标题： 基于张量-列车分解的结构化协方差估计 显示英文标题

标题： Structured covariance estimation via tensor-train decomposition

摘要： 我们考虑从独立同分布的高维随机向量样本中进行协方差估计的问题。 为了避免维度灾难，我们对协方差矩阵$\Sigma$的结构施加了额外的假设。 更准确地说，我们研究当$\Sigma$可以通过张量列车（TT）格式中较小矩阵的双Kronecker积之和来近似的情况。 我们的设置自然扩展了广为人知的Kronecker和和CANDECOMP/PARAFAC模型，但允许跨模式的更丰富的交互。 我们提出了一种基于TT-SVD和高阶正交迭代（HOOI）的迭代多项式时间算法，该算法适应Tucker-2混合结构。 我们推导了在考虑隐藏的Kronecker乘积和张量列车结构的情况下，协方差估计精度的非渐近无维度界限。 我们的方法的效率通过数值实验得到了说明。 显示英文摘要

摘要： We consider a problem of covariance estimation from a sample of i.i.d. high-dimensional random vectors. To avoid the curse of dimensionality we impose an additional assumption on the structure of the covariance matrix $\Sigma$. To be more precise we study the case when $\Sigma$ can be approximated by a sum of double Kronecker products of smaller matrices in a tensor train (TT) format. Our setup naturally extends widely known Kronecker sum and CANDECOMP/PARAFAC models but admits richer interaction across modes. We suggest an iterative polynomial time algorithm based on TT-SVD and higher-order orthogonal iteration (HOOI) adapted to Tucker-2 hybrid structure. We derive non-asymptotic dimension-free bounds on the accuracy of covariance estimation taking into account hidden Kronecker product and tensor train structures. The efficiency of our approach is illustrated with numerical experiments.