Subject Clustering by an Improved IF-PCA Algorithm

Many thanks for your detailed feedback on our manuscript.

You brought up many good points:

1. It is not clear how to examine if a low-dimensional manifold structure exists for a practical problem.

2. How should one select parameters in practice without any knowledge about the potential number of clusters etc.?

3. Give a clear recommendation on which variant of IF-PCA+ to use for a particular problem.

4. Please provide some discussion on the data size that the proposed method can handle in terms of n and p since in many genomic data, p can be in millions.

5. How does zero inflation affect the proposed clustering methods?

6. What are the numbers in Table 1?

7. Why use 4 in the max(4,K) adjustment for PCA clustering step instead of a higher number?

8. Does IF-PCA+ only work for non-negative data?

9. The writing needs to be improved for more clear presentation.

Response to your main points:

1. Thank you for your insightful comment regarding the key assumption of a low-dimensional nonlinear structure in the context of IF-PCA+. We agree that the effectiveness of IF-PCA+ hinges on the presence of such a structure. In practice, one may use elbow points of Scree Plots to check if the data have a low-dimensional structure. Alternatively, one may test the quality of manifold fitting using spectral embedding methods (e.g., diffusion maps), which visually or quantitatively reveal whether the data conforms to a low-dimensional structure. Last but not least, domain knowledge can also inform researchers whether a low-dimensional manifold structure is expected, as is often the case in genomic or single-cell data.

2. Thank you for highlighting the practical challenges associated with selecting tuning parameters. To address this concern, we have added a section in the Appendix to provide practical (and theoretical) guidelines for parameter selection, particularly when there is no prior knowledge about the dataset.

3. Thank you for your suggestion to provide clear recommendations on the use of different variants of IF-PCA+ for specific problems. In response, we have revised the corresponding subsection to clarify that the full IF-PCA+ is generally recommended for most scenarios due to its robust performance.

4. Thank you for highlighting the importance of discussing the data sizes that the proposed method can handle. To address this concern, we have added details on the computational complexity of the proposed method in the revised manuscript. This discussion, combined with the provided runtime analysis, allows readers to estimate the scalability of the method for specific n and p values. While we do not explicitly state hardware requirements, the complexity analysis and the runtime data provide practical insights into the feasibility of applying our method to large genomic datasets.

5. Thank you for raising the important question regarding zero inflation in single-cell data and its impact on the proposed clustering methods. Although not explicitly stated in the initial manuscript, our simulation for DMF in the Appendix demonstrates that an increase in noise levels leads to a decrease in neighborhood accuracy, ultimately resulting in worse clustering performance. Additionally, Figure 2 illustrates that although the increase in zero-inflation (dropout) noise level from 0% to 30% substantially reduces neighborhood accuracy, the general trend of performance does not change: our method consistently outperforms others. To address your concern, we have added this discussion to the Appendix in the revised manuscript, providing further insights on how zero inflation (and other noises) affects our methods.

6. Thank you for pointing out the need for clarification regarding the numbers in Table 1. In the revised manuscript, we have added more detail to describe what these numbers represent and how methods were tuned to obtain these results.

7. Thank you for your question regarding the use of \max(4, K) top left singular vectors for the PCA clustering step. In the revised manuscript, we have added a detailed discussion in the Appendix to explain this choice.

8. Thank you for your comment regarding the applicability of IF-PCA+ to non-negative data. Yes IF-PCA+ does work with non-negative data by simply skipping the log-transformation step. In the revised manuscript, we have made this clear.

9. Thank you for your feedback regarding the clarity of the writing and for pointing out specific areas for improvement. In the revised manuscript, we thoroughly reviewed and improved the writing for a clearer presentation.

Thank you again for your time and valuable feedback!

Many thanks for your thoughtful feedback on our manuscript.

You brought up five points:

1. The Methods section lacks a comprehensive description of prior work (e.g., ysl23, yao2, IF-PCA), making it difficult to follow. Some abbreviations (e.g., KS on line 209) are undefined, and citing experimental results (lines 174-185, 288-289) within the algorithm description seems unprofessional and unnecessary.

2. Comparisons use different languages (Python vs. R), leading to biased runtime results; theoretical complexity analysis would be fairer. Additionally, runtime results (lines 316-321) belong in the results section, ideally in a table for clarity.

3. Some abbreviations of comparative algorithms (Xs) are unclear and are only explained in the appendix. These explanations should be moved to the main text to improve readability and make the content flow more smoothly.

4. The parameters tuning part remains challenging.

5. Table 4 shows a large regret gap between IF-PCA+ and IF-PCA, which needs further discussion. Table 5’s IF-sf results reach 1.0 in settings A and B without a clear explanation, raising questions about robustness.

Response to your main points:

1. We appreciate your comments about the comprehensiveness of prior work descriptions, the clarity of abbreviations, and the placement of experimental results.

• Comprehensive Description of Prior Work: We acknowledge your concern about the descriptions of prior work, such as ysl23, Yao2, and IF-PCA. While we have not made significant changes to this section, we believe that the existing descriptions provide sufficient context to understand the basis for our proposed method. Furthermore, detailed discussions of these prior methods are available in their respective cited references, which we encourage readers to consult for additional clarity.

• Abbreviations: In response to your feedback, we have ensured that all abbreviations, including “KS” on line 209, are now properly defined when first introduced. We believe this change improves the overall readability and accessibility of the paper.

• Experimental Results in the Methods Section: While we understand your concern that including experimental results in the algorithm description may appear unconventional, we have retained these results as they directly support and validate specific methodological choices. For example, the cited results demonstrate why certain steps in the algorithm are necessary. We believe this integration provides readers with immediate insight into the rationale behind our approach, streamlining the presentation.

2. Thank you for pointing out the concern regarding comparisons using different programming languages, as well as the placement and format of runtime results. We agree that theoretical complexity analysis provides a fairer basis for comparison and have included a discussion in Section 2.4 with detailed analysis in the Appendix. Regarding the presentation of runtime results, we chose to retain the existing format in the Methods section as it is directly related to the description of the algorithms and their computational differences. That said, we appreciate the suggestion of presenting these results in a table and will consider this for future refinements.

3. Thank you for pointing out the issue with unclear abbreviations of comparative algorithms (Xs) and suggesting that their explanations be moved to the main text. To address this concern, we have removed most of the unclear abbreviations (Xs) from the manuscript. For the remaining abbreviations, we have added explanations directly in the main text where they are first introduced.

4. Thank you for your feedback regarding the challenges in the parameter tuning section. We acknowledge that parameter tuning can be complex, especially for methods involving multiple components like ours. To address this concern, we have included additional explanations and guidelines for parameter selection in the revised manuscript's Appendix, particularly in scenarios where there is no prior information about the dataset. This additional explanation will make the process clearer and more accessible to readers.

5. Thank you for pointing out the need for further discussion regarding Tables 4 and 5. To address your concerns, we have expanded the discussion for Table 4 to clarify the reasons behind the regret gap between IF-PCA+ and IF-PCA, which is likely due to overfitting, and provide an explanation of where IF-PCA+ sees an improvement over IF-PCA in microarray datasets. For Table 5, we have included more explanation on why an accuracy of 1.0 in settings A and B is impressive and added more interpretation to the result.

Thank you again for your time and valuable feedback!

Many thanks for your positive feedback on the originality and significance of our paper and for carefully pointing out the grammatical errors in our manuscript.

You brought up the point that:

1. "The clarity of presentation of this paper needs to be improved. There are a lot of English mistakes … These should be carefully addressed prior to the publication of this paper."

2. In line 270, how to select the tuning parameters $n_0$, $knn_f$ and $knn_s$?

Response to your main point:

1. We have addressed these issues by thoroughly revising the manuscript to enhance language clarity and grammatical accuracy. Additionally, we have included new simulations, expanded explanations, and further discussions to improve the flow and coherence of the paper.

2. We have included additional explanations and guidelines for parameter selection in the revised manuscript's Appendix, particularly in scenarios where there is no prior information about the dataset. This additional explanation will make the process clearer and more accessible to readers.

Thank you again for your time and valuable feedback!

Thanks for your valuable comments.

You brought up points that:

1. What is 'subject clustering', as opposed to 'clustering'?

2. Manifold fitting, both sample-wise and feature-wise, are new to me. Is manifold fitting just another name for manifold learning? Please explain these concepts prior to using them.

3. What is a 'nonlinear dataset' (line 98 and elsewhere)

4. Some of the references are incorrect, please correct.

5. Please put all the background material in a background section. The methods can then simply describe your method, highlighting the differences with previous work.

Response to your main points:

1. In the context of our work, "subject clustering" refers to grouping individual biological subjects (e.g., cells, patients, or tissue samples) based on their features or characteristics, as opposed to clustering features or other variables.

2. Thank you for your question regarding manifold fitting and its distinction from manifold learning. While both manifold learning and manifold fitting focus on identifying the underlying geometric structures in data, traditional manifold learning methods primarily address the problem of "manifold embedding," which involves projecting data onto a lower-dimensional space. In contrast, manifold fitting focuses on projecting data onto a low-dimensional manifold that resides within the original high-dimensional space. We have added explanations of the concept in the methods section to enhance clarity.

3. Thank you for your question regarding the term "nonlinear dataset." A dataset is considered linear if its data matrix can be expressed as $X=L M+noise$ , where $L \in \mathbb{R}^{n, k}$ and $M \in \mathbb{R}^{k, p}$, representing a linear model. IF-PCA has been proven effective for such models. However, if this representation is not feasible, the dataset is classified as nonlinear, following a nonlinear model. We have revised the manuscript to avoid confusion.

4. Thank you for pointing out the issue with incorrect references. We have carefully reviewed and corrected all references in the revised manuscript to ensure accuracy. This includes verifying citation details such as author names, titles, publication years, and journal information.

5. Thank you for your feedback regarding the clarity of the method description and your suggestion to streamline the discussion by reducing background information.
   We acknowledge your concern and agree that excessive background information can distract from the main focus. However, manifold fitting is a relatively new concept that plays a key role in our approach. To avoid detracting from the primary discussion on the improvement to IF-PCA, we chose to briefly introduce manifold fitting in the Methods section rather than elaborating on it in the Introduction. This ensures that the focus remains on the main contributions of the paper while providing sufficient context for understanding the method. To address your concern, we have worked to simplify and condense the background content, ensuring it remains concise and essential without overcompromising the paper's readability or flow.

Thank you again for your time and valuable feedback!