Sharp rates of convergence for the tensor graphical Lasso estimator

Building models and methods for complex data is an important task for many scientific and application areas. 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 sharp 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. This closes the gap between the low single-sample error for the two-way model empirically observed in GZH 2019 and the theoretical bounds therein. The two-way regime is particularly significant since it is the setting of common and generic applications in practice. Normality is not needed in our proofs; instead, we consider subgaussian ensembles and derive tight concentration of measure bounds, using tensor unfolding techniques. The proof techniques may be of independent interest to the analysis of tensor-valued data.