# Topological Constraints on Homeomorphic Auto-Encoding

arXiv: https://arxiv.org/abs/1812.10783 · HTML: https://mlsanity.com/paper/1812.10783/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Dec 2018.

## Built on by (most cited first, 2 of 2)

- [On the Latent Holes of VAEs for Text Generation](https://mlsanity.com/paper/2110.03318/) (2021)
- [Topological Obstructions and How to Avoid Them](https://mlsanity.com/paper/2312.07529/) (2023)

## Abstract (arXiv)

When doing representation learning on data that lives on a known non-trivial manifold embedded in high dimensional space, it is natural to desire the encoder to be homeomorphic when restricted to the manifold, so that it is bijective and continuous with a continuous inverse. Using topological arguments, we show that when the manifold is non-trivial, the encoder must be globally discontinuous and propose a universal, albeit impractical, construction. In addition, we derive necessary constraints which need to be satisfied when designing manifold-specific practical encoders. These are used to analyse candidates for a homeomorphic encoder for the manifold of 3D rotations SO(3).