# Quaternion Product Units for Deep Learning on 3D Rotation Groups

arXiv: https://arxiv.org/abs/1912.07791 · HTML: https://mlsanity.com/paper/1912.07791/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2019.

## Built on by (most cited first, 7 of 7)

- [Consistent 3D Hand Reconstruction in Video via self-supervised Learning](https://mlsanity.com/paper/2201.09548/) (2022)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation](https://mlsanity.com/paper/2502.14637/) (2025)
- [Dual Quaternion Rotational and Translational Equivariance in 3D Rigid Motion Modelling](https://mlsanity.com/paper/2310.07623/) (2023)
- [Disentangling Geometric Deformation Spaces in Generative Latent Shape Models](https://mlsanity.com/paper/2103.00142/) (2021)
- [DifAttack++: Query-Efficient Black-Box Adversarial Attack via Hierarchical Disentangled Feature Space in Cross-Domain](https://mlsanity.com/paper/2406.03017/) (2024)
- [Design equivariant neural networks for 3D point cloud](https://mlsanity.com/paper/2205.00630/) (2022)

## Abstract (arXiv)

We propose a novel quaternion product unit (QPU) to represent data on 3D rotation groups. The QPU leverages quaternion algebra and the law of 3D rotation group, representing 3D rotation data as quaternions and merging them via a weighted chain of Hamilton products. We prove that the representations derived by the proposed QPU can be disentangled into "rotation-invariant" features and "rotation-equivariant" features, respectively, which supports the rationality and the efficiency of the QPU in theory. We design quaternion neural networks based on our QPUs and make our models compatible with existing deep learning models. Experiments on both synthetic and real-world data show that the proposed QPU is beneficial for the learning tasks requiring rotation robustness.