In this paper, we first derive the channel capacity of the full-duplex spatial modulation (FD-SM) system and its upper and lower bounds. Furthermore, different from the traditional optimization-driven decision, we use the data-driven prediction method to solve the transmit antenna selection (TAS) problem in the FD-SM system. Specifically, two novel TAS methods based on the support vector machine (SVM) and deep neural network (DNN) are proposed for reducing the effect of residual self-interference (RSI) on the FD-SM system performance. In our design, we propose a novel feature extraction method based on the principal component analysis (PCA) to help the proposed classifiers improve training. Our simulation results show that our data-driven TAS schemes can approach the optimal performance achieved by exhaustive search while significantly reducing complexity.