# Torsion in Persistent Homology and Neural Networks

arXiv: https://arxiv.org/abs/2506.03049 · HTML: https://mlsanity.com/paper/2506.03049/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## Builds on (most cited first, 10 of 10)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [Edge Contractions and Simplicial Homology](https://mlsanity.com/paper/1304.0664/) (2013)

## Abstract (arXiv)

We explore the role of torsion in hybrid deep learning models that incorporate topological data analysis, focusing on autoencoders. While most TDA tools use field coefficients, this conceals torsional features present in integer homology. We show that torsion can be lost during encoding, altered in the latent space, and in many cases, not reconstructed by standard decoders. Using both synthetic and high-dimensional data, we evaluate torsion sensitivity to perturbations and assess its recoverability across several autoencoder architectures. Our findings reveal key limitations of field-based approaches and underline the need for architectures or loss terms that preserve torsional information for robust data representation.