Node-Level Topological Representation Learning on Point Clouds

Vincent Peter Grande, Michael T Schaub

Keywords: Topological Data Analysis, TDA, Hodge Laplacian, Higher-Order Networks, Simplicial Complexes, Algebraic Topology, Differential Geometry, Point Clouds, Persistent Homology

Abstract: Topological Data Analysis (TDA) allows us to extract powerful topological, and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise.

Primary Area: Other (please use sparingly, only use the keyword field for more details)

Submission Number: 16537