Rethinking Gradient Step Denoiser: Towards Truly Pseudo-Contractive Operator
Authors: Shuchang Zhang
Yaoyun Zeng
Kangkang Deng
Hongxia Wang
Venue: NeurIPS 2025 poster ()
Keywords: Plug-and-Play prior
Regularization by Denoising
Pseudo-contractive denoiser
Image restoration
Primary area: optimization

Abstract: Learning pseudo-contractive denoisers is a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising  (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail to guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink
gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and pseudo-contractive operators. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED
via fixed-point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive property and, when integrated into RED-PRO, provides a favorable trade-off between interpretability and empirical performance on inverse problems.

Rethinking Gradient Step Denoiser: Towards Truly Pseudo-Contractive Operator
Shuchang Zhang
Yaoyun Zeng"
College of Science
College of Science
National University of Defense Technology
National University of Defense Technology Changsha, 410073
Changsha, 4 410073
zhangshuchang19@ne 9unudt.. edu. cn
yaoyun yaoyun zeng@nut. edu.cn cn
Kangkang Deng
Hongxia Wang
College of Science
College of Science
National University of Defense Technology Changsha, 410073
National University of Defense Technology
Changsha, 410073
freedengg2088gmail.com
wanghongxia@nudt: edu. cn
Abstract
Learning pseudo-contractive denoisers a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and ceuudo-contacttve operators. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED via fixed-point point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive prop- erty and, when integrated into RED-PRO, provides favorable trade-off between interpretability and empirical performance on inverse problems.
1
Introduction
The pseudo-contractive operators constitute a wide class of operators that arise in iterative methods for solving fixed point problems contractive (d-PC) there exists a constant d (-o0, 1] such that for any x, E H. it holds that 12 31
[17]. Here, recall that operator T' H + H. is d-pseudo-
(1) where H. is Hilbert space. When < 1, the operator T is called d-strictly PC (d-SPC) op-
[x(x) T < | y1 + d d]((x T(x)) - -y-
and d = -1 [6. The pseudo-cone contractce assumption has played an important role in the convergence
erator [7]. The operator T nonexpansive and firmly nonexpansive (FNE) when
analysis of PnP methods [60] 53
51 56
6] 46 52 10 and RED
"Equal contribution
CCorespooning author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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framework [5 49
21 42 Fimmly nonexpansive (FNE) denoisers 59
48
57
10 averaged non-
52, and contractive all belong to the class of deeudo-contrativv operators [7]. The demicontractive assumption (see Definition 3.T) further generalizes the SPC concept, enabling the inclusion of a broader range of denoisers. Based on the fixed-point projection of demicontractive denoisers, Cohen et al. proposed the RED-PRO model [21] to theoretically bridge between PnP and RED priors. The indicator of the fixed-point set can serve as regularizer and plays increasingly important role in solving inverse problems. It is critically important to develop an efficient technique for testing the demicontractivity of given mapping [21]. Of course, obtaining pseudo-contracee denoiser also ensures the demicontractive property, as pseudo-contric contractive denoisers are a subclass of demicontractive operators shown in Figure However, ensuring nonexpansivity, more generally, a Lipschitz constraint on artificial neural networks is not easy in practice [88] Therefore, how to design an efficient training framework that enables neural networks learn theoretically guaranteed denoising mappings under the weak assumption, such as (T). is highly challenging problem. In this paper, will try answer the open question by establishing a theoretical connection between the GS denoiser and the pseudo-contractive denoiser. The GS denoiser is all you need- parameterizing convex potential functions with ICNNS is sufficient without complex training techniques.
expansive denoisers [56
47
30
46 nonexpansive denoisers [53 50 14
40
residuals or operators [5
42
35
Recently, spectral methods have been proposed to address the aforementioned problem [51 48 35 61].
The SPC-DRUNet trained by spectral methods has achieved the sfate-of-the-ar performance in image restoration problems [6]]. Spectral methods typically use the power iteration (PI) method to compute the spectral norm of the Jacobian matrix, which is then either normalized by dividing the convolutional kernels by the spectral norm [51] or incorporated into the loss function penalty during training 48 33 61] Ryu et al. (51) utilize a PI method, treating the convolution as linear operator that performs matrix-vector product. To balance speed and precision, PI only runs one iteration. They enforced the contractiveness of the residual - To of the denoiser To by real spectral normalization (RealSN), which normalized the spectral norm of each layer. To obtain FNE denoiser Te, which is equivalent to the nonexpansiveness of Qo = 2T I (see [16] Theorem 2.2. 10] and [7] Proposition 4.4]). Pesquet al. (88 added the penalty term of the spectral norm to the loss function. i.c.
x + > maxx (1, 1 (()) = Ex~p(x) o [T + - x
where p(x) is the data distribution, JQ (x) is the Jacobian of Qe at point X = ox (1 () n)(e E [0, the noise follows Gaussian distribution with mean and standard deviation al, A > 0 is penalization parameter, and the parameter E (0, 1) controls the penalty term. Then the denoiser Tg resolvent of maximally monotone operator (MMO). Later, the GS denoiser To = I Viaa was proposed 20, 34], where Vie : R + R U }co} is parameterized by the differentiable neural network such as ICNN (2 and DRUNet [66]. Based on the GS denoiser, Hurault et al. proposed the proximal DRUNet (Prox-DRUNet) 35 which requires that V is T-contractive with < 1. They fine-tuned the previously trained GS denoiser by the spectral method with the following loss:
[()) Txx- [ + n)
x|| +A ma (x + n)ll. e]-
During training, the spectral norm | (x + n)ll, is estimated with 50 iterations. Wei et al. trained the d-SPC (d < 1) denoiser by the spectral method with the following loss [61]
(0) = -xx=-(xx. [|To(x + n) - -x|² 2 A maxx [IddT + (1 Jr (x n)ll, 11.,,111) e).
However, these spectral methods constrain the spectral norm using a finite number of training samples
{x, ) (noisy and clean image pairs), rather than all samples from the entire space. It unable to accurately constrain the global spectral norm such as maXxER" 1|2./T (x) Ill. in [48], which may violate the SPC property. Moreover, PI methods have notable drawbacks: they can be computationally expensive (61), are not fully deterministic due to random initialization, and intermediate iterations do not guarantee a strict upper bound on the spectral norm
Amos et al. 12 introduced ICNNs, which guarantee the convexity by imposing non-negative con- straints on network weights and employing convex, non-decreasing activation functions. The gradients of ICNNs possess universal approximation capabilities (33) making them particularly valuable in data-driven optimal transport 44
[3] The GS denoiser leverages ICNNS to learn gradients of convex
implicit regularizers [20). Fang et al. (28) further proposed learned proximal networks (LPN), which

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are parameterized by the gradient of ICNN, In this work, we rethink the GS denoiser Ta = V Vele. where 10 is an ICNN, that can serve as truly SPC denoiser. We are the first to establish the essential connection between SPC denoisers and GS denoisers. The main contributions of this work are summarized as follows:
1. Theoretical contributions. We rethink that the GS denoiser corresponds to truly SPC denoiser, as demonstrated in Proposition 4.1 We also theoretically propose another novel construction method for truly pscudo-contractive denoisers, presented in Proposition 43 This work is the first to theoretically confirm the feasibility of training truly SPC denoisers and to practically address the unresolved challenge of training demicontractive denoisers posed by the RED-PRO model [2T]
2. Algorithms and applications. We integrate the learned truly SPC denoiser into the RED- PRO model and propose Algorithm|] for solving imaging inverse problems. Theorem 4.6 further establishes the convergence objective functions in RED-PRO, complementing the prior work [2] that focused solely sequence convergence.
3. Experimental validation. Through extensive numerical experiments, we validate the SPC property of the GS denoiser compared to spectral methods [48] 61]. Our results highlight
the ability of the method to balance interpretability and performance in addressing inverse problems.
Related works
Regularization models are important in image restoration problems, which can be formulated as follows:
min xER"
f(x) Ag(x),
where f is the data fidelity, g is the regularizer, and À > 0 is a regularization parameter. For example, the data fidelity fxx ~ ||xx - yyl corresponds to = Ax + with given linear operator A and Gaussian noise
2.1 PnP methods
PnP methods that combine splitting algorithms with denoiser priors have been widely applied in
practical problems (60
53
1
37
62
28
63, 38 and have achieved state-of-the-art performance in
inverse imaging tasks 666.
26
35
58. Venkatakrishnan et al. (60] first proposed the PnP method
using the alternating direction method of multipliers (ADMM). PnP methods solve the problem (2)
by replacing the proximal operator prox, g (x) with denoisers, such as non-local means (NLM) [13
and block-matching 3D filtering (BM3D)
24 within ADMM
forward-backwarr splitting (FBS),
also known as the proximal gradient method 9 Zhang et al. [66] extended PnP methods using
trained DNNs achieve state-of-the-an performance in image restoration [66]. The convergence of PnP methods has been extensively studied. Sreehari et al. established theoretical conditions for PnP- ADMM, requiring that the Jacobian VD, be doubly stochastic and symmetric matrix with all real eigenvalues in the range (0, 1] 53 Buzzard et al. provided Consensus Equilibrium interpretation on denoiser priors [1]. Chen
18 analyzed exxentpoint convergence under bounded denoisers, while Ryu et al. [51] proved the fixed-point convergence of PnP-FBS and PnP-ADMM using the Banach contraction principle, assuming strongly convex data fidelity and nonexpansiveness of the residual of DnCNN [67]. Diffusion models (DMs) 322 can also act as efficient PnP priors, which
have been widely used in physical sciences such as black hole imaging problems 63 68 and image
restoration 69
2.2 RED framework
Romano et al. [50] introduced the well-known RED framework, which constructs an explicit ob- jective function and flexibly incorporates various denoisers, such as NLM [13), BM3D (24]. or trainable nonlinear reaction diffusion (TNRD) 19]. The RED framework has been widely used in computational imaging [54 55
Reehorst and Schniter et al. [49] highlighted that some existing denoisers not satisfy the assumptions RED, and provided the SMD (score-matching by denoising) interpretation. To explore the relationship between PnP priors and RED, the RED-PRO
42

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framework was proposed from the perspective of fixed- fixed-point point projection. Since the fixed-poing set Fix(T) fx E Tx) = x} closed and convex [2T] Theorem 3.8], RED-PRO reformulates RED as convex optimization problem for image restoration via fixed-point projection, the regularizer g in is the indicator of the fixed-point set Fix(T), i.e.,
00 if Fix(T),
g(x) = 00; otherwise.
Building on idea of RED, the GS denoiser Ta = Vite was proposed (20 where the is differentiable function. Based the GS denoiser, Hurault al. [35 established the convergence theory of PnP methods. He et al. proposed simultaneous local and nonlocal RED (SLN-RED for image restoration [29]. To avoid tuning the regularization parameter, Cascarano et al. proposed the constrained RED called CRED [5] based on the discrepancy principle [27]
Preliminaries
Cohen al. first introduced the following demicontractive assumption denoisers 21 Here is the definition of demicontractive operators.
Definition 3.1. The mapping T + is d-demicontractive with d < if for any x E and a E Fix(T' it holds that
[Tx) z|
zd x||
The d-SPC operator is d-demicontraactvee operator.
Let C1, C2. C3, C4 denote the classes of all operators H + H satisfying the assumptions of demicontractive, SPC, NE, and FNE, respectively. For example, consider the inclusion defined as follows. Let T E C. be arbitrary. Then, for all x.y E H, it holds that
(x] - ( 1 < ||x y|1 - ||(x T(x)) - (y - T(y))l = IT(x) - T(y)ll ||x yll,
which means 7 C3. Therefore, C4 C3. The relationship between different classes of operators is shown in Figure] |
C1
C2
Ca
C1
Figure 1: Relationship between different classes of operators.
The operator T: H - H is called conically X-averaged for À 5 if there exists nonexpansive operator U such that T (1 -)] / + XU, where denotes the identity. In particular, when (0. the operator is X-averaged. class that plays important role in fixed-point algorithms (22 232.]. If U is FNE, then T = (1 ^) + AU X-relaxed FNE (A-RFNE).
Next, we give some equivalent relationships between d-SPC operators, conically averaged operators, and RFNE operators demonstrated in Proposition 3.2. We give the proof in Appendix A Proposition 3.2 (177). Let X
TLa- Then the following statements equivalent:
(i) Let T be d-SPC (d 1).

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(ii) T is a conically X-averaged operator.
(iii) T is a 2X-RFNE operator.
The following Proposition 3.3 serves key connection for exploring how to learn truly pseudo- contractive denoiser. Please see the proof in Appendix B
Proposition 3.3 (1)) Let R - T and p then T : H. is p-RFNE if and only for all x, E H.
(x y, vx) - R(y))
[[(x) R(y)
The residual It = I-T's is also called - --cocoercive in 3 Figure 2 shows all equivalence relationships
between x SPC operator, conically X-averaged operator, 2A-RENE operator, and _L -cocoercive residual, where X = 1-d
2X
- T
cocoercive 2A operator
T is conically >-averaged operator
T is a x SPC operator
T is 2X-RFNE operator
Figure Equivalent relationships.
In the following Proposition we introduce important property of a-averaged operators for the
convergence rate of objective functions about the RED-PR PRD model. The proof is given in Appendix C
Proposition 3.4 (165). Let T be a-averaged operator with (0, 1), then T is 1-a 1-a. strongly guasi-nonexxaansiee operator, i.e., for any E Fix(T), il holds that
[T(x) - z||² < x z||²
a xx - T(x)
(4)
Finally, give several equivalent characterizations of the T-smoothness property the entire space H. Here is the definition of T-smoothness.
Definition 3.5 (l9). Let L 0. The function 7 IR U foo} is L-smooth it is differentiable over H. and satisfies
VV((x) Vh(y)] S L|Ix - yll. Vx,y E H.
(5)
Theorem 3.6 (1)). Let h H + R U {oo} be convex function, differentiable over H, and let L O. Then the following claims are equivalent:
(i) h is L-smooth
(ii) For all x,y E H.
(h(()) - vh(y),x-y) [V/(x)= -
(6)
4 Learned truly SPC denoiser
4.1 The GS denoiser is all you need
In this section, we will prove that the GS denoiser Tó V in which v/ is an ICNN, can precisely correspond to truly SPC denoiser. In [20]. although Cohen et al. proposed the GS denoiser early, they failed to explore its connection with the SPC denoiser. The pursuit learning
SPC denoisers, once perceived as distant, is now within reach. Two inequalities (3) and 6 form crucial bridge, enabling us to establish the essential connection between the GS denoiser and the SPC denoiser. We give the following result theoretically guarantee the SPC property of the GS denoiser.

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Proposition 4.1. Consider a scalar-valued (K + 11)111lyyeeee neural network vo : RR" - R defined by (x) wo ZK b and the recursion
Φ H + b1), Ak: = Oi WWWAR + H&x + by), k= 2,3 K.
where = {w, b, JK (HaJR= (bs)k-1) are learnable parameters,
R
convex, non-decreasing and continuously differentiable scalar function, which operates pointwise. Assume that all entries of WK and w are non-negaiie, and let the Lg-smooth, then the GS denoiser To = V La-2 SPC operator.
Le
Proof. Since We and w are non-negative, it follows that th is convex from 2 Proposition 1]. Since to is Lg-smooth, by (6). we have
1 [V)o(()
(V)x(()) Vialy), x- y)
V Vitaa (y)]).
Lg
Let Tg = 7 V the, by Proposition 3.2 and recall 33) in Proposition
3.3 we directly derive I-d Lo-
i.e. d L Therefore, the denoiser Ta is LPC operator, which completes the proof. C Ly
Le
Given the ICNN the we train the GS denoiser To = Via with the following loss function
(()) = Pxx-ppxx. [x + n) - x||².
As previously proposed in (20). the training process is straightforward and does not require any additional spectral norm penalty.
Our Proposition 4.1 is the first to realize a truly Le L operator via the ICNN GS denoiser, whose assumption is weaker than FNE and nonexpansive, and thus easier satisfy in practice. Once the GS denoiser meets the - condition, the RED-PRO framework automatically guarantees Le sequence convergence and objective convergence rate, shown in Theorems 4.5 and 4.6, without requiring stronger FNE or nonexpansive assumptions. The result of Proposition 4.1 can further benefit existing PnP/RED thcoretical works by enabling the ICNN GS denoiser satisfy stronger FNE or averaged nonexpansive assumptions in two ways:
Controlling Lg S 1, c.g., by normalizing convolution kernels via spectral methods or penalizing the network's ' Lipschitz constant in the loss function, so that the FNE and nonexpansive assumptions required in (57 Assumption 2], [53 Theorem III.1], and 49 Lemma 5] are met;
Estimating Lg via the power method and tuning the weight w < 2 so that T. = wTp (1 / is " averagged operator, thus satisfying the averaged operator assumptions in [56] Assumption 2(b)] and [47 Theorem 3.5, Theorem 3.6]
Therefore, Proposition 4.1 provides valuable practical guidance for existing PnP/RED theoretical works that require FNE, and averaged nonexpansive assumption.
Remark 4.2. Although we use the same GS denoiser as in (20), the difference is that we are the first to theoretically address the relationship between d-SPC denoisers and GS denoisers, and also practically demonstrate the feasibility of training SPC denoisers that fully satisfy the demicontractive condition required by RED-PRO 21.. We believe that this theoretical finding is valuable.
35 our approach, like 20), directly embeds the underlying mathematical structure, i.e., and (6) into the denoiser, thereby naturally satisfying the SPC property. This eliminates the need for adding extra penalty terms in the loss function and overcomes the limitation that spectral methods cannot constrain the global spectral norm.
In contrast spectral methods 5T
However, as pointed out in 20). one limitation is that the non-negative weights may constrain the expressivity of ICNNs. We are now theoretically give another alternative construction method for any truly pseudo-contractive neural networks.
Proposition 4.3. Let Rg R - Rh" be a T.g -ipisicci continuous convolutional neural networks. Denote Rs Ro Lg then the denoiser To = I - Ro is a pseudo-contractivv operator, i.e.,
[T (x) 2x) < xx yl1 + |1(x Tx(x)) (y ( -To()))1??

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Proof. Since Ro is Lo-Liscchitz continuous, for any x,y E H. we have
[[) - Ro(y))l - Lg ||x - yll,
by and Sauthy-Schwarz inequality, we have
[[(o((x) Ro (x(y), x y)1l S Lo |x - yyl?.
We can derive that
(Ro(x) - Ro(y), x y) + Lu |x y|1² 2 O.
Thus, we have
(Ra Ro(y), - (Rg(x) Ro(y), x y) + LB ||x - y|1
By Example 20.8], it follows that T. = Ro is pseudo-contractive operator.
The core problem in Proposition 4.3 is how to training Lipschitz-constrrineed neural networks. Ryu et al. normalized each convolutional kernel KC with estimated spectral norm [[C]],,, [[]]] - which can
obtain 1-Lipschitz CNN. Anil et al. 3 proposed that by combining GroupSort activation functions with orthonormal weight matrices, can construct networks that provably 1-Lipschitz and capable of approximating any 1-Lipschitz function arbitrarily well. These methods can be used to train the w.Lipschitz-constraiiiee
neural networks Rg in Proposition 4.3 In this case, the Lipschitz constant Lg is equal then Rg Rs I, and Rs Ro can be viewed a residual connection, which is used to fit the noise distribution n. That is, Ro is obtained by minimizing the following loss function:
min [(0) = (=- Ro(( + n) 1'')
and the pseudo-contractive denoiser Te = Rg constructed. Morcover, Delattre et al. 25 controlled L-Lipschitz convolutional kernel K2 j s (), the training loss becomes: [())
[trog Σj= (K,) with
Creg(() = (KI(K)))L (oNK)i))ocx(K)))>) (k((K))>Lr
where ØGI denotes the spectral norm computed by Gram iteration (GI), which is more efficient and accurate than the power method, and - 1z>1 indicates if L, and O otherwise.
Although the above methods yields pseudo-contractive denoisers by training Lipschitz-constrained neural networks, such networks may practically suffer from limited expressive capacity 36).
Remark 4.4. According Proposition 4.3 long accurately compute the Lipschitz constant of the neural network, we can construct the pseudo-contractive neural network. The recent work [25] has shown that is feasible to accurately obtain the Lipschitz constant of neural networks. We believe that more expressive truly pseudo-contractive neural networks with inherent interpretability shown in Proposition 4.3 will be developed in the future.
4.2 RED-PRO with the learned SPC denoiser
Based on the fixed-point projection of the demicontractive denoiser, Cohen et al. proposed the following RED-PRO model
min f(x),
(8) yl The
Fix
20 where the denoiser Ta is assumed to be d-demicontractive (d < 1) and f(x) = 1 || hybrid steepest descent algorithms 64
651 is used solve 88 i.c.
= Tw (z k- pa ff(z*11)
(9)
where T,, wTy (1 II, and { heN is diminishing, i.e., Σken P4 = +00, lim s00 2a
and w E (0, 1_d) for d-demicontractive denoiser T. Proposition 3.2 shows that Tie is wvl-averaged and thus always nonexpansive for < 1- d. Algorithm shows the detailed steps of RED-PRO with the learned truly SPC denoiser Té, which is written into the following compact form,
Sshows
x" (x
14 V fT. 11)
(10)
7

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where Tu = wTg + (1 - I I. In fact, the above iteration 10 is equivalent to the iteration 9
Algorithm 1 RED-PRO with the learned truly SPC denoiser
Require: initialization x" E IR" Akk = (1+k)a= w E (0, Lo ), and the GS denoiser To = I Vi/0.
2
1: for k = 1... K do
2:
y = wx)x =
3:
x y Pa ()y*)
4: end for
Ensure: x"
The known Theorem 4.5
provides the convergence guarantee of Algorithm T to an optimal solution of ( with the learned truly d-SPC denoiser Tg. Compared to RED-PRO [2I] Theorem 4.3), we can extend the interval of from (0, 1 (0, d), and explore that depends the Lg-smooth property of the ICNN. We further complement the convergence rate of the objective function for
RED-PRO in Theorem 4.6 The proof is given in the Appendix
Theorem 4.5 (165 21 Let To = I Vihy be a continuous d-SPC denoiser, and f(x) be proper convex l.s.c. differentiable function with L-Lipschitz gradient. Then the sequence {x" JeEN generated solution of (8).
by Algorithm converges
Theorem 4.6. Let bx"hee and {y" be sequences generated by Algorithm 1 Assume that = arg minnxeFix(T)) f(x) is the solution set of the RED-PRO model and the sequence {x" jkEN is bounded, then
(i) For any x E S and k 2 1, there exist D1, D2 > 0 such that
D}
cD
uk S ck +
k
where us min((V x') S < 2k}.
(ii) For any x S and l 2 we have
Dj chila
cD ka
f(y f(x)) <
where Kbeat arg min f(y').
Remark 4.7. The boundedness of (x* JEEN in Theorem 4.6 is straightforward to verify. Specifically,
we can replace Ti with P(o.1j" o T., in Algorithm where P(o.1]" denotes the metric projection
onto the unit hypercube [0. 1|m Thus the sequence {y* IkeN is bounded such that y E [0, 1]".
Since x y - M4 V (y**,, for any z E Fix(T), we have | 2|| < y x|| + V (>)()) Therefore, the sequence (x*) } is also bounded.
Remark 4.8. Compared to the convergence results of RED-PRO in 21 Theorem 4.3 and Theorem 4.4], we provide new complementary analysis by establishing the outer convergence rate of the objective function in Theorem 4.6 (ii).
5 Experiments
In this section, we present some experiments to evaluate the performance of the learned truly SPC denoiser. We benchmark it against state-of-the-an spectral methods, including MMO 48 and SPCNet [6]]. Specifically, we validate the SPC property of the learned denoiser. RED-PRO with the learned truly SPC denoiser has the theoretical guarantee, which be applied complicated imaging inverse problems. Our primary focus is on achieving theoretical interpretability rather than pursuing state-of-the-an performance.
5.1 Implementation details
We compare spectral methods with the DrCNN architecture. SPCNet 6 adopts 0.5 for MNIST
and CelebA datasets (39 43), and d 0.8 for BSD400 [45), while the MMO method 48 uses

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= -1 in T.) Spectral methods are configured with À = 10-3, = =0 0.1, and 20 PI iterations for CelebA and BSD400, or 30 iterations for MNIST. The ICNN models start with an initial learning rate of 10-3, decaying to 10-² after half the epochs. The DrCNN models begin with 10-2 decaying to x 10-5 mid-training.
For the MNIST dataset, ICNN is implemented with four convolutional layers, each containing 64 hidden neurons and softplus activation function ((z) 1 log(1 e with 10. For BSD400 and CelebA, ICNN uses 256 hidden neurons with 100. All models are trained for Gaussian denoising with noise level of 5/255 and batch size of 128. Training spans 50 epochs for MNIST and BSD400, and 30 epochs for CelebA. All experiments are conducted on one NVIDIA A800 GPU using the PyTorcch framework.
5.2
Validation of SPC property
We test the SPC property of the learned GS denoiser T3 = I - Te with ICNN Ve. MMO [48] and SPCNet 61 on two MNIST and testl2 datasets. We calculate the maximum d defined by
à IT(x) Tyy] x y/1 11x -xx- T(x)) (y (
where y - n, (N0 (1). Noise levels uniformly sampled 11 points in the interval [10-5, 10-2). As shown in Figure 3 the SPCNet [6] obtains maximum d that exceeds 0.8 and 0.5 at a noise level 10-5, whereas the MMO method [48] yields a d value that exceeds 1. Therefore, denoisers trained by spectral methods violate the SPC property.
10
3000
-30
MNIST
(b) test
Figure 3: Validation of the truly SPC property two different datasets.
34
0,018
Objetivv fundion
0.3
0.000
0.104
116
29
0.4
0.2
20
40
50
100
20
80
G11
100
Iteeatinn
Iteration
(b)
Figure 4: Convergence of Algorithm D on one image in the CelebA dataset. (a) Objective function. (b) PSNR.

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5.3 Competitive results with theoretical guarantees
1 lly Ax and PSNR (dB) throughout the In Figure 4 we show the trend of the fidelity term 20 iterations with different w. We provide additional results Appendix E
In the following, we demonstrate the effectiveness of RED-PRO with the SPC denoiser in inverse problems, highlighting the ability achieve competitive results while strictly satisfying theoretical constraints. Compared to non-SPC methods such as DPIR [66], which achieve high PSNR in only iterations but do not converge with more iterations (see Appendix G.5 in [35.. As shown in Table D RED-PRO can offer both competitive performance and guaranteed convergence. We also provide visual PSNR curves shown in Figure to clearly demonstrate that non-SPC methods may not converge.
340
06.
222
DUCO0
DUISS
boodor
(a) Man
(b) House
Figure 5: PSNR curves RED-PRO with the learned truly SPC denoiser and non-SPC methods on two images in the Gaussian deblurring task.
Table Comparison of RED-PRO with various non-SPC denoisers on the Gaussian deblurring task. All non-SPC and the ICNN GS denoisers use Algorithm with the same hyperparameters.
Methods
Parrot
House
Boat
Couple
Man
SPC-DrCNN
61
25.73
31.43
28.53
28.17
29.58
MMO
25.87
31.58
28.57
28.27
29.60
DnCNN
67
25.30
26.74
26.23
25.98
26.47
DRUNet
[66
27.00
30.44
29.15
28.64
29.85
GS denoiser
35
27.38
32.58
29.41
29.08
29,91
Ours
27.17
32.40
29.54
29.17
30.25
Conclusion
In this paper, we proposed novel perspective to construct truly SPC denoiser by directly embedding the underlying mathematical structure into the neural network architecture. Our theoretical analysis shown in Proposition 4.1 rethink rethiink that the known GS denoiser (20] built upon an ICNN, definitely satisfies the d-SPC property, which plays crucial role in convergence in PnP methods and RED framework. Unlike spectral methods that require additional penalty terms and suffer from high computational cost due to costly PI iterations, our method naturally guarantees interpretability while offering the advantage in terms of time complexity. Furthermore, our theoretical insight shown in Proposition 4.3 can pave the way for future research in developing interpretative neural networks for imaging inverse problems.
10

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Acknowledgments
We sincerely thank the Associate Professor Hui Zhang for valuable discussions. We sincerely thank four anonymous reviewers for their valuable and constructive feedback, which has greatly improved our work. This work was supported by the following grants: the National Key Research and Development Program of China (No. 2020YFA071 (300YAA071304)) 13504), the National Natural Science Foundation of China (Grant No. 12471401), and the National Natural Science Foundation of China Young Scientists Fund (Grant No. 12401419).
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NeurIPS Paper Checklist
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: we rethink that the GS denoiser corresponds to truly SPC denoiser, as demonstrated in Proposition 4.1 This work is the first to theoretically confirm the feasibility of training truly SPC denoisers and to practically address the unresolved challenge of training demicontractive denoisers posed by the RED-PRO model 2
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Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: one limitation is that the non-negative weights may constrain the expressivity of ICNNs.
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20

---

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22

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A The proof of Proposition 3.2
Let --2. We first show the equivalence between conically X-averaged and d-SPC operators. Let
S (1 --d + dI, then
[15(x) - S(y)]l = II(1 d)(T(x) T(y)) d(x y) = (1 - d) [[7(x) T(y)ll + - y11
d(1 d) ||x T(x)) (y - -(y)))
S is nonexpansive + [|5(x) - S(y)1 - ||x y11²
(1-d) d) [x] - T(y) d |||
- yyl
- d(1 d) 11(x - T(x)) - (y -T(y))l1 S ||x - yl1
[T(x) T(y)ll < |x x ||1l +d]((x - -T(x)) - (y TTy)]
+ T is dPSPC
By the above equivalence, we obtain T = (1 - X)I + AS is conically X-averaged, where S is nonexpansive.
(i) = (ii): Take = 11, then the FNE operator V is equivalent to j-averaged operator. There
exists a NE operator S such that V = St1 then S = 2V - I. Substituting S = 2V I into
2
T= A)I AS yields
T = (1 2 + 2XV,
hence T rel)-delaxed FNE (i.e. 2X-RENE). The conically X-averaged operator equivalent the 2X-FNE operator.
B
The proof of Proposition 3.3
Let T = pS (1- (1-x)] for a FNE operator S. Since S is FNE, then
[|5(x) S(x) - xx
yl1 [|(x S(x) (y - S(y))1
Since
[[S(x) (103)-S(y)) S(y)l = [1(5(x) x) + (x + (y S(y))11
= =[((x)) - x|| + -|x + fly - 5(y)
2(5(xx) x, x, 2(5(x) x,y - S(y)) + + y,y- S(y))
and
((x))(x)) - (y - -(y))] = =1|x - S(()) - 2(x 5(x), S(y)) + S
From
[15(x) S(y) < yl12 11x S(x)) (y 1(
we obtain
||x - 5(x)1 + +lly -ly-- S())11 < (x S(x), S(y)) (S(x) - x,x y) y)
(S(x) - x, y - s(y)) (x- -x-y.y -S(y))
(S(y) y, S(x) x) (x s.x)((xx x)
+ (S(x) - x, 5(y) -y) + (x - y, s(y) -y)
-(S()) S(x) - x) + (S(x) - S(y) - y),
i.e.
(S(y) - x, S(x) - x) + (S(x) -,
y, S(y) - y) 2 ||5(x) - x + S - 2
(11)
Since S(x) x (T(()) x) and S(y) (T((y) y). then |I is equivalent to
(S(y) x, T(x) - x) + -,T()) - -y) N _ [[((x) - x|| + [[ y²² (12)
23

---

Let R = I - T. then
(5(y)-xx T(x) x) = -(S(y)- -y) (y -y-x -x), x)
= (x - y, R(x)) + 1 + (R(x), R(y)).
(13)
Similarly, we have
(S(x) y, Tyy - y) = (y x, R(y)) + (B(x), R(y)).
(14)
1
By 12 and above two equations
13) and 14). we have
(x - y, R(x) R(y)) 2 1 [[((x) R(
C
The proof of Proposition 3.4
Since T is a-averaged, by Proposition 3.2 (ii), then we have 1
I-d = a = d o-1
iee. the
a
operator T is 1-a -SPC operator,
[T(x) - T(y)] S ||x - y/ 1-a a
- ((xx-((x)) - (y T(x))1]
(15)
a
Let z Fix in [5]. we finally obtain
[T(()) 211 < ||x 2/|²
a [[T(x) -x122
D The proof of Theorem 4.6
Proof. (i) For any z E Fix(T), since y = (w and Tw is -na averrged,, by 4 we have
l 2|| S | * - 2||
1- w | Tw (x-
w
follows that
xx* 2||² 2
11y* 146 V f(yx)) 21
1| z1(² 2m fy + 2 Vf(
$ ||x0 - =11² 1 - - w || Ten (x*-1) 2
21y (y* 2, ()(x))) + 2 [
(16)
According to the known conditions, there exist D1, D2 > 0 such that
||y* 2|| < || 2| D1,
/ D2-
If uæ < 0, then the inequality holds. Otherwise, applying 16 with z replaced by x', we obtain, for
any 1.
||x² - -x|1 < | - x112 241,
V y - xx) + D2
arrange the above inequality, we obtain
2μ, (Vf(y') x') < || x' ||
|| ||x'=x|i
-x'1" + 12
Since M* 2 My for all l: S < 2k - 1, summing it for all j = k, k + 1, ... 2k 1, yields
2k-1 1
2k 1
2 Σ
M3 (VI(y'),'' -xx < 1|x - x112 | - x + Σ
2 D
n
j=k
j=k
2h-1
< DD + D3
Σ
14
2
j=
$ SD + (2
k 1)T
D3c22
< <D + (1 + k))aai
24

---

By the definition of uy and us 2 0, it follows that
2k
2k
2 Σ ( j=
-x)
2uk
Hj 2 22akt1ak11
=(2k)) -uu
uu 2 ck-
uuk
==
then
D cki +
cD
Dt cki
cD2 ka
Uk -
(1+ + 2k
k2)a
(ii) In order to prove the rate terms of the outer objective function S. we will use (i) and for simplicity define & = arg
xx) k 2k}. We apply the sub- sub-gradient
inequality on the convex function f to obtain
DJ
cD2
f(ybest) f(x) f(y²) f(x) (Vf(y*). x) 2 cki k
the first inequality follows from f(y S f(y"), and the last inequality follows from (i).
E Other experiments about convergence
We evaluate the convergence of Algorithm D on the CelebA dataset for the Gaussian deblurring
task with Øuu 1, Tnoise 0.02. We set K 100, Pa 9 255 (k + 1)-0.1 We first compute the spectral norm Jv (x*) I|. where {x* K
k== is the iterative sequence. Here we run the PI method with 200 iterations. As shown in Figure 6 we can estimate the tight Lipschitz constant La m
v (x)lle 2 2.5, then 2/L0 < 0.8, w E (0, 0.8).
2.5
2.4
2.1
2.0
2.0
1.9
1.9
1.8
1.8
20
40
(20
100
20
40
60
80
100
Iteration
Iteration
(a)
(b)
Figure 6: Spectral norm
|| (x+) Il, at the k-th iterative point x² on two images.
We provide additional experiments demonstrate the convergence of Algorithm I four images in the CelebA dataset. The objective function and PSNR trends during iterations are shown in Figure 7
Hyperparameter Setting
We manually set the step size Pa [1+k)= and the weight of Algorithm T to achieve the best performance on the CelebA dataset. All hyperparameters are set to be the same for all images. The
hyperparameters are summarized in Table 2
Table 2: Parameter setting for CelebA dataset.
Parameter
Oblu
Onoise
.02
Coour
Onotse
.04
Obiur 2, (Toossc
.02 de
1, Anoise .04
2550 nouse
2550
14 2550m
20 255dmnies
0.1
0.1
0.09
0.25
0.5
0.75
0.75
0.75
25

---

0.010
0.5
30.0
0.5
8.016
20.5
0.014
29.0
Obyeclve
0.013
28.0
0.5
0.012
0.4
0.011
0
20
40
(3)
50
100
20
100
Iteration
(a)
(b)
36
0.015
0.4
Ooyccive function
0.014
PSNR
0.013
133
0.6
32
0.5
0.012
31
8.011
30
20
100
40
60
(c)
(d)
33
0.015
Obyccive furction
0.014
M = 0.3
0.4
32
0.013
0.5
0.012
0.4
0.3
0.011
Iteration
Iteration
(c)
(f)
0.6
0.015
35
Objective function
0.014
0.5
34
0.2
PSN
0.013
0.002
0.3
0.011
0.2
40
60
80
100
Iteration
Iteration
(g)
Figure 7: Convergence of Algorithm D on other images in the CelebA dataset. (a) Objective function. (b) PSNR.
26

---

G Compared with state-of-the-art methods
We first train the SPC denoiser using the CelebA dataset and then apply Algorithm perform deblurring. We compare with diffusion-basse method DifiPRR 69)] PnP-PGD [44, and DPIR [66]] As demonstrated Table we evaluate the efficacy of RED-PRO with the learned truly SPC denoiser across a range of blur intensities, noise levels, and evaluation metrics. We give the hyperparameter
setting RED-PRO in Table [ Appendix F We provide visual comparisons in Figure g DPIR achieves the best results, our method ranks second, and DiffPIR effectively restores fine details but occasionally alters facial expressions.
Table 3: Deblurring results CelebA 20 samples.
METHOD
alaur
Gooise
Datur
Cnoise
Obter
2, nosis 02
Tllar
2, Onoise
PSNR(T)
SSIM(1)
PSNR(1)
SSIM(1)
PSNR(1)
SSIM(1)
PSNR(1)
SSIM(1)
DimPIR
30.8
.66 .03
.82
28.6
27.6
.05
PaP-PGD 34
31.4
1,9
.87 .02
27.6 0.9
.71 005
29.9
.85 + 00
28.8 - 2.0
.81 .05
DPIR [60
33.2 + 300
.92 = .03
31.8 2.6
.89 00
30.1
88 + .05
29.1 = 2.2
88 .05
(a) Clean image
(b) Degraded image
(c) DiffPIR (30.07 dB)
(d) PnP-PGD (30.53 dB)
DPIR (31.95 dB)
Ours (30.93 dB)
Figure 8: Visual comparison on CelebA for Gaussian deblurring with Tblur 1, Tnoise 0.02.
27

---

We consider sparse-view computed tomography (CT) measurement model:
min XCFi N
Σ
|Ax b;ll; 1=1
where b; E I is the measured sinogram for the i-th projection, and A; is m X discretized Radon transform matrix. For RED-PRO, we use jas [|]]"[1+ where
[Ai, A2.... AN]",
and w = 0.1. The GS denoiser is trained on the public Mayo-CT dataset ?? J. We simulate CT sinograms using parallel-beam geometry with 200 angles and 400 detectors. We compare with FBP and the recommended method in (38). T presents the results for CT reconstruction. Despite RED-PRO's theoretical guarantees, empirical performance in CT reconstruction may be inferior to that of PnP-ADMM.
Table 4: Numerical results for CT reconstruction the Mayo-CT dataset, computed over 128 test images.
Method
PSNR
SSIM
FBP
20.233 + 0.034
0.1763 + 0.0138
RED-PRO
30.057 + 0.488
0.8190 + 0.0075
PnP-ADMM 28
34.216 + 0.597
0.8938 + 0.0077
28


Reviews:
[{"review_id": "KHYcxKb5f2", "reviewer": "Reviewer_1SLa", "summary": "This method combines gradient step denoisers with input convex neural networks (ICNNs) to structurally ensure the pseudo-contractive property. It avoids additional spectral norm penalties while guaranteeing the theoretical convergence of the RED-PRO model. Experiments demonstrate stable performance in image deblurring tasks, balancing interpretability and effectiveness.", "questions": "1. The experiments are primarily conducted on relatively simple datasets such as MNIST and CelebA. Would the method still perform effectively on more complex or diverse datasets, such as ImageNet or real-world natural scenes?\n\n2. Could the authors consider including experiments in the domain of medical image reconstruction, such as MRI or CT, to further demonstrate the generalizability of their method to high-impact inverse problems?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper is the first to prove that gradient step denoisers parameterized by ICNNs are inherently d-strictly pseudo-contractive.\n2. The learned SPC denoiser is integrated into the RED-PRO framework, and both sequence and objective convergence are established.\n3. Experiments demonstrate that the learned denoiser strictly satisfies the SPC condition across noise levels and achieves competitive or superior performance in image deblurring compared to PnP-BM3D and PnP-DnCNN.\n\nWeaknesses:\n1. The denoiser is constructed using ICNNs) to ensure convexity and, consequently, the pseudo-contractive property. However, due to the architectural constraints of ICNNs—specifically the requirement for non-negative weights and monotonic activation functions—the expressive capacity of the model is inherently limited. This may hinder its ability to capture complex image structures or non-convex priors, especially in high-complexity tasks such as natural image denoising or super-resolution.\n2. Although the proposed method avoids frequent power iteration steps used in spectral normalization, it still requires estimating the global Lipschitz constant of the denoising network.\n3. The experiments are conducted on relatively simple datasets such as MNIST and CelebA. While CelebA contains real facial images, the overall image diversity and structural complexity are limited.", "was_revised": true, "final_justification": "The authors provided additional experiments on CT. The scale of the experiments is relatively small, and as other reviewers have pointed out, the compared algorithms are also relatively simple. I would like to keep the current score.", "rebuttal": "# To Reviewer 1SLa\nWe sincerely appreciate your insightful feedback and positive evaluation of our work. We have addressed your concerns regarding theoretical guarantees, experimental validation, and practical applicability as follows:\n## Reply to Weaknesses\n- **Expressive Capacity of ICNNs:** We acknowledge that the non-negative weight and monotonic activation constraints may limit the model’s ability to capture highly complex image structures or non-convex priors. Our primary goal is to ensure the truly pseudo-contractive property for theoretical guarantees, which we believe is a significant contribution. And we believe the trade-off between interpretability and expressiveness is a promising direction for future research.\n- **Lipschitz Constant Estimation:** While our method circumvents frequent power iterations, the estimation of the global Lipschitz constant remains challenging. We are actively exploring efficient estimation techniques and plan to incorporate these in our future work.\n- **Experiments Complexity:** We selected MNIST and CelebA to demonstrate the theoretical and practical feasibility of our approach. We agree that evaluating on more complex and diverse datasets would strengthen our findings. We have already included comparisons with various methods on natural images in **Table 1**. \n\n## Reply to Question\n**A1.** We agree that evaluating our method on more complex and diverse datasets, such as ImageNet or real-world natural scenes, would provide a more compelling demonstration of its effectiveness. \n\nWe train an FNE DnCNN using the spectral method [40], which is employed in PnP-FBS and RED, as detailed in the following **Table 1**. In contrast, RED-PRO uses the GS denoiser described in Proposition 4.1. The quantitative results in **Table 1** show that RED-PRO outperforms comparable methods on the House and Butterfly images.\n\n**Table 1**. Numerical results of different algorithms on the deblurring task with a Gaussian blur kernel ($\\sigma_{\\mathbf{A}} = 1.6$) and noise level $\\sigma =2.55$. The best two results are highlighted using bold and underline, respectively.\n\n| Methods               | House  | Boats  | Butterfly | Couple | Peppers |\n|---------------------|--------|--------|-----------|--------|---------|\n| PnP-FBS [40]         | 30.49  | 27.36  | 25.51    | 26.90  | 26.00   |\n| RED [2]             | $\\underline{32.36}$  | **29.56**  | $\\underline{28.35}$ | **29.19**  | $\\underline{27.18}$ |\n| CRED [12]            | 32.28  | 29.44  | 28.70     | 29.11  | **27.46**   |\n| RED-PRO                | **32.40**  | $\\underline{29.54}$  | **28.75** | $\\underline{29.17}$  | 27.17 |\n\n**A2.** We also appreciate your suggestion to explore medical image reconstruction tasks, such as MRI or CT. We consider a sparse-view computed tomography (CT) measurement model defined as follows: \n$$ \\min_{\\mathbf{x}\\in \\mathrm{Fix}(T_\\theta)} \\frac{1}{N}\\sum_{i=1}^N || \\mathbf{A}_i \\mathbf{x}-\\mathbf{b}_i ||^2, \n$$ \nwhere $\\mathbf{b}_i \\in \\mathbb{R}^m$ is the measured sinogram for the $i$-th projection, and $\\mathbf{A}_i$ is an $m \\times n$ discretized Radon transform matrix. For RED-PRO, we set the paremeter $\\mu_k =\\frac{2}{|| \\mathbf{A} ||^2(1+k)^{0.01}}$, where $\\mathbf{A} =[\\mathbf{A}_1,\\mathbf{A}_2,\\ldots, \\mathbf{A}_N]^\\mathrm{T}$, and $w=0.1$. The GS denoiser is trained on the publicly available Mayo-CT dataset. We simulate CT sinograms using a parallel-beam geometry with 200 angles and 400 detectors. The results for CT reconstruction are presented in **Table 2**.\n\n**Table 2**. Numerical results for CT reconstruction\non the Mayo-CT dataset, computed over 128 test images. \n\n| Method | PSNR                | SSIM                |\n|--------|---------------------|---------------------|\n| FBP    | 20.233 ± 0.034      | 0.1763 ± 0.0138     |\n| RED-PRO   | 30.057 ± 0.488      | 0.8190 ± 0.0075     |"}, {"review_id": "gzafY8W7KI", "reviewer": "Reviewer_Ln5k", "summary": "The RED-PRO algorithm offers a robust approach to solving inverse problems by incorporating a denoiser within a fixed-point iteration, with convergence proven for the broad family of demicontractive operators, in particular pseudo-contractive denoisers. However, the practical construction of pseudo-contractive denoisers has presented a significant open challenge. This paper directly addresses this by adopting gradient-driven, or gradient-step, denoisers. The main contribution is a theoretical proof establishing that these gradient-driven denoisers are indeed pseudo-contractive, thereby guaranteeing RED-PRO's convergence when they are employed. Furthermore, the authors present an additional theoretical result detailing how to construct a pseudo-contractive denoiser from a Lipschitz network. Experimental validation in image deblurring supports these theoretical claims, confirming the pseudo-contractive nature of the trained denoisers, the convergence of RED-PRO, and showcasing their favorable performance in PSNR and SSIM.", "questions": "1.  Please clarify in the experimental section whether the denoiser used is gradient-based or constructed using Proposition 4.3. \n2. Elaborate on the advantages and disadvantages of gradient-step denoisers compared to denoisers built upon Proposition 4.3.\n3. Extend the experiments to include other inverse problems, such as super-resolution, and an application where convergence, stability, or interpretability is crucial (e.g., medical imaging).", "limitations": "While the authors prioritize theoretical interpretability over state-of-the-art performance, the paper should still discuss the limitations and applicability of gradient-step denoisers within the RED-PRO algorithm. This discussion should compare them against other state-of-the-art approaches in both academic and real-world settings, detailing their respective pros and cons, performance gaps, and potential future strategies for improvement.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\nThis paper is well-written, clearly presenting the problem it addresses along with the necessary background and relevant literature. The theoretical derivations, which form the core contribution, are clear and well-supported. The significance of this theoretical work lies in the fact that it directly leads to a practical algorithm for inverse problems that offers provable stability and convergence, a characteristic that can be crucial in many real-world applications.\n\nWeaknesses:\n1. The paper's theoretical strength lies in its straightforward connection of two prior works to achieve provable convergence. However, this simplicity simultaneously suggests a limited degree of independent originality.\n2. The current experiments, limited to image deblurring, could be expanded. Extending them to other well-known inverse problems would significantly strengthen the case for RED-PRO with SPC denoisers as a general solution. \n3. Continuing the previous point, in common academic inverse problems like image super-resolution, approaches such as diffusion models often exhibit significantly superior performance to RED-PRO. Therefore, I recommend including experiments in applications where stability is crucial, thereby highlighting RED-PRO's relevance and utility in practical scenarios.\n4. Proposition 4.3, though linked to the paper's broader discussion, does not seem to fit seamlessly within the focus of the paper, which can slightly obscure the main contribution.", "was_revised": false, "final_justification": "", "rebuttal": "# To Reviewer Ln5k\nWe sincerely appreciate your constructive and positive feedback.  We have addressed your main concerns regarding theoretical originality, experimental validation, and practical relevance as detailed below.\n## Reply to Weaknesses\n- **Theoretical Originality:** We acknowledge that our theoretical contribution primarily establishes a straightforward connection to prior works, and we appreciate your feedback regarding the level of originality. Existing theoretical studies on PnP/RED methods typically assume that the denoiser is either FNE, nonexpansive, or averaged nonexpansive, all of which are special cases of pseudo-contractive denoisers. While most studies employ spectral methods to construct denoisers intended to satisfy these assumptions, but in practice, they often do not meet the theoretical conditions. Our work is the first to provide a theoretical proof that the gradient-step denoiser corresponds to a pseudo-contractive operator, which is crucial for enhancing the theoretical understanding of PnP/RED methods.\n- **Experimental Expansion:** In response to your suggestion, we have expanded our experimental validation to include CT reconstruction and super-resolution tasks in **Table 2** (see rebuttal to Reviewer 1SLa) and **Table 3**, respectively. These experiments demonstrate the broader applicability and robustness of our method.\n- **Proposition 4.3:** This Proposition provides a theoretical framework for constructing pseudo-contractive denoisers with $d=1$. To practically implement this, it may be necessary to combine the method from reference [20]. We acknowledge that Proposition 4.3 may not seamlessly fit within the paper's focus. We will revise the manuscript to better integrate this proposition and clarify its role in the overall contribution. \n\n## Reply to Question\n- **A1:** In our experiments, we use the GS denoiser obtained from Proposition 4.1, rather than the denoiser constructed using Proposition 4.3.\n- **A2:** Proposition 4.3 provides a theoretical construction for pseudo-contractive denoisers with $d=1$, potentially providing greater expressive capacity. However, it requires computing the network's Lipschitz constant, which can be challenging. In contrast, the gradient-step denoiser obtained from Proposition 4.1 is a $\\frac{L_\\theta-2}{L_\\theta}$-SPC denoiser. It is practical and easy to implement using existing ICNN architectures, though it has limited expressive capacity due to architectural constraints. \nIn summary, denoisers from Proposition 4.3 may be more expressive but less practical, while gradient-step denoisers from Proposition 4.1 are more straightforward to implement but have limited expressive capacity.\n- **A3:** We have included detailed experiments on other inverse problems, such as super-resolution (see **Table 3**) and CT reconstruction (see **Table 2** in rebuttal to Reviewer 1SLa), to demonstrate the versatility and effectiveness of our method.\n\n**Table 3**. Numerical results of different methods on the $\\times 2$ super-resolution task with noise level $\\sigma=2.55$.  The best two results are highlighted using bold and underline, respectively.\n\n| Methods             | House  | Boats  | Couple | Peppers | Man   |\n|--------------------|--------|--------|--------|---------|-------|\n| PnP-FBS [40]        | 27.76  | 25.35  | 25.08  | 24.13   | 26.17 |\n| RED [2]            | **31.58**  | $\\underline{28.35}$ | 27.91  | 26.45   | 29.10 |\n| CRED [12]           | 31.41  | 28.33  | $\\underline{27.94}$  | **26.66**   | 29.08 |\n| RED-PRO               | $\\underline{31.53}$ | **28.43** | **28.00** | $\\underline{26.49}$ | **29.31** |\n\n## Reply to Limitations\nThank you for your suggestion. We have provided comparison experiments with state-of-the-art methods in the following **Tables 4** and **Table 5**. These results illustrate the respective advantages and disadvantages, as well as performance gaps, between our approach and other methods. Our findings demonstrate that we can effectively achieve the trade-off between interpretability and performance.\n\n**Table 4**. Deblurring results of different methods on CelebA over 20 samples.\n\n| Method  | PSNR ($\\sigma_{blur}=1$, $\\sigma_{noise}=.02$) | SSIM ($\\sigma_{blur}=1$, $\\sigma_{noise}=.02$) | PSNR ($\\sigma_{blur}=1$, $\\sigma_{noise}=.04$) | SSIM ($\\sigma_{blur}=1$, $\\sigma_{noise}=.04$) | PSNR ($\\sigma_{blur}=2$, $\\sigma_{noise}=.02$) | SSIM ($\\sigma_{blur}=2$, $\\sigma_{noise}=.02$) | PSNR ($\\sigma_{blur}=2$, $\\sigma_{noise}=.04$) | SSIM ($\\sigma_{blur}=2$, $\\sigma_{noise}=.04$) |\n|--------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|-----------------------------------------------|\n| Blurred and Noisy  | 27.0 ± 1.6  | .80 ± .03 | 24.9 ± 1.0 | .63 ± .05 | 24.0 ± 1.7   | .69 ± .04  | 22.8 ± 1.3  | .54 ± .04 |\n| PnP-BM3D [1] | 31.0 ± 2.7   | .88 ± .04  | 29.5 ± 2.2  | .84 ± .05  | 28.5 ± 2.2  | .82 ± .05 | 27.6 ± 2.0 | .79 ± .05 |\n| PnP-DnCNN [2] | 32.3 ± 2.6 | .90 ± .03  | $\\underline{30.9 ± 2.1}$   | .87 ± .04  | 29.5 ± 2.0| .84 ± .04 | 28.3 ± 1.8 | .79 ± .05|\n| DiffPIR [60] | 30.81 ± 2.0 | .86 ± .03  | 29.5 ± 1.8   | .82 ± .03  | 28.6 ± 2.0 | .80 ± .05 | 27.6 ± 1.8 | .77 ± .05|\n| PnP-PGD [28] | 31.4 ± 1.9 | .87 ± .02  | 27.6 ± 0.9   | .71 ± .05  | $\\underline{29.9 ± 2.3}$ | .85 ± .05 |$\\underline{28.8 ± 2.0}$ | .81 ± .05|\n| DPIR [57] | **33.2± 3.0** | **.92 ± .03**  | **31.75 ± 2.6**| **.89 ± .04**  | **30.1 ± 2.5** | **.86 ± .05** | **29.1 ± 2.2** |**.83 ± .05**|\n| RED-PRO | $\\underline{32.4 ± 2.8}$| $\\underline{.92 ± .03}$| 30.8 ± 2.3 |$\\underline{.88 ± .03}$| 29.3 ± 2.3  |$\\underline{.86 ± .04}$|28.4 ± 2.0|$\\underline{.83 ± .04}$|\n\n**Table 5**. Comparison with state-of-the-art methods. Test grayscale images are degraded by the Gaussian kernel with the noise level $\\sigma = 2.55$.\n\n| Method         | Pepper | Craft | C.man | Couple | Man   | House  | Starfish | Butterfly | Boat   |\n|----------------|--------|-------|-------|--------|-------|--------|----------|-----------|--------|\n| PnP-DRS [29]    | **29.37**| $\\underline{26.91}$ | $\\underline{27.46}$ | $\\underline{29.70}$  | 29.59 | *33.25* | **29.94** | **30.66** | 29.46  |\n| PnP-PGD [28]    | 27.29  | 26.52 | 27.01 | 29.45  | 29.38 | 33.07   | 29.54    | 29.99     | 29.01  |\n| DPIR [57]       | $\\underline{28.47}$| **27.02** | **27.63** | **30.24** | **30.89** | **33.56** | $\\underline{29.62}$   | $\\underline{30.43}$ | **30.54** |\n| RED-PRO| 27.17  | 26.02 | 26.34 | 29.17  | $\\underline{30.25}$ | 32.40   | 28.14    | 28.75 | $\\underline{29.54}$ |"}, {"review_id": "NsiwPbOjSh", "reviewer": "Reviewer_ma6n", "summary": "This paper considers the problem of learning a denoiser which possesses the pseudo-contractive property, for use in solving inverse problems via the plug-and-play (PnP) and regularization-by-denoising (RED) techniques. This property (or special cases of it) on a denoiser appears in various convergence analyses of PnP and RED methods. Learning denoisers with this property is a challenging task, and the paper describes the limitations of spectral methods that have been proposed to solve this. The key contribution of this paper is that a gradient step (GS) denoiser, using the gradient of an $L_\\theta$-smooth input-convex neural network (ICNN), is a strictly pseudo-contractive (SPC) operator with parameter depending on $L_\\theta$. It also shows that a Lipschitz continuous neural network is pseudo-contractive and presents a quantitative convergence result for RED-PRO with these gradient step denoisers. Experiments are performed: (1) testing the pseudo-contractive property of two denoisers trained using the spectral method and one ICNN GS denoiser, finding that the spectral method denoisers violate the SPC property whilst the GS denoiser does not; (2) exploring the convergence of RED-PRO with an ICNN GS denoiser; and (3) testing RED-PRO with an ICNN GS denoiser vs. two PnP methods for deblurring CelebA images.", "questions": "* How does the theoretical analysis of PnP/RED methods with pseudo-contractive denoisers depend on $d$? Crucially, do the values of $d = (L_\\theta-2)/L_\\theta$ and $d = 1$ in Propositions 4.1 and 4.3 significantly affect the ability of these Propositions in establishing any meaningful guarantees for the resulting PnP/RED method(s)? \n* For the ICNN GS denoiser, does the $L_\\theta$-smoothness need to be carefully controlled (for example, if Proposition 4.1 needs to guarantee that the ICNN GS denoiser must be non-expansive or even FNE)—and if so, how practical is this?—or can $L_\\theta$ be as large as you like whilst still achieving good guarantees? \n* For Proposition 4.3, why restrict $R_\\theta$ to be a CNN and, much more importantly, when is $T_\\theta$ meaningfully a “denoiser”? For example, if $R_\\theta = 0$, then $T_\\theta = (1-L)I$ for any choice of $L \\geq 0$, which isn’t really a denoiser. If $R_\\theta$ is bounded but varies quickly, and hence has Lipschitz constant much higher than its bound, then $T_\\theta$ will just be a small perturbation of $(1-L_\\theta)I$, and so also be no use at denoising; that [20] only gives an upper bound on $L_\\theta$ will amplify this issue. What properties must $R_\\theta$ have for $T_\\theta$ to be a useful denoiser? \n* Given the size of the errors in the PSNR and SSIM relative to the advantage of RED-PRO vs. PnP-DnCNN, does RED-PRO really “outperform” PnP-DnCNN for any parameters? What does a qualitative comparison of the deblurred image reveal? Although I appreciate that this is theoretical work and not trying to  achieve state-of-the-art, PnP-DnCNN is 8 years old; can you explain why one should expect practical benefits from using RED-PRO with an ICNN GS denoiser?\n* Two confusions/typos. Line 262: Isn’t softplus $\\frac{1}{\\beta} \\log (1 + e^{\\beta x})$? Is this a typo or did you use a different activation to softplus? Line 272: Should this say “exceeds 1” not “exceeds -1”?", "limitations": "* Unclear whether the $d$ values for pseudo-contractivity guaranteed by the key results produce useful guarantees from the theory. If they do, authors should make this clear. If they do not, authors should discuss, e.g., how the $L$-smoothness can be controlled to ensure suitable $d$ values. \n* Unclear under which conditions Proposition 4.3 yields a useful denoiser. \n* Experimental deblurring results do not show any significant advantage.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Quality \n\n## Strengths \n\n* Key result is of potential practical interest. \n* Theorems all appear to be correct, with valid, straightforward proofs. \n* Related work is adequately described. \n\n## Weaknesses\n\n* The key result (Proposition 4.1) is a fairly simple consequence of known results. \n* Their alternative construction (Proposition 4.3) does not indicate how to learn a pseudo-contractive denoiser: the obtained operator T may not do anything resembling denoising. \n* The link to the core motivation is not established: the existing theory for PnP/RED which supposedly uses these properties is not clearly stated, making it hard to judge the value of the properties established by these propositions. \n* Estimating and controlling the L-smoothness/Lipschitz parameter may be important for making use of these results, yet is mentioned only off-hand. \n* The inverse problem considered is not especially complex, and the results show no significant improvement for RED-PRO (any gains over PnP are much smaller than the errors). No example outputs are given, preventing qualitative assessment. \n* Authors do not discuss limitations. \n\n# Clarity\n\n## Strengths \n\n* Proofs are clearly written.\n* Figures/tables are easy to interpret.\n* Contributions are clear. \n\n## Weaknesses\n\n* Far too many technical details are frontloaded in the introduction, obscuring the story of the work, and technical definitions are dispersed between introduction and Preliminaries. Clarity would be greatly enhanced by focusing paragraph 1 on the story of this work: what the problem is, why it matters, and how this work answers it, rather than immediately throwing a lot of technical terms and definitions at the reader, which can be moved to Preliminaries. \n* Technical details of spectral methods don’t belong in the introduction, move to Related Works. Focus on the ideas, advantages, and disadvantages of those methods in the introduction.\n* “The pseudo contractive assumption has played an important role in the convergence analysis of PnP methods [50, 42, 10, 40, 46, 49, 31, 36, 47, 37, 24, 3, 7, 51, 35, 41, 6]  and RED framework [39, 38, 18, 33]” some of these references do not appear to be  relevant to this claim. Clarify this. \n* Given the centrality of the claim to the work, it should be crystal clear why pseudo-contractivity is important for PnP and RED methods. Include in Related Work any theory which you will later rely on to motivate results in the Preliminaries and/or Section 4, so that it is very clear what the properties established in Propositions 4.1 and 4.3 entail. For example, it is unclear why Proposition 3.3 is a key motivation for anything. \n* Some awkward grammar/typos, e.g., lines 77, 200, 205, and 291.\n\n# Significance\n\n## Strengths \n\n* The key results show a straightforward way to guarantee a property related to desired properties in practice. \n\n## Weaknesses\n\n* As mentioned above, the authors could do more to clarify the significance of these results.\n* No significant advantage was observed in inverse problem experiments\n\n# Originality\n\n## Strengths \n\n* To my knowledge of the literature, the key results are novel\n\n## Weaknesses\n\n* The key results are largely simple consequences of known results.", "was_revised": true, "final_justification": "The authors have addressed the bulk of my concerns in their rebuttals. I raise my score to 4, but no higher, because without reading the revised manuscript in full it is impossible to tell if my clarity concerns have been fully addressed.", "rebuttal": "# To Reviewer ma6n\nIn summary, we believe our work provides valuable new theoretical insights and establishes a principled framework for designing interpretable denoisers with provable guarantees. We sincerely hope that our clarifications and new experimental results highlight the theoretical and practical significance of our contributions and effectively address your concerns. And we would greatly appreciate your reconsideration.\n## Reply to Weaknesses of Quality\n- **A1:** Although Proposition 4.1 is a direct consequence of established results, most existing theoretical analyses of PnP/RED methods, to our knowledge, depend on restrictive assumptions regarding the denoiser, such as nonexpansiveness or averaged nonexpansiveness, which are often not satisfied in practice. Our work is the first to reveal the connection between GS denoisers and truly pseudo-contractive denoisers. This connection enables us to obtain truly SPC denoisers without relying on spectral methods, which is crucial for the theoretical analysis of PnP/RED methods. Moreover, Theorem 4.6 provides the convergence rate of the objective function for the RED-PRO method, which is a new result in this field.\n- **A2:** Proposition 4.3 presents a theoretical framework for constructing pseudo-contractive denoisers. The condition required for $R_\\theta$ is $L_\\theta$-Lipschitz continuity. In practice, a meaningful $\\tilde{R}_ {\\theta}$ needs to be learned from data. The learning process involves minimizing the loss function:\n\n$$\n\\min_{\\theta} \\{ \\mathcal{L}(\\theta) = \\mathbb{E}_{\\mathbf{x}\\sim p(\\mathbf{x}), \\mathbf{n}\\sim \\mathcal{N}(\\mathbf{0},\\sigma^2\\mathbf{I})} || T _{\\theta} (\\mathbf{x}+\\mathbf{n})-\\mathbf{x} ||^2 \\},\n$$\n\nwhere $T_{\\theta}  = I-\\tilde{R}_ {\\theta}, \\tilde{R}_ {\\theta} =R_ {\\theta}+L_{\\theta} I $.\n\nReference [20] provides an effective method to estimate the Lipschitz constant of a network, which can be combined with Proposition 4.3 to learn pseudo-contractive denoisers effectively.\n- **A3:** The existing theory for some PnP/RED methods relies on assumptions such as nonexpansive, FNE, or averaged nonexpansive operators, but it is actually difficult to obtain denoisers that strictly satisfy these mathematical properties in practice. For example, [37, Assumption 2] requires the residual $R_\\sigma = I - D_\\sigma$ to be FNE, so is $D_\\sigma$; [42, Theorem III.1] requires the denoiser to be a proximal mapping for some proper closed convex function, which is also FNE; [38, Lemma 5] requires the denoiser to be nonexpansive; [36, Theorem 3.5, Theorem 3.6] and [46, Assumption 2(b)] require the denoiser to be $\\theta$-averaged with $\\theta \\in (0,1)$; [33, Assumption 2] requires the denoiser to be contractive. All these definitions fall under the category of pseudo-contractive operators. One of our core contributions is Proposition 4.1, which proves that GS denoisers based on ICNNs are truly SPC denoisers. We will clarify in future revisions how Proposition 4.1 has practical significance in the theory of PnP/RED methods.\n- **A4:** Figure 5 illustrates the use of the power method for estimating the global lower bound of $L_\\theta$, which can assist in tuning $w$. Reference [20] provides a method for estimating the Lipschitz constant of a network, but how to precisely and efficiently control the Lipschitz constant of neural networks remains an open problem. We will continue to investigate this problem in our future work.\n- **A5:** Thank you for your comment. Our work is mainly focused on theoretical analysis and the connection between GS denoisers and pseudo-contractive operators, rather than on proposing a new state-of-the-art method for complex inverse problems. We have added CT reconstruction and super-resolution experiments in **Table 2** (see rebuttal to Reviewer 1SLa) and **Table 3** (see rebuttal to Reviewer Ln5k), respectively. Additionally, **Tables 4 and 5** (see rebuttal to Reviewer Ln5k) provide comparison experiments with state-of-the-art methods. We acknowledge that pursuing interpretability may sacrifice some performance, and we will consider how to better balance the two aspects in future work. Due to this year's conference rebuttal rules, we will provide visual results in the revised manuscript.\n- **A6**：We discussed some limitations briefly in lines 199–201, highlighting the trade-off between performance and interpretability. In the revised manuscript, we will provide a more detailed analysis.\n\n## Reply to Weaknesses of Clarity\n- **A7:** Thank you for your suggestion. Our motivation starts from the drawbacks of spectral methods. In the revised manuscript, we will consider moving the technical details of spectral methods to the Related Work Section to improve readability. Additionally, we will clarify how the PnP/RED theory relies on the mathematical properties of pseudo-contractivity, further highlighting the value of our work.\n- **A8**: We explained in lines 26–28 which operator assumptions are used in these references, as they are relevant to this field. We also provide some specific examples in **A2** illustrating how these mathematical properties are used in related works. \n- **A9**: We believe there may be some misunderstanding regarding our motivation. The starting point of our work is that spectral methods cannot yield truly SPC denoisers, while existing PnP and RED theories rely on this property for theoretical guarantees. Proposition 3.3 is important because it helps us reveal the connection between GS denoisers and truly SPC denoisers. In the revised manuscript, we will clarify the role of Proposition 3.3, rather than describing it as a \"key motivation\" to avoid any misunderstanding.\n- **A10**: Thank you. We will carefully proofread and correct the issues in lines 77, 200, 205, 291, and throughout the manuscript.\n\n## Reply to Weaknesses of Originality\n- **A11**：First, $d$ only affects parameter selection, specifically $w \\in (0, 1-d) = (0, \\frac{2}{L_\\theta})$ (see line 226). Choose $w \\in (0, 1-d)$ appropriately so that $ T_w = wT_ \\theta + (1-w)I $ is a nonexpansive operator. It does not affect the main theoretical results, such as the convergence rate in Theorem 4.6. Theoretical analysis of PnP/RED methods relies on $d$-SPC assumptions such as nonexpansive ($d=0$), FNE ($d=-1$), and $\\alpha$-averaged nonexpansive ($d=\\frac{\\alpha-1}{\\alpha}$) operators. Therefore, achieving the truly pseudo-contractivity property is important for the theoretical understanding of PnP/RED methods. Our theoretical results, i.e., Theorem 4.6,  are mainly based on $d$-SPC denoisers. For the case $d=1$, Proposition 4.3 provides only a theoretical construction of pseudo-contractive denoisers.  Ishikawa iteration presented in reference [51, Theorem 3] can be used to ensure convergence of PnP methods.\n- **A12**: In our work, we don't precisely control the smoothness constant $L_\\theta$. Theoretically, $L_\\theta = 2$ corresponds to a nonexpansive operator and $L_\\theta = 1$ to a FNE operator. However, how to precisely control the network's Lipschitz constant $L_\\theta$ remains a very important open problem; in our experiment shown in Figure 4, we only provide a rough estimate. As far as we know, reference [20] provides a recent method to estimate an upper bound for the network's Lipschitz constant, which can help prevent $L_\\theta$ from becoming too large. Achieving precise control of $L_\\theta$ to obtain desired operators such as FNE or nonexpansive is still challenging and worth further exploration in the future. In our experiments, we mainly tune the weight $w$ to ensure that $T_w$ remains non-expansive. Since $w \\in (0, \\frac{2}{L_\\theta})$, a larger $L_\\theta$ requires a smaller $w$ to ensure good theoretical guarantees. \n- **A13**: Because [20] provides a way to control the Lipschitz constant of CNNs, which are widely used in image processing. Theoretically, we aim to design new pseudo-contractive denoisers. When $R_\\theta$ is trained in a data-driven manner, $T_\\theta$ will be a meaningful denoiser. For further explanation, please see our response to **A2**.\n- **A14**: No, RED-PRO does not always outperform PnP-DnCNN. We have tuned $w$ and step size $\\mu_k$ to achieve better performance than PnP-DnCNN in some cases. We will provide some qualitative results to observe their difference in the revised paper. The practical benefit of using RED-PRO with an ICNN GS denoiser is that the ICNN GS denoiser is guaranteed to satisfy the strict pseudo-contractivity (SPC) property, which ensures the theoretical convergence of RED-PRO. This provides a principled way to design denoisers that are both interpretable and compatible with the convergence guarantees of RED-PRO.\n- **A15**: We will correct two confusions/typos about the softplus function and the expression “exceeds -1”.\n## Reply to limitations\n- **A16**: For Proposition 4.1, we have $d = \\frac{L_\\theta-2}{L_\\theta}$, and we only need to tune the weight $w \\in (0,1-d) =  (0, \\frac{2}{L_\\theta})$ in Algorithm 1. The experiments in Figure 5 show that the obtained $d$ values can indeed provide convergence guarantees. However, as $L_\\theta$ increases, $w$ must be smaller. In future work, we will explore how to obtain better $d$ values by accurately controlling the network's $L$-smoothness.\n- **A17**: Proposition 4.3 provides a theoretical condition by requiring the Lipschitz constant of the network to be controlled in advance. To obtain a useful denoiser in practice, it is necessary to train the denoiser in a data-driven manner.\n- **A18**: We acknowledge that the experimental deblurring results do not show a significant advantage. Our main contribution is to provide new theoretical insights. Additional experiments, such as those in **Tables 4 and 5** (see rebuttal to the **Reviewer Ln5k**), show that RED-PRO with truly SPC denoisers can achieve a balance between theoretical guarantees and practical performance compared to state-of-the-art methods."}, {"review_id": "IATOFSyQQh", "reviewer": "Reviewer_Ygzw", "summary": "This paper proposes a Gradient-Step denoiser with an ICNN as a learned potential in order to restore image with Plug-and-Play algorithm. The authors show that this Gradient-Step denoiser is a strictly pseudo-contractive operator, which allow to obtain convergence of the RED-PRO algorithm. This Gradient-Step denoiser does not require spectral norm penality in the training loss making the training of the denoiser more efficient.", "questions": "Major questions :\n- The denoiser is train for one level of noise, can you propose strategy to train a denoiser with various level of noise, as DRUNet ?\n- Why the Proposition 4.3 is restrict to convolutional neural networks ? I do not see where you use this convolutional structure in the proof.\n- Why is it usefull to construct pseudo-contractive operators with Proposition 4.3 ? What is the difference with strictly pseudo-contractive operators in particular from a convergence point of view ?\n- Do you observe instabilities when the constant $w$ violate your conditions ? Typically, $w > 0.8$ in Figure 5.\n\nMinor questions:\n- In the proof of Proposition 3.3, I do not see why the first and the second inequality are equivalent. Can you detail this computation ?\n- In the proof of Proposition 3.3, you do not prove the reciprocal or exhibit that all arguments are equivalences.", "limitations": "The limitations of this work are not clearly exhibit. There is no comparison with state of the art methods in the experimental part. The question of expressivity of ICNN is not discuss in detail.", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 3, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Strength:\nThe paper explains clearly and early the main point. The various operator conditions, FNE, SPC, PC, RFNE..., are well explained and their introduction is pedagogic (e.g. Figure 1). Reducing the time of training without spectral norm penality might interests practionners.\n\nWeaknesses:\n- There are numerous references mistakes and imprecisions:\n\t- For strictly pseudocontractive operator in the beggining of the introduction that have been studied for decades, you need to cite the first works such that [1,2]. With only the citation of [3], it makes the reader believe that these operators have been introduced last year.\n\t- For conically $\\lambda$-averaged and nonexpansive, the references seems not very explored.\n\t- There is no precise references (which proposition or theorem ara copied) for books or article, especially for Proposition 3.2, 3.3, 3.4 or Theorem 3.6. Non precise reference makes it hard to look at.\n\t- There is no reference given for the different dataset.\n\t- In line 155, no reference is given for cocoercive operators.\n\t- In line 117, there is a wrong citation. DiffPIR is cited to support application not limited to image inverse problem. However, the method DiffPIR has been designed specifically for image inverse problem and is only applied to image inverse problem in the original paper that is cited.\n\t- In line 132, there is a missing citation. In fact \"He et al.\" is cited, however in the references, no paper writing by He is provided.\n- Some of the assumptions are not clearly stated. In Theorem 4.6, you suppose that the step-size are of the form $\\mu_k = \\frac{c}{(1+k)^\\alpha}$ without saying it in the paper. It only appears in the proof in Appendix. This assumption must be clearly stated. In particular, it is need to write what is the condition on $\\alpha$.\n- The experimental validation is not strong. In line 91, it is claim that \"extensive numerical experiments\" are provided. However, only deblurring experiments are performed with few kernels of blur. Morevover, in Table 1, there is no comparison with state of the art PnP restoration method, such as using a DRUNet [4] or a Gradient-Step DRUNet denoiser [5], or the DiffPIR methode [6]. Then, RED-PRO do not outperform significatively PnP-DnCNN, restricting the impact of this work. Finally, no image are provided, even in Appendix, making it hard to have a quanlitative comparison between methods.\n- There are imprecision in the proofs. In the proof of Proposition 3.2, in line 484, $d$ is suppose to be $-1$ where the Proposition claim that this is true for any $d < 1$. Moreover, according to Figure 2, Proposition 3.2 is not well formulate. In fact, it is not under the $d$-SPC that (i) is equivalent to (ii), but $d$-SPC is equivalent to (i) is and to (ii). I suggest to reformulate Proposition 3.2 to solve this error of logic.\n- There is unprecise claims, which are false in the current version :\n\t- In line 74, you claim that gradient of ICNN possess universal approximation capabilities. However, the gradient of convex function are always monotone operators, so can not approximate every functions.\n\t- In equation (2), you claim that image restoration problems can be reformulate as a convex optimization problem. However, many image inverse problems have no-convex data-fidelity (e.g. phase retrieval) or non-convex regularization (e.g. Gradient-Step denoiser).\n\n[1] On the Mann iteration process in a Hilbert space. Hicks, T.L., Kubicek, J.D., J. Math. Anal. Appl. 59, 498–504 (1977)\n[2] Construction of fixed points of nonlinear mappings in Hilbert space, FE Browder, WV Petryshyn (1967)\n[3] Andrzej Cegielski. Strict pseudocontractions and demicontractions, their properties, and applications. Numerical Algorithms, 95(4):1611–1642, Apr 2024.\n[4] Plug-and-Play Image Restoration With Deep Denoiser Prior, Kai Zhang; Yawei Li; Wangmeng Zuo; Lei Zhang; Luc Van Gool; Radu Timofte\n[5] Gradient step denoiser for convergent plug-and-play, Samuel Hurault, Arthur Leclaire, Nicolas Papadakis\n[6] Denoising Diffusion Models for Plug-and-Play Image Restoration, Yuanzhi Zhu, Kai Zhang, Jingyun Liang, Jiezhang Cao, Bihan Wen, Radu Timofte, Luc Van Gool", "was_revised": true, "final_justification": "I have raised my score to 3 to thank the authors efforts for answering my concerns.\n\nI will still recommand to reject the paper because there is too many things to modify to the current version of the paper. The experiments are clearly unsufficient and it is hard to judge additionnal experiments only based on table (without looking closely to all the details). Moreover, as my review shows, I have detected numerous errors/imprecisions particularly in the references or in the statement of the theoretical results. Therefore I am not confident that I saw all major errors and that the results presented in the paper are correct.", "rebuttal": "# To Reviewer Ygzw\nWe sincerely appreciate your detailed and critical feedback. We have carefully addressed each of your concerns and made substantial improvements to the manuscript, including adding new experiments, clarifying theoretical contributions, and correcting technical details. We would greatly appreciate your reconsideration.\n## Reply to Reference Corrections\n1. Strictly Pseudocontractive Operators: We acknowledge that reference [2] you mentioned corresponds to our citation [8]. We will add reference [1] about Krasnosel'ski $\\breve{ı}$-Mann (KM) iterations and restructure the introduction to present the earliest foundational works first, providing proper historical context.\n2. Conically-averaged and nonexpansive References: We will add comprehensive citations about these operators, including:\n- Ref1. Bartz S, Dao M N, Phan H M. Conical averagedness and convergence analysis of fixed point algorithms[J]. Journal of Global Optimization, 2022, 82(2): 351-373\n- Ref2. Baillon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)\n- Ref3. Bauschke, H.H., Borwein, J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)\n- Ref4. Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425(1), 55–70 (2015)\n- Ref5. Bauschke, H.H., Noll, D., Phan, H.M.: Linear and strong convergence of algorithms involving averaged\nnonexpansive operators. J. Math. Anal. Appl. 421(1), 1–20 (2015)\n- Ref6. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)\n- Ref7. Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425(1), 55–70 (2015)\n3. Precise Citations: Proposition 3.2 (i) follows from [Proposition 2.2, Ref1]. The equivalence of (i) and (ii) can be found in [14], page 1614, last sentence of the first paragraph. Proposition 3.3 is from [13, Corollary 2.2.3]. Proposition 3.4 is from [13, Corollary 2.2.3] and [56, Proposition 1 (b)]. Theorem 3.6 is based on [5, Theorem 5.8].\n4. Dataset References: We add the following citations.\n- MNIST: LeCun Y, Bottou L, Bengio Y, et al. Gradient-based learning applied to document recognition[J]. Proceedings of the IEEE, 2002, 86(11): 2278-2324.\n- CelebA: Liu Z, Luo P, Wang X, et al. Deep learning face attributes in the wild[C]//Proceedings of the IEEE International Conference on Computer Vision. 2015: 3730-3738.\n- BSD400: D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human\nsegmented natural images and its application to evaluating segmentation\nalgorithms and measuring ecological statistics,” in Proceedings Eighth\nIEEE International Conference on Computer Vision. ICCV 2001, vol. 2,\n2001, pp. 416–423 vol.2.\n5. Specific Corrections:\nLine 155: The definition of cocoercive operator can be found in [5, Section 4.2, Definition 4.10].\nLine 117: We will correct the statement regarding DiffPIR.\nLine 132: We will add \"He L, Zhang Q, Yang X, et al. SLN-RED: \nRegularization by simultaneous local and nonlocal denoising for image restoration[J]. IEEE Signal Processing Letters, 2023, 30: 578-582.\"\n## Reply to assumptions on step-size\nWe have mentioned the form of $\\mu_k  =\\frac{c}{(1+k)^\\alpha}$ in Algorithm 1, not only in the Appendix. And $\\alpha$ should satisfy that $\\alpha>0$. \n## Reply to experiments and limitations\nOn the one hand, we provide a comparison with SOTA methods. Please refer to **Tables 4 and 5** at the end of the rebuttal to the 2nd **Reviewer Ln5k**. Although non-negative weights may limit ICNN expressivity, they ensure a balance between theoretical guarantees and empirical performance.\n\nOn the other hand, our main contribution is theoretical: we take an important first step by showing how to construct truly SPC denoisers in Proposition 4.1, which provides new insights for designing interpretable and potentially more useful denoisers in the future. While our experimental validation is preliminary, we believe our work opens up new research directions and has the potential for significant impact.\n## Reply to Proposition 3.2\nWe set $d=-1$ as a special case to illustrate the equivalence between FNE and $1/2$-averaged operators. Then, we show that any nonexpansive operator $S$  can be written as a $2$-RFNE operator, i.e., $S= 2V-I$, where $V$ is FNE. By using the equivalence between conically averaged operators $T= (1-\\lambda) I+\\lambda S$ ($\\lambda =\\frac{1}{1-d}$) and $d$- SPC operators, by substituting $S = 2V - I$ we obtain $T = 2\\lambda V+(1-2\\lambda) I$, which can extend the result to any $d<1$. We will improve the exposition in the revised version. We will further clarify the logic of Proposition 3.2 by presenting the equivalence in parallel among $d$-SPC, conically $\\lambda$-averaged, and $2\\lambda$-RFNE, rather than the equivalence caused by the $d$-SPC condition, which may lead to misunderstanding.\n## Reply to imprecise claims\n- Universal approximation capability:  We will revise the statement in line 74 to clarify that the universal approximation capability refers to monotone operators.\n- Model (2): We will revise the description of $f$ and $g$ in model (2) to cover more general inverse problems.\n## Reply to Major questions:\n- **A1**: According to (7), it is possible to train a denoiser that handles multiple noise levels by randomly and uniformly sampling the noise level during training.\n- **A2 and A3**: The reason we mentioned CNNs is that [20] provides a method to control the Lipschitz constant of CNNs, so we presented a theoretical construction based on this, making general CNNs satisfy the pseudo-contractive assumption. ICNNs use non-negative weight constraints to ensure convexity, but this may reduce the expressive ability. We propose Proposition 4.3 to design new pseudo-contractive denoisers with stronger expressive power, but this is only a theoretical construction, and we plan to explore it in future work. Pseudo-contractive denoisers with $d=1$ have weaker conditions than strictly pseudo-contractive ones, but require the Ishikawa iteration to guarantee convergence, as shown in [51, Theorem 3.3]. Our theoretical analysis requires the denoiser to be  $d$-SPC.\n- **A4**: Yes, we do. Algorithm 1 requires $0 < w < 1-d$. Choosing a larger $w$ outside $(0,1-d)$ may violate the condition of Algorithm 1 and lead to instability in experiments.\n## Reply to Minor questions\n- **A5**: Since\n$$\\begin{aligned}\n \\|\\| S(\\mathbf{x}) - S(\\mathbf{y}\n) \\|\\|^2 & =\\|\\| (S(\\mathbf{x})-\\mathbf{x})+(\\mathbf{x}-\\mathbf{y})+(\\mathbf{y}- S(\\mathbf{y})) \\|\\|^2  \n\\\\\\\\\n& = \\|\\|S(\\mathbf{x})-\\mathbf{x} \\|\\|^2+\\|\\| \\mathbf{x}\n-\\mathbf{y}\\|\\|^2+\\|\\| \\mathbf{y}-S(\\mathbf{y}) \\|\\|^2+2\\langle S(\\mathbf{x})-\\mathbf{x},\\mathbf{x}-\\mathbf{y}\\rangle \\\\\\\\\n&+2\\langle S(\\mathbf{x})-\\mathbf{x},\\mathbf{y}-S(\\mathbf{y})\\rangle+2\\langle\\mathbf{x}-\\mathbf{y},\\mathbf{y}-S(\\mathbf{y})\\rangle.\n\\end{aligned}$$\nand $$\n\\|\\| (\\mathbf{x}-S(\\mathbf{x})) - (\\mathbf{y}-S(\\mathbf{y}))\\|\\|^2=\\|\\| \\mathbf{x}-S(\\mathbf{x}) \\|\\|^2-2\\langle \\mathbf{x}-S(\\mathbf{x}), \\mathbf{y}-S(\\mathbf{y})\\rangle+\\|\\| \\mathbf{y}-S(\\mathbf{y}) \\|\\|^2.$$\nFrom $$\n \\|\\| S(\\mathbf{x})-S(\\mathbf{y}) \\|\\|^2\\leq \\|\\| \\mathbf{x}-\\mathbf{y} \\|\\|^2 -\\|\\| (\\mathbf{x}-S(\\mathbf{x}))-(\\mathbf{y}-S(\\mathbf{y})) \\|\\|^2,$$\nwe obtain \n$$\\begin{aligned}\n\\|\\|\\mathbf{x}-S(\\mathbf{x}) \\|\\|^2+\\|\\| \\mathbf{y}-S(\\mathbf{y}) \\|\\|^2 \n&\\leq\\langle \\mathbf{x}-S(\\mathbf{x}),\\mathbf{y}-S(\\mathbf{y})\\rangle -\\langle S(\\mathbf{x})-\\mathbf{x},\\mathbf{x}-\\mathbf{y}\\rangle-\\langle S(\\mathbf{x})-\\mathbf{x},\\mathbf{y}-S(\\mathbf{y})\\rangle-\\langle \\mathbf{x}-\\mathbf{y},\\mathbf{y}-S(\\mathbf{y})\\rangle\n\\\\\\\\\n&=\\langle S(\\mathbf{y})-\\mathbf{y}, S(\\mathbf{x})-\\mathbf{x}\\rangle-\\langle\\mathbf{x}-\\mathbf{y},S(\\mathbf{x})-\\mathbf{x}\\rangle +\\langle S(\\mathbf{x})-\\mathbf{x}, S(\\mathbf{y})-\\mathbf{y}\\rangle+\\langle\\mathbf{x}-\\mathbf{y},S(\\mathbf{y})-\\mathbf{y}\\rangle \n\\\\\\\\\n&=\\langle S(\\mathbf{y})-\\mathbf{x},S(\\mathbf{x})-\\mathbf{x}\\rangle+\\langle S(\\mathbf{x})-\\mathbf{y},S(\\mathbf{y})-\\mathbf{y}\\rangle.\\end{aligned}$$\nFinally, we show that these two inequalities are equivalent.\n- **A6**: We check all the derivations in the proof of Proposition 3.3 are equivalences. Please refer to [13, Corollary 2.2.3] or [14, Proposition 2.2] for a detailed justification. Proposition 3.3 is correct.\n## Reply to Paper Formatting Concerns\nThank you again for your careful and professional review. Some formatting concerns may be due to misunderstandings.\n- When $0<\\alpha<1$, the term \"$\\alpha$-averaged\" is commonly used in the literature[13,14,56, Ref1], rather than \"conically $\\alpha$-averaged\", although $\\alpha$-averaged is a special case of conically $\\alpha$-averaged. Therefore, there is no issue of notation inconsistency in this regard.\n- We confirm that in Proposition 4.1, $\\phi: \\mathbb{R} \\to \\mathbb{R}$.\n- We respectfully disagree with the reviewer’s concern. Iterations (13) and (14) are equivalent, as clearly shown in the previous RED-PRO work published in SIIMS [18]. \n- We will correct \"$T_w$ is $w\\theta$-averaged\" to \"$T_w$ is $\\frac{w}{1-d}$-averaged\".\n- We will use $\\mu_k$ to replace $\\eta_k$ in the proof of Theorem 4.6.  \n- We will recall the definition of $u_k$ in line 502 for clarity.\n- We will correct the inequality in line 504 to \"$\\eta_k\\geq \\eta_j$\".\n- We respectfully disagree. In our proof, $\\mathbf{z}$ is introduced as an arbitrary fixed point and $\\mathbf{x}'$ as the optimal solution. Replacing $\\mathbf{z}$ with $\\mathbf{x}'$ in the formula below line 502 is appropriate.\n- We will correct \"MNIST\" throughout the manuscript.\n- We will correct the activation function $\\phi(x) = \\frac{1}{\\beta}\\log\\(1+\\mathrm{e}^{\\beta x}\\)$.\n- We will update Figure 2 to use \"$1-\\frac{1}{\\lambda}$ is-SPC\" to clarify the relationship between $d$ and $\\lambda$."}]

Author rebuttal:

Meta-review: This paper studies denoising algorithms for general inverse problems, focusing on the RED (and RED-PRO) approach. In this setting, it is important for the employed denoisers to be pseudo-contractive. While other partially successful methods exist, the authors show here that gradient-step (GS) denoisers (introduced not long ago and parametrized by ICNNs) are indeed pseudo-contractive. With this results, the authors then integrate (GS) within RED-PRO, with the respective convergence guarantees. Experimental results are provided on image deblurring. 

**Strengths**
- The result presented in the paper (linking GS denoisers with pseudo-contractivity) is new and interesting.
- Nice and elegant integration with RED-PRO with provable guarantees.
- Experiments validate their SPC property, and show compelling results.

**Weaknesses**
- By constraining the denser to be a GS (parametrized by an ICCN), the performance of the model is limited (unsurprisingly).
- While the result is novel, it follows rather straightforwardly from known results.
- There were some inconsistencies on the technical details, raised by Reviewer Ygzw’s.
- The experimental study was limited to a single inverse problem (deblurring) and on a simple datasets.

**Discussion and conclusion**

The discussions between reviewers and authors was productive. The authors have included further experimental results on two other tasks (CT reconstruction and super-resolution), including other (newer and stronger) methods for comparison, significantly expanding on their numerical component. The new results indicate that this method trades performance for theoretical guarantees, as expected. The technical details and innovations that were not completely clear were clarified by the authors, and references and clarification comments were (promised to be) added. 

Three reviewers recommend acceptance (two strong, one moderate), while one reviewer maintains a weak reject on concerns over technical rigor and experimental sufficiency. If the reviewers implement the changes that were raised through the rebuttal process, I am confident that the technical rigor and the experimental section will be much improved, and that this paper will be an interesting addition to the community.
Decision: Accept (poster)