The Higher-Dimensional Skeletonization Problem

Data is often given in a form of a point cloud. Its shape plays an important role --- for example, it is good to know if data lies along a line or if it can be divided into several components, called clusters, based on similarity. One of the most popular tools for measuring the shape of a point cloud is called persistent homology. While there have been plenty of papers written about the algebraic foundations of persistent homology and stability results, not much work has been done on visualizing homology generators within these point clouds at different scales. Vitaliy Kurlin defined a homologically persistent skeleton, which depends only on a point cloud and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. In this paper we generalize his results to higher dimensions.