  Over-the-air computation (AirComp) is emerging as a promising technology for
wireless data aggregation. However, its performance is hampered by users with
poor channel conditions. To mitigate such a performance bottleneck, this paper
introduces an active reconfigurable intelligence surface (RIS) into the AirComp
system. Specifically, we begin by exploring the ideal RIS model and propose a
joint optimization of the transceiver design and RIS configuration to minimize
the mean squared error (MSE) between the target and estimated function values.
To manage the resultant tri-convex optimization problem, we employ the
alternating optimization (AO) technique to decompose it into three convex
subproblems, each solvable optimally. Subsequently, we investigate two specific
cases and analyze their respective asymptotic performance to reveal the
superiority of the active RIS in mitigating the MSE relative to its passive
counterpart. Lastly, we adapt our transceiver and RIS configuration design to
account for the self-interference of the active RIS. To handle the resultant
highly non-convex problem, we further devise a two-layer AO framework.
Simulation results demonstrate the superiority of the active RIS in enhancing
AirComp performance compared to its passive counterpart.
