## 3 Conventional Data Detection Algorithm Based on Manifold Optimization
### Problem Formulation and Solving
 \tag{43}\]  
The Euclidean gradient of \(F(\mathbf{\hat{S}})\) can be calculated as  
\[\text{Grad}\,F(\mathbf{\hat{S}})=-\mathbf{\tilde{H}}^{T}\mathbf{\tilde{Y}}+\sqrt{N_{t}N_{ r}N_{f}K\sigma_{x}^{2}}\mathbf{\tilde{H}}^{T}\mathbf{\tilde{H}}\mathbf{\hat{S}}. \tag{44}\]  
With (44), we can apply the Riemannian conjugate gradient algorithm to solve (43). Let \(\mathbf{\hat{\hat{S}}}\) denote the solution of (43). Then, according to (32), (34), (37), and (42), we can obtain estimation of the transmitted signal as  
\[\mathbf{\hat{x}}_{n,k}=\text{hard}\Big{\{}[\mathbf{\hat{x}}_{n,k}]_{[1:N_{t}]_{:}}: \,+\,\text{j}\cdot[\mathbf{\hat{x}}_{n,k}]_{[N_{t}+1:2N_{t}]_{:}}:\,\Big{\}}, \tag{45}\]  
where \(\mathbf{\hat{x}}_{n,k}\) denotes the estimation of the transmitted signal after real-virtual separation. From (43) and (44), it is observed that the objective function and gradient function contain channel matrix \(\mathbf{\tilde{H}}\). Here, we assume that the channels can be estimated based on a small number of pilots by, for example, using GAN-based channel estimation methods [17; 18]. Let \(\mathbf{\hat{H}}_{n,k,\text{ini}}\) denote the initial estimation of \(\mathbf{H}_{n,k}\). According to (29) and (39), we can obtain the initial estimation of \(\mathbf{\hat{H}}\), i.e., \(\mathbf{\hat{H}}_{\text{ini}}\). Given \(\mathbf{\hat{H}}_{n,k,\text{ini}}\), we can also use ZF detection and hard-decisioning to obtain the initial estimation of the transmitted signal as  
\[\mathbf{\hat{x}}_{n,k,\text{ini}}=\text{hard}\Big{\{}(\mathbf{\hat{H}}_{n,k,\text{ini }}^{H}\mathbf{\hat{H}}_{n,k,\text{ini}})^{-1}\mathbf{\hat{H}}_{n,k,\text{ini}}^{H}\bm {y}_{n,k}\Big{\}}. \tag{46}\]  
After that, we can obtain the initial estimation of \(\mathbf{\hat{S}}\), i.e., \(\mathbf{\hat{\hat{S}}}_{\text{ini}}\), according to (32), (34), (37), and (42). The whole procedure of the proposed method is summarized in Algorithm 2. ```
1:Input:\(\left\{\left\{\mathbf{y}_{n,k}\right\}_{n=1}^{N_{f}}\right\}_{k=1}^{K}\)
2:Output:\(\left\{\left\{\mathbf{\hat{x}}_{n,k}\right\}_{n=1}^{N_{f}}\right\}_{k=1}^{K}\)
3:(1) Obtain the initial channel estimation \(\mathbf{\hat{H}}_{n,k,\text{ini}}\) by using GAN-based methods;
4:(2) Obtain the initial estimation of the transmitted signal \(\mathbf{\hat{x}}_{n,k,\text{ini}}\) according to (46);
5:(3) Obtain \(\mathbf{\hat{H}}_{\text{ini}}\) according to (29) and (39), \(\mathbf{\tilde{Y}}\) according to (33), and (35), (38), and \(\mathbf{\hat{\hat{S}}}_{\text{ini}}\) according to (32), (34), (37), and (42);
6:(4) Solve (43) by using the Riemannian conjugate gradient algorithm to obtain the solution \(\mathbf{\hat{S}}\);
7:(5) Obtain \(\mathbf{\hat{x}}_{n,k}\) according to (32), (34), (37), and (42), and then the estimated transmission signal \(\mathbf{\hat{x}}_{n,k}\) according to (45). ### Improved Algorithm with Less Complexity

Note that the dimensions of \(\mathbf{\hat{H}}_{\text{ini}}\), \(\mathbf{\hat{Y}}\), and \(\mathbf{\hat{\xi}}_{\text{ini}}\) may be large. Thus, directly solving (43) will result in high computational complexity. To reduce the complexity, we can divide the data frame into \(N_{B}=N_{f}/B\) subframes, each containing \(B\) OFDM symbols, and bringing subframes one by one to solve the problem. Since the reduction in the dimension of \(\mathbf{\hat{H}}_{\text{ini}}\), \(\mathbf{\hat{Y}}\), and \(\mathbf{\hat{\xi}}_{\text{ini}}\) from \(2 KN_{r}N_{f}\times 2 KN_{l}N_{f}\) to \(2 KN_{r}B\times 2 KN_{l}B\), from \(2 KN_{r}N_{f}\times 2 KN_{r}N_{f}\) to \(2 KN_{r}B\times 2 KN_{r}B\), and from \(2 KN_{l}N_{f}\times 2 KN_{r}N_{f}\) to \(2 KN_{l}B\times 2 KN_{r}B\), respectively, the computational complexity will be greatly reduced. However, directly reducing the matrix dimension will also deteriorate the algorithm performance. To maintain the performance of the algorithm, we further propose an improved algorithm that reuses the pilot symbols.