Bipath Persistence as Zigzag Persistence

TL;DR

This paper establishes a connection between bipath persistence and zigzag persistence, enabling the transfer of computational techniques and stability results from zigzag to bipath persistence in topological data analysis.

Contribution

It shows bipath persistence modules can be decomposed via infinite zigzag modules, allowing the application of zigzag techniques and stability results to bipath persistence.

Findings

Bipath persistence decomposes via infinite zigzag modules.

Techniques from zigzag persistence are applicable to bipath persistence.

Bipath persistence inherits stability properties from zigzag persistence.

Abstract

Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which persistence modules always decompose into intervals, by a recent result of Aoki, Escolar and Tada: these are standard single-parameter persistence, zigzag persistence, and bipath persistence. No other setting offers such guarantees. We show that a bipath persistence module can be decomposed via a closely related infinite zigzag persistence module, understood as a covering. This allows us to translate techniques of zigzag persistence, like recent advancements in its efficient computation by Dey and Hou, to bipath persistence. In addition, and again by the relation with the infinite zigzag, we can define an interleaving and bottleneck distance on bipath persistence. In turn, the algebraic stability of zigzag persistence implies the algebraic stability of bipath persistence.