While iterative matrix inversion methods excel in computational efficiency, memory optimization, and support for parallel and distributed computing when managing large matrices, their limitations are also evident in multiple-input multiple-output (MIMO) fading channels. These methods encounter challenges related to slow convergence and diminished accuracy, especially in ill-conditioned scenarios, hindering their application in future MIMO networks such as extra-large aperture array. To address these challenges, this paper proposes a novel matrix regularization method termed symmetric rank-$1$ regularization (SR-$1$R). The proposed method functions by augmenting the channel matrix with a symmetric rank-$1$ matrix, with the primary goal of minimizing the condition number of the resultant regularized matrix. This significantly improves the matrix condition, enabling fast and accurate iterative inversion of the regularized matrix. Then, the inverse of the original channel matrix is obtained by applying the Sherman-Morrison transform on the outcome of iterative inversions. Our eigenvalue analysis unveils the best channel condition that can be achieved by an optimized SR-$1$R matrix. Moreover, a power iteration-assisted (PIA) approach is proposed to find the optimum SR-$1$R matrix without need of eigenvalue decomposition. The proposed approach exhibits logarithmic algorithm-depth in parallel computing for MIMO precoding. Finally, computer simulations demonstrate that SR-$1$R has the potential to reduce the required iteration by up to $35\%$ while achieving the performance of regularized zero-forcing.

翻译：尽管迭代矩阵求逆方法在处理大规模矩阵时在计算效率、内存优化以及支持并行与分布式计算方面表现优异，但其在多输入多输出（MIMO）衰落信道中的局限性也显而易见。这些方法面临收敛速度慢和精度下降的挑战，尤其在病态场景下，阻碍了其在未来MIMO网络（如超大规模孔径阵列）中的应用。为应对这些挑战，本文提出一种新颖的矩阵正则化方法，称为对称秩-1正则化（SR-$1$R）。该方法通过向信道矩阵添加一个对称秩-1矩阵来增强其性质，主要目标是使所得正则化矩阵的条件数最小化。这显著改善了矩阵条件，使得正则化矩阵能够实现快速且精确的迭代求逆。随后，通过对迭代求逆结果应用Sherman-Morrison变换，即可获得原始信道矩阵的逆矩阵。我们的特征值分析揭示了通过优化SR-$1$R矩阵所能达到的最佳信道条件。此外，本文提出了一种幂迭代辅助（PIA）方法，无需特征值分解即可找到最优SR-$1$R矩阵。所提方法在MIMO预编码的并行计算中展现出对数级算法深度。最后，计算机仿真表明，SR-$1$R在达到正则化迫零性能的同时，有望将所需迭代次数减少高达$35\%$。