Persistent de Rham-Hodge Laplacians in the Eulerian representation

Recently, topological data analysis (TDA) has become a trending topic in data science and engineering. However, the key technique of TDA, i.e., persistent homology, is defined on point cloud data, which restricts its scope. In this work, we propose persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL) for abbreviation, for the TDA on manifolds with boundaries, or volumetric data. Specifically, we extended the evolutionary de Rham-Hodge theory from the Lagrangian formulation to the Eulerian formulation via structure-persevering Cartesian grids, and extended the persistent Laplacian on point clouds to persistent (de Rham-)Hodge Laplacian on nested families of manifolds with appropriate boundary conditions. The proposed PHL facilitates the machine learning and deep learning prediction of volumetric data. For a proof-of-principle application of the proposed PHL, we propose a persistent Hodge Laplacian learning (PHLL) algorithm for data on manifolds or volumetric data. To this end, we showcase the PHLL prediction of protein-ligand binding affinities in two benchmark datasets. Our numerical experiments highlight the power and promise of PHLL.