单纯复形上信号的多尺度变换
Multiscale Transforms for Signals on Simplicial Complexes
AI总结：
本文提出将图信号多尺度变换推广至单纯复形上的信号，利用Hodge拉普拉斯算子分层划分单纯形并构建局部基函数，在合成与真实数据集上验证了其有效性。
摘要：
我们之前开发的多尺度图基字典/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——是为分析给定图节点上记录的数据而设计的。在本文中，我们提出它们的推广，用于分析记录在边、面（即三角形）或更一般地，单纯复形（例如流形的三角网格）的$\kappa$维单纯形上的数据。关键思想是使用Hodge拉普拉斯算子及其变体对给定单纯复形中的一组$\kappa$维单纯形进行分层划分，然后在这些划分的子集上构建局部基函数。我们在说明性合成示例和从合著/引文数据集及海洋洋流/流动数据集生成的现实世界单纯复形上展示了它们对数据表示的有用性。
英文摘要：
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 $κ$-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 $κ$-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.
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