  Due to their ability to create favorable line-of-sight (LoS) propagation
environments, intelligent reflecting surfaces (IRSs) are regarded as promising
enablers for future millimeter-wave (mm-wave) wireless communication. In this
paper, we investigate channel estimation for IRS-assisted mm-wave
multiple-input multiple-output (MIMO) {\color{black}wireles}s systems. By
leveraging the sparsity of mm-wave channels in the angular domain, we formulate
the channel estimation problem as an $\ell_1$-norm regularized optimization
problem with fixed-rank constraints. To tackle the non-convexity of the
formulated problem, an efficient algorithm is proposed by capitalizing on
alternating minimization and manifold optimization (MO), which yields a locally
optimal solution. To further reduce the computational complexity of the
estimation algorithm, we propose a compressive sensing- (CS-) based channel
estimation approach. In particular, a three-stage estimation protocol is put
forward where the subproblem in each stage can be solved via low-complexity CS
methods. Furthermore, based on the acquired channel state information (CSI) of
the cascaded channel, we design a passive beamforming algorithm for
maximization of the spectral efficiency. Simulation results reveal that the
proposed MO-based estimation (MO-EST) and beamforming algorithms significantly
outperform two benchmark schemes while the CS-based estimation (CS-EST)
algorithm strikes a balance between performance and complexity. In addition, we
demonstrate the robustness of the MO-EST algorithm with respect to imperfect
knowledge of the sparsity level of the channels, which is crucial for practical
implementations.
