# TetraSphere: A Neural Descriptor for O(3)-Invariant Point Cloud Analysis

arXiv: https://arxiv.org/abs/2211.14456 · HTML: https://mlsanity.com/paper/2211.14456/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 54 references that are in the map.
- First posted on arXiv 26 Nov 2022.

## Builds on (most cited first, 30 of 37)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Revisiting Point Cloud Classification: A New Benchmark Dataset and Classification Model on Real-World Data](https://mlsanity.com/paper/1908.04616/) (2019)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Walk in the Cloud: Learning Curves for Point Clouds Shape Analysis](https://mlsanity.com/paper/2105.01288/) (2021)
- [Revisiting Point Cloud Shape Classification with a Simple and Effective Baseline](https://mlsanity.com/paper/2106.05304/) (2021)
- [Image-to-Lidar Self-Supervised Distillation for Autonomous Driving Data](https://mlsanity.com/paper/2203.16258/) (2022)
- [Deep Projective 3D Semantic Segmentation](https://mlsanity.com/paper/1705.03428/) (2017)
- [Rotation Invariant Convolutions for 3D Point Clouds Deep Learning](https://mlsanity.com/paper/1908.06297/) (2019)
- [A Rotation-Invariant Framework for Deep Point Cloud Analysis](https://mlsanity.com/paper/2003.07238/) (2020)
- [Quaternion Equivariant Capsule Networks for 3D Point Clouds](https://mlsanity.com/paper/1912.12098/) (2019)
- [Rotation-Invariant Local-to-Global Representation Learning for 3D Point Cloud](https://mlsanity.com/paper/2010.03318/) (2020)
- [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)
- [3D-Rotation-Equivariant Quaternion Neural Networks](https://mlsanity.com/paper/1911.09040/) (2019)
- [Learning to Orient Surfaces by Self-supervised Spherical CNNs](https://mlsanity.com/paper/2011.03298/) (2020)
- [Investigating the Impact of Multi-LiDAR Placement on Object Detection for Autonomous Driving](https://mlsanity.com/paper/2105.00373/) (2021)
- [The Devil is in the Pose: Ambiguity-free 3D Rotation-invariant Learning via Pose-aware Convolution](https://mlsanity.com/paper/2205.15210/) (2022)
- [Endowing Deep 3D Models with Rotation Invariance Based on Principal Component Analysis](https://mlsanity.com/paper/1910.08901/) (2019)

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

- [On Learning Deep O(n)-Equivariant Hyperspheres](https://mlsanity.com/paper/2305.15613/) (2023)
- [Towards Training-Free Open-World Classification with 3D Generative Models](https://mlsanity.com/paper/2501.17547/) (2025)

## Abstract (arXiv)

In many practical applications, 3D point cloud analysis requires rotation invariance. In this paper, we present a learnable descriptor invariant under 3D rotations and reflections, i.e., the O(3) actions, utilizing the recently introduced steerable 3D spherical neurons and vector neurons. Specifically, we propose an embedding of the 3D spherical neurons into 4D vector neurons, which leverages end-to-end training of the model. In our approach, we perform TetraTransform--an equivariant embedding of the 3D input into 4D, constructed from the steerable neurons--and extract deeper O(3)-equivariant features using vector neurons. This integration of the TetraTransform into the VN-DGCNN framework, termed TetraSphere, negligibly increases the number of parameters by less than 0.0002%. TetraSphere sets a new state-of-the-art performance classifying randomly rotated real-world object scans of the challenging subsets of ScanObjectNN. Additionally, TetraSphere outperforms all equivariant methods on randomly rotated synthetic data: classifying objects from ModelNet40 and segmenting parts of the ShapeNet shapes. Thus, our results reveal the practical value of steerable 3D spherical neurons for learning in 3D Euclidean space. The code is available at https://github.com/pavlo-melnyk/tetrasphere.