Torsion in Persistent Homology and Neural Networks

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.