In this work, we consider both channel estimation and reflection design problems in point-to-point reconfigurable intelligent surface (RIS)-aided millimeter-wave (mmWave) MIMO communication systems. First, we show that by exploiting the low-rank nature of mmWave MIMO channels, the received training signals can be written as a low-rank multi-way tensor admitting a canonical polyadic CP decomposition. Utilizing such a structure, a tensor-based RIS channel estimation method (termed TenRICE) is proposed, wherein the tensor factor matrices are estimated using an alternating least squares method. Using TenRICE, the transmitter-to-RIS and the RIS-to-receiver channels are efficiently and separately estimated, up to a trivial scaling factor. After that, we formulate the beamforming and RIS reflection design as a spectral efficiency maximization problem. Due to its non-convexity, we propose a heuristic non-iterative two-step method, where the RIS reflection vector is obtained in a closed form using a Frobenius-norm maximization (FroMax) strategy. Our numerical results show that TenRICE has a superior performance, compared to benchmark methods, approaching the Cramer-Rao lower bound with a low training overhead. Moreover, we show that FroMax achieves a comparable performance to benchmark methods with a lower complexity.
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