## 3 Preliminaries  
### Introduction to Tensors  
This section briefly introduces the definitions, notation, and tensor operations used in this study. Table 2 lists mathematical symbols, definitions, and commonly used abbreviations. \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}\cdots\times n_{N}}\) represents a multiway array of the \(N\)th order, also referred to as an N-way array or \(N\)th-order tensor. In a specific instance in which N = 3, \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) is defined as a third-order tensor. A slice is a data partition that occurs when one mode index is fixed, whereas the indices in the other two modes vary. The slices of a third-order tensor are illustrated in Figure 1. Fibers of the third-order tensor were obtained by maintaining all except one dimension constant, as shown in Figure 2. \begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline \multirow{2}{*}{**Paper**} & \multirow{2}{*}{**Year**} & \multirow{2}{*}{**Multivariate**} & \multirow{2}{*}{**Univariate**} & \multicolumn{4}{c}{**Corrupted Type**} & \multirow{2}{*}{**Outlier**} & \multirow{2}{*}{**Weighted**} & \multirow{2}{*}{**Method**} \\ \cline{5-6} \cline{8-10}  & & & & & & & & & & \\ \hline
[11] & 2021 & & X & X & X & & & & TRPCA \\ \hline
[12] & 2021 & X & & X & & & X & & TRPCA-TNN \\ \hline
[13] & 2022 & & X & X & X & & & X & WRPCA \\ \hline Our & 2024 & X & & X & X & X & X & X & WRPTCA-WTNN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Recent works on RPCA in IoTs. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(\mathcal{G}\), \(\mathcal{G}\), \(\mathcal{G}\) & scalar, vector, matrix, and tensor formats \\ \(\mathcal{G}_{i}\) & \(i\)th singular value \\ \(\mathcal{G}(i,:,:)\), \(\mathcal{G}(:,i:)\), \(\mathcal{G}(:,:,i)\) & \(i\)th horizontal slice, \(i\)th lateral slice, and \(i\)th frontal slice \\ \(\mathcal{G}^{(i)}\) & \(\mathcal{G}^{(i)}=\mathcal{G}(:,:i:)\), \(i\)th frontal slice \\ \(\mathcal{\tilde{G}}\) & Fourier transformation of \(\mathcal{G}\), \(\mathcal{\tilde{G}}=fft(\mathcal{G},[|,3)\) \\ \(||G||_{F}\) & Frobenius norm \(||G||_{F}=\sqrt{\sum_{i,j}G(i,j)^{2}}\) \\ \(||G||_{*}\) & nuclear norm \(||G||_{*}=\sum_{i=1}^{m_{1}(n_{1},n_{2})}\sigma_{i}\) \\ \(||G||_{*,\mu}\) & weighted nuclear norm \(||G||_{*,\mu}=\sum_{i=1}^{m_{1}(n_{1},n_{2})}\sigma_{i}\) \\ \(||\mathcal{G}||_{F}\) & tensor Frobenius norm \(||\mathcal{G}||_{F}=\sqrt{\sum_{i,j}\mathcal{G}(i,j,k)^{2}}\) \\ \(||\mathcal{G}||_{*}\) & tensor nuclear norm \(||\mathcal{G}||_{*}=\sum_{j=1}^{n_{3}}\sum_{i=1}^{min(n_{1},n_{2})}\sigma_{ij}\) \\ \(||\mathcal{G}||_{*,\mu}\) & weighted tensor nuclear norm \(||\mathcal{G}||_{*,\mu}=\sum_{j=1}^{n_{3}}\sum_{i=1}^{min(n_{1},n_{2})}\sigma_{ ij}\omega_{ij}\) \\ \(rank(\mathcal{G})\) & rank of matrix \(\mathcal{G}\) \\ \(rank(\mathcal{G})\) & tubal rank of tensor \(\mathcal{G}\) \\ t-SVD & tensor singular value decomposition \\ ADMM & alternative direction method of multipliers \\ WTNN & weighted tensor nuclear norm \\ \(\Omega\) & position of missing elements in \(\mathcal{G}\) \\ \(\mathcal{P}_{\Omega}\) & projection on \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical notations and definitions. Figure 1: Slices of third-order tensor \(\mathcal{G}\): (**a**) frontal slices; (**b**) lateral slices; (**c**) horizontal slices. For the tensor \(\mathcal{G}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), the \(i\)th frontal slices are indicated by \(\mathcal{G}^{(i)}\), where \(i\) ranges from one to \(n_{3}\). The operation \(unfold\) is described as follows:  
\[unfold(\mathcal{G})= \begin{bmatrix}G^{(1)}\\ G^{(2)}\\ \vdots\\ G^{(n_{3})}\end{bmatrix}  \tag{1}\]  
Furthermore, the \(fold\) operation is as follows:  
\[fold\left( \begin{bmatrix}G^{(1)}\\ G^{(2)}\\ \vdots\\ G^{(n_{3})}\end{bmatrix} \right)=\mathcal{G} \tag{2}\]  
The block circulant matricization operator, denoted by \(circ\), was defined in [21].