## APPLICATIONS OF PERSISTENT HOMOLOGY IN FINITE METRIC SPACES AND CLUSTERING

A. M. RASHAD

Department of Mathematics, Faculty of Science, Aswan University, Egypt.

H. N. ALAA *

Department of Mathematics, Faculty of Science, Aswan University, Egypt.

M. S. MOHAMMED

Faculty of Fine Arts-Luxor, Luxor University, Egypt.

*Author to whom correspondence should be addressed.

### 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.

#### 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

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