## 2 Theory and Method
### Construction of Low-Rank Tensor Completion Model
#### 2.2.2 Construction of Low-Rank Tensor Completion Model  
The low-rank matrix completion (LRMC) model uses the low-rank structure at the bottom to restore the incomplete matrix [21], defining the rank of the tensor \(\mathcal{X}\in\mathbb{R}^{l_{1}l_{2}l_{3}}\) as \(\text{rank}(\mathcal{X})\) and the objective function of low-rank tensor completion as:  
\[\left\{ \begin{array}{c}\underset{\mathcal{X}}{\text{min}}\text{rank}( \mathcal{X})\\ s.t.\mathcal{X}_{\Omega}=\mathcal{T}_{\Omega}\end{array} \right. \tag{1}\]  
where \(\mathcal{X}\in\mathbb{R}^{l_{1}l_{2}l_{3}}\) is a tensor to be solved; \(\mathcal{T}\in\mathbb{R}^{l_{1}l_{2}l_{3}}\) is a missing tensor; \(\Omega\) is the observed landslide data in \(T\). Then, the tensor kernel norm is replaced by the rank minimum [22], and the concept of tensor kernel norm is defined:  
\[\left\|\ \mathcal{X}\ \right\|_{*}=\sum\limits_{i=1}^{3}\alpha_{i}\left\|\ X_{(i)} \ \right\|_{*} \tag{2}\]  
where \(\alpha_{i}\) is the weighting factor and \(X_{(i)}\) is the matrix expanded along the \(i\)-th mode for tensor \(\mathcal{X}\). The three pattern expansion matrices of the data tensor \(\mathcal{X}\in\mathbb{R}^{l_{1}l_{2}l_{3}}\) are shown in Formula (3):  
\[\left\{ \begin{aligned} X_{(1)}&\in\mathbb{R}^{l_{1} \times(l_{2}\times l_{3})}\\ X_{(2)}&\in\mathbb{R}^{l_{2}\times(l_{1}\times l_{3})}\\ X_{(1)}&\in\mathbb{R}^{l_{3}\times(l_{1}\times l_{2})} \end{aligned} \right. \tag{3}\]  
where \(X_{(1)}\), \(X_{(2)}\), \(X_{(3)}\) are the module expansion matrix for the staggered sampling of three types of expansion patterns, respectively. The features of different orders between data are fused with each other through three module expansion matrices, effectively ensuring the high-precision completion of data [23]. Below is the singular value decomposition of each pattern expansion matrix [20]:  
\[\left\{ \begin{array}{c}X_{(1)}=U_{l_{1}\times l_{1}}\underset{l_{1}\times l _{2}l_{3}}{\overset{(1)}{\sum}}M^{T}_{l_{2}l_{3}\times l_{2}l_{3}}\\ X_{(2)}=V_{l_{2}\times l_{2}}\underset{l_{2}\times l_{1}l_{3}}{\overset{(2 )}{\sum}}L^{T}_{l_{1}l_{3}\times l_{1}l_{3}}\\ X_{(3)}=W_{l_{3}\times l_{3}}\underset{l_{3}\times l_{1}l_{2}}{\overset{(3 )}{\sum}}K^{T}_{l_{1}l_{2}\times l_{1}l_{2}}\end{array} \right. \tag{4}\]  
where \(U\), \(V\) and \(W\) are left singular matrices; \(M\), \(L\) and \(K\) are right singular matrices; and \(\Sigma\) is a diagonal matrix. It is defined as:  
\[\sum{{}^{(q)}}=diag\Big{(}\sigma_{1}{{}^{(q)}},\sigma_{2}{{}^{(q)}},\cdots, \sigma_{n_{q}}{{}^{(q)}}\Big{)},q=1,2,3 \tag{5}\]  
where \(n_{q}\) is the total number of singular values of the \(q\)-th module expansion matrix. Based on the definition of tensor kernel norm, the objective function is:  
\[\left\{ \begin{array}{c}\underset{\mathcal{X}}{\text{min}}\ \underset{i=1}{\overset{3}{\sum}}\ \alpha_{i}||X_{(i)}||\\ s.t.\mathcal{X}_{\Omega}=\mathcal{T}_{\Omega}\end{array} \right. \tag{6}\]  
where the matrix core norm is defined as [24]:  
\[\left\|\ X\ \right\|_{*}=\sum\limits_{k=1}^{j}\sigma_{k}(X) \tag{7}\]where \(\left\|\cdot\right\|_{*}\) is the kernel norm of \(X;j\) is the rank of matrix \(X;\sigma_{k}\) is the \(k\)-th singular value of the matrix \(X\) in order of size. Because of the existence of interdependent matrix kernel norm terms, the objective function of low-rank tensor completion is difficult to be solved by ordinary methods. Alternating direction method of multipliers (ADMM) [23; 25] is a more widely used optimization method for constrained problems in machine learning. It is an extension of the augmented Lagrange method. ADMM algorithm provides a framework for solving constrained optimization problems with linear equations, which is convenient for disassembling the original optimization problem into several relatively solvable sub-optimization problems for iterative solutions. With continuous research, the low-rank tensor completion method is further developed. The purpose of reducing the computational complexity is to use QR decomposition instead of singular value decomposition.