🤖 AI Summary
This work identifies a critical topological distortion issue in the Mapper algorithm from Topological Data Analysis (TDA) when employing non-hierarchical clustering methods—such as k-means—as the local clustering step. Such methods can arbitrarily alter the homology groups of the resulting Mapper complex, thereby violating topological fidelity to the underlying data. Through rigorously constructed counterexamples and theoretical analysis, we establish—for the first time—that widely used clustering algorithms lack topological stability within the Mapper framework, and their induced distortions are inherently unbounded. In contrast, connectivity-preserving hierarchical methods—e.g., single-linkage clustering—are shown to be fundamentally better aligned with Mapper’s topological modeling objectives. Our findings provide a crucial theoretical caution for reliable Mapper deployment and articulate explicit topological criteria for clustering method selection. This advances robustness research in TDA by grounding algorithmic design in formal topological guarantees.
📝 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.