SIAM Review, Volume 66, Issue 3, Page 575-601, May 2024.
As key subjects in spectral geometry and combinatorial graph theory, respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian share similarities in revealing the topological dimension and geometric shape of data and in their realization of diffusion and minimization of harmonic measures. It is believed that they also both associate with vector calculus, through the gradient, curl, and divergence, as argued in the popular usage of “Hodge Laplacians on graphs” in the literature. Nevertheless, these Laplacians are intrinsically different in their domains of definitions and applicability to specific data formats, hindering any in-depth comparison of the two approaches. For example, the spectral decomposition of a vector field on a simple point cloud using combinatorial Laplacians defined on some commonly used simplicial complexes does not give rise to the same curl-free and divergence-free components that one would obtain from the spectral decomposition of a vector field using either the continuous Hodge Laplacians defined on differential forms in manifolds or the discretized Hodge Laplacians defined on a point cloud with boundary in the Eulerian representation or on a regular mesh in the Eulerian representation. To facilitate the comparison and bridge the gap between the combinatorial Laplacian and Hodge Laplacian for the discretization of continuous manifolds with boundary, we further introduce boundary-induced graph (BIG) Laplacians using tools from discrete exterior calculus (DEC). BIG Laplacians are defined on discrete domains with appropriate boundary conditions to characterize the topology and shape of data. The similarities and differences among the combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined. Through an Eulerian representation of 3D domains as level-set functions on regular grids, we show experimentally the conditions for the convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.

中文翻译：

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组合拉普拉斯算子和霍奇拉普拉斯算子：异同


《SIAM 评论》，第 66 卷，第 3 期，第 575-601 页，2024 年 5 月。

分别作为谱几何和组合图论的关键学科，（连续）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数和几何形状以及实现谐波测度的扩散和最小化方面具有相似之处。据信，它们还通过梯度、旋度和散度与矢量微积分相关，正如文献中“图上的霍奇拉普拉斯算子”的流行用法所争论的那样。然而，这些拉普拉斯算子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，阻碍了这两种方法的深入比较。例如，使用在一些常用的单纯复形上定义的组合拉普拉斯算子对简单点云上的向量场进行谱分解不会产生与从向量场的谱分解中获得的相同的无旋度和无散度分量。使用在流形中的微分形式上定义的连续霍奇拉普拉斯算子或在欧拉表示中具有边界的点云上或在欧拉表示中的规则网格上定义的离散霍奇拉普拉斯矢量场。为了便于比较和弥合组合拉普拉斯算子和霍奇拉普拉斯算子之间的差距，以实现带边界的连续流形的离散化，我们进一步引入了使用离散外微积分 (DEC) 工具的边界诱导图 (BIG) 拉普拉斯算子。 BIG 拉普拉斯算子是在离散域上定义的，具有适当的边界条件来表征数据的拓扑和形状。然后检查组合拉普拉斯、BIG 拉普拉斯和霍奇拉普拉斯之间的异同。 通过将 3D 域欧拉表示为规则网格上的水平集函数，我们通过实验展示了 BIG 拉普拉斯特征值与基本形状的霍奇拉普拉斯特征值收敛的条件。