  In this paper, the Pareto-optimal beam structure for multi-user
multiple-input multiple-output (MIMO) interference channels is investigated and
a necessary condition for any Pareto-optimal transmit signal covariance matrix
is presented for the K-pair Gaussian (N,M_1,...,M_K) interference channel. It
is shown that any Pareto-optimal transmit signal covariance matrix at a
transmitter should have its column space contained in the union of the
eigen-spaces of the channel matrices from the transmitter to all receivers.
Based on this necessary condition, an efficient parameterization for the beam
search space is proposed. The proposed parameterization is given by the product
manifold of a Stiefel manifold and a subset of a hyperplane and enables us to
construct a very efficient beam design algorithm by exploiting its rich
geometrical structure and existing tools for optimization on Stiefel manifolds.
Reduction in the beam search space dimension and computational complexity by
the proposed parameterization and the proposed beam design approach is
significant when the number of transmit antennas is larger than the sum of the
numbers of receive antennas, as in upcoming cellular networks adopting massive
MIMO technologies. Numerical results validate the proposed parameterization and
the proposed cooperative beam design method based on the parameterization for
MIMO interference channels.
