Evolutionary de Rham-Hodge method

2019/12/28 by Jiahui Chen, Rundong Zhao, Chen, Jiahui +6 · 9 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Cell Image Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Topological and Geometric Data Analysis #math.DG

paper · pdf · doi:10.48550/arxiv.1912.12388

28 pages, 15 figures

arxiv created 2019/12/28 · openalex publication_date 2019/12/28 · arxiv updated 2020/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The de Rham-Hodge theory is a landmark of the 20^th Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacian operators are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1, and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the utility and usefulness of the proposed method for data representation and shape analysis.

Citations

• Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
• A Topological View of Unsupervised Learning from Noisy Data
• Can One Hear the Shape of a Drum?
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• PETLS: PErsistent Topological Laplacian Software
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• Weighted Hodge Laplacians on Manifolds with Boundary

Related

• Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning
• Discretization of Riemannian manifolds applied to the Hodge Laplacian
• Hodge theory in the Sobolev topology for the de Rham complex
• Hodge Laplacians and Hodge Diffusion Maps
• Combinatorial and Hodge Laplacians: Similarity and Difference