# Capsule Network Projectors are Equivariant and Invariant Learners

arXiv: https://arxiv.org/abs/2405.14386 · HTML: https://mlsanity.com/paper/2405.14386/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 40 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Objaverse: A Universe of Annotated 3D Objects](https://mlsanity.com/paper/2212.08051/) (2022)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Kubric: A scalable dataset generator](https://mlsanity.com/paper/2203.03570/) (2022)
- [Masked Siamese Networks for Label-Efficient Learning](https://mlsanity.com/paper/2204.07141/) (2022)
- [On Feature Decorrelation in Self-Supervised Learning](https://mlsanity.com/paper/2105.00470/) (2021)
- [Semi-Supervised Learning of Visual Features by Non-Parametrically Predicting View Assignments with Support Samples](https://mlsanity.com/paper/2104.13963/) (2021)
- [Stacked Capsule Autoencoders](https://mlsanity.com/paper/1906.06818/) (2019)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Improving Transferability of Representations via Augmentation-Aware Self-Supervision](https://mlsanity.com/paper/2111.09613/) (2021)
- [Self-supervised learning of Split Invariant Equivariant representations](https://mlsanity.com/paper/2302.10283/) (2023)
- [Efficient-CapsNet: Capsule Network with Self-Attention Routing](https://mlsanity.com/paper/2101.12491/) (2021)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [Capsule Routing via Variational Bayes](https://mlsanity.com/paper/1905.11455/) (2019)
- [EquiMod: An Equivariance Module to Improve Self-Supervised Learning](https://mlsanity.com/paper/2211.01244/) (2022)
- [Hierarchical Graph Capsule Network](https://mlsanity.com/paper/2012.08734/) (2020)
- [Learning with Capsules: A Survey](https://mlsanity.com/paper/2206.02664/) (2022)
- [Pose-Aware Self-Supervised Learning with Viewpoint Trajectory Regularization](https://mlsanity.com/paper/2403.14973/) (2024)
- [Why Capsule Neural Networks Do Not Scale: Challenging the Dynamic Parse-Tree Assumption](https://mlsanity.com/paper/2301.01583/) (2023)
- [ProtoCaps: A Fast and Non-Iterative Capsule Network Routing Method](https://mlsanity.com/paper/2307.09944/) (2023)
- [Pushing the Limits of Capsule Networks](https://mlsanity.com/paper/2103.08074/) (2021)
- [Masked Capsule Autoencoders](https://mlsanity.com/paper/2403.04724/) (2024)

## Built on by (most cited first, 2 of 2)

- [Contrastive Learning Via Equivariant Representation](https://mlsanity.com/paper/2406.00262/) (2024)
- [EquiCaps: Predictor-Free Pose-Aware Pre-Trained Capsule Networks](https://mlsanity.com/paper/2506.09895/) (2025)

## Abstract (arXiv)

Learning invariant representations has been the long-standing approach to self-supervised learning. However, recently progress has been made in preserving equivariant properties in representations, yet do so with highly prescribed architectures. In this work, we propose an invariant-equivariant self-supervised architecture that employs Capsule Networks (CapsNets), which have been shown to capture equivariance with respect to novel viewpoints. We demonstrate that the use of CapsNets in equivariant self-supervised architectures achieves improved downstream performance on equivariant tasks with higher efficiency and fewer network parameters. To accommodate the architectural changes of CapsNets, we introduce a new objective function based on entropy minimisation. This approach, which we name CapsIE (Capsule Invariant Equivariant Network), achieves state-of-the-art performance on the equivariant rotation tasks on the 3DIEBench dataset compared to prior equivariant SSL methods, while performing competitively against supervised counterparts. Our results demonstrate the ability of CapsNets to learn complex and generalised representations for large-scale, multi-task datasets compared to previous CapsNet benchmarks. Code is available at https://github.com/AberdeenML/CapsIE.