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-

Network inference in matrix-variate Gaussian models with non-independent noise

Inferring a graphical model or network from observational data from a large number of variables is a well studied problem in machine learning and computational statistics. In this paper we consider a version of this problem that is relevant to the analysis of multiple phenotypes collected in genetic studies. In such datasets we expect correlations between phenotypes and between individuals. We model observations as a sum of two matrix normal variates such that the joint covariance function is a sum of Kronecker products. This model, which generalizes the Graphical Lasso, assumes observations are correlated due to known genetic relationships and corrupted with non-independent noise. We have developed a computationally efficient EM algorithm to fit this model. On simulated datasets we illustrate substantially improved performance in network reconstruction by allowing for a general noise distribution

antGLasso: An Efficient Tensor Graphical Lasso Algorithm

The class of bigraphical lasso algorithms (and, more broadly, 'tensor'-graphical lasso algorithms) has been used to estimate dependency structures within matrix and tensor data. However, all current methods to do so take prohibitively long on modestly sized datasets. We present a novel tensor-graphical lasso algorithm that analytically estimates the dependency structure, unlike its iterative predecessors. This provides a speedup of multiple orders of magnitude, allowing this class of algorithms to be used on large, real-world datasets.Comment: 9 pages (21 including supplementary material), 8 figures, submitted to the GLFrontiers workshop at NeurIPS 202

Sharper rates of convergence for the tensor graphical Lasso estimator

Many modern datasets exhibit dependencies among observations as well as variables. This gives rise to the challenging problem of analyzing high-dimensional matrix-variate data with unknown dependence structures. To address this challenge, Kalaitzis et. al. (2013) proposed the Bigraphical Lasso (BiGLasso), an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs. Subsequently, Greenewald, Zhou and Hero (GZH 2019) introduced a multiway tensor generalization of the BiGLasso estimator, known as the TeraLasso estimator. In this paper, we provide sharper rates of convergence in the Frobenius and operator norm for both BiGLasso and TeraLasso estimators for estimating inverse covariance matrices. This improves upon the rates presented in GZH 2019. In particular, (a) we strengthen the bounds for the relative errors in the operator and Frobenius norm by a factor of approximately $\log p$; (b) Crucially, this improvement allows for finite-sample estimation errors in both norms to be derived for the two-way Kronecker sum model. The two-way regime is important because it is the setting that is the most theoretically challenging, and simultaneously the most common in applications. Normality is not needed in our proofs; instead, we consider sub-gaussian ensembles and derive tight concentration of measure bounds, using tensor unfolding techniques. The proof techniques may be of independent interest