The reflection distance between zigzag persistence modules

Abstract

By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the ${\ell }^1$-bottleneck distance between their respective persistence diagrams.

30 pages, 2 figures. Final version; to appear in the Journal of Applied and Computational Topology