Interference alignment in single-beam MIMO networks via homotopy continuation

In this paper we consider the application of a homotopy-continuation based method for finding interference alignment (IA) solutions for the deterministic K-user multiple-input multiple-output (MIMO) channel, when all users wish to send one stream of data. Homotopy continuation is based on the idea of deforming a start system, whose solution can easily be found, to reach the target system that we want to solve. For the IA problem we show that a good initial system is obtained by considering a rank-one approximation of the original MIMO interference channels. Specifically, as long as the original system is feasible, a rank-one approximation of the MIMO channels allow us to find a closed-form interference-free solution. The proposed algorithm is shown to have a lower complexity than previous methods with comparable sum-rate performance. Furthermore, it is also shown that the trivial system (rank-one MIMO channels) and target system (full-rank MIMO channels) have exactly the same number of solutions. Exploiting this equivalence, an efficient method to enumerate all the IA solutions that exist in a single-beam MIMO network is proposed.