⚡ electrical engineering

The manifold of unitary and symmetric matrices: characterization, Riemannian optimization and application to BD-RIS design

This paper provides a rigorous geometric characterization of the manifold of unitary and symmetric matrices to develop efficient Riemannian optimization algorithms for designing beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) in MIMO systems.

Original authors: Ignacio Santamaria, Carlos Beltrán, Eduard Jorswieck, Mohammad Soleymani, Jesus Gutiérrez

Published 2026-04-27

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CC BY 4.0

Original authors: Ignacio Santamaria, Carlos Beltrán, Eduard Jorswieck, Mohammad Soleymani, Jesus Gutiérrez

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a conductor of a massive, high-tech orchestra. This orchestra isn't playing violins; it’s playing radio waves.

In the world of 6G communications, we want to send data from a phone to a base station as clearly as possible. But sometimes, the "music" (the signal) gets messy because of walls, buildings, or distance. To fix this, scientists use something called a Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS).

Think of a BD-RIS as a "smart mirror" placed in a room. Unlike a normal mirror that just reflects light, this smart mirror can change how it reflects radio waves. It can catch a signal, twist it, flip it, and aim it perfectly toward your phone.

The Problem: The "Perfect Mirror" is Hard to Build

The math required to design this "smart mirror" is incredibly difficult. To make the mirror work perfectly, the settings (the "scattering matrix") must follow two very strict rules:

It must be Unitary: This is like saying the mirror can’t "eat" the signal. It must reflect all the energy it receives without losing it (no energy leakage).
It must be Symmetric: This is a rule of physics called "reciprocity." It means the mirror treats a signal going from A to B the same way it treats a signal going from B to A.

In math terms, finding the best settings for this mirror is like trying to find the highest peak in a mountain range while being forced to walk only on a very specific, narrow, winding path. If you step off the path, your "mirror" breaks the laws of physics.

The Solution: A New Map and a Faster Way to Climb

The authors of this paper have done three brilliant things to solve this:

1. They mapped the "Hidden Path" (The Manifold)
Previously, engineers were trying to find the peak by wandering around the whole mountain and then "forcing" themselves back onto the path whenever they strayed. This is slow and often leads them to the wrong hill.
The authors mathematically defined the exact "path" (which they call the $U_s$ manifold). They created a mathematical "map" that allows an algorithm to stay on the path naturally, making the search much more efficient.

2. They invented two ways to "Climb" (The Algorithms)
They proposed two ways to reach the peak:

The "Steady Hiker" (Line Search): This hiker takes a step in the direction of the peak and then carefully checks, "Is this step too big? Did I go too far?" and adjusts accordingly.
The "Phase Specialist" (Phase Optimization): This is a much faster climber. Instead of looking at the whole mountain, this climber looks at each individual "tuning knob" on the mirror one by one and twists them to the perfect position. It’s like tuning a guitar string by string rather than trying to tune the whole orchestra at once.

3. They discovered the "Shortcut" (Low-Rank Solution)
This is perhaps the most "magic" part of the paper. They realized that if you have a massive mirror with 1,000 elements, you don't actually need to control all 1,000 of them individually to get a perfect signal.
They proved that you can get the same amazing performance by only focusing on a small, "low-rank" subset of the mirror. It’s like realizing that to direct a spotlight, you don't need to move every single atom in the bulb; you only need to move the main lens. This makes the math much faster and much cheaper to run on real hardware.

Why does this matter to you?

As we move toward 6G, our world will be filled with these "smart surfaces"—on walls, in ceilings, and on buildings. This paper provides the mathematical "brain" that allows these surfaces to work instantly and efficiently, ensuring your high-speed internet stays lightning-fast, even in the most difficult environments.

Technical Summary: The Manifold of Unitary and Symmetric Matrices

This paper presents a rigorous geometric characterization of the manifold of unitary and symmetric matrices ( $\mathcal{U}_s$ ) and leverages this characterization to develop efficient Riemannian optimization algorithms. The primary motivation is the design of Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RISs), where the scattering matrix must be both unitary (to ensure passivity/losslessness) and symmetric (to satisfy reciprocity).

1. The Problem

Existing optimization methods for BD-RIS design suffer from two main drawbacks:

Suboptimality: Some methods relax the symmetry constraint or use projection methods that do not strictly respect the manifold's geometry.
Complexity: Other methods use the Takagi factorization ( $\Theta = Q\Sigma Q^T$ ) to transform the problem into the unitary manifold $\mathcal{U}$ . However, this increases the dimensionality of the search space and transforms simple quadratic cost functions into complex quartic ones, leading to higher computational costs and more local minima.

The authors aim to develop optimization algorithms that operate directly on $\mathcal{U}_s$ , maintaining the correct constraints while minimizing complexity.

2. Methodology

The authors employ Riemannian Manifold Optimization (MO), treating $\mathcal{U}_s$ as a smooth geometric object rather than a constrained Euclidean space.

A. Geometric Characterization:
The paper provides the first complete mathematical description of $\mathcal{U}_s$ , including:

Tangent Space ( $T_U\mathcal{U}_s$ ): They derive that the tangent space at a point $U$ can be parameterized using a real symmetric matrix $R$ , specifically $B = jQR^T Q^T$ (where $U = QQ^T$ is the Takagi decomposition).
Geodesics: They derive closed-form expressions for geodesics (the shortest paths on the manifold), showing that moving along a geodesic corresponds to updating the phases of the Takagi factors.
Retraction and Cayley Transform: They introduce a retraction mapping to project Euclidean steps back onto the manifold and revisit the Cayley transform to show it provides a diffeomorphism between real symmetric matrices and a subset of $\mathcal{U}_s$ .

B. Proposed Algorithms:
Two Riemannian optimization algorithms are developed:

Line Search (LS): A standard gradient ascent approach where a single step size $\mu$ is optimized along the geodesic using a bisection method.
Phase Optimization (PO): A more advanced approach that optimizes the phases of the diagonal matrix in the geodesic parameterization one by one (block coordinate ascent). This expands the search space and typically accelerates convergence.

3. Key Contributions

Mathematical Foundation: Rigorous derivation of the tangent space, geodesics, and retraction for the $\mathcal{U}_s$ manifold.
New Optimization Framework: The development of the LS and PO algorithms tailored specifically for $\mathcal{U}_s$ .
Low-Rank Theory: A significant theoretical contribution proving that when the number of RIS elements $M$ exceeds the number of antennas ( $N_t + N_r$ ), the optimal scattering matrix is low-rank. This allows the optimization to be performed in a much smaller $r \times r$ space (where $r = N_t + N_r$ ), drastically reducing complexity.

4. Results and Applications

The framework is applied to three critical MIMO communication metrics:

Sum Channel Gain Maximization: Maximizing the Frobenius norm of the equivalent channel.
Achievable Rate Maximization: Maximizing the logarithmic capacity of the link.
Minimum Mean-Square Error (MSE) Minimization: Minimizing the error between transmitted and received signals.

Key Findings:

Performance: The proposed MO algorithms (especially the PO version) outperform existing suboptimal and Takagi-based approaches in terms of both accuracy and convergence speed.
Complexity: The Low-Rank PO algorithm is highly efficient. While full-rank algorithms scale cubically with the number of RIS elements ( $M^3$ ), the low-rank version's complexity is essentially independent of $M$ , depending instead on the number of antennas ( $N^3$ ).
Robustness: The algorithms perform well even when the direct link is blocked, a common scenario in RIS-assisted communications.

5. Significance

This work bridges the gap between abstract differential geometry and practical wireless communications. By providing the mathematical tools to optimize directly on the $\mathcal{U}_s$ manifold, the authors enable the design of more powerful, reciprocal, and efficient BD-RIS hardware. The discovery of the low-rank property is particularly significant for real-world implementation, as it suggests that high-performance wave manipulation can be achieved with a reduced number of reconfigurable components (e.g., using $q$ -stem topologies).