Geometry-Informed Optimization of Binary RIS Configurations for Communication and Sensing

Practical Reconfigurable Intelligent Surfaces (RISs) often support only a small number of phase states, making their configuration inherently discrete. For a 1-bit RIS with $N$ elements, direct optimization requires searching among $2^N$ binary configurations. This work shows that this exponential configuration space is not unstructured. By reformulating 1-bit RIS optimization as the maximization of the norm of a signed sum of channel-dependent vectors, we prove that every globally optimal configuration must be induced by the signs of their projections onto a common direction. This geometric characterization restricts the class of configurations that can contain global optima and leads to different algorithmic consequences depending on the signal-space dimension. For general Multiple-Input-Multiple-Output (MIMO) systems, we develop a geometry-informed sampling method that evaluates only structurally admissible configurations. For Single-Input-Single-Output (SISO) systems, the same principle reduces to a two-dimensional angular partition, allowing the complete candidate set to be characterized and the global optimum to be recovered through polynomial-time enumeration, by evaluating at most $N+1$ out of the $2^N$ configurations. Finally, we apply the same binary optimization principle to an Integrated Sensing and Communication (ISAC) scenario, where communication enhancement and target localization reduce to the same underlying geometric problem. The proposed framework therefore provides a unified approach for exploiting the structure of practical 1-bit RIS configurations across communication and sensing functionalities.