The rising field of Topological Data Analysis (TDA) provides a new approach to learning from data through persistence diagrams, which are topological signatures that quantify topological properties of data in a comparable manner. For point clouds, these diagrams are often derived from the Vietoris-Rips filtration—based on the metric equipped on the data—which allows one to deduce topological patterns such as components and cycles of the underlying space. In metric trees these diagrams often fail to capture other crucial topological properties, such as the present leaves and multifurcations. Prior methods and results for persistent homology attempting to overcome this issue mainly target Rips graphs, which are often unfavorable in case of non-uniform density across our point cloud. We therefore introduce a new theoretical foundation for learning a wider variety of topological patterns through any given graph. Given particular powerful functions defining persistence diagrams to summarize topological patterns, including the normalized centrality or eccentricity, we prove a new stability result, explicitly bounding the bottleneck distance between the true and empirical diagrams for metric trees. This bound is tight if the metric distortion obtained through the graph and its maximal edge-weight are small. Through a case study of gene expression data, we demonstrate that our newly introduced diagrams provide novel quality measures and insights into cell trajectory inference.

的上升场拓扑数据分析（TDA）提供了一种新的方法来从通过数据中学习的持久性的图，其是拓扑签名以可比较的方式数据的定量表达拓扑性质。对于点云，这些图通常来自Vietoris-Rips过滤（基于数据所配备的度量标准），从而可以推断拓扑模式，例如基础空间的组成和循环。在公制树中这些图通常无法捕获其他关键的拓扑特性，例如当前的叶子和分叉。尝试克服此问题的持久同源性的现有方法和结果主要针对Rips图，在点云上密度不均匀的情况下，这通常是不利的。因此，我们引入了新的理论基础，可以通过任何给定的图来学习更多的拓扑模式。给定定义余辉图以概括拓扑模式（包括归一化的中心性或离心率）的特定强大功能，我们证明了新的稳定性结果，明确地限制了瓶颈距离度量树的真实图和经验图之间。如果通过图形获得的度量失真及其最大边缘权很小，则此边界很严格。通过基因表达数据的案例研究，我们证明了我们新引入的图提供了新颖的质量度量和对细胞轨迹推断的见解。