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.

English Abstract

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.

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