## 2 Materials and Methods
### Methods
#### 2.2.2 Pretreatment of Data Tensioning
EWLRTC-TT adopts the automatic update weight \(W_{k}\) of the k-th mode matrix, then the goal can be formulated as:  
\[\min_{U_{k},V_{k},\sim\mathcal{X},W_{k}}\mathrm{J}=\sum_{k=1}^{N-1}\parallel W _{k}\odot\left(U_{k}V_{k}^{\mathrm{T}}-\mathcal{X}_{[k]}\right)\parallel_{F}^ {2}\mathrm{s.t.}\mathcal{X}_{\Omega}=\mathcal{T}_{\Omega} \tag{17}\]  
\(W_{k}\) no longer remains static at \(0\) and \(1\), instead using the estimation of missing positions, \(w_{ij}\in(0,1)\) to fill it, if element \(x_{ij}\) is known, \(w_{\bar{\jmath}}=1\). Figure 3: Flowchart of matrix decomposition method based on diagonal block matrix. This article constructs a maximization objective function. Based on the research in [30], the optimal convex approximation tensor kernel norm can replace tensor rank. Combining the characteristics of tensor rank, the following optimization problem is obtained:  
\[\min_{X_{[k]},\mathcal{H}}\sum_{k=1}^{N-1}\alpha_{k}\parallel X_{[k]}\parallel_ {*}+\lambda\parallel\mathcal{H}\parallel_{1}\ \ \ s.t.\ \ \mathcal{D}_{\Omega}(y)=\mathcal{P}_{\Omega}(\mathcal{Z}), \mathcal{Z}=\mathcal{X}+\mathcal{H},\boldsymbol{\alpha^{T}}\boldsymbol{1}=1, \boldsymbol{\alpha}\geq 0 \tag{18}\]  
\(\parallel\cdot\parallel_{*}\) represents tensor kernel norms, \(\alpha_{k}\) represents the weight of \(X_{[k]}\); the matrix flattens along the k-th mode. \(\Omega\) is a partial observation set with unknown noise, \(\mathcal{U}\) is the number of damaged objects with missing entries, \(\mathcal{Z}\) is the recovery object, and \(l_{1}\) norms are used to separate sparse components from observations. Generate adaptive weights based on the change in matrix rank of Equation (18) above, and add the \(l_{2}\) norm term:  
\[\min_{U_{k},V_{k},\mathcal{L}}\sum_{k=1}^{N-1}\parallel M_{k}\odot\left(U_{k} V_{k}^{T}-\mathcal{X}_{[k]}\right)\parallel_{F}^{2}-y\parallel\boldsymbol{ \alpha}\parallel_{2}^{2},\ s.t.\ \ \ \mathcal{X}_{\Omega}=\mathcal{T}_{\Omega} \boldsymbol{\alpha^{T}}\boldsymbol{1}=1,\boldsymbol{\alpha}\geq 0 \tag{19}\]  
where \(\gamma,\lambda>0\), the first term of Equation (19) and comes from Equation (18), which is summarized and weighted \(\alpha\) relevant information. Flat matrix \(X_{[k]}\). The larger the kernel norm, the greater the weight forced to maintain more basic tensor data information. The new term in Equation (19) is the penalty term used to smoothen the weight distribution. To ensure the identifiability of the solution, the \(l_{2}\) norm penalty of \(U_{k}\) and \(V_{k}\) is introduced into the above formula. \[\min_{U_{k},V_{k}}\parallel W_{k}\odot\left(\mathcal{X}_{<k>}-U_{k}V_{k}^{ \prime}\right)\parallel_{p}-y\parallel\boldsymbol{\alpha}\parallel_{2}^{2}+ \frac{\lambda_{u}}{2}\parallel U_{k}\parallel_{2}^{2}+\frac{\lambda_{v}}{2} \parallel V_{k}\parallel_{2}^{2} \tag{20}\]  
Due to the use of \(\ell_{2}\) norm penalty, we update the weight \(W_{k}\) of the k-th mode through convex function. This function follows the equation used in:  
\[W_{k}=c\sqrt{\exp\Bigl{(}-\Xi\big{|}\mathcal{X}_{\langle k\rangle}-U_{k}V_{k} ^{\prime}\big{|}\Bigr{)}}. \tag{21}\]  
where hyperparameter \(c\) and \(\Xi\) is a normal number. Therefore, by iteratively calculating \(U_{k},V_{k}\) and \(W_{k}\), we can guarantee the (local) optimal solution. In order to deal with the identifiability of the permutation of the decomposition technique proposed in this paper, the r column vector is rearranged \(\left(\mathbf{p}_{1}^{k}+\mathbf{q}_{1,r}^{k},\mathbf{p}_{2}^{k}+\mathbf{q}_{2 }^{k},\ldots,\mathbf{p}_{r}^{k}+\mathbf{q}_{r}^{k}\right)\), and we can get the following equation:  
\[\sum_{k=1}^{d}\parallel\mathbf{p}_{\cdot 1}^{k}+\mathbf{p}_{\cdot 1}^{k} \parallel_{2}^{2}\geq\sum_{k=1}^{d}\parallel\mathbf{p}_{2}^{k}+\mathbf{p}_{ \cdot 2}^{k}\parallel_{2}^{2}\geq\cdots\geq\sum_{k=1}^{d}\mathbf{p}_{r}^{k}+ \mathbf{q}_{r}^{k}\parallel_{2}^{2} \tag{22}\]  
This is similar to applying a descending order of eigenvalues in matrix decomposition.