Joint estimation of inverse covariance matrices lying in an unknown subspace

We consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices. We perform the estimation using groups of measurements with different covariances. Assuming the inverse covariances span a low-dimensional subspace, our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation. We develop a novel optimization algorithm discovering and exploiting the underlying low-dimensional subspace. We provide a computationally efficient algorithm and derive a tight upper performance bound. Numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm.