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.

翻译：构建复杂数据的模型和方法是许多科学及应用领域的重要任务。现代数据集中常同时存在观测值之间和变量之间的依赖关系，这引出了对未知依赖结构的高维矩阵变量数据进行分析的挑战性问题。为应对这一挑战，Kalaitzis等人(2013)提出了基于图的笛卡尔积的矩阵正态精度矩阵估计量——双图Lasso(BiGLasso)。随后，Greenewald、Zhou和Hero(GZH 2019)将BiGLasso估计量推广为多路张量形式，即TeraLasso估计量。本文给出了BiGLasso和TeraLasso两种估计量在Frobenius范数和算子范数下估计逆协方差矩阵的锐利收敛速率，改进了GZH 2019中的速率结果。具体而言：(a)我们将算子范数和Frobenius范数下相对误差的界提升了约$\log p$倍；(b)关键的是，这一改进使得两种范数下的有限样本估计误差均可推导出针对二路Kronecker和模型的结果，从而弥合了GZH 2019中经验观察到的二路模型低单样本误差与理论界之间的差距。二路模型特别重要，因为它是实际常见且通用的应用场景。本证明无需正态性假设，转而考虑次高斯系综并利用张量展开技术推导紧的测度集中界。该证明方法对张量值数据的分析可能具有独立参考价值。