Interference Alignment as a Rank Constrained Rank Minimization

TL;DR

This paper formulates interference alignment in MIMO channels as a rank minimization problem and proposes a convex relaxation approach that effectively achieves interference alignment, outperforming previous methods in many cases.

Contribution

It introduces a novel rank constrained rank minimization formulation for interference alignment and develops a convex relaxation method inspired by compressed sensing and matrix completion theories.

Findings

The proposed algorithm often achieves perfect interference alignment.

The convex relaxation outperforms previous approaches in several scenarios.

The method can be tuned for multi-cell interference channels.

Abstract

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.