Multiscale Transforms for Signals on Simplicial Complexes

Abstract

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.

Dirac signal processing of higher-order topological signals

“…On one side it has been recently shown that topological signals undergo collective phenomena that display a rich interplay between topology and dynamics [13,14,[21][22][23][24][25][26][27]. On the other side topological signal data can be processed by machine learning and signal processing algorithm [12,15,16,28], including neural networks [29][30][31] which make use of algebraic topology.…”

“…In the last years signal processing on non-Euclidean domains such as graphs has received large attention [32]. Using variants of the Hodge Laplacian operators [19,33], these ideas have been extended to signal processing on simplicial complexes and other topological spaces [12,15,[34][35][36][37][38][39] and found their way into extensions of graphs neural networks [28,29,[40][41][42][43] Further current approaches build both on the Hodge Laplacians or the Magnetic Laplacian [44][45][46] to effectively extract information from higher-order or directed network data. However, despite great progress achieved so far, most current signal processing algorithms focus only on the filtering topological signals of a given dimension, e.g., flows on edges.…”