Point-Level Topological Representation Learning on Point Clouds

2024-06-04

AI Summary

Point cloud classification suffers from a disconnect between global topological features and local point-level discriminative capacity, while remaining vulnerable to noise and non-uniform sampling. Method: We propose TOPF—the first end-to-end differentiable framework for point-wise topological feature extraction—integrating discrete differential geometry (e.g., discrete curvature) with algebraic topology (local persistent homology, Euler transform) to construct geometric-topological joint representations, and leveraging graph neural networks for feature propagation and aggregation. Contribution/Results: Unlike conventional topological data analysis (TDA) methods that yield only global descriptors, TOPF generates learnable, semantically meaningful topological embeddings for each point. Evaluated on both synthetic and real-world datasets, TOPF achieves significant improvements in point-level classification accuracy and demonstrates strong robustness against noise and sampling non-uniformity.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Data transparency and provenance

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.

Problem

• Point Cloud Data
• Advanced Spatial Feature Learning
• Machine Learning Classification

Innovation

• Algebraic Topology
• Differential Geometry
• Point Cloud Feature Learning