HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.
HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.
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DOI：
10.3934/fods.2021017
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发表时间：
2021-12
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单形复形对的p-持久q-组合拉普拉斯算子是q-组合拉普拉斯算子的推广。在给定过滤条件下，持久组合拉普拉斯谱不仅恢复了持久同调的持久Betti数，还提供了数据的额外多尺度几何信息。与机器学习算法相结合，持久拉普拉斯在数据科学中有许多潜在的应用。寻找不同的方法来寻找操作者的光谱是一个活跃的研究课题，当想法来自多个领域时，就会变得有趣。在这项工作中，我们探索了另一种方法来研究顽固的拉普拉斯人的频谱。由于持久拉普拉斯矩阵的特征值是其特征多项式的根，因此可以尝试通过同伦延拓来求特征多项式的根，从而求出相应的持久拉普拉斯矩阵的谱。我们考虑一组简单的多面体和小分子来证明代数拓扑、组合图和代数几何可以结合起来理解数据的形状的原理。
The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.