Beyond-Diagonal RIS: Adversarial Channels and Optimality of Low-Complexity Architectures

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) have recently gained attention as an enhancement to conventional RISs. BD-RISs allow optimizing not only the phase, but also the amplitude responses of their discrete surface elements by introducing adjustable inter-element couplings. Various BD-RIS architectures have been proposed to optimally trade off between average performance and complexity of the architecture. However, little attention has been paid to worst-case performance. This paper characterizes novel sets of adversarial channels for which certain low-complexity BD-RIS architectures have suboptimal performance in terms of received signal power at an intended communications user. Specifically, we consider two recent BD-RIS models: the so-called group-connected and tree-connected architecture. The derived adversarial channel sets reveal new surprising connections between the two architectures. We validate our analytical results numerically, demonstrating that adversarial channels can cause a significant performance loss. Our results pave the way towards efficient BD-RIS designs that are robust to adversarial propagation conditions and malicious attacks.