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1. On Graphical Modeling of High-Dimensional Long-Range Dependent Time Series

   https://doi.org/10.1109/IEEECONF56349.2022.10052091  

   Tugnait, Jitendra K. (October 2022, 2022 56th Asilomar Conference on Signals, Systems, and Computers)

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary, multivariate long-range dependent (LRD) Gaussian time series. In a time series graph, each component of the vector series is represented by a distinct node, and associations between components are represented by edges between the corresponding nodes. In a recent work on graphical modeling of short-range dependent (SRD) Gaussian time series, the problem was cast as one of multi-attribute graph estimation for random vectors where a vector is associated with each node of the graph. At each node, the associated random vector consists of a time series component and its delayed copies. A theoretical analysis based on short-range dependence has been given in Tugnait (2022 ICASSP). In this paper we analyze this approach for LRD Gaussian time series and provide consistency results regarding convergence in the Frobenius norm of the inverse covariance matrix associated with the multi-attribute graph. 

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2. On Sparse High-Dimensional Graph Estimation from Multi-Attribute Data

   https://doi.org/10.1109/IEEECONF60004.2024.10942781  

   Tugnait, Jitendra K (October 2024, IEEE)

   We consider the problem of inferring the conditional independence graph (CIG) of high-dimensional Gaussian vectors from multi-attribute data. Most existing methods for 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 provide a unified theoretical analysis of multi-attribute graph learning using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the precision matrix to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. 

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3. Delay Embedding for Matrix Graphical Model Learning from Dependent Data

   https://doi.org/10.1109/ICASSP48485.2024.10447097  

   Tugnait, Jitendra K (April 2024, IEEE)

   We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional, stationary matrix-variate Gaussian time series. The correlation function of the matrix series is Kronecker-decomposable. Unlike most past work on matrix graphical models, where independent and identically distributed (i.i.d.) observations of matrix-variate are assumed to be available, we allow time-dependent observations. We follow a time-delay embedding approach where with each matrix node, we associate a random vector consisting of a scalar series component and its time-delayed copies. A group-lasso penalized negative pseudo log-likelihood (NPLL) objective function is formulated to estimate a Kronecker-decomposable covariance matrix which allows for inference of the underlying CIG. The NPLL function is bi-convex and the Kronecker-decomposable covariance matrix is estimated via flip-flop optimization of the NPLL function. Each iteration of flip-flop optimization is solved via an alternating direction method of multipliers (ADMM) approach. Numerical results illustrate the proposed approach which outperforms an existing i.i.d. modeling based approach as well as an existing frequency-domain approach for dependent data, in correctly detecting the graph edges. 

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4. Conditional Independence Graph Estimation from Multi-Attribute Dependent Time Series

   https://doi.org/10.1109/MLSP58920.2024.10734734  

   Tugnait, Jitendra K (September 2024, IEEE)

   Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. All existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We illustrate our approach using numerical examples utilizing both synthetic and real data. 

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5. Multi-Attribute Graph Estimation With Sparse-Group Non-Convex Penalties

   https://doi.org/10.1109/ACCESS.2025.3566959  

   Tugnait, Jitendra K (January 2025, IEEE Access)

   We consider the problem of inferring the conditional independence graph (CIG) of high-dimensional Gaussian vectors from multi-attribute data. Most existing methods for 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 provide a unified theoretical analysis of multi-attribute graph learning using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and sparse-group non-convex (log-sum and smoothly clipped absolute deviation (SCAD) penalties) penalty/regularization functions. An alternating direction method of multipliers (ADMM) approach coupled with local linear approximation to non-convex penalties is presented for optimization of the objective function. For non-convex penalties, theoretical analysis establishing local consistency in support recovery, local convexity and precision matrix estimation in high-dimensional settings is provided under two sets of sufficient conditions: with and without some irrepresentability conditions. We illustrate our approaches using both synthetic and real-data numerical examples. In the synthetic data examples the sparse-group log-sum penalized objective function significantly outperformed the lasso penalized as well as SCAD penalized objective functions with F1 -score and Hamming distance as performance metrics. 

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