Stratifying the space of barcodes using Coxeter complexes

TL;DR

This work addresses the challenge of performing statistics on the space of barcodes by introducing a Coxeter-geometry–based stratification of $\mathcal{B}_n$. It defines Coxeter coordinates $(\bar{b},\|v_b\|,b_\theta,\bar{d},\|v_d\|,d_\theta)$ that pair barcode averages and dispersions with permutation data, and shows that $\mathcal{B}_n$ is stratified by marked double cosets of parabolic subgroups of $\operatorname{Sym}_n$. The top strata are indexed by permutations, and the construction yields natural invariants that refine permutation-type classification, enabling permutation-statistics–driven analysis of barcodes. Moreover, the authors define metrics $\tilde{d}_B$ and $\tilde{d}_W$ on $\mathcal{B}_n$ that correspond to modified bottleneck and Wasserstein distances under the Coxeter coordinate description, bridging geometric and statistical perspectives. This framework opens avenues for size-aware barcode statistics, potential extensions to merge trees, and new geometric interpretations of barcode invariants."

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.

Figures (8)

• Figure 1: The permutohedron permutohedron of order $4$ is a polyhedral decomposition of the sphere where each vertex corresponds to an element of the symmetric group $\operatorname{Sym}_{ 4 }$. Its $1$-skeleton is the Cayley graph of $\operatorname{Sym}_{ 4 }$ (see also \ref{['fig_cayley_s4']}).
• Figure 2: Two barcodes with the same associated permutation (the identity $[1234]$) but with large differences in their birth and death values.
• Figure 3: The permutohedron of order $4$ (black) is the dual of the Coxeter complex $\Sigma( {\operatorname{Sym}_{ 4 }} )$ (grey).
• Figure 4: (A) A barcode with $4$ bars. (B) The same barcode with a different indexing where the bars are ordered by increasing birth times.
• Figure 5: (Figure from TRN) The Cayley graph of $\operatorname{Sym}_{ 4 }$ generated by the three transpositions $(12),(23),(34)$. Four barcodes are drawn next to the extremities of the graphs (permutations $[1234],[2134],[2143],[1243]$) to illustrate a typical barcode corresponding to each permutation.

Theorems & Definitions (37)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Example 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7
• Definition 2.8
• Remark 2.9