• DOI:10.1007/S10440-012-9769-0
• Corpus ID: 4809929

Stable Comparison of Multidimensional Persistent Homology Groups with Torsion

@article{Frosini2010StableCO,
  title={Stable Comparison of Multidimensional Persistent Homology Groups with Torsion},
  author={Patrizio Frosini},
  journal={Acta Applicandae Mathematicae},
  year={2010},
  volume={124},
  pages={43-54}
}

• P. Frosini
• Published 19 December 2010
• Mathematics
• Acta Applicandae Mathematicae

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 ℝn-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.