  Increasing data rate in wireless networks can be accomplished through a
two-pronged approach, which are 1) increasing the network flow rate through
parallel independent routes and 2) increasing the user's link rate through
beamforming codebook adaptation. Mobile relays are utilized to enable achieving
these goals given their flexible positioning. First at the network level, we
model regularized Laplacian matrices, which are symmetric positive definite
(SPD) ones representing relay-dependent network graphs, as points over
Riemannian manifolds. Inspired by the geometric classification of different
tasks in the brain network, Riemannian metrics, such as Log-Euclidean metric
(LEM), are utilized to choose relay positions that result in maximum LEM.
Simulation results show that the proposed LEM-based relay positioning algorithm
enables parallel routes and achieves maximum network flow rate, as opposed to
other metrics (e.g., algebraic connectivity).
  Second at the link level, we design unique relay-dependent beamforming
codebooks aimed to increase data rate over the spatially-correlated fading
channels between a given relay and its neighboring users. To do so, we propose
a geometric machine learning approach, which utilizes support vector machine
(SVM) model to learn an SPD variant of the user's channel over Riemannian
manifolds. Consequently, LEM-based Riemannian metric is utilized for
classification of different channels, and a matched beamforming codebook is
constructed accordingly. Simulation results show that the proposed
geometric-based learning model achieves the maximum link rate after a short
training period.
