Persistent spectral graph

• Published in: International journal for numerical methods in biomedical engineering Vol. 36; no. 9; pp. e3376 - n/a
• Main Authors: Wang, Rui, Nguyen, Duc Duy, Wei, Guo‐Wei
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.09.2020; Wiley Subscription Services, Inc
• Subjects: Chains; Combinatorial analysis; Data Analysis; Datasets; Eigenvalues; Filtration; Fullerenes; Homology; persistent spectral analysis; persistent spectral graph; persistent spectral theory; Predictions; Protein B; Protein structure; Proteins; Proteins - chemistry; Spectra; spectral data analysis; Spectral theory; Stability analysis; Topology; persistent spectral graph; spectral data analysis; persistent spectral analysis; persistent spectral theory
• ISSN: 2040-7939; 2040-7947; 2040-7947
• DOI: 10.1002/cnm.3376

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 shapes 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, that is, 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 powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non‐harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down. 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. We show that a full set of topological persistence could be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. The proposed method is found to provide excellent predictions of the protein B‐factors and fullerene stability for which current popular biophysical models break down.