I am a PhD. in pure mathematics.

  1. Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
  2. I am reading the interesting paper, but could not get how the Fourier transform is extended to the graph Fourier transform as illustrated on page 23. Indeed, why should the matrix V be considered as the extension of (discrete) Fourier transform?
  • $\begingroup$ Thanks, It will be done just right now. Should it be deleted? $\endgroup$ – Ali Bagheri May 20 '20 at 11:37
  • $\begingroup$ This might be better on the maths site than on dsp $\endgroup$ – Neil_UK May 20 '20 at 13:06
  • $\begingroup$ yeah, i kinda don't grok this question at all. $\endgroup$ – robert bristow-johnson Jun 13 '20 at 5:01
  • $\begingroup$ @robertbristow-johnson This is about "signal processing on graphs", a new area of SP. As I understand it, the idea is to extend SP concepts like frequency transforms, spectra, densities, convolution, etc and apply them to graphs instead of signals. The hypothesis is that this will allow people to find interesting things about the graphs (just like, for example, the DFT allows us to find interesting things about a signal). $\endgroup$ – MBaz Jun 13 '20 at 14:20
  • $\begingroup$ According to page 23, $\mathbf{V}$ diagonalizes $\mathbf{S}$. Applications employ Fourier series/transforms to diagonalize operations (such as differentiation) that involve translation. It allows the user to work with a simpler diagonal operator and then move back to the original basis when the work is complete. The eigenvectors of $\mathbf{V}$ seem to be viewed as orthogonal modes of oscillations on the graph, much as eigenfunctions of the Lapacian on $\mathbb{R}^n$ are interpreted as modes of oscillation of a field permeating $\mathbb{R}^n$. It is in that sense that they are Fourier-like. $\endgroup$ – Joe Mack Jun 21 '20 at 16:08

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