Manifold Optimization Empowered Two-Timescale Channel Estimation for RIS-Assisted Systems

In this paper, we propose a manifold optimization empowered two-timescale channel estimation (MO-TTCE) scheme for reconfigurable intelligent surface (RIS)-assisted multi-user millimeter wave massive MIMO systems. Since the base station (BS) and RIS remain relatively stationary while the user equipments (UEs) are mobile, the coherence time of the RIS-BS channel is significantly longer than that of the UE-related channels, namely the UE-RIS and UE-BS channels. Therefore, it is sufficient to estimate the RIS-BS channel only once in the large timescale, while frequently estimating the UE-related channels in the small timescale. We leverage the sparse and low-rank properties of the RIS-BS channel and transform the channel estimation problem into a series of sparse low-rank matrix recovery (SLRMR) problems, specifically $\ell _{1}$-norm regularized constrained optimization problems with the feasible region being a complex bounded-rank (CBR) matrix set. To ensure the differentiability of the objective function, we employ a differentiable Huber-$\gamma$ function as a substitute for the $\ell _{1}$-norm. To handle the non-convex nature of the CBR matrix set, we treat the complex fixed-rank (CFR) matrix set as a CFR manifold and consider the CBR matrix set as a collection of CFR manifolds, thereby employing manifold optimization techniques to solve the problem. Furthermore, in the small timescale, we utilize the downlink pilot transmission and uplink feedback scheme to simultaneously estimate all UE-RIS channels. Simulation results show that the proposed MO-TTCE scheme can enhance the accuracy of channel estimation and reduce pilot overhead.