  Millimeter wave (mmWave) cell-free massive MIMO (CF mMIMO) is a promising
solution for future wireless communications. However, its optimization is
non-trivial due to the challenging channel characteristics. We show that mmWave
CF mMIMO optimization is largely an assignment problem between access points
(APs) and users due to the high path loss of mmWave channels, the limited
output power of the amplifier, and the almost orthogonal channels between users
given a large number of AP antennas. The combinatorial nature of the assignment
problem, the requirement for scalability, and the distributed implementation of
CF mMIMO make this problem difficult. In this work, we propose an unsupervised
machine learning (ML) enabled solution. In particular, a graph neural network
(GNN) customized for scalability and distributed implementation is introduced.
Moreover, the customized GNN architecture is hierarchically
permutation-equivariant (HPE), i.e., if the APs or users of an AP are permuted,
the output assignment is automatically permuted in the same way. To address the
combinatorial problem, we relax it to a continuous problem, and introduce an
information entropy-inspired penalty term. The training objective is then
formulated using the augmented Lagrangian method (ALM). The test results show
that the realized sum-rate outperforms that of the generalized serial
dictatorship (GSD) algorithm and is very close to the upper bound in a small
network scenario, while the upper bound is impossible to obtain in a large
network scenario.
