The graph Laplacian is a fundamental object in the analysis of and optimization on graphs. This operator can be extended to a simplicial complex $K$ and therefore offers a way to perform ``signal processing" on $p$-(co)chains of $K$. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes $K\hookrightarrow L$ connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we expand the scope of the persistent Laplacian by generalizing it to a pair of simplicial complexes connected by a simplicial map $f: K \to L$. Such simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets i.e. cases in which it does not hold that $K\subset L$. However, the simplicial map setting is more challenging than the inclusion setting since the underlying algebraic structure is more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint PSD operator to a given subspace. As a consequence, we prove that persistent Betti numbers of a simplicial map can be recovered by persistent Laplacians. We then propose an algorithm for finding the matrix representations of persistent Laplacians which in turn yields a new algorithm for computing persistent Betti numbers of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under simplicial maps and establish monotonicity results for their eigenvalues.

翻译：图拉普拉斯算子是图分析与优化中的基本对象。该算子可扩展至单纯复形$K$，从而为$K$上的$p$-(余)链提供“信号处理”途径。近期，针对通过包含关系$K\hookrightarrow L$相连的一对单纯复形，持久拉普拉斯算子的概念被提出并研究，进一步拓展了拉普拉斯型算子的应用范围。本文通过将持久拉普拉斯算子推广至由单纯映射$f: K \to L$连接的一对单纯复形，扩展了其适用范围。此类单纯映射场景频繁出现，例如当粗化单纯表示与原始表示相关联时，或当两个单纯复形由不同点集生成时（即不满足$K\subset L$的情形）。然而，由于底层代数结构更为复杂，单纯映射场景比包含场景更具挑战性。我们提出了持久拉普拉斯算子至单纯场景的自然推广。为揭示其内在结构并设计计算算法，我们利用持久拉普拉斯算子与矩阵Schur补之间的关系，关键步骤是将Schur补视为将自伴半正定算子函子式限制到给定子空间的方法。由此证明，单纯映射的持久贝蒂数可通过持久拉普拉斯算子恢复。进而提出持久拉普拉斯算子矩阵表示的计算算法，该算法同时提供了计算单纯映射持久贝蒂数的新方法。最后，我们研究单纯映射下单纯塔上的持久拉普拉斯算子，并建立其特征值的单调性结果。