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Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures

Zheyu Wu, Bruno Clerckx

TL;DR

This work develops an architecture-independent optimization framework for beyond-diagonal RIS (BD-RIS) by modeling BD-RIS topologies through the admittance Y-parameter and the scattering Θ-parameter, with Θ = (I + iZ0Y)^{-1}(I − iZ0Y) and Y = iB (B real, symmetric). It then introduces a novel partially proximal ADMM (pp-ADMM) approach that transforms matrix-inversion constraints into bilinear forms using auxiliary variables, enabling efficient, near closed-form updates for sum-rate maximization and transmit power minimization in MU-MISO and, more generally, MU-MIMO systems. The framework is demonstrated to outperform existing methods in terms of the trade-off between performance and computational complexity, and it enables architecture-wide comparisons among single-, group-, tree-, and fully-connected BD-RIS. A key finding is that tree-connected BD-RIS, optimal in single-user cases, is not necessarily optimal in multi-user scenarios, motivating future work to identify true optimum BD-RIS architectures for multi-user networks. The paper further generalizes the approach to broader utility functions and multi-user MIMO, illustrating wide applicability of the architecture-agnostic design.

Abstract

Reconfigurable intelligent surfaces (RISs) are envisioned as a promising technology for future wireless communication systems due to their ability to control the propagation environment in a hardware- and energy-efficient way. Recently, the concept of RISs has been extended to beyond diagonal RISs (BD-RISs), which unlock the full potential of RISs thanks to the presence of tunable interconnections between RIS elements. While various algorithms have been proposed for specific BD-RIS architectures, a universal optimization framework applicable to arbitrary architectures is still lacking. In this paper, we bridge this research gap by proposing an architecture-independent framework for BD-RIS optimization, with the main focus on sum-rate maximization and transmit power minimization in multiuser multi-input single-output (MU-MISO) systems. Specifically, we first incorporate BD-RIS architectures into the models by connecting the scattering matrix with the admittance matrix and introducing appropriate constraints to the admittance matrix. The formulated problems are then solved by our custom-designed partially proximal alternating direction method of multipliers (pp-ADMM) algorithms. The pp-ADMM algorithms are computationally efficient, with each subproblem either admitting a closed-form solution or being easily solvable. We further explore the extension of the proposed framework to general utility functions and multiuser multi-input multi-output (MU-MIMO) systems. Simulation results demonstrate that the proposed approaches achieve a better trade-off between performance and computational efficiency compared to existing methods. We also compare the performance of various BD-RIS architectures in MU-MISO systems using the proposed approach, which has not been explored before due to the lack of an architecture-independent framework.

Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures

TL;DR

This work develops an architecture-independent optimization framework for beyond-diagonal RIS (BD-RIS) by modeling BD-RIS topologies through the admittance Y-parameter and the scattering Θ-parameter, with Θ = (I + iZ0Y)^{-1}(I − iZ0Y) and Y = iB (B real, symmetric). It then introduces a novel partially proximal ADMM (pp-ADMM) approach that transforms matrix-inversion constraints into bilinear forms using auxiliary variables, enabling efficient, near closed-form updates for sum-rate maximization and transmit power minimization in MU-MISO and, more generally, MU-MIMO systems. The framework is demonstrated to outperform existing methods in terms of the trade-off between performance and computational complexity, and it enables architecture-wide comparisons among single-, group-, tree-, and fully-connected BD-RIS. A key finding is that tree-connected BD-RIS, optimal in single-user cases, is not necessarily optimal in multi-user scenarios, motivating future work to identify true optimum BD-RIS architectures for multi-user networks. The paper further generalizes the approach to broader utility functions and multi-user MIMO, illustrating wide applicability of the architecture-agnostic design.

Abstract

Reconfigurable intelligent surfaces (RISs) are envisioned as a promising technology for future wireless communication systems due to their ability to control the propagation environment in a hardware- and energy-efficient way. Recently, the concept of RISs has been extended to beyond diagonal RISs (BD-RISs), which unlock the full potential of RISs thanks to the presence of tunable interconnections between RIS elements. While various algorithms have been proposed for specific BD-RIS architectures, a universal optimization framework applicable to arbitrary architectures is still lacking. In this paper, we bridge this research gap by proposing an architecture-independent framework for BD-RIS optimization, with the main focus on sum-rate maximization and transmit power minimization in multiuser multi-input single-output (MU-MISO) systems. Specifically, we first incorporate BD-RIS architectures into the models by connecting the scattering matrix with the admittance matrix and introducing appropriate constraints to the admittance matrix. The formulated problems are then solved by our custom-designed partially proximal alternating direction method of multipliers (pp-ADMM) algorithms. The pp-ADMM algorithms are computationally efficient, with each subproblem either admitting a closed-form solution or being easily solvable. We further explore the extension of the proposed framework to general utility functions and multiuser multi-input multi-output (MU-MIMO) systems. Simulation results demonstrate that the proposed approaches achieve a better trade-off between performance and computational efficiency compared to existing methods. We also compare the performance of various BD-RIS architectures in MU-MISO systems using the proposed approach, which has not been explored before due to the lack of an architecture-independent framework.

Paper Structure

This paper contains 40 sections, 6 theorems, 100 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System Model
  4. Different RIS Architectures
  5. Single-Connected RIS
  6. Fully-Connected RIS
  7. Group-Connected RIS
  8. Tree-Connected RIS
  9. Problem Formulation
  10. Sum-rate Maximization
  11. Transmit Power Minimization
  12. Sum-Rate Maximization problem
  13. Equivalent Reformulation
  14. pp-ADMM Algorithm for Solving \ref{['eqn:reformulate2']}
  15. Update of $\mathbf{y}$
  16. ...and 25 more sections

Key Result

Theorem 1

Assume that the sequences $\{\mathbf{U}^t\}_{t\geq 0}$ and $\{\boldsymbol{\lambda}^t\}_{t\geq 0}$ generated by the pp-ADMM algorithm in ppADMM are bounded. Then there exists $\rho_0>0$ such that when $\rho>\rho_0$, any limit point $\left(\mathbf{y}^*,\boldsymbol{\gamma}^*,\mathbf{W}^*,\mathbf{B}^{*}

Figures (8)

  • Figure 1: Different BD-RIS architectures, where (a) single-connected RIS, (b) fully-connected RIS, (c) group-connected RIS, (d) tree-connected RIS.
  • Figure 2: An illustration of the BD-RIS aided MU-MISO system for simulation.
  • Figure 3: Sum-rate versus number of RIS elements ($N=K=4, G_s=4$).
  • Figure 4: CPU time versus number of RIS elements for sum-rate maximization ($N=K=4, G_s=4$).
  • Figure 5: Transmit power versus number of RIS elements ($N=K=4, G_s=4$).
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more