Robust Beamforming for Active RIS-Assisted Over-the-Air Computation with Partial Channel State Information

In this paper, we propose a robust beamforming scheme for an active reconfigurable intelligent surface (RIS)-assisted over-the-air computation system, where only partial channel state information (CSI) is available. The partial CSI means deterministic line-of-sight components and Gaussian distribution statistics of non-line-of-sight (NLoS) components for individual transmitter-RIS link and RIS-receiver link. To minimize the average computation mean square error, the technical challenge is the robust RIS reflection matrix optimization considering the unknown information of the NLoS components. The optimization problem, after equivalent transformation using majorization-minimization-based procedure, is a quadratically constrained quadratic programming with additional maximum allowable amplification factor constraints, whose semi-closed-form solution has not been documented. In this paper, the semi-closed-form solution is skillfully derived using the property of matrix inversion and cofactor expansion. Furthermore, by incorporating the optimal receiving beamforming expression into the robust RIS reflection matrix optimization, the conventional triple alternating optimization is effectively reduced to a double one. Numerical results illustrate that our proposed active RIS-assisted system outperforms the conventional passive RIS-assisted one, when the transmission power of sensors is low.