into neural network architectures using representations of groups. However, there exist so-called “quantum symmetries” that cannot be understood formally by groups. In this talk, we show how to construct neural networks that are equivariant to compact matrix quantum groups using Woronowicz’s version of Tannaka-Krein duality. We go on to characterise the linear weight matrices that appear in these neural networks for a class of compact matrix quantum groups known as “easy”.  In particular, we show that every compact matrix group equivariant neural network is a compact matrix quantum group equivariant neural network.

The Royal Institution of Great Britain, London, UK

Date: Thursday, 29th February 2024, 2:00pm GMT

Exploring Group Equivariant Neural Networks Using Set Partition Diagrams

Fortieth International Conference on Machine Learning, Honolulu, Hawaii, United States

Date: Wednesday, July 26th, 5:12pm HST (Thursday, July 27th, 4:12am BST)

Brauer's Group Equivariant Neural Networks

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$ for three symmetry groups that are missing from the machine learning literature: $O(n)$, the orthogonal group; $SO(n)$, the special orthogonal group; and $Sp(n)$, the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$ when the group is $O(n)$ or $SO(n)$, and in the symplectic basis of $\mathbb{R}^{n}$ when the group is $Sp(n)$.

Date: Wednesday, June 21st, 10am BST

Exploring Group Equivariant Neural Networks Using Set Partition Diagrams

What do jellyfish and an 11th century Japanese novel have to do with neural networks? In recent years, much attention has been given to developing neural network architectures that can efficiently learn from data with underlying symmetries. These architectures ensure that the learned functions maintain a certain geometric property called group equivariance, which determines how the output changes based on a change to the input under the action of a symmetry group. In this talk, we will describe a number of new group equivariant neural network architectures that are built using tensor power spaces of $R^n$ as their layers. We will show that the learnable, linear functions between these layers can be characterised by certain subsets of set partition diagrams. This talk will be based on several papers that are to appear in ICML 2023.