# A Category-theoretical Meta-analysis of Definitions of Disentanglement

arXiv: https://arxiv.org/abs/2305.06886 · HTML: https://mlsanity.com/paper/2305.06886/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 45 references that are in the map.
- First posted on arXiv 11 May 2023.

## Builds on (most cited first, 25 of 25)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Robustly Disentangled Causal Mechanisms: Validating Deep Representations for Interventional Robustness](https://mlsanity.com/paper/1811.00007/) (2018)
- [Learning Deep Disentangled Embeddings with the F-Statistic Loss](https://mlsanity.com/paper/1802.05312/) (2018)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [Weakly Supervised Disentanglement with Guarantees](https://mlsanity.com/paper/1910.09772/) (2019)
- [Measuring Disentanglement: A Review of Metrics](https://mlsanity.com/paper/2012.09276/) (2020)
- [On the Transfer of Disentangled Representations in Realistic Settings](https://mlsanity.com/paper/2010.14407/) (2020)
- [Leveraging Unlabeled Data for Crowd Counting by Learning to Rank](https://mlsanity.com/paper/1803.03095/) (2018)
- [Theory and Evaluation Metrics for Learning Disentangled Representations](https://mlsanity.com/paper/1908.09961/) (2019)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Graph Neural Networks are Dynamic Programmers](https://mlsanity.com/paper/2203.15544/) (2022)
- [Self-Supervised Learning Disentangled Group Representation as Feature](https://mlsanity.com/paper/2110.15255/) (2021)
- [Unsupervised Model Selection for Variational Disentangled Representation Learning](https://mlsanity.com/paper/1905.12614/) (2019)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Natural Graph Networks](https://mlsanity.com/paper/2007.08349/) (2020)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Progressive Learning and Disentanglement of Hierarchical Representations](https://mlsanity.com/paper/2002.10549/) (2020)
- [Transformation Properties of Learned Visual Representations](https://mlsanity.com/paper/1412.7659/) (2014)
- [Disentanglement Analysis with Partial Information Decomposition](https://mlsanity.com/paper/2108.13753/) (2021)
- [Compositional Deep Learning](https://mlsanity.com/paper/1907.08292/) (2019)
- [Knowledge Representation in Bicategories of Relations](https://mlsanity.com/paper/1706.00526/) (2017)
- [How to Not Measure Disentanglement](https://mlsanity.com/paper/1910.05587/) (2019)

## Built on by (most cited first, 3 of 3)

- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Disentangled Representations for Causal Cognition](https://mlsanity.com/paper/2407.00744/) (2024)
- [Disentangling Hyperedges through the Lens of Category Theory](https://mlsanity.com/paper/2510.16289/) (2025)

## Abstract (arXiv)

Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more theoretical research is needed to fully understand the defining properties of disentanglement and how different definitions relate to each other. This paper presents a meta-analysis of existing definitions of disentanglement, using category theory as a unifying and rigorous framework. We propose that the concepts of the cartesian and monoidal products should serve as the core of disentanglement. With these core concepts, we show the similarities and crucial differences in dealing with (i) functions, (ii) equivariant maps, (iii) relations, and (iv) stochastic maps. Overall, our meta-analysis deepens our understanding of disentanglement and its various formulations and can help researchers navigate different definitions and choose the most appropriate one for their specific context.