Graph skeletonization of high-dimensional point cloud data via topological method

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 behind a point cloud dataset (PCD) embedded in high-dimensional space. 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 (in the sense that the input points are within a small bounded distance to a true hidden graph). 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 triangulation of low-dimensional Euclidean domain. 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, this generalization allows us to use the idea of sparsified weighted Rips filtration (developed by Buchet et al) to develop a new graph reconstruction algorithm for noisy point cloud data (PCD) (which do not need to be embedded). The new algorithm is robust to background noise and non-uniform distribution of input points. We provide various experimental results to show the efficiency and effectiveness of our new graph reconstruction algorithm for PCDs.