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1. Estimation of High-Dimensional Differential Graphs from Multi-Attribute Data

   https://doi.org/10.1109/ICASSP49357.2023.10095585  

   Tugnait, Jitendra K. ( June 2023 , ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   We consider the problem of estimating differences in two Gaussian graphical models (GGMs) which are known to have similar structure. The GGM structure is encoded in its precision (inverse covariance) matrix. In many applications one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data. Existing methods for differential graph estimation are based on single-attribute models where one associates a scalar random variable with each node. In multi-attribute graphical models, each node represents a random vector. In this paper, we analyze a group lasso penalized D-trace loss function approach for differential graph learning from multi-attribute data. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. Theoretical analysis establishing consistency in support recovery and estimation in high-dimensional settings is provided. We illustrate our approach using a numerical example where the multi-attribute approach is shown to outperform a single-attribute approach. 

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2. Estimation of Differential Graphs via Log-Sum Penalized D-Trace Loss

   https://doi.org/10.1109/SSP53291.2023.10208014  

   Tugnait, Jitendra K. ( July 2023 , 2023 IEEE Statistical Signal Processing Workshop (SSP))

   We consider the problem of estimating differences in two Gaussian graphical models (GGMs) which are known to have similar structure. The GGM structure is encoded in its precision (inverse covariance) matrix. In many applications one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data. Most existing methods for differential graph estimation are based on a lasso penalized loss function. In this paper, we analyze a log-sum penalized D-trace loss function approach for differential graph learning. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. Theoretical analysis establishing consistency in estimation in high-dimensional settings is provided. We illustrate our approach using a numerical example where log-sum penalized D-trace loss significantly outperforms lasso-penalized D-trace loss as well as smoothly clipped absolute deviation (SCAD) penalized D-trace loss. 

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3. Learning High-Dimensional Differential Graphs From Multi-Attribute Data

   https://doi.org/10.1109/TSP.2023.3343553  

   Tugnait, Jitendra K. ( December 2023 , IEEE Transactions on Signal Processing)

   We consider the problem of estimating differences in two Gaussian graphical models (GGMs) which are known to have similar structure. The GGM structure is encoded in its precision (inverse covariance) matrix. In many applications one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data. Existing methods for differential graph estimation are based on single-attribute (SA) models where one associates a scalar random variable with each node. In multi-attribute (MA) graphical models, each node represents a random vector. In this paper, we analyze a group lasso penalized D-trace loss function approach for differential graph learning from multi-attribute data. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. Theoretical analysis establishing consistency in support recovery and estimation in high-dimensional settings is provided. Numerical results based on synthetic as well as real data are presented. 

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4. A Multi-Task Learning Formulation for Survival Analysis

   https://doi.org/10.1145/2939672.2939857  

   Li, Yan ; Wang, Jie ; Ye, Jieping ; Reddy, Chandan K. ( January 2016 , Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining)

   Predicting the occurrence of a particular event of interest at future time points is the primary goal of survival analysis. The presence of incomplete observations due to time limitations or loss of data traces is known as censoring which brings unique challenges in this domain and differentiates survival analysis from other standard regression methods. The popularly used survival analysis methods such as Cox proportional hazard model and parametric survival regression suffer from some strict assumptions and hypotheses that are not realistic in most of the real-world applications. To overcome the weaknesses of these two types of methods, in this paper, we reformulate the survival analysis problem as a multi-task learning problem and propose a new multi-task learning based formulation to predict the survival time by estimating the survival status at each time interval during the study duration. We propose an indicator matrix to enable the multi-task learning algorithm to handle censored instances and incorporate some of the important characteristics of survival problems such as non-negative non-increasing list structure into our model through max-heap projection. We employ the L2,1-norm penalty which enables the model to learn a shared representation across related tasks and hence select important features and alleviate over-fitting in high-dimensional feature spaces; thus, reducing the prediction error of each task. To efficiently handle the two non-smooth constraints, in this paper, we propose an optimization method which employs Alternating Direction Method of Multipliers (ADMM) algorithm to solve the proposed multi-task learning problem. We demonstrate the performance of the proposed method using real-world microarray gene expression high-dimensional benchmark datasets and show that our method outperforms state-of-the-art methods. 

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5. 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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