# Equivariant Networks for Pixelized Spheres

arXiv: https://arxiv.org/abs/2106.06662 · HTML: https://mlsanity.com/paper/2106.06662/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 60 references that are in the map.
- First posted on arXiv 12 Jun 2021.

## Builds on (most cited first, 30 of 43)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Joint 2D-3D-Semantic Data for Indoor Scene Understanding](https://mlsanity.com/paper/1702.01105/) (2017)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Spherical CNNs on Unstructured Grids](https://mlsanity.com/paper/1901.02039/) (2019)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)

## Built on by (most cited first, 16 of 16)

- [Bending Reality: Distortion-aware Transformers for Adapting to Panoramic Semantic Segmentation](https://mlsanity.com/paper/2203.01452/) (2022)
- [Behind Every Domain There is a Shift: Adapting Distortion-aware Vision Transformers for Panoramic Semantic Segmentation](https://mlsanity.com/paper/2207.11860/) (2022)
- [Deep Learning for Omnidirectional Vision: A Survey and New Perspectives](https://mlsanity.com/paper/2205.10468/) (2022)
- [Elite360D: Towards Efficient 360 Depth Estimation via Semantic- and Distance-Aware Bi-Projection Fusion](https://mlsanity.com/paper/2403.16376/) (2024)
- [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)
- [A Survey of Representation Learning, Optimization Strategies, and Applications for Omnidirectional Vision](https://mlsanity.com/paper/2502.10444/) (2025)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networks](https://mlsanity.com/paper/2210.08001/) (2022)
- [Making Vision Transformers Truly Shift-Equivariant](https://mlsanity.com/paper/2305.16316/) (2023)
- [Truly Scale-Equivariant Deep Nets with Fourier Layers](https://mlsanity.com/paper/2311.02922/) (2023)
- [M"obius Convolutions for Spherical CNNs](https://mlsanity.com/paper/2201.12212/) (2022)
- [Deep Neural Networks with Efficient Guaranteed Invariances](https://mlsanity.com/paper/2303.01567/) (2023)
- [Spherical Feature Pyramid Networks For Semantic Segmentation](https://mlsanity.com/paper/2307.02658/) (2023)

## Abstract (arXiv)

Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.