Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal

TL;DR

This paper explores the spectral properties of the Hodge Laplacian, revealing that small eigenvalues encode different topological and geometric information depending on their associated eigenmodes, and introduces methods to analyze and utilize this spectrum.

Contribution

It introduces persistent eigenvector similarity, enabling tracking of harmonic, curl, and gradient eigenvectors across scales, and proposes new spectral clustering and classification techniques.

Findings

Eigenvalues relate to different topological features.

Persistent eigenvector similarity tracks eigenvectors across scales.

New spectral clustering and classification methods are proposed.

Abstract

The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge Laplacian has come into focus as a generalisation of the ordinary Laplacian for higher-order graph models such as simplicial and cellular complexes. Akin to the traditional analysis of graph Laplacians, many authors analyse the smallest eigenvalues of the Hodge Laplacian, which are connected to important topological properties such as homology. However, small eigenvalues of the Hodge Laplacian can carry different information depending on whether they are related to curl or gradient eigenmodes, and thus may not be comparable. We therefore introduce the notion of persistent eigenvector similarity and provide a method to track individual harmonic, curl, and gradient eigenvectors/-values through the so-called persistence filtration, leveraging the full information contained in the Hodge-Laplacian spectrum across all possible scales of a point cloud. Finally, we use our insights (a) to introduce a novel form of Hodge spectral clustering and (b) to classify edges and higher-order simplices based on their relationship to the smallest harmonic, curl, and gradient eigenvectors.