SIAM Review, Volume 66, Issue 3, Page 573-573, May 2024.
In this issue the Education section presents “Combinatorial and Hodge Laplacians: Similarities and Differences,” by Emily Ribando-Gros, Rui Wang, Jiahui Chen, Yiying Tong, and Guo-Wei Wei. Combinatorial Laplacians and their spectra are important tools in the study of molecular stability, electrical networks, neuroscience, deep learning, signal processing, etc. The continuous Hodge Laplacian allows one, in some cases, to generate an unknown shape from only its Laplacian spectrum. In particular, both combinatorial Laplacians and continuous Hodge Laplacian are useful in describing the topology of data; see, for instance, [L.-H. Lim, “Hodge Laplacians on graphs,” SIAM Rev., 62 (2020), pp. 685--715]. Since nowadays computations frequently involve these Laplacians, it is important to have a good understanding of the differences and relations between them. Indeed, though the Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data, at the same time they are intrinsically different in their domains of definitions and applicability to specific data formats. To facilitate comparisons, the authors introduce boundary-induced graph (BIG) Laplacians, the purpose of which is “to put the combinatorial Laplacians and Hodge Laplacian on equal footing.” BIG Laplacian brings, in fact, the combinatorial Laplacian closer to the continuous Hodge Laplacian. In this paper similarities and differences between combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are examined. Some elements of spectral analysis related to topological data analysis (TDA) are also provided. TDA and connected ideas have recently gained a lot of interest, and so this paper is timely. It is written in a way that should make it accessible for early career researchers; the reader should already have a good understanding of some notions of graph theory, spectral geometry, differential geometry, and algebraic topology. The paper is not self-contained and eventually could be used by group-based research projects in a Master's program for advanced mathematics students.

中文翻译：

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 教育


《SIAM 评论》，第 66 卷，第 3 期，第 573-573 页，2024 年 5 月。

本期教育版块介绍了“组合拉普拉斯算子和霍奇拉普拉斯算子：异同”，作者：Emily Ribando-Gros、Rui Wang、Jiahui Chen、Yiying Tong 和Guo-Wei Wei。组合拉普拉斯算子及其谱是研究分子稳定性、电网络、神经科学、深度学习、信号处理等的重要工具。在某些情况下，连续霍奇拉普拉斯算子允许人们仅从其拉普拉斯谱生成未知形状。特别是，组合拉普拉斯算子和连续霍奇拉普拉斯算子在描述数据拓扑方面都很有用。例如，参见[L.-H. Lim，“图上的霍奇拉普拉斯算子”，SIAM Rev.，62 (2020)，第 685--715 页]。由于当今的计算经常涉及这些拉普拉斯算子，因此很好地理解它们之间的差异和关系非常重要。事实上，尽管霍奇拉普拉斯算子和组合拉普拉斯算子在揭示数据的拓扑维度和几何形状方面有相似之处，但同时它们在定义领域和对特定数据格式的适用性方面存在本质上的不同。为了便于比较，作者引入了边界诱导图（BIG）拉普拉斯算子，其目的是“将组合拉普拉斯算子和霍奇拉普拉斯算子置于同等地位”。事实上，BIG 拉普拉斯使组合拉普拉斯更接近连续霍奇拉普拉斯。本文研究了组合拉普拉斯算子、BIG 拉普拉斯算子和霍奇拉普拉斯算子之间的异同。还提供了与拓扑数据分析 (TDA) 相关的频谱分析的一些元素。 TDA 和互联思想最近引起了人们的广泛关注，因此这篇论文恰逢其时。 它的编写方式应该使早期职业研究人员能够理解；读者应该已经很好地理解了图论、谱几何、微分几何和代数拓扑的一些概念。该论文不是独立的，最终可以被高等数学学生硕士课程中基于小组的研究项目使用。