A Feasibility Test for Linear Interference Alignment in MIMO Channels with Constant Coefficients

TL;DR

This paper develops a polynomial-time feasibility test for linear interference alignment in MIMO channels with constant coefficients, using advanced algebraic geometry and differential topology techniques to analyze the solution space.

Contribution

It introduces a novel mathematical framework that generalizes previous results and provides a practical, exact test for IA feasibility in complex MIMO systems.

Findings

Feasibility depends on the algebraic dimension of the solution variety.

The test reduces to checking the rank of a specific matrix.

The feasibility problem is in the BPP complexity class.

Abstract

In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input multiple-output (MIMO) channels with constant coefficients for any number of users, antennas and streams per user; and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. Specifically, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians) and the solution set (channels, decoders and precoders satisfying the IA polynomial equations), not only as algebraic sets but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial (BPP) complexity class.