Search for: All records

We consider the problem of estimating the structure of an undirected weighted sparse graph underlying a set of signals, exploiting both smoothness of the signals as well as their statistics. We augment the objective function of Kalofolias (2016) which is motivated by a signal smoothness viewpoint and imposes a Laplacian constraint, with a penalized log-likelihood objective function with a lasso constraint, motivated from a statistical viewpoint. Both of these objective functions are designed for estimation of sparse graphs. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the augmented objective function. Numerical results based on synthetic data show that the proposed approach improves upon Kalofolias (2016) in estimating the inverse covariance, and improves upon graphical lasso in estimating the graph topology. We also implement an adaptive version of the proposed algorithm following adaptive lasso of Zou (2006), and empirically show that it leads to further improvement in performance. 

We consider the problem of estimating the structure of an undirected weighted sparse graphical model of multivariate data under the assumption that the underlying distribution is multivariate totally positive of order 2, or equivalently, all partial correlations are non-negative. Total positivity holds in several applications. The problem of Gaussian graphical model learning has been widely studied without the total positivity assumption where the problem can be formulated as estimation of the sparse precision matrix that encodes conditional dependence between random variables associated with the graph nodes. An approach that imposes total positivity is to assume that the precision matrix obeys the Laplacian constraints which include constraining the off-diagonal elements of the precision matrix to be non-positive. In this paper we investigate modifications to widely used penalized log-likelihood approaches to enforce total positivity but not the Laplacian structure. An alternating direction method of multipliers (ADMM) algorithm is presented for constrained optimization under total positivity and lasso as well as adaptive lasso penalties. Numerical results based on synthetic data show that the proposed constrained adaptive lasso approach significantly outperforms existing Laplacian-based approaches, both statistical and smoothness-based non-statistical. 

We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the sparse inverse power spectral density (PSD) of the data via optimization of a sparse-group lasso based penalized log-likelihood cost function that is formulated in the frequency-domain. The CIG is then inferred from the estimated inverse PSD. Optimization in the previous approach was performed using an alternating minimization (AM) approach whose performance depends upon choice of a penalty parameter. In this paper we investigate an alternating direction method of multipliers (ADMM) approach for optimization to mitigate dependence on the penalty parameter. We also investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using synthetic and real data. Comparisons with the "usual" i.i.d. modeling of time series for graph estimation are also provided. 

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 consider a sparse-group smoothly clipped absolute deviation (SG-SCAD) penalty instead of sparse-group lasso (SGL) penalty to regularize the problem. We analyze an SG-SCAD-penalized log-likelihood based objective function to establish consistency of a local estimator of inverse covariance. A numerical example is presented to illustrate the advantage of SG-SCAD-penalty over the usual SGL-penalty. 

We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A p-variate Gaussian time series graphical model associated with an undirected graph with p vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph. A sparse-group lasso-based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the inverse power spectral density (PSD) of the data via optimization of a sparse-group lasso based penalized log-likelihood cost function that is formulated in the frequency-domain. The CIG is then inferred from the estimated inverse PSD. In this paper we establish sufficient conditions for consistency of the inverse PSD estimator resulting from the sparse-group graphical lasso-based approach. 

We consider the problem of estimating the conditional independence graph (CIG) of a sparse, high-dimensional proper complex-valued Gaussian graphical model (CGGM). For CGGMs, the problem reduces to estimation of the inverse covariance matrix with more unknowns than the sample size. We consider a smoothly clipped absolute deviation (SCAD) penalty instead of the ℓ 1 -penalty to regularize the problem, and analyze a SCAD-penalized log-likelihood based objective function to establish consistency and sparsistency of a local estimator of inverse covariance in a neighborhood of the true value. A numerical example is presented to illustrate the advantage of SCAD-penalty over the usual ℓ 1 -penalty.