# Steerable 3D Spherical Neurons

arXiv: https://arxiv.org/abs/2106.13863 · HTML: https://mlsanity.com/paper/2106.13863/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 2 Jun 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (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)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Quaternion Equivariant Capsule Networks for 3D Point Clouds](https://mlsanity.com/paper/1912.12098/) (2019)
- [Embed Me If You Can: A Geometric Perceptron](https://mlsanity.com/paper/2006.06507/) (2020)

## Built on by (most cited first, 3 of 3)

- [TetraSphere: A Neural Descriptor for O(3)-Invariant Point Cloud Analysis](https://mlsanity.com/paper/2211.14456/) (2022)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Learning Positive-Incentive Point Sampling in Neural Implicit Fields for Object Pose Estimation](https://mlsanity.com/paper/2602.19937/) (2026)

## Abstract (arXiv)

Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of neurons with spherical decision surfaces and operates on point clouds. Such spherical neurons are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Focusing on 3D geometry, we exploit the isometry property of spherical neurons and derive a 3D steerability constraint. After training spherical neurons to classify point clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate the neurons and construct rotation-equivariant spherical filter banks. We then apply the derived constraint to interpolate the filter bank outputs and, thus, obtain a rotation-invariant network. Finally, we use a synthetic point set and real-world 3D skeleton data to verify our theoretical findings. The code is available at https://github.com/pavlo-melnyk/steerable-3d-neurons.