# Clebsch-Gordan Transformer: Fast and Global Equivariant Attention

arXiv: https://arxiv.org/abs/2509.24093 · HTML: https://mlsanity.com/paper/2509.24093/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 14 references that are in the map.
- First posted on arXiv 28 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Demystify Mamba in Vision: A Linear Attention Perspective](https://mlsanity.com/paper/2405.16605/) (2024)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [Graph Networks with Spectral Message Passing](https://mlsanity.com/paper/2101.00079/) (2020)
- [SE(3)-Hyena Operator for Scalable Equivariant Learning](https://mlsanity.com/paper/2407.01049/) (2024)

## Built on by (most cited first, 1 of 1)

- [The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups](https://mlsanity.com/paper/2606.20547/) (2026)

## Abstract (arXiv)

The global attention mechanism is one of the keys to the success of transformer architecture, but it incurs quadratic computational costs in relation to the number of tokens. On the other hand, equivariant models, which leverage the underlying geometric structures of problem instance, often achieve superior accuracy in physical, biochemical, computer vision, and robotic tasks, at the cost of additional compute requirements. As a result, existing equivariant transformers only support low-order equivariant features and local context windows, limiting their expressiveness and performance. This work proposes Clebsch-Gordan Transformer, achieving efficient global attention by a novel Clebsch-Gordon Convolution on SO(3) irreducible representations. Our method enables equivariant modeling of features at all orders while achieving O(N log N) input token complexity. Additionally, the proposed method scales well with high-order irreducible features, by exploiting the sparsity of the Clebsch-Gordon matrix. Lastly, we also incorporate optional token permutation equivariance through either weight sharing or data augmentation. We benchmark our method on a diverse set of benchmarks including n-body simulation, QM9, ModelNet point cloud classification and a robotic grasping dataset, showing clear gains over existing equivariant transformers in GPU memory size, speed, and accuracy.