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In the tensor data analysis, the Kronecker covariance structure plays a vital role in unsupervised learning and regression. Under the Kronecker covariance model assumption, the covariance of an  M -way tensor is parameterized as the Kronecker product of  M  individual covariance matrices. With normally distributed tensors, the key to high-dimensional tensor graphical models becomes the sparse estimation of the  M  inverse covariance matrices. Unable to maximize the tensor normal likelihood analytically, existing approaches often require cyclic updates of the  M  sparse matrices. For the high-dimensional tensor graphical models, each update step solves a regularized inverse covariance estimation problem that is computationally nontrivial. This computational challenge motivates our study of whether a noncyclic approach can be as good as the cyclic algorithms in theory and practice. To handle the potentially very high-dimensional and high-order tensors, we propose a separable and parallel estimation scheme. We show that the new estimator achieves the same minimax optimal convergence rate as the cyclic estimation approaches. Numerically, the new estimator is much faster and often more accurate than the cyclic approach. Moreover, another advantage of the separable estimation scheme is its flexibility in modeling, where we can easily incorporate user-specified or specially structured covariances on any modes of the tensor. We demonstrate the efficiency of the proposed method through both simulations and a neuroimaging application.  Supplementary materials  for this article are available online.

Fast and Separable Estimation in High-Dimensional Tensor Gaussian Graphical Models