## 5 Conclusions  
In this paper, a data security training framework based on symmetric projection space and adversarial training is proposed, aimed at balancing the conflict between data privacy protection and model utility. In traditional privacy protection methods, such as homomorphic encryption and differential privacy, a certain trade-off in computational efficiency or model utility is often required to ensure data privacy. In contrast, the proposed method in this paper utilizes nonlinear mapping to compress high-dimensional data into a low-dimensional space, reducing the computational cost of redundant information while preserving key data, thus achieving a balance between privacy protection and data utility. Experimental results show that the proposed method outperforms traditional privacy protection methods across multiple tasks. In the financial time-series data task, the model using projection loss shows significant advantages in metrics such as precision, recall, and accuracy, especially in enhancing model utility while ensuring privacy protection. In image data tasks, the projection loss function also demonstrates strong performance, not only improving accuracy but also achieving higher values in metrics such as mAP. Furthermore, the experiments indicate that the proposed method maintains good computational efficiency across different hardware platforms, especially on high-performance GPU platforms, showing substantial advantages.  
The trade-off between privacy protection and model utility is a complex task, and in practical applications, privacy protection often needs to be compatible with efficient model performance. By introducing symmetric projection and adversarial training mechanisms, this paper effectively reduces the computational overhead caused by privacy protection while maintaining the efficiency and practicality of the model. When dealing with different data types and tasks, the proposed method not only provides strong privacy protection but also ensures efficient training and inference performance, demonstrating its broad applicability across various fields. In summary, the framework proposed in this paper provides an effective solution for privacy protection tasks, optimizing model utility while ensuring data privacy. With further optimization and adjustments, the method can play an important role in more complex and diverse application scenarios, offering a theoretical foundation and practical reference for the balance between data privacy protection and machine learning model utility in the future.  
Conceptualization, Q.L. (Qianqian Li), S.Z., X.Z., Y.J. and C.L.; Data curation, Q.L. ([PERSON]) and C.H.; Formal analysis, J.S. and Q.L. ([PERSON]); Funding acquisition, C.L.; Investigation, J.S. and Y.Y.; Methodology, Q.L. ([PERSON]), S.Z. and X.Z.; Project administration, C.L.; Resources, Q.L. ([PERSON]) and C.H.; Software, Q.L. ([PERSON]), S.Z., X.Z. and Y.Y.; Supervision, Y.J. and C.L.; Validation, J.S.; Visualization, C.H., Y.Y. and Y.J.; Writing--original draft, Q.L. ([PERSON]), S.Z., J.S., Q.L. ([PERSON]), C.H., Y.Y., Y.J. and C.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China grant number 61202479.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.