Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A noncommutative approach to the graphon Fourier transform

Mahya Ghandehari, Jeannette Janssen, Nauzer Kalyaniwalla

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.

A noncommutative approach to the graphon Fourier transform

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.

Paper Structure

This paper contains 13 sections, 15 theorems, 97 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and background
  3. Signal processing on graphs
  4. Fourier analysis on compact (not necessarily Abelian) groups
  5. Fourier and inverse Fourier transforms for compact groups.
  6. Graph limit theory
  7. Cayley graphons
  8. Spectral decomposition of graphons
  9. Graphon signal processing
  10. Interpretation of Theorem \ref{['thm:convergence']} and its applications
  11. Application: Filter Design
  12. Signal processing on Cayley graphons
  13. Acknowledgements

Key Result

Proposition \oldthetheorem

(Properties of Fourier transform on compact groups) Let $\mathbb G$ be a compact group. For the Fourier and inverse Fourier transforms defined above we have:

Figures (2)

  • Figure 1: Cayley graphon on the 1-dim torus with parameters $d=0.2, p=0.08$ and a graph sampled from it.
  • Figure 2: Graph Fourier coefficients 2 and 3 for the different samples.

Theorems & Definitions (39)

  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 29 more