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A new non-archimedean metric on persistent homology

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Abstract

In this article, we define a new non-archimedean metric structure, called cophenetic metric, on persistent homology classes of all degrees. We then show that zeroth persistent homology together with the cophenetic metric and hierarchical clustering algorithms with a number of different metrics do deliver statistically verifiable commensurate topological information based on experimental results we obtained on different datasets. We also observe that the resulting clusters coming from cophenetic distance do shine in terms of different evaluation measures such as silhouette score and the Rand index. Moreover, since the cophenetic metric is defined for all homology degrees, one can now display the inter-relations of persistent homology classes in all degrees via rooted trees.

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Notes

  1. The source code and the data of the numerical experiments we conducted in the paper can be found on the authors’ GitHub page at https://github.com/ismailguzel/TDA-HC .

  2. The computational tools we use in this section are as follows: To compare dendrograms, we use tools come from the dendextend (Galili 2015) and vegan (Oksanen et al. 2019) packages of the R programming language (R Core Team 2021). For the map of Turkey, we used Generic Mapping Tools (Wessel et al. 2019). To compute cophenetic distance matrix we used SageMath Developers et al. (2020). In order to compute and visualize clusters, we used python programming language (Van Rossum and Drake 2009) and its scikit-learn library (Pedregosa et al. 2011).

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Acknowledgements

The first author was supported by Research Fund Project Number TDK-2020-42698 of the Istanbul Technical University.

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  1. Department of Mathematics Engineering, Istanbul Technical University, Istanbul, Turkey

    İsmail Güzel & Atabey Kaygun

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  1. İsmail Güzel
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  2. Atabey Kaygun
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Correspondence to İsmail Güzel.

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Güzel, İ., Kaygun, A. A new non-archimedean metric on persistent homology. Comput Stat 37, 1963–1983 (2022). https://doi.org/10.1007/s00180-021-01187-z

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