# Commutative Lie Group VAE for Disentanglement Learning

arXiv: https://arxiv.org/abs/2106.03375 · HTML: https://mlsanity.com/paper/2106.03375/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 53 references that are in the map.
- First posted on arXiv 7 Jun 2021.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (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)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Deep Convolutional Inverse Graphics Network](https://mlsanity.com/paper/1503.03167/) (2015)
- [Fader Networks: Manipulating Images by Sliding Attributes](https://mlsanity.com/paper/1706.00409/) (2017)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Attribute2Image: Conditional Image Generation from Visual Attributes](https://mlsanity.com/paper/1512.00570/) (2015)
- [The Hessian Penalty: A Weak Prior for Unsupervised Disentanglement](https://mlsanity.com/paper/2008.10599/) (2020)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Learning Disentangled Joint Continuous and Discrete Representations](https://mlsanity.com/paper/1804.00104/) (2018)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)
- [Theory and Evaluation Metrics for Learning Disentangled Representations](https://mlsanity.com/paper/1908.09961/) (2019)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Symmetry-Based Disentangled Representation Learning requires Interaction with Environments](https://mlsanity.com/paper/1904.00243/) (2019)
- [Progressive Learning and Disentanglement of Hierarchical Representations](https://mlsanity.com/paper/2002.10549/) (2020)
- [Learning Hierarchical Features from Generative Models](https://mlsanity.com/paper/1702.08396/) (2017)
- [HexaConv](https://mlsanity.com/paper/1803.02108/) (2018)
- [Where and What? Examining Interpretable Disentangled Representations](https://mlsanity.com/paper/2104.05622/) (2021)

## Built on by (most cited first, 27 of 27)

- [Disentangled Representation Learning](https://mlsanity.com/paper/2211.11695/) (2022)
- [Lost in Latent Space: Disentangled Models and the Challenge of Combinatorial Generalisation](https://mlsanity.com/paper/2204.02283/) (2022)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [Scalable Infomin Learning](https://mlsanity.com/paper/2302.10701/) (2023)
- [Learning Lie Group Symmetry Transformations with Neural Networks](https://mlsanity.com/paper/2307.01583/) (2023)
- [Topological Obstructions and How to Avoid Them](https://mlsanity.com/paper/2312.07529/) (2023)
- [Sequential Representation Learning via Static-Dynamic Conditional Disentanglement](https://mlsanity.com/paper/2408.05599/) (2024)
- [Identifying Metric Structures of Deep Latent Variable Models](https://mlsanity.com/paper/2502.13757/) (2025)
- [Learning Conditionally Independent Transformations using Normal Subgroups in Group Theory](https://mlsanity.com/paper/2504.04490/) (2025)
- [Algebras of actions in an agent's representations of the world](https://mlsanity.com/paper/2310.01536/) (2023)
- [CFASL: Composite Factor-Aligned Symmetry Learning for Disentanglement in Variational AutoEncoder](https://mlsanity.com/paper/2401.08897/) (2024)
- [Concept-free Causal Disentanglement with Variational Graph Auto-Encoder](https://mlsanity.com/paper/2311.10638/) (2023)
- [DiViD: Disentangled Video Diffusion for Static-Dynamic Factorization](https://mlsanity.com/paper/2507.13934/) (2025)
- [Disentangled Representation Learning via Modular Compositional Bias](https://mlsanity.com/paper/2510.21402/) (2025)
- [Discovering Data Manifold Geometry through Geometric Properties](https://mlsanity.com/paper/2602.02611/) (2026)
- [Refining the Information Bottleneck via Adversarial Information Separation](https://mlsanity.com/paper/2602.06549/) (2026)
- [Generative Modeling in Protein Design: Neural Representations, Conditional Generation, and Evaluation Standards](https://mlsanity.com/paper/2603.26378/) (2026)
- [ContraFeat: Contrasting Deep Features for Semantic Discovery](https://mlsanity.com/paper/2212.07277/) (2022)
- [ProtoVAE: Prototypical Networks for Unsupervised Disentanglement](https://mlsanity.com/paper/2305.09092/) (2023)
- [Independent Modular Networks](https://mlsanity.com/paper/2306.01316/) (2023)
- [Multiple Invertible and Partial-Equivariant Function for Latent Vector Transformation to Enhance Disentanglement in VAEs](https://mlsanity.com/paper/2502.03740/) (2025)
- [Harmonizing Generalization and Personalization in Ring-topology Decentralized Federated Learning](https://mlsanity.com/paper/2504.19103/) (2025)
- [Knowledge Abstraction for Knowledge-based Semantic Communication: A Generative Causality Invariant Approach](https://mlsanity.com/paper/2507.17784/) (2025)
- [Causal Representation Learning from Network Data](https://mlsanity.com/paper/2509.01916/) (2025)
- [Commutator-Induced Uncertainty in VAEs](https://mlsanity.com/paper/2605.23449/) (2026)
- [Unsupervised Disentanglement Without Compromises : How Functional Orthogonality Enforces Identifiability](https://mlsanity.com/paper/2606.21385/) (2026)
- [Decafs: Disentangled Conditional adversarial Flows](https://mlsanity.com/paper/2607.18755/) (2026)

## Abstract (arXiv)

We view disentanglement learning as discovering an underlying structure that equivariantly reflects the factorized variations shown in data. Traditionally, such a structure is fixed to be a vector space with data variations represented by translations along individual latent dimensions. We argue this simple structure is suboptimal since it requires the model to learn to discard the properties (e.g. different scales of changes, different levels of abstractness) of data variations, which is an extra work than equivariance learning. Instead, we propose to encode the data variations with groups, a structure not only can equivariantly represent variations, but can also be adaptively optimized to preserve the properties of data variations. Considering it is hard to conduct training on group structures, we focus on Lie groups and adopt a parameterization using Lie algebra. Based on the parameterization, some disentanglement learning constraints are naturally derived. A simple model named Commutative Lie Group VAE is introduced to realize the group-based disentanglement learning. Experiments show that our model can effectively learn disentangled representations without supervision, and can achieve state-of-the-art performance without extra constraints.