PERSISTENT HOMOLOGY METHOD OF TOPOLOGICAL DATA ANALYSIS

Authors

  • І.А. Юрчук

DOI:

https://doi.org/10.18372/2310-5461.23.7397

Keywords:

persistent homology method, Vietoris-Rips complex

Abstract

Let be a finite sample from some unknown space . Let us find a structure of using the persistent homology method based on a group of homology which is an algebraic invariant of a topological space suggesting the global structure of the space based on its local setting. First, let construct a filtration which is based on Vietoris-Rips complex, where its vertices are a point of sample . Next, we compute the persistent homology for that consists of all the groups of homologies which have appeared in and stay "alive" in , where . By their boundary and its reduced matrices representation, the persistent Betty numbers are obtained and a persistence diagram is constructed based on the latest a conclusion about a structure of space is obtained. Main concepts and theorems concerning filtration and persistent  homology can be found in [ 1-8].

In this work the flowchart of algorithms of construction of the boundary and its reduced matrices of filtration are obtained which allow us to create a software in any programming language. The time of such algorithms equals , where has different values depending on algorithm.

References

Zomorodian А. Computing persistent homology / А. Zomorodian, G. Carlsson // Disc. Comput.Geom. — 2005. — V. 33. — No 2. — P. 249–274.

Zomorodian A. Advances in Applied and Computational Topology / Afra Zomorodian. — NY: AMS, 2012. — P. 232. — (AMS Short Course on Computational Topology; V. 70).

Ghrist R. Barcodes: The persistent topology of data // Bull.Amer.Math.Soc. — 2008. — V. 45. — № 1. — P. 61–75.

Silva V. Homological sensor networks / V. Silva, R. Ghrist // Not. Amer. Math. Soc. — 2007. — V. 54. — № 1. — P. 10–17.

Dey T. Optimal homologous cycles, total unimodularity and linear programming / T. Dey, A. N. Hirani, B. Krishnamoorthy // SIAM Jour. Comp. — 2011.— V. 40. — № 4. — P. 1026–1040.

Cohen-Steiner D. Vines and vineyards by updating persistence in linear time / D. Cohen-Steiner, H. Edelsbrunner, D. Morozov // Proc.22nd Annu. Sympos. Comput. Geom. — 2006. — Р. 119–134.

Edelsbrunner H. Topological persistence and simplification / H. Edelsbrunner, D. Letscher, A. Zomorodian // Disc. Comput. Geom. — 2002. — V. 28. — № 4. — Р. 511–533.

Carlsson G. Topology and data // Bull. Amer. Math. Soc. — 2009. — V. 46. — № 2. — Р. 255–308.

Morozov D. Dionysus. [Electronic Resourse]. — Mode of access: URL: http://www.mrzv.org/software/dionysus/

Sexton H. JPlex / H. Sexton, M.V. Johansson // [Electronic Resourse]. — Mode of access: URL:

http://www.comptop.stanford.edu/programs/jplex

Downloads

Published

2014-05-28

How to Cite

Юрчук, І. (2014). PERSISTENT HOMOLOGY METHOD OF TOPOLOGICAL DATA ANALYSIS. Science-Based Technologies, 23(3), 289–294. https://doi.org/10.18372/2310-5461.23.7397

Issue

Vol. 23 No. 3 (2014)

Section

Information and Communication Systems and Networks

License

The scientific journal adheres to the principles of Open Access and provides free, immediate, and permanent access to all published materials without financial, technical, or legal barriers for readers.

All articles are published in Open Access under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

Copyright

Authors who publish their works in the journal:

  • retain the copyright to their publications;

  • grant the journal the right of first publication of the article;

  • agree to the distribution of their materials under the CC BY 4.0 license;

  • have the right to reuse, archive, and distribute their works (including in institutional and subject repositories), provided that proper reference is made to the original publication in the journal.