Multiscale Basis Dictionaries on Simplicial Complexes

Applied Mathematics
Event time: 
Tuesday, November 1, 2022 - 4:00pm
Naoki Saito
Speaker affiliation: 
UC Davis
Event description: 

Abstract: Our previous multiscale graph basis dictionaries (e.g., 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 work, we propose their generalization for analyzing data recorded on edges or on faces (i.e., triangles) 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 edges or faces, and then build localized basis functions on those subsets. We plan to demonstrate their usefulness for data approximation on simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.