Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
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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 d T that represents a possible solution to this problem. Indeed, d T 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 ℝ n -valued filtering functions. Furthermore, we prove a result showing the relationship between d T 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.
KeywordsMultidimensional persistent homology Shape comparison Matching distance Natural pseudo-distance
Mathematics Subject Classification55N35 68U05
Research partially supported by DISTEF. The author thanks Sara, and all his friends in the “Coniglietti” Band.
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