A Comparative Study of Persistence and Mapper in Topological Data Analysis
TL;DR
Together, persistent homology and Mapper provide complementary topological perspectives on the same data: persistent homology identifies a quantitatively distinct topological structure between the benign and malignant groups (quantified via the 1-Wasserstein distance), while Mapper localizes where this difference manifests in the data as transition zones and class-dominated subgraphs.
Abstract
Topological data analysis offers techniques for uncovering structure in high-dimensional datasets, with persistent homology and the Mapper algorithm being two widely used methods. Persistent homology captures topological features across filtration scales, while Mapper constructs a graph-based summary of the data at a chosen resolution. In this paper, we apply both methods to the Wisconsin Breast Cancer Dataset in order to compare the type of structural information each method provides. Rather than treating persistent homology and Mapper as competing tools, we frame them as complementary approaches. Persistent homology summarizes the underlying shape of the data across different scales, while Mapper highlights clustering structure, local connectivity, and regions where benign and malignant samples exhibit overlapping characteristics in the feature space. We further support the visual persistent homology analysis using Wasserstein distances between persistent diagrams. Together, persistent homology and Mapper provide complementary topological perspectives on the same data: persistent homology identifies a quantitatively distinct topological structure between the benign and malignant groups (quantified via the 1-Wasserstein distance), while Mapper localizes where this difference manifests in the data as transition zones and class-dominated subgraphs. Neither method alone provides both the global algebraic summary and the local graph-level resolution that the combined analysis yields.