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.
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A closed-form symmetric low-rank BD-RIS scattering matrix of rank 2r achieves the same determinant as the optimal unitary BD-RIS and near-optimal rates, implementable with only 2r-1 stems.
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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.
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Optimal symmetric low-rank BD-RIS configuration maximizing the determinant of a MIMO link
A closed-form symmetric low-rank BD-RIS scattering matrix of rank 2r achieves the same determinant as the optimal unitary BD-RIS and near-optimal rates, implementable with only 2r-1 stems.