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Instability of Mapper Type Algorithms for Topological Data Analysis Public Deposited

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  • Mapper is a tool designed by Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson for Topological Data Analysis (TDA) that constructs a simplicial complex from a finite subset of a metric space, which has been met with great success. Due to the many moving parts of Mapper and the potential it has shown in TDA, questioning its stability is natural. In this expository paper we follow the work of Francisco Belchí, Jacek Brodzki, Matthew Burfitt, and Mahesan Niranjan in their paper A Numerical Measure on the Instability of Mapper-Type Algorithms, with our own additions. In this paper we define a machine learning technique known as clustering and provide a definition of the Mapper algorithm. Following this, we outline a possible measure of instability proposed by Belchí, Brodzki, Burfitt, and Niranjan, based on the work of Shai Ben-David and Ulrike von Luxburg on clustering instability. Finally, we present a theorem by Belchí et. al. that provides a bound on the proposed instability measure.
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