How can we naturally order and organize graph Laplacian eigenvectors? Proc. 2018 IEEE Workshop on Statistical Signal Processing, pp.483-487, 2018.
When attempting to develop wavelet transforms for graphs and networks,
some researchers have used graph Laplacian eigenvalues and eigenvectors in place
of the frequencies and complex exponentials in the Fourier theory for regular
lattices in the Euclidean domains. This viewpoint, however, has a fundamental
flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted
as the frequencies of the corresponding eigenvectors.
In this paper, we discuss this important problem further and propose a new
method to organize those eigenvectors by defining and measuring "natural"
distances between eigenvectors using the Ramified Optimal Transport Theory
followed by embedding them into a low-dimensional Euclidean domain.
We demonstrate its effectiveness using a synthetic graph as well as a dendritic
tree of a retinal ganglion cell of a mouse.
Keywords: Graph Laplacian eigenvectors, ramified optimal transport, multidimensional scaling
Get the full paper (via arXiv:1801.06782): PDF file.
Get the official version via doi:10.1109/SSP.2018.8450808.
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