  The sparsity of millimeter wave (mmWave) channels in the angular and temporal
domains is beneficial to channel estimation, while the associated channel
parameters can be utilized for localization. However, line-of-sight (LoS)
blockage poses a significant challenge on the localization in mmWave systems,
potentially leading to substantial positioning errors. A promising solution is
to employ reconfigurable intelligent surface (RIS) to generate the virtual
line-of-sight (VLoS) paths to aid localization. Consequently, wireless
localization in the RIS-assisted mmWave systems has become the essential
research issue. In this paper, a multiple measurement vector (MMV) model is
constructed and a two-stage channel estimation based localization scheme is
proposed. During the first stage, by exploiting the beamspace sparsity and
employing a random RIS phase shift matrix, the channel parameters are
estimated, based on which the precoder at base station and combiner at user
equipment (UE) are designed. Then, in the second stage, based on the designed
precoding and combining matrices, the optimal phase shift matrix for RIS is
designed using the proposed modified temporally correlated multiple sparse
Bayesian learning (TMSBL) algorithm. Afterwards, the channel parameters, such
as angle of reflection, time-of-arrival, etc., embedding location information
are estimated for finally deriving the location of UE. We demonstrate the
achievable performance of the proposed algorithm and compare it with the
state-of-the-art algorithms. Our studies show that the proposed localization
scheme is capable of achieving centimeter level localization accuracy, when LoS
path is blocked. Furthermore, the proposed algorithm has a low computational
complexity and outperforms the legacy algorithms in different perspectives.
