# Disentangling by Subspace Diffusion

arXiv: https://arxiv.org/abs/2006.12982 · HTML: https://mlsanity.com/paper/2006.12982/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 28 of 28)

- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [The Origins of Representation Manifolds in Large Language Models](https://mlsanity.com/paper/2505.18235/) (2025)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [Disentanglement of Correlated Factors via Hausdorff Factorized Support](https://mlsanity.com/paper/2210.07347/) (2022)
- [Cross-Entropy Is All You Need To Invert the Data Generating Process](https://mlsanity.com/paper/2410.21869/) (2024)
- [Do Not Escape From the Manifold: Discovering the Local Coordinates on the Latent Space of GANs](https://mlsanity.com/paper/2106.06959/) (2021)
- [Symmetry-Based Representations for Artificial and Biological General Intelligence](https://mlsanity.com/paper/2203.09250/) (2022)
- [Towards Building A Group-based Unsupervised Representation Disentanglement Framework](https://mlsanity.com/paper/2102.10303/) (2021)
- [Learning disentangled representations via product manifold projection](https://mlsanity.com/paper/2103.01638/) (2021)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [From superposition to sparse codes: interpretable representations in neural networks](https://mlsanity.com/paper/2503.01824/) (2025)
- [Unsupervised Learning of Equivariant Structure from Sequences](https://mlsanity.com/paper/2210.05972/) (2022)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [GeomCA: Geometric Evaluation of Data Representations](https://mlsanity.com/paper/2105.12486/) (2021)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [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)
- [Algebras of actions in an agent's representations of the world](https://mlsanity.com/paper/2310.01536/) (2023)
- [Learning to Transfer: A Foliated Theory](https://mlsanity.com/paper/2107.10763/) (2021)
- [On a Built-in Conflict between Deep Learning and Systematic Generalization](https://mlsanity.com/paper/2208.11633/) (2022)
- [A Short Survey of Systematic Generalization](https://mlsanity.com/paper/2211.11956/) (2022)
- [Parametrizing Product Shape Manifolds by Composite Networks](https://mlsanity.com/paper/2302.14665/) (2023)
- [Robust GAN inversion](https://mlsanity.com/paper/2308.16510/) (2023)
- [From Points to Spheres: A Geometric Reinterpretation of Variational Autoencoders](https://mlsanity.com/paper/2507.17255/) (2025)
- [Disentangled Representation Learning through Unsupervised Symmetry Group Discovery](https://mlsanity.com/paper/2603.11790/) (2026)

## Abstract (arXiv)

We present a novel nonparametric algorithm for symmetry-based disentangling of data manifolds, the Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER provides a partial answer to the question posed by Higgins et al. (2018): is it possible to learn how to factorize a Lie group solely from observations of the orbit of an object it acts on? We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known and each factor manifold has nontrivial holonomy -- for example, rotation in 3D. Our algorithm works by estimating the subspaces that are invariant under random walk diffusion, giving an approximation to the de Rham decomposition from differential geometry. We demonstrate the efficacy of GEOMANCER on several complex synthetic manifolds. Our work reduces the question of whether unsupervised disentangling is possible to the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.