2023/08/18 by Ippei Obayashi, Michio Yoshiwaki · 3 citations
Computer Science · Mathematics · Medicine · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Advanced Neuroimaging Techniques and Applications
paper · pdf · doi:10.1007/s00454-023-00544-7
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 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 \mathbb Z <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> 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 \mathbb R3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . The experiment suggests that, in practical terms, changes in the field coefficient will not change persistence diagrams when the data are in \mathbb R3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> .