We introduce, for the first time, a cohomology-based Gromov-Hausdorff ultrametric method to analyze 1-dimensional and higher-dimensional (co)homology groups, focusing on loops, voids, and higher-dimensional cavity structures in simplicial complexes, to address typical clustering questions arising in molecular data analysis. The Gromov-Hausdorff distance quantifies the dissimilarity between two metric spaces. In this framework, molecules are represented as simplicial complexes, and their cohomology vector spaces are computed to capture intrinsic topological invariants encoding loop and cavity structures. These vector spaces are equipped with a suitable distance measure, enabling the computation of the Gromov-Hausdorff ultrametric to evaluate structural dissimilarities. We demonstrate the methodology using organic-inorganic halide perovskite (OIHP) structures. The results highlight the effectiveness of this approach in clustering various molecular structures. By incorporating geometric information, our method provides deeper insights compared to traditional persistent homology techniques.

翻译：我们首次引入了一种基于上同调的Gromov-Hausdorff超度量方法，用于分析一维及更高维度的（上）同调群，重点关注单纯复形中的环、空洞及高维腔体结构，以解决分子数据分析中典型的聚类问题。Gromov-Hausdorff距离用于量化两个度量空间之间的不相似性。在此框架中，分子被表示为单纯复形，并计算其上同调向量空间以捕获编码环和腔体结构的内在拓扑不变量。这些向量空间配备了合适的距离度量，从而能够计算Gromov-Hausdorff超度量以评估结构差异。我们以有机-无机卤化物钙钛矿（OIHP）结构为例展示了该方法。结果突显了该方法在聚类多种分子结构方面的有效性。通过整合几何信息，我们的方法相较于传统的持续同调技术能够提供更深入的见解。