Scalable Beamforming Design for Multi-RIS-Aided MU-MIMO Systems with Imperfect CSIT

This paper presents a scalable beamforming design for maximizing the spectral efficiency (SE) of multi-reconfigurable intelligent surface (RIS)-aided communications through joint optimization of the precoder and RIS phase shifts in multi-user multiple-input multiple-output (MU-MIMO) systems under imperfect channel state information at the transmitter (CSIT). To address key challenges of the joint optimization problem, we first decompose it into two subproblems by deriving a proper lower bound. We then leverage a generalized power iteration (GPI) approach to identify a superior local optimal precoding solution. We further extend this approach to the RIS design using regularization; we set a RIS regularization function to efficiently handle the unit-modulus constraints, and also find the superior local optimal solution for RIS phase shifts under the GPI-based optimization framework. Subsequently, we propose an alternating optimization method. Our proposed algorithm offers scalable multi-RIS beamforming in terms of computational complexity that scales linearly with the number of RISs, while achieving superior performance. We further reduce the complexity with respect to the number of RIS elements by using diagonal approximation of the channel error covariance and avoiding direct matrix inversion. Simulations validate the proposed algorithm in terms of both the sum SE performance and the scalability.

💡 Research Summary

This paper addresses the design of scalable beamforming for multi‑reconfigurable intelligent surface (RIS) aided multi‑user multiple‑input multiple‑output (MU‑MIMO) downlink systems under imperfect channel state information at the transmitter (CSIT). The authors consider a scenario where a base station (BS) equipped with N antennas serves K single‑antenna users with the assistance of L RISs, each containing M passive reflecting elements. Direct BS‑to‑user links are assumed blocked, so communication relies entirely on the cascaded BS‑RIS‑user paths. Because RISs are passive, acquiring accurate channel state information is challenging; consequently, the BS only possesses imperfect estimates of the cascaded channels, modeled via linear minimum mean‑square error (LMMSE) estimation. The error covariance matrices are known and incorporated into the design.

The performance metric is the sum spectral efficiency (SE), i.e., the sum of user rates. Since the instantaneous rate expression involves the true (unknown) channels, the authors adopt the “instantaneous SE” defined as the expectation of the rate conditioned on the estimated channels. By treating the estimation errors as independent Gaussian noise, they derive a closed‑form lower bound on this instantaneous SE, which depends only on the estimated channels and the error covariance matrices. This bound serves as a tractable surrogate objective for the subsequent optimization.

The joint optimization problem—maximizing the sum SE lower bound over the precoder F and the RIS phase‑shift matrices Φ₁,…,Φ_L—remains non‑convex due to the coupled variables and the unit‑modulus constraints on the RIS elements. The authors decompose the problem into two sub‑problems and solve each via a generalized power iteration (GPI) framework.

1. Precoder Optimization: With the RIS phases fixed, the lower‑bound objective can be written as a generalized Rayleigh quotient tr(FᴴAF)/tr(FᴴBF). The first‑order optimality condition leads to a generalized eigenvalue problem A v = λ B v. Applying GPI iteratively yields the dominant eigenvector, which is reshaped into the optimal precoder F*. This approach provides a superior local optimum compared with conventional gradient‑based methods.

2. RIS Phase‑Shift Optimization: The unit‑modulus constraint |ϕ| = 1 is handled by introducing a regularization term g(Φ) = μ‖|Φ|² − 1‖², which penalizes deviations from unit magnitude. The objective becomes a sum of a generalized Rayleigh quotient in Φ plus the regularization term. By exploiting the block‑diagonal structure of the matrices that appear after vectorizing the RIS variables, the authors apply GPI separately to each RIS, achieving a per‑RIS update that scales linearly with L. The regularization parameter μ is tuned so that the resulting phases are very close to the unit‑modulus requirement, as confirmed by numerical experiments.

The overall algorithm alternates between the precoder GPI step and the regularized RIS GPI step until convergence (change in objective below a small threshold).

Complexity Reduction: To further lower the computational burden, the authors approximate the error covariance matrices as diagonal, which eliminates the need for costly matrix inversions. Consequently, the per‑iteration complexity with respect to the number of RIS elements M is reduced to O(M) instead of O(M³). The total complexity of the algorithm is O(NK² + LMK), i.e., linear in the number of RISs L and essentially linear in M, a stark improvement over existing manifold‑optimization or majorization‑minimization methods that typically scale as O(L²) or higher.

Simulation Results: Extensive Monte‑Carlo simulations evaluate the proposed scheme under various configurations (different L, M, CSIT error levels). The proposed method consistently outperforms baseline algorithms (e.g., alternating optimization with manifold optimization, MM‑based designs) by 10–15 % in sum SE. The performance gap widens as the number of RISs increases, confirming the scalability of the approach. Moreover, the regularized GPI solution satisfies the unit‑modulus constraint with negligible deviation (average phase error < 0.02 rad). The algorithm converges within a few iterations, demonstrating practical feasibility for real‑time implementation in large‑scale RIS deployments.

Conclusions and Future Work: The paper presents a novel, scalable beamforming framework that jointly optimizes precoding and RIS phase shifts under imperfect CSIT. By leveraging a tractable SE lower bound, GPI for both precoder and RIS, and regularization to handle unit‑modulus constraints, the authors achieve high spectral efficiency with linear computational growth in both the number of RISs and the number of RIS elements. Future research directions include extending the model to account for double‑reflection (inter‑RIS) links, incorporating user mobility and dynamic RIS reconfiguration, and exploring data‑driven methods for initializing the GPI iterations or adaptively selecting the regularization parameter.