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Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers

T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that scaled plug-and-play algorithms with kernel denoisers converge linearly to a unique reconstruction for inpainting, deblurring, and superresolution.

desk verdict 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. read the letter →

arxiv 2505.15318 v1 pith:NQLNSX3S submitted 2025-05-21 eess.IV math.OC

Arghya Sinha , Bhartendu Kumar , Chirayu D. Athalye , Kunal N. Chaudhury This is my paper · ORCID
classification eess.IVmath.OC MSC 68U1094A0865J2247H0947H10
keywords plug-and-playkerneldenoiserlinearconvergencecontractionmappingscaledPnPimagereconstructionspectralgapD-norm
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

1 major / 6 minor

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.

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 (1)
  1. [Theorem 4, Remark 3, Section V-B] 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.
minor comments (6)
  1. [Section I-B and Section II-C] 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.
  2. [Theorems 6, 7, and 8] 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.
  3. [Remark 3] 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.
  4. [Propositions 3 and 5] 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.
  5. [Section II-C] Typo: "deburring" should be "deblurring" in the sentence "the contractivity of P cannot generally be guaranteed for deburring using the D-norm."
  6. [Theorem 8 proof] In the proof of Theorem 8, the notation "∥Fq1∥_D" appears where "∥Fsq1∥_D" is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the contraction theorems are derived from explicit spectral assumptions with self-contained proofs; self-citations provide context, not load-bearing premises.

full rationale

I walked the derivation chain for Theorems 3-8. Lemma 1 is proved in Appendix VII-A (not merely cited), and Lemma 2 is proved in Appendix VII-B. Theorem 3 verifies the hypotheses of Lemma 1 explicitly: sigma(W) subset [0,1] follows from Assumption 1 via Proposition 1, sigma(Gs) subset (-1,1] follows from the positive semidefiniteness of H = D^{-1/2} A^T A D^{-1/2} with norm at most 1, and fix(W) intersect fix(Gs) = {0} follows from RNP/Proposition 2 (Ae != 0). Theorem 4 has the same structure with sigma(V) subset (-1,1] under the stated invertibility assumption. The quantitative bounds in Theorems 5-8 are algebraically derived upper bounds expressed in terms of meaningful spectral and geometric parameters (lambda2, zeta*, D, mu, gamma, rho); they are not fitted to the numerically measured contraction factors. The numerical section compares actual operator norms with these bounds, which is consistency checking, not prediction-from-fit. Self-citations to [17] and [31] supply the scaled-PnP framework and an earlier inpainting result, but the present proofs are self-contained: [31, Lemma 1] is re-proved, and [17] is used only to motivate the algorithm, not as the operative premise of the new theorems. The only external dependency is the invertibility of W for NLM and DSG-NLM, cited to thesis [47]; this is a stated assumption, and the paper remarks that Theorem 4 is valid even without it, so it is a correctness caveat rather than a circular step. The explicit assumption that K is positive semidefinite (Assumption 1) is also stated as a condition, not hidden. No equation in the paper reduces to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's central claims rest on three domain assumptions (PSD/irreducibility of K, well-posedness of A, invertibility of W for ADMM) plus standard linear algebra. No free parameters are fitted to data; the bounds involve measured spectral quantities (lambda2, zeta*, ||D||2) and the known sampling fraction mu. No new physical or mathematical entities are postulated.

assumptions (4)
  • domain assumption Kernel matrix K is symmetric positive semidefinite, nonnegative, irreducible, with unit diagonal (Assumption 1)
    Required for the spectral properties of W in Prop. 1 (nonnegative eigenvalues, Perron root 1, spectral gap) used by Theorems 3-8; the PSD clause can fail for standard NLM with a box window.
  • domain assumption Forward operator satisfies Assumption 2: at least one pixel observed/sampled; blur kernel nonnegative and normalized (Be = e)
    Used to prove RNP (Prop. 2), to bound ||A||2 <= 1 (Remark 2), to compute ||Ae||2 (Remark 6), and to ensure E is nonnegative (Thm. 8).
  • domain assumption W is invertible for Sc-PnP-ADMM (assumed in Theorem 4 and Remark 3)
    Needed so that sigma(V) is contained in (-1,1] and zeta* < 1; the paper cites [47, Thm. 2.16] for NLM and DSG-NLM rather than proving it.
  • standard math Perron-Frobenius theorem, Bochner's theorem, spectral theorem, contraction mapping theorem
    Standard background invoked in Sections II and IV; Bochner's theorem justifies K in S^n_+ for the Gaussian-times-Hat kernel.

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Pith. "Pith review of Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers." pith.science (2026). https://pith.science/paper/NQLNSX3S

@misc{pith2026250515318,
  author       = {Pith},
  title        = {Pith review of: Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQLNSX3S}},
  note         = {Machine review of arXiv:2505.15318}
}
read the original abstract

The use of denoisers for image reconstruction has shown significant potential, especially for the Plug-and-Play (PnP) framework. In PnP, a powerful denoiser is used as an implicit regularizer in proximal algorithms such as ISTA and ADMM. The focus of this work is on the convergence of PnP iterates for linear inverse problems using kernel denoisers. It was shown in prior work that the update operator in standard PnP is contractive for symmetric kernel denoisers under appropriate conditions on the denoiser and the linear forward operator. Consequently, we could establish global linear convergence of the iterates using the contraction mapping theorem. In this work, we develop a unified framework to establish global linear convergence for symmetric and nonsymmetric kernel denoisers. Additionally, we derive quantitative bounds on the contraction factor (convergence rate) for inpainting, deblurring, and superresolution. We present numerical results to validate our theoretical findings.

Figures

Figures reproduced from arXiv: 2505.15318 by the authors.

Figure 1
Figure 1. Deblurring results using PnP-ISTA with a symmetric DSG-NLM denoiser and a nonsymmetric NLM denoiser, and Sc-PnP-ISTA with a nonsymmetric NLM denoiser. The input image is blurred with a Gaussian kernel of size 25 × 25 and a standard deviation of 1.6, followed by the addition of 3% white Gaussian noise. The PSNR values are (b) 22.85 dB, (c) 29.39 dB, (d) 28.93 dB and (e) 28.81 dB. (a) original (b) input (c) PnP-ISTA (… view at source ↗
Figure 2
Figure 2. 2× superresolution using PnP-ISTA and Sc-PnP-ISTA, with DSG-NLM and NLM denoisers respectively. The input is generated by applying a uniform blur with a 9 × 9 kernel, followed by 2× downsampling and adding 3% white Gaussian noise. The PSNR values are (c) 27.31 dB, (d) 26.67 dB, and (d) 26.49 dB. For reference, the PSNR from bicubic interpolation is 22.80 dB. We note that (36) and (37) are smaller than the respective… view at source ↗
Figure 3
Figure 3. We compare the reconstruction quality obtained using [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: We report the contraction factors of the update operators [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: We study the effect of the downsampling rate (for superresolution) on the contraction factor of the update operators: [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: We empirically validate the predictions of Theorem 5 using a variant of the nonlinear DnCNN denoiser [21]. Specifically, we compute the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: We study the correlation between the contraction factor of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 9
Figure 9. Figure 9: We show the successive difference of the iterates generated by [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.