# Homogeneous vector bundles and G-equivariant convolutional neural networks

arXiv: https://arxiv.org/abs/2105.05400 · HTML: https://mlsanity.com/paper/2105.05400/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 56 references that are in the map.
- First posted on arXiv 12 May 2021.

## Builds on (most cited first, 27 of 27)

- [Scaling Up Visual and Vision-Language Representation Learning With Noisy Text Supervision](https://mlsanity.com/paper/2102.05918/) (2021)
- [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)
- [EfficientNetV2: Smaller Models and Faster Training](https://mlsanity.com/paper/2104.00298/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [CvT: Introducing Convolutions to Vision Transformers](https://mlsanity.com/paper/2103.15808/) (2021)
- [An Introduction to Convolutional Neural Networks](https://mlsanity.com/paper/1511.08458/) (2015)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Large Kernel Matters -- Improve Semantic Segmentation by Global Convolutional Network](https://mlsanity.com/paper/1703.02719/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [VNect: Real-time 3D Human Pose Estimation with a Single RGB Camera](https://mlsanity.com/paper/1705.01583/) (2017)
- [Fully Convolutional Adaptation Networks for Semantic Segmentation](https://mlsanity.com/paper/1804.08286/) (2018)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Gauge equivariant neural networks for quantum lattice gauge theories](https://mlsanity.com/paper/2012.05232/) (2020)
- [Theoretical Aspects of Group Equivariant Neural Networks](https://mlsanity.com/paper/2004.05154/) (2020)
- [Covariance in Physics and Convolutional Neural Networks](https://mlsanity.com/paper/1906.02481/) (2019)
- [Discrete Rotation Equivariance for Point Cloud Recognition](https://mlsanity.com/paper/1904.00319/) (2019)

## Built on by (most cited first, 21 of 21)

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulation](https://mlsanity.com/paper/2310.16050/) (2023)
- [A case for using rotation invariant features in state of the art feature matchers](https://mlsanity.com/paper/2204.10144/) (2022)
- [Equivariance versus Augmentation for Spherical Images](https://mlsanity.com/paper/2202.03990/) (2022)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [Banana: Banach Fixed-Point Network for Pointcloud Segmentation with Inter-Part Equivariance](https://mlsanity.com/paper/2305.16314/) (2023)
- [Optimization Dynamics of Equivariant and Augmented Neural Networks](https://mlsanity.com/paper/2303.13458/) (2023)
- [EFEM: Equivariant Neural Field Expectation Maximization for 3D Object Segmentation Without Scene Supervision](https://mlsanity.com/paper/2303.15440/) (2023)
- [ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds](https://mlsanity.com/paper/2111.15341/) (2021)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Equivariant Light Field Convolution and Transformer](https://mlsanity.com/paper/2212.14871/) (2022)
- [Equivariant Manifold Neural ODEs and Differential Invariants](https://mlsanity.com/paper/2401.14131/) (2024)
- [Symmetries in Overparametrized Neural Networks: A Mean-Field View](https://mlsanity.com/paper/2405.19995/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres](https://mlsanity.com/paper/2503.09456/) (2025)
- [Learning Generalizable Shape Completion with SIM(3) Equivariance](https://mlsanity.com/paper/2509.26631/) (2025)
- [Learning from Frustration: Torsor CNNs on Graphs](https://mlsanity.com/paper/2510.23288/) (2025)
- [Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras](https://mlsanity.com/paper/2303.07157/) (2023)
- [In Search of Projectively Equivariant Networks](https://mlsanity.com/paper/2209.14719/) (2022)
- [Steerable Neural ODEs on Homogeneous Spaces](https://mlsanity.com/paper/2605.11133/) (2026)

## Abstract (arXiv)

G-equivariant convolutional neural networks (GCNNs) is a geometric deep learning model for data defined on a homogeneous G-space M. GCNNs are designed to respect the global symmetry in M, thereby facilitating learning. In this paper, we analyze GCNNs on homogeneous spaces M = G/K in the case of unimodular Lie groups G and compact subgroups K ≤ G. We demonstrate that homogeneous vector bundles is the natural setting for GCNNs. We also use reproducing kernel Hilbert spaces to obtain a precise criterion for expressing G-equivariant layers as convolutional layers. This criterion is then rephrased as a bandwidth criterion, leading to even stronger results for some groups.