Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

Erosion distance for generalized persistence modules

Pages: 233 – 254

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a14


Ville Puuska (Faculty of Information Technology and Communication Sciences, Tampere University, Tampere, Finland)


The persistence diagram of Cohen–Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance for persistence diagrams, the erosion distance. Motivated by this work, we extend the erosion distance to a distance of rank invariants of generalized persistence modules by using the generalization of the interleaving distance of Bubenik, de Silva, and Scott as a guideline. This extension of the erosion distance also gives, as a special case, a distance for multidimensional persistent homology groups with torsion introduced by Frosini. We show that the erosion distance is stable with respect to the interleaving distance, and that it gives a lower bound for the natural pseudo-distance in the case of sublevel set persistent homology of continuous functions.


persistence module, persistent homology

2010 Mathematics Subject Classification


Copyright © 2019, Ville Puuska. Permission to copy for private use granted.

Received 2 February 2018

Received revised 11 September 2018

Published 20 November 2019