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

TL;DR

This paper introduces persistent Hodge Laplacians in Eulerian representation for manifold topological learning, enabling persistent homology analysis directly on manifold data without remeshing, with applications to protein-ligand binding prediction.

Contribution

The paper proposes a novel persistent Hodge Laplacian in Eulerian form for manifold data, improving numerical stability and applicability in machine learning tasks.

Findings

Effective in predicting protein-ligand binding affinities

Avoids remeshing issues in manifold data analysis

Shows promising results on benchmark datasets

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 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.