标题： 欧拉表示中流形拓扑学习的持久德·拉姆-霍奇拉普拉斯算子 显示英文标题

标题： Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning

摘要： 最近，拓扑数据分析已成为数据科学和工程领域的热门话题。 然而，拓扑数据分析的关键技术，即持续同调，是定义在点云数据上的，这无法直接用于流形上的数据。 尽管早期的德拉姆-霍奇理论处理的是流形上的数据，但由于在拉格朗日表示中对涉及的流形进行重新网格化导致的数值不一致，使其在机器学习应用中不太方便。 在本工作中，我们引入了持续德拉姆-霍奇拉普拉斯算子，或简称为持续霍奇拉普拉斯（PHL），用于流形拓扑学习。 我们的PHL是在欧拉表示中通过结构保持的笛卡尔网格构建的，避免了多尺度流形上的数值不一致。 为了促进流形拓扑学习，我们提出了一个用于流形或体积数据的持续霍奇拉普拉斯学习算法。 作为所提出流形拓扑学习模型的原理性应用，我们考虑了使用两个基准数据集预测蛋白质-配体结合亲和力。 我们的数值实验突显了所提出方法的力量和前景。 显示英文摘要

摘要： Recently, topological data analysis has become a trending topic in data science and engineering. However, the key technique of topological data analysis, i.e., persistent homology, is defined on point cloud data, which does not work directly for data on manifolds. Although earlier evolutionary de Rham-Hodge theory deals with data on manifolds, it is inconvenient for machine learning applications because of the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation. In this work, we introduce persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL) as an abbreviation, for manifold topological learning. Our PHLs are constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multiscale manifolds. To facilitate the manifold topological learning, we propose a persistent Hodge Laplacian learning algorithm for data on manifolds or volumetric data. As a proof-of-principle application of the proposed manifold topological learning model, we consider the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlight the power and promise of the proposed method.