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.

翻译：在不同拓扑对象中识别并比较拓扑特征（尤其是环结构）仍然是持久同调与拓扑数据分析领域的一项基础性挑战。本研究提出了一个通过两种互补方法构建环社区的新框架。首先，基于树状图的方法利用合并树算法，从持久区间构建同调类的层次化表示。通过引入合并树上的Wasserstein距离作为度量树状图差异的指标，建立了与层次聚类框架的理论联系。仿真研究表明，树状图表征在识别环社区方面具有显著的区分能力。其次，通过扩展分层梯度采样方法，同时学习多个滤波函数，这些函数能够生成可忠实重构不同环集的环重心函数。每个滤波函数所能重构的环集合构成了互不重叠且能划分所有环空间的环社区。综合来看，这两种方法将环匹配问题转化为层次聚类与拓扑优化的双重框架，为识别拓扑对象组内及组间相似拓扑结构提供了理论严谨的方法体系。