Title: A Latent Multilayer Graphical Model For Complex, Interdependent Systems
Venue: NeurIPS 2025
Accepted: yes
Keywords: multilayer networks

neuroimaging

graphical models

latent models

graph learning
Abstract: Networks have been extensively used and have provided novel insights across a wide variety of research areas. However, many real-world systems are, in fact, a ``network of networks'', or a multilayer network, which interact as components of a larger multimodal system. A major difficulty in this multilayer framework is the estimation of interlayer edges or connections.  In this work, we propose a new estimation method, called multilayer sparse + low-rank inverse covariance estimation (multiSLICE), which estimates the interlayer edges. multiSLICE bridges latent variable Gaussian graphical methods with multilayer networks, offering a flexible framework for modeling processes with irregular sampling and heterogeneous graph structures.  We develop an effective algorithm to compute the estimator. We also establish theoretical conditions for the recoverability of the joint space, analyze how inter-layer interactions influence joint parameter estimation, and provide theoretical bounds on their relationships. Finally, we rigorously evaluate our method on both simulated and multimodal neuroimaging data, demonstrating improvements over state-of-the-art approaches.  Finally, all the relevant R code implementing the method in the article is available on [GitHub](https://github.com/mondrus96/multiSLICE).


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the problem of recovering dependencies among individual systems in multimodal data by modeling them as a multilayer graph, where each system is represented by a latent-variable Gaussian graphical model (GGM). The primary contribution is a statistical model that estimates interlayer edges across disjoint GGMs that vary in structure, number of nodes, and sample sizes.\n\nThe authors provide theoretical guarantees for the recoverability of the joint space, exploring how interlayer interactions impact joint parameter estimation. They also derive theoretical bounds on these relationships. To the best of their knowledge, this is the first model capable of estimating interlayer connections in a multilayer network that handles heterogeneous dimensions, sample sizes, and scales to multiple modalities.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths:**\n\nThe paper is well written and well motivated, with a clear problem formulation. The proposed framework is flexible and shows superior empirical performance compared to baseline methods with insightful ablation studies. It includes theoretical recovery guarantees for the joint parameters. The approach can also handle heterogeneous systems and is scalable to multiple modalities.\n\n### **Weaknesses:**\n\nThe paper makes several assumptions which may limit generalizability in real-world data. These assumptions include: \n1. the individual models are disjoint Gaussians \n2. known intralayer connectivity and assumes that data is generated from a self-sufficient model with that known structure. From my understanding from Line 138, the data is assumed to be not influenced by the interlayer latent dependencies. \n3. prior knowledge of the maximum number of variables (p), including latent ones\n4. decomposability of the joint likelihood into layer-wise summations for tractability. The decomposition is made with few justifications. It would have been more useful if the authors could provide more convincing justifications for this design choice if there were any.\n\nWhile the paper provides a solid solution to the problem at hand, in all honestly, I am unsure whether it fits into the landscape of NeurIPS since the authors solely employ traditional statistical methods and the paper lacks application of neural networks. The authors should enhance the paper's relevancy and impact by demonstrating applications in deep learning contexts to align with contemporary research directions.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nYes. The limitations have been discussed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of learning the structure of multi-layer network systems by modeling them as Gaussian graphical models, where network connectivity is encoded in the sparsity pattern of the precision (inverse covariance) matrix. The authors formulate the learning task as a regularized maximum likelihood estimation problem, incorporating sparsity-inducing penalties tailored for multi-layer network structures. They derive sample complexity results that establish the number of samples needed for accurate structure recovery and provide high probability bounds on the estimation error of the precision matrix. Additionally, they propose a numerical optimization algorithm to solve the regularized problem efficiently and validate their approach on two benchmark datasets, demonstrating effective structure recovery.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nOverall, the paper is clearly written with technically sound mathematical analyses and derivations. While the results align with expectations from existing graphical model literature, the extension to multi-layer networks is well motivated and fairly presented. The results give insights into the interdependency between variables among layers, sparse edges, and the number of samples, thereby further broadening the statistical learning theory of Gaussian graphical models.\n\nWeaknesses:\nThe single biggest weakness is that the contribution feels like a “textbook exercise” to existing theory. I use quotes here to emphasize that while the effort is non-trivial, the motivation for this study feels weak, and the results are expected. I could easily imagine giving this problem as a mini project in a class on non-asymptotic statistics for networks. In the forest of sparsity-regularized likelihoods grown over the last two decades, the current value addition is more in highlighting the need for structure learning and then planting the right tree rather than planting a tree and trying to just the need with very laborious math. However, the motivation itself comes across as purely academic extension, and the final dataset results feel like a throw-off – following the standard template of works in this area.\n\nOn the technical side, since all norms and the cost function are very well-behaved, and the authors resort to fairly standard proof techniques, it is not clear nor well crafted what the theoretical implications really are. Simulation results do not meaningfully (or qualitatively) justify the derived sample complexity statements. It is also unclear if the considered datasets actually adhered to the hypotheses laid out in the theorems. One suggestion is to include a simple but powerful synthetic experiment, similar to what Martin Wainwright did in his 2009 LASSO paper in IEEE IT. Those experiments were simple but spectacularly teased out theoretical results. Once that type of experimental validation is established, the reader gains credibility in your method before seeing brain or cute-cats-running datasets. This is particularly important here because the authors impose their method on certain datasets without any empirical validation that these datasets demanded the proposed setup.\n\nQUESTIONS:\n1) What is the formal definition of Gaussian distribution on multi-layer networks? If the definition is same as the standard multivariate Gaussian with sparsity pattern of inverse covariance matrix reflecting network structure. Then, the notion of multi-layer network structure is not particularly an interesting probabilistic graphical model. \n2) Is it possible to cast this multi-layer network based graphical model in terms of matrix-variate Gaussian distribution? If so, how? If not why? \n3) Suggestions for improving simulations were already provided in the weakness section. \n4) What is \\mathcal{R}(\\hat{L})=r$ constraint in Eq (2)? Seems like this quantity is not defined in the paper.\n\nLIMITATIONS:\nThe authors, in passing, address as a limitation the fact that their work does not consider the time-varying nature of networks. Personally, I would not consider this as a limitation per se but rather as a natural future avenue. Second, in the checklist, the authors note (paraphrased) that “theorems have assumptions, and those directly state constraints.” I do not completely agree with this viewpoint. Every mathematical theorem has constraints (isn't that the nature of abstract reasoning?). These are not limitations in my view, and hence, I suggest authors to give more serious thought. On societal impact, the authors state there is no potential negative impact. I find this surprising since their simulation work relates to brain studies, where issues such as interpretability, misuse, and privacy often arise.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents multiSLICE, a novel framework for learning multilayer Gaussian graphical models where each layer can have a different number of variables and samples. It models each layer’s precision matrix as a sum of a sparse matrix (intralayer structure) and a shared low-rank matrix (interlayer structure). The two-stage algorithm estimates sparse and latent components per layer, then combines them using a matrix completion approach. The method is theoretically grounded, handles unaligned data, and outperforms prior methods on both simulated and neuroimaging data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. Addresses a significant challenge in heterogeneous multilayer graph learning. The method handles differing numbers of variables and samples across layers—an important and previously under-addressed problem. This makes it widely applicable to real-world multimodal settings, such as neuroscience or genomics.\n2. Novel integration of sparse and low-rank modeling with matrix completion. While individually inspired by existing techniques, the combination of latent GGMs, sparse precision estimation, and low-rank matrix completion is original in this context. The model provides both within-layer network structure and explicit interlayer edge estimation.\n3. Scalable, theoretically grounded algorithm. The two-stage procedure (SLICE + matrix assembly) is computationally tractable and supported by recovery guarantees under reasonable assumptions. The design allows each layer to be processed separately, enabling parallelism and better scalability to many layers.\n4. Strong empirical performance across tasks. MultiSLICE consistently outperforms baselines in recovering both sparse and latent structure in simulations. On real neuroimaging data, it produces networks with higher modularity and lower latent entropy, indicating more meaningful and interpretable structure.\n\nWeaknesses:\n\n\nThe paper does not discuss how to choose key parameters like the latent rank or sparsity penalties in practice. This limits immediate usability and may affect performance if the method is misconfigured.\nIncomplete comparisons with recent baselines.\n\nQUESTIONS:\nPlease describe how hyper-parameters are selected\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces multiSLICE, a latent-variable Gaussian graphical model tailored for multilayer networks. The key idea is to decompose each layer’s precision matrix into a sparse intra-layer component plus a shared low-rank component spanning all layers. This low-rank latent factor effectively captures inter-layer connections (influences common across modalities) while the sparse part captures layer-specific associations (e.g. brain-region connections within each modality). Notably, multiSLICE can handle heterogeneous graph structures, without forcing resampling or alignment. The authors propose an efficient two-stage algorithm: first, estimate each layer’s sparse + low-rank structure separately via an EM-like procedure (Algorithm 1), then latent components via a block-wise SVD matrix completion step (Algorithm 2). Finally, the paper empirically validates multiSLICE on both synthetic data (varying number of layers and latent rank) and real multimodal neuroimaging data (MEG + fMRI + sMRI), showing improved accuracy over several state-of-the-art multilayer GGM baselines. Proofs are provided in the supplementary.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n-\tCombines sparse + low-rank modeling in a multilayer graphical model formulation.\n-\tAuthors provide a comprehensive theoretical background.\n-\tEmpirical results (synthetic and real neuroimaging data) show superior performance vs. strong baselines.\n\nWeaknesses: \n-\tNotation and matrix completion step could be explained with more intuition and clearly, especially in Algorithm 2, where the authors use block-wise SVD to combine latent components from different layers. It would be helpful to elaborate how this procedure reconstructs a coherent global low-rank matrix from layer-wise estimates and how it addresses issues arising from partial observability or node mismatch across layers.\n\nQUESTIONS:\nHow might the multiSLICE framework be adapted for time-varying or longitudinal multilayer data?\n\nIt would be insightful to see how multiSLICE performs in a scenario where those methods are applicable. For example, could you run an additional simulation where all layers have equal n and p, and compare multiSLICE with baselines in that controlled setting? This would show that multiSLICE is not only superior in heterogeneous regimes but also competitive in homogeneous ones.\n\nThe method introduces a rank parameter r for L. How should these be chosen in practice? providing guidance or an automatic strategy (even in supplementary) for selecting these hyperparameters would make the method easier to use.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}