## 4 Results and Discussion
### Ablation Experiment on Different Loss Functions Results  
The purpose of this experiment was to evaluate the effectiveness of different loss functions in privacy-preserving tasks, particularly in time-series and image classification tasks. By training the models with cross-entropy loss, focal loss, and projection loss, the experiment aimed to explore the differences between these loss functions in terms of precision, recall, F1 score, accuracy, and mAP. These metrics are key indicators for evaluating the performance of classification and detection tasks and provide insight into how different loss functions influence the model's effectiveness. The ablation experiment helps to understand the contribution of each loss function to the model's training process and final results, offering a theoretical basis for further optimization of the model. The experimental results in the tables clearly show that the projection loss function exhibits significant advantages across different tasks, especially in more challenging scenarios. Based on the experimental results shown in Tables 6 and 7 for both time-series and image data, the model using the projection loss function outperforms those using cross-entropy loss and focal loss on all evaluation metrics. For time-series data, the cross-entropy loss model has a precision of 0.77, recall of 0.72, F1 score of 0.74, and accuracy of 0.75. In contrast, the focal loss improves these metrics with a precision of 0.86, recall of 0.81, F1 score of 0.83, and accuracy of 0.84. The model using the projection loss function, however, shows a much more significant improvement with a precision of 0.95, recall of 0.91, F1 score of 0.93, and accuracy of 0.93, demonstrating that the projection loss not only maintains high privacy protection but also improves model performance effectively. In the image data task, a similar trend is observed. The cross-entropy loss achieves a precision of 0.71, recall of 0.67, F1 score of 0.69, and accuracy of 0.69. The focal loss performs better, with a precision of 0.82, recall of 0.78, F1 score of 0.80, and accuracy of 0.80. The projection loss, however, achieves a precision of 0.93, recall of 0.90, F1 score of 0.91, accuracy of 0.91, and mAP@50 and mAP@75 of 0.91 and 0.90, respectively, showing the strong performance of the projection loss in image classification and detection tasks. From a theoretical perspective, the cross-entropy loss function is widely used for multi-class classification tasks but does not account for class imbalance, leading to lower recall and F1 score in imbalanced datasets. The focal loss addresses this issue by adjusting the weights of easy-to-classify and hard-to-classify samples, aiming to reduce the influence of easy-to-classify samples on the loss function and focus more on harder samples. Although focal loss improves model performance on imbalanced data, it still struggles with retaining data privacy and critical information features. In contrast, the projection loss function combines reconstruction error and privacy loss, effectively preserving the critical features of the data while ensuring privacy protection. Mathematically, the projection loss uses a nonlinear mapping to compress the data into a low-dimensional space, reducing the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Precision** & **Recall** & **F1 Score** & **Accuracy** \\ \hline Cross-Entropy Loss & 0.77 & 0.72 & 0.74 & 0.75 \\ Focal Loss & 0.86 & 0.81 & 0.83 & 0.84 \\ Projection Loss & 0.95 & 0.91 & 0.93 & 0.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Ablation experiment on different loss functions for financial data. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Model** & **Precision** & **Recall** & **F1 Score** & **Accuracy** \\ \hline Cross-Entropy Loss & 0.71 & 0.67 & 0.69 & 0.69 & 0.68 & 0.67 \\ Focal Loss & 0.82 & 0.78 & 0.80 & 0.80 & 0.79 & 0.78 \\ Projection Loss & 0.93 & 0.90 & 0.91 & 0.91 & 0.91 & 0.90 \\ \hline \hline \end{tabular}
\end{table}