Point-Level Topological Representation Learning on Point Clouds
This work addresses the gap in applying topological data analysis to point-level tasks for researchers and practitioners in fields like computer vision and 3D data processing, though it appears incremental as it builds on existing TDA tools.
The paper tackled the problem of extracting point-level topological features from point clouds, which is needed for machine learning applications like classification, by proposing a novel method using discrete variants of algebraic topology and differential geometry, and verified its effectiveness on synthetic and real-world data with robustness under noise and heterogeneous sampling.
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 and heterogeneous sampling.