  We consider the problem of estimating the channel in reconfigurable
intelligent surface (RIS) assisted millimeter wave (mmWave) systems. We propose
two variational expectation maximization (VEM) based algorithms for channel
estimation in RIS-aided wireless systems. The first algorithm is a structured
mean field-based sparse Bayesian learning (SM-SBL) algorithm that exploits the
doubly-structured sparsity and the individual sparsity of the elements of the
channel. To exploit the sparsities, we propose a column-wise coupled Gaussian
prior. We next design the factorized mean field-based algorithm based on the
prior we propose. This algorithm called the factorized mean field SBL (FM-SBL)
algorithm, addresses the time complexities of the SM-SBL algorithm without
sacrificing channel estimation accuracy. We show using extensive numerical
investigations that the i) proposed SM-SBL and FM-SBL algorithms outperform
several existing algorithms and ii) FM-SBL has lower time complexity than the
SM-SBL algorithm.
