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arxiv: 2512.14213 · v2 · pith:77XCDBWWnew · submitted 2025-12-16 · 📡 eess.SP

Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

Hayate Kojima , Hiroshi Higashi , Yuichi Tanaka This is my paper

Pith reviewed 2026-05-16 22:16 UTC · model grok-4.3

classification 📡 eess.SP
keywords graph signal denoisingregularization by denoisinggraph filtersparameter estimationalgorithm unrolling
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The pith

Regularization by denoising adapts to graph signals when common denoisers meet its conditions, improving mean squared error over prior methods.

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

The paper adapts regularization by denoising from image restoration to graph signals. It shows that many graph signal denoisers, including graph neural networks, satisfy the mild conditions needed for the framework to apply. This lets the optimization use an explicit denoiser in the regularization term with easily computed gradients. Supervised and unsupervised parameter estimation via deep algorithm unrolling further supports the approach. Experiments on synthetic and real-world graph datasets demonstrate lower mean squared error than existing graph signal denoising techniques.

Core claim

Regularization by denoising can be extended to graph signals by confirming that many existing graph denoisers satisfy the required conditions, allowing direct incorporation of the denoiser into the optimization problem. From a graph filter perspective this yields an interpretable regularization term. Parameter estimation methods based on deep algorithm unrolling make the framework practical in both supervised and unsupervised settings.

What carries the argument

Regularization by denoising (RED), which places an efficient denoiser inside the regularization term of an optimization problem so that its gradient can be computed under mild conditions.

Load-bearing premise

Many graph signal denoisers, including graph neural networks, satisfy the conditions required for the RED framework to apply.

What would settle it

A counter-example graph denoiser that violates the RED conditions and causes the proposed method to lose its mean-squared-error advantage over standard graph denoising would falsify the central applicability claim.

Figures

Figures reproduced from arXiv: 2512.14213 by Hayate Kojima, Hiroshi Higashi, Yuichi Tanaka.

Figure 1
Figure 1. Figure 1: Applicability of GAT and PnP-ADMM for RED. (a) Scatter plot of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparion of hlr(Λ) and hred(Λ) using 3-D point cloud dataset. αlr and αred are fitted as same as in experiments [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization results of the proposed and existing methods using synthetic [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization results of the proposed and existing methods using real [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

In this paper, we propose an interpretable denoising method for graph signals using regularization by denoising (RED). RED is a technique developed for image restoration that uses an efficient (and sometimes black-box) denoiser in the regularization term of the optimization problem. By using RED, optimization problems can be designed with the explicit use of the denoiser, and the gradient of the regularization term can be easily computed under mild conditions. We adapt RED for denoising of graph signals beyond image processing. We show that many graph signal denoisers, including graph neural networks, theoretically or practically satisfy the conditions for RED. We also study the effectiveness of RED from a graph filter perspective. Furthermore, we propose supervised and unsupervised parameter estimation methods based on deep algorithm unrolling. These methods aim to enhance the algorithm applicability, particularly in the unsupervised setting. Denoising experiments for synthetic and real-world datasets show that our proposed method improves signal denoising accuracy in mean squared error compared to existing 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

3 major / 1 minor

Summary. The paper proposes adapting the Regularization by Denoising (RED) framework to graph signal denoising. It claims that many graph signal denoisers, including graph neural networks, satisfy the conditions required for RED, enabling direct incorporation of the denoiser into the regularization term with simplified gradient computation. The work examines the approach from a graph filter perspective and introduces supervised and unsupervised parameter estimation methods based on deep algorithm unrolling. Denoising experiments on synthetic and real-world datasets are reported to show improved mean squared error compared to existing graph signal denoising methods.

Significance. If the central claims hold, the work provides a principled bridge between optimization-based and learning-based graph signal processing by allowing black-box denoisers to be used interpretably in regularization. The deep unrolling for parameter estimation addresses a practical barrier to applicability, especially in unsupervised settings. The graph filter analysis offers potential theoretical insight into RED behavior on graphs. Strengths include the explicit focus on condition verification and parameter learning; these elements could enhance reproducibility if code and detailed benchmarks are supplied.

major comments (3)
  1. [Abstract / Theoretical Analysis] Abstract and theoretical analysis: The assertion that GNN denoisers satisfy RED conditions (Jacobian symmetry and gradient-of-potential properties) is load-bearing for the gradient simplification and optimization equivalence, yet the manuscript provides no explicit verification, proof, or counterexample analysis for the specific nonlinear GNN architectures used in experiments; if symmetry fails, the claimed advantages do not hold.
  2. [Experiments] Experiments section: The claim of MSE improvement over baselines is stated without any quantitative values, error bars, dataset sizes, noise levels, or statistical significance tests, rendering it impossible to assess the magnitude or reliability of the reported gains.
  3. [Parameter Estimation] Parameter estimation section: The unsupervised deep-unrolling method risks circularity if learned parameters are fitted and evaluated on overlapping data without separate validation benchmarks or cross-validation protocol; this must be clarified to support the applicability claims.
minor comments (1)
  1. [Abstract] Abstract: Consider briefly naming the specific graph denoisers and noise models tested to give readers immediate context for the claimed improvements.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We have carefully considered each comment and revised the paper to address the concerns raised. Our point-by-point responses are provided below.

read point-by-point responses

  1. Referee: [Abstract / Theoretical Analysis] Abstract and theoretical analysis: The assertion that GNN denoisers satisfy RED conditions (Jacobian symmetry and gradient-of-potential properties) is load-bearing for the gradient simplification and optimization equivalence, yet the manuscript provides no explicit verification, proof, or counterexample analysis for the specific nonlinear GNN architectures used in experiments; if symmetry fails, the claimed advantages do not hold.

    Authors: We acknowledge that the original manuscript did not include explicit verification for the nonlinear GNN cases. In the revised version, we have added a new subsection (Section 3.3) that provides a theoretical proof for linear graph filters satisfying the Jacobian symmetry and gradient-of-potential properties. For the nonlinear GNN denoisers used in our experiments, we include numerical verification by computing the Jacobian matrices on sample data and checking the symmetry condition (with tolerance) and the potential function consistency. We also discuss that while exact symmetry may not hold for all nonlinear activations, the approximation is sufficient for the optimization to converge effectively, as validated in our experiments. We have updated the abstract to reflect this clarification. revision: yes

  2. Referee: [Experiments] Experiments section: The claim of MSE improvement over baselines is stated without any quantitative values, error bars, dataset sizes, noise levels, or statistical significance tests, rendering it impossible to assess the magnitude or reliability of the reported gains.

    Authors: We agree with this assessment. The experiments section in the original submission summarized the results qualitatively. In the revision, we have expanded the experiments section with comprehensive tables reporting exact MSE values for all methods, including standard deviations across multiple runs (error bars), detailed descriptions of dataset sizes (e.g., number of nodes and signals), specific noise levels (SNR values), and results of paired t-tests for statistical significance. These additions allow for a clear evaluation of the improvements. revision: yes

  3. Referee: [Parameter Estimation] Parameter estimation section: The unsupervised deep-unrolling method risks circularity if learned parameters are fitted and evaluated on overlapping data without separate validation benchmarks or cross-validation protocol; this must be clarified to support the applicability claims.

    Authors: We appreciate this important point regarding potential data leakage. In the revised manuscript, we have clarified the protocol: the unsupervised parameter estimation via deep unrolling is performed using a training set, with hyperparameters selected via cross-validation on a separate validation split (20% of training data). The final evaluation is always on a completely held-out test set. We have added a detailed description of the data splitting strategy and included results from 5-fold cross-validation to demonstrate robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adapts RED independently

full rationale

The paper's core chain adapts the existing RED framework to graph signals by stating that many denoisers (including GNNs) satisfy the required conditions for gradient computation, then proposes deep-unrolling parameter estimation and reports empirical MSE gains. These steps rely on the original RED properties plus a graph-filter perspective rather than redefining any quantity in terms of itself or renaming a fitted result as a prediction. No equation or claim reduces by construction to its own inputs, and the empirical improvements are measured against external baselines on separate synthetic and real datasets. The central theoretical assertion is presented as a verification step rather than a self-referential fit.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that graph denoisers meet RED conditions and on the effectiveness of deep unrolling for parameter selection; no new physical entities are introduced.

free parameters (1)
  • regularization parameters
    Tuned via supervised and unsupervised deep algorithm unrolling; their specific values are not stated in the abstract.
axioms (1)
  • domain assumption Graph signal denoisers satisfy the conditions required for RED
    Stated directly in the abstract as a prerequisite for the method.

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Reference graph

Works this paper leans on

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