Applied harmonic analysis on graphs and networks, Bulletin of the Japan Society for Industrial and Applied Mathematics, vol.25, no.3, pp.102-111, 2015 (in Japanese). Invited Paper.
In recent years, the advent of new sensor technologies and social network
infrastructure has provided huge opportunities and challenges for analyzing
data recorded on such networks.
For analyzing data recorded on regular lattices, computational harmonic analysis
tools such as the Fourier and wavelet transforms have well-developed theories
and proven track records of success.
It is therefore quite important to extend such tools from the classical setting
of regular lattices to the more general setting of graphs and networks.
In this article, we first review basics of Laplacian matrices of a graph
whose eigenpairs are often interpreted as the frequencies and the Fourier
basis vectors on a given graph. We point out, however, that such an
interpretation is misleading unless the underlying graph is unweighted
path or cycle. We then discuss our recent effort of constructing
multiscale basis dictionaries on a graph including the Hierarchical
Graph Laplacian Eigenbasis Dictionary and the Generalized Haar-Walsh Wavelet Packet Dictionary, which are viewed as the generalization of the classical
hierarchical block DCTs and the Haar-Walsh wavelet packets for the graph setting.
Get the full paper: PDF file.
Get the official version via http://doi.org/10.11540/bjsiam.25.3_102.
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