# Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2606.02758 · HTML: https://mlsanity.com/paper/2606.02758/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Categorical Equivariant Deep Learning: Category-Equivariant Neural Networks and Universal Approximation Theorems](https://mlsanity.com/paper/2511.18417/) (2025)
- [Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories](https://mlsanity.com/paper/2604.20797/) (2026)
- [The Role of Fibration Symmetries in Geometric Deep Learning](https://mlsanity.com/paper/2408.15894/) (2024)
- [Graph Neural Networks in the Wilson Loop Representation of Abelian Lattice Gauge Theories](https://mlsanity.com/paper/2605.03901/) (2026)
- [Second-order variational problems on Lie groupoids and optimal control applications](https://mlsanity.com/paper/1506.08580/) (2015)
- [Group Invariant Global Pooling](https://mlsanity.com/paper/2305.19207/) (2023)

## Built on by (most cited first, 1 of 1)

- [Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains](https://mlsanity.com/paper/2609.25987/) (2026)

## Abstract (arXiv)

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.