# Commutativity and Disentanglement from the Manifold Perspective

arXiv: https://arxiv.org/abs/2210.07857 · HTML: https://mlsanity.com/paper/2210.07857/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 14 Oct 2022.

## Builds on (most cited first, 11 of 11)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Inductive Biases for Deep Learning of Higher-Level Cognition](https://mlsanity.com/paper/2011.15091/) (2020)
- [Deep Learning of Representations: Looking Forward](https://mlsanity.com/paper/1305.0445/) (2013)
- [Self-Supervised Learning Disentangled Group Representation as Feature](https://mlsanity.com/paper/2110.15255/) (2021)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [The Sparse Manifold Transform](https://mlsanity.com/paper/1806.08887/) (2018)
- [Evaluating the Disentanglement of Deep Generative Models through Manifold Topology](https://mlsanity.com/paper/2006.03680/) (2020)
- [Disentangling images with Lie group transformations and sparse coding](https://mlsanity.com/paper/2012.12071/) (2020)

## Abstract (arXiv)

In this paper, we interpret disentanglement as the discovery of local charts of the data manifold and trace how this definition naturally leads to an equivalent condition for disentanglement: commutativity between factors of variation. We study the impact of this manifold framework to two classes of problems: learning matrix exponential operators and compressing data-generating models. In each problem, the manifold perspective yields interesting results about the feasibility and fruitful approaches their solutions. We also link our manifold framework to two other common disentanglement paradigms: group theoretic and probabilistic approaches to disentanglement. In each case, we show how these frameworks can be merged with our manifold perspective. Importantly, we recover commutativity as a central property in both alternative frameworks, further highlighting its importance in disentanglement.