Pith. sign in

REVIEW 3 major objections 4 minor 177 references

This paper shows that a permutation-based weighted-aggregation denoiser can be made globally nonexpansive (Lipschitz constant at most 1) by construction, and that the resulting plug-and-play reconstruction operator is contractive, guarantee

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Permutation-symmetrized positive-weight aggregation creates trainable denoisers that are globally nonexpansive and make plug-and-play reconstruction a contraction.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection Genuinely new mechanism for global nonexpansivity, but the boundary convention in the implementation must be clarified before the contraction guarantee is believable. the 3 major comments →

arxiv 2607.23347 v1 pith:XSN5JRPL submitted 2026-07-25 eess.IV cs.CV

Trainable Nonexpansive Denoisers for Contractive Image Reconstruction

Arghya Sinha , Aditya Banerjee , Trishit Mukherjee , Kunal N. Chaudhury This is my paper
classification eess.IV cs.CV
keywords nonexpansive denoiserplug-and-play reconstructioncontractive fixed-point iterationpermutation-based aggregationinverse-consistency conditionimage deblurringsuperresolutionconvergence guarantee
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Plug-and-play image reconstruction treats a denoiser as an implicit regularizer, but the convergence of the fixed-point iteration depends on the denoiser's Lipschitz properties, which existing deep denoisers enforce only on training samples, not globally. This paper constructs a denoiser whose global Lipschitz constant is at most one for every parameter value by keeping all nonlinearity in the prediction of positive weights and making the actual denoising step a linear, normalized aggregation over permuted copies of the image. The authors prove that, under mild conditions on the permutation set, the network's outputs, and the forward operator, plugging this denoiser into half-quadratic splitting makes the reconstruction operator a contraction. Contractivity implies the algorithm converges to a unique fixed point regardless of initialization, and experiments on deblurring and superresolution show performance competitive with softly constrained baselines that offer no such global guarantee.

Core claim

For a fixed reference image ξ, the proposed denoiser D(·;ξ) is linear in the input image, while remaining nonlinear in ξ through the weight-predicting network Nθ. By imposing an inverse-consistency condition (ICC) — computing the weight for permutation π⁻¹ as the permuted copy of the weight for π — the aggregation operator K(·;ξ) becomes symmetric for every ξ. A symmetrization step then yields Dsym, a symmetric, entrywise nonnegative, stochastic operator with spectral norm exactly 1, i.e., globally nonexpansive. The main theorem (Theorem 3.1) shows that the reconstruction operator Tξ = Dsym ∘ prox_{ρf} is contractive, meaning ∥Dsym (I+ρAᵀA)⁻¹∥₂ < 1, under assumptions that include transitivit

What carries the argument

The central object is the weighted aggregation operator K(x;ξ) = Σ_{π∈G} Nθ(ξ, π·ξ) ⊙ (π·x), normalized element-wise by C(ξ) = Σ_{π∈G} Nθ(ξ, π·ξ), where G is an inversion-closed, transitive set of lattice permutations (in practice, local translations T[7]). The load-bearing identity is the inverse-consistency condition Nθ(ξ, π·ξ) = π·Nθ(ξ, π⁻¹·ξ), which makes K symmetric. The symmetrization φ (defined in Eq. 17) converts the normalized denoiser into Dsym, a symmetric, stochastic matrix with spectral norm 1. Theorem 3.1 then proves that composing Dsym with the proximal operator of the quadratic data-fidelity term yields a contraction, combining a Perron–Frobenius argument for the simplicity o

Load-bearing premise

The proof treats the weight for π⁻¹ as a permuted copy of the weight for π via a computational shortcut, an identity that is not intrinsic to a generic trained CNN—if inverse pairs are evaluated separately, or order-2 permutations are included, the symmetry of K and the contractivity conclusion can fail.

What would settle it

For a fixed trained network, forward operator A, and reference ξ, numerically compute the matrix norm ∥Dsym(·;ξ)(I+ρAᵀA)⁻¹∥₂; the paper predicts it is strictly less than 1, so finding any ξ or network instance with norm ≥ 1—or observing two different initializations converge to different fixed points—refutes the contractivity claim. A code-level check is simpler: replace the permuted-copy weight with a genuine evaluation of Nθ(ξ, π⁻¹·ξ) and repeat the reconstruction; any change in the limiting image indicates Assumption (A3) was not actually satisfied.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • With the reference image fixed after warm-up, the reconstruction iteration converges to the same unique fixed point from any initialization, including all-zeros, random noise, the observed image, or the true image.
  • Because the nonexpansive property is architectural rather than penalty-based, the Lipschitz guarantee holds for all inputs and all parameter values, ruling out divergence on out-of-distribution data where empirically constrained baselines can fail.
  • The denoiser can be reused across different forward operators without retraining; contractivity requires only that the forward operator be non-annihilating (A e ≠ 0), which holds for standard deblurring and superresolution operators.
  • On standard CBSD10 benchmarks, the method's PSNR falls within roughly 0.3–0.8 dB of the strongest convergent baselines while providing a strictly stronger, global convergence guarantee.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The inverse-consistency construction could be ported to any input-dependent linear aggregation scheme (e.g., non-local means or attention) to obtain certified nonexpansive variants, since the symmetry argument only requires an inversion-closed, transitive permutation structure and positive weights.
  • A natural testable extension is an adaptive-reference schedule: re-freeze the reference only when the iterate change is small, which might improve adaptation while retaining contraction in practice, although the paper's theorem does not cover the adaptive phase.
  • The sharpest stress test of the proof is to remove the ICC shortcut and evaluate the network on inverse pairs directly; if the reconstructed images differ materially, the implementation–theory alignment is carrying the guarantee rather than the network itself.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a plug-and-play reconstruction framework in which the denoiser is a normalized, permutation-aggregated linear operator whose weights are produced by a small CNN. By imposing an inverse-consistency condition (ICC) on the CNN outputs, the unnormalized aggregation operator becomes symmetric; a subsequent symmetrization step yields a symmetric, entrywise nonnegative, stochastic operator Dsym. Under assumptions of positivity, transitivity of the permutation set, and a nonannihilating forward operator, the paper claims that Dsym composed with the proximal operator of the quadratic data-fidelity term is a strict contraction, so the fixed-point iteration converges from any initialization. Experiments report competitive PSNR on deblurring and superresolution benchmarks compared with recently proposed convergent PnP methods.

Significance. If the claims hold, the paper offers a useful structural route to globally nonexpansive trainable denoisers: the Lipschitz control is by construction and does not depend on the training distribution. This is in contrast to sample-based spectral-regularization baselines, and the paper is honest that its guarantee covers the symmetrized operator and the frozen-reference phase. The theoretical development is explicit, with proofs in the appendix and a code link, and the experimental comparisons are against relevant convergent baselines. The main value is the combination of a simple permutation-based architecture with a parameter-independent contractivity guarantee, which would be a meaningful contribution to convergent deep image reconstruction.

major comments (3)
  1. [Section 4, Assumption (A4), Theorem 3.1] The paper states that the translation set T[7] 'satisfies the permutation assumptions' because translations act transitively on the lattice. This is only true if Ω is treated as a torus with periodic boundary conditions. A finite-lattice shift is a bijection only under circular wrap-around; with zero-padded or replicated shifts, each π is not a permutation of Ω, the pullback (7) is not a group action, and the inverse steps in Lemma 3.3 and the transitivity condition (A4) fail. The manuscript must state the boundary convention explicitly and confirm that the implementation uses circular shifts. Otherwise Theorem 3.1 does not apply to the reported experiments.
  2. [Section 3, Eq. (15), Algorithm 1] The theoretical operator K in Eq. (12) is defined in terms of the CNN Nθ satisfying the ICC (A3). In Algorithm 1, ICC is instead imposed by defining winv = π^{-1}·w for the inverse permutation and never querying Nθ for π^{-1}. This creates an effective operator that coincides with (12) only if the CNN itself satisfies (15). Since a generic trained CNN will not satisfy ICC as a function on X×X, the proof of Lemma 3.3 applies to the algorithm's effective map, not to the map (12) with an arbitrary Nθ. The paper should define the effective network output, e.g., Ñθ, and state that all theoretical statements are for that effective map, so that the theorem and the implementation refer to the same object.
  3. [Sections 2, 3, and Algorithm 2] The contraction guarantee is established for Dsym = φ(D), not for the denoiser D in Algorithm 1. The text itself notes that the normalization in D can destroy symmetry and that ∥D(·;ξ)∥2 can exceed 1. Nevertheless, the standalone denoising results in Figure 2 and Table 4 are reported for Algorithm 1's D, not for Dsym. In addition, the warm-up phase of Algorithm 2 updates the reference ξ, so the contraction theorem does not cover the initial N_warm iterations; the paper acknowledges this but the abstract and contribution statements should consistently scope 'globally nonexpansive denoiser' and 'globally convergent reconstruction' to the symmetrized, frozen-reference operator to avoid overclaiming.
minor comments (4)
  1. [Section 4] The claim that T[R] contains no nontrivial order-2 elements depends on the lattice dimensions being larger than 2R. On a torus of side length ≤ 2R, a shift by half the side length is an involution. Since Algorithm 1 skips all involutions, the effective permutation set is smaller than G; please state the size condition and note that the analysis applies to the reduced set.
  2. [Section 2] G is repeatedly called a 'group of permutations,' but T[R] is only inversion-closed and is not closed under composition. The assumptions actually use a set of permutations with inversion closure and transitivity of the generated action. The terminology should be corrected or clarified.
  3. [Appendix, Theorem 3.1 proof] In the proof, the reference to Eq. (29) is ambiguous; the displayed expression for K_{ij} should be numbered or cited accurately. This is a minor presentation issue but helps the reader verify the irreducibility argument.
  4. [Equation (17)] The notation C^{1/2}DC^{-1/2}(x) is dense and appears before its componentwise convention is fully explained. A short example or an explicit matrix expression would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the nonexpansive/contractive guarantees are architecturally enforced and proved from stated assumptions.

full rationale

The paper's central derivation is self-contained. Nonexpansivity of Dsym is not a fitted prediction; it is imposed by the explicit symmetrization formula (17), which by construction yields a symmetric, entrywise nonnegative, stochastic operator, and Lemma 3.4 then verifies these properties from the definition. Contractivity in Theorem 3.1 follows from a spectral argument: Dsym has eigenvalues in [-1,1], transitivity (A4) gives Perron-Frobenius irreducibility so the eigenvalue 1 is simple, and (A5) rules out E u = u, yielding ||Dsym E||_2 < 1. No parameter is fit to a subset of data and then reported as a prediction of a closely related quantity; the trained network weights affect denoising quality and PSNR but not the validity of the contraction bound. The ICC condition (A3) is enforced by construction in Algorithm 1 (w_inv = pi^{-1} . w), and the paper explicitly states it is imposed rather than derived from data. The only self-citation (Gavaskar et al., 2021, Chandhury coauthor) appears in a contextual list of prior PnP dynamical-systems analyses and is not load-bearing for the main theorem. Concerns about boundary conventions or exact implementation-theory alignment of ICC are correctness/robustness issues, not circularity, because they concern whether the implemented operator satisfies the stated assumptions, not whether the proof reduces to its own conclusion.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new physical entities are proposed. The theoretical core rests on standard linear algebra plus the paper-specific assumptions A2-A5 and the ICC implementation shortcut. Fitted network parameters and hyperparameters affect empirical performance, not the validity of the guarantee.

free parameters (5)
  • CNN parameters θ
    Trained by MSE on noisy/clean image pairs. The convergence guarantee holds for all θ, but denoising performance depends on the fit.
  • Reference noise multiplier δ = 1.2
    Chosen so the reference image is noisier during training; improves reconstruction-time behavior. Not fitted to the convergence claim.
  • Permutation radius R = 7
    T[7] contains 225 permutations and is used in all experiments; affects receptive field and compute time.
  • Warm-up iterations N_warm = 20
    Recommended default; ablation shows insensitivity. The contraction guarantee applies only after the reference is frozen.
  • HQS penalty ρ
    Fidelity weight in prox_ρf; not specified per experiment in the text, presumably tuned. Affects contraction rate and reconstruction quality.
axioms (5)
  • domain assumption A2 strict positivity: Nθ(x,y) > 0 for all x,y
    Needed for elementwise normalization and for nonnegativity of Dsym; enforced by a trainable positive activation, with no discussion of floating-point underflow.
  • ad hoc to paper A3 Inverse-Consistency Condition (ICC), eq. (15)
    Central symmetry-enabling assumption. Algorithm 1 imposes it by substituting winv = π^{-1}·w instead of evaluating Nθ on the inverse pair.
  • domain assumption A4 transitivity of permutation set G
    Used to prove Dsym is irreducible via Perron-Frobenius, giving λ2^2 < 1. T[7] is claimed to satisfy this.
  • domain assumption A5 nonannihilating forward model Ae ≠ 0
    Used to show ∥Eu∥_2 < 1; holds for blur and superresolution operators assumed in experiments.
  • standard math Perron-Frobenius theorem and spectral theorem for symmetric matrices
    Invoked in the proof of Lemma 3.4 and Theorem 3.1 to bound the second eigenvalue and spectral norm.

reviewed 2026-07-31 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Trainable Nonexpansive Denoisers for Contractive Image Reconstruction." pith.science (2026). https://pith.science/paper/XSN5JRPL

@misc{pith2026260723347,
  author       = {Pith},
  title        = {Pith review of: Trainable Nonexpansive Denoisers for Contractive Image Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSN5JRPL}},
  note         = {Machine review of arXiv:2607.23347}
}
Share X Bluesky LinkedIn Reddit HN
abstract

Trainable denoisers with Lipschitz control have become central to convergent image reconstruction. However, training neural networks that simultaneously offer strong denoising performance and global Lipschitz guarantees is challenging. Existing approaches enforce Lipschitz control only empirically, providing no guarantees beyond the training data. In this work, we show that by exploiting the action of permutations on the image lattice, we can constrain a neural architecture that is globally nonexpansive (Lipschitz bound $\leqslant 1$). We integrate the proposed denoiser with forward imaging operators to develop a reconstruction mechanism that is provably contractive and therefore globally convergent. Experiments on standard inverse problems, such as superresolution and deblurring, demonstrate that our reconstruction performance is competitive with softly constrained baselines while providing Lipschitz guarantees.

Figures

Figures reproduced from arXiv: 2607.23347 by Aditya Banerjee, Arghya Sinha, Kunal N. Chaudhury, Trishit Mukherjee.

Figure 1
Figure 1. Figure 1: What does the network Nθ learn? Once trained, the network learns to highlight regions that are truly similar. Each panel shows two images: the noisy image with noise level σ on the left, and a colormap on the right. The colormap shows the contribution, determined by Nθ, of all pixels reached by the permutations to denoising the central red pixel. When the noise is zero, the contributions from all pixels ot… view at source ↗
Figure 2
Figure 2. Figure 2: Denoising at noise σ = 50/255. are linear in x, while remaining nonlinear in ξ. We will use this structure in two ways: the nonlinearity in ξ provides expressive, data-adaptive weighting, while the linearity in x enables tight operator control and, ultimately, convergence guarantees for the reconstruction map. A natural question is what Nθ learns in (12). As illustrated in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 3
Figure 3. Figure 3: Visual comparison on the “man” image under motion blur (kernel 3) and noise level ν = 0.03. Our denoiser suppresses artifacts while avoiding oversmoothing more effectively than CoCo-DRUNet (Wei et al., 2025) and IHQS (Wei et al., 2024). 4. Implementation Algorithm 1 implements the denoiser in (12) by aggregating permuted versions of the input image. The construction is designed to satisfy Assumptions (A1)–… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Instability of IHQS+SPC-DRUNet (Wei et al., 2024) and CoCo-ADMM (Wei et al., 2025) in 3× superresolution PnP framework of coral image of CBSD68. The corresponding norm plot is provided in the Appendix. 5. Experiments We evaluate the performance of our proposed denoiser in terms of denoising (Algorithm 1) and as an implicit regu￾ [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Instability IHQS+SPC-DRUNet (Wei et al., 2024) and CoCo-ADMM (Wei et al., 2025) in 3× superresolution PnP framework of coral image of CBSD68 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Consequence of Contractivity: Motion deblurring task on the castle image under blur kernel 4 with noise level ν = 0.01. For all initializations, the iterates converge to the same reconstruction. This illustrates the independence of Algorithm 2 from the initialization x0, for a fixed reference ξ, owing to the contractivity of Algorithm 2. (a) clean (b) CoCo-PGD (28.51) (c) IHQS+SPC-DRUNet (28.73) (d) GSPnP … view at source ↗
Figure 8
Figure 8. Figure 8: Superresolution (2×) of the “coral” image (CBSD68) under kernel 1 and noise level ν = 0.01. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

177 extracted references · 1 linked inside Pith

  1. [1]

    Oracle inequalities and minimax rates for nonlocal means and related adaptive kernel-based methods

    Arias-Castro, E., Salmon, J., and Willett, R. Oracle inequalities and minimax rates for nonlocal means and related adaptive kernel-based methods. SIAM Journal on Imaging Sciences, 5 0 (3): 0 944--992, 2012

    2012

  2. [2]

    Bauschke, H. H. and Combettes, P. L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, 2011

    2011

  3. [3]

    Bouman, C. A. Foundations of Computational Imaging: A Model-Based Approach. SIAM, 2022

    2022

  4. [4]

    A non-local algorithm for image denoising

    Buades, A., Coll, B., and Morel, J.-M. A non-local algorithm for image denoising. Proc. CVPR, 2: 0 60--65, 2005

    2005

  5. [5]

    It has potential: G radient-driven denoisers for convergent solutions to inverse problems

    Cohen, R., Blau, Y., Freedman, D., and Rivlin, E. It has potential: G radient-driven denoisers for convergent solutions to inverse problems. Proc. NeurIPS, pp.\ 18152--18164, 2021

    2021

  6. [6]

    Image denoising by sparse 3- D transform-domain collaborative filtering

    Dabov, K., Foi, A., Katkovnik, V., and Egiazarian, K. Image denoising by sparse 3- D transform-domain collaborative filtering. IEEE Trans. Image Process., 16 0 (8): 0 2080--2095, 2007

    2080

  7. [7]

    Improving lipschitz-constrained neural networks by learning activation functions

    Ducotterd, S., Goujon, A., Bohra, P., Perdios, D., Neumayer, S., and Unser, M. Improving lipschitz-constrained neural networks by learning activation functions. Journal of Machine Learning Research, 25 0 (65): 0 1--30, 2024

    2024

  8. [8]

    Dummit, D. S. and Foote, R. M. Abstract algebra. Wiley, 2004

    2004

  9. [9]

    Image denoising: The deep learning revolution and beyond—a survey paper

    Elad, M., Kawar, B., and Vaksman, G. Image denoising: The deep learning revolution and beyond—a survey paper. SIAM J. Imaging Sci., 16 0 (3): 0 1594--1654, 2023

    2023

  10. [10]

    G., Athalye, C

    Gavaskar, R. G., Athalye, C. D., and Chaudhury, K. N. On plug-and-play regularization using linear denoisers. IEEE Trans. Image Process., 30: 0 4802--4813, 2021

    2021

  11. [11]

    A neural-network-based convex regularizer for inverse problems

    Goujon, A., Neumayer, S., Bohra, P., Ducotterd, S., and Unser, M. A neural-network-based convex regularizer for inverse problems. IEEE Trans. Comput. Imaging, 9: 0 781--795, 2023

    2023

  12. [12]

    Learning weakly convex regularizers for convergent image-reconstruction algorithms

    Goujon, A., Neumayer, S., and Unser, M. Learning weakly convex regularizers for convergent image-reconstruction algorithms. SIAM J. Imaging Sci., 17 0 (1): 0 91--115, 2024

    2024

  13. [13]

    Convolutional proximal neural networks and plug-and-play algorithms

    Hertrich, J., Neumayer, S., and Steidl, G. Convolutional proximal neural networks and plug-and-play algorithms. Linear Algebra Appl., 631: 0 203--234, 2021

    2021

  14. [14]

    Horn, R. A. and Johnson, C. R. Matrix Analysis. Cambridge University Press, 2012

    2012

  15. [15]

    Gradient step denoiser for convergent plug-and-play

    Hurault, S., Leclaire, A., and Papadakis, N. Gradient step denoiser for convergent plug-and-play. Proc. ICLR, 2022 a

    2022

  16. [16]

    Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization

    Hurault, S., Leclaire, A., and Papadakis, N. Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization. Proc. ICML, pp.\ 9483--9505, 2022 b

    2022

  17. [17]

    Denoising diffusion restoration models

    Kawar, B., Elad, M., Ermon, S., and Song, J. Denoising diffusion restoration models. Proc. NeurIPS, pp.\ 23593--23606, 2022

    2022

  18. [18]

    Moreau, J. J. Proximit\' e et dualit\' e dans un espace H ilbertien. Bull. Soc. Math. Fr., 93: 0 273--299, 1965

    1965

  19. [19]

    Learning maximally monotone operators for image recovery

    Pesquet, J.-C., Repetti, A., Terris, M., and Wiaux, Y. Learning maximally monotone operators for image recovery. SIAM J. Imaging Sci., 14 0 (3): 0 1206--1237, 2021

    2021

  20. [20]

    Reehorst, E. T. and Schniter, P. Regularization by denoising: Clarifications and new interpretations. IEEE Trans. Comput. Imaging, 5 0 (1): 0 52--67, 2018

    2018

  21. [21]

    The little engine that could: Regularization by Denoising ( RED )

    Romano, Y., Elad, M., and Milanfar, P. The little engine that could: Regularization by Denoising ( RED ). SIAM J. Imaging Sci., 10 0 (4): 0 1804--1844, 2017

    2017

  22. [22]

    I., Osher, S., and Fatemi, E

    Rudin, L. I., Osher, S., and Fatemi, E. Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom., 60 0 (1-4): 0 259--268, 1992

    1992

  23. [23]

    Plug-and-play methods provably converge with properly trained denoisers

    Ryu, E., Liu, J., Wang, S., Chen, X., Wang, Z., and Yin, W. Plug-and-play methods provably converge with properly trained denoisers. Proc. ICML, pp.\ 5546--5557, 2019

    2019

  24. [24]

    V., Wohlberg, B., Buzzard, G

    Sreehari, S., Venkatakrishnan, S. V., Wohlberg, B., Buzzard, G. T., Drummy, L. F., Simmons, J. P., and Bouman, C. A. Plug-and-play priors for bright field electron tomography and sparse interpolation. IEEE Trans. Comput. Imaging, 2 0 (4): 0 408--423, 2016

    2016

  25. [25]

    N., Kaiser, L

    Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L. u., and Polosukhin, I. Attention is all you need. Proc. NeurIPS, 2017

    2017

  26. [26]

    V., Bouman, C

    Venkatakrishnan, S. V., Bouman, C. A., and Wohlberg, B. Plug-and-play priors for model based reconstruction. In 2013 IEEE global conference on signal and information processing, pp.\ 945--948. IEEE, 2013

    2013

  27. [27]

    and Scaman, K

    Virmaux, A. and Scaman, K. Lipschitz regularity of deep neural networks: analysis and efficient estimation. Proc. NeurIPS, 2018

    2018

  28. [28]

    Non-local neural networks

    Wang, X., Girshick, R., Gupta, A., and He, K. Non-local neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp.\ 7794--7803, 2018

    2018

  29. [29]

    Learning pseudo-contractive denoisers for inverse problems

    Wei, D., Chen, P., and Li, F. Learning pseudo-contractive denoisers for inverse problems. Proc. ICML, 2024

    2024

  30. [30]

    Learning cocoercive conservative denoisers via helmholtz decomposition for poisson imaging inverse problems

    Wei, D., Chen, P., Xu, H., Yao, J., Li, F., and Zeng, T. Learning cocoercive conservative denoisers via helmholtz decomposition for poisson imaging inverse problems. Proc. NeuRIPS, 2025

    2025

  31. [31]

    Basis prediction networks for effective burst denoising with large kernels

    Xia, Z., Perazzi, F., Gharbi, M., Sunkavalli, K., and Chakrabarti, A. Basis prediction networks for effective burst denoising with large kernels. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp.\ 11844--11853, 2020

    2020

  32. [32]

    Beyond a G aussian denoiser: Residual learning of deep CNN for image denoising

    Zhang, K., Zuo, W., Chen, Y., Meng, D., and Zhang, L. Beyond a G aussian denoiser: Residual learning of deep CNN for image denoising. IEEE Trans. Image Process., 26 0 (7): 0 3142--3155, 2017 a

    2017

  33. [33]

    Learning deep cnn denoiser prior for image restoration

    Zhang, K., Zuo, W., Gu, S., and Zhang, L. Learning deep cnn denoiser prior for image restoration. Proc. CVPR, pp.\ 3929--3938, 2017 b

    2017

  34. [34]

    Plug-and-play image restoration with deep denoiser prior

    Zhang, K., Li, Y., Zuo, W., Zhang, L., Van Gool, L., and Timofte, R. Plug-and-play image restoration with deep denoiser prior. IEEE Trans. Pattern Anal. Mach. Intell., 44 0 (10): 0 6360--6376, 2021

    2021

  35. [35]

    Zhang, K., Li, Y., Liang, J., Cao, J., Zhang, Y., Tang, H., Fan, D.-P., Timofte, R., and Gool, L. V. Practical blind image denoising via S win- C onv- UN et and data synthesis. Mach. Intell. Res., 20 0 (6): 0 822--836, 2023

    2023

  36. [36]

    Residual non-local attention networks for image restoration

    Zhang, Y., Li, K., Li, K., Zhong, B., and Fu, Y. Residual non-local attention networks for image restoration. In Proceedings of the International Conference on Learning Representations (ICLR), 2019

    2019

  37. [37]

    2015 , journal=

    On the convergence of the iterates of the ``fast iterative shrinkage/thresholding algorithm'' , author =. 2015 , journal=

    2015

  38. [38]

    2014 , publisher=

    Proximal algorithms , author=. 2014 , publisher=

    2014

  39. [39]

    Non-local sparse models for image restoration , author=. Proc. ICCV , pages=. 2009 , organization=

    2009

  40. [40]

    and Hunt, Bobby R

    Andrews, Harry C. and Hunt, Bobby R. , title =. 1977 , publisher =

    1977

  41. [41]

    Nature , volume=

    Deep Learning , author=. Nature , volume=

  42. [42]

    Tuning-free plug-and-play proximal algorithm for inverse imaging problems , author =. Proc. ICML , pages=

  43. [43]

    Variational networks: An optimal control approach to early stopping variational methods for image restoration , author=. J. Math. Imaging Vis. , volume=. 2020 , publisher=

    2020

  44. [44]

    Zhang, Kai and Li, Yawei and Liang, Jingyun and Cao, Jiezhang and Zhang, Yulun and Tang, Hao and Fan, Deng-Ping and Timofte, Radu and Gool, Luc Van , title =. Mach. Intell. Res. , volume=. 2023 , publisher=

    2023

  45. [45]

    Dong, Chao and Loy, Chen Change and Tang, Xiaoou , title=. Proc. ECCV , year=

  46. [46]

    Nonlinear total variation based noise removal algorithms , author=. Phys. D: Nonlinear Phenom. , volume=. 1992 , publisher=

    1992

  47. [47]

    IEEE Trans

    A neural-network-based convex regularizer for inverse problems , author=. IEEE Trans. Comput. Imaging , volume=

  48. [48]

    Geman, Stuart and Graffigne, Christine , journal=. Markov

  49. [49]

    Huang, Jia-Bin and Singh, Abhishek and Ahuja, Narendra , title =. Proc. CVPR , month =

  50. [50]

    Tachella, Julia and Chen, Dongdong and Hurault, Samuel and Terris, M and Wang, Andrew , journal=

  51. [51]

    A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics , year=

    Martin, David and Fowlkes, Charless and Tal, Doron and Malik, Jitendra , journal=. A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics , year=

  52. [52]

    2016 , publisher=

    Variational Methods in Image Processing , author =. 2016 , publisher=

    2016

  53. [53]

    Restormer:

    Syed Waqas Zamir and Aditya Arora and Salman Khan and Munawar Hayat and Fahad Shahbaz Khan and Ming-Hsuan Yang , journal=. Restormer:

  54. [54]

    Learning a deep convolutional network for image super-resolution , author=. Proc. ECCV , pages=. 2014 , organization=

    2014

  55. [55]

    , author=

    A Wavelet Tour of Signal Process. , author=. 1999 , publisher=

    1999

  56. [56]

    and Chaudhury, Kunal N

    Sinha, A. and Chaudhury, Kunal N. , journal=. On the strong convexity of. 2024 , publisher=

    2024

  57. [57]

    2012 , publisher=

    Digital Picture Processing: An Introduction , author=. 2012 , publisher=

    2012

  58. [58]

    Total Generalized Variation , author=. SIAM J. Imaging Sci. , volume=

  59. [59]

    Digital image smoothing and the sigma filter , author=. Comput. Gr. Image Process. , volume=. 1983 , publisher=

    1983

  60. [60]

    On instabilities of deep learning in image reconstruction and the potential costs of

    Antun, Vegard and Renna, Francesco and Poon, Clarice and Adcock, Ben and Hansen, Anders C , journal=. On instabilities of deep learning in image reconstruction and the potential costs of

  61. [61]

    SIAM Review , volume=

    The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems , author=. SIAM Review , volume=

  62. [62]

    2022 , publisher=

    Computer Vision: Algorithms and Applications , author=. 2022 , publisher=

    2022

  63. [63]

    IEEE Trans

    Kernel regression for image processing and reconstruction , author=. IEEE Trans. Image Process. , volume=. 2007 , publisher=

    2007

  64. [64]

    Bilateral filtering for gray and color images , author=. Proc. ICCV , pages=

  65. [65]

    IEEE Signal Process

    Fast high-dimensional kernel filtering , author=. IEEE Signal Process. Lett. , volume=

  66. [66]

    Averaged deep denoisers for image regularization , author=. J. Math. Imaging Vis. , pages =. 2024 , publisher=

    2024

  67. [67]

    Plug-and-play methods provably converge with properly trained denoisers , author=. Proc. ICML , pages=

  68. [68]

    Hyperspectral image fusion using fast high-dimensional denoising , author=. Proc. IEEE ICIP , pages=

  69. [69]

    , journal=

    Reddy, P and Chaudhury, Kunal N. , journal=. Learning to optimally sample in

  70. [70]

    IEEE Trans

    Plug-and-play regularization using linear solvers , author=. IEEE Trans. Image Process. , volume=

  71. [71]

    Plug-and-play

    Chan, Stanley H and Wang, Xiran and Elgendy, Omar A , journal=. Plug-and-play. 2017 , publisher=

    2017

  72. [72]

    Backtracking strategies for accelerated descent methods with smooth composite objectives , author=. SIAM J. Optim. , volume=

  73. [73]

    2011 , publisher=

    Convex Analysis and Monotone Operator Theory in Hilbert Spaces , author=. 2011 , publisher=

    2011

  74. [74]

    1983 , publisher=

    Matrix Computations , author=. 1983 , publisher=

    1983

  75. [75]

    Johnstone, P. R. and Moulin, P. , journal=. A

  76. [76]

    , journal=

    Milanfar, P. , journal=

  77. [77]

    IEEE Trans

    Plug-and-play priors for bright field electron tomography and sparse interpolation , author=. IEEE Trans. Comput. Imaging , volume=

  78. [78]

    IEEE Signal Process

    Plug-and-play methods for magnetic resonance imaging: Using denoisers for image recovery , author=. IEEE Signal Process. Mag. , volume=

  79. [79]

    and Boley, D

    Tao, S. and Boley, D. and Zhang, S. , journal=. Local linear convergence of

  80. [80]

    An inertial forward-backward algorithm for monotone inclusions , author=. J. Math. Imaging Vis. , volume=. 2015 , publisher=

    2015

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on July 31, 2026.