Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper provides convergence results attached with invex regularizers. Few non-convex regularizers with no global convergence guarantees profited from this analysis through invexity. Authors establish convergence guarantees to global optima for various advanced image reconstruction techniques with invex regularizers.  Authors prove few functions as invex functions and provide the proximal operators corresponding to them. These invex regularizers were utilized for various imaging tasks.

strengths_and_weaknesses: Strengths:
1. Invex regularizers give a decent theoretical analysis of a few non-convex regularizers.
2. Experiments included various image restoration tasks such as compressed image recovery and image denoising.

Acknowledgments:
1. Authors included more comparisons with TVAL3, ReconNet, and ISTA-Net as part of the revision.
2. More Image reconstruction examples are provided in the supplementary document as part of the revision.

Issues:
1. The revised paper is more than ten pages long. I believe it has to be limited to 9 pages until acceptance. 
questions: Suggestions: (I am not asking authors to work on this right away. These are merely suggestions for a camera-ready version.)
1. The paper presentation could be improved. A comparative plot between all invex functions would've clarified things.
2. A visual aid with optimization landscape of Eq (11) i.e., $\| x - u \|^2$ + Invex function, will give more insights into the convexity of the problem, and comparison with $\| x - u \|^2 + \ell_1$ will help us understand why invex regularizers are performing better than convex regularizers. 
3. Convergence analysis has been provided in theory. It would be better if the comparison of the algorithms has been provided in practice (experimental validation). plots with reconstruction MSE vs iterations, MSE vs time would be suggestible.
4. Proximal operators of non-convex penalties are known to have longer computation time. So, a comparison of the proposed methods running times will establish solid ground for the paper.
limitations: The authors discussed the limitations of their work.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors consider the task of image reconstruction with regularization. Beginning at least a few decades past, regularizers have been adapted by various communities first from L2 quadratic penalties to L1-like edge-preserving penalties. More recently, motivated by compressed sensing researchers have used L1 or even non-convex penalties. Non-convex penalties are more difficult to analyze from a theoretical perspective, but empirically have been observed to produce the best results for image reconstruction tasks.

In this manuscript the authors propose to focus on classes of regularizers that are invex functions. Invex functions are functions where "at any point where the derivative vanishes (stationary point), it is a global minimizer of the function." The manuscript's contributions are three-fold: 1) the manuscript provides the first list of regularizers with proved invexity for image reconstruction, 2) the manuscript establishes convergence guarantees for three types of image reconstruction problems with invex regularizers and 3) the manuscript provides empirical data on the performance of the regularizers.
strengths_and_weaknesses: **Strengths**
- The authors present convergence theory for three problems of substantial interest to the reconstruction community: denoising, compressed sensing, and plug-and-play reconstruction. Each of these has been applied in practical settings. The paper includes global convergence guarantees for each setting.
- The invex approach seems to be a new and interesting perspective into inverse problems. In particular, the quasi-norm of Equation (6) has long been of great interest to the image reconstruction community due to its relations to compressed sensing, so it is encouraging to see that the current manuscript's theory can be applied to this case.
- In addition to theoretical results, the paper also includes empirical results for each of the regularizers under consideration.

**Weaknesses**
- Some properties of H in Theorem 4 seem inconsistent with how it is used in the subsequent algorithms. For example, Theorem 4 states that H is unitary, but H is elsewhere specified to have full row rank. The specific properties of H necessary for each theorem setting should be further clarified.
- The presentation of Table 3 is somewhat confusing.
- The variations in invex regularizer performance is not discussed.
questions: 1. How can the H matrix in Theorem 4 be unitary if m < n? If m < n, H cannot have an orthonormal basis for vectors of size n.
2. Could the annotation of Table 3 be improved? This is one of the most important results of the paper. I would bold the best-performing PSNR value in each row as is standard practice at machine learning conferences rather than the regularizer class. Also, the performance of Eq. (6) should be discussed.
3. How were the datasets used? Did the authors merge all datasets under consideration? These details should be further specified for the experiments.
limitations: The authors discussed limitations under the relevant theorems and lemmas, but for practitioners I think it would be better to have a dedicated Limitations section. A few items that might be considered for such a section:

- The plug-and-play result only shows that the optimization sequences converges to a close estimate of the solution.
- The theory does not seem to consider noisy measurements.
- The reconstruction tasks under consideration seem to be limited to deconvolution.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This manuscript revisits the concept of invexity and provides the first list of proved invex regularizers for improving image reconstruction. Convergence conditions are analyzed and experiments are conducted on practical applications.
strengths_and_weaknesses: Strength:
1.	The paper is well-written with compact structure and clear interpretations.
2.	Some useful regularizers in imaging applications are formally studied in a new perspective of invex function. Besides, a simple and valid way of constructing an invex function is provided, which may potentially inspire later researchers.
3.	Experimental comparisons indicate that the proposed invex regularizers have potential to improve performance in practical applications. 
Weakness:
The proposed new invex function doesn’t achieve the best efficiency compared with the existing non-convex functions.

questions: 1.	Can you compare the performance with some CNN-based baselines?
2.	For different regularizers, were the parameters (e.g. \lambda in eq.12) chosen as the same value?

limitations: No potential negative societal impact is mentioned. Both theoretical analysis and practical applications of invex functions are provided. I believe the proposed finite list will inspire exploration in related fields.
review_title: null

Meta-review: metareview: This paper addresses image reconstruction problems exploiting invex regularizers (which are not necessarily convex). For many modern signal processing applications, invexity of the cost is proved. Many examples are considered, and an extensive comparison with state-of-the art methods are provided in an application section. 
As all reviewers point out unanimously, the paper is very well presented, original and overall of very high quality.  It is therefore a clear accept. But I recall the authors the strict rule on the number of pages for the final version (10 pages).
confidence: Certain
Decision: Accept