{"id":"4b74d5eb-352b-42ac-8954-3719d66b3d06","arxiv_id":"2509.09793","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A practical study showing that Gradient-Step and Proximal Plug-and-Play algorithms reproduce their promised convergence, with sigma, lambda, and tau0 as interpretable but tuning-sensitive parameters.","lead":"This paper reproduces and stress-tests two convergent Plug-and-Play denoising frameworks, the Gradient-Step and Proximal denoisers, on super-resolution, deblurring, and inpainting. It maps how the denoiser strength, regularization weight, and step size affect reconstruction quality and convergence speed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convergence guarantee for the Prox denoiser hinges on an unverified contraction property: Theorems 3.1–3.3 require L<1 (or L<1/2) for ∇gσ, but loss (21) only soft-penalizes the Jacobian spectral norm and no measured Lipschitz constant is reported.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the convergence theorems for the Prox-PnP algorithms require a strict contraction (L<1 or L<1/2) for ∇gσ, while the training loss (21) only softly penalizes the spectral norm and the paper never reports a measured Lipschitz constant. This is the decisive gap between the theory, which restates valid theorems from [9, 10], and the experiments, which use a trained network whose contraction property is unverified. Without this verification, the central practical claim that the algorithms are 'convergent' when instantiated with the Prox-DRUNet is conditional, not demonstrated. The paper is otherwise a careful reproduction and parameter study, with the theory restated correctly and experiments that qualitatively match the predicted residual behavior. Other weaknesses noted by the reader, such as parameter selection on the evaluation set, lack of error bars in restoration curves, and absence of released code or weights, are real but secondary; they affect reproducibility and statistical strength rather than the logical validity of the convergence claim. Because the reader's verdict is already CONDITIONAL and the concern we identify is the same unverified contraction property, our stress-test does not change the verdict. The appropriate action is to require the authors to report a measured Lipschitz constant or otherwise justify that the trained Prox-DRUNet satisfies the contraction bound before the convergence guarantee is claimed for the deployed model.","tokens_in":15396,"tokens_out":2933,"duration_ms":26110,"concrete_test":"Estimate the Lipschitz constant of ∇gσ for the trained Prox-DRUNet by computing the spectral norm of J(Id−Dσ)(x) = ∇²gσ(x) via power iteration on a held-out set of images and noise levels (e.g. 1000 random test points spanning the operating range). Report the maximum and the empirical distribution of the estimated spectral norm. If max |||J(Id−Dσ)||| ≥ 1 (or ≥ 1/2 for the DRS variant), the assumptions of Theorems 3.1–3.3 are violated and the convergence guarantee does not apply to the deployed model. As a complementary check, run Prox-PnP-PGD without any backtracking on a deblurring problem satisfying λ>L_f and verify that Fλ,σ is non-increasing over 1000 iterations; if it is not, the contraction assumption is violated in practice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that a practitioner can train the Prox-DRUNet, plug it into PGD or DRS, and obtain an algorithm that provably converges to a stationary point of the explicit objective F = 1/λ f + φσ. That claim depends on Proposition 3.1 and Theorems 3.1–3.3, all of which assume that ∇gσ is L-Lipschitz with L<1 (and L<1/2 for Prox-PnP-DRS). The proximal denoiser is trained with loss (21), which adds the term µ max(|||J(Id−Dσ)|||, 1−ε) to the denoising loss. This is a soft penalty: it does not enforce a hard bound, and even if the penalty were driven to zero, it would only control the spectral norm at training samples, not globally on R^n. The paper never reports the achieved value of the spectral norm, the chosen ε, or any estimate of the Lipschitz constant of ∇gσ for the trained Prox-DRUNet. The text itself flags the gap: Section 3.1 says it is 'sufficient to ask for ∇gσ to be a contraction,' and Remark 2 merely says it is 'reasonable to assume' Lipschitzness based on the activation functions. Thus the convergence guarantee is not established for the deployed model; it holds only conditionally on an unverified contraction property. The reported O(1/√K) residual decay and monotone F decrease in Section 5 are consistent with the theory, but they do not by themselves certify that the assumption is satisfied for all inputs encountered.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15834,"tokens_out":6241,"duration_ms":48947,"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":[{"comment":"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.","section":"§3.1, loss (21), and Theorems 3.1–3.3"},{"comment":"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.","section":"§3.3, Theorem 3.3, and §5.2 inpainting"},{"comment":"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.","section":"§2.4, Algorithm 1, Algorithm 2, Algorithm 4"}],"minor_comments":[{"comment":"The name 'Douglas-Rashford Splitting' should be 'Douglas-Rachford Splitting'.","section":"§3.2"},{"comment":"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.","section":"§5.2.3"},{"comment":"The phrase 'Mean denoising PSNRevolution' should read 'Mean denoising PSNR evolution'.","section":"Figure 3 caption"},{"comment":"The parameter list includes z0 and τ0, but the algorithm body does not use them; add a proper initialization or remove the unused parameters.","section":"Algorithm 2"}],"recommendation":"major_revision","confidential_remarks":"This is largely an empirical reproducibility and parameter-sensitivity study of the authors' own prior results. The main risk is that the paper's central 'convergent' claim for Prox-PnP is not backed by any measured contraction property of the trained Prox-DRUNet. I would ask the editor to require either empirical Lipschitz certificates or a change of training to a hard constraint, and to fix the pseudo-code inconsistencies before acceptance. The recalled theorems themselves appear sound, so the paper is salvageable within its current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead this if you care about whether the convergent Plug-and-Play line actually holds up in practice. It is a reproduction and parameter-sensitivity study of the authors' own Gradient-Step and Proximal denoisers, not a new method or a new theorem. That is fine: the restated convergence theorems are accurate, and the experiments confirm the predicted O(1/√K) residual decay and monotone decrease of the explicit objective. The parameter scans for σ, λ, and τ0 give genuinely useful guidance — σ should scale with the noise level via a coefficient roughly in {2,3,5}, and an overly large initial step size causes early stopping. I also credit the paper for openly documenting that these PnP methods fail on large-hole inpainting; that is the right kind of negative result.\n\nThe main soft spot is the one flagged in the stress-test note. Theorems 3.1–3.3 require ∇gσ to be L-Lipschitz with L<1 (or L<1/2 for the DRS variant), but the proximal denoiser is trained with loss (21), which only soft-penalizes the spectral norm via max(|||J(Id−Dσ)|||, 1−ε), estimated by power iteration. The paper never reports the achieved spectral norm or ε, and it never measures a Lipschitz constant on the trained network. So the phrase \"convergent Plug-and-Play\" for the deployed Prox-DRUNet is conditional on an unverified contraction property. The paper's Remark 2 says it is \"reasonable to assume\" — that is an expectation, not a verification. This is a genuine gap, but it is the same gap as in the prior work [9,10]; this paper neither introduces it nor closes it. Read the experiments as consistent with the theory, not as a certification of it.\n\nMinor issues: parameters are tuned on the evaluation set without a clear held-out split, restoration curves lack error bars (the denoising section does report mean ± std), and the paper points to DeepInv without a direct link to the exact code or weights used here.\n\nWho is this for? Practitioners wanting a reliable recipe for convergent PnP on moderately ill-posed inverse problems, and researchers curious about where the theory-practice gap sits. It deserves a serious referee — the experiments are reproducible in spirit and the negative result is worth publishing. My recommendation: peer review, with the request that the authors report measured Lipschitz information and clarify the soft-penalty threshold.\n\nBest,\n\n[Your name]","headline":"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.","tokens_in":16321,"tokens_out":2387,"would_cite":true,"duration_ms":19776,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","90C26","65K10","94A08"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Plug-and-Play","gradient-step denoiser","proximal denoiser","Douglas-Rachford splitting","proximal gradient descent","nonconvex regularization","image restoration","convergence guarantees"],"falsifier":"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.","tokens_in":15194,"feed_emoji":"🖼️","tokens_out":8526,"duration_ms":69527,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two denoisers turn plug-and-play imaging into convergent optimization","feed_subtitle":"The explicit objective lets practitioners monitor convergence, tune σ and λ, and trust the restored image.","key_machinery":"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.","core_discovery":"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 $tau_0$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the gradient-step denoiser, its training loss, and the GS-PnP convergence theorems recalled as Theorems 2.1 and 2.2.","marker":"[9]"},{"why":"Supplies Proposition 3.1 and the Prox-PnP convergence theorems recalled as Theorems 3.1-3.3.","marker":"[10]"},{"why":"Supplies the DRUNet architecture and the combined natural-image training set used for both denoisers.","marker":"[27]"},{"why":"Supplies the power-iteration spectral penalty used in the proximal denoiser training loss.","marker":"[17]"},{"why":"Supplies the backtracking sufficient-decrease condition used in GS-PnP to enforce the step-size bound.","marker":"[3]"},{"why":"Supplies the closed-form proximity operators used for the super-resolution and deblurring data-fidelity terms.","marker":"[28]"}],"fun_headline_variants":["Explicit objectives make plug-and-play denoisers provably convergent","Denoisers as gradients: plug-and-play now converges","Gradient-step identity unlocks convergent plug-and-play","Plug-and-play denoisers get explicit objectives and convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit objectives make plug-and-play denoisers provably convergent","Denoisers as gradients: plug-and-play now converges","Gradient-step identity unlocks convergent plug-and-play","Plug-and-play denoisers get explicit objectives and convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1337,"prompt_tokens":881,"completion_tokens":456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":389}},"tokens_in":497,"tokens_out":456,"duration_ms":4406,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:58:55.969468+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Zhang, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the DRUNet architecture and the combined natural-image training set used for both denoisers."},{"cited_title":"Pesquet, A","cited_arxiv_id":null,"evidence_quote":"Supplies the power-iteration spectral penalty used in the proximal denoiser training loss."},{"cited_title":"Hurault, A","cited_arxiv_id":null,"evidence_quote":"Supplies the gradient-step denoiser, its training loss, and the GS-PnP convergence theorems recalled as Theorems 2.1 and 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 3.1 and the Prox-PnP convergence theorems recalled as Theorems 3.1-3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closed-form proximity operators used for the super-resolution and deblurring data-fidelity terms."}],"review_version":2}