## 4 Semi-Blind Receiver
Thus, vectors \(vec\left(\mathbf{\hat{S}}^{(0)}\right)\) and \(vec\left(\mathbf{\hat{H}}^{(0)}\right)\) can be estimated as:  
\[vec\left(\mathbf{\hat{S}}^{(0)}\right)=\alpha\mathbf{V}_{\cdot 1}^{*},\;\;\;vec \left(\mathbf{\hat{H}}^{(0)}\right)=\frac{\sigma_{1}}{\alpha}\mathbf{U}_{\cdot 1} \tag{29}\]  
where \(\alpha\) is the scalar factor. \(vec\left(\mathbf{\hat{S}}^{(0)}\right)\) and \(vec\left(\mathbf{\hat{H}}^{(0)}\right)\) are determined up to this scalar factor. In practical communication systems, this scalar factor \(\alpha\) can be removed by setting \(s_{1,1}=1\). Thus, the value \(\alpha\) in this paper is equal to \(\frac{1}{v_{1,1}}\) when \(s_{1,1}=1\). Note that we can also choose Equation (14) to implement the above optimal initialization procedure. Define a parameter vector stacking all the unknowns as:  
\[\mathbf{u}=\left[\mathbf{u}_{\mathbf{S}}^{T},\ \mathbf{u}_{\mathbf{H}}^{T}\right]^{T} \in\mathbb{C}^{Q\times 1} \tag{30}\]  
where \(\mathbf{u}_{\mathbf{S}}=vec\left(\mathbf{S}^{T}\right)\in\mathbb{C}^{NR\times 1}\), \(\mathbf{u}_{\mathbf{H}}=vec\left(\mathbf{H}^{T}\right)\in\mathbb{C}^{M_{D}M_{ \mathrm{S}}\times 1}\), and \(Q=NR+M_{D}M_{\mathrm{S}}\). The cost function to be minimized is given by:  
\[ \begin{split}\phi\left(\mathbf{u}\right)&=\frac{1}{ 2}\sum_{m_{D}=1}^{M_{D}}\sum_{n=1}^{N}\sum_{p=1}^{p}\left|\tilde{y}_{m_{D},n,p} \left(\mathbf{u}\right)-y_{m_{D},n,p}\right|^{2}\\ &=\frac{1}{2}\sum_{l=1}^{L}\left|z_{l}\left(\mathbf{u}\right) \right|^{2}=\frac{1}{2}\mathbf{z}^{H}\left(\mathbf{u}\right)\mathbf{z}\left( \mathbf{u}\right)\end{split}  \tag{31}\]  
where \(\tilde{y}_{m_{D},n,p}\left(\mathbf{u}\right)\) is the typical element of the tensor \(\mathbf{\tilde{Y}}\left(\mathbf{u}\right)\in\mathbb{C}^{M_{D}\times N\times P}\), which denotes the output tensor in absence of noise. \(\mathbf{z}\left(\mathbf{u}\right)=vec\left(\mathbf{\tilde{Y}}\left(\mathbf{ u}\right)\right)-vec\left(\mathbf{\tilde{Y}}\right)=\mathbf{\tilde{y}}\left( \mathbf{u}\right)-\mathbf{y}\in\mathbb{C}^{NPM_{D}\times 1}\) denotes the vector of residuals and \(L=NPM_{D}\). Let the \(\mathbf{J}\in\mathbb{C}^{L\times Q}\) be the Jacobian matrix of \(\mathbf{z}\left(\mathbf{u}\right)\) with respect to \(\mathbf{u}\), and \(\mathbf{g}\) be the gradient of \(\phi\left(\mathbf{u}\right)\) with respect to \(\mathbf{u}\). \(\mathbf{J}\) and \(\mathbf{g}\) are respectively defined by:  
\[J_{l,q}=\partial z_{l}\left(\mathbf{u}\right)/\partial\mathbf{u}_{q}=\partial \tilde{y}_{l}\left(\mathbf{u}\right)/\partial\mathbf{u}_{q} \tag{32}\]  
\[\mathbf{g}=\partial\phi\left(\mathbf{u}\right)/\partial\mathbf{u}=\mathbf{J} ^{H}\left(\mathbf{u}\right)\mathbf{z}\left(\mathbf{u}\right). \tag{33}\]  
The optimized LM algorithm consists in optimizing \(\mathbf{u}^{(0)}\), and estimating \(\mathbf{u}^{(i+1)}\) at the \((i+1)\)-th iteration from \(\mathbf{u}^{(i)}\) at the \(i\)-th iteration via \(\mathbf{u}^{(i+1)}\) = \(\mathbf{u}^{(i)}+\Delta\mathbf{u}^{(i)}\). The step \(\Delta\mathbf{u}^{(i)}\in\mathbb{C}^{NPM_{D}\times 1}\) is updated by solving the following modified normal equations:  
\[\left(\mathbf{J}^{(i)}{}^{H}\mathbf{J}^{(i)}+\lambda^{(i)}\mathbf{I}_{Q} \right)\Delta\mathbf{u}^{(i)}=-\mathbf{g}^{(i)} \tag{34}\]  
where \(\lambda^{(i)}\) is the damping parameter to ensure that \(\Delta\mathbf{u}^{(i)}\) is a descent direction. The whole procedure of the optimized LM algorithm used in our semi-blind receiver is listed in Algorithm 1.