When Will a Sequence of Points in a Riemannian Submanifold Converge?

Round 1

Reviewer 1 Report

The paper addresses the problem on the convergence of a sequence of points
xn on a a Riemannian manifold X, provided some adequate properties of the set
of the cluster points are satised, in order to emphasize convergence properties
of some iterative algorithms for constrained optimization problems, applied to
deep learning. Some examples are also in view, and open problems are raised.

The author has brushed up the manuscript. I recommend the publication.

Author Response

Thank you very much.

Reviewer 2 Report

The critiques and recommendations made by the reviewers were not addressed by the author, and the resubmitted version of the manuscript is practically equal to the original one.

The only differences between the two versions of the manuscript are the inclusion of a new keyword (Nash conjecture), the addition of "when X is unbounded" at the end of the second sentence of Example 2, on page 4, and a final paragraph of Remarks, which does not add anything to the text, as it directs the reader to yet another manuscript (ref. [14]) that has not yet been accepted or published.

Bearing in mind that, in addition to the aforementioned manuscript, 10 other references (out of a total of 20) are from arXiv and not from peer-reviewed journals, and also considering that the author did not comply with (and did not take seriously) the reviewers' criticisms and recommendations, nor convincingly justified his refusal to improve the manuscript, which I consider unacceptable and inadmissible, I am forced to consider that the manuscript, as it stands, is not in condition to be accepted for publication in the MDPI Mathematics journal.

Author Response

Thank you for your comments. Here I briefly reply to your points. 

Point 1: I think I addressed all reviewers' comments. The other reviewers are happy with my revision. 

Point 2: I added only a few, but I think they address the reviewers' comments. 

Point 3: I wrote (in the letter to editors when submitting before), concerning the fact that preprints can maybe be very valuable and trustable, and gave example of Adadelta paper. I also explained that my papers are under review. Anyway, in this revision I update information on some of the papers which are now published online. 

I hope this answer replies your main concerns. 

 

 

 

 

Reviewer 3 Report

In the present paper under review, author studied the convergence behavior for iterative algorithms for (constrained) optimization problems, with many applications in Deep Learning.

As an observation, please give some more information regarding Example 1. …. This is the case in [9]. …please give page, theorem etc

As aspect, the paper would be more intelligible if the author use Introduction and some sections for the theory studied in the paper.

The author refers to the book [9] K. Lange, Optimization, 2nd edition, Springer texts in statistics, New York 2013. Please be more specific in explanations, giving page, no. of theorem, context, etc

It would be very nice if the authors would give some suggestions about the answers of Question 1 and Question 2.

Author Response

Thank you very much. 

I now have added details pertaining your comments. 

 

Reviewer 4 Report

This manuscript is an expository paper. The author does not prove any result and he presents two open problems. In my opinion, this subject is very interesting and has many applications, so the paper deserves to be published. The manuscript is well written and easily to read.

Author Response

Thank you very much. 

Round 2

Reviewer 2 Report

Some minor changes made to the text of the manuscript (including the alteration of the final of the first paragraph, the modification of the title of the first section to "Motivation", the division of the remaining text into two new sections, "Convergence results" and "Riemannian manifold optimisation", as well as the improvement of the "Remarks" section with the inclusion of the questions stated by the autor) made the text of the manuscript better structured.

However, in the second paragraph of the "Remarks" section, the author mentions "two questions" (which appear in the original manuscript and its version 1), but forgot to include the second question.

It should also be noted the addition of a few new bibliographic references, although it is evident the lack of standardization of the references (which indicates that the author did not use BibTeX), especially regarding the way of indicating the volume, number and title of journals (in full or abbreviated) and the correct capitalization of words in book titles and conference proceedings. In addition, it is necessary to make some changes regarding the comment on the reference [20].

Thus, I recommend the provisional acceptance of the manuscript for publication, subject to the inclusion of the following modifications and corrections:

1. Inclusion of Question 2 that appears in previous versions of the manuscript ("Are there Riemmannian manifolds which are nor non-expandingly homeomorphically compactible for which the conclusions of Theorem 0.2 hold?"), as well as a short paragraph with some additional comments on it, as was done in relation to Question 1.

2. Amendment of the opening paragraph of the Remarks section, where the author states that "In a very recent paper [17], the author has been able to extend previously mentioned results (...). There the readers can consult more general algorithms (...)", since the reference [17] is an unpublished manuscript that has not been peer reviewed and is not available to the reader.

3. Exclusion of the comment in parentheses associated with the reference [20], which must be transferred to the text.

4. Inclusion of the article mentioned in the comment currently associated with the reference [20] as a new reference, as indicated below:

Truong, T.T., Nguyen, H. Backtracking gradient descent method and some applications in large scale optimisation. Part 2: Algorithms and experiments. Appl. Math. Optim. (2020). doi:10.1007/s00245-020-09718-8

5. Finally, a rigorous standardization of references must be made, with the adoption of the referencing style recommended by the journal.

Author Response

Thank you for the feedbacks. 

For your points #1 and #2: The mentioned preprint is now posted online at arXiv:2008.11091. Questions 1 and 2 in the previous version were resolved in that preprint, and hence I deleted them from the revised version of the submission, and instead include a current interesting open problem.  I think that is the most reasonable manner to proceed, given also that arXiv:2008.11091 contains also description of the algorithm in Theorem 1.2.

For your point #3, #4, #5: I revised as you suggested, as best as I can, using the standard abbreviation of the journals' names.   

I upload a pdf file of the revision below. 

 

 

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The manuscript, in general, is well written and is relevant and appropriate to the theme of the special issue on "Riemmanian geometry of submanifolds".

The following points should be considered by the author when resubmitting his manuscript for publication:

The discussion around the general result on page 4 (Theorem 2) should be more detailed.

Grammar and especially punctuation should be improved in some sentences, in order to enhance the readability of the text.

The use of round brackets in citations is not usual and can cause some confusion to the reader. It is convenient to replace them with square brackets.

 

Reviewer 2 Report

See the attached file.

Comments for author File: Comments.pdf

Reviewer 3 Report

The manuscript deals with studies whether a sequences of points form a Riemannian manifold converges or not. The paper starts with an introduction on large scale optimization and iterative numerical methods which are related to it. It continues with a discussion on critical points of a GD method and with some issues on the connectedness of the set of cluster points in certain conditions. The main result is Theorem 0.2, stating convergence/divergence properties related to a sequence constructed by the use of backtracking GD method. Open questions are formulated. The paper is nice and brings outcome to the field; its approach is an original one. There are some minor typos (eg use $\{x_n\}$ all over the place) which has to be corrected. I recommend the publication after this correction is made.