{"id":"136de11e-351e-4dc9-8d97-ccbea3f16bd1","arxiv_id":"2607.14894","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For PnP-PGD, residual reconstruction error is bounded by average squared mismatch between the deployed denoiser and the target proximal map, motivating proximal-matching few-shot adaptation that outperforms MSE adaptation under domain shift.","lead":"This paper studies what happens when a plug-and-play image denoiser trained on one domain is used on another, and shows how to adapt it with very few target images. It derives a bound connecting reconstruction error to \"proximal mismatch\" and proposes matching the denoiser's proximal behavior instead of its mean-squared error, with gains up to 10 dB in severe domain shift.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"GS target denoiser's global contractivity is not verified; if it fails, the theorem's assumptions do not hold for the GS experiments.","rationale":"The reader identified Assumption 3.3 as the weakest assumption, including the GS image-domain and contractivity conditions. I agree that this is the most fragile load-bearing element, but I narrow it to a concrete technical gap: the global contractivity of the GS target reference is asserted, not verified, and the empirical Jacobian estimate bL_max does not certify the L<1 condition globally. If the global Lipschitz constant exceeds 1, the proximal-subproblem convexity/smoothness used in Lemmas 3.1–3.3 fails, so Theorem 3.1 does not apply to the GS experiments. This is more pressing than the 'true target unknown' issue, because the latter is a modeling interpretation while the former is a checkable condition on a denoiser whose proximal interpretation the theory explicitly rests on. The LPN case is much safer: the α-strong convexity of Ψ_θ in (23) guarantees global proximal structure and H_k smoothness for any θ, so the theorem holds for LPNs without further verification. Given the soundness of the LPN analysis and the strong empirical results for both families, the overall verdict should remain CONDITIONAL (UNCHANGED), with the condition being that GS contractivity must be checked or enforced. The reader's broader rationale about the target reference not being the true prior is valid but secondary; my check would focus on the GS global Lipschitz bound or, alternatively, a direct evaluation of (17) on trajectories, which would settle whether the theorem's assumptions hold in practice.","tokens_in":18236,"tokens_out":13807,"duration_ms":118408,"concrete_test":"Compute a certified upper bound on the global Lipschitz constant of ∇g_σ,* for the GS target reference used in Tables 2–3 (e.g., via LipSDP or semidefinite based Lipschitz estimation on the network weights). If the certified L is <1, the concern is resolved. If L≥1, evaluate the key inequality (17) of Assumption 3.3 along actual PnP-PGD trajectories; if it fails at any visited point, the GS stationarity bound is not established, and the paper should either enforce contractivity on the target reference (e.g., spectral normalization) or restrict its theoretical claims to LPNs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1's stationarity bound depends on Assumption 3.3, which for GS denoisers requires the target reference D_σ,* = ∇h_σ,* to be a global proximal map with inverse 1/(1-L)-Lipschitz, i.e., ∇g_σ,* must be globally L-Lipschitz with L<1 (Section 3.4.2, Proposition B.2). The paper never verifies this global contractivity for the target-trained GS reference; it only reports bL_max, an empirical local Jacobian estimate evaluated on adaptation points (Tables 2 and 3). For a neural-network potential, the global Lipschitz constant can easily exceed 1 even when pointwise Jacobian estimates are below 1. If L≥1 globally, the induced regularizer φ_σ,* may not exist on all of R^n, H_k may be nonconvex or nonsmooth, and the derived constants L_H=1/(1-L) and the descent/stationarity lemmas (3.2, 3.3) are not justified. Since GS is one of the two families used to validate the central claim that proximal-matching adaptation reduces mismatch, an unverified global contractivity condition makes the theoretical support for the GS half of the paper conditional at best.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies plug-and-play proximal gradient descent (PnP-PGD) when the deployed denoiser is mismatched to the target domain. It defines proximal mismatch as the squared output difference between the deployed denoiser and a target reference proximal map D_* = prox_{R_*}, and shows (Theorem 3.1) that the average squared gradient of the target objective is bounded by O(1/K) plus a constant times the average squared mismatch, under smoothness and regularity assumptions on the target proximal subproblems. The bound motivates a proximal-matching (PM) adaptation objective, which is implemented for learned proximal networks (LPNs) and gradient-step (GS) denoisers. Experiments on Gaussian deblurring and 4x super-resolution under a BreCaHAD-to-CelebA domain shift report that PM adaptation improves PSNR over MSE adaptation, especially in the few-shot regime, and reduces a family-matched trajectory gap relative to a target-trained reference.","tokens_in":18672,"tokens_out":4423,"duration_ms":43144,"significance":"If the assumptions hold, Theorem 3.1 is a clean and useful result: it quantifies how persistent denoiser mismatch degrades stationarity in PnP-PGD and justifies proximal matching as an adaptation principle. The proof is self-contained, the mismatch measure d_k is defined rather than fitted, and there is no circular derivation. The two denoiser families (LPN and GS) are well chosen because they realize proximal structure in different ways. The experimental findings, if validated, are practically significant: the reported gains over MSE adaptation in the one-shot and five-shot regimes are substantial. The main weaknesses are that the GS half of the validation depends on an unverified global contractivity condition, and the experimental gap metric compares against a target-trained denoiser of the same family rather than the true D_*, so the empirical claims are conditional on a surrogate reference.","major_comments":[{"comment":"The GS proximal interpretation requires that ∇g_{σ,⋆} is globally L-Lipschitz with L<1; otherwise the induced regularizer may not exist, H_k may be nonconvex or nonsmooth, and L_H=1/(1-L) in Eq. (26) is invalid. The paper only reports bL_max, an empirical local Jacobian estimate evaluated on adaptation points. For a neural-network potential, global Lipschitz constants can exceed 1 even when pointwise Jacobian estimates are below 1. Since GS is one of the two families used to validate the central claim, this unverified global condition makes the theoretical support for the GS experiments conditional. The authors should either verify the global contractivity bound for their trained potentials, restrict the GS empirical claims to a version where the condition is checked, or clearly state that Theorem 3.1 applies to GS only when the global condition holds.","section":"Section 3.4.2 / Proposition B.2 / Tables 2 and 3"},{"comment":"The trajectory gap is computed against a target-trained denoiser of the same family, i.e., D_{θ⋆}, not against the true D_* = prox_{R_*}. Thus the experiments measure agreement with a surrogate reference, not with the proximal map of the target regularizer. The observation that lower gap correlates with higher PSNR is therefore an within-family consistency result, not a direct empirical confirmation of Theorem 3.1. The paper acknowledges this in the conclusion, but the abstract and Section 1 state more strongly that 'proximal matching adaptation reduces the proximal mismatch'; that claim should be qualified. A test with a synthetic or analytically known R_* would strengthen the validation, or the wording should be adjusted to 'mismatch relative to the family-matched target-trained reference.'","section":"Section 5.1, gap definition; Section 6"},{"comment":"Assumption 3.3 also requires that at every deployed iterate the gradient-dominance inequality (17) holds, and for GS denoisers that every deployed output lies in Im(D_{σ,⋆}). For LPNs this is automatic, but for GS it is not verified. The paper does not report any check of these conditions along actual PnP-PGD trajectories. Since the theorem's conclusion depends on them, the experimental validation should include at least trajectory-level diagnostics (e.g., whether \\hat D_σ(z_k) stays in the image of the target map, or whether the inequality in (17) is satisfied). Without such checks, the applicability of Theorem 3.1 to the GS experiments is not established.","section":"Assumption 3.3 / Section 3.4.2"}],"minor_comments":[{"comment":"No code or detailed training configuration is provided. Releasing code and specifying hyperparameters (e.g., λ_con, L_max, γ annealing schedule, number of corruptions M) would substantially improve reproducibility.","section":"General"},{"comment":"The proximal-matching objectives in Eqs. (37) and (41) depend on a bandwidth γ, but the annealing schedule is not described. Please specify how γ is set or annealed in the experiments.","section":"Section 4.2/4.3"},{"comment":"The caption states that SSIM and gap are reported in units of 10^{-2}, but the figures use different scales. Clarify the units consistently. Also, the bL_max values are described as 'empirical local Jacobian estimates'; state explicitly how they are computed.","section":"Tables 2 and 3"},{"comment":"There are typographical issues: 'asproximal mismatch' in the abstract, broken author name 'V o' in the references, and some missing spaces. Also, reference 'Vert et al., 2026' appears to be a future work; verify its status.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The theoretical result is sound and the adaptation principle is interesting. The main reasons for major revision are the unverified global contractivity assumption for GS denoisers and the surrogate-reference issue in the experimental gap metric. Both are fixable within the manuscript's scope: the authors can add a contractivity verification or explicitly restrict the GS claims, and they can reframe the experimental validation as testing adaptation to a family-matched reference rather than to the true D_*. I do not see a load-bearing mathematical error in the main derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main thing you should know: this paper gives a clean formulation of proximal mismatch in PnP-PGD and shows, across two denoiser families and two inverse problems, that proximal-matching adaptation beats MSE adaptation in the few-shot regime. The theory is a careful specialization of inexact proximal-gradient analysis (Schmidt et al. 2011) with the error term made explicit as squared proximal mismatch. That is not a new technique, but the framing is useful and the bound is derived cleanly with explicit constants.\n\nWhat the paper does well is the experiments. They train on BreCaHAD, adapt on 1 to 50 CelebA images, and test on deblurring and 4x SR. The pattern is consistent: in the 1-5 shot regime, PM keeps or improves PSNR while MSE adaptation often collapses below the source model. The GS results are particularly striking, with 10+ dB gaps at n=1 and n=5. They also report a trajectory mismatch metric and show it correlates with reconstruction quality, which supports the claim that reducing proximal mismatch is the right target.\n\nSoft spots, in order of seriousness. First, the GS half of the paper depends on global contractivity of the target gradient field (L < 1), but they only verify a local Jacobian estimate on adaptation points. For a neural network, the global Lipschitz constant can easily exceed 1 even if pointwise Jacobians are below 1. If that happens, the induced regularizer may not be well-defined, and L_H = 1/(1-L) is meaningless. The stress-test concern is correct. This does not invalidate the empirical comparison, but it makes the theorem's support for the GS experiments conditional at best. They should either enforce global contractivity (e.g., spectral normalization) or restrict the theoretical claims to LPNs. Second, the target reference D* is always a target-trained denoiser from the same family, not the true proximal map of the data distribution. They acknowledge this in the conclusion, and it is an honest limitation, but it means both the adaptation loss and the evaluation metric are family-internal. Third, no code or seed statistics are provided, and there is no comparison to existing test-time adaptation baselines like Shoushtari et al. 2024. These do not kill the paper, but they narrow the claim.\n\nOverall: this is a solid empirical contribution with a correct but standard theoretical scaffold. The few-shot proximal-matching result is worth refereeing. I would send it out, with the GS contractivity question as the main thing for reviewers to push on.","headline":"A useful, well-run empirical study of proximal-matching adaptation for PnP, wrapped in a correct but not novel theory that has a genuine unverified assumption in the GS experiments.","tokens_in":19050,"tokens_out":2012,"would_cite":false,"duration_ms":21447,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68U10","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Domain shift in plug-and-play reconstruction reduces to a measurable 'proximal mismatch' that directly controls convergence.","keywords":["plug-and-play","proximal mismatch","convergence analysis","domain adaptation","image reconstruction","learned proximal networks","gradient-step denoisers"],"falsifier":"Compute the stationarity gap of PnP-PGD on a controlled inverse problem with a known target regularizer, using a deployed denoiser deliberately mismatched to it; if the empirical average squared stationarity error does not grow approximately linearly with the average squared proximal mismatch as predicted, Theorem 3.1 is wrong. Equivalently, construct a denoiser with zero proximal mismatch at all query points that still fails to drive the stationarity measure to zero, disproving the direct implication.","tokens_in":18137,"feed_emoji":"🖼️","tokens_out":6122,"duration_ms":49782,"temperature":0.7,"pith_summary":"This paper studies plug-and-play proximal gradient descent (PnP-PGD) when the deployed denoiser—trained on one image domain—is used on a different target domain. It defines 'proximal mismatch' as the squared difference between the deployed denoiser and the target-domain reference proximal map at every PnP query point. Its main theorem bounds the average squared gradient of the target objective by O(1/K) plus a constant times the average squared proximal mismatch. This means persistent domain shift leaves an irreducible stationarity error, and reducing the average mismatch is the right adaptation target. The paper then shows that fine-tuning a denoiser by proximal matching—rather than by denoising MSE—reduces that mismatch and improves reconstruction substantially in few-shot settings, sometimes by more than 10 dB.","feed_headline":"Proximal mismatch—not MSE—drives PnP domain shift","feed_subtitle":"Ties reconstruction error to the gap from the target proximal map; matching it wins with a handful of images.","key_machinery":"The key object is the proximal mismatch d_k = || D_hat(z_k) - D_*(z_k) ||^2 between the deployed denoiser and the target proximal map at each PnP query point. It converts the deployed update into an inexact proximal step for the target objective. The analysis then uses the target proximal subproblem H_k(u) = (1/2)||u - z_k||^2 + R_*(u); its smoothness (L_H) and a gradient error inequality (Assumption 3.3) let the mismatch control both the descent loss and the stationarity residual. For Learned Proximal Networks and Gradient-Step denoisers, the paper verifies these conditions via explicit smoothness constants (1/alpha and 1/(1-L), respectively).","core_discovery":"The central claim is that the effect of domain shift on PnP-PGD is captured, to first order, by the sequence of proximal mismatch values d_k = || D_hat(z_k) - D_*(z_k) ||^2, where D_* is the proximal map of the target regularizer R_*. Theorem 3.1 shows that under smoothness and regularity assumptions, the average squared stationarity gap of the target objective decays as O(1/K) plus C_1 times the average squared mismatch. If the mismatch is square-summable the iterates converge to stationarity; if it is persistent, the stationarity error persists. This turns an abstract prior mismatch into a measurable quantity along the trajectory and provides a principled adaptation objective.","pith_inferences":["The paper does not explore how to estimate the target proximal map when no target-trained reference is available; a natural extension is self-supervised or trajectory-based proximal estimation that directly minimizes the mismatch term.","A natural extension is to sample adaptation inputs from the actual PnP trajectory rather than only independent noisy images, which would directly reduce the mismatch term appearing in the bound.","The proximal-matching idea may transfer beyond imaging to other plug-and-play inexact proximal algorithms (e.g., RED or ADMM variants); a testable prediction is that reconstruction quality should correlate monotonically with the average squared mismatch, with the correlation strength depending on the constants in the bound."],"forward_implications":["If the bound holds, a denoiser that is well-matched to the target proximal map will make PnP-PGD converge to a stationary point of the target objective; any persistent mismatch creates a floor on achievable stationarity.","The bound identifies average squared proximal mismatch along the PnP trajectory as the quantity to monitor and reduce during adaptation, rather than standalone denoising MSE.","Proximal-matching adaptation consistently outperforms MSE-based adaptation in the few-shot regime across two denoiser families and two inverse problems, with the largest gains (over 10 dB in severe shifts) at one or five target images.","In the matched case (deployed equals target reference), the bound recovers the standard O(1/K) stationarity rate of proximal gradient descent."],"fun_headline_variants":["Proximal mismatch, not MSE, explains PnP domain shift","Match the proximal map: better few-shot PnP adaptation","Domain shift in PnP? Measure proximal mismatch","Proximal matching outperforms MSE adaptation in PnP","PnP domain adaptation: proximal mismatch is the key"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the target domain has a well-defined proximal reference map D_* = prox_{R_*} and that every PnP query point satisfies the smoothness and unique-minimizer conditions of Assumption 3.3; the paper verifies these for ideal LPN and GS denoisers but not for the actually adapted ones.","fun_headline_variants_meta":{"raw":{"variants":["Proximal mismatch, not MSE, explains PnP domain shift","Match the proximal map: better few-shot PnP adaptation","Domain shift in PnP? Measure proximal mismatch","Proximal matching outperforms MSE adaptation in PnP","PnP domain adaptation: proximal mismatch is the key"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1161,"prompt_tokens":795,"completion_tokens":366,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":539,"tokens_out":366,"duration_ms":3503,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T00:46:02.496932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the stationarity gap of PnP-PGD on a controlled inverse problem with a known target regularizer, using a deployed denoiser deliberately mismatched to it; if the empirical average squared stationarity error does not grow approximately linearly with the average squared proximal mismatch as predicted, Theorem 3.1 is wrong. Equivalently, construct a denoiser with zero proximal mismatch at all query points that still fails to drive the stationarity measure to zero, disproving the direct implication.","supporting_citations":[],"review_version":1}