  Passive intelligent reconfigurable surfaces (IRS) are becoming an attractive
component of cellular networks due to their ability of shaping the propagation
environment and thereby improving the coverage. While passive IRS nodes
incorporate a great number of phase-shifting elements and a controller entity,
the phase-shifts are typically determined by the cellular base station (BS) due
to its computational capability. Since the fine granularity control of the
large number of phase-shifters may become prohibitive in practice, it is
important to reduce the control overhead between the BS and the IRS controller.
To this end, in this paper we propose a low-rank approximation of the
near-optimal phase-shifts, which would incur prohibitively high communication
overhead on the BS-IRS controller links. The key idea is to represent the
potentially large IRS phase-shift vector using a low-rank tensor model. This is
achieved by factorizing a tensorized version of the IRS phase-shift vector,
where each component is modeled as the Kronecker product of a predefined number
of factors of smaller sizes, which can be obtained via tensor decomposition
algorithms. We show that the proposed low-rank models drastically reduce the
required feedback requirements associated with the BS-IRS control links. Our
simulation results indicate that the proposed method is especially attractive
in scenarios with a strong line of sight component, in which case nearly the
same spectral efficiency is reached as in the cases with near-optimal
phase-shifts, but with a drastically reduced communication overhead.
