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1. 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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2. Simultaneous Change Point Inference and Structure Recovery for High Dimensional Gaussian Graphical Models

   Liu, B. ; Zhang, X. ; Liu, Y. ( October 2021 , Journal of machine learning research)

   In this article, we investigate the problem of simultaneous change point inference and structure recovery in the context of high dimensional Gaussian graphical models with possible abrupt changes. In particular, motivated by neighborhood selection, we incorporate a threshold variable and an unknown threshold parameter into a joint sparse regression model which combines p l1-regularized node-wise regression problems together. The change point estimator and the corresponding estimated coefficients of precision matrices are obtained together. Based on that, a classifier is introduced to distinguish whether a change point exists. To recover the graphical structure correctly, a data-driven thresholding procedure is proposed. In theory, under some sparsity conditions and regularity assumptions, our method can correctly choose a homogeneous or heterogeneous model with high accuracy. Furthermore, in the latter case with a change point, we establish estimation consistency of the change point estimator, by allowing the number of nodes being much larger than the sample size. Moreover, it is shown that, in terms of structure recovery of Gaussian graphical models, the proposed thresholding procedure achieves model selection consistency and controls the number of false positives. The validity of our proposed method is justified via extensive numerical studies. Finally, we apply our proposed method to the S&P 500 dataset to show its empirical usefulness. 

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3. Heterogeneous Graphical Model for Non-Negative and Non-Gaussian PM2.5 data

   https://doi.org/10.1111/rssc.12575  

   Zhang, Jiaqi ; Fan, Xinyan ; Li, Yang ; Ma, Shuangge ( June 2022 , Journal of the Royal Statistical Society Series C: Applied Statistics)

   Abstract Studies on the conditional relationships between PM2.5 concentrations among different regions are of great interest for the joint prevention and control of air pollution. Because of seasonal changes in atmospheric conditions, spatial patterns of PM2.5 may differ throughout the year. Additionally, concentration data are both non-negative and non-Gaussian. These data features pose significant challenges to existing methods. This study proposes a heterogeneous graphical model for non-negative and non-Gaussian data via the score matching loss. The proposed method simultaneously clusters multiple datasets and estimates a graph for variables with complex properties in each cluster. Furthermore, our model involves a network that indicate similarity among datasets, and this network can have additional applications. In simulation studies, the proposed method outperforms competing alternatives in both clustering and edge identification. We also analyse the PM2.5 concentrations' spatial correlations in Taiwan's regions using data obtained in year 2019 from 67 air-quality monitoring stations. The 12 months are clustered into four groups: January–March, April, May–September and October–December, and the corresponding graphs have 153, 57, 86 and 167 edges respectively. The results show obvious seasonality, which is consistent with the meteorological literature. Geographically, the PM2.5 concentrations of north and south Taiwan regions correlate more respectively. These results can provide valuable information for developing joint air-quality control strategies. 

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4. Thresholded graphical lasso adjusts for latent variables

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

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

   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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5. 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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