nLab zigzag persistence module

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Representation theory

Contents

Idea

Where a linear persistence module, as traditionally considered in persistent homology theory, is a diagram of vector space/modules of the directed form

in a zig-zag persistence module one allows the linear maps to go in either direction, such as in regular zigzags

or in more general zigzags such as

In the language of quiver theory, zigzag persistence modules are precisely the quiver representations of A-type quivers. But in quiver representation theory the classical Gabriel theorem says (see there) that for all possible zig-zag patterns as above the indecomposable quiver representations are still “interval modules”, ie. those for which the V iV_i are zero except in some interval where they are 1-dimensional and connected by (zigzags of) identity maps.

This means that the key property of persistence modules – namely that they are entirely characterized by persistence barcodes, i.e. by multisets of such intervals – immediately generalizes to zigzag persistence modules.

A typical type of a zigzag persistence module appearing in the practice of topological data analysis consists of homology groups H(X i)H(X_i) of stages X i1X iX i+1\cdots \hookrightarrow X_{i - 1} \hookrightarrow X_i \hookrightarrow X_{i + 1} \hookrightarrow \cdots of a filtered topological space, as usual, but evaluating now on the zigzag of inclusions into the unions

of consecutive filter stages:

It is claimed (…) that such zigzag persistence modules still retain the same persistent information of interest, but are more robust.

Examples

Well modules

As the level varies, the well groups of a continuous function naturally form a zigzag persistence module, called a well module (Edelsbrunner, Morozov & Patel 2011group#EMP11)).

Properties

References

The concept of zigzag persistence modules as a tool in topological data analysis and their relation to quiver representation theory (including a re-proof of Gabriel's theorem for the case of A-type quivers) is due to:

Application to level sets:

Review:

The algebraic stability theorem for zigzag persistence modules:

Last revised on May 20, 2022 at 05:49:21. See the history of this page for a list of all contributions to it.