# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 3 Proposed Method  
Tikhonov regularization, also known as ridge regularization, is a method which inserts a regularization term to the solution for solving ill-posed problems or preventing overfitting in linear regression. Penalizing the magnitude of the coefficients in the loss function we are attempting to minimize is the core idea behind Tikhonov regularization. In its simplest form, Tikhonov regularization penalizes solutions with large magnitudes by adding a regularization term to the objective function to be reduced [38]. The solution is more resilient to varying conditions thanks to this regularization term. Several applications in engineering and physics lead to the following types of linear least-squares problems:  
\[min\ x_{i+1}\in R^{d_{2}}\ \|Ax_{i+1}-b^{\delta}\|,\quad AeR^{d_{1}xd_{2}},\quad b ^{\delta}eR^{d_{1}} \tag{4}\]  
Matrix \(A\) is of ill-determined rank, that is, its singular values gradually decay to zero without a noticeable gap, and the measured data, tainted by an unknown error \(e\in R^{d_{1}}\) of norm constrained by \(\delta>0\), are represented by \(b^{\delta}\). The matrix \(A\in R^{d_{1}xd_{2}}\) has \(d_{1}\) rows representing the number of measurements and \(d_{2}\) columns corresponding to the number of unknowns or variables, which are the components of the vector \(x_{i+1}\). Least-squares problems, also referred to as discrete ill-posed problems, require this kind of matrix. An exact approximation of the minimal norm solution \(x^{+}=A^{+}b\) for the error-free least-squares problem associated with (4) is sought after. \(A^{+}\) represents the pseudoinverse of Moore-Penrose in this case. Due to the error in \(b^{\delta}\) and the clustering of A's singular values near the origin, the solution \(A^{+}b^{\delta}\) of (4) is usually not a reasonable approximation of \(x^{+}\). Changing the minimization problem to a nearby problem whose solution is less vulnerable  
Figure 2: LCG algorithm in \(i^{th}\) layer. to the error in \(b^{\delta}\) is one method to overcome this problem (4). This substitution is sometimes referred to as regularization. \[min\ x_{i+1}\in R^{d_{2}}\left\{\left\|Ax_{i+1}-b^{\delta}\right\|^{2}+\lambda \|x_{i+1}-x_{i}\|^{2}\right\} \tag{5}\]  
The regularization parameter \(\lambda>0\) in this case controls how sensitive the solution of (5) is to the error e in \(b^{\delta}\) as well as how near the solution is to the target vector \(x^{+}\). It is generally known that by substituting an appropriate regularization matrix for the Tikhonov minimization problem (5), it is frequently possible to increase the quality of the \(x^{+}\) approximation determined by Tikhonov regularization \(L\). \[min\ x_{i+1}\in R^{d_{2}}\left\{\left\|Ax_{i+1}-b^{\delta}\right\|^{2}+\lambda \|L(x_{i+1}-x_{i})\|^{2}\right\} \tag{6}\]  
where the regularization matrix \(L\) e \(R^{d_{3}\mathrm{x}d_{2}}\) typically has dimensions such that \(d_{3}\) may vary depending on the specific regularization approach, although it is often equal to \(d_{2}\) in the case of square matrices. The regularization matrix \(L\) encodes the extra restrictions or previous knowledge about the solution x in (6), and \(\lambda\) is a regularization parameter that governs the trade-off between fitting the data and meeting the regularization term. The process of choosing the regularization matrix \(L\) and regularization parameter \(\lambda\), which requires domain expertise and careful tuning, is critical to obtaining precise and reliable solutions to ill-posed inverse problems. The matrix \(L\) is typically an \(N\times N\) matrix, where \(N\) represents the number of unknown variables, whereas parameter \(\lambda\) is usually a constant scalar. Basically, there are two main approaches for selecting the regularization matrix \(L\). The first one is the non-derivative method which forms \(L\) as \(L=DV^{T}\). Here, \(V^{T}\) is derived from the singular value decomposition (SVD) of matrix \(A\), and \(D\) is a diagonal matrix of singular values.