Homotopy continuation for vector space interference alignment in MIMO X networks

In this paper we propose an algorithm to design interference alignment (IA) precoding and decoding matrices for MIMO X networks (XN). The proposed algorithm is rooted in the homotopy continuation techniques commonly used to solve systems of nonlinear equations. Homotopy methods find the solution of a target system by smoothly deforming the known solutions of a start system which can be trivially solved. The key observation leading to a simple start system is realizing that the inverse IA problem, i.e, finding the channels that satisfy the IA conditions given a set of precoders and decoders, is linear and, therefore, a convenient trivial system. Once the start system has been solved, standard prediction and correction techniques are applied to track the solution all the way to the target system. Our results show that the proposed algorithm is able to consistently find solutions achieving the maximum number of degrees of freedom (DoF) whereas alternating minimization techniques, which typically work well for the interference channel (IC), repeatedly fail for the XN. Further, the algorithm provides insights into the feasibility of alignment in MIMO X networks for which theoretical results are scarce.