Mathematics Colloquia and Seminars

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More and more data are collected in a distributed and irregular manner. They are not organized such as familiar digital signals and images sampled on regular lattices. Examples include from sensor networks, social networks, webpages, biological networks, and so on. Moreover, constructing a graph from a usual signal or image and analyzing it can also be very useful (e.g., the nonlocal mean denoising algorithm of Buades, Coll, and Morel). Hence, it is very important to transfer harmonic and wavelet analysis techniques originally developed on the usual Euclidean spaces to graphs and networks. In this talk, I will first discuss my own work with Ernest Woei on analysis of the eigenvalue distribution of the Laplacian of graphs representing neuronal dendritic trees and the surprising behavior of the corresponding eigenvalues and eigenfunctions. Then, I will review some of the very recent works by: 1) Hammond, Vandergheynst, and Gribonval on constructing wavelets from graph Laplacians; and 2) Coifman, Gavish, and Nadler on tensor-product Haar-like analysis of digital databases.