In 5G and beyond radios, the increased bandwidth, the fast-changing waveform scenarios, and the operation of large array multiple-input multiple-output (MIMO) transmitter architectures have challenged both the polynomial and the artificial neural network (ANN) MIMO adaptive digital predistortion (DPD) schemes. This article proposes training data selection methods and dimensionality reduction techniques that can be combined to enable relevant reductions of the DPD training time and the implementation complexity for MIMO transmitter architectures. In this work, the combination of an efficient uncorrelated equation selection (UES) mechanism together with orthogonal least squares (OLS) is proposed to reduce the training data length and the number of basis functions at every behavioral modeling matrix in the polynomial MIMO DPD scheme. For ANN MIMO DPD architectures, applying UES and principal component analysis (PCA) is proposed to reduce the input dataset length and features, respectively. The UES-OLS and the UES-PCA techniques are experimentally validated for a MIMO test setup with strong power amplifier (PA) input and output crosstalk.