The Space of Barcode Bases for Persistence Modules

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.

arXiv comment29 pages

References in corpus (5)

• Matrix Method for Persistence Modules on Commutative Ladders of Finite Type
• U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology
• Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
• Matroids and Canonical Forms: Theory and Applications
• Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology

Cited by in corpus (3)

• Stratifying the space of barcodes using Coxeter complexes
• Harder-Narasimhan Filtrations of Persistence Modules
• Parallel computation of interval bases for persistence module decomposition