Field choice problem in persistent homology

TL;DR

This paper investigates how the choice of coefficient field affects persistent homology diagrams, providing conditions for independence, an efficient verification algorithm, and empirical evidence that diagrams are stable across fields in 3D data.

Contribution

It introduces necessary and sufficient conditions for coefficient field independence in persistent homology and proposes an efficient algorithm to verify this independence.

Findings

Persistence diagrams are rarely affected by coefficient field changes in 3D data.

The dependency on the coefficient field is linked to torsion in relative homology.

In practical 3D cases, changing the field coefficient does not alter the persistence diagram.

Abstract

This paper tackles the problem of coefficient field choice in persistent homology. When we compute a persistence diagram, we need to select a coefficient field before computation. We should understand the dependency of the diagram on the coefficient field to facilitate computation and interpretation of the diagram. We clarify that the dependency is strongly related to the torsion of $\mathbb{Z}$ relative homology in the filtration. We show the sufficient and necessary conditions of the independence of coefficient field choice. An efficient algorithm is proposed to verify the independence. In a numerical experiment with the algorithm, a persistence diagram rarely changes even when the coefficient field changes if we consider a filtration in $\mathbb{R}^3$. The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in $\mathbb{R}^3$.