Abstract

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.

摘要

在张量数据分析中，克罗内克协方差结构在无监督学习和回归中起着至关重要的作用。在 Kronecker 协方差模型假设下，M路张量的协方差被参数化为M个独立协方差矩阵的 Kronecker乘积。对于正态分布的张量，高维张量图模型的关键变成了对M个逆协方差矩阵的稀疏估计。无法通过分析最大化张量正态似然，现有方法通常需要对M进行循环更新稀疏矩阵。对于高维张量图形模型，每个更新步骤都解决了计算上不重要的正则化逆协方差估计问题。这一计算挑战促使我们研究非循环方法在理论和实践中是否可以与循环算法一样好。为了处理潜在的非常高维和高阶张量，我们提出了一种可分离的并行估计方案。我们表明，新估计器实现了与循环估计方法相同的极小极大最优收敛速度。在数值上，新的估计器比循环方法更快且通常更准确。此外，可分离估计方案的另一个优点是它在建模方面的灵活性，我们可以轻松地将用户指定的或特殊结构的协方差合并到张量的任何模式上。我们通过模拟和神经成像应用证明了所提出方法的效率。本文的补充材料可在线获取。