  We consider multi-antenna wireless systems aided by large intelligent
surfaces (LIS). LIS presents a new physical layer technology for improving
coverage and energy efficiency by intelligently controlling the propagation
environment. In practice however, achieving the anticipated gains of LIS
requires accurate channel estimation. Recent attempts to solve this problem
have considered the least-squares (LS) approach, which is simple but also
sub-optimal. The optimal channel estimator, based on the minimum
mean-squared-error (MMSE) criterion, is challenging to obtain and is non-linear
due to the non-Gaussianity of the effective channel seen at the receiver. Here
we present approaches to approximate the optimal MMSE channel estimator. As a
first approach, we analytically develop the best linear estimator, the LMMSE,
together with a corresponding majorization-minimization based algorithm
designed to optimize the LIS phase shift matrix during the training phase. This
estimator is shown to yield improved accuracy over the LS approach by
exploiting second-order statistical properties of the wireless channel and the
noise. To further improve performance and better approximate the
globally-optimal MMSE channel estimator, we propose data-driven non-linear
solutions based on deep learning. Specifically, by posing the MMSE channel
estimation problem as an image denoising problem, we propose two convolutional
neural network (CNN) based methods to perform the denoising and approximate the
optimal MMSE channel estimation solution. Our numerical results show that these
CNN-based estimators give superior performance compared with linear estimation
approaches. They also have low computational complexity requirements, thereby
motivating their potential use in future LIS-aided wireless communication
systems.
