  This paper investigates a communication-efficient split learning (SL) over
multiple-input multiple-output (MIMO) communication system. In particular, we
mathematically decompose the inter-layer connection of a neural network (NN) to
a series of linear precoding and combining transformations using over-the-air
computation (OAC), which synergistically form a linear layer in NNs. The
precoding and combining matrices are trainable parameters in such a system,
whereas the MIMO channel is implicit. The proposed system eliminates the
implicit channel estimation through exploiting the channel reciprocity and
properly casting the backpropagation process, significantly saving the system
costs and further improving the overall efficiency. The practical constellation
diagrams are used as the activation function to avoid sending arbitrary analog
signals as in the traditional OAC system. Numerical results are illustrated to
demonstrate the effectiveness of the proposed scheme.
