Torsion in Persistent Homology and Neural Networks

TL;DR

This paper investigates the significance of torsion in persistent homology within neural networks, revealing that standard field-based TDA methods often miss torsional features, which are crucial for robust data representation.

Contribution

It demonstrates the limitations of field-based TDA in capturing torsion and proposes the need for new architectures or loss functions to preserve torsional information.

Findings

Torsion can be lost or altered during encoding in autoencoders.

Standard decoders often fail to reconstruct torsional features.

Torsion sensitivity varies with data perturbations and architecture.

Abstract

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.