# Quantifying and Learning Linear Symmetry-Based Disentanglement

arXiv: https://arxiv.org/abs/2011.06070 · HTML: https://mlsanity.com/paper/2011.06070/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2020.

## Built on by (most cited first, 9 of 9)

- [Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions](https://mlsanity.com/paper/2207.12067/) (2022)
- [Equivariant Representation Learning via Class-Pose Decomposition](https://mlsanity.com/paper/2207.03116/) (2022)
- [Latent Traversals in Generative Models as Potential Flows](https://mlsanity.com/paper/2304.12944/) (2023)
- [Topological Obstructions and How to Avoid Them](https://mlsanity.com/paper/2312.07529/) (2023)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Measuring Orthogonality in Representations of Generative Models](https://mlsanity.com/paper/2407.03728/) (2024)
- [Equivariant Representation Learning in the Presence of Stabilizers](https://mlsanity.com/paper/2301.05231/) (2023)
- [Disentangled Representation Learning through Unsupervised Symmetry Group Discovery](https://mlsanity.com/paper/2603.11790/) (2026)
- [Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations](https://mlsanity.com/paper/2608.24762/) (2026)

## Abstract (arXiv)

The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes the notion of linearly disentangled representations, but there is currently no metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to compare to previous understandings of disentanglement. We propose DLSBD, a mathematically sound metric to quantify LSBD, and provide a practical implementation for SO(2) groups. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to learn LSBD representations. We demonstrate the utility of our metric by showing that (1) common VAE-based disentanglement methods don't learn LSBD representations, (2) LSBD-VAE as well as other recent methods can learn LSBD representations, needing only limited supervision on transformations, and (3) various desirable properties expressed by existing disentanglement metrics are also achieved by LSBD representations.