A Three Axis Evaluation Framework for Mapper Algorithms

Abstract

Mapper is a well-known tool in topological data analysis, which visualizes and summarizes high-dimensional data. However, its output is sensitive to choices of lens functions, cover parameters, and clustering strategies, making evaluation challenging. Most works that have attempted to evaluate the Mapper algorithm have done so visually. In this paper, we review a roadmap for assessing Mapper algorithms along three complementary axes: stability, cluster quality, and topological shape preservation. We analyze Mapper and its variants on synthetic datasets and the UCI Digits dataset. These modes include topological explosion at high resolutions. Our findings indicate that these axes of evaluation are often in tension and that no single Mapper variant performs optimally across all three. This review provides practical guidelines for choosing Mapper variants and identifies open challenges toward a principled Mapper analysis.