Stable comparison of multidimensional persistent homology groups with torsion

Frosini, Patrizio (2010) Stable comparison of multidimensional persistent homology groups with torsion. [Preprint]
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Abstract

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 vector-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.

Abstract
Document type
Preprint
Creators
CreatorsAffiliationORCID
Frosini, Patrizio
Keywords
Multidimensional persistent homology, shape comparison, matching distance, natural pseudo-distance
Subjects
DOI
Deposit date
21 Dec 2010 14:16
Last modified
16 May 2011 12:16
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