In this paper, we consider deep unfolding the standard iterative conjugate gradient (CG) algorithm to solve a linear system of equations. Instead of being adjusted with known rules, the parameters are learned via backpropagation to yield the optimal results. However, the proposed unfolded CG (UCG) is extended wherein a scalar parameter is substituted by a matrix-parameter to augment the degrees of freedom per layer. Once the training is completed, the UCG has revealed to require far a smaller number of layers than the number of iterations needed using the standard iterative CG. It is also shown to be very robust to noise and outperforms the standard CG in low signal to noise ratio (SNR) region. A key merit of the proposed approach is the fact that no explicit training data is dedicated to the learning phase as the optimization process relies on the residual error which is not explicitly expressed as a function of the desired data. As an example, the proposed UCG is applied to solve the reciprocity calibration problem encountered in massive MIMO (Multiple-Input Multiple-Output) systems.

在本文中，我们考虑深度展开标准迭代共轭梯度（CG）算法来求解线性方程组。无需通过已知规则进行调整，而是通过反向传播学习参数以产生最佳结果。但是，对提出的展开式CG（UCG）进行了扩展，其中标量参数被矩阵参数替代，以增加每层的自由度。一旦训练完成，与使用标准迭代CG相比，UCG所需要的层数要少得多。它也被证明对噪声非常鲁棒，并且在低信噪比（SNR）区域中优于标准CG。所提出的方法的主要优点在于，由于优化过程依赖于残余误差，因此没有明确的训练数据专用于学习阶段，而残余误差并未明确表示为所需数据的函数。例如，所提出的UCG被用于解决大规模MIMO（多输入多输出）系统中遇到的互易性校准问题。