Low-Rank Modeling of Continuous Threat Regions for Cooperative Secure Beamforming in UAV Networks

Open wireless propagation makes unmanned aerial vehicle (UAV) links vulnerable to eavesdroppers distributed over roads, building clusters, or other continuous regions. This paper proposes a low-rank threat-subspace method for cooperative secure beamforming from distributed ground transmitters to a legitimate UAV. Steering vectors sampled over one or multiple azimuth–elevation threat regions are concatenated into a training matrix, whose dominant left singular vectors compactly represent regional exposure. The legitimate steering vector is projected onto the orthogonal complement of this subspace and power-normalized. We prove global optimality of the normalized projection for every feasible retained rank, derive a leakage bound from the first discarded singular value, and introduce uncertainty padding for independently mismatched region boundaries. Simulations evaluate disconnected and volumetric regions, deterministic geometries, Rician scattering, channel and phase errors, non-colluding and colluding eavesdroppers, and covariance-reconstruction and sampled peak-leakage baselines. In the default setting, six modes retain 98% of the sector energy, and the proposed method reduces average leakage to −26.25 dB, compared with −19.41 dB for pointwise nulling and −9.89 dB for maximum-ratio transmission. Independent boundary-error tests show that interval padding stabilizes leakage at the cost of desired gain, while multi-region tests quantify the progressive increase in effective rank. The results establish both the applicability limits and the low-overhead advantages of spatial-structure-based secure beamforming.