# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 3 Proposed Method
### Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm  
We utilize Tikhonov regularization in the CG algorithm with deep unfolding to improve performance on different types of channels, addressing issues such as noise sensitivity and ensuring convergence in high-dimensional MIMO systems. The CG technique effectively tackles the complexity of high-dimensional MIMO systems, improving performance and signal estimates when paired with Tikhonov regularization. When incorporated into model-driven MIMO detection frameworks such as LCG, Tikhonov regularization appears to be an effective method for improving robustness and performance. This method involves adding a term \(L\) to the system matrix in the context of MIMO detection. In this work, the proposed method is dynamically adjusting the detection strength by the model during training thanks to the trainable parameter regularization matrix \(L\), alpha, and beta. It is noteworthy that, unlike the original Tikhonov regularization approach shown in (6), which represents the parameters \(\lambda\) and \(L\) individually, our proposed method combines them into a single matrix \(L\), which is the product of the scalar \(\lambda\) and the matrix \(L\). By treating the multiplication of these elements as a single matrix, the network streamlines the computation and improves the flexibility of the model for different channel conditions. The pseudocode of the proposed method is shown in Algorithm 2. The DU-TCG algorithm enhances the CG method with deep unfolding by incorporating Tikhonov regularization and an optimization step using trainable parameters \(\alpha\), \(\beta\), and \(L\). These parameters are optimized with an Adam optimizer [40; 41] during each iteration to minimize the loss function, playing crucial roles in updating the residual, refining the search direction, and applying regularization. The algorithm begins by initializing matrices and vectors as in the standard CG method. The parameters \(\alpha\) and \(\beta\) are initially set to \(0\), while the \(L\) matrix is initialized as an identity matrix of size \(N_{T}\times N_{T}\). In each iteration, the residual \(r_{i}\) is updated with \(\alpha\), and the search direction \(p_{i}\) is refined with \(\beta\). A new solution estimate \(\widetilde{x}\) is computed using Tikhonov regularization matrix \(L\), with a loss function measuring the difference between \(\widetilde{x}\) and \(x_{i}\). The process iteratively refines the solution by updating trainable parameters until the final transmitted signal vector estimate \(\hat{s}\) is obtained. The iterations of the Tikhonov-regularized CG algorithm are unrolled to create the proposed Deep-Unfolded Tikhonov-Regularized Conjugate Gradient (DU-TCG), a deep learning architecture in which each layer is associated with an algorithm iteration as shown in Figure 3.  
Using this method, the model can be trained to find the best parameters for improved performance in a range of channel situations. Along with the \(\alpha\) and \(\beta\) parameters, the DU-TCG algorithm adds a new trainable parameter called the regularization, \(L\), which is in matrix form. This combination allows for dynamic modification of the regularization strength during training and simplifies computation as it is processed by the network as a single matrix. To guide the training process, DU-TCG's loss function is designed to assess the difference between the transmitted signal and the predicted output. For this purpose, the mean squared error (MSE) loss function is used as shown in Algorithm 2. The mean squared difference between the expected and actual values is quantified by the MSE loss function, giving the network a specific target to minimize during training. The network increases the resilience and accuracy of detection by iteratively updating the parameters \(\alpha\), \(\beta\), and \(L\) to minimize this loss. The \(i^{th}\) iteration of the CG algorithm corresponds to the \(i^{th}\) layer  
Figure 3: DU-TCG algorithm in \(i^{th}\) layer. of DU-TCG detector.