  Given a sample of i.i.d. high-dimensional centered random vectors, we
consider a problem of estimation of their covariance matrix $\Sigma$ with an
additional assumption that $\Sigma$ can be represented as a sum of a few
Kronecker products of smaller matrices. Under mild conditions, we derive the
first non-asymptotic dimension-free high-probability bound on the Frobenius
distance between $\Sigma$ and a widely used penalized permuted least squares
estimate. Because of the hidden structure, the established rate of convergence
is faster than in the standard covariance estimation problem.
