This paper studies the problem of learning the large-scale Gaussian graphical models that are multivariate totally positive of order two ($\text{MTP}_2$). By introducing the concept of bridge, which commonly exists in large-scale sparse graphs, we show that the entire problem can be equivalently optimized through (1) several smaller-scaled sub-problems induced by a \emph{bridge-block decomposition} on the thresholded sample covariance graph and (2) a set of explicit solutions on entries corresponding to bridges. From practical aspect, this simple and provable discipline can be applied to break down a large problem into small tractable ones, leading to enormous reduction on the computational complexity and substantial improvements for all existing algorithms. The synthetic and real-world experiments demonstrate that our proposed method presents a significant speed-up compared to the state-of-the-art benchmarks.
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Summary
The authors show that, in Gaussian MTP2 distributions, bridges in the graph structure have a closed form solution. They use this observation to suggest practical solutions that can be applied whenever such models are being fit.
Strengths
A nice, more or less self-contained theoretical work that unifies and extends some existing lines of research into MTP2 Gaussian distributions.
Weaknesses
Some room for improvement in terms of presentation and some minor typos. It's harder for me to judge practical utility. While it is true that MTP2 distributions have some practical utility (I've used them myself), it isn't clear that many real datasets meet that requirement (even approximately).
Questions
- "calculating the thresholded graph, bridges, and clusters, is negligible" Sure, but I feel that a precise statement and citation is needed here. In particular, you still have to compute the sample covariance matrix for this. - I actually had to read it a couple times to understand the exact procedure that was being proposed. I suggest adding some clarifying details to help the reader. - What is the primary limitation that makes this result only apply to MTP2 distributions? Is it possible that something similar could be true more generally? I didn't really get a sense of why MTP2 was required in the main text. - Do you need Gaussian distributions here? Can something be done for more general MTP2 distributions, e.g., those that can still be represented as pairwise graphical models? Misc. typos (only a few listed): - "which severs as a common assumption" - "make possibilities of solving..." (revise) - "As thresholded graph plays" - "garnered considerable attentions" - "Bridge is one of the important concepts" - "closed-form solutions in literature heavily" - "seems inapplicable for dense graph"
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness
3 good
Presentation
2 fair
Contribution
3 good
Limitations
Some discussion of limitations, though it would have been nice to see a more clearly identified set of problems yet to be solved.
Summary
This paper studies a graphical lasso problem where the precision matrix is restricted to be symmetric M-matrix and the associated GMRF graph has a special structure. Specifically, the authors consider the situation when the graph allows bridge-block decomposition so that vertices can be partitioned into k parts by cutting k-1 "bridges". Under this situation, the authors show that the original problem can be decomposed into k subproblems, illustrated in theorems in section 3.2. Since the existing algorithm's complexity quickly increases as the dimensionality increases, decomposition into subproblems offers large computational benefits. Authors compare the performance of four different algorithms, using synthetic and real data, with and without leveraging the bridge-block decomposition and show that bridge-block decomposition provides huge computational benefits if the given graph has many edges that are “bridges”.
Strengths
Authors provide a closed form of the elements of precision matrix corresponding to bridges, and make a connection with existing literature when underlying GMRF graph is an acyclic graph. This method is useful when one wants to solve MTP2 constrained graphical lasso problem with a penalty parameter that allows block-bridge decomposition. This work is resembles the existing literature on graphical lasso (without MTP2 constraint), such as Witten et al (2011); Mazumder and Hastie (2012), where graphical lasso problem can be decomposed to the smaller subproblems when graph has many connected components, and adds a contribution in the context of MTP2 constrained graphical lasso when graph has many bridges. Witten, D. M., Friedman, J. H., & Simon, N. (2011). New insights and faster computations for the graphical lasso. Journal of Computational and Graphical Statistics, 20(4), 892-900. Mazumder, R., & Hastie, T. (2012). Exact covariance thresholding into connected components for large-scale graphical lasso. The Journal of Machine Learning Research, 13(1), 781-794.
Weaknesses
Simulation settings and real data examples have some discrepancy from usual (graphical) lasso settings and its motivation. The main goal of graphical lasso is to discover the underlying conditional independency structure from the multivariate Gaussian data with various choice of penalty parameter (aka lasso path diagram). However authors first fix the graph in their settings such as preferential attachment graph and stochastic block model and choose penalty parameter according to the graph. It is understandable in the simulation study setting to show the computational benefits with block-bridge decomposition, but fixing the graph in the real data is not convincing, and also it is not clearly stated why MTP2 graphical lasso is appropriate to this crop image data problem (is it reasonable to assume that crop image data comes from high-dimensional multivariate Gaussian distribution? what is the interpretation of the resulting graph? why MTP2 constraint is necessary / plausible in this problem?) In practice, graphical lasso is ran under various settings of penalty parameters and chosen appropriately such as cross validation. The existing graphical lasso decomposition (Witten et al (2011); Mazumder and Hastie (2012)) are useful since graphical lasso with high penalty parameter often leads to a graph with many connected components. In similar spirit, I suggest authors to illustrate how often the graph that allows block bridge decomposition appears by the choice of penalty parameter.
Questions
- Does the main result, decomposition into smaller graphical lasso subproblems under block-bridge decomposition, holds without MTP2 constraint? - If not, please clarify the the role of the MTP2 constraint in the main results; does MTP2 constraint is necessary for the assumption 2.1. to be hold? - What happens if graph has many connected components (without bridges) in MTP2 constrained graphical lasso problem? Does similar decomposition holds? - Section 4.1: Sample size $n$ is 10 times the dimension $p$? This setting completely violates the main motivation of graphical lasso where $\ell_1$ penalty has been proposed since MLE does not exist when $n<p$. - Section 4.3: Shouldn't be data $y_i$ is 24000-dimensional with $i=1,\dots,46$ observations?
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness
2 fair
Presentation
2 fair
Contribution
2 fair
Limitations
The main results are meaningful but simulation results and real data settings are less convincing. No potential negative societal impact.
Summary
The paper studies the problem of estimating the precision matrix, which is the inverse of the correlation matrix, of a given Gaussian random vector $y$. The precision matrix $\Theta$ is assumed to satisfy a technical condition called MTP2 which states that $\Theta$ is symmetric and $\Theta_{i,j} \le 0$. This seems to be a well motivated assumption from various applications. The contribution of this paper is a technique for estimating $\Theta$ as follows: given a predicted sparsity pattern on $\Theta$ in the form of a graph $G$ , the natural optimization problem for estimating $\Theta$ can be solved by first solving the optimization problem on smaller 'blocks' and combining them across 'bridges'. They are defined as follows. Bridges are single edge cuts in $G$ and the resulting connected vertices are called blocks.
Strengths
The paper shows that given a block bridge decomposition, the optimization problem of estimating $\Theta$ can be efficiently solved by first solving the problem on the individual blocks and then combining the solution across bridges. The paper gives an explicit formula for doing so. In the case where $G$ is sparse, this can represent significant computational savings over estimating the entire precision matrix at once. Furthermore, since the work provides a structural theorem, any optimization algorithm can be used in conjunction with their observation.
Weaknesses
I am not familiar with the literature but it seems like a big assumption to know the threshold graph explicitly. What happens if this graph is unknown? It seems to be more natural that the graph is unknown and one must estimate it. An intermediate setting which also seems interesting is in the case where we know a noisy approximation to the block bridge structure. How do the proposed methods perform under such noisy information? Are the derived formulas robust? What is the motivation for even assuming the block bridge structure? I can see social networks being one motivation but it would be more convincing if there were experiments on (real) social networks.
Questions
What happens to the SBM experiments if the edge probabilities between the different blocks are increased? The quality of the method should deteriorate as the probabilities increase, since the block structure increasingly deteriorates. It would be interesting to see how much the proposed method can tolerate.
Rating
5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
2 fair
Contribution
2 fair
Limitations
No ethical concerns.
Summary
This paper studies the problem of learning Gaussian Graphical Models (GGMs) satisfying a certain positive associativity condition among the variables, namely that the precision matrix has nonnegative off-diagonal elements. This condition is known as being "multivariate totally positive of order two", or MTP$_2$, and has applications in ML (where it corresponds to attractive Markov random fields), finance, and more. MTP$_2$ GGMs come with the benefit that the traditional optimization procedure used to estimate the precision matrix of a GGM, namely the graphical Lasso, takes on a particularly simple and smooth form. Prior work had shown various polynomial-time convergence guarantees for the graphical Lasso under the MTP$_2$ assumption, but these are still not very suitable for practical applications (scaling with the dimension $p$ as $O(p^3)$ or $O(p^4)$). Other prior work had shown a closed-form solution for the graphical Lasso under the assumption that the "thresholded sample covariance graph" (an object defined in terms of the sample covariance and the regularization parameters used in the graphical Lasso), or thresholded graph for short, is acyclic. The main contribution of this paper is essentially to generalize the latter closed-form result in terms of the "bridge-block decomposition" of the thresholded graph. The bridge-block decomposition of a graph is essentially a partition of the graph into components connected only by "bridge" edges. Formally, a bridge edge is an edge such that deleting it increases the number of connected components in a graph; it is effectively "the only edge" bridging two different components (see Fig 1). The paper's main theorem (Thm 3.3) essentially says the following: to solve the graphical Lasso for an MTP$_2$ GGM, compute the bridge-block decomposition of the thresholded graph, run the graphical Lasso for each component separately, and stitch them together using a simple closed-form formula. Moreover, this theorem also readily recovers as a special case the prior closed-form result for acyclic thresholded graphs. This is because in an acyclic graph, every edge is a bridge, and the bridge-block decomposition is particularly simple. Thus the main result amounts to a divide-and-conquer recipe for learning MTP$_2$ GGMs, and the authors show various numerical experiments suggesting the practical superiority of this method over all prior methods (which operate on the entire graph). A key benefit is that the subproblems may be solved using any graphical Lasso implementation whatsoever, and potentially in parallel.
Strengths
Disclaimer: I am not very familiar with the literature in this area, and my review should be taken as that of a relative outsider. The paper's main result is both an interesting structural result about MTP$_2$ GGMs as well as a genuinely practical algorithmic advance in learning such models. From a conceptual point of view, the idea of leveraging the bridge-block decomposition seems novel. The overall result seems like a useful and nice contribution to the literature on this problem, and to the extent that one considers MTP$_2$ GGMs significant, one should consider this result significant as well. The paper is largely clear and easy to follow (modulo some occasionally confusing bits; see the Questions section). It does a good job of setting up the main problem as well as the necessary context. I did not manage to verify the proofs in detail, but they seemed fairly clean, relying on an analysis of the KKT conditions of the graphical Lasso as well as some clever algebraic manipulation.
Weaknesses
I think the main things to really evaluate about this paper are its novelty and significance. As an outsider to this area, I find this hard to accurately gauge, but I think the paper scores well on these fronts. I do think the paper could benefit from a better conceptual overview of the main proof and the role of the bridge-block decomposition. The context and benefits are discussed adequately, but the key ideas in the proof do not come through very well, and the main proof seemed slightly magical to me. Why would one have expected the bridge-block decomposition to help? Was its role surprising?
Questions
I think the biggest question I have is about the conceptual role of the bridge-block decomposition, as explained above. A few other questions: - One of the key parts of the eventual proof of Thm 3.3 is Lemma A.2, which is simply described as following from the KKT conditions. Presumably the authors mean the KKT conditions associated with Problem (5)? Since it plays a relatively important conceptual role, I feel this could definitely use more elaboration. - What exactly is the time required to compute the bridge-block decomposition? This is described as negligible in Sec 3.1, but is it e.g. $O(p^2)$? Similarly, what about the quantities used in the closed-form formula? Currently the only discussion of the overall asymptotic running time appears in the first bullet of the "Proposed Framework" list on page 5. - In the experimental section, it would be helpful to say more about it is natural to synthesize/define $\Theta$ and $\Lambda$ in the specific way described in the first two paragraphs of Section 4.1, especially for readers who are unfamiliar with the prior work mentioned in those paragraphs. A couple other nits regarding the presentation: - It is not immediately obvious how Corollary 3.5 follows from Thm 3.3, and this could use a line of explanation. - The notation $\mathbb{S}^p$ is not formally defined in the paper, and MTP$_2$'s expansion is only mentioned in the abstract.
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
4 excellent
Contribution
3 good
Limitations
The paper could use a couple additional lines in the final section about the technical limitations of this work and what the major next steps could be. I am not aware of any significant potential negative societal impact of this theoretical work.
Summary
The paper focuses on the problem of learning large-scale Gaussian graphical models (GGMs) that are multivariate totally positive of order two (MTP2). The high-dimensional, sparse MTP2 GGMs are not easily manageable due to their size and complexity. The authors propose a novel approach, introducing the concept of a "bridge", to optimize the entire problem into several smaller, more manageable sub-problems and a set of closed-form solutions. The approach is based on the bridge-block decomposition of the thresholded sample covariance graph, which leads to reductions in computational complexity and improvements in existing algorithms.
Strengths
The proposed bridge-block decomposition framework on Gaussian graphical models seems novel. The problem is motivated nicely, and according to the authors, the framework could significantly reduce computational and memory cost. The proposed method seems to subsume various network structures, including the BA graph and the SBM, which are common models used in network analysis. Experimental results are provided, and the computational results look promising.
Weaknesses
As the authors mentioned in the paper, the proposed method might not generalize to dense cases. Still I feel in many settings like BA and SBM, sparsity is a reasonable assumption.
Questions
Is there any way to characterize the effect of the proposed approach in terms of some global graphical properties, for example, the edge expansion? Asking this because bridge is related.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
3 good
Contribution
3 good
Limitations
See Above.
Thanks for the detailed response. I have some additional questions. > Reply: In the context of graphical models, every node corresponds to a data, while the edges represent the conditional dependencies between the data. As a result, our network comprises 24000 nodes. It is important to clarify that in this scenario, yi does not represent a feature, but rather a signal consisting of 46 observations. This yields a configuration where p=24000 and n=46. At the bottom of page 8 (why there are no line numbers?), the current manuscript reads "Our goal is to perform graph-based clustering for the indexed data ${y_1, . . . , y_{24000}}$ using MTP2 GGMs, where $y_i \in \mathbb{R}^{46}$." This sentence implies $p=46$ and $n=24000$, and this is what I asked. Please cross-check with the notations above eq. (1) in page 2. > (Section 2.2) This paper advances prior research in two ways. Firstly, we extend the closed-form solutions beyond the acyclic graph structure to encompass any edge corresponding to a bridge. > (Global response) Our research offers two main theoretical advancements. The first proposes an explicit form for the inverse of $\Theta$. I feel these statements are too strong. As described in Figure 2, when graph allows block-bridge decomposition, what authors show is the decomposition of the big problem into the smaller subproblems, and closed form solutions for off-blockdiagonal entries (orange entries of Fig. 2). Authors claim that this paper "extend the closed-form solutions beyond the acyclic graph structure" of Fattahi and Sojoudi (2019, JMLR, ref. 27). I think this may confuse readers, since what authors show is mainly the decomposition, not the explicit solution that is similar to equation (11) of Fattahi and Sojoudi (2019). I believe authors should use terms like "closed from" or "explicit from" with extra caution in this regard, and I strongly suggest to revise sentence "we extend the closed-form solutions beyond the acyclic graph structure to encompass any edge corresponding to a bridge" and other sentences with similar context. I appreciate authors running additional simulations when $p>>n$. Authors use preferential attachment (Barabasi-albert) graph and stochastic blockmodel in simulation studies to illustrate block-bridge decomposition, which often has many bridges due to its construction. While those graph arises from completely different context (e.g. community detection problems), and I understand this is for the illustrative purposes, but are there any references that using those preferential attachement graph or SBM in the graphical lasso settings like in this paper? > (Section 2.1) This paper considers estimating the precision matrix $\Theta$ given $n$ **independent and identically distributed** observations ${y_1, . . . , y_n}$ that follow an MTP2 Gaussian distribution" > (section 4.3) Though it is not the focus of this paper to discover insights into the estimated MTP2 GGMs for better understanding the inherent nature of the data. I thank authors to run graphical lasso and present additional real data analysis results (fig.5 and fig. 6). However I am still not sure crop image dataset is best suited for the proposed method. Each node corresponds to each pixel of satellite image, and there are $n=46$ images $y_1,\dots,y_{46}$ which are time-varying measurements, illustrating the temporal evolution of the observed area. This implies data are temporally correlated, not iid. How big is the time interval (montly, yearly)? What is being measured? (pixel greyscale value? please clarify and describe at least minimally, not just referring ref.47) Most importantly, how should we interpret the conditional dependency structure of this real data analysis result? If estimated MTP2 GMM does not give any further understanding of the crop image data, I believe this data should not be used in the real data analysis. I have increased my score from 3 to 4 based on the additional results, but still lean to a rejection due to above concerns.
Thank you for your valuable comments and for raising the score of our paper. We're more than happy to further discuss and address your concerns. ## Reply to Post-Rebuttal Questions Part 1 > The current manuscript reads "Our goal is to perform graph-based clustering for the indexed data $y_1,\dots,y_{24000}$ using MTP2 GGMs, where $ y_i\in\mathbb R^{46}$." This sentence implies $p=46$ and $n=24000$. __Reply__: Thank you for pointing out the ambiguity in our manuscript. We apologize for any confusion. To clarify, we propose to revise the sentence as follows: "Our aim is to apply graph-based clustering to the time series data that contains 46 observations, denoted as $\mathbf y_1,\dots, \mathbf y_{46}$, where $\mathbf y_i\in \mathbb R^{24000}$." This representation aligns with the conventions typically adopted in the realm of graphical models. We hope this modification adequately addresses your concern. We are committed to improving our manuscript based on your feedback. ## Reply to Post-Rebuttal Questions Part 2 > Authors claim that this paper "extend the closed-form solutions beyond the acyclic graph structure" of Fattahi and Sojoudi (2019, JMLR, ref. 27). I think this may confuse readers, since what authors show is mainly the decomposition, not the explicit solution that is similar to equation (11) of Fattahi and Sojoudi (2019). __Reply__: We deeply appreciate your constructive suggestions. We agree with the reviewer's comment that more caution is needed when using the terms "closed-form" and "explicit solution" to avoid mixing up the results from our paper and the paper of Fattahi and Sojoudi. To address your concern, we propose to revise the sentence in our manuscript as follows: "While previous studies have offered an explicit solution for $\Theta_{ij}$ in the case of acyclic thresholded graphs, we reveal that the explicit solution for $\Theta_{ij}$ consistently applies to every $(i,j)$ pair acting as a bridge in non-acyclic graphs." To provide further clarification, it's noteworthy that the JMLR paper outlines an explicit computation method for deriving each entry of the solution associated with the acyclic graph structure. However, our paper illustrates that the solution can be obtained using a decomposed approach based on block-bridge decomposition. In this approach, the entries corresponding to bridges have an explicit computation method, similar to the results presented in the JMLR paper, and the decomposed subproblems can be solved independently. Importantly, our results are more general: the JMLR paper necessitates an acyclic graph structure, which is a specific case in our study. This follows from the fact that all edges in an acyclic graph are bridges, which have explicit solutions as demonstrated in our theory. Your feedback is highly valuable in helping us improve the clarity and precision of our manuscript. We would make revisions accordingly and ensure a detailed comparison of our results with those of Fattahi and Sojoudi's work in our revised manuscript. > Are there any references that using those preferential attachement graph or SBM in the graphical lasso settings like in this paper? __Reply__: Yes, both the Barabási–Albert (BA) and Stochastic Block Models (SBM) are frequently used in the context of graphical models, including graphical lasso settings. Below, we provide several relevant references: __BA graph:__ 1. Liu, H., & Wang, L. (2017). TIGER: A Tuning-Insensitive Approach for Optimally Estimating Gaussian Graphical Models. 2. Ying, J., Cardoso, J. V. de M., & Palomar, D. (2020). Nonconvex Sparse Graph Learning under Laplacian Constrained Graphical Model. Advances in Neural Information Processing Systems, 33, 7101-7113. 3. Li, R., et al. (2023). Graph Learning for Latent-Variable Gaussian Graphical Models under Laplacian Constraints. Neurocomputing, 532, 67-76. __SBM graph:__ 1. Mohan, K., et al. (2014). Node-based learning of multiple Gaussian graphical models. The Journal of Machine Learning Research, 15(1), 445-488. 2. Pircalabelu, E., & Claeskens, G. (2020). Community-Based Group Graphical Lasso. The Journal of Machine Learning Research, 21(1), 2406-2437. 3. Ying, J., Cardoso, J. V. de M., & Palomar, D. P. (2023). Adaptive Estimation of Graphical Models under Total Positivity. International Conference on Machine Learning. BA models play an important role in network science, which generate random scale-free networks using a preferential attachment mechanism such that new nodes tend to link to nodes that have higher degree in the evolution. Scale-free networks are well-suited to model the Internet, the world wide web, protein interaction networks, citation networks, and most social and online networks. Stochastic block models serve as fundamental tools in network science, creating random networks based on community structures, where nodes within the same group are more likely to form connections.
## Reply to Post-Rebuttal Questions Part 3 > Each node corresponds to each pixel of satellite image, and there are $n=46$ images $y_1,\dots,y_{46}$ which are time-varying measurements, illustrating the temporal evolution of the observed area. This implies data are temporally correlated, not iid. __Reply__: Thank you for your insightful comment. We agree with your observation that the nodes, representing each pixel of the satellite image, are indeed temporally correlated due to the time-varying nature of the measurements. This introduces a temporal correlation in the data, which inevitably deviates from the assumption of independent and identically distributed (iid) data. Despite the inherent temporal correlation in time series data, it's a common practice in the realm of Gaussian graphical models to approximate the data as being independent and identically distributed (i.i.d). This approximation, while not strictly accurate, has been successfully implemented in various studies such as those by Liu et al. (2012), Wang et al. (2020), and Agrawal et al. (2022). These works utilize Gaussian graphical models for financial time series analysis, and despite the fact that stock prices don't perfectly conform to the i.i.d assumption, they are effectively modeled and interpreted within this framework. This demonstrates the practical utility of such models, suggesting that our approach can still provide valuable insights and meaningful results, even when acknowledging the presence of temporal correlation. It is worth mentioning that estimating time-varying graphical models is indeed an intriguing research direction in the context of time series. Such an approach becomes particularly relevant when there's a need to understand the evolving interactive relationships among a set of random variables. While this specific aspect is not the primary focus of our current paper, we acknowledge its importance and potential for future research. We greatly appreciate your insightful feedback. We plan to incorporate a more detailed discussion on this matter in our revised manuscript. Should you have any further queries or comments, please do not hesitate to reach out to us. __Reference__: -- Liu, H., Han, F., & Zhang, C. (2012). Transelliptical graphical models. Advances in Neural Information Processing Systems, 25. -- Wang, Y., Roy, U., & Uhler, C. (2020). Learning high-dimensional Gaussian graphical models under total positivity without adjustment of tuning parameters. International Conference on Artificial Intelligence and Statistics. PMLR. -- Agrawal, R., Roy, U., & Uhler, C. (2022). Covariance matrix estimation under total positivity for portfolio selection. Journal of Financial Econometrics, 20(2), 367-389. > How big is the time interval (montly, yearly)? What is being measured? (pixel greyscale value? please clarify and describe at least minimally, not just referring ref.47) __Reply__: Images in the dataset are captured at five-day intervals. The recorded spectral information from these images represents variations in "colours" (referring to different spectral bands) for each pixel over the study period. Hence, each data point in this dataset corresponds to a time series of the spectral changes observed at a specific geographical location over time. > Most importantly, how should we interpret the conditional dependency structure of this real data analysis result? If estimated MTP2 GMM does not give any further understanding of the crop image data, I believe this data should not be used in the real data analysis. __Reply__: Thank you for your insightful question. The estimated graph is statistically meaningful. We observed that the bulk of edges are located within the same type of crop, while the edges between nodes associated with different crops are relatively sparse. This finding is beneficial for clustering processes and aligns with our anticipations, as a stronger positive dependency is often exhibited within the same class, while the dependency among different classes tends to be significantly weaker. In addition to revealing clustering patterns, our graph can reflect more intricate insights. For example, in Figure 6a of our manuscript, we noticed a substantial density of edges between two crop types, 'temporary meadow' and 'pasture' (colored in gold '#B79F00' and cyan '#00BFC4' respectively), indicating a conditional dependency significantly stronger than those found between other categories. This observation aligns with our perception. Therefore, the conditional dependency structure that we have inferred possesses the ability to represent the inherent interrelationships among different crops. We believe this provides valuable insight to further understanding of the crop image data. We appreciate your consideration and look forward to any further questions or comments you may have.
Thanks for the detailed answers. I believe that incorporation of authors comments to the revised manuscript, especially for the real data analysis section, will make a significant improvement of the paper in my view. I increased my score from 4 to 6, provided that detailed description and interpretation of the real data analysis will be added in the revised manuscript. Adding color labels in figure 6(a) would be a plus.
Dear Reviewer, We greatly appreciate your expert review and the time you've dedicated to improving our manuscript. We are committed to revising our paper in response to your valued feedback. Best regards The Authors
I am satisfied with the responses, thank you. It would be great to incorporate some of this into the final revision.
Dear Reviewer, We sincerely appreciate the time and effort you've dedicated to reviewing our paper. Your constructive comments have been instrumental in refining our paper. Thank you once again for your invaluable input and support. Best regards, The Authors
Response to rebuttal
Thank you for responding to the review. It seems that knowing $\Lambda$ is still an assumption. For now I will maintain my score.
Dear Reviewer, We appreciate your feedback. In practice, the regularization matrix $\boldsymbol{\Lambda}$ can be effectively computed based on certain initial estimates. In our paper, we suggest the use of Nie et al.'s (2016) method for efficiently deriving these initial estimates. __Reference__: — Feiping Nie, Xiaoqian Wang, Michael Jordan, and Heng Huang. The constrained Laplacian rank algorithm for graph-based clustering. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016.
Decision
Accept (poster)