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Addressing the disruptive capacity requirements of 5G networks calls for a thorough exploration of multiple technological solutions. Two promising approaches are i) the use of multiple-input multiple-output (MIMO) technologies that enable multiplicative capacity gains, and ii) the exploration of new frequency bands to enlarge available bandwidth. While millimeter spectrum, one of the main bands under study, poses significant challenges due to its cumbersome propagation characteristics (particularly severe path loss and channel sparsity), its 10-fold frequency increase favors the deployment of reduced-size large antenna arrays for massive MIMO. However, the high cost and power consumption of its required signal mixers and analog-to-digital converters precludes mmWave beamforming from being performed entirely at baseband using digital precoders. A possible cost-effective alternative is the hybrid precoding transceiver architecture, which combines digital and analog precoders. In this article, we exploit the sparse nature of the channel to unveil an advantage in the design of the hybrid precoder. Specifically, by reformulating the hybrid precoder design as a matrix factorization problem and adopting an atomic norm minimization approach, we propose a new hybrid precoding algorithm that takes advantage of the sparse nature of the mmWave channel to approach the performance of the optimal fully digital precoder. Simulation results confirm that the proposed algorithm can approach the performance achieved by unconstrained digital beamforming solutions.