Graph skeletonization of high-dimensional point cloud data via topological method (2109.07606v4)

Abstract: Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton of a high-dimensional point cloud dataset (PCD). Previously, there has been much work to recover a hidden graph from a low-dimensional density field, or from a relatively clean high-dimensional PCD. Our proposed approach builds upon the recent line of work on using a persistence-guided discrete Morse (DM) theory based approach to reconstruct a geometric graph from a density field defined over a low-dimensional triangulation. In particular, we first give a very simple generalization of this DM-based algorithm from a density-function perspective to a general filtration perspective. On the theoretical front, we show that the output of the generalized algorithm contains a so-called lexicographic-optimal persistent cycle basis w.r.t the input filtration, justifying that the output is indeed meaningful. On the algorithmic front, the generalization allows us to combine sparsified weighted Rips filtration to develop a new graph reconstruction algorithm for noisy point cloud data. The new algorithm is robust to background noise and non-uniform distribution of input points, and we provide various experimental results to show its effectiveness.