## 3 RMSprop SGD-Based LFT Model
### Algorithm Design and Analysis
Detailed dataset specifications are provided in Table 2.

The undamaged structural state of the scaled bridge model was selected, with synchronized timestamps identified to construct a third-order tensor from SHM data across varying time points, locations, and working conditions. Given the dataset's measurement completeness, random tensor entries are artificially removed at missing ratios of 0.1, 0.3, 0.5, 0.7, and 0.9 to simulate data loss scenarios. The dataset is partitioned into training and test sets following an 8:2 ratio. #### 4.1.2 Evaluation Metrics

In this experiment, the accuracy of the imputation algorithm is primarily evaluated based on the test set's performance. To quantify model performance and generalization capability, we selected two evaluation metrics: root mean square error (RMSE) and mean absolute error (MAE). These metrics, respectively, reflect the discrepancy between completed values and actual measurements. While RMSE demonstrates higher sensitivity to large errors, MAE exhibits relatively lower sensitivity to outliers and better represents the average error magnitude. Together, these metrics not only indirectly reflect the model's ability to capture latent features, but also directly demonstrate its effectiveness in missing data imputation. The corresponding formulas are as follows for Equation (16) [25]:

\[RMSE=\sqrt{\sum_{y_{ijk}\in V}{\left({y_{ijk}-\hat{y}_{ijk}}\right)^{2}/\left| V\right|}};MAE=\sum_{y_{ijk}\in V}{\left|{y_{ijk}-\hat{y}_{ijk}}\right|/\left| V\right|} \tag{16}\]

where V denotes the test set. The experiment also tracked the number of iterations required to reach minimum accuracy, demonstrating the model's optimization efficiency and convergence rate during training. #### 4.1.3 Model Settings

This experiment involved five models, with detailed specifications provided in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Condition** & **Scale** & **Frequency (HZ)** & \
\begin{tabular}{c} **Time Point** \\ **Count** \\ \end{tabular} &
\begin{tabular}{c} **Location** \\ **Count** \\ \end{tabular} \\ \hline
0.3 kg car passing & 0–64 s & 200 (low frequency) & 12,544 & 18 \\
0.15 kg car passing & 0–70 s & 200 (low frequency) & 13,824 & 18 \\
0.45 kg car passing & 0–44 s & 200 (low frequency) & 8704 & 18 \\ Half-point excitation & 0–83 s & 200 (low frequency) & 16,384 & 18 \\ Quarter-point excitation & 0–83 s & 200 (low frequency) & 16,384 & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental dataset specifications. \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Description** \\ \hline LFT & SGD-based LFT model \\ RLFT & RMSprop-incorporated SGD-based LFT model \\ ALS & Alternating least squares \\ MF & Matrix factorization \\ LSTM & Long short-term memory network \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of models in the experiment. 4. MF: It completes missing data by first consolidating raw data into a matrix format, then decomposes it into a product of two low-rank matrices. Missing values are inferred by leveraging patterns and structures in observed data. 5. LSTM: It captures long-range dependencies in time-series data through memory cells and gating mechanisms, then utilizes these learned temporal features to predict and impute missing data points. This paper presents a missing data imputation model based on LSTM, which utilizes a single-layer unidirectional LSTM network as its core architecture for processing time-series data. The model constructs training samples through a sliding window method, with input dimensions of (1729, 50, and 90), and outputs the completed acceleration data (1729 and 90). The network includes an LSTM layer with 64 hidden units, complemented by a dropout rate of 0.2 and L2 regularization of 1e-3 to prevent overfitting, followed by a 32-dimensional ReLU fully connected layer and a linear output layer to accomplish feature transformation.