Cycles Communities from the Perspective of Dendrograms and Gradient Sampling

TL;DR

This paper introduces new methods for identifying and comparing cycle communities in topological data analysis using dendrograms and gradient sampling, enabling hierarchical clustering and cycle reconstruction.

Contribution

It presents a novel dendrogram-based approach with Wasserstein distance for cycle community detection and extends stratified gradient sampling for cycle reconstruction, advancing topological analysis techniques.

Findings

Dendrogram representations effectively distinguish cycle communities.

Wasserstein distance on merge trees improves cycle comparison.

Gradient sampling accurately reconstructs cycle sets.

Abstract

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.