标题： 张量图拉索（TeraLasso） 显示英文标题

标题： Tensor Graphical Lasso (TeraLasso)

摘要： 本文介绍了一种多路张量广义的Bigraphical Lasso（BiGLasso），它使用了两路稀疏Kronecker和多元正态模型来简洁地建模基于图笛卡尔积的矩阵数据的条件依赖关系。我们称这种广义为{\bf 泰}张量广{\bf ra}图Lasso（TeraLasso）。通过理论和实例演示，我们证明了TeraLasso模型可以从具有空间、时间和复制等多维坐标的高维变量的有限数据样本中准确且可扩展地估计。分别建立了BiGLasso和TeraLasso精度矩阵估计器及其支持（非稀疏性）集估计器的统计一致性及收敛速度。我们提出了一个可扩展的复合梯度下降算法，并分析了计算收敛速度，表明复合梯度下降算法保证以几何速率收敛到TeraLasso目标函数的全局最小值。最后，我们利用气象数据集的模拟和实验数据展示了TeraLasso的应用，表明我们可以从高维复杂的数据集中准确估计精度矩阵并恢复有意义的条件依赖图。 显示英文摘要

摘要： This paper introduces a multi-way tensor generalization of the Bigraphical Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal model for the precision matrix to parsimoniously model conditional dependence relationships of matrix-variate data based on the Cartesian product of graphs. We call this generalization the {\bf Te}nsor g{\bf ra}phical Lasso (TeraLasso). We demonstrate using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway coordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the BiGLasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set, respectively. We propose a scalable composite gradient descent algorithm and analyze the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate the TeraLasso using both simulation and experimental data from a meteorological dataset, showing that we can accurately estimate precision matrices and recover meaningful conditional dependency graphs from high dimensional complex datasets.