## 3 Materials and Methods
### Proposed Method
#### 3.3.4 Projection Loss Function  
Common loss functions are designed to optimize the model's predictive accuracy. However, in privacy-preserving tasks, traditional loss functions have certain limitations when it comes to achieving privacy protection goals. Specifically, directly optimizing traditional loss functions may lead to privacy leaks when dealing with sensitive data, or while the generated data may be accurate, it might be overfitted. To address this issue, a new projection loss function is designed, as shown in Algorithm 2. The projection loss function consists of two components: reconstruction error and privacy loss. The reconstruction error primarily measures the similarity between the generated data and the original data, while privacy loss ensures that the generated data meet the privacy protection requirements. The formula for the projection loss function can be expressed as:\[\mathcal{L}_{\text{projection}}=\alpha\mathcal{L}_{\text{recon}}+\beta\mathcal{L}_{ \text{privacy}}, \tag{25}\]  
where \(\alpha\) and \(\beta\) are hyperparameters that control the weights of the reconstruction error and privacy loss, respectively. In this paper, the hyperparameters \(\alpha\) and \(\beta\) are used to adjust the weights of the reconstruction error (\(\mathcal{L}_{\text{recon}}\)) and the privacy loss (\(\mathcal{L}_{\text{privacy}}\)). After initial experimentation and tuning, the values of \(\alpha\) and \(\beta\) are set to 0.5 and 0.5, meaning equal weights for both. To further optimize the model's performance, the values of \(\alpha\) and \(\beta\) are fine-tuned using a cross-validation strategy, to adapt to different datasets and tasks. Specifically, the reconstruction error \(\mathcal{L}_{\text{recon}}\) can be expressed using mean squared error, which measures the difference between the generated data and the original data:  
\[\mathcal{L}_{\text{recon}}=\frac{1}{n}\sum_{i=1}^{n}|\mathbf{X}_{i}-\mathbf{X }_{i}^{\prime}|^{2}, \tag{26}\]  
The reconstruction error measures the similarity between the generated data and the original data, and the objective is to optimize this term to preserve the key information of the data. Privacy loss \(\mathcal{L}_{\text{privacy}}\) is then measured through the output of the discriminator \(D\), which evaluates the privacy of the data. The goal is to minimize the probability that the generated data are classified as \"real\" by the discriminator. The privacy loss can be expressed as:  
\[\mathcal{L}_{\text{privacy}}=-\mathbb{E}_{\mathbf{X}^{\prime}\sim p_{\text{ gm}}}[\log(1-D(\mathbf{X}^{\prime}))], \tag{27}\]  
where \(D(\mathbf{X}^{\prime})\) represents the output of the discriminator for the generated data \(\mathbf{X}^{\prime}\). The goal is to make the generated data difficult for the discriminator to distinguish, thus ensuring the privacy of the generated data. By optimizing the privacy loss, the generated data must not only retain enough information for training and inference but also avoid leaking sensitive information. The projection loss function is combined with the symmetric projection space extractor, forming a complete privacy protection framework. In the symmetric projection space extractor, the original data are projected into the low-dimensional space \(\mathbf{Z}\) via a nonlinear mapping. This process effectively compresses the redundant parts of the data and avoids information loss through symmetry design. The optimization of the projection loss function ensures the balance between privacy protection and data utility in the generated data. Specifically, after the data are mapped into the low-dimensional space by the symmetric projection space extractor, it is passed through the generator for generation. The generated data are then evaluated by the discriminator and ultimately optimized through the projection loss function to ensure that the generated data achieves an optimal balance between privacy protection and utility.