nLab zigzag persistence module

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 $V_i$ 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 $H(X_i)$ of stages $\cdots \hookrightarrow X_{i - 1} \hookrightarrow X_i \hookrightarrow X_{i + 1} \hookrightarrow \cdots$ 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

• stability of persistence diagrams

• diamond principle

Related concepts

• well group

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:

• Gunnar Carlsson, Vin de Silva, Zigzag Persistence, Found Comput Math 10 (2010) 367–405 $[$arXiv:0812.0197, doi:10.1007/s10208-010-9066-0$]$

Application to level sets:

• Gunnar Carlsson, Vin de Silva, Dmitriy Morozov, Zigzag persistent homology and real-valued functions, in: SCG ‘09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$doi:10.1145/1542362.1542408$]$

• Gunnar Carlsson, Vin de Silva, Sara Kališnik, Dmitriy Morozov, Parametrized Homology via Zigzag Persistence, Algebr. Geom. Topol. 19 (2019) 657-700 $[$arXiv:1604.03596, doi:10.2140/agt.2019.19.657$]$

Review:

• Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs 209 AMS (2015) $[$pdf, ISBN:978-1-4704-3443-4$]$

The algebraic stability theorem for zigzag persistence modules:

• Magnus Bakke Botnan, Michael Lesnick, Algebraic Stability of Zigzag Persistence Modules, Algebr. Geom. Topol. 18 (2018) 3133-3204 $[$arXiv:1604.00655, doi:10.2140/agt.2018.18.3133$]$