Title: Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Imaging Inverse Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: Poisson inverse problems

plug-and-play methods

cocoercive conservative

Helmholt decomposition

global convergence
Abstract: Plug-and-play (PnP) methods with deep denoisers have shown impressive results in imaging problems. They typically require strong convexity or smoothness of the fidelity term and a (residual) non-expansive denoiser for convergence. These assumptions, however, are violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a cocoercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising performance. By leveraging the generalized Helmholtz decomposition, we introduce a novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to ensure cocoerciveness. We prove that CoCo denoiser is a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods to a stationary point of this restoration model is established. Extensive experimental results demonstrate that our approach outperforms closely related methods in both visual quality and quantitative metrics.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to build better denoising priors for image reconstruction in Poisson inverse problems while maintaining a proximal structure that provides strong theoretical guarantees on convergence. Specifically, relaxes the current strong convergence requirements for denoisers for non-smooth likelihoods (such as those in Poisson inverse problems) in order to be less restrictive, potentially allowing better denoising performance. To this end, the paper introduces a new assumption, called $\\gamma$-cocoercivity, which can be made weaker than firm non-expansiveness to allow more freedom for denoiser learning. Further, it requires that the denoiser be a conservative operator in order to avoid wasteful optimization steps. With these assumptions, the authors prove that the denoiser is a proximal operator of an $L$-Lipschitz function. In order to encourage such cocoerciveness and conservativeness properties, they introduce appropriate regularization terms in the denoiser training. They further prove the global convergence of ADMM and proximal gradient descent using this trained denoiser to a stationary point of the Poisson inverse problem. Results on photon-limited inverse problems show the benefits of designing less restricted denoisers while maintaining a guaranteed convergence structure.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The idea of relaxing the current requirements on denoisers while still maintaining their proximal nature, making them interpretable as solutions to a minimization problem and providing strong convergence guarantees, is promising.\n- The only essential change made to denoiser training is the addition of the two spectral norm terms, which can be used with any denoising architecture to encourage it to be CoCo.\n- Using a CoCo denoiser is shown to be especially effective for Poisson inverse problems, where the likelihood is not guaranteed to be smooth globally. The competitive or superior results with this method, especially at with low signal, reflect the benefits of this property.\n\nWeaknesses:\n- For most of the theoretical development in the paper, it is assumed that $D_\\sigma$ is a CoCo operator and guarantees for the convergence of CoCo-ADMM and COCO-PGD are derived. However, the training for $D_\\sigma$ only *encourages* CoCo nature, doesn’t *enforce* it. It is possible to check the $\\gamma$-cocoerciveness via the spectral norm $\\|\\|2\\gamma J-I\\|\\|$ during training—however, it is not possible to check if the denoiser is “conservative enough”. The guarantees *require* conservativeness, however—so it is unclear, if even though the norm $\\|\\|J-J^T\\|\\|$ is close to zero, if the guarantee holds enough for it to be useful in reality. A natural next question to ask would be—how much could we relax the conservativeness to still ensure good enough convergence guarantees / proximal nature?\n- In the evaluation on line 270, the paper claims that the PSNR values in Table 1 validate that “cocoercive and conservative properties are less restrictive for deep denoiser”. I’m not sure why this performance improvement can necessarily be thought of as an indication of lesser restriction—the performance improvement could also have been because of the different loss function.\n- Appendix 13 claims that this is useful for real-time applications, even though 256x256 images take 5-10 seconds to process—I’m not sure if this claim makes sense.\n\nWriting suggestions:\n- Line 108: “epansive\" $\\rightarrow$ “expansive”\n- Line 167: Why is there an expectation operator?\n- Lines 204, 275, 283: Does the equation number need to be (18)?\n\nQUESTIONS:\nPlease see the weaknesses above. Specifically, please clarify:\n- the *enforce* v/s *encourage* difference between the theoretical analysis and the training strategy\n- if there is a way to quantify how non-conservative a denoiser can be to still be a “good enough” proximal operator\n- why the improved performance Table 1 validates the decreased restrictiveness.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates plug-and-play (PnP) priors in the case of linear inverse problems with Poisson noise, which gives rise to a data fidelity term that is neither strongly convex nor L-smooth. It proposes to use co-coercive conservative (CoCo) denoisers in a PnP fashion, i.e., using a particular learning denoising network like the proximal operator of a classical optimization approach. CoCo denoisers are those that (implicitly) are the gradient of an underlying cost function and are $\\gamma$-cocoercive (a property slightly stronger than being $1/\\gamma$ Lipschitz). The authors prove convergence of the resulting PnP approach to a stationary point. As the constraints on the proposed denoiser are less restrictive, it yields a (slightly) better performance in the numerical results.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is mostly well-written and well-motivated. The idea to impose the weakest possible constraints on the denoiser to maximize its performance while remaining in the regime of a provably convergent algorithm is very good, and the numerical results support the intuition that there can be gains in performance (although they remain very small). \n\nDespite an extensive paper with a rich supplementary material and new theory, I also see some weaknesses. \n(1) To my mind, the paper jumps a little between great mathematical generality (e.g., talking about Frechet differentials, suggesting that $V$ could be infinite dimensional) and implicitly making assumptions without introducing them, e.g. the Helmholtz decompositions needs a smoothness assumption, any time the transpose symbol is used, I assume we are in a finite dimensional setting or is this denoting the adjoint operator? Is the spectral norm the operator norm in the infinite-dimensional setting? Do all convergence results really hold in the infinite-dimensional setting as well? What assumptions do we need to make about $K$ in this case? If the convergence analysis holds in the finite-dimensional case only, I highly recommend writing the entire paper under this assumption and simplifying all parts that can be simplified. Moreover, the notion of the subdifferential and its Lipschitz continuity (Theorem 2.2.) should be introduced. \n(2) The denoiser is conservative if the Jacobian is symmetric, i.e., $J=J^T$. The authors propose to regularize $||J-J^T||_*$, but it seems like any norm could be used. This is not discussed in the paper. \n(3) In practice, the necessary assumptions are implemented via soft penalties only, such that all provable statements are lost. The demonstration that the assumptions are met is demonstrated \"over 100 patches\". Rather than being met on average, I recommend reporting the highest norm after actively optimizing for an $x$ at which a high norm is attained (similar to an adversarial example). \n(4) The penalties require the computation and subsequent differentiation of the spectral norm of the Jacobian of the denoiser. Given that the denoiser is high-dimensional (mapping images to images), the Jacobian is extremely high-dimensional. While the appendix provides details about the computation, I think it is important to include a short statement on the computational expense of the training of the denoiser. \n(5) Given the small differences in performance of the different methods, it is important to confirm that all training procedures are identical and to possibly report standard deviations.\n\nQUESTIONS:\n- Please address the question of the setting (finite vs. infinite-dimensional) of (1) under weaknesses and possibly adapt the paper accordingly.  \n- Please answer the question why you chose the spectral norm to enforce symmetry of J. \n- Do the values in table 1 (right) change if one optimizes for a point $x$ at which the norms are high?\n- How large is the computational overhead in training the denoisers with spectral penalties involving its Jacobian?\n- Have all (competing) methods been fine-tuned equally? Did you consider multiple runs with different random seeds?\n\nLIMITATIONS:\nI think two limitations could be stressed (as they do not appear in Section 6).\n- All convergence guarantees are lost when only using a soft penalty on the denoiser. \n- The fact that the approach (with strictly enforced constraints) is nice, but the interpretability gain of minimizing a cost function seems limited. The approach converges to a stationary point of a nonconvex cost function that involves a regularizes of which we only know the prox-operator (parameterized as a complex neural network). Advantages from classical (convex) approaches (such as error estimates) are lost.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new class of deep denoisers, called CoCo denoisers, for solving Poisson inverse problems via plug-and-play (PnP) methods. Unlike prior works that require restrictive non-expansiveness, the proposed CoCo denoisers are both cocoercive and conservative, making them more flexible while still ensuring theoretical convergence. The authors introduce a novel training scheme using spectral and Hamiltonian regularization to enforce these properties. Theoretical analysis shows CoCo denoisers are proximal operators of weakly convex functions, and experiments demonstrate strong performance on Poisson image restoration tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n(1) The idea of combining cocoerciveness and conservativeness in denoiser design is quite novel. Using Helmholtz decomposition to interpret the denoising behavior adds a fresh perspective that's both theoretically grounded and intuitive.\n\n(2) The authors provide solid theoretical results, proving that their denoiser acts as a proximal operator of a weakly convex function. This gives convergence guarantees even under the nonconvex and nonsmooth setting of Poisson inverse problems.\n\n(3) The experimental results are convincing. On several Poisson imaging tasks, the proposed methods outperform strong baselines like DPIR and Prox-DRS in both PSNR and SSIM.\n\nWeaknesses\n\n(1) While the paper proposes cocoerciveness as a relaxed alternative to non-expansiveness, it is unclear why this particular relaxation is theoretically sufficient and practically better. What exactly fails when the operator is not cocoercive, and how tight is this condition?\n\n(2) The theoretical guarantee that the CoCo denoiser is a proximal operator relies on both cocoerciveness and exact conservativeness (i.e., symmetry of the Jacobian). In practice, both are only enforced softly during training. Does this soft enforcement suffice for convergence? What happens when these conditions are only approximately satisfied?\n\n(3) The use of Helmholtz decomposition to interpret denoiser geometry is elegant, but relies on assumptions that may not hold in practice. Is the decomposition stable across different architectures and noise levels? How reliable is the connection between the anti-symmetric part and “useless” Hamiltonian directions in high-dimensional deep networks?\n\n(4) The convergence analysis (e.g., in Theorem 4.1) introduces a parameter $t<t_0(\\gamma)$ where $t_0$ is the root of a cubic equation. However, the paper does not give much intuition on what range of $t$ works best in practice, or how sensitive the algorithm is to the choice of $\\gamma$ and $t$. A more detailed discussion or ablation would help.\n\n(5) Although the paper presents a rich and rigorous theoretical framework, the presentation is overwhelmingly formal and math-heavy, with long sections dominated by formulas and lemmas. It may be very difficult for readers—especially those not from optimization theory—to grasp the main ideas. Some intuitive explanation, diagrams, or simplified takeaways would be highly beneficial.\n\nQUESTIONS:\nMy concerns are outlined in the Weaknesses section. To be responsible, I must admit that I am not fully familiar with some of the theoretical components presented in this paper, and thus I am not in a position to make a definitive judgment. To be cautious, I am currently assigning a borderline rating. However, if the authors can address some of my questions during the rebuttal phase, I would be open to raising my rating.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work is part of the broader topic of constraining denoisers used in plug-and-play algorithms for solving inverse problems in order to recover convergence guarantee.\nPrevious convergence guarantees for PnP methods often come with constraints on the non-expansiveness and the conservativeness of the learned denoiser, with a price on the empirical performance compared to unconstrained denoisers.\nBesides, most existing work assume gradient Lipschitzness of the datafit which is not applicable regarding Poisson noisy degradations with non smooth KL datafit.\nThe goal of this paper is to replace the non-expansiveness enforcement by a weaker cocoercivity constraint while having a setup that can effectively handle Poisson inverse problems.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nThe overall topic of recovering classical optimization convergence guarantees by enforcing some well-chosen properties on the learned denoiser is interesting. Among them, cocoercivity is a new and appropriate choice. The emprical results do improve on the other mtehods for Poisson restoration. \n\n\n**Weaknessess**\n\nWhile the general idea of the paper is novel and interesting; I feel that at the end it reduces to some soft enforcement of the constraints on the denoisers which are already seen in the literature:\n- cocoercivity comes down to relaxing by a parameter $\\gamma$ the firmly non-expansiveness soft contraint of Terris and al [66]\n- enforcing conversativeness of the denoiser (i.e. having a denoiser which is a gradient) can also be found in the literature (RED, Gradient Step denoisers)\nBesides, the proof techniques for convergence strongly rely on the prox characterization of [22] which has already been used a lot in Hurault's works.\n\nQUESTIONS:\n1. A first thing that particularly struggles me is the absence of reference to the paper  Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems (Hurualt and al.). While the methods from this paper do are cited, the reference is lacking. Besides, I think the positionning of this paper should be clarified: as far as I understand, Hurault's paper also use a conservative denoiser (it is modeled as a gradient by design) and show that their denoiser is a (Bregman) prox. The main difference is that what is done in this paper is that it uses Gaussian denoisers and classical Euclidian prox (compared to Bregman prox): can the authors provide more comments on their method compared to the choice done in Hurault's work which uses a more well suited Bregman geometry.\n\n2. I may have missed something but is the Helmholtz decomposition really needed ? I feel like the result needed for convergence guarantees is just Proposition 2. of [22] which basically states that being the prox of a weakly convex function is equivalent to be Lipschitz + be some gradient of a convex potential. I don't see how Helmholtz decomposition relates with being the gradient of a *convex* potential.\n\n3. As the authors leverage prox characterization of [22], the authors should be able to compute the implicit regularization which would give the authors a way to plot the objective function decrease (equation (15)) during the iterations of the algorithm.\n\n4. On the algorithms:\n-  CoCo-PEGD: the authors actually smooth the datafidelity term (ensuring it has Lipschitz gradient): this seems quite contractory regarding the first motivation of the work which is to handle non-smooth datafits.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-07-03"}