## 3 Conventional Data Detection Algorithm Based on Manifold Optimization
### Problem Formulation and Solving
 \tag{30}\]  
Since \(\tilde{\mathbf{y}}_{n,k}\tilde{\mathbf{y}}_{n,k}^{T}\) is independent of the transmitted signal, and the traces of \(\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}\tilde{\mathbf{y}}_{n,k}^{T}\) and \(\tilde{\mathbf{y}}_{n,k}\tilde{\mathbf{x}}_{n,k}^{T}\tilde{\mathbf{H}}_{n,k}^{T}\) are equal, we transform the objective function into  
\[F(\tilde{\mathbf{x}}_{1,1},\cdots,\tilde{\mathbf{x}}_{N_{f},K})=\sum_{n=1}^{N_{f}}\sum _{k=1}^{K}\text{trace}\bigg{\{}-\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}\tilde {\mathbf{y}}_{n,k}^{T}+\frac{1}{2}(\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}\tilde{ \mathbf{x}}_{n,k}^{T}\tilde{\mathbf{H}}_{n,k}^{T})\bigg{\}}. \tag{31}\]  
By defining  
\[\tilde{\mathbf{X}}_{n}=\text{diag}\big{\{}\big{[}\tilde{\mathbf{X}}_{n,1}\quad\tilde{ \mathbf{X}}_{n,2}\quad\cdots\quad\tilde{\mathbf{X}}_{n,K}\big{]}\big{\}}\in\mathbb{R} ^{2 KN_{l}\times KN_{r}}, \tag{32}\]  
\[\tilde{\mathbf{Y}}_{n}=\text{diag}\big{\{}\big{[}\tilde{\mathbf{Y}}_{n,1}\quad\tilde{ \mathbf{Y}}_{n,2}\quad\cdots\quad\tilde{\mathbf{Y}}_{n,K}\big{]}\big{\}}\in\mathbb{R} ^{2 KN_{l}\times KN_{l}}, \tag{33}\]  
where  
\[\tilde{\mathbf{X}}_{n,k}=\mathbf{1}_{1\times N_{r}}\otimes\tilde{\mathbf{x}}_{n,k}\in \mathbb{R}^{2N_{l}\times N_{r}}, \tag{34}\]  
\[\tilde{\mathbf{Y}}_{n,k}=\mathbf{1}_{1\times N_{r}}\otimes\tilde{\mathbf{y}}_{n,k}\in \mathbb{R}^{2N_{l}\times N_{r}}, \tag{35}\]  
\(\mathbf{1}_{m\times n}\) denotes the \(m\times n\) all-one matrix, and \(\otimes\) denotes the Kronecker product, we can express (31) in a more concise form as  
\[F(\tilde{\mathbf{X}}_{1},\cdots,\tilde{\mathbf{X}}_{N_{f}})=\sum_{n=1}^{N_{f}}\text{ trace}\bigg{\{}-\tilde{\mathbf{H}}_{n}\tilde{\mathbf{X}}_{n}\tilde{\mathbf{Y}}_{n}^{T}+\frac{1}{2} \tilde{\mathbf{H}}_{n}\tilde{\mathbf{X}}_{n}\tilde{\mathbf{X}}_{n}^{T}\tilde{\mathbf{H}}_{n}^ {T}\bigg{\}}. \tag{36}\]  
By further defining matrices  
\[\tilde{\mathbf{X}}=\text{diag}\Big{\{}\Big{[}\tilde{\mathbf{X}}_{1}\quad\tilde{\mathbf{X }}_{2}\quad\cdots\quad\tilde{\mathbf{X}}_{N_{f}}\Big{]}\Big{\}}\in\mathbb{R}^{2 KN _{l}N_{f}\times KN_{r}N_{f}}, \tag{37}\]  
\[\tilde{\mathbf{Y}}=\text{diag}\Big{\{}\Big{[}\tilde{\mathbf{Y}}_{1}\quad\tilde{\mathbf{Y}} _{2}\quad\cdots\quad\tilde{\mathbf{Y}}_{N_{f}}\Big{]}\Big{\}}\in\mathbb{R}^{2 KN _{r}N_{f}\times KN_{r}N_{f}}, \tag{38}\]  
\[\tilde{\mathbf{H}}=\text{diag}\Big{\{}\Big{[}\tilde{\mathbf{H}}_{1}\quad\tilde{\mathbf{H}} _{2}\quad\cdots\quad\tilde{\mathbf{H}}_{N_{f}}\Big{]}\Big{\}}\in\mathbb{R}^{2 KN_{l}N _{f}\times 2 KN_{l}N_{f}}, \tag{39}\]  
we can obtain the final objective function as  
\[F(\tilde{\mathbf{X}})=-\text{trace}\Big{\{}\tilde{\mathbf{H}}\tilde{\mathbf{X}}\tilde\mathbf{{Y}}^{T}\Big{\}}+\frac{1}{2}\text{trace}\Big{\{}\tilde{\mathbf{H}}\tilde{\mathbf{X}} \tilde{\mathbf{X}}^{T}\tilde{\mathbf{H}}^{T}\Big{\}}. \tag{40}\]  
Consequently, we construct  
\[ \begin{split}&\min_{\tilde{\mathbf{X}}}F(\tilde{\mathbf{X}})=-\text{ trace}\Big{\{}\tilde{\mathbf{H}}\tilde{\mathbf{X}}\tilde{\mathbf{Y}}^{T}\Big{\}}+\frac{1}{2}\text{ trace}\Big{\{}\tilde{\mathbf{H}}\tilde{\mathbf{X}}\tilde{\mathbf{X}}^{T}\tilde{\mathbf{H}}^{T} \Big{\}},\\ &\text{s.t.}\big{\|}\tilde{\mathbf{X}}\big{\|}_{F}=\sqrt{N_{t}N_{r}N_{f} K\sigma_{x}^{2}}.\end{split}  \tag{41}\]As in Section 3, by defining the matrix  
\[\mathbf{\hat{S}}=\frac{1}{\sqrt{N_{t}N_{r}N_{f}K\sigma_{x}^{2}}}\mathbf{\hat{X}}, \tag{42}\]  
we can transform (41) into the following Riemannian optimization problem:  
\[ \begin{split}&\min_{\mathbf{\hat{S}}}F(\mathbf{\hat{S}})=-\text{ trace}\Big{\{}\mathbf{\tilde{H}}\mathbf{\hat{S}}\mathbf{\hat{Y}}^{T}\Big{\}}+\frac{1}{2}\sqrt{N_{t}N_{r}N _{f}K\sigma_{x}^{2}}\text{trace}\Big{\{}\mathbf{\tilde{H}}\mathbf{\hat{S}}\mathbf{\hat{Y}}^ {T}\mathbf{\tilde{H}}^{T}\Big{\}},\\ &\text{s.t.}\big{\|}\mathbf{\hat{S}}\big{\|}_{F}=1.\end{split}