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Discrete Integral Operators on Graphs and Multiscale Transforms on Simplicial Complexes

In this dissertation, 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.
Next, we present extensions of two powerful multiscale graph signal transforms for analyzing signals defined on the k-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 k-Laplacian Eigen Transform (k-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 k-Generalized Haar-Walsh Transform (k-GHWT).The key idea is to use the Hodge Laplacians and their variants for hierarchical bipartitioning of the k-dimensional simplices in a given simplicial complex, and then building localized basis functions on these partitioned subsets.

ZOOM INFO: https://ucdavis.zoom.us/j/97117581537?pwd=YjVLM0djeXVRVTlTQmZ4b1dncVlXdz09