发表机构
National Key Laboratory for Novel Software Technology, Nanjing University; School of Artificial Intelligence, Nanjing University（南京大学新型软件技术全国重点实验室; 南京大学人工智能学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对大型点云持续同调计算昂贵的问题，提出基于Hodge拉普拉斯算子的地标选择方法，通过量化点的调和参与度选取拓扑相关地标，在合成与真实数据上更准确地恢复原始点云PH。
AI 中文摘要
持续同调（Persistent Homology, PH）是拓扑数据分析中用于点云的重要工具，但对于大型数据集而言，其计算成本可能过高。一种实用的近似方法是在较小的代表性点（称为地标）子集上计算PH。现有的地标选择方法主要解决诸如异常值污染等挑战，而非恢复完整数据集的拓扑特征。为了选择拓扑相关的地标，我们提出了一种基于Hodge拉普拉斯算子的地标选择方法。该方法根据每个点的调和参与度为其分配一个拓扑相关性分数，并选择得分高的点作为地标。通过量化每个点在全局同调结构中的参与程度，我们的方法在合成数据集和真实世界数据集上，比几何基线和基于局部PH的方法更准确地恢复了原始点云的PH。
英文摘要
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.
