## 3 RMSprop SGD-Based LFT Model
### Algorithm Design and Analysis
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Missing Rate** & **Time Cost RMSE (ms)** & **Time Cost MAE (ms)** \\ \hline \multirow{4}{*}{RLFT} & 0.1 & 6797 & 8312 \\  & 0.3 & 5227 & 5895 \\  & 0.5 & 4606 & 5930 \\  & 0.7 & 2906 & 3213 \\  & 0.9 & 624 & 630 \\ \hline \multirow{4}{*}{LFT} & 0.1 & 7833 & 7833 \\  & 0.3 & 6549 & 6549 \\  & 0.5 & 4283 & 4283 \\  & 0.7 & 2600 & 2600 \\  & 0.9 & 872 & 872 \\ \hline \multirow{4}{*}{ALS} & 0.1 & 9966 & 9966 \\  & 0.3 & 7699 & 7699 \\  & 0.5 & 5337 & 5337 \\  & 0.7 & 3301 & 3301 \\  & 0.9 & 1067 & 1067 \\ \hline \multirow{4}{*}{MF} & 0.1 & 5206 & 5206 \\  & 0.3 & 4475 & 4475 \\ \cline{1-1}  & 0.5 & 3055 & 3055 \\ \cline{1-1}  & 0.7 & 1772 & 1772 \\ \cline{1-1}  & 0.9 & 580 & 580 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Time cost values and convergence iteration counts for all models. RLFT exhibits superior accuracy compared to the LSTM model. It reduces RMSE by 58.11%, 59.84%, 59.33%, 61.95%, and 62.58% and MAE by 96.00%, 96.12%, 96.21%, 96.30%, and 96.34% across different missing rates. Furthermore, RLFT converges more quickly than LSTM, which requires 1000 iterations to reach optimal RMSE and MAE values. This evidence firmly establishes RLFT's better performance in spatiotemporal feature capture compared to LSTM models. 4. The RLFT model demonstrates unique advantages in computational efficiency. As shown in Figure 6 and Table 6, RLFT exhibits significantly lower time costs compared to (1) the ALS model across all missing rates; (2) the LFT model at missing rates of 0.1, 0.3, and 0.9; and (3) the LSTM model at a missing rate of 0.9. However, RLFT requires more computation time than (1) the MF model, (2) the LFT model at missing rates of 0.5 and 0.7, and (3) the LSTM model at missing rates of 0.1, 0.3, 0.5, and 0.7. Notably, RLFT converges in only 507, 493, 616, 473, and 226 iterations across different missing rates, while all other models reach the preset maximum of 1000 iterations. This demonstrates RLFT's superior training efficiency through faster convergence. Furthermore, RLFT outperforms all comparative models in both RMSE and MAE metrics. Therefore, when comprehensively considering both convergence speed and prediction accuracy, the RLFT model demonstrates unique advantages in computational efficiency, delivering more reliable prediction results within significantly reduced timeframes. This distinctive characteristic enables substantial savings in actual training time when processing large-scale or complex datasets. ### Effect of R

For the task of missing data imputation on HDI tensors, the LF dimension R affects the performance of the model based on LFT. In this section, we focus on the impact of R on the predictive accuracy of the RLFT model. Figure 7 illustrates the variation in the RLFT model as R increases from 3 to 9. The key findings are as follows:

The predictive accuracy of the RLFT model exhibits a distinct trend as the rank R varies. As depicted in Figure 7, when the rank increases from 3 to 5, both the RMSE and MAE of the model decrease. Specifically, at rank 3, the RMSE is 7.273 \(\times\) 10\({}^{-4}\) and the MAE is 4.665 \(\times\) 10\({}^{-4}\). In contrast, at rank 5, the RMSE reduces to 7.185 \(\times\) 10\({}^{-4}\) and the MAE to 4.640 \(\times\) 10\({}^{-4}\), corresponding to predictive accuracy improvements of 1.21% and 0.54%, respectively. However, as the rank increases from 7 to 9, both RMSE and MAE exhibit an upward trend. This suggests that the model achieves optimal fitting at rank 5. Ranks that are either too high or too low may lead to suboptimal model performance. Consequently, selecting an appropriate rank value is crucial for optimizing the performance of the RLFT model in practical applications. Figure 7: The effect of R on the predictive accuracy of the RLFT model. ## 5 Conclusions

This paper proposes an RLFT model that incorporates the RMSprop mechanism to achieve rapid convergence and adaptive learning rate adjustment during training.