reviews: 
Comments: This paper introduces a novel relaxed formulation of Input-Convex Neural Networks (ICNNs), expressed as a convex minimization problem over intermediate activations. This approach is applied as a regularizer for image inverse problems. The authors solve the resulting convex optimization problem using the Primal-Dual Hybrid Gradient (PDHG) method. The ICNN regularizer is trained in a task-specific manner within an adversarial framework. The effectiveness of this approach is demonstrated on three image inverse problems: salt-and-pepper noise removal, image inpainting, and sparse-view computed tomography (CT) reconstruction.
Strengths: - The paper is well-structured and articulates its contributions clearly, making it accessible and easy to follow for readers.
- The reformulation of the ICNN as a convex minimization problem, combined with its solution through the Primal-Dual Hybrid Gradient (PDHG) algorithm, is innovative and shows strong potential for practical applications.
- The formulation and theoretical backing of the ICNN as a convex minimization problem are solidly presented, providing a strong theoretical foundation for the proposed approach.
- While the focus is on image inverse problems, the proposed ICNN could potentially be extended to other fields, such as optimal transport, where a deep convex potential is beneficial, showing the paper's broad applicability.
Weaknesses: - **Limited Comparison in Experiments**: The experimental section only compares the proposed method with a subgradient algorithm applied to the same objective function, which limits the scope of the evaluation. It would strengthen the findings if the method were compared against a standard ICNN approach or a similar architectures without constraints, as this would better illustrate the relevance and advantages of the proposed modifications.

- **Insufficient Training Details**: The paper lacks clarity on the training process for the ICNN-based regularization. Key aspects, such as the training methodology, hyperparameter choices, and datasets, are either under-explained or absent, which may impact the reproducibility and clarity of the results.

- **Rationale for Adversarial Training Choice**: The use of an adversarial framework for training the regularization introduces task dependency, which limits the general applicability of the approach. The rationale behind this choice is not sufficiently justified, especially given that adversarial training can be complex and potentially less efficient compared to alternative approaches. Exploring or explaining why more standard training schemes, such as training by denoising, were not pursued could improve the paper’s rigor and scope. Denoising-based training could make the regularizer more versatile and less sensitive to specific tasks, potentially offering broader applicability.

- **Clarity in Problem Motivation**: While the limitations of ICNNs are well discussed in the abstract, they are not sufficiently detailed in the introduction. Expanding on these limitations in the introduction would provide clearer context and stronger motivation for the proposed method.

- **Simplistic Architecture**: The proposed ICNN architecture is relatively simple, with only two layers. Recent works have explored deeper ICNN architectures that could improve performance in complex tasks, and the paper would benefit from a comparison or discussion on how a deeper architecture might impact results.
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 3