Field Choice Problem in Persistent Homology

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 dependence of the diagram on the coefficient field to facilitate computation and interpretation of the diagram. We clarify that the dependence is strongly related to the torsion part of 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. A slight modification of the standard persistence algorithm gives the verification algorithm. In a numerical experiment with the algorithm, a persistence diagram rarely changes even when the coefficient field changes if we consider a filtration in R-3. The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in R-3.

Automated summary

This paper addresses the issue of coefficient field choice in persistent homology. It is important to understand the dependence of the persistence diagram on the coefficient field in order to effectively compute and interpret the diagram. The paper examines the relationship between the dependence and the torsion part of Z relative homology in the filtration and presents sufficient and necessary conditions for the independence of coefficient field choice. An efficient algorithm is proposed to verify the independence, and experimental results show that persistence diagrams rarely change when the coefficient field changes in the case of a filtration in R-3.