Variational Learning Algorithms for Channel Estimation in RIS-Assisted mmWave Systems

We consider the problem of estimating channel in reconfigurable intelligent surface (RIS) assisted millimeter wave (mmWave) systems. We propose two variational expectation maximization (VEM) based channel estimation algorithms, which exploit the angular domain sparsity of RIS-assisted mmWave channel. To fully capture this sparsity, both within and across UEs, we construct a novel column-wise coupled Gaussian prior. The first proposed structured-mean-field-based VEM (SMF-VEM) algorithm uses the proposed prior, and calculates the posterior distribution of the unknown channel by assuming that it belongs to a set of multivariate distributions. This algorithm inverts a high-dimensional matrix in its posterior update, and consequently does not scale well for a large number of RIS elements and base station antennas, which are commonly used in practical systems. The second proposed fast mean field-based VEM (FMF-VEM) algorithm reduces complexity by assuming a fully-factorized posterior. It also bounds the variational objective to remove the residue coupling between the channel and phase matrices. Using extensive numerical investigations for a practical RIS mmWave system, and by using multiple metrics, we show that the proposed i) SMF- and FMF-VEM algorithms outperform several of their state-of-the-art counterparts; and ii) FMF-VEM has a much lower time complexity than SMF-VEM.