# Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space

arXiv: https://arxiv.org/abs/2310.02970 · HTML: https://mlsanity.com/paper/2310.02970/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 86 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 30 of 63)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)

## Built on by (most cited first, 30 of 41)

- [The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domains](https://mlsanity.com/paper/2410.24169/) (2024)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)
- [Space-Time Continuous PDE Forecasting using Equivariant Neural Fields](https://mlsanity.com/paper/2406.06660/) (2024)
- [Probing Equivariance and Symmetry Breaking in Convolutional Networks](https://mlsanity.com/paper/2501.01999/) (2025)
- [Geometry-aware RL for Manipulation of Varying Shapes and Deformable Objects](https://mlsanity.com/paper/2502.07005/) (2025)
- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [Flopping for FLOPs: Leveraging equivariance for computational efficiency](https://mlsanity.com/paper/2502.05169/) (2025)
- [STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulation](https://mlsanity.com/paper/2505.18647/) (2025)
- [SE(3)-Hyena Operator for Scalable Equivariant Learning](https://mlsanity.com/paper/2407.01049/) (2024)
- [Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networks](https://mlsanity.com/paper/2406.04052/) (2024)
- [A Cosmic-Scale Benchmark for Symmetry-Preserving Data Processing](https://mlsanity.com/paper/2410.20516/) (2024)
- [Learning Symmetries via Weight-Sharing with Doubly Stochastic Tensors](https://mlsanity.com/paper/2412.04594/) (2024)
- [Zatom-1: Towards a Multimodal Foundation Model for 3D Molecules and Materials](https://mlsanity.com/paper/2602.22251/) (2026)
- [Geometric Hyena Networks for Large-scale Equivariant Learning](https://mlsanity.com/paper/2505.22560/) (2025)
- [E(n) Equivariant Message Passing Cellular Networks](https://mlsanity.com/paper/2406.03145/) (2024)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [Quick ViTs: Speeding up Vision Transformers through Equivariance](https://mlsanity.com/paper/2505.15441/) (2025)
- [A Circular Argument : Does RoPE need to be Equivariant for Vision?](https://mlsanity.com/paper/2511.08368/) (2025)
- [The NGT200 Dataset: Geometric Multi-View Isolated Sign Recognition](https://mlsanity.com/paper/2409.15284/) (2024)
- [AdS-GNN -- a Conformally Equivariant Graph Neural Network](https://mlsanity.com/paper/2505.12880/) (2025)
- [On the Fourier analysis in the SO(3) space : EquiLoPO Network](https://mlsanity.com/paper/2404.15979/) (2024)
- [Harmformer: Harmonic Networks Meet Transformers for Continuous Roto-Translation Equivariance](https://mlsanity.com/paper/2411.03794/) (2024)
- [Equivariant spatio-hemispherical networks for diffusion MRI deconvolution](https://mlsanity.com/paper/2411.11819/) (2024)
- [Symmetries-enhanced Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2501.01136/) (2025)
- [Universal Collection of Euclidean Invariants between Pairs of Position-Orientations](https://mlsanity.com/paper/2504.03299/) (2025)
- [EqCollide: Equivariant and Collision-Aware Deformable Objects Neural Simulator](https://mlsanity.com/paper/2506.05797/) (2025)
- [On Equivariant Model Selection through the Lens of Uncertainty](https://mlsanity.com/paper/2506.18629/) (2025)

## Abstract (arXiv)

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions ℝ3, position and orientations ℝ3 × S2, and the group SE(3) itself. Among these, ℝ3 × S2 is an optimal choice due to the ability to represent directional information, which ℝ3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.