# Spin-Weighted Spherical CNNs

arXiv: https://arxiv.org/abs/2006.10731 · HTML: https://mlsanity.com/paper/2006.10731/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2020.

## Built on by (most cited first, 30 of 50)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptors](https://mlsanity.com/paper/2109.00182/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [SGAT4PASS: Spherical Geometry-Aware Transformer for PAnoramic Semantic Segmentation](https://mlsanity.com/paper/2306.03403/) (2023)
- [Equivariance versus Augmentation for Spherical Images](https://mlsanity.com/paper/2202.03990/) (2022)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021)
- [360BEV: Panoramic Semantic Mapping for Indoor Bird's-Eye View](https://mlsanity.com/paper/2303.11910/) (2023)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Group Equivariant Subsampling](https://mlsanity.com/paper/2106.05886/) (2021)
- [M"obius Convolutions for Spherical CNNs](https://mlsanity.com/paper/2201.12212/) (2022)
- [Complementary Bi-directional Feature Compression for Indoor 360deg Semantic Segmentation with Self-distillation](https://mlsanity.com/paper/2207.02437/) (2022)
- [Attention on the Sphere](https://mlsanity.com/paper/2505.11157/) (2025)
- [EqNIO: Subequivariant Neural Inertial Odometry](https://mlsanity.com/paper/2408.06321/) (2024)
- [OSLO: On-the-Sphere Learning for Omnidirectional images and its application to 360-degree image compression](https://mlsanity.com/paper/2107.09179/) (2021)
- [Equivariant Wavelets: Fast Rotation and Translation Invariant Wavelet Scattering Transforms](https://mlsanity.com/paper/2104.11244/) (2021)
- [Neural Isometries: Taming Transformations for Equivariant ML](https://mlsanity.com/paper/2405.19296/) (2024)
- [SE(3) Equivariant Ray Embeddings for Implicit Multi-View Depth Estimation](https://mlsanity.com/paper/2411.07326/) (2024)
- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [Inferring dark matter substructure with astrometric lensing beyond the power spectrum](https://mlsanity.com/paper/2110.01620/) (2021)

## Abstract (arXiv)

Learning equivariant representations is a promising way to reduce sample and model complexity and improve the generalization performance of deep neural networks. The spherical CNNs are successful examples, producing SO(3)-equivariant representations of spherical inputs. There are two main types of spherical CNNs. The first type lifts the inputs to functions on the rotation group SO(3) and applies convolutions on the group, which are computationally expensive since SO(3) has one extra dimension. The second type applies convolutions directly on the sphere, which are limited to zonal (isotropic) filters, and thus have limited expressivity. In this paper, we present a new type of spherical CNN that allows anisotropic filters in an efficient way, without ever leaving the spherical domain. The key idea is to consider spin-weighted spherical functions, which were introduced in physics in the study of gravitational waves. These are complex-valued functions on the sphere whose phases change upon rotation. We define a convolution between spin-weighted functions and build a CNN based on it. The spin-weighted functions can also be interpreted as spherical vector fields, allowing applications to tasks where the inputs or outputs are vector fields. Experiments show that our method outperforms previous methods on tasks like classification of spherical images, classification of 3D shapes and semantic segmentation of spherical panoramas.