# Learning SO(3) Equivariant Representations with Spherical CNNs

arXiv: https://arxiv.org/abs/1711.06721 · HTML: https://mlsanity.com/paper/1711.06721/

## Key facts

- Cited by 248 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2017.

## Built on by (most cited first, 30 of 248)

- [Perceiver: General Perception with Iterative Attention](https://mlsanity.com/paper/2103.03206/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Self-Supervised Pretraining of 3D Features on any Point-Cloud](https://mlsanity.com/paper/2101.02691/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Model-Based Domain Generalization](https://mlsanity.com/paper/2102.11436/) (2021)
- [DualPoseNet: Category-level 6D Object Pose and Size Estimation Using Dual Pose Network with Refined Learning of Pose Consistency](https://mlsanity.com/paper/2103.06526/) (2021)
- [3D AffordanceNet: A Benchmark for Visual Object Affordance Understanding](https://mlsanity.com/paper/2103.16397/) (2021)
- [PanoFormer: Panorama Transformer for Indoor 360 Depth Estimation](https://mlsanity.com/paper/2203.09283/) (2022)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [CSP: Self-Supervised Contrastive Spatial Pre-Training for Geospatial-Visual Representations](https://mlsanity.com/paper/2305.01118/) (2023)
- [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)
- [360MonoDepth: High-Resolution 360deg Monocular Depth Estimation](https://mlsanity.com/paper/2111.15669/) (2021)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Wide-Baseline Relative Camera Pose Estimation with Directional Learning](https://mlsanity.com/paper/2106.03336/) (2021)
- [Spherical Channels for Modeling Atomic Interactions](https://mlsanity.com/paper/2206.14331/) (2022)
- [LED2-Net: Monocular 360 Layout Estimation via Differentiable Depth Rendering](https://mlsanity.com/paper/2104.00568/) (2021)
- [VI-Net: Boosting Category-level 6D Object Pose Estimation via Learning Decoupled Rotations on the Spherical Representations](https://mlsanity.com/paper/2308.09916/) (2023)
- [Equivariant Adaptation of Large Pretrained Models](https://mlsanity.com/paper/2310.01647/) (2023)
- [Deep Learning for Omnidirectional Vision: A Survey and New Perspectives](https://mlsanity.com/paper/2205.10468/) (2022)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [BiFuse++: Self-supervised and Efficient Bi-projection Fusion for 360 Depth Estimation](https://mlsanity.com/paper/2209.02952/) (2022)
- [Robust (Controlled) Table-to-Text Generation with Structure-Aware Equivariance Learning](https://mlsanity.com/paper/2205.03972/) (2022)
- [Deep Learning on Monocular Object Pose Detection and Tracking: A Comprehensive Overview](https://mlsanity.com/paper/2105.14291/) (2021)

## Abstract (arXiv)

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.