# Beyond Uniform Local Isometry and Topology: FactoMap for Disentangled Representations

arXiv: https://arxiv.org/abs/2608.24762 · HTML: https://mlsanity.com/paper/2608.24762/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 29 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [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)
- [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)
- [Weakly-Supervised Disentanglement Without Compromises](https://mlsanity.com/paper/2002.02886/) (2020)
- [On the Fairness of Disentangled Representations](https://mlsanity.com/paper/1905.13662/) (2019)
- [Learning Disentangled Representations for Recommendation](https://mlsanity.com/paper/1910.14238/) (2019)
- [Disentanglement via Latent Quantization](https://mlsanity.com/paper/2305.18378/) (2023)
- [Disentanglement of Correlated Factors via Hausdorff Factorized Support](https://mlsanity.com/paper/2210.07347/) (2022)
- [Isometric Autoencoders](https://mlsanity.com/paper/2006.09289/) (2020)
- [Flow Factorized Representation Learning](https://mlsanity.com/paper/2309.13167/) (2023)
- [Quantifying and Learning Linear Symmetry-Based Disentanglement](https://mlsanity.com/paper/2011.06070/) (2020)
- [Statistical and structural identifiability in representation learning](https://mlsanity.com/paper/2603.11970/) (2026)
- [Constructing VAE Latent Spaces with Prescribed Topology](https://mlsanity.com/paper/2606.07058/) (2026)

## Abstract (arXiv)

Many disentanglement methods represent generative factors using Euclidean product coordinates, although the underlying factor spaces may wrap, collapse, or have position-dependent geometry. We introduce factor-space structure, combining factor domains, generator-induced identifications, and position-dependent scales to distinguish topologically equivalent spaces with different factor geometries. We show that statistically independent factors need not be geometrically separable: hue and scale produce effects that grow at different rates, yielding anisotropy that no fixed rescaling removes. We propose the Factor-Space Topographic Map (FactoMap), which learns interpretable prototypes indexed by a factor-space lattice. Topographic learning transfers the lattice's periodicity, collapses, and non-uniform extent to the representation. Experiments show that matching this structure preserves factor continuity and enables disentanglement of the underlying factors.