# Group Equivariant Capsule Networks

arXiv: https://arxiv.org/abs/1806.05086 · HTML: https://mlsanity.com/paper/1806.05086/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2018.

## Built on by (most cited first, 30 of 31)

- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptors](https://mlsanity.com/paper/2109.00182/) (2021)
- [Exploring Complementary Strengths of Invariant and Equivariant Representations for Few-Shot Learning](https://mlsanity.com/paper/2103.01315/) (2021)
- [Self-Contrastive Learning with Hard Negative Sampling for Self-supervised Point Cloud Learning](https://mlsanity.com/paper/2107.01886/) (2021)
- [Efficient-CapsNet: Capsule Network with Self-Attention Routing](https://mlsanity.com/paper/2101.12491/) (2021)
- [Enabling equivariance for arbitrary Lie groups](https://mlsanity.com/paper/2111.08251/) (2021)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Equivariance and Invariance Inductive Bias for Learning from Insufficient Data](https://mlsanity.com/paper/2207.12258/) (2022)
- [Capsule Network is Not More Robust than Convolutional Network](https://mlsanity.com/paper/2103.15459/) (2021)
- [Lie Group Decompositions for Equivariant Neural Networks](https://mlsanity.com/paper/2310.11366/) (2023)
- [Learning with Capsules: A Survey](https://mlsanity.com/paper/2206.02664/) (2022)
- [Unsupervised Object Representation Learning using Translation and Rotation Group Equivariant VAE](https://mlsanity.com/paper/2210.12918/) (2022)
- [Spiking CapsNet: A Spiking Neural Network With A Biologically Plausible Routing Rule Between Capsules](https://mlsanity.com/paper/2111.07785/) (2021)
- [XnODR and XnIDR: Two Accurate and Fast Fully Connected Layers For Convolutional Neural Networks](https://mlsanity.com/paper/2111.10854/) (2021)
- [Steerable Equivariant Representation Learning](https://mlsanity.com/paper/2302.11349/) (2023)
- [Noncommutative C^*-algebra Net: Learning Neural Networks with Powerful Product Structure in C^*-algebra](https://mlsanity.com/paper/2302.01191/) (2023)
- [Self-Supervised Multi-View Learning via Auto-Encoding 3D Transformations](https://mlsanity.com/paper/2103.00787/) (2021)
- [Autoequivariant Network Search via Group Decomposition](https://mlsanity.com/paper/2104.04848/) (2021)
- [Self-Supervised Graph Representation Learning via Topology Transformations](https://mlsanity.com/paper/2105.11689/) (2021)
- [3DConvCaps: 3DUnet with Convolutional Capsule Encoder for Medical Image Segmentation](https://mlsanity.com/paper/2205.09299/) (2022)
- [Topographic VAEs learn Equivariant Capsules](https://mlsanity.com/paper/2109.01394/) (2021)
- [Capsule networks with non-iterative cluster routing](https://mlsanity.com/paper/2109.09213/) (2021)
- [Transformation Coding: Simple Objectives for Equivariant Representations](https://mlsanity.com/paper/2202.10930/) (2022)
- [Capsule Network based Contrastive Learning of Unsupervised Visual Representations](https://mlsanity.com/paper/2209.11276/) (2022)
- [Robustcaps: a transformation-robust capsule network for image classification](https://mlsanity.com/paper/2210.11092/) (2022)
- [Iterative collaborative routing among equivariant capsules for transformation-robust capsule networks](https://mlsanity.com/paper/2210.11095/) (2022)
- [Learning Latent Part-Whole Hierarchies for Point Clouds](https://mlsanity.com/paper/2211.07082/) (2022)
- [Capsule network with shortcut routing](https://mlsanity.com/paper/2307.10212/) (2023)
- [The Convergence of Dynamic Routing between Capsules](https://mlsanity.com/paper/2501.06240/) (2025)

## Abstract (arXiv)

We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea. Our work can be divided into two contributions. First, we present a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as invariance of output activations, hold under certain conditions. Second, we connect the resulting equivariant capsule networks with work from the field of group convolutional networks. Through this connection, we provide intuitions of how both methods relate and are able to combine the strengths of both approaches in one deep neural network architecture. The resulting framework allows sparse evaluation of the group convolution operator, provides control over specific equivariance and invariance properties, and can use routing by agreement instead of pooling operations. In addition, it is able to provide interpretable and equivariant representation vectors as output capsules, which disentangle evidence of object existence from its pose.