# Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning

arXiv: https://arxiv.org/abs/2409.07327 · HTML: https://mlsanity.com/paper/2409.07327/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 179 of its 217 references that are in the map.
- First posted on arXiv 11 Sep 2024.

## Builds on (most cited first, 30 of 179)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [KPConv: Flexible and Deformable Convolution for Point Clouds](https://mlsanity.com/paper/1904.08889/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [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)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [OctNet: Learning Deep 3D Representations at High Resolutions](https://mlsanity.com/paper/1611.05009/) (2016)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)

## Built on by (most cited first, 2 of 2)

- [Bridging Symmetry and Robustness: On the Role of Equivariance in Enhancing Adversarial Robustness](https://mlsanity.com/paper/2510.16171/) (2025)
- [GMR-Conv: An Efficient Rotation and Reflection Equivariant Convolution Kernel Using Gaussian Mixture Rings](https://mlsanity.com/paper/2504.02819/) (2025)

## Abstract (arXiv)

Euclidean deep learning is often inadequate for addressing real-world signals where the representation space is irregular and curved with complex topologies. Interpreting the geometric properties of such feature spaces has become paramount in obtaining robust and compact feature representations that remain unaffected by nontrivial geometric transformations, which vanilla CNNs cannot effectively handle. Recognizing rotation, translation, permutation, or scale symmetries can lead to equivariance properties in the learned representations. This has led to notable advancements in computer vision and machine learning tasks under the framework of geometric deep learning, as compared to their invariant counterparts. In this report, we emphasize the importance of symmetry group equivariant deep learning models and their realization of convolution-like operations on graphs, 3D shapes, and non-Euclidean spaces by leveraging group theory and symmetry. We categorize them as regular, steerable, and PDE-based convolutions and thoroughly examine the inherent symmetries of their input spaces and ensuing representations. We also outline the mathematical link between group convolutions or message aggregation operations and the concept of equivariance. The report also highlights various datasets, their application scopes, limitations, and insightful observations on future directions to serve as a valuable reference and stimulate further research in this emerging discipline.