## 4 The Proposed Model
The inputs for WTSVT include the input tensor \(\mathcal{M}\) and the weighted vector \(\omega\), and \(\beta\) is the penalty factor. The notation \(fft\) represents the FFT used to compute the discrete Fourier transform (DFT) of a sequence. In contrast, \(ifft\) represents the inverse fast Fourier, which essentially reverses the FFT process. First, the input tensor is transformed using \(fft\). Next, the loop of Algorithm 2 begins, and each frontal slice of the input tensor is applied (\(SVD\)). The process \(SVD\) breaks down a matrix into three separate matrices--a left singular matrix, a diagonal matrix of singular values, and a matching singular matrix. The \(diag\) pertains to the diagonal elements of the singular matrix. The operation \(diag(\mathbf{D})-\frac{\omega}{\beta}\) takes each diagonal component, converts it into a vector, and subtracts it, with each value in vector \(\omega\) already divided to \(\beta\). Subsequently, the \(diag(diagD)\) converts \(diagD\) into a diagonal matrix with shrunken singular values. The weight shrinks the diagonal elements in the singular matrix D. After completing \(SVD\) for all frontal slices, \(ifft\) is used with the three components to obtain results. ```
Input:\(\mathcal{B}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) with missing values, weighted vector \(\mathbf{\omega}\) Initialize:\(\mathcal{A}_{0}=\mathcal{B}\), \(\mathcal{W}_{0}=\mathcal{B}\), \(\mathcal{U}_{0}=\mathcal{O}\), \(\epsilon=1\times 10^{-8}\), \(\rho=1.1\), \(\beta_{0}=1\times 10^{-1}\) while not converged do  Update \(\mathcal{W}_{i+1}=\textbf{WTSVT}(\mathcal{A}_{i}+\frac{\mathcal{U}_{i}}{ \beta_{i}},\omega,\beta_{i})\)  Update \(\mathcal{A}_{i+1}=\mathcal{W}_{i+1}-\frac{1}{\beta_{i}}\mathcal{U}_{i}\), \(\mathcal{P}_{\Omega}(\mathcal{A}_{i+1})=\mathcal{P}_{\Omega}(\mathcal{B})\)  Update \(\mathcal{U}_{i+1}=\mathcal{U}_{i}+\beta_{i}(\mathcal{A}_{i+1}-\mathcal{W}_{i+ 1})\)  Update \(\beta_{i+1}=\rho\)\(\beta_{i}\)  Check the convergence criteria \(\|\mathcal{A}_{i+1}-\mathcal{W}_{i}\|_{F}^{2}\leq\epsilon\) or \(\|\mathcal{W}_{i}-\mathcal{W}_{i+1}\|_{F}^{2}\leq\epsilon\) endwhile Output: low-rank tensor without missing value \(\mathcal{A}\)
```  
**Algorithm 3** Weighted tensor completion. The solution to Equation (18) is obtained through WTSVT (Algorithm 2), and Equation (19) represents the least-squares projection constrained by the problem [42]. The \(\delta\)-indicator function maps the elements of a subset of a set to one and all other elements to zero. Following MATLAB R2021a notation, the symbols \(\mathcal{U}(:)\) and \(\mathcal{G}(:)\) signify the tensor-to-vector conversion. To recover the missing values, the ADMM is implemented, the details of which are provided in Algorithm 3. The input to Algorithm 3 includes a tensor with the missing values \(\mathcal{B}\) and weighted vector \(\mathbf{\omega}\). First, we initialize the tensors \(\mathcal{A}\), \(\mathcal{W}\), and \(\mathcal{U}\) as \(\mathcal{B}\), \(\mathcal{B}\), and \(\mathcal{O}\), respectively. Here, \(\mathcal{A}\) is the output vector without any missing values, \(\mathcal{W}\) is the tensor that helps the algorithms ensure the constraint, and \(\mathcal{U}\) is the dual factor of the ADMM, which represents the augmented Lagrange penalty parameter. The step size is denoted as \(\beta\). An \(n_{1}\times n_{2}\times n_{3}\)-tensor with all its elements set to zero is denoted as \(\mathcal{O}\). The other parameters \(\epsilon\) and \(\rho\) are the threshold stopping algorithm and the degree of increase in the step size \(\beta\) in each iteration, respectively. Starting with the loop in Algorithm 3, \(\mathcal{W}\) is updated using WTSVT in Algorithm 2. Furthermore, \(\mathcal{A}\) is updated; however, the constraint \(\mathcal{P}_{\Omega}(\mathcal{A}_{i+1})=\mathcal{P}_{\Omega}(\mathcal{B})\) must be ensured. The dual factor \(\mathcal{U}\) and step size \(\beta\) are also updated after that.