FGFRFT: Fast Graph Fractional Fourier Transform via Exact Spectral Splitting and Fourier-Series Approximation

TL;DR

This paper introduces FGFRFT, a fast and accurate method for computing the graph fractional Fourier transform that significantly reduces computational complexity while maintaining high fidelity, enabling efficient applications in signal processing on graphs.

Contribution

The paper proposes an exact spectral splitting approach combined with Fourier-series approximation to accelerate GFRFT computation, addressing spectral singularities and ensuring differentiability.

Findings

FGFRFT reduces online complexity from O(N^3) to O(2LN^2).

Experiments demonstrate high approximation accuracy and substantial acceleration.

FGFRFT performs well in image and point-cloud denoising tasks.

Abstract

The graph fractional Fourier transform (GFRFT) for unitary graph Fourier transform (GFT) matrices can be interpreted through the scalar function $e^{j\alpha\theta}$ on the unit circle. Under the principal branch, its Fourier-series representation encounters an intrinsic obstruction at the spectral point $\lambda=-1$ for non-integer orders. To address this issue, we propose a fast graph fractional Fourier transform (FGFRFT) based on exact spectral splitting: the $\lambda=-1$ component is treated exactly, and the complementary component is approximated by a truncated Fourier series in integer powers of the GFT matrix. This construction yields an offline--online implementation that reduces the online complexity of repeated operator updates from $O(N^3)$ to $O(2LN^2)$ for truncation order $L$, while preserving differentiability with respect to the transform order. We further derive truncation-error bounds, approximate unitarity and additivity, and reconstruction-error bounds. Experiments on approximation accuracy, transform-order learning, image denoising, and point-cloud denoising show that FGFRFT provides substantial online acceleration while remaining close to the exact GFRFT under the tested settings.