A generalization of the persistent Laplacian to simplicial maps
Aziz Burak Gülen, Facundo Mémoli, Zhengchao Wan, Yusu Wang
备注：
This is the full version of a paper accepted to SoCG 2023. It contains 4 appendix sections to provide proofs, extra examples and other related results to the SoCG version
图的拉普拉斯算子是图的分析和优化的基本对象。该算子可以扩展到单纯复形$K$，因此提供了一种在$K$的$p$ -（共）链上执行"信号处理”的方法。最近，持久拉普拉斯算子的概念被提出，并研究了一对由包含关系连接的单纯复形$K\hookrightarrow L$，进一步拓宽了基于拉普拉斯算子的使用。在本文中，我们扩大了持久拉普拉斯算子的范围，将其推广到由单纯映射$f: K \to L$连接的一对单纯复形。这种单纯映射设置经常出现，例如，当将粗化单纯表示与原始表示相关联时，或者当两个单纯复形被不同的点集所跨越时的情况，即不保持该$K\subset L$的情况。然而，单纯映射设置比包含设置更具挑战性，因为底层的代数结构更复杂。我们提出了一个自然的推广的持久拉普拉斯单纯设置。为了深入了解它背后的结构，以及开发一种算法来计算它，我们利用持久拉普拉斯算子和矩阵的舒尔补之间的关系。关键的一步是将Schur补视为将自伴PSD算子限制到给定子空间的函子方式。因此，我们证明了单纯映射的持久贝蒂数可以由持久拉普拉斯算子恢复。然后，我们提出了一个算法，找到持久的拉普拉斯算子的矩阵表示，从而产生一个新的算法计算持久的贝蒂数的单纯映射。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。
英文摘要：
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.
The Smith normal form of Laplacian matrices of simplicial annuli and high dimensional trees
Banco de México（墨西哥银行）；Facultad de Ciencias Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Cinvestav（墨西哥国立理工学院先进研究中心）；Facultad de Economía, Universidad Autónoma de San Luis Potosí（圣路易斯波托西自治大学经济学院）；Worcester Polytechnic Institute（伍斯特理工学院）
Signless Laplacian Spectral Radius and Link Homology of Simplicial Complexes
Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院纯数学中心）；School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）
Eigenvalue bounds for combinatorial Laplacians and an application to random complexes
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；School of Mathematics, East China University of Science and Technology（华东理工大学数学系）
Random Walks on Simplicial Complexes and Harmonics
本文引入了单纯复形上具有吸收状态的随机游动。给定一个$d$维的单纯复形，定义了一个具有吸收状态的随机游动，它与$1 \leq k \leq d$的$d$维Laplacian的谱有关，并且与范涌定义的图上的局部随机游动有关.我们还研究了随机游动在单纯复形的半监督学习问题的应用。具体来说，我们考虑一个标签传播算法的定向边缘，这适用于推广的部分标记分类问题的图。
