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 vertices 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-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.