PERSISTENT HOMOLOGY METHOD OF TOPOLOGICAL DATA ANALYSIS
DOI:
https://doi.org/10.18372/2310-5461.23.7397Keywords:
persistent homology method, Vietoris-Rips complexAbstract
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.
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