In this paper, we study digital predistortion (DPD) based linearization with specific focus on millimeter wave (mmW) active antenna arrays. Due to the very large channel bandwidths and beam-dependence of nonlinear distortion in such systems, we propose a closed-loop DPD learning architecture, look-up table (LUT) based memory DPD models, and low-complexity sign-based estimation algorithms, such that even continuous DPD learning could be technically feasible. To this end, three different learning algorithms - sign, signed regressor, and sign-sign - are formulated for the LUT-based DPD models, such that the potential rank deficiencies, experienced in earlier methods, are avoided. Then, extensive RF measurements utilizing a state-of-the-art mmW active antenna array system at 28 GHz are carried out and reported to validate the methods. Additionally, the processing and learning complexities of the considered methods are analyzed, which together with the measured linearization performance figures allow to assess the complexity-performance tradeoffs. Overall, the results show that efficient mmW array linearization can be obtained through the proposed methods.