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图优化中的基本对象。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”方法。近年来，针对由包含关系连接的单纯复形对 $K\hookrightarrow L$，持久拉普拉斯算子被提出并研究，进一步拓展了基于拉普拉斯算子的应用范畴。本文通过将持久拉普拉斯算子推广至由单纯映射 $f: K \to L$ 连接的单纯复形对，扩大了其适用范围。此类单纯映射场景频繁出现，例如在关联粗化单纯表示与原始表示时，或两个单纯复形由不同点集生成（即不满足 $K\subset L$ 的情况）。然而，由于底层代数结构更为复杂，单纯映射设置比包含设置更具挑战性。我们给出了持久拉普拉斯算子向单纯映射情形的自然推广。为揭示其内在结构并设计相应算法，我们利用了持久拉普拉斯算子与矩阵舒尔补之间的关系。关键步骤是将舒尔补视为将自伴半正定算子函数性约束至给定子空间的方法。由此，我们证明了单纯映射的持久贝蒂数可通过持久拉普拉斯算子恢复。进而提出求解持久拉普拉斯算子矩阵表示的算法，该算法同时为计算单纯映射持久贝蒂数提供了新途径。最后，我们研究了单纯映射下单纯塔上的持久拉普拉斯算子，并建立了其特征值的单调性结果。