## 2 Methods
### Dual-Driven Learning-Based MIMO Signal Detection
In the OAMP algorithm, \(\mathbf{W}_{t}=\zeta_{t}\mathbf{\hat{W}}_{t}\), where \(\zeta_{t}\) is the decorrelated coefficient given by \(\zeta_{t}\)=\(\frac{2N_{\text{t}}}{\text{tr}\left(\mathbf{\hat{W}}_{t}\mathbf{ \overline{H}}\right)}\) and \(\mathbf{\hat{W}}_{t}\) is the LMMSE matrix defined as \(\mathbf{\hat{W}}_{t}=\frac{\mathbf{\overline{H}}^{\text{T}}}{\left(\mathbf{ \overline{H}}^{\text{T}}+\frac{\sigma_{\mathbf{n}}^{2}}{2v_{t}^{2}}\mathbf{I} \right)}\)[24]. The error variance estimator \(\tau_{t}^{2}\) can be calculated as follows [24]:  
\[\tau_{t}^{2}=v_{t}^{2}(\zeta_{t}-1). \tag{8}\]  
In OAMP-Net, there are four trainable variables denoted as \(\mathbf{\Theta}_{\text{ON}}=\{\gamma_{t},\theta_{t},\phi_{t},\xi_{t}\}\), which are optimized to enhance the detection performance in each LE and NLE iteration [26]. The calculation of \(v_{t}^{2}\) in Equation (7) remains the same in both the OAMP algorithm and the OAMP-Net network. Equation (5) in the OAMP-Net network can be rewritten as follows [25; 26]:  
\[\mathbf{r}_{t}=\mathbf{\widehat{s}}_{t}+\gamma_{t}\frac{2N_{\text{t}}}{\text {tr}\left(\frac{\mathbf{\overline{H}}^{\text{T}}\mathbf{\overline{H}}}{ \left(\mathbf{\overline{H}}^{\text{T}}+\frac{\sigma_{\mathbf{n}}^{2}}{2v_{t }^{2}}\mathbf{I}\right)}\right)}\frac{\mathbf{\overline{H}}^{\text{T}}}{\left( \mathbf{\overline{H}}\mathbf{\overline{H}}^{\text{T}}+\frac{\sigma_{\mathbf{ n}}^{2}}{2v_{t}^{2}}\mathbf{I}\right)}\left(\mathbf{\overline{y}}-\mathbf{ \overline{H}}\mathbf{\widehat{s}}_{t}\right)}. \tag{9}\]  
The error variance estimator \(\tau_{t}^{2}\) of \(\mathbf{p}_{t}\) in the OAMP-Net network is calculated as [25; 26]  
\[\tau_{t}^{2}=v_{t}^{2}\left(\theta_{t}^{2}\frac{2N_{\text{t}}}{\text{tr} \left(\frac{\mathbf{\overline{H}}^{\text{T}}\mathbf{\overline{H}}}{\left( \mathbf{\overline{H}}^{\text{T}}+\frac{\sigma_{\mathbf{n}}^{2}}{2v_{t}^{2}} \mathbf{I}\right)}\right)}-2\theta_{t}+1\right). \tag{10}\]  
The NLE estimator in the OAMP-Net network, denoted as \(\mathbf{\widehat{s}}_{t+1}=\eta_{t}\big{(}\mathbf{r}_{t},\tau_{t}^{2};\phi_{t },\xi_{t}\big{)}\), can be represented by [25; 26]  
\[\eta_{t}\big{(}\mathbf{r}_{t},\tau_{t}^{2};\phi_{t},\xi_{t}\big{)}=\phi_{t}( \mathbb{E}[\mathbf{\overline{s}}\mid\mathbf{r}_{t},\tau_{t}]-\xi_{t}\mathbf{r}_ {t}). \tag{11}\]Equations (9)-(11) represent key processes in each iteration of the model-driven OAMP-Net. Now, let us introduce the data-driven symbol correction network. The calculations of \(\mathbf{r}_{t}\) in Equation (9) and \(\tau_{t}^{2}\) in Equation (10) are modified versions of Equations (5) and (8), respectively, where trainable variables \(\{\gamma_{t},\theta_{t}\}\) are introduced. Notably, when \(\gamma_{t}=1\) and \(\theta_{t}=1\), the calculations remain the same as in the OAMP algorithm and the OAMP-Net network. These trainable variables, \(\gamma_{t}\) and \(\theta_{t}\), control the update step sizes of \(\mathbf{r}_{t}\) and \(\tau_{t}^{2}\). Consequently, they can influence the convergence of the LE estimator and the overall detection performance. In Equation (11), we observe that \(\mathbf{r}_{t}\) and \(\tau_{t}^{2}\) serve as the prior mean and variance of \(\overline{\mathbf{s}}\), respectively. To ensure the convergence and optimal performance of the OAMP-Net detector, it is necessary to minimize estimation errors in the LE estimator, which are essentially calculations of \(\mathbf{r}_{t}\). Therefore, prior to passing \(\mathbf{r}_{t}\) to the NLE estimator, we insert a data-driven neural network to correct the values of \(\mathbf{r}_{t}\). This data-driven symbol correction network maps the estimated \(\mathbf{r}_{t}\) to a corrected version, \(\hat{\mathbf{r}}_{t}\), which is closer to the true value. In the symbol correction network, we treat the errors of the LE estimator outputs as residual noise. By training the symbol correction network, it can reconstruct the residual noise.