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

Abstract

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 introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning. Our PHLs were constructed in the Eulerian representation via structure-persevering Cartesian grids, avoiding the numerical inconsistency over the multi-scale manifolds. To facilitate the manifold topological learning, we proposed 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 considered the prediction of protein-ligand binding affinities with two benchmark datasets. Our numerical experiments highlighted the power and promise of the proposed method.