UC Davis

As network analysis tools have become more powerful over the last decade, the need has arisen for new multiscale signal processing tools, both for robust understanding of the intrinsic geometry underlying data, and for higher-order signals, incorporating not only data which lives on the vertices of a graph, but perhaps on its edges, or the faces of a triangular mesh of a manifold. In this disserta- tion, first we develop a discrete integral operator suitable for spectral embedding and partitioning of graphs, by carefully studying the development of graph Laplacian techniques from their continuous analogues on domains, and applying those developments to integral operators which commute with the continuous Laplacian. We demonstrate the operator’s effectiveness at distinguishing underly- ing geometry in scattered data, and efficient sparse techniques for performing partitioning using it. Next, we present extensions of two powerful multiscale graph signal transforms for analyzing signals defined on the κ-dimensional simplices of a simplicial complex. The previous Hierarchical Graph Laplacian Eigen Transform (HGLET) generalizes the block DCT to the graph setting, and our Hierarchical κ-Laplacian Eigen Transform (κ-HGLET) generalizes further to the simplicial complex setting. Likewise, for the previous Generalized Haar-Walsh Transform (GHWT) which generalizes the Haar-Walsh wavelet packet transform, we propose the κ-Generalized Haar-Walsh Transform (κ-GHWT). The key idea is to use the Hodge Laplacians and their variants for hierarchical bipar- titioning of the κ-dimensional simplices in a given simplicial complex, and then building localized basis functions on these partitioned subsets. We demonstrate the usefulness of the κ-HGLET and κ-GHWT on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset.