# Efficient Generalized Spherical CNNs

arXiv: https://arxiv.org/abs/2010.11661 · HTML: https://mlsanity.com/paper/2010.11661/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2020.

## Built on by (most cited first, 29 of 29)

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [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)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [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)
- [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [Attention on the Sphere](https://mlsanity.com/paper/2505.11157/) (2025)
- [In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?](https://mlsanity.com/paper/2210.03773/) (2022)
- [How can spherical CNNs benefit ML-based diffusion MRI parameter estimation?](https://mlsanity.com/paper/2207.00572/) (2022)
- [Differentiable and accelerated spherical harmonic and Wigner transforms](https://mlsanity.com/paper/2311.14670/) (2023)
- [SE(3) Equivariant Ray Embeddings for Implicit Multi-View Depth Estimation](https://mlsanity.com/paper/2411.07326/) (2024)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)
- [Frequency and Generalisation of Periodic Activation Functions in Reinforcement Learning](https://mlsanity.com/paper/2407.06756/) (2024)
- [3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction](https://mlsanity.com/paper/2411.00543/) (2024)
- [Direction of Arrival Estimation of Sound Sources Using Icosahedral CNNs](https://mlsanity.com/paper/2203.16940/) (2022)
- [Holographic-(V)AE: an end-to-end SO(3)-Equivariant (Variational) Autoencoder in Fourier Space](https://mlsanity.com/paper/2209.15567/) (2022)
- [A Feedforward Unitary Equivariant Neural Network](https://mlsanity.com/paper/2208.12146/) (2022)
- [A Unified Learning Model for Estimating Fiber Orientation Distribution Functions on Heterogeneous Multi-shell Diffusion-weighted MRI](https://mlsanity.com/paper/2303.16376/) (2023)
- [Higher Order Gauge Equivariant CNNs on Riemannian Manifolds and Applications](https://mlsanity.com/paper/2305.16657/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Hierarchical Spherical CNNs with Lifting-based Adaptive Wavelets for Pooling and Unpooling](https://mlsanity.com/paper/2205.15571/) (2022)
- [Spherical Feature Pyramid Networks For Semantic Segmentation](https://mlsanity.com/paper/2307.02658/) (2023)
- [Neural Operators for Biomedical Spherical Heterogeneity](https://mlsanity.com/paper/2601.03561/) (2026)
- [A library for differentiable signal processing and machine learning on the sphere](https://mlsanity.com/paper/2609.39737/) (2026)

## Abstract (arXiv)

Many problems across computer vision and the natural sciences require the analysis of spherical data, for which representations may be learned efficiently by encoding equivariance to rotational symmetries. We present a generalized spherical CNN framework that encompasses various existing approaches and allows them to be leveraged alongside each other. The only existing non-linear spherical CNN layer that is strictly equivariant has complexity O(C2L5), where C is a measure of representational capacity and L the spherical harmonic bandlimit. Such a high computational cost often prohibits the use of strictly equivariant spherical CNNs. We develop two new strictly equivariant layers with reduced complexity O(CL4) and O(CL3 log L), making larger, more expressive models computationally feasible. Moreover, we adopt efficient sampling theory to achieve further computational savings. We show that these developments allow the construction of more expressive hybrid models that achieve state-of-the-art accuracy and parameter efficiency on spherical benchmark problems.