Designing interference alignment algorithms by algebraic geometry analysis

Recent results have shown that interference alignment (IA) achieves the optimal throughput scaling law w.r.t. SNR. However, except for a few special cases, finding transceivers to achieve IA in MIMO interference networks is still an open problem. This problem is challenging due to the non-convex nature of the interference optimization problem. Inspired by recent success in using tools from algebraic geometry to analyze the feasibility conditions of IA, in this work, we adopt algebraic geometry analysis to guide IA algorithm design. Specifically, we first explore the relation between algebraic independence and feasibility of polynomial equation sets, and transform the IA problem into an equivalent polynomial form, whose policy space is convex. Then we reformulate the transformed IA problem into an interference optimization problem. By exploiting the connection between algebraic independence and full rankness of Jacobian matrix, we prove that in the interference optimization problem, there is no performance gap between local and global optimums when IA is feasible. This property enables us to easily design IA algorithms by adopting existing local search algorithms. Combining the propositions obtained in this work and those obtained in the authors' prior work on IA feasibility, we have established a unified algebraic framework for both IA feasibility analysis and algorithm design.