Stratifying the space of barcodes using Coxeter complexes

TL;DR

This paper introduces a geometric group theory approach to stratify the space of barcodes, providing new coordinates and metrics that classify barcodes by permutation type, averages, and standard deviations of birth and death times.

Contribution

It develops a novel stratification of barcode space using Coxeter complexes, extending existing invariants and defining new metrics based on this structure.

Findings

Stratification of barcode space into regions with same permutation type and statistics.

Introduction of new barcode invariants extending previous work.

Metrics on barcode space aligning with modified bottleneck and Wasserstein metrics.

Abstract

We use tools from geometric group theory to produce a stratification of the space $\mathcal{B}_n$ of barcodes with $n$ bars. The top-dimensional strata are indexed by permutations associated to barcodes as defined by Kanari, Garin and Hess. More generally, the strata correspond to marked double cosets of parabolic subgroups of the symmetric group $Sym_n$. This subdivides $\mathcal{B}_n$ into regions that consist of barcodes with the same averages and standard deviations of birth and death times and the same permutation type. We obtain coordinates that form a new invariant of barcodes, extending the one of Kanari-Garin-Hess. This description also gives rise to metrics on $\mathcal{B}_n$ that coincide with modified versions of the bottleneck and Wasserstein metrics.