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.

翻译：我们考虑从独立同分布的高维随机向量样本中估计协方差的问题。为避免维度灾难，我们对协方差矩阵$\Sigma$的结构施加额外假设。具体而言，我们研究$\Sigma$可通过张量列车（TT）格式下较小矩阵的双重Kronecker积之和来近似的情况。我们的设定自然扩展了广泛使用的Kronecker和与CANDECOMP/PARAFAC模型，但允许模态间更丰富的交互作用。我们提出一种基于TT-SVD和适用于Tucker-2混合结构的高阶正交迭代（HOOI）的迭代多项式时间算法。通过考虑隐藏的Kronecker积和张量列车结构，我们推导出协方差估计精度的非渐近无维度界限。数值实验证明了我们方法的有效性。