  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.
