Field choice problem in persistent homology

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 $\mathbb{R}^3$.