A Notion of Harmonic Clustering in Simplicial Complexes

Abstract

We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of, graph spectral clustering. The algorithm involves only sparse eigenproblems, and is therefore computationally efficient. We believe that it has broad application as a way to extract features from simplicial complexes that often arise in topological data analysis.

References in corpus (4)

• Hierarchical structures of amorphous solids characterized by persistent homology
• Clique topology reveals intrinsic geometric structure in neural correlations
• Isoperimetric Inequalities in Simplicial Complexes
• Geometric Data Analysis Across Scales via Laplacian Eigenvector Cascading

Cited by in corpus (5)

• Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
• Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal
• Quantifying the structural stability of simplicial homology
• The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian
• Parallel computation of interval bases for persistence module decomposition