## 2 System Model
### Multi-User UAV Communication Channel Model with RIS
Meanwhile, Equation (32) can continue to be expressed as a Tensor-train decomposition of the _Kruskal_ form as follows:  
\[\mathcal{Y}=\mathcal{T};U_{1},U_{2},U_{3}+\mathcal{N} \tag{33}\]  
\[U_{1} \tag{34}\]  
\[U_{2}=\left[\widetilde{\kappa}_{1}(t),\cdots\widetilde{\kappa}_{Q}(t)\right] \in\mathbb{C}^{N^{T}_{RF}\times Q} \tag{35}\]  
\[U_{3}=\left[\widetilde{\alpha}_{T}\Big{(}\gamma_{1}{}^{T}(t)\Big{)},\cdots \widetilde{\alpha}_{T}\Big{(}\gamma_{Q}{}^{T}(t)\Big{)}\right]\in\mathbb{C}^{ T\times Q} \tag{36}\]  
The FCTPM based on tensor decomposition is utilized by the power method (PM) for the auxiliary unfolding matrix decomposition. FCTPM is based on a combination of Tensor-train decomposition and power. The FCTPM approximates the most dominant rank-negative-one tensor of the input tensor by the Tensor-train decomposition. Each unfolding matrix is also decomposed using a rank-negative-one matrix decomposition. This property allows each unfolding matrix to ignore the concatenation between other unfolding matrices. Therefore, a cyclic update method is proposed to consider the connectivity between the unfolding matrices. After all of the left singular vectors have been computed, the order of the update steps for the right singular vectors needs to be changed. FCTPM utilizes the PM for the auxiliary unfolding matrix decomposition. Therefore, the PM ensures the convergence of the decomposition factors. The cyclic updating method enhances the connectivity between the decomposition factors. The FCTPM consists of two main components, namely the rank-negative-one tensor approximated by the Tensor-train decomposition of the auxiliary expansion matrix and the PM of determining the left and right singular vectors obtained from the randomly initialized vectors. The core tensor \(\mathcal{G}^{(n)}\) calculation in Tensor-train decomposition is based on low-rank approximation of the auxiliary matrix of \(N\)-dimensional input tensor \(\mathcal{Y}\)\(\in\)\(\mathbb{R}^{1_{1}\times\cdots I_{N}}\), and \(R_{N}\) denotes the Tensor-train rank in Tensor-train decomposition of the received signal tensor \(\mathcal{Y}\). In FCTPM, a rank-negative-one approximation is made to the high-dimensional input tensor using the Tensor-train decomposition. The received signal at the user is compressed into a low-order core tensor \(\mathcal{G}^{(1)}\), \(\mathcal{G}^{(2)}\), and \(\mathcal{G}^{(3)}\) of the concatenation. The received signal tensor \(\mathcal{Y}\) is written in the following form:\[ \begin{split}\mathcal{Y}=\sum_{r_{1}=1}^{R_{1}}&\cdots \sum_{r_{2}=1}^{R_{2}}\mathcal{G}_{r_{0},r_{1}}^{(1)}\circ\mathcal{G}_{r_{1},r _{2},r_{2}}^{(2)}\circ\mathcal{G}_{r_{2},r_{3}}^{(3)}+\mathcal{N}\\ &=\mathcal{G}_{1,\cdot\cdot\cdot}^{(1)}\circ\mathcal{G}_{1, \cdot\cdot\cdot}^{(2)}\circ\mathcal{G}_{1,\cdot\cdot\cdot}^{(3)}+\mathcal{N} \\ &=g^{(1)}\ \circ g^{(2)}\circ g^{(3)}+\mathcal{N}\end{split}  \tag{37}\]  
where \(\mathcal{G}^{(n)}\) denotes the core tensor of size \(R_{n-1}\times I_{n}\times R_{n}\) of the core tensor of the rank-negative-one Tensor-train, with all elements of the Tensor-train rank being 1. Then, \(\mathcal{G}^{(n)}\) size become \(1\times I_{n}\times 1\) the core tensor of the rank-negative-one Tensor-train. Furthermore, if the core tensor \(\mathcal{G}^{(n)}\) of dimensionality is compressed, the core tensor can be regarded as factored vector data \(g^{(n)}e\mathbb{R}^{I_{n}}\) where \(g^{(n)}\) is the rank-negative-one Tensor-train core tensor \(\mathcal{G}_{1,\cdot\cdot\cdot}^{(n)}\) of the first-order factor vector. To solve the factor vector \(g^{(n)}\), Equation (32) is expressed as the following Equation (38):  
\[ \begin{split}&\min_{g^{(1)},g^{(3)}}\left\|\mathcal{Y}-dg^{(1)} \circ g^{(2)}\circ g^{(3)}\right\|^{2}\\ & s.\ t.\quad d>0,{g^{(n)}}^{T}g^{(n)}=1(n=1,2,3)\end{split}  \tag{38}\]  
where \(d\) is the principal singular value of the approximate rank-negative-one tensor.