Due to the closely-spaced antenna elements in largearray or massive MIMO transmitters, antenna crosstalk is inevitable. This imposes additional challenges when seeking to linearize the power amplifiers at the transmitter through digital predistortion (DPD). In the commonly applied indirect learning architecture (ILA), the antenna crosstalk is known to result in a large amount of additional basis functions (BFs) in order to account for all the coupling signal terms and achieve good linearization. In this article, we propose a novel closed-loop DPD architecture and associated parameter learning algorithms that can provide efficient linearization of digital MIMO transmitters under antenna crosstalk. The proposed solution does not need extra basis functions, and is thus shown to provide large benefits in terms of computational complexity compared to existing state-of-the-art. Comprehensive numerical results are also provided, showing excellent linearization performance outperforming the existing reference methods.