nLab persistence diagram

Idea

In persistent homology a persistence diagram or barcode is a way to encode the isomorphism class of a persistence module in terms of a multiset of pairs of numbers $a \leq b$ which stand for “features” reflected in the module which “appear” at stage $a$, “persist” until stage $b$ and then “disappear”.

Concretely, Gabriel's theorem for A-type quivers (and analogously its generalizations to infinite linear diagrams and/or infinite-dimensional modules) says that every (zig-zag) persistence module (over some ground field $\mathbb{K}$), such as

is the direct sum (in the category of quiver representations) of interval modules $I_{a,b}$ of the form:

Hence the multiset of pairs $(a,b)$ labeling its inverval summands completely characterizes the given persistence module, up to isomorphism.

Regarding these pairs as intervals, the collection they form is naturally envisioned as a collection of “bars” and then known as the barcode of the persistence module.

Regarding these pairs instead as points in the plane, $(a,b) \in \mathbb{R}^2$, the multi-subset which they form is called the persistence diagram of the persistence module.

Remark

While for a given quiver-shape, persistence diagrams/barcodes carry the same information as the isomorphism classes of the persistence modules of this shape, it makes sense to compare persistence diagrams arising from quivers of different shape. For example, the diamond principle gives conditions under which persistence modules whose arrow directions differs in two consecutive positions still have bijective persistence diagrams.

Properties

• stability of persistence diagrams

References

General

See also the references at persistent homology.

Introduction and survey:

• Robert Ghrist, Barcodes: The Persistent Topology of Data, Bull. Amer. Math. Soc. 45 (2008), 61-75 (doi:10.1090/S0273-0979-07-01191-3, pdf)

• 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$]$

• Dan Burghelea, “Barcodes” for continuous maps and a brief introduction to Alternative Morse Theory, arXiv:2305.19828

Kernel methods and topological machine learning

On kernel methods applicable to persistence diagrams/barcodes for making topological data analysis amenable to “topological” machine learning:

• Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt, A Stable Multi-Scale Kernel for Topological Machine Learning, NIPS’15: Proceedings of the 28th International Conference on Neural Information Processing System, 2 (2015 3070–3078 $[$arXiv:1412.6821$]$

• Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich Bauer, Statistical Topological Data Analysis – A Kernel Perspective, in: Advances in Neural Information Processing Systems (NIPS 2015) $[$ISBN:9781510825024, doi:10.5555/2969442.2969582$]$

• Bastian Rieck, Filip Sadlo, Heike Leitte, Topological Machine Learning with Persistence Indicator Functions, In: Topological Methods in Data Analysis and Visualization V TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$arXiv:1907.13496, doi:10.1007/978-3-030-43036-8_6$]$

• Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, Persformer: A Transformer Architecture for Topological Machine Learning $[$arXiv:2112.15210$]$