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REVIEW 3 major objections 4 minor 28 references

From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A denoiser trained as a gradient step or proximal operator can be plugged into PGD or Douglas-Rachford splitting to produce iterates that provably converge to a stationary point of an explicit objective.

desk verdict A useful, honest parameter study of the authors' own convergent PnP algorithms; the theory is restated accurately, but the trained Prox denoiser's contraction property is never measured, so the 'convergent' label stays conditional. read the letter →

arxiv 2509.09793 v1 pith:XBQ52DV3 submitted 2025-09-11 cs.LG

Vincent Herfeld , Baudouin Denis de Senneville , Arthur Leclaire , Nicolas Papadakis This is my paper · ORCID
classification cs.LG MSC 68T0790C2665K1094A08
keywords Plug-and-Playgradient-stepdenoiserproximalDouglas-Rachfordsplittinggradientdescentnonconvexregularizationimagerestorationconvergenceguarantees
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

The paper establishes a practical recipe for Plug-and-Play image restoration with convergence guarantees. It studies two denoisers: the Gradient-Step denoiser $D_\sigma = \mathrm{Id} - \nabla g_\sigma$, trained so that $g_\sigma$ is an explicit regularizer, and the Proximal denoiser, which under a contraction condition is exactly $\mathrm{Prox}_{\varphi_\sigma}$ for an explicit potential $\varphi_\sigma$. When either denoiser is substituted into proximal gradient or Douglas-Rachford splitting, the paper's recalled theorems say the iterates decrease a monitorable objective $F$, drive the residual to zero at rate $O(1/\sqrt{K})$, and converge to a stationary point. The experiments confirm the announced residual rate and show how $\sigma$, $\lambda$, and $tau_0$ steer the result, which matters because practitioners can now reproduce convergent PnP without an opaque implicit prior.

What carries the argument

The load-bearing object is the Gradient-Step denoiser $D_\sigma = \mathrm{Id}-\nabla g_\sigma$, built from $g_\sigma(x)=\frac{1}{2}\|x-N_\sigma(x)\|^2$; computing it costs one forward and one backward pass through $N_\sigma$, and because it is a gradient field the implicit prior becomes explicit. The companion identity is Proposition 3.1: if $\nabla g_\sigma$ is $L$-Lipschitz with $L<1$, then $D_\sigma = \mathrm{Prox}_{\varphi_\sigma}$ for the explicit weakly convex potential given above, so a contracted gradient-step denoiser is a true proximity operator. The contraction itself is trained by softly penalizing the spectral norm of $J(\mathrm{Id}-D_\sigma)$ with $\max(\||J(\mathrm{Id}-D_\sigma)\||,1-\varepsilon)$ via power iteration. These identities are what carry the argument: they convert the Plug-and-Play fixed-point iteration into a proximal splitting of a known functional, bringing standard nonconvex optimization convergence machinery to bear.

What would settle it

Take the released Prox-DRUNet checkpoint, compute the power-iteration estimate of $\||J(\mathrm{Id}-D_\sigma)\||$ on a batch of natural images, and compare it with 1 and with 1/2. If the estimate exceeds those thresholds at points the algorithms visit, the hypotheses of Theorems 3.1-3.3 fail there, and one can then check whether the residual still follows the announced $O(1/\sqrt{K})$ rate and whether $F$ remains nonincreasing.

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Extended reading notes

Core claim

The central claim is that the gradient-step identity $D_\sigma = \mathrm{Id} - \nabla g_\sigma$, with $g_\sigma(x)=\frac{1}{2}\|x-N_\sigma(x)\|^2$, turns a trained denoiser into a gradient of an explicit regularizer, so the Plug-and-Play fixed point solves the explicit problem $\min \frac{1}{\lambda}f + g_\sigma$. When $\nabla g_\sigma$ is $L$-Lipschitz with $L<1$, the same operator is exactly $\mathrm{Prox}_{\varphi_\sigma}$ for the explicit weakly convex potential $\varphi_\sigma(x)=g_\sigma(D_\sigma^{-1}(x))-\frac{1}{2}\|D_\sigma^{-1}(x)-x\|^2+K$, and this unlocks Douglas-Rachford variants whose convergence needs no condition on $\lambda$. With these identities in place, the convergence theorems in Sections 2.3 and 3.3 guarantee monotone decrease of a computable objective, residual convergence at rate $O(1/\sqrt{K})$, and stationarity of cluster points; the paper's experiments confirm the rate and map out the role of $\sigma$, $\lambda$, and $tau_0$.

Load-bearing premise

The convergence theorems assume the trained network's $\nabla g_\sigma$ is $L$-Lipschitz with $L<1$ (or $L<1/2$ for Prox-PnP-DRS), but the training loss only softly penalizes the spectral norm of $J(\mathrm{Id}-D_\sigma)$ and the paper reports no measured value of $L$ for the deployed Prox-DRUNet.

Editorial extensions

If this is right

  • GS-PnP iterates make the explicit objective $F=\frac{1}{\lambda}f+g_\sigma$ nonincreasing and drive $\|x^{k+1}-x^k\|$ to $0$; every cluster point is stationary, and with the Kurdyka-Lojasiewicz property the whole sequence converges.
  • Prox-PnP-PGD and Prox-PnP-DRSdiff converge under $L<1$ and $\lambda>L_f$, with residual rate $\min_{k\le K}\|u^k-v^k\|=O(1/\sqrt{K})$; Prox-PnP-DRS additionally handles nondifferentiable $f$ for any $\lambda>0$, at the price of $L<1/2$ and convexity of $\mathrm{Im}(D_\sigma)$.
  • Since $F$ is computable, users can watch $F$ and residuals during iterations; the experiments show the announced $O(1/\sqrt{K})$ decrease on super-resolution, deblurring, and random inpainting.
  • The parameter study gives practical guidance: $\sigma$ should scale with noise level, intermediate $\lambda$ balances detail against artifacts, and $tau_0$ values near $\lambda$ are fastest without overshooting.
  • Large-hole inpainting fails: these Plug-and-Play priors can smooth but cannot hallucinate structured content.

Reading between the lines

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

  • The paper leaves implicit that the hard contraction bound is not certified after training; before trusting the theorems for a deployed Prox-DRUNet, one should measure the spectral norm of $J(\mathrm{Id}-D_\sigma)$ on the image manifold actually visited by the algorithm.
  • Because the proximal denoiser differs from the gradient-step denoiser only by the spectral penalty, the natural next step is a hard-Lipschitz architecture that guarantees $L<1$ by construction rather than by soft penalty.
  • A testable extension: the explicit potentials $g_\sigma$ and $\varphi_\sigma$ could be differentiated with respect to $\sigma$ and $\lambda$, allowing automatic parameter schedules instead of the current grid tuning.
  • The large-hole inpainting failure suggests these priors are local smoothing operators; combining the proximal denoiser with a generative model for hallucination is an obvious extension the paper does not make.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the Gradient-Step Denoiser (GS) and the Proximal Denoiser (Prox) and their use in Plug-and-Play image restoration. It recalls convergence theorems from the authors' prior work [9,10], gives training losses and implementation details for a DRUNet parameterization, and presents experiments on super-resolution, deblurring, and random inpainting. The paper reports that GS-PnP and Prox-PnP algorithms produce explicit objectives F = (1/λ)f + gσ or F = (1/λ)f + φσ, that residuals decay as O(1/√K), and that the parameters σ, λ, and τ0 have interpretable and reproducible effects. It also documents a limitation: these methods cannot perform large-area inpainting because they cannot hallucinate content.

Significance. If the practical claims hold, the paper gives practitioners two convergent Plug-and-Play algorithm families with explicit objective values that can be monitored, which is a meaningful step beyond black-box PnP methods. The recalled theorems are stated accurately, the residual curves are consistent with the announced O(1/√K) rate, and the experiments cover three inverse problems with concrete advice on parameter choices. The use of the public DeepInv library and detailed training prescriptions strengthens reproducibility. The main weakness is that the contraction assumption L<1 (or L<1/2) underpinning the Prox-PnP convergence guarantees is only softly penalized in training and never measured, so the 'convergent' label is conditional on an unverified property of the deployed network.

major comments (3)
  1. [§3.1, loss (21), and Theorems 3.1–3.3] The convergence guarantees for Prox-PnP require ∇gσ to be L-Lipschitz with L<1 (and L<1/2 for Theorem 3.3), but the training loss (21) only adds a soft penalty µ·max(|||J(Id−Dσ)|||,1−ε) with µ=10⁻³. A soft penalty does not enforce a hard global Lipschitz bound, and the paper reports neither the achieved penalty values, nor the chosen ε, nor any empirical estimate of |||J(Id−Dσ)||| for the trained Prox-DRUNet. Remark 2 merely says it is 'reasonable to assume' Lipschitzness. Therefore the theorems as stated do not apply to the deployed model unless the contraction property is verified; the residual plots in Section 5.2 are consistent with the theory but do not certify the assumption. Please add a direct measurement of the Jacobian spectral norm over the iteration path and test data, or modify the training to enforce a certified hard constraint.
  2. [§3.3, Theorem 3.3, and §5.2 inpainting] Theorem 3.3 additionally assumes that Im(Dσ) is convex and that L<1/2. The manuscript does not verify either condition for the trained network, and the α=1/2 averaging used for inpainting in Section 5.2 is not connected to the convexity-of-image assumption. Without such verification, the convergence conclusion for Prox-PnP-DRS with nondifferentiable f is not established for the implemented model.
  3. [§2.4, Algorithm 1, Algorithm 2, Algorithm 4] The backtracking condition in Algorithm 1 appears inverted relative to Section 2.4: Eq. (18) reduces τ when F(x^k)−F(T(x^k)) < (γ/τ)||x^k−T(x^k)||², whereas Algorithm 1 reduces τ when F(x^k)−F(x^{k+1}) ≤ (γ/τ)||x^k−x^{k+1}||². In addition, Algorithm 2 lists z0 and τ0 but never initializes x0, and Algorithm 4 uses Prox_{τf}(z0) without defining τ. These inconsistencies directly affect the reproducibility of the proposed algorithms and should be corrected in a revision.
minor comments (4)
  1. [§3.2] The name 'Douglas-Rashford Splitting' should be 'Douglas-Rachford Splitting'.
  2. [§5.2.3] The text says GS-PnP converges for τ < 1/(λ L), but Theorem 2.1 states the condition τ < λ/L; the two should be made consistent.
  3. [Figure 3 caption] The phrase 'Mean denoising PSNRevolution' should read 'Mean denoising PSNR evolution'.
  4. [Algorithm 2] The parameter list includes z0 and τ0, but the algorithm body does not use them; add a proper initialization or remove the unused parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: convergence theorems are transparently imported from prior work by overlapping authors as parameter-free results with explicit assumptions, and the experimental claims are not fitted inputs renamed as predictions.

full rationale

The paper does not derive its convergence guarantees from the training objective; it explicitly recalls them from prior work. Theorem 2.1 and 2.2 are introduced as 'the main theorems established in the paper [9]', and Theorems 3.1–3.3 are presented as results from [10] (e.g., 'Theorem 4.1 in [10]'). Although [9,10] share two co-authors with the present paper, the cited results are stated as parameter-free mathematical theorems with explicit assumptions (L-Lipschitz gradient of gσ, L<1 or L<1/2, boundedness and KL conditions), not as fitted constants, and the present paper tests the induced algorithms on fresh experiments instead of using the theorems to fit the observed curves. The Prox-DRUNet is trained with loss (21), which soft-penalizes the spectral norm of J(Id-Dσ), and no measured Lipschitz constant is reported; thus the convergence guarantee for the deployed network is conditional on an unverified contraction property. This is a correctness or verification gap, not a circularity: the theorem is not equivalent to the training loss by construction, and the paper does not claim the penalty enforces the hard bound. Denoising PSNR on the training objective is reported descriptively and is not used as a predictive test of convergence. Residual decay at the announced O(1/sqrt(K)) rate is observed experimentally and is consistent with the recalled theorems, but consistency of data with a theorem is not a case of the theorem reducing to its inputs. No equation is shown to be identical to another by definition, and no fitted parameter is renamed as a prediction. The self-citations are load-bearing in the sense that the proofs live in [9,10], but those proofs are independent mathematical results with stated assumptions rather than unverified appeals to authority, so the appropriate finding is no significant circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central experimental claims rest on the recalled convergence theorems from the authors' prior work, on the training losses (12) and (21), on the explicit potentials g_sigma and phi_sigma, and on several tuned algorithm hyperparameters. No new physical entities are introduced; phi_sigma is a mathematical potential constructed from g_sigma. The main uncharged premise is that the trained networks satisfy the Lipschitz and contraction conditions required by the theorems.

free parameters (6)
  • lambda (regularization weight) = 0.065 (GS-PnP), 1/0.99 (Prox-PnP), 1000 (large inpainting)
    Balances fidelity and the explicit prior in F = 1/λ f + g; tuning strongly affects PSNR in Section 5.2.2.
  • sigma (denoiser noise level) = sigma = c*nu with c in {2,3,5} recommended; c up to 10 tested
    Sets denoiser strength; the optimal value scales with observation noise but is tuned per problem in Section 5.2.1.
  • tau0 (initial GS-PnP step size) = tau0 = lambda recommended; 0.1/lambda and 2/lambda tested
    Initial step size controlling convergence speed and stability; too large causes early stopping with poor images in Section 5.2.3.
  • mu (Lipschitz penalty weight) = 1e-3
    Weight of the spectral-norm penalty in the proximal training loss (21); chosen without reported ablation.
  • beta (DRS averaging parameter) = not stated
    Averaging parameter in the Douglas-Rachford schemes (24) and (28); the numerical value is not reported.
  • epsilon in Lipschitz penalty = not reported
    Margin in max(...,1-epsilon) from loss (21); controls strict contraction but no value is given.
assumptions (6)
  • standard math Classical proximal splitting and Kurdyka-Lojasiewicz theory for nonconvex optimization
    Theorems 2.1 to 3.3 are imported from [9,10] and rely on these standard tools; the paper cites [2,8] for KL details.
  • domain assumption Observation model y = Ax0 + w with known linear A and white Gaussian noise w
    Section 1 sets the inverse problem; all theory and experiments assume this model and a known forward operator.
  • domain assumption Convolutions use periodic boundary conditions
    Remark 4 states this assumption; it makes the data-fidelity proximity operator available in closed form.
  • ad hoc to paper DRUNet parameterization with ELU or Softplus activations yields differentiable N_sigma with Lipschitz gradient
    Remark 2 says it 'becomes reasonable' to assume ∇g_sigma is L-Lipschitz, but no proof or measured constant is given.
  • ad hoc to paper Soft spectral-norm penalty with power iteration enforces the contraction condition L < 1 or L < 1/2
    Loss (21) penalizes max(|||J(Id-D_sigma)|||,1-epsilon); it does not hard-constrain L, and no post-training L is reported.
  • domain assumption F is bounded below and iterates are bounded, with coercivity enforced by projection to C = [-1,2]^n
    Section 2.3 adds a projection term to make F coercive and says it does not activate in practice; boundedness is still an assumption in the theorems.

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Cite this review

Pith. "Pith review of From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms." pith.science (2026). https://pith.science/paper/XBQ52DV3

@misc{pith2026250909793,
  author       = {Pith},
  title        = {Pith review of: From the Gradient-Step Denoiser to the Proximal Denoiser and their associated convergent Plug-and-Play algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBQ52DV3}},
  note         = {Machine review of arXiv:2509.09793}
}
read the original abstract

In this paper we analyze the Gradient-Step Denoiser and its usage in Plug-and-Play algorithms. The Plug-and-Play paradigm of optimization algorithms uses off the shelf denoisers to replace a proximity operator or a gradient descent operator of an image prior. Usually this image prior is implicit and cannot be expressed, but the Gradient-Step Denoiser is trained to be exactly the gradient descent operator or the proximity operator of an explicit functional while preserving state-of-the-art denoising capabilities.

Figures

Figures reproduced from arXiv: 2509.09793 by the authors.

Figure 1
Figure 1. Architecture of the DRUNet denoiser [27]. Original image from their work. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Denoising results for different noise levels with the GS-DRUNet architecture trained as the [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Mean denoising PSNR evolution w.r.t σ. We average PSNR results over the images of set3C. We see that for low noise levels both denoisers have similar performance. For larger noise levels, the GS-denoiser is slightly more efficient, except for noise level highly out of the range used for training (σ ≥ 100/255), which seems to indicate that the Prox-denoiser generalizes better. 11 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Filters used for the super-resolution problem. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Kernels used for the blurring operator. These were originally shared in [11]. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Mean PSNR evolution, for different noise levels (ν = 0.01, 0.03 and 0.05), over iterations of GS-PnP for ×2 super-resolution with varying parameter c ∈ {0.1, 1, 2, 3, 5, 10} such that we used σ = cν. The mean is computed over every filter from [PITH_FULL_IMAGE:figures…
Figure 7
Figure 7. Figure 7: Mean PSNR and residual evolution over iterations of each algorithm for ×4 super-resolution with varying parameter c and ν = 0.05 such that σ = cν. The mean is computed over every anti-aliasing filter from [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Deblurring results after 1000 iterations of the Prox-PnP-DRSdiff algorithm with varying [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Random masked inpainting results after 200 iterations of the GS-PnP algorithm with varying [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 13
Figure 13. Figure 13: After 200 iterations, the only effect of the algorithm is to slightly change the regions color to [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 10
Figure 10. Figure 10: Mean PSNR and residuals evolution over iterations of each algorithm for deblurring with varying λ parameter and ν = 0.05. The mean is computed over every motion blur filter from [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Mean PSNR and residuals evolution over iterations of GS-PnP for ×2 super-resolution with varying initial step size τ0. The mean is computed over every filter from [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: ×2 Super-resolution results from the GS-PnP algorithm with varying initial step size τ0 with ν = 0.03. We can see the effect of the choice of τ0, for a value that is small the resulting image (12a) is close to best performance (12b) but the stopping criterion is reach…
Figure 13
Figure 13. Figure 13: Large area inpainting example. PnP methods cannot generate structured information and [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

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