Optimization of Passive Beyond-Diagonal RIS via Relaxation, Randomization, and Autoencoding

We consider beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) whose elements are connected in groups and aim at optimizing their configuration to maximize the achievable rate of the cascade channel. We propose two suboptimal approaches (i.e., semidefinite programming (SDP) and projected gradient ascent (PGA) solutions) to first find the BD-RIS configuration that maximizes the composite channel trace and then locally maximizes the achievable rate by a randomization approach. We impose a constraint on the choice of the coefficients to ensure that the BD-RIS is passive, i.e., it does not emit more power than that received. Still, our solution has a high communication overhead for a large number of connections among the BD-RIS elements. We then propose a dynamic mapping between the BD-RIS configuration and a small number of control variables. The mapping is provided by the encoder part of an autoencoder, trained to minimize a suitable loss function on the optimal configurations in the specific deployment. We also design the BD-RIS configuration directly in the latent space of the autoencoder, reducing the complexity. By simulations in a typical cellular communication scenario, we show that the group-connected BD-RIS can achieve up to 95% of the rate obtained for a fully-connected BD-RIS with two orders of magnitude lower complexity, while the autoencoder compression and configuration optimization in the latent space reduces the control rate by 90% with negligible rate loss.