  We show that the maximization of the sum degrees-of-freedom for the static
flat-fading multiple-input multiple-output (MIMO) interference channel is
equivalent to a rank constrained rank minimization problem (RCRM), when the
signal spaces 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 spaces spanning all available spatial dimensions. That way, we
reformulate all 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 introduce tractable constraints that are
asymptotically equivalent to the rank constraints for the initial problem. We
also show that our heuristic relaxation can be tuned for the multi-cell
interference channel. Furthermore, 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 zero-forcing matrices
for interference alignment.
