Estimation of rank deficient covariance matrices with Kronecker structure

Given a set of observations, the estimation of covariance matrices is required in the analysis of many applications. To this end, any know structure of the covariance matrix can be taken into account. For instance, in case of separable processes, the covariance matrix is given by the Kronecker product of two factor matrices. Assuming the covariance matrix is full rank, the maximum likelihood (ML) estimate in this case leads to an iterative algorithm known as the flip-flop algorithm in the literature. In this work, we first generalize the flip-flop algorithm to the case when the covariance matrix is rank deficient, which happens to be the case in several situations. In addition, we propose a non-iterative estimation approach which incurs in a performance loss compared to the ML estimate, but at the expense of less complexity.