# Quaternion Equivariant Capsule Networks for 3D Point Clouds

arXiv: https://arxiv.org/abs/1912.12098 · HTML: https://mlsanity.com/paper/1912.12098/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2019.

## Built on by (most cited first, 30 of 41)

- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptors](https://mlsanity.com/paper/2109.00182/) (2021)
- [Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimation](https://mlsanity.com/paper/2111.00190/) (2021)
- [Equivariant Point Cloud Analysis via Learning Orientations for Message Passing](https://mlsanity.com/paper/2203.14486/) (2022)
- [Projective Manifold Gradient Layer for Deep Rotation Regression](https://mlsanity.com/paper/2110.11657/) (2021)
- [Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization](https://mlsanity.com/paper/2111.12878/) (2021)
- [RIConv++: Effective Rotation Invariant Convolutions for 3D Point Clouds Deep Learning](https://mlsanity.com/paper/2202.13094/) (2022)
- [Leveraging SE(3) Equivariance for Learning 3D Geometric Shape Assembly](https://mlsanity.com/paper/2309.06810/) (2023)
- [Self-Supervised Category-Level Articulated Object Pose Estimation with Part-Level SE(3) Equivariance](https://mlsanity.com/paper/2302.14268/) (2023)
- [Towards Efficient Graph Convolutional Networks for Point Cloud Handling](https://mlsanity.com/paper/2104.05706/) (2021)
- [E2PN: Efficient SE(3)-Equivariant Point Network](https://mlsanity.com/paper/2206.05398/) (2022)
- [Unsupervised Motion Representation Learning with Capsule Autoencoders](https://mlsanity.com/paper/2110.00529/) (2021)
- [PRIN/SPRIN: On Extracting Point-wise Rotation Invariant Features](https://mlsanity.com/paper/2102.12093/) (2021)
- [Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing](https://mlsanity.com/paper/2405.15389/) (2024)
- [Field Convolutions for Surface CNNs](https://mlsanity.com/paper/2104.03916/) (2021)
- [ART-Point: Improving Rotation Robustness of Point Cloud Classifiers via Adversarial Rotation](https://mlsanity.com/paper/2203.03888/) (2022)
- [Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant](https://mlsanity.com/paper/2505.20280/) (2025)
- [ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation](https://mlsanity.com/paper/2502.14637/) (2025)
- [Learning with Capsules: A Survey](https://mlsanity.com/paper/2206.02664/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [Deformation Robust Roto-Scale-Translation Equivariant CNNs](https://mlsanity.com/paper/2111.10978/) (2021)
- [PaRot: Patch-Wise Rotation-Invariant Network via Feature Disentanglement and Pose Restoration](https://mlsanity.com/paper/2302.02535/) (2023)
- [End-to-end Learning the Partial Permutation Matrix for Robust 3D Point Cloud Registration](https://mlsanity.com/paper/2110.15250/) (2021)
- [A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks](https://mlsanity.com/paper/2410.04692/) (2024)
- [3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction](https://mlsanity.com/paper/2411.00543/) (2024)
- [Steerable 3D Spherical Neurons](https://mlsanity.com/paper/2106.13863/) (2021)
- [Learning a Task-specific Descriptor for Robust Matching of 3D Point Clouds](https://mlsanity.com/paper/2210.14899/) (2022)
- [Stable and Consistent Prediction of 3D Characteristic Orientation via Invariant Residual Learning](https://mlsanity.com/paper/2306.11406/) (2023)

## Abstract (arXiv)

We present a 3D capsule module for processing point clouds that is equivariant to 3D rotations and translations, as well as invariant to permutations of the input points. The operator receives a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end transformation equivariance through a novel dynamic routing procedure on quaternions. Further, we theoretically connect dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving iterative re-weighted least squares (IRLS) problems with provable convergence properties. It is shown that such group dynamic routing can be interpreted as robust IRLS rotation averaging on capsule votes, where information is routed based on the final inlier scores. Based on our operator, we build a capsule network that disentangles geometry from pose, paving the way for more informative descriptors and a structured latent space. Our architecture allows joint object classification and orientation estimation without explicit supervision of rotations. We validate our algorithm empirically on common benchmark datasets.