## 2 Preliminaries
### Problem Formulation  
A multi-dimensional representation is constructed for a structural state in SHM data by modeling these relationships as a third-order tensor, capturing high-dimensional features across various time points, locations, and working conditions. For an incomplete tensor \(\mathbf{Y}\), each element \(y_{ijk}\) represents the acceleration measurement recorded at the \(j\)th sensor location and \(i\)th time point under the \(k\)th working condition.  
Let \(\Lambda\) and \(\Gamma\) denote the known and unknown datasets of \(\mathbf{Y}\), respectively. The canonical polyadic (CP) tensor-based latent factor (LF) decomposition method [19; 20; 21; 22; 23; 24; 25; 26; 27] constructs an approximation \(\hat{\mathbf{Y}}\) of the original tensor \(\mathbf{Y}\), as shown in Equation (1) [19]:  
\[\hat{\mathbf{Y}}=\sum_{r=1}^{R}\mathbf{X_{r}} \tag{1}\]  
where each component tensor \(\mathbf{X}_{r}\) is defined as follows:  
The tensor \(\mathbf{Y}\) is decomposed into \(R\) rank-one tensors \(\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{R}\). Each rank-one tensor \(\mathbf{X}_{r}\in\mathbb{R}^{|I|\times|I|\times|K|}\) can be expressed as the outer product of three LF vectors \(\mathbf{t_{r}}\), \(\mathbf{l_{r}}\), and \(\mathbf{w_{r}}\) with lengths \(|I|\), \(|J|\), and \(|K|\), respectively, as shown in Equation (2) [19]:  
\[\mathbf{X_{r}}=\mathbf{t_{r}}\circ\mathbf{l_{r}}\circ\mathbf{w_{r}} \tag{2}\]  
Each element in \(\mathbf{X_{r}}\) is given by Equation (3) [20]:  
\[a_{ijk}=t_{i}lr_{j}w_{kr} \tag{3}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\mathbf{Y}\) & A time point–location–working condition tensor. \\ T, L, W & LF matrices for time points, locations and working conditions. \\ \(I\), \(J\), \(K\) & Sets of time points, locations, and working conditions. \\ \(y_{ijk}\) & A single element in \(\mathbf{Y}\). \\ \(\hat{\mathbf{Y}}\) & Low-rank approximation to \(\mathbf{Y}\) based on T, L, and W. \\ \(\hat{y}_{ijk}\) & A single element in \(\hat{\mathbf{Y}}\). \\ \(R\) & Rank of \(\hat{\mathbf{Y}}\). \\ \(\mathbf{t_{r}}\),\(\mathbf{l_{r}}\),\(\mathbf{w_{r}}\) & LF vectors for the \(r\)th rank-one tensor in \(\hat{\mathbf{Y}}\). \\ \(t_{ir}\), \(l_{ir}\), \(w_{kr}\) & Single LF in T, L, and W. \\ \(\mathbb{R}\) & Real number domain. \\ \(\Lambda\) & Known dataset of \(\mathbf{Y}\). \\ \(\Gamma\) & Unknown dataset of \(\mathbf{Y}\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbols used and their descriptions.  
The \(r\)th LF vectors from matrices T, L, and W collectively form the \(r\)th rank-one tensor \(\mathbf{X}_{r}\), yielding the approximation \(\hat{\mathbf{Y}}\). Each element \(\hat{y}_{ijk}\) in \(\hat{\mathbf{Y}}\) is computed as Equation (4) [20]:  
\[\hat{y}_{ijk}=\sum_{r=1}^{R}l_{it}l_{jr}s_{kr} \tag{4}\]  
Considering only the data in \(\Lambda\), the discrepancy between the corresponding elements of \(\mathbf{Y}\) and \(\hat{\mathbf{Y}}\) is measured using the Euclidean distance metric. To prevent overfitting, regularization terms are incorporated, yielding the objective loss function (Equation (5)) [21]:  
\[\varepsilon=\sum_{y_{ijk}\in\Lambda}\left((y_{ijk}-\hat{y}_{ijk})^{2}+\lambda_ {t}\sum_{r=1}^{R}t_{ir}^{2}+\lambda_{l}\sum_{r=1}^{R}l_{jr}^{2}+\lambda_{w} \sum_{r=1}^{R}w_{kr}^{2}\right) \tag{5}\]  
where \(\lambda_{t}\), \(\lambda_{l}\), and \(\lambda_{w}\) denote the regularization coefficients for the latent feature matrices T, L, and W, respectively.