Natural Methods of Unsupervised Topological Alignment
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors• What is the main question addressed by the research?
The manuscript touches a wide range of topics. However, the main one is probably the main topic is the so-called unsupervised toplogical aligment. This is a very popular topic in the modern research, and one can easily find dosens of article of last 3 years, devoted to the topic.
The structure of a network is reconstructed by an incomplete set of data.
• Do you consider the topic original or relevant to the field? Does it address a specific gap in the field? Please also explain why this is/ is not the case.
This is exacly what I need the authors to clarify. There are many new works in the area... Well. the autthors' approach is original and does not resemble to what I have seen before. However, it would be much better, if the authors indicate which specific gap they fill or which specific real life problem they solve.
• What does it add to the subject area compared with other published material?
In my opinion, the authors' approach to study the coupled graph Laplcian method(section 3.1) and also general kernel methods, Subsection 3.2.2 and and later on, is quite original and deserves to appear in a good journal).
• What specific improvements should the authors consider regarding the methodology?
The methodology itself looks OK but it is very unclear for the people who are not involved in the specific area.
• Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed? Please also explain why this is/is not the case.
Frankly speaking, the phrase concerning hypercomplex numbers and Clifford algebras in the Conclusion must be clarified or deleted. I do not understand it.
• Are the references appropriate?
Yes, they look appropriate. However, I do not know all the literature about this problem, the complete list would be really huge.
• Any additional comments on the tables and figures.
No tables or fugures are there.
The main results of the paper look quite interesting. They are new and allow some opportunities to model resl-life problems.
However, I have some major remarks listed below.
1) The paper involves an impresive amount of approaches including Manifold learning, kernel based method, MicrOmix approach, SNE method and others. However, the place of the current manuscript in the modern theory is unclear. The Introduction gives the main idea of the theory but does not answer the question.
2) The applications are briefly mentionned (subsection 3.3) but this is very vague. I strongly recommend to include some calculations related to a particular problem or, at least, make more precise ideas of how to apply the results of the paper.
3) The use terminology is sometimes incorrect. For example,
non-square matrices do not form the ring (page 3, line 1). The whole paper must be checked.
4) English must be significantly updated. For instance, AI tools can be used.
Therefore although I support the idea of publication of the manuscript in Mathematics, I suggest the major revision.
Author Response
Response letter to the referee 1.\\
\vspace{0.5 cm}
Manuscript ID
mathematics-3962116. Natural methods of the unsupervised topological alignment.
\end{center}
\vspace{0.5 cm}
\begin{center}
{\bf Remarks and comments}
\end{center}
\noindent Referee 1:\\
\noindent $\bullet$ What is the main question addressed by the research? \\
\noindent The manuscript touches a wide range of topics. However, the main one is probably the main topic is the so-called unsupervised topological alignment. This is a very popular topic in the modern research, and one can easily find dozens of article of last 3 years, devoted to the topic.\\
\noindent The structure of a network is reconstructed by an incomplete set of data.\\
\noindent $\bullet$ Do you consider the topic original or relevant to the field? Does it address a specific gap in the field? Please also explain why this is/ is not the case.\\
\noindent This is exactly what I need the authors to clarify. There are many new works in the area... Well. the authors' approach is original and does not resemble to what I have seen before. However, it would be much better, if the authors indicate which specific gap they fill or which specific real life problem they solve.\\
\noindent $\bullet$ What does it add to the subject area compared with other published material?\\
\noindent In my opinion, the authors' approach to study the coupled graph Laplcian method(section 3.1) and also general kernel methods, Subsection 3.2.2 and and later on, is quite original and deserves to appear in a good journal).\\
\noindent $\bullet$ What specific improvements should the authors consider regarding the methodology?\\
\noindent The methodology itself looks OK but it is very unclear for the people who are not involved in the specific area.\\
\noindent $\bullet$ Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed? Please also explain why this is/is not the case.\\
\noindent Frankly speaking, the phrase concerning hypercomplex numbers and Clifford algebras in the Conclusion must be clarified or deleted. I do not understand it.\\
\noindent $\bullet$ Are the references appropriate?\\
\noindent Yes, they look appropriate. However, I do not know all the literature about this problem, the complete list would be really huge.\\
\noindent $\bullet$ Any additional comments on the tables and figures.\\
\noindent No tables or figures are there.\\
The main results of the paper look quite interesting. They are new and allow some opportunities to model resl-life problems.\\
However, I have some major remarks listed below.\\
1) The paper involves an impressive amount of approaches including Manifold learning, kernel based method, MicrOmix approach, SNE method and others. However, the place of the current manuscript in the modern theory is unclear. The Introduction gives the main idea of the theory but does not answer the question.\\
2) The applications are briefly mentionned (subsection 3.3) but this is very vague. I strongly recommend to include some calculations related to a particular problem or, at least, make more precise ideas of how to apply the results of the paper. \\
3) The use terminology is sometimes incorrect. For example,
non-square matrices do not form the ring (page 3, line 1). The whole paper must be checked. \\
4) English must be significantly updated. For instance, AI tools can be used. \\
Therefore although I support the idea of publication of the manuscript in Mathematics, I suggest the major revision.\\
\begin{center}
{\bf Replies and comments}
\end{center}
\noindent Author:
Dear referee, I am sincerely grateful to you for the made remarks. However, let us consider them consistently.\\
\noindent Referee 1:
\noindent $\bullet$ Do you consider the topic original or relevant to the field? Does it address a specific gap in the field? Please also explain why this is/ is not the case.\\
\noindent This is exactly what I need the authors to clarify. There are many new works in the area... Well. the authors' approach is original and does not resemble to what I have seen before. However, it would be much better, if the authors indicate which specific gap they fill or which specific real life problem they solve.\\
\noindent Author:
Dear referee, thank you for the clarifying remark, the specific gap we have filled relates to the way of constructing a unified harmonious theory in comparison with the previously made attempts. In order to give more detailed explanation, we represent the additional description of the paper main idea.
In this paper, we study methods for aligning topological structures in Hilbert spaces, the central idea of which is to create a mapping of various, in particular heterogeneous, sets onto a space in which images can be in some sense comparable. In particular, the problem of the alignment of manifolds (Manifold Alignment) is considered in the context of the analysis of heterogeneous data. The key problem is the lack of a single feature space for analyzing data of different natures and, as a result, lying on various low-dimensional manifolds in their corresponding high-dimensional initial spaces. The central hypothesis of the study is that for a set of such data, a common latent space can be constructed in which the geometric relationships between the elements of the set (for example, pairwise distances or angles) will reflect the relationships between the corresponding objects from different data sources. This will allow information transfer between sets when solving various application problems.
The question of the relationship of images of heterogeneous data sets has been studied by many researchers, this relationship was usually established using technical methods that ensure the dependence of images in one sense or another. In this paper, the main idea is to create an algebraic structure that by virtue of the mathematical origin guarantees the connection between the images of the data set elements.
The main distinguishing feature of the developed methods is the naturalness property of the mathematical construction used, which harmoniously combines heterogeneous preimages and images belonging to a common latent space in the mapping. For example, a mapping of two heterogeneous sets whose elements can be put into correspondence with a real axis can be represented naturally as a mapping of some subset of the complex plane. This idea can be developed by considering hypercomplex numbers as a tool for describing a case corresponding to several heterogeneous sets, i.e. the set of quaternions correspond to four heterogenous data sets, etc. It becomes clear that a classified algebraic structure with well-known properties and considered as a technical base of the mapping provides the connection of heterogeneous data sets in a natural way from the point of view of theoretical mathematics (and, consequently, applications).
Here, I should stress that in many papers the method of coupling images of heterogenous data sets is based on invention of a special technique involving artificial constructions at list not reflecting at most contradicting to the mathematical nature of the mapping, i.e. saying "mathematical nature" we mean abstract classical mathematical notions such as the space, the algebraic structure, etc. It is clear that dealing with the classical mathematical notions we have an opportunity to develop the theory harmoniously using well-known methods and putting them as a fundamental base inventing new ones that can be in their own turn classified in accordance with the given stable classical notions. Quite the contrary is the case when we involve artificial technical formulas allowing us to achieve a concrete goal but likely preventing the development of the theory in general, for in this case we know nothing on the consequent influence of the used construction on the scheme of reasonings and it is rather reasonable, due to the technical obstacles, to expect the end of the theory.
The advantages in terms of applications, particularly in biology, include the following. The application of the operator theory becomes more natural and harmonious, since unitary spaces, involved due to the choice of the scalar field - the complex numbers with a wider range of properties, are considered in comparison with Euclidean ones.\\
\noindent Referee 1:
\noindent $\bullet$ Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed? Please also explain why this is/is not the case.\\
\noindent Frankly speaking, the phrase concerning hypercomplex numbers and Clifford algebras in the Conclusion must be clarified or deleted. I do not understand it.\\
\noindent Author:
Dear referee, thank you for the remark. I completely agree with you. As far as I understand, you mean the following extract from the conclusion section\\
"Thus, the quaternion structure can be considered as further generalization that leads to the hypercomplex numbers and the most abstract mathematical object Clifford algebra".
Here, I should point out that the note on these structures is given before paragraph 3.3. The explanation is very simple, the complex number has two components, i.e. the real component and the imaginary component, the quaternion has four components, the octonion which set is known as the Cayley algebra has eight components and so on. Recall that quaternions and octonions and others constructed in the analogous way numbers (see the detailed definition) are generally called by hypercomplex numbers. Thus, we can consider an arbitrary number of data sets and put them into correspondence with the hypercomplex number having the corresponding number of components.
The Clifford algebra appears in the context as an algebraic structure generalizing the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The detailed information can be found in:
Clifford, W.K. (1873). Preliminary sketch of bi-quaternions. Proc. London Math. Soc. 4., Clifford, W.K. (1882). Tucker, R. (ed.). Mathematical Papers. London: Macmillan.
I do believe that the remark on the hypercomplex numbers and Clifford algebras in the Conclusion is rather reasonable since it shows a prospective way of further abstract mathematical generalizations. However, dear referee, if you find such arguments insufficient, I will follow your wise advise and delete the information on the Clifford algebras.\\
\noindent Referee 1:
1) The paper involves an impressive amount of approaches including Manifold learning, kernel based method, MicrOmix approach, SNE method and others. However, the place of the current manuscript in the modern theory is unclear. The Introduction gives the main idea of the theory but does not answer the question.\\
\noindent Author:
Dear referee, thank you very much for the remark, however the reply was in particular given above. Thus, for the sake of formality, I just should repeat it adding some concrete arguments.
In many papers the method of coupling images of heterogenous datasets is based on invention of a special technique involving artificial constructions at list not reflecting at most contradicting to the mathematical nature of the mapping, i.e. saying "mathematical nature" we mean abstract classical mathematical notions such as the space, the algebraic structure, etc. It is clear that dealing with the classical mathematical notions we have an opportunity to develop the theory harmoniously using well-known methods and putting them as a fundamental base inventing new ones that can be in their own turn classified in accordance with the given stable classical notions. Quite the contrary is the case when we involve artificial technical formulas allowing us to achieve a concrete goal but likely preventing the development of the theory in general, for in this case we know nothing on the consequent influence of the used construction on the scheme of reasonings and it is rather reasonable, due to the technical obstacles, to expect the end of the theory.
Moreover, the offered method allows the convenient tuning of parameters clearly reflecting the share of each dataset in the the image under the mapping (see paragraph 3.1.2).
The methods of coupling data sets and their images can be considered in detailed. In the framework of MMD method the coupling of the datasets images is due to the auxiliary constructed matrix obtained by virtue of the optimization process (see paragraph 3.2.2). In the framework of the Cupeled NMF method the coupling matrix $A$ is involved, however it depends on the concrete application and does not reflect or relate to the mathematical nature of the mapping. Many other well-known methods such as the classical graph Laplacian method even do not contain the idea of coupling the images of data sets. At the same time we represent the idea which can be universally implemented in many mathematical constructions related to the issue due to the natural harmonious mathematical approach. The considered generalizations of the graph Laplacian and MMD methods demonstrate the efficiency of the idea. \\
\noindent Referee 1:
2) The applications are briefly mentioned (subsection 3.3) but this is very vague. I strongly recommend to include some calculations related to a particular problem or, at least, make more precise ideas of how to apply the results of the paper.\\
\noindent Author: Thank you for the remark. Generally, the application appeals to the problems of the manifold alignment when two o more datasets should be coupled, we considered two fundamental theoretical methods, however the main idea can be put as a base in study of an arbitrary method if a corresponding technical support is available.\\
The aim of this paper is to develop methods for the diagonal integration of multimodal biological data for a comprehensive study of the biological processes in particular the aging process. The prospective project based on the paper fundamental results plans to develop new mathematical methods of diagonal integration that take into account the identified limitations of existing approaches. We propose to combine and jointly analyze seven different types of data: genomic, transcriptomic, proteomic, metabolomic, and lipidomic data. The aging process acts as a suprasystem factor for diagonal integration as a fundamental biological phenomenon that manifests itself at all levels of the molecular organization. A comprehensive analysis of the integrated data will make it possible to identify new pharmaceutical targets for therapeutic interventions in the aging process, identify multidimensional biomarkers of aging with increased predictive power, and establish causal relationships between molecular changes at various levels of biological organization in the context of age-associated processes.\\
\noindent Referee 1:
3) The use terminology is sometimes incorrect. For example,
non-square matrices do not form the ring (page 3, line 1). The whole paper must be checked.\\
\noindent Author:
Dear referee, thank you very much for the significant remark. I have proofread the paper and corrected uncertainties. \\
\noindent Referee 1:
4) English must be significantly updated. For instance, Al tools can be used.\\
\noindent Author:
Dear referee, thank you very much for the remark, I have corrected misprints and updated stylistically some places, I have proofread the paper one more time, many typos have been corrected. However, if you have concrete remarks regarding the grammar, please produce them.\\
\noindent Author:
Dear referee, I highly appreciate your attention and very grateful to you for the remarks which allow me to observe the matter from another point of view and 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 AuthorsTo achieve this, the authors formulate this coupled clustering problem as an optimization problem and present a method for solving it called non-negative coupled matrix factorization, taking several references.
In this sense, it is not clear how they present their contribution, since it is stated that the article presents a new algorithm for the unsupervised topological alignment of single-cell multi-omics integration. The method does not require any information about the correspondence between cells or between measurements. This does not clarify what part they will take as a basis or whether they will work on an alternative or complementary line of work.
The references are current, except for a couple of works, which are understood to be classics.
The background development is very robust from a mathematical point of view; however, it is necessary to clarify the form and stages of the work to achieve the objective.
I believe it is necessary to address the applications discussed in the introduction, which could contribute to a better unders
Comments on the Quality of English LanguageIt is always advisable to give the document a general review to ensure that it is better written. Check for any typos.
Author Response
Response letter to referee 2.\\
\vspace{0.5 cm}
Manuscript ID
mathematics-3962116. Natural methods of the unsupervised topological alignment.
\end{center}
\vspace{0.5 cm}
\begin{center}
{\bf Remarks and comments}
\end{center}
\noindent Referee 2:
1. To achieve this, the authors formulate this coupled clustering problem as an optimization problem and present a method for solving it called non-negative coupled matrix factorization, taking several references.\\
In this sense, it is not clear how they present their contribution, since it is stated that the article presents a new algorithm for the unsupervised topological alignment of single-cell multi-omics integration. The method does not require any information about the correspondence between cells or between measurements. This does not clarify what part they will take as a basis or whether they will work on an alternative or complementary line of work.\\
2. The references are current, except for a couple of works, which are understood to be classics.\\
The background development is very robust from a mathematical point of view; however, it is necessary to clarify the form and stages of the work to achieve the objective.\\
3. I believe it is necessary to address the applications discussed in the introduction, which could contribute to a better unders\\
4. It is always advisable to give the document a general review to ensure that it is better written. Check for any typos.\\
\begin{center}
{\bf Replies and comments}
\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. To achieve this, the authors formulate this coupled clustering problem as an optimization problem and present a method for solving it called non-negative coupled matrix factorization, taking several references.\\
In this sense, it is not clear how they present their contribution, since it is stated that the article presents a new algorithm for the unsupervised topological alignment of single-cell multi-omics integration. The method does not require any information about the correspondence between cells or between measurements. This does not clarify what part they will take as a basis or whether they will work on an alternative or complementary line of work.\\
\noindent Author:
Dear referee, thank you for the useful remark, I will try to explain. In this paper, we study methods for aligning topological structures in Hilbert spaces, the central idea of which is to create a mapping of various, in particular heterogeneous, sets onto a space in which images can be in some sense comparable. In particular, the problem of the alignment of manifolds (Manifold Alignment) is considered in the context of the analysis of heterogeneous data. The key problem is the lack of a single feature space for analyzing data of different natures and, as a result, lying on various low-dimensional manifolds in their corresponding high-dimensional initial spaces. The central hypothesis of the study is that for a set of such data, a common latent space can be constructed in which the geometric relationships between the elements of the set (for example, pairwise distances or angles) will reflect the relationships between the corresponding objects from different data sources. This will allow information transfer between sets when solving various application problems.
The question of the relationship of images of heterogeneous data sets, as you rightly pointed out, has been studied by many researchers, this relationship was usually established using technical methods that ensure the dependence of images in one sense or another.
{ \it The main idea of the paper is to create an algebraic structure that by virtue of the mathematical origin guarantees the relationship between the images of the data set elements.
The main distinguishing feature of the developed methods is the naturalness property of the mathematical construction used, which harmoniously combines heterogeneous preimages and images belonging to a common latent space in the framework of the mapping.}
For example, a mapping of two heterogeneous sets whose elements can be put into correspondence with a real axis can be represented naturally as a mapping of some subset of the complex plane. This idea can be developed by considering hypercomplex numbers as a tool for describing a case corresponding to several heterogeneous sets, i.e. the set of quaternions correspond to four heterogenous data sets, etc. It becomes clear that a classified algebraic structure with well-known properties and considered as a technical base of the mapping provides the connection of heterogeneous data sets in a natural way from the point of view of theoretical mathematics and consequently, applications.
In many papers the method of coupling images of heterogenous datasets is based on invention of a special technique involving artificial constructions at list not reflecting at most contradicting to the mathematical nature of the mapping, i.e. saying "mathematical nature" we mean abstract classical mathematical notions such as the space, the algebraic structure, etc. It is clear that dealing with the classical mathematical notions we have an opportunity to develop the theory harmoniously using well-known methods and putting them as a fundamental base inventing new ones that can be in their own turn classified in accordance with the given stable classical notions. Quite the contrary is the case when we involve artificial technical formulas allowing us to achieve a concrete goal but likely preventing the development of the theory in general, for in this case we know nothing on the consequent influence of the used construction on the scheme of reasonings and it is rather reasonable, due to the technical obstacles, to expect the end of the theory.
Moreover, the offered method allows the convenient tuning of parameters clearly reflecting the share of each data set in the the image under the mapping (see paragraph 3.1.2).
The methods of coupling data sets and their images can be considered in detailed. In the framework of MMD method the coupling of the data sets images is due to the auxiliary constructed matrix obtained by virtue of the optimization process (see paragraph 3.2.2). In the framework of the Cupeled NMF method, the coupling matrix $A$ is involved, however it depends on the concrete application and does not reflect or relate to the mathematical nature of the mapping. Many other well-known methods such as the classical graph Laplacian method even does not consider the idea of coupling the images of data sets. At the same time we represent the idea which can be universally implemented in many mathematical constructions related to the issue due to the natural harmonious mathematical approach. The considered generalizations of the graph Laplacian and MMD methods demonstrate the efficiency of the idea.\\
\noindent Referee 2:
2. The references are current, except for a couple of works, which are understood to be classics.\\
The background development is very robust from a mathematical point of view; however, it is necessary to clarify the form and stages of the work to achieve the objective.\\
\noindent Author:
Dear referee, thank you very much for the remark and worm words, I have tried to follow your advise and represent the detailed clarification of forms and stages of the work.\\
\noindent Referee 2:
3. I believe it is necessary to address the applications discussed in the introduction, which could contribute to a better understanding.\\
\noindent Author:
Dear referee, thank you very much for the remark. I have considered more comprehensively the applications discussed in the introduction section and hope that they can reveal the main idea of the paper. \\
\noindent Referee 2:
4. It is always advisable to give the document a general review to ensure that it is better written. Check for any typos.\\
\noindent Author:
Dear referee, thank you for the remark! The paper has been properly proofread as a result some places were stylistically improved and almost all typos were checked.\\
\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 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 AuthorsThis paper proposes the "natural coupling structure" in the complex Hilbert space, mapping heterogeneous datasets (such as single-cell multi-omics data) to the real and imaginary parts of complex numbers to achieve mathematical coupling. By combining the Kullback-Leibler divergence to optimize the local structure preservation of low-dimensional embeddings, the theory of Reproducing Kernel Hilbert Space (RKHS) is extended to the complex space, designing a target function including the maximum mean discrepancy (MMD), distortion term, and penalty term, and proposing the generalized graph Laplacian and kernel methods, which are applied to the alignment of heterogeneous data without feature correlations. However, the paper has shortcomings in theoretical details, insufficient experimental verification, and improper expression, specifically:
1. In Section 3.1.3 when combining the probabilistic model of t-SNE with the generalized graph Laplacian embedding, the selection strategy of the variance \(\sigma_s\) of the Gaussian function is not explained, and the "unified symbol writing" section does not clarify the definitions of \(P^{X_1}\) and \(Q^{Y_1}\) and their relationship with the coupling structure in the previous section, resulting in unclear connection between probability optimization and topological alignment.
2. No specific conception of the Clifford algebra theory framework (such as the definition of the reproducing kernel in the hypercomplex space) is presented, remaining at the conceptual level, unable to accurately describe the expansion direction of the Clifford algebra.
3. The alignment effect of the "coupled Laplacian mapping" on scRNA-seq and scATAC-seq data has not been verified, nor has the computational efficiency of the kernel method under different data volumes been tested.
4. Missing or incorrect formula numbers: For example, in Section 3.1.1, "Problem (6)" and "Decomposition (7)" are not labeled with numbers beside the formulas, and the formula position of "Problem (17)" in Section 3.1.2 does not correspond to the description.
5. The subscript of the weighting matrix in Section 3.1.2 is incorrect.
6. Reference 6 (MUON method) lacks the specific page number, and reference 15 (Jie Liu et al. paper) lacks the journal name.
7. It is recommended to supplement complete experimental verification: Use at least one publicly available single-cell multi-omics dataset (such as GSE164378), design a comparative experiment (compared with methods such as Pamona, SCIM, etc.), provide quantitative indicators (such as ARI, NMI, RMSE, etc.), and visualization results (such as UMAP embedding graph, confusion matrix).
Author Response
Response letter to the referee 3.\\
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Manuscript ID
mathematics-3962116. Natural methods of the unsupervised topological alignment.
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\begin{center}
{\bf Remarks and comments}
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\noindent Referee 3:
This paper proposes the "natural coupling structure" in the complex Hilbert space, mapping heterogeneous datasets (such as single-cell multi-omics data) to the real and imaginary parts of complex numbers to achieve mathematical coupling. By combining the Kullback-Leibler divergence to optimize the local structure preservation of low-dimensional embeddings, the theory of Reproducing Kernel Hilbert Space (RKHS) is extended to the complex space, designing a target function including the maximum mean discrepancy (MMD), distortion term, and penalty term, and proposing the generalized graph Laplacian and kernel methods, which are applied to the alignment of heterogeneous data without feature correlations. However, the paper has shortcomings in theoretical details, insufficient experimental verification, and improper expression, specifically:\\
1. In Section 3.1.3 when combining the probabilistic model of t-SNE with the generalized graph Laplacian embedding, the selection strategy of the variance \(\sigma_s\) of the Gaussian function is not explained, and the "unified symbol writing" section does not clarify the definitions of $(P^{X_1})$ and $(Q^{Y_1})$ and their relationship with the coupling structure in the previous section, resulting in unclear connection between probability optimization and topological alignment.\\
2. No specific conception of the Clifford algebra theory framework (such as the definition of the reproducing kernel in the hypercomplex space) is presented, remaining at the conceptual level, unable to accurately describe the expansion direction of the Clifford algebra.\\
3. The alignment effect of the "coupled Laplacian mapping" on scRNA-seq and scATAC-seq data has not been verified, nor has the computational efficiency of the kernel method under different data volumes been tested.\\
4. Missing or incorrect formula numbers: For example, in Section 3.1.1, "Problem (6)" and "Decomposition (7)" are not labeled with numbers beside the formulas, and the formula position of "Problem (17)" in Section 3.1.2 does not correspond to the description.\\
5. The subscript of the weighting matrix in Section 3.1.2 is incorrect.\\
6. Reference 6 (MUON method) lacks the specific page number, and reference 15 (Jie Liu et al. paper) lacks the journal name.\\
7. It is recommended to supplement complete experimental verification: Use at least one publicly available single-cell multi-omics dataset (such as GSE164378), design a comparative experiment (compared with methods such as Pamona, SCIM, etc.), provide quantitative indicators (such as ARI, NMI, RMSE, etc.), and visualization results (such as UMAP embedding graph, confusion matrix).\\
\begin{center}
{\bf Replies and comments}
\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. In Section 3.1.3 when combining the probabilistic model of t-SNE with the generalized graph Laplacian embedding, the selection strategy of the variance $\sigma_s$ of the Gaussian function is not explained, and the "unified symbol writing" section does not clarify the definitions of $(P^{X_1})$ and $(Q^{Y_1})$ and their relationship with the coupling structure in the previous section, resulting in unclear connection between probability optimization and topological alignment.\\
\noindent Author:
Thank you for the remark, the selection strategy of the variance $\sigma_s$ of the Gaussian function is explained in the paper [25]. Since, we are rather satisfied with the symmetric case then from the one hand there is no need to represent lengthy reasonings devoted to the issue. From the other hand, I share your point of view and in this regard will try to represent short description devoted to the selection strategy of the variance $\sigma_s.$ \\
In accordance with [25], the remaining parameter $\sigma_{s}$ of the Gaussian is centered over each
high-dimensional datapoint, $\mathbf{x}_{s}.$ It is not likely that there is a single value of $\sigma_{s}$ that is optimal for all
datapoints in the dataset because the density of the data is likely to vary. In dense regions, a smaller
value of $\sigma_{s}$ is usually more appropriate than in sparser regions. Any particular value of $\sigma_{s}$ induces a
probability distribution, $P_{s}$ over all of the other datapoints. This distribution has an entropy which
increases as $\sigma_{s}$ increases. In accordance with [14], SNE performs a binary search for the value of $\sigma_{s}$ that produces a $P_{s}$ with
a fixed perplexity [25] that is specified by the user. The perplexity is defined as
$$
Perp (P_s):=2^{H(P_{s}) },
$$
where $H(P_{s})$ is the Shannon entropy defined as follows
$$
H(P_{s}):=-\sum\limits_{j=1}^{n}p_{j|s} \mathrm{log}_{2}p_{j|s},\;
p_{ j|s}=
\frac{e^{-\| \mathbf{x}_{s}-\mathbf{x}_{j} \|/2\sigma_{s}^{2}}}{\sum\limits_{m\neq s}e^{-\| \mathbf{x}_{s}-\mathbf{x}_{m} \|/2\sigma_{s}^{2}}}.
$$
Thus, in accordance with [25] the perplexity is specified by the user ( is chosen "by hand" [14] ) what gives a key to obtain the corresponding set $\{\sigma_{s}\}_{1}^{n}.$ However, as it was mentioned above, in the symmetric case due to the given construction the parameter $\sigma_{s}$ takes a constant value ($\sigma_{s}=1/\sqrt{2}$ according to [25]). In this regard, we have the following remark [25] "Setting the variance in the low-dimensional Gaussians to another value that is different from $1/\sqrt{2}$ only results in a rescaled version of the final
map. Note that by using the same variance for every datapoint in the low-dimensional map, we lose the property
that the data is a perfect model of itself if we embed it in a space of the same dimensionality, because in the highdimensional
space, we used a different variance $\sigma_{s}$ in each Gaussian."
Thus, we conclude that in the non symmetric case the usage of the constant value of the parameter $\sigma_{s}$ is not appropriate.\\
I have also rewritten the formulas containing Kullback-Leibler divergence symbol following your remark and trying to use the unified form of writing.\\
The relationship of the probabilistic approach (in particular the definitions of $(P^{\mathrm{X}_1})$ and $(Q^{\mathrm{Y}_1})$) with the coupling structure given in the paragraph 3.1.2 "previous section" is considered in the context in the beginning of the paragraph 3.1.3 "It is remarkable that Theorem 3 creates a prerequisite for further optimization problem. In this regard, we can observe the Kullback-Leibler divergence as the most suitable tool practically showing the preservation between the elements characteristics." Here, we should note that in accordance with Theorem 3 the solution of the problem (17) is not unique. This allows us to apply the probabilistic approach choosing, the most suitable from the probabilistic point of view, mapping from the given ones due to the uncertainty of the graph Laplacian method manifesting in Theorem 3. More detailed, I should represent Theorem 3 statement.\\
... consider the following problem
\begin{equation}\label{14a}
\underset{ A^{\ast}D A=e^{i\theta} I}{ \mathrm{argmin}} \left|\sum\limits_{s,j=1}^{n}\|\mathbf{a}_{s}-\mathbf{a}_{j}\|_{\mathbb{C}^{m}}^{2}W_{sj}\right|,\;A\in \mathbb{C}^{m\times n}.
\end{equation}
Eventually, we can resume the following theorem.
\begin{teo}\label{T3} The solution of problem \eqref{14a} is represented by a set $\{\mathbf{y}_{j} \}_{1}^{n}\subset \mathbb{C}^{m}$ satisfying the following conditions
$$
\{\mathbf{y}^{\cdot}_{j} \}_{1}^{m}\subset \mathfrak{N}_{m},\; Y^{\ast}Y=I.
$$
Moreover the following coupled mapping is defined
$$
\mathbf{u}_{j}\rightarrow \mathrm{Re}\,\mathbf{y}_{j} ,\;\mathbf{v}_{s}\rightarrow \mathrm{Im}\,\mathbf{y}_{j},\;j=1,2,...,n.
$$
\end{teo}
\begin{proof} The proof follows immediately from Lemma1, Lemma 2.
\end{proof}
Here, it is clear that generally the solution satisfying the conditions $ \{\mathbf{y}^{\cdot}_{j} \}_{1}^{m}\subset \mathfrak{N}_{m},\; Y^{\ast}Y=I$ is not unique.
We should recall that
$$
\mathfrak{N}_{m}, \; m=1,2,...,\xi
$$
denotes the subspace generated by the generalized eigenvectors $\mathbf{e}_{ 1 },\mathbf{e}_{2},...,\mathbf{e}_{m}.$
Thus, together with the theoretical meaning of the Kullback-Leibler divergence this fact represents a substantiation of usage of the probabilistic approach. \\
\noindent Referee 3:
2. No specific conception of the Clifford algebra theory framework (such as the definition of the reproducing kernel in the hypercomplex space) is presented, remaining at the conceptual level, unable to accurately describe the expansion direction of the Clifford algebra.\\
\noindent Author:
Dear referee, thank you for the remark.
The explanation is very simple, the complex number has two components, i.e. the real component and the imaginary component, the quaternion has four components, the octonion, which set is known as Cayley algebra, has eight components and so on. Recall that quaternions and octonions and others constructed in the analogous way numbers (see the detailed definition) are generally called by hypercomplex numbers. Thus, we can consider an arbitrary number of data sets and put them into correspondence with the hypercomplex number having the corresponding number of components.
The Clifford algebra appears in the context as an algebraic structure generalizing the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The detailed information can be found in:\\
Clifford, W.K. (1873). "Preliminary sketch of bi-quaternions". Proc. London Math. Soc. 4., Clifford, W.K. (1882). Tucker, R. (ed.). Mathematical Papers. London: Macmillan.
I do believe that the remark on the hypercomplex numbers and Clifford algebras in the Conclusion is rather reasonable since it shows a prospective way of further abstract mathematical generalizations. However, dear referee, if you find such arguments insufficient, I will follow your wise advise and delete the information on the Clifford algebras.\\
\noindent Referee 3:
3. The alignment effect of the "coupled Laplacian mapping" on scRNA-seq and scATAC-seq data has not been verified, nor has the computational efficiency of the kernel method under different data volumes been tested.\\
\noindent Author:
Dear referee, thank you for the valuable remark. Generally, the application appeals to the problems on the manifold alignment when two o more data sets should be coupled, we considered two fundamental theoretical methods. However the main idea is valuable itself and can be put as a base in study of an arbitrary method if a corresponding technical support is available.
In order to give a partial reply, I should notice that the efficiency of the naturally coupled method cannot be measured due to the simple calculations for the natural origin of the method is the measure of efficiency itself. At the same time the question on the comparison with other methods is very comprehensive due to the produced argument and this is why require a proper study in the separate paper. Here, we just produce a mathematical fundamental support for further comprehensive investigation.
\\
\noindent Referee 3:
4. Missing or incorrect formula numbers: For example, in Section 3.1.1, "Problem (6)" and "Decomposition (7)" are not labeled with numbers beside the formulas, and the formula position of "Problem (17)" in Section 3.1.2 does not correspond to the description.\\
\noindent Author:
Thank you for the valuable remark, all formula numbers have been checked and the unnecessary numbers were deleted. Now, the formula position of "Problem (17)" ("Problem (11)") in Section 3.1.2 does correspond to the description, please check it and confirm or please explain the made remark.\\
\noindent Referee 3:
5. The subscript of the weighting matrix in Section 3.1.2 is incorrect.\\
\noindent Author:
Dear referee, thank you for the remak, however either I cannot understand the remark or the subscripts of the matrices in Section 3.1.2 are correct. I would be grateful if you are so kind to explain your opinion.\\
\noindent Referee 3:
6. Reference 6 (MUON method) lacks the specific page number, and reference 15 (Jie Liu et al. paper) lacks the journal name.\\
\noindent Author:
Thank you for the remark, the reference [6] has been corrected in accordance with the information given on the journal page (in some journals there is no page numbering), the reference [15] has been checked and the following official name of the journal has been used (Algorithms Bioinform.).\\
\noindent Referee 3:
7. It is recommended to supplement complete experimental verification: Use at least one publicly available single-cell multi-omics dataset (such as GSE164378), design a comparative experiment (compared with methods such as Pamona, SCIM, etc.), provide quantitative indicators (such as ARI, NMI, RMSE, etc.), and visualization results (such as UMAP embedding graph, confusion matrix).\\
\noindent Author:
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 tho 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.
\newpage
\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 improve the paper significantly.
\vspace{0.1 cm}
Sincerely yours Ph.D. Maksim V. Kukushkin
Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsSee the Referee Report.
Comments for author File: Comments.pdf
Author Response
Response letter to the referee 4.\\
\vspace{0.5 cm}
Manuscript ID
mathematics-3962116. Natural methods of the unsupervised topological alignment.
\end{center}
\vspace{0.5 cm}
\begin{center}
{\bf Remarks and comments}
\end{center}
\noindent Referee 4:
There are some questions that the authors should answer in a clear and precise
manner.\\
(1) What do the authors want to achieve with this research? Furthermore,
the idea of "what they want to obtain and where they intend to conclude
is not clearly stated".\\
(2) In page 16 and throughout the rest of the text, what is the significance of
the operator $H$ being normal? Please explain briefly.\\
(3) In page 12, what is the significance and application of Theorem 3?\\
(4) In page 13, why is the phrase "Since we are only interested in modeling
pairwise similarities" important. Please explain.\\
(5) In page 13, what is its purpose and importance of the following statement.
"In Accordance with the description given in [25], we conclude that if the
map elements $y_{s}$ and $y_{s}$ correctly model the similarity between the high-
dimensional data elements $x_s$ and $x_j,$ the conditional probabilities $p_{j|s}$
and $q_{j|s}$ will be equal". Explain briefly.\\
(6) In page 16, give an example of a Hilbert space RKSH.\\
(7) Can you clearly explain the aim and applications of this article?.\\
\begin{center}
{\bf Replies and comments}
\end{center}
\noindent Author:
Dear referee, I am sincerely grateful to you for the made remarks. However, let us consider them consistently.\\
\noindent Referee:
(1) What do the authors want to achieve with this research? Furthermore,
the idea of "what they want to obtain and where they intend to conclude
is not clearly stated".\\
\noindent Author:
Dear referee, thank you for the valuable remark! Frankly speaking, since the issue is rather comprehensive it is not so easy to make a clear brief statement (see below).
Here, I should recall that the investigation is in the framework of the issue devoted to the topological manifold alignment. Recall that manifold alignment is a class of algorithms that create a mapping of heterogeneous data sets into a common, lower-dimensional latent space. The central point is to find a mapping that reveals the entire structure of the initial manifold.
Now, I should remark that the specific gap in the theory we have filled relates to the way of constructing a unified harmonious theory in comparison with the previously made, I may say artificial attempts. In order to give more detailed explanation, I represent the additional description of the paper main idea.
In this paper, we study methods for aligning topological structures in Hilbert spaces, the central idea of which is to create a mapping of various, in particular heterogeneous, sets onto a space in which images can be in some sense comparable. In particular, the problem of the alignment of manifolds (Manifold Alignment) is considered in the context of the analysis of heterogeneous data. The key problem is the lack of a single feature space for analyzing data of different natures and, as a result, lying on various low-dimensional manifolds in their corresponding high-dimensional initial spaces. The central hypothesis of the study is that for a set of such data, a common latent space can be constructed in which the geometric relationships between the elements of the set (for example, pairwise distances or angles) will reflect the relationships between the corresponding objects from different data sources. This will allow information transfer between sets when solving various application problems.
The question of the relationship of images of heterogeneous data sets has been studied by many researchers, this relationship was usually established using technical methods that ensure the dependence of images in one sense or another. In this paper, the main idea is to create an algebraic structure that by virtue of the mathematical origin guarantees the connection between the images of the data set elements.
The main distinguishing feature of the developed methods is the naturalness property of the mathematical construction used, which harmoniously combines heterogeneous preimages and images belonging to a common latent space in the mapping. For example, a mapping of two heterogeneous sets whose elements can be put into correspondence with a real axis can be represented naturally as a mapping of some subset of the complex plane. This idea can be developed by considering hypercomplex numbers as a tool for describing a case corresponding to several heterogeneous sets, i.e. the set of quaternions correspond to four heterogenous data sets, etc. It becomes clear that a classified algebraic structure with well-known properties and considered as a technical base of the mapping provides the connection of heterogeneous data sets in a natural way from the point of view of theoretical mathematics (and, consequently, applications).
Here, I should stress that in many papers the method of coupling images of heterogenous data sets is based on invention of a special technique involving artificial constructions at list not reflecting at most contradicting to the mathematical nature of the mapping, i.e. saying "mathematical nature" we mean abstract classical mathematical notions such as the space, the algebraic structure, etc. It is clear that dealing with the classical mathematical notions we have an opportunity to develop the theory harmoniously using well-known methods and putting them as a fundamental base inventing new ones that can be in their own turn classified in accordance with the given stable classical notions. Quite the contrary is the case when we involve artificial technical formulas allowing us to achieve a concrete goal but likely preventing the development of the theory in general, for in this case we know nothing on the consequent influence of the used construction on the scheme of reasonings and it is rather reasonable, due to the technical obstacles, to expect the end of the theory.
The advantages in terms of applications, particularly in biology, include the following. The application of the operator theory becomes more natural and harmonious, since unitary spaces, involved due to the choice of the scalar field - the complex numbers with a wider range of properties, are considered in comparison with Euclidean ones.\\
{\it Brief statement.} In accordance with the given above, the clear brief statement of the research purposes and achievements can be expressed nearly in the following form. We want to find a natural mathematical structure coupling the images and the preimages of the heterogenous data sets under a mapping aligning topological structures in Hilbert spaces. The coupling should preserve and convey the properties of the geometric objects what is obtained due to the fact that a mapping is defined on the sets of elements supplied with the structure of the complex numbers field. Thus, the idea of preservation of geometrical objects is entirely based upon the statement that the mapping being continuous in the sense of the norm conveys some geometrical properties of the preimages.\\
\noindent Referee:
(2) In page 16 and throughout the rest of the text, what is the significance of
the operator $H$ being normal? Please explain briefly.\\
\noindent Author:
Dear referee, thank you very much for the significant remark. Hear, we have a simple misprint, instead of "It is clear that the operator $H$ is normal" must be "It is clear that the operator $W$ is normal."
However, the idea of the generalized eigenvalue problem is based upon the assumptions regarding the operator $L$ what leads us to the assumptions on the corresponding operator $W.$ We restrict the sefadjoint property but consider more general normal property. Having constructed the operator $W$ we involve the operators $\alpha H$ and $\beta H$ as its real and imaginary components, i.e.
$$
W=\alpha H+i \beta H.
$$
The necessary and sufficient condition of the normal property is the commutative property of the operator Hermitian components. However in the considered case, the normal property of the operator $W$ can be established directly due to the simple calculation.\\
\noindent Referee:
(3) In page 12, what is the significance and application of Theorem 3?\\
\noindent Author:
The significance of Theorem 3 is the opportunity to obtain a mapping with the desired properties, moreover in accordance with Theorem 3, we conclude that there is a number of mappings having the equal properties from the considered point of view. The latter fact gives us the opportunity to improve the resulting mapping in the form of the consequent application of the graph Laplacian method and afterwards the probabilistic approach.
The application of Theorem 3 is considered above in the reply on remark (1), however since this theorem represents an accurately formulated theoretical result making a comprehensive description, we can likely extend the application to other applied sciences.\\
\noindent Referee:
(4) In page 13, why is the phrase "Since we are only interested in modeling
pairwise similarities" important. Please explain.\\
\noindent Author:
Dear referee, thank you for the remark. Here, I represent the authentic variant of the prase "Since we are only interested in modeling pairwise
similarities, we set the value of $p_{s|s}$ to zero."
The pairwise similarities means the opportunity of the elements having the different indexes being understood similar due to the property of the following construction
$$
p_{ j|s}=
\frac{e^{-\| \mathbf{x}_{s}-\mathbf{x}_{j} \|/2\sigma_{s}^{2}}}{\sum\limits_{m\neq s}e^{-\| \mathbf{x}_{s}-\mathbf{x}_{m} \|/2\sigma_{s}^{2}}}.
$$
It is clear that the closer the different elements to each other in the sense of the norm the closer the value $p_{ j|s}$ to the one. Thus the value $p_{ j|s}$ can be interpreted as a measure of the elements similarity.\\
\noindent Referee:
(5) In page 13, what is its purpose and importance of the following statement.
"In Accordance with the description given in [25], we conclude that if the
map elements $y_{s}$ and $y_{s}$ correctly model the similarity between the high-
dimensional data elements $x_s$ and $x_j,$ the conditional probabilities $p_{j|s}$
and $q_{j|s}$ will be equal". Explain briefly.\\
\noindent Author:
Dear referee, thank you for the remark, I represent the complete updated extract from the paper in order to make some clarifications. Below, I resume a brief explanation. \\
"For the low-dimensional counterparts $\mathbf{y}_{s}$ and $\mathbf{y}_{j}$ of the
high-dimensional data elements $\mathbf{x}_{s}$ and $\mathbf{x}_{j},$ it is easy to compute a similar conditional probability,
which we denote by $q_{j|s}.$ We set the variance of the Gaussian function that is employed in the calculation
of the conditional probabilities $q_{j|s}$ to the value $2^{-1/2}.$ Therefore, we model the similarity of map element $\mathrm{y}_{j}$ to map
element $y_{s}$ by
$$
q_{ j|s}:=
\frac{e^{-\| \mathbf{y}_{s}-\mathbf{y}_{j} \| }}{\sum\limits_{m\neq s}e^{-\| \mathbf{y}_{s}-\mathbf{y}_{m} \| }}.
$$
Analogously to the previous case we set $q_{s|s}=0.$ In accordance with the concept represented in [25], we assume that if the map elements $\mathbf{y}_{s}$ and $\mathbf{y}_{j}$ correctly model the similarity between the high-dimensional data elements
$\mathbf{x}_{s}$ and $\mathbf{x}_{j},$ the conditional probabilities $p_{ j|s}$ and $q_{ j|s}$ will be equal. Having taken into account this concept,
the SNE method aims to find a low-dimensional data representation that minimizes the divergence between $p_{ j|s}$
and $q_{ j|s}.$ A natural measure of the faithfulness with which $q_{ j|s}$ models $p_{ j|s}$ is the Kullback-Leibler
divergence
$$
\mathrm{KL}(P_{s} ||Q_{s} )=\sum\limits_{j=1}^{n}p_{ j|s}\log \frac{p_{ j|s}}{q_{ j|s}},
$$
where $P_{s}$ represents the conditional probability distribution corresponding to the data element $\mathbf{x}_{s}$ over all other data elements."\\
It is clear that the minimization of the Kullback-Leibler
divergence allows to find a suitable in the sense of the relation between conditional probabilities low-dimensional data representation. \\
{\it Brief explanation.} Apparently, we can also conclude that the probabilistic approach is based on finding a low-dimensional data representation in accordance with the property of the conditional probabilities being as close to each other as possible.\\
\noindent Referee:
(6) In page 16, give an example of a Hilbert space RKSH.\\
\noindent Author:
Dear referee, thank you for the remark. The most known representatives of RKHS are the Bergman space and the Dirichlet space - the spaces of functions holomorphic in the unit disc. I have made the corresponding changes in the text.\\
\noindent Referee:
(7) Can you clearly explain the aim and applications of this article?.\\
\noindent Author:
Dear referee since the remark is similar to remark (1), I can refer to the extended version of the reply given above, however the brief statement is represented below.
From the mathematical point of view, the aim of the paper is to find a natural mathematical structure coupling the images and the preimages of the heterogenous datasets under a mapping aligning topological structures in Hilbert spaces. The coupling should preserve and convey the properties of the geometric objects what is obtained due to the fact that a mapping is defined on the sets of elements supplied with the structure of the field of complex numbers. Thus, the idea of preservation of geometrical objects is entirely based upon the statement that the mapping being continuous in the sense of the norm conveys the geometrical properties of preimages. The advantages in terms of applications, particularly in biology, include the following. The application of the operator theory becomes more natural and harmonious, since unitary spaces, involved due to the choice of the scalar field - the complex numbers, with a wider range of properties are considered in comparison with Euclidean ones.
From the point of view of applications, the aim of this paper is to develop methods for the diagonal integration of multimodal biological data for a comprehensive study of the biological processes in particular the aging process. The prospective project based on the paper fundamental results plans to develop new mathematical methods of diagonal integration that take into account the identified limitations of existing approaches. We propose to combine and jointly analyze seven different types of data: genomic, transcriptomic, proteomic, metabolomic, and lipidomic data. The aging process acts as a suprasystem factor for diagonal integration as a fundamental biological phenomenon that manifests itself at all levels of the molecular organization. A comprehensive analysis of the integrated data will make it possible to identify new pharmaceutical targets for therapeutic interventions in the aging process, identify multidimensional biomarkers of aging with increased predictive power, and establish causal relationships between molecular changes at various levels of biological organization in the context of age-associated processes.\\
\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 improve the paper significantly.
\vspace{0.1 cm}
Sincerely yours Ph.D. Maksim V. Kukushkin
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsI am satisfied with the corrected version of the manuscript.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe introduction was expanded.
The language was revised.
The wording of the document was supplemented, clarifying some points.
The changes were addressed, highlighting them in a different color and making them easier to follow
The mathematical notation of the suggested points was clarified
The aspects proposed in the summary are consistent with the development, and the conclusions highlight the findings
