Stable topological signatures for metric trees through graph approximations

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