Latent Topology and Geometry in some Riemannian Autoencoders

Code accompanying the Diploma thesis
"Exploratory Analysis of Latent Topology and Geometry in some Riemannian Autoencoders"

The project investigates how different autoencoder architectures learn latent representations that reflect the topology and geometry of the underlying data manifold.

In particular, the thesis studies whether autoencoders can

1. learn latent representations that are topologically and geometrically aligned with the data manifold
2. learn a meaningful parametrization of the manifold through the decoder, revealing information about the geometry of the underlying data manifold

It further investigates the impact of topological regularization on these properties.

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Overview

The repository contains implementations and experimental tools for analyzing the latent topology and geometry of autoencoders.

The following model classes are implemented:

• Euclidean Autoencoders (AE)
• Euclidean Variational Autoencoders (VAE)
• Manifold Autoencoders (Manifold-AE) with non-Euclidean latent spaces
• Manifold Variational Autoencoders (Manifold-VAE) with non-Euclidean latent spaces

The code allows experiments on synthetic datasets with known topology, enabling quantitative evaluation of how well the learned latent space reflects the true structure of the data.

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The lib/ module provides utilities for

• training the different autoencoder architectures
• generating synthetic datasets with known topology
• estimating geometric quantities such as curvature
• analyzing the topology of latent representations
• visualizing latent embeddings and decoder maps

The notebooks/ directory contains the experiments and visualizations used in the thesis.

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Installation

The project uses a conda environment.

conda env create -f conda-env.yml
conda activate TopoGeoAEs

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Running Experiments

Experiments are provided as Jupyter notebooks in:

experiments/notebooks/

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Dependencies and Related Work

This project builds on the following implementations:

Spherical VAEs
https://github.com/nicola-decao/s-vae-pytorch

Toroidal VAEs and geometric analysis tools
https://github.com/geometric-intelligence/neurometry

Topological Autoencoders with persistence regularization
https://github.com/BorgwardtLab/topological-autoencoders

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Results

The full thesis is available here:

latent_geometry_and_topology_in_autoencoders_thesis

Parts of this work were accepted as an extended abstract and poster tracks at the NeurIPS 2025 "NeurReps" and "UniReps" workshops

Extended abstracts:

paper/unireps_extended_abstract.pdf

paper/neurreps_extended_abstract.pdf

Poster:

paper/neurreps_poster.pdf

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Citation

If you use this code, please cite the Diploma thesis:

@mastersthesis{samuelgraepler2025TopoGeoAEs,
  title={Exploratory Analysis of Latent Topology and Geometry in some Riemannian Autoencoders},
  author={Samuel Graepler},
  year={2025}
}

For the citation of the extended abstracts please use:

@inproceedings{
graepler2025on,
title={On the Impact of Topological Regularization on Geometrical and Topological Alignment in Autoencoders: An Empirical Study},
author={Samuel Graepler and Nico Scherf and Anna Wienhard and Diaaeldin Taha},
booktitle={UniReps: 3rd Edition of the Workshop on Unifying Representations in Neural Models},
year={2025},
url={https://openreview.net/forum?id=5i6cA8XS3T}
}

@inproceedings{
graepler2025on,
title={On the Impact of Topological Regularization on Geometrical and Topological Alignment in Autoencoders: An Empirical Study},
author={Samuel Graepler and Nico Scherf and Anna Wienhard and Diaaeldin Taha},
booktitle={NeurIPS 2025 Workshop on Symmetry and Geometry in Neural Representations},
year={2025},
url={https://openreview.net/forum?id=d5MaJiYmUB}
}