持久拉普拉斯算子：性质、算法和含义
Persistent Laplacians: properties, algorithms and implications
摘要：
我们提出了一个深入的研究的理论性质和设计有效的算法的\emph{persistent Laplacian}，扩展的标准组合拉普拉斯设置对（或者，更一般地，序列）的单纯复形$K \hookrightarrow L$，其由Lieutier等人和Wang等人独立地引入。特别地，与非持久情况类似，首先证明了$q$次持久拉普拉斯$Δ_q^{K,L}$的零性等于包含$(K \hookrightarrow L)$的$q$次持久贝蒂数.然后，我们提出了一个初始算法，找到一个矩阵表示的$Δ_q^{K,L}$，这本身有助于解释持久的拉普拉斯算子。我们展示了一个新的持久的拉普拉斯算子和Schur补矩阵的概念之间的关系，有几个重要的影响。在图的情况下，它既揭示了与有效阻力概念的联系，又导致了契格不等式的持久版本。这种关系还产生了一种额外的、非常简单的算法，用于寻找第$q$个持久拉普拉斯算子（的矩阵表示），这反过来又产生了一种新颖且根本不同的算法，用于计算一对$(K,L)$的第$q$个持久贝蒂数，这可以比标准算法显着更有效。最后，我们研究了单纯滤波的持久拉普拉斯算子，并给出了其特征值的新稳定性结果。我们的工作带来的方法，从谱图论，电路理论，持久的同源性与拓扑视图的组合拉普拉斯算子的单纯复形。
英文摘要：
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.
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