Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning
This work addresses a bottleneck in topological data analysis for manifold data, enabling machine learning applications in fields like computational biology, though it appears incremental as it builds on existing de Rham-Hodge theory.
The authors tackled the problem of applying topological data analysis to manifold data by introducing persistent de Rham-Hodge Laplacians in Eulerian representation, which avoids numerical inconsistencies from remeshing, and demonstrated its effectiveness in predicting protein-ligand binding affinities on benchmark datasets.
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.