## 2 Methods
### Dual-Driven Learning-Based MIMO Signal Detection  
Our proposed approach is a dual-driven learning framework for MIMO signal detection. This framework leverages the architecture of the model-driven OAMP-Net network \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(N_{\text{t}}\) & the number of transmitting antennas (also the number of UAVs) \\ \(N_{\text{r}}\) & the number of receiving antennas \\ \(\mathbf{y}_{k}\) & received signal vector at the \(k\)-th subcarrier \\ \(\mathbf{H}_{k}\) & channel for the \(k\)-th subcarrier \\ \(\mathbf{s}_{k}\) & symbols for the \(k\)-th subcarrier \\ \(\mathbf{n}_{k}\) & complex Gaussian noise vector at the \(k\)-th subcarrier \\ \(\mathbf{\hat{s}}_{k}\) & detected symbols at the \(k\)-th subcarrier \\ \(\mathbf{\hat{y}}\) & received signal vector \\ \(\mathbf{\overline{H}}\) & channel \\ \(\mathbf{\overline{s}}\) & symbols \\ \(\mathbf{\overline{n}}\) & complex Gaussian noise vector \\ \(\mathbf{r}_{t}\) & output of the linear estimator at the \(t\)-th iteration \\ \(\mathbf{\hat{s}}_{t}\) & output of the nonlinear estimator at the \(t\)-th iteration \\ \(\mathbf{W}_{t}\) & decorrelated matrix in the \(t\)-th iteration \\ \(\eta_{t}(\cdot)\) & nonlinear divergence-free estimator \\ \(\sigma_{n}^{2}\) & noise variance of \(\mathbf{\overline{n}}\) \\ \(\mathbf{p}_{t}\) & linear estimation error vector at the \(t\)-th iteration \\ \(\mathbf{q}_{t}^{2}\) & nonlinear estimation error vector at the \(t\)-th iteration \\ \(\tau_{t}^{2}\) & error variance estimators \(\tau_{t}^{2}\) of \(\mathbf{p}_{t}\) \\ \(\
u_{t}^{2}\) & error variance estimators \(\upsilon_{t}^{2}\) of \(\mathbf{q}_{t}\) \\ \(\zeta_{t}\) & decorrelated coefficient at the \(t\)-th iteration \\ \(\mathbf{W}_{t}\) & LMMSE matrix \\ \(\mathbf{\Theta}_{\text{ON}}=\{\gamma_{t},\theta_{t},\phi_{t},\zeta_{t}\}\) & four trainable variables in OAMP-Net \\ \(\mathbf{\hat{r}}_{t}\) & noise-cleaned output of the linear estimator at the \(t\)-th iteration \\ \(\mathcal{F}_{\text{NN}}(\cdot)\) & the function expression of the symbol correction network \\ \(L_{d}\) & the number of layers in the symbol correction network \\ \(\mathbf{z}^{[l]}\) & output vector of the \(l\)-th layer in the symbol correction network \\ \(\mathbf{w}^{[l]}\) & weights matrix of the \(l\)-th layer in the symbol correction network \\ \(\mathbf{x}^{[l]}\) & the input vector of the \(l\)-th layer in the symbol correction network \\ \(\mathbf{b}^{[l]}\) & the bias term of the \(l\)-th layer in the symbol correction network \\ \(\mathbf{\Theta}_{\text{NN}}\) & symbol correction network parameters \\ \(\epsilon=1\times 10^{-10}\) & constant \\ \(\mathbf{n}_{\text{F}}^{\epsilon}\) & complex Gaussian noise vector with \(\sigma_{\text{c}}^{2}\) as the noise variance \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of key notations in this article. and incorporates a data-driven symbol correction network. By integrating this component, we aim to enhance the performance and convergence of the OAMP-Net network. As shown in Figure 3, we unfold the architecture of the model-driven OAMP-Net network and introduce a data-driven symbol correction network within the framework. The data-driven symbol correction network is trained extensively to reduce the errors associated with the estimated symbols passed through each iteration of the OAMP-Net network. This correction mechanism ensures an improved performance and convergence of the OAMP-Net network. By combining the strengths of both the model-driven and data-driven approaches, our dual-driven learning framework provides a novel solution for MIMO signal detection. This allows for the efficient utilization of the OAMP-Net network while leveraging the capabilities of the data-driven symbol correction network to refine the estimated symbols in each iteration. We can rewrite Equation (1) in a real-valued form, since both the OAMP-Net and the symbol correction network operate in the real-valued domain:  
\[\left[