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Riemannian optimization on the manifold of unitary and symmetric matrices with application to BD-RIS-assisted systems

Ignacio Santamaria, Mohammad Soleymani, Eduard Jorswieck, Jesus Gutierrez, Carlos Beltran

TL;DR

The paper addresses optimization over the manifold $\mathcal{U}_s$, the set of unitary and symmetric matrices, by deriving its tangent space and geodesics via Takagi factorization. It then develops a parameter-free Riemannian MO algorithm that exploits a geodesic-based diagonal phase parametrization to update $\Theta\in\mathcal{U}_s$ without adaptation parameters. Applied to BD-RIS-assisted MIMO, the method achieves the same rate performance as Takagi-based approaches but with significantly reduced computational cost, validated through simulations showing fast convergence. The work contributes a rigorous geometric foundation and a scalable optimization tool for reciprocity-constrained reconfigurable surfaces and potentially other unitary-symmetric design problems.

Abstract

In this paper, we rigorously characterize for the first time the manifold of unitary and symmetric matrices, deriving its tangent space and its geodesics. The resulting parameterization of the geodesics (through a real and symmetric matrix) allows us to derive a new Riemannian manifold optimization (MO) algorithm whose most remarkable feature is that it does not need to set any adaptation parameter. We apply the proposed MO algorithm to maximize the achievable rate in a multiple-input multiple-output (MIMO) system assisted by a beyond-diagonal reconfigurable intelligent surface (BD-RIS), illustrating the method's performance through simulations. The MO algorithm achieves a significant reduction in computational cost compared to previous alternatives based on Takagi decomposition, while retaining global convergence to a stationary point of the cost function.

Riemannian optimization on the manifold of unitary and symmetric matrices with application to BD-RIS-assisted systems

TL;DR

The paper addresses optimization over the manifold , the set of unitary and symmetric matrices, by deriving its tangent space and geodesics via Takagi factorization. It then develops a parameter-free Riemannian MO algorithm that exploits a geodesic-based diagonal phase parametrization to update without adaptation parameters. Applied to BD-RIS-assisted MIMO, the method achieves the same rate performance as Takagi-based approaches but with significantly reduced computational cost, validated through simulations showing fast convergence. The work contributes a rigorous geometric foundation and a scalable optimization tool for reciprocity-constrained reconfigurable surfaces and potentially other unitary-symmetric design problems.

Abstract

In this paper, we rigorously characterize for the first time the manifold of unitary and symmetric matrices, deriving its tangent space and its geodesics. The resulting parameterization of the geodesics (through a real and symmetric matrix) allows us to derive a new Riemannian manifold optimization (MO) algorithm whose most remarkable feature is that it does not need to set any adaptation parameter. We apply the proposed MO algorithm to maximize the achievable rate in a multiple-input multiple-output (MIMO) system assisted by a beyond-diagonal reconfigurable intelligent surface (BD-RIS), illustrating the method's performance through simulations. The MO algorithm achieves a significant reduction in computational cost compared to previous alternatives based on Takagi decomposition, while retaining global convergence to a stationary point of the cost function.
Paper Structure (11 sections, 5 theorems, 13 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 5 theorems, 13 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Manifold optimization on $\mathcal{U}_s$
  3. The manifold $\mathcal{U}$
  4. The manifold $\mathcal{U}_s$
  5. Manifold optimization algorithm
  6. Application
  7. Rate optimization of BD-RIS-assisted MIMO links
  8. Computational cost
  9. Simulation results
  10. Conclusions
  11. Appendix: Proof of Proposition 1

Key Result

Theorem 1

Let ${\bf A} = {\bf A}^T$ be an $n \times n$ complex symmetric matrix. Then, there exist an $n \times n$ unitary matrix ${\bf Q} \in \mathcal{U}$ and an $n\times n$ diagonal matrix $\bm{ \Sigma} = \operatorname{diag}(\sigma_1,\ldots,\sigma_n)$ with $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_n \

Figures (2)

  • Figure 1: Rate vs. number of BD-RIS elements achieved with different algorithms.
  • Figure 2: a) Convergence curves of the proposed algorithm starting from different random initializations, b) Run time (in log scale) vs. number of BD-RIS elements.

Theorems & Definitions (9)

  • Theorem 1: Takagi factorization Takagi
  • Proposition 1: tangent space and geodesics
  • proof
  • Proposition 2: Retraction to $\mathcal{U}_s$
  • proof
  • Proposition 3: Projection to the tangent space
  • proof
  • Proposition 4: Convergence
  • proof