FGFRFT：通过精确频谱分裂和傅里叶级数近似实现的快速图分数学傅里叶变换
FGFRFT: Fast Graph Fractional Fourier Transform via Exact Spectral Splitting and Fourier-Series Approximation
AI总结：
本文提出FGFRFT，通过精确频谱分裂和傅里叶级数近似，将重复操作的在线复杂度从O(N^3)降至O(2LN^2)，同时保持对变换阶数的可微性，并在不同任务中验证了其加速效果和精度。
摘要：
本文提出FGFRFT，通过精确频谱分裂和傅里叶级数近似，将重复操作的在线复杂度从O(N^3)降至O(2LN^2)，同时保持对变换阶数的可微性，并在不同任务中验证了其加速效果和精度。
英文摘要：
The graph fractional Fourier transform (GFRFT) for unitary graph Fourier transform (GFT) matrices can be interpreted through the scalar function $e^{jαθ}$ on the unit circle. Under the principal branch, its Fourier-series representation encounters an intrinsic obstruction at the spectral point $λ=-1$ for non-integer orders. To address this issue, we propose a fast graph fractional Fourier transform (FGFRFT) based on exact spectral splitting: the $λ=-1$ 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.
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