Replication-based Outer bounds and the Optimality of "Half the Cake" for Rank-Deficient MIMO Interference Networks

In order to gain new insights into MIMO interference networks, the optimality of $\sum_{k=1}^K M_k/2$ (half the cake per user) degrees of freedom is explored for a $K$-user multiple-input-multiple-output (MIMO) interference channel where the cross-channels have arbitrary rank constraints, and the $k^{th}$ transmitter and receiver are equipped with $M_k$ antennas each. The result consolidates and significantly generalizes results from prior studies by Krishnamurthy et al., of rank-deficient interference channels where all users have $M$ antennas, and by Tang et al., of full rank interference channels where the $k^{th}$ user pair has $M_k$ antennas. The broader outcome of this work is a novel class of replication-based outer bounds for arbitrary rank-constrained MIMO interference networks where replicas of existing users are added as auxiliary users and the network connectivity is chosen to ensure that any achievable scheme for the original network also works in the new network. The replicated network creates a new perspective of the problem, so that even simple arguments such as user cooperation become quite powerful when applied in the replicated network, giving rise to stronger outer bounds, than when applied directly in the original network. Remarkably, the replication based bounds are broadly applicable not only to MIMO interference channels with arbitrary rank-constraints, but much more broadly, even beyond Gaussian settings.