Interference Alignment as a Rank Constrained Rank Minimization

We show that the maximization of the sum degrees-of-freedom for the static flat-fading multiple-input multiple-output (MIMO) interference channel (IC) is equivalent to a rank constrained rank minimization problem (RCRM), when the signal subspaces span all available dimensions. The rank minimization corresponds to maximizing interference alignment (IA) so that interference spans the lowest dimensional subspace possible. The rank constraints account for the useful signal subspaces spanning all available spatial dimensions. That way, we reformulate the IA requirements to requirements involving ranks. Then, we present a convex relaxation of the RCRM problem inspired by recent results in compressed sensing and low-rank matrix completion theory that rely on approximating rank with the nuclear norm. We show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and replace the rank constraints with asymptotically equivalent and tractable ones. We then tune our heuristic relaxation for the multicell interference channel. We experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and receive matrices for interference alignment.