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Design natural Graph Wavelet by the ``distance'' between graph Laplacian eigenvectors

Student-Run Research Seminar

Speaker: Haotian Li, UC Davis
Location: 2112 MSB
Start time: Wed, Dec 12 2018, 12:10PM

Our main goal is to design a natural Wavelet-like basis on graph which can be used to better analyze and understand graph signals or data, such as dendrite neuron, Facebook social networks data, e.t.c. To accomplish that, our first step is to naturally order the graph laplacian eigenvectors, which usually can be interpreted as the ``Fourier modes'' on graphs. One of current popular schemes, Spectral Graph Wavelet Transform (SGWT) of Hammond et al., designs graph wavelet based on organizing the graph laplacian eigenvectors by their corresponding eigenvalues. However, this ordering only works well for basic graphs, e.g., undirected unweighted paths and cycles. So the idea of natural ordering is grouping eigenvectors based on their ``behaviors'' on graph. Prof. Naoki Saito proposed a scheme to do that is by defining a proper ``distance'' between those eigenvectors such that similar behavior ones are close and distinct behavior ones are far. We have tried several ways to define such distance, i.e. ramified optimal transport (ROT) method, absolute gradient method and time-stepping diffusion method (TSDM). Then based on these distances, we can design a natural graph wavelet.



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