{"id":"41003f29-48da-495a-a062-22453af5cff1","arxiv_id":"2505.15318","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Scaled plug-and-play iterations with nonsymmetric kernel denoisers provably converge linearly, with explicit contraction-factor bounds for inpainting, deblurring, and superresolution.","lead":"This paper proves that a rescaled version of plug-and-play image reconstruction converges at a guaranteed geometric rate when the denoiser is a nonsymmetric kernel filter such as NLM. It also derives formulas predicting the convergence speed for inpainting, deblurring, and superresolution, and it shows that the original unscaled version generally cannot be made contractive in the same norm.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the contraction proofs for Theorems 3-8 are internally consistent and the main fragility (K PSD) is explicitly acknowledged as a stated assumption.","rationale":"The reader's verdict ACCEPT is justified. The paper's central claim depends on Assumption 1(1), which is explicitly stated and flagged by the authors as excluding box-window NLM. The invertibility of W in Theorem 4 is a conditional assumption, with external support. I did not find a counterexample to Lemma 1 or a missed case in the spectral arguments. The quantitative bounds are close to 1 but strictly below 1 given positive spectral gap and RNP. Therefore no load-bearing concern that would change the verdict emerged.","tokens_in":21926,"tokens_out":27029,"duration_ms":238558,"concrete_test":"Run the released code on a small (e.g., 32x32) deblurring instance with a Hat-window NLM denoiser, compute the smallest eigenvalue of K to confirm PSD, and use the power method to independently verify ||Ps||_D < 1 for a few values of gamma in (0,2); this checks the entry assumption and the main contraction bound simultaneously.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the central chain: Lemma 1 follows from Prop. 3 because eigenvalues in (-1,1] make equality in the nonexpansive bound equivalent to being a fixed point; Theorem 3 applies Lemma 1 with M=W, N=Gs, using Assumption 1 for sigma(W) subset [0,1], sigma(Gs) subset (-1,1] via H in [0,1], and RNP to ensure fix(W) intersect fix(Gs) = {0}. The D-self-adjointness of W and Gs is correct, and Theorem 4 has the same structure with invertibility excluding -1 from sigma(V). Lemma 2's quantitative bound is valid, and the subsequent substitutions in Theorems 5-8 are algebraically consistent. The only substantive caveats are those already stated by the authors: K is required to be PSD (Hat-window NLM, not box-window NLM) and Theorem 4 assumes W invertible, with invertibility for NLM/DSG-NLM cited to a thesis. These are explicit conditions on the theorem statements, not hidden assumptions, and I found no step in the proofs that would make the central claim fail within the stated scope.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies plug-and-play (PnP) iteration for linear inverse problems with kernel denoisers of the form W = D^{-1}K. The authors first present a counterexample showing that, for a nonsymmetric kernel denoiser, the standard PnP-ISTA update operator P = W(I - γA^T A) need not be contractive in the D-norm for deblurring. They then analyze scaled PnP variants (Sc-PnP-ISTA and Sc-PnP-ADMM) that arise naturally in the D-inner-product space. Under Assumption 1 (K positive semidefinite, nonnegative, irreducible, unit diagonal) and Assumption 2 (nonnegative normalized blur, at least one observed pixel/sample), Theorem 3 proves a strict contraction bound ∥Ps∥_D < 1 for Sc-PnP-ISTA for inpainting, deblurring, and superresolution, yielding global linear convergence; Theorem 4 proves an analogous contraction for Sc-PnP-ADMM when W is invertible. Theorems 5–8 give quantitative contraction-factor bounds in terms of the second eigenvalue λ2 (or ζ*), the measurement fraction μ, and the algorithmic parameters γ and ρ. Numerical experiments validate the predicted dependence of the contraction factors on these parameters.","tokens_in":22123,"tokens_out":37159,"duration_ms":280444,"significance":"If the results are correct, they provide a general linear-convergence guarantee for PnP with nonsymmetric kernel denoisers, extending prior symmetric-denoiser results [31] to a broader and practically relevant class. The main lemmas are proved from first principles, and the bounds involve no fitted parameters: they depend only on spectral quantities of the denoiser and the forward operator. The counterexample in Section II-C is instructive, and the paper provides code and empirical validation, including a check with a nonlinear CNN denoiser. The work is a solid contribution to the PnP convergence literature.","major_comments":[{"comment":"The ADMM linear-convergence results depend on the invertibility of W, since they require σ(V) ⊂ (-1,1] and ζ* < 1. Remark 3 asserts that NLM and DSG-NLM are invertible, citing [47, Thm. 2.16], and further asserts without proof that Theorem 4 remains valid even when W is singular. These assertions are load-bearing for the claim that Sc-PnP-ADMM converges for the concrete denoisers NLM and DSG-NLM. Please provide a short proof of the invertibility claim (or state it explicitly as an assumption in Theorems 4, 6, 7, and 8), and either prove or remove the singular-W extension statement.","section":"Theorem 4, Remark 3, Section V-B"}],"minor_comments":[{"comment":"The statement that the counterexample shows contractivity \"cannot be guaranteed ... even if we work with a different norm\" is stronger than what is proved; the computation only shows failure of the D-norm. Please rephrase to restrict the claim to the D-norm, or add a spectral-radius argument showing nonexistence of any norm.","section":"Section I-B and Section II-C"},{"comment":"These theorem statements do not explicitly assume W invertible, yet without invertibility the bounds become trivial (equal to 1) and are not contraction factors. Please add the invertibility hypothesis to the statements or add an explicit caveat that the bounds are strict only under invertibility.","section":"Theorems 6, 7, and 8"},{"comment":"The assertion that Theorem 4 remains valid for singular W is stated without a proof. Since this is a nontrivial extension of the lemma, either provide a proof or soften the claim.","section":"Remark 3"},{"comment":"The proofs of Propositions 3 and 5 are omitted. They are short eigendecomposition arguments and are used in the main lemmas; please include one-line proofs for self-containedness.","section":"Propositions 3 and 5"},{"comment":"Typo: \"deburring\" should be \"deblurring\" in the sentence \"the contractivity of P cannot generally be guaranteed for deburring using the D-norm.\"","section":"Section II-C"},{"comment":"In the proof of Theorem 8, the notation \"∥Fq1∥_D\" appears where \"∥Fsq1∥_D\" is intended.","section":"Theorem 8 proof"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound and within scope. The central contraction proofs are correct as stated. The main reservation is that the ADMM results depend on invertibility of W, with the concrete denoiser claims relying on an external thesis; this should be made self-contained or more explicitly conditional. The counterexample claim about 'a different norm' is slightly overstated but does not affect the main theorems. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it does what it claims: it proves global linear convergence of scaled PnP-ISTA and scaled PnP-ADMM for nonsymmetric kernel denoisers (NLM-type) on inpainting, deblurring, and superresolution, and it gives explicit contraction-factor bounds. Second, the proof strategy is clean and the assumptions are honestly flagged. I checked the central chain—Lemma 1, RNP, the D-self-adjointness of W and Gs, and the substitutions in Theorems 5–8—and found no hidden circularity. The counterexample in Section II-C is genuinely new and useful: it kills the hope that the old D-norm contractivity for inpainting would extend to deblurring with vanilla PnP, which motivates the scaled algorithms. The quantitative bounds in Theorems 5–8 are modest but real; they depend on measurable spectral quantities (lambda2, zeta*, mu, ||D||) rather than fitted parameters. The numerical experiments are appropriate: they validate the monotone trends predicted by the bounds (e.g., contraction factor decreasing with more measurements or wider spectral gap), and the code is available. So the paper deserves a serious referee and, I think, acceptance after minor revisions.\n\nNow the soft spots, in proportion. The load-bearing assumption is K positive semidefinite (Assumption 1(1)), which the authors explicitly note excludes the standard box-window NLM; their guarantees cover the hat-window variant. That is a stated limitation, not a hidden one, but it does narrow the practical reach of the title's claim about \"kernel denoisers.\" Theorem 4 assumes W invertible, and invertibility for NLM/DSG-NLM is cited to an external PhD thesis rather than proved here—that's acceptable but worth a line in the paper. The proofs of Proposition 3 and Proposition 5 are skipped as \"trivial,\" which they are, but for a paper whose contribution is rigor, including those one-line eigendecomposition arguments would make it self-contained. The bounds are also admittedly loose—the 1/n factor in Theorem 8 that does not vanish when D=I is a known artifact, and the authors say the actual contraction factors are barely below 1. That does not undermine the result, but it means the practical guidance from the bounds is qualitative rather than tight. The D-norm is denoiser-dependent, so the contraction statement lives in a norm that changes with the denoiser; that is inherent to the approach and clearly explained.\n\nWho gets value from this: anyone working on PnP convergence theory, especially people who want a non-symmetric, non-trained denoiser with a rigorous linear-rate guarantee. It is not a paper about state-of-the-art reconstruction quality—the authors are explicit that DSG-NLM reconstruction is better and NLM is mainly cheaper. I would take it to a reading group, I would cite it, and I would send it to reviewers. My recommendation: accept after minor revisions, mainly the small self-containedness fixes.","headline":"A solid, honest extension of the PnP convergence line: scaled PnP with nonsymmetric kernel denoisers gets a genuine contraction proof and quantitative bounds, with the main caveats (K PSD, W invertible for ADMM) stated openly by the authors.","tokens_in":22731,"tokens_out":1302,"would_cite":true,"duration_ms":13230,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68U10","94A08","65J22","47H09","47H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that scaled plug-and-play algorithms with kernel denoisers converge linearly to a unique reconstruction for inpainting, deblurring, and superresolution.","keywords":["plug-and-play","kernel denoiser","linear convergence","contraction mapping","scaled PnP","image reconstruction","spectral gap","D-norm"],"falsifier":"Take any kernel denoiser satisfying Assumption 1 and any of the three forward operators, and compute ||P_s||_D for a step size γ in (0, 2); the theorem predicts the value is always below 1, so a single instance at or above 1 would refute the central contractivity claim. A sharper test would use a box-window NLM, which violates the positive-semidefinite assumption, to see whether the bound can fail when the assumption is dropped.","tokens_in":21701,"feed_emoji":"🖼️","tokens_out":7091,"duration_ms":57828,"temperature":0.7,"pith_summary":"The paper extends linear-convergence guarantees for plug-and-play (PnP) image reconstruction from symmetric denoisers to the broader, cheaper class of nonsymmetric kernel denoisers such as NLM. It proves that a scaled version of PnP-ISTA, in which gradient steps are taken with respect to a norm adapted to the denoiser, is globally contractive for inpainting, deblurring, and superresolution for any step size in (0,2). A companion result gives the same guarantee for scaled PnP-ADMM, provided the denoiser is invertible. The paper also derives explicit upper bounds on the contraction factor in terms of the denoiser's spectral gap, the sampling rate, and the algorithm parameters, and validates the bounds numerically. A reader should care because this turns an empirical workhorse into an algorithm with a provable geometric rate of convergence and no dependence on initialization.","feed_headline":"Proof: scaled plug-and-play converges for kernel denoisers","feed_subtitle":"A denoiser-adapted norm restores contraction, extending guarantees past symmetric denoisers","key_machinery":"The key object is the D-norm, defined by ⟨x, y⟩_D = x^T D y with D = diag(Ke), the row-sum matrix of the kernel. A kernel denoiser W is generally not self-adjoint in the Euclidean inner product, but it is self-adjoint in ⟨·, ·⟩_D; this makes the D-norm the natural geometry for these nonsymmetric denoisers. Working in this geometry changes the PnP update: the gradient of the data-fidelity term is replaced by $D^{{-1}}$∇f, giving scaled PnP-ISTA, and the proximal operator is defined with the D-norm, giving scaled PnP-ADMM. The contraction argument is carried by a lemma stating that if M and N are self-adjoint operators with spectra in (-1, 1] and disjoint fixed-point sets, then ||MN|| < 1; applying it with M = W and N = I - $γD^{{-1}}$A^T A, or with M = F_s and N = 2W - I for ADMM, yields the contractivity of the update operators.","core_discovery":"The central claim is that for any kernel denoiser W = $D^{{-1}}$K whose kernel matrix K is nonnegative, irreducible, has unit diagonal, and is positive semidefinite, the scaled PnP-ISTA update P_s = W(I - $γD^{{-1}}$A^T A) satisfies ||P_s||_D < 1 for inpainting, deblurring, and superresolution whenever 0 < γ < 2. Because the D-norm is a norm, the contraction mapping theorem then gives global linear convergence of the iterates to a unique reconstruction. The proof rests on writing P_s as a product of two operators that are self-adjoint in the D-inner product, have spectra contained in (-1, 1], and have disjoint fixed-point spaces; a lemma shows any such pair has product norm strictly below 1. The same lemma, applied to operators built from Douglas-Rachford splitting, yields contractivity of scaled PnP-ADMM when W is invertible. Quantitative contraction-factor bounds follow by refining the lemma with eigenvector estimates, giving rates that depend on the spectral gap of W, the fraction of observed or sampled pixels, and the parameters γ and ρ.","pith_inferences":["The same contraction mechanism should apply to any linear denoiser that is primitive and has the same spectral structure as a kernel denoiser, since the proof uses only the properties in Proposition 1 and the fixed-point space span(e).","The bounds suggest a practical tuning strategy: choose denoiser bandwidth to widen the spectral gap while monitoring reconstruction quality, since the theory predicts faster convergence before quality degrades.","The numerical Jacobian experiment with a nonlinear trained denoiser hints that a similar qualitative pattern may hold locally for nonlinear denoisers, but no global guarantee follows from this paper; that remains an open question.","Because the positive-semidefiniteness assumption excludes box-window NLM, a natural extension would determine whether a box-window kernel can be modified or symmetrized while preserving the contraction bound."],"forward_implications":["Sc-PnP-ISTA with any kernel denoiser satisfying Assumption 1 converges globally at a geometric rate to a unique reconstruction for inpainting, deblurring, and superresolution, for every step size 0 < γ < 2.","Sc-PnP-ADMM gives the same guarantee for every ρ > 0 whenever the kernel denoiser is invertible, as it is for NLM and DSG-NLM.","The contraction-factor bounds decrease as the fraction of observed pixels (inpainting) or sampled pixels (superresolution) increases, so more measurements provably speed up convergence.","A larger spectral gap of the denoiser, obtained by increasing the bandwidth h, reduces the contraction bound and speeds convergence, at the price of reconstruction quality if h is pushed too far.","The standard unscaled PnP-ISTA is not contractive in the D-norm for nonsymmetric kernel denoisers in deblurring; the paper's counterexample shows that scaling is genuinely needed."],"supporting_citations":[{"why":"Supplies the prior contractivity results for symmetric denoisers and for inpainting with kernel denoisers that this work extends to nonsymmetric denoisers.","marker":"[31]"},{"why":"Introduces scaled PnP and the D-inner-product viewpoint that makes kernel denoisers self-adjoint.","marker":"[17]"},{"why":"Introduces the plug-and-play framework and the DSG-NLM symmetrization protocol used as the symmetric baseline.","marker":"[2]"},{"why":"Provides the kernel-denoiser representation and spectral properties on which Proposition 1 relies.","marker":"[36]"},{"why":"Bochner's theorem from this reference is used to prove positive semidefiniteness of the hat-window kernel matrix.","marker":"[42]"},{"why":"Supplies Perron-Frobenius and irreducibility facts underlying the spectral-gap and fixed-point arguments.","marker":"[41]"},{"why":"Cited for the invertibility of NLM and DSG-NLM kernel matrices used in the ADMM contractivity theorem.","marker":"[47]"},{"why":"Provides the contraction mapping theorem that converts contractivity in the D-norm into global linear convergence.","marker":"[34]"}],"fun_headline_variants":["Proof: scaled PnP ensures linear convergence for kernel denoisers","Unified linear convergence proof for plug-and-play with kernels","Nonsymmetric kernel denoisers now covered by PnP convergence proof","Contraction mapping yields rates for PnP with kernel denoisers","Scaling restores PnP convergence for all kernel denoisers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantees depend on the kernel matrix K being positive semidefinite, a property that holds for the hat-window version of NLM but is explicitly not guaranteed for a box-window NLM; the ADMM result additionally assumes the denoiser operator is invertible, which is cited to an external thesis rather than proved here.","fun_headline_variants_meta":{"raw":{"variants":["Proof: scaled PnP ensures linear convergence for kernel denoisers","Unified linear convergence proof for plug-and-play with kernels","Nonsymmetric kernel denoisers now covered by PnP convergence proof","Contraction mapping yields rates for PnP with kernel denoisers","Scaling restores PnP convergence for all kernel denoisers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000345,"raw_usage":{"total_tokens":1900,"prompt_tokens":959,"completion_tokens":941,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":847}},"tokens_in":575,"tokens_out":941,"duration_ms":8705,"temperature":1.0,"reasoning_tokens":847,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:20:33.763678+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any kernel denoiser satisfying Assumption 1 and any of the three forward operators, and compute ||P_s||_D for a step size γ in (0, 2); the theorem predicts the value is always below 1, so a single instance at or above 1 would refute the central contractivity claim. A sharper test would use a box-window NLM, which violates the positive-semidefinite assumption, to see whether the bound can fail when the assumption is dropped.","supporting_citations":[{"cited_title":"On the contractivity of plug-and-play operators,","cited_arxiv_id":null,"evidence_quote":"Supplies the prior contractivity results for symmetric denoisers and for inpainting with kernel denoisers that this work extends to nonsymmetric denoisers."},{"cited_title":"On plug-and-play regularization using linear denoisers,","cited_arxiv_id":null,"evidence_quote":"Introduces scaled PnP and the D-inner-product viewpoint that makes kernel denoisers self-adjoint."},{"cited_title":"Plug-and-play priors for bright field electron tomography and sparse interpolation,","cited_arxiv_id":null,"evidence_quote":"Introduces the plug-and-play framework and the DSG-NLM symmetrization protocol used as the symmetric baseline."},{"cited_title":"Symmetrizing smoothing filters,","cited_arxiv_id":null,"evidence_quote":"Provides the kernel-denoiser representation and spectral properties on which Proposition 1 relies."},{"cited_title":"Katznelson, An Introduction to Harmonic Analysis","cited_arxiv_id":null,"evidence_quote":"Bochner's theorem from this reference is used to prove positive semidefiniteness of the hat-window kernel matrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Perron-Frobenius and irreducibility facts underlying the spectral-gap and fixed-point arguments."},{"cited_title":"Provably Convergent Algorithms for Denoiser-Driven Image Regularization,","cited_arxiv_id":null,"evidence_quote":"Cited for the invertibility of NLM and DSG-NLM kernel matrices used in the ADMM contractivity theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the contraction mapping theorem that converts contractivity in the D-norm into global linear convergence."}],"review_version":1}