  We consider the problem of channel estimation and joint active and passive
beamforming for reconfigurable intelligent surface (RIS) assisted millimeter
wave (mmWave) multiple-input multiple-output (MIMO) orthogonal frequency
division multiplexing (OFDM) systems. We show that, with a well-designed
frame-based training protocol, the received pilot signal can be organized into
a low-rank third-order tensor that admits a canonical polyadic decomposition
(CPD). Based on this observation, we propose two CPD-based methods for
estimating the cascade channels associated with different subcarriers. The
proposed methods exploit the intrinsic low-rankness of the CPD formulation,
which is a result of the sparse scattering characteristics of mmWave channels,
and thus have the potential to achieve a significant training overhead
reduction. Specifically, our analysis shows that the proposed methods have a
sample complexity that scales quadratically with the sparsity of the cascade
channel. Also, by utilizing the singular value decomposition-like structure of
the effective channel, this paper develops a joint active and passive
beamforming method based on the estimated cascade channels. Simulation results
show that the proposed CPD-based channel estimation methods attain mean square
errors that are close to the Cramer-Rao bound (CRB) and present a clear
advantage over the compressed sensing-based method. In addition, the proposed
joint beamforming method can effectively utilize the estimated channel
parameters to achieve superior beamforming performance.
