Given a sample of i.i.d. high-dimensional centered random vectors, we consider a problem of estimation of their covariance matrix $Σ$ with an additional assumption that $Σ$ 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 $Σ$ 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.