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1. Thresholded graphical lasso adjusts for latent variables

   https://doi.org/10.1093/biomet/asac060  

   Wang, Minjie; Allen, Genevera I (November 2022, Biometrika)

   Summary Structural learning of Gaussian graphical models in the presence of latent variables has long been a challenging problem. Chandrasekaran et al. (2012) proposed a convex program for estimating a sparse graph plus a low-rank term that adjusts for latent variables; however, this approach poses challenges from both computational and statistical perspectives. We propose an alternative, simple solution: apply a hard-thresholding operator to existing graph selection methods. Conceptually simple and computationally attractive, the approach of thresholding the graphical lasso is shown to be graph selection consistent in the presence of latent variables under a simpler minimum edge strength condition and at an improved statistical rate. The results are extended to estimators for thresholded neighbourhood selection and constrained $$\ell_{1}$$-minimization for inverse matrix estimation as well. We show that our simple thresholded graph estimators yield stronger empirical results than existing methods for the latent variable graphical model problem, and we apply them to a neuroscience case study on estimating functional neural connections. 

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2. Attributed Graph Clustering: an Attribute-aware Graph Embedding Approach

   Akbas, Esra; Zhao, Peixiang (January 2017, 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining)

   Graph clustering is a fundamental problem in social network analysis, the goal of which is to group vertices of a graph into a series of densely knitted clusters with each cluster well separated from all the others. Classical graph clustering methods take advantage of the graph topology to model and quantify vertex proximity. With the proliferation of rich graph contents, such as user profiles in social networks, and gene annotations in protein interaction networks, it is essential to consider both the structure and content information of graphs for high-quality graph clustering. In this paper, we propose a graph embedding approach to clustering content-enriched graphs. The key idea is to embed each vertex of a graph into a continuous vector space where the localized structural and attributive information of vertices can be encoded in a unified, latent representation. Specifically, we quantify vertex-wise attribute proximity into edge weights, and employ truncated, attribute-aware random walks to learn the latent representations for vertices. We evaluate our attribute-aware graph embedding method in real-world attributed graphs, and the results demonstrate its effectiveness in comparison with state-of-the-art algorithms. 

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3. Regression of binary network data with exchangeable latent errors

   https://doi.org/10.1017/nws.2023.12  

   Marrs, Frank W; Fosdick, Bailey K (September 2023, Network Science)

   Undirected, binary network data consist of indicators of symmetric relations between pairs of actors. Regression models of such data allow for the estimation of effects of exogenous covariates on the network and for prediction of unobserved data. Ideally, estimators of the regression parameters should account for the inherent dependencies among relations in the network that involve the same actor. To account for such dependencies, researchers have developed a host of latent variable network models; however, estimation of many latent variable network models is computationally onerous and which model is best to base inference upon may not be clear. We propose the probit exchangeable (PX) model for undirected binary network data that is based on an assumption of exchangeability, which is common to many of the latent variable network models in the literature. The PX model can represent the first two moments of any exchangeable network model. We leverage the EM algorithm to obtain an approximate maximum likelihood estimator of the PX model that is extremely computationally efficient. Using simulation studies, we demonstrate the improvement in estimation of regression coefficients of the proposed model over existing latent variable network models. In an analysis of purchases of politically aligned books, we demonstrate political polarization in purchase behavior and show that the proposed estimator significantly reduces runtime relative to estimators of latent variable network models, while maintaining predictive performance. 

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4. Outcome-guided disease subtyping by generative model and weighted joint likelihood in transcriptomic applications

   https://doi.org/10.1214/23-AOAS1865  

   Li, Yujia; Liu, Peng; Wang, Wenjia; Zong, Wei; Fang, Yusi; Ren, Zhao; Tang, Lu; Celedón, Juan C; Oesterreich, Steffi; Tseng, George C (September 2024, The Annals of Applied Statistics)

   With advances in high-throughput technology, molecular disease subtyping by high-dimensional omics data has been recognized as an effective approach for identifying subtypes of complex diseases with distinct disease mechanisms and prognoses. Conventional cluster analysis takes omics data as input and generates patient clusters with similar gene expression pattern. The omics data, however, usually contain multifaceted cluster structures that can be defined by different sets of genes. If the gene set associated with irrelevant clinical variables (e.g., sex or age) dominates the clustering process, the resulting clusters may not capture clinically meaningful disease subtypes. This motivates the development of a clustering framework with guidance from a prespecified disease outcome, such as lung function measurement or survival, in this paper. We propose two disease subtyping methods by omics data with outcome guidance using a generative model or a weighted joint likelihood. Both methods connect an outcome association model and a disease subtyping model by a latent variable of cluster labels. Compared to the generative model, weighted joint likelihood contains a data-driven weight parameter to balance the likelihood contributions from outcome association and gene cluster separation, which improves generalizability in independent validation but requires heavier computing. Extensive simulations and two real applications in lung disease and triple-negative breast cancer demonstrate superior disease subtyping performance of the outcome-guided clustering methods in terms of disease subtyping accuracy, gene selection and outcome association. Unlike existing clustering methods, the outcome-guided disease subtyping framework creates a new precision medicine paradigm to directly identify patient subgroups with clinical association. 

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5. T-LoHo: A Bayesian Regularization Model for Structured Sparsity and Smoothness on Graphs

   Lee, Changwoo; Zhao Tang Luo; and Huiyan Sang (December 2021, Advances in neural information processing systems)

   Graphs have been commonly used to represent complex data structures. In models dealing with graph-structured data, multivariate parameters may not only exhibit sparse patterns but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high-dimensional parameters with graphical relations, referred to as the Tree-based Low-rank Horseshoe (T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultaneously. The T-LoHo prior can be embedded in many high-dimensional hierarchical models. To illustrate its utility, we apply it to regularize a Bayesian high-dimensional regression problem where the regression coefficients are linked by a graph, so that the resulting clusters have flexible shapes and satisfy the cluster contiguity constraint with respect to the graph. We design an efficient Markov chain Monte Carlo algorithm that delivers full Bayesian inference with uncertainty measures for model parameters such as the number of clusters. We offer theoretical investigations of the clustering effects and posterior concentration results. Finally, we illustrate the performance of the model with simulation studies and a real data application for anomaly detection on a road network. The results indicate substantial improvements over other competing methods such as the sparse fused lasso. 

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