Dimension-free Bounds for Covariance Estimation with Tensor-Train Structure
Abstract
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 . To be more precise, we study the case when 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.
Cite
@article{arxiv.2510.08174,
title = {Dimension-free Bounds for Covariance Estimation with Tensor-Train Structure},
author = {Artsiom Patarusau and Nikita Puchkin and Maxim Rakhuba and Fedor Noskov},
journal= {arXiv preprint arXiv:2510.08174},
year = {2026}
}