Multiscale Transforms for Signals on Simplicial Complexes

TL;DR

This paper extends multiscale graph signal transforms to simplicial complexes, enabling analysis of data on higher-dimensional simplices like edges and faces using Hodge Laplacians, with applications to synthetic and real-world datasets.

Contribution

It generalizes existing graph transforms to simplicial complexes by leveraging Hodge Laplacians for hierarchical partitioning and basis construction.

Findings

Effective data representation on simplices demonstrated.

Applicable to synthetic and real-world datasets.

Enhanced analysis of complex structured data.

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 $\kappa$-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.