Our previous multiscale graph basis dictionaries/graph signal transforms -- Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives -- were developed for analyzing data recorded on nodes of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally $\kappa$-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of $\kappa$-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.

翻译：我们先前开发的基于节点数据的多尺度图基字典/图信号变换——广义Haar-Walsh变换(GHWT)、分层图拉普拉斯特征变换(HGLET)、自然图小波包(NGWPs)及其相关方法——主要用于分析给定图上节点记录的数据。本文提出将这些方法推广至边、面(即三角形)，或更一般地，单纯复形(如流形的三角网格)中$\kappa$维单纯形上记录的数据分析。核心思路是利用霍奇拉普拉斯算子及其变体对给定单纯复形中的$\kappa$维单纯形集合进行分层划分，并在此基础上构建局部化基函数。通过合著/引文数据集和洋流/流场数据集生成的模拟示例和真实世界单纯复形，我们验证了这些方法在数据表示中的有效性。