To Reviewer Ygzw

We 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.

Reply to Reference Corrections

1. 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.
2. Conically-averaged and nonexpansive References: We will add comprehensive citations about these operators, including:

• 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
• 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)
• Ref3. Bauschke, H.H., Borwein, J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
• Ref4. Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425(1), 55–70 (2015)
• Ref5. Bauschke, H.H., Noll, D., Phan, H.M.: Linear and strong convergence of algorithms involving averaged nonexpansive operators. J. Math. Anal. Appl. 421(1), 1–20 (2015)
• Ref6. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)
• Ref7. Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425(1), 55–70 (2015)

3. 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].
4. Dataset References: We add the following citations.

• 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.
• 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.
• BSD400: D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,” in Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, vol. 2, 2001, pp. 416–423 vol.2.

5. Specific Corrections: Line 155: The definition of cocoercive operator can be found in [5, Section 4.2, Definition 4.10]. Line 117: We will correct the statement regarding DiffPIR. Line 132: We will add "He L, Zhang Q, Yang X, et al. SLN-RED: Regularization by simultaneous local and nonlocal denoising for image restoration[J]. IEEE Signal Processing Letters, 2023, 30: 578-582."

Reply to assumptions on step-size

We 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 $\alpha>0$.

Reply to experiments and limitations

On 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.

On 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.

Reply to Proposition 3.2

We 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.

Reply to imprecise claims

• Universal approximation capability: We will revise the statement in line 74 to clarify that the universal approximation capability refers to monotone operators.
• Model (2): We will revise the description of $f$ and $g$ in model (2) to cover more general inverse problems.

Reply to Major questions:

• 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.
• 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.
• 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.

Reply to Minor questions

• A5: Since

\|\| S(\mathbf{x}) - S(\mathbf{y} ) \|\|^2 & =\|\| (S(\mathbf{x})-\mathbf{x})+(\mathbf{x}-\mathbf{y})+(\mathbf{y}- S(\mathbf{y})) \|\|^2 \\\\ & = \|\|S(\mathbf{x})-\mathbf{x} \|\|^2+\|\| \mathbf{x} -\mathbf{y}\|\|^2+\|\| \mathbf{y}-S(\mathbf{y}) \|\|^2+2\langle S(\mathbf{x})-\mathbf{x},\mathbf{x}-\mathbf{y}\rangle \\\\ &+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. \end{aligned}$$ and $$ \|\| (\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.$$ From $$ \|\| S(\mathbf{x})-S(\mathbf{y}) \|\|^2\leq \|\| \mathbf{x}-\mathbf{y} \|\|^2 -\|\| (\mathbf{x}-S(\mathbf{x}))-(\mathbf{y}-S(\mathbf{y})) \|\|^2,$$ we obtain $$\begin{aligned} \|\|\mathbf{x}-S(\mathbf{x}) \|\|^2+\|\| \mathbf{y}-S(\mathbf{y}) \|\|^2 &\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 \\\\ &=\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 \\\\ &=\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}$$ Finally, we show that these two inequalities are equivalent. - **A6**: We check all the derivations in the proof of Proposition 3.3 are equivalences. Please refer to [13, Corollary 2.2.3] or [14, Proposition 2.2] for a detailed justification. Proposition 3.3 is correct. ## Reply to Paper Formatting Concerns Thank you again for your careful and professional review. Some formatting concerns may be due to misunderstandings. - When $0<\alpha<1$, the term "$\alpha$-averaged" is commonly used in the literature[13,14,56, Ref1], rather than "conically $\alpha$-averaged", although $\alpha$-averaged is a special case of conically $\alpha$-averaged. Therefore, there is no issue of notation inconsistency in this regard. - We confirm that in Proposition 4.1, $\phi: \mathbb{R} \to \mathbb{R}$. - We respectfully disagree with the reviewer’s concern. Iterations (13) and (14) are equivalent, as clearly shown in the previous RED-PRO work published in SIIMS [18]. - We will correct "$T_w$ is $w\theta$-averaged" to "$T_w$ is $\frac{w}{1-d}$-averaged". - We will use $\mu_k$ to replace $\eta_k$ in the proof of Theorem 4.6. - We will recall the definition of $u_k$ in line 502 for clarity. - We will correct the inequality in line 504 to "$\eta_k\geq \eta_j$". - We respectfully disagree. In our proof, $\mathbf{z}$ is introduced as an arbitrary fixed point and $\mathbf{x}'$ as the optimal solution. Replacing $\mathbf{z}$ with $\mathbf{x}'$ in the formula below line 502 is appropriate. - We will correct "MNIST" throughout the manuscript. - We will correct the activation function $\phi(x) = \frac{1}{\beta}\log$1+\mathrm{e}^{\beta x}$$. - We will update Figure 2 to use "$1-\frac{1}{\lambda}$ is-SPC" to clarify the relationship between $d$ and $\lambda$.

I thank the authors for these responses and details.

I suggest to add all the corrected references in the paper.

I suggest to add the stepsize definition in Theorem 4.6.

Having proper theoretical guarantees for image restoration is crucial in many applications. However, if this is at the cost of a significant decrease of perfomances with respect with state-of-the-art methods, it is not useful. The experiments presented in the paper are clearly not sufficient to deduce the performances of the proposed methods. Concerning table 3-4-5, I truly appreciate the effort to provide additional experiments. I am surprised by the performance of DiffPIR, which denoiser have been used ? Also there is no comparison with the gradient-step denoiser, which is the closest method. Only "easily" inverse problem are tackle in these experiments, with low level of noise, low blur or low super-resolution factor. Therefore, it is hard to conclude the performance of your method. Finally, no images are provided (I know that it is not possible in the rebuttal format). I believe that a paper that is tackling the problem of image restoration should provide image for a qualitative comparisons.

I thank the authors for the correction of Proposition 3.2.

About typos disagreements :

• I keep my comment, iteration (13) and (14) are not the same. If we name $y_k = x_k - \mu_k \nabla f(x_k)$ in equation (13), then $y_k$ is verifying (14).
• Could you precise in line 502 that you take $z = x'$ for clarity ?

Thanks for this clear response. I will raise my score to 3 to thank the authors efforts for answering my concerns.

I will still recommand a "weak reject" because there is too many things to modify to the current version of the paper (especially on experiments). Moreover, as my review shows I have detected numerous errors/imprecisions and it is highly possible that I missed others.

Thanks for the discussion !

To Reviewer ma6n

In 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.

Reply to Weaknesses of Quality

• 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.
• 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:

$$\min_{\theta} \{ \mathcal{L}(\theta) = \mathbb{E}_{\mathbf{x}\sim p(\mathbf{x}), \mathbf{n}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})} || T _{\theta} (\mathbf{x}+\mathbf{n})-\mathbf{x} ||^2 \},$$

where $T_{\theta} = I-\tilde{R}_ {\theta}, \tilde{R}_ {\theta} =R_ {\theta}+L_{\theta} I$.

Reference [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.

• 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.
• 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.
• 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.
• 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.

Reply to Weaknesses of Clarity

• 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.
• 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.
• 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.
• A10: Thank you. We will carefully proofread and correct the issues in lines 77, 200, 205, 291, and throughout the manuscript.

Reply to Weaknesses of Originality

• 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.
• 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.
• 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.
• 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.
• A15: We will correct two confusions/typos about the softplus function and the expression “exceeds -1”.

Reply to limitations

• 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.
• 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.
• 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.

I thank the authors for their thorough response to my review. I will here follow up on where my most significant concerns remain.

• A2/A4/A13: This point deserves to be fleshed out more in the paper. Proposition 4.3, as written, tells us how to turn an L-Lipschitz NN into a pseudo-contractive operator, but this need not be a denoiser. I thank you for clarifying in your reply that you intend to get a denoiser by training the resulting operator via a denoising loss, but this training will require considering the Lipschitz constant of the underlying $R_\theta$ neural network during training. The discussion of how to get a pseudo-contractive denoiser out of Proposition 4.3 would therefore be greatly enhanced by a discussion of training Lipschitz-constrained neural networks in this setting, going beyond the simple reference to [20]. For example, there are known challenges regarding expressivity, see "Sorting Out Lipschitz Function Approximation" by Cem Anil, James Lucas, Roger Grosse (2019) and "Limitations of the Lipschitz constant as a defense against adversarial examples" by Todd Huster, Cho-Yu Jason Chiang, Ritu Chadha (2018).
• A3/A7/A11/A12: The heart of my issue is this: Proposition 4.1 shows that the ICNN GS denoiser will be SPC with $d = (L_\theta-2)/L_\theta$. But as you say in your reply, the existing theory for PnP/RED relies on properties that are more specific than simply being SPC--being SPC is a necessary but not a sufficient condition for this theory. As you note, this necessary condition is not achieved by other methods, but the way things are currently being presented in your paper, it is highly unclear to the reader what theory the GS denoiser gains automatically from being $d$-SPC (like Theorem 4.6) vs. what will require careful control of $L_\theta$ (e.g., if FNE were required, then one would need $L_\theta \leq 1$). Tidying this up (and adding discussion/references for how $L_\theta$ might be controlled, even if actually doing so is beyond the scope of this work).
• A14: My concerns are two-fold: Firstly, you should not describe RED-PRO as outperforming another method if the difference in performance is much smaller than the error bars, as this miscommunicates the (statistical) significance of the performance. (Ideally, one would perform a significance test of RED-PRO vs. PnP-DnCNN/etc., and report it as outperforming only when results were statistically significantly better.) In the results in the paper, and from the above Tables, it seems that RED-PRO rarely if ever has significant improved performance, and this should be communicated clearly. Secondly, therefore, ideally your experiments should seek to highlight "why should I use RED-PRO?" in some other way--perhaps focusing on the convergence property, showcasing cases where the non-SPC methods fail to converge. Alternatively, since as you say state-of-the-art experimental performance are not this work's focus, you could reframe your experiments as showcasing how competitive results can be attained whilst still satisfying the constraints of theory.

If these concerns are met, I will consider raising my score to a 4. Please reply with excerpts of what you add with respect to these concerns, so I can get a sense of what the revised manuscript would look like.

To Reviewer Ln5k

We sincerely appreciate your constructive and positive feedback. We have addressed your main concerns regarding theoretical originality, experimental validation, and practical relevance as detailed below.

Reply to Weaknesses

• 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.
• 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.
• 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.

Reply to Question

• A1: In our experiments, we use the GS denoiser obtained from Proposition 4.1, rather than the denoiser constructed using Proposition 4.3.
• 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. In 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.
• 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.

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.

Reply to Limitations

Thank 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.

Table 4. Deblurring results of different methods on CelebA over 20 samples.

Table 5. Comparison with state-of-the-art methods. Test grayscale images are degraded by the Gaussian kernel with the noise level $\sigma = 2.55$.

To Reviewer 1SLa

We 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:

Reply to Weaknesses

• 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.
• 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.
• 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.

Reply to Question

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.

We 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.

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.

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:

$$$$

where $\mathbf{b}_i \in \mathbb{R}^m$ 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.

Table 2. Numerical results for CT reconstruction on the Mayo-CT dataset, computed over 128 test images.