The persistent homotopy type distance

We introduce the persistent homotopy type distance $d_{\mathrm{HT}}$ to compare two real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of $d_{\mathrm{HT}}$ is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopy equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that $d_{\mathrm{HT}}$ still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the $L^{\infty}$ distance and the natural pseudo-distance $d_{\mathrm{NP}}$. From a different standpoint, we prove that $d_{\mathrm{HT}}$ extends the $L^{\infty}$ distance and $d_{\mathrm{NP}}$ in two ways. First, we show that, appropriately restricting the category of objects to which $d_{\mathrm{HT}}$ applies, it can be made to coincide with the other two distances. Finally, we show that $d_{\mathrm{HT}}$ has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.

Keywords

bottleneck distance between persistence diagrams, natural pseudo-distance, interleaving distance, stability, merge trees

2010 Mathematics Subject Classification

18A23, 55N35, 55P10, 68U05

Full Text (PDF format)

This work started during a visit of the third author to the first author at the University of Bologna in 2014 which was partially supported by INdAM-GNSAGA. The first two authors partially carried out this research within the activities of ARCES “E. De Castro”, University of Bologna. The third author has been partially supported by NSF under grants DMS-1547357, CCF-1526513, and IIS-1422400. The authors thank Francesca Cagliari for her helpful advice about the categorical setting and Michael Lesnick for useful discussions that helped us to focus Section 5.

Received 21 May 2018

Received revised 11 October 2018

Accepted 22 November 2018

Published 13 March 2019