Efficient Spatial Channel Estimation in Extremely Large Antenna Array Communication Systems: A Subspace Approximated Matrix Completion Approach

Matrix completion techniques are widely employed to estimate the channel matrix from partially observed channel measurements. However, its computational complexity is cubic in the number of antennas, which is non-scalable for extremely large-scale antenna array (ELAA) communication systems. To address this issue, in this paper the ELAA channel matrix completion is reformulated as a proximal gradient descent (PGD) problem, where the subgradient of the nuclear norm is computed by singular value thresholding (SVT) operator and the proximal gradient of the $L_{1}$ norm is derived using a softthresholding operator. To mitigate the computational overhead caused by subspace orthogonalization in the SVT operation, a novel subspace-approximated (SA)-SVT-PGD algorithm is designed. This algorithm exploits the subspace similarity of the channel’s Gram matrix in consecutive PGD iterations and enables concurrent subspace orthogonalization during PGD updates. By eliminating the specific nested loop for the subspace orthogonalization, the computational complexity of the SA-SVT-PGD algorithm is proportional to the product between the number of antennas and the square of rank of the channel matrix. Simulation results demonstrate that the SA-SVT-PGD algorithm can reduce the convergence time by 71.7% compared with the traditional SVT-PGD algorithm.