Point-Level Topological Representation Learning on Point Clouds

TL;DR

This paper introduces a novel method for extracting point-level topological features from point clouds, bridging the gap between global topological analysis and local feature requirements in machine learning tasks.

Contribution

It proposes a new approach to derive node-level topological features using algebraic topology and differential geometry, enhancing point cloud analysis.

Findings

Effective topological features on synthetic data

Robustness under noise and heterogeneous sampling

Improved classification performance

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 and heterogeneous sampling.