🤖 AI Summary
Addressing the core challenges in persistent homology—namely, cycle feature identification, cross-dataset cycle matching, and cycle community construction—this paper proposes a dual-path framework. First, it introduces a novel Wasserstein distance metric integrated with merge trees to construct discriminative cycle dendrogram representations. Second, it designs Stratified Gradient Sampling—a hierarchical, multi-filter function co-learning strategy—that enables non-overlapping, exhaustive partitioning of cycle communities. By optimizing cycle centroid functions and generating topology-driven filter functions, the model achieves precise reconstruction of multiple independent cycles and structural alignment across objects on synthetic data. The framework significantly enhances interpretability and transferability of the cycle space, enabling robust topological analysis across diverse datasets while preserving geometric and topological fidelity.
📝 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.