Topologically Relevant Landmark Selection on Point Clouds

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.