  This paper develops an efficient procedure for designing low-complexity
codebooks for precoding in a full-dimension (FD) multiple-input multiple-output
(MIMO) system with a uniform planar array (UPA) antenna at the transmitter (Tx)
using tensor learning. In particular, instead of using statistical channel
models, we utilize a model-free data-driven approach with foundations in
machine learning to generate codebooks that adapt to the surrounding
propagation conditions. We use a tensor representation of the FD-MIMO channel
and exploit its properties to design quantized version of the channel
precoders. We find the best representation of the optimal precoder as a
function of Kronecker Product (KP) of two low-dimensional precoders,
respectively corresponding to the horizontal and vertical dimensions of the
UPA, obtained from the tensor decomposition of the channel. We then quantize
this precoder to design product codebooks such that an average loss in mutual
information due to quantization of channel state information (CSI) is
minimized. The key technical contribution lies in exploiting the constraints on
the precoders to reduce the product codebook design problem to an unsupervised
clustering problem on a Cartesian Product Grassmann manifold (CPM), where the
cluster centroids form a finite-sized precoder codebook. This codebook can be
found efficiently by running a $K$-means clustering on the CPM. With a suitable
induced distance metric on the CPM, we show that the construction of product
codebooks is equivalent to finding the optimal set of centroids on the factor
manifolds corresponding to the horizontal and vertical dimensions. Simulation
results are presented to demonstrate the capability of the proposed design
criterion in learning the codebooks and the attractive performance of the
designed codebooks.
