How Can We Naturally Order and Organize Graph Laplacian Eigenvectors?

Abstract

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.

Random-Walk Laplacian for Frequency Analysis in Periodic Graphs

“…With regards to L (rw) and L (n) , they share the same set of eigenvalues as they are similar matrices. As opposed to {λ i } N i=1 in (12), the upper bound of the eigenvalues of L (rw) and L (n) is graph independent, i.e.,…”

“…With regards to the frequency analysis of a discrete signal, it is widely accepted that the Discrete Fourier Transform (DFT) provides the frequency components of a signal ordered from low-pass to high-pass components. However, the frequency analysis of graph signals and, in particular, the definition and ordering of frequencies of graph signals from low-pass (or smooth) to high-pass modes still causes certain controversy [ 12 ]. The Graph Fourier Transform (GFT) generalizes the concept of Fourier transform to graphs and is given by the eigenvectors of the so-called graph-shift operator [ 13 ].…”

On the Dual Geometry of Laplacian Eigenfunctions

“…Given a Graph G = (V, E), define, if possible, an analogous geometry on its eigenvectors. This is an absolutely fundamental problem, we refer to [1,2,5,7,8,9,10,12,15,17,18,23,25,28,29,30] for recent examples. Moreover, it is not expected that this is always (or even generically) possible -even in Euclidean space, one would expect that eigenfunctions on generic domains do not have any distinguishing features except for their eigenvalue; this vague statement is made precise in different ways in the study of quantum chaos [16,21].…”

Natural Graph Wavelet Packet Dictionaries

“…This includes: the Spectral Graph Wavelet Transform [16]; Diffusion Wavelets [7]; extensions to spectral graph convolutional networks [29]. However, these dictionaries do not fully address the relationships among eigenvectors [5,31,49], which should be utilized for graph dictionary construction; instead, they focus on the eigenvalue distributions to organize the corresponding eigenvectors (although there are some works, e.g., [43,44,55], which recognized the graph structures strongly influence the eigenvector behaviors). These relationships among eigenvectors can result from eigenvector localization in different clusters, differing scales in multi-dimensional data, etc.…”

“…These relationships among eigenvectors can result from eigenvector localization in different clusters, differing scales in multi-dimensional data, etc. These notions of similarity and difference between eigenvectors, while studied in the eigenfunction literature [5,31,49], have yet to be incorporated into building localized dictionaries on graphs.…”