  The WMMSE beamforming algorithm is a popular approach to address the NP-hard
weighted sum rate (WSR) maximization beamforming problem. Although it
efficiently finds a local optimum, it requires matrix inverses,
eigendecompositions, and bisection searches, operations that are problematic
for real-time implementation.
  In our previous work, we considered the MU-MISO case and effectively replaced
such operations by resorting to a first-order method. Here, we consider the
more general and challenging MU-MIMO case. Our earlier approach does not
generalize to this scenario and cannot be applied to replace all the
hard-to-parallelize operations that appear in the MU-MIMO case. Thus, we
propose to leverage a reformulation of the auxiliary WMMSE function given by Hu
et al. By applying gradient descent and Schulz iterations, we formulate the
first variant of the WMMSE algorithm applicable to the MU-MIMO case that is
free from matrix inverses and other serial operations and hence amenable to
both real-time implementation and deep unfolding. From a theoretical viewpoint,
we establish its convergence to a stationary point of the WSR maximization
problem. From a practical viewpoint, we show that in a deep-unfolding-based
implementation, the matrix-inverse-free WMMSE algorithm attains, within a fixed
number of iterations, a WSR comparable to the original WMMSE algorithm
truncated to the same number of iterations, yet with significant implementation
advantages in terms of parallelizability and real-time execution.
