The Space of Barcode Bases for Persistence Modules

TL;DR

This paper introduces a new algorithm for computing barcodes of persistence modules that also tracks the change of basis, providing a detailed understanding of the space of barcode bases and their transformations.

Contribution

It offers an explicit characterization of the group of transformations between barcode bases and extends the algorithm to zigzag modules.

Findings

Characterization of the transformation group between barcode bases

Algorithm for computing barcodes with basis change tracking

Representation of module maps via partial matchings under certain conditions

Abstract

The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules.