# Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions

arXiv: https://arxiv.org/abs/2207.12067 · HTML: https://mlsanity.com/paper/2207.12067/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 52 references that are in the map.
- First posted on arXiv 25 Jul 2022.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [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)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Deep Convolutional Inverse Graphics Network](https://mlsanity.com/paper/1503.03167/) (2015)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [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)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Causal Curiosity: RL Agents Discovering Self-supervised Experiments for Causal Representation Learning](https://mlsanity.com/paper/2010.03110/) (2020)
- [Symmetry-Based Disentangled Representation Learning requires Interaction with Environments](https://mlsanity.com/paper/1904.00243/) (2019)
- [Towards Building A Group-based Unsupervised Representation Disentanglement Framework](https://mlsanity.com/paper/2102.10303/) (2021)
- [Group invariance principles for causal generative models](https://mlsanity.com/paper/1705.02212/) (2017)
- [Representing Closed Transformation Paths in Encoded Network Latent Space](https://mlsanity.com/paper/1912.02644/) (2019)
- [A theory of independent mechanisms for extrapolation in generative models](https://mlsanity.com/paper/2004.00184/) (2020)
- [Linear Disentangled Representations and Unsupervised Action Estimation](https://mlsanity.com/paper/2008.07922/) (2020)
- [Independently Controllable Factors](https://mlsanity.com/paper/1708.01289/) (2017)
- [Quantifying and Learning Linear Symmetry-Based Disentanglement](https://mlsanity.com/paper/2011.06070/) (2020)

## Built on by (most cited first, 18 of 18)

- [Sensorimotor World Models: Perception for Action via Inverse Dynamics](https://mlsanity.com/paper/2606.20104/) (2026)
- [A Generative Model of Symmetry Transformations](https://mlsanity.com/paper/2403.01946/) (2024)
- [A Measure-Theoretic Axiomatisation of Causality](https://mlsanity.com/paper/2305.17139/) (2023)
- [Flow Equivariant World Models: Memory for Partially Observed Dynamic Environments](https://mlsanity.com/paper/2601.01075/) (2026)
- [Learning Lie Group Symmetry Transformations with Neural Networks](https://mlsanity.com/paper/2307.01583/) (2023)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [Pooling Image Datasets With Multiple Covariate Shift and Imbalance](https://mlsanity.com/paper/2403.02598/) (2024)
- [Learning Actionable World Models for Industrial Process Control](https://mlsanity.com/paper/2503.01411/) (2025)
- [Structured Latent Dynamics in Wireless CSI via Homomorphic World Models](https://mlsanity.com/paper/2603.20048/) (2026)
- [Homomorphic Self-Supervised Learning](https://mlsanity.com/paper/2211.08282/) (2022)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Algebras of actions in an agent's representations of the world](https://mlsanity.com/paper/2310.01536/) (2023)
- [Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions](https://mlsanity.com/paper/2510.21706/) (2025)
- [Learning Nonlinear Causal Reductions to Explain Reinforcement Learning Policies](https://mlsanity.com/paper/2507.14901/) (2025)
- [Using Forwards-Backwards Models to Approximate MDP Homomorphisms](https://mlsanity.com/paper/2209.06356/) (2022)
- [SEER-ZSL: Semantic Encoder-Enhanced Representations for Generalized Zero-Shot Learning](https://mlsanity.com/paper/2312.13100/) (2023)
- [Multiple Invertible and Partial-Equivariant Function for Latent Vector Transformation to Enhance Disentanglement in VAEs](https://mlsanity.com/paper/2502.03740/) (2025)
- [Disentangled Representation Learning through Unsupervised Symmetry Group Discovery](https://mlsanity.com/paper/2603.11790/) (2026)

## Abstract (arXiv)

How can agents learn internal models that veridically represent interactions with the real world is a largely open question. As machine learning is moving towards representations containing not just observational but also interventional knowledge, we study this problem using tools from representation learning and group theory. We propose methods enabling an agent acting upon the world to learn internal representations of sensory information that are consistent with actions that modify it. We use an autoencoder equipped with a group representation acting on its latent space, trained using an equivariance-derived loss in order to enforce a suitable homomorphism property on the group representation. In contrast to existing work, our approach does not require prior knowledge of the group and does not restrict the set of actions the agent can perform. We motivate our method theoretically, and show empirically that it can learn a group representation of the actions, thereby capturing the structure of the set of transformations applied to the environment. We further show that this allows agents to predict the effect of sequences of future actions with improved accuracy.