  Downlink beamforming is a key technology for cellular networks. However,
computing the transmit beamformer that maximizes the weighted sum rate subject
to a power constraint is an NP-hard problem. As a result, iterative algorithms
that converge to a local optimum are used in practice. Among them, the weighted
minimum mean square error (WMMSE) algorithm has gained popularity, but its
computational complexity and consequent latency has motivated the need for
lower-complexity approximations at the expense of performance. Motivated by the
recent success of deep unfolding in the trade-off between complexity and
performance, we propose the novel application of deep unfolding to the WMMSE
algorithm for a MISO downlink channel. The main idea consists of mapping a
fixed number of iterations of the WMMSE algorithm into trainable neural network
layers, whose architecture reflects the structure of the original algorithm.
With respect to traditional end-to-end learning, deep unfolding naturally
incorporates expert knowledge, with the benefits of immediate and well-grounded
architecture selection, fewer trainable parameters, and better explainability.
However, the formulation of the WMMSE algorithm, as described in Shi et al., is
not amenable to be unfolded due to a matrix inversion, an eigendecomposition,
and a bisection search performed at each iteration. Therefore, we present an
alternative formulation that circumvents these operations by resorting to
projected gradient descent. By means of simulations, we show that, in most of
the settings, the unfolded WMMSE outperforms or performs equally to the WMMSE
for a fixed number of iterations, with the advantage of a lower computational
load.
