## 3 RMSprop SGD-Based LFT Model
### Algorithm Design and Analysis  
To elucidate the operational principles of the RLFT model, the workflow of the RLFT algorithm is summarized as follows:  
According to Algorithm 1, the computational complexity of the RLFT algorithm is characterized, as shown in Equation (14) [24]:  
\[C=\Theta(2\times(|I|+|J|+|K|\times R)+N\times|\Lambda|\times 2R)\approx\Theta(N \times|\Lambda|\times R) \tag{14}\]  
Regarding its storage complexity, this is primarily determined by (a) the known dataset \(\Lambda\), (b) three LF matrices, and (c) three RMSprop correlation matrices. The storage complexity of the RLFT algorithm is formalized below using Equation (15) [24]:  
\[S=\Theta((|I|+|J|+|K|)\times R+|\Lambda|)+(|I|+|J|+|K|)\times R\approx\Theta(( |I|+|J|+|K|)\times R+|\Lambda|) \tag{15}\]All the above complexity calculations disregard constant factors and lower-order terms. The above computational analysis demonstrates that the RLFT model achieves high efficiency in both computation and storage. ```
Operation
``` Input: \(\Lambda\), \(R\), \(\lambda_{t}\), \(\lambda_{u}\), \(\eta_{ir}\), \(\eta_{jr}\), \(\eta_{kr}\), \(\gamma\), \(\delta\), and iterations - Initializing the latent factor matrix: T, L, and W ```
Initializing the RMSprop correlation matrix: \(A_{t}\), \(A_{l}\), and \(A_{uv}=0\) for iteration = 1 to iterations (Maximum number of iterations \(N\)) \(\times N\) for each \(y_{ijk}\in\Lambda\) \(\hat{y}_{ijk}=\sum_{r=1}^{R}t_{ir}^{s-1}t_{ir}^{s-1}w_{kr}^{s-1},err=y_{ijk}- \hat{y}_{ijk}\) \(\Theta(R)\) for \(r=1\) to \(R\) do \(A_{lv_{ir}}^{s}=\gamma A_{lv_{ir}}^{s-1}+(1-\gamma)(\lambda_{t}t_{ir}^{s-1}-l_ {ir}^{s-1}w_{kr}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{s-1}))^{2}\) \(t_{ir}^{s}\gets t_{ir}^{s-1}-\frac{\eta_{ir}}{\sqrt{A_{lv_{ir}}^{s-1}}} \cdot(\lambda_{t}t_{ir}^{s-1}-l_{ir}^{s-1}w_{kr}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{s- 1}))\) \(A_{lv_{ir}}^{s}=\gamma A_{lv_{ir}}^{s-1}+(1-\gamma)(\lambda_{l}t_{ir}^{s-1}-t_ {ir}^{s-1}w_{kr}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{s-1}))^{2}\) \(l_{ir}^{s}\gets l_{ir}^{s-1}-\frac{\eta_{ir}}{\sqrt{A_{lv_{ir}}^{s-1}}} \cdot(\lambda_{t}l_{ir}^{s-1}-t_{ir}^{s-1}w_{kr}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{ s-1}))\) \(A_{uv_{ir}}^{s}=\gamma A_{uv_{ir}}^{s-1}+(1-\gamma)(\lambda_{uv}w_{kr}^{s-1}-t_ {ir}^{s-1}l_{ir}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{s-1}))^{2}\) \(w_{kr}^{s}\gets w_{kr}^{s-1}-\frac{\eta_{kr}}{\sqrt{A_{bv_{ir}}^{s-1}}} \cdot(\lambda_{uv}w_{kr}^{s-1}-t_{ir}^{s-1}l_{ir}^{s-1}(y_{ijk}-\hat{y}_{ijk}^{ s-1}))\) end for - end for - - Output LF matrix: T, L, and W
```

**Algorithm 1** RLFT Algorithm

## 4 Experimental Results and Analysis

This section systematically evaluates the proposed RLFT model through comprehensive experiments. We first describe the experimental setup including datasets, evaluation metrics, and implementation details (Section 4.1), then present comparative results with state-of-the-art methods (Section 4.2). The analysis validates RLFT's advantages in both accuracy and computational efficiency. ### General Settings

The experimental configuration is organized into three components: (1) benchmark datasets for performance validation, (2) quantitative metrics for objective assessments, and (3) implementation parameters ensuring reproducible results. This standardized setup guarantees a fair comparison across all evaluated methods. #### 4.1.1 Datasets

To comprehensively evaluate the performance of the RLFT model, we conducted experiments using the Yunnan Hechonggou scaled bridge model dataset. This dataset records acceleration vibration signals from a 1:20 scale model of the Heichonggou Bridge, featuring high fidelity and comprehensive annotations. Serving as an excellent benchmark for SHM data imputation tasks, it significantly enhances the model's capability to characterize structural dynamic responses and improve the accuracy of missing data reconstruction.