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FGFRFT: Fast Graph Fractional FourierTransform via Fourier Series Approximation

Ziqi Yan, Sen Shi, Feiyue Zhao, Manjun Cui, Yangfan He, Zhichao Zhang

TL;DR

A fast GFRFT (FGFRFT) algorithm for unitary GFT matrices based on Fourier series approximation and an efficient caching strategy is proposed, which significantly accelerates computation and achieves superior performance compared with the GFRFT.

Abstract

The graph fractional Fourier transform (GFRFT) generalizes the graph Fourier transform (GFT) but suffers from a significant computational bottleneck: determining the optimal transform order requires expensive eigendecomposition and matrix multiplication, leading to $O(N^3)$ complexity. To address this issue, we propose a fast GFRFT (FGFRFT) algorithm for unitary GFT matrices based on Fourier series approximation and an efficient caching strategy. FGFRFT reduces the complexity of generating transform matrices to $O(2LN^2)$ while preserving differentiability, thereby enabling adaptive order learning. We validate the algorithm through theoretical analysis, approximation accuracy tests, and order learning experiments. Furthermore, we demonstrate its practical efficacy for image and point cloud denoising and present the fractional specformer, which integrates the FGFRFT into the specformer architecture. This integration enables the model to overcome the limitations of a fixed GFT basis and learn optimal fractional orders for complex data. Experimental results confirm that the proposed algorithm significantly accelerates computation and achieves superior performance compared with the GFRFT.

FGFRFT: Fast Graph Fractional FourierTransform via Fourier Series Approximation

TL;DR

A fast GFRFT (FGFRFT) algorithm for unitary GFT matrices based on Fourier series approximation and an efficient caching strategy is proposed, which significantly accelerates computation and achieves superior performance compared with the GFRFT.

Abstract

The graph fractional Fourier transform (GFRFT) generalizes the graph Fourier transform (GFT) but suffers from a significant computational bottleneck: determining the optimal transform order requires expensive eigendecomposition and matrix multiplication, leading to complexity. To address this issue, we propose a fast GFRFT (FGFRFT) algorithm for unitary GFT matrices based on Fourier series approximation and an efficient caching strategy. FGFRFT reduces the complexity of generating transform matrices to while preserving differentiability, thereby enabling adaptive order learning. We validate the algorithm through theoretical analysis, approximation accuracy tests, and order learning experiments. Furthermore, we demonstrate its practical efficacy for image and point cloud denoising and present the fractional specformer, which integrates the FGFRFT into the specformer architecture. This integration enables the model to overcome the limitations of a fixed GFT basis and learn optimal fractional orders for complex data. Experimental results confirm that the proposed algorithm significantly accelerates computation and achieves superior performance compared with the GFRFT.
Paper Structure (20 sections, 30 equations, 7 figures, 6 tables, 3 algorithms)

This paper contains 20 sections, 30 equations, 7 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GFT
  4. GFRFT
  5. Fourier Series Approximation Theory
  6. FGFRFT
  7. Fourier Series Expansion of GFRFT Eigenvalues
  8. FGFRFT Definition
  9. Complexity Analysis
  10. GFRFT Algorithm
  11. FGFRFT Algorithm
  12. Analysis of Approximation Accuracy and Computational Efficiency
  13. Experimental Settings and Evaluation Metrics
  14. Impact of Truncation Order $L$ on Approximation Accuracy
  15. Comparison of Computational Efficiency on Large-Scale Graphs
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Overall framework of the paper
  • Figure 2: The trend of MSE with respect to truncation order $L$.
  • Figure 3: The trend of MAE with respect to truncation order $L$.
  • Figure 4: The trend of NMSE with respect to truncation order $L$.
  • Figure 5: Visualizations of the training process showing Loss curves and transform order update trajectories under different network layer counts.
  • ...and 2 more figures