To beam or not to beam? Beamforming with submodularity-inspired group sparsity

In this paper, we address the beamforming problem, which asks to choose the best subset of antennas and their corresponding amplitudes and phases to match a given beam pattern. To solve this problem, we propose an optimization formulation that can efficiently solve large scale problems, and is versatile in its ability to express a variety of meaningful subset selection scenarios. Focusing on the case of antennas with fixed positions, without assuming any geometric structure, we show how to cast the beamforming problem as a regularized least squares problem. Drawing inspiration from subset selection problems in submodular optimization, we select meaningful submodular set functions and use their Lovàsz extensions as convex regularizers promoting antenna selection in a useful manner, as demonstrated in a number of presented scenarios.