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In this paper, we investigate the fundamental problem of wireless link scheduling in device-to-device (D2D) networks, through the lens of Riemannian geometry. Our goal is to find a novel metric to characterize interference among D2D pairs, which can pave the way towards efficient and fast scheduling algorithms. Towards achieving this goal, we first model the connectivity pattern of each D2D pair, including its interference links, as a positively-shifted Laplacian matrix, which is a symmetric positive definite (SPD) one. Noting that SPD matrices constitute a non-Euclidean manifold, we represent each of the D2D pairs as a point on the SPD (i.e., conic) manifold, which is analyzed via Riemannian geometry. Accordingly we employ Riemannian metrics (e.g., Log-Euclidean metric "LEM"), which are suitable measures of distances on manifolds, to characterize the interference among D2D points on the SPD manifold.To validate the effectiveness of the proposed LEM-based interference measure, we propose a sequential link selection algorithm that schedules D2D pairs in a descending order of their signal-to-noise ratio (SNR), while keeping their LEM distances towards the already-scheduled pairs on the Riemannian manifold to be greater than a certain LEM threshold. Such LEM-based condition is equivalent to limiting the interference from potential D2D pairs to be below certain threshold. We show that the proposed LEM-based scheduling algorithm achieves sum rate of more than 86% of state-of-the-art ones (e.g., FPLinQ [1]), while only requiring spatial locations of D2D pairs, as opposed to requiring full channel state information (CSI).

Wireless Link Scheduling via Interference-aware Symmetric Positive Definite Connectivity Manifolds