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

Area of Science:

• Computational topology
• Data science
• Scientific computing

Background:

• Topological data analysis (TDA) and persistent homology are powerful tools, but limited to point cloud data.
• Existing methods for manifold data, like evolutionary de Rham-Hodge theory, suffer from numerical inconsistencies in machine learning contexts.
• There is a need for robust TDA methods applicable to data residing on manifolds.

Purpose of the Study:

• To develop a novel topological learning framework for data defined on manifolds.
• To address the limitations of existing persistent homology methods for manifold-structured data.
• To enable consistent and efficient topological analysis of manifold data in machine learning.

Main Methods:

• Introduction of the persistent de Rham-Hodge Laplacian (PHL) for manifold topological learning.
• Construction of PHLs in the Eulerian representation using structure-persevering Cartesian grids.
• Development of a persistent Hodge Laplacian learning algorithm for manifold and volumetric data.

Main Results:

• PHLs avoid numerical inconsistencies associated with remeshing in Lagrangian representations.
• The proposed method facilitates manifold topological learning on multi-scale manifolds.
• Successful application in predicting protein-ligand binding affinities using benchmark datasets.

Conclusions:

• The persistent Hodge Laplacian offers a robust and numerically consistent approach for topological learning on manifolds.
• This method expands the applicability of TDA to complex, manifold-structured datasets.
• The framework shows significant potential for applications in computational biology and other scientific domains.