2010/01/01 by Patrizio Frosini, Frosini, Patrizio · 2 citations
Computer Science · Engineering · Mathematics · Medicine · #Algebraic Topology (math.AT) #Alzheimer's disease research and treatments #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Hydrocarbon exploration and reservoir analysis #I.2.10 #I.3.5 #I.5.1 #Primary 55N35 #Secondary 68U05 #Topological and Geometric Data Analysis #acm:55N35 #acm:68U05 #cs.CG #math.AT #msc:55N35 #msc:68U05
paper · pdf · doi:10.48550/arxiv.1012.4169
10 pages, 3 figures
openalex publication_date 2010/01/01 · arxiv created 2010/12/19 · arxiv updated 2010/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance dT that represents a possible solution to this problem. Indeed, dT is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between dT and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.