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

Technical Summary: Geometry-Informed Optimization of Binary RIS Configurations

Problem Statement
Practical Reconfigurable Intelligent Surfaces (RISs) often utilize low-resolution phase shifters, typically limited to 1-bit (two discrete states: $0^\circ$ and $180^\circ$). For an RIS with $N$ elements, the configuration space consists of $2^N$ binary states. Direct optimization to maximize performance metrics (such as received signal strength or sensing accuracy) via exhaustive search is computationally prohibitive due to this exponential growth. While existing literature often assumes continuous phase shifts and applies quantization post-optimization, this approach frequently results in significant performance degradation when phase resolution is low. Conversely, direct discrete search methods often rely on heuristic or unstructured exploration, failing to exploit the inherent structure of the optimal solution space.

Methodology
The authors propose a framework that reformulates the 1-bit RIS phase optimization problem as a geometric problem. Instead of treating the binary space as unstructured, the paper demonstrates that globally optimal configurations share a specific geometric property.

1. Mathematical Reformulation:
   The problem of maximizing the received signal-to-noise ratio (SNR) is transformed into maximizing the norm of a signed sum of channel-dependent vectors. Specifically, the objective is to find a binary vector $\mathbf{x} \in \{+1, -1\}^K$ that maximizes $\|\sum_{k} x_k \mathbf{a}_k\|$, where $\mathbf{a}_k$ are vectors derived from the channel state information (CSI) and the direct link.

2. Geometric Characterization (Lemma 1):
   The core theoretical contribution is the proof that any globally optimal configuration $\mathbf{x}^*$ is induced by the signs of the projections of the vectors $\mathbf{a}_k$ onto a common direction vector $\mathbf{v}$. Mathematically, $x_k^* = \text{sgn}(\mathbf{a}_k \cdot \mathbf{v})$. This implies that the set of potentially optimal configurations is restricted to those generated by varying the direction $\mathbf{v}$ in the signal space.

3. Algorithmic Approaches:

   • General MIMO Case (Geometry-Informed Sampling): For high-dimensional signal spaces, the authors propose an algorithm that samples direction vectors $\mathbf{v}$ from a distribution and constructs binary configurations using the projection rule. This restricts the search budget exclusively to "structurally admissible" candidates, avoiding the vast majority of suboptimal configurations.
   • SISO Case (Exact Polynomial-Time Solution): In Single-Input-Single-Output systems, the vectors lie in a 2D real space ($\mathbb{R}^2$). The authors show that the optimal configurations correspond to the sectors formed by lines orthogonal to the vectors $\mathbf{a}_k$. By enumerating these sectors (at most $N+1$ distinct configurations), the global optimum can be found exactly in polynomial time ($O(K^2)$ or $O(K \log K)$ with parallelism), rather than through random sampling.

4. Application to ISAC:
   The framework is extended to Integrated Sensing and Communication (ISAC) scenarios. The RIS elements are partitioned into sets for communication and sensing. The authors demonstrate that the subproblems for enhancing communication signals and illuminating targets for localization both reduce to the same underlying signed-vector maximization problem, allowing the same optimization algorithms to be applied to both functionalities.

Key Contributions

• Geometric Proof: Proved that every globally optimal 1-bit RIS configuration is induced by the signs of projections of channel vectors onto a common direction.
• Efficient MIMO Algorithm: Developed a geometry-informed sampling method that significantly reduces the search space for general MIMO systems by focusing only on structurally valid configurations.
• Exact SISO Algorithm: Derived a polynomial-time exact algorithm for SISO systems by characterizing the complete set of candidate configurations via 2D angular partitioning.
• Unified ISAC Framework: Showed that communication enhancement and target localization in ISAC share the same discrete optimization structure, enabling a unified design approach.

Simulation Results
The paper evaluates the proposed methods through numerical simulations in a 3D environment:

• LOS Maximization: The geometry-informed sampling method was compared against uninformed direct binary sampling and an ideal continuous-phase benchmark. With a fixed sampling budget of 1,000 candidates, the proposed method achieved gains of at least 5.36 dB over direct sampling. It approached the performance of the ideal continuous-phase RIS within 3.12 dB, demonstrating that effective exploration of the binary space can recover a substantial fraction of continuous-phase performance.
• ISAC Trade-offs: In ISAC scenarios with RIS elements partitioned between communication and sensing, the results showed a non-linear trade-off. Increasing the fraction of elements dedicated to communication improved received power steadily but did not immediately degrade sensing performance (localization error and miss rate) until a critical threshold was reached. This suggests an operating region where communication can be enhanced with minimal impact on sensing.

Significance
The paper claims that the discrete nature of practical RIS hardware does not render phase optimization intractable. Instead, the discrete constraints reveal an exploitable geometric structure. By leveraging this structure, the proposed framework provides a unified approach for optimizing practical 1-bit RIS configurations across both communication and sensing functionalities, offering a computationally efficient alternative to exhaustive search or continuous-phase approximations.