Geometric Obstructions in Finsler Spaces and Torsion-Free Persistent Homology

TL;DR

This paper explores the application of Topological Data Analysis in Finsler spaces, focusing on representability as an obstruction to spurious features, and introduces a matrix decomposition method to analyze persistent homology modules.

Contribution

It introduces a novel connection between Finsler geometry and persistent homology, providing a method to identify genuine topological features in high-dimensional data.

Findings

Persistent homology over rac{p}{p} detects only genuine holes.

Decomposition of integer matrices helps find suitable prime p.

Aligns integer homology with rational homology in Finsler spaces.

Abstract

We relate the novel concept of Topological Data Analysis in Finsler space with representability property, which is a natural obstruction to prevent spurious features in high dimensions. We use decomposition of integer matrix in order to find suitable prime integer $p$ such that persistent homology module over $\mathbb{Z}_p$ encompasses only the holes associated to the free part, in agreement with the rational case.