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Fourier transform

As explained in Graph Neural Networks in Action

A Fourier transform expresses a signal as a combination of components associated with different frequencies.

Expresses a signal as components associated with different frequencies. For graphs, the signal is a vector or matrix of node features, allowing frequency-domain analysis of graph structure. section 3.4.1

A graph-based workflow proceeds as follows:

  • Represent node features as a graph signal. section 3.4.1
  • Transform the graph signal into a frequency representation. section 3.4.1
  • Use spectral convolution ingredients such as the adjacency matrix, Laplacian matrix, and Laplacian eigenvectors. section 3.4.3
  • Apply the inverse Fourier transform to recover a graph signal from its frequency representation. section 3.4.1
Figure 3.13 A random positive graph signal on the vertices of the graph. The height of each vertical bar represents the signal value at the node where the bar originates.
figure

Visualizes a graph signal by drawing each node feature as the height of a bar at its originating node. section 3.4.1

How Graph Neural Networks in Action covers it

How deep it goes
Defines forward and inverse transforms and names spectral-convolution matrices, but stops short of derivation or implementation code.
Examples
Uses a graph whose node-value bars form a visual graph signal and provides spectral-convolution equations as formal examples.
What it assumes
Assumes graph matrices and node features, while introducing graph signals and their transform between node and frequency representations.
What you can do after
Evaluate graph signals in frequency representations and understand their role in spectral graph convolution.
How it connects
Connects graph signals with frequency domains, inverse Fourier transforms, Laplacian matrices, eigenvectors, and spectral convolution.
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