# Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions

arXiv: https://arxiv.org/abs/2209.13603 · HTML: https://mlsanity.com/paper/2209.13603/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 35 references that are in the map.
- First posted on arXiv 27 Sep 2022.

## Builds on (most cited first, 24 of 24)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Matterport3D: Learning from RGB-D Data in Indoor Environments](https://mlsanity.com/paper/1709.06158/) (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)
- [Simple Copy-Paste is a Strong Data Augmentation Method for Instance Segmentation](https://mlsanity.com/paper/2012.07177/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Joint 2D-3D-Semantic Data for Indoor Scene Understanding](https://mlsanity.com/paper/1702.01105/) (2017)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [HoHoNet: 360 Indoor Holistic Understanding with Latent Horizontal Features](https://mlsanity.com/paper/2011.11498/) (2020)
- [Spherical CNNs on Unstructured Grids](https://mlsanity.com/paper/1901.02039/) (2019)
- [Tangent Images for Mitigating Spherical Distortion](https://mlsanity.com/paper/1912.09390/) (2019)
- [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications](https://mlsanity.com/paper/1810.12186/) (2018)
- [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)
- [Pano3D: A Holistic Benchmark and a Solid Baseline for 360o Depth Estimation](https://mlsanity.com/paper/2109.02749/) (2021)
- [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)
- [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)
- [M"obius Convolutions for Spherical CNNs](https://mlsanity.com/paper/2201.12212/) (2022)

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

- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [A Survey of Representation Learning, Optimization Strategies, and Applications for Omnidirectional Vision](https://mlsanity.com/paper/2502.10444/) (2025)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [Attention on the Sphere](https://mlsanity.com/paper/2505.11157/) (2025)
- [A Unified Model for Compressed Sensing MRI Across Undersampling Patterns](https://mlsanity.com/paper/2410.16290/) (2024)
- [Differentiable and accelerated spherical harmonic and Wigner transforms](https://mlsanity.com/paper/2311.14670/) (2023)
- [Elite360M: Efficient 360 Multi-task Learning via Bi-projection Fusion and Cross-task Collaboration](https://mlsanity.com/paper/2408.09336/) (2024)
- [Physics-Aware Neural Operators for Direct Inversion in 3D Photoacoustic Tomography](https://mlsanity.com/paper/2509.09894/) (2025)
- [Generalized Spherical Neural Operators: Green's Function Formulation](https://mlsanity.com/paper/2512.10723/) (2025)
- [Examining Fast Radiatively Driven Responses Using Machine-Learning Weather Emulators](https://mlsanity.com/paper/2602.16090/) (2026)
- [E(3) × SO(3)-Equivariant Networks for Spherical Deconvolution in Diffusion MRI](https://mlsanity.com/paper/2304.06103/) (2023)
- [Unified Spherical Frontend: Learning Rotation-Equivariant Representations of Spherical Images from Any Camera](https://mlsanity.com/paper/2511.18174/) (2025)
- [Rotation-Invariant Spherical Watermarking via Third-Order SO(3) Representation Coupling](https://mlsanity.com/paper/2605.26702/) (2026)
- [Panoramic Scene Understanding: A Survey from Distortion-Aware Engineering to Sphere-Native Modeling](https://mlsanity.com/paper/2606.27745/) (2026)
- [S-R2D2: a spherical extension of the R2D2 deep neural network series paradigm for wide-field radio-interferometric imaging](https://mlsanity.com/paper/2503.01462/) (2025)
- [Resolution-Agnostic Neural Operators for Multi-Rate Sparse-View CT](https://mlsanity.com/paper/2512.12236/) (2025)
- [A neural operator view on U-Nets for inverse imaging problems](https://mlsanity.com/paper/2608.05839/) (2026)
- [Dandelion: A Spherical Flower for Neural Simulation of Planetary Dynamics](https://mlsanity.com/paper/2608.27521/) (2026)
- [SONAR: A Structure-Consistent Neural Operator for Null-Space-Aware Sparse View CT Reconstruction](https://mlsanity.com/paper/2609.13688/) (2026)
- [A library for differentiable signal processing and machine learning on the sphere](https://mlsanity.com/paper/2609.39737/) (2026)
- [SO(3)-RoPE for Spherical Transformers](https://mlsanity.com/paper/2610.06229/) (2026)

## Abstract (arXiv)

No existing spherical convolutional neural network (CNN) framework is both computationally scalable and rotationally equivariant. Continuous approaches capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO) group convolution that is simultaneously equivariant and computationally scalable to high-resolution. While our framework can be applied to any compact group, we specialize to the sphere. Our DISCO spherical convolutions exhibit SO(3) rotational equivariance, where SO(n) is the special orthogonal group representing rotations in n-dimensions. When restricting rotations of the convolution to the quotient space SO(3)/SO(2) for further computational enhancements, we recover a form of asymptotic SO(3) rotational equivariance. Through a sparse tensor implementation we achieve linear scaling in number of pixels on the sphere for both computational cost and memory usage. For 4k spherical images we realize a saving of 109 in computational cost and 104 in memory usage when compared to the most efficient alternative equivariant spherical convolution. We apply the DISCO spherical CNN framework to a number of benchmark dense-prediction problems on the sphere, such as semantic segmentation and depth estimation, on all of which we achieve the state-of-the-art performance.