# Learning Partial Equivariances from Data

arXiv: https://arxiv.org/abs/2110.10211 · HTML: https://mlsanity.com/paper/2110.10211/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 61 references that are in the map.
- First posted on arXiv 19 Oct 2021.

## Builds on (most cited first, 30 of 36)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Temporal Ensembling for Semi-Supervised Learning](https://mlsanity.com/paper/1610.02242/) (2016)
- [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)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Fast AutoAugment](https://mlsanity.com/paper/1905.00397/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [CKConv: Continuous Kernel Convolution For Sequential Data](https://mlsanity.com/paper/2102.02611/) (2021)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Faster AutoAugment: Learning Augmentation Strategies using Backpropagation](https://mlsanity.com/paper/1911.06987/) (2019)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Attentive Group Equivariant Convolutional Networks](https://mlsanity.com/paper/2002.03830/) (2020)
- [Group Equivariant Stand-Alone Self-Attention For Vision](https://mlsanity.com/paper/2010.00977/) (2020)

## Built on by (most cited first, 30 of 45)

- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Making Vision Transformers Truly Shift-Equivariant](https://mlsanity.com/paper/2305.16316/) (2023)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Almost Equivariance via Lie Algebra Convolutions](https://mlsanity.com/paper/2310.13164/) (2023)
- [A Generative Model of Symmetry Transformations](https://mlsanity.com/paper/2403.01946/) (2024)
- [Denoising Hamiltonian Network for Physical Reasoning](https://mlsanity.com/paper/2503.07596/) (2025)
- [Approximate Equivariance in Reinforcement Learning](https://mlsanity.com/paper/2411.04225/) (2024)
- [Equivariance-aware Architectural Optimization of Neural Networks](https://mlsanity.com/paper/2210.05484/) (2022)
- [Approximately Equivariant Neural Processes](https://mlsanity.com/paper/2406.13488/) (2024)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Color Equivariant Convolutional Networks](https://mlsanity.com/paper/2310.19368/) (2023)
- [Latent Field Discovery In Interacting Dynamical Systems With Neural Fields](https://mlsanity.com/paper/2310.20679/) (2023)
- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [Equivariance Is Not All You Need: Characterizing the Utility of Equivariant Graph Neural Networks for Particle Physics Tasks](https://mlsanity.com/paper/2311.03094/) (2023)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries](https://mlsanity.com/paper/2312.12223/) (2023)
- [Data Augmentation: A Fourier Analysis Perspective](https://mlsanity.com/paper/2606.24418/) (2026)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [Partially Equivariant Reinforcement Learning in Symmetry-Breaking Environments](https://mlsanity.com/paper/2512.00915/) (2025)
- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [An Analysis of Causal Effect Estimation using Outcome Invariant Data Augmentation](https://mlsanity.com/paper/2510.25128/) (2025)

## Abstract (arXiv)

Group Convolutional Neural Networks (G-CNNs) constrain learned features to respect the symmetries in the selected group, and lead to better generalization when these symmetries appear in the data. If this is not the case, however, equivariance leads to overly constrained models and worse performance. Frequently, transformations occurring in data can be better represented by a subset of a group than by a group as a whole, e.g., rotations in [-90circ, 90circ]. In such cases, a model that respects equivariance textitpartially is better suited to represent the data. In addition, relevant transformations may differ for low and high-level features. For instance, full rotation equivariance is useful to describe edge orientations in a face, but partial rotation equivariance is better suited to describe face poses relative to the camera. In other words, the optimal level of equivariance may differ per layer. In this work, we introduce textitPartial G-CNNs: G-CNNs able to learn layer-wise levels of partial and full equivariance to discrete, continuous groups and combinations thereof as part of training. Partial G-CNNs retain full equivariance when beneficial, e.g., for rotated MNIST, but adjust it whenever it becomes harmful, e.g., for classification of 6 / 9 digits or natural images. We empirically show that partial G-CNNs pair G-CNNs when full equivariance is advantageous, and outperform them otherwise.