$k$-means considered harmful: On arbitrary topological changes in Mapper complexes

TL;DR

This paper highlights how common clustering choices in Mapper construction can cause significant distortions in the topological features, affecting the reliability of data analysis results.

Contribution

It demonstrates the impact of arbitrary clustering choices on Mapper complexes and warns against their uncritical use in topological data analysis.

Findings

Widespread clustering methods can distort Mapper features

Certain clustering choices lead to arbitrary topological changes

The paper provides examples of distortions in Mapper complexes

Abstract

The Mapper construction is one of the most widespread tools from Topological Data Analysis. There is an unfortunate trend as the construction has gained traction to use clustering methods with properties that end up distorting any analysis results from the construction. In this paper we will see a few ways in which widespread choices of clustering algorithms have arbitrarily large distortions of the features visible in the final Mapper complex.