Persistent spectral graph.,International Journal for Numerical Methods in Biomedical Engineering

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Persistent spectral graph.
International Journal for Numerical Methods in Biomedical Engineering ( IF 2.2 ) Pub Date : 2020-08-17 , DOI: 10.1002/cnm.3376
Rui Wang ₁ , Duc Duy Nguyen ₁ , Guo-Wei Wei _{1,

2,

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Affiliation

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.

中文翻译：

持久光谱图。

持久同源性仅限于纯粹的拓扑持久性，而多尺度图仅考虑几何信息。这项工作引入了持久性谱理论来创建一个统一的低维多尺度范式，用于揭示拓扑持久性并从高维数据集中提取几何形状。对于点云数据集，过滤过程用于生成一系列链复合体以及相关的单纯复合体和链族，从中我们构建持久组合拉普拉斯矩阵。我们证明，可以从持久组合拉普拉斯矩阵的调和持久谱（即具有零特征值的谱）完全恢复一整套拓扑持久性。然而，由过滤引起的拉普拉斯矩阵的非调和谱为数据分析、建模和预测提供了另一个强大的工具。在这项工作中，通过使用谐波光谱和非谐波持久光谱来预测富勒烯的稳定性，而后者的光谱被成功设计用于分析富勒烯的结构和模型蛋白质的灵活性，这不能直接从当前的持久同源性中提取。研究发现，所提出的方法可以对当前流行的生物物理模型失效的蛋白质 B 因子提供出色的预测。

更新日期：2020-08-17

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