# Unsupervised Learning of Group Invariant and Equivariant Representations

arXiv: https://arxiv.org/abs/2202.07559 · HTML: https://mlsanity.com/paper/2202.07559/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 70 references that are in the map.
- First posted on arXiv 15 Feb 2022.

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [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)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [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)
- [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)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (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)
- [Graph Matching Networks for Learning the Similarity of Graph Structured Objects](https://mlsanity.com/paper/1904.12787/) (2019)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)

## Built on by (most cited first, 30 of 39)

- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Self-supervised learning of Split Invariant Equivariant representations](https://mlsanity.com/paper/2302.10283/) (2023)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [A General Theory of Correct, Incorrect, and Extrinsic Equivariance](https://mlsanity.com/paper/2303.04745/) (2023)
- [Equivariant Representation Learning via Class-Pose Decomposition](https://mlsanity.com/paper/2207.03116/) (2022)
- [Neural Fourier Transform: A General Approach to Equivariant Representation Learning](https://mlsanity.com/paper/2305.18484/) (2023)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [A Generative Model of Symmetry Transformations](https://mlsanity.com/paper/2403.01946/) (2024)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [Pose-Aware Self-Supervised Learning with Viewpoint Trajectory Regularization](https://mlsanity.com/paper/2403.14973/) (2024)
- [SELU: Self-Learning Embodied MLLMs in Unknown Environments](https://mlsanity.com/paper/2410.03303/) (2024)
- [Self-Supervised Learning for Place Representation Generalization across Appearance Changes](https://mlsanity.com/paper/2303.02370/) (2023)
- [Topological Obstructions and How to Avoid Them](https://mlsanity.com/paper/2312.07529/) (2023)
- [Self-Supervised Detection of Perfect and Partial Input-Dependent Symmetries](https://mlsanity.com/paper/2312.12223/) (2023)
- [Holographic-(V)AE: an end-to-end SO(3)-Equivariant (Variational) Autoencoder in Fourier Space](https://mlsanity.com/paper/2209.15567/) (2022)
- [Non-parametric Hypothesis Tests for Distributional Group Symmetry](https://mlsanity.com/paper/2307.15834/) (2023)
- [PESTO: Real-Time Pitch Estimation with Self-supervised Transposition-equivariant Objective](https://mlsanity.com/paper/2508.01488/) (2025)
- [CFASL: Composite Factor-Aligned Symmetry Learning for Disentanglement in Variational AutoEncoder](https://mlsanity.com/paper/2401.08897/) (2024)
- [Interpreting Equivariant Representations](https://mlsanity.com/paper/2401.12588/) (2024)
- [Capsule Network Projectors are Equivariant and Invariant Learners](https://mlsanity.com/paper/2405.14386/) (2024)
- [Constraint Decoupled Latent Diffusion for Protein Backmapping](https://mlsanity.com/paper/2410.13264/) (2024)
- [Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions](https://mlsanity.com/paper/2510.21706/) (2025)
- [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025)
- [Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models](https://mlsanity.com/paper/2604.09769/) (2026)
- [Domain Generalization In Robust Invariant Representation](https://mlsanity.com/paper/2304.03431/) (2023)
- [LMDM:Latent Molecular Diffusion Model For 3D Molecule Generation](https://mlsanity.com/paper/2412.04242/) (2024)
- [Multiple Invertible and Partial-Equivariant Function for Latent Vector Transformation to Enhance Disentanglement in VAEs](https://mlsanity.com/paper/2502.03740/) (2025)

## Abstract (arXiv)

Equivariant neural networks, whose hidden features transform according to representations of a group G acting on the data, exhibit training efficiency and an improved generalisation performance. In this work, we extend group invariant and equivariant representation learning to the field of unsupervised deep learning. We propose a general learning strategy based on an encoder-decoder framework in which the latent representation is separated in an invariant term and an equivariant group action component. The key idea is that the network learns to encode and decode data to and from a group-invariant representation by additionally learning to predict the appropriate group action to align input and output pose to solve the reconstruction task. We derive the necessary conditions on the equivariant encoder, and we present a construction valid for any G, both discrete and continuous. We describe explicitly our construction for rotations, translations and permutations. We test the validity and the robustness of our approach in a variety of experiments with diverse data types employing different network architectures.