## 3 Materials and Methods
### Proposed Method
#### 3.3.2 Symmetric Projection Space Extractor  
The symmetric projection space extractor is a key module in the privacy protection framework proposed in this study. Its primary task is to map high-dimensional sensitive data into a lower-dimensional latent space through nonlinear transformations, while ensuring data privacy protection during the projection process by utilizing symmetry design, as shown in Figure 4.  
Specifically, this module is designed to learn a mapping function that reduces the input data from their original high-dimensional space to a low-dimensional abstract space, preserving essential information while minimizing the risk of sensitive information leakage. The structure of the symmetric projection space extractor is built upon a deep neural network, primarily composed of several fully connected (FC) layers, with ReLU activation functions employed to enhance the network's nonlinear representational power. The input to the network is high-dimensional data \(\mathbf{X}\in\mathbb{R}^{n\times d}\), where \(n\) represents the number of samples and \(d\) denotes the feature dimension of the input data. The output is a low-dimensional latent representation \(\mathbf{Z}\in\mathbb{R}^{n\times k}\), where \(k\) is the dimensionality of the latent space and is typically chosen such that \(k<d\). The detailed architecture of the symmetric projection space extractor is outlined as follows:  
1. Input layer: The input data \(\mathbf{X}\) are mapped through a fully connected layer with output dimension \(h_{1}\). Assuming the input feature dimension is \(d\), the parameters of the first layer are \(W_{1}\in\mathbb{R}^{d\times h_{1}}\) and the bias is \(b_{1}\in\mathbb{R}^{h_{1}}\), with the activation function being ReLU: \[\mathbf{H}_{1}=\mathrm{ReLU}(\mathbf{X}W_{1}+b_{1}),\] (20) where \(\mathbf{H}_{1}\in\mathbb{R}^{n\times h_{1}}\). Hidden layers: Data are transformed through multiple hidden layers, with the output dimension gradually decreasing. The parameters of the \(i\)-th layer are \(W_{i}\in\mathbb{R}^{h_{i-1}\times h_{i}}\), the bias is \(b_{i}\in\mathbb{R}^{h_{i}}\), and the activation function remains ReLU: \[\mathbf{H}_{i}=\text{ReLU}(\mathbf{H}_{i-1}W_{i}+b_{i}),\] (21) where \(\mathbf{H}_{i}\in\mathbb{R}^{n\times h_{i}}\). 3. Output layer: The final layer maps the data to the low-dimensional latent space \(\mathbf{Z}\in\mathbb{R}^{n\times k}\), where \(k\) is the latent space dimension. The output layer's parameters are \(W_{\text{out}}\in\mathbb{R}^{h_{m}\times k}\), the bias is \(b_{\text{out}}\in\mathbb{R}^{k}\), and the activation function is a linear function to ensure continuous output: \[\mathbf{Z}=\mathbf{H}_{m}W_{\text{out}}+b_{\text{out}},\] (22) where \(\mathbf{Z}\) is the low-dimensional latent representation. The core idea of the symmetric projection space extractor mapping process gradually reduces the dimensionality of the data using multiple fully connected layers within the deep neural network. By progressively compressing the feature dimensions, the network ensures that the final low-dimensional representation \(\mathbf{Z}\) effectively retains the key information of the input data, while redundant information is gradually discarded. The network's parameters are optimized via backpropagation to minimize the loss function. A specific loss function, such as the projection loss function introduced in this study, combines reconstruction error and privacy loss to optimize the balance between data privacy and model utility. During the design of the symmetric projection space extractor, symmetry constraints are applied to ensure the smoothness and robustness of the mapping. The principle of symmetry design is to share or constrain network parameters so that the data transformation during projection has a similar effect on each dimension of the input data. This avoids the overcompression of certain dimensions, while preserving other dimensions, resulting in more balanced feature compression and information retention.