论文标题
持续的拉普拉斯人：属性，算法和含义
Persistent Laplacians: properties, algorithms and implications
论文作者
论文摘要
我们对理论属性进行了详尽的研究，并设计了\ emph {持久性拉普拉斯}的有效算法，这是标准组合拉普拉斯式的扩展到对成对的设置（或更常规的序列）的设置（或者更常规的序列）。以及Wang等人。特别是，与非持久性案例相比，我们首先证明了$ q $ -th持久的laplacian $Δ_Q^{k，l} $等于$ q $ - Q $ -th persistent betti betti the包含$（k \ jkingrightarrow l）$。然后，我们提出了一种初始算法，用于查找$Δ_Q^{k，l} $的矩阵表示形式，该算法本身有助于解释持久的laplacian。我们表现出持久的拉普拉斯主义者与矩阵补充的概念之间的新颖关系，该矩阵具有一些重要的含义。在图案例中，它俩都发现了与有效阻力概念的链接，并导致了Cheeger不平等的持续版本。这种关系还产生了一种额外的，非常简单的算法，用于查找$ q $ -th持续的laplacian，进而导致一种小说和根本不同的算法来计算$ q $ -th persistent betti persistent betti的betti betti number（k，k，l）$比标准算法更有效的效率。最后，我们研究了持久的拉普拉斯人的简单过滤，并为其特征值提供了新的稳定性结果。我们的工作从光谱图理论，电路理论和持久同源性带来了方法，以及对简单复合物的组合拉普拉斯式的拓扑视图。
We present a thorough study of the theoretical properties and devise efficient algorithms for the \emph{persistent Laplacian}, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes $K \hookrightarrow L$, which was independently introduced by Lieutier et al. and by Wang et al. In particular, in analogy with the non-persistent case, we first prove that the nullity of the $q$-th persistent Laplacian $Δ_q^{K,L}$ equals the $q$-th persistent Betti number of the inclusion $(K \hookrightarrow L)$. We then present an initial algorithm for finding a matrix representation of $Δ_q^{K,L}$, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional, very simple algorithm for finding (a matrix representation of) the $q$-th persistent Laplacian which in turn leads to a novel and fundamentally different algorithm for computing the $q$-th persistent Betti number for a pair $(K,L)$ which can be significantly more efficient than standard algorithms. Finally, we study persistent Laplacians for simplicial filtrations and present novel stability results for their eigenvalues. Our work brings methods from spectral graph theory, circuit theory, and persistent homology together with a topological view of the combinatorial Laplacian on simplicial complexes.
