Topologically Relevant Landmark Selection on Point Clouds

2026-08-04

AI Summary

This study addresses the computational expense of persistent homology (PH) for large-scale point clouds and the inability of existing landmark selection methods to recover global topological features. To overcome these limitations, this work proposes a Hodge Laplacian-based landmark selection method that, for the first time, defines landmark scores from the perspective of global topological relevance rather than geometric distribution. By integrating persistent homology, Hodge Laplacian theory, and harmonic analysis techniques, the proposed approach constructs a topological relevance model that quantifies each point’s contribution to the overall homological structure, thereby identifying critical landmarks. Experiments on both synthetic and real-world datasets demonstrate that this method approximates the persistent homology of the original point cloud more accurately than geometric baselines and local PH approaches, effectively overcoming the limitations of conventional techniques.

Abstract

Persistent Homology (PH) is an important tool in Topological Data Analysis for point clouds, but can be prohibitively expensive for large datasets. A practical approximation is to compute PH on a smaller subset of representative points, known as landmarks. Existing landmark selection methods mainly address challenges such as outlier contamination rather than restoring the topological features of the full dataset. To select topologically relevant landmarks, we propose a landmark selection method based on Hodge Laplacian. It assigns each point a topological relevance score based on its harmonic participation and selects points with high scores as landmarks. By quantifying each point's participation in global homological structures, our method restores the PH of the original point cloud more accurately than geometric baselines and a method built upon local PH on both synthetic and real-world datasets.

Problem

• Persistent Homology
• Landmark Selection
• Point Clouds
• Topological Data Analysis
• Hodge Laplacian

Innovation

• Hodge Laplacian
• Landmark Selection
• Persistent Homology
• Topological Data Analysis
• Harmonic Participation