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arxiv: 2505.22175 · v2 · submitted 2025-05-28 · 📡 eess.SP

Algorithm Unrolling-based Denoising of Multimodal Graph Signals

Hayate Kojima , Keigo Takanami , Junya Hara , Yukihiro Bandoh , Seishi Takamura , Hiroshi Higashi , Yuichi Tanaka This is my paper

Pith reviewed 2026-05-19 13:49 UTC · model grok-4.3

classification 📡 eess.SP
keywords multimodal graph signalsgraph denoisinggraph learningalgorithm unrollingalternating minimizationprimal-dual splittingtwofold graphssignal restoration
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The pith

Unrolled alternating minimization jointly restores multimodal graph signals and learns their twofold graph structure.

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

The paper develops a denoising approach for multimodal graph signals, which capture multiple data types at each sensor point along with spatial and modality correlations. It models the data as signals on a twofold graph whose edges are unknown in advance and must be estimated together with the denoising process. The core procedure alternates between a graph-learning subproblem solved by primal-dual splitting and a closed-form signal update step. These iterations are unrolled into a trainable deep network whose parameters are optimized on training data. Experiments on synthetic and real-world multimodal sets show the joint approach outperforms both model-based graph denoising and existing deep-learning methods.

Core claim

The proposed method solves signal denoising and twofold graph learning jointly via alternating minimization. Graph learning subproblems are handled by the primal-dual splitting algorithm while the signal restoration step has a closed-form solution. The resulting iterative procedure is unrolled and its parameters are learned from data, allowing the network to estimate the underlying twofold graph during denoising without requiring it to be supplied in advance.

What carries the argument

Unrolled alternating minimization that interleaves primal-dual splitting graph learning subproblems with closed-form signal updates.

If this is right

  • Denoising becomes possible without any prior knowledge of the twofold graph edges.
  • The same framework applies directly to sensor-network data that record multiple modalities at each location.
  • Joint learning of structure and signal yields measurable gains over both purely model-based and purely data-driven denoising pipelines.
  • Unrolling converts the alternating optimization into an efficient, trainable network suitable for repeated use on similar data.

Where Pith is reading between the lines

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

  • The same unrolling strategy could be tested on graph models that include higher-order or directed relations if suitable subproblem solvers can be derived.
  • In deployed sensor systems the learned graphs might themselves become useful outputs for downstream tasks such as anomaly detection.
  • Performance may degrade when modality and spatial correlations are only weakly coupled, suggesting a need for explicit regularization terms that the current scheme does not include.

Load-bearing premise

The true relationships among multimodal observations can be captured by one twofold graph that the specific alternating scheme with primal-dual splitting and closed-form updates can recover from noisy data.

What would settle it

A multimodal dataset in which the ground-truth relationships require a graph model other than a single twofold graph or in which the alternating minimization fails to recover accurate edges, yet the method still shows no performance gain over baselines.

Figures

Figures reproduced from arXiv: 2505.22175 by Hayate Kojima, Hiroshi Higashi, Junya Hara, Keigo Takanami, Seishi Takamura, Yuichi Tanaka, Yukihiro Bandoh.

Figure 1
Figure 1. Figure 1: A multimodal graph signal on a twofold graph. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the proposed architecture. III. MULTIMODAL GRAPH SIGNAL DENOISING WITH SIMULTANEOUS GRAPH LEARNING As mentioned above, existing graph signal denoising meth￾ods assume the underlying graph is given prior to denois￾ing. They also assume signals are single-modal. However, they are not often the case. In this section, we propose a denoising method for multimodal graph signals by learning twofold gr… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of synthetic multimodal graph signals. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Denoising results for synthetic dataset ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of learned spatial graph Ws for synthetic dataset. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 0.0 0.2 0.4 0.6 0.8 1.0 (a) Ground-truth 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 0.0 0.2… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of learned modality graph Wm for synthetic dataset. 0 2 4 6 8 Layer 0.00 0.25 0.50 0.75 1.00 1.25 1.50 αs βs γs αm βm γm [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Values of the learned parameters for each layer [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Layer-wise root mean squared error (RMSE) for different noise levels [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Denoising results for real-world dataset ( [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Visualization of learned spatial graph Ws for real-world dataset. 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 0.0 0.2 0.4 0.6 0.8 1.0 (a) Layer 1 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 0.0 0.2 0.4 0.6 0.8 1.… view at source ↗
Figure 11
Figure 11. Figure 11: Visualization of learned modality graph Wm for real-world dataset. 2) Learned Graphs: The adjacency matrices of the graphs estimated by the proposed method are shown in Figs. 10 and 11. Similar to the synthetic data, the graphs obtained in the first layer have low edge weights which results in global denoising. In the following layers, edge weights are almost consistent until the seventh or eighth layers:… view at source ↗
read the original abstract

We propose a denoising method for multimodal graph signals by an alternating minimization scheme that sequentially solves signal restoration and graph learning problems. Many complex-structured data, i.e., those on sensor networks, can capture multiple modalities at each measurement point, referred to as modalities. They are also assumed to have an underlying structure or correlations in modality as well as space. Such multimodal data are regarded as graph signals on a twofold graph and they are often corrupted by noise. Furthermore, their spatial/modality relationships are not always given a priori: We need to estimate twofold graphs during a denoising algorithm. In this paper, we consider a signal denoising method on twofold graphs, where graphs are learned simultaneously. Specifically, the graph learning subproblems are solved using the primal-dual splitting (PDS) algorithm, while the signal update has a closed-form solution. Parameters in this iterative algorithm are learned from training data by unrolling the iteration with deep algorithm unrolling. Experimental results on synthetic and real-world data demonstrate that the proposed method outperforms existing model- and deep learning-based graph signal denoising methods.

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.

Referee Report

2 major / 1 minor

Summary. The paper proposes a denoising method for multimodal graph signals on twofold graphs via an alternating minimization scheme. Graph learning subproblems are addressed using the primal-dual splitting (PDS) algorithm while the signal restoration step admits a closed-form solution; all iteration parameters are learned from data by unrolling the algorithm. Experiments on synthetic and real-world multimodal data report outperformance relative to existing model-based and deep-learning graph signal denoising methods.

Significance. If the central claim holds, the work contributes a hybrid model-based/data-driven framework for joint signal denoising and twofold graph learning, which is relevant for sensor-network applications with spatial and modality correlations. The explicit use of PDS for the graph subproblems paired with a closed-form signal update is a concrete strength that preserves some interpretability while allowing data-driven tuning of step sizes and regularizers. The experimental validation across synthetic and real multimodal datasets further supports practical relevance.

major comments (2)
  1. §3.2 (unrolled PDS description): No verification is given that the learned step sizes and regularization parameters satisfy the step-size or Lipschitz-constant conditions required for convergence of the original primal-dual splitting algorithm applied to the graph-learning subproblems. Because the central claim rests on the unrolled alternating scheme jointly recovering a twofold graph and denoised signal, violation of these conditions could mean the finite unrolling produces iterates that do not correspond to any stationary point of the original objective, rendering the reported gains potentially artifacts of a non-convergent surrogate rather than a principled estimator.
  2. §5 (experimental results): The performance tables and figures provide no error bars, standard deviations across multiple runs, details on training/validation splits, or ablation studies on unrolling depth and the twofold-graph assumption. These omissions directly affect the verifiability of the outperformance claim over baselines on both synthetic and real multimodal data.
minor comments (1)
  1. Abstract and §2: The term 'twofold graph' is used without a concise definition of its two edge sets (spatial and modality) on first appearance; a brief parenthetical or reference would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify and strengthen our manuscript. We provide point-by-point responses to the major comments below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [—] §3.2 (unrolled PDS description): No verification is given that the learned step sizes and regularization parameters satisfy the step-size or Lipschitz-constant conditions required for convergence of the original primal-dual splitting algorithm applied to the graph-learning subproblems. Because the central claim rests on the unrolled alternating scheme jointly recovering a twofold graph and denoised signal, violation of these conditions could mean the finite unrolling produces iterates that do not correspond to any stationary point of the original objective, rendering the reported gains potentially artifacts of a non-convergent surrogate rather than a principled estimator.

    Authors: We appreciate the referee's emphasis on theoretical convergence. In the unrolled framework, parameters are optimized end-to-end via back-propagation on training data to minimize a task-specific loss; they are therefore not constrained to obey the step-size or Lipschitz conditions of the original PDS iteration. The resulting finite-depth network is a learned estimator whose effectiveness is validated empirically rather than by inheritance of the unrolled algorithm's fixed-point guarantees. Our experiments on synthetic and real multimodal data show consistent gains over both model-based and deep-learning baselines. In the revision we will expand Section 3.2 with a brief discussion of this distinction and add a short empirical study of iterate stability under the learned parameters. revision: partial

  2. Referee: [—] §5 (experimental results): The performance tables and figures provide no error bars, standard deviations across multiple runs, details on training/validation splits, or ablation studies on unrolling depth and the twofold-graph assumption. These omissions directly affect the verifiability of the outperformance claim over baselines on both synthetic and real multimodal data.

    Authors: We agree that these omissions limit the strength of the experimental claims. In the revised manuscript we will augment Section 5 with (i) error bars and standard deviations computed over at least five independent runs with different random seeds, (ii) explicit descriptions of the training/validation/test splits for both synthetic and real-world datasets, and (iii) ablation experiments that vary the unrolling depth and compare performance with and without the twofold-graph modeling assumption. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic unrolling defines independent method

full rationale

The paper proposes a concrete alternating minimization algorithm for joint signal denoising and twofold graph learning, with PDS subproblems for graphs and closed-form signal updates, then unrolls it for parameter learning from training data. This construction is self-contained and does not reduce any claimed result or prediction to its own inputs by definition, fitted parameters renamed as outputs, or load-bearing self-citation chains. Performance claims rest on separate experimental comparisons rather than any tautological derivation step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the existence of an underlying twofold graph that can be recovered by the described alternating procedure and on the suitability of the chosen PDS and closed-form updates for the subproblems.

free parameters (1)
  • unrolling depth and layer-wise parameters
    Learned from training data; number of iterations and per-layer weights are not fixed a priori.
axioms (1)
  • domain assumption Multimodal data lie on a twofold graph whose spatial and modality edges can be learned jointly with the signal.
    Stated in the abstract as the modeling premise for the denoising task.

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    Multimodadl Graph Signal Denoising With Simultaneous Graph Learning using Deep Algorithm Unrolling,

    K. Takanami, Y . Bandoh, S. Takamura, and Y . Tanaka, “Multimodadl Graph Signal Denoising With Simultaneous Graph Learning using Deep Algorithm Unrolling,” in2023 IEEE Int Conf Image Process ICIP, Oct. 2023, pp. 1555–1559

    work page 2023

  2. [2]

    A survey on sensor networks,

    I. Akyildiz, W. Su, Y . Sankarasubramaniam, and E. Cayirci, “A survey on sensor networks,”IEEE Commun. Mag., vol. 40, no. 8, pp. 102–114, Aug. 2002

    work page 2002

  3. [3]

    Wireless sensor networks: A survey,

    I. F. Akyildiz, W. Su, Y . Sankarasubramaniam, and E. Cayirci, “Wireless sensor networks: A survey,”Comput. Netw., vol. 38, no. 4, pp. 393–422, Mar. 2002

    work page 2002

  4. [4]

    Wireless sensor networks for healthcare: A survey,

    H. Alemdar and C. Ersoy, “Wireless sensor networks for healthcare: A survey,”Comput. Netw., vol. 54, no. 15, pp. 2688–2710, Oct. 2010

    work page 2010

  5. [5]

    Sensor networks: An overview,

    M. Tubaishat and S. Madria, “Sensor networks: An overview,”IEEE Potentials, vol. 22, no. 2, pp. 20–23, Apr. 2003

    work page 2003

  6. [6]

    An Overview of IoT Sensor Data Processing, Fusion, and Analysis Techniques,

    R. Krishnamurthi, A. Kumar, D. Gopinathan, A. Nayyar, and B. Qureshi, “An Overview of IoT Sensor Data Processing, Fusion, and Analysis Techniques,”Sensors, vol. 20, no. 21, p. 6076, Jan. 2020

    work page 2020

  7. [7]

    Sensor network data denoising via recursive graph median filters,

    D. B. Tay, “Sensor network data denoising via recursive graph median filters,”Signal Processing, vol. 189, p. 108302, Dec. 2021

    work page 2021

  8. [8]

    Graph signal processing: Overview, challenges, and ap- plications,

    A. Ortega, P. Frossard, J. Kovacevic, J. M. F. Moura, and P. Van- dergheynst, “Graph signal processing: Overview, challenges, and ap- plications,”Proc. IEEE, vol. 106, no. 5, pp. 808–828, May 2018

    work page 2018

  9. [9]

    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 Process. Mag., vol. 40, no. 4, pp. 49–60, Jun. 2023

    work page 2023

  10. [10]

    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 Process. Mag., vol. 37, no. 6, pp. 14–30, Jan. 2020

    work page 2020

  11. [11]

    Multimodal Graph Signal Denoising Via Twofold Graph Smoothness Regularization with Deep Algorithm Unrolling,

    M. Nagahama and Y . Tanaka, “Multimodal Graph Signal Denoising Via Twofold Graph Smoothness Regularization with Deep Algorithm Unrolling,” inICASSP 2022 - 2022 IEEE Int. Conf. Acoust. Speech Signal Process. ICASSP, May 2022, pp. 5862–5866

    work page 2022

  12. [12]

    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 Process. Mag., vol. 30, no. 3, pp. 83–98, May 2013

    work page 2013

  13. [13]

    Graph Filters for Signal Processing and Machine Learning on Graphs,

    E. Isufi, F. Gama, D. I. Shuman, and S. Segarra, “Graph Filters for Signal Processing and Machine Learning on Graphs,”IEEE Trans. Signal Process., vol. 72, pp. 4745–4781, 2024

    work page 2024

  14. [14]

    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. Signal Inf. Process. Netw., vol. 2, no. 2, pp. 137–148, Jun. 2016

    work page 2016

  15. [15]

    Graph spectral image smoothing using the heat kernel,

    F. Zhang and E. R. Hancock, “Graph spectral image smoothing using the heat kernel,”Pattern Recognition, vol. 41, no. 11, pp. 3328–3342, Nov. 2008. 3Letg:R N →R∪ {∞}be a proper lower semicontiunous convex function. The proximal operator prox µg :R N →R N ofgwith a parameter θ >0is defined by prox θg (x) = arg min y{g(y) + 1 2θ ∥x−y∥ 2 2}. If proxθg can be ...

    work page 2008

  16. [16]

    Reconstruction of Time-Varying Graph Signals via Sobolev Smoothness,

    J. H. Giraldo, A. Mahmood, B. Garcia-Garcia, D. Thanou, and T. Bouw- mans, “Reconstruction of Time-Varying Graph Signals via Sobolev Smoothness,”IEEE Trans. Signal Inf. Process. Netw., vol. 8, pp. 201– 214, 2022

    work page 2022

  17. [17]

    Graph Wavelet-Based Point Cloud Geometric Denoising with Surface- Consistent Non-Negative Kernel Regression,

    R. Watanabe, K. Nonaka, E. Pavez, T. Kobayashi, and A. Ortega, “Graph Wavelet-Based Point Cloud Geometric Denoising with Surface- Consistent Non-Negative Kernel Regression,” inICASSP 2023 - 2023 IEEE Int. Conf. Acoust. Speech Signal Process. ICASSP, Jun. 2023, pp. 1–5

    work page 2023

  18. [18]

    Total generalized variation for graph signals,

    S. Ono, I. Yamada, and I. Kumazawa, “Total generalized variation for graph signals,” in2015 IEEE Int. Conf. Acoust. Speech Signal Process. ICASSP, Apr. 2015, pp. 5456–5460

    work page 2015

  19. [19]

    Robust Low-Rank Latent Feature Analysis for Spatiotemporal Signal Recovery,

    D. Wu, Z. Li, Z. Yu, Y . He, and X. Luo, “Robust Low-Rank Latent Feature Analysis for Spatiotemporal Signal Recovery,”IEEE Trans. Neural Netw. Learn. Syst., vol. 36, no. 2, pp. 2829–2842, Feb. 2025

    work page 2025

  20. [20]

    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 Trans. Signal Process., vol. 63, no. 17, pp. 4609–4624, Sep. 2015

    work page 2015

  21. [21]

    Graph Signal Diffusion Model for Collaborative Filtering,

    Y . Zhu, C. Wang, Q. Zhang, and H. Xiong, “Graph Signal Diffusion Model for Collaborative Filtering,” inProc. 47th Int. ACM SIGIR Conf. Res. Dev. Inf. Retr., ser. SIGIR ’24. New York, NY , USA: Association for Computing Machinery, Jul. 2024, pp. 1380–1390

    work page 2024

  22. [22]

    Survey of Graph Neural Networks and Applications,

    F. Liang, C. Qian, W. Yu, D. Griffith, and N. Golmie, “Survey of Graph Neural Networks and Applications,”Wirel. Commun. Mob. Comput., vol. 2022, no. 1, p. 9261537, 2022

    work page 2022

  23. [23]

    Acceleration Algorithms in GNNs: A Survey,

    L. Ma, Z. Sheng, X. Li, X. Gao, Z. Hao, L. Yang, X. Nie, J. Jiang, W. Zhang, and B. Cui, “Acceleration Algorithms in GNNs: A Survey,” IEEE Trans. Knowl. Data Eng., vol. 37, no. 6, pp. 3173–3192, Jun. 2025

    work page 2025

  24. [24]

    Semi-supervised classification with graph convolutional networks,

    T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” in2017 Int Conf Learn Represent ICLR. OpenReview.net, 2017

    work page 2017

  25. [25]

    Deep Learning with Graph Convolutional Networks: An Overview and Latest Applications in Computational Intelligence,

    U. A. Bhatti, H. Tang, G. Wu, S. Marjan, and A. Hussain, “Deep Learning with Graph Convolutional Networks: An Overview and Latest Applications in Computational Intelligence,”Int. J. Intell. Syst., vol. 2023, no. 1, p. 8342104, 2023

    work page 2023

  26. [26]

    A Survey on Graph Neural Networks and Graph Transformers in Computer Vision: A Task-Oriented Perspective,

    C. Chen, Y . Wu, Q. Dai, H.-Y . Zhou, M. Xu, S. Yang, X. Han, and Y . Yu, “A Survey on Graph Neural Networks and Graph Transformers in Computer Vision: A Task-Oriented Perspective,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 46, no. 12, pp. 10 297–10 318, Feb. 2024

    work page 2024

  27. [27]

    GCN-Denoiser: Mesh Denoising with Graph Convolutional Networks,

    Y . Shen, H. Fu, Z. Du, X. Chen, E. Burnaev, D. Zorin, K. Zhou, and Y . Zheng, “GCN-Denoiser: Mesh Denoising with Graph Convolutional Networks,”ACM Trans. Graph., vol. 41, no. 1, pp. 8:1–8:14, Feb. 2022

    work page 2022

  28. [28]

    GNF-Net: An Adaptive Mesh Denoising Method with GCN-Based Guided Normal Filtering,

    J. Yan, L. Liang, Y . Yang, and S. Huang, “GNF-Net: An Adaptive Mesh Denoising Method with GCN-Based Guided Normal Filtering,” in2024 IEEE Int. Conf. Syst. Man Cybern. SMC, Oct. 2024, pp. 4516–4521

    work page 2024

  29. [29]

    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 Trans. Signal Process., vol. 72, pp. 2308–2322, 2024

    work page 2024

  30. [30]

    A review of graph neural networks: Concepts, architectures, techniques, challenges, datasets, applications, and future directions,

    B. Khemani, S. Patil, K. Kotecha, and S. Tanwar, “A review of graph neural networks: Concepts, architectures, techniques, challenges, datasets, applications, and future directions,”Journal of Big Data, vol. 11, no. 1, p. 18, Jan. 2024

    work page 2024

  31. [31]

    Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising,

    S. Chen, Y . C. Eldar, and L. Zhao, “Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising,”IEEE Trans. Signal Process., vol. 69, pp. 3699–3713, 2021

    work page 2021

  32. [32]

    Interpolation and denoising of graph signals using plug-and-play ADMM,

    Y . Yazaki, Y . Tanaka, and S. H. Chan, “Interpolation and denoising of graph signals using plug-and-play ADMM,” inSpan Classnocaseieee- Icasspspan, 2019, pp. 5431–5435

    work page 2019

  33. [33]

    Learning fast approximations of sparse coding,

    K. Gregor and Y . LeCun, “Learning fast approximations of sparse coding,”Proc 27th Int Conf Mach. Learn., pp. 399–406, 2010

    work page 2010

  34. [34]

    Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing,

    V . Monga, Y . Li, and Y . C. Eldar, “Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing,”IEEE Signal Process. Mag., vol. 38, no. 2, pp. 18–44, Mar. 2021

    work page 2021

  35. [35]

    Graph signal restoration using nested deep algorithm unrolling,

    M. Nagahama, K. Yamada, Y . Tanaka, S. H. Chan, and Y . C. Eldar, “Graph signal restoration using nested deep algorithm unrolling,”IEEE Trans. Signal Process., vol. 70, pp. 3296–3311, 2022

    work page 2022

  36. [36]

    Graph-Based Denoising for Respiration and Heart Rate Estimation During Sleep in Thermal Video,

    C. Yang, M. Hu, G. Zhai, and X.-P. Zhang, “Graph-Based Denoising for Respiration and Heart Rate Estimation During Sleep in Thermal Video,” IEEE Internet Things J., vol. 9, no. 17, pp. 15 697–15 713, Sep. 2022

    work page 2022

  37. [37]

    How to learn a graph from smooth signals,

    V . Kalofolias, “How to learn a graph from smooth signals,” inProc. 19th Int. Conf. Artif. Intell. Stat.PMLR, May 2016, pp. 920–929

    work page 2016

  38. [38]

    Learning laplacian matrix in smooth graph signal representations,

    X. Dong, D. Thanou, P. Frossard, and P. Vandergheynst, “Learning laplacian matrix in smooth graph signal representations,”IEEE Trans. Signal Process., vol. 64, no. 23, pp. 6160–6173, Dec. 2016. IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS, VOL. XX, 2026 12

    work page 2016

  39. [39]

    Learning graphs from data: A signal representation perspective,

    X. Dong, D. Thanou, M. Rabbat, and P. Frossard, “Learning graphs from data: A signal representation perspective,”IEEE Signal Process. Mag., vol. 36, no. 3, pp. 44–63, May 2019

    work page 2019

  40. [40]

    Bilateral filtering for gray and color images,

    C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” inSixth Int Conf Comp Vis. ICCV, Jan. 1998, pp. 839–846

    work page 1998

  41. [41]

    The optimal hard threshold for singular values is $4/\sqrt 3$,

    M. Gavish and D. L. Donoho, “The optimal hard threshold for singular values is $4/\sqrt 3$,”IEEE Trans. Inf. Theory, vol. 60, no. 8, pp. 5040– 5053, Aug. 2014

    work page 2014

  42. [42]

    Connecting the Dots: Identifying Network Structure via Graph Signal Processing,

    G. Mateos, S. Segarra, A. G. Marques, and A. Ribeiro, “Connecting the Dots: Identifying Network Structure via Graph Signal Processing,” IEEE Signal Process. Mag., vol. 36, no. 3, pp. 16–43, May 2019

    work page 2019

  43. [43]

    Large scale graph learning from smooth signals,

    V . Kalofolias and N. Perraudin, “Large scale graph learning from smooth signals,” in2019 Int Conf Learn Represent ICLR, 2019

    work page 2019

  44. [44]

    Time-varying graph learning based on sparseness of temporal variation,

    K. Yamada, Y . Tanaka, and A. Ortega, “Time-varying graph learning based on sparseness of temporal variation,” in2019 IEEE Int Conf Acoust Speech Signal Process ICASSP. IEEE, May 2019, pp. 5411– 5415

    work page 2019

  45. [45]

    Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems,

    N. Komodakis and J.-C. Pesquet, “Playing with duality: An overview of recent primal-dual approaches for solving large-scale optimization problems,”IEEE Signal Process. Mag., vol. 32, no. 6, pp. 31–54, Jan. 2015

    work page 2015

  46. [46]

    K. M. Abadir and J. R. Magnus,Matrix Algebra. Cambridge University Press, Aug. 2005

    work page 2005

  47. [47]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in3rd Int. Conf. Learn. Represent. ICLR 2015 San Diego CA USA May 7-9 2015 Conf. Track Proc., Y . Bengio and Y . LeCun, Eds., 2015

    work page 2015

  48. [48]

    GSPBOX: A toolbox for signal processing on graphs,

    N. Perraudin, J. Paratte, D. Shuman, L. Martin, V . Kalofolias, P. Van- dergheynst, and D. K. Hammond, “GSPBOX: A toolbox for signal processing on graphs,” Mar. 2016

    work page 2016

  49. [49]

    Rue and L

    H. Rue and L. Held,Gaussian Markov Random Fields: Theory and Applications. Chapman and Hall/CRC, Feb. 2005

    work page 2005

  50. [50]

    Representations of piecewise smooth signals on graphs,

    S. Chen, R. Varma, A. Singh, and J. Kova ˇcevi´c, “Representations of piecewise smooth signals on graphs,” in2016 IEEE Int Conf Acoust Speech Signal Process ICASSP, Mar. 2016, pp. 6370–6374

    work page 2016

  51. [51]

    Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion,

    P. Vincent, H. Larochelle, I. Lajoie, Y . Bengio, and P.-A. Manzagol, “Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion,”J Mach. Learn. Res., Dec. 2010

    work page 2010

  52. [52]

    Asymptotic behaviors of support vector machines with gaussian kernel,

    S. S. Keerthi and C.-J. Lin, “Asymptotic behaviors of support vector machines with gaussian kernel,”Neural Comput., vol. 15, no. 7, pp. 1667–1689, Jul. 2003

    work page 2003

  53. [53]

    Convergence rates of efficient global optimization algo- rithms,

    A. D. Bull, “Convergence rates of efficient global optimization algo- rithms,”J Mach. Learn. Res., vol. 12, no. 88, pp. 2879–2904, 2011

    work page 2011