Componentwise Perturbation Analysis of the Singular Value Decomposition of a Matrix

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

COMMENTS:
• The work is interesting for specialists and written correctly.
• This paper presents a rigorous perturbation analysis of orthogonal matrices, singular subspaces, and singular values of a full-column order real matrix.
• It is proved that the SVD problem is well posed only in the case of distinct (simple) singular values (Theorem 1; Example 1).
• Interesting analysis (Theorems 2 and 3; appropriate asymptotic estimation).
• The paper presents the results of numerical experiments; Examples 2-5.

I SUGGEST ADDITIONS:
• In equality (1), make it clear that we are considering a real matrix. Present the SVD distribution when the order A=r (for didactic reasons). Add that we will further consider the case of r=n (full column row).
• The proof of Theorem 1 uses the determinant of a special block matrix. For didactic reasons, I suggest adding a note (e.g. in the introduction) about calculating the determinant from a block matrix in the general case (see e.g. Gantmacher 1953, 1988).
• It may be worth supplementing the literature list with the following items:

[1] Gantmacher, F.R. (1988), Matric Theory, ed. 4, Nauka, Moscow [1 edition: 1953–1954].
[2] Wilkinson, J.H. (1965), The algebraic eigenvalue problem, Clarendon Press, Oxford 1965

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript, the authors considered a rigorous perturbation analysis of the singular value decomposition of a full column rank matrix. They showed the SVD perturbation problem is well posed only in case of distinct singular values.  They obtained the local bounds for the sensitivity of the singular subspaces measured by the angles between the unperturbed and perturbed subspaces. They also presented an iterative scheme find global bounds. In conclusion, this is an interesting manuscript. I'd like to recommend it for publication, however, the following points should be revised carefully: (1) The motivation of this manuscript is not clear from the viewpoint of applications. (2) What the matrix means at Page 5? (3) What's the color means in the matrix? (4) More details are needed for equation (21). (5) To avoid the possible conflict and further support this manuscript, the following recent related references are needed: [H. Harcha, O. Chakrone, N. Tsouli, On the nonlinear eigenvalue problems involving the fractional p-Laplacian operator with singular weight, J. Nonlinear Funct. Anal. 2022 (2022) 40] and [Z. Tu, Multitemporal image cloud removal using group sparsity and nonconvex low-rank approximation, J. Nonlinear Var. Anal. 7 (2023), 527-548.]. (6) Finally, there are many typos and misprins, which should be corrected point by point.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. The authors in point 9.2 on the page 26 state, among others, that the global component disturbance limits of the U and V matrices can be found using non-linear estimators of the δWU and δWV matrices, defined by (15) and (16), respectively, and then that such estimates correct the linear estimates of the parameters relevant at each iteration step in a manner similar to that presented in [18], and then point out that, like asymptotic estimates, global bounds can only be found by knowing the norm of δA, without providing sufficient discussion to show sufficiency of the method proposed in the article.

2. There is no convincing justification for choosing the convergence of specifically determined relative errors (unmarked lines between lines 295-396 on page 30) interpreted for example 1 as in Fig. 3 and 4, among many other possible error definitions, such as MAPE, MSE, etc. Moreover, the indicated convergence on the above-mentioned drawings should be shown in an additional drawing as it is currently unreadable.

3. The research results in examples 2-5 are definitely not comprehensively discussed and discussed, which needs to be supplemented. A cartoon illustration never replaces text.

4. I also do not find an appropriate comparison of the obtained research results to the results of this type of research available in the literature on the subject, as well as the selection of a criterion for comparing the results.

5. The conclusion should include your own research results in relation to the state of the literature on the subject or with an indication of originality if such results are not available in the literature, including to what extent the main research objective and additional objectives related to it were achieved and what are the further directions of research .

6. There is also a lack of selection of a computer environment for research, why MATLAB and not another computing environment, including how accurate the calculations were?

7. I did not check the correctness of the derived dependencies and solved issues, assuming that the authors used appropriate functions available in various MATLAB toolboxes for this purpose, although there is no mention of this in the references.

8. The article lacks uniform markings, e.g. those used in the MATLAB convention or according to other accepted ones, which applies, among others, to: marking matrices, scalars, etc.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

1. It would be appropriate to discuss this fragment in more detail, first showing what it is like in the literature on the subject and what the authors experimentally contribute in this regard.

2. The intuitive choice of only the relative error is highly unscientific here. You should show what other types of errors would look like. Moreover, I do not share the authors' optimism, hence I am asking again to show the course of the error in terms of its convergence, the current drawing interpretation does not show details, including boundaries, etc.

3. Therefore, perhaps it is appropriate to provide at least the results of comparative research resulting from the examples; the complete abandonment of a comprehensive arrangement of examples 2-5 weakens the article greatly.

4. Therefore, it would be necessary to clearly show the criterion and discuss the obtained results of comparing your own achievements with those presented in work [28], not only in the drawings!

5. OK

6. In such a case, it should be clearly stated that the research was assumed to be carried out in the MATLAB environment without prior selection of an appropriate computing environment, hence the accuracy of the calculations may be checked in other computing environments in further research.

7. OK, but this should be written clearly, including whether you have also developed your own GUI.

8. This situation does not justify the need for uniformity of the markings used, including: distinguishing matrices from scalars, etc.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Please see the attached file.

Author Response File: Author Response.pdf