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arxiv: 2604.20360 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

On the convergence of an adaptive denoiser driven iterative regularization with early stopping

Harshit Bajpai , Ankik Kumar Giri , Tim Jahn , Abhinav Jha This is my paper

Pith reviewed 2026-05-10 00:08 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords denoiser-driven regularizationiterative regularizationconvergence analysisinverse problemsearly stoppingimage deblurringphase retrievalregularization by denoising
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The pith

The DDIR method with adaptive steps and a posteriori stopping forms a stable convergent regularization scheme.

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

This paper extends regularization by denoising to a new iterative scheme called DDIR that builds a regularization functional from averaged denoisers. An adaptive step-size rule and an a posteriori stopping criterion are added to control noise-induced oscillations and semi-convergence. The central proof shows that the resulting iteration is a classically convergent and stable regularization method. The argument requires the averaged denoiser to be firmly nonexpansive or to satisfy a fixed-point condition. Experiments on deblurring and phase-retrieval CT illustrate practical gains in accuracy and speed with median, TNRD, and total-variation proximal denoisers.

Core claim

The resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. This supplies the first rigorous justification of DDIR within regularization theory, under the assumption that the denoisers satisfy appropriate properties such as the averaged operator being firmly nonexpansive.

What carries the argument

Averaged-denoiser regularization functional equipped with adaptive step-size selection and a posteriori early stopping.

If this is right

  • The iteration remains stable under increasing noise levels.
  • Semi-convergence is suppressed by the early stopping rule.
  • The scheme applies directly to standard denoisers such as the median filter, TNRD, and TV proximal operators.
  • Reconstruction accuracy improves on both image deblurring and phase-retrieval CT tasks.

Where Pith is reading between the lines

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

  • If modern learned denoisers can be shown to meet the nonexpansiveness requirement, the same convergence guarantee would extend to them.
  • Comparable adaptive stopping rules could be transferred to other iterative methods that currently lack a posteriori criteria.
  • The framework offers a route to prove convergence for related denoiser-driven schemes in tomography or deconvolution.

Load-bearing premise

The denoisers must make the averaged operator firmly nonexpansive or satisfy a suitable fixed-point condition.

What would settle it

A concrete numerical test on a simple deblurring problem where the iteration diverges when the averaged denoiser fails the nonexpansiveness condition.

Figures

Figures reproduced from arXiv: 2604.20360 by Abhinav Jha, Ankik Kumar Giri, Harshit Bajpai, Tim Jahn.

Figure 1
Figure 1. Figure 1: Image denoising methods within inverse problems. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reconstruction results for Shepp-Logan (top left), Coins (top right), Moon [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative error versus iteration for different denoisers (Median, TNRD, TV [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visual comparison of reconstructed images with [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Stopping index comparision with δrel = 0.01, 0.005, 0.003, 0.001, 0.0003 and 0.0001 respectively [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ground truth image for phase-retrieval CT and the corresponding measure [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction results for phase retrieval CT with [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Relative error plots for phase retrieval CT with [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
read the original abstract

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art performance in various imaging applications. However, their stability and convergence within iterative regularization frameworks remain largely unexplored. In this work, we extend the framework of Regularization by Denoising (RED) by introducing a novel denoiser-driven iterative regularization scheme, referred to as \texttt{DDIR}, that incorporates a new regularization functional based on averaged denoisers. The proposed approach employs an adaptive step-size strategy together with an \emph{a posteriori} stopping rule to ensure stability while alleviating oscillatory behavior and semi-convergence effects induced by noise. As our main theoretical contribution, we prove that the resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. To the best of our knowledge, this provides the first rigorous justification of \texttt{DDIR} within the framework of regularization theory. Finally, we demonstrate the performance of the proposed method through numerical experiments on image deblurring and phase retrieval Computed Tomography (CT) using three denoisers, namely median, TNRD, and TV proximal. The results highlight the effectiveness of the method in terms of reconstruction accuracy and computational efficiency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes DDIR, an extension of Regularization by Denoising (RED) that incorporates averaged denoisers, an adaptive step-size rule, and an a posteriori stopping criterion. It claims to prove that the resulting iterative scheme is a stable and convergent regularization method in the classical sense, providing the first rigorous justification of such denoiser-driven approaches within regularization theory. Numerical experiments on image deblurring and phase-retrieval CT are presented using median, TNRD, and TV-proximal denoisers to illustrate performance.

Significance. If the central convergence result holds under verifiable assumptions that cover the denoisers actually used, the work would supply a missing theoretical foundation for denoiser-driven iterative methods, allowing them to be analyzed as classical regularization schemes with explicit stability guarantees. The combination of adaptive stepping and early stopping to control semi-convergence is a practically relevant contribution, though its theoretical coverage of modern learned denoisers remains the key open point.

major comments (3)
  1. [§3, Theorems 3.1–3.3] §3 (convergence analysis, Theorems 3.1–3.3): the stability and convergence proofs require the averaged denoiser operator to be firmly nonexpansive (or to satisfy an equivalent fixed-point condition). This property is verified for the TV proximal operator but is neither stated nor demonstrated for the TNRD learned denoiser or the median filter employed in the experiments of §5. Consequently the theorems do not apply to the reported numerical results.
  2. [§4] §4 (adaptive step-size and stopping rule): the a-posteriori stopping criterion is asserted to prevent semi-convergence, yet the proof supplies no explicit discrepancy principle or residual bound that would guarantee the stopping index remains finite and independent of the noise level for denoisers that only approximately satisfy the nonexpansiveness assumption.
  3. [§5] §5 (numerical experiments): the tables and figures present reconstruction errors for three denoisers but contain no baseline comparisons against established RED or proximal-gradient methods, nor any statistical error bars or multiple noise realizations, weakening the claim that the adaptive scheme improves both accuracy and efficiency.
minor comments (2)
  1. [§2] Notation for the averaged denoiser (e.g., the operator D_α) is introduced without a clear reference to its precise definition in the preceding RED literature; a short reminder equation would improve readability.
  2. [§5] Figure captions for the deblurring and CT examples do not list the exact noise levels or blur kernels used, making direct reproduction of the reported PSNR/SSIM values difficult.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our manuscript. We address each of the major comments in detail below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§3, Theorems 3.1–3.3] §3 (convergence analysis, Theorems 3.1–3.3): the stability and convergence proofs require the averaged denoiser operator to be firmly nonexpansive (or to satisfy an equivalent fixed-point condition). This property is verified for the TV proximal operator but is neither stated nor demonstrated for the TNRD learned denoiser or the median filter employed in the experiments of §5. Consequently the theorems do not apply to the reported numerical results.

    Authors: We thank the referee for pointing this out. The proofs in Theorems 3.1–3.3 are derived under the assumption that the averaged denoiser is firmly nonexpansive, which is a standard condition in the analysis of RED-type methods and is used throughout §3. This assumption is verified explicitly for the TV proximal operator. For the median filter, it can be shown to be nonexpansive as it is a proximal operator of a suitable functional in discrete settings, though we did not include the verification. Regarding the TNRD denoiser, as a learned network, it approximately satisfies the condition in practice, but the rigorous proof requires the exact property. We will revise the manuscript to explicitly state the assumption in the theorems and add a paragraph in §5 clarifying that the numerical experiments with median and TNRD serve to demonstrate practical performance, while the convergence guarantees hold when the firmly nonexpansive condition is satisfied. This addresses the concern without altering the main claims. revision: partial

  2. Referee: [§4] §4 (adaptive step-size and stopping rule): the a-posteriori stopping criterion is asserted to prevent semi-convergence, yet the proof supplies no explicit discrepancy principle or residual bound that would guarantee the stopping index remains finite and independent of the noise level for denoisers that only approximately satisfy the nonexpansiveness assumption.

    Authors: The adaptive step-size and a posteriori stopping rule are analyzed in §4 under the firmly nonexpansive assumption on the averaged denoiser. The stopping criterion is based on a discrepancy principle that ensures the residual is controlled, leading to a finite stopping index independent of the noise level when the assumption holds exactly. For denoisers that only approximately satisfy nonexpansiveness, the theoretical guarantee on finiteness may not be strict, but our numerical results indicate that the rule still prevents semi-convergence effectively. In the revision, we will include an explicit statement of the discrepancy principle used and add a note on the limitation for approximate cases, while emphasizing that the main result is for the exact setting. revision: yes

  3. Referee: [§5] §5 (numerical experiments): the tables and figures present reconstruction errors for three denoisers but contain no baseline comparisons against established RED or proximal-gradient methods, nor any statistical error bars or multiple noise realizations, weakening the claim that the adaptive scheme improves both accuracy and efficiency.

    Authors: We acknowledge that the experimental section would benefit from additional comparisons and statistical analysis. In the revised version, we will incorporate baseline comparisons with the original RED method and standard proximal gradient algorithms. Furthermore, we will report results averaged over multiple independent noise realizations, including error bars to indicate variability, and provide a more detailed discussion on how the adaptive scheme enhances both reconstruction accuracy and computational efficiency compared to fixed-step alternatives. revision: yes

Circularity Check

0 steps flagged

No significant circularity; convergence proof is a standard derivation under explicit operator assumptions

full rationale

The paper's central result is a mathematical proof that DDIR with adaptive step-size and a-posteriori stopping forms a convergent regularization method when the averaged denoiser is firmly nonexpansive or satisfies a fixed-point condition. This is derived from the properties of the new regularization functional based on averaged denoisers and standard arguments in regularization theory, without reducing any claimed prediction or theorem to a fitted parameter from the same data or to a self-referential definition. The assumptions are stated as prerequisites rather than derived from the target result, the numerical experiments on median/TNRD/TV denoisers are presented separately as validation, and no load-bearing self-citation chain or ansatz smuggling is required for the proof steps themselves. The derivation remains self-contained against external benchmarks in variational regularization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof must rest on standard properties of proximal operators and denoisers (firm nonexpansiveness, averaged mappings) plus domain assumptions on the forward operator in inverse problems; no new entities are postulated and no free parameters are fitted to data in the abstract description.

axioms (2)
  • domain assumption Denoisers satisfy averaged nonexpansiveness or fixed-point conditions sufficient for the convergence analysis
    Invoked to extend the RED framework and prove stability of the iterative scheme
  • standard math The forward operator in the inverse problem is linear and bounded
    Standard assumption in regularization theory for well-posedness

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [1]

    Arridge, P

    S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch¨ onlieb. Solving inverse problems using data- driven models.Acta Numer., 28:1–174, 2019

    work page 2019

  2. [2]

    Aspri, S

    A. Aspri, S. Banert, O. ¨Oktem, and O. Scherzer. A data-driven iteratively regularized landweber iteration.Numer. Funct. Anal. Optim., 41(10):1190–1227, 2020

    work page 2020

  3. [3]

    Aspri, Y

    A. Aspri, Y. Korolev, and O. Scherzer. Data driven regularization by projection.Inverse Problems, 36(12):125009, 2020

    work page 2020

  4. [4]

    Bajpai and A

    H. Bajpai and A. K. Giri. Graph laplacian assisted regularization method under noise level free heuristic and statistical stopping rule.arXiv preprint arXiv:2601.12792, 2026

    work page arXiv 2026

  5. [5]

    Bajpai, G

    H. Bajpai, G. Mittal, and A. K. Giri. Stochastic data-driven bouligand–landweber method for solving non-smooth inverse problems.J. Inverse Ill-Posed Probl., 33(2):153–170, 2025

    work page 2025

  6. [6]

    Bajpai, G

    H. Bajpai, G. Mittal, and A. K. Giri. On the convergence of iterative regularization method assisted by the graph laplacian with early stopping.SIAM J. Imaging Sci., 19(2):643–676, 2026

    work page 2026

  7. [7]

    Barbano, J

    R. Barbano, J. Antoran, J. Leuschner, J. M. Hern´ andez-Lobato, B. Jin, and Z. Kereta. Image reconstruction via deep image prior subspaces.Transact. mach. learn. res., 2024

    work page 2024

  8. [8]

    Bianchi, D

    D. Bianchi, D. Evangelista, S. Aleotti, M. Donatelli, E. L. Piccolomini, and W. Li. A data- 26H. BAJPAI, A. K. GIRI, T. JAHN AND A. JHA dependent regularization method based on the graph laplacian.SIAM J. Sci. Comput., 47(2):C369–C398, 2025

    work page 2025

  9. [9]

    Bianchi, G

    D. Bianchi, G. Lai, and W. Li. Uniformly convex neural networks and non-stationary iterated network tikhonov (inett) method.Inverse Problems, 39(5):055002, 2023

    work page 2023

  10. [10]

    Buades, B

    A. Buades, B. Coll, and J.-M. Morel. A review of image denoising algorithms, with a new one. Multiscale Model. Simul., 4(2):490–530, 2005

    work page 2005

  11. [11]

    Chambolle

    A. Chambolle. An algorithm for total variation minimization and applications.J. Math. Imaging Vis., 20(1):89–97, 2004

    work page 2004

  12. [12]

    Chen and T

    Y. Chen and T. Pock. Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration.IEEE Trans. Pattern Anal. Mach. Intell., 39(6):1256–1272, 2016

    work page 2016

  13. [13]

    Clason and V

    C. Clason and V. H. Nhu. Bouligand–landweber iteration for a non-smooth ill-posed problem. Numer. Math., 142(4):789–832, 2019

    work page 2019

  14. [14]

    Cohen, M

    R. Cohen, M. Elad, and P. Milanfar. Regularization by denoising via fixed-point projection (red-pro).SIAM J. Imaging Sci., 14(3):1374–1406, 2021

    work page 2021

  15. [15]

    Dabov, A

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

    work page 2080

  16. [16]

    Daubechies, M

    I. Daubechies, M. Defrise, and C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.Comm. Pure Appl. Math., 57(11):1413–1457, 2004

    work page 2004

  17. [17]

    Ebner and M

    A. Ebner and M. Haltmeier. Plug-and-play image reconstruction is a convergent regularization method.IEEE Trans. Image Process., 33:1476–1486, 2024

    work page 2024

  18. [18]

    M. Elad, B. Matalon, and M. Zibulevsky. Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization.Appl. Comput. Harmon. Anal., 23(3):346–367, 2007

    work page 2007

  19. [19]

    H. W. Engl, M. Hanke, and A. Neubauer.Regularization of inverse problems, volume 375. Springer Science & Business Media, 1996

    work page 1996

  20. [20]

    Fannjiang and T

    A. Fannjiang and T. Strohmer. The numerics of phase retrieval.Acta Numer., 29:125–228, 2020

    work page 2020

  21. [21]

    M. Hanke. Accelerated landweber iterations for the solution of ill-posed equations.Numer. Math., 60(1):341–373, 1991

    work page 1991

  22. [22]

    Hanke, A

    M. Hanke, A. Neubauer, and O. Scherzer. A convergence analysis of the landweber iteration for nonlinear ill-posed problems.Numer. Math., 72(1):21–37, 1995

    work page 1995

  23. [23]

    Harrach, T

    B. Harrach, T. Jahn, and R. Potthast. Beyond the bakushinkii veto: regularising linear inverse problems without knowing the noise distribution.Numer. Math., 145(3):581–603, 2020

    work page 2020

  24. [24]

    Hauptmann, S

    A. Hauptmann, S. Mukherjee, C.-B. Sch¨ onlieb, and F. Sherry. Convergent regularization in inverse problems and linear plug-and-play denoisers.Found. Comput. Math., 25(4):1087– 1120, 2025

    work page 2025

  25. [25]

    Huang, Q

    J. Huang, Q. Jin, X. Lu, and L. Zhang. On early stopping of stochastic mirror descent method for ill-posed inverse problems.Numer. Math., 157(2):539–571, 2025

    work page 2025

  26. [26]

    Huang, G

    T. Huang, G. Yang, and G. Tang. A fast two-dimensional median filtering algorithm.IEEE Trans. Acoust., Speech, Signal Process., 27(1):13–18, 1979

    work page 1979

  27. [27]

    Ieee.A non-local algorithm for image denoising, volume 2, 2005

    work page 2005

  28. [28]

    Jahn and B

    T. Jahn and B. Jin. On the discrepancy principle for stochastic gradient descent.Inverse Problems, 36(9):095009, 2020

    work page 2020

  29. [29]

    Jahn and B

    T. Jahn and B. Jin. Early stopping of untrained convolutional neural networks.SIAM J. Imaging Sci., 17(4):2331–2361, 2024

    work page 2024

  30. [30]

    Jin and P

    B. Jin and P. Maass. Sparsity regularization for parameter identification problems.Inverse Problems, 28(12):123001, 2012

    work page 2012

  31. [31]

    Kaltenbacher, A

    B. Kaltenbacher, A. Neubauer, and O. Scherzer.Iterative regularization methods for nonlinear ill-posed problems. Walter de Gruyter, 2008

    work page 2008

  32. [32]

    H. Li, J. Schwab, S. Antholzer, and M. Haltmeier. Nett: solving inverse problems with deep neural networks.Inverse Problems, 36(6):065005, 2020

    work page 2020

  33. [33]

    S. Lunz, O. ¨Oktem, and C.-B. Sch¨ onlieb. Adversarial regularizers in inverse problems.Advances in neural information processing systems, 31, 2018

    work page 2018

  34. [34]

    Milanfar

    P. Milanfar. A tour of modern image filtering: New insights and methods, both practical and theoretical.IEEE signal processing magazine, 30(1):106–128, 2012

    work page 2012

  35. [35]

    V. Morozov. On the solution of functional equations by the method of regularization.Doklady Akademii Nauk SSSR, 167(3):510, 1966

    work page 1966

  36. [36]

    Natterer.The mathematics of computerized tomography

    F. Natterer.The mathematics of computerized tomography. SIAM, 2001

    work page 2001

  37. [37]

    Natterer and F

    F. Natterer and F. W¨ ubbeling.Mathematical methods in image reconstruction. SIAM, 2001

    work page 2001

  38. [38]

    Obmann, L

    D. Obmann, L. Nguyen, J. Schwab, and M. Haltmeier. Augmented nett regularization of inverse problems.J. Phys. Commun., 5(10):105002, 2021. DDIR: DENOISER DRIVEN ITERATIVE REGULARIZATION27

    work page 2021

  39. [39]

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

    work page 2018

  40. [40]

    R. T. Rockafellar and R. J.-B. Wets.Variational Analysis. Springer Science & Business Media, 2009

    work page 2009

  41. [41]

    Romano, M

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

    work page 2017

  42. [42]

    Scherzer, M

    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen.Variational methods in imaging, volume 167. Springer, 2009

    work page 2009

  43. [43]

    S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg. Plug-and-play priors for model based reconstruction. In2013 IEEE global conference on signal and information processing, pages 945–948. IEEE, 2013

    work page 2013

  44. [44]

    C. R. Vogel.Computational methods for inverse problems. SIAM, 2002

    work page 2002

  45. [45]

    H. Wang, T. Li, Z. Zhuang, T. Chen, H. Liang, and J. Sun. Early stopping for deep image prior.Transact. mach. learn. res., 2023, 2023

    work page 2023

  46. [46]

    Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity.IEEE Trans. Image Process., 13(4):600–612, 2004

    work page 2004

  47. [47]

    Zhang, W

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

    work page 2017

  48. [48]

    Z. Zhou. On the convergence of a data-driven regularized stochastic gradient descent for non- linear ill-posed problems.SIAM J. Imaging Sci., 18(1):388–448, 2025. Appendix A. Additional experimental results.In this appendix, we provide supplementary numerical evaluations to further validate the performance of the proposedDDIRmethod. 28H. BAJPAI, A. K. GIR...

    work page 2025