# Probing Equivariance and Symmetry Breaking in Convolutional Networks

arXiv: https://arxiv.org/abs/2501.01999 · HTML: https://mlsanity.com/paper/2501.01999/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 67 references that are in the map.
- First posted on arXiv 1 Jan 2025.

## Builds on (most cited first, 30 of 56)

- [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)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [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)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (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)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [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)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [3D Shape Generation and Completion through Point-Voxel Diffusion](https://mlsanity.com/paper/2104.03670/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [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)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)

## Built on by (most cited first, 12 of 12)

- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)
- [Controlled Generation with Equivariant Variational Flow Matching](https://mlsanity.com/paper/2506.18340/) (2025)
- [Transformers Discover Molecular Structure Without Graph Priors](https://mlsanity.com/paper/2510.02259/) (2025)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [ToolMol: Evolutionary Agentic Framework for Multi-objective Drug Discovery](https://mlsanity.com/paper/2605.12784/) (2026)
- [AdS-GNN -- a Conformally Equivariant Graph Neural Network](https://mlsanity.com/paper/2505.12880/) (2025)
- [Learning Generalizable Shape Completion with SIM(3) Equivariance](https://mlsanity.com/paper/2509.26631/) (2025)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields](https://mlsanity.com/paper/2603.08758/) (2026)
- [Discretizing Group-Convolutional Neural Networks for 3D Geometry in Feature Space](https://mlsanity.com/paper/2605.15368/) (2026)
- [When Does Equivariance Help? Canonical Alignment in Neural Fluid Surrogates](https://mlsanity.com/paper/2605.18816/) (2026)
- [Sample Complexity of Equivariant Reinforcement Learning](https://mlsanity.com/paper/2609.36421/) (2026)

## Abstract (arXiv)

In this work, we explore the trade-offs of explicit structural priors, particularly group equivariance. We address this through theoretical analysis and a comprehensive empirical study. To enable controlled and fair comparisons, we introduce textttRapidash, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit textitsymmetry breaking via geometric reference frames consistently improves performance, while textitbreaking equivariance through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection.