Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Imaging Inverse Problems
Authors: Deliang Wei
Peng Chen
Haobo Xu
Jiale Yao
Fang Li
Tieyong Zeng
Venue: NeurIPS 2025 poster ()
Keywords: Poisson inverse problems
plug-and-play methods
cocoercive conservative
Helmholt decomposition
global convergence
Primary area: optimization
TL;DR: This paper studies convergent plug-and-play methods with cocoercive conservative denoisers for Poisson inverse problems.

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.

Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Imaging Inverse Problems
Deliang Wei
Peng Chen
School of Mathematical Sciences, East China Normal University, Shanghai 200241, China 52215500006estu ecnu. edu. cn
School of Mathematical Sciences, East China Normal University. Shanghai 200241, China 522655000058stu ecnu. edu. cn
Haobo Xu School of Mathematical Sciences, East China Normal University. Shanghai 200241, China 52275500009estu, ecnu, edu. cn
Jiale Yao
School of Mathematical Sciences, East China Normal University, Shanghai 200241, China 522855000198stu ecnu. edu.
Fang Li" School of Mathematical Sciences,
Tieyong Zeng
Institute for Advanced Study,
Key Laboratory of MEA (Ministry of Education). Shanghai Key Laboratory of PMMP,
Beijing Normal-Hong Kong Baptist University,
Zhuhai 51 519000, Guangdong, China School of Mathematics and Statistics, Guangzhou Nanfang College, Guangzhou 510970, Guangdong Province. China tieyongzengêuic edu. cn
East China Normal University,
Shanghai 200241, China fli 1iimmathh.. ecnu. edu. cn
Abstract
Plug-and- alng-and-play (PnP) methods with deep denoisers have shown impressive results imaging problems. They typically require strong convexity smoothness of the fi- delity term and (residual) non-expansive denoiser for convergence. These assump- tions, however, violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a co- coercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising performance. By leveraging the generalized Helmholtz decomposition, introduce novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to encourage cocoerciveness. We prove that CoCo denoiser a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods 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. A test code is provided for reproducibility? ?
"Corresponding author https://ithub.com./ Com/Fiezzzizzz/CoooPPP
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Introduction
Image restoration is fundamental task in the computer vision field. Although most works assume that the noise follows Gaussian distribution 43 12] under low-light condition, such as hyperspectral imaging [53] 17,, medical imaging [67], and limited photon imaging [3] images are inevitably
corrupted by Poisson noises. In this paper, we focus on solving Poisson inverse problems by Plug-and-Plas (PnP) methods.
The following formula describes the Poisson corruption procedure:
f Poisson(pKu) u)/p,
where u is the potential image, f is the noisy image, the peak value is the average number of photons recieved per pixel, and K a known linear operator such as identity blur kernel, or Radon transform. A small corresponds to a severe noise environment. In order to recover u from f, a variational approach considered:
arg min F(u) + G(u), G(u) = A(1, Ku flog Ku), weV
(2)
where V is the underlying Hilbert space endowed with the inner product (-, F' denotes the prior regularization term, and the data fidelity is the Bregman divergence, with > being the balancing parameter. Typical choices for F includes total variation [59] and its variants [33 [1], weighted
nuclear norm [23], and group- based low rank prior [44]. First order methods are employed to find é, such the proximal gradient descent method (PGD):
k+1 Prox (u* - B - VG(""))
(3)
and the alternating direction method with multipliers (ADMM) 10
++1 Prox & (ot b²),
k+1 Prox e (u + b*),
(4)
3
b++1 = 6² + x+1 gh+1
where B 0 is balancing parameter. For closed, convex and proper (CCP) function F
V + R = (-00, oo). the proximal operator Proxf V - V is defined as: Proxf(y)
arg min F(x) + b f|l yl
zEV
Plug-and-play methods. By replacing Prox g (.) with off-the-shelf Gaussian denoiser D (+).) where a denotes the denoising strength, Venkatakrishnan et al. introduced the plug-and-play ADMM (PnP-ADMM) algorithm [69]. Since then, PnP methods have achieved surprisingly remarkable recovery results in many inverse problems, including bright field electron tomography (64), diffraction
tomography 65.. camera image processing [25] low-dose CT imaging 9 72). Poisson image
denoising [3 27 deblurring [36], inpainting
77 and super-resolution 37 32
Challenges. Despite the impressive recovery performance, the convergence analysis of PnP methods remains challenging, especially in the context of Poisson inverse problems. The standard convergence of ADMM and PGD relies on the convexity of both F and G. However, convex prior term F often leads to limited recovery performance. When F is weakly convex, additional assumptions on G, such strong convexity or smoothness, required. Unfortunately. these assumptions do not hold in Poisson inverse problems. Moreover, in general, a deep denoiser is not a proximal operator, which further complicates the situation.
In order to get a proximal Do. should satisfy some Lipschitz propertics such as non-expansiveness, and be the (sub)gradient of some CCP potential ó [47 222 In the following, briefly discuss approaches to satisfy these conditions, as well as some related convergent PnP methods.
Lipschitz property. There are mainly two methods to ensure the Lipschitz property: real spectral normalization (RealSN) and spectral regularization (SR). Ryu al. enforced contractive I D, by applying RealSN to the denoiser, which normalized the spectral norm of each layer [60]. However, RealSN is computationally expensive, and was specifically designed for denoisers with cascade residual learning structures, such as DrCNN 774, making it unsuitable for other networks like UNet
[58 A more flexible approach SR technique introduced by Terris et al. (66). order to get
firmly non-expansive D,, they added spectral term min(1 2 1 }|+} to the original loss

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function, such that [2J-II| S 1, where J = VD, is the Jacobian matrix of Do, and e E (0. 1) pre-defined parameter. With this spectral constraint, D, is fimmly non-expansive, and thus becomes the resolvent of some implicitly maximally monotone operator (RMMO), incorporated into the Mauglas-Racffrre splitting (DRS) method. RMMO-DRS only requires a convex fidelity, making it suitable for Poisson inverse problems. Inspired by SR. Wei et al. proposed to train strictly pseudo-contractive (SPC) denoiser, termed SPC-DRUNet [71]. By incorporating the Ishikawa process and half-quadratic splitting (HQS), they developed the PnPI-HQS method. The convergence of PnPI-HQS only requires a convex fidelity term G. which makes it suitable for Poisson problems. SPC is much weaker than firm or residual con-expansiveness. However, SPC-DRUNet is in general not a proximal operator, and PnPI-HQS does not solve any optimization problem.
Conservativeness. To ensure the conservative property, many works attempt to explicitly define a prior function Φ such that the denoiser its gradient or proximal operator. Romano et al. proposed the regularization by denoising (RED) 57 The RED prior term takes the form of ((x)) 2 (2, x Da (z)). Under the assumptions that D, locally homogeneous, and that V D, is symmetric with spectral radius less than one, Romano et al. proved that D,, = 1 Vo. Yet the assumptions might be impractical for deep denoisers as reported by Reehorst & Schniter [54). Instead of training a Gaussian denoiser D,,, Cohen et al. [66 parameterized e with neural network. D, Vé. Unfortunately, as verified by Salimans& & Ho [1], and Hurault et al. [28]. directly modeling & with a neural network leads to poor performance. To tackle this, Hurault et al. [28] introduced the gradient step (GS) denoiser, where D, = V with d(x) = }/1==11
Go (z). go (x) = ||= - No (z)]1², where N₂ is a neural network. Then D, Vg = N₂ + J (I-No), where JN is the Jacobian matrix of No. No uses the light DRUNet architecture (73.. and such D,7 is called GS-DRUNet. Following this, they then proposed ProX-DRUNet, which trains a GS-DRUNet with a non-expansive residual I - D2 via SR [29]. They proved that PRO-DD acts as a proximal operator of some weakly convex prior F. The DDS algorithm is given by plugging L D. +(1 (11-14 L) as the denoiser into DRS. Prox-DRS converges for proper closed fidelity G with L E [0, 0.5), making it applicable for Poisson inverse problems. However, this great work still requires non-expansive residual, which alters the denoising performance, especially for large noises. To apply GS-DRUNet in Poisson inverse problems, Hurault al. proposed to train Bregman denoiser to remove Gamma noise (27]. Two convergent algorithms B-RED and B-PnP are derived. However, experimental results indicate that the Gamma denoiser struggles to remove large Gamma noise, thereby limiting the restoration performance.
Motivations. As discussed above, in order a get a proximal denoiser, it needs to be Lipschitz and conservative.
Typical Lipschitz conditions, such as non-expansiveness and residual non-expansiveness, are restric- tive and lead to compromised performance. Weaker assumptions, like (strictly) pscudo-contractive conditions, can not guarantee proximal denoiser. This limitation motivates us to explore a more suitable assumption that ensures the denoiser remains proximal while maintaining good denoising performance.
For the conservativeness, existing approaches typically construct an explicit potential function ø. While this idea is intuitively appealing and convenient for optimization, it raises question: Is there any alternative approach to promote the conservativeness, without requiring an explicit prior, by directly regularizing the denoiser, without significantly changing its network structure?
Contributions. To address the issues outlined above, this paper introduces a novel training strategy for learning a proximal denoiser. Leveraging the SR technique and the generalized Helmholtz decomposition, we propose to train a cocoercive conservative (CoCo) denoiser. CoCo denoiser proves to be proximal operator of some implicit non-convex prior function. Cocoerciveness is weaker constraint on the denoiser, compared to existing constraints like firm or residual non-exxansiveness. As result, the CoCo denoiser not only achieves superior denoising performance but also enhances PnP recovery in Poisson inverse problems. Overall, our main contributions are fourfold:
We introduce novel assumption, cocoercivenes,, for the deep denoisers. Cocoerciveness is strictly weaker assumption than non-epansive type assumptions, and therefore, less restrictive for denoisers. We shed new light the conservative assumption. by studying the denoising geometry. Using the generalized Helmholtz. decomposition, a denoiser is implicitly decomposed into a conservative part and Hamiltonian part. We show intuitively and rigorously that an ideal denoiser should be conservative with no Hamiltonian part.
We prove that a CoCo denoiser is proximal operator of some implicit prior. An effective training
3

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strategy is proposed to promote these properties. A Poisson inverse model with an implicit prior is derived. The global convergence of PnP methods with CoCo denoisers is established.
2
CoCo denoisers
In this section, we propose CoCo denoisers. First, we introduce the cocoercive assumption, and its spectral distribution on the complex plane. Next, by the generalized Helmholtz decomposition, we show intuitively that an ideal denoiser should be conservative, with no Hamiltonian part. Then, we prove that a CoCo denoiser D, is the proximal operator of some weakly convex and smooth prior function F + R == R U {oo}, D = Proxg. Finally, a restoration model with an implicit weakly convex prior given.
2.1 Cocoercive denoisers
Let V = R" be a real Hilbert space with inner product (s,)). and the induced norm || = v (xr, v). Let D V - V be an operator An operator D : V - V is said to be 7-cocoercive (7 E [0, oo)), )), if
Yx, y E V, there holds:
v D(x) D(w)) 2 >1 D(x) D
Cocoereive operators are important class of monotone operators. Many operators cocoercive operators. Below are two typical cocoercive operators:
D is firmly non-expansive: (Vz, y E V)
(ar D(x) - D(v)) 2 DD(x) - D
(6)
Such D is 1-cocoercive.
D is residual -on-aaaansive: (Vx,y V) [I (x) (I- D) (v)l S || -yl--
Such D is 0.5-cocoereive.
When y 0, cocoercive assumption make denoisers Lipschitz: by
inequality,
(x y, D(x) Dvv) | alll D(x) Dly)
Therefore, by 5). D(z)) (v)i < + |z y/1. that is, D is 1 -ippcchit When y 0.5, both the operator and its residual part expansive. In general, smaller y corresponds less contraint, and therefore a more powerful denoiser.
Spectral analysis on D. Note that (Va, y E V)
5) can be equivalently transformed into the following inequality:
(| D 1) (Ir) (2> D 1) (v)ll || vll-
Based on the mean value theorem [8 is transformed into:
|| J (z) ||. - 11 Vr E V,
(10)
where [(x) is the Fréchet differential, that is the Jacobian matrix when V is finite dimensional, of D the point æ. and || ||, denotes the spectral norm. Since the spectral radius p(-) is always no larger than the spectral norm || ll.: we have that:
p(2 [(x) 1) < Vr V...
(11)
Pleasee note that, although we limit V to be finite dimensional for easy understanding, many theoretical analysis hold general infinite dimensional real Hilbert space. To clarify, the spectrum set of an operator in a finite space is just the eigenvalue set. The transpose of a matrix in a finite dimensional space corresponds to the adjoint operator in an infinite space. || ||- is the spectral norm in a finite space, and denotes the operator norm in an infinite space. The degradation operator K in an infinite space assumed to bounded linear operator, that E B[]]
An operator in general may not be single-valued However, since the operator D in this paper is a denoiser. its output is unique given fixed input. Thus, only consider the single-valued operator here.

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(II)
Figure 1: Left: Spectrum distributions of the Fréchet differential (Jacobian matrix) the complex
plane under different assumptions. (a) Firmly non-e non-expansiveness Sp(J) c {2 E C [2e - 1 S 1};
(b) Non-expansiveness, Sp(J) C (= C: : |=| S 1): (c) Residual non-expansiveness, Sp(J) C {z
Sp(J) C (= E C + 1 < 2}: (e) 0.25- < 1); (f) Conservativeness, Sp(J) C R. In general, larger region means less restrictive assumption. The spectrum of r-CoCo denoisers (7 = 0.25) lies inside the interval (f) (c) [0, 4]. Spectrum outside E corresponds the Hamiltonian part of the denoiser, and does not contribute to the denoising performance.
C - < 1]; (d) 2 3-strictly pseudo-contractivenes.
cocoerciveness, Sp(J)
{= E C: |0.5z
Right: two-dimensional illustration of the Helmholtz decomposition of denoiser D. denotes the clean image point, denotes the Gaussian noise, and z + & denotes the noisy image. The arrow "->" represents the denoising direction. (i) Denoising field (ii) Conservative field De. (iii) Hamiltonian field D,,.
Let Sp(-) denote the spectrum set, that is, the eigenvalue set when V is finte dimensional:
Sp(J) = f= e C: :I - is not invertible).
(12)
11 implies Sp(J(x)) C (= - 1}.
-
Lemma 2.1 gives equivalent condition of cocoereiveness in (5p-() and thereby enables the training through SR.
Lemma 2.1 (Proof Appendix A.I.p. Let D
+ V, and VD be its Fréchel differential. Then D) is 7-cocoercive (7 E [0, 2o)). if and only if 27 ((x - T| S 1 for any E V.
Since the spectral radius no larger than the spectral norm, we are able to plot the spectral distribution on the complex plane C in Fig. A larger region corresponds to less restrictive constraint, and thus a better denoising performance. Figs. (a), (c), (e) show that firm non-expansiveness, and residual non-expansiveness special cases of cocoerciveness. Therefore, cocoeriveness is weaker assumption for deep denoisers, and yields better denoising performance.
2.2 Conservative denoisers
We consider the conservative assumption intuitively. Let D [V[ be denoiser. D is said to be conservative, if there is a potential function Φ : V- RR such that D = Vń. Geometrically. D is mapping from V to V, thus can be viewed vector field.
By the generalized Helmholtz decomposition g 4 any vector field D can be decomposed into conservative field D.. and Hamiltonian field Dr,, such that D, is curl-free and is the gradient of potential function d, and D, is divergence-free:
D D Dr. = Vo. div(D,)
(13)
However, the decomposition D D. Dr. implicit. In order characterize this decomposition. we consider the Jacobian matrix J. The generalized Helmholtz decomposition can be rewritten as J = S A, where S J+ symmetric, and A J_JT is anti-symmetric. S and A correspond to D. and D,, respectively: S = V DD,
In Fig. I we consider a two-dimensional case. I E V is a clean image, & is the Gaussian noise with zero mean and standard derivation a, I + E is the noisy image. ||++ + x = [[][[|]] = 2o the distance between the clean image and the noisy image in V. A denoiser D is expected to reduce the
distance as Fig. (i). That is D(x + {) - xill < || + E æ:| The arrow " "->" in Fig. I denotes
the denoising direction vector 7, 7 D(x - {) (x E 77 can be decomposed into ne and 7/h
5

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Figs. (ii)-ii... Te points to the clean image, and thus represents the correct denoising direction. eircles around z, and D cannot remove noises.
In general real Hilbert space, given point æ; Vy E V, define the Hamiltonian functional H(y) = 2 (²) x=1². When D is Hamiltonian field, S 0. Then, the vector D(y) preserves the level set of H. because:
(D(v) vyH(())) = (D()) z, A (D(() x)) = 0.
Note that when x is clean image, H is a typical loss function for denoiser. Therefore, Hamiltonian field does not contribute to the denoising. Thus we penalize this useless part D₁ by penalizing A. The
following relations are useful:
conservative > D₁ = = A
|| ||+ =
In order to penalize DR. we only need to minimize self-adjoint, Sp(J) C IR, see Fig. (f).
|| _ || Spectrally speaking, when J _
2.3
Characterization on CoCo denoisers
In the context of PnP, the denoiser D, is expected be proximal. Building upon prior findings by [22 (see Appendix[A.2] we present the following Theorem 2.2 A more general characterization on
CoCo denoisers is given in Theorem A.. in Appendix A.3
Theorem 2.2 (Proof in Appendix A.4). Ler Do C² CV B 1 gz. Da satisfies that:
D, is conservative;
Dr is y-cocoercive with (0,
Let D = tD, +(1 I) E (0, 1). Ir holds that:
there exists r-weakly convex function F V + R, r(f) = 3 D4o--p such that (xr
Prox (z), Vx E V:
OF is L-Lipschitz where
L(t) { 3 2 r()), r(t) 3 b-tt
ff and E [0, 1) (Case 1); ift and t E [0, (Case 2).
(14)
Remark 2.3. D is an averaged version of D.- As reported by 29 when D2 is residual non- expansive, PnP with D out-performs PnP with D, many inverse problems. Remark 2.4. If 0.5, D, is residual non-expansive, D' is then residual t-Lipschitz, and ( is I#7 -eeklly convex, with 3 3- being --- This special case recovers the result by [29].
Remark 2.5. By Theorem 2.2 we know that D is a proximal operator weakly convex function F We arrive the following implicit non-convex restoration model:
8
arg min P(u) + G(u; f). C(u; A(1, Ku WEV
log Ku).
(15)
3 Training strategy
Let 0 be the parameter weights in D, to be optimized, π be the distribution of the training set of clean images, and [amin Fmax] be the interval of the noise level.
Based on Lemma 2.T and the pioneer works by 666 3I]. we encourage the cocoerciveness by the SR technique. The spectral regularization term T., takes the form of:
(0) E,, max y ||+, c)
(16)
where 0 is the network weights, e (0, 1) is parameter that controls the constraint. x n, U U'aamin 0 o0, I). J(z { D. + 0).
To encourage conservative denoiser, we propose the Hamiltonian regularization term L5:
L B,. || J ||.-
(17)
Combining [1 and (16), the overall loss function of D, is
Loss(0) E (x + 0) æ111 anLn agL.,
(18)

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01,49 Q2 > O are balancing parameters. The first term in (97 ensures that D, is a Gaussian denoiser, the second tern makes D, conservative, and the third term results in y-cocoercive denoiser. By minimizing Loss(0), properly regularized CoCo denoiser is obtained. Detailed calculation of [1 1)]l+ and || - J || is given in Appendix A.5
4 PnP methods with CoCo denoisers
In order to solve the implicit nonconvex restoration model (15), we first consider PnP-ADMM with CoCo denoisers, termed CoCo-ADMM. Then, we replace the fidelity term G in 15 by its Moreau envelope, and derive the proximal envelope gradient descent method (PEGD) with CoCo denoisers (CoCo-PEGD). The convergence provided.
CoCo-ADMM, We first consider CoCo-ADMM iterations:
w+1 Prox g (1** bb),,
x+1 Dd (u + b²). g+1 b + wktl gktl
(19)
where D' is defined in Theorem 2.2 and 3 = I. The PnP-ADMM algorithm in 19 with y-CoCo denoiser referred as 7--Coo-ADMMM, or CoCo-ADMM for short.
When the denoiser D, E [V is CoCo denoiser satisfying the conditions in Theorem 2.2 and F verifies the Kurdyka-Lojasiewiez (KL) property [ the global convergence of PnP-ADMM in (19) can be established as follows.
+ be coercive weakly convex KL. function in Theorem 2.2 such that D E Prox 3. G + R is lower semi-continuous and convex. y E (0, 1). t E [0, to), where to = to(7) is the positive root of the equation
Theorem 4.1 (Proof in Appendix A.6). Ler
(2 - 27) + 712 + 24l - =
(20)
Then, the sequence {(u*, ,,,,,,))) bb) generated by 19 converges globally to a point (u", v², bb),, and that u² = v" is stationary point of the model (5)
Remark 4.2. The KL property has been widely used to study the convergence of optimization algorithms in the nonconvex and nonsmooth setting [ Many functions, in particular all the real semi-algebraic functions, satisfy this property.
Remark 4.3. We consider several special cases of Theorem 4. Let D, be conservative. When 7 0.5, D, is residual non-expansive, and to R 0.3761. This recovers the case by (27): when 0.25, both D and its residual [ T- D, can be expansive, but 0.5 D, I is non-expansive. In this case, to 0.3333. When 7 1, D. is firmly non-expansive and is proximal operator of some CCP function by Moreau's 's theorem. In this case, to is no more needed to ensure the convergence, since this is the standard case of ADMM. In experiments, we use D (u( + 6*) as the final output.
CoCo-PEGD. Now consider the PGD algorithm with CoCo denoiser. The standard PGD takes the form of:
++1 Prox (u'-B-'VG(u')
(21)
When F is r-weakly convex, VG is Lipsccht,, and 8 8- max Lotr Le }, the iteration 21 converges, see 27
However, the Poisson inverse problems, G is not smooth, because VG is not Lipschitz. In order to apply the PGD algorithm, smooth the fidelity G in [1) and (2 by replacing it with its Moreau envelope "G (a E (0, oo)): "G(u) = min, G(u) + 2a
a L-Lischhitz Since there are already step parameter 3 and balancing parameter À in G, throughout this paper, we set a 1. Now the iteration becomes:
Since is CCP, aG is differentiable 77: V=G(u) = (1 (1- Prox,)), and that V is
u A+I
Prox (u² 3 ---'v'G(u")
(22)
u alternates between a proximal operator Prox g, and a gradient descent step on the envelope
of G. 22 is referred to the proximal envelope gradient descent method (PEGD). Similar to
7

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CoCo-ADMM 19 we replace Prox with D',, and arrive at CoCo-PEGD:
u k+ =D' (u B
(23)
By Theorem 2.2
D'
Prox g. with F being r-weakly convex. Then by the Theorem in 27
CoCo-PEGD converges when f=1 max |+T. 13. This convergence result is summarized in
Theorem 4.4
By Theorem 4.4 when 3 > 1, CoCo-PPGD converges.
Theorem 4.4 (Proof in Appendix A.7]. Let F V R be a coercive r-weakly convex KL. functinn
in Theorem 2.2 such that D' Prox 5. G V IR is CCP y E [0.25, 1). t E (0, 11. 0 < B-1
max
2 ( 14r
}. Then the sequence {u*) generated in 23 converges to stationary point of:
û E arg min F(u) + f).
(24)
wEV
5
Experiments
All the experiments are conducted under Linux system, Python 3.8.12 and Pytorch 1.10.2 with a RTX 3090 GPU.
Training details. For Da,, we select DRUNet (73,, which combines residual learning [24 and UNet architecture [58]. DRUNet takes both the noisy image and the noise level a as input, making convenient for PnP image restoration.
To train a CoCo denoiser, we collect 800 images from the DIV2K dataset 30 as the training set and used batch size of 32 and patch size 128. We add Gaussian noise with randomly generated standard deviation values in the range of [ōmin: max [0, 50] to the clean image, Adam optimizer applied to train the model with learning rate |r 10 4 We set a1 1, 02 0.01, and 0.1 to ensure the regularity conditions. To accurately evaluate the spectral norms (zx) - (x)]l= and [1 --)))=))ll ||,, we use the power iterative method 21 with 30 iterations to ensure the convergence.
Denoising performance. We evaluate the Gaussian denoising performances of the proposed denoiser CoCo-DRUNet. strictly ereudd-contracteo DRUNet (SPC-DRUNet) (7T], resolvent of maximally monotone operator (RMMO) [51] which is firmly non-expansive, GS-DRUNet (28. Prox-DRUNet
29.. the standard DRUNet, FFDNet 775), and DrCNN 774.
The PSNR values are given in Table I A "-cocoercive conservative denoiser is referred to as 7-CoCo- DRUNet. We see in Table that compared with DrCNN, FFDNet, and the regularized denoisers, 0.25-oCC-DRRNNe has competitive denoising performance. This is because that: cocoercive and conservative properties are less restrictive for deep denoisers; denoiser trained with different loss function. Both 0.50-CoCo-DRUNC and Prox-DRUNet are conservative with non-expane -xpaaninssvv residual, and therefore share similar denoising performance.
Table 1: Left: Average denoising PSNR performance of different denoisers on CBSD68, for various noise levels σ; Right: Mean symmetry error J J ||~ (N 1) and maximal values of the norm [2y 1| (N 30) for various noise levels and 7 = 0.50, 0.25.
==
25
1.5
25
40
Norns
FFDNet 33.86
31.18
28.81
DrCNN
33.88
31.20
28.89
DRUNet
4.1c+0
4.2c+1
1.8c+1
|| J
DRUNer
34.14
31.54
29.33
D.50-CeCo-DRUNNe
8.6e-3
2.5e.4
7.1e-4
[ J
RMMO
32,21
29.99
27.87
0. Co25-CoCo-DRUNet
3.1c-4
1.8c-4
3.9c-4
[ ||+
GS-DRUNet
33.56
31.01
28.81
Prox DRUNet
33.18
30.60
28.38
DRUNer
3:285
4.343
6.283
SPC-DRUNet
33.90
31.29
29.10
0.50-CoCo-DRUNet
0.994
0.992
0.972
| J- [|]
0.50 CoCo DRUNer
33.38
30.65
28.25
0.25-CoCo-DRUNet
0.986
0.969
0.982
00.5J
0.25 D35-CCCo-DRUNG
31.38
Assumption validations. In the experiments, the symmetric Jacobian and non-expansive residual are softly constrained by the loss function (97) with trade-off parameters a1, 0r2. We validate the
conditions Table 1 We use the symmetry error || [(x) - J (a)]l. over 100 different patches
with N = 1 as in Algorithm[ for better demonstration. A smaller error denotes higher Jacobian
symmetry. The cocoerciveness validated by calculating || J(x) - x:|-- If | ((x) - xl1- S the denoiser is y-cocoercive.
8

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As shown in Table II DRUNet without regularization terms has an expansive residual part. The regularized CoCo-DRUNet is cocoercive. Besides, the proposed Hamiltonian regularization term indeed encourages symmetric Jacobian: in Table [ we see that Hamiltonian regularization reduces the symmetry error. validates the effectiveness of the proposed training strategy. Ablation study on the parameters (1 and (22 in (97) are provided on Appendix A.16
PnP restoration. We apply CoCo-ADMM and CoCo-PEGD multiple Poisson inverse problems including: photon limited deconvolution, single photon imaging in a real-world low-light setting in Appendix AA.10] low-dose CT reconstruction tasks in Appendix A. Poisson denoising in Appendix A.13] Computational time, performances under extreme conditions, and blind deblur and denoise results, are reported Appendices A..1 A.15 A.17 and A.18 respectiivy..
When K in 2 is not the identity matrix, in general, Proxg has no closed-form solution. Since G is convex, we use ADMM to solve Proxg efficiently. For details, please refer Appendix A.8
We choose ? 0.25 since D.25-CoCo-DRUNet has a satisfactory performance. According to Theorems 4.1 4.4 CoCo-ADMR and CoCo-PEGD are guaranteed to converge with < 0.3333. For both CoCo-ADMM and PEGD, we need to calculate the proximal operator Proxg. We detail the calculations for each task in Appendix A.8 For each method, we fine tune the parameters to achieve the best quantitive PSNR values. The proposed methods are initialized with the observed image, that is 0 vd f,b0 =.
Table 2: Left: Average deconvolution PSNR and SSIM performance by different methods on CBSD68 with Levin's 's 8 kernels and Poisson noises with peak value 50 and 100. Right: Average low-dose CT reconstruction PSNR and SSIM performance by different methods on Mayo's s dataset with Poisson noises with peak value 500, and p 100.
=
100
50
PSNR
SSIM
PSNR
SSIM
P 500
P 100
DPIR
26.51
0.7419
25.38
0,6910
PSNR
SSIM
PSNR
SSIM
RMMO-DRS
25.94
0.7019
25.10
0.6546
FBP
28.76
0.5212
24.10
0.2974
Prox-DRS
25.68
0.6764
25.21
0.6572
DPS
23.65
0.6062
23.10
0.5816
PWLS-TGV
33.16
0.8461
30.43
0.7750
PWLS- PWLS-CSCGR
35.45
0.8909
33.78
0.8534
DifIPIR
24.82
0.6429
24.08
0.6074
UNet
36.93
0.9139
35.19
0.8875
SNORE
26.33
0.7158
25.21
0.6719
WNet
37.12
0.9266
35.98
0.9130
PnPI-HQS
26.42
0.7158
25.61
0.6956
CoCo-ADMM
37.63
0.9403
36.68
0,9194
B-RED
0.6794
23.78
0.6603
CoCo-PEGD
37.72
0.9391
36.43
0.9159
CoCo-ADMM
26.89
0.7358
26.00
0.7026
CoCo-PEGD
26.79
0.7323
25.90
0.6958
(a) Blur
(b) DPIR
(c) RMMO-DRS (d) Prox-DRS
(e) DPS
DifPIR
(g) SNORE
CON HORRE
(h) PnPI-HQS B-RED()) CoCo-ADMM CoCD (1) Clean
(m) PSNR (n) Relative emror
Figure 2: Deconvolution results by different methods on the image 'Butterfly' from Set3e with kernel 2 and p = 50 Poisson noises. (a) Blur image. (b) DPIR, PSNR=25.00d0 (c) RMMO-DRS, PSNR=21.55 1..21.866BB. (d) ProX-DRS, PSNR-22.57dB3 (e) DPS, PSNR=21 .4dd.. (f) Diff- PIR, PSNR=22. PSNR=23.34d0 34dB. (i) CoCo-ADMM, PSNR=25.05 25dB. (k) CoGo- PSNR-25. SENR-25.17dB, (l) Clean image. (m) PSNR curves. (n) Relative error curves. r-axis denotes the iteration number.
(g) SNORE, PSNR=25. 14dB. (h) PnPI-HQS. PSNR=23.C 56dB. (i) B-RED,
Photon limited deconvolution. In this task, K 15 the blur kernel. We use 8 real-world camera shake kernels by (39) Fig. 4 in Appendix A.9
We compare our methods with some close related PnP methods, including DPIR 73 which applies PnP-HQS method with decreasing step size. Please note that DPIR is not guaranteed to converge;

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RMMO-DD DRS [66], which uses the DRS method with a firmly non-expansive denoiser RMMO; B-RED 27. which uses the gradient descent method with a Bregman denoiser; PnPI-HQS [7]]
which uses the Ishikawa HQS method with a strictly pseudo-contractive denoiser, SNORE [5 which proposes the novel stochastic denoising regularization by iteratively adding Gaussian noises.
We also compare two state-of-the-an diffusion-basee methods: DPS [T5] and DiffPIR 77 The
average PSNR and SSIM values on CBSD68 are summarized in Table
E
E
Ela
Eda
Ela
(a) Reflectivity
(b) Photon TV
(c) BM3D-VST
(d) DrCNN
(e) VDIR
(I) VBDNet
E
E
Parh 1955
E
F.S
(g) DPIR
(h) RMMO-DRS
(i) PnPI-HQS
(i) CoCo-ADMM (k) CoCo-PEGD (I) Relative error
Figure 3: Single photon imaging results real-world low light setting by different methods.
We show the visual results in Fig. In Fig. 2 (a), the image 'Butterfly" is severely degraded. In this setting, many methods fail to recover the clear edges, see Figs. (c)- Compared with DPIR and SNORE, the enlarged parts by CoCo-ADMM and CoCo-PEGI are closer to the potential clean image, see Figs. 2 (b), (c), (h), (i).
Conclusion and Limitation
This paper introduces a novel cocoercive conservative assumption on the denoiser. Cocoerciveness is weaker than the existing assumptions, and less restrictive for deep denoiser. Conservativeness is analyzed geometrically by the generalized Helmholtz decomposition on the Fréchet differential. We propose a novel training strategy that incorporates a Hamiltonian regularization term and spectral regularization term, which encourages a cocoercive conservative (CoCo) Gaussian denoiser. Theoretically, CoCo denoiser is proved be a proximal operator of an implicit weakly convex prior function. The global convergence results of PnP methods to stationary point are given. The results can be naturally generalized other inverse problems with a convex fidelity term and an implicit weakly convex prior term. Extensive experimental results demonstrate that the proposed CoCo-PnP methods achieve competitive performance in terms of both visual quality and quantitative measures.
The main limitation of this paper is that the proposed CoCo denoiser exhibits slightly worse Gaussian denoising performance compared to the non-regularized denoiser, see Table 6 This is because the proposed regularized denoiser sacrifices little denoising performance in order to achieve guaranteed theoretical results as stated in Theorems 2.2 4.4 Another limitation is that the proposed learning strategy can only encourage the desired mathematical properties, not enforce them. In experiments, we do observe that sometimes, even after long time spectral regularization training, the trained denoiser still violates the cocoercive property on some particular images. For conservativity. we find that when = 30, the symmetry error compared with standard DRUNet smaller, but not significantly smaller. How to enforce such properties without greatly compromising the denoising performance is still an open problem. Since the paper mainly focuses on the theoretical analysis of PnP methods, will address these limitations in future works.
Acknowledgments
Fang Li is supported by National Key R&D Program of China (Nos. 2021 10A10000),))0, Funda- mental Research Funds for the Central Universities, Natural Science Foundation of Shanghai (No. 227R 4195000, Science and Technology Commission Shanghai Municipality (No. 22D72229014) and Natural Science Foundation of Chongqing, China (No. CSTB2023NSCQ-MSXX00 Zeng is supported in part by BNBU Research Grant (No. UICRO100031, U1CR0700108-25 and UICR0900003) Beijing Normal-Hon Hormal-Hong Kong Baptist University, Zhuhai, PR China.
Ticyong
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A
Technical Appendices and Supplementary Material
Contents
Introduction
2
CoCo denoisers
Cocoercive denoisers
2.2
Conservative denoisers
2.3
Characterization CoCo denoisers
3
Training strategy
4
PnP methods with CoCo denoisers
5
Experiments
Conclusion and Limitation
10
Technical Appendices and Supplementary Material
23
Proof of Lemma 2.]]
24
Characterizations the denoiser
24
A.3
More general characterization on the CoCo denoiser
24
A.4
Proof of Theorem 2.2
................................
25
A.5
Calculation for (2
and
................................
28
A.6
Proof of Theorem 4.1
................................
29
A.7
Proof of Theorem
32
A.8
Proximal operator the fidelity for each task
................................
33
A.9
Detailed Deconvolution results on each kernel
................................
33
A. 10 Single photon imaging
34
A. 11 Ablation study on and in CoCo-PnI
34
12 Low-dose CT reconstruction
34
A.13 Poisson denoising results
35
A. 14 Computational cost
35
15 Performances under extreme conditions
36
A. 16 Ablation study on parameters ai and 022
36
A. Blind Deblur Results
37
A. 18 Blind Denoise Results
38
23

---

A.1 Proof of Lemma 2.1
In order to prove Lemma we will mainly use Proposition 3.1 by 6
Proposition A.1 (Proposition 3.1 by [6). Let C be nonempty open corvex subset of V, let B be a real Banach space, and let D : C + be continuously Fréchet differentiable on C. Then D is non-expansive if and only [(x)]l+ vr C,J(z) = D()).
Now we prove Lemma
Proof. In Lemma 2.1 we let C B V, thus C' is open convex, and B is real Banach space. In
order to prove Lemma 2.1 we only need to prove that y-cocoercive (7 E [0, oo)), if and only Vx, y E V
[1 D- - o (α) (24 D (v)ll = || -yl-
(25)
Note that (25 equivalent
D(z) - D(y)] + || - y/ -- - y, (xx) - D(v)) s || y11,
(26)
and equivalent to
>|1 D(x) - D < fo - y. D(x) - D(v)),
(27)
which means D is y-cocoercive.
A.2
Characterizations the denoiser
The first characterization of proximal operators of convex, closed, and proper functions is due to
Moreau's 's theorem (7).
Theorem A.2
A map D: V -> V is proximal operator of proper, closed, convex function F:V. (+00), if and only if the holds that:
47
there exists a convex,
vb such that D(x) E @b(x), V;
D is nonexpansive, i.e.,
D(x) D(y)l < |a vll, EV.
(28)
22
generalized Moreau's theorem to the proximal operators of potentially nonconvex functions.
Theorem A.3 ((22). Let D V V and L > 0. The following are equivalent:
there is F= V + RU (+x) such that D(x) E <Proxp(x), Vx E V, and x + F(x)
(1 - 1) ||
2 closed and convex:
is L-Lipschitz, and that there exists a closed, convex function vp such that D(x)) e @v)()) Vx V.
When D ECC. [V], Gribonval and Nikolova also provided the following characterization.
Theorem A.4 (22). Let D: V, and D E [V]. The following properties are equivalent:
D is a proximal operator of function F - R (+00):
there exists a convex CV] function vb such that D(x) = Vz
the differential J(z)= = ((x) is symmetric positive semi-definite for all a V.
A.3 More general characterization on the CoCo denoiser
The following theorem is tight use of Theorem A.4
Theorem A.S. Let D, E C* 3 J. D, satisfies that:
D, is conservative;
Dr is ry-cocerrcive with (0, oc).
Then, there exists a function F V + R. such that F is r-weakly convex, where B(1 --), and that (r()) E Prox (I), Vx E V.

---

Note that here is a little abuse of notation here: when 7 > 1, < 0. "r-weakly convexity with negative r" actually means "(-rn-stoongly convexity".
Proof. When Do is 7-cocoercive with ? > 0, by Lemma 2.1 Vr E V. [(a) - -||- < 1. Therefore, (xx) is positive semi-definite for any x E V. Since D. conservative, has no Hamiltonian part, J(x) (x), Vo E Therefore, J(ar) is symmetric semi-positive for any a V. By Poincaré's lemma (see Theorem 6.6.3 in (20)), there exists convex [2[V] function &i such that Do (a) = Vz)( Vx E V.
By Theorem A.4. there exists function F' V - R. such that D2 E Proxg, where we let 3 = # for convenience. Now we prove that F is weakly convex. Recall the resolvent form of proximal operator: Prox- (1+ Given any z, E V. choose arbitrary u, E V. Dr(x) E (I+ (x), Dv(()) E (I (1+ (v), then v) E OF(c).
8(x u) E
(29)
Since D, --cooeerivve, by definition
{I y, (x) Dv(1)) 2 >1 (x - (
(30)
By substituting u, v in 30 have
r - y, u v) 2 > |u - 1
(xr u) (y - 0) +
v), u - v) 2 2 // r/
(3(x u) B(y v) + B(u - v), 1 0) 2 87 1 1
(31)
(3(r - u) - -B(y - v), u - 2 8 -1)
1) m 2
(B(x u) B(y v), u 0) + 7))
o 2 0.
Recall (29). we know that F is r-weklly convex with = 8 -7).
A.4 Proof of Theorem 2.2
Please note that, when 7 2 D. is firmly non-expansive and conservative, and therefore is proximal operator of some convex function. In this case, r(t) could be negative: the "negative weakly convexity" actually means convexity. Since a convex implicit prior function F' is out of interest in this paper, Theorem 2.2 focuses on the weakly convex case, that is, E 1).
Y cocoorcioee Vz, E V, we Proof. Since D, is 7-cocoercive with > 0, D' is naturally ++(1-17 have
(Do(z) - D. (
2
(1' (1-1)11)))00 -1--))))))))---00 -(1 [[]-)]]) (v), v)
2
- -1-))))))00)) (11 1)1)
(01)))(00) o (2)
tDDs, (z) - f(( (1-1))lz y/1 2 > (x) (2) - (1 t)(x y)ll
(32)
Denote D')))) - (v), b y for convenience. Then we have
t(a,b) f(1 - -t]lbb12 (t+ (1 - ())
2 2|||a||2_ 1- - (a, b) + 711
1)
1/
2 2 (t(1 - t) + +1
(33)
Now note that
(a, b) (z) (v(,, 2) + (1 +)| 2112 Since De is y-cocoercive with 7 > 0, we know that D2 is Lipschitz. Therefore, by Cauchy- Schwarz inequality.
(34)
1
(D>(x) - (v).). 1) 5 7 | - 2
(35)
That
(a, b) 711 - ())
< ( (4+1-) ||b112
(36)
S < - 1)|1612 + 7(1 - 1)1111112
25

---

By substituting (6) into
33 we have
(+=)(1tt)(a,bb
2 ?||a|1 7
(a,
2 + 7 1 |all?
(37)
which means that Dz 1 cocoercive. Since D2 conservative, D2 is also conservative. By Theorem A.5 we know that there exists r-weakly convex function F: V + R where
7 7 - 1)) (1- r r= r=)(t)= =
t- t-yt t +
such that D'2 E Prox 4
Now prove that OF is L-Lipschitz. We consider two seperately.
Case 1: t 2 1-2% and t E [0, 1). 2-
In this case, we need to prove that is L(t) = 3 Lipschitz. Given Vz, y E V, choose arbitrary =
u, v, such that, u = (1) E (z)) (v) E (I+) (v), then
3(x - u) E
u) E OF(v).
(38)
Note that a U. y. In order prove OF is - - Lipschitz we need to prove
t²
[| - u) - Bly - 1)
S Lt) u r11 = 32
||u 01?
(1- 1)2
22
113(b 1)
$ 8 (1 - t)2 |aal33
(39)
1
1b- Σ (1 f)2 la|2
Since a = u - --= (x)
(v) = 1(D) (x) Dz(y)) + (1 - t) - -)) 20
= x =.y,
y. we have
a- b- b-b=t(D,(s) Do - t(x y).
(40)
Now we only need to prove
| (x - Do (3)12 2t -0 y/1
f2
s
t)- (z) D2 28(1 -1 (z) 2
Dz()), y) + (1 1) -
--- y/1²),
(41)
which is equivalent to prove
1 - 2t
Dx(x)= - (1)l) (z) Do(v), v)-
(42)
2
Since t
1-27 2-27*
we have that
1 1- 2t
1 1- 1-
2-
2 2 2
27
2 - 21
2 - 1-27
2(2 - 2v) - 2(1 2%)
7
(43)
We already have
> Do Do (v) < (z) - v).
(44)
Thus.
1 2t
(2(x) - (1) < -
D.(1) < (D) (D()), -
2).
(45)
2 - 21
1-2x Case 2: t 2-27
and E [0, 1).
26

---

In this case, we need to prove that OF is L(t) r(t) = B 147-17 Lipschitz. Similarly in (39 we need to prove
|1 - u) - Bly - w]12 s [T)())) u/13= 32 (1 7)22
(t+ - tyy2 ||ue - -11?
((1 ( 1 17)3 1|a1?.
(46)
||b all²
Since a = u 4-- -- (x) D2, = 1(D, Dy) + (1 () - y), b b=bb=2 ----- =3
3/ we have
- b= (-bbt(D)(z) - -D(y)) - 1(x - y).
(47)
Now we only need to prove
(,))) Do 2
21
(v), - 5) + + y11
f1 ((1-7) 7)
(++7-1 ty)3 (e (v)) 2t(1 - -t) (2) Do (y), y) + - -1))t)²11= 011²)
(48)
When = 0, it holds naturally. When f 1), and t S 1-2y it is equivalent prove 2-2y"
|| [D, (z) - Do 12 2(D. (x) (v), y) + || y||²
(1 71-
ty) (+
- - Dr (v). v) (1 +) 81(3)
(++r- -
(49)
which is equivalent prove
- (10 1x- y11
$ (1 ( D.
(1- (1-))) - y/1
(50)
that is to prove
[[t + 1-) [
[z) Do (v)l
-
() 7} (x) (v), - -v)
(51)
s - (1--0000))--((----1))
y/l².
Now we check each coefficient, and estimate each term carefully. For the coefficient of [z) (o(y) we have
2-(1 7) + 0
(52)
For the coefficient of -D.( obviously nossiiiieive Besides, have that
- +2 + - -)) (1)01))))11)))) -
- -2 (t-
7)
(53)
(2 -
Since D, --cocoercive, we have that
[- - 22 - 2>²] (D, DD, x y) -
DD (54)
For the coefficient of || y1 note that when (0, t E 00, and t 5
1-22
we have
2-27'
27
(1- (t + ty)² = (2 - 2) - 27 + 1 2 (2 - 2) 2 27
27 + 1 2 0. (55)
Combining (52
(55) (55) prove 51 we only need to prove that
[2>(1 - 7) (2 - 2² - 2 (x)x) 2y) < - 2)t - 2 + -|]||=--1 y
= -[(2>3 27 - 2
2>
|D_ (z) D. (2)
< <1 1 2)t - 2 +1][| ||11--11?
(56)
Since D, -Lipschit we only need prove that
1
[1273
2 (o)2) 72 27
< (2 - 2)t - 27
(57)
(2y 2) + - 27
< (27 - 2)t - 27 +
This completes the proof.
Please note that any Lipschitz operator must be single-valued Thus, there is actually only one element in (z()), given any E V.
27

---

A.5
Calculation for [1 - T)I| and
||
JJ-J"] ||+
We mainly use the power iterative method 21 calculate the spectral norm of given operator A.
Algorithm 1 Power iterative method
Given oº with ||
= 1, A N
for 1: .N do 2"
A g"
q" 1=
end for Return AN = (g
By Algorithm|| I we can compute the spectral norm of A. Note that in Algorithm we need to calculate the matrix-vecte vectorr product A given any E V.
Given denoiser D,, Vr V, J(x) (D) is the Jacobian matrix at the point To specify, let x = [1 T2, ,.....,,aa Fna T E V = Rh, and = [3n, V.......... Mn] = D (x) be the output after D,,
Then the Jacobian matrix J(x*) given point x² is:
0y1
dy1
0y1
dr
Or
3zn
Oyz
0yz
Oyz
J(x')
dr1
Or
3zn
(58)
:
Oyn
Oyn
Oyn
0x
ixn
2=2"
We start with a = [a1, a....., an JT (æ)), where linear algebra, know that
[U1, U2. Un] is any vector. By basic
Oy
a: =
Σ j=1
Gjj =i= = 11.=12...,,2 n,
(59)
which implies
a a= J= = Oly, v)
(60)
Or
The calculation of b = J(r) is bit more complex. We will use the so-called double back- propagation technique: since b = J(r),, have
Oyk
by Σ
Vk
112,....,
(61)
Or:
wr Let t = (.)).. Thus t, = Σ Oys Dz,
Now consider w [Wi, 02.
wi.
f=1
0tt, v) Ow
Let
then
Olt, v)
عys Or:
2 dw
Σ i=1
U1
wi
(62)
Ir
1=1
The kth element of is:
Oyi Wi Dr:
11
Oyj 0x
2k
Σ U1
U;
Vk = 1.2....,
(63)
Owk
28

---

That is.
J(x)e att, u)
O
((y.m) Or
1)
Vw
IR
(64)
Ow
dw
The AutoGrad toolbox in Pytorch 50 can be:
allows the calculation for B(.)
The pseudo-codes in Pytorch
0.
Calculate J(xVv
def _jacobian_ vec x, v):
w = torch. ones ones_like (x, gequire_grad=rree
torch autograd, gya x, w, create Sreate_graph-Tnne [0]
return torch autograd (t, w, v, create Greate_graph-Tuse [0]
Calculate J^T(x)v
def _jacobian trt__transsss_eeee (yc x, v): return torch. autograd gr, x, v, create_graph=Trug
By 660) and (64. one can calculate --2)l and || - 3"11 ||- by Algorithm
A.6 Proof of Theorem
We will make use of the Lyapunov function Lg for 199 according
40 68
29
8 2 u1l
(()) G(u, B(b, v)
(65)
We will first prove in part that important value for Lş(u, (u. v, b) positive whenever E (0, 40), where in is the positive root of the characteristic equation in (20). Then, we will prove in part 2 that Lg is non-increasing with the iteration number k. Finally, will prove in part that CoCo-ADMM iteration in (19) converges globally to stationary point of (15)
Proof. Let h(t) (2-27)t² + yt2 2y -y where y E (0, 1). Note that is obviously smooth, and h(0) = < 0, ((oo) co. Also note that when t >> 0,
(1()) = (2 1122-249 2+) + 2yl + 27 > 0.
(66)
Therefore, there exists unique to 0, such that h(to) 0. h(t) t e [0, to).
0ift to. and h(t) 0 if
Part 1:
We consider a characteristic value for (u (u, v, b) in 655: 2
1
By Theorem 2.2
if t E [0, 1) and t
2
1-27, 2-27"
we have that r B 1-7
and I. = B 1- Thus we
have
8 r
2
B
3(t 7t)
At
2
3
2 + 7 1r)
(1-t)²
3
t 1-76
212
1
2
t t+ 1 tr
(1 f)2
3
(67)
=
2(1 11) 1/2
(7(1 t)2 2t2
+ (t+
-tt))
= o(t+>-tr(11 t)2 ((2 12-229 2x) + yt² + 2yt - 7)
When 0 < < lo where lo is the positive root of the characteristic equation in 20 holds.
12
0
2
7
29

---

Ift E (0, and S
1-27,
we have that L =
-,,4++4700 t-2t_
Note that in this case, L = S 1-7' $11-1
2-2y"
247-1y"
Thus,
B
r
1
2
2
8
8
8(t
7t)
At²
2
2
2(t + br)
(1 - t)2
(68)
y - ty) (1-))1111)) -t) ((2 27)x²² + gt + 2yt 7)
2 Therefore, also have that when 0 < to,
2
L2 3
0 holds.
Part 2:
Now we prove that (u(, vf b²)) non-increasing. Before that, we show two important formulas.
For *+1 by the first-order optimal condition, we know that
mb = -B - #+1 - -b²) @F(cttt)
(69)
Similarly, for t have
-B( v + b²) aG((ttt
(70)
In order to prove that (u wtt (a) is non-increasing, we decompose , ,b, La(u (u++1 vl1 bh+1) into two parts:
Lu w/ B²) ,²) Lo(uR, w.. bb L (ut ot bb) Lg(ubt1 buk,b") Lu(ut gk+t bb+1),
L Lst oktt1 1, (111
(71)
and estimate them separately follows. By the convexity of G and the iteration form in have
19
Lo(u", vt
o0,bb% b²)
Ls(u++1 ,²,
3 11=² 1/122 - f) + 3(b 2² - utt) +
3
- =1 - 211
2
xt G(u - G f) (-B(ukt] - vt + +b²), -b - + Blu - n+ utt x
"21l** - 8
B
2 butt 1
3
2 0 + B(1 - kt1 u - xt1)
+ (ut - u k+1 u² + nt - 20
2
4 + b+1
3(u -t1 wt ++1 bb +
2
Ilue - 11
(72)
By the r-weakly convexity of F, Vx,y E V. E (F(y), we have:
F(x) F(y) 2 (fy, x v)
r | y/12
(73)
2
By the L-Lipschitz property of OF, Vz, y V. V. fy E 0F()), we have:
L
F(z) F(v) - (fy, x v)
æ v112.
(74)
2
30

---

Combining (73 and (7), we can obtain that
Lo(uttl rk b²) Lu(ut vh+1 .b*11) F(u*) - -Fuutt)) ) 3(6", w - -00 -3(bh ,u - -cht1)
2 111l//+1 - 1² -
-
8
12
2
- F(v*) - F(uht1) + 3(b,, u - -1²) - 3(1 uht - rbtt) - 3(u wht1 wht1 jltz)
+ $2 |1 - 1/t2112 + 3(1 -1++1 - ol)
= F(r²) - F(vEt1) + +,((6, , ghtl r²) - 31 112
B
$2 11 - -0+111 + 0( - -+ x - -²t
+ 3
F(1²) - F(uct1) -186 okt1 - - -B1lut+1 ++1 - +
11 11
2
+3(u +1 -btt1, , -ty
B
1 112 B11u 14t1118
1110 u 112
2
1
1 - - -3116* - 11
=
2
2
B
I.2
z
110. -1tt112=
8 11
112
2
B r L2
2 110 u
(75)
Note that the second '=' comes from the cosine rule, the first '>" follows from the r-weakly convexity of F in (73), and the second 'z' results from the I--Lipschitz of OF in (74
Combining (72) and 75) we get
3
L2 12) 110 12 0,
La(u", * byt) Lu kt bttt
l w 12
2
(76)
that is, Lo(ut, ot bb) is non-increasing.
Now we prove that (ur, bb)}}}} is bounded. Note that F and G are coercive on V. As result,
F(u*) G(u:; f) > too.
(77)
Along with 36* (F and the property that OF L-Lipschitz, we arrive at
Lø(u,, wh Ba) = =((u²) + G(u": f) + 3(b", ud ) +
2 1 111
(78)
L II 112
B | - 11.
2 F(u*) + G(u+; f)
+
2
2
Note that L = At T-
and that t < 0.5. Thus, L < B. Therefore,
3 g11ut - 0/113
F(u)) + G(u";1) - I|ue - 00113 + 2
2
(79)
F(u)) ; f) (3 - L)llu* - rx11 2 -O0.
Since F(u) + G(u; f) is coercive on V, ut rt be are bounded. Part
Define h+1 as follows:
g+l = [3 - 8 + rt x11)), g(r - u k+1), 3(b - 8O)].
(80)
Define (L, v. = [0,Lg, O, Lo, OgL.). By the formulas 69 77p. we know that
gt E aL st1 bk+1)..
(81)
31

---

Note that
||@(6*+1 6t to - )|1 S 3116 - - - - --lu--of 11+ -
(82)
and that
18(0 - u(tt)|| =B1bb bz+1 S L 11+1
(83)
we arrive at
| || - ClIu ul
(84)
where
C 3= + 3.
(85)
Now finally prove Theorem 4.1 By Part 2, {(ut uh Da1} is bounded. So there is sub-sequence {(u"", o**,, b"*)) such that (u"*, 2² 6dh) + (u', ,0",b", when + too Since La(u*, (u", ,8,, is lower bounded and non-increaseee increasing,, we have that I|u 2 || - + as k + too. Besides, since qe E ŌLg v 6*) and || & ClI - r |]. we know that O. Thus, 0 OLa(u", ,², b","", and (u", u², ,=,b"))) 6-)) is a stationary point Lp.
||a"1| + 0, |1ght
Since G, is KL, we conclude that Lg is also KL. Then, by the proof of Theorem 2.9 in 22 f(u*,, un " b²*)] converges globally (u", bb")..
Since (u", oz, ;b)) is stationary point of Lg. and as a result, q' = 0, that is q' [0, 3(0" ==),)) 0] = 0.
(86)
Therefore, v² By CoCo-ADMM iteration in (I9). know that u* Prox g (u* 6*),
(87)
u* (u b*) E Proxg (u* + +b))
Equivalently,
5b* aG(u": f),
Bb* OF(("),
(88)
0 E OF(")) + f
(89)
Therefore, u* is stationary point of 15).
A.7 Proof of Theorem
The convergence of PGD with weakly convex prior function has been extensively studied. For details, one can refer to 27 For example, (27) prove that
Theorem A.6 (Theorem by (77). Let F, H be proper, closed, bounded from below, and H is differentiable with Li-Liscchitz gradient, and F is r-weakly convex. Then for T < max Ln Ln }. the iterates
++1 Prox,F o( -TVH)(u²*)
(90)
verify
(F(1)) + H(u))) is non-increasinn and converges.
All cluster points of the sequence n are stationary points of F + H.
If the sequence 4 is bounded and if F + H verifies the KL property at the cluster points of ut then ut converges stationary point of F H.
By Theorem A.6 we can prove Theorem 4.4
Proof. Let F be the proper, closed, weakly convex prior function. Let IG be the Moreau envelope of G, that is
(uu) = ='G(u) == min G(v) - u||2.
(91)
Then !G is diffentiable:
V+G I ProxG.
(92)
Since proper, closed, and convex, its proximal operator firmly non-expansive. Therefore, V is also firmly non-expansive, thus 1-Lipschitz. Since F and G is coercive and KL, F G also coercive and KL. Therefore, x² bounded. By Theorem A.6 f < 2 1}. CoCo-PEGD
converges to stationary point of F F+ 'G.
32

---

A.8 Proximal operator on the fidelity for each task
When K * I, in general, there no closed-form solution for Prox f Proxg (z).
Given f V, we say
f. = arg min A(1, Kr - flo Kx) + _ 11 2112.
(93)
93
solved by ADMM with iterations and as follows:
x+1 (B + pk K) (32 pk (yy w²)),
y t arg min A fog ) + Kx ft + w w1
(94)
2
w w + Kr y²+
When K is blur kernel, a-subproblem can be solved by the fast Fourier transformation as in 49
When K = is the Radon transform, one can calculate (8 + pk K) by the conjugate gradient method. There is a closed form solution for the y-subproblem by 34
2¹ + (=t)² + 4pAf
(95)
2p
where
p (Kxxt w²) A1
(96)
We initialize z. = w = 0, and output å = =x²" We set T 10 ensure the convergence of
the a-subproblemm
A.9
Detailed Deconvolution results on each kernel
See Tables
Figure 4: Eight blur kernels from 39
Table 3: Average Deconvolution PSNR and SSIM performance by different methods on CBSD68 dataset with Poisson noise with peak value p 100.
p=100
kernel1
kerne12
kernel3
kerne4
kerne15
kerneló
kernel7
kernel8
Average
DPIR
26.24
25.97
26.66
25.65
27.61
27.42
26.52
26.00
26.51
0.7321
0.7209
0.7439
0.7016
0.7870
0.7780
0.7462
0.7257
0.7419
RMMO-DRS
25.60
25.39
26.07
25.19
26.92
26.59
26.06
25.67
25.94
0.6895
0.6804
0.7057
0.6665
0.7426
0.7296
0.7089
0.6921
0.7019
Prox-DRS
25.32
25.03
25.82
24.80
26.75
26.39
25.94
25.40
25.68
0.6546
0.6488
0.6842
0.6336
0.7256
0,7050
0.6924
0.6668
0.6764
DPS
23.40
23.23
24.19
22.86
24.74
24.06
23.60
23.10
23.65
0.5941
0.5882
0.6303
0.5699
0.6550
0.6265
0.6039
0.5815
0.6062
DiftPIR
24.46
24.45
25.03
24.17
25.66
25.27
24.93
24.56
24.82
0.6250
0.6251
0.6510
0.6085
0.6838
0.6664
0.6509
0.6325
0.6429
SNORE
25.75
25.99
26.56
25.70
27.13
26.85
26.55
26.09
26.33
0.6985
0.6993
0.7202
0.6856
0.7485
0.7405
0.7249
0.7085
0.7158
PaPI-HQS
26.25
25.95
26.32
25.70
27.32
27.15
26.55
26.15
26.42
0.7024
0.6930
0.7124
0.6797
0.7570
0.7466
0.7269
0.7084
0.7158
B-RED
23.89
23.55
24.09
23.33
25.04
24.61
24.33
23.88
24.09
0.6206
0.6188
0.6537
0.5988
0.6824
0.6584
0.6434
0.6200
0.6370
CoCo-ADMM
26.66
26.46
26.93
26.23
27.78
27.52
26.96
26.59
26.89
0.7270
0.7183
0.7343
0.7075
0.7707
0.7630
0.7404
0.7254
0.7358
33

---

Table 4: Average Poisson deblurring PSNR and SSIM performance by different methods on CBSD68 dataset with Poisson noise with peak value p = 50.
p=50
kernel1
kerne12
kernel3
kernel4
kerne15
kernel6
kernel7
kernel8
Average
DPIR
25.10
24.83
25,66
24.50
26.45
26.20
25.41
24.89
25.38
0.6773
0.6656
0.6988
0,6480
0.7390
0.7261
0.6981
0.6752
0.6910
RMMO-DRS
24.86
24.73
25.39
24.51
25.84
25.52
25.12
24.83
25.10
0.6439
0.6419
0.6690
0.6292
0.6826
0.6697
0.6549
0.6457
0.6546
Prox-DRS
24.89
24.69
25.45
24.43
26.14
25.81
25.36
24.92
25.21
0.6363
0.6319
0.6676
0.6164
0.7024
0.6829
0.6714
0.6484
0.6572
DPS
22.92
22.76
23.72
22.34
24.13
23.48
22.98
22.49
23.10
0.5724
0.5668
0.6088
0.5475
0.6266
0.5990
0.5758
0.5561
0.5816
DifIPIR
23.79
23.73
24.43
23.45
24.91
24.45
24.16
23.77
24.08
0.5915
0.5894
0.6220
0,5744
0.6478
0.6263
0.6134
0.5941
0.6077
SNORE
24.57
25.06
25.80
24.75
25.78
25.34
25,34
25.01
25,21
0.6522
0.6617
0.6893
0.6490
0.6967
0.6863
0.6749
0.6648
0.6719
PnPI-HQS
25.54
25.29
25.80
25.02
25.60
26.37
25.83
25.41
25.61
0.6972
0.6705
0.6943
0,6563
0.7359
0.7231
0.7039
0,6837
0.6956
B-RED
23.49
23.41
23.86
22.99
24.69
24.49
23.86
23.46
23.78
0.6083
0.6055
0.6436
0.5854
0.6692
0.6441
0.6264
0.6022
0.6232
CoCo-ADMM
27.74
25.57
26.16
25.32
26.89
26.52
26.09
25.67
26.00
0.6906
0.6817
0.7072
0.6701
0.7412
0.7289
0.7098
0.6911
0.7026
A.10 Single photon imaging
We test the proposed methods in a low-light real-world scenario: single photon imaging task by time-correlated Linglephhoon avalanche diode (SPAD) camera [62]. In this task, SPAD array is used to track periodic light pulses in flight, and each detector only recieves about 1 signal photon per pixel average (p = 1). By the uncertainty principle (66 [56), since the momentum of the photons known, the position the photons cannot be accurately recorded. Therefore, the arrival time of each photon in the SPAD array is random variable, and be modelled by Poisson process. By the filtered histogram method, a reflectivity image is obtained, see Fig. 3 (a)-
We compare the visual denoising performance on Fig. (a) by some state-of-the-at Poisson noise removal methods: Photon TV using TV prior specialized for this problem [62]; BM3D-VST, BM3D method with variance stabilization transform designed for Poisson noises
3 DnCNN [74
trained as a Poisson denoiser; VDIR, variational inference network [63]; VBDNet, a variational bayesian deep network for blind poisson denoising [41]. We also compare three PnP methods, DPIR,
RMMO-DRS and PnPI-HQS, as introduced before.
We show the visual results in Fig. 3 It can be seen in Figs. (b)-(c) that, non-deep methods provide blurred results with unclear edges. End-to-end deep learning methods provide over-smoothed results, and cannot restore the fine details in the flower, see Figs. (d)-(f). Compared with other PnP methods, CoCo methods can recover the fine textures the flower, as well as the sharp edges in the equation, see Figs. 3 (g)k Since there is reference image in this real-world scenario, we only report the relative error curves to validate the convergence in Fig. (1).
A.11 Ablation study on 7 and in CoCo-PnP
We report the PSNR values by CoCo-PnP with different y and in the deblurring task in Table 5 can seen that, the proposed methods are sensitive to y, because determines the denoising performance, but not sensitive to t.
A.12 Low-dose CT reconstruction
In order to further illustrate the effectiveness of the proposed methods, we consider the sparse-view low-dose CT reconstruction task. In this task, K R is the Radon transform, and f in Eq. (T) is the down-sampled data the Radon field. The Radon field data is corrupted by the Poisson noise.
'The data kindly provided
22 oggghhhwwwwooooooteeeeiittitttir
34

---

Table 5: PSNR results by CoCo-PuP with different y and t when deblurring the image '0005' from CBSD68 and kernel 8 from Levin's ' dataset. The Poisson noise level is 50. We run the algorithm for 500 iterations ensure the convergence. When 7 0.25, (7) 1/3, thus t 1/3. When 0.5, (o(7) 2 0.3761, thus S 0.3761.
7 = 0.25
0.3333
0.300
0.200
0.100
0.001
CoCo-ADMM PSNR
25.33
25.31
25.30
25.24
25.16
CoCo-PEGD
PSNR
25.32
25.30
25.27
25.22
24.98
7 0.50
0.3761
0.300
0.200
0.100
0.001
CoCo-ADMM PSNR
24.88
24.87
24.79
24.64
24.47
CoCo-PEGD
PSNR
24.84
24.82
24.71
24.54
24.35
For the test set, we select ten typical images from "the NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge" [45 see Fig.
For the comparison methods, TGV, penalized weighted least-squares (PWLS) with total generalized variation (TGV) prior [88;: PWLS-CSCGR, PWLS with convolutional sparse coding and gradient regularization [5: UNet, which take the result by FBP as the input [31]; WNet, which uses two UNets to construct the data from both the image and Radon domain [14]
FBP, the conventional filtered back projection method; PWLS-
We show the reconstruction results with 60 projection views and Poisson noises with peak value p 100, 500 in Table It can be seen that, compared with two iterative PWLS-based methods, and two end-to-end deep learning methods, the proposed CoCo methods have a significant improvement PSNR and SSIM values.
Figure 5: CT test images.
A.13 Poisson denoising results
We apply CoCo-ADMM and CoCo-PEGD to Poisson denoising problems. In this task, K is the identity. The average PSNR and SSIM values CBSD68 are summarized in Table It can be that compared with other convergent methods, the proposed methods achieve the best PSNR and SSIM values. It validates the effectiveness of CoCo methods.
A.14 Computational cost
In 7 we give the average computation time in seconds, well the memory cost in MB by different methods when deblurring 256 256 image. It can be seen that the proposed methods are the most efficient with least memory cost.
35

---

(a) FBP
(b) PWLS-TGV (c) PWLS-CSCGR
(d) UNet
WNet
PHLS-TOY CoCO KONI
PAI SCSCER
Coco ADMM
RCCD
(f) CoCo-ADMM (g) CoCo-PEGD(h) High-dose reference
(i) PSNR
iJ Relative error
Figure Sparse view CT reconstruction results with 60 views and Poisson noises (peak=500). (a) FBP, PSNR=30.91dl dB. (b) PWLS-TGV.
1NNR=36.12BB (c) PWLS-CSCCH PSNR=37.8066
(d) UNet, WWNR=38.54dB (e) WNet, PSNR=38. (f) CoCo-ADMM, PSNR=40.46dH (g)
CoCo-PEGD, PSNR=40.28dB. (h) High-dose reference. (i) PSNR curves. (j) Relative error curves.
Table 6: Average denoising PSNR and SSIM performance by different methods CBSD68 with peak value p = 30 and p = 20.
30
20
PSNR
SSIM
PSNR
SSIM
DPIR
29.85
0.8654
28.68
0.8300
RMMO-DRS
27.90
0,8008
27.74
0.7887
Prox-DRS
28.99
0.8087
27.94
0.7724
SNORE
29.47
0.8474
28.34
0.8226
PaPI-HQS
29.90
0.8613
28.63
0.8156
B-RED
28.80
0.8041
26.26
0.7362
CoCo-ADMM
29.99
0.8604
28.76
0.8329
CoCo-PEGD
29.99
0.8625
28.77
0.8332
A.15 Performances under extreme conditions
In Fig. 7 show example when deblurring image under severe noisy condition. It can be seen in Fig. (a) that the image severely corrupted. We compare results by different methods. be that CoCo-ADMM and CoCo-PEGD provide sharper edges. Even in this extreme case, the proposed methods are still convergent, see Figs. (m)-(n).
A.16 Ablation study the parameters a1 and 02
Loss(0) = B|| Dz (x + 0) =||1 - an J J ||- + 02 max
J- -e..
(97)
In the loss function 97 controls the penalty strength of the Hamiltonian term to encourage con-
servativeness, and 0r controls the penalty strength of the spectral term to encourage cocoerciveness.
As shown in Table 8 DRUNet without regularization terms has an expansive residual part. When 0n gets bigger, the mean symmetry error gets lower. When 0g is smaller than 1e-2, the regularized denoiser may not be cocoercive.
B-RED
Prox-DRS
SNORE
DifIPIR
DPS
CoCo-ADMM
CoCo-PEGD
Iteration
300
200
100
1000
100
Time
42.5s
13.6s
74.1s
11.8s
177s
5.24s
9.08s
Memory
4946MB
3982MB
3624MB
2998MB
4694MB
2304MB
2304MB
Table Average computation time in seconds, and memory cost in MB by different methods when deblurring 256 x 256 image.
36

---

(a) Blur
(b) DPIR (c) RMMO-DRS (d) ProX-DRS
(c) DPS
DiftPIR
L SNORE
(h) PnPI-HQS (i) B-RED (j) CoCo-ADMM CoCo--EGG (i) Clean
(m) PSNR (n) Relative error
Figure 7: Deconvolution results by different methods on the image '0005' from CBSD68
with kernel 8 and p 10 Poisson noises. (a) Blur image. (b) DPIR. PSNR=22. 2922. (c)
RMMO-DRS, PSNR=22.23dH (d) Prox- DRRD PSNR=22.5400 54dB. (e) DPS, PSNR=20. 70dB. (f) Diff- PIR. PSNR=21.666 PSNR=21.15dB. (j) CoCo-ADMM, +SNR=23.04dB (k) CoCo-PEGD, PSNR=23.C 2ENR==2.02dB.. (I) Clean
(g) SNORE, PSNR=23. d. (h) PnPI-HQS. PSNR=23.11 14d.. (i) B-RED,
image. (m) PSNR curves. (n) Relative enror curves. r-axis denotes the iteration number.
Therefore, set the penalty parameters large enough to encourage the properties want. However, when a gets too large, the denoising performance may be sacrificed. We empirically set a₁ = 1, 02 le-2 in the experiments.
Table Mean symmetry error | J ||- with N and maximal values of the norm || J I||
with N = 30 on CBSD68 for various noise levels a and 7 0.50, 0.25.
25
40
Norms
01
022
DRUNet
4.1c+0
4.2c+1
1.8c+1
| _
1.50-CoCo-DRUNeL
8.6e-3
2.5e-4
7.1e-4
[ J
le-2
1.50-CoCo-DRES
2.9c-1
2.6c-2
9.5c-2
J J
1c-2
1c-2
D.25-CoCo-DRUNet
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DPIR
RMMO
CoCo-ADMM
CoCo-PEGD
Figure 8: Blind decovolution results by different methods.
A.17
Blind Deblur Results
In this section, we evaluate the performance of our proposed methods on real-world blind deblurring. We select 18 images from the dataset by Lai al. [35), which captures challenging real-world blurry scenes. Four state-of ofee--ff-hhe tet-thean art methods are benchmarked: Deblur-INR (76). Lv et al. [42]. self-supervised method that jointly estimates clean images and blur kernels, while Lv et al. 42 applies total variation regularization refine latent image estimation in a blind deblurring framework. contrast, DPIR, RMMO, and our proposed C.Co-ADMMCCCC-PPPA
DPIR (77), and RMMO (66). Deblur-INR
require pre-estimated blur kernels input. To ensure fairness, all
37

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INR
el.
DPIR
RMMO
CoMO-ADMM
CoCo-PEGD 0.5474
Average
0.2990
0.3444
0.3629
0.2664
0.5461
Table 9: Average CLIP-IQA results across all deblurred images
kemel-deepndent methods utilize blur kernels generated by the blind deblurring framework of Liu al. [42]. Visual comparisons in Fig. g demonstrate that CoCo-ADMM and CoCo-PEGD produce sharper textures and fewer artifacts than baseline methods.
For quantitative evaluation, we employ the CLIP-IQA metric [70], a non-reference image quality assessment tool that leverages the pretrained vision-language representation of CLIP (Contrastive Language-Imase Pretraining) to evaluate perceptual quality without requiring pristine reference images. The average CLIP-IQA scores across all deblurred images are summarized in Table 9 Despite relying on the same estimated kemelss as DPIR and RMMO, our methods achieve superior performance, highlighting their robustness kernel estimation errors and enhanced restoration capability.
A.18 Blind Denoise Results
Noisy
Zoomed
DRUNet
PnPI-HCIS
DPIR
RMMO
CoCo-AUMM
CoCo-PEGD
Figure 9: Blind noise removal results by different methods.
Noisy
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PnPi-HQS
DPIR
RMMO
CoCo-ADMM
CoCo-PEGD
Figure 10: Blind noise suppression results in CT images by different methods.
In this section, evaluate the performance of our proposed denoising framework on benchmark datasets that capture real-world noise characteristics. For medical imaging, we employ the Low-Dose CT dataset from Mayo Clinic [46), which provides paired low-dose CT scans with Poisson noise.
For natural images, we utilize the DND dataset [52), comprising 50 real noisy photographs corrupted by mixture of Poisson and Gaussian noise. Following the standard evaluation protocol, perceptual quality is quantified using the CLIP-IQA metric 770
To ensure fair comparison, we benchmark proposed methods (CoCo-ADMM and CoCo-PEGD) against five representative techniques: DRUNet[77), DPIR [77], RMMO (66] and PnPI-HQS 771. Quantitative results in TO demonstrate that our proposed methods achieve state-of-the-ar performance across both imaging domains.
38

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DRUNet
DPIR
RMMO
PnPI-HQS
CoCo-ADMM
CoCo-PEGD
Color images
0.4464
0.4509
0.4561
0.4593 0.4011
0.4586
0.4596
CT
0.4154
0.4258
0.4398
0.4707
0.4730
Table 10: Average CLIP-IQA results for blind image denoising.
39


Reviews:
[{"review_id": "dOVxJPewhD", "reviewer": "Reviewer_8mzP", "summary": "This 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.", "questions": "1. 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.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I have raised my score to weak accept following the rebuttal. The authors provided convincing explanations for previously unclear points. I believe the main strength of the paper lies in its specialization to Poisson inverse problems, which is a relevant contribution. I do not provide a higher score as I believe the soft regularisations used to enforce co-coercivity and conservativity of the denoiser are not particulary novel and in practice do not provide any guarantees (these constraints are softly enforced on training points and not on the whole space). Moreover, the main novelty of the approach essentially lies in relaxing the denoiser to be the prox of a weakly convex penalty instead of being the prox of a convex function: while this is an interesting direction, it still builds on several known ideas and similar formulations from prior work.", "rebuttal": "Dear reviewer 8mzP, \n\nThank you so much for your professional feedback and suggestions on the originality and theory. We sincerely appreciate your positive and negative comments, which are extremely helpful. We have carefully considered your comments and are committed to refining our paper. Below, you will find our detailed response as we strive for excellence.\n\n### Responses to weaknesses:\n> 1.) Cocoercivity comes down to relaxing by a parameter $\\gamma$ the firmly non-expansiveness soft constraint of Terris and al. [66].\n\nCocoercivity is a much broader (and weaker) concept than firm non-expansiveness, and residual non-expansive. In particular, if $\\gamma=1$, $\\gamma$-cocoercivity is equivalent to firm non-expansiveness; if $\\gamma=0.5$, $\\gamma$-cocoercivity is equivalent to residual non-expansiveness. Terris et al. proposed to softly enforce the firm non-expansiveness of the denoiser. By doing so, the denoiser is proved to be a resolvent of some monotone operator prior. If further assumed conservativity, this means that the learned prior is convex. However, as reported by many past papers [28,71], enforcing such constraints significantly compromises the denoising performance and PnP restoration. This paper relaxes this contraint to cocoercivity. If the denoiser is $\\gamma$-cocoercive with $\\gamma\\in(0,1)$, the denoiser is actually a resolvent of some weakly monotone operator prior. If further assumed conservativity, this means that the learned prior is weakly convex. Overall, this paper adopted the soft constraints technique by Terris et al., and extended it to a weaker concept. \n\nPlease note that, compared to the pioneer work by Terris and Pesquet et al., this paper has the following significant improvements:\n\n(1) The learned prior is weakly convex, enabling much better PnP restoration performance. In Table 2, compared to RMMO-DRS, the proposed methods has about 1dB increase in PSNR. \n\n(2) We proved the convergence of PnP-ADMM with an implicit weakly convex prior, and a non-strongly convex, non-smooth data fidelity prior. \n\n> 2.) Enforcing conservativeness of the denoiser (i.e. having a denoiser which is a gradient) can also be found in the literature (RED, Gradient Step denoisers). Besides, the proof technique for convergence strongly rely on the prox characterization of [22] which has already been used a lot in Hurault's works.\n\nGradient Step denoiser (GS-DRUNet) is indeed a great way to enforce the conservativeness. However, please note that, the training and the inference procedure of GS-DRUNet both need to calculate Jacobian-vector product, because there is always a Jacobian term in GS-DRUNet. This increases the training cost. \n\nThis paper wants to propose an alternative way to learn a conservative denoiser. We introduced the Helmholtz decomposition to study the denoising geometry, and showed that the Hamiltonian part is useless for denoising, and thus an ideal denoiser should be conservative. Besides, this decomposition also inspires us how to train a conservative denoiser without modifying the network architechture as in GS-DRUNet.\n\nFor the proof, we do utilize the great work [22] by Gribonval and Nikolova. Although Hurault has also use it to characterize the denoiser, there is an important difference between our paper and Hurault's paper. The Lipschitz assumption on the denoiser in Hurault's paper is residual non-expansiveness. This assumption is still restrictive, especiall when the noise level is large. In our paper, the assumption is relaxed to be cocoercive with even small $\\gamma$. This relaxation $\\textbf{fully}$ utilized the result in [22]. Mathematically speaking, since the `Lipschitz + be some gradient of a convex potential' can be equivalently rewritten as 'the denoiser is conservative and cocoercive', the proposed CoCo denoiser is a $\\textbf{tight}$ utilization of the characterization in [22], and therefore provides better PnP performance. \n\n\n\n\n### Responses to questions: \n\n1. We apologize for this careless mistake: in fact, in line 85-88 in the introduction, we have already talked about Bregman-PnP by Hurault et al.. However, we cited the wrong paper. We actually wanted to cite the paper `Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems' published in Neurips. Besides, in the experiments, we have already compared the proposed method with Bregman-PnP (termed as B-PnP in the paper), see Table 2 and Figure 2. \n\nBesides, thank you for your clarification! We totally agree with your statement that, we proposed to use a Gaussian denoiser as a prox rather than a Bregman prox. Compared to Bregman-PnP, the proposed CoCo-PnP has the following advantages:\n\n(1) In the context of PnP, people use Gaussian denoisers mainly because that Gaussian denoisers are easier to get than other type denoisers. However, as stated in the introduction, there is a lacking literature on using Gaussian denoisers as prox to solve Poisson inverse problems. Our paper aims to solve this problem with Gaussian denoiser as a proximal operator of some implicit weakly convex prior. \n\n(2) As reported in Pages 10 and 30 in the Appendix of the paper `Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems', for large noise levels, Bregman denoiser does not perform well. Besides, when applied to deblurring with Poisson noises, Bregman-PnP does not outperform existing methods. We have also observed this in the experiments, see Table 2 and Figure 2 for example. Compared to Bregman-PnP (termed B-PnP in our paper), the proposed B-PnP performs significantly better. This suggests that using Gaussian denoisers in Poisson inverse problems is more suitable.\n\n2. The Helmholtz decomposition is useful to not only interpret the denoising geometry, but also inspire us the design the loss functions, such that the learned denoiser is encouraged to be conservative. Proposition 2 does only require the denoiser to be Lipschitz and be some gradient of a convex potential. But, please note that, these two requirements are equivalent to that `the denoiser is conservative and cocoercive'. For the proof, please see line 596-600 in our paper, in which we have proved that the denoiser is a gradient of some convex potential. \n\nIntuitively speaking, the `convexity' of the potential mainly comes from the cocoercivity of the denoiser. Please note that cocoercivity actually requires the denoiser to be a monotone operator. The gradient of a convex potential is a special case of monotone operators. \n\n3. Unfortunately, we are not able to compute it. This is because that, we can not access the convex potential of the denoiser. Different from the great pioneering works by Hurault et al., the proposed denoiser has an implicit potential. According to Theorem 3 in [22], or Proposition 3.1 by Hurault et al. in [29], in order to compute the prior, we need to access the potential. Therefore, we chose to report the PSNR and relative error curves in Figures 2-3, 6-7. These figures validates the convergence.\n\n4. For a differentiable data fidelity term, in the context of operator splitting algorithms, there are basically two ways to use it. We can use the fidelity in a backward way, that is, to use its proximal operator like in PnP-ADMM. Since the fidelity is convex, the proximal operator of such fidelity has good property: it is firmly non-expansive. We can also use the fidelity in a forward way, that is, to use its gradient like in PGD or FBS. But since the fidelity is not smooth, we have to smooth it by considering its Moreau envelope. We chose to include this algorithm not only for completedness of the two ways to use the fidelity, but also that because we wanted to show that, sometimes, the first motivation can not be solved by simply smoothing the non-smooth fidelity. In fact, as reported in Table 2, CoCo-ADMM performs better than CoCo-PEGD: this is because that the fidelity is changed to its envelope, and there can not fully reflect the Poisson distribution.\n\n\n\nThank you again for such professional comments on our paper. Please let us know if you have more suggestions. We are very willing to revise our paper based on your comments."}, {"review_id": "k2QASqbjol", "reviewer": "Reviewer_UHTR", "summary": "This 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.", "questions": "My 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.", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "First of all, we must reiterate that we are not very familiar with some of the theoretical aspects presented in this paper. We can only evaluate the work based on the parts we understand. We believe the paper holds certain theoretical value. However, its weaknesses are also clear: the theoretical presentation is almost entirely based on formulas and theorems, lacking visual or intuitive illustrations, which makes it somewhat difficult to follow for readers who are not well-versed in this field. That said, we acknowledge that conducting in-depth theoretical research is a challenging task and should be encouraged. Therefore, we have decided to raise our rating to \"borderline accept,\" while keeping our confidence level unchanged. We kindly ask the area chair to consider our decision as a partial reference only.", "rebuttal": "Dear reviewer UHTR, \n\nThank you so much for your valuable feedback and insightful comments. We sincerely appreciate your encouraging words and constructive critique; both are extremely valuable. We have given careful consideration to your points and are focused on refining the paper. Our detailed responses are provided below as part of our pursuit of excellence.\n\n### Responses to weaknesses:\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\nSpecial thanks for this comment! This question is very valuable, and we will discuss this not only here, but also in the revised paper. \n\nThere basically two ways to use a prior in the context of PnP. If we use it in a forward way, for example to use it in RED, to the best of our knowledge, the weakest assumption is pseudo-contractiveness as in [71]. But if we use it in a backward way, for example this paper uses it as a prox, the weakest assumption (in terms of a weakly convex prior) has been proven to be `Lipschitz + be the gradient of some convex potential' in [22]. [22] established the equivalence between 'the denoiser is a proximal operator of some weakly convex prior', and 'the denoiser is Lipschitz and is the gradient of some convex potential'. Please note that, 'the denoiser is Lipschitz and is the gradient of some convex potential' can be equivalently rewritten as 'the denoiser is conservative and cocoercive': cocoerciveness ensures that the denoiser is Lipschitz and monotone; combined with conservativeness, the denoiser is then a gradient of a convex potential. The gradient of a convex potential is a special case of monotone operators. \n\nThe proof for this equivalence was not included in the paper. Currently, in Theorem A.5, we have proved that if the denoiser is conservative and cocoercive, it is a prox of some weakly convex prior. We will include the equivalence in the revised version. \n\n> What exactly fails when the operator is not cocoercive, and how tight is this condition?\n\nAs stated above, if we want the denoiser to be a prox of some weakly convex prior, the cocoercive and conservativeness are required. The condition is tight, and actually sufficient and necessary.\n\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\nWe softly enforce cocoerciveness and conservativeness by spectral regularization technique. The established convergence requires the two properties. In experiments, we found that, if the denoiser is not conservative enough, the convergence can still be stable with fine-tuned parameters. This may be because that the Hamiltonian part of the denoiser is useless (and therefore sometimes harmless) for the denoising performance.  We will include this in the revised paper.\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\nThe decomposition is stable with different architechtures and noise levels, as long as the network output is differentiable with respect to the input, such that $J$ is computable. When the dimension is high, the connection between the anti-symmetric part and “useless” Hamiltonian directions is actually illustrated briefly in the equation blow Line 172. This means that the inner product of $D(y)-x$ and $\\nabla_y \\mathcal{H}(y)$ is zero, and thus the two vector is orthogonal. This further means that, if we consider the continuous dynamics, $\\dot{y}=-\\nabla_y \\mathcal{H}(y)$, $\\mathcal{H}(y(t))\\equiv Constant$. That is, the dynamics preserve the Hamiltonian functional, if the denoiser only has Hamiltonian part. Since the Hamiltonian functional is a typical denoising loss, this means that a Hamiltonian field does not contribute to the denoising. \n\nWe will include more discussion on high dimensional cases in the revised paper.\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\nSince the rebuttal window does not allow PDF or inserting images, we only report the PSNR here. We have conducted oblation study with different $\\gamma$ and $t$. We consider deblurring for the image `0005.png' from CBSD68 and kernel 8 from Levin's dataset. We set the Poisson noise level to be $50$. We run the algorithm for 500 iterations to ensure the convergence. \n\nWhen $\\gamma=0.25$, $t_0(\\gamma)=\\frac{1}{3}$. Thus $t<\\frac{1}{3}$. \n\n\n$ $|  $t$   | $0.3333$ |  $0.3$ | $0.2$ | $0.1$ | $0.001$ \n:----: | :----: | :----: | :----:| :----: | :----: | :----:  \nCoCo-ADMM |PSNR    | 25.33    | 25.31 |   25.30 | 25.24 | 25.16   \nCoCo-PEGD |PSNR    | 25.32    | 25.30 |  25.27 | 25.22 | 24.98   \n\nWhen $\\gamma=0.5$, $t_0(\\gamma)\\approx 0.3761$. Thus $t\\le 0.3761$. \n\n$ $|  $t$   | $0.3761$ |  $0.3$ | $0.2$ | $0.1$ | $0.001$ \n:----: | :----: | :----: | :----: | :----: | :----: | :----:  \nCoCo-ADMM |PSNR    | 24.88    |  24.87 | 24.79 | 24.64 | 24.47   \nCoCo-PEGD |PSNR    | 24.84    |   24.82 | 24.71 |24.54 | 24.35   \n\nFrom the two tables, it is clear that when $t$ is not too small, the PnP performance is not sensitive to the choice of $t$. However, different $\\gamma$ has significant influence on the PSNR performance. This is because that a larger $\\gamma$ means a more restrictive assumption on the denoiser, and therefore results in a worse performance. In practice, we recommend to set $t\\approx t_0(\\gamma)$, with $\\gamma=0.25$. No significant denoising improvement is observed when $\\gamma<0.25$.\n\n\n\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\nThank you for this valuable feedback. After thorough reading our own paper, we have to admit that, the paper is indeed a little math-heavy. We will revise the paper carefully, and add some intuitive explanation, diagrams, simplified takeaways, and proof sketches to make the paper easier to read. \n\n\n\nPlease let us know if you have more suggestions to polish the paper and raise the rating. We are very willing to revise the paper based on your comments to further improve its quality."}, {"review_id": "tlBf1KUgYu", "reviewer": "Reviewer_G4Uk", "summary": "This 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.", "questions": "- 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?", "limitations": "I 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.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The 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.", "was_revised": true, "final_justification": "The authors have addressed my concerns well and promised corresponding adaptations for the possible camera-ready version. I am increasing my rating.", "rebuttal": "Dear reviewer G4Uk, \n\nWe sincerely appreciate your thoughtful review and constructive feedback. Your comments are highly valuable, and we have carefully considered each point in revising the paper. Our detailed responses are provided below.\n\n### Responses to weaknesses:\n\n\nThank you for this valuable comments. Please note that, we fully agree with your suggestion that ``I highly recommend writing the entire paper under this assumption and simplifying all parts that can be simplified.'' We have revised the paper carefully, such that all settings are in a finite-dimensional space. \n\nBut now, please allow us to respond to your questions. As answered below, most of the theoretical results in the paper still $\\textbf{hold}$ in an infinite-dimensional setting. \n\n\n\n> 1.1)  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?\n\nThrough out all theoretical parts in the paper, we were assuming a general real Hilbert space, which might be infinite-dimensional. We will include the smooth assumption for the Helmholtz decomposition. The transpose symbol means the adjoint operator in a Hilbert space $V$. When the space is finite dimensional like $\\mathbb{R}^n$, the adjoint operator is the transpose. \n\nWhen comes to training, we will have to consider a finite dimensional space of course. So we agree that the paper should be written in a finite dimensional setting.\n\n\n\n> 1.2)  Is the spectral norm the operator norm in the infinite-dimensional setting?  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.\n\nIn an infinite-dimensional setting, $\\|\\cdot\\|_*$ is the operator norm. When the space is finite-dimensional, it is just the spectral norm, that is, the largest singular value. \n\n\n> 1.3) Do all convergence results really hold in the infinite-dimensional setting as well?\n\nThank you for this valuable comments. We have revised the paper, such that all results are included in a finite dimensional space.\n\n\n> 1.4) What assumptions do we need to make about $K$ in this case?\n\nWhen $V$ is just a general real Hilbert space, $K$ is a bounded linear operator, that is $K\\in \\mathscr{B}[V]$. When considering a finite dimensional space, this is just a matrix. \n\n\n> 1.4) Moreover, the notion of the subdifferential and its Lipschitz continuity (Theorem 2.2.) should be introduced. \n\nWe will include the subdifferential and its Lipschitz continuity in the revised version. Thank you so much for such a detailed comment.\n\n\n\n\n> 2.) The denoiser is conservative if the Jacobian is symmetric, i.e., $J=J^\\mathrm{T}$. The authors propose to regularize $\\|J-J^\\mathrm{T}\\|_*$, but it seems like any norm could be used. This is not discussed in the paper. \n\nSpecial thanks for this valuable feedback! We choose to use the spectral norm, mainly because that it is has low computational cost. Due to the power iterative method, one can calculate the spectral norm very easily, without actually calculate the Jacobian, see Appendix A.5 for details. For other norms, we do not know whether there exist similiar algorithms to calculate them. We will include this discussion in the revised paper. \n\n\n\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\nNow the highest norm for $\\|J-J^\\mathrm{T}\\|_*$ is reported below. Please note that the highest norm for cocoercivity has already been reported in Table 1, and thus omitted here.\n\n\n$ $|  15  | 25 |  40 | Norms\n:----: | :----: | :----: | :----: | :----: \nDRUNet | 3.697e+2  |   8.107e+4  | 2.077e+4  | $\\|J-J^\\mathrm{T}\\|_*$  \n0.5-CoCo-DRUNet | 1.841e-5  |  3.135e-6   | 1.978e-5  | $\\|J-J^\\mathrm{T}\\|_*$  \n0.5-CoCo-DRUNet |  6.629e-5 |   2.754e-6  |  7.827e-4 | $\\|J-J^\\mathrm{T}\\|_*$  \n\nThis table will be included in the revised paper upon acceptance.\n\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\nThank you for this comment. Since the rebuttal window does not allow extra PDF or images, we report by words the computational cost. We first pre-train a DRUNet without spectral regularization. After that, we post-train the DRUNet with CoCo regularization. This is for saving training time. \n\nCompared with the training for DRUNet without any spectral regularization, for each epoch, the computational time for training CoCo is about $12$ times larger than training DRUNet without spectral regularizations. For the GPU memory cost, DRUNet is about $2080$ MB, and CoCo is about $9360$ MB. \n\n\n\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\nWe will state in the revised paper, that all training procedures for all denoisers are identical. In the revised paper, we will also report standard deviations with different random seed for denoising performance comparison. \n\n\n### Responses to questions:\n\n> Please address the question of the setting (finite vs. infinite-dimensional) of (1) under weaknesses and possibly adapt the paper accordingly.\n\nPlease refer to the responses to the weaknesses 1.1.\n\n> Please answer the question why you chose the spectral norm to enforce symmetry of $J$. \n\nPlease refer to the responses to the weaknesses 2.\n\n> Do the values in table 1 (right) change if one optimizes for a point $x$ at which the norms are high?\n\nUnfortunately, yes. Thus we agree with your suggestions, and report the maximum norms, see the responses to the weaknesses 3.\n\n> How large is the computational overhead in training the denoisers with spectral penalties involving its Jacobian?\n\nUnfortunately, pretty large: the computational cost for CoCo is 12 times larger than the training for DRUNet, see the responses to the weaknesses 4. As a result, we pre-trained a DRUNet first, and then post-train it with CoCo regularizations to save the training time.\n\n\n> Have all (competing) methods been fine-tuned equally? Did you consider multiple runs with different random seeds?\n\nWe have tried our best to fine tune each method, such that they reach the best PSNR values. For the random seed, for a fair comparison, we set the same random seed for all methods.\n\n### Responses to limitations:\n\nThanks for your suggestions on the limitations. We will address them in Section 6 in the revised paper. \n\n\n\nPlease let us know if you have more suggestions to polish the paper and raise the rating. We are very willing to revise the paper based on your comments to further improve its quality."}, {"review_id": "GkIYo2C81o", "reviewer": "Reviewer_aRzT", "summary": "This 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.", "questions": "Please 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.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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)?", "was_revised": true, "final_justification": "All my questions have been satisfactorily answered in the rebuttal, and I believe that this work merits acceptance to NeurIPS due to its strong mathematical foundation, high impact potential and excellent work quality.", "rebuttal": "Dear reviewer aRzT, \n\nWe sincerely appreciate your thoughtful review and constructive feedback. Your comments are highly valuable, and we have carefully considered each point in revising the paper. Our detailed responses are provided below.\n\n### Responses to weaknesses:\n\n> 1.)  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 $\\|J-J^\\mathrm{T}\\|_*$ 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\n\nWe will use the word ``encourage`` rather than ``enforce``, because that spectral regularization can not guarantee the properties. \n\nIn the experiments, we found that, if the denoiser is not conservative enough, the convergence can still be stable with fine-tuned parameters. This may be because that the Hamiltonian part of the denoiser is useless (and therefore sometimes harmless) for the denoising performance. \n\nTo the best of our knowledge, there is no way to quantify how a non-conservative denoiser can still be ``good enough`` for being a proximal operator.\n\nWe will include the discussions in the revised paper.\n\n\n\n> 2.)  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\nWe agree with your comment. We will discuss carefully about the performance improvements. The performance improvement could also have been because of the different loss function.\n\n\n> 3.) 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\nWe will remove the `real-time' expression in the revised paper.\n\n\n\nPlease let us know if you have more suggestions. We are very willing to revise the paper based on your comments to further improve its quality."}]

Author rebuttal:

Meta-review: **Meta-Review:**

This paper presents a novel approach to Poisson inverse problems using cocoercive conservative (CoCo) denoisers in plug-and-play (PnP) methods. The authors offer rigorous theoretical guarantees and demonstrate strong empirical performance, particularly in Poisson image restoration tasks.

### **Strengths:**

* **Novelty and Theoretical Contribution:** The use of cocoercivity as a relaxation of non-expansiveness for denoisers in PnP methods is a significant theoretical advancement. The convergence proof for PnP with weakly convex priors is well-founded.
* **Empirical Validation:** The method shows strong performance in comparison to closely related methods, particularly in challenging scenarios like Poisson inverse problems.
* **Well-Structured Paper:** The methodology is clearly presented with extensive experimental results supporting the theoretical claims.

### **Weaknesses:**

* **Soft Enforcement of Conditions:** The soft enforcement of cocoercivity and conservativeness raises concerns about the **practical sufficiency** of the convergence guarantees. The authors addressed this in the rebuttal, but further clarification is needed on how this affects real-world performance.
* **Helmholtz Decomposition:** While insightful, its **applicability across architectures and noise levels** in high-dimensional settings could be further explored.
* **Interpretation of Performance Gains:** Performance improvements could also be attributed to factors like **loss function changes**, which the authors should clarify further.
* **Math-Heavy Presentation:** The paper could benefit from more **intuitive explanations** and **diagrams** to make the theoretical framework more accessible.

### **Discussion & Rebuttal:**

The authors responded well to reviewer concerns, providing **clarifications** on the theoretical guarantees, the Helmholtz decomposition, and experimental design. However, **further refinement in presentation** (especially for non-experts) and additional validation are still needed.

### **Final Recommendation:**

**Accept**, but with **strong recommendations for revisions**. The theoretical contributions and empirical results are strong, but addressing the practical implications of the **soft enforcement** of conditions and providing more accessible explanations and comparisons will enhance the paper’s impact.
Decision: Accept (poster)