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1. Graph Learning From Multivariate Dependent Time Series Via A Multi-Attribute Formulation

   https://doi.org/10.1109/ICASSP43922.2022.9746603  

   Tugnait, Jitendra K. (May 2022, ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. In a time series graph, each component of the vector series is represented by distinct node, and associations between components are represented by edges between the corresponding nodes. We formulate the problem 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. We present an alternating direction method of multipliers (ADMM) solution to minimize a sparse-group lasso penalized negative pseudo log-likelihood objective function to estimate the precision matrix of the random vector associated with the entire multi-attribute graph. The time series CIG is then inferred from the estimated precision matrix. A theoretical analysis is provided. Numerical results illustrate the proposed approach which outperforms existing frequency-domain approaches in correctly detecting the graph edges. 

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2. 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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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. Provable Subspace Identification Under Post-Nonlinear Mixtures

   Lyu, Qi; Fu, Xiao (December 2022, Advances in neural information processing systems)

   Unsupervised mixture learning (UML) aims at identifying linearly or nonlinearly mixed latent components in a blind manner. UML is known to be challenging: Even learning linear mixtures requires highly nontrivial analytical tools, e.g., independent component analysis or nonnegative matrix factorization. In this work, the post-nonlinear (PNL) mixture model---where {\it unknown} element-wise nonlinear functions are imposed onto a linear mixture---is revisited. The PNL model is widely employed in different fields ranging from brain signal classification, speech separation, remote sensing, to causal discovery. To identify and remove the unknown nonlinear functions, existing works often assume different properties on the latent components (e.g., statistical independence or probability-simplex structures). This work shows that under a carefully designed UML criterion, the existence of a nontrivial {\it null space} associated with the underlying mixing system suffices to guarantee identification/removal of the unknown nonlinearity. Compared to prior works, our finding largely relaxes the conditions of attaining PNL identifiability, and thus may benefit applications where no strong structural information on the latent components is known. A finite-sample analysis is offered to characterize the performance of the proposed approach under realistic settings. To implement the proposed learning criterion, a block coordinate descent algorithm is proposed. A series of numerical experiments corroborate our theoretical claims. 

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5. Joint inference of multiple graphs from matrix polynomials

   Novarro, M.; Wang, Y.; Marques, A.G.; Uhler, C.; Segarra, S. (January 2022, Journal of machine learning research)

   Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. Graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime. 

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