## 4 Semi-Blind Receiver  
The ALS algorithm is a classical solution for fitting tensor models. However, it is well known that the ALS algorithm exhibits a convergence problem when collinearity is present in one or more modes [27; 28]. The LM algorithm is successfully used to fit the PARAFAC and PARATUCK2 models, adapted to collinearity problems, and provide quadratic convergence [19; 20]. As an iterative algorithm, the LM algorithm is also sensitive to initialization. Thus, the optimization of the initial value is important to improving the performance of the LM algorithm. In this section, a novel semi-blind receiver based on the optimized LM algorithm is developed for joint symbol and channel estimation. The basic principle of the optimized LM algorithm is to first resort to a LSK approximation problem [29; 30], based on the singular value decomposition (SVD) of rank-1 matrix to initialize the symbol and channel matrices, and then update these two matrices at the same time in each iteration. Finally, the modified singular value projection (SVP) based algorithm [31; 32] is used to further improve the performance of channel estimation. The proposed optimal initialization method is based on the Kronecker least squares algorithm, which exploits SVD-based rank-one approximations to get an initial estimation of \(\mathbf{S}\) and \(\mathbf{H}\) from their Kronecker matrix product. By post-multiplying Equation (14) with \(\mathbf{F}_{3}^{\dagger}\), we get \(\mathbf{Z}=\left(\mathbf{\hat{S}}^{(0)}\otimes\mathbf{\hat{H}}^{(0)}\right)= \mathbf{Y}_{3}\mathbf{F}_{3}^{\dagger}\in\mathbb{C}^{NM_{D}\times RM_{S}}\), where \(\mathbf{\hat{S}}^{(0)}\) and \(\mathbf{\hat{H}}^{(0)}\) are initial estimates of \(\mathbf{S}\) and \(\mathbf{H}\). According to the Theorem 2.1 in [29], we have:  
\[\left\|\mathbf{Z}-\mathbf{\hat{S}}^{(0)}\otimes\mathbf{\hat{H}}^{(0)}\right\| _{F}^{2}=\left\|\mathbf{\Xi}-vec\left(\mathbf{\hat{H}}^{(0)}\right)\left(vec \left(\mathbf{\hat{S}}^{(0)}\right)\right)^{T}\right\|_{F}^{2} \tag{25}\]  
where \(\mathbf{\Xi}=unvec(\mathbf{\Delta})\in\mathbb{C}^{M_{D}M_{S}\times NR}\) is a rank-one matrix, and \(\mathbf{\Delta}\in\mathbb{C}^{NM_{D}RM_{S}\times 1}\) is, given that:  
\[\mathbf{\Delta}=\left[ \begin{array}{c}vec(\mathbf{Z}(1:M_{D},\;1:M_{S}))\\ \vdots\\ vec(\mathbf{Z}((N-1)M_{D}+1:NM_{D},\;1:M_{S}))\\ \vdots\\ vec(\mathbf{Z}(1:M_{D},\;(R-1)M_{S}+1:RM_{S}))\\ \vdots\\ vec(\mathbf{Z}((N-1)M_{D}+1:NM_{D},\;(R-1)M_{S}+1:RM_{S})).\end{array} \right] \tag{26}\]  
In this case, the Kronecker product matrix \(\mathbf{Z}\) has been rearranged into a rank-one matrix \(\mathbf{\Xi}\). Applying SVD to the rank-one matrix \(\mathbf{\Xi}\), the vectors \(vec\left(\mathbf{\hat{S}}^{(0)}\right)\) and \(vec\left(\mathbf{\hat{H}}^{(0)}\right)\) can be estimated by using a rank-one approximation method, i.e., by computing its largest singular value and the corresponding left and right singular vectors. \(\mathbf{\hat{S}}^{(0)}\) and \(\mathbf{\hat{H}}^{(0)}\) are determined up to a scaling factor, which can be removed by setting \(s_{1,1}=1\) as in [27; 30]. The detailed process is shown below. By applying SVD to the rank-one matrix \(\mathbf{\Xi}\), we have:  
\[\mathbf{\Xi}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{H} \tag{27}\]  
where \(\mathbf{\Sigma}\in\mathbb{C}^{M_{D}M_{S}\times NR}\) is a diagonal matrix containing singular values of \(\mathbf{\Xi}\), \(\mathbf{U}\in\mathbb{C}^{M_{D}M_{S}\times M_{D}M_{S}}\) and \(\mathbf{V}\in\mathbb{C}^{NR\times NR}\) are unitary matrices. Using the rank-one approximation of \(\mathbf{\Xi}\), we have:  
\[\mathbf{\Xi}\approx\mathbf{U}_{1}\sigma_{1}\mathbf{V}_{1}^{H} \tag{28}\]  
where \(\sigma_{1}\) is the largest singular value, and \(\mathbf{U}_{1}\) and \(\mathbf{V}_{1}\) are the corresponding left and right singular vectors.