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Frames for signal processing on Cayley graphs

Kathryn Beck, Mahya Ghandehari, Skyler Hudson, Jenna Paltenstein

TL;DR

This paper provides an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group.

Abstract

The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph's symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.

Frames for signal processing on Cayley graphs

TL;DR

This paper provides an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group.

Abstract

The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis that better reflects the graph's symmetries. In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a weighted Cayley graph. Our method applies to all weighted Cayley graphs, regardless of whether they are quasi-Abelian, and offers detailed descriptions of eigenvalues and eigenvectors derived from the coefficient functions of the representations of the underlying group. Next, we turn our attention to constructing frames on Cayley graphs. Frames are overcomplete spanning sets that ensure stable and potentially redundant systems for signal reconstruction. We use our proposed eigenbases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.
Paper Structure (15 sections, 53 equations, 3 figures, 2 tables)

This paper contains 15 sections, 53 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main contribution
  3. Organization of the paper
  4. Notations and background
  5. Representation theory of finite groups
  6. Eigen-decompositions of (weighted) Cayley graph adjacency matrices
  7. Construction of frames for graph signals
  8. Special frames for Cayley graphs
  9. Cayley frames with special properties
  10. Comparison with previous work: frames for the permutahedron
  11. Future work
  12. Proof of Proposition \ref{['prop:eigenvector']}
  13. Representations of $\mathbb{S}_n$
  14. Representations of $\mathbb{S}_3$ in matrix form and their matrix coefficients
  15. Representations of $\mathbb{S}_4$ in matrix form and their matrix coefficients

Figures (3)

  • Figure 1: The Cayley graph ${\mathbb P}_3$ and its adjacency matrix $A_3$.
  • Figure 2: The permutahedron $\mathbb P_4$ is the Cayley graph on the group $\mathbb{S}_4$ with generating set $S=\{(12), (23), (34)\}$.
  • Figure 3: The Cayley graph of $\mathbb{S}_4$ with generating set $S=\{(12), (23), (34), (12)(34)\}.$

Theorems & Definitions (5)

  • proof
  • proof
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