Generalized Permutants and Graph GENEOs

Round 1

Reviewer 1 Report

I have identified several serious areas where further improvements can be made to enhance the overall quality of the manuscript.

1.      To improve the clarity and impact of your work, it is recommended that you add a succinct yet compelling statement highlighting the novelty of your research in the abstract section.

2.      The introduction section lacks adequate motivation and fails to clearly articulate the contributions of your study. I suggest that you incorporate a detailed pointwise explanation to provide a strong rationale for your research and precisely state the specific contributions made in this paper.

3.      At the beginning of section 2, it would be beneficial to provide a comprehensive discussion on why this particular section was included in the paper, as well as an overview of the topics covered within this section.

4.      To maintain academic integrity and facilitate ease of reference, ensure that all definitions are appropriately cited. Additionally, kindly assign equation numbers to each equation presented in the paper.

5.      At the outset of section 3, the authors make a reference to "this paper," which appears to be incorrect. Please provide a detailed explanation of the content and objectives of section 3 to rectify this issue.

6.      To enhance the coherence of the paper, consider adding a logical and well-structured introduction to section 4.

7.      Provide a comprehensive explanation of Figure 2 to aid readers in better understanding its significance and relevance to the study.

8.      Address the missing figures in section 5.1 and supplement the section with a detailed theoretical discussion to support your findings.

9.      Ensure that definitions 35, 36, 37, and 38 are clearly articulated, as they appear to play a significant role akin to a conclusion in the paper.

10.  A detailed, theoretical discussion is necessary to expound on the reasons for creating section 5.2 and its relevance to the overall study.

11.  Incorporate a clear statement outlining the main objective of your work in section 6. Additionally, discuss future directions to emphasize the potential impact and significance of your research.

12.  To improve the readability and coherence of the paper, consider refining its presentation in a well-structured manner.

13.  Eliminate unnecessary references and ensure that all cited sources are relevant and logically sound.

14.  Revise the keywords of the paper to adhere to formal correctness and accurately represent the content of your research.

15.  Replace the mention of "see ref [20, 26, 27]" on page 3 with a detailed discussion to elaborate on the relevant aspects of the paper.

a lot of issues in many sentences, please resolve all the problems.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The article is very nice and the topic of the article is very interesting. The article can be accepted in its own form..

Reviewer 3 Report

1.     The introduction is relevant and theory based. It is suggested to add the findings in Introduction section.

2.     Improve the quality of figures as they are not appearing clear with proper citations and arrange the table uniformly.

3.     It is also suggested to arrange the equations in manuscript uniformly and also explain their components.

4.     It is also advised to add the main outcome measures in Discussion it is suggested to add the observations that you have studied from the process. It is suggested to state the limitations after the analysis of your work. The conclusion of a paper needs to summarize the content and purpose of the paper.

5.     The reference section needs improvement as the format is not uniform. Increase the numbers of relevant references from good journals to validate your work.

6.     The manuscript needs to be arranged as per the format and is acceptable for publication after these following changes.

The article needs to be revised for typing errors and grammatical errors.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 4 Report

1- The authors should ask the help of native English speaking proof reader,
because there are too many typo and linguistic mistakes that should be
fixed.
2- Abstract to modify: the abstract should contain Objectives,
Methods/Analysis, Findings, and Novelty /Improvement. It is suggested to
present the abstract in one 200 words paragraph.
3- The introduction is poorly written and it does not properly refer to
previously published studies. The authors need to carefully review the
published literature, identify the gaps in the literature, and propose their
approach to fill the gap on the areas of  bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs).
4- Literature review is not enough. It is important to add some recent work
(2018-2023) to the literature review. At least 8 new references should be
added to article on the areas of bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs)..
5- We prefer if you use the third person singular, instead of the first
person singular or plural (e.g. "we").
6- An editable version  should be added.
7- Almost all references are not international and valid. Authors should use
from international resources. Maximum 4 local references are allowed.
8- Much more explanations and interpretations should be added for the
result, which are not enough on the areas of bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs)..
9- It is suggested to compare the results of the present study with previous
studies and analyze their results completely on the areas of bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs)..
10- It is suggested to organize the conclusion section much better. This
section should be presented in one or two 250-300 word paragraphs.
11- The citation of references in the text doesn’t follow the format
requested by the journal. References must be numbered in order.
12- The manuscript npvelty and contribution should be improved.

The authors must provide solid english revision.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 5 Report

It is a revised version. It can be accepted now.

It is good.

Reviewer 6 Report

Referee Report:

The manuscript titled "Generalized Permutants and Graph GENEOs" discusses the development of a compositional and geometric theory of Group Equivariant Non-Expansive Operators (GENEOs) for Geometric Deep Learning. The paper aims to generalize the notions of permutant and permutant measure to a map between perception pairs, with the objective of extending the application domain of the theory to graphs. The author claims to have developed a coherent theory for both vertex-weighted and edge-weighted graphs, providing simple examples to showcase the potential of GENEOs and permutants in graph theory.

This paper does not meet the standards required for publication in the field of Mathematics and Machine Learning. The results presented in the paper are not strong, and the proofs are described as straightforward. Consequently, the paper lacks the necessary depth and novelty to make a significant contribution to the field. Hence, I recommend rejecting the manuscript.

The English need minor revision.

Round 2

Reviewer 1 Report

The revised version is ok from my side.

Minor editing of the English language required