## 3 RMSprop SGD-Based LFT Model
### Algorithm Design and Analysis
The training process employs the Adam optimizer for 1000 iterations, with a batch size of 32 and equipped with an early stopping mechanism, aiming to minimize the loss to achieve the imputation of multivariate time series. The experimental parameter settings are as follows: 1. All five models adopt a rank of five and a maximum iteration count of 1000. 2. As detailed in Section 3, the performance of the RLFT model primarily depends on the learning rate, the regularization coefficient, and the decay rate. Through parameter tuning, we set the learning rate to \(1\times 10^{-6}\), the regularization coefficient to \(1\times 10^{-5}\), and the decay rate to 0.99 for optimal performance. 3. A comparative summary of model parameter configurations is presented in Table 4. 4. The training process incorporated an early stopping mechanism. Training terminated if either * The accuracy difference between consecutive iterations fell below \(1\times 10^{-10}\); * The gap between the current iteration count and the iteration achieving minimum accuracy exceeded 50. This strategy prevented overfitting while improving computational efficiency. ### Comparison Results

This section evaluates all five models under missing data ratios of 0.1, 0.3, 0.5, 0.7, and 0.9 to comprehensively assess their performance across varying levels of data incompleteness. The experiment includes the following comparisons:

1. We compare the RLFT model with its LFT baseline to validate the improvement from RMSprop optimization. 2. The evaluation contrasts LF decomposition with CP decomposition in data imputation tasks to assess their respective technical merits. 3. We examine the performance differences between LF decomposition and MF to demonstrate the proposed method's advantages in handling complex data structures. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Learning Rate** & **Regularization Coefficient** \\ \hline LFT & \(2\times 10^{-6}\) & \(1\times 10^{-5}\) \\ ALS & - & \(1.5\times 10^{-10}\) \\ MF & \(1\times 10^{-5}\) & \(1\times 10^{-5}\) \\ LSTM & \(4\times 10^{-7}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative model parameter settings. and 4.709 \(\times\) 10\({}^{-4}\), showing improvements of 21.12%, 60.40%, 84.31%, 91.58%, and 98.35% over LFT. RLFT demonstrates faster convergence than the LFT at baseline. Figure 5 and Table 5 show that RLFT requires only 507, 493, 616, 473, and 226 iterations to reach minimum RMSE and 620, 556, 793, 523, and 228 iterations for minimum MAE at the respective missing rates. In contrast, LFT consistently requires a maximum of 1000 iterations to achieve both minimum RMSE and MAE across all five missing rate scenarios. 3. Compared with other models,

* RLFT achieves higher imputation accuracy than the CP decomposition-based ALS model. For RMSE, RLFT reduces values by 85.52%, 89.44%, 94.48%, 95.98%, and 97.40% at different missing rates compared to ALS. For MAE, the reductions are 81.71%, 84.69%, 91.11%, 91.18%, and 94.27%, respectively. Moreover, RLFT converges faster than ALS, which requires a maximum of 1000 iterations to reach the minimum RMSE and MAE across all missing rates. These results strongly demonstrate the superiority of latent feature decomposition over traditional CP decomposition for missing data imputation tasks;
* RLFT shows better imputation accuracy than the MF model. The RMSE improvements are 39.38%, 76.85%, 92.34%, 97.65%, and 99.22%, while MAE shows reductions of 49.15%, 80.96%, 93.79%, 98.07%, and 99.36% at various missing rates. RLFT also converges faster than MF, which needs 1000 iterations to achieve the minimum RMSE and MAE in all cases. This conclusively proves that latent feature decomposition outperforms matrix factorization techniques, particularly in handling complex data structures;