Optimal symmetric low-rank BD-RIS configuration maximizing the determinant of a MIMO link

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) significantly improve wireless performance by allowing tunable interconnections among elements, but their design in multiple-input multiple-output (MIMO) systems has so far relied on complex iterative algorithms or suboptimal approximations. This work introduces a simple yet powerful approach: instead of directly maximizing the achievable rate, we maximize the absolute value of the determinant of the equivalent MIMO channel. We derive a closed-form symmetric unitary scattering matrix whose rank is exactly twice the channel's degrees of freedom ($2r$). Remarkably, this low-rank solution achieves the same determinant value as the optimal unitary BD-RIS. Using log-majorization theory, we prove that the rate loss relative to the optimal unitary BD-RIS vanishes at high signal-to-noise ratio (SNR) or when the number of BD-RIS elements becomes large. Moreover, the proposed solution can be perfectly implemented using a $q$-stem BD-RIS architecture with only $q=2r-1$ stems, requiring a minimum number of reconfigurable circuits. The resulting Max-Det solution is orders of magnitude faster to compute than existing iterative methods while achieving near-optimal rates in practical scenarios. This makes high-performance BD-RIS deployment feasible even with large surfaces and limited computational resources.

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

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.

Interference Leakage Minimization in RIS-assisted MIMO Interference Channels

We address the problem of interference leakage (IL) minimization in the $K$-user multiple-input multiple-output (MIMO) interference channel (IC) assisted by a reconfigurable intelligent surface (RIS). We describe an iterative algorithm based on block coordinate descent to minimize the IL cost function. A reformulation of the problem provides a geometric interpretation and shows interesting connections with envelope precoding and phase-only zero-forcing beamforming problems. As a result of this analysis, we derive a set of necessary (but not sufficient) conditions for a phase-optimized RIS to be able to perfectly cancel the interference on the $K$-user MIMO IC.