A noncommutative approach to the graphon Fourier transform

TL;DR

This paper extends the concept of the graph Fourier transform to graphons, demonstrating convergence of graph Fourier transforms in graph sequences and applying the theory to Cayley graphons for unified signal processing.

Contribution

It generalizes the graph Fourier transform to graphons and proves convergence in graph sequences, enabling unified signal analysis on large, similar graphs.

Findings

Graph Fourier transforms of converging graph sequences approach the graphon Fourier transform.

Application to Cayley graphons shows the utility of group-based eigen-decomposition.

Provides a framework for signal processing on large, structured graphs.

Abstract

Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon. Theoretically, graphons are limit objects of converging sequences of graphs. Our work extends previous research aiming to provide a common scheme for signal analysis of graphs that are similar in structure to a graphon. We extend a previous definition of graphon Fourier transform, and show that the graph Fourier transforms of graphs in a converging graph sequence converge to the graphon Fourier transform of the limiting graphon. We then apply this convergence result to signal processing on Cayley graphons. We show that Fourier analysis of the underlying group enables the construction of a suitable eigen-decomposition for the graphon, which can be used as a common framework for signal processing on graphs converging to the graphon.