On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and Its Application to Multi-Omics Data Integration

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please find the attached report 

Comments for author File: Comments.pdf

Author Response

 Response letter to the reviewer 1.\\
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  Manuscript ID
mathematics-4369188.  On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and  its Application  to  Multi-omics Data Integration.
\end{center}

\vspace{0.5 cm}

 

\noindent Referee 1:

After reading the paper entitled: ”On the Extremal Trace Problem on SetsHomeomorphic
to the Stiefel Manifoldand its Application to Multi-omics Data
Integration”. In this manuscript, the extremal trace problem for the coupled
Laplacian on the setshomeomorphic to the Stiefel manifold defined on the complex
Euclidean space with application in biologically meaningful integration.
The manuscript is well written and I have to say that the paper can be recommended
for possible publication only after the authors have addressed the
following comments and questions:\\

1. The abstract should include a comparison with previous studies and explicitly
state that the present work is a continuation of the research previously conducted
by the same group of authors.

2. The Introduction contains a list of numbers that appears without sufficient
explanation. The purpose of presenting these values is unclear, and the authors
are encouraged to provide the necessary context and discuss their relevance to the
current work.

 

 

 

 

 

 

 

 

 

 

3. Although Theorem 1 is clearly stated, its proof involves extensive calculations
that could be significantly shortened. The authors are encouraged to reduce
the computational details and adopt a more compact derivation of the solution
set.

4. Corollary 1 is described by the authors as presenting a ”relationship”; however,
it appears to be merely an equation rather than a genuine mathematical
relationship requiring separate emphasis. The authors should reconsider the terminology used and clarify whether the statement merits being formulated as a
corollary. Also, Lemma3 presents inequality

5. The authors state that ”In Section 4, we finalize the theoretical results, observe
the biological part of the investigation, and discuss prospective generalizations.”
However, Section 4 corresponds to the Conclusion and does not appear
to contain these elements in the manner described. The authors should revise
this statement to accurately reflect the actual content and organization of the
manuscript..\\

\noindent Author:  Dear referee, I am sincerely grateful to you for   the made remarks. However, let us consider them consistently.\\

 

 \begin{center}
 {\bf Author's replies}
\end{center}

 

 \noindent Referee 1:

 1. The abstract should include a comparison with previous studies and explicitly
state that the present work is a continuation of the research previously conducted
by the same group of authors. \\

  \noindent Author: Dear referee, thank you for the remark!  I have made the corresponding changes.\\

\newpage
 \noindent Referee 1:

 2. The Introduction contains a list of numbers that appears without sufficient
explanation. The purpose of presenting these values is unclear, and the authors
are encouraged to provide the necessary context and discuss their relevance to the
current work.\\

 \noindent Author: Dear referee,  thank you for the remark! The references given in the introduction section have been carefully verified  and the additional description has been  added. As for their relevance  to the
current work they represent a scientific background related to   the methods of the  manifold alignment and we should demonstrate them in order to implement a comprehensive analysis and take into account a scientific contribution in the direction. Thus, in the introduction section, we consider previously elaborated methods showing the relevance of the scientific direction and  trying to integrate our work into the constructed theory.

Additionally, I should remark that the so-called {\it list of numbers} appears in the introduction section three times.

 The first list ({\it  Nowadays a number of tools have been developed for integrating multi-omics data: mixOmics [25]...}) is supplied with the following   analysis ({\it First, most of them are based on the search for statistical correlations between datasets. This approach allows us to identify jointly varying features, but does not answer the question of the nature of the detected relationships...}).

 The second  list ({\it The following papers consider  a method of coupling    several types of measurements of observed data in a single cell [28,14,16,8,10].}) represents the papers that the authors have found significant for    the  given scientific  direction and therefore worth citing. The list is supplied with  not formal comments reflecting the limitations (from the authors' point of view) that can be eliminated in the present work.

 The third  list ({\it Generally, the elements of the transition matrix correspond to  the biological characteristics, for instance  profiles of scATAC-seq   [27,26,18,24]}) appears in the context of discussion on the transition matrix related to the central idea of the present paper - coupling of the heterogenous datasets. Thus, we observe the previously made attempts  to elaborate methods of coupling heterogenous datasets.
 \\

 

  \noindent Referee 1:

 3. Although Theorem 1 is clearly stated, its proof involves extensive calculations
that could be significantly shortened. The authors are encouraged to reduce
the computational details and adopt a more compact derivation of the solution
set.\\

 \noindent Author: Dear referee, thank you for the remark!   Some reasonings of Theorem 1  containing calculations based upon the trace properties  have been shortened. At the same time, I have added  some comments excluding any kind of misunderstanding   and helping the reader to study the proof without  tense.  Here, I should note that reducing the reasonings we must keep a balance in order not to make the proof unreadable.    \\

 \noindent Referee 1:

4. Corollary 1 is described by the authors as presenting a ”relationship”; however,
it appears to be merely an equation rather than a genuine mathematical
relationship requiring separate emphasis. The authors should reconsider the terminology used and clarify whether the statement merits being formulated as a
corollary. Also, Lemma 3 presents inequality\\

  \noindent Author: Dear referee, thank you for a good remark and clarification.  I have made the corresponding  changes. \\

 

 \noindent Referee 1:

5. The authors state that ”In Section 4, we finalize the theoretical results, observe
the biological part of the investigation, and discuss prospective generalizations.”
However, Section 4 corresponds to the Conclusion and does not appear
to contain these elements in the manner described. The authors should revise
this statement to accurately reflect the actual content and organization of the
manuscript.\\

  \noindent Author: Dear referee, thank you for the remark!  The conclusion section have been carefully revised  along with the mentioned  statement and  as a result appropriate  changes providing more detailed information have been made.    \\

 

 

 

\noindent Author:  Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to  improve the paper significantly.

\vspace{0.1 cm}

Sincerely yours Ph.D. Maksim V. Kukushkin

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

See the report

Comments for author File: Comments.pdf

Author Response

\begin{document}

 \begin{center}
  Response letter to the reviewer 2.\\
\vspace{0.5 cm}
  Manuscript ID
mathematics-4369188.  On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and  its Application  to  Multi-omics Data Integration.
\end{center}

\vspace{0.5 cm}

 

\noindent Referee 2:

General assessment
The manuscript studies an extremal trace problem for a coupled graph Laplacian on sets homeomorphic to a Stiefel manifold, with motivation from multi-omics data integration. The topic is relevant to the scope of Mathematics, particularly in relation to spectral graph theory, optimization, matrix analysis, and mathematical approaches to data integration.
The paper presents a substantial theoretical development. The formulation of the minimum and maximum trace problems, the stationary point analysis through Lagrange multipliers, and the Hessian-based classification are mathematically interesting. The biological motivation is relevant, although it should be presented more cautiously because the application is mainly conceptual in the current version.
In my opinion, the manuscript can be considered for publication after minor revision. The required changes are achievable and mainly concern clarity, presentation, and a more precise description of the connection with multi-omics data integration.
Strengths\\

The mathematical problem is nontrivial and fits the scope of the journal.\\

The use of coupled graph Laplacians and Stiefel-type constraints provides an interesting framework.\\

The stationary point classification and Hessian analysis are valuable components of the paper.\\

The biological motivation may attract readers interested in mathematical approaches to multi-omics integration.\\

The manuscript has a clear theoretical direction and can be improved without a complete restructuring.\\

Comments for minor revision

1. Clarify the exact contribution of the paper. The authors should add a short paragraph at the end of the introduction listing the main contributions. This would help distinguish the present paper from previous work on coupled graph Laplacian methods and manifold alignment.

2. Moderate the application claim in the title, abstract, and conclusion. The manuscript proposes a connection to multi-omics data integration, but the application is mainly conceptual. The authors should avoid suggesting that a full biological validation has already been performed unless such results are added.

3. Explain the construction of the coupled matrix more clearly. The paper should include a concise explanation of how the matrix W would be constructed from heterogeneous datasets, and what the real and imaginary parts represent in the multi-omics context.

4. Improve the readability of the assumptions. The assumptions on $W, D, L, T_\zeta$, and the admissible set should be summarized more clearly. This would make the mathematical framework easier to follow.

5. Expand a few compressed proof steps. Some parts of the proofs use phrases such as "by direct calculation" or "left to the reader." The authors do not need to rewrite all proofs, but they should expand the most important steps in the derivation of the stationary system and the classification of stationary points.\\

6. Clarify the role of exceptional points. The discussion involving exceptional points and fixed points should be stated carefully. If a generic or probabilistic argument is used, the authors should clearly identify the assumptions under which it is valid.\\

7. Add a short algorithmic summary. A concise step-by-step description of the proposed mathematical procedure would help applied readers understand how the theory could be used in practice.\\

8. Improve language and style. The manuscript contains some long sentences and broad statements, especially in the biological motivation. The authors should revise the English for clarity, precision, and conciseness.\\

9. Strengthen the conclusion. The conclusion should clearly separate what has been proved in the manuscript from what is proposed as future biological or computational work.\\

 

Optional suggestions\\

The authors may consider adding a small synthetic numerical example to illustrate the construction of $W, D, L, T_\zeta.$  This is not strictly necessary for a theoretical paper, but it would improve readability.\\

If the phrase "application to multi-omics data integration" is retained in the title, the authors should provide at least a concise illustrative pipeline showing how the method would be applied to such data.\\

A more cautious title could be: "An Extremal Trace Problem for Coupled Graph Laplacians on Stiefel-Type Manifolds with Motivation from Multi-omics Data Integration."\\

Final recommendation\\

I recommend Minor Revision. The manuscript is mathematically interesting and appears suitable for publication after the authors address the clarification and presentation issues listed above.
The requested changes are achievable: they mainly concern clarification of novelty, moderation of the application claim, improved explanation of assumptions, limited expansion of proof details, and language polishing.
\newpage

\begin{center}
{\bf Author's replies}
\end{center}

\noindent Author:

 Dear referee, I am sincerely grateful to you for   the made remarks. However, let us consider them consistently.\\

\noindent Referee 2:\\

1. Clarify the exact contribution of the paper. The authors should add a short paragraph at the end of the introduction listing the main contributions. This would help distinguish the present paper from previous work on coupled graph Laplacian methods and manifold alignment.\\

\noindent Author: Dear referee, thank you for the remark. The detailed description indicating advantages of the developed theory from the applied point of view is given at the end of the introduction section as well as the  comprehensive analysis of the pure mathematical  approach.\\

 

 

\noindent Referee 2:\\

2. Moderate the application claim in the title, abstract, and conclusion. The manuscript proposes a connection to multi-omics data integration, but the application is mainly conceptual. The authors should avoid suggesting that a full biological validation has already been performed unless such results are added.\\

\noindent Author: Thank you for the remark. This work is the first in a series of papers devoted to the development of new mathematical approaches to the integration of biological data from various omics. At the end of 2024, our laboratory received a government assignment on the topic "Searching for genes associated with the aging process and age-associated diseases using the diagonal integration of multiomics data." The project is planned to be completed by 2027. The interim results of the project include measures to protect intellectual property and the development of a platform for the diagonal integration of biological data from various omics. According to the terms of the state assignment, we do not have the right to publish the calculation results until a certain date, and at the moment we cannot publish the results of diagonal integration, since the mathematical approach alone is not enough to carry out the calculations. After we formulate a mathematical approach, a team of developers joins the work, which translates the mathematical vision of solving the problem into program code, after which the corresponding data will be submitted to the input program. Then the results of the analysis  given by the available biological instruments  and the instrument corresponding to the developed   mathematical approach  will be compared.
There is also a legal reason for not being able to publish the calculation results right now. In order to register a program code or a new approach, we must first register and protect the approach, and only after that we will be able to submit the next article, which will describe all this in more detail. In general, it turns out that this is a project work that involves step-by-step progress with consistent results.

 

Yet at the same time, we have not claimed that the theoretical results have been already  experimentally  verified, however  we propose a harmonious theory especially elaborated for a concrete biological problem that is based, in its formulation, on the hard scientific labor of the research group including experiments, measurements, and observation.\\

 

 

 

 \noindent Referee 2:\\

 3. Explain the construction of the coupled matrix more clearly. The paper should include a concise explanation of how the matrix W would be constructed from heterogeneous datasets, and what the real and imaginary parts represent in the multi-omics context. \\

 

\noindent Author: Dear referee,  thank you for an  extremely valuable   remark.  The detailed explanation concerning the construction of the matrix W is represented in the introduction section (paragraph  1).   \\

 

 

\noindent Referee 2:\\

  4. Improve the readability of the assumptions. The assumptions on $W, D, L, T_\zeta$, and the admissible set should be summarized more clearly. This would make the mathematical framework easier to follow. \\

\noindent Author: Dear referee,  thank you for a valuable   remark. I completely agree with you. In order to make the notations clearer I have added an  additional list of notations in the beginning   of the section 3.1.  \\

 

 

 

\noindent Referee 2:\\

5. Expand a few compressed proof steps. Some parts of the proofs use phrases such as "by direct calculation" or "left to the reader." The authors do not need to rewrite all proofs, but they should expand the most important steps in the derivation of the stationary system and the classification of stationary points.\\

  \noindent Author: Dear referee,  thank you for a valuable   remark.  I have verified the pointed out steps of the proof and partially expended them (if they are really complicated)  partially supplied with references and explanations.\\

 

\noindent Referee 2:\\

6. Clarify the role of exceptional points. The discussion involving exceptional points and fixed points should be stated carefully. If a generic or probabilistic argument is used, the authors should clearly identify the assumptions under which it is valid.\\

\noindent Author: Dear referee,  thank you for a  remark.    The exceptional points are considered  in the paragraph 3.1, page 13, where I refer  to  to the monograph by Kato T. I have added the   word {\bf therefore} in order to stress the implication and in this way to show more clearly applicability of  results by Kato T.,  see the extract given below.

{\it  Note that the operator  function
$$
\mathfrak{Re}\{e^{-iz}L\}  =e^{-iz}L+e^{ iz}L^{\ast},\;z\in \mathbb{C}
$$
is holomorphic with respect to the parameter $z$ and therefore its restriction to the real half-axis is analytic. Thus, we can  apply    results by Kato    related to  the perturbation of the finite dimensional operator, Chapter II, \S 1.3, \S1.4,  [17]. In order  to avoid misunderstanding, that may happen while using terminology, we want to show an obvious representation that holds in the sufficiently small $\varepsilon$ - neighborhood of the arbitrary chosen point $\theta_{0}\in \mathbb{R},$  we have
$$
\mathfrak{Re}\{e^{-i\theta}L\}=\mathfrak{Re}\{e^{-i\theta_{0}}L\}+
 (\theta-\theta_{0}) \mathfrak{Re}^{(1)}\{e^{-i\theta_{0}}L\}+\ldots+\frac{(\theta-\theta_{0})^{q}}{q!}\mathfrak{Re}^{(q)}\{e^{-i\theta_{0}}L\}+\ldots \,,
$$
where
$$
\mathfrak{Re}^{(q)}\{e^{-i\theta_{0}}L\} =\left\{ \begin{aligned}
 (-i)^{k}\mathfrak{Im} \{e^{-i\theta_{0}}L\},\;\, q=2k+1,  \\
 (-1)^{k }\mathfrak{Re} \{e^{-i\theta_{0}}L\},\;\,q=2k ,\,k\in \mathbb{N}_{0}   \\
\end{aligned}
 \right.  .
$$
{\bf Therefore}, in accordance with   [17, p.67], the eigenvalues $\mu_{j}(\theta),\,\theta\in \mathbb{R},\,j=1,2,\dots,n$ have  constant  geometric multiplicities   except for a set of the {\bf exceptional points} that  is finite on any compact segment.}

The role of the exceptional points is completely clear from Lemma 3 (Lemma 4 in the revised version).  As for the probabilistic approach then a transparent explanation, where  the assumptions under which it is valid are clearly identified,              is given at the end of the section main results. For you convenience,   I represent it below.\\

{\it Eventually, the remaining problem is the location of exceptional points $\mathbb{D}$ with respect to the fixed points $\mathbb{S}_{p}.$  In accordance with Lemma 3 (Lemma 4 in the revised version), we have  $ \mathrm{card}\, \mathbb{S}_{p}<\infty,$   whereas according to [17, p.67],  we have  $\mathrm{card} ( \mathbb{D} \cap  \mathbb{J}) <\infty.$ Using a probabilistic  approach, we conclude that   the geometric probability of the random event of coincidence of the exceptional points and the fixed points equals  zero in the case when the initial matrix $W$ is formed from the elements of   experimental origin.  This clarification creates a complete theoretical foundation for   prospective applications. This can be  formulated in the form of a hypothesis.}\\

 

However, this hypothesis reflects exclusively  applied point of view that is preferable if we consider a final result. At the same time from the mathematical point of view, we are rather satisfied with the compleat description of the cardinality of the set of the exceptional points. Additionally, I may notice that the theoretical and applied  understanding of the set of the exceptional  points coincide  in some sense since the first one appeals to zero Lebesgue measure and the second one appeals to zero probability.\\

 

\noindent Referee 2:\\

  7. Add a short algorithmic summary. A concise step-by-step description of the proposed mathematical procedure would help applied readers understand how the theory could be used in practice. \\

\noindent Author: Dear referee,  thank you for a    remark. I have made the corresponding changes and added a Subsection  3.3 at the and of the paper.   \\

\noindent Referee 2:\\

8. Improve language and style. The manuscript contains some long sentences and broad statements, especially in the biological motivation. The authors should revise the English for clarity, precision, and conciseness.\\

\noindent Author: Dear referee,  thank you for a  valuable remark. I have tried to follow it as much as I could  in order  not to contradict  stylistic peculiarities related to a biological terminology. I  have made the corresponding changes.\\

 

\noindent Referee 2:\\

  9. Strengthen the conclusion. The conclusion should clearly separate what has been proved in the manuscript from what is proposed as future biological or computational work. \\

\noindent Author: Dear referee,  thank you for a   remark. I have added a  description reflecting the pointed out issue   to the conclusion section.\\

 

\noindent Referee 2:\\

Optional suggestions\\

 The authors may consider adding a small synthetic numerical example to illustrate the construction of $W, D, L, T_\zeta.$  This is not strictly necessary for a theoretical paper, but it would improve readability. \\

 If the phrase "application to multi-omics data integration" is retained in the title, the authors should provide at least a concise illustrative pipeline showing how the method would be applied to such data.\\

A more cautious title could be: "An Extremal Trace Problem for Coupled Graph Laplacians on Stiefel-Type Manifolds with Motivation from Multi-omics Data Integration." \\

 

 

 

 

 

 

\noindent Author: Dear referee, I have found the optional suggestions rather reasonable and interesting in particular the one related to the numerical example illustrating the construction of the used matrices. As for the changes in the title, I also find your clarification  corresponding to the  academician standard.  As far as I know, in accordance with the journal rules, there is a kind of little  stylistic  freedom regarding  the abstract and the title. In this case the phrase {\it and its application } can be treated as a completely classical term allowing a large auditory to perceive the idea without tense.  I want to preserve the current title but if you persist I will change it after discussion with coauthors, who have various opinions on this matter, and  requesting a formal permission from the editorial board.       \\

 

 

 

 

 

 

 

 

 

 

 

 

 

\noindent Author:\\

 Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to   improve the paper significantly.

\vspace{0.1 cm}

Sincerely yours Ph.D. Maksim V. Kukushkin

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

 

Reviewer Report

Title: On the Extremal Trace Problem on Sets Homoeomorphic to the Stiefel Manifold and its Application to Multi-omics Data Integration

The manuscript presents a theoretical study of an extremal trace problem on sets homoeomorphic to the Stiefel manifold and discusses its application to multi-omics data integration. The topic is interesting and combines optimization, manifold theory, graph Laplacian methods, and biological data analysis. The mathematical framework appears rigorous, and the paper contains several novel theoretical results. However, before publication, the manuscript would benefit from several minor revisions to improve clarity, readability, and presentation.

Minor Corrections

1. The manuscript contains numerous grammatical and stylistic issues. A thorough English language revision is recommended.
2. The abstract should briefly mention the main theoretical findings and provide more specific information about the proposed biological application.
3. The Introduction is relatively long and contains extensive discussion of biological background. It could be shortened and focused more directly on the research gap and contributions.
4. A separate paragraph clearly highlighting the novelty and main contributions of the work should be added at the end of the Introduction.
5. Since many symbols and matrices are introduced, a notation table would improve readability for readers from interdisciplinary backgrounds.
6. The application to multi-omics integration remains largely conceptual. A more detailed explanation or illustrative example would strengthen the practical relevance of the work.
7. A schematic diagram showing the proposed manifold alignment framework and data integration process would help readers understand the methodology.
8. Ensure consistency in reference style, abbreviations, and citation formatting throughout the manuscript.
9. The discussion could include a clearer comparison with existing manifold alignment and graph Laplacian approaches to emphasize advantages and limitations.
10. The conclusion should briefly summarize the main mathematical results and discuss potential future research directions and applications.

Mathematical Corrections

11. The notation used for complex conjugation, adjoint operators, and transpose operations should be reviewed carefully. In several places, symbols such as , , and appear together, which may create ambiguity for readers. A clear distinction and consistent notation throughout the manuscript are recommended.
12. The derivation leading from the stationary condition

to the generalized eigenvalue problem should be explained in greater detail. Providing intermediate steps would improve readability and help readers verify the mathematical arguments more easily.

Alternative Equation-Specific Comment

In Equation (2), where the quadratic form of the coupled graph Laplacian is derived, the authors may provide a brief proof or additional justification for the relation

 

)

since this identity plays a fundamental role in the subsequent theoretical development.

• In the proof of Theorem 1, the transition from the Lagrangian formulation to equation (6) could be elaborated further to ensure that the application of matrix differential calculus and Wirtinger derivatives is completely transparent.

Recommendation

The manuscript contains valuable theoretical contributions and is suitable for publication after minor revision addressing the above comments.

 

Author Response

\begin{document}

 \begin{center}
  Response letter to the rviewer 3.\\
\vspace{0.5 cm}
  Manuscript ID
mathematics-4369188.  On the Extremal Trace Problem on Sets Homeomorphic to the Stiefel Manifold and  its Application  to  Multi-omics Data Integration.
\end{center}

\vspace{0.5 cm}

 

\noindent Referee 3:

The manuscript presents a theoretical study of an extremal trace problem on sets homoeomorphic to the Stiefel manifold and discusses its application to multi-omics data integration. The topic is interesting and combines optimization, manifold theory, graph Laplacian methods, and biological data analysis. The mathematical framework appears rigorous, and the paper contains several novel theoretical results. However, before publication, the manuscript would benefit from several minor revisions to improve clarity, readability, and presentation.\\

{\bf Minor Corrections}\\

1. The manuscript contains numerous grammatical and stylistic issues. A thorough English language revision is recommended.

2. The abstract should briefly mention the main theoretical findings and provide more specific information about the proposed biological application.

3. The Introduction is relatively long and contains extensive discussion of biological background. It could be shortened and focused more directly on the research gap and contributions.

4. A separate paragraph clearly highlighting the novelty and main contributions of the work should be added at the end of the Introduction.

5. Since many symbols and matrices are introduced, a notation table would improve readability for readers from interdisciplinary backgrounds.

6. The application to multi-omics integration remains largely conceptual. A more detailed explanation or illustrative example would strengthen the practical relevance of the work.

  7. A schematic diagram showing the proposed manifold alignment framework and data integration process would help readers understand the methodology.

8. Ensure consistency in reference style, abbreviations, and citation formatting throughout the manuscript.

9. The discussion could include a clearer comparison with existing manifold alignment and graph Laplacian approaches to emphasize advantages and limitations.

10. The conclusion should briefly summarize the main mathematical results and discuss potential future research directions and applications.\\

{\bf Mathematical Corrections}\\

The notation used for complex conjugation, adjoint operators, and transpose operations should be reviewed carefully. In several places, symbols such as ,... , and appear together, which may create ambiguity for readers. A clear distinction and consistent notation throughout the manuscript are recommended.
The derivation leading from the stationary condition to the generalized eigenvalue problem should be explained in greater detail. Providing intermediate steps would improve readability and help readers verify the mathematical arguments more easily.\\

{\bf Alternative Equation-Specific Comment}\\

In Equation (2), where the quadratic form of the coupled graph Laplacian is derived, the authors may provide a brief proof or additional justification for the relation (?) since this identity plays a fundamental role in the subsequent theoretical development.\\

In the proof of Theorem 1, the transition from the Lagrangian formulation to equation (6) could be elaborated further to ensure that the application of matrix differential calculus and Wirtinger derivatives is completely transparent.\\

{\bf Recommendation}\\

The manuscript contains valuable theoretical contributions and is suitable for publication after minor revision addressing the above comments.\\

\begin{center}
{\bf Author's replies}
\end{center}

\noindent Author:

 Dear referee, I am sincerely grateful to you for   the made remarks. However, let us consider them consistently.\\

\noindent Referee 3:\\

1. The manuscript contains numerous grammatical and stylistic issues. A thorough English language revision is recommended.\\

 

\noindent Author: Dear referee, thank you for the remark, I have proofread the paper one more time in order to verify the style of writing.  In some  places I have made stylistic changes and grammatical corrections.   \\

 

\noindent Referee 3:\\

2. The abstract should briefly mention the main theoretical findings and provide more specific information about the proposed biological application.\\

 

\noindent Author: Dear referee, thank you for the remark, I have made the corresponding changes. \\

 

\newpage

\noindent Referee 3:\\

3. The Introduction is relatively long and contains extensive discussion of biological background. It could be shortened and focused more directly on the research gap and contributions.\\

 

\noindent Author: Dear referee, thank you for the remark, I have made the corresponding changes. \\

 

 

\noindent Referee 3:\\

4. A separate paragraph clearly highlighting the novelty and main contributions of the work should be added at the end of the Introduction.\\

 

\noindent Author: Dear referee, thank you for the remark, I have added a subsection at the end of the introduction section. \\

\noindent Referee 3:\\

5. Since many symbols and matrices are introduced, a notation table would improve readability for readers from interdisciplinary backgrounds.\\

 

\noindent Author: Dear referee, thank you for the remark, I have made the corresponding changes. A list of notations is presented in the beginning of the Subsection 3.1.  \\

 

\noindent Referee 3:\\

6. The application to multi-omics integration remains largely conceptual. A more detailed explanation or illustrative example would strengthen the practical relevance of the work.\\

 

\noindent Author: Thank you for the remark. This work is the first in a series of papers devoted to the development of new mathematical approaches to the integration of biological data from various omics. At the end of 2024, our laboratory received a government assignment on the topic "Searching for genes associated with the aging process and age-associated diseases using the diagonal integration of multiomics data." The project is planned to be completed by 2027. The interim results of the project include measures to protect intellectual property and the development of a platform for the diagonal integration of biological data from various omics. According to the terms of the state assignment, we do not have the right to publish the calculation results until a certain date, and at the moment we cannot publish the results of diagonal integration, since the mathematical approach alone is not enough to carry out the calculations. After we formulate a mathematical approach, a team of developers joins the work, which translates the mathematical vision of solving the problem into program code, after which the corresponding data will be submitted to the input program. Then the results of the analysis  given by the available biological instruments  and the instrument corresponding to the developed   mathematical approach  will be compared.
There is also a legal reason for not being able to publish the calculation results right now. In order to register a program code or a new approach, we must first register and protect the approach, and only after that we will be able to submit the next article, which will describe all this in more detail. In general, it turns out that this is a project work that involves step-by-step progress with consistent results.

 

Yet at the same time, we have not claimed that the theoretical results have been already  experimentally  verified, however  we propose a harmonious theory especially elaborated for a concrete biological problem that is based, in its formulation, on the hard scientific labor of the research group including experiments, measurements, and observation.\\

 

\noindent Referee 3:\\

7. A schematic diagram showing the proposed manifold alignment framework and data integration process would help readers understand the methodology.\\

 

\noindent Author: Dear referee, thank you for the remark. I have made the corresponding changes and added  Subsection  3.3 at the and of the paper. \\

 

 

\noindent Referee 3:\\

8. Ensure consistency in reference style, abbreviations, and citation formatting throughout the manuscript.\\

 

\noindent Author: Dear referee, thank you for the remark, I have made a careful  verification of consistency in reference style, abbreviations, and citations  throughout the manuscript.  \\

 

\noindent Referee 3:\\

9. The discussion could include a clearer comparison with existing manifold alignment and graph Laplacian approaches to emphasize advantages and limitations.\\

 

\noindent Author: Dear referee, thank you for the remark, I have added an paragraph   in the introduction section  including  a clearer comparison with existing manifold alignment and graph Laplacian approaches to emphasize more convexly  advantages and limitations.   \\

 

\noindent Referee 3:\\

10. The conclusion should briefly summarize the main mathematical results and discuss potential future research directions and applications.\\

 

\noindent Author: Dear referee, thank you for the remark, I have made the corresponding changes. \\

 

\noindent Referee 3:\\

In Equation (2), where the quadratic form of the coupled graph Laplacian is derived, the authors may provide a brief proof or additional justification for the relation (?) since this identity plays a fundamental role in the subsequent theoretical development.\\

\noindent Author: Dear referee, thank you for the remark, the following lemma reflects the well-known relation (I could not manage to find who the author is), however the proof is  presented in the expanded variant in the  paper [22], thus I have included  the corresponding  reference.  For your convenience, I represent  the lemma below in terms of the paper. \\

\begin{lem}\label{L1} Assume that $W\in \mathbb{C}^{ n\times n }$ is symmetric   then  the following relation holds
$$
  \mathrm{tr}\{ A ^{ \ast } L  A \} =\frac{1}{2} \sum\limits_{s,j=1}^{n}\|\mathbf{a}^{\cdot}_{s}-\mathbf{a}^{\cdot}_{j}\|_{\mathbb{C}^{m}}^{2}W_{sj}.
$$
\end{lem}
\begin{proof}
It is clear that
\begin{equation}\label{1y}
\mathrm{tr}\{ A ^{ \ast } L  A \}=\sum\limits_{q=1}^{m} (A ^{ \ast } L  A)_{qq}=\sum\limits_{q=1}^{m} ( L \mathbf{a}_{ q },    \mathbf{a}_{ q })_{\mathbb{C}^{n}}.
\end{equation}
On the other hand
$$
\mathrm{Re} \sum\limits_{s,j=1}^{n}|a_{sq}-a_{jq}|^{2}W_{sj}=
\mathrm{Re}\left\{\sum\limits_{s,j=1}^{n}\left(|a_{sq}|^{2}+|a_{jq}|^{2}-2\mathrm{Re}\{a_{sq}\bar{a}_{jq}\}\right)W_{sj}\right\}=
$$
\begin{equation*}\label{1b}
= 2\mathrm{Re}\left\{\sum\limits_{s=1}^{n}\mathrm{Re}D_{ss} |a_{sq}|^{2} - \sum\limits_{s,j=1}^{n} a_{sq}\bar{a}_{jq} \mathrm{Re}W_{sj}\right\}
=2 (\mathbf{a}_{q},  \mathfrak{Re} L \mathbf{a}_{q})_{\mathbb{C}^{n}}=2 \mathrm{Re} (L \mathbf{a}_{q},  \mathbf{a}_{q})_{\mathbb{C}^{n}}.
 \end{equation*}
Analogously, we get
$$
\mathrm{Im} \sum\limits_{s,j=1}^{n}|a_{sq}-a_{jq}|^{2}W_{sj}=\mathrm{Im}\left\{ \sum\limits_{s,j=1}^{n}\left(|a_{sq}|^{2}+|a_{jq}|^{2}-2\mathrm{Re}\{a_{sq}\bar{a}_{jq}\}\right)W_{sj}\right\}=
$$
\begin{equation*}\label{2b}
 =\mathrm{Re}\left\{\sum\limits_{s=1}^{n}\mathrm{Im}D_{ss} |a_{sq}|^{2} - \sum\limits_{s,j=1}^{n} a_{sq}\bar{a}_{jq} \mathrm{Im}W_{sj}\right\} = 2\mathrm{Re}(\mathbf{a}_{q},   \mathfrak{Im } L\mathbf{a}_{q})_{\mathbb{C}^{n}}=2\mathrm{Im}(L\mathbf{a}_{q},      \mathbf{a}_{q})_{\mathbb{C}^{n}}.
\end{equation*}
Combining the above  relations, we get
 $$
\sum\limits_{q=1}^{m}       ( L\mathbf{a}_{ q },       \mathbf{a}_{ q })_{\mathbb{C}^{n}}=\frac{1}{2}\sum\limits_{q=1}^{m} \sum\limits_{s,j=1}^{n}W_{sj}|a_{sq}-a_{jq} |^{2}= \frac{1}{2} \sum\limits_{s,j=1}^{n}W_{sj}\sum\limits_{q=1}^{m}|a_{sq}-a_{jq} |^{2}=\frac{1}{2} \sum\limits_{s,j=1}^{n}\|\mathbf{a}^{\cdot} _{s}-\mathbf{a}^{\cdot} _{j}\|_{\mathbb{C}^{m}}^{2}W_{sj}.
$$
Using formula \eqref{1y}, we obtain   the desired result.
\end{proof}

\noindent Referee 3:\\

In the proof of Theorem 1, the transition from the Lagrangian formulation to equation (6) could be elaborated further to ensure that the application of matrix differential calculus and Wirtinger derivatives is completely transparent.\\

\noindent Author: Dear referee, following your remark, I have supplied the reasonings devoted to the proof of relation (6) with the comments, making them completely transparent and     excluding misunderstanding of any kind.\\

\noindent Author:\\

 Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to see the matter from another point of view and in this way to   improve the paper significantly.

\vspace{0.1 cm}

Sincerely yours Ph.D. Maksim V. Kukushkin

 

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