# Equivariance versus Augmentation for Spherical Images

arXiv: https://arxiv.org/abs/2202.03990 · HTML: https://mlsanity.com/paper/2202.03990/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 26 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 21 of 21)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [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)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Orientation-aware Semantic Segmentation on Icosahedron Spheres](https://mlsanity.com/paper/1907.12849/) (2019)
- [Spin-Weighted Spherical CNNs](https://mlsanity.com/paper/2006.10731/) (2020)
- [Efficient Generalized Spherical CNNs](https://mlsanity.com/paper/2010.11661/) (2020)
- [Theoretical Aspects of Group Equivariant Neural Networks](https://mlsanity.com/paper/2004.05154/) (2020)
- [Surface Networks via General Covers](https://mlsanity.com/paper/1812.10705/) (2018)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [Rotation-Equivariant Deep Learning for Diffusion MRI](https://mlsanity.com/paper/2102.06942/) (2021)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Convolutions on Spherical Images](https://mlsanity.com/paper/1905.08409/) (2019)
- [Sample Efficient Semantic Segmentation using Rotation Equivariant Convolutional Networks](https://mlsanity.com/paper/1807.00583/) (2018)
- [Training or Architecture? How to Incorporate Invariance in Neural Networks](https://mlsanity.com/paper/2106.10044/) (2021)
- [Spherical Transformer: Adapting Spherical Signal to CNNs](https://mlsanity.com/paper/2101.03848/) (2021)

## Built on by (most cited first, 24 of 24)

- [FAENet: Frame Averaging Equivariant GNN for Materials Modeling](https://mlsanity.com/paper/2305.05577/) (2023)
- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [Optimization Dynamics of Equivariant and Augmented Neural Networks](https://mlsanity.com/paper/2303.13458/) (2023)
- [Emergent Equivariance in Deep Ensembles](https://mlsanity.com/paper/2403.03103/) (2024)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Almost Equivariance via Lie Algebra Convolutions](https://mlsanity.com/paper/2310.13164/) (2023)
- [Ensembles provably learn equivariance through data augmentation](https://mlsanity.com/paper/2410.01452/) (2024)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Equivariant Neural Tangent Kernels](https://mlsanity.com/paper/2406.06504/) (2024)
- [Learning Polynomial Problems with SL(2,ℝ) Equivariance](https://mlsanity.com/paper/2312.02146/) (2023)
- [SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres](https://mlsanity.com/paper/2503.09456/) (2025)
- [Do we need equivariant models for molecule generation?](https://mlsanity.com/paper/2507.09753/) (2025)
- [SEAL - A Symmetry EncourAging Loss for High Energy Physics](https://mlsanity.com/paper/2511.01982/) (2025)
- [Training Dynamics of Learning 3D-Rotational Equivariance](https://mlsanity.com/paper/2512.02303/) (2025)
- [Near Optimal Reconstruction of Spherical Harmonic Expansions](https://mlsanity.com/paper/2202.12995/) (2022)
- [Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold](https://mlsanity.com/paper/2307.12517/) (2023)
- [Learning Coordinate-based Convolutional Kernels for Continuous SE(3) Equivariant and Efficient Point Cloud Analysis](https://mlsanity.com/paper/2603.17538/) (2026)
- [On the Equivariant Learning of the Q-tensor Order Parameter](https://mlsanity.com/paper/2605.27679/) (2026)
- [Equivariance and Augmentation for Bayesian Neural Networks](https://mlsanity.com/paper/2606.26273/) (2026)
- [Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform](https://mlsanity.com/paper/2607.21271/) (2026)
- [Boosting Data Augmentation with Stochastic Weight Averaging](https://mlsanity.com/paper/2608.14373/) (2026)
- [EMERGE: Resolution-Agnostic Point Cloud Generation with Equivariant Graph-Based Diffusion](https://mlsanity.com/paper/2609.26039/) (2026)

## Abstract (arXiv)

We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems. The equivariant spherical networks used in the experiments are available at https://github.com/JanEGerken/sem_seg_s2cnn .