# APPLICATIONS OF PERSISTENT HOMOLOGY IN FINITE METRIC SPACES AND CLUSTERING

## Main Article Content

## Abstract

Finite data sets are considered as metric spaces with defining the dissimilarity measure between the data points of them. In this work, we describe the persistent homology flowchart (PHF) in detail and study the persistence barcodes of Vietoris-Rips. PHF starts with a finite metric space and ends with a persistence diagram encoding features of the input data set. We compute and -dimensional persistence diagrams and barcodes of the -point metric space, . Finally, as an application on PH flowchart, we interpret the single linkage clustering of via -dimensional persistence diagram , we then deduce that persistent homology generalizes clustering.

Keywords:

Persistent homology, metric spaces, clustering and diagrams.

## Article Details

How to Cite

*Asian Journal of Mathematics and Computer Research*,

*27*(1), 14-27. Retrieved from https://ikprress.org/index.php/AJOMCOR/article/view/4959

Section

Original Research Article

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Carlsson G, Mémoli F. Classifying clustering schemes. Foundations of Computational Mathematics. 2013;13(2):221-252.

Jacobson N. Basic Algebra II: Second Edition. Dover Books on Mathematics. Dover Publications; 2012.

Spivak DI. Category theory for the sciences. The MIT Press. MIT Press; 2014.

Carlsson G, Mémoli F. Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 2010;11:1425-1470.

Munkres JR. Elements of algebraic topology. Avalon Publishing; 1996.

Carlsson G. Topology and data. Bulletin of the American Mathematical Society. 2009;46(2):255-308.

Edelsbrunner H, Harer J. Computational topology – an introduction. American Mathematical Society; 2010.

Edelsbrunner H. A short course in computational geometry and topology. Springer Briefs in Applied Sciences and Technology. Springer, Cham; 2014.

Ghrist RW. Elementary applied topology, volume 1. Createspace Seattle; 2014.

Oudot S. Persistence theory: From quiver representations to data analysis, volume 209 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI; 2015.

Carlsson G. Topological pattern recognition for point cloud data. Acta Numerica. 2014;23:289368.

Crawley-Boevey W. Decomposition of pointwise finite dimensional persistence modules; 2012.