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1. Graph Learning from Multi-Attribute Smooth Signals

   https://doi.org/10.1109/MLSP49062.2020.9231563  

   Tugnait, Jitendra K. (September 2020, 2020 IEEE 30th International Workshop on Machine Learning for Signal Processing (MLSP))

   null (Ed.)

   We consider the problem of estimating the structure of an undirected weighted graph underlying a set of smooth multi -attribute signals. Most existing methods for graph estimation are based on single-attribute models where one associates a scalar data variable with each node of the graph, and the problem is to infer the graph topology that captures the relationships between these variables. An example is image graphs for grayscale texture images for modeling dependence of a pixel on neighboring pixels. In multi-attribute graphical models, each node represents a vector, as for example, in color image graphs where one has three variables (RGB color components) per pixel node. In this paper, we extend the single attribute approach of Kalofolias (2016) to multi -attribute data. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function to infer the graph topology. Numerical results based on synthetic as well as real data are presented. 

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2. On High-Dimensional Graph Learning Under Total Positivity

   https://doi.org/10.1109/IEEECONF53345.2021.9723191  

   Tugnait, Jitendra K. (October 2021, 22021 55th Asilomar Conference on Signals, Systems, and Computers)

   We consider the problem of estimating the structure of an undirected weighted sparse graphical model of multivariate data under the assumption that the underlying distribution is multivariate totally positive of order 2, or equivalently, all partial correlations are non-negative. Total positivity holds in several applications. The problem of Gaussian graphical model learning has been widely studied without the total positivity assumption where the problem can be formulated as estimation of the sparse precision matrix that encodes conditional dependence between random variables associated with the graph nodes. An approach that imposes total positivity is to assume that the precision matrix obeys the Laplacian constraints which include constraining the off-diagonal elements of the precision matrix to be non-positive. In this paper we investigate modifications to widely used penalized log-likelihood approaches to enforce total positivity but not the Laplacian structure. An alternating direction method of multipliers (ADMM) algorithm is presented for constrained optimization under total positivity and lasso as well as adaptive lasso penalties. Numerical results based on synthetic data show that the proposed constrained adaptive lasso approach significantly outperforms existing Laplacian-based approaches, both statistical and smoothness-based non-statistical. 

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3. Nonparanormal graph quilting with applications to calcium imaging

   https://doi.org/10.1002/sta4.623  

   Chang, Andersen; Zheng, Lili; Dasarathy, Gautam; Allen, Genevera I. (September 2023, Stat)

   Abstract Probabilistic graphical models have become an important unsupervised learning tool for detecting network structures for a variety of problems, including the estimation of functional neuronal connectivity from two‐photon calcium imaging data. However, in the context of calcium imaging, technological limitations only allow for partially overlapping layers of neurons in a brain region of interest to be jointly recorded. In this case, graph estimation for the full data requires inference for edge selection when many pairs of neurons have no simultaneous observations. This leads to the graph quilting problem, which seeks to estimate a graph in the presence of block‐missingness in the empirical covariance matrix. Solutions for the graph quilting problem have previously been studied for Gaussian graphical models; however, neural activity data from calcium imaging are often non‐Gaussian, thereby requiring a more flexible modelling approach. Thus, in our work, we study two approaches for nonparanormal graph quilting based on the Gaussian copula graphical model, namely, a maximum likelihood procedure and a low rank‐based framework. We provide theoretical guarantees on edge recovery for the former approach under similar conditions to those previously developed for the Gaussian setting, and we investigate the empirical performance of both methods using simulations as well as real data calcium imaging data. Our approaches yield more scientifically meaningful functional connectivity estimates compared to existing Gaussian graph quilting methods for this calcium imaging data set. 

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4. High-Dimensional Inference for Cluster-Based Graphical Models

   Eisenach, C; Bunea, F; Ning, Y; Dinicu, C (January 2020, Journal of machine learning research)

   Motivated by modern applications in which one constructs graphical models based on a very large number of features, this paper introduces a new class of cluster-based graphical models, in which variable clustering is applied as an initial step for reducing the dimension of the feature space. We employ model assisted clustering, in which the clusters contain features that are similar to the same unobserved latent variable. Two different cluster-based Gaussian graphical models are considered: the latent variable graph, corresponding to the graphical model associated with the unobserved latent variables, and the cluster-average graph, corresponding to the vector of features averaged over clusters. Our study reveals that likelihood based inference for the latent graph, not analyzed previously, is analytically intractable. Our main contribution is the development and analysis of alternative estimation and inference strategies, for the precision matrix of an unobservable latent vector Z. We replace the likelihood of the data by an appropriate class of empirical risk functions, that can be specialized to the latent graphical model and to the simpler, but under-analyzed, cluster-average graphical model. The estimators thus derived can be used for inference on the graph structure, for instance on edge strength or pattern recovery. Inference is based on the asymptotic limits of the entry-wise estimates of the precision matrices associated with the conditional independence graphs under consideration. While taking the uncertainty induced by the clustering step into account, we establish Berry-Esseen central limit theorems for the proposed estimators. It is noteworthy that, although the clusters are estimated adaptively from the data, the central limit theorems regarding the entries of the estimated graphs are proved under the same conditions one would use if the clusters were known in advance. As an illustration of the usage of these newly developed inferential tools, we show that they can be reliably used for recovery of the sparsity pattern of the graphs we study, under FDR control, which is verified via simulation studies and an fMRI data analysis. These experimental results confirm the theoretically established difference between the two graph structures. Furthermore, the data analysis suggests that the latent variable graph, corresponding to the unobserved cluster centers, can help provide more insight into the understanding of the brain connectivity networks relative to the simpler, average-based, graph. 

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5. Do-calculus enables estimation of causal effects in partially observed biomolecular pathways

   https://doi.org/10.1093/bioinformatics/btac251  

   Mohammad-Taheri, Sara; Zucker, Jeremy; Hoyt, Charles Tapley; Sachs, Karen; Tewari, Vartika; Ness, Robert; Vitek, Olga (June 2022, Bioinformatics)

   Abstract Motivation Estimating causal queries, such as changes in protein abundance in response to a perturbation, is a fundamental task in the analysis of biomolecular pathways. The estimation requires experimental measurements on the pathway components. However, in practice many pathway components are left unobserved (latent) because they are either unknown, or difficult to measure. Latent variable models (LVMs) are well-suited for such estimation. Unfortunately, LVM-based estimation of causal queries can be inaccurate when parameters of the latent variables are not uniquely identified, or when the number of latent variables is misspecified. This has limited the use of LVMs for causal inference in biomolecular pathways. Results In this article, we propose a general and practical approach for LVM-based estimation of causal queries. We prove that, despite the challenges above, LVM-based estimators of causal queries are accurate if the queries are identifiable according to Pearl’s do-calculus and describe an algorithm for its estimation. We illustrate the breadth and the practical utility of this approach for estimating causal queries in four synthetic and two experimental case studies, where structures of biomolecular pathways challenge the existing methods for causal query estimation. Availability and implementation The code and the data documenting all the case studies are available at https://github.com/srtaheri/LVMwithDoCalculus. Supplementary information Supplementary data are available at Bioinformatics online. 

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