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The Graph Fractional Fourier Transform in Hilbert Space

Yu Zhang, Bing-Zhao Li

TL;DR

The paper develops the Graph Fractional Fourier Transform in Hilbert Space (HGFRFT), a fractional-domain generalization of graph signal processing that jointly analyzes Hilbert-space-valued vertex signals. It defines HGFRFT via an $(\alpha,\beta)$-ordered tensor-product basis, proves fundamental properties (nestability, associativity, additivity, commutativity, invertibility, separability, unitarity), and builds filtering and sampling theories in the fractional domain. The authors demonstrate concrete benefits through simulations on product graphs, SEIRS epidemic models, radar on graphs, and digital image denoising, highlighting improved energy concentration, flexible frequency analysis, and robust reconstruction. This framework enables processing of non-Euclidean data in continuous domains and enhances joint time-vertex analyses, with potential impact across networks, sensing, and imaging applications where Hilbert-space-valued signals arise.

Abstract

Graph signal processing (GSP) leverages the inherent signal structure within graphs to extract high-dimensional data without relying on translation invariance. It has emerged as a crucial tool across multiple fields, including learning and processing of various networks, data analysis, and image processing. In this paper, we introduce the graph fractional Fourier transform in Hilbert space (HGFRFT), which provides additional fractional analysis tools for generalized GSP by extending Hilbert space and vertex domain Fourier analysis to fractional order. First, we establish that the proposed HGFRFT extends traditional GSP, accommodates graphs on continuous domains, and facilitates joint time-vertex domain transform while adhering to critical properties such as additivity, commutativity, and invertibility. Second, to process generalized graph signals in the fractional domain, we explore the theory behind filtering and sampling of signals in the fractional domain. Finally, our simulations and numerical experiments substantiate the advantages and enhancements yielded by the HGFRFT.

The Graph Fractional Fourier Transform in Hilbert Space

TL;DR

The paper develops the Graph Fractional Fourier Transform in Hilbert Space (HGFRFT), a fractional-domain generalization of graph signal processing that jointly analyzes Hilbert-space-valued vertex signals. It defines HGFRFT via an -ordered tensor-product basis, proves fundamental properties (nestability, associativity, additivity, commutativity, invertibility, separability, unitarity), and builds filtering and sampling theories in the fractional domain. The authors demonstrate concrete benefits through simulations on product graphs, SEIRS epidemic models, radar on graphs, and digital image denoising, highlighting improved energy concentration, flexible frequency analysis, and robust reconstruction. This framework enables processing of non-Euclidean data in continuous domains and enhances joint time-vertex analyses, with potential impact across networks, sensing, and imaging applications where Hilbert-space-valued signals arise.

Abstract

Graph signal processing (GSP) leverages the inherent signal structure within graphs to extract high-dimensional data without relying on translation invariance. It has emerged as a crucial tool across multiple fields, including learning and processing of various networks, data analysis, and image processing. In this paper, we introduce the graph fractional Fourier transform in Hilbert space (HGFRFT), which provides additional fractional analysis tools for generalized GSP by extending Hilbert space and vertex domain Fourier analysis to fractional order. First, we establish that the proposed HGFRFT extends traditional GSP, accommodates graphs on continuous domains, and facilitates joint time-vertex domain transform while adhering to critical properties such as additivity, commutativity, and invertibility. Second, to process generalized graph signals in the fractional domain, we explore the theory behind filtering and sampling of signals in the fractional domain. Finally, our simulations and numerical experiments substantiate the advantages and enhancements yielded by the HGFRFT.
Paper Structure (29 sections, 7 theorems, 59 equations, 29 figures, 2 tables, 2 algorithms)

This paper contains 29 sections, 7 theorems, 59 equations, 29 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Signals in Hilbert Space
  4. Graph Fourier Transform in Hilbert Space
  5. Graph Fractional Fourier Transform
  6. Graph Fractional Fourier Transform in Hilbert Space
  7. Definition
  8. Properties
  9. Examples
  10. Filtering
  11. Shift Invariant Filters in the Fractional Domain
  12. Fractional Order Convolution Filters
  13. $\alpha, \beta$-Bandlimited Signals and Bandpass Filters
  14. Sampling
  15. Graph Fractional Sampling in Hilbert Space
  16. ...and 14 more sections

Key Result

Proposition 1

If the adjacency matrix $\mathbf{A}$ is similarly constructed as compact self-adjoint operators, then $\mathcal{F}^{\alpha,\beta}_f$ reduces to the two-dimensional DFRFT of orders $\alpha$ and $\beta$ in Hilbert space.

Figures (29)

  • Figure 1: Chirp signals on the U.S. graph.
  • Figure 2: Simulation graph: 4-node path graph, 4-node ring graph and 16-node product graph.
  • Figure 3: The fractional graph signal in traditional GSP.
  • Figure 4: Graph signals on the product graph.
  • Figure 5: Sampling node locations (shown in black).
  • ...and 24 more figures

Theorems & Definitions (30)

  • Definition 1
  • Remark 1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • Remark 2
  • proof
  • ...and 20 more