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

Feiyue Zhao; Manjun Cui; Mingzhi Wang; Sen Shi; Yangfan He; Zhichao Zhang; Ziqi Yan

arxiv: 2602.20870 · v2 · pith:PUZEKH5Hnew · submitted 2026-02-24 · 📡 eess.SP

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

Ziqi Yan , Mingzhi Wang , Sen Shi , Feiyue Zhao , Manjun Cui , Yangfan He , Zhichao Zhang This is my paper

Pith reviewed 2026-05-21 13:05 UTC · model grok-4.3

classification 📡 eess.SP

keywords graph fractional Fourier transformfast algorithmspectral splittingFourier series approximationunitary GFTcomplexity reductiongraph signal processingdenoising

0 comments

The pith

A fast graph fractional Fourier transform splits the spectrum exactly at eigenvalue -1 and approximates the rest with a truncated Fourier series to reduce repeated-update cost from cubic to quadratic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an efficient way to compute the graph fractional Fourier transform when the fractional order is non-integer. It isolates the eigenvalue -1 component of the unitary graph Fourier transform matrix for exact treatment and approximates the remaining spectrum through a short series of integer matrix powers. This construction supports an offline-online scheme that lowers the cost of updating the operator when the order changes repeatedly. The approach also supplies error bounds and keeps the result differentiable with respect to the order, which matters for applications that need to adjust the transform on the fly.

Core claim

For unitary graph Fourier transform matrices the fractional transform is realized by the scalar map e^{j α θ} on the unit circle. The principal-branch Fourier series representation is obstructed at the spectral point λ = -1 for non-integer α. The method therefore splits the spectrum exactly at this point, applies the exact operator to the -1 component, and replaces the complementary part by a truncated Fourier series in powers of the GFT matrix. The resulting offline-online implementation reduces online complexity of repeated operator updates from O(N^3) to O(2 L N^2) while preserving differentiability in the fractional order α.

What carries the argument

Exact spectral splitting at λ = -1 combined with truncated Fourier-series approximation of the complementary powers of the GFT matrix.

If this is right

Truncation-error bounds are derived for the series approximation.
Approximate unitarity and additivity of the resulting transform are established.
Reconstruction-error bounds are obtained for the overall operator.
Experiments demonstrate substantial online acceleration on image denoising and point-cloud denoising tasks while remaining close to the exact GFRFT.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Preserved differentiability with respect to the order could support gradient-based optimization of the fractional parameter inside larger learning pipelines.
The same splitting idea might be reused for other fractional operators built from unitary matrices on graphs or other domains.
For very large graphs the offline stage that precomputes the needed matrix powers could be accelerated further by fast matrix exponentiation or randomized techniques.

Load-bearing premise

The graph Fourier transform matrix is unitary, so that the scalar function on the unit circle is well-defined and the principal-branch Fourier series can be applied after isolating the -1 component.

What would settle it

Measure the wall-clock time of repeated operator updates on an N-node graph for increasing truncation order L and check whether the observed scaling is O(L N^2) rather than O(N^3), or compute the realized approximation error on a chosen graph and non-integer order and test whether it exceeds the derived truncation bound.

Figures

Figures reproduced from arXiv: 2602.20870 by Feiyue Zhao, Manjun Cui, Mingzhi Wang, Sen Shi, Yangfan He, Zhichao Zhang, Ziqi Yan.

**Figure 1.** Figure 1: Overall framework of the paper Furthermore, to maintain consistency with the definition of classical fractional Fourier transform (FRFT) based on hyper-differential operators and to support arbitrary realnumber orders α ∈ R, literature [25] proposes another equivalent definition: F α G = exp −j απ 2 π(D2 G + FGD2 GF −1 G ) − 1 2 IN , (6) where the operator D2 G is given by: D2 G = 1 2π j2 π log(F… view at source ↗

**Figure 2.** Figure 2: The trend of MSE with respect to truncation order [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: The trend of MAE with respect to truncation order [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: The trend of NMSE with respect to truncation order [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Visualizations of the training process showing Loss [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Visual comparison of image denoising performance on sample images from the Set12 dataset. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Visual comparison of 3D point cloud denoising [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

FGFRFT gives a practical speed-up for repeated GFRFT calls by splitting at λ=-1 but depends on eigenvalues lying on the unit circle.

read the letter

This paper's main contribution is a fast approximation for the graph fractional Fourier transform that exactly handles the eigenvalue at -1 and approximates the complementary spectrum with a truncated Fourier series. The result is an offline-online scheme that brings online complexity down to O(2 L N^2) for repeated operator applications. The new element is that exact split at λ = -1 to bypass the obstruction in the principal-branch Fourier series for non-integer orders. They derive the corresponding truncation-error bounds along with approximate unitarity, additivity, and reconstruction bounds. The experiments on approximation accuracy and on image and point-cloud denoising indicate that the method stays close to the exact GFRFT while delivering the promised speed-up. It does well by focusing on a concrete implementation detail that matters for pipelines where the transform order changes often, and by preserving differentiability with respect to the order. The soft spot is the unit-circle assumption. The construction interprets the GFRFT via e^{j α θ} on the unit circle after splitting. Yet unitarity of the GFT matrix only ensures orthonormal eigenvectors; it does not force the eigenvalues of the shift operator to have modulus one. When |λ_k| ≠ 1 the true fractional operator includes magnitude scaling, and the integer-power approximation would then miss that part. The paper might limit itself to graphs where the shift is normal or unitary, but that needs to be clear in the text. If the stress-test concern holds, the error bounds and closeness claims would apply only under that extra condition. This work is for people in graph signal processing who need efficient fractional transforms inside larger tasks such as denoising or order learning. A reader interested in fast algorithms for GSP would get practical value from the complexity analysis and the numerical results. I recommend putting it through peer review. The idea is specific and the performance claims are worth checking against the derivations and the full set of experiments.

Referee Report

1 major / 3 minor

Summary. The manuscript presents FGFRFT, a fast implementation of the graph fractional Fourier transform for unitary GFT matrices. It uses exact spectral splitting to handle the λ=-1 eigenvalue separately and approximates the rest with a truncated Fourier series in powers of the GFT matrix. This enables an offline-online scheme reducing online complexity to O(2LN^2), with derived bounds on errors, unitarity, additivity, and reconstruction, and experimental validation on various tasks.

Significance. If the derived bounds hold and the approximation remains close to the exact GFRFT, this method provides a valuable tool for efficient computation of GFRFTs in applications where the order is varied repeatedly, such as in learning-based graph signal processing. The preservation of differentiability is particularly useful. The approach builds on standard techniques but applies them effectively to this setting, with the exact splitting addressing the principal branch obstruction at λ=-1.

major comments (1)

[Abstract, first paragraph] Abstract, first paragraph: the construction interprets the GFRFT via e^{j α θ} on the unit circle after splitting at λ=-1. Because the GFT matrix is stipulated to be unitary, its eigenvalues necessarily satisfy |λ_k|=1 by the spectral theorem for unitary operators; the skeptic's concern about moduli of eigenvalues of the shift operator S therefore does not apply to the fractionalization step performed on the GFT matrix itself. A one-sentence clarification in the definition section confirming this point would eliminate any ambiguity.

minor comments (3)

[Abstract] The stated online complexity O(2 L N^2) should be accompanied by a brief accounting of the factor of 2 (e.g., separate handling of the exact -1 component and the series terms).
[Experiments] Experiments section: consider adding a table that directly compares the derived truncation-error bounds against observed approximation errors for several values of L and α; this would make the tightness of the bounds immediately verifiable.
Notation: ensure that the symbol for the GFT matrix (presumably U) is introduced before its powers appear in the Fourier-series approximation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below and will make the suggested revision.

read point-by-point responses

Referee: [Abstract, first paragraph] Abstract, first paragraph: the construction interprets the GFRFT via e^{j α θ} on the unit circle after splitting at λ=-1. Because the GFT matrix is stipulated to be unitary, its eigenvalues necessarily satisfy |λ_k|=1 by the spectral theorem for unitary operators; the skeptic's concern about moduli of eigenvalues of the shift operator S therefore does not apply to the fractionalization step performed on the GFT matrix itself. A one-sentence clarification in the definition section confirming this point would eliminate any ambiguity.

Authors: We thank the referee for highlighting this point. As the manuscript stipulates that the GFT matrix is unitary, the spectral theorem for unitary operators directly implies that its eigenvalues satisfy |λ_k|=1, which is the basis for interpreting the GFRFT via e^{jαθ} on the unit circle after the exact split at λ=-1. The potential concern about eigenvalue moduli for the (non-unitary) shift operator S does not apply to the fractionalization performed on the GFT matrix. To eliminate any ambiguity, we will add a one-sentence clarification in the definition section of the revised manuscript explicitly confirming this property. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard spectral methods

full rationale

The paper's central construction applies exact spectral splitting at λ=-1 followed by classical Fourier-series truncation of e^{jαθ} on the remaining spectrum of a unitary GFT matrix. This yields the claimed offline-online complexity reduction O(2 L N^2) directly from precomputing integer powers of the GFT matrix and online summation; no fitted parameters, self-referential definitions, or load-bearing self-citations appear in the derivation. The approach is self-contained against external benchmarks such as direct matrix powering and standard Fourier approximation theory, with truncation-error bounds derived from classical remainder estimates rather than from the target result itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach assumes a unitary GFT matrix and relies on the classical Fourier series of the exponential function on the circle after spectral splitting; the only explicit tunable quantity is the truncation length L.

free parameters (1)

truncation order L
Integer length of the Fourier-series sum; controls the accuracy-speed trade-off and appears in the stated complexity O(2 L N²).

axioms (1)

domain assumption The graph Fourier transform matrix is unitary
Required for the scalar-function interpretation e^{j α θ} on the unit circle and for the principal-branch series to be applicable after splitting.

pith-pipeline@v0.9.0 · 5752 in / 1357 out tokens · 56058 ms · 2026-05-21T13:05:02.790477+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

eigenvalues of the GFT matrix are distributed on the unit circle... λ_k = e^{j θ_k} ... fractional power ... e^{j α θ_k}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 1 internal anchor

[1]

The emerging field of signal processing on graphs: Extending high dimensional data analysis to networks and other irregular domains,

D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Van- dergheynst, “The emerging field of signal processing on graphs: Extending high dimensional data analysis to networks and other irregular domains,”IEEE Signal Processing Magazine, vol. 30, no. 3, pp. 83–98, 2013

work page 2013
[2]

Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure,

A. Sandryhaila and J. M. F. Moura, “Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure,”IEEE Signal Processing Magazine, vol. 31, no. 5, pp. 80–90, 2014

work page 2014
[3]

Graph signal denoising via trilateral filter on graph spectral domain,

M. Onuki, S. Ono, M. Yamagishi, and Y . Tanaka, “Graph signal denoising via trilateral filter on graph spectral domain,”IEEE Trans- actions on Signal and Information Processing over Networks, vol. 2, no. 2, pp. 137–148, 2016

work page 2016
[4]

Graph signal processing – Part I: Graphs, graph spectra, and spectral clustering,

L. Stankovic, D. Mandic, M. Dakovic, M. Brajovic, B. Scalzo, and T. Constantinides, “Graph signal processing – Part I: Graphs, graph spectra, and spectral clustering,”arXiv preprint arXiv:1907.03467, 2019

work page arXiv 1907
[5]

Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,

S. K. Narang and A. Ortega, “Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,”IEEE Transactions on Signal Processing, vol. 61, no. 19, pp. 4673–4685, 2013

work page 2013
[6]

Discrete signal processing on graphs: sampling theory,

S. Chen, R. Varma, A. Sandryhaila, and J. Kova ˇcevi´c, “Discrete signal processing on graphs: sampling theory,”IEEE Transactions on Signal Processing, vol. 63, no. 24, pp. 6510–6523, Dec. 2015

work page 2015
[7]

Perfect reconstruction two-channel wavelet filter banks for graph structured data,

S. K. Narang and A. Ortega, “Perfect reconstruction two-channel wavelet filter banks for graph structured data,”IEEE Transactions on Signal Processing, vol. 60, no. 6, pp. 2786–2799, 2012

work page 2012
[8]

Towards a sampling theorem for signals on arbitrary graphs,

A. Anis and A. Ortega, “Towards a sampling theorem for signals on arbitrary graphs,” inProceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2014, pp. 3864–3868

work page 2014
[9]

A survey and taxonomy of graph sampling,

P. Hu and W. C. Lau, “A survey and taxonomy of graph sampling,” Computer Science, Aug. 2013

work page 2013
[10]

Signal recovery on graphs: Variation minimization,

S. Chen, A. Sandryhaila, J. M. F. Moura, and J. Kova ˇcevi´c, “Signal recovery on graphs: Variation minimization,”IEEE Transactions on Signal Processing, vol. 63, no. 17, Sept. 2015

work page 2015
[11]

Signal recovery on graphs: fundamental limits of sampling strategies,

S. K. Narang and A. Ortega, “Signal recovery on graphs: fundamental limits of sampling strategies,”IEEE Transactions on Signal and Information Processing over Networks, vol. 2, no. 4, pp. 539–554, Dec. 2016

work page 2016
[12]

Windowed fractional Fourier transform on graphs: Properties and fast algorithm,

F. Yan and B. Li, “Windowed fractional Fourier transform on graphs: Properties and fast algorithm,”Digital Signal Processing, vol. 118, p. 103210, Nov. 2021

work page 2021
[13]

Windowed fractional Fourier transform on graphs: Fractional translation operator and Hausdorff-Young in- equality,

F. Yan, W. Gao, and B. Li, “Windowed fractional Fourier transform on graphs: Fractional translation operator and Hausdorff-Young in- equality,” inProceedings of APSIPA Annual Summit and Conference (APSIPA ASC), 2020, pp. 255–259

work page 2020
[14]

Spectral graph fractional Fourier transform for directed graphs and its application,

F. Yan and B. Li, “Spectral graph fractional Fourier transform for directed graphs and its application,”Signal Processing, vol. 210, p. 109099, Sep. 2023

work page 2023
[15]

The graph fractional Fourier transform in Hilbert space,

Y . Zhang and B. Li, “The graph fractional Fourier transform in Hilbert space,”IEEE Transactions on Signal and Information Processing over Networks, vol. 11, pp. 242–257, 2025

work page 2025
[16]

Generalized sampling of graph signals with the prior information based on graph fractional Fourier transform,

D. Wei and Z. Yan, “Generalized sampling of graph signals with the prior information based on graph fractional Fourier transform,”Signal Processing, vol. 214, p. 109263, Jan. 2024

work page 2024
[17]

Hermitian random walk graph Fourier transform for directed graphs and its applications,

D. Wei and S. Yuan, “Hermitian random walk graph Fourier transform for directed graphs and its applications,”Digital Signal Processing, vol. 155, p. 104751, Dec. 2024

work page 2024
[18]

Wiener filtering in joint time- vertex fractional Fourier domains,

T. Alikas ¸ifo˘glu, B. Kartal, and A. Koc ¸, “Wiener filtering in joint time- vertex fractional Fourier domains,”IEEE Signal Processing Letters, vol. 31, pp. 1319–1323, 2024

work page 2024
[19]

The fractional Fourier transform on graphs,

Y . Wang, B. Li, and Q. Cheng, “The fractional Fourier transform on graphs,” inProceedings of APSIPA Annual Summit and Conference (APSIPA ASC), 2017, pp. 105–110

work page 2017
[20]

The fractional fourier transform on graphs: Sampling and recovery,

Y . Wang and B. Li, “The fractional fourier transform on graphs: Sampling and recovery,” in2018 14th IEEE International Conference on Signal Processing (ICSP), 2018, pp. 1103–1108

work page 2018
[21]

Optimal fractional Fourier filtering for graph signals,

C. Ozturk, H. M. Ozaktas, S. Gezici, and A. Koc ¸, “Optimal fractional Fourier filtering for graph signals,”IEEE Transactions on Signal Processing, vol. 69, pp. 2902–2912, 2021

work page 2021
[22]

Graph chirp signal and graph fractional vertex–frequency energy distribution,

M. Cui, Z. Zhang, and W. Yao, “Graph chirp signal and graph fractional vertex–frequency energy distribution,”IEEE Transactions on Signal Processing, vol. 73, pp. 5286–5302, 2025

work page 2025
[23]

Multi-dimensional graph fractional fourier transform and its application to data compression,

F.-J. Yan and B.-Z. Li, “Multi-dimensional graph fractional fourier transform and its application to data compression,”Digital Signal Processing, vol. 129, p. 103683, 2022

work page 2022
[24]

Multiple-parameter graph fractional fourier transform: Theory and applications,

M. Cui, Z. Zhang, and W. Yao, “Multiple-parameter graph fractional fourier transform: Theory and applications,” 2025. [Online]. Available: https://arxiv.org/abs/2507.23570

work page arXiv 2025
[25]

Graph fractional Fourier transform: A unified theory,

T. Alikas ¸ifo˘glu, B. Kartal, and A. Koc ¸, “Graph fractional Fourier transform: A unified theory,”IEEE Transactions on Signal Process- ing, vol. 72, pp. 3834–3850, 2024

work page 2024
[26]

Trainable joint time-vertex fractional Fourier transform,

Z. Yan and Z. Zhang, “Trainable joint time-vertex fractional Fourier transform,”Digital Signal Processing, vol. 173, p. 105909, 2026

work page 2026
[27]

Angular graph fractional Fourier transform: Theory and application,

F. Zhao, Y . He, and Z. Zhang, “Angular graph fractional Fourier transform: Theory and application,”arXiv preprint arXiv:2511.16111, 2025

work page arXiv 2025
[28]

Graph Embedding in the Graph Fractional Fourier Transform Domain

C. Sheng, Z. Zhang, and W. Yao, “Graph embedding in the graph fractional fourier transform domain,” 2025. [Online]. Available: https://arxiv.org/abs/2508.02383

work page internal anchor Pith review Pith/arXiv arXiv 2025
[29]

Graph fractional Hilbert transform: Theory and application,

D. Li and Z. Zhang, “Graph fractional Hilbert transform: Theory and application,”Fractal and Fractional, vol. 10, no. 2, 2026

work page 2026
[30]

Specformer: Spectral graph neural networks meet transformers,

D. Bo, C. Shi, L. Wang, and R. Liao, “Specformer: Spectral graph neural networks meet transformers,” inProceedings of International Conference on Learning Representations (ICLR), 2023

work page 2023
[31]

Discrete signal processing on graphs,

A. Sandryhaila and J. M. F. Moura, “Discrete signal processing on graphs,”IEEE Transactions on Signal Processing, vol. 61, no. 7, pp. 1644–1656, 2013

work page 2013
[32]

Signless Laplacian spec- trum of a graph,

A. H. Ghodrati and M. A. Hosseinzadeh, “Signless Laplacian spec- trum of a graph,”Linear Algebra and its Applications, vol. 682, pp. 257–267, 2024

work page 2024
[33]

Graph signal processing over a probability space of shift operators,

F. Ji, W. P. Tay, and A. Ortega, “Graph signal processing over a probability space of shift operators,”IEEE Transactions on Signal Processing, vol. 71, pp. 1159–1174, 2023

work page 2023
[34]

Fractional graph spectral filtering based on unified graph representation matrix,

F. Zhao and Z. Zhang, “Fractional graph spectral filtering based on unified graph representation matrix,”IEEE Signal Processing Letters, 2025

work page 2025
[35]

J. R. Partington,Fourier Transforms and Complex Analysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006

work page 2006
[36]

A. V . Oppenheim, A. S. Willsky, and S. H. Nawab,Signals & systems. Pearson Educaci ´on, 1997

work page 1997
[37]

E. M. Stein and R. Shakarchi,Fourier analysis: an introduction. Princeton University Press, 2011, vol. 1

work page 2011
[38]

K. W. Cattermole,The Fourier transform and its applications. IET, 1965

work page 1965
[39]

USC-SIPI image database,

“USC-SIPI image database,” https://sipi.usc.edu/database/, accessed: 2026

work page 2026
[40]

JPEG pleno database: microsoft voxelized upper bodies-a voxelized point cloud dataset,

L. Charles, Q. Cai, E. S. Orts, and A. C. Philip, “JPEG pleno database: microsoft voxelized upper bodies-a voxelized point cloud dataset,” 2021

work page 2021

[1] [1]

The emerging field of signal processing on graphs: Extending high dimensional data analysis to networks and other irregular domains,

D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Van- dergheynst, “The emerging field of signal processing on graphs: Extending high dimensional data analysis to networks and other irregular domains,”IEEE Signal Processing Magazine, vol. 30, no. 3, pp. 83–98, 2013

work page 2013

[2] [2]

Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure,

A. Sandryhaila and J. M. F. Moura, “Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure,”IEEE Signal Processing Magazine, vol. 31, no. 5, pp. 80–90, 2014

work page 2014

[3] [3]

Graph signal denoising via trilateral filter on graph spectral domain,

M. Onuki, S. Ono, M. Yamagishi, and Y . Tanaka, “Graph signal denoising via trilateral filter on graph spectral domain,”IEEE Trans- actions on Signal and Information Processing over Networks, vol. 2, no. 2, pp. 137–148, 2016

work page 2016

[4] [4]

Graph signal processing – Part I: Graphs, graph spectra, and spectral clustering,

L. Stankovic, D. Mandic, M. Dakovic, M. Brajovic, B. Scalzo, and T. Constantinides, “Graph signal processing – Part I: Graphs, graph spectra, and spectral clustering,”arXiv preprint arXiv:1907.03467, 2019

work page arXiv 1907

[5] [5]

Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,

S. K. Narang and A. Ortega, “Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,”IEEE Transactions on Signal Processing, vol. 61, no. 19, pp. 4673–4685, 2013

work page 2013

[6] [6]

Discrete signal processing on graphs: sampling theory,

S. Chen, R. Varma, A. Sandryhaila, and J. Kova ˇcevi´c, “Discrete signal processing on graphs: sampling theory,”IEEE Transactions on Signal Processing, vol. 63, no. 24, pp. 6510–6523, Dec. 2015

work page 2015

[7] [7]

Perfect reconstruction two-channel wavelet filter banks for graph structured data,

S. K. Narang and A. Ortega, “Perfect reconstruction two-channel wavelet filter banks for graph structured data,”IEEE Transactions on Signal Processing, vol. 60, no. 6, pp. 2786–2799, 2012

work page 2012

[8] [8]

Towards a sampling theorem for signals on arbitrary graphs,

A. Anis and A. Ortega, “Towards a sampling theorem for signals on arbitrary graphs,” inProceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2014, pp. 3864–3868

work page 2014

[9] [9]

A survey and taxonomy of graph sampling,

P. Hu and W. C. Lau, “A survey and taxonomy of graph sampling,” Computer Science, Aug. 2013

work page 2013

[10] [10]

Signal recovery on graphs: Variation minimization,

S. Chen, A. Sandryhaila, J. M. F. Moura, and J. Kova ˇcevi´c, “Signal recovery on graphs: Variation minimization,”IEEE Transactions on Signal Processing, vol. 63, no. 17, Sept. 2015

work page 2015

[11] [11]

Signal recovery on graphs: fundamental limits of sampling strategies,

S. K. Narang and A. Ortega, “Signal recovery on graphs: fundamental limits of sampling strategies,”IEEE Transactions on Signal and Information Processing over Networks, vol. 2, no. 4, pp. 539–554, Dec. 2016

work page 2016

[12] [12]

Windowed fractional Fourier transform on graphs: Properties and fast algorithm,

F. Yan and B. Li, “Windowed fractional Fourier transform on graphs: Properties and fast algorithm,”Digital Signal Processing, vol. 118, p. 103210, Nov. 2021

work page 2021

[13] [13]

Windowed fractional Fourier transform on graphs: Fractional translation operator and Hausdorff-Young in- equality,

F. Yan, W. Gao, and B. Li, “Windowed fractional Fourier transform on graphs: Fractional translation operator and Hausdorff-Young in- equality,” inProceedings of APSIPA Annual Summit and Conference (APSIPA ASC), 2020, pp. 255–259

work page 2020

[14] [14]

Spectral graph fractional Fourier transform for directed graphs and its application,

F. Yan and B. Li, “Spectral graph fractional Fourier transform for directed graphs and its application,”Signal Processing, vol. 210, p. 109099, Sep. 2023

work page 2023

[15] [15]

The graph fractional Fourier transform in Hilbert space,

Y . Zhang and B. Li, “The graph fractional Fourier transform in Hilbert space,”IEEE Transactions on Signal and Information Processing over Networks, vol. 11, pp. 242–257, 2025

work page 2025

[16] [16]

Generalized sampling of graph signals with the prior information based on graph fractional Fourier transform,

D. Wei and Z. Yan, “Generalized sampling of graph signals with the prior information based on graph fractional Fourier transform,”Signal Processing, vol. 214, p. 109263, Jan. 2024

work page 2024

[17] [17]

Hermitian random walk graph Fourier transform for directed graphs and its applications,

D. Wei and S. Yuan, “Hermitian random walk graph Fourier transform for directed graphs and its applications,”Digital Signal Processing, vol. 155, p. 104751, Dec. 2024

work page 2024

[18] [18]

Wiener filtering in joint time- vertex fractional Fourier domains,

T. Alikas ¸ifo˘glu, B. Kartal, and A. Koc ¸, “Wiener filtering in joint time- vertex fractional Fourier domains,”IEEE Signal Processing Letters, vol. 31, pp. 1319–1323, 2024

work page 2024

[19] [19]

The fractional Fourier transform on graphs,

Y . Wang, B. Li, and Q. Cheng, “The fractional Fourier transform on graphs,” inProceedings of APSIPA Annual Summit and Conference (APSIPA ASC), 2017, pp. 105–110

work page 2017

[20] [20]

The fractional fourier transform on graphs: Sampling and recovery,

Y . Wang and B. Li, “The fractional fourier transform on graphs: Sampling and recovery,” in2018 14th IEEE International Conference on Signal Processing (ICSP), 2018, pp. 1103–1108

work page 2018

[21] [21]

Optimal fractional Fourier filtering for graph signals,

C. Ozturk, H. M. Ozaktas, S. Gezici, and A. Koc ¸, “Optimal fractional Fourier filtering for graph signals,”IEEE Transactions on Signal Processing, vol. 69, pp. 2902–2912, 2021

work page 2021

[22] [22]

Graph chirp signal and graph fractional vertex–frequency energy distribution,

M. Cui, Z. Zhang, and W. Yao, “Graph chirp signal and graph fractional vertex–frequency energy distribution,”IEEE Transactions on Signal Processing, vol. 73, pp. 5286–5302, 2025

work page 2025

[23] [23]

Multi-dimensional graph fractional fourier transform and its application to data compression,

F.-J. Yan and B.-Z. Li, “Multi-dimensional graph fractional fourier transform and its application to data compression,”Digital Signal Processing, vol. 129, p. 103683, 2022

work page 2022

[24] [24]

Multiple-parameter graph fractional fourier transform: Theory and applications,

M. Cui, Z. Zhang, and W. Yao, “Multiple-parameter graph fractional fourier transform: Theory and applications,” 2025. [Online]. Available: https://arxiv.org/abs/2507.23570

work page arXiv 2025

[25] [25]

Graph fractional Fourier transform: A unified theory,

T. Alikas ¸ifo˘glu, B. Kartal, and A. Koc ¸, “Graph fractional Fourier transform: A unified theory,”IEEE Transactions on Signal Process- ing, vol. 72, pp. 3834–3850, 2024

work page 2024

[26] [26]

Trainable joint time-vertex fractional Fourier transform,

Z. Yan and Z. Zhang, “Trainable joint time-vertex fractional Fourier transform,”Digital Signal Processing, vol. 173, p. 105909, 2026

work page 2026

[27] [27]

Angular graph fractional Fourier transform: Theory and application,

F. Zhao, Y . He, and Z. Zhang, “Angular graph fractional Fourier transform: Theory and application,”arXiv preprint arXiv:2511.16111, 2025

work page arXiv 2025

[28] [28]

Graph Embedding in the Graph Fractional Fourier Transform Domain

C. Sheng, Z. Zhang, and W. Yao, “Graph embedding in the graph fractional fourier transform domain,” 2025. [Online]. Available: https://arxiv.org/abs/2508.02383

work page internal anchor Pith review Pith/arXiv arXiv 2025

[29] [29]

Graph fractional Hilbert transform: Theory and application,

D. Li and Z. Zhang, “Graph fractional Hilbert transform: Theory and application,”Fractal and Fractional, vol. 10, no. 2, 2026

work page 2026

[30] [30]

Specformer: Spectral graph neural networks meet transformers,

D. Bo, C. Shi, L. Wang, and R. Liao, “Specformer: Spectral graph neural networks meet transformers,” inProceedings of International Conference on Learning Representations (ICLR), 2023

work page 2023

[31] [31]

Discrete signal processing on graphs,

A. Sandryhaila and J. M. F. Moura, “Discrete signal processing on graphs,”IEEE Transactions on Signal Processing, vol. 61, no. 7, pp. 1644–1656, 2013

work page 2013

[32] [32]

Signless Laplacian spec- trum of a graph,

A. H. Ghodrati and M. A. Hosseinzadeh, “Signless Laplacian spec- trum of a graph,”Linear Algebra and its Applications, vol. 682, pp. 257–267, 2024

work page 2024

[33] [33]

Graph signal processing over a probability space of shift operators,

F. Ji, W. P. Tay, and A. Ortega, “Graph signal processing over a probability space of shift operators,”IEEE Transactions on Signal Processing, vol. 71, pp. 1159–1174, 2023

work page 2023

[34] [34]

Fractional graph spectral filtering based on unified graph representation matrix,

F. Zhao and Z. Zhang, “Fractional graph spectral filtering based on unified graph representation matrix,”IEEE Signal Processing Letters, 2025

work page 2025

[35] [35]

J. R. Partington,Fourier Transforms and Complex Analysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006

work page 2006

[36] [36]

A. V . Oppenheim, A. S. Willsky, and S. H. Nawab,Signals & systems. Pearson Educaci ´on, 1997

work page 1997

[37] [37]

E. M. Stein and R. Shakarchi,Fourier analysis: an introduction. Princeton University Press, 2011, vol. 1

work page 2011

[38] [38]

K. W. Cattermole,The Fourier transform and its applications. IET, 1965

work page 1965

[39] [39]

USC-SIPI image database,

“USC-SIPI image database,” https://sipi.usc.edu/database/, accessed: 2026

work page 2026

[40] [40]

JPEG pleno database: microsoft voxelized upper bodies-a voxelized point cloud dataset,

L. Charles, Q. Cai, E. S. Orts, and A. C. Philip, “JPEG pleno database: microsoft voxelized upper bodies-a voxelized point cloud dataset,” 2021

work page 2021