Pith. sign in

REVIEW 3 major objections 3 minor 45 references

Optimal Sampling and Reconstruction of Graph Signals in the Fractional Fourier Domain

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Sweeping the fractional order lowers graph-signal reconstruction error

desk verdict Mathematically clean GFRFT-domain sampling framework whose headline 'optimal order' experiment is an oracle selection on the true signal; the conditional theory is valid but the practical claim is unproven. read the letter →

arxiv 2607.15602 v1 pith:537OYUOP submitted 2026-07-17 eess.SP

Xiaopeng Cheng , Zhichao Zhang , Yangfan He This is my paper · ORCID
classification eess.SP
keywords graphsignalprocessingfractionalFouriertransformsamplingreconstructionorderselectionenergyconcentrationprojectionresidualgeneralized
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Graph signals are often sampled and reconstructed in the graph Fourier transform (GFT) domain, but the GFT basis is fixed by the graph and many real signals are not concentrated in it. This paper proposes instead to work in the graph fractional Fourier transform (GFRFT) domain, where a fractional order tunes the spectral representation continuously. The paper argues that choosing the order that best concentrates a signal's energy reduces the low-dimensional projection residual, and it proves a sufficient condition under which the fractional-domain reconstruction has strictly smaller expected error than any GFT-domain reconstruction. Experiments on simulated and real traffic-sensor signals show that the optimized order usually lowers reconstruction error relative to the GFT case. The contribution matters because it turns spectral-domain choice into a tunable parameter inside a unified sampling-correction-reconstruction framework.

What carries the argument

The central object is the GFRFT matrix F_a = V Λ_F^a V^{-1}, fractional powers of the GFT eigenvalues. The scalar a is an adjustable spectral-domain parameter: a = 1 is the GFT; other values rotate the spectral basis continuously. For fixed K, each a defines a K-dimensional fractional spectral subspace with projector P_{K,a}; the energy concentration ratio ρ_K(a) = ||P_{K,a}x||^2/||x||^2 captures how much signal energy is kept. Theorem 1 carries the argument: it decomposes expected reconstruction error into projection residual ||r_{K,a}||^2, residual-leakage coefficient τ_a = ||A^†_a S^*_a (I - P_{K,a})||_2, and noise amplification ν_a = σ^2||A^†_a||_F^2, and states an explicit condition und

What would settle it

Take a graph-signal dataset, split into training/validation/test sets, choose the fractional order using only validation data, and compare test-set reconstruction error against the GFT case a = 1. If the validation-selected order does not beat a = 1 consistently, the central claim is falsified. Alternatively, use a pure graph-Fourier eigenvector as the signal: for that signal the GFT is already maximally concentrating, so ρ_K(a) ≤ ρ_K(1) for all a, and any reported improvement would contradict Theorem 1.

Watch

Extended reading notes

Core claim

The paper claims that the graph fractional Fourier transform (GFRFT) with a carefully chosen fractional order gives a better low-dimensional spectral representation for sampling and reconstruction than the fixed GFT. There often exists an order a such that the K-dimensional fractional subspace captures more energy and leaves a smaller projection residual r_{K,a} than the GFT subspace. Theorem 1 makes this precise: if (1 + τ_a^2)||r_{K,a}||^2 + ν_a < ||r_{K,1}||^2, then the expected error of the fractional-domain least-squares reconstruction is strictly below any reconstruction confined to the K-dimensional GFT subspace. τ_a is residual leakage through sampling inversion; ν_a is noise amplifi

Load-bearing premise

The experiments choose the optimal fractional order using the true complete signal to compute reconstruction error, so the reported 'optimal' gains are in-sample oracle results; when the true signal is unknown, the selection rule in Eq. (32) cannot be applied, and the practical advantage over the GFT is not established.

Editorial extensions

If this is right

  • For signals whose energy is spread across the GFT spectrum, searching over fractional orders can reduce reconstruction error at the same sampling rate.
  • The optimal fractional order is not fixed: it depends on the signal, the sampling count, the spectral response, and the prior, so the method is genuinely adaptive.
  • The gain is conditional: Theorem 1 shows that an order which improves energy concentration can still hurt if it increases residual leakage or noise amplification after sampling inversion.
  • The same unified sampling-correction-reconstruction chain covers subspace, smoothness, and stochastic priors, so the order-selection rule applies across different recovery models.
  • On real traffic data, the stochastic unconstrained variant shows the largest gains, suggesting that mismatches between the signal covariance and the GFT spectrum are common in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The experiments select the optimal order by comparing reconstructions against the ground-truth signal (Eq. 32 and Algorithm 1), so the reported improvements are oracle gains; a practical version would need a validation-based or unsupervised order-selection rule, and the observed margins may shrink.
  • The fractional order can be viewed as a cheap way to adapt the spectral dictionary to each signal; the same tuning idea could be transferred to graph filtering, denoising, or classification tasks.
  • Theorem 1 suggests a practical diagnostic: before reconstructing, compute ρ_K(a), τ_a, and ν_a on a candidate grid and flag orders that satisfy the inequality; this could predict when GFRFT will help without knowing the test signal.
  • The large improvement of the stochastic unconstrained method at a* ≈ 0.04 on traffic data hints that real graph signals have low-rank covariance structure poorly aligned with the Laplacian eigenvectors; testing on more datasets would show whether such extreme orders recur.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proposes a graph signal sampling and reconstruction framework in the graph fractional Fourier transform (GFRFT) domain. The fractional order a is treated as an adjustable spectral-domain parameter, and the authors formulate a unified sampling-correction-reconstruction pipeline under subspace, smoothness, and stochastic priors, with both unconstrained and predefined reconstruction modes. They provide theoretical analysis showing that a suitable fractional subspace can reduce projection residual, while residual leakage and noise amplification must be controlled; Theorem 1 gives a sufficient condition for the fractional-domain reconstruction to beat the GFT-domain reconstruction. Experiments on simulated and real traffic signals compare a fixed order a=1 with an 'optimal' order a* selected by minimizing the reconstruction error against ground truth.

Significance. The theoretical framework is systematic and the algebraic derivations in the appendices are internally consistent; Theorem 1 provides a valid conditional sufficient condition. The idea of adapting the spectral representation to the signal via a fractional transform is of potential interest. However, the key empirical claim that GFRFT-domain methods 'generally achieve better recovery performance' is not supported because the optimal order is selected using ground truth, turning all reported improvements into in-sample oracle gains. Without an operational selection rule that uses only the observed samples, the practical significance of the framework remains unestablished.

major comments (3)
  1. [§III-F, Eq. (32), Algorithm 1] The optimal order a* is defined as the minimizer of ||x - x_hat_a||^2 using the true signal x, and Algorithm 1 selects a* by evaluating reconstruction error against ground truth. The same MSE is then reported in Tables II–V as the improvement over a=1. This makes the measured improvements in-sample oracle results; they are a consequence of the selection rule, not evidence that the method can outperform GFT in a practical reconstruction task where x is unknown on unobserved nodes. The abstract's claim of 'generally better recovery performance' is therefore not supported.
  2. [§IV-B, Theorem 1] Theorem 1 gives a sufficient condition (51) involving ||r_{K,a}||, tau_a, and nu_a. These quantities depend on the unknown signal x and on the chosen fractional subspace, and the paper provides no way to estimate them or to choose a* from the observed samples y_a alone. The theorem thus characterizes when a hindsight oracle would benefit, but it does not supply an operational order-selection rule. To connect theory to practice, the authors need either a data-driven estimator of the condition or a practical criterion based only on y.
  3. [§V-C, Table IV] The real-data experiment uses one traffic signal at a single time step, with no error bars, repeated trials, or statistical significance tests. Many reported improvements are below 0.01 dB (e.g., SS PD G2: 0.0010 dB; ST PD G2: 0.0008 dB) and are likely within sampling and measurement variability. This does not establish that GFRFT-domain methods 'generally' outperform GFT on real data. Please provide confidence intervals, multiple time steps, or a significance analysis, and also show the a=1 baseline in the sensitivity analysis of Table V.
minor comments (3)
  1. [§IV-A, Definition 1] The symbol U_{K,a} is used both for the K-dimensional subspace and for an orthonormal basis of that subspace. This is confusing; please use distinct notation, e.g., V_{K,a} for the basis.
  2. [Appendix A, Prop. 2 proof] Typographical error: 'aaainduced' should be 'induced'.
  3. [§V, Tables II–V] The tables do not report standard deviations or the number of random trials. Fixed random seeds ensure reproducibility, but variability across seeds or sampling sets is not quantified.

Circularity Check

2 steps flagged · score 7.0 of 10

Empirical GFRFT advantage is selected with ground-truth x, making reported gains a min-over-grid artifact rather than a prediction.

  1. self definitional [Section III-F, Eq. (32) and Section V-A, Eq. (58); Algorithm 1]
    "a∗ = arg min a∈A ∥x− exa∥2 2 . (32) ... ∆MSE = MSEa=1 −MSE a=a∗ . Here, MSE a=a∗ denotes the minimum error obtained over the searched fractional orders."

    Equation (32) defines the 'optimal' fractional order as the minimizer of the same reconstruction MSE that is later reported as the improvement. Since the candidate set explicitly contains a=1, MSE_{a=a*} is the minimum over the grid, so ΔMSE≥0 by construction. The empirical statement that GFRFT 'generally achieves better recovery' is therefore not an independent measurement but a logical consequence of the selection rule.

  2. fitted input called prediction [Algorithm 1 (Require line and steps 7-10); Section III-F; Section V-C]
    "Require: Graph signal x, candidate order set A ... 7: Compute the reconstruction error Ea = 10 log10 (1/N ∥x− exa∥2 2) ... In supervised experiments, the ground truth graph signal is available, and the reconstruction error can be directly evaluated."

    Algorithm 1 selects a* by evaluating reconstruction error against the true signal x, which is exactly the quantity a reconstruction method is supposed to recover from the partial observations y. The paper acknowledges ground truth is used, but gives no rule to choose a* from y alone. Thus the reported gains are oracle in-sample fits, not predictions of a deployable GFRFT method; in real graph-signal reconstruction x is unknown on unobserved nodes, so the comparison is self-referential.

full rationale

The theoretical core (Propositions 1-3, Theorem 1) is not circular: the reconstruction formulas follow from least-squares/linear estimation, and Theorem 1 is a conditional sufficient condition (if a fractional subspace has enough energy concentration relative to leakage and noise, then its MSE is smaller). However, the paper's central empirical claim—that GFRFT domain sampling and reconstruction generally outperform GFT—is validated by Eq. (32), which defines a* as the minimizer of the evaluation metric, and by Algorithm 1, which uses the true signal x to pick the best grid point. Because the candidate set includes a=1, the reported ΔMSE is guaranteed nonnegative, so the 'improvement' is a mathematical artifact of the selection procedure. No cross-validation or leave-out rule based only on y is provided, and the one-step real-data experiment also uses this oracle selection. This is not a self-citation issue; the problem is that the load-bearing experimental comparison reduces by construction to fitting the order to the test signal. Score 7: significant circularity in the empirical evaluation, while the theoretical analysis retains independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the GFRFT definition and rank/known-prior assumptions carried from prior literature, plus an oracle-fitted fractional order in the experimental validation. No new physical entities are introduced.

free parameters (3)
  • optimal fractional order a* = varies per experiment, e.g., 0.26, 0.54, 0.04, 0.88, 1.00
    Chosen by Eq. (32) as argmin over a candidate set of reconstruction MSE on the same signal being evaluated; this makes the reported improvement an in-sample oracle result.
  • candidate order grid step = 0.02 over [0,1]
    Hand-chosen discretization; determines the resolution of a* and therefore the achievable improvement.
  • experimental constants = epsilon=0.3 in G1; sigma=0.10 noise level
    Arbitrary constants used in the spectral responses and noisy simulations; they shape all numerical results but are not central to the theoretical claim.
assumptions (5)
  • domain assumption GFRFT matrix definition F_a = V Lambda_F^a V^{-1} with a chosen branch of the complex fractional power
    Section II-B adopts this definition from prior GFRFT literature; the subsequent sampling and reconstruction formulas all depend on it.
  • domain assumption Full column rank of the effective sampling matrices S_a^* M (or A_a) for all orders considered
    Required for Propositions 1-3 and Theorem 1; the paper does not verify rank conditions in the experiments.
  • domain assumption Signal priors: known generator matrix M under subspace prior, bounded variation under smoothness prior, known covariance Gamma_x under stochastic prior
    Section II-C states these priors; in the experiments, the construction of M and Gamma_x is not fully specified, so the prior assumptions cannot be independently checked.
  • ad hoc to paper U_{K,a} is the K-dimensional fractional spectral subspace spanned by 'selected dominant spectral components'
    Definition 1 assumes a selection criterion (largest magnitude? lowest graph frequency?) is given, but no criterion is stated. The energy concentration ratio rho_K(a) and the projection residual r_{K,a} depend on this unstated choice.
  • standard math Noise is zero-mean with covariance sigma^2 I and is independent of the signal
    Used in Theorem 1 and Proposition 3 (Eqs. 27, 48). Standard but asserted rather than derived.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal Sampling and Reconstruction of Graph Signals in the Fractional Fourier Domain." pith.science (2026). https://pith.science/paper/537OYUOP

@misc{pith2026260715602,
  author       = {Pith},
  title        = {Pith review of: Optimal Sampling and Reconstruction of Graph Signals in the Fractional Fourier Domain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/537OYUOP}},
  note         = {Machine review of arXiv:2607.15602}
}
read the original abstract

Graph signal sampling and reconstruction are commonly formulated in the graph Fourier transform (GFT) domain. However, the reconstruction performance may be limited when practical graph signals are not sufficiently concentrated in the GFT spectrum. To address this issue, this paper proposes a graph signal sampling and reconstruction framework based on the graph fractional Fourier transform (GFRFT) domain. The fractional order is introduced as an adjustable spectral domain parameter, and the optimal order is selected to provide a more suitable representation domain for a given graph signal and sampling model. Under a unified sampling reconstruction formulation, subspace, smoothness, and stochastic priors are incorporated, and both unconstrained and predefined reconstruction mechanisms are considered, leading to several fractional domain sampling and reconstruction methods. Furthermore, the theoretical analysis shows that the optimal GFRFT domain can provide a more suitable low-dimensional spectral representation by improving energy concentration and reducing projection residual. The effects of residual leakage and noise amplification are further considered to explain how this representation advantage is translated into reconstruction error reduction. Experimental results show that, GFRFT domain sampling and reconstruction generally achieve better recovery performance than GFT domain methods.

Figures

Figures reproduced from arXiv: 2607.15602 by the authors.

Figure 1
Figure 1. Overall framework of the GFRFT domain sampling reconstruction method. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Visualization of the simulated graph signals under noiseless and noisy settings. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Representative visualization results for the noiseless simulated experiments. The first row shows the near-subspace case under the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Representative visualization results for the noisy simulated experiments. The first row shows the near-subspace case under the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 5 linked inside Pith

  1. [1]

    Graph signal processing: Overview, challenges, and applications,

    A. Ortega, P. Frossard, J. Kova ˇcevi´c, J. M. Moura, and P. Vandergheynst, “Graph signal processing: Overview, challenges, and applications,”Pro- ceedings of the IEEE, vol. 106, no. 5, pp. 808–828, 2018

    2018

  2. [2]

    Graph signal processing: History, development, impact, and outlook,

    G. Leus, A. G. Marques, J. M. Moura, A. Ortega, and D. I. Shuman, “Graph signal processing: History, development, impact, and outlook,” IEEE Signal Processing Magazine, vol. 40, no. 4, pp. 49–60, 2023

    2023

  3. [3]

    Discrete signal processing on graphs,

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

    2013

  4. [4]

    Ortega,Introduction to Graph Signal Processing

    A. Ortega,Introduction to Graph Signal Processing. Cambridge University Press, 2022

    2022

  5. [5]

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

    A. Sandryhaila and J. M. 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

    2014

  6. [6]

    The emerging field of signal processing on graphs: Ex- tending 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: Ex- tending high-dimensional data analysis to networks and other irregular domains,”IEEE Signal Processing Magazine, vol. 30, no. 3, pp. 83–98, 2013

    2013

  7. [7]

    Fast graph Fourier transforms based on graph symmetry and bipartition,

    K.-S. Lu and A. Ortega, “Fast graph Fourier transforms based on graph symmetry and bipartition,”IEEE Transactions on Signal Processing, vol. 67, no. 18, pp. 4855–4869, 2019

    2019

  8. [8]

    Graph Fourier transform: A stable approximation,

    J. Domingos and J. M. Moura, “Graph Fourier transform: A stable approximation,”IEEE Transactions on Signal Processing, vol. 68, pp. 4422–4437, 2020

    2020

Show all 45 references
  1. [9]

    Generic reversible visible watermarking via regularized graph Fourier transform coding,

    W. Qi, S. Guo, and W. Hu, “Generic reversible visible watermarking via regularized graph Fourier transform coding,”IEEE Transactions on Image Processing, vol. 31, pp. 691–705, 2021

    2021

  2. [10]

    On the shift operator, graph frequency, and optimal filtering in graph signal processing,

    A. Gavili and X.-P. Zhang, “On the shift operator, graph frequency, and optimal filtering in graph signal processing,”IEEE Transactions on Signal Processing, vol. 65, no. 23, pp. 6303–6318, 2017

    2017

  3. [11]

    Sampling, filtering and sparse approximations on combinatorial graphs,

    I. Z. Pesenson and M. Z. Pesenson, “Sampling, filtering and sparse approximations on combinatorial graphs,”Journal of Fourier Analysis and Applications, vol. 16, no. 6, pp. 921–942, 2010

    2010

  4. [12]

    Semi-supervised classification with graph convolutional networks,

    T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,”arXiv preprint arXiv:1609.02907, 2016

    2016 arXiv

  5. [13]

    Inductive representation learning on large graphs,

    W. Hamilton, Z. Ying, and J. Leskovec, “Inductive representation learning on large graphs,”Advances in Neural Information Processing Systems, vol. 30, 2017

    2017

  6. [14]

    Sampling signals on graphs: From theory to applications,

    Y . Tanaka, Y . C. Eldar, A. Ortega, and G. Cheung, “Sampling signals on graphs: From theory to applications,”IEEE Signal Processing Magazine, vol. 37, no. 6, pp. 14–30, 2020

    2020

  7. [15]

    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, 2015

    2015

  8. [16]

    Efficient sampling set selection for bandlimited graph signals using graph spectral proxies,

    A. Anis, A. Gadde, and A. Ortega, “Efficient sampling set selection for bandlimited graph signals using graph spectral proxies,”IEEE Transactions on Signal Processing, vol. 64, no. 14, pp. 3775–3789, 2016

    2016

  9. [17]

    Sampling of graph signals with successive local aggregations,

    A. G. Marques, S. Segarra, G. Leus, and A. Ribeiro, “Sampling of graph signals with successive local aggregations,”IEEE Transactions on Signal Processing, vol. 64, no. 7, pp. 1832–1843, 2015

    2015

  10. [18]

    Sampling in Paley-Wiener spaces on combinatorial graphs,

    I. Pesenson, “Sampling in Paley-Wiener spaces on combinatorial graphs,”Transactions of the American Mathematical Society, vol. 360, no. 10, pp. 5603–5627, 2008

    2008

  11. [19]

    Graph signal processing: Modulation, convo- lution, and sampling,

    J. Shi and J. M. Moura, “Graph signal processing: Modulation, convo- lution, and sampling,”arXiv preprint arXiv:1912.06762, 2019

    1912 arXiv

  12. [20]

    Recovery of missing sensor data by reconstructing time-varying graph signals,

    A. Mondal, M. Das, A. Chatterjee, and P. Venkateswaran, “Recovery of missing sensor data by reconstructing time-varying graph signals,” in 2022 30th European Signal Processing Conference. IEEE, 2022, pp. 2181–2185

    2022

  13. [21]

    Graph signal reconstruction techniques for iot air pollution monitoring platforms,

    P. Ferrer-Cid, J. M. Barcelo-Ordinas, and J. Garcia-Vidal, “Graph signal reconstruction techniques for iot air pollution monitoring platforms,” IEEE Internet of Things Journal, vol. 9, no. 24, pp. 25 350–25 362, 2022

    2022

  14. [22]

    Piecewise stationary modeling of random processes over graphs with an application to traffic prediction,

    A. Hasanzadeh, X. Liu, N. Duffield, and K. R. Narayanan, “Piecewise stationary modeling of random processes over graphs with an application to traffic prediction,” in2019 IEEE International Conference on Big Data, 2019, pp. 3779–3788

    2019

  15. [23]

    Spectral domain sampling of graph signals,

    Y . Tanaka, “Spectral domain sampling of graph signals,”IEEE Trans- actions on Signal Processing, vol. 66, no. 14, pp. 3752–3767, 2018

    2018

  16. [24]

    Two-channel critically sampled graph filter banks with spectral domain sampling,

    A. Sakiyama, K. Watanabe, Y . Tanaka, and A. Ortega, “Two-channel critically sampled graph filter banks with spectral domain sampling,” IEEE Transactions on Signal Processing, vol. 67, no. 6, pp. 1447–1460, 2019

    2019

  17. [25]

    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, 2018, pp. 1103–1108

    2018

  18. [26]

    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 Pro- cessing, vol. 69, pp. 2902–2912, 2021

    2021

  19. [27]

    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 Processing, vol. 72, pp. 3834–3850, 2024

    2024

  20. [28]

    Joint time-vertex fractional Fourier transform,

    T. Alikas ¸ifo˘glu, B. Kartal, E. ¨Ozg¨unay, and A. Koc ¸, “Joint time-vertex fractional Fourier transform,”Signal Processing, vol. 233, p. 109944, 2025

    2025

  21. [29]

    FGFRFT: Fast graph fractional Fourier transform via ex- act spectral splitting and Fourier-series approximation,

    Z. Yan, M. Wang, S. Shi, F. Zhao, M. Cui, Y . He, and Z. Zhang, “FGFRFT: Fast graph fractional Fourier transform via ex- act spectral splitting and Fourier-series approximation,”arXiv preprint arXiv:2602.20870, 2026

    2026 arXiv

  22. [30]

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

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

    2023

  23. [31]

    The graph fractional Fourier transform in Hilbert space,

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

    2025

  24. [32]

    The shannon sampling theorem—its various extensions and applications: A tutorial review,

    A. J. Jerri, “The shannon sampling theorem—its various extensions and applications: A tutorial review,”Proceedings of the IEEE, vol. 65, no. 11, pp. 1565–1596, 1977

    1977

  25. [33]

    Y . C. Eldar,Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, 2015

    2015

  26. [34]

    Beyond bandlimited sampling,

    Y . C. Eldar and T. Michaeli, “Beyond bandlimited sampling,”IEEE Signal Processing Magazine, vol. 26, no. 3, pp. 48–68, 2009

    2009

  27. [35]

    Generalized graph sig- nal sampling by difference-of-convex optimization,

    K. Yamashita, K. Naganuma, and S. Ono, “Generalized graph sig- nal sampling by difference-of-convex optimization,”arXiv preprint arXiv:2306.14634, 2023

    2023 arXiv

  28. [36]

    Generalized sampling on graphs with sub- space and smoothness priors,

    Y . Tanaka and Y . C. Eldar, “Generalized sampling on graphs with sub- space and smoothness priors,”IEEE Transactions on Signal Processing, vol. 68, pp. 2272–2286, 2020

    2020

  29. [37]

    Sampling method for generalized graph signals with pre-selected vertices via dc optimization,

    K. Yamashita, K. Naganuma, and S. Ono, “Sampling method for generalized graph signals with pre-selected vertices via dc optimization,” IEEE Open Journal of Signal Processing, vol. 7, pp. 314–323, 2026

    2026

  30. [38]

    Generalized graph signal sampling and reconstruction,

    X. Wang, J. Chen, and Y . Gu, “Generalized graph signal sampling and reconstruction,” in2015 IEEE Global Conference on Signal and Information Processing, 2015, pp. 567–571

    2015

  31. [39]

    Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,

    Y . C. Eldar and M. Unser, “Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,”IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2636–2651, 2006

    2006

  32. [40]

    A minimum squared-error framework for generalized sampling,

    Y . C. Eldar and T. G. Dvorkind, “A minimum squared-error framework for generalized sampling,”IEEE Transactions on Signal Processing, vol. 54, no. 6, pp. 2155–2167, 2006

    2006

  33. [41]

    Graph signal sampling under stochastic priors,

    J. Hara, Y . Tanaka, and Y . C. Eldar, “Graph signal sampling under stochastic priors,”IEEE Transactions on Signal Processing, vol. 71, pp. 1421–1434, 2023

    2023

  34. [42]

    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, 2024

    2024

  35. [43]

    Diffusion convolutional re- current neural network: Data-driven traffic forecasting,

    Y . Li, R. Yu, C. Shahabi, and Y . Liu, “Diffusion convolutional re- current neural network: Data-driven traffic forecasting,”arXiv preprint arXiv:1707.01926, 2017

    2017 arXiv

  36. [44]

    Signal recovery on graphs: Variation minimization,

    S. Chen, A. Sandryhaila, J. M. Moura, and J. Kova ˇcevi´c, “Signal recovery on graphs: Variation minimization,”IEEE Transactions on Signal Processing, vol. 63, no. 17, pp. 4609–4624, 2015

    2015

  37. [45]

    Kernel based reconstruction for generalized graph signal processing,

    X. Jian, W. P. Tay, and Y . C. Eldar, “Kernel based reconstruction for generalized graph signal processing,”IEEE Transactions on Signal Processing, vol. 72, pp. 2308–2322, 2024

    2024

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.