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

TL;DR

This work introduced persistent de Rham-Hodge Laplacian, or persistent Hodge Laplacian (PHL), as an abbreviation for manifold topological learning, and considered the prediction of protein-ligand binding affinities with two 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.

Figures (10)

• Figure 1: The chain complex of a single-cell grid formed by the boundary operator: from the face, to its edges, and to their vertices.
• Figure 2: An example of the primal and dual grid cells for the 2D case. The top row highlights the primal cells, and the bottom row presents their corresponding dual cells.
• Figure 3: Distinction of normal supports (left) and tangential supports (right) for primal $1$-forms in a 2D Cartesian grid.
• Figure 4: An example of a nested sequence of sub-cell complexes in a 2D Cartesian grid under the normal boundary condition, illustrating the inclusion of normal supports for $0$, $1$, and $2$ discrete differential forms for an evolution of manifolds. Here the manifolds are represented by the bounded regions of the blue isocurves of a level set function.
• Figure 5: First row: Snapshots of evolving manifolds for the Bimba model. Second row: Changes in Betti numbers $\beta_0$, $\beta_1$, $\beta_2$ and the first non-zero eigenvalues in T, C, N along $20$ evenly spaced isovalues from $0$ to $0.2$. Here the first shape in the top first row corresponds to isovalue $0$ and the last shape in the first row corresponds to isovalue $0.2$. $\lambda^T_1$, $\lambda^C_1$ and $\lambda^N_1$ are the first non-zero eigenvalues in the set T, C, N, respectively. The signed distance function generated from the original Bimba model is used as the level set function.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4