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：为复杂数据构建模型与方法是众多科学与应用领域的重要任务。现代数据集通常呈现观测值与变量间的双重依赖关系，这使得分析具有未知依赖结构的高维矩阵型数据成为一项挑战性课题。针对此问题，Kalaitzis等人(2013)基于图的笛卡尔积提出了双图Lasso(BiGLasso)估计量——一种用于矩阵正态分布精度矩阵的估计方法。随后，Greenewald、Zhou与Hero(简称GZH 2019)将BiGLasso估计量推广为多路张量形式，称为TeraLasso估计量。本文给出了BiGLasso与TeraLasso两类估计量在Frobenius范数和算子范数下估计逆协方差矩阵的最优收敛速度，改进了GZH 2019中的速率结果。具体而言：(a) 我们将算子范数与Frobenius范数相对误差的界提高了约$\log p$倍；(b) 更重要的是，这一改进使得两种范数下双路Kronecker和模型的有限样本估计误差得以推导，弥合了GZH 2019中实证观察到的双路模型低单样本误差与理论界之间的差距。双路模型具有特殊重要性，因为它是实际应用中常见且通用的场景。我们的证明无需正态性假设，而是考虑次高斯分布族，并利用张量展开技术推导出紧的测度集中界。这些证明技术对张量型数据分析可能具有独立的理论价值。