Tensor Graphical Lasso (TeraLasso)

This paper introduces a multi-way tensor generalization of the Bigraphical Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal model for the precision matrix to parsimoniously model conditional dependence relationships of matrix-variate data based on the Cartesian product of graphs. We call this generalization the {\bf Te}nsor g{\bf ra}phical Lasso (TeraLasso). We demonstrate using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway coordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the BiGLasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set, respectively. We propose a scalable composite gradient descent algorithm and analyze the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate the TeraLasso using both simulation and experimental data from a meteorological dataset, showing that we can accurately estimate precision matrices and recover meaningful conditional dependency graphs from high dimensional complex datasets.Comment: accepted to JRSS-

Group Symmetry and non-Gaussian Covariance Estimation

We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their success is an extended form of convexity over the geodesics in the manifold of positive definite matrices. A modern approach to improve estimation accuracy is to exploit prior knowledge via additional constraints, e.g., restricting the attention to specific classes of covariances which adhere to prior symmetry structures. In this paper, we prove that such group symmetry constraints are also geodesically convex and can therefore be incorporated into various non-Gaussian covariance estimators. Practical examples of such sets include: circulant, persymmetric and complex/quaternion proper structures. We provide a simple numerical technique for finding maximum likelihood estimates under such constraints, and demonstrate their performance advantage using synthetic experiments

Sparse Identification and Estimation of Large-Scale Vector AutoRegressive Moving Averages

The Vector AutoRegressive Moving Average (VARMA) model is fundamental to the theory of multivariate time series; however, in practice, identifiability issues have led many authors to abandon VARMA modeling in favor of the simpler Vector AutoRegressive (VAR) model. Such a practice is unfortunate since even very simple VARMA models can have quite complicated VAR representations. We narrow this gap with a new optimization-based approach to VARMA identification that is built upon the principle of parsimony. Among all equivalent data-generating models, we seek the parameterization that is "simplest" in a certain sense. A user-specified strongly convex penalty is used to measure model simplicity, and that same penalty is then used to define an estimator that can be efficiently computed. We show that our estimator converges to a parsimonious element in the set of all equivalent data-generating models, in a double asymptotic regime where the number of component time series is allowed to grow with sample size. Further, we derive non-asymptotic upper bounds on the estimation error of our method relative to our specially identified target. Novel theoretical machinery includes non-asymptotic analysis of infinite-order VAR, elastic net estimation under a singular covariance structure of regressors, and new concentration inequalities for quadratic forms of random variables from Gaussian time series. We illustrate the competitive performance of our methods in simulation and several application domains, including macro-economic forecasting, demand forecasting, and volatility forecasting