## 2 System Model
### Multi-User UAV Communication Channel Model with RIS
According to FCTPM, the higher-order tensor is decomposed in the Tensor-train decomposition by a low-rank decomposition of the auxiliary expansion matrix of the input tensor. The solution to the factor vector \(g^{(n)}\) can be solved in a uniform manner by expressing:  
\[ \begin{split}&\min_{g^{(N-1)},g^{(N)}}\left\|Y_{(N-1)}-d^{(N-1)}g^{ (N-1)}{g^{(N)}}^{T}\right\|_{F}^{2}\\ & s.\ t.\quad d>0,{g^{(N-1)}}^{T}g^{(N-1)}=1,\ g^{(N)}{}^{T}g^{(N )}=1\end{split}  \tag{39}\]  
where \(Y_{(N-1)}=\mathit{reshape}\left(v^{(N-2)},\left[I_{N-1},I_{N}\right]\right),\ d^{(N-1)}\) is the principal singular value after reshaping the matrix. The principal singular value of the rank-negative-one tensor can be calculated as \(d=\prod_{i=1}^{N-1}d^{(i)}\), which is the corresponding factor vector. \(g^{(N-1)}\) and \(g^{(N)}\) are the corresponding factor vectors, from which the factor vector can then be composed to derive the corresponding factor matrix. The solution process is repeated until the right singular vector in the decomposition process \(v^{(N-1)}\) is the last factor vector \(g^{(N)}\). Therefore, the first factor vector \(g^{(1)}\) can be expressed as follows:  
\[ \begin{split}&\min_{g^{(1)},g^{(1)}}\left\|Y_{(1)}-d^{(1)}g^{(1)}{v^ {(1)}}^{T}\right\|_{F}^{2}\\ & s.\ t.\quad d^{(1)}>0,{g^{(1)}}^{T}g^{(1)}=1,\ {v^{(1)}}^{T}v^{(1)}=1\end{split}  \tag{40}\]  
where \(Y_{(1)}\) is modulo 1 expansion matrix, \(d^{(1)}\) is the principal singular value of \(Y_{(1)}\), and \(g^{(1)}\) represents the first factor vector of the rank-negative-one Tensor-train decomposition. \(v^{(1)}\) is the right singular vector of \(Y_{(1)}\). After obtaining \(g^{(1)}\), \(v^{(1)}\) is expressed as the modulo 2 expansion matrix form of the received signal tensor \(\mathcal{Y}\). The second factor vector \(g^{(2)}\) is calculated as:  
\[Y_{(2)}=\mathit{reshape}\left(v^{(1)},\left[I_{2},\prod_{j=3}^{N}I_{j}\right]\right) \tag{41}\]  
The rank-negative-one matrix decomposition for \(Y_{(2)}\) is expressed as the following equation:  
\[ \begin{split}&\min_{g^{(2)},g^{(2)}}\left\|Y_{(2)}-d^{(2)}g^{(2)}{v^ {(2)}}^{T}\right\|_{F}^{2}\\ & s.\ t.\quad d^{(2)}>0,{g^{(2)}}^{T}g^{(2)}=1,\ {v^{(2)}}^{T}v^{(2)}=1\end{split}  \tag{42}\]where \(d^{(2)}\) is the principal singular value of the tensor \(\mathcal{Y}\) modulo 2 matrices \(Y_{(2)}\), and \(g^{(2)}\) represents the second factor vector of the rank-negative-one Tensor-train decomposition. Similarly, the third factor vector can be obtained. This computation can be performed repeatedly until the right singularity vector of the decomposition process is the last factor vector. To ensure convergence of the factorization, a matrix factorization method PM is incorporated. In solving the first-factor matrix, the PM method initializes the factor vector \(g^{(1)}\) to a random unit vector satisfying \(\left\|g^{(1)}\right\|_{2}=1\). After that, the vector-matrix operation is performed iteratively until the factor vectors satisfy the pre-set specific termination conditions, as shown below:  
\[v^{(1)}=\frac{Y_{(1)}^{T}g^{(1)}}{\left\|Y_{(1)}^{T}g^{(1)}\right\|_{2}}e \mathbb{R}^{\prod_{j=2}^{N_{2}}j} \tag{43}\]  
\[g^{(1)}=\frac{Y_{(1)}v^{(1)}}{\left\|Y_{(1)}v^{(1)}\right\|_{2}}e\mathbb{R}^{ I_{1}} \tag{44}\]  
where \(g^{(1)}\) and \(v^{(1)}\) are \(Y_{(1)}\) the left and right singular vectors, respectively. The main singular value \(d^{(1)}\) associated with it can be expressed as \(d^{(1)}=g^{(1)}{}^{T}Y_{(1)}v^{(1)}\). The left singular vector calculation follows immediately after the right singular vector calculation. Assuming the existence of an orthogonal basis \(\overline{U}e\mathbb{R}^{I_{1}\times R}\) and \(\overline{V}e\mathbb{R}^{I_{2}\cdots I_{N}\times R}\), the unfolding matrix \(Y_{(1)}\) satisfies the following equation:  
\[Y_{(1)}=\sum_{j=1}^{R}d_{j}\overline{U}_{:j}\overline{V}_{:,j}^{T} \tag{45}\]  
where \(d_{j}\) is the singular value of the singular vector, which corresponds to \(Y_{(1)}\).