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arXiv · 1912.04135

Persistent spectral graph

Rui Wang·Duc Duy Nguyen·Guo-Wei Wei
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Abstract

Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for revealing topological persistence and extracting geometric shape from high-dimensional datasets. For a point-cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, i.e., the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non-harmonic spectra of the Laplacian matrices induced by the filtration offer another power tool for data analysis, modeling, and prediction. In this work, non-harmonic persistent spectra are successfully devised to analyze the structure and stability of fullerenes and predict the B-factors of a protein, which cannot be straightforwardly extracted from the current persistent homology. Extensive numerical experiments indicate the tremendous potential of the proposed persistent spectral analysis in data science.

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Rui Wang, Duc Duy Nguyen, Guo-Wei Wei. 2019-12-09. Persistent spectral graph. https://arxiv.org/abs/1912.04135

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