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

arXiv: https://arxiv.org/abs/2408.00220 · HTML: https://mlsanity.com/paper/2408.00220/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 72 references that are in the map.
- First posted on arXiv 1 Aug 2024.

## Builds on (most cited first, 7 of 7)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Geometric Graph Learning with Extended Atom-Types Features for Protein-Ligand Binding Affinity Prediction](https://mlsanity.com/paper/2301.06194/) (2023)
- [A generalization of the persistent Laplacian to simplicial maps](https://mlsanity.com/paper/2302.03771/) (2023)

## Built on by (most cited first, 3 of 3)

- [Manifold Topological Deep Learning for Biomedical Data](https://mlsanity.com/paper/2503.00175/) (2025)
- [Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions](https://mlsanity.com/paper/2506.21352/) (2025)
- [Combinatorial Network-Based Manifold Topological Deep Learning for Image Analysis](https://mlsanity.com/paper/2609.25453/) (2026)

## Abstract (arXiv)

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.