The unitary symmetric matrix manifold is geometrically characterized with tangent space, retraction, and geodesics, enabling Riemannian line-search and phase-optimization algorithms that outperform prior BD-RIS methods and exploit low-rank structure when elements exceed antennas.
Boumal,An Introduction to Optimization on Smooth Manifolds
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The manifold of unitary and symmetric matrices: characterization, Riemannian optimization and application to BD-RIS design
The unitary symmetric matrix manifold is geometrically characterized with tangent space, retraction, and geodesics, enabling Riemannian line-search and phase-optimization algorithms that outperform prior BD-RIS methods and exploit low-rank structure when elements exceed antennas.